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Force Generation by Microtubule End-Binding Proteins

Permanent Link: http://ufdc.ufl.edu/UFE0019600/00001

Material Information

Title: Force Generation by Microtubule End-Binding Proteins
Physical Description: 1 online resource (178 p.)
Language: english
Creator: Caro, Luz Elena
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: adenomatous, apc, binding, brownian, cilia, clasp, clip, coli, eb1, end, fire, flagella, force, gdp, gtp, hill, hydrolysis, kinetochore, load, lock, microtubule, mitosis, motor, p150glued, polyposis, ratchet, sleeve, thermodynamic, tracking
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Microtubules are cytoskeletal filaments essential for multiple cell functions, including motility of microorganisms and cell division. Of particular interest is how these biological polymers generate the forces required for movement of chromosomes during mitosis and for formation of cilia and flagella. Defective microtubule-based force generation can lead to various pathological complications; therefore, an understanding of microtubule force generation is important for cancer research and biotechnology. The mechanism by which elongating microtubules generate force is unknown. Several proteins, including End-Binding Protein 1 (EB1) and adenomatous polyposis coli (APC), specifically localize to microtubule elongating ends where the microtubule is tightly bound to a motile object and generating force. The role of these end-tracking proteins is not fully understood, but they likely modulate microtubule-motile surface interactions, and may aid in force production. The objective of my research is to elucidate the role of polymerizing microtubules and end-binding proteins, specifically EB1, in force-dependent processes by formulating a model that explains their interaction and role in force generation. The commonly assumed Brownian Ratchet model describing the forces caused by elongating microtubules cannot easily explain force generation during rapid elongation and strong attachment of the microtubule to the motile object. I propose a novel mechanism in which EB1 proteins behave as end-tracking motors that have a higher affinity for guanosine triphosphate-bound tubulin than guanosine diphosphate-bound tubulin, thereby allowing them to convert the chemical energy of microtubule-filament hydrolysis to mechanical work. These microtubule end-tracking motors are predicted to provide the required forces for cell motility and persistent attachment between the motile surface and polymerizing microtubules. I have developed mechanochemical models that characterize the kinetics of these molecular motors based on experimentally determined binding parameters and thermodynamic constraints. These models account for the association of EB1 to tethered and untethered elongating microtubule ends, in the absence or presence of force, and with or without EB1 binding to solution-phase tubulin. These models explain the observed exponential profile of EB1 on untethered filaments and predict that affinity-modulated end-tracking motors should achieve higher stall forces than with the Brownian Ratchet system, while maintaining a strong, persistent attachment to the motile object.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Luz Elena Caro.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Dickinson, Richard B.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019600:00001

Permanent Link: http://ufdc.ufl.edu/UFE0019600/00001

Material Information

Title: Force Generation by Microtubule End-Binding Proteins
Physical Description: 1 online resource (178 p.)
Language: english
Creator: Caro, Luz Elena
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: adenomatous, apc, binding, brownian, cilia, clasp, clip, coli, eb1, end, fire, flagella, force, gdp, gtp, hill, hydrolysis, kinetochore, load, lock, microtubule, mitosis, motor, p150glued, polyposis, ratchet, sleeve, thermodynamic, tracking
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Microtubules are cytoskeletal filaments essential for multiple cell functions, including motility of microorganisms and cell division. Of particular interest is how these biological polymers generate the forces required for movement of chromosomes during mitosis and for formation of cilia and flagella. Defective microtubule-based force generation can lead to various pathological complications; therefore, an understanding of microtubule force generation is important for cancer research and biotechnology. The mechanism by which elongating microtubules generate force is unknown. Several proteins, including End-Binding Protein 1 (EB1) and adenomatous polyposis coli (APC), specifically localize to microtubule elongating ends where the microtubule is tightly bound to a motile object and generating force. The role of these end-tracking proteins is not fully understood, but they likely modulate microtubule-motile surface interactions, and may aid in force production. The objective of my research is to elucidate the role of polymerizing microtubules and end-binding proteins, specifically EB1, in force-dependent processes by formulating a model that explains their interaction and role in force generation. The commonly assumed Brownian Ratchet model describing the forces caused by elongating microtubules cannot easily explain force generation during rapid elongation and strong attachment of the microtubule to the motile object. I propose a novel mechanism in which EB1 proteins behave as end-tracking motors that have a higher affinity for guanosine triphosphate-bound tubulin than guanosine diphosphate-bound tubulin, thereby allowing them to convert the chemical energy of microtubule-filament hydrolysis to mechanical work. These microtubule end-tracking motors are predicted to provide the required forces for cell motility and persistent attachment between the motile surface and polymerizing microtubules. I have developed mechanochemical models that characterize the kinetics of these molecular motors based on experimentally determined binding parameters and thermodynamic constraints. These models account for the association of EB1 to tethered and untethered elongating microtubule ends, in the absence or presence of force, and with or without EB1 binding to solution-phase tubulin. These models explain the observed exponential profile of EB1 on untethered filaments and predict that affinity-modulated end-tracking motors should achieve higher stall forces than with the Brownian Ratchet system, while maintaining a strong, persistent attachment to the motile object.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Luz Elena Caro.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Dickinson, Richard B.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0019600:00001


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FORCE GENERATION BY MICROTUBULE END-BINDING PROTEINS


By

LUZ ELENA CARO













A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007




































O 2007 Luz Elena Caro





























To my supportive parents, Ruby and Armando Caro,
encouraging siblings, Maritza, Mando, and Cesar,
and my eternal best friend, Antonio.









ACKNOWLEDGMENTS

I acknowledge the support of my advisor, Dr. Richard B. Dickinson, whose patience,

guidance, and motivation provided the necessary tools for my successful and rewarding graduate

experience. The helpful comments and constructive criticisms from my committee members

were greatly appreciated. I thank Dr. Anuj Chauhan for the continued encouragement and

support throughout my professional career. The insightful career advice and assistance provided

by Dr. Jennifer Curtis was greatly appreciated. I recognize Dr. Anthony Ladd for introducing me

to the exciting research at the University of Florida. The assistance provided by the Chemical

Engineering faculty and staff was invaluable for my experience at the University of Florida.

I thank Dr. Daniel Purich for his biochemistry and professional advice which helped me to

develop my research skills. I am grateful for the expertise of the members in the biochemistry

group (under advisement of Dr. Daniel Purich) which helped me gain the proper biochemistry

understanding needed for my graduate research; I thank Dr. William Zeile, Dr. Joseph Phillips,

Dr. Fangliang Zhang.

I thank my group members who helped me learn experimental techniques and exchanged

ideas pertaining to research; my graduate experience was enriched by their companionship and

support: Kimberly Interliggi, Colin Sturm, Gaurav Misra, Jeff Sharp, Huilian Ma, Adam

Feinburgh.












TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....


LIST OF TABLES ........._.___..... .__. ...............8....


LIST OF FIGURES .............. ...............9.....


LIST OF TERM S ........._ _. ........_. ...............11....


AB S TRAC T ............._. .......... ..............._ 17...


CHAPTER


1 INTRODUCTION ................. ...............19.......... ......


1.1 M icrotubules .............. ...............20....
1.2 End-Tracking Proteins .............. ...............23....
1.2.1 EB 1 .............. ........ ...... ... ... .......2
1.2.2 Adenomatous Polyposis Coli (APC)............... ...............25.
1.2.3 Ciliary and Flagellar Movement .............. ...............26....
1.3 Force Generation Models............... ...............27.
1.3.1 Brownian Ratchet Models ................. ...............27........... ...
1.3.2 Sleeve M odel ................. ...............28........... ....
1.3.3 Kinetochore M otors .............. ...............29....
1.3.4 Filament End-Tracking Motors ................. ...............29................
1.4 Thermodynamic Driving Force ................. ...............30................
1.5 Sum m ary .................. ...............3.. 1..............
1.6 Outline of Dissertation ................ ...............31........... ...


2 MICROTUBULE END-TRACKING MODEL ................ ...............37................


2.1 EB 1 End-Tracking Motors ................. ...............37........... ...
2.2 Microtubule Growth Model .............. ...............38....
2.2.1 Parameter Estimations ............... ... ..... ........ .. ...............42.....
2.2.2 Elongation Rate in the Absence of External Force ................ ............... .....43
2.2.3 Force effects on elongation rate ................ ...............44...............
2.3 Summary .............. ...............45....


3 PROTOFILAMENT END-TRACKING MODEL WITH MONOVALENT EBl..............__53


3.1 Non-Tethered Protofilament Growth ................ ........... ............... 53. ....
3.1.1 Thermodynamics of EB l-tubulin interactions ................. ................ ...._.54
3.1.2 Kinetics of EB l-tubulin interactions. ......___ ..... .._._. ....._.... ......5
3.1.3 Parameter Estimations ........._..... ...._... ...............56....
3.1.4 Results ........._..... ...._... ...............58...












3.2 Tethered ProtoHilament Growth .............. ...............58....
3.2.2 M odel .............. .. ...............59...
3.2.3 Parameter Estimations ............. ..... ._ ...............61.....
3.2.4 Results ........._..... ...._... ...............63.....

3.3 Sum m ary .............. .. ............. ..........6
3.3.1 Non-Tethered ProtoHilaments ................. ...............64........... ....
3.3.2 Tethered ProtoHilaments ................. ...............65......_.._ ....


4 PROTOFILAMENT END-TRACKING MODEL WITH DIVALENT EBl1................... .....73


4.1 Non-Tethered ProtoHilament Growth ................ ...............73................
4.1.1 Kinetics of EB l-Tubulin Interactions ................ ...............73........... ..

4. 1.2 EB1 Occupational Probability Model .............. ...... ...............75
4.1.3 Average Fraction of EBli-bound Subunits at Equilibrium ................... .............77
4. 1.4 Average Fraction of EBli-bound subunits during protoHilament growth ..........77
4.1.5 Parameter Estimations .....__................. ...............78......
4.1.6 Results ................. .......... .............. ..... ..........8
4.1.6.1 Occupational probability ................ .. .... .......... ........8
4.1.6.2 Average fraction of EBli-bound subunits at equilibrium ................... .82
4.1.6.3 Average fraction of EBli-bound subunits during protoHilament

grow th ........._.._..... ......... ._ ...............82.....
4.2 Tethered ProtoHilament Growth Model ...._.._.._ ........._.._......_ ...........8
4.2.1 Kinetics of EB l-Tubulin Interactions ...._.._.._ ........... ......_.._.........8
4.2.2 ProtoHilament End-Tracking Model .............. ...............87....
4.2.3 Parameter Estimations ........._.._.... ...............87.._.._........
4.2.4 Results ........._..._... ...............87.._.._.. ......

4.3 Sum m ary .............. .. ............. ..........9
4.3.1 Non-Tethered ProtoHilaments ................. ...............92........... ....
4.3.2 Tethered ProtoHilaments ................. ...............93......_.._ ....


5 CILIARY PLUG MODEL ............. ........... ...............115...


5.1 M odel .............. ........... ...............115...
5.2 Parameter Estimations ............. ...... ._ ...............118...
5.3 Results ............. ...... ...............119....
5.4 Summary ............. ...... __ ...............120...


6 DI SCUS SSION ............. ..... ._ .............. 124...


6.1 Possible Roles of End-Tracking Motors in Biology ..................._ ..............124
6.2 Microtubule End-Tracking Model .............. ...............126....
6.3 ProtoHilament End-Tracking Models .............. ...............127....
6.4 Future W ork ................. ...............128...............


APPENDIX


A PARAMETER ESTIMATIONS .............. ...............130....











A. 1 Concentrations of EB 1 Species in Solution ................ ... ... .......... .. .............. ...13
A.2 Occupation Probability of Monovalent EB 1 Binding to Non-Tethered
Protofil1am ent ................ ...... .__ ...... ..._ _..... ... ..... ...... ........ 3
A.3 Occupation Probability of Monovalent EB 1 Binding to Tethered Protofilament ......132
A.4 Occupation Probability of Divalent EB 1 Binding to Tethered Protofilament ............137

B MATLAB CODES .............. ...............145....

B.1 13-Protofilament Microtubule Model ................. ............. ............... 145 ...
B.2 Protofilament Growth Model with Monovalent EB 1 .............. ......... .. ..............147
B.2.1 Occupational Probability of Monovalent EB 1 on a Non-Tethered
Protofil1am ent................. ....... ............... ........ ............4
B.2.2 Occupational Probability of Monovalent EB 1 on a Tethered Protofilament..149
B.3 Protofilament Growth Model with Divalent EBl1 .............. ....... .._ .................1 52
B.3.1 Occupational Probability of Divalent EB 1 on a Non-Tethered
Protofil1am ent ............... .... ............__.. ........_ __. .. .. .. ........5
B.3.2 Average Fraction of divalent EBli-bound Protomers on Side of
Protofil1am ent ............... .... ....__ ......__ ................5
B.3.3 Average Fraction of EBli-bound protomers during protofilament growth .....157
B.3.4 Tethered Protofilament Growth with Divalent EB 1 .............. ...............160
B.4 Ciliary Plug M odel ................. ...............170......... .....

LIST OF REFERENCES ................. ...............172................

BIOGRAPHICAL SKETCH ................. ...............178......... ......










LIST OF TABLES


Table page

1-1 Thermodynamic equations characterizing the multiple steps for GDP to GTP
conversion ........... ..... .._ ...............35...

1-2 Equilibrium constants used in energy equations ......___. .... ... .._ .. ......_.......3

4-1 Protofilament stall forces at varying values of KT and affinity modulation factors.
Stall forces (in units of pN) correspond to the data represented in Figure 4-12. .............109











LIST OF FIGURES


Figure page

1-1 Microtubule structure................ ..............3

1-2 Chromosomal binding site of microtubules ......... ........ ................. ...............33

1-3 EB 1 binding to microtubule lattice. .............. ...............34....

1-4 Concentration of EB 1 along length of microtubule. ...._._._._ .... ... .___ ........._.......34

2-1 Model for microtubule force generation by EB 1 end-tracking motor. .............. .... ...........47

2-2 Reaction mechanisms of EB 1 end-tracking motor. .......____ ..... ._ .............. ..48

2-3 Force dependence on EB 1 binding and equilibrium surface position. ............. .............49

2-4 Microtubule elongation in the absence of external force ................. ........................49

2-5 Distribution of protoHilament lengths for microtubule end-tracking model .................. ....50

2-6 Effect of applied force on MT elongation rate............... ...............5...1

2-7 Thermodynamic versus simulated stall forces ................. ...............52...............

3-1 Schematic of non-tethered, monovalent EB 1 end-tracking motor mechanisms. ..............67

3-2 Various pathways of non-tethered monovalent EB 1 binding to protoHilament. ................67

3-3 Choosing an optimal K~value for monovalent EBl1.....__.__. .... ... ._ ........._._......68

3-4 EB1 density profie on a non-tethered microtubule protoHilament with monovalent
E B 1. ............. ...............69.....

3-5 Effect of K; on profie of monovalent EB 1 occupational probability.. ............_ .............70

3-6 Schematic of tethered, monovalent EB 1 end-tracking motor mechanisms.. ...................70

3-7 Force effects on a tethered protoHilament with monovalent EBl. ............. ...............71

3-8 Divalent EB 1 represented as divalent end-tracking motor. ..........__.... ..._ ............72

4-1 Mechanisms of a non-tethered, divalent end-tracking motor.. ..........._._ ........._.. ......95

4-2 Mechanisms of equilibrium, side binding of EB 1 to protoHilament.. ................ ...............97

4-3 Choosing an optimal K~value for divalent EBl1.....__.....___ ..............__.......9











4-4 Effect of ko,, on optimalK; ........._._.._......_.. ...............98...

4-5 EB 1 equilibrium binding............... ...............99

4-6 Occupational probability of EB 1 along length of protoHilament.. ................ .................100

4-7 Time averaged EBli-bound tubulin fraction at equilibrium ................. .....................101

4-8 Time averaged fraction of EBli-bound subunits during protoHilament growth. ...............102

4-9 Mechanisms of tethered, protoHilament end-tracking model with divalent EBl1.............103

4-10 Mechanisms of tubulin addition to linking protein-bound protoHilament. ................... ....105

4-11 Force-velocity profies for tethered protofilaments bound to divalent EB 1
end-tracking motors ................. ...............106........ ......

4-12 Stall forces versus affinity modulation factor at various KT values ........._..... ..............109

4-13 Effect of f KT, and F on pathways taken ................. ...._ ...............110

4-14 Percent of time protoHilament bound and unbound to motile surface ................... ...........11 1

4-15 State of the terminal subunit (Sl) when f-1 and f-1000 ........._.__..... ..._._............1 12

4-16 Fraction of S1 subunits bound and unbound from motile surface .........._.... ..............113

4-17 Average state of unbound linking protein. ......___ ... ....._ ....___ ...........1

5-1 EM image of a ciliary plug at the end of a ciliary microtubule ................ ................. .120

5-2 Schematic of ciliary plug inserted into the lumen of a cilia/flagella microtubule ...........121

5-3 Mechanism of the ciliary/flagellar end-tracking motor ........... ..... .___ ..............121

5-4 Force effects on ciliary microtubules............... .............12

5-5 Ciliary plug movement .............. ...............123....









LIST OF TERMS

ADP: Adenosine diphosphate

APC: Adenomatous Polyposis Coli

ATP: Adenosine triphosphate

a: Width of protofilament

Cyf: Effective concentration of a free subunit of filament-bound EB 1

CT: Effective concentration of tracking unit at protofilament plus-end

d:. Size of tubulin protomer

dbE : State of tubulin protomer attached to the subunit on the minus-side of a
double- bound EB 1 dimer

dbE : State of tubulin protomer attached to the subunit on the plus-side of a
double- bound EB 1 dimer

[E]o: Total intracellular EB 1 concentration

[E]: Concentration of EB 1 in solution

EBl: End Binding Protein 1

Esp: Hookean Spring energy

Df : Protofilament diffusivity

dt: Time steps taken in simulation

F: Force applied to microtubule plus-end

Fsranl: Stall force maximum achievable force

f Energy captured from hydrolysis that is used for affinity modulation

GDP: Guanosine diphosphate nucleotide

GTP: Guanosine triphosphate nucleotide

[GDP]: Concentration of guanosine diphosphate nucleotide

[GTP]: Concentration of guanosine triphosphate nucleotide

K;: Equilibrium dissociation constant for tubulin in solution binding to EB 1











K; ':Equilibrium dissociation constant for tubulin addition to track-bound

protofilament (Kl' E k r/kf')

K3: Equilibrium dissociation constant for EB 1 subunit binding to track-bound

filament-T-GDP (K3 k -side'/k+side')

K: Ratio of forward and reverse rate of EB 1 subunit binding to protofilament

plus-end (K k +/k-)

K': Equilibrium dissociation constant for EB 1 subunit binding to

protofilament plus-end (K' k -/k+)

Kd: Equilibrium dissociation constant for EB 1 (or TE) binding to filament-

bound T-GDP (Kd k-side/konside)

Kd*: Equilibrium dissociation constant for EB 1 (or TE) in solution binding to

T-GTP at protofilament plus-end

Kp,: Equilibrium dissociation constant for reversible phosphate binding to T

protomers

KT: Equilibrium dissociation constant for track binding to solution-phase EB 1

(or TE or TTE)

Kx: Equilibrium dissociation constant for the GTP/GDP exchange reaction

k : Forward rate constant for tubulin in solution binding to EB 1

k l: Reverse rate constant for tubulin in solution binding to EB 1

k : Forward rate constant for subunit of filament-bound EB 1 binding to

T-GTP at protofilament plus-end

k side: Forward rate constant for subunit of filament-bound EB 1 binding to

filament-bound T-GDP

k_: Reverse rate constant for EB 1 (or TE) in solution binding to T-GTP at









protofilament plus-end

k side: Reverse rate constant for EB 1 (or TE) binding to filament-bound T-GDP

ksT: Thermal energy (Boltzmann constant, k, x absolute temperature, T)

k: Forward and reverse rate constants for tubulin in solution binding to

protofilament plus-end

ke,o: Initial forward rate constant for tubulin in solution binding to

protofilament plus-end

kfE: Forward rate constant for EBli-bound tubulin in solution binding to T-GTP

at protofilament plus-end

kMT: Kinetochore-bound microtubule

kobs: Observed decay constant of EB 1 on MT

korS: Dissociation rate constant for EB 1 dimer from protofilament

kon:Forward rate constant for EB 1 (or TE) in solution binding to T-GTP at

protofilament plus-end

ko;2side: Forward rate constant for EB 1 (or TE) binding to filament-bound T-GDP

k,: Reverse rate constant for tubulin in solution binding to protofilament

plus-end

k,E: Reverse rate constant for EBli-bound tubulin in solution binding to T-GTP

at protofilament plus-end

kT: Forward rate constant for track binding to solution-phase EB 1 (or TE or

TTE)

kT : Reverse rate constant for track binding to solution-phase EB 1 (or TE or

TTE)

L: Length of protofilament in ciliary plug

LLF: Lock, Load, and Fire model









MT: Microtubule

N: Total number of protomers in a protofilament

N,: Number of protofilaments tethered to motile obj ect

n: Position of tubulin protomer bound to track

ns: Number of protomers between EB 1 subunit at equilibrium position and
final binding position

Peq:Equilibrium fraction of EBli-bound protomers on protofilament

P,: Phosphate

p: Probability of protomer bound to EB 1 subunit

spend: Probability of EB 1 binding to the protofilament plus-end

peq: Equilibrium probability of protomer bound to EB 1 subunit

pside: Equilibrium probability of EB 1 binding to filament-bound T-GDP

q : Probability of protomer in state dbE+

q : Probability of protomer in state dbE-

q: Probability of protomer attached to double-bound EB 1 subunit

qeq: Equilibrium probability of protomer attached to double-bound EB 1

subunit

Sl: State of terminal protomer in protofilament

S2: State of penultimate protomer in protofilament

TAC: Tip-Attachment Complex model

Tb: Tubulin protomer

[Tb]: Tubulin protomer concentration

[Tb],: Critical tubulin concentration for free MT plus-end

[Tb]Ec: Critical TE concentration for free MT plus-end

TE: EBl-bound tubulin protomer

T-GDP: GDP-bound tubulin protomer










T-GDP concentration

Critical tubulin concentration for T-GDP at MT plus-end

GTP-bound tubulin protomer

Track bound to protofilament-bound EB 1

Track bound to protofilament-bound TE

Track bound to protofilament-bound dbE+

Track (tethering protein bound to motile surface)

Track bound to EB 1 in solution

Track bound to TE in solution

Track bound to TTE in solution

Time required for tubulin addition and filament-bound GTP hydrolysis

EB1 bound to two tubulin protomers

Total simulation time

Probability of protomer being unbound from EB 1

Equilibrium probability of protomer being unbound from EB 1

Irreversible velocity

Reversible velocity

Maximum expected velocity

Probability of protomer bound to TE

Equilibrium probability of protomer bound to TE

Protofilament end position

Equilibrium surface position

Equilibrium position of protofilament-bound EB 1 subunit

Transition state distance


[T-GDP] :

[T-GDP](-),:

T-GTP :

Tk2:

Tk3:

Tk4:

Tk:

Tk-E:

Tk-TE:

Tk-TTE:

Tm:

TTE:

t:

u:

ueq:

v:

vr:



w:

weq:

x:

z:

ze:

A:


AG:


Net free energy change of the tubulin cycle










AGo:


AGr-nloss :


AG(+add:

AGexchange :


AGhydrolysis:

AGPI-release :

3:





r:

r:


p :

z


Initial free energy change of the tubulin cycle


Free energy of T-GDP dissociation from MT minus-end


Free energy of T-GTP addition to MT plus-end


Free energy of GDP/GTP exchange in solution


Free energy of MT-bound GTP hydrolysis


Free energy of MT-bound phosphate (Pi) release


Viscous drag coefficient


Hookean spring constant


Viscosity


Stiffness of MT protofilament under compression


Surface density of EB 1 on motile obj ect


Time required for ciliary plug to shift and rebind









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

FORCE GENERATION BY MICROTUBULE END-BINDING PROTEINS

By

Luz Elena Caro

December 2007

Chair: Richard B. Dickinson
Major: Chemical Engineering

Microtubules are cytoskeletal filaments essential for multiple cell functions, including

motility of microorganisms and cell division. Of particular interest is how these biological

polymers generate the forces required for movement of chromosomes during mitosis and for

formation of cilia and flagella. Defective microtubule-based force generation can lead to various

pathological complications; therefore, an understanding of microtubule force generation is

important for cancer research and biotechnology.

The mechanism by which elongating microtubules generate force is unknown. Several

proteins, including End-Binding Protein 1 (EB l) and adenomatous polyposis coli (APC),

specifically localize to microtubule elongating ends where the microtubule is tightly bound to a

motile obj ect and generating force. The role of these end-tracking proteins is not fully

understood, but they likely modulate microtubule-motile surface interactions, and may aid in

force production.

The obj ective of my research is to elucidate the role of polymerizing microtubules and end-

binding proteins, specifically EB1, in force-dependent processes by formulating a model that

explains their interaction and role in force generation. The commonly assumed Brownian

Ratchet model describing the forces caused by elongating microtubules cannot easily explain









force generation during rapid elongation and strong attachment of the microtubule to the motile

object. I propose a novel mechanism in which EB 1 proteins behave as end-tracking motors that

have a higher affinity for guanosine triphosphate-bound tubulin than guanosine diphosphate-

bound tubulin, thereby allowing them to convert the chemical energy of microtubule-filament

hydrolysis to mechanical work. These microtubule end-tracking motors are predicted to provide

the required forces for cell motility and persistent attachment between the motile surface and

polymerizing microtubules.

I have developed mechanochemical models that characterize the kinetics of these

molecular motors based on experimentally determined binding parameters and thermodynamic

constraints. These models account for the association of EB1 to tethered and untethered

elongating microtubule ends, in the absence or presence of force, and with or without EB 1

binding to solution-phase tubulin. These models explain the observed exponential profile of EB 1

on untethered filaments and predict that affinity-modulated end-tracking motors should achieve

higher stall forces than with the Brownian Ratchet system, while maintaining a strong, persistent

attachment to the motile obj ect.









CHAPTER 1
INTTRODUCTION

Forces produced by microtubule polymerization are required for chromosomal movement

during mitosis and ciliary/flagellar formation (Dentler and Rosenbaum, 1977; Inoue and Salmon,

1995; Dogterom and Yurke, 1997). End-tracking proteins (a.k.a. tip-tracking proteins), such as

end-binding protein 1 (EBl) and adenomatous polyposis coli (APC), have previously been

shown to bind specifically to the polymerizing microtubule plus-end where the microtubule is

tightly bound, at the kinetochore and at the tips of growing cilia/flagella (Allen and Borisy, 1974;

Dentler, 1981; Severin et al., 1997), suggesting a possible role for end-tracking proteins in force

generation at these sites. A few models demonstrate how end-binding proteins may be involved

in microtubule force-generation, suggesting that end-tracking proteins bind weakly to the

microtubule plus-end and serve as a linker between the MT and a motile surface (e.g.,

kinetochore) (Hill, 1985; Inoue and Salmon, 1995; Rieder and Salmon, 1998; Maddox et al.,

2003; Maiato et al., 2004). However, these models cannot explain the strong attachment of the

microtubule to a motile obj ect during elongation, nor the energetic and mechanism of the

interaction between the end-binding proteins and a motile surface. The obj ective of my thesis

research was to help elucidate the role of microtubule elongation mediated by end-binding

proteins in force generation. Our models explain and characterize the interaction of end-binding

proteins with growing microtubule ends and their role in force generation

Understanding the functions of microtubules and end-tracking proteins in cellular motility

and cell proliferation is of great importance to the medical field, particularly in the area of cancer

research. For example, the end-tracking protein APC not only plays a potentially key role in

microtubule-chromosome attachment during cell division, but it also suppresses excessive cell

production that could lead to colon cancer. Cells with a specific mutation in APC, which prevent









its binding to microtubules and EB 1, lead to aneuploid progency and an absence of APC's tumor

suppression function (Fodde et al., 2001b; Kaplan et al., 2001). By providing insight into the

potential function of these proteins and the interaction among them is just one example of how

our research can provide a significant impact in cell biology.

1.1 Microtubules

Microtubules (MTs) are versatile polymers that occur in nearly every eukaryotic cell.

They provide form and support in cells, aid in mitosis, guide transport of organelles, and enable

cell motility (Olmsted and Borisy, 1973; Yahara and Edelman, 1975; Dentler and Rosenbaum,

1977). Microtubules are hollow, tubular structures composed of 8-nm a/P-tubulin heterodimers;

where the P-subunit can bind to either a guanosine triphosphate (GTP) or guanosine diphosphate

(GDP) nucleotide (Farr et al., 1990). Tubulin bound to GTP assembles head-to-tail to form the

13 asymmetric, linear protofilaments of a microtubule (Figure 1-1) (Chretien et al., 1995;

Chretien and Fuller, 2000). Each protofilament has the same polarity, with a P-tubulin at one end

(minus or slow-growing end), and an a-tubulin at the other (plus or fast-growing end) (Chretien

et al., 1995; Chretien and Fuller, 2000). The structural polarity of the microtubules is important

in their growth and ability to participate in many cellular functions. During microtubule

polymerization (MT growth), GTP-tubulin protomers add to the plus-end of a MT, the subunits

then hydrolyze their bound GTP and subsequently release the hydrolyzed phosphate. During

depolymerization (MT shortening), GDP-tubulin subunits are released from the MT minus-ends

at a very rapid rate (Desai and Mitchison, 1997).

The elongation velocity of a microtubule during polymerization, v, is reported as 167 nm/s

for the free, microtubule plus-end during mitosis (Piehl and Cassimeris, 2003). Assuming









irreversible elongation at the MT plus-end in vivo, this value can be used to estimate the plus-end

protofilament effective on-rate constant of tubulin ( f) association,


rd -[Tbh] (1-1)

where d is the length of a protomer and [Tb] the intracellular GTP-tubulin concentration (~10

CIM; (Mitchison and Kirschner, 1987); yielding k= 2 CIM- -s- The critical concentration for

tubulin polymerization in vitro is [Tb], = 5 CIM, which can be used to calculate an effective

tubulin off-rate (k,) of 10.6 s^l, assuming

[Tb]e = ky / k, (1-2)

This calculated off-rate is used to determine the reversible, elongation rate of the microtubule

plus-end (85 nm/s) by

v, = (kf [Tbj- kr)-d. (1-3)

For the purposes of model comparisons in subsequent chapters, these rate constants for binding

and dissociation of tubulin are assumed, and v and v,- are taken as nominal velocities of

irreversible and reversible elongation, respectively, of MT plus-ends without the involvement of

end-tracking proteins.

Microtubule polymerization/depolymerization provides the forces required for cilia and

flagella assembly as well as chromosomal alignment during mitosis. During microtubule

elongation in both processes, the plus-end of the MT remains attached to the motile obj ect (i.e.,

the ciliary/flagellar assembly plug or kinetochore, respectively) (Allen and Borisy, 1974;

Dentler, 1981; Severin et al., 1997). Microtubule assembly is known to play a key role

throughout mitosis, the process of division and separation of the two identical daughter

chromosomes (Inoue, 1981; Salmon, 1989; Rieder and Alexander, 1990) In an early stage of









mitosis, replicated chromosomes (sister chromatids) are attached to each other at the centromere,

which also serves as the binding site for the proteinaceous kinetochore structure (Figure 1-2)

(Lodish et al., 1995). The outer plate of kinetochores contains proteins that bind to kinetochore

microtubules (kMTs). Other types of microtubules are also involved in mitosis, including astral

and polar MTs. However, kMTs are of particular interest because of their role in properly

arranging cellular chromosomes by generating force at the kinetochore (Lodish et al., 1995).

The six phases of mitosis include prophase, prometaphase, metaphase, anaphase, and

telophase. During prometaphase, kMTs of different length emanate away from opposite poles of

the cell, and bind their plus-end to the kinetochores of chromosomes. By rapid addition and loss

of tubulin protomers at the kinetochore, the kMTs oscillate back and forth (directional

instability), generating the force required to balance the length of kMTs on opposite ends of each

chromosome (Skibbens et al., 1993). These oscillations eventually results in the alignment of the

chromosomes at the spindle equator (congression). In metaphase, kMTs from opposite poles

experience a net polymerization at the kinetochore and net depolymerization at the poles (MT

flux) (Maddox et al., 2003), exerting tension on each of the chromosomes (Inoue, 1982). As a

result, the chromosomes maintain their alignment along the equatorial plane. The spindle

checkpoint then ensures chromosomes are properly attached to the spindle before anaphase onset

by releasing an inhibitory signal that delays anaphase if kinetochores are unattached (Rieder et

al., 1994; 1995). The kMTs proceed to depolymerize while still attached to the kinetochores

during anaphase (Coue et al., 1991), pulling the sister chromatids apart and moving them toward

opposite poles for cellular division (cytokinesis).

Kinetochores stabilize microtubules against disassembly by attaching specifically to

elongating GTP-rich MT plus-ends (Severin et al., 1997). A complex of proteins is required for









kinetochore capture by kMTs, but their interaction have not been elucidated (Mimori-Kiyosue

and Tsukita, 2003). If one of the kinetochore-associated proteins could recognize and track the

GTP-rich end, this protein could potentially provide a mechanism that would couple kinetochore

movement to force generated by MT polymerization during mitosis. Several proteins that

localize at the kinetochore-kMT attachment during mitosis have also been shown to bind to the

plus-ends of MTs (end-tracking proteins), suggesting their likely role in such a mechanism. Of

particular interest here are the plus-end tracking proteins EB 1 and adenomatous polyposis coli.

Although the interaction among these two proteins and the protein/microtubule relationship is

still unclear, a significant amount of recent research demonstrates their potential role in

kinetochore motility and kMT attachment, as discussed presently.

1.2 End-Tracking Proteins

Several MT end-tracking proteins are thought to facilitate force generation by microtubule

polymerization (Schuyler and Pellman, 2001). End-tracking proteins localize to the MT plus-end,

and when fluorescently labeled, they mark the translating polymerizing ends of MTs. Recent

studies demonstrate the ability of a variety of proteins to track the ends of growing MTs,

including CLIPs, CLASPs, pl50glue"d, APC, EB 1, etc. It is suggested these end-tracking proteins

aid in control of MT dynamics and in attachment of MTs to a motile surface (i.e., the cell

membrane or kinetochore) in several organisms, such as fungi and humans.

1.2.1 EB1

Of particular interest here is the EB 1 protein, because it was found to localize at points

where polymerizing MTs generate force (mitosis, cell growth, flagellar movement, etc). EBl's

specific localization suggests that EB 1 has a role in modulating the attachment of the MTs to

motile surfaces and regulating MT dynamics at the attachment site to generate the forces during

these cellular processes.









EB1 is a dimeric, 30-kDa leucine zipper protein (Mimori-Kiyosue et al., 2000) with two

MT binding domains. EB 1 binds to microtubules throughout the cell cycle. During mitosis and

cell growth, EB 1 specifically localizes to the GTP-rich tubulin protomers (polymerizing unit) at

the polymerizing plus-ends of microtubules. EB 1 quickly disappears from the plus ends of

depolymerizing GDP-rich MT' s, indicating that the higher EB 1 density at plus ends requires

polymerization and/or a GTP-rich MT end. This association/dissociation behavior suggests that

EB1 has a role in targeting the MTs to a motile surface and/or regulating MT dynamics at the

attachment site (cell membrane or kinetochore) (Mimori-Kiyosue and Tsukita, 2003). This

hypothesis is supported by demonstrating that EBli-null Drosophila cells cause mitotic defects

including mis-positioning of kinetochores during congression (Rogers et al., 2002). Specific

localization of EB 1 to GTP-rich MT plus-ends is not understood, but may be the result of direct

binding of EB 1 to the GTP-stabilized conformation of the MT plus end, co-polymerization with

tubulin in solution, or recruitment by other proteins, and dissociation from GTP-bound MT

subunits (Figure 1-3).

A study performed by Tirnauer et al. (2002b) provided important quantitative data that can

be used to evaluate and provide parameters for models of EB1 interactions with MTs. As shown

in Figure 1-4. They observed an exponentially decreasing density of EB 1 from the MT tips to a

uniform density on the MT sides, with 4.2x greater EB 1 density relative to the sides. They

measured the equilibrium dissociation constant of EB 1 to MT sides in vitro to be Kd = 0.5 C1M.

Equilibrium binding EB 1 from the cytoplasm to MT sides also explains the faint uniform

fluorescence of EB 1 along the side of polymerizing MTs in vivo (Mimori-Kiyosue et al., 2000).

The above results suggested that EB 1 may either bind near plus ends with higher affinity

than MT sides, or it could copolymerize with tubulin at plus ends before release at filament sides,









which would require association between EB 1 and tubulin in solution. However, direct binding

between EB 1 and tubulin protomers in solution is uncertain. Vincent Gache's (2005) group used

sucrose gradient centrifugation to Eind that bovine brain T-GTP did not bind to an EB 1 construct.

Contrary to his finding, Juwana et al. (1999) demonstrated that recombinant EB 1 co-precipitates

with purified bovine tubulin an immunoaffinity assay, despite the 100-time lower concentration

of EB1 than T-GTP (Tirnauer et al., 2002a; Tirnauer et al., 2002b). However, other studies

report no interaction between purified EB 1 and tubulin. For example, Ligon et al. (2006)

showed that full length human EB 1 did not bind to a tubulin-affinity matrix. Nevertheless, lack

of binding in vitro does not rule out EBli's interaction with tubulin protomers in vivo, which may

require cytoplasmic components or conditions not present in these in vitro experiments.

Consistent with this possibility, EB 1 and tubulin alone in vitro do not promote microtubule

polymerization (Nakamura et al., 2001; Tirnauer et al., 2002b; Ligon et al., 2003).

1.2.2 Adenomatous Polyposis Coli (APC)

EB1 may be recruited to the MT plus-ends by other proteins such as APC. APC is a

dimeric tumor suppressor protein that plays an important protein role in preventing colon cancer.

APC is known to co-localize and interact with both EB 1 and polymerizing microtubule plus-ends

at the kinetochore and at the cell cortex (Juwana et al., 1999). Like EB 1, APC falls off the

microtubule upon plus-end depolymerization. The C-terminal domain of APC (C-APC) is

responsible for its association with EB 1 and microtubules (Bu and Su, 2003), which is

diminished upon phosphorylation of APC. In the absence of C-APC, there is an ineffective

connection between kMTs and the kinetochore (Fodde et al., 2001a; Kaplan et al., 2001; Green

and Kaplan, 2003), about 75% of cells exhibit failed chromosome congression (Green and

Kaplan, 2003), and chromosome segregation is defective (which may be responsible for colon

cancer) (Fodde et al., 2001a; Kaplan et al., 2001; Green and Kaplan, 2003). Studies also indicate









that neither the microtubule-binding domain nor the EB 1 binding domain of APC can be

compromised to obtain proper chromosomal segregation. In the absence of the EBli-binding

domain, APC localizes non-specifically to MTs (Askham et al., 2000), and in the presence of

only the EB 1 binding domain, APC distributes throughout the entire cell without binding to

microtubules or kinetochores (Green and Kaplan, 2003). These observations suggest that APC

may modulate plus end attachment of EB 1 to kMTs, help kMTs target the kinetochore, and (in

association with EBl) aid in regulating kMT polymerization during mitosis.

Other kinetochore-associated proteins (pl50Glued, CLIP-170, and CLASPs) also have direct

interactions with EB 1, have the ability to bind to the MT plus-end, and are located at the

kinetochore-MT interface (Folker et al., 2005; Hayashi et al., 2005; Mimori-Kiyosue et al.,

2005). Therefore, these components may also be involved in activation of EB1 at the MT tip

and/or linking the EBli-bound MT plus-end to the kinetochore.

1.2.3 Ciliary and Flagellar Movement

Another example of force generation mediated by polymerizing microtubules that remain

attached to the motile obj ect is ciliary/flagellar formation and regeneration. During formation of

these organelles, membrane-bound capping structures (or MT "plugs") are persistently

associated with the plus-ends of polymerizing MTs during MT assembly and disassembly

(Suprenant and Dentler, 1988). These capping structures consist of (a) a plug-like unit that

inserts into the lumen of the microtubule, and (b) plate-like structure that j oins the plug to the

membrane. Interestingly, components of the capping structure have been found to resemble

proteins within the kinetochore, as indicated by their antigenic cross-reactivity (Miller et al.,

1990), and these findings suggest that the kinetochore and ciliary/flagellar capping structures

may interact with polymerizing microtubules in a similar manner. In this regard, EB 1

co-localizes with the plus-ends of microtubules within cilia/flagella as well as those attached to









kinetochores (Pedersen et al., 2003; Schroder et al., 2007; Sloboda and Howard, 2007). Depleted

or mutated EB 1 microtubule ends significantly reduces the efficiency of primary cilia assembly

in fibroblasts (Schroder et al., 2007). Because the sites of EB 1 localization are involved in force

generation in the above organelles, the MT end-tracking properties of EB 1 are likely to play a

role in MT elongation-dependent force generation.

1.3 Force Generation Models

Although much progress was made identifying microtubule-associated proteins and their

locations, how MT elongation is coupled to force generation has not been determined. Various

force-generating models have been considered, including force from microtubule

polymerization, force from motor-protein activity, and force from affinity modulation (Mitchison

and Salmon, 2001).

1.3.1 Brownian Ratchet Models

It is commonly assumed that the Brownian Ratchet model describes the protrusive forces

caused by elongating microtubules (Peskin et al., 1993; Mogilner and Oster, 1996). The

thermodynamic driving force in this model is the free energy change of protomer addition to free

protofilament ends (Hill, 1981; Theriot, 2000). An essential feature of this model is that thermal

fluctuations open a gap between the free protoHilament plus end and the motile surface to allow

addition of each new protomer. Because the protoHilaments must freely fluctuate from the

surface, the Brownian ratchet mechanism therefore cannot easily explain force generation during

rapid elongation and strong attachment of the elongating microtubule end to the motile obj ect.

The thermodynamic stall force associated with the Brownian Ratchet model is limited by

the free energy of protomer addition, and is given by

Fazz, =(N,, kBTd)1n ([Tb /[Tbje), (1-4)










where N, = 13 is the number of protofilaments, kBT = 4.14 pN-nm is the thermal energy

(Boltzmann constant x absolute temperature), and d= 8 nm is longitudinal dimer-repeat distance.

Under typical intracellular tubulin (Tb) concentrations of 10-15 CIM and a plus-end critical

concentration [Tb], = 5 CIM (Walker et al., 1988), then Fsran = ~5-7 pN, or ~ 0.5 pN per

protofilament.

1.3.2 Sleeve Model

The Hill "sleeve" model couples polymerization of MTs with the force generated at the

anti-pole ward moving kinetochore in cells. The model assumes MTs are inserted into a sleeve

and tubulin dimers are added to the growing MT through the center of the sleeve. Movement of a

MT through the sleeve as it grows is accounted for by a random-walk approach, where the free-

energy source is the binding of GTP-tubulin protomers to MT ends (Hill, 1985).

The Tip Attachment Complex model (TAC) incorporates the idea of a "sleeve" in order to

model force-generation by MT polymerization in the presence of linker proteins. In TAC

models, the tip of the microtubule inserts into a "sleeve" containing linker proteins that bind

weakly to the subunits at/near the ends of MTs, and are assumed to grow freely by means of a

Brownian ratchet mechanism (Inoue and Salmon, 1995). The weak-binding properties of the

TAC linker proteins are assumed to allow the TAC to advance with the growing MT tip without

hindering elongation. Therefore, the assumed bonds between the TAC linker proteins and MT

have the seemingly contradictory properties of being strong enough to sustain attachment of the

motile obj ect, while at the same time being weak enough for their rapid unbinding/rebinding to

permit unhindered elongation. In contrast, the models we are proposing suggest that linker

proteins behave as end-tracking motors that have unique binding proteins allowing them to









maintain a strong, persistent attachment between the protofilament and a motile surface during

MT elongation.

1.3.3 Kinetochore Motors

Several researchers have proposed a motor-induced force-generation model. One such

model is known as the "reverse Pac-Man" mechanism (Maddox et al., 2003), where plus-end

directed motors move kinetochores anti-pole ward during plus-end kMT polymerization (Inoue

and Salmon, 1995). The plus-end directed motor protein, CENP-E, was assumed to play this

role because of its localization to the kinetochore and its role in sensing kMT attachment at the

kinetochore (Abrieu et al., 2000). However, recent experimental evidence shows that the CENP-

E protein is not required for chromosome congression (McEwen et al., 2001). This result does

not dismiss the possibility that MT motors contribute to anti-poleward kinetochore motility in the

cell; there are other kinetochore-associated motor proteins (i.e., MCAK) of unknown function.

One recent force-generation model, the "slip-clutch" model, integrates both the reverse Pac-Man

and lateral- TAC mechanisms. This model represents the polymerization state of the kinetochore

by a "slip-clutch" mechanism involving molecular motors and linkerr" proteins that are attached

to the kinetochore and bind along the wall of MTs. The energetic of such a mechanism have not

yet been analyzed, but it suggests that the proteins involved provide force at the kinetochore, and

prevent strong forces from pulling MT plus ends out of their kinetochore attachment sites

(Maddox et al., 2003).

1.3.4 Filament End-Tracking Motors

Dickinson & Purich (2002) first proposed a model for actin-based motility whereby end-

tracking proteins tethered elongating filaments to motile obj ects and facilitated force generation.

In this mechanochemical model for actin subunit addition, surface-bound end-tracking proteins

bind preferentially to newly added ATP-bound terminal subunits on each subfilament and release









from ADP-bound penultimate subunits. This cycle facilitates force generation of persistently

tethered filaments by capturing the free energy of ATP hydrolysis in the monomer addition

cycle. The ATP hydrolysis-driven processive tracking on the filament end gives the end-binding

protein the characteristics of a molecular motor. We later proposed that the interaction of

microtubule end-tracking proteins with terminal GTP subunits could similarly explain force

generation and persistent attachment of MT's at motile obj ects (Dickinson et al., 2004). The

models presented in this thesis are quantitative extensions of that initial model.

1.4 Thermodynamic Driving Force

The thermodynamic advantage of GTP-driven affinity modulated interactions can be seen

by accounting for the free energy requirements of the tubulin polymerization cycle. The net free

energy of the tubulin cycle (AG) is partitioned among the five key steps of the tubulin cycle:

tubulin addition (polymerization), filament GTP hydrolysis, phosphate (Pi) release,

depolymerization, and GDP/GTP exchange in solution (Figure 1-5). The net free energy of this

cycle is the sum of these individual free energies

AG =G+)add + AG' hyroyii~s + A~e -releise + AG(-)loss + A~exchangSe, (1 -5)

this is equal to the net free energy of GTP hydrolysis:

AG = AGo k,Tln([GTPI/[GDP][f D, (1-6)

where AGo 11k,T is the standard-state free energy change for tubulin in vivo (Howard, 2001).

The free energy changes for the individual assembly steps are listed in Table 1-1, where Kp, is the

equilibrium dissociation constant of reversible phosphate binding to GDP-tubulin protomers and

Kx is the equilibrium constant for the GTP/GDP exchange reaction. Based on literature values

(Table 1-2), the free energy from the combined filament-bound hydrolysis and phosphate-release

steps account for ~1 1 kBT Of energy, which is nearly half of the total energy of the tubulin cycle










(AG 22 kBT; (Howard, 2001), and is significantly greater than the free energy of monomer

addition at the MT plus-ends (~5.8 kBT). Hence, considerably greater forces can be expected by

exploiting the ability of end-trackers like EB 1 that bind preferentially to T-GTP protomers,

thereby providing a pathway for harnessing the energy released by MT-bound GTP hydrolysis to

facilitate protomer addition and resultant force generation.

1.5 Summary

Microtubule polymers play an essential role in force generated during cell division, ciliary

movement, and many other cell processes. The polarity of MTs is key features that allow them

to provide guided transport and to target specific proteins, such as end-tracking proteins, EB 1

and APC. EB 1 is known to specifically localize to the GTP-rich end of MTs when MTs are

polymerizing at the leading edge of growing cells and when MTs are polymerizing at the

kinetochore during mitosis. These properties suggest a critical role of EB1 force generation by

MTs. Prior force-generation mechanisms involving end-binding proteins and MTs have been

proposed including TAC and models involving ATP-driven MT motors kinesis and dynein,

which move on MT sides. This thesis explores the hypothesis that end-tracking motor facilitate

plus-end attachment and force generation, by harnessing the energy nucleotide triphosphate

(NTP) hydrolysis and converting it to mechanical work. The key feature of this model is that the

end-tracking proteins binding specifically to the NTP-bound monomers on the filaments, a

feature correlates well with the properties of the MTs and their corresponding end-tracking

proteins.

1.6 Outline of Dissertation

The layout of this dissertation is as follows. Chapter 2 describes a preliminary

mechanochemical MT end-tracking model which was first developed to demonstrate how end-

tracking proteins on a motile obj ect (e.g., kinetochore) can facilitate MT attachment, elongation









and force generation. This model demonstrates the principles of filament end-tracking and force

generation and assumes EB 1 is immobilized at the motile obj ect, but it does not account for the

interaction of EB1 from solution with MTs. End-tracking models based on interactions of

monovalent or divalent solution-phase EB 1 with MT protofilaments are modeled in Chapters 3

and 4, respectively. Chapter 3 first treats the simpler case of monovalent EB 1 to illustrate how

the exponential EB 1 density on MT tips results from affinity modulated interactions and how

simply allowing EB 1 to bind reversible to flexible proteins (e.g., APC) in the kinetochore

comprises an end-tracking motors. Chapter 4 then addresses the more realistic (and complex)

case of divalent EB 1, which makes similar predictions at the monovalent case, but predicts

enhanced processivity due to EBli's divalent interactions with the MT lattice. Both Chapters 3

and 4 discuss the growth of a single protofilament allowing EB 1 binding, the probabilistic model

used to determine optimal kinetic parameters, and stochastic simulations of protofilament growth

against a load. Chapter 5 explores an MT model with EB 1 end-tracking from a rigid "plug,"

reflecting ciliary /flagellar growth. Finally, Chapter 6 summarizes the work completed and

suggests future directions.









GTP-bound tubulin
(~T-GTP)


GDP-bound tubulin
(Tl-G DP)



} Protofilament
S(-) end


GTP GDP


(+) end


Figure 1-1. Microtubule structure. Tubulin bound to GTP polymerize into 13-protofilament
polymers: microtubules. Because tubulin is a heterodimer, the microtubule has a
structural polarity with a plus and minus end. During MT polymerization, T-GTP
binds to the MT plus-end, which induces hydrolysis of the penultimate tubulin
subunit causing filament-bound GTP to be converted to GDP. T-GDP dissociates
from the minus end.


8


Figure 1-2. Chromosomal binding site of microtubules. Two sister chromatids bind at the
centromere to form a chromosome. Kinetochore microtubules bind to the
chromosome in the kinetochore at the centromere [Reprinted with permission from
Lodish, H. 1995. Molecular Cell Biology (Figure 23-28, p. 1094). New York, New
York.]










ks' 1 k+ i'


EB1


OR


Figure 1-3. EB 1 binding to microtubule lattice. EB 1 has equal association and dissociation rates
on GDP-bound microtubule lattice. EB1 may bind directly to the microtubule plus-
end or copolymerize with tubulin in solution first.









OcEZI, ~MT Tip Background


0 46
U)ES?'Distance (pm)
Figre1-. onentraton fE1aoglnt fmcouueymauigtefursec



FigurE B 1-.concentration o along the protof iamet. [Reprne wit peasrmission fluromcn

Tirnauer, J. 2002. EB l-microtubule interactions in Xenopus egg extracts: role of EB1
in microtubule stabilization and mechanisms of targeting to microtubules. Molecular
Biology of the Cell. (Pg. 3622, Figure 4).]


OO T-GTP
SO T-GDP





























Reaction Coordinate

Figure 1-5. Thermodynamics of GDP to GTP tubulin exchange cycle. The free energy change
of the tubulin cycle, AG, is -22 ksT, which is partitioned among the various steps:
polymerization, hydrolysis and phosphate release, depolymerization, and GTP-GDP
exchange in solution.


Table 1-1. Thermodynamic equations characterizing the multiple steps for GDP to GTP
conversion


lG


- 22 k,"T


L, Hydrolysis + Pi Release




III~ -GTPDepolymerization

~OO
90 T-GDP \yExchange


Definition


Equation
AG(+)add = -k,Tln([Tbj/[Tbje)

AGP,-reles = kg Tln([ ]/K,

Acc-3;oss = k,Tln([T GD)P]/[T GDP],-a )

AGexc~hange~ = RkTIn([Tbj- [GD~P]/[T GDP]- [GTPI)
- kTln(Kx )

[T GDP },[T~


Addition of T-GTP to MT plus-end

Phosphate (Pi) release
Loss of T-GDP from MT minus-end

GDP/GTP Exchange


Hydrolysis of MT-bound GTP in terms of
free energy of other steps


Polymerization









Table 1-2. Equilibrium constants used in energy equations
Symbol Reaction Value Reference
Kx GTP/GDP exchange 3.00 (Zeeberg and Caplow, 1979)
K, Pi binding to filaments 25.00 mM (Carlier et al., 1988)
[Tb]c T-GTP addition to MT plus-end 0.03 C1M (Howard, 2001)
[T-GDP](-)c T-GDP addition to MT minus-end 90.00 CIM (Howard, 2001)
*Calculated from the ratio of measured equilibrium dissociation constants of nucleotide binding
to the protomer, i.e., Kx= KGDP/KGTP









CHAPTER 2
MICROTUBULE END-TRACKING MODEL

This chapter describes a preliminary model that simulates the growth of a 13-protofilament

microtubule (MT) in the presence of surface-tethered EB 1 end-tracking motors. While this

model does not account for EB 1 binding from solution, it does illustrate the principle of MT

end-tracking and force generation on a motile object. As described in the previous chapter, the

key feature of the EB 1 end-tracking motor is that it captures filament-bound GTP hydrolysis

energy and converts it to mechanical work. In the model presented here, EBlI's dimeric structure

allows it to maintain persistent attachment of the MT plus-end and the motile surface (i.e., a

processive motor) and it is expected to allow for larger stall forces than the Brownian Ratchet

Model. EB1 is modeled as a Hookean spring whose binding to the MT depends on its

Gaussian-based probability density, which is a function of EBlI's equilibrium and binding

positions. An external load applied to the motile surface affects the probability of EB1 binding

and the velocity and maximum achievable force of the microtubule. The velocity as a function

of applied force and the resulting stall forces are simulated and analyzed.

2.1 EB1 End-Tracking Motors

The preferred binding of EB 1 to MT plus-ends is reminiscent of the interaction between

end-tracking proteins and actin in the actoclampin end-tracking motor model and suggests that

end-binding proteins may behave as end-tracking motors. To explore this possibility, we model

EB1 as a protein tethered to a motile surface on one end and interacting with the MT plus-end

through its MT-binding domain on the other end. There are two key features of a MT

end-tracking motor: affinity-modulated interaction driven by hydrolysis of GTP on the filament

end, and multiple or multivalent interactions with the filament end to maintain its possession to

the motile surface. EB 1 is assumed to bind preferentially to filament GTP subunits and release









from GDP subunits, thereby capturing some of the available hydrolysis energy, stabilizing GTP-

bound terminal subunits, and increasing the net free energy of protomer addition. Because EB 1

dimers are multivalent and multiple EB 1 molecules can interact with each MT end, the

end-tracking motors to maintain a strong interaction with the protomiament even when other

end-tracking units release, thereby allowing the motor can advance processively along the

polymerizing MT end. This processive action is driven by GTP hydrolysis and is the primary

characteristic of other molecular motors, such as kinesin, except in this case hydrolysis occurs on

the MT rather than on the MT-binding protein.

2.2 Microtubule Growth Model

Our preliminary MT end-tracking model illustrated in Figure 2-1 simulates the growth of a

13-protoHilament microtubule bound to surface-tethered EB 1 motors and analyzes the force

effects on the growth of the microtubule. By capturing part of the filament-bound GTP

hydrolysis energy and converting it to mechanical work, the resulting stall force is expected to

exceed that of the Brownian Ratchet Mechanism, which is driven solely by free energy of

monomer addition. The model assumes that EB 1 is tethered to the motile obj ect and does not

bind to tubulin protomers from solution, although solution phase EB 1 exists in the cytoplasm and

likely interacts with tubulin in solution (see Chapter 1). These complications are addressed in

the subsequent chapters. The key reactions for the present model are shown in Figure 2-2 and

include several possible end-tracking "stepping motor" pathways for two EB 1 dimeric subunits

(referred to hereafter as EB 1 "heads") operating at the plus-end of each MT protofilaments.

Considering Stage A, where only one EB 1 head is bound to terminal GTP-bound subunit, as the

beginning of the cycle, monomers can add directly from solution (Reaction 1), which triggers

hydrolysis on the now penultimate subunit, resulting in Stage B. The second EB 1 head then

binds the new terminal subunit (Reaction 2, resulting in Stage C). The first EB 1 head then









releases from the penultimate subunit (Reaction 3) to restore Stage A, with the net effect of the

cycle of having added one protomer. We also allow for binding of the second EB 1 head in the

wrong direction (Reaction 5) or T-GTP addition when both heads are bound (Reaction 4), either

of which results resulting in Stage D. Note that of the two EB 1 heads remains associated with

terminal T-GTP until hydrolysis of its GTP is induced when a new tubulin protomer adds to the

protofilament end and/or when the second EB 1 head binds the newly added Tb protomer.

Hydrolysis weakens the "older" EBl1-MT bond, thereby releasing that EB 1 head to bind to the

next added Tb protomer in the cycle. Because at least one EB 1 head should be bound at any time

during the end-tracking cycle, the protofilament remains associated with the motile obj ect (i.e.,

the motor is processive). (Long-term processivity may not be essential when there is a high

density of EB 1 molecules on the surface near the protofilament; even if both heads are released,

other EB 1 molecules would quickly capture the protofilament end.)

In the absence of hydrolysis-induced affinity modulation, the principle of detailed balance

would fix the relation among the various equilibrium dissociation constants in Reactions 1-3

shown in Figure 2-2, such thatK,K, = [TbjeK,, where K, = k,. /kf, K? = k_ /k and


K, = k "'~ / k, are the equilibrium dissociation constants for Reactions 1, 2, and 3, respectively.

However, affinity modulation is assumed to increase k~~~ thereby increasing K3 by a factor f

such thatK,K, = [Tb]U,K / f The value off reflects the portion of the GTP hydrolysis energy

that can be transduced into work in each end-tracking cycle. Because hydrolysis and protomer

addition are the two sources of energy used for force generation in this mechanism, the

thermodynamic stall force is characterized by

F,,,, = (N -k,T/d)1n([Tbj-KK/K, )=(N-k, T/d)(In(hl[Tbj/Tbj)+In~f)). (2-1)









The first term on the right hand represents the contribution of tubulin addition without GTP

hydrolysis (same as that of a free MT in the Brownian Ratchet model). The second term

corresponds to the benefit of having GTP-hydrolysis-driven affinity modulation. For example,

f = 1000 corresponds to ~7 kBT additional energy captured per cycle, putting Fsran at ~54 pN, a

value that is much higher than the ~7-pN stall force predicted for a Brownian Ratchet driven

solely by the free energy of protomer addition (c.f., Eq. 1-4).

While Eq. 2-1 provides a thermodynamic limit, MT growth by the end-tracking cycle may

kinetically stall at a lower force, whose value can be determined by stochastic simulation. To

simulate the elongation of the 13 protofilaments of an EBli-bound microtubule, we made several

simplifying assumptions about the binding properties of EB 1 to the protofilament lattice. We

assume that only one EB 1 dimer operates processively on each of the 13 protofilaments at any

one time. Any lateral effects among adjacent protofilaments on their elongation are neglected.

Because the EB 1 dimer has flexible segments between its coiled-coil region and its two

MT-binding heads (Honnappa et al., 2005), we modeled each EB 1 head as a Hookean springs

with spring constant y. The contribution of the spring energy, Es given by


E=
2 (2-2)

where z and ze are the instantaneous and equilibrium positions, respectively, of EBli's MT

binding domain. All EB 1 molecules bound to the motile obj ect are assumed to have the same

equilibrium position, hence ze determines the position of the translating motile obj ect relative to

the MT (assumed fixed in space). Assuming EB 1 is present at the motile obj ect with a mean

lateral spacing, p, the effective local concentration of EB 1 at the motile obj ect is










thus C,fs(z) = p p(z) where p(z) is the Boltzmann' s distribution of the EB 1 binding position,

i.e.,

e Y""22,
p(z) = (2-3)


where ks is Boltzmann's constant, and Tis the absolute temperature. The position-dependent

binding rate constant k (z) (s- ) of the EB 1 head to the MT lattice at distance : is taken as

k, (z)= konCfe,(z) or:


k, (z)= ko,,p2 (2-4)


where p is the EB 1 spacing distance, kl is the forward association rate constant (CIM- s^ or

nm3/S) for EB 1 binding to a T-GDP subunit from solution. Because binding sites are at discrete

positions spaced by distance d = 8 nm, then : = nd in Eq. 2-4.

While it is possible that a stressed bond may have an increased or decreased dissociation

rate (i.e., "slip" bond or "catch" bond, respectively) under several piconewtons of force (Bell,

1978; Dembo, 1994); (Dembo et al., 1988), we assume the simplest case where the EB 1 bonds

are neither catch nor slip under the forces involved here, and force is not assumed to not affect

the dissociation rate constants of EB1 releasing from MT sides.

The characteristic time for forces to relax between transitions is ~ 6/137, where 3is the

viscous drag coefficient 3 (drag force/velocity) of the motile object propelled by the MT. For a

~100-nm motile object, this time would be ~10-100 Cls, much faster than the cycle time for

protomer addition. Therefore, we assume the instantaneous position of the motile obj ect remains

in mechanical quasi-equilibrium with the external load, F, such that its position ze is determined

by the balance of spring forces due to the bound EB 1 heads. The equation for the external load is









given by Equation 2-5; the position of bound EB 1 heads in each stage can be determined from

Figure 2-3.

F = y(n~d ze)+ CY((n 1)dJ- ze)+ C y((2ni -1)d -z,) + Cy((2ni- 3)d- ze)
state A state B state C state D (2-5)

Solving Equation 2-5 for ze thus allows p(z) and the resulting transition probabilities for

transition between states (k-At) and a time step of At to be calculated for each EB 1 head at each

time point in the simulation. In the simulation results shown in Figures 2-4, 2-5, 2-6, and 2-7, At

was taken to 2 Cls. This time increment was chosen to be ten percent of the inverse of the largest

kinetic constant to ensure that the kinetics of all reactions was accounted for.

2.2.1 Parameter Estimations

The key parameters in this model include [Tb], [Tb],, 7, p, and the kinetic rate constants

shown in Figure 2-2. The intracellular tubulin concentration [Tb] was assumed to be 10 C1M

(Mitchison and Kirschner, 1987). We use the value of the plus-end critical concentration [Tb], =

5 C1M estimated by Walker et al. (1988) from the ratio of on- and off-rate constants for

elongation (8.9 CIM- s^ and 44 s^l, respectively). The macroscopic on-rate constant (8.9 IM- s^l)

from MT elongation rate measurements reflects the collective assembly of the 13 protofilaments

on the MT tip; however, the growth rule for individual protofilaments is uncertain. We therefore

made the simplest assumption that each protofilaments operates independently and elongates

reversibly with on-rate constant kf- 8.9/13 CIM- s^ or 0.68 C1M- s^ and k,= 44/13 s^l or 3.4 s^l

The MT reversible elongation speed used in the model was assumed to that determined by Piehl

and Cassemeris (167 nm/s) and not the velocity calculated by the on and off-rates from Walker et

al. The spring constant yof an EB 1 head was estimated as y = ker d where cr~ 10 nm is the

estimated standard deviation in the z-position of an EB 1 head based on EM micrographs









(Honnappa et al., 2005). The spacing p= 7.5 nm was chosen assuming EB 1 dimers are closely

packed on the motile obj ect. The association rate constant for an EB 1 dimer on a MT-bound

T-GDP subunit, kon = 25 CIM- s^ = 5 x10 nm3/S, was assumed by taking a typical association rate

constant for protein binding in solution (Eigen and Hammes, 1963). The off-rate constant koyS=

0.24 s^l for an EB 1 dimer from MT GDP-subunits was calculated from the measured velocity

and the exponential decay-length of EB 1 dissociating from the wall of a polymerizing MT

(Tirnauer et al., 2002a). However, this value reflects the probability of both EB 1 heads being

released simultaneously, which is assumed to be proportional to the off-rate of one EB 1 head, k_

side, multiplied by the probability of the other head being dissociated, which is K3/(1+K3), Such

that kag= k sde K/1K), hr 3- k"'" k and k is calculated at z-ze = d/2 from


Equation 2-4. The primary simulation parameter was the total simulation time, t, which was set

at 4 seconds. For f-1 to f-10,000, data points for F > 20 pN were obtained using a simulation

time of 24 seconds to allow sufficient time for the microtubule to equilibrate.

2.2.2 Elongation Rate in the Absence of External Force

A typical simulated traj ectory for a surface-tethered, polymerizing microtubule in the

absence of external load is given in Figure 2-4. Assuming an affinity modulation of 1000, and

choosing optimal values for kon and p (Appendix B.1 contains the MATLABe code), the

resulting MT position increases linearly with time. The tubulin on-rate was chosen to yield the

experimentally determined velocity of 167 nm/s for microtubules during mitosis (Piehl and

Cassimeris, 2003). Figure 2-5 is a representation of the protofilament lengths and average

equilibrium surface position corresponding to (t = 4s, v=165 nm/s). As seen in Figure 2-5, the

maximum difference between the shortest and longest filaments is four subunits. This small

difference reflects how the end-tracking model can also ensure high fidelity: the protofilaments









do not advance too far past one another during polymerization. This diagram shows that the

end-tracking motors also maintain the average equilibrium position near the filament ends. The

equilibrium surface position, z, is not located at the average filament end position since it is

dependent on the individual springs' binding location.

2.2.3 Force effects on elongation rate

To analyze the effect of applied force on the polymerization rate of EB 1 tethered

microtubules, F, was varied over a range of -4 pN to 34 pN. Figure 2-6 shows that the speed of

MT polymerization decreased with increasing external load for all values of f calculated. When

the end-tracking protein was not affinity-modulated (f= 1), the velocity decreased linearly with

increasing external force. As fwas increased, the end-tracking motor was able to capture some

of the filament hydrolysis energy to elongate more rapidly under significant forces, with the

velocity depending approximately exponentially on the compressive force. Negative (tensile)

forces applied to the surface increased the polymerization rate of growing MTs slightly until the

maximum rate was reached. Moreover, tensile forces increased the probability of EB 1 binding

to the GTP-bound filament, and promoted the forward MT assembly process. Although large

tensile forces should dissociate the filament end-tracking motors from the MT and thereby

detach the MT from motile object, the possible of complete dissociation of the EB 1 molecule

was allowed in our simulations.

As the modulation factor increased from 1 to 10, the dependence of velocity on the force

resulted in a faster elongation (Fig. 2-5) and a greater maximum achievable force. Once the

modulation factor became greater than 10, there was no significant effect of fon the

polymerization rate, and the microtubule achieved similar stall forces. These observations can

be explained by the force-limitations on the reaction kinetics. By increasingJ; the forward









reaction in step 2 is favored, increasing the rate of polymerization. Once becomes greater than

10, the forward reaction in both steps 2 and 5 become essentially irreversible (Equation 1-1).

Further increasing the modulation factor has minimal effect on the rate of reaction, MT

polymerization, and stall force.

The kinetic stall force for each simulation was taken as the force at which the speed of the

MT is less than 0. 1% of the velocity when there is no force (F=0). The thermodynamic stall

forces predicted for the microtubules at various values of were calculated from Equation 2-5,

and are compared to these simulated stall forces in Figure 2-7. The simulated and calculated

results are comparable when the EB 1 motor has little affinity modulation (from f-1 to f-10); for

~f-1, the thermodynamic and simulated stall force is approximately 7 pN. However, as f

increases, the simulated stall force deviates from the expected thermodynamic limit. This

phenomenon can be explained by the kinetic and thermodynamic properties. When the

end-tracking motors are not affinity modulated (at f-1) the critical tubulin concentration for MT

assembly remains relatively large, and the velocity is thermodynamically limited; once the

thermodynamic stall force is achieved, the MT will experience negative velocities, or net

depolymerization. At larger modulation factors (f > 1), the effective critical concentration is

reduced and the MT dynamics are kinetically, rather than thermodynamically, limited, and the

velocity can be approximated by a force-dependent exponential equation (Figure 2-5). That is,

for large values of f MTs are predicted to kinetically stall at much lower forces than the

thermodynamic stall force.

2.3 Summary

This preliminary model simulates the growth of a 13-protofilament MT bound to

surface-tethered EB 1 motors and serves to demonstrate the principles of force generation by

processive MT end-tracking motors. The key features of these end-tracking motors are (1) their









ability to capture filament-bound GTP hydrolysis and convert them to mechanical work (2) their

dimeric structure which allow them to maintain persistent attachment of the MT plus-end to the

motile surface. EB1 was modeled as a Hookean spring whose association rate with the MT is

governed by the probability density function of the spring and varies depending on an external

applied force. The dissociation rate of EB1 from the MT was determined by its affinity to

T-GTP versus T-GDP subunits, or affinity modulation factor, f: The resulting velocity as a

function of applied force was determined at varying values of f The model demonstrates EBlI's

ability to maintain fidelity of the MT, with a maximum difference in protoHilament length of four

subunits. In addition, an increasing affinity modulation of EB 1 results in an increase in stall

force, with a maximum stall force that is significantly greater than that predicted by the

Brownian Ratchet mechanism.

The primary limitation of this model is that EB 1 does not bind to tubulin in solution, nor

does it account for solution-phase EBl1. The proposed co-factor assisted end-tracking model not

only addresses the importance of a co-factor such as APC, which could be critical in the

monomer addition step, but also the issue of solution phase EB 1 and tubulin binding. This

solution binding may be essential, and is addressed further in Chapters 3 and 4.














O


T-GTP
Newvly Hydrolyzed ~T-GDP
T--GDP

EB1 dimner


Figure 2-1. Model for microtubule force generation by EB 1 end-tracking motor. Model that
represents the distal attachment of tubulin protomers at the MT plus-end. A uniform
density of EB 1 dimers on the motile obj ect links the MT protofilaments to the
surface.












1~ k S Newly Hydrolyzed T-GDP

k, av Z' EB1 dimer


k side'




St C (4) rStag eD








Figure 2-2. Reaction mechanisms of EB 1 end-tracking motor. Mechanism of the EB 1
end-tracking motor on the plus-end of one of the MT protofilaments (from upper
left): One EB 1 head is initially bound to the terminal GTP-tubulin subunit. Step 1: A
tubulin protomer binds to the filament end from solution. Step 2: The second EB 1
head binds to the newly added terminal subunit. The complex can now follow two
different pathways, 3 or 4. Step 3: Binding of second EB 1 head to MT end induced
hydrolysis of penultimate subunit and attenuates affinity of EB 1 bound to the
penultimate subunit; this EB 1 head is released from the MT and the protofilament is
returned to its original state. Step 4: A tubulin protomer adds to the filament end
from solution, inducing hydrolysis of the penultimate subunit. Step 5: Affinity of
distal EB 1 to hydrolyzed subunit is attenuated and is released from the MT.




























Figure 2-3. Force dependence on EB 1 binding and equilibrium surface position. Binding position
of EB 1 dimers in each stage, where n represents the position of the bound tubulin
subunit along the protofilament.


700

600

500

400

300

200


F=0
f= 1000


1 2 3 4
Time (s)


Figure 2-4. Microtubule elongation in the absence of external force. (A) The position of a
13-protofilament MT tethered to a surface by EB 1 end-tracking motors is plotted as a
function of time. No external forces are applied to the surface, the modulation factor
is set to 1000, and optimal values of kon, cr, and L are used (See Appendix B.1i). The
average velocity of 172 nm/s is near the set value of 167 nm/s.


SaeA







SaeC





n n-1


Stage D





n-1 n-2













S85




Ecr 82

819




80


HProtomers
Avg z














1 2 3 4 5 6 7 8 9 10 11 1213
Protofilament Number


Figure 2-5. Distribution of protofilament lengths for microtubule end-tracking model. Filament
lengths for microtubule described in Figure 1 (t = 4s, v = 660nm/s ). The maximum
difference between the shortest and longest filaments is four subunits. The solid blue
line represents the average equilibrium position of the microtubule.















* f=10000
* f=1000


15 20 25 30


35 40


-100


Figure 2-6. Effect of applied force on MT elongation rate. The dependence of velocity on force is
presented for models with various modulation factors: 1, 5, 10, 100, 1,000, and
10,000. For f~l, the velocity decreases linearly with force, shown by the fitted line.
For f5S to f 10,000, the velocity decreases exponentially with increasing force. The
data is fitted to a three-parameter exponential equation represented by the solid line.
The stall force for each simulation is estimated as the force at which the velocity is
less than 0. 1% of the velocity when F=0. The simulation time was set at 4s. For f5S
to f 10,000, data points for F > 20 pN were obtained using a simulation time of 24
seconds to allow sufficient time for the microtubule to equilibrate.


Velocity vs Force


Force (pN)










Stall Force vs Hydrolysis Factor


8 0 ---------------

70 a

60
2
S50


Thermodynamic
Simulated
Fitted (Thermodynamic)
Filted (Simulated)


100
Hydrolysis Factor (f)


"1000


10000


Figure 2-7. Thermodynamic versus simulated stall forces. The thermodynamic stall force was
calculated for the various MT end-tracking motor represented in Figure 4 by using
equation 2-5. Comparison of the calculated and simulated stall forces is shown.
When the hydrolysis affects the microtubule dynamics very little (f-1 to f-5) the
model provides a good prediction for the EB 1 end-tracking model. At higher f
values, the data deviates from thermodynamic predictions. The simulated stall force
only slightly increases once becomes greater than 100. The solid lines represent
logarithmic fit to each of the data presented.









CHAPTER 3
PROTOFILAMENT END-TRACKING MODEL WITH MONOVALENT EB 1

The microtubule end-tracking model developed in Chapter 2 neglected solution-phase End

Binding protein 1 (EBl1) and binding to microtubules and tubulin protomers. While not essential

for end-tracking, binding of EB 1 from solution is evident in the exponential profiles of bound

EB 1 at elongating free plus-ends and the apparent equilibrium density of EB 1 along the length of

the microtubule (MT) (Chapter 1 and Figure 1-3). To account for binding solution-phase EB 1,

we first developed a simplified model that simulates the growth of a single protofilament in the

presence of a monovalent EB 1 protein... A model for more complex and realistic case of dimeric

EB1 binding is presented in the next chapter.

In the previous chapter, it was assumed that the EB 1 protein behaves as an end-tracking

motor, with preferential binding to T-GTP over T-GDP, and an affinity modulation factor greater

than 1. In this chapter, the assertion that EB 1 has a higher affinity for GTP subunits is supported

by showi ng that the ob served 4.2 tip -to- si de rati o of EB 1 den sity requi re s GTP-hy droly si s-driven

affinity modulated binding. We do so by first modeling free filament growth with EB 1 binding,

but without attachment of EB 1 a motile obj ect, and comparing the predicted EB 1 density along

the length of the MT to the experimental results. We then allow EB 1 to interact with a linker

protein at the motile obj ect (e.g., Adenomatous Polyposis Coli, APC) and predict the resulting

MT dynamics and force generation. The force-velocity relationship of this the end-tracking

model is then compared to those of the simple Brownian Ratchet mechanism.

3.1 Non-Tethered Protofilament Growth

We first consider growth of a single microtubule protofilament in the presence of a

solution-phase monovalent EB 1 and then show in Section 3.2 how linking the growing tip to a









surface containing a flexible binding protein for EB 1 forms an end-tracking motor similar to that

described in Chapter 2.

The various reactions considered in the free MT model are shown in Figure 3-1. Tubulin

protomers (Tb) can add directly to filament ends (equilibrium dissociation constant [Tb],), or

they can first bind to EB 1 (E) in solution (K;) then add as an EB l-tubulin complex ([Tb]cE) .I

either pathway, tubulin addition is assumed to be followed by prompt GTP hydrolysis on the

penultimate subunit. Because EB 1 is assumed to have a lower affinity for GDP subunits

(equilibrium dissociation constant Kd > Kd*), the energy provided by GTP hydrolysis later

releases the EB 1 from non-terminal subunits.

3.1.1 Thermodynamics of EB1-tubulin interactions

As described in Chapter 1, free energy of the direct binding pathway is given simply by

[Tb]
AG(+)add = -k,Tln~
[Tb], (3-1)

but the free energy of the net reaction involving binding and release of EB 1 is

[Tb] [E] [E]
AG(+)add = -k,Tln kTln + +k TIn (3 -2)
[Tb], Kd Kd

hence more negative than that of monomer addition. In this way, EB 1 binding temporarily

stabilizes the protofilament plus-end and facilitates the net reaction of monomer addition. The

principle of detailed balance requires that Eq. 3-2 holds whether E binds first to Tb in solution or

E binds to the terminal subunit following monomer addition. Eq. (3-2) can be re-written

[Tb]
AG(+)add = -k,Tln k, T n f (3-3)
[Tb],

where f= KdKd* is the affinity modulation factor. Like in the previous chapter, at f- 1 would

represent the case in which EB 1 binds to both GTP- as well as GDP-tubulin with equal affinity.









A value greater than one means signifies that the affinity of EB 1 to GTP is greater than its

affinity to GDP.

To determine the solution phase concentration of E and TE, we assume [E] is determined

by equilibrium binding with Tb and to sides of the MTs within the cell (at subunit concentration

[MT]). As derived in the Appendix (A.1i), this assumption yields

[E]o
[E]=~

K, Kd (3-4)

and

[E]o [Tb] /K,
[ TE ]
,[Tb] [MT]
K, Kd (3-5)

3.1.2 Kinetics of EB1-tubulin interactions

We have assumed that the affinity modulation factor must be greater than 1 for EB 1 to

track on the plus-ends of protofilaments (i.e., EB 1 must have a higher affinity for GTP rather

than GDP). To test this assertion, we developed a probabilistic model accounting various EB 1

binding pathways shown in Figure 3-2: EB 1 binds directly to the GTP-rich protofilament

plus-end, EB 1 associates with T-GDP on the side of the protofilament, or EB 1 copolymerize

with tubulin in solution. This model was used to predict the probability of EB 1 binding to the

protofilament plus-end, pend, and the equilibrium probability of EB 1 binding protofilament sides,

peq. The value of peq WAS obtained by the steady-state of a differential equation describing the

probability of EB 1 binding to an MT side (far from the plus end) as a function of concentrations

and reactions rates, which is given by

dp, =-"T"-kE- dp+(k,E[TE]+k [E] 1-p)
dt (3-6)









The equations specific for solving the probability of EB 1 binding to the plus-end and side of the

protofilament are given by

= k [E~c -k p, kTbjk,E TE] p,_z p,) + kiEP + ki.cjp X,~ -p) (3-7)


dp3 = k [E],c, -kp,- p,[Tbj+kiE TE]c, +[T ktczp, [TbjeKdk,Ep102, (3-8)
dt K, f

respectively, where c, 1-p, (see full derivation in Appendix A.2). Here, the index i represents

the subunit on the protofilament numbered from the plus-end. These two differential equations

were numerically integrated (fourth-order Runge-Kutta method; Appendix B.3 contains the

Matlab code) under set parameters in order to calculate the occupational probability (p,) of EB 1

along the length of a free protofilament (equivalent to the EB 1 binding density).

3.1.3 Parameter Estimations

The key parameters in this model, which include [Tb], [Tb]c, [MT], [E]o,J; and the kinetic

rate constants, were obtained from literature or calculated based on known values. The

intracellular tubulin and microtubule concentration, [Tb] and [MT], was assumed to be 10 CIM

(Mitchison and Kirschner, 1987), the value of the plus-end critical concentration [Tb], = 5 CIM

was estimated by Walker et al. (Walker et al., 1988), and the total intracellular concentration of

EB1 was estimated as 0.27 CIM (Tirnauer et al., 2002b). Unless otherwise indicated, the value

f=103 was chosen for the affinity modulation factor, which reflects 7 ksT of the available GTP

hydrolysis energy captured for affinity modulation. (As shown below, many predicted properties

because asymptotically independent of f> 1). The rate constants, kfand k, were assumed to be

the same for free MT elongation as from Eq. 1-1 and Eq. 1-2, respectively, assuming a maximum

velocity v=170 nm/s (Piehl and Cassimeris, 2003). The dissociation constant for EB 1 to the

GDP-bound side (Kd) of protofilaments was taken as 0.5 CIM, based on an in vitro study on EBl1-









MT binding interactions (Tirnauer et al., 2002b). The rate equations for the on rates of EB 1 to

the MT plus-end and sides are assumed to be equal and are based on the observed decay rate

constant of EB 1 form MT sides determined in a study by Tirnauer et al (Tirnauer et al., 2002b).

The off-rates for EB 1 on both the sides and plus-end of the protofilament are a fraction,J; less

than their on-rates. This on-rate constant of TE binding to plus ends is assumed equivalent to that

for Tb, (i.e., =Ekf).

Experimental data has not validated a dissociation constant for the binding of EB 1 and

tubulin in solution, K;, so its optimal value was determined from peq (at steady-state, Eq. 3-9)

and spend (from Eq. 3-8).


pe (3-9)
eqK
1+d
[E]

Assuming a value for f pend WaS calculated for various values of K;. The ratio of pend to peq

was determined at each chosen K; value to determine which K; resulted a pend Peq ratio of 4.2 at

steady-state. This procedure was repeated for various values of f and the resulting K; values are

shown in Figure 3-3. When fis equal to one or two, the EB 1 binding ratio remains below the

expected 4.2 value at all values of K;. This result suggests that EB 1 must have a greater affinity

for T-GTP than T-GDP (i.e., fmust be greater than two) in order for EB 1 to accumulate at the

plus-ends of protofilaments as seen experimentally. The optimal value of K; (for greater than

two) increases with increasing values of f At larger values off(f-5 and f-10), the optimal value

for K; is approximately 0.21 CIM. Increasing past 10 does not provide any additional effect on

K;. By increasingJ; the binding reaction of EB 1 (to either the plus-end or the side of the

protofilament) is favored. Once becomes greater than ~10, these forward reactions essentially

become irreversible and the probability of EB 1 binding to the protofilament end is no longer










dependent on K; Thus, further increasing fhas minimal effect on the net rate of tubulin

addition. Assuming an affinity modulation factor of 1000, the optimal value for K; (0.21 CIM)

was determined from the results obtain in Figure 3-3.

3.1.4 Results

Figure 3-4 shows the steady-state EB 1 binding density profies for f=1 and f =1000. When

~f-1 the steady state occupational probability is uniform along the length of the protoHilament.

The slightly higher EB 1 density at the plus-end reflects some benefit of copolymerization with

tubulin. However, EB 1 is predicted to have a much larger density at the plus-end when fis large.

The EB 1 density decreases exponentially along the length of the protoHilament, consistent with

experimental observations (Figure 1-4). This finding supports our assertation EB 1 must have a

significantly higher affinity for GTP-bound tubulin in order to track on the GTP-rich

protofilament plus-ends.

The effect of K; on the EB 1, monovalent occupational probability profile is demonstrated

in Figure 3-5. The model was simulated at K; values from 0.01 CIM to 1 CIM at f-1000. When

K; is small (e.g., 0.01 CIM), EB 1 preferentially binds to tubulin protomers in solution, therefore

EB1 has a high occupational probability at the plus-end of a protofilament, which decreases

along the length of the protofilament. This decay profile flattens out as K; increases; at K; =1

CIM the profile is similar to the profile of f-1 in Figure 3-4. This behavior is expected because at

larger values of K;, EB 1 has a large off-rate from tubulin protomers in solution; therefore it can

bind along the entire length of the protofilament.

3.2 Tethered Protofilament Growth

Similar to the above model of untethered protofilament, this model simulates the growth of

microtubules that bind to monovalent EB 1 motors, but also introduces a linking protein (e.g.,










APC) that tethers the protoHilament via EB 1 to a motile surface. Here, we assume reversible

binding of the linking protein on the motile obj ect to EB 1 from solution or on the MT lattice.

Otherwise, the assumptions and parameter values from the previous model were applied in this

model .

This model has the similar pathways as seen in the non-tethered model (Figure 3-6). In

Mechanism A, EB 1 binds directly to the protoHilament; in Mechanism B, EB 1 copolymerizes

with tubulin, and Mechanism C (not shown) is a combination of A and B, but it also allows EB 1

to dissociate from the tethering protein. The terminal subunit of a protoHilament is assumed not

to dissociate when bound to an EB 1 molecule.

Consider the initial configuration of each cycle as the state with the EB 1 motor bound at

the protoHilament plus-end. When tubulin adds to the protoHilament, hydrolysis of the

penultimate subunit that is bound to the motor is induced, and the motor' s reduced affinity for

the protofilament causes EB 1 to dissociate from either the protofilament (A and B) or the

tethering protein (C). In mechanism A, the motor can directly rebind to the protofilament plus-

end; whereas the motor in B has to copolymerize with tubulin in solution first, and the motor in

C has to wait until EBli-binds to the protofilament before either of the two motors can attach to

the protofilament plus end. In each of the mechanisms, once the motor rebinds, the surface

advances. These motors can continue to act processively on the end of the microtubule to

generate force and propel the surface forward.

3.2.2 Model

To simulate this monovalent EB 1 molecular motor, a probabilistic model similar to the

non-tethered monovalent end-tracking model was derived to simulate the EB 1 fluorescence

along a protofilament based on the probability of EB 1 and the tethering protein (Tk) making

transitions between different binding states. The relevant probabilities considered were:









p, = probability of EB 1 bound to the protofilament

q, = probability of Tk-E bound to the protofilament

w = probability of Tk bound to TE in solution

v = probability of Tk bound to E in solution

y = probability of Tk being unbound

The probability of Tk being unbound, y, is represented by:

y=1-w-v-[q (3-10)

Similar to the derivation of Equations 3-7 and 3-8, the transition probabilities between

states can be obtained from reaction rate constants for each pathway (Figure 3-6). The resulting

differential equations for the probabilities of EB 1 and Tk-E binding to the protofilament (in

terms of the kinetic rates) are given by Equations 3-11 and 3-12, respectively, where u, =1- q,- p,,

kT is on-rate of the linking protein binding to solution-phase EB 1, and Cgfis the effective local

concentration of the linking protein near the protofilament.


-= ko,, [E]u, kp, k,~,C f,yp + k q,

+(k,[Tb]+k,E[TE]+k,E q~)f -1-p,) jk,.EP1+k,u, +k,'yEl 1 Pr+1 (3-11)

dq,
-= k,~,C f,, yp, k ,q, + k C f~vu, k q

+(k,[Tb]+k,E[TE]+k,E(~) l ,~ u --, +k,"EP1+k,u, +k,"yEq 1 y+1 (3-12)

The differential equations for the probability of the track binding to either TE (3-13) or EB 1

(3-14) in solution were also determined by the reaction rates and corresponding probability for

that reaction.


dw= kT [TE]y kr wi + k,[Tb]v k, w kfE eg,1w + kY 41
dt (3-13)










= k,.[E]y-k, v-k,[Tb]v +k,~l w koCs,-2vu + k-,29, (3-14)


These ordinary differential equations were solved using a fourth-order Runge-Kutta method in

Matlab in order to determine the occupational probability of EB 1 along the length of a

protofilament, as well as the effect of force on the velocity of the filament. The velocity of the

protofilament is obtained to the steady-state net rate of the tubulin addition and dissociation

pathways:


V =d [Tb]e (3-15)

+kfE TE~]+ CI l)e-F Tb]cE end +end ;

There are two ways tubulin can add to the protofilament plus-end, directly with an on-rate ofkf

or copolymerizing with EB 1 with an on-rate of kfE, hence, there are two rates of tubulin addition

included in the equation. Assuming direct tubulin addition, the first term of the equation

accounts for the effect of applied force on direct tubulin addition (e-Fd) and the dependence of the

forward rate on the critical concentration when the protofilament plus-end is not bound to EBli.

The second term represents the case when tubulin copolymerizes with EBli. This part of the

equation accounts for: the effect of applied force on both direct TE addition and Tk-TE addition,

and the dependence of the forward rate on the critical concentration [Tb]cE when the

protofilament plus-end is bound to EBli. The probabilities, qend and pend, WeTO SOlVed from

Equations 3-11 and 3-12 for the protofilament plus end.

3.2.3 Parameter Estimations

The protein concentrations used for the simulations in this section are the same as those in

the monovalent, non-tethered case. The kinetic rate constants were calculated from detailed

balance. The on-rates for an EB 1 subunit (or head) to the protofilament side (konside) and to the









protofilament plus-end (kon) were calculated based on the observed decay rate of an EB 1 dimer

from the MT side, kagf= 0. 11 s- EB 1 in solution can bind one of its heads to the side of a

protofilament with a rate of konside- [E], and dissociates from the protofilament with a rate of kside

At equilibrium, the decay rate of an EB 1 dimer from the MT side, koyS, is equal to the sum of

these two terms:

key = konslde [E]+k~slde (3-16)

Rearranging this equation gives


kosd = k o'(3-17)
onon =[E] +Kd

where Kd side/konside. The off-rate constant for EB 1 from the GDP-bound tubulin subunits, k

side, iS calculated from Kd. The off-rate of EB 1 from the protofilament plus-end, k_, is the equal to

the off-rate of EB 1 from divided by a factor of f

The linking protein was assumed to be a flexible, spring-like tethering region with position

fluctuations (cr) of 10 nm. The resulting effective concentration of the linking protein near the

protofilament is estimated like Cey, for a 3-D normal distribution on a half-sphere. The normal

Gaussian distribution of the spring is given by Equation 3-18, where yis the spring constant and

is equal to k T/c?,

1 yd2"
C, = eex p(-8
cri~ 2k,TA

The surface area of the binding location, A,, is estimated as half a sphere (2nd) since the linking

protein can only bind to the one half of the microtubule at a time. This value is analogous to p-2

in the EB 1 effective concentration calculated in Chapter 2.









3.2.4 Results

Figure 3-7A shows the predicted density of EB 1 along the length of the protofilament

(zero represents the plus-end). The protein species considered are the non-tethered EB 1 protein

in solution, the EB 1 tethered to the protofilament, and the sum of the two species. The density of

the tethered EB 1 species shows a high concentration of EB 1 at the protofilament plus-end which

decreases along the length of the protofilament. This decay behavior is expected; it requires more

energy for the linking protein (spring) to maintain attachment at distances from the protofilament

plus-end, and EB 1 is expected to have preferential binding to the protofilament plus-end due to

its affinity modulation. The unattached EB 1 protein does not seem to bind significantly at the

protofilament plus end, most likely because the on-rate of linking protein to EB 1 at this location

is much greater than its dissociation rate. The non-tethered EB 1 experiences a small peak in

probability near the protofilament plus-end, likely because it was initially GTP-bound, and

eventually dissociates from T-GDP.

The force-velocity profile for these mechanisms is shown in Figure 3-7B, which compares

a protofilament whose driving force is the monovalent EB 1 motor to where the driving force is

solely free monomer addition. At an affinity modulation factor of 1000, the end-tracking model

provides a higher maximum achievable force (~1.2 pN) demonstrating its advantage over the

thermal ratchet model, whose stall force is 0.4 pN. However the advantage is modest because of

the monovalent nature of this end-tracking motor requires it to detach from the protofilament

during the cycle, thereby still permit tubulin dissociation, which is energetically favored while

EB 1 is unbound. However, it is known that EB 1 is actually a dimer (Figure 3-8), with two MT

binding domains, which may facilitate processivity by allowing one EB 1 head to remain bound

while the other head releases. EB1 may therefore behave as a divalent motor, which would

provide the end-tracking model with the advantage to allow rapid MT polymerization while










maintaining a persistent attachment between the MT and the motile surface. This idea is explored

in Chapter 4.

3.3 Summary

This chapter described two models that simulate the growth of a single protofilament in the

presence of a monovalent EB 1 protein to determine the advantages of the mechanochemical

process over a simple monomer addition-driven (Brownian ratchet) mechanism. They key

characteristic of these models is that they account for the reaction between solution phase EB 1

and tubulin protomers. In the previous chapter, it was assumed that the EB 1 protein behaves as

an end-tracking motor, with preferential binding to T-GTP over T-GDP, and an affinity

modulation factor greater than one. In this chapter, this assumption is supported by our finding

that affinity modulation is necessary to achieve the observed high density of EB 1 at filament

ends relative to filament sides.

3.3.1 Non-Tethered Protofilaments

The first model presented eliminates any force effects by allowing free filament growth

and EB 1 binding, and assumes that neither the EB 1 nor the protofilament are tethered to a motile

surface. Although experimental results are not conclusive as to whether EB 1 binds to T-GTP in

solution, this model accounts for several reaction pathways to allow EB 1 to bind with tubulin in

solution as well as filament-bound tubulin. The dissociation constant for EB 1 and free T-GTP

was taken as that need to provide a 4.2 ratio of EBli-bound subunits at the protofilament plus-end

versus protofilament sides, which would correlate well with experimental results. Large affinity

modulation factors resulted in an equilibrium value for the tubulin-EB 1 dissociation constant,

and are therefore optimal for simulation purposes. Regardless of the value for other key kinetic

rates (i.e., on-rate of tubulin, kf, or on-rate of EB 1 on protofilament sides, k ), it is required for

EB1 to have a larger affinity for T-GTP rather T-GDP (f >1) in order to achieve the 4.2 ratio.










This result supports the assertation that EB 1 has an affinity-modulated interaction with tubulin,

which is not accounted for in the Brownian ratchet mechanisms, but is the key characteristic of

the end-tracking model.

This model predicts the density of EB 1 bound along the length of a protofilament, and

compares the results from affinity modulation to a mechanism with no affinity modulation. The

optimal, equilibrium EB l-tubulin dissociation constant was used to calculate the binding

probability of EB 1 to the plus-end and sides of a microtubule protofilament. The results of the

model demonstrate that the mechanism with no affinity modulation results in a near-uniform

EB 1 density along the entire length of the protofilament. However, large affinity modulation

results in a greater EB 1 binding at the protofilament plus-end that decays along the length of the

protofilament, a prediction which agrees to experimental results showing the decay of

fluorescent EB 1 on a non-tethered microtubule.

To simplify this complex model, several assumptions were made. First, tubulin addition

induces filament-bound hydrolysis at the protofilament plus-end. The affinity modulation is

assumed to affect only the off-rates of the protein interactions and not the on-rates. Because

EB1 stabilizes the protofilament end, it is assumed that the terminal subunit of a protofilament

cannot dissociate if bound to an EB 1 molecule. All protein concentrations are considered to be

constant.

3.3.2 Tethered Protofilaments

We have previously proposed a potential role of EB 1 acting as a co-factor protein in

end-tracking mechanisms (Dickinson et al., 2004). Consistent with this proposition, the second

model allows EB 1 binding to be translated to MT force generation by introducing a linking

protein that attaches the monovalent EB 1 protein to a motile surface. To simulate this

monovalent EB 1 molecular motor, a model similar to the non-tethered, monovalent end-tracking









model was used. The reaction mechanisms considered were the same with exception of

association and dissociation of EB 1 from the surface linking protein. As a result, these motors

act processively on the end of the microtubule to generate force and propel the surface forward.

The occupational probability of all EB 1 species (tethered and non-tethered) demonstrates

that at large affinity modulation factors there is a high occupation of total EB 1 at the

protofilament plus-end, which decays along the length of the protofilament. This decayed

concentration of EB 1 along the protofilament is comparable to the decay profile shown by

Tirnauer et al. (Tirnauer et al., 2002b) for EB 1 on free-growing protofilaments. More

importantly was the effect of force on the end-tracking model. This model demonstrates the

potential of the monovalent end-tracking motor to provide a higher maximum achievable force

(~1.2 pN) than the thermal ratchet model (0.4 pN). However, the advantage is not that

significant because it was assumed that EB 1 is a monovalent protein instead of its true

configuration as a divalent protein.

For simplification, this modeling approach neglected the potential energy exerted by

compression and extension of the spring-like linking protein, particularly when a load is applied

to the motile surface. When a load is introduced, there is an associated change in the kinetic

reactions between the end-tracking complex and the protofilament that would affect the

occupational probability ofEB1 and the force-velocity profile. The subsequent tethered

protofilament growth model with divalent end-tracking EB 1 motors will account for the force

effects on the linking protein.














Stabilization


.K*


Release


Figure 3-1. Schematic of non-tethered, monovalent EB 1 end-tracking motor mechanisms.
Tubulin protomers (Tb) can add directly to filament ends with an equilibrium
dissociation constant [Tb],, or they can first bind to EB 1 (E) in solution (with
dissociation constant K;) then add as an EB l-tubulin complex ([Tb]cE). GTP
hydrolysis on the penultimate subunit occurs upon tubulin addition to the
protofilamnent plus-end. Kd i s the EB1I dissociation from the protofilament plus-end
and Kd* is the dissociation constant for EB 1 from T-GDP, where f Kd/Kd -


TE


Figure 3-2. Various pathways of non-tethered monovalent EB 1 binding to protofilament. EB 1
can bind directly to the GTP-rich protofilament plus-end, or EB 1 can associate with
T-GDP on the side of the protofilament, or EB 1 can copolymerize with tubulin in
solution.


*) T-GTP
O T-GDP


Tb [Tb]c


I~X>3=


fK


1P"v"


kp sidek side
k Er onid
kE
















Sf=50
I~ -f=100
B qh- ~ ~ef=1 000

$ lt~9~c I ~9~~aK = 0.21 IrM








0.01 0.1 1 10
Tubulin-EB1 Eq. Dissoc. Constant, K1 (prM)


Figure 3-3. Choosing an optimal K~value for monovalent EBli. The experimentally determined
ratio of EB 1 binding to the tip versus the side of a protofilament (4.2) is represented
by the dotted line. Each curve represents a different affinity modulation factor value,
J; and the data points correspond to the EB 1 binding ratio at various values of K; and
a kon of 2.1 IM- s^l. The value ofK; required to achieve a tip-to-side binding ratio of
4.2 for f< 50 increases with increasing f: The optimal value of K; chosen was 0.21
CIM where f >10. The simulation time was 1000 seconds.











0 04


0.03 -




a 0.025-


W 0.02-


S0.015-
O

0.01


0.005
0 50 100 150 200 250 300 350 400

Tubulin Sub~unit

Figure 3-4. EB 1 density profile on a non-tethered microtubule protofilament with monovalent
EBli. Considering the various mechanisms of EB 1 binding, the occupation probability
for both fIl and f 1000 are shown. At/1~ the steady state occupational probability
is uniform along the length of the protofilament. When/f1000, EB 1 has a high
occupational probability at the plus-end, which decreases along the length of the
protofilament.

























0 -------
0 100 200 300
Tubulin Subunit


Figure 3-5. Effect ofK; on profile of monovalent EB 1 occupational probability. Considering the
various mechanisms of EB1 binding, the occupation probability for K; from 0.01 to 1
CIM at f-1000 is shown. When K; is 0.01 CIM, EB 1 has a high occupational
probability at the plus-end, which decreases along the length of the protofilament.
This decay profile flattens out as K; increases; at K~ =1 CIM the profile is similar to
the profile of f-1 in Figure 3-4.


r T-GTP
O *T"-GDP


Figure 3-6. Schematic of tethered, monovalent EB 1 end-tracking motor mechanisms. Tubulin
protomers can add directly to filament ends, or they can first bind to surface-tethered
EB 1 in solution then add as an EB l-tubulin complex. GTP hydrolysis on the
penultimate subunit occurs upon tubulin addition to the protofilament plus-end. EB1
is allowed to dissociate from surface linking protein (pathway not shown here).


I)


~/J~


I


SEB1

~Jr protein~"


1 i~c -Mch

Mech B










A








O


-- -Brownlan
Ratchet
-+ End-Tracking












1 1,5 2 2.5


i 0a,5


Figure 3-7. Force effects on a tethered protofilament with monovalent EBli. A) Occupational
probability of EB 1 along length of protoHilament (zero represents plus-end) at f 1000.
Two protein species considered: un-tethered EB 1 and tethered EB 1 on protoHilament.
Occupation of EB 1 for each species decreases along length of protoHilament. B)
Force-Velocity profie. Maximum achievable force for end-tracking model at/f~1000
(~1.2 pN) exceeds that of Brownian Ratchet model (0.4 pN)


Tfubulin Subunit


Force (pN)















(C)- 5



Coiled coil reg on


(N) MT-binding domains



"/~


SGTP- tubulin
O GDP- tubulin
~hEB1


Figure 3-8. Divalent EB 1 represented as divalent end-tracking motor. A) Depiction of EB 1
structure characterized from crystal structures. The C-terminus is represented by (C)
and the N-terminus is represented by (N). Reprinted by permission from Macmillan
Publishers Ltd: [Nature] (Honnappa et al., 2005) copyright (2005). B) Schematic of
end-tracking motor complex comparable to crystal structure of EBli.









CHAPTER 4
PROTOFILAMENT END-TRACKING MODEL WITH DIVALENT EB 1

The protofilament end-tracking model described in Chapter 3 is a simplified model that

does not account for the divalent structure of EBl. This chapter discusses the more realistic

models developed that simulate the growth of a protofilament in the presence of divalent EB 1

end-tracking motors. Similar to Chapter 3, we first model the growth of an untethered

protofilament in the presence of solution-phase EB 1 end-tracking motors. The model accounts

for the solution binding of tubulin and EB 1, and predicts the EB 1 density along a polymerizing

protofilament, with a 4.2 tip-to-side ratio of EB l. The second model allows EB 1 to bind to a

motile surface via a tethering protein. The resulting protofilament dynamics were analyzed, and

the force-dependent velocity was compared to that of the Brownian Ratchet mechanism.

4.1 Non-Tethered Protofilament Growth

As in the models from previous chapters, EB1 is assumed to preferentially binds to T-GTP

rather than T-GDP. For simplicity, we assume that if one subunit ("head") is bound to the

protofilament, the remaining unbound head can only bind to an adj acent tubulin subunit.

4.1.1 Kinetics of EB1-Tubulin Interactions

The reactions considered in this protofilament model are shown in Figures 4-1 and 4-2. We

assume for all pathways that GTP hydrolysis of the penultimate subunit occurs immediately after

tubulin addition (O'Brien et al., 1987; Schilstra et al., 1987; Stewart et al., 1990) and that

solution-phase EB 1 can exist in three forms: unbound (E), bound to one tubulin protomer (TE),

or bound to two tubulin protomers (TTE). To determine the concentrations of these three

species, we determined [E] by assuming equilibrium binding with Tb and microtubule sides, and

equilibrium binding of TE and Tb. EB 1 is a homodimer, so it is assumed that both tubulin

binding domains ("heads") are identical and non-cooperative; this property allows EB 1 to bind to









its first or second tubulin protomer with an equilibrium dissociation constant, K;. As derived in

Appendix A.1i, these assumptions results in the following concentrations:

[E]o
[E] = 2

1 +Tb]K [MZT]K
K, Kd(4-1)

2[E][Tb]
[ TE ] =
K, (4-2)


[TTE] = [E[2b
K, (4-3)

Figure 4-1A shows the two methods in which unbound EB 1 in solution can bind to a

protofilament, by adding directly to filament ends (kon) after tubulin addition (kf), or by first

binding to tubulin in solution (k;) then adding as an EB l-tubulin complex (kfE). Since unbound

EB1 can bind to Tb in two identical ways, the on-rate for these binding steps is doubled (2k; or

2kon). Once EB 1 is bound, there are two pathways that result in attachment of EBli's second head

to the protofilament plus-end (Figure 4-1B). One pathway involves direct binding of the EB 1

head to the terminal, GTP-bound subunit (k ) after tubulin addition (kf). The other pathway

allows the EB 1 head to bind to solution-phase Tb (k;) and facilitate tubulin addition by shuttling

it to the protofilament plus-end (kfE "''). The value for kfE "'' accounts for the on-rate of TE and

the local, effective concentration of the unbound EB 1 head (CeyS), which is represented by

Equation 4-4.

kE'"= kEC'# (4-4)

The terminal two subunits at this stage are both bound to the same EB 1 protein. The state of the

terminal subunit is referred to as dbEt (double-bound to EB 1 on plus-end) and the penultimate

subunit is in state dbE- (double-bound to EB 1 on minus-end). Figure 4-1C shows how TE in









solution can bind the protoHilament: binding directly to the protoHilament plus-end (kon) after

tubulin addition (kf), or by first binding to tubulin in solution (k;) then adding as an EB l-tubulin

complex (2kfE).

Figure 4-2 shows the various mechanisms by which EB 1 can bind to the GDP-rich

protofilament side. As shown in the Eigure, free EB 1 can bind directly from solution to a T-GDP

subunit (pathway A), and subsequently bind its unbound head the neighboring subunit (pathway

E). Additionally, TE can bind to the side of a protofilament with an on-rate of kogside (pathway

C).

4.1.2 EB1 Occupational Probability Model

A probabilistic model similar to the tethered, divalent end-tracking model (Chapter 3) was

developed to simulate the pathways shown in Figure 4-1. This model determines the EB 1

density along a protofilament based on the probability of each tubulin subunit being in a specific

EB1 binding state. The relevant probabilities considered were:

p, : probability of subunit i bound to EB 1 head (other head unbound)

w, : probability of subunit i bound to TE

q, : probability of subunit i in state dbft

q,~ : probability of subunit i in state dbK

u, : probability of subunit i being unbound

The probability of the subunit being unbound, u,, is represented by Equation 4-5.

u, = 1 q,' + q, p, w, 45

The probability of Tb being in any one of these binding states is based on its reaction for that

pathway, the probability of the reaction, and the corresponding protein concentrations. The

probabilities are defined by a set of ordinary differential equations (in terms of the kinetic rates),









and are represented by Equations 4-6 through 4-10 (Appendix A.3), where R+ and R_ are defined


by Equations 4-11 and 4-12.


2ko,,[E]u, k p, + k, w,


kl [Tb]pl + k qll k plul + k ~+q


k plul '


(4-6)


+ R (p,_z pi ) +R (pl ,


kw, w +R w,_,


w, )+R (w +


ks,,[TE]u, kw, + k, [Tb]p,


(4-7)


q,


-k ~ ', q-k ,q' +k ~p u +k pu ,u +R qy~,_ -q, )+R qI 1


(4-8)


= -k ~,q, -k ~ q +k ;p ,_zzi+k ~ p~~u +R q,_z


k p + kw, +k ql + k q k pl, u k pl,_u 21


(4-9)


(4-10)


ko,, [E]u, k,, [TE]u'


+ R (u,l u, ) + R (u,+


R_ = kf[Tb] +kfE([TE] + [TTE])+ kfEC I

R_ = ku,u + k~'(, E 1 1 1 ,


(4-11)


(4-12)


At equilibrium, these probabilities reduce to Equations 4-13 through 4-16, where qeq

and K-k side/k side

2ko,, [E ]
Peq k usq


w pe
eg K "


4eq 4eq





(4-13)



(4-14)


f eq = 2 -K -p g2< <


(4-15)


q,)+R q










[il T ] 2ko,"' [ E] [1 IT] ] 2ko,"' [ E] ko ,,""' [E ]
+1 d +1 + +1 +1+1K
K, k K k si"' k ,,"
U ,e(4-16)
8 ko, [E ]
k Sl"

The results of these equations were used to analyze the distribution of the divalent, EB 1

end-tracking motors on the non-tethered protofilament, and determine the equilibrium EB 1

concentration along the protofilament, Peq.


PeI', P, 'e, e, / 2 (4-17)

4.1.3 Average Fraction of EB1-bound Subunits at Equilibrium

A stochastic model was developed that determines the average binding fraction of EB 1

along a non-growing protofilament in order to test the previous probabilistic model and compare

the results. The pathways considered for this model are those where EB 1 binds to the side of the

protofilament and not the plus-end, which are shown in Figure 4-2. To model these pathways,

the state of each tubulin subunit was analyzed. During the simulation, the state of each subunit

in the protofilament was initially unbound from EBli. (The Matlab code can be found in

Appendix B.3.2). The transition probability in time At for each pathway reaction was analyzed; if

that reaction occurred, then the state of the tubulin subunit would change to its new state. The

EBli-binding state of each subunit was used to determine the fraction of EBli-bound subunits in

the protofilament; this fraction was averaged over time for a total simulation time of 40 seconds.

4.1.4 Average Fraction of EB1-bound subunits during protofilament growth

A stochastic model was also used to calculate the time-averaged fraction of EBli-bound

subunits during protofilament growth. The pathways considered in this model are those shown

in Figure 4-2., as well as the association and dissociation of tubulin from the protofilament

plus-end (Figure 4-1). The stochastic model used to simulate these pathways is very similar to









the previous model; it utilizes the same kinetic parameters, and the state of the tubulin subunits

was determined from reaction rate for each pathway. (Appendix B.3.3 contains the Matlab code

for this model.) The EBli-binding state of each subunit was determined for each time step and

the fraction of EBli-bound subunits in the protofilament was averaged over time for a total

simulation time of 40 seconds.

4.1.5 Parameter Estimations

Several key kinetic parameters listed in Figures 4-1 and 4-2 have not yet been determined

experimentally, including the dissociation constant for Tb and EB 1 in solution (K;) the EB 1

on-rate for protofilament-bound T-GTP (kon), the EB 1 off-rate from protofilament-bound T-GDP

(ksi'de), and the value ofK. To solve for these parameters, it was first assumed the protofilament

was at equilibrium (i.e., it does not polymerize). At equilibrium, the fraction of filament-bound

subunits attached to EB 1 is given by Equation 4-18 (see appendix A.3 for full derivation).

[E]o [E] 1
p (4-18)
[E]o Kd,eff
+1
[ MT] ]to

When half of the protofilament is saturated with EB 1, the effective equilibrium dissociation

constant of EB 1 and the protofilament (K, ...1 is given by Equation 4-19, where ul/z- ueq

([E]-[E]o/2).


K -(4-19)
def kside 2kside

kszd ksde 1/ 2
,,1/2*

Under this constraint, um/ is given by Equation 4-20, where k side=konsideCe~ffOr the protofilament

plus end, and kside= k side/K










K [E]o +1 + K- E +1I +8K2 [E]o

ul/2 =(4-20)
2[E]o
4K2
C,

Assuming [Tb]=0, [E]o=0.27 CIM, [E]= [E]o/2, and Cey-153 C1M, Equations 4-19 and 4-20 and the

experimentally determined value for Kd~,ef Of 0.44 CIM (Tirnauer et al., 2002b) where used to

determine the value of K as 37. With this known value of K, k side can be represented as a

function of K; from Equation 4-21i, where kofyis the known off-rate of dimeric EB 1 from a

protofilament (koyS=0.26 s l, (Tirnauer et al., 2002b)). Appendix A.3 contains the derivations for

these equations and parameter calculations.



kslede(K1 )= kov 1+ "',K (4-21)
[Tb]


To determined the optimal value of K; (that provides a 4.2 tip-to-side EB 1 binding ratio), the

probabilistic model discussed in section 4.1.2 was simulated under different values of fand K;

(Appendix B.3.1 contains the Matlab code). Figure 4-3 shows the results of these simulations.

When fis equal to one, the EB 1 binding ratio remains below the expected 4.2 value at all values

of K;. This result suggests that EB 1 must have a greater affinity for T-GTP than T-GDP (i.e., f

must be greater than one) in order for EB 1 to accumulate at the plus-ends of protofilaments as

seen experimentally. The optimal value of K; for greater than one increases with increasing

values of f At larger values of f(50 and 500), the optimal value for K; is approximately 0.65

CIM. Increasing fpast 50 does not provide any additional effect on K;. The reasoning behind the

trends in these results lies in the reaction rates. Both k E and k_ are inversely proportional to f so

an increase in causes the forward reactions to be favored in the mechanisms corresponding to









these rates (EB 1 binding to the protofilament plus-end). But, as approaches infinity, it reaches

a point in which the k;E and k_ become zero and no increase in fwill favor the forward reaction

further. The optimal values of K; and chosen for all simulations used in this chapter were: K; =

0.65 CIM and f>50.

The value of kon used to determine K; in the previous analysis was estimated as 1 IM- s^l,

but this value has not been experimentally determined. To ensure that the value of kon chosen

does not affect the binding ratio of EB 1 (or K;), we analyzed the effect of kon on K; The model

from section 4.1.2 was simulated for affinity modulation factors from 1 and 50, and the value of

kon was varied from 0. 1 CIM- s- to 10 IM- s^l. The resulting EB 1 tip-to-side binding ratios for

these conditions are shown in Figure 4-4. When f-1 and K; is less than 0.1 CIM, there is no affect

of kon on the binding ratio, and when K; is greater than 0. 1 CIM, there is a minimal effect of kon.

In either case, the binding ratio still fails to obtain the optimal value of 4.2. The optimal value of

K; (0.65 CIM) is not affected by the value of kon for f-50, therefore an average value of 5 CIM- s^

for kon was chosen to be used for all further simulations.

The kinetic parameters used in this chapter that were also used in Chapter 3 were determined

the same way. Additionally, the on-rate of EB1 and tubulin (k;) was assumed to be a typical

value for protein-protein binding interactions, 10 IM- s^l. Consequently, the value of ky- was

determined from the optimal K; value (K;=kl-/k;). The on-rate constant for TE and TTE to the

protofilament plus end, kfE, was assumed to be equal to the on-rate of tubulin addition, k. The

off-rate constant of TE and TTE to the protofilament plus-end, k; was calculated based on

detailed balance, and is represented by Equation 4-22.


kEE
K, ko; (4-22)









4.1.6 Results

4.1.6.1 Occupational probability

The ordinary differential equations that define the probabilities of EB 1 binding (Eq 4-6 to

Eq. 4-10) were solved to determine the expected equilibrium fraction of EBli-bound protomers in

the protofilament (Figure 4-5). This fraction was evaluated by determining Peq frOm Equation

4- 17 at various values of K; The percent of the protofilament bound to EB 1 increases with

increasing K; At large values of K; (K; > 10), the EB 1 binding fraction reaches an equilibrium,

with approximately 40% of the protofilament bound. The value of K; used for simulations (0.65

CIM) corresponds to about 2.5% of EB 1 bound to the protofilament at equilibrium. The data for

this plot was recreated for/f1000; since the equilibrium binding probability of EB 1 is not

dependent on the affinity modulation factor, the results were the same as for/1~ (data not

shown).

The set of ordinary differential equations in Equations 4-6 to 4-10 were numerically

integrated and solved at a set value ofK=37 and K;=0.65 CIM- s^ using a fourth-order

Runge-Kutta method in Matlab. The occupational probability of EB1 along the length of the

protofilament for/1~ and f 1000 is shown in Figure 4-6. When f 1, the EB 1 density is nearly

constant along the length of the protofilament at 2.5%. This behavior is expected since at/1~

EB1 does not have preferential binding to GTP or GDP-bound subunits, and this is the

equilibrium EB 1 binding fraction determined earlier (Figure 4-5). At/f1000, the probability at

the protofilament end is 0. 107, which decreases along the length of the protofilament to a value

of 0.025. It is expected that there is a higher occupational probability at the protofilament plus

end since EB 1 has a higher affinity for T-GTP versus T-GDP when#f1000. This decay profile is

comparable with the experimental results shown in Figure 1-4. The ratio of the occupational









probability at the plus and minus end of the protofilament for f-1000 is about 4.3, which is

similar to the tip-to-side ratio of 4.2 observed in experiments. For both f-1 and f-1000. The

occupational probability at the first tubulin subunit is significantly less than that of the rest of the

filament because the value of q,- is zero for the first subunit.

4.1.6.2 Average fraction of EB1-bound subunits at equilibrium

The time averaged fraction of EBli-bound protomers at f-1 and f-1000 and optimal values

for K; (0.65 CIM) and kon (5 1M- s^l) are displayed in Figure 4-7. The calculated equilibrium

fraction of 0.024 from the probabilistic model is also shown on the plot for comparison with

results from the stochastic model. Since this model only considers the binding of EB1 to

filament-bound T-GDP, the time averaged fraction along the length of the protofilament is

similar for f-1 and f-1000. In both cases, the fraction fluctuates around the equilibrium value of

0.024. Since it is assumed that the affinity modulation factor only affects the off-rates and not

the on-rates on EB 1, TE, or TTE to the protofilament plus-end, the only rates affected by fare k_

and krE. These rates do not correspond to any of the pathways considered for this model;

therefore, it is expected that there be a similarity between the two curves generated at f-1 and

~f-1000. The data for this plot was recreated for a simulation time of 20 seconds, which resulted

in no noticeable difference in the plots (data not shown).

4.1.6.3 Average fraction of EB1-bound subunits during protofilament growth

Figure 4-8 displays the average fraction of EBli-bound subunits on a protofilament when

polymerization is allowed to occur, at affinity modulation factors f= 1 and f= 1000. The

equilibrium percentage of EB 1 bound (2.4%) for K;=0.65 C1M and kon=5 C1M- s^ is shown for

comparison. The results show that the fraction corresponding to f-1 is overall slightly smaller

than that of f-1000, but both results show a larger fraction of bound EB 1 at the plus-end. This









result is expected since when/f1000 the rate of EB 1 addition at the protoHilament plus-end is

increased. When/f1000, there is a sharp decrease in the EBli-bound fraction along the length of

the protoHilament. This behavior is due to the large affinity modulation factor, which results in a

lower affinity for EB 1 to GDP-bound tubulin subunits and hence significantly reduced off-rates

from the protomiament plus-end (k,E and k_). Conversely, when f 1 the EBl-bound fraction of

subunits along the side of the protomiament fluctuates around the equilibrium value; since EB 1

has no preferential binding to GTP- or GDP-bound subunits, there are significant amounts of

EB 1 bound along the side of the protofilament.

4.2 Tethered Protofilament Growth Model

This model, similar to the tethered-protofilament model discussed in section 3.2, considers

the growth of a single microtubule protofilament in the presence of solution-phase, divalent EBli.

A flexible binding protein provides as a link between EB 1 and a motile surface, allowing EB 1 to

behave as an end-tracking motor. The various reaction mechanisms considered for this tethered,

protofilament model are those previously shown in Figures 4-1 and 4-2, and the binding

pathways involving the surface-bound tethering protein (Figures 4-9 and 4-10). The tethering

protein was modeled as a Hookean spring which exerts energy on the motile surface under a

load. The spring is defined by its spring constant, 7, which is given by


uk,T
o- (4-23)

The thermal energy is given by kBT, and a represents the variance in its position fluctuations.

The effective concentration of the linking protein near the protofilament is obtained from normal

Gaussian distribution of the spring given by Eq. 3-17 in section 3.2.3. The assumptions and

parameter values used are the same as those in section 4.1i.









4.2.1 Kinetics of EB1-Tubulin Interactions

Figures 4-9 A and B shows the two methods by which EB 1 can bind to the surface-bound

tethering protein, hereafter abbreviated Tk. Tk can associate with EB 1 in solution (pathway A) or

with protofilament-bound EB 1 (pathway B). Although pathway A only shows the reaction

between free EB 1 and Tk, this reaction can occur with TE or TTE under the same on- and

off-rates.

In pathway B, Tk binds to EB 1 on the protofilament (this could also be filament-bound TE

or doubly bound EBli). The forward kinetic rate of this reaction, kT ', is represented by Equation

4-24, and is proportional to the forward rate of Tk binding to EB 1, kT, the effective concentration

of Tk at the protofilament plus-end, CT, and the effects of the transition state and spring energies.


k,'= k,Czer(n-1)dA/2kBTe -((n-1)d)'2 2kB (4-24)

The er(n-1)da/2kBT term represents the contribution of the transition state effects from force, where

A is the transition state distance. The subunit position on the protofilament (n) is equal to one at


the plus-end and increases toward the minus-end of the filament. The e- 11 1)d)22kB7te

corresponds to the effect of stretching the tethering protein (or spring) from its initial position to

its binding position on the protofilament. Since the tethering protein's unbound, equilibrium

position is one, the number of subunits between an unbound, surface-tethered EB 1 protein and its

equilibrium binding position on the protofilament is n-1. Hence, the displacement distance of the

spring is given by (n-1) -d where d is the length of a subunit (8 nm).

The effects of the transition state energy is associated with a bond under tension; therefore

it also affects the reverse rate constants. Since the dissociation pathway in 4-9 B allows the EB 1

spring to return to its equilibrium position, the only energy associated with the reverse rate, kTm ',

is that of the transition state:










k, '= k, e' (n-1dA 2kB'


(4-25)


The mechanisms by which surface-tethered EB 1 can bind to the protomiament are shown

in Figures 4-9 C and D. Surface-bound TE can attach to the protomiament with a forward rate of

kon and a reverse rate of k_ '(pathway C), given by Equations 4-26 and 4-27, respectively.

kon '= konfr es (n-)dA 2k^BTe 11)d ) 2kB' (4-26)

k '= kes (n-1)dA, 2kB' (4-27)

The contribution of the transition state and spring energies are equal to that in equations 4-24 and

4-25. If the surface is initially tethered to unbound EB 1 (pathway D), the on-rate of EB 1 to the

filament is twice that of kon since EB 1 is a homodimer that can bind with either one of its heads

equally. For either pathway, C or D, the on-rate of EB 1 in solution to a protomiament-bound

subunit changes depending on whether EB 1 binds to the terminal tubulin subunit (kon) or a

subunit on the side of the filament (kogside)

EB1 bound to the protofilament by only one of its heads has the potential to "walk" along

the protofilament toward the plus- or minus-end. These two potential pathways are shown in

Figure 4-9 E and F. If the EB 1 motor walks in the plus direction (pathway E), no energy is

exerted on the spring and the rates of reaction are those for a single EB 1 head binding to the

protofilament subunit. However, these rates will depend on whether the EB 1 head binds to the

terminal subunit (k+ and k_) or to the side of the filament (k side and ksi'de). If the EB 1 motor

walks in the minus-direction (pathway F), the kinetic rates will be affected by the transition and

spring energies. These rate equations are described in Equations 4-28 and 4-29.

k side'= k side v(n- 1 I 2)d) 2?kB' (4-28)

k side = k sid~e (n-1)dA, 2kB (4-29)









Figure 4-10 shows the various ways tubulin can add to the protofilament plus-end that

involve the linking protein. Tubulin can add directly to the plus-end of the surface-tethered

protofilament (mechanism A), tubulin can be transferred to the protofilament plus-end by

surface-tethered TE (Tk-TE) or TTE (Tk-TTE) as seen in pathways B and C, respectively, or the

protofilament-tethered TE can shuttle tubulin to the protofilament end (mechanism D). Only the

on-rates, not the reverse rates, for these reactions are affected by the interaction with the

tethering protein. For all four pathways (A-D), when there is an applied force against the surface

(in the opposite direction of protofilament growth), F, the tubulin on-rates are reduced by a

factor ofe-Fd/kBT

For direct tubulin addition in pathway A, the forward rate, kf' is proportional to the on-rate

of tubulin addition, kf, and the effect of force, as shown in Equation 4-30. There is also an effect

of the spring energy due to the insertion of tubulin and extension of the spring.


ki'= kie- 11 1)d)22kBT -Fd/kB (4-30)

Tubulin transferred to the protofilament end by Tk-TE or Tk-TTE (pathways B and C) are

both proportional to the on-rate of TE (or TTE) to the protofilament plus end, kfE, as seen in

Equation 4-31. There is also an effect from CT and any load applied to the motile surface. The

on-rate of tubulin in these pathways is twice that of kfE because of EBli's dimeric structure.


kf' kf -F/k (4-31)

Tubulin shuttled to the protofilament plus-end by filament-bound EB 1 (pathway D) has a

corresponding on-rate of kfE "'. This forward rate (Equation 4-32) has an effect from applied

force, spring energy, and from the local effective concentration of the EB 1 head, Ceyf. This local

concentration is estimated based on the 3-D normal distribution on a half-sphere (see section

4.2.3).










kiE"= k," iE -y((1d'z,2)d 2kB -d kB' (4-32)

4.2.2 Protofilament End-Tracking Model

Considering the above reactions, stochastic models were performed to analyze the behavior of

divalent, EB1 end-tracking motors operating on a single, growing microtubule protofilament.

The model used to simulate the various reaction pathways is very similar to the model in section

4. 1.4, where the pathway taken by the EB 1 motor was determined by the probability of the

corresponding kinetic reaction occurring (Appendix B.3.4 contains Matlab code).

4.2.3 Parameter Estimations

The kinetic parameters used in this chapter section that were also used in section 4. 1 were

determined the same way. The on-rate of Tk and EB 1 binding (kr) was estimated as 5 C1M- s^

and the off-rate, kT was calculated from the value of KT provided (Kr-k//kT). The value for v

used to calculate the tubulin on-rate from Equation 1-1 was 170 nm/s (Piehl and Cassimeris,

2003), which was assumed to be the irreversible elongation at the protofilament plus-end. CT

was estimated as 100C1M. It was assumed that the bond between EB 1 and the protofilament is a

slip bond (i.e., tension force on the motile surface would increase the dissociate rate of EB 1 to

the microtubule). The transition state distance for this slip bond was estimated as 20 percent of a

typical bond length, or 1 nm.

4.2.4 Results

In the presence of a force, F, the surface-tethered protofilament polymerized in the direction

of the surface. The effect of force on the velocity of the protofilament was analyzed. The values

for F were varied, which consequently affected the kinetic rate equations and corresponding

probability for the pathways that are dependent on force. The resultant protofilament velocity

was determined by dividing the total length of tubulin dimers added to the protofilament










plus-end (17add-d) by the total simulation time, t. This model also provided the state of the

terminal subunit, position of the linking protein, the time-averaged fluorescence along the

protofilament, and time spent in each pathway.

Figure 4-11 shows the force-velocity profiles for a polymerizing protofilament with

surface-tethered EB 1 end-tracking motors. To analyze the effect of the affinity modulation

factor n the velocity profile, several affinity modulation factors were considered (Figures 4-11

A-E). Regardless of the value of f the velocity decreased as the force increased because the

force is opposite the direction of growth. For f-1 and f-10, velocities at forces greater than the

stall force (force at which the velocity is zero) were negative; at larger values of fthe velocity

decayed slower and approached zero as the force increased.

These figures also show the effect ofKT on the velocity profile. Since KT is the

dissociation constant for EB 1 and Tk, it represents the strength of the interaction between the

protofilament and the motile surface, and the protofilament cannot attach to the surface if EB 1 is

not bound to Tk. For all values ofJ fa KT value of 10 CIM resulted in a maximum velocity of

approximately 80 nm/s. This value is similar to the expected reversible elongation speed of the

protofilaments is 85 nm/s (Equation 1-3), based on the rates determined for tubulin

polymerization and depolymerization.

At decreasing values of Kr, the velocity at F=0 decreased, which is possibly due to the

tether between the protofilament and the motile surface. At lower values ofKT this interaction is

less likely to dissociate, therefore more energy is required to insert a tubulin at the plus-end. At

all affinity modulation factors, the value of KT did not affect the stall forces. However, it is

expected that as KT increases, the protofilament will spend less time attached to the surface and

will not be able to generate significant forces against a load. At KT =10 1M- s^1, the stall force









increased with increasing values offJ from 0.37 pN (at f-1) to~-1.7 pN (at f-1000), as seen in

Figure 4-11 F.

Figure 4-12 summarizes the effect of KT and fon the stall force, with the corresponding

data in Table 4-1. The thermodynamic stall forces are shown for comparison to the simulation

results. It is clear from the diagram that KT has little effect on the stall forces. When f-1, there is

no affinity modulation and EB 1 binds to T-GTP and T-GDP with equal affinity, hence the model

is comparable with the Brownian Ratchet Mechanism. Therefore, it is not surprising that the

resulting maximum achievable force at f-1 correlates well with the thermodynamics values, and

is equal to that of the Brownian Ratchet Model, 0.37 pN. For f>1, the stall forces were lower

than the predicted thermodynamic values. At increasing values of the affinity modulation factor,

the simulated stall forces increasingly deviated from the thermodynamic values. The reason the

reactions stalled at forces lower than the thermodynamic limit is that there are parallel pathways

of tubulin addition/dissociation (i.e., the direct tubulin addition/dissociation pathway and the

end-tracking pathway), and the net tubulin dissociation is favored thermodynamically for the

direct pathway and at higher forces. Increasing fpast a value of 1000 did not provide any

additional effect on the stall force. Both krE and k_ are inversely proportional to f so an increase

in f favors the forward reactions for the mechanisms corresponding to these rates. But, as f

approaches infinity, it reaches a point in which the krE and k_ become zero and no increase in f

will favor the forward reaction further.

To determine how this end-tracking mechanism mediates tubulin addition and to

understand the effects of f and KT on the velocity profiles (Figures 4- 11 and 4-12), the frequency

of the different pathways possible for association or dissociation of tubulin were measured and

the resulting percentages are displayed in Figure 4-13. For an affinity modulation factor of










1000, when the F=0 and KT =0. 1 C1M, the protofilament spent 47% of its time in free tubulin

association at the protofilament plus-end. But, when the force was increased to 2.1 pN (near the

stall force), the percentage in the forward and reverse pathways were equal (50%), which

explains the zero velocity at this force (Figure 4-11 D). An increase in KT from 0.1 to 10 CIM

when f-1000 and F=0, resulted in a larger percentage of time spent associating tubulin (47%

versus 58%, respectively), which explains why the initial velocity was slightly higher when KT

=10 CIM (Figure 4-1 1 D). The same result was found when f-1; at F=0 pN, the percentage spent

in the forward pathway at Kr-10 CIM (78%) was significantly greater than at K-0. 1 CIM (32%),

and resulted in a higher initial velocity at Kr-10 CIM (Figure 4-11 A). When comparing the two

affinity modulation factors (at Kr-10 CM and F=2. 1 pN) the time spent associating tubulin at f-1

was 93%, which was higher than when f-1000 (85%). This result explains why lower values off

resulted in negative velocities at large forces (Figure 4-11 F).

The percent of time the protofilament spent bound and unbound to the motile surface is

shown in Figure 4-14. The unbound percentage increased with larger values of KT or F. Also,

when a protofilament was surface-tethered, it was usually bound at its terminal or penultimate

subunit. The forward rate equation in Equation 4-33 shows that when the linking protein binds

to EB 1 on the terminal subunit (n=1), the on-rate is proportional to kTCT. But when n is greater

than one, the on-rate, kT ', is reduced to nearly zero. Therefore, no matter what the value of kT,

the linking protein either binds to terminally bound EB 1 or most likely it does not bind to any

filament-bound EB l.

The state of the terminal subunit in the filament was determined for each simulation to

analyze the effect of f and KT on the EB 1 binding behavior. The fraction of time spent in each

state is shown in Figure 4-15, where states Tk2, Tk3, and Tk4 represent states in which the









linking protein is bound to EB 1, TTE or dbE+ on the protofilament, respectively. For all

variations of KT, F, and f most of the time the terminal subunit was in the unbound state (state

1). When f-1, the terminal subunit was in state Tk3 or Tk4 a significant fraction of time; when

~f-1000, S1 was in state 4 and Tk4 a large amount of time. The most significant difference in the

state of the filament is when KT is 0. 1 versus when KT is 10 (for both values of f); the larger KT

value resulted in more subunits being unbound from EB l.

A graphical representation of the bound versus unbound fraction of terminal subunits for

each combination of f KT, and F is shown in Figure 4-16. The most significant result is that

when f-1000, the unbound fraction decreased with increasing force, but when f-1 the unbound

fraction increased with increasing force. This result has a significant implication for the role of

the motor. When the force was increased at f-1, the frequency EBli-bound tubulin addition

decreased (Figure 4-13). However, with large affinity modulation (f-1000), the frequency of

tubulin addition occurs increased. Therefore, at large forces, affinity modulation allows EB 1 to

facilitate tubulin addition and maintain a persistent attachment to the motile surface.

When the linking protein is unbound from the protofilament, the state of the linking protein

varied depending on the force and the affinity modulation factor (Figure 4-17). When the force

was zero the state of the linking protein was mostly either unbound or bound to E or TE, which

makes it easier to bind to the protofilament. When the force increased to 2. 1 pN and f-1000,

most of the linking protein are mostly bound to TE. When f-1 and force is 2. 1 pN, most of the

linking proteins were unbound or bound to TTE, which makes it easier to bind to the

protofilament.

4.3 Summary

To account for the dimeric structure of EB 1, this chapter discusses the models we have

developed that simulate the growth of one protofilament in the presence of either tethered or









non-tethered, divalent EB 1 end-tracing motors that processively linking protein the plus-ends of

protofilaments. Because EB 1 is divalent, even if one of its heads dissociates upon hydrolysis of

its bound tubulin, the other EB 1 head can remain bound to the protofilament. Hence, the

divalent end-tracking model has an advantage over the monovalent end-tricking model and the

Brownian ratchet mechanisms by maintaining a high EB 1 concentration at the protofilament

plus-end and allowing rapid MT polymerization

4.3.1 Non-Tethered Protofilaments

This model assumes that EB 1 is not tethered to a motile surface, but is allowed to bind to

tubulin in solution. By allowing tubulin addition (to the protofilament plus-end or side) from

solution or by copolymerization with EB 1, protofilament-bound tubulin can be in various

EBli-binding states. The probability of tubulin being in any one of these states was used to

determine the optimal dissociation constant for EB 1 and tubulin in solution (K;) that would result

in the 4.2 binding ratio. As in the monovalent case, the Brownian ratchet mechanism was not

able to obtain the expected 4.2 EB 1 binding ratio at any value of K;. EB l-tubulin interactions

with large affinity modulation resulted in an optimal value for K; of 0.65 CIM, which was used to

determine the occupational probability of EB 1 along the length of the protofilament. The results

of this analysis demonstrates the advantage of the end-tracking model over the Brownian ratchet

mechanism to preferentially bind to the protofilament plus-end and provide a decay behavior as

seen in experiments. In addition, the model is able to simulate the occupational probability

providing a 4.2 ratio of EB 1 binding at the plus-end versus the side of the protofilament.

We also created a model that analyzes the average, equilibrium fraction of EB 1 bound to

the protofilament. This model only allows EB 1 to bind to the sides of a protofilament (rich in

GDP-bound subunits) and prevented the protofilament from growing. The results of this model









show that the affinity modulation of EB 1 does not affect this side-binding behavior. The

resulting fraction of subunits bound to EB 1 was 2.6%, which is close to the expected equilibrium

value of 2.4%. The same analysis was performed for growing protoHilaments. The resulting

average EBli-bound fraction of subunits shows a slightly larger fraction of EB 1 binding at the

plus-end when the affinity modulation factor is greater.

4.3.2 Tethered Protofilaments

The tethered protofilament end-tracking model simulates EB 1 end-tracking motors

operating on a growing protofilament plus-end, and introduces a co-factor protein that tethers

EB1 to a motile surface. Unlike the monovalent end-tracking model, this model allows

association and dissociation of the tethering protein to the motile surface and of EB 1 to the

tethering protein. The tethering protein was modeled as a Hookean spring, which translates its

potential energy from mechanical work at the protofilament plus-end. This model also accounts

for any transition state effects on the on-and off-rates due to binding between surface-tethered

EB1 and the protofilament.

The force-velocity relationships developed from this model were compared to the

Brownian ratchet mechanism. Under no affinity modulation, the model predicts values

consistent with the thermodynamics values and comparable to the Brownian Ratchet mechanism,

with a resulting stall of 0.37 pN. The end-tracking model provides a stall force up to 5 times

greater than that of the Brownian Ratchet mechanism. Depending on the affinity of the

interaction between EB 1 and tubulin, the resulting stall force in the end-tracking model can

range from 0.72 pN to 1.95 pN. However, as affinity modulation increases, the resulting stall

forces deviate from the stall forces predicted by thermodynamics because the net tubulin

dissociation is favored thermodynamically for the direct pathway and at higher forces. The effect

of the dissociation rate of EB 1 from the linking protein does not affect the stall force of the









end-tracking model, but it does affect the maximum protofilament velocity. We show that an

increase in KT results in an increase in the maximum velocity, and vice-versa. A KT value of 10

CIM allows the protofilament to grow at a rate of 80 nm/s which is comparable to the calculated

value of 85 nm/s for reversible elongation.

At large forces (2.1 pN), the end-tracking model is able to maintain a persistent attachment

of the protofilament plus-end (specifically the terminal and penultimate subunits) to the motile

surface (71% of time); whereas the protofilament in the Brownian ratchet model spends most of

the time (36%) un-tethered. This result suggests that the EB 1 end-tracking motors are able to

maintain persistent attachment of the protofilament end to the motile surface, translating its

filament-bound hydrolysis energy to mechanical work and allowing the protofilament to grow

even under large loads.









Figure 4-1. Mechanisms of a non-tethered, divalent end-tracking motor. A) Top-left: EB 1 and
T-GTP free in solution. EB 1 binds to the protofilament in two ways: after tubulin
addition (clockwise) or copolymerizing with tubulin (counter-clockwise). Clockwise:
T-GTP adds to the protofilament end (kf) and induces hydrolysis of the penultimate
tubulin subunit; EB 1 binds to T-GTP at the protofilament end (2kon).
Counter-clockwise: EB1 and T-GTP bind in solution (2kl); Together, EB1 and
T-GTP add to the protofilament end (k E). B) Top-let EB1 intite bound: to the-- '
GTP-rich protofilament plus-end. Free tubulin in solution binds to the protofilament
in two ways: directly from solution (clockwise) or facilitated by the EB 1 motor
(counter-clockwi se). Clockwi se: Tubulin in solution adds to the protofilament end
(kf), which induces hydrolysis of the EBl-bound, penultimate tubulin subunit. The
unbound EB 1 head binds to the GTP-bound protofilament end (k,).
Counter-clockwise: The free EB 1 head binds to tubulin in solution (k;) and shuttles
the protomer to the protofilament end (kyE '). C) Top-Left: TE and T-GTP free in
solution. TE binds to the protofilament in two ways: after tubulin addition
(clockwise) or copolymerize as TTE (counter-clockwise). Clockwise: T-GTP binds
to protofilament end (kf), inducing hydrolysis of the penultimate subunit. TE binds to
the T-GTP protofilament end (kon). Counter-clockwise: TE binds to T-GTP in
solution (k;). TTE binds to the protofilament end (kfE) and induces hydrolysis of the
penultimate tubulin subunit.







k,

k,


kE

kr


k,

k,


kE"'

kE
r

k,

k,


kk


2kE
kf


) T-GDP


< *
0'0

2k k _,

G)


2kOn k


Etc~l=!


~f~iX_7L?


kk


g 00


SCOCC)


A EB1 T-GTP










A


2konslde a

k s'de


sdE


Ssiddee


konslde

k s'de


Figure 4-2. Mechanisms of equilibrium, side binding of EB1 to protofilament. Off-rates of EB 1
binding to protofilament-bound GDP affected by affinity modulation factor. Tubulin
addition and dissociation pathway neglected for this equilibrium mechanism.


100


o
rJ
m
PC:
cn
c
rr
E
m io
o
s
ul
.B
r
m
w


1 C
0.01


0.1 1
Tubulin-EB1 Eq. Dissoc. Constant, Ky (pM)


Figure 4-3. Choosing an optimal K~value for divalent EBli. The experimentally determined ratio
of EB 1 binding to the tip versus the side of a protofilament (4.2) is represented by the
dotted line. Each curve represents a different affinity modulation factor value, J and
the data points correspond to the EB 1 binding ratio at various values of K; and a kon
of 1 IM- s^l. The value of K; required to achieve a tip-to-side binding ratio of 4.2 for
50 > f> 1 increases with increasing f: The optimal value of K; chosen was 0.65 CIM
where f >10. The simulation time was 40 seconds.


~k~3;~?~


~ 000


G~x~;~5


~y CCO












1000


-e-f-1. kon 0.1
-a-f-1,kon =1


Ti a- f-1 koin=10D
0" 1 00 -*-S, ko n -0 .1

w~ f-5 0, k2n=10

m 10. -- ----- Rati o =42





01


0.01 0.1 1 10

Tubulin-EB1 Eq~.Diss~c. Constant, K,{prM)

Figure 4-4. Effect of kon on optimal~K;. The experimentally determined ratio of EB1 binding to
the tip versus the side of a protofilament (4.2) is represented by the dotted line. For
affinity modulation factors of I and 50, the value of kon was varied from 0. 1 to 10
IM- s^l, and K; from 0.01 to 5 CIM. The simulation time was 40 seconds. The optimal
value of K; is not significantly affected by the value of kon, and the optimal K;
remains at 0.65 CIM when f=50.
















rro 0.024






0.0001
0.01 0.1 0.65 1 10 100

Tubulin-EB1I Eq. Dissoc. Constant, K (ptM)


Figure 4-5. EB 1 equilibrium binding. Data shown in figure is for a kon value of 5 CIM- s^ and a
simulation time of 40 seconds. At equilibrium, the percent of the protofilament
bound to EB 1 increases with increasing K; At large values of K; (K;>10), the
protofilament reaches an equilibrium with approximately 40% of EB 1 bound. The
value of K; used for simulations (0.65 CIM) corresponds to an expected 2.4% of EB 1
bound to the protofilament.











0.11


0.1



E 0.08-

0.07~ -i ..000

S0.06-





O
0.03 #

0.02

0.01
U 50 100 150 200 250 300 350 400
Protomer Number


Figure 4-6. Occupational probability of EB1 along length of protofilament. Zero on the x-axis
represents the protofilament plus (growing) end. Simulation time used was 40
seconds. Values for other variables: K;=0.65 CIM, kon 5 1M- s^l. Probability of EB 1
when/1~ is nearly constant along the length of the protofilament. When/f1000,
occupational probability at protofilament end (0.107) is ~4.2 times higher than f~l
(0.025); the probability decays along the length of the protofilament.











I 1 I 1 I 1 1 1 (


-~ f=1

-f =1000

- Equilibrium


0.8 .


0.6

0.6


04-

0.3

0.2


dn.


nh nh~ ~n


0 20 40 60 830 100 1~20 140O 160 130


200


hJh..,n .rr2 ,.


n h ..,nR,.


Subunit


Figure 4-7. Time averaged EBli-bound tubulin fraction at equilibrium. Zero on the x-axis
represents the protofilament plus (growing) end. Results for both f-1 and f-1000
shown. Values for other variables: K;=0.65 C1M, kon=5 IM- s^l. Equilibrium
EBl-bound fraction represented by solid line at 0.024. Simulation time used was 80
seconds and N=200















S0.16 f =1000
ILL
P 0.14 Equilibrium



a 0.08-




01.02





U 20 40 60 80 100 120 `140 160 180 2003
Subunit


Figure 4-8. Time averaged fraction of EBli-bound subunits during protofilament growth. Zero on
the x-axis represents the protofilament plus (growing) end. Results for both fIl and
~fl000 shown. Values for other variables: K;=0.65 CIM, kon=5 1M- s^l. Equilibrium
EBl-bound fraction represented by solid line at 0.024. Simulation time used was 80
seconds and N=200.









Figure 4-9. Mechanisms of tethered, protofilament end-tracking model with divalent EB l. A)
Pathway for tethering protein to bind to EB 1 (or TE) in solution. B) Tethering
protein binds to protofilament-bound EB 1 (or TE). Energy is exerted by the spring,
which is accounted for in the on- and off-rates (kT and kr-'). C) Surface-tethered TE
binds to protofilament plus-end with on rate of kon'. D) Surface tethered EB 1 has
twice the on-rate due to EBli's dimeric structure. E) EB 1 can "walk" along the
protofilament in the plus direction (E) or minus direction (F). EB1 walking toward
the minus-direction exerts no force on the spring, and has an on and off rate of k side
and ksi'de, TOSpectively. EB 1 walking in the minus-direction exerts a force on the
spring, which is accounted for in the on and off rates (kside' and kside ')





kT

kT~


n-ll n Xn+lXn+2


kT'

k,.


kan'

k.


2kon

k.'


k side

k side'


O T-GDP


~W)

OCOO


0000


IrvL~
OOCO




OCOO


A EB1


T-GTP





k;

k,


kE


2hF'
c----------


kF"

kE
r


O T-GDP


Figure 4-10. Mechanisms oftubulin addition to linking protein-bound protofilament. A) Tubulin
can add to the plus-end of a surface-tethered protofilament. B, C) Tubulin bound to
Tk-E or Tk-TE attaches to the protofilament end. Tubulin bound to Tk-TE has two
configurations with which it can bind. D) Tubulin addition is facilitated by the
filament-bound EB 1 end-tracking motor. The forces exerted on the spring are
accounted for in the forward rate constants for each mechanism.


HIY.
000


B


~000


rZEB1


T-GTP











Figure 4-11. Force-velocity profiles for tethered protofilaments bound to divalent EB 1
end-tracking motors. The effect of force and velocity of both the Brownian Ratchet
and End-Tracking models are shown. Simulation time used was 40 seconds. Values
for other variables: k;=10 CIM- s^, K;=0.65C1M, kon 5 C1M- s^. KT values were varied
(0.1, 1, 5, and 10 CIM) in A E to analyze the effects on the stall force. A)?1~ B)
?10l C)?100o D)?1000o E)?10l,000 F) Force-velocity profiles shown for varying
values of f(1,10, 100,1000, 10000) when KT =10 C1M.









* Kr0.1 p~M
.7 Ky1pM
SKe5p1M
*Kr=lOpM


A
80 *

S60 a
Z=40 "
20 +-


-20 0


J= 1


0.1 0.2 0.3 0.4 *


f= 10


.1:h 2 0

0
0 0.2
-20

-40'

C 80
f= 100
60

403

n 8 **


* 1.2* *1.4


I 1.2 1.4 1.6 1.8 2 2.2


-20 0 0.2 0.4 0.6 0.8


Force (pN)









* Ke0-.1gM~v
. Ky'ICM
ii K-eSpM
. KEC~=Olp


f = 1,000


S60 *

40
3 2
> **?I


20 ( 0.2 0.4 0.6 0.8
E 0

80 f = 10,000

60

40

S20


-00 0.2 0.4 0.6 0.8


1 1.2 1.4 1.6i 1.8 2


2.2 2.~4


* Ke0t~.1pM
. KelpI~M
K?-SpM
KT KelPM


1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6


80 i KYT = 10 rM


* 010
Sf=100
* f=C1000
* f'-10,000


-20 0 0.2 0.


1.4 1.6 1.8 2 2.2


Force (pN)
































Hydrolysis Factor (f)


Figure 4-12. Stall forces versus affinity modulation factor at various KT values. Resulting stall
forces for data in Figures 3-19 A-E. Thermodynamnic represents the thermodynamic
values at the various values of the affinity modulation factor, based on ksT=4. 14
pN-nm, d = 8 nm, [Tb] = 10 CIM, and [Tb], = 5 CIM. Data for this figure can be found
in Table 4-1.


Table 4-1. Protofilament stall forces at varying values ofKT and affinity modulation factors.
Stall forces (in units of pN) correspond to the data represented in Figure 4- 12.
f Thermodynamic Kr = 0.1 C1M KT = 1 C1M KT = 5 C1M KT = 10 C1M
1 0.36 0.37 0.37 0.38 0.36
10 1.55 0.72 0.73 0.73 0.72
100 2.74 1.22 1.2 1.22 1.21
1,000 3.93 1.69 1.65 1.66 1.78
10,000 5.13 1.78 1.7 1.95 1.72
Thermodynamic values show the expected thermodynamic stall forces when ksT= 4.14 pN-nm,
d= 8 nm, [Tb] = 10 CIM, and [Tb], = 5 CIM.


+t Thermodynamic
+eKe0=.1pM
5 KTelpM
+ KeSpM
-+- Kr= 10pMh~


1 10 100 1000


10000
















0.8

ce



O






0.2




0.0


J=lo 100 = 100 J=lo 00 =lo 100 f= 1 J= ] J 1 J
K,= 0.1pM K,= .1 pM K, = 10pM K r = 10 pM Ky = 0.1 M K r= 0.1 p K, = 10 M K,= 10 RM
F = 0 pN F = 2.13 pN F = 0pN F = 2.13 pNu F = 0 pN F = 2.13 pN F = 0pN F= 2.13 pN
Variiables


Figure 4-13. Effect of f KT, and F on pathways taken. For each affinity modulation factor value
(1 and 1000), forces of 0 and 2. 13 pN were analyzed for both a KT value of 0. 1 and 10
CtM. For each f KT, F combination, the percentage of time the protofilament
advanced along a pathway that resulted in association or dissociation of a tubulin
protomer is shown. Pathways occurring less than 5% of the time are not shown.












1.0


Unboulng
M Bounglo 51
Boulnd to S2


0.8 ~


0.6


0.4


0.2


J= 1000 J`= 1000 /= 100 Jr= 100 /= 1 /= 1 J= 1
K,= 0.1M K,= 0.1 pM K, = 10 pM K, = 10 M K, = 0.1 M K,= 0.1 pM Kr = 10 M
F=0pN F=2.13pN F=0pN F =2.13 pN F=0pN F =2.13 pN F=0pN
Variables


J= 1
K,= I0 pM
F= 2. 13pN


Figure 4-14. Percent of time protofilament bound and unbound to motile surface. The
percentages listed are based on the same combinations of f, KT, and F values
considered in Figure 4-13. Protofilaments bound to the motile surface were
consistently tethered at the terminal tubulin subunit (Sl) or the second tubulin subunit
(S2). Percentage of time bound protofilaments were tethered to either S1 or S2 shown
for each combination of f, KT, and F values.





















_I II ~ L ~.L L1


KO=0 rJM F=0pN
H17 1p~l~M, F=2.13pN
O1 KFlopM, F=0pN
O Ke=0rl~p. F=2.13pN


n_


Tk4


1 2 3 4 Tk2 Tk3
State! of TermnlPxerooe


Figure 4-15. State of the terminal subunit (Sl) when f-1 and f-1000. KT values of0. 1 and 10
CIM and F values of 0 and 2. 1 pN were analyzed. The fraction of time the terminal
tubulin subunit (Sl) remained in each of each of the 7 different states is shown. The
various states the protofilament subunits include: 1-unbound, 2-bound to E, 3-bound
to TE, 4-bound to dbE Tk2-bound to E tethered to linking protein, TK3-bound to
TE tethered to linking protein, Tk4-bound to dbE' tethered to linking protein.


h.


I I r
















0.8



0.6 -






0.4



0.2


f =1000 f =1000 f =1000
KT= 0.I1pM KT= 0.1pM KT= 10pM
F = 0pN F = 2.13 pu F = 0pN


M nou nd
I LInbound


KT = 10pM KT= 10pMI
F = 0pN F= 2.13pN


f =1000 f = f = 1
KT= 10pM KT= 0.1pM~ KT= 0.1pM
F =2.13 pN F = 0pN F =2.13 pN
Variables


Figure 4-16. Fraction of S1 subunits bound and unbound from motile surface. Based on data
presented in Figure 4-14, subunits in states 1-4 were considered unbound and
subunits in states Tk2,Tk3, Tk4 were considered bound. All eight different variable
combinations of f KT, and F are presented.












H Tk
STk-E
OTk-TE
OTk-TTE


0.8 I


-~


o l M1 1 I r I I 1 1 I I 1 l l
/= 1000 J= 1000 f =1000 = 1000 /= 1 f = 1 f=I 1 = I
K,= 0.1pM K y= 0.1 M KY,= 10 pM K, = 10 M K1 = 0.1 RM K,= 0.1 M KI = 10 pM K,= 10plM
F =0pN F=2.13pN F=0OpN F =2.13 pN F=0pN F =2.13 pN F =0pN F= 2. 13pN

Variables



Figure 4-17. Average state of unbound linking protein. The fraction of time the linking protein
spent in each of its unbound states is shown: Tk unbound, Tk-E linking protein
bound to EB1, Tk-TE linking protein bound to TE, Tk-TTE linking protein bound
to TTE. Each of the eight combinations of variables (A KT, F) was considered.









CHAPTER 5
CILIARY PLUG MODEL

Cilium is a motile organelle made up of an array of MTs. The plus-ends of ciliary MTs are

attached to the cell membrane by MT-capping structures, which are located at the site of tubulin

addition (Figure 5-1, (Suprenant and Dentler, 1988)). As the protomiaments polymerize, the cap

remains tethered to the filament end and pushes the cell membrane forward. As mentioned

earlier, EB 1 has also been localized at the plus ends of ciliary microtubules. EB 1 tends to

localize at sites of MT force generation, therefore it was assumed that EB 1 may be behaving

end-tracking motor, similarly to the end-tracking motors in cell division and cellular growth.

This chapter discussed the EB 1 end-tracking model developed for the ciliary plug.

Essentially, the plug is the end-tracking motor, which is behaves similar to the Lock, Load and

Fire Mechanism (Dickinson and Purich, 2002). The advancement of the plug at the microtubule

plus-end occurs in three steps: tubulin addition, filament-bound GTP hydrolysis, and the shifting

and rebinding of the ciliary plug (e.g., EBl1) to the filament end. The key parameters used to

simulate this model are the diffusivity of the microtubule in the medium, the length of the ciliary

plug, the expected microtubule velocity, and the force applied against the plug. The force-

velocity relationship for the microtubule is analyzed to determine the maximum achievable force

the microtubule can withstand with the EB 1 end-tracking motor.

5.1 Model

The physical characteristics of the plug make this model complex, but for simplification,

the EB 1 end-tracking motor is represented as a plug with multiple tubulin binding sites. The

plug is inserted into the microtubule; hence, it was assumed that the plug creates a region where

the protoHilaments are separated from one another. The length of this region is labeled by a










distance L (See Figure 5-2), where the 13 protoHilaments are assumed to be independent of one

another, and each of the motors operates on a single protoHilament

The three steps of the end-tracking motor are represented in Figure 5-3. The first step in

this end-tracking model is addition of tubulin to the filament end, which induces hydrolysis of

the penultimate subunit. Because the EB 1 plug has a low affinity for T-GDP, the motor rebinds

to the GTP-rich, filament plus-end causing the plug to advance. This model is very similar to the

Lock, Load and Fire Mechanism (LLF) proposed in 2002 (Dickinson and Purich, 2002).

The total time to complete one cycle is T,;;plus z. T,;;is the time it takes for the filament to

add a dimer and undergo hydrolysis (Equation 5-1), time required for the plug to shift and

relock to the new dimer following hydrolysis, and d represents the length of a tubulin protomer.

We anticipate that elongation of unloaded protofilaments is rate-limited by Tm (confirmed

below), in which case Tm can again be estimated from v;;; (167 nm/s), i.e.,


T, = =0.05s (5-1)
max

Following the approach of Dickinson & Purich (2002), the mean shift time ris taken as the time

required for the protofilament end to diffuse a distance d and rebind and can be solved by the

differential equation given by (Eq. 5-2) (Gardiner, 1986):

D, dr dr2
F-;+D = -1
kT dx' f dx'?
(5-2)

where x is the protofilament end position, F is the force subj ected to the protofilament end, and

the protofilaments fluctuate in position with a characteristic diffusivity, Df. This diffusivity is

dependent on the drag coefficient, 3, and hence becomes a function of the length of the

independent protofilament (Equation 5-3 and 5-4)



































This equation was used to generate force-velocity profiles for the ciliary plug model.

Based on the compression stiffness of a protofilament, ic, and the thermal energy, kB ] the

stepwise motion of a microtubule with the ciliary end-tracking motor attached was also

simulated. The filament end position, x, was governed by Equation 5-7.

x ac-TI (5-7)

In the simulations for this model, the 13 protofilaments were initiated at different, random

lengths. Prior to polymerization, the length of one of the protofilaments was set so that the

initial, equilibrium force was balanced. For a protofilament to go through one cycle of tubulin

addition, GTP hydrolysis, surface advancement, the probability of the cycle occurring was

evaluated. This probability was determined based on the rate of the cycle reaction, 1/T,; T. If any


k,T

D,=

log(L /a)

By varying the length of the protofilament, we can also analyze the dependence ofr and

elongation rate on force. To determine the force dependence of r on force, the differential

equation in 5-2 was solved and is represented in Equation 5-5.

d2 ksT dF k,T k,T
z(F) exD i i!
DdF; k,T dF; dF;


(5-3)



(5-4)


With r as a function of force, the equation that governs the velocity becomes a function of force,

and is represented by Equation 5-6.


5-( 5)


d
T, + r(F)


v(F)


(5-6)









of the protofilaments underwent shifting and rebinding, the new plug equilibrium position was

determined by zeq=K x/Em,, where K, is either the stretch of compression stiffness of each

protofilament; the protofilament has a stretch stiffness if its length is less than the equilibrium

value, and it is under compression when its length is greater than the equilibrium value. The

resulting force on each filament is equal to its stiffness times the displacement of the

protofilament from equilibrium, and the overall force on the ciliary plug is the sum of these

individual forces.

5.2 Parameter Estimations

The key parameters used for this model were either based on literature values or estimated.

The width of the protomer, a, was calculated as 5.15 nm from a=2nR/N where R is the radius of

the protofilament and is equal to 11.48 nm for a 14-protofilmaent microtubule (Mickey and

Howard, 1995). The length of the ciliary plug (L), or region where protofilaments are assumed

to be independent, is estimated to be 75 nm from the EM image of the ciliary plug in Figure 5-1.

The viscosity of the fluid used to calculate the drag coefficient was assumed to be that of water,

10-9 pN-s/nm2 (Boal, 2002).

Assuming each of the protofilaments to be a semi-flexible rod, their filament compression

(e) and stretch stiffness (r) were calculated. The compression stiffness is defined by the

persistence length of the filament (h), the thermal energy, and the length of the filament, and is

represented by Equation 5-8 (Howard, 2001).

k,T i2
L4 (5-8)









The persistence length is represented by Equation 5-9, where B is the bending modulus of the

filament (B = 1.2 x 10-26 N-m2, (Mickey and Howard, 1995)). The resulting value for the

compression stiffness is 1.1 pN/nm.

S= B /kg T (5-9)

The stretch stiffness is proportional to Young's modulus (Y=1.9 GPa, (Howard, 2001)), the cross

sectional area of the microtubule (A=190 nm2, (Gittes et al., 1993)), and the length of the rod

being stretched, L:

ic= Y -A/L (5-10)

The resulting stretch stiffness for a protofilament was determined to be 370 pN/nm.

5.3 Results

Figure 5-4 shows the force effects on ciliary microtubules. In Figure 5-4A, the mean time

to shift as a function of force is shown for various protofilament lengths. Regardless of the

force, there is little effect of length on the cycle time. The time required for tubulin addition and

filament-bound GTP hydrolysis remains constant and is force-independent, so the cycle time of

the filaments is initially governed by T,;. As the load on the filament increases, the model is

governed by the time it takes for the plug to advance (r). The effect of force and corresponding

cycle time on the protofilament velocity is shown in Figure 5-4B. Again, the lengths of the

protofilaments have little effect on the velocity of the microtubule. As the cycle time increases

with increasing forces, the velocity exponentially decays to its maximum achievable force, or

stall force (Fsran). The approximate stall for the microtubules simulated is approximately 12 pN.

The position versus time data is represented by Figure 5-5A, where the x-axis is

representative of the end position of the ciliary plug. This figure shows how the ciliary plug

advances as a steady rate for a short time then jumps to a new position. The size of this jump is









usually about 8nm, which is the size of the tubulin dimer. The reasoning for the step size is that

the end-tracking motor for each protofilament must fill with tubulin before the ciliary plug can

advance. The histogram in Figure 5-5B shows the number of protofilaments at each length

greater than the equilibrium position. In this simulation, the protofilament end positions relative

to the equilibrium position range from -5 to 55 nm after the simulation time of 1.6 s from Figure

5-5A.

5.4 Summary

The ciliary plug was simulated as an EB 1 end-tracking motor similar to the end-tracking

motors described in the Lock, Load and Fire Mechanism (Dickinson and Purich, 2002). The

primary steps of this model are tubulin addition, filament-bound GTP hydrolysis and the shifting

and rebinding of the ciliary plug (e.g., EBl) to the filament end. By analyzing the force-velocity

profile of this mechanism, we found the stall force to be approximately 12 pN at various

protofilament lengths, which is significantly greater than the stall force of 4.8 pN predicted by

the Brownian ratchet mechanism. The results also shows the strong dependence of the stall force

on the time for shift/rebinding of the ciliary plug to the filament end. The velocity profile shows

the ability of the end-tracking motor to maintain fidelity of the microtubule by allowing the plug

to advance only once all protofilaments are the same length.












Figure 5-1. EM image of a ciliar~y pl ;ug atthe nd f ailiry icroubule. The average length of
the plug is approximately 75 nm. [Reproduced from The Journal of Cell Biology,
1988, 107: 2259-2269. Copyright 1988 The Rockefeller University Press]












~


Independent
Protofilaments

Figure 5-2. Schematic of ciliary plug inserted into the lumen of a cilia/flagella microtubule. The
microtubules behave as independent protoHilaments for a distance L from the MT
plus-end. Red represents the GTP-bound tubulin subunits.


T~ubulin


T-GTP

T-G DP

End-Tracking
M oto r/ E B


Figure 5-3. Mechanism of the ciliary/flagellar end-tracking motor. In the first step, tubulin adds
to the MT plus-end into the end-tracking complex. This binding induces hydrolysis
attenuating the affinity of the complex to the protomiament. The surface advances to
the filament end.


Addition I

~~m ~ -,m
Surface't GTP
Advances Hydrolysis I( _













103
IM L=50 nm
-c L=75nmn
L= 1OO nmn Tm +
8 E.10 -L=125 nm

.- T,





10-5


35
e -L=25 nm
30 -- -L= 50nm
c 1L=75 nm
25 1- -L100 nm
-L= 125 nm
c 20-



S10-




O 2 4 6 8 10 12
Force on Protofilament (pN)

Figure 5-4. Force effects on ciliary microtubules. Initially the filament is governed by the time it
takes for monomer addition & hydrolysis; however as the load on the filament
increases, the model is governed by the time it takes for the plug to advance.













280 -,- -' -
S264-

o 248-




1.2 1.25 1. .5 1.4 1.45 1.5 1.55
Time (s)






oei




-5 5 15 25 35 45 55
Protofilament Length (nm)


Figure 5-5. Ciliary plug movement. A) Position versus Time Based on the compression
stiffness r, and the thermal energy kB T, the stepwise motion of a ciliary plug with the
end-tracking motor attached is shown. This motion shows how the ciliary plug
advances as a steady rate for a short time then jumps to a new position. B) The
histogram shows the length of the protofilaments.









CHAPTER 6
DISCUSSION

6.1 Possible Roles of End-Tracking Motors in Biology

The role of nucleotide hydrolysis in cytoskeletal molecular motor action is well-established

for myosin, kinesin, and dynein. The affinity of a myosin head for the actin filament lattice is

modulated by ATP hydrolysis, and dynein and kinesin act analogously in their binding and

release from the microtubule lattice. In defining a new class of cytoskeletal filament

end-tracking motors, we previously described how an microtubule filament end-tracking motors

can exploit nucleotide hydrolysis to generate significantly greater force than that predicted by a

free-filament thermal ratchet (i.e., the elastic Brownian ratchet mechanism), and these ideas were

generalized based on thermodynamic considerations (Dickinson et al., 2004). In this report, we

used known kinetic properties of EB 1 binding and MT plus-end elongation to examine whether a

hypothetical end-tracking motor consisting of affinity-modulated interactions of EB 1 at MT and

protofilament ends can propel obj ects (e.g., MT-attached kinetochores or MT-attached ciliary

plugs) at typically observed velocities while operating against appreciable loads.

While there is no direct evidence that force production by polymerizing MT's is governed

by an end-tracking motor mechanism, several experimental observations suggest that the

properties and interactions of EB 1 are compatible with such a model. Kinetochores, for example,

are known to selectively bind EB 1 by means of APC and/or other adapter proteins (Folker et al.,

2005; Hayashi et al., 2005; Mimori-Kiyosue et al., 2005). Kinetochores also stabilize MTs

against disassembly by preferentially attaching to GTP-containing P-subunits of tubulin subunits

situated at or near MT plus-ends, and this property is likely to be the consequence of EBl1's

ability to attach to polymerizing GTP-rich MT subunits and to dissociate from GDP-containing

subunits, thereby providing a thermodynamic driving force for localization at or near the MT










plus-end. Capture of EBli-rich MTs by kinetochores may allow those EB 1 molecules combining

with APC to self-assemble into an end-tracking motor unit that links force generation to MT

polymerization and hydrolysis of MT-bound GTP. It is known that in the absence of the

EBl/APC complex, chromosomes fail to align at the metaphase-plate, presumably due to

disrupted MT polymerization and kinetochore attachment. The distal tips of ciliary/flagellar MTs

are likewise decorated with EB 1 proteins during formation and regeneration, suggesting EB 1

may serve a similar role in forming an end-tracking motor there and playing a role in elongation-

dependent force generation. In fact, it has previously been suggested that the plug-like structures

found at the plus-ends of MTs in regenerating Chlamydmona~s flagella appear to be "MT

assembly machines". The analogous geometry of the ciliary/flagellar plug and the

tubule-attachment complex in the kinetochore would allow plug- and kinetochore-bound EB 1 to

interact with their MT partners as an end-tracking motor. This proposal does not preclude the

action of other ATP hydrolysis-dependent motors. For example, although the kinesin-like

protein NOD lacks residues known to be critical for kinesin function, microtubule binding

activates NOD's ATPase activity some 2000-fold, a property that (Matthies et al., 2001)

suggested may allow chromosomes to be transiently attached to MTs without producing vectorial

transport.

The Brownian Ratchet mechanism proposed for force generation by MTs in TAC models

(Inoue and Salmon, 1995)) does not allow a strong association between the filaments and the

motile obj ect, and cannot predict substantial force generation at low protomer concentrations.

End-Binding Protein 1 (EBl1) has previously been shown to bind specifically to the polymerizing

microtubule plus-end where the microtubule is tightly bound, suggesting a possible role in force

generation at these sites. We propose that end-binding proteins specificallyy EBl) behave as









molecular motors that modulate the interaction between MTs and the motile obj ect, and generate

the forces required for MT-based motility.

Although the importance of EB 1 and its potential to behave as and end-tracking motors has

been discussed thoroughly in this research study, the models developed can be used to

understand force generating mechanisms involving other end-tracking proteins and their

potential to act as motors (e.g., CLASPS, Clip-170). Adenomatous Polyposis Coli (or APC),

which has an important role in preventing colon cancer, is like EB 1 in that it is found at the tips

of microtubules where microtubules bind to the chromosome at the kinetochore. It therefore also

has the potential to behave like an end-tracking motor.

6.2 Microtubule End-Tracking Model

We developed and analyzed a preliminary EB 1 filament end-tracking model for MTs to

determine the advantages of the mechanochemical process over the monomer-driven ratchet

mechani sms. The two important properties of this model are (a) maintenance of a tight,

persistent (processive) attachment at elongating MT plus-ends by means of EBlI's multivalent

affinity-modulated interactions; and (b) a mechanism for the assembly of the MT end by EB 1

dimers bound on the motile obj ect, thus affording a high-fidelity pathway for assembling

tethered MT's.

For simplicity, our model only considers simple reactions of the EB 1 filament end-tracking

motors. The details of the key assumptions applied to facilitate our analysis of this mechanism

do not compromise key results of high force generation and processivity. For example, the effect

of interactions among protofilaments on EBli-associated MT assembly was neglected. Although

we accounted for EB 1 flexibility, we neglected any contribution of the flexibility of the

protofilaments themselves in net compliance of the EBl-protofilament interaction. We

previously suggested that EB 1 may be a polymerization cofactor acting together with APC to









end-track MT protofilaments (see Mechanism-C in (Dickinson et al., 2004). However, in view

of the recent finding that EB 1 is a stable, two-headed dimer (Honnappa et al., 2005), we now

explain how such multivalency would allow EB 1 alone to operate as the end-tracking motor (like

Mechanism-A in Dickinson et al. (2004)). Either mechanism could capture energy from GTP

hydrolysis and potentially translate it to mechanical work.

We simulated the kinetics of the latter mechanism by characterizing each reaction step

based on its corresponding kinetic rate constant, with force-dependence of elongation arising

from the dependence of probability of the flexible EB 1 head binding at a specific MT lattice

position. With hydrolysis-driven affinity-modulation factor greater than 10, our model

recapitulates experimental, irreversible polymerization rates for free MTs of 170 nm/s. In the

presence of an opposing force, the collective action of hydrolysis-mediated motors on an MT's

thirteen protofilaments can yield kinetic stall forces of approximately 30 pN. This value is

considerably larger than the ~7-pN achievable maximum force provided by the energy of

monomer addition alone (i.e., without the benefit of GTP hydrolysis) in a Brownian Ratchet

mechani sm.

6.3 Protofilament End-Tracking Models

The microtubule end-tracking model developed neglected solution-phase End Binding

protein 1 (EBl1) and binding to microtubules and tubulin protomers. To account for binding

solution-phase EB1, we developed simplified models that simulated the growth of a single

protofilament in the presence of EB1 end-tracking motors. The properties of all protofilament

the end-tracking models were compared to those of the simple Brownian Ratchet mechanism.

Two of the models consider only free-protofilament growth operating with either monovalent or

divalent EB 1 proteins. The simulations for both models included a probabilistic analysis to

determine the expected EB 1 occupancy along the length of the protofilament. The results









confirm the assumption that GTP-driven affinity modulated binding of the EB 1 end-tracking

proteins is required in order to provide a 4.2 tip-to-side binding ratio as observed in experiments.

We also developed two other protofilament models that allow EB 1 to interact with a linker

protein on a motile obj ect (e.g., Adenomatous Polyposis Coli, APC), one model contained

monovalent EB 1 and the other had divalent EBl1. By applying a load on the motile surface, we

analyzed the resulting MT dynamics and force generation. The force-velocity profiles show that

the divalent, EB 1 end-tracking model provides an great advantage over the monovalent end-

tracking model as well as a Brownian Ratchet mechanism. The divalent end-tracking motors are

able to provide processive end-tracking and persistent attachment to the motile surface during

protofilament polymerization. These divalent motor characteristics allow the protofilament to

obtain much higher stall forces than predicted by the monovalent case or by a system with no

affinity modulation (e.g., Brownian Ratchet model).

6.4 Future Work

Further analysis of a 13-protofilament, microtubule end-tracking model should be

considered. It is suggested to develop a stochastic model that includes all mechanisms discussed

in the tethered-protofilament model with divalent EB l. For simplifications it could be assumed

that all protofilament behave independently, but whose individual EB 1 end-tracking motors each

contribute to the equilibrium position of the motile surface, much like the ciliary plug model.

Based on results from the protofilament models, it is expected that the MT end-tracking model

will predict greater stall forces than that of the Brownian ratchet model at large affinity

modulation values.

Although much of the literature supports our proposed EB 1 end-tracking mechanism, there

remains definitive experimental literature that confirms this model. Future studies could clarify

some of the assumptions made, and help to better characterize the mechanism by which EB 1









associates with the MT plus-end. Of particular interest is whether EB 1 together with growing

MTs can generate the force predicted by our simulations while remaining persistently attached to

the motile obj ect. This hypothesis might be tested by adding EBli-coated beads to a solution of

tubulin and MTs and with fluorescence microscopy determine if the MT binds to the beads and

remains persistently attached as it polymerizes. Using optical trapping techniques, the

velocity-force relationships could also be determined. This technique would provide more

accurate stall force estimations for comparison with the simulated results.









APPENDIX A
PARAMETER ESTIMATIONS

A.1 Concentrations of EB1 Species in Solution

Monovalent EBl: The reaction equations and corresponding equilibrium equations

considered for monovalent EB 1 binding to tubulin protomers (Tb) and microtubule sides (MT) in

solution are:


E+Tb++TE K; = [E][Tb]/[TE] (A-1)


E +M2T ++MT -E Kd= [E][MT]/[MT-E] (A-2)

The total concentration of EB1 is represented in all states is

[E]o = [E] + [TE] + [MT-E] (A-3)

Substituting A-1 and A-2 into A-3 gives Equation A-4.

[E]o = [E] + [E][Tb]/K;+ [E][MT]/Kd (A-4)

such that solving for [E] yields.

[E],
[E] = o (A-5)
,[Tb] [MT]
K, Kd

Divalent EBl: For divalent EB 1, two tubulin protomers can bind to each EB 1 molecule

(E) to form TE or TTE. The two binding sites are assumed identical and non-cooperative. Here,

the relevant reaction and equilibrium equations are

E + Tb t, TE K; =2- [E][Tb]/[TE] (A-6)

Tb + TE t, TTE K; = [Tb][TE]/(2- [TTE]) (A-7)


E +M2T ++MT E Kd= [E][MT]/[MT-E] (A-8)

[E]o = [E] + [TE] + [TTE] + [MT-E] (A-9)









Combining A-6 and A-7 yields

[TE][Tb~] [E][Tb]2
[ TTE ] = -
2K, $2(A-10)

Combining Eqs. A-9 and A-10 yields

[E]o = [E] + 2[E] [Tb]/K;+[E][Tb]2/K;2 + [E][MT]/Kei (A-11)

hence

[E],
[E]= (A-12)

K, KdK

from which [TE] and [TTE] can be calculated using Eqs. A-6 and A-7.

A.2 Occupation Probability of Monovalent EB1 Binding to Non-Tethered Protofilament

The probability of tubulin being bound to EB 1 is given by the following:

d = k [El],u, -k, ,p, + k, [Tb]+k,E TIE]us, p,_z- p,)+ k,zu, +k,.E 1 7+1~ -p7
dr (A-13)

The dimensionless relationships in Equations A-14 to A-17 can be substituted into Equation A-

13:

k ,l[E]I
95 a '(A-14)
'k,[T]


a, (A-15)
Sk,[T]

k,E[T-E ]
f= (A-16)
k,[T]


T-,, ,E1+kcT K )11T,- (A-17)
k,[T] [T] [T] k, K, f









The resulting equations represent the differential equation for the probability of EB 1 binding to

the protoHilament side (A-18) and plus-end (A-19), where u-1-p,.

dp' =,c, -a, p, +(1+ f)(p,_z p,)+T,' (p,,1 -p) (A-18)
dzd


dp, =Td 1~' 1 191 1 1Tr,0-1lp 192 Tr1-10 (A-19)



A.3 Occupation Probability of Monovalent EB1 Binding to Tethered Protofilament

This model determines the EB 1 fluorescence along the protofilament based on the

probability of each tubulin protomer being in a specific EB 1 binding state. The binding states

considered were:

pi = probability of EB 1 bound to tubulin protomer in protoHilament

qi = probability of Tk-E bound to tubulin in protoHilament

w = probability of Tk bound to TE in solution

v = probability of Tk bound to E in solution

y = probability of Tk being unbound



The probability of Tk being unbound, y, is represented by Equation A-20.

y = 1-w -v -q (A-20)

The differential equations for the probabilities of EB 1 and Tk-E binding to the protoHilament are


-= ko,,[ E]u, k p, k,~,C,f, yp, + k ,q,

+ (k, [Tb] + kE[TE] + k*)(,E ef ,P- -pk.EP1 io+ k'u,+kp,E 1 )+ (A-21)










kr,2Cey;,JP~, kr,z 9 + konCs,fvu, kq,


2 kEP1 p + k~u, + k",E 1 2+1


+ (ks[T] + kfE[TE] + kE effi, 2-41


(A-22)


where u, =1- q,- p,,,. The differential equations for the probability of the track binding to either

TE or EB 1 are given below:


- k,[TES]y -k,r w+ k,[T]v -k,M w- kfCey,llw+ k, ql


k, [E]y-k,v vk,[T]v +k, w- ko,,2Ced,i2vuz + k ;I9,


(A-23)


(A-24)


To de-dimensionalize time in these differential equations, the variable Tr~ was introduced, which

is defined by Equation A-25.

-1k, E 1 + k,u,
T-l
k, [T](A-25)


1c k,
[T] k,


-1 c1
= and TI,_,'
[T]


K,
dEquation A-25 gets reduced to A-26.
K,f f


Setting Tr o


T, -Toouz Tnpl


(A-26)


Dividing Equations A-21 to A-24 by kf [T], results in the following differential equations with

dimensionless time:


k, [T]


dp,k [ E ]
u -U
dz k, [T] '


k, Kd
p
k, [T] '


+(+kfE [T -E] k E eff ,r r1
k, [T] k, [T]


YPfTrll r,1-11 *ITr,0-1I 1 r,- 1P+1


(A-27)


k, C[T] f~









dq, k,, c,~, k,, K,~, k C7,, k, Kd
yp3 q, + vu, q
dz k, [T] 'k, [T] k, [T] 'k, [T]

+(+kE [T -E] kE C 7(,-1,+r~l +T7, u 1-1,~ 1 1 1 ,1-1 71
k, [T] k,- [T] (A-28)

dwo k, [ T -E] k, K, k, k, K, kE f1-
dz k, [T] k, [T] k, k, [T] k, [T (A-29)4

dv k, [E] k, K, k, k, K k, k Kd,;
y --v v + w-1 vu +C q,
dz k, [T] k, [T] k, k, [T] k, [T] k, [T] (-0

To evaluate the probabilities ofE or Tk-TE binding to the terminal subunit in the protofilament,

these probabilities were re-written for the case when i 1l:

dp,k [ E] k_ Kd k,, Csl kfE C~f,1
Ill 13 Y~1 P1 Wpl
dz k, [T] k, [T] k, [T] k, [T]
(A-31)
kt~ K,; 1 kE [T E ] (1-p,) + Tu, +T,~, q p-p, T,,p(1- p
k, [T] k, [T]


dq k, Ct k, C s, kfE qCf ,1
yp, + vu1 + w(1-q4 )
dz k, [T] k, [T] k, [T]
+ D1 Il P 9 (A-32)
k, [T] k, [T] k, [T] [] k, [T] K, f
k ETcK
J d,1 q1 -2)
k, [ T ]K, f

To solve for the probabilities (p,, q,, w, v) from Equations A-27 to A7-30, the equations were de-

dimensionalized by using the following dimensionless parameters:

k, K, k,
k, [T] kf

k, kE
Sk, k, cp = eA(ld-0 0.2









The resulting de-dimensionalized differential equations are:


dp3, [ E] Kd cef,1R~, y Kt
= a u, -a p p 7 7q
dz [T] [T] [T] [T]
[T -E] C
+ (1+S +S aw~M)(p,_z p) (A-33)
[T] [T]
To K, To To K
+~[T 3 d #1 + IllS d,, 1 7+
[T]K~f[T] [T] K,f

dq, Cq K, C Kd
=Y 7 W,7, -- 7 7ql + a avu, a q
dz [T] [T] [T] [T]
[T-E] C
+ (1+ 3 + 3 "f. aw)(q,_z q,) (A-34)
[T] [T]
To K To To K
[T] K,f [T] []~

dw [ T E ] K, K, ef,1
=7 ~~ y- !wq- '- w+ ,,q,p (A-35)
dt [T] [T] [T] [T]


dv [ E] K, K, C Kd
= 7 y v qv + r w -P e9,1~vu + P q
dz [T] [T] [T] [T] [T] f
(A-36)
-C~Cf, PLU C aK
[T] ,, [T]

dp [ E] Kd qif ,1 qC
dz [T] [T] [T] [T]
(A-3 7)

[T] [nliiT] [T] [T] K,f f1

dq, C,, C,, C,,
=Y "fg~yp,+p efglvu1+S ef~w(1-ql)
dz [T] [T] [T]
-YK, Kd/ f [T-E] T T] Kf A 8
7 +P p+1+3 z3 d 2(-8
[T] [T] [T] [] []K
T, K,
-S
[ T] K, f










For further simplification, more dimensionless parameters were substituted into these differential

equati ons :

Sk[E] K, CefSo
k, [T] [7 []

Kd T
Pu F r("d-zm)/2o'


Both a and 6, represent the effect of force on the probabilities for binding, based on a normal

Gaussian distribution with a variance, #, and filament end position, zm. The rewritten

differential equations are as follows:


S= aeul appl yXBI 07, + yE U
dz
(A-39)
+ (1+3- -+ SX00 0.") Pz1 P,y)+Sy p 1 + lyu, + Sly eq,(p1 -



dq,
S= YXB0 0 P, yp89, + aXBl vul apg~ql
dz
(A-40)
+ (1+S-+70.)219) Sy p+u+S q q1 92


at 7-~+ -g -700wS q (A-41)


dv .
= TE/ P yv- 1+ rfw a 70,pvu + a eq,
dz f
(A-42)

1>1 1>1


dp a nu, a p1 yX8,pyp, p, S00 o*PI
dz f
(A-43)
+Y~pq, +3 (-p) u S ep










d = yXB,p yp, + a *X0, vu, + S00fow(1 -q, )
dz
(A-44)

fy +a* 9,++ ,+pz S ,q

These differential equations (A-39 to A-44) were solved with given kinetic parameters in Matlab

(Appendix B.2.2) to determine the probability of the various EB 1 binding interactions with the

protomers in a protofilament.

A.4 Occupation Probability of Divalent EB1 Binding to Tethered Protofilament

This model determines the EB 1 fluorescence along the protofilament based on the

probability of each tubulin protomer being in a specific EB 1 binding state. The binding states

considered were:

p, : probability of protomer i bound to EB 1 subunit (other subunit unbound)

w, : probability of protomer i bound to TE

q, : probability of protomer i in state dbE

q,~ : probability of protomer i in state dbE-

u, : probability of protomer i being unbound

The probability of the protomer being unbound, u,, is represented by

u~= 1-y~ q 92 P,-n,~ (A-45)


= 2kon [E]ul k p, + k, wl k, [T]p, + k q,_z k pu, u (-46


dt (A-47)









dq,
= k 9 -s9+ su + tu (A-48)



dq,
= k,9 1 k P z, k -tu- (A-49)
+R (q-1 -q, +R q -9

-= k p, + kw, + k ql + k ql k P,zu k ~pzuA-0

2kon [E]u, kon[T -E]u, + R (u-, u,)+R (u,, u,)

R = kl[T] +k, E([T -EF] +[T -E-T])+ kE efn 1 (A-51)

R_= kiu + k~u,E +I1 1 1 (A-52)

The probabilities for the terminal protomer subunit are listed below:

= 2kon [E]u, k p, + k w, k,' [T]p, +k~q2 k pzu2




+ k E[T -E~. ](1 p,) (kS[T] + kfE[T -E-T])pi
+/(a ,u + k,E 1' p2 -k,E1 2)"i~?kt,l i (A-53)




dq,
= -k- 2q k q, + k, p2 1, + k+2p 12

[T]+kE([T "-E+ [T-E T])b +kE;l eff y 1 k (A-55)
+ u," + k, E(/ 1 1 2)








du,
= k_,p, +k ,w, +k_,,q,' k ~p2u 1-2kon[E]u, kon[T -E]u,

+ ki[T](1 -un) (kE([T -E] +kfE[T -E -T]) 1,
-k~u(1-u)+kE 1 1 2(A-.56)

Since an EB 1 bound to the protofilament plus-end cannot bind in the negative direction,

dq, /dt = 0. At equilibrium (the protofilament does not polymerize), when i=1,

qeq 'e eq-+ these differential equations reduce to:

0 = 2kon[ ]E ]uq k _peq + k, weq k'[ T ]peq + k _qeq k peq eq, (A-57)

0 = kon[T' -E]ueq -k~weq + k, [T']peq~ -kl weq (A-58)

0 = -kqeq + 2k peq eq~ (A-59)

0 = k peq + k weq + k qeq -2k peq1, e -2kon[E]ueqo kon[T E]mue (A-60)

And the following holds true:

ueq 1 eq eq eq", (A-61)

Solving Equation A-59 gives:

k _qeq = 2k peq elq (A-62)

Substituting this relationship into the other three equilibrium equations results in:

0 =2kon~[E~]ueq -k +k, [T])peq +k weq (A-63)

0 = k peq+k weq (2kon [E]+ kon [T~ -E eql (A-64)

0 = kon[TE]ueq -k_ +k1 eq +k, [T]peq (A-65)

From these three relationships, the equations for weq, and peq WeTO SOlVed for:









kon [E] (k +k '[T])
w, = -2 uq + ne
k, k, (A-66)




I k +k +[T]}I
Sk +k e
(A-67)

At equilibrium, the following relationship is true:

[T][E]
[T -E]= 2
K, (A-68)

Plugging this into Equation A-67, gives:



-+ 1 +, IT


2kon [E]k k
Peq eq


k k +k[T


(A-69)

The terms in backets on, the top qand bottom arest idntcl thereforlwgeiat A69reucs o

2kon[E],[] ~ 2k,2n~~

[Tb]]
eq K eq
k
(A-72)









Solving for Zieq giVeS:


+8k
k


2ko,,[E]
k


[T]+1) 2ko,, [E]+ + [T 1 2ko,,[E]+ '
K, k K, k
4k, 2ko,, [E ]
k k


(A-74)


To determine the value ofKl, we first assume [Tb]=0, which reduces Eq. A-75 to:


"e, ([T] = 0)


S2ko, [E] 2ko, [E] k 2k[E
k kkk
k, 2ko,, [E ]
k k


(A-75)


The fraction of filament-bound protomers attached to EB 1 at equilibrium defined by total


amount of EB 1 minus the amount of EB 1 in solution:


(A-76)


This is also equivalent to:


[M~T] or
p = E We, / 2+ p,,)

=-;t~k 2ko,, (1


Solving for p gives:


p)


(A-77)


1

2k ~lin~


(A-78)


The effective equilibrium dissociation constant ofEB1 and the protofilament (K, ...) is defined

as:


E, [E]
E,


k
[MT],,
k









1
Kd,eff kk~" ,



Simplifying Equation A-78 gives:

1
K
d,eff
+1
[M~T] or


(A-79)


(A-80)


When half of the protofilament is saturated (p=-1/2), Kd,ef is given by Equation A-81 and ul/


ueq ([E]-[E]o/2).


K
d,ef f


k


(A-81)


Under this constraint, um/ is given by Equation A-82,


ul/2 equ, ([E] = Eo /2)


Skonn r +o `I konn ro kon, konC, Eo
Sk k k k
ko ko C,Eo
4 "
k k


(A-82)


Since k =kon*Cgt and k side"_konside CyrifOr the protofilament plus end, Eqluation A-83 can be

rewritten as:


k o+1 + I'k E +1 +k E
~kC kC k kC,
k k Eo
4'
k kCF


ul/2 equ, ([E] = Eo /2)


(A-83)


Substituting the definition of K, where K


k./k gives:










+1 + K +1 +8-K E


4-K
fs (A-84)

Assuming [Tb]=0, [E]o=0.27C1M, [E]= [E]o/2, and C,f-153 CIM and the experimentally

determined value for Kdaggfof 0.44C1M (Tirnauer et al., 2002b) where used to determine the value

of K as 37. With K k, / k_ Zieq can be calculated for a given [T
K,



2KE [T 2KE ,E
C,+ + +1 + +16K2f E


u=' E,


(A-85)

Determination of k: At steady-state, the measured off-rate k ,s = 0.26 s' is related to the

k_ by:

k (p, + aie) k
k,
p,, + x~, +qg, /2 1+Klie

1+ [T]K
K' (A-86)

Rearranging Eq. A-86 gives Equation A-87.



k = k, 1 'K + "'7
[T1









The only free parameter is K1 for the probabilities and variables. To find the value of K1, the

probabilistic model is used to determine the value that provides the following ratio

( p, + ~i, + 4, / 2 + q,/2)
pqz qq2 =4.2 (A-88)












APPENDIX B
MATLAB CODES


B.1 13-Protofilament Microtubule Model


This stochastic program simulates a 13-protofilament MT polymerizing against a motile


surface with a constant load. The values of f and sigma can be varied to determine the resulting


velocity of the microtubule and each of the protofilament end positions. The kinetic parameters


were estimated or used from literature values.

% This program simulates a 13-protofilament MT polymerizing against a motile surface with a
constant load.

clear all;
hold off;
Tc = 5.3; % Tubulin critical concentration (free filament) (uM)
Tb = 15; % Tubulin dimer concentration (uM)

d=8; % size of tubulin dimer (nm)
nf=13; % Number of protofilaments
kT = 4.14; % Thermal energy (pN-nm) Howard 7 2001
sigma=10; % 5 (nm)
kappa=kT/(sigma^2) ; % (pN/nm)
f= 1000; % Affinity modulation (Kd reduced by a factor of f)
v=167; % velocity of MT end growth (nm/s)
Lhalf=700; % half length of MT (nm)
koff=v/Lhalf; % off rate of both EB1 arms coming off 2 adjacent T-GDP's (SA-1)
kD=0.5; % Dissociation rate constant for both EB1 arms on taxol-stabilized MT
wal(uM)
L=10; % (nm)
rho=1/(LA2) ; % (nmA-2)
%conversion=1.66e6; % conversion from uM to nmA3
%ks ol1=kof f/ (kD*/ 3 ;
ksol=5e7; % ksol: on-rate for both EB1 arms on MT in
solution (nmA3/s)
klon=8.9/13; % kl+: kon for tubulin dimer on MT
(1/(uM-s))
kloff=44/13; % kl-: koff for tubulin dimer on MT
(1/s)
k30n=((rho~ksol)/((sigma)*(2*pi)^0.5)); % k3+: kon for 1 EB1 arm on MT wall (1/s)
(1.6*10A3)
k30ff=(1/2)*koff+(1/2)*((ko ffA2)+(4*k30nnof)05 % k3-: koff for 1 EB1 arm from MT wall

k2on=k30n; % k2+: kon for 1 EB1 arm between T-GTP/T-
GDP (1/s) (1.6*10A3)
kZ of f=k 30f f/ f; % k2-: koff for 1 EB1 arm between T-
GTP/T-GDP(1/s) (0.02)
dt=4.8e-6; % dt should be at least 0.1 x 1/fastest time constant (s)
tim=1; % initialize time
nt=round(tim/dt); % Number of time steps
nshow0=1000; % Initial value for dummy index used to minimize number of 'n'
and'z0's displayed
nshow=nshow0; % Let nshow equal 100 for first iteration
nplot=round(nt/nshow0); % set number of time steps that will be stored
t=(1:nplot)*nshow0*dt; % calculate time from number of time steps taken (has 1000 elements)

F=0; % Constant force applied to surface (pN)
q=0; % Initial value for dummy variable used in "position" loop
z0=zeros(1,nf); % Initial filament equilibrium position
z=(1/nf)*sum(z0)-F/(nf~kappa); % Initial position of motile surface
n=zeros(1,nf); % Initial number of tubulin dimers on protofilament
ps=ones(1,nf); % Initial state of each protofilament
position=zeros(1,nplot); % Initial vector of z for each time step
% state 1: one EB1 arm bound between terminal T-GTP/T-GDP
% State 2: One EB1 arm bound between terminal 2 T-GDP's
% state 3: one EB1 arm bound between terminal 2 T-GDP's, One arm btwn terminal T-GTP/T-GDP











% state
% state
% state


4: one EB1 arm bound between terminal 2 T-GDP's, One arm btwn lagging 2 T-GDP's
5: one EB1 arm bound between lagging 2 T-GDP's
6: one EB1 arm bound between lagging 2 T-GDP's, One arm btwn terminal T-GTP/T-GDP


rnl=rand(nf,nshow0) ; % Generate a (nf x nt) matrix of random numbers (from 0 to 1) for loop.
% Generating random numbers OUTSIDE of loop makes program faster to run
rn2=rand(nf,nshow0) ; rn3=rand(nf,nshow0) ; rn4=rand(nf,nshow0);
rn5=rand(nf,nshow0) ; rn6=rand(nf,nshow0) ;
rn7=rand(nf,nshow0) ; rn8=rand(nf,nshow0) ;
rn9=rand(nf,nshow0) ; rnl0=rand(nf,nshow0);

jshow=1; jstore=1; % Indices for storing, plotting data
beta=dA2/2/sigma^2; % shortcut parameter in calculating fordependence.
np=round(nt/nshow0) ; % number of plotted points
zp = zeros(1,np); t=zp; % storage vectors for position plot


for j=1:nt
itst = [rnl(:,jshow)' rn2(:,jshow)' rn3(:,jshow)' rn4(:,jshow)' rn5(:,jshow)' rn6(:,jshow)' rn7(:,jshow)' rn8(:,jshow)' rn9(:,jshow)' rnl0(:,jshow)' % 10 x nf matrix containing ones where transition occurs
% Note z is dimensionless, scaled by d

n=n+itst(1,:)-itst(3,:)+itst(8,:)-itst(10:);

tnum = itst.*(ones(nf,1)*[2 3 1 3 4 1 2 4 2 3])';


% from 1 to 100,000
% 1to 2
=1) % 1to 3
% 2 to 1
=2) % 2 to 3
-2) % 2 to 4
% 3 to 1
% 3 to 2
% 3 to 4
% 4 to 2
% 4 to 3


% update numbers
% matrix for updating ps


ps = ps.*(1-(max(tnum)>0)) + max(tnum).*(max(tnum)>0); % update ps based on
transitions (note if one filament makes two transitions in one step (shouldn't happen often) only
one leading to larger ps value is used.

fz = (ps==1).*n+(ps--2).*(n-1)+(ps==3).*(-+)+p=4(n-n-+-2 % Dimenionless forces
(correct for multiple springs in states 3,4


ns = sum((ps-
z=sum(fz)/ns


-1)+(ps==2)+(ps==3)*2+(ps==4)*2);
-F/d/ns/kappa; % New Equlibrium position dimensionlesss)


jshow=jshow+1; % update jshow


if j==nshow
j/nt
n;
z0;
nshow=nshow+nshow0;
%bar(n);
%drawnow;

rnl=rand(nf,nshow0);
rn2=rand(nf,nshow0);
rn5=rand(nf,nshow0);
rn7=rand(nf,nshow0);
rn9=rand(nf,nshow0);

jshow=1; %
zp(jstore)=z; %
t(jstore)=j"dt; %
jstore=jstore+1; %
end
end % end "j" loop


% when j is a multiple of 100*time step
% display percent of the loop performed
% display number of dimers added (vector)
% display equilibrium position for protofilament (vector)
% new value for loop
% Bar plot of subfilament lengths


% regenerate numbers for running loop to make program faster to run
rn3=rand(nf,nshow0); rn4=rand(nf,nshow0);
rn6=rand(nf,nshow0);
rn8=rand(nf,nshow0);
rnl0=rand(nf,nshow0);

reset jshow for new random numbers
Position of motile surface at each storer' time step
store data for only those positions plotted
update index for storing data


position=z~d
ksol
sigma
dt
subplot(2,1,1)
plot(t,zp~d,'r')
xlabel('Time (sec) ')
ylabel('surface Position (nm) ')
title('surface Position vs Time')
axis([0 1 -2 300])
%axis([0 dt~nt min(zp)*d max(zp)*d])


% Plot Surface Position (z) vs. Time (t)
% Label x-axis
% Label y-axis
% Label title of plot

% set axis plotting range











zproto=n~d;
subplot(2,1,2)
bar(zproto,'b')
hold on
zi=[position,position,position,position,poiinpsto~oiinpsto~oiinpsto~o
ition,position,position];
plot(zi,'r')
xlabel('Protofilament') % Label x-axis
ylabel('Filament End Position (nm) ') % Label y-axis


B.2 Protofilament Growth Model with Monovalent EB1

B.2.1 Occupational Probability of Monovalent EB1 on a Non-Tethered Protofilament

This probabilistic model simulates the free growth of a single protofilament in the presence


of monovalent, EB 1 end-tracking motors. The value of the affinity modulation factor can be


varied to determine the resulting EB 1 density along a protofilament. The kinetic parameters

were estimated or used from literature values.

% Probabalistic model free MT's
% simulates free MT's in presence of EB1
% Monovianet EB1
% Plots: Occupation Probability vs subunit

clear all;
n=400; % number of subunits to simulate
tspan=[0 1000];

j=1:n;
x0=zeros(n,1) ;
% Parameters

% Fixed parameters

Tb = 10; % uM tubulin dimer concentration
MT = 10; % uM microtubule concentration
V = 170; % nm/s; % elongation speed
d = 8; % nm; % subunit length
% kf = 0.68; % uM^-1s^-1 on-rate for tubulin
% kr = 3.38; % s^-2

kf = v/d/Tb; % uM^-1s^-1 on-rate for tubulin -taken assuming irreversible
% elongation at observed elongation speed
TC = 5; % uM plus-end critical concentration
kr = kf*Tc; % s^-1 off-rate, assuming TC=kr/kf;

thalf=2.6; % s
kobs=log(2)/thalf; % s^-1 observed decay rate constant of EB1 from MT sides
EBltot = 0.27; % uM Total EB1 concentration
K1 = .2; % uM Equilibrium dissocation constant of Tb for E in solution -
%determined be value need for 4.2:1 tip-to nsa o-midl conenraMt ion
Kd = 0.5; %uM Equilibrium dissocatincntn fEfrM ie
E = EBltot/(1+Tb/K1+MT/Kd) ; %uM Equilibrium value of EB1 concentration
TE = Tb/K1*E; %uM Equilibrium value of EB1-Tb concentration

klus_side=kobs/(E+Kd); % uM^-1 s^-1 on-rate constant for EB1 to MT side --
kinus_side=kplus_side*Kd; % s^-1 off-rate constant for EB1 from MT side

enh = kf/kplus_side % End binding Rate enhancement factor

% Roughly estimated parameters










f = 1000; % affinity modulation factor
siga=1; % nm stdev of EB1 position fluctuations
kplus=enh~kplussside; % uM^-1 s^-1 on-rate constant for EB1 to terminal subunit
% assumed same as on side
kminus=enhqkminus_side/f; % s^-1 off-rate constant for EB1 from MT tip
kfE = kf; % binding of TE to end;

% Dimensionless parameters
a = kminus_side/(kf*Tb);
as = kminus/(kf*Tb);

g = kplus_side*E/(kf*Tb) ;
gs = kplus*E/(kf*Tb) ;
b=kfE*TE/kf/Tb;

% Fed parameters
kT=4.14;
sigEB1 = 10; % nm stdev of EB1 linkage position
amp=100; % EB1 local concentration increase factor
dla1; % nm characteristic interaction distance
%st:FiffE1=kT/sigEB1^2;
% EB1 linkage stiffness
av = aqones(n,1); av(1)=as;
gv = gqones(n,1); gy(1)=gs;
% gv = gy.*amp.*exp(-(0:n-1).^2Q~.*(./iEB)2).*x((:-1*ldlt/igB^2'
% av=ay.*exp((0:n-1)*dqdelta/sigEB1^2)';
Trinv0=TC/[Tb];
Trinv1=TC/[Tb]*Kd/K1/f~kfE/kf;

[tout, xout]=ode23s(@(t,x0) sfrate(t,x0,ay,gy,b,Trinv0,Trinv1),tspan~ x)
nt=1ength(tout); nmid=round(nt/2);
pmid=xout(nmid,j);
pend=xout(nt,j);
fluor=pend;
fluorm=pmid;
plot(1:n,pmid,'k:',1:n,pend,'k') ;
xlabel('subunit')
ylabel('occupation probability')

function f=sfrate(t,x,ay,gy,b,Trinv0,Trinv1)

n=1ength(x) ;
j=1:n;
p=x(j) ;
u=1-p;
f=ze ros (n ,1) ;
Trinv=Trinv0*u(1)+Trinv1*p(1);

f(1) = gy(1)*u(1)-av(1)*Qp(1) -p(1)+b~u(1)+Tri nv0*u (1)*p(2)-Trinv1*p (1)*u(2) ;
i=2:n-1;

/Ti)= gy(i) .*u(i)-av(i) .*p(i)+(1+b)*(p~-)p(i-)pi)Tiv(~+)pi

% /In)=0 ;
/In)= /In-1) ;












B.2.2 Occupational Probability of Monovalent EB1 on a Tethered Protofilament

This probabilistic model simulates the growth of a surface-tethered protofilament in the


presence of monovalent, EB 1 end-tracking motors. The value of the affinity modulation factor

can be varied to determine the resulting EB 1 density along a protofilament. The kinetic


parameters were estimated or used from literature values.

% "brunode.m"
% Probabalistic model tethered MT's
% simulates tethered MT's in presence of EB1
% Monovianet EB1
% Plots: Occupation Probability vs Subunit

clear all;
n=200; % number of subunits to simulate
tspan=[0 1000];

j=1:n;
x0=zeros(2*n+2,1);
% Parameters

% Fixed parameters
kT = 4.1; % pN-nm thermal energy
T = 10; % uM tubulin dimer concentration
MT = 10; % uM microtubule concentration
V = 170; % nm/s; % elongation speed
d = 8; % nm; % subunit length

kf = v/d/T; % uM^-1s^-1 on-rate for tubulin -taken assuming irreversible elongation
at observed elongation speed
TC = 5; % uM plus-end critical concentration
kr = kf*Tc; % s^-1 off-rate, assuming TC=kr/kf;

thalf=2.6; % s
kobs=log(2)/thalf; % s^-1 observed decay rate constant of EB1 from MT sides
EBltot = 0.27; % uM Total EB1 concentration
K1 = .16; % uM Equilibrium dissocation constant of T for E in solution -- determined
be value need for 4.2:1 tip-to-middle concentration
Kd = 0.5; %uM Equlibrium dissocation constant of E for MT sides
E = EBltot/(1+T/K1+MT/Kd) ; %uM Equilibrium value of EB1 concentration
TE = T/K1*E; %uM Equilibrium value of EB1-Tb concentration

kplus_side=kobs/(E+Kd); % uM^-1 s^-1 on-rate constant for EB1 to MT side --
kminus_side=kplus_side*Kd; % s^-1 off-rate constant for EB1 from MT side

% Roughly estimated parameters
f = 1; % affinity modulation factor

TCE = TC*Kd/K1/f;

sigm = ; nm stdev of EB1 position fluctuations
Delt = 0; % bond distance5; n
kplus=kplus_side; % uM^-1 s^-1 on-rate constant for EB1 to terminal subunit --
assumed same as on side
kminus=kminus_side/f; % % s^-1 off-rate constant for EB1 from MT tip
kfE = kf; % binding of TE to end;

ceff0 = 2/(2*pi)^(3/2)/sigma^3; % concentration in nm^-3 -- based on 3-D normal
distribution on half-sphere










Ceff0 = ceff0/(6.022e23)*1e27/1000*1e6; % cone in uM: nm^-3 x (1
(1827 nm^3/m^3) x (1 m^3/1000 L) x (10^6 uM/M)

% Varied parameters


mol/ 6.022e23) x


Kt = 5; % uM
Force =1*kT~log(T/Tc)/d; %


load in pN positive if compressive, negative if tensile


%% Dimensionless parameters
alpha=kplus_side/kf; %
alpha_s =kplus/kf; %
gamma = 1; %kt/kf;
delta = 1; %kfE/kf
eta = 1; %kl/kf;
chi = Ceff0/T;
beta = Kt/T;
mui = Kd/T=T/T;
xi = K1/T;
epsilon=E/T;
deld = Delta/d;
kappa = kT/sigma^2; kappad= kappaqd^2/kT;
Fd = Force~d/kT;

pars = [alpha alpha_s gamma delta eta chi beta mu psi xi epsilon f kappad deld Fd];


alpha=pars(1)
alpha_s =pars(2)
gamma=pars(3)
delta=pars(4)
eta = pars(5)
chi = pars(6)
beta = pars(7)
mu = pars(8)
psi = pars(9)
xi = pars(10)

kappad = pars(13) % kappad = kappaqd^2/kT;
deld = pars(14) % deld = Delta/d;
Fd = pars(15) % Fd = F~d/kT neq if under compression
F~d/kT+kappa~i~d*(d/kT) = Fd +kappad*1


Fnet~d/kT =


[tout, xout]=ode23s(@(t,x0) bfrate(t,x0,pars),tspan,x0);
nt=1ength(tout); nmid=round(nt/2);
pmid=xout(nmid,2*j-1) ;
pend=xout(nt,2*j-1);
qmid= xout(nmid,2*j) ;
qend=xout(nt,2*j);
fluor=pend+qend;
fluorm=pmid+qmid;
plot(1:n,pmid,'k:',1:n,qmid,'b:',1:n,pend',1nqd'b1:flo'g)
xlabel('subunit')
ylabel('occupation probability')
w = xout(nt,2*n+2);
v = xout(nt,2*n+1);

im = sum(qend.*(1:1ength(qend)))/sum(qend)
FT = (im-1)*kappad;
Fnetd = Fd+FT;

Vl = kf*T*(exp(-Fd)-TC/T*(-qn(1-(qend(1))+pn))
V2 = kfE*((TE+Ceff0*wqep(F)*TE~exp(-F)-TE(ed1+ed1)
V=V1+V2
relv=v/(kf*T-kr)










Attachprob = 1-(1-sum(qend))^13

function ff=bfrate(t,x,pars)

n=(length(x)-2)/2;
j=1:n;
p=x(2*]-1) ;
q=x(2*]) ;
u=1-p-q;
v=x(2*n+1) ;
w=x(2*n+2) ;
y=1-v-w-sum(q) ;
fp=zeros(n,1) ;
fq=zeros(n,1) ;
ff=zeros(2*n+2,1);

alpha=pars(1) ;
alpha_s =pars(2) ;

eta = pars(5);
chi = pars(6) ;
beta = pars(7) ;
mui = pars(8); p r()
xi = pars(10) ;
eps = pars (11);
f= pars(12) ;
kappad = pars(13); % kappad = kappaqd^2/kT;
deld = pars(14); % deld = Delta/d;
Fd = pars(15); % Fd = F~d/kT pos if under compression Fnet~d/kT =
F*d/kT~kappa*(i-(im-1))*d^2/kT) = Fd +kappad~i
meani = 0;

if sum(q)>0
meani = sum(q.*j ')/sum(q);
end

FT = (meani-1)*kappad;
Fnetd = Fd+FT;

%afac=exp(Fnetd) ;
%psi = psiqafac;
% alpha=alphaqexp(Fnetd); alpha_s=alpha_sqafac; gamma=gammaqafac; eta=etaqafac;
im =1; %im = Fd/kappad+meani; % mean subunit position for unstressed trackers

afac=exp(Fd);
pi=piqafac;
apaaphaqafac; alpha_s=alpha_sqafac; gamma=gammaqafac; eta=etaqafac;

phi=exp(abs(j-im)*kappadqdeld)';
theta=exp(-kappad/2*(j-im).^2)';
phi0 = exp(abs(im)*kappadqdeld) ;
theta0 = exp(-kappad/2*(im).^2);

chiv=chiqtheta(j).*phi(j);
chiv0 = chiqtheta0*phi0;
linel = alpha_s*(epsqu(1)-mu/f~p(1))-gamma~ chiv1)yv(1 p(1)-deltaqchiv0*wqp(1);
line2 = gammaqbetaqphi(1)*q(1)+delta~eps/xi*(1-
p(1))+psl*(u(1)+deltaqmu/f/xiqphi~ q(1)q1)*(
fp(1) = linel+line2-deltaqpsiqmu/xi/fqpl qlp(1*-p2)
linel = (chiv(1)*(gamma~y~p(1)-lpa~svulph)sv) (1)chv*h*w1-1)
line2 = -(gamma~beta~phi(1)+alpha_s~mu/f~phi(1)+1+delta ep/)*1;
line3 = psl*(u(1)+deltaqmu/xi/fqplp q(1))*()detdsim/iflh()*()(1q1)
fq(1) =linel+line2+line3;
i=2:n-1;











linel = alpha.*(epsqu(i)-muqp(i))-gammaqchiv(i). *y*~)gmabt~h~)*~)
line2 = (1+delta*(eps/xi+chiv0*w))*(p(i-1)-p(i));
line3 = psi*(deltaqmu/xi/f*(p(1)-theta(1~q~ )*q1)u1)*pi1)pi
fp(i) = linel+line2+line3;
linel = gamma*(chiv(i).*y.*p(i)-beta~phii) Q(i).qi)apa(hvi.v*~)
muqphi(i).*q(i));
line2 = (1+delta~eps/xi+deltaqchiv0*w)*~(i-)qi--q);
line3 = psi*(deltaqmu/xi/f*(p(1)-phi(1~ql)*q(1+u1*qi1-q);
fq(i)=linel+line2+line3;
ff(1)=fp(1) ;
ff(2)=fq(1) ;
ff(2*i-1)=fp~i);
ff(2*i) = fq(i);
linel = gamma*(epsqy-betaqv)+eta*(xiwwv-v)-alpa _s(hv1*u1)m/ph1*q1;
line2 = alphaqsum(muqphi(2:n).*q(2:n)-vqchiv2:n.*(2:))
ff(2*n+1) = linel+line2;

ff(2*n+2)= gamma*(eps/xiqy-beta~w)+eta*(vx~w-xiw-ela(hi0w
psi*mu/xi/f*phi~l*(1)*q1)
Vl = (1+delta~eps/xi+deltaqchiv0*w);
V2 = psi*(deltaqmu/xi/f*(p(1)-phi(1~ql)*q(1+u1)
linel = alpha.*(epsqu(n)-muqp(n))-gammaqchih).y*pn+gma~eanh)n.q.)
line2 = (1+delta*(eps/xi+chiv0*w))*(p(n-1)-p(n));
fp(n) = linel+line2;
linel = gamma*(chiv(n).*y.*p(n)-betaphi phiQ(n).qn)apa(hvn.v*~)
muqphi(n).*q(n));
line2 = (1+delta~eps/xi+deltaqchiv0*w)*~(n-)qn--qn)
fq(n)=linel+line2;
ff(2*n-1)=fp~n);
ff(2*n)=fq~n) ;



B.3 Protofilament Growth Model with Divalent EB1

B.3.1 Occupational Probability of Divalent EB1 on a Non-Tethered Protofilament

This probabilistic model simulates the free growth of a single protofilament in the presence

of divalent, EB 1 end-tracking motors. The value of the affinity modulation factor can be varied

to determine the resulting EB 1 density along a protofilament. The kinetic parameters were

estimated or used from literature values.

% Probabilistic model -- free-ended MT's
% Simulates free-ended MT's in presence of EB1
% Divalent EB1
% Inputs: kon
% Outputs: EB1 Tip: Side Binding Ratio
% Plots: Occupation Probability vs Subunit
n=400; % number of subunits to simulate
tspan=[0 140];
j=1:n;
x0=zeros(3*n,1);
% Parameters













% Fixed parameters

Tb = 10; % uM tubulin dimer concentration
MT = 10; % uM microtubule concentration
V = 170; % nm/s; % elongation speed
d = 8; % nm; % subunit length
% kf = 0.68; % uMA-1sA-1 on-rate for tubulin
% kr = 3.38; % sA-2

kf = v/d/Tb; % uMA-1sA-1 on-rate for tubulin -taken assuming irreversible
elongation at observed elongation speed
Tc = 5; % uM plus-end critical concentration
kr = kf*TC; % SA-1 off-rate, assuming Tc=kr/kf;

thalf=2.6; % s
kobs=log(2)/thalf; % sA-1 observed decay rate constant of EB1 from MT sides
EB1 = 0.27; % uM Total EB1 concentration

sigma = 10 ; % nm stdev of EB1 position fluctuations
ceff = 2*exp(-(8/10)^2/2)/(2*pi)^(3/2)/sigma^3;
% concentration in nmA-3 -- based on 3-D normal distribution on half-
sphere
Ceff = ceff/(6.0Z~22e23)e27/1000*1e6;
% nm-3 x (1 mol/ 6.022e23) x (1e27 nmA3/mA3) x (1 mA3/1000 L) x (10^6
uM/M)
K = 37; % equals kplus_side/kminus_side, valued required for Kdeff = 0.44

%%% Guessed parameters

Kd1 = .65; % Kd, Dissociation constant for EB1 subunit and Tb, Kdl=klm/k1 (uM) value for
typical monovalent protein
%Kdl=Kdlvec(i run);
k1 = 10; % on-rate for EB1 subunit and Tb (uMA-1*sA-1), value for typical protein-protein
binding
klm = kl*Kdl; % off-rate for EB1 subunit and Tb (sA-1)

%% Equlibria

E = EB1/((1+(Tb/Kdl))A2); % [EBl], Concentration of EB1 dimer in sol'n
TE = 2*E*Tb/Kdl; % [EB1-Tb], concentration of EB1 dimer bound to 1 tubulin protomer
TTE = Tb*TE/(2*Kdl); % [EB1-TbA2], Concentration of EB1 dimer bound to 2 tubulin protomers

a=4*KA2*E/Ceff;
b=(1+Tb/Kdl)*2*K*E/Ceff+1;

u_eq = (-b+sqrt(bA2+4*a))/2/a
p_eq=2*K*E/Ceff~u_eq;
q_eq = 2*K~p_eq~u_eq;
pi_eq = Tb/Kdl~p_eq;

fl_eq=p_eq+pi _eq+q_eq/2 %% Equilibrium flourescence conce
check = u_eq+p_eq+q_eq+pi_eq %%% should equal one

%% Determine kminus_side, kplus_side, kon

kmi nus_si de = kobs*(1+K~u_eq/(1+Tb/Kdl)) ; %% Based on FRAP half-life
kplus_side = K~kminus_side; %% by definition
kon_side = kplus_side/Ceff;
f= 1000 ; % affinity modulation factor
% f=fve c(i run) ;

%% Mixed model -- chose kon, calculated koff from f
kon = 5; % uM-1s-1
fon = kon/kon_side; %% accelerated on-rate at end
kplus = kon*Ceff;
foff = fon/f;
kmi nus=fon/f~kmi nus_si de ; %% correspondi ng change off-rate at end

%% other parameters

kfE = kf % on-rate constant of TE and TTE to MT end uM-1s-1;
krE = kfE*TC/Kdl~kminus/kon % off-rate constant of TE or TTE

%% Dimensionless parameters
pars =[kminus_side kminus kplus_side kplus kf kr krE kfE k1 klm Tb E TE TTE Ceff];
x0(3*j-1)=p_eq; %initial conditions
x0(3*j)=q_eq/2; x0(4)=0;
x0(3*j-2)=pi_eq;

[tout, xout]=ode23s(@(t,x0) dfrate(t,x0,pars),tspan,x0);












nt=1ength(tout); nmid=round(nt/2);
pimid=xout(nmid,3*j-2) ;
pmid=xout(nmid,3*j-1) ;
qpmid=xout(nmid,3*j) ;
piend=xout(nt,3*j-2) ;
pend=xout(nt,3*j-1) ;
qpend=xout(nt,3*j) ;
qmend=[0 qpend(1:n-1)];
qmmid=[0 qpmid(1:n-1)];
fluor=pend+.5*qpend+.5*qmend+piend;
fluorm=pmid+.5*qmmid+.5*qpmid+pimid;

pp=polyfit(20:150, log(fluor(20:150)-fl_eq),1);
fitf=exp(pp(2)+pp(1)*(1:n)) ;
plot(1:n,fluor,'go' ,1:n,fitf+fl_eq);
xlabel('subunit')
ylabel('occupation probability')
tip_ratio=real(exp(pp(2))+fl_eq)/fl_eq
fl_eq

function f=dfrate(t,x,pars)


%pars = [kminus_side kminus
kminus_side=pars(1) ;
kminus=pars(2) ;
kplus_side=pars(3) ;
kplus=pars(4) ;
kf=pars(5) ;
kr=pars(6) ;
krE=pars(7) ;
kfE=pars(8) ;
kl=pars(9) ;
klm=pars(10) ;
Tb=pars(11) ;
E=pars(12) ;
TE=pars(13) ;
TTE=pars(14) ;
Ceff=pars(15) ;
kon_side=kplus_side/Ceff;
kon=kplus/Ceff;


kplus_side kplus kf kr krE kfE k1 klm Tb E TE TTE Ceff];


n=1ength(x)/3;
j=1:n;
pid=x(3*j-2) ;
p=x(3*j-1) ;
qp=x(3*j) ;
qm=[0 qp(1:n-1) ']';
u=1-p-qp-qm-pid;
kony = ones(n,1)*kon_side;
kpy = konv*Ceff;
kmy=ones(n,1)*kminus_side;


konv(1)=kon;
kmy(1)=kminus;


fp=zeros(n,1) ; fqp=fp; fqm=fp; fpi=fp; f=zeros(3*n,1) ;
Rp=kf*Tb+kfE*(TE+2*TTE)+kfE*Ceff~pid(1);
Rm=kr~u(1)+krE*(pid(1)+p(1)+qp(1));

i=2:n-1;

tmpl=2*konv(i)*E.*u(i)-kmy(i).*p(i)+klm"pdi-lT~~)kyi.q~-)kyi1 *~)*~+)
tmp2=kmy(i+1).*qm(i+1)-kpy(i-1).*p(i).*ui-+R*p-1pi)Rmpi1-pi;

fp(i) = tmpl+tmp2;
fqp(i)=-kmy(i+1).*qm(i+1)-kmy(i).*qp(i)+kyi.pi1.ui+yi1.pi.ui1+R*qi--
qp(i))+Rm*(qp(i+1)-qp~i)) ;
fpi(i)=konv(i)*TE.*u(i)-kmy(i).*pid(i)+kl*bpi-l~i~)R*pdi1-i~)+m(i~+)
pid(i)) ;
tmpl=2*kon*E~u(1)-kminus~p(1)+klm~pid(1)-k*bp1+mnssd~m2-pu~iep1*()
tmp2=kfE*TE*(1-p(1))-(k"bZkf*"Tb2kE*TE*()(ru1+rE(i()q())p2)kEp1*1p1)












fp(1)=tmpl+tmp2;
tmpl = kon*TE~u(1)-kminus~pid(1)+kl*Tb~p(1)-klm pi1)
tmp2 = 2*kfE*TTE*(1-pid(1))-(k"bkf*"TbkE*TE*pi(1)
kfE*Ceff~pid(1)+krE~q qp)(1r~)+(krEu1+r~()*pd2-r~i(1*1pd2)

fpi(1)=tmpl+tmp2;
tmpl= -kminus_side~qm(2)-kminus~qp(1)+kplus~p(2)u1+pu~iep1*()
tmp2= -(kf*Tb+kfE*TE+2*kfE*TTE)*qp(1)+kfE*Ceffpi1-
krE~qp(1)+(kr~ulu kE(1i~)+krE(pid()+p1)*q2;
fqp(1) =tmpl+tmp2;

fp(n)=0;
fqp(n)=0;
fpi(n)=0;

f(3*j-2)=fpi ;
f(3*j-1)=fp;
f(3*j)=fqp;



B.3.2 Average Fraction of divalent EB1-bound Protomers on Side of Protofilament


This stochastic model simulates the side-binding of divalent EB 1 on a non-growing


protofilament. The value of the affinity modulation factor can be varied to determine the time-


averaged fluorescence of EB 1 along the length of the protofilament and the state of the subunits


in the protofilament. The kinetic parameters were estimated or used from literature values.

% Probabalistic model
% Simulates free-ended MT's in presence of EB1
% Divalent EB1
% Inputs: f, Kd1, kon
% Outputs: Time Avg Fluorescence
% Plots: Time Avg Fluorescence vs subunit
clear all;
n=400; % number of subunits to simulate 400
tspan=[0 40]; 40
j=1:n;
x0=zeros(3*n,1) ;

% Determine Parameters

% Fixed parameters
Tb = 10; % uM tubulin dimer concentration
Tc = 5; % uM plus-end critical concentration
EB1 = .27; % uM EB1 concentration
d = 8; % nm; % subunit length
V = 170; % nm/s; % elongation speed
kf = v/d/Tb; % uMA-1sA-1 on-rate for tubulin -taken assuming irreversible elongation at
observed elongation speed
kr = kf*TC; % SA-1 off-rate, assuming Tc=kr/kf;
thalf=2.6; % s
kobs=log(2)/thalf; % sA-1 decay rate constant of EB1 from MT sides

sigma = 10 ; % nm stdev of EB1 position fluctuations
ceff = 2*exp(-(8/10)^2/2)/(2*pi)^(3/2)/sigma^3; % concentration in nmA-3 -- based on 3-D normal
distribution on half-sphere












Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm-3 x (1 mol/ 6.022e23) x (1e27 nmA3/mA3) x (1 mA3/1000
L) x (10^6 uM/M)

K = 37; % equals kplus_side/kminus_side, valued required for Kdeff = 0.44


%%% Guessed parameters

Kd1 = 0.65; % Kd, Dissociation constant for EB1 subunit and Tb, Kdl=klm/k1 (uM) value for
typical monovalent protein
%Kdl=Kdlvec(i run);
k1 = 10; % on-rate for EB1 subunit and Tb (uMA-1*sA-1), value for typical protein-protein
binding
klm = kl*Kdl; % off-rate for EB1 subunit and Tb (sA-1)


%% Equlibria

E = EB1/((1+(Tb/Kdl))A2); % [EBl], Concentration of EB1 dimer in sol'n
TE = 2*E*Tb/Kdl; % [EB1-Tb], concentration of EB1 dimer bound to 1 tubulin protomer
TTE = Tb*TE/(2*Kdl) ; % [EB1-TbA2], Concentration of EB1 dimer bound to 2 tubulin protomers

a=4*KA2*E/Ceff;
b=(1+Tb/Kdl)*2*K*E/Ceff+1;

u_eq = (-b+sqrt(bA2+4*a))/2/a
p_eq=2*K*E/Ceff~u_eq;
q_eq = 2*K~p_eq~u_eq;
pi_eq = Tb/Kdl~p_eq;

fl_eq=p_eq+pi _eq+q_eq/2 %% Equilibrium flourescence conce
check = u_eq+p_eq+q_eq+pi_eq %% should equal one


%% Determine kminus_side, kplus_side, kon

kmi nus_si de = kobs*(1+K~u_eq/(1+Tb/Kdl)) ; %% Based on FRAP half-life
kplus_side = K~kminus_side; %% by definition
kon_side = kplus_side/Ceff;
f= 1000 ; % affinity modulation factor
% f=fve c(i run) ;


%% Mixed model -- chose kon, calculated koff from f
kon = 10; % uM-1s-1
fon = kon/kon_side; %% accelerated on-rate at end
kplus = kon*Ceff;
foff = fon/f;
kmi nus=fon/f~kmi nus_si de ; %% correspondi ng change off-rate at end


%% other parameters

kfE = 10 % on-rate constant of TE and TTE to MT end uM-1s-1;
krE = kfE*TC/Kdl~kminus/kon % off-rate constant of TE or TTE

% Initial conditions

N=40; % 50-number of subunits to simulate
S=ones(1,N); %s = 1 if unocupplied; 2 if bound to E, 3 if bound to TEE, 4 if bound to +side of
doubly bound, 5 if bound to -side
tim=10; % 200-run time (s)
chartime=1/max([kfE*TTE kfE*TE kf*Tb kr krE kon*Tb kl*Tb k1 kplus kplus_side kminus
kminus_side]); %Characteristic time
dt = chartime/5; %simulation time increment
nt=round(tim/dt) ;
rnsidel=rand(nt,N) ;
radd = rand(nt,1) ;
previt=0; FLav=0*S;

for it=1:nt

5% side binding
tstl=(s==1)& rnsidel(it,:)<2*kon_side*E~dt; % binds E
tst2= (S==1)& rnsidel(it,:)<(kon_side*TE~dt+ 2*kon_side Edt &~tstl; % or binds TE

tst3=(s==2)& rnsidel(it,:) tst4=(((S==2)&[0 S(1:N-1) 1]l) & rnsidel(it,:)<(kplus_side~dt + kminus_side~dt)) &~tst3; %
bind plus side











tstS=(((s==2)&[S(2:N)==1 0]) & rnsidel(it,:)<(kplus_side~dt + kplus_side~dt+kminus_side~dt))
&~(tst3|tst4); % bind minus side
tst6=((s==2)& rnsidel(it,:)<(kl*T~dt + 2*kplus_side~dt+kminus_side~dt)) &~(tst3|tst4|tst5); %
bind T

tst7=(S==3)& rnsidel(it,:) tst8=(S==4)& rnsidel(it,:) tst9=(S==5)& rnsidel(it,:) tstl0=((s==3) & rnsidel(it,:)<(kminus_side~dt+klm~dt)) &~(tst7); %dissociate TE from side

otst=(tstl+tst2+tst3+tst4+tstS+tst6+tst7+tt+s9tt0
if sum(otst)>0

ntav=it-previt;

FL=(S==2)+(S==3)+.5*(S==5)+.5*(S==4);
FLav=(previt*FLav+ntav*FL)/(ntav+previt);
previt=it;
ifndl=find(tstl) ;
s(ifndl)=2;
ifnd2=find(tst2) ;
s(ifnd2)=3;
ifnd3=find(tst3) ;
s(ifnd3)=1;
ifnd4=find(tst4) ;
s(ifnd4)=5; S(ifnd4-1)=4;
ifnd5=find(tst5) ;
s(ifnd5)=4; S(ifnd5+1)=5;
ifnd6=find(tst6) ;
s(ifnd6)=3;
ifnd7=find(tst7) ;
s(ifnd7)=2;
ifnd8=find(tst8) ;
s(ifnd8)=1; s(ifnd8+1)=2;
ifnd9=find(tst9) ;
s(ifnd9)=1; s(ifnd9-1)=2;
ifnd10=find(tstl0) ;
S(ifndl0)=1;
it

FLav
plot(1:N,FLav,[1 N],[fl_eq fl_eq]); axis([0 N 0 1]); drawnow;
end
end

B.3.3 Average Fraction of EB1-bound protomers during protofilament growth


This stochastic model simulates the time averaged fluorescence of EB1 along a non-


tethered, single microtubule protofilament in the presence of divalent EB 1 during protofilament


growth. The value of the affinity modulation factor can be varied to determine its affect on the


EB1 fluorescence. The state of the subunits in the protofilament can also be determined.

% Simulates free-ended MT's in presence of EB1
% stochastic model
% Divalent EB1
% Inputs: f, Kd1, kon
% Outputs: Velocity, State of subunits, Time Avg Fluorescence
% Plots: Time Avg Fluorescence vs subunit

clear all;
tic;
N=200; % number of subunits to simulate 200
tim=40; % run time (s)40
axmax=.2; % max y-axis.

% Determine Parameters

% Fixed parameters













T = 10; % uM tubulin dimer concentration
V = 170; % nm/s; % elongation speed
d = 8; % nm; % subunit length

kf = v/d/T; % uMA-1sA-1 on-rate for tubulin -taken assuming irreversible elongation at
observed elongation speed
Tc = 5; % uM plus-end critical concentration
kr = kf*TC; % SA-1 off-rate, assuming Tc=kr/kf;

thalf=2.6; % s
kobs=log(2)/thalf; % sA-1 decay rate constant of EB1 from MT sides
EB1 = 0.27; % uM EB1 concentration

sigma = 10 ; % nm stdev of EB1 position fluctuations
ceff = 2*exp(-(8/10)^2/2)/(2*pi)^(3/2)/sigma^3; % concentration in nmA-3 -- based on 3-D normal
distribution on half-sphere
Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm-3 x (1 mol/ 6.022e23) x (1e27 nmA3/mA3) x (1 mA3/1000
L) x (10^6 uM/M)
K = 37; % equals kplus_side/kminus_side, valued required for Kdeff = 0.44


%%% Guessed parameters

Kd1 = .65; % Kd, Dissociation constant for EB1 subunit and T, Kdl=klm/k1 (uM) value for
typical monovalent protein
%Kdl=Kdlvec(i run) ;
k1 = 10; % on-rate for EB1 subunit and T (uMA-1*SA-1), Value for typical protein-protein
binding
klm = kl*Kdl; % off-rate for EB1 subunit and T (SA-1)

%% Equlibria

E = EB1/((1+(T/Kdl))A2); % [EBl], Concentration of EB1 dimer in sol'n
TE = 2*E*T/Kdl; % [EB1-T], Concentration of EB1 dimer bound to 1 tubulin protomer
TTE = T*TE/(2*Kdl); % [EB1-TA2], Concentration of EB1 dimer bound to 2 tubulin protomers

a=4*KA2*E/Ceff;
b=(1+T/Kdl)*2*K*E/Ceff+1;

u_eq = (-b+sqrt(bA2+4*a))/2/a;
p_eq=2*K*E/Ceff~u_eq;
q_eq = 2*K~p_eq~u_eq;
pi_eq = T/Kdl~p_eq;

fl_eq=p_eq+pi _eq+q_eq/2 ; %% Equilibrium flourescence conce
check = u_eq+p_eq+q_eq+pi_eq; %%% should equal one


%% Determine kminus_side, kplus_side, kon

kminus_side = kobs*(1+K~u_eq/(1+T/Kdl)); %% Based on FRAP half-life
kplus_side = K~kminus_side; %% by definition
kon_side = kplus_side/Ceff;
f= 1 ; % affinity modulation factor
% f=fve c(i run) ;


%% Mixed model -- chose kon, calculated koff from f

kon = 5; % uM-1s-1
fon = kon/kon_side; %% accelerated on-rate at end
kplus = kon*Ceff;
foff = fon/f;
kmi nus=fon/f~kmi nus_si de ; %% correspondi ng change off-rate at end


%% other parameters

kfE = kf; % on-rate constant of TE and TTE to MT end uM-1s-1;
%kfE = le-8;
krE = kfE*TC/Kdl~kminus/kon; % off-rate constant of TE or TTE


% Initial conditions

S=ones(1,N); %s = 1 if unocupplied; 2 if bound to E, 3 if bound to TEE, 4 if bound to +side of
doubly bound, 5 if bound to -side
chartime=1/max([kfE*TTE kfE*TE kf*T kr krE kon*T kl*T k1 kplus kplus_side kminus kminus_side]) ;
%Characteristic time
dt = chartime/10; %simulation time increment











nt=round(tim/dt) ;
rnsidel=rand(nt,N) ;
radd = rand(nt,1) ;
roff=rand(nt,1) ;
previt=0; FLav=0*S;
nadd=1;

konv=[kon kon_side~ones(1,N-l1];
kminusv=[kminus kminus_side~ones(1,N-1)];
kplusv=[kplus kplus_side~ones(1,N-1)];
klv=kl~ones(1,N) ;
klmy=klm~ones(1,N) ;
for it=1:nt

5% side binding

tstl=(s==1)& rnsidel(it,:)<2*konv*E~dt; % binds E
tst2= (S==1)& rnsidel(it,:)<(konv*TE~dt+2*konv*E~dt) &~tstl; % or binds TE

tst3=(s==2)& rnsidel(it,:) tst4=(((S==2)&[0 S(1:N-1) 1]l) & rnsidel(it,:)<(kplusv~dt + kminusv~dt)) &~tst3; % bind plus
side
tstS=(((s==2)&[S(2:N)==1 0]) & rnsidel(it,:)<(kplusv~dt + kplusv~dt+kminusv~dt))
&~(tst3|tst4); % bind minus side
tst6=((s==2)& rnsidel(it,:)<(klv*T~dt + 2*kplusv~dt+kminusv~dt)) &~(tst3|tst4|tst5); % bind T

tst7=(s==3)& rnsidel(it,:) tst8=(S==4)& rnsidel(it,:) tst9=(S==5)& rnsidel(it,:) tstl0=((s==3) & rnsidel(it,:)<(kminusv~dt+klmy~dt)) &~(tst7); %dissociate TE from side

otst=(tstl+tst2+tst3+tst4+tstS+tst6+tst7+s8t9ttl) % Anything happen?

if sum(otst)>0

Sold=s; % store old
ntav=it-previt; % number of additional steps in average

FL=(S==2)+(S==3)+.5*(S==5+5)+.5*(==4) % EB1 fluorsescence
FLav=(previt*FLav+ntav*FL)/(ntav+previt); % Update time-averaged fluorescence
previt=it; % update
ifndl=find(tstl) ;
s(ifndl)=2;
ifnd2=find(tst2) ;
s(ifnd2)=3;
ifnd3=find(tst3) ;
s(ifnd3)=1;
ifnd4=find(tst4) ;
s(ifnd4)=5; S(ifnd4-1)=4;
ifnd5=find(tst5) ;
s(ifnd5)=4; S(ifnd5+1)=5;
ifnd6=find(tst6) ;
s(ifnd6)=3;
ifnd7=find(tst7) ;
s(ifnd7)=2;
ifnd8=find(tst8) ;
s(ifnd8)=1; s(ifnd8+1)=2;
ifnd9=find(tst9) ;
s(ifnd9)=1; s(ifnd9-1)=2;
ifnd10=find(tstl0) ;
S(ifndl0)=1;

s=s(1:N) ;
FLav;
plot(1:N,FLav,[1 N],[fl_eq fl_eq],'r'); axis([0 N 0 axmax]); drawnow;
Veloc = nadd/it/dt~d;
end

% Tubulin addition

tal = radd(it) ta2 = radd(it)<(kfE*TE~dt~kf*T~dtt) &~ tal; % add TE
ta3 = radd(it)<(2*kfE*TTE~dt~kfE*TETE*df~t~k*~t &~ (tallta2); % add TTE
ta4 = radd(it)<(kfE*Ceff~dt~kfE*2*TTE~dt~kfE*TE~tk*~t*s1=3 &~(tallta2|ta3);

if tallta2|ta3|ta4
ntav=it-previt;











FL=(S==2)+(S==3)+.5*(S==5)+.5*(S==4);
FLav=(previt*FLav+ntav*FL)/(ntav+previt);
previt=it;
nadd=nadd+1;
s(2:N)=S(1:N-1) ;
if tal
s(1)=1;
elseif ta2
s(1)=2;
elseif ta3;
s(1)=3;
elseif ta4
s(1)=4; S(2)=5;
end
end

% Tubulin removal

ta5 = roff(it)<(s(1)==1)*kr~dt;
ta6 = roff(it)<(s(1)==2)*krE~dt;
ta7 = roff(it)<(s(1)==3)*krE~dt;
ta8 = roff(it)<(s(1)==4)*krE~dt;
if ta5|ta6|ta7|ta8
Sold=s;
ntav=it-previt;

FL=(S==2)+(S==3)+.5*(S==5)+.5*(S==4);
FLav=(previt*FLav+ntav*FL)/(ntav+previt);
previt=it;
nadd=nadd-1;
S(1:N-1)=S(2:N) ;
if s(N-1)==4
s(N)=5;
else
S(N) = S(N-1) ;
end
if ta8
s(1)=3;
end
end

if s(1)==5
[tal ta2 ta3 ta4 ta5 ta6 tal ta8];
otst;
pause
end
it/nt
end

% Velocity(irun)=veloc;


B.3.4 Tethered Protofilament Growth with Divalent EB1


This stochastic model simulates the growth of a single microtubule protofilament in the


presence of divalent, EB 1 end-tracking motors and an applied force. The value of the affinity


modulation factor, applied force, and Kr can be varied to determine the resulting velocity. This


model also provides the state of the terminal subunit, position of the tracking unit, the time


average fluorescence along the protofilament, and time spent in each pathway. The kinetic


parameters were estintat~ed or used from literature values.


% stochastic model
% Simulates tethered MT's in presence of EB1
% Divalent EB1
% Inputs: f, Kd1, kon












% outputs: Velocity, State of subunit, Position of Track
% Plots: Time Avg Fluorescence vs subunit

%clear all;
tlC

rnsidel=0;
radd = 0;
roff=0;
rndT=0;

tim=1; % run time (s)
axmax=1; % max y-axis.

% Determine Parameters

% Fixed parameters
T = 10 ; % uM tubulin dimer concentration
V = 170; % nm/s; % elongation speed
d = 8; % nm; % subunit length

kf = v/d/T; % uMA-1sA-1 on-rate for tubulin -taken assuming irreversible elongation at
observed elongation speed
Tc = 5; % uM plus-end critical concentration
kr = kf*TC; % SA-1 off-rate, assuming Tc=kr/kf;

thalf=2.6; % s
kobs=log(2)/thalf; % sA-1 decay rate constant of EB1 from MT sides
EB1 = 0.27; % uM EB1 concentration

sigma = 10 ; % nm stdev of EB1 position fluctuations
ceff = 2*exp(-(8/10)^2/2)/(2*pi)^(3/2)/sigma^3; % concentration in nmA-3 -- based on 3-D normal
distribution on half-sphere
Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm-3 x (1 mol/ 6.022e23) x (1e27 nmA3/mA3) x (1 mA3/1000
L) x (10^6 uM/M)
K = 37; % equals kplus_side/kminus_side, valued required for Kdeff = 0.44

Kd1 = .65; % Kd, Dissociation constant for EB1 subunit and T, Kdl=klm/k1 (uM) value for
typical monovalent protein
%Kdl=Kdlvec(i run) ;

k1 = 10; % on-rate for EB1 subunit and T (uMA-1*SA-1), Value for typical protein-protein
binding
klm = kl*Kdl; % off-rate for EB1 subunit and T (SA-1)


%% Equlibria

E = EB1/((1+(T/Kdl))A2); % [EBl], Concentration of EB1 dimer in sol'n
TE = 2*E*T/Kdl; % [EB1-T], Concentration of EB1 dimer bound to 1 tubulin protomer
TTE = T*TE/(2*Kdl); % [EB1-TA2], Concentration of EB1 dimer bound to 2 tubulin protomers

a=4*KA2*E/Ceff;
b=(1+T/Kdl)*2*K*E/Ceff+1;

u_eq = (-b+sqrt(bA2+4*a))/2/a;
p_eq=2*K*E/Ceff~u_eq;
q_eq = 2*K~p_eq~u_eq;
pi_eq = T/Kdl~p_eq;

fl_eq=p_eq+pi _eq+q_eq/2 ; %% Equilibrium flourescence cone
check = u_eq+p_eq+q_eq+pi_eq; %%% should equal one


%% Determine kminus_side, kplus_side, kon

kmi nus_si de = kobs*(1+K~u_eq/(1+T/Kdl)) ; %% Based on FRAP half-life
kplus_side = K~kminus_side; %% by definition
kon_side = kplus_side/Ceff;

%f= 1; % affinity modulation factor
f-fvec(i run) ;

%% Mixed model -- chose kon, calculated koff from f
kon = 5; % uM-1s-1
fon = kon/kon_side; %% accelerated on-rate at end
kplus = kon*Ceff;
foff = fon/f;
kmi nus=fon/f~kmi nus_si de ; %% correspondi ng change off-rate at end













%% Other parameters

kfE = kf; % on-rate constant of TE and TTE to MT end uM-1s-1;
krE = kfE*TC/Kdl~kminus/kon; % off-rate constant of TE or TTE

% Initial conditions

S=ones(1,N); %s = 1 if unocupplied; 2 if bound to E, 3 if bound to TEE, 4 if bound to +side of
doubly bound, 5 if bound to -side
previt=0; FLav=0*S;
nadd=1;

konv=[kon kon_side~ones(1,N-l1];
kminusv=[kminus kminus_side~ones(1,N-1)];
kplusv=[kplus kplus_side~ones(1,N-1)];
klv=kl~ones(1,N) ;
klmy=klm~ones(1,N) ;

%KT =10;% 5; % eq. dissoc const. for tracker binding to EB1
KT=KTvec(i run) ;
kfT = 5; %
krT = KT~kfT;

%%% Tracking unit parameters
sigT = 10; % nm; tracking unit stdev
kT=4.1; %pN-nm
d=8; %nm spacing
gamT = kT/sigTA2; % pN/nm Tracking unit stiffness
CpTO = 100; %% uM effective concentration of Tracking unit end MT end
delta = 1; % nm -- transition state distance

%q=0;
q=qvec(i run) ;
Force = q~log(T/Tc)*kT/d; %pN
Ffac = exp(-Force~d/kT) ;

konTv = kony.*CpTO.*exp(-gamT*((1:N)-)-1.A*A22kTgmT(1:)1)dlt//k) % Effect of
stretching on Tracker-bound Eb1 binding
kfEp = kfE*CpTO; % forward rate for transfer of tubulin from tracking unit (based on detailed
balance)
kfTv = kfT.*CpTO.*e xp (-gamT ( (1:N) -1) A2*dA2/2/kT+gamT ( (1:N) -1)*"delta~d/kT) ; % Effect of
stretching on Tracker binding to MT-bound EB1

Track = 0;
Trackdist=0*(1:N) ;

charti me=1/max([kfEp sum(konTy) sum(kfTy) kfT*TTE kfT*TE kfT*E kfE*TTE kfE*TE kf*T kr krE kon*T
kl*T k1 kplus kplus_side kminus kminus_side]); %Characteristic time
dt = chartime/20; %simulation time increment
nt=round(tim/dt) ;
rnsidel=rand(nt,N) ;
radd = rand(nt,1) ;
roff=rand(nt,1) ;
rndT=rand(nt,1) ;


for it=1:nt; %% start time loop

5% side binding

tstl=(s==1)& rnsidel(it,:)<2*konv*E~dt; % binds E
tst2= (S==1)& rnsidel(it,:)<(konv*TE~dt+2*konv*E~dt) &~tstl; % or binds TE

tst3=(s==2)& rnsidel(it,:) tst4=(((S==2)&[0 S(1:N-1) 1]l) & rnsidel(it,:)<(kplusv~dt + kminusv~dt)) &~tst3; % bind plus
side
tstS=(((s==2)&[S(2:N)==1 0]) & rnsidel(it,:)<(kplusv~dt + kplusv~dt+kminusv~dt))
&~(tst3|tst4); % bind minus side
tst6=((s==2)& rnsidel(it,:)<(klv*T~dt + 2*kplusv~dt+kminusv~dt)) &~(tst3|tst4|tst5); % bind T
to E

tst7=(s==3)& rnsidel(it,:) tst8=(s==4)& rnsidel(it,:) tst9=(s==5)& rnsidel(it,:) tstl0=((s==3) & rnsidel(it,:)<(kminusv~dt+klmy~dt)) &~(tst7); %dissociate TE from side

otst=(tstl+tst2+tst3+tst4+tstS+tst6+tst7+s8t9ttl) % Anything happen?











if sum(otst)>0

Sold=s; % store old
ntav=it-previt; % number of additional steps in average

FL=(abs(s)==2)+(ass-3+5(abs(s)==5)+.5*(abs(s)==4) % EB1 fluorsescence
FLav=(previt*FLav+ntav*FL)/(ntav+previt); % Update time-averaged fluorescence
previt=it; % update


ifndl=find(tstl) ;
s(ifndl)=2;
ifnd2=find(tst2) ;
s(ifnd2)=3;
ifnd3=find(tst3) ;
s(ifnd3)=1;
ifnd4=find(tst4) ;
s(ifnd4)=5; S(ifnd4-1)=4;
ifnd5=find(tst5) ;
s(ifnd5)=4; S(ifnd5+1)=5;
ifnd6=find(tst6) ;
s(ifnd6)=3;
ifnd7=find(tst7) ;
s(ifnd7)=2;
ifnd8=find(tst8) ;
s(ifnd8)=1; s(ifnd8+1)=2; S=S(1:N);
ifnd9=find(tst9) ;
s(ifnd9)=1; s(ifnd9-1)=2;
ifnd10=find(tstl0) ;
S(ifndl0)=1;
if ifnd3>0 & ifnd3(1)==1 ifnd8>0 & ifnd8(1)==1|
51(1,irun)=s1(1,irun)+1;
elseif ifndl>0 & ifndl(1)==1| ifnd7>0 & ifnd7(1)=
51(2,irun)=s1(2,irun)+1;
elseif ifnd2>0 & ifnd2(1)==1| ifnd6>0 & ifnd6(1)=
51(3,irun)=s1(3,irun)+1;
elseif ifnd4>0 & ifnd4(1)==2| ifnd5>0 & ifnd5(1)=
51(4,irun)=s1(4,irun)+1;
end


ifndl0>0 & ifndl0(1)=

:1| ifnd9>0 & ifnd9(1)


if Track<1 %% start unbound tracker loop
% Tubulin addition
tal = radd(it) ta2 = radd(it)<(kfE*Ffac*TE~dt+kf*Ffac*T~dt) &~ tal; % add TE
ta3 = radd(it)<(2*kEFa*T"kfEfE*Ffac*TE~dt~kf*Ffac*TTdt &~ (tallta2); % add TTE
ta4 = radd(it)<(kfE*Ffac*Ceff~dt+kfE*Ffac*2TEdtkEFacT"tTfFa*Td)(s1=3
&~(tallta2|ta3) ;
if tallta2|ta3|ta4
ntav=it-previt;

FL=(abs(s)==2)+(ass-3+5(abs(s)==5)+.5*(abs(s)==4)
FLav=(previt*FLav+ntav*FL)/(ntav+previt);
previt=it;
nadd=nadd+1;
S(2:N)=S(1:N-1) ;
if tal
s(1)=1;
count(1,irun)=count(1,irun)+1;
s1(1,irun)=s1(1,irun)+1;
elseif ta2
s(1)=2;
count(2,irun)=count(2,irun)+1;
s1(2,irun)=s1(2,irun)+1;
elseif ta3;
s(1)=3;
count(3,irun)=count(3,irun)+1;
s1(3,irun)=s1(3,irun)+1;
elseif ta4
s(1)=4; S(2)=5;
count(4,irun)=count(4,irun)+1;
s1 (4,i run) =s1(4,i run+1


% Tubulin removal
ta5 = roff(it)<(s(1)==1)*kr~dt;
ta6 = roff(it)<(s(1)==2)*krE~dt;
ta7 = roff(it)<(s(1)==3)*krE~dt;












ta8 = roff(it)<(s(1)==4)*krE~dt;

if ta5|ta6|ta7|ta8
Sold=s;
ntav=it-previt;
FL=(abs(s)==2)+(ass-3+5(abs(s)==5)+.5*(abs(s)==4)
FLav=(previt*FLav+ntav*FL)/(ntav+previt);
previt=it;
nadd=nadd-1;
S(1:N-1)=S(2:N) ;
if s(N-1)==4
s(N)=5;
else
S(N) = S(N-1) ;
end

if ta5
count(5,irun)=count(5,irun)+1;
elseif ta6
count(6,irun)=count(6,irun)+1;
elseif tal
count(7,irun)=count(7,irun)+1;
elseif ta8
s(1)=3;
count(8,irun)=count(8,irun)+1;
end

if s(1)==1;
s1(1,irun)=s1(1,irun)+1;
elseif s(1)==2;
s1(2,irun)=s1(2,irun)+1;
elseif s(1)==3;
s1(3,irun)=s1(3,irun)+1;
elseif s(1)==4;
s1(4,irun)=s1(4,irun)+1;
end
end

%% Track =0 Tracking unit unbound

tTa = (Track==0)& rndT(it) tTb = ((Track==0)& rndT(i t)<(kfT*TE~dt~kfT*E~dt)) &~ tTa; % Bind TE
tTC = ((Track==0)& rndT(i t)<(kfT*TTE~dt~kfT*TEdt+kf"dt)kT*)d) &~ (tTaltTb) ; % Bind TTE
totA = Cumsum((S==2).*kfTy~dt);
tTd = ((Track==0) & rndT(it) < (totA(N) +kfT*TTE~dt+kfT*TE~dt+kfT*E~dt) ) &~ (tTaltTbltTC) ; %
Binding tracking unit to EB1 on MT
totA2 = cumsum((S==3) .*kfTy~dt) ;
tTe = ((Track==0) & rndT(it) < (totA2 (N)+totA(N)+kfT*TTE~dt+kfT*TE~dt+kfT*EEdt) &~
(tTaltTbltTCltTd); % Binding tracking unit to TE on MT
totA3 = Cumsum((S==5) .*kfTy~dt) ;
tTf = ((Track==0) & rndT(it) < (totA3(N)+totA2 (N)+totA(N)+kfT*TTE~dt+kfT*TE~dt+kfT*EEdt) &~
(tTaltTbltTCltTdltTe); % Binding tracking unit to doubly bound TE on MT

%% Track = -1 Tracking unit bound w/ E
tT1 = (Track==-1)& rndT(it)<2*kl*T~dt; % Binding tracking unit to 1st tubulin
totB = Cumsum((S==1).*2.*konTy~dt); %
tT2 = ((Track==-1)& rndT(it)<(totB(N)+2*kl*T~dt)) &~ tT1; % Binds MT
tT3 = ((Track==-1) & rndT(it) < (krT~dt + totB(N)+2*kl*T~dt)) &~ (tT1|tT2); % release EB1


%% Track = -2 Tracking unit bound w/ TE
tT4 = (Track==-2)& rndT(it) tT5 = (Track==-2)& rndT(it)<(klm~dt+kl*T~dt) &~tT4; % dissociate 1st tubulin
totC = cumsum((S==1).*konTy~dt);
tT6 = ((Track==-2)& rndT(it)<(totC(N)+klm~dt+kl*T~dt)) &~ (tT4|tT5); % Binds free MT site
tT7 = ((Track==-2) & rndT(it)<(krT~dt+totc(N)+klm~dt+kl*T~dt))& (tT4|tT5|tT6) ; % release TE
tT8 = ((Track =2) & rndT(it)<(kfEp*Ffac~dt+krT~dt+totc(N)+kl mdklTt)&
(tT4|tT5|tT6|tT7); % transfer T to end

%% Track = -3 Tracking unit bound w/ TTE
tT9 = (Track==-3)& rndT(it)<2*klm~dt; % release one tubulin
tT10 = ((Track==-3)& rndT(it)<(krT~dt+2*klm~dt)) &~ tT9; % release TTE
tT11 = ((Track==-3)& rndT(it)<(2*kfEp*Ffac~dt+krT~dt+2*klmmdt) &~ (tT9|tT10); % transfer
tubulin to end

if tTa
Track=-1;
elseif tTb
Track =-2;
elseif tTC












Track = -3;
elseif tTd
%% find which subunit bound
i fnd=fi nd (rndT(i t) <(totA+kfT*TTE~dt~kfT*TEdt+kt~fT"T*E d));
Track=min(ifnd);
S(Track)=-2;
if Track==1;
sl(6, irun)=s1(6, irun)+1;
end
elseif tTe
%% find which subunit bound
i fnd=fi nd(rndT(i t)<(totA2+totA(NU)+kfT*TTE~dt~kfT*TEdt+ktfkfT*Etdt));
Track=min(ifnd);
S(Track)=-3;
if Track==1;
51(7, irun)=s1(7, irun)+1;
end
elseif tTf
%% find which subunit bound
i fnd=fi nd (rndT(i t)<(totA3+totA2 (N)+totA(N) +kfT*TTE~dt~kfT*TEdt+ktfkfT*Etdt));
Track=min(ifnd);
if Track>1
S(Track)=-5;
S(Track-1) = -4;
end
if Track-1==1;
51(8, irun)=s1(8, irun)+1;
end
elseif tT1
Track = -2;
elseif tT2
%% find which subunit bound
ifnd=find(rndT(it)<(totB+2*kl*T~dt));
Track=min(ifnd);
S(Track) = -2;
if Track==1;
51(6, irun)=s1(6, irun)+1;
end
elseif tT3
Track = 0;
elseif tT4
Track = -3;
elseif tT5
Track = -1;
elseif tT6
% find which subunit bound
ifnd=find(rndT(it)<(totC~kmd+klmT~dt)k*~t
Track=min(ifnd);
S(Track) = -3;
if Track==1;
sl(7,i run)=s1(7,i run)+1;
end
elseif tT7
Track =0;
elseif tT8
Track =1;
S(2:N)=S(1:N-1);
s(1)=-2;
nadd=nadd+1;
count (9, irun)=count (9, i run)+1
51(6, irun)=s1(6, irun)+1;
elseif tT9
Track = -2;
elseif tT10
Track = 0;
elseif tT11
Track =1;
S(2:N)=S(1:N-1);
s(1)=-3;
nadd=nadd+1;
count(10,irun)=count(10,irun)+1;
sl(7,i run)=s1(7,i run)+1;
end

if Track==-3;
T1(1, irun)=T1(1,i run)+1;
el sei f Track==-2;
T1(2,irun)=T1(2,irun)+1;
el sei f Track==-1;
T1(3, irun)=T1(3,i run)+1;
elseif Track==0;












T1(4,irun)=T1(4,irun)+1;
else
T1(Track,irun)=T1(Track,irun)+1;
end

elseif Track>0
Trackdist(Track)=Trackdist(Track)+1;
ffac = exp(-gamT*(Track-.5)*dA2/kT) ; % rate factor due to stretching tracking unit upon
addition

% Tubulin addition
tal = radd(it) ta2 = radd(it)<(ffac~kfE*Ffac*TE~dt+ ffac~kf*FfaTdt &~ tal; % add TE
ta3 = radd(it)<(ffac*2*kfE*Ffac*TTE~dt+ ffac~kfE*fcT~tfa~fFa*~t &~ (tallta2); %
add TTE
ta4 =
radd(it)<(ffac~kfE*Ffac*Ceff~dt+ ffac~kfE*Fa**T~tfa~f*fcT~tfa~fFa*~t*a
s(s(1))==3) &~(tallta2|ta3); % add from TE-bound MT

if tallta2|ta3|ta4
ntav=it-previt;

FL=(abs(s)==2)+(ass-3+5(abs(s)==5)+.5*(abs(s)==4)
FLav=(previt*FLav+ntav*FL)/(ntav+previt);
previt=it;
nadd=nadd+1;
Track=Track+1;
S(2:N)=S(1:N-1) ;
if tal
s(1)=1;
count(11,irun)=count(11,irun)+1;
s1(1,irun)=s1(1,irun)+1;
elseif ta2
S(1)=2;
count(12,irun)=count(12,irun)+1;
s1(2,irun)=s1(2,irun)+1;
elseif ta3;
s(1)=3;
count(13,irun)=count(13,irun)+1;
sl(3,irun)=s1(3,irun)+1;
elseif ta4
if Track --2
S(1) = -4; S(2)=-5; % bound tracking unit
sl(8,irun)=s1(8,irun)+1;
else
S(1)=4; S(2)=5; % unbound tracking unit
s1(4,irun)=s1(4,irun)+1;
end
count(14,irun)=count(14,irun)+1;
end

if Track==-2;
T1(2,irun)=T1(2,irun)+1;
elseif Track==-1;
T1(3,irun)=T1(3,irun)+1;
elseif Track==0;
T1(4,irun)=T1(4,irun)+1;
else
T1(Track,irun)=T1(Track,irun)+1;
end

end

% Tubulin removal

ta5 = roff(it)<(s(1)==1)*kr~dt;
ta6 = roff(it)<(abs(s(1))==2)*krE~dt;
ta7 = roff(it)<(abs(s(1))==3)*krE~dt;
ta8 = roff(it)<(abs(s(1))==4)*krE~dt;
if ta5|ta6|ta7|ta8
Sold=s;
ntav=it-previt;

FL=(abs(s)==2)+(ass-3+5(abs(s)==5)+.5*(abs(s)==4)
FLav=(previt*FLav+ntav*FL)/(ntav+previt);
previt=it;
nadd=nadd-1;
if Track ==
if ta6
Track = -2;
elseif ta7












Track = -3;
end
else
Track=Track-1;
end

S(1:N-1)=S(2:N) ;

if s(N-1)==4
s(N)=5;
elseif s(N-1)==-4
s(N)=-5;
else
S(N) = S(N-1) ;
end

if ta5
count(15,irun)=count(15,irun)+1;
elseif ta6
count(16,irun)=count(16,irun)+1;
elseif tal
count(17,irun)=count(17,irun)+1;
elseif ta8
if s(1)---5;
s(1)=-3;
else
s(1)=3;
end
count(18,irun)=count(18,irun)+1;
end

if s(1)==1;
sl(1,irun)=s1(1,irun)+1;
elseif s(1)==2;
sl(2,irun)=s1(2,irun)+1;
elseif s(1)==3;
sl(3,irun)=s1(3,irun)+1;
elseif s(1)==4;
s1(4,irun)=s1(4,irun)+1;
elseif s(1)==-1;
sl(5,irun)=s1(5,irun)+1;
elseif s(1)==-2;
sl(6,irun)=s1(6,irun)+1;
elseif s(1)==-3;
sl(7,irun)=s1(7,irun)+1;
elseif s(1)==-4;
sl(8,irun)=s1(8,irun)+1;
end

if Track==-3;
T1(1,irun)=T1(1,irun)+1;
elseif Track==-2;
T1(2,irun)=T1(2,irun)+1;
elseif Track==-1;
T1(3,irun)=T1(3,irun)+1;
elseif Track==0;
T1 (4,i run) =T1 (4,i run+1
else
T1(Track,irun)=T1(Track,irun)+1;
end
end

if Track>0

5% Tracking unit-E detachment
kminusy =[kminus kminus_side.*ones(1,N-1)];
ffac2 = exp(gamT*(Track-1)*d~delta/kT);
ffac3 = exp(-gamT*(Track-.5)*dA2/kT);


5% Doubly bound EB1
tst1 = (s(Track) = -5)& rndT(it) head (doubly bound)
tstla = ((s(Track) = -5)& rndT(it)<(krT~ffac2*dt+kminusv(Track) *ffa2d) &~ tstl; %
detachment of tracking from doubly bound EB1
tstlb = ((s(Track) = -5) & rndT(it)< (kminusv(Track-
(Track>1))*dt+krT~ffac2*dt+kminusv(Track )fa2d) &~ (tstlltstla); % detachment of plus-side
EB1 head (doubly bound)

5% bound T-E
tst2 = (S(Track) == -3)& rndT(it)











tst2a = ((S(Track) == -3)& rndT(it)<(krT~ffac2*dt+kminusv(Track) *ffa2d) &~ tst2; % detach
Track from TE
tst2b = ((s(Track) == -3)& rndT(it)<(klm~dt+krT~ffac2*dt+kminusv(Tr ac)fa2d) &~
(tst2|tst2a) ; % dissociate T

5% bound E
tst3 = (S(Track) == -2)& rndT(it) tst3a = ((S(Track) == -2)& rndT(it)<(krT~ffac2*dt + kminusv(Track)*ffac2*dt)) &~ tst3; %
detach Tracker
tst3b = ((s(Track) == -2)& rndT(it)<(kl*TE~dt+krT~ ffac2*dt+kminusv( Trc)fa2d) &~
(tst3|tst3a) ; % add T
tst3c = ((S(Track) == -2)& S(Track+1)==1 &
rndT(it)<(kplus_side~ffac3*ffac2*dt+kl *TE~tkTfa2d~mns(rc)fa2d) &~
(tst3|tst3altst3b); % bind second head in minus-direction

if Track>1
tst3d = ((s(Track-1)==1 & s(Track) == -2)& rndT(it)<(kplusv(Track-
1)*dt+kplus_side~ffac3*ffac2*dt+kl *TE~d~r~fc*tkiuvTak*fc*dt) &~
(tst3|tst3altst3bltst3c); % bind second head in plus-direction
else
tst3d=0;
end

if tst1
Track=Track-1;
S(Track) = -2;
S(Track+1) = ;
if Track==1;
sl(6,irun)=s1(6,irun)+1;
end
elseif tstla
Track_old=Track;
S(Track) = 5;
S(Track-1)=4;
Track = 0;
if Track ;old-1==1;
s1(4,irun)=s1(4,irun)+1;
end
elseif tstlb
S(Track) = -2;
S(Track-1) = ;
if Track-1==1;
s1(1,irun)=s1(1,irun)+1;
end
elseif tst2
Track_old=Track;
S(Track) = 1;
Track = -2;
if Track_old==1;
s1(1,irun)=s1(1,irun)+1;
end
elseif tst2a
Track_old=Track;
S(Track) = 3;
Track = 0;
if Track old==1;
sl(3,irun)=s1(3,irun)+1;
end
elseif tst2b
S(Track) = -2;
if Track==1;
sl(6,irun)=s1(6,irun)+1;
end
elseif tst3
Track_old=Track;
S(Track) = 1;
Track = -1;
if Track_old==1;
s1(1,irun)=s1(1,irun)+1;
end
elseif tst3a
Track_old=Track;
S(Track) = 2;
Track = 0;
if Track ;old==1;
sl(2,irun)=s1(2,irun)+1;
end
elseif tst3b
S(Track) = -3;
if Track==1;












sl(7,irun)=s1(7,irun)+1;
end
elseif tst3c
Track_old=Track;
S(Track+1) = -5;
S(Track) = -4;
Track=Track+1;
if Track_old==1;
sl(8,irun)=s1(8,irun)+1;
end
elseif tst3d
Track_old=Track;
S(Track) = -5;
S(Track-1) = -4;
if Track_old-1==1;
sl(8,irun)=s1(8,irun)+1;
end
end

if Track==-3;
T1(1,irun)=T1(1,irun)+1;
elseif Track==-2;
T1(2,irun)=T1(2,irun)+1;
elseif Track==-1;
T1(3,irun)=T1(3,irun)+1;
elseif Track==0;
T1(4,irun)=T1(4,irun)+1;
else
T1(Track,irun)=T1(Track,irun)+1;
end
end % Ends "if Track>0" Loop
end %Ends "if Track<1, elseif Track>0 Loop"

irun
percent=(nt*(irun-1)+it)/(nt~nrun)
veloc = nadd/it/dt~d;

end % Ends "for it=1:nt" time loop

kon_vect=kon~vector;
Kdlv=Kdl~vector;
timy=tim~vector;
Nv=N~vector;
F = (qvec~log(T/Tc)*kT/d) ;
Velocity(irun)=veloc;
matrix=[kon_vect',Kd~v',KTvec',fvec',qve c'Feoiyiyv]

xlswrite('sim_track_M.xls', matrix','matrix'); % save position & time data in EXCel
xlswrite('sim_track_M.xls', count,'count'); % save position & time data in EXCel
xlswrite('sim_track_M.xls', 51,'s'); % save


The following is a macro that runs the stochastic model above at various values off; KT, and q.

% This macro runs track for multiple parameter sets
% Inputs: f, KT, q
% Outputs: Velocity, state of subunit, location of tracking unit

clear all;
tic;

fvec=[1000*ones(1,20)]; % f
qvec=[0,0.25,0.5,1,2,3,4,5,6,7,0,0.25,0.5,1,2,3,4,5,6,7];
KTvec=[0.1*ones(1~O,10oe(1,1*ns,10)];

N=40; % number of subunits to simulate
velocity=0*fvec;
nrun=1ength(fvec) ;
vector=ones(1,nrun) ;
count=zeros(18,nrun) ;
sl=zeros(8,nrun) ;
T1=zeros(N+4,nrun) ;

for irun=1:nrun;
track
end

Velocity(irun)=veloc;
kon_vect=kon~vector;











Kdlv=Kdl~vector;
timy=tim~vector;
Nv=N~vector;

F = (qvec~log(T/Tc)*kT/d) ;
matrix=[kon_vect',Kd~v',KTvec',fvec',qve c'Feoiyiyv]

xlswrite('sim_track_M.xls', matrix','matrix'); % save position & time data in EXCel
xlswrite('sim_track_M.xls', count,'count'); % save position & time data in EXCel
xlswrite('sim_track_M.xls', 51,'s'); % save state of subunit data in EXCel

Track
Velocity
time=toc/3600

B.4 Ciliary Plug Model


This model simulates a 13-protofilament MT polymerizing in a ciliary plug against a


motile surface with a constant load. The value of the applied force and protofilament length can


be varied to determine the trajectory of the ciliary plug (position versus time) and the resulting


velocity. The kinetic parameters were estimated or used from literature values.




% Simulates MT-based motility in ciliary plugs based on the LLF model
% Trajectory between steps not simulated (fast version)

clear all;
hold off;
% Filament Parameters
kT=4.14; % Thermal energy (pN-nm)
nf = 13; % No. filaments
kappa = 0.15; % Filament compression stiffness (pN/nm)
Df = 4e6; % Filament diffusivity (nmA2/s)
deltaf = kT/Df; % Filament Drag (pN-s/nm)
v=167; % Expected velocity (nm/s)
d = 8 ; % subunit length (nm)
Tmin = d/v;% Mean time to load (s)
Kappa2 = 60; % filament stretch stiffness (pN/nm)
pn=1. ; % Positioning error (nm)
%% Simulation setup
z~f=rand(1,nf)*100; % random initial distribution of filament lengths
z0f(1)=-kappa/kappa2*sum(z0f(2:nf)); % set filament 1 position to balance forces
dt=.005*Tmin; % simulation time increment
nt = 2^18; % total time steps
zp = 0*(1:nt); t=0; z=0; % Initialize t=time; z= position of motile surface
nplot = 2A4; dnplot=nplot; % Time steps between plotting
ih=1; nh=nt/nplot; zh=zeros(1,nh); th=zh; zhn=zh; % Plotting storage vectors/variables
nbp=10*round(Tmin/(dnplot~dt)) % sets plotting range based on number of expected steps
runb=rand(dnplot,nf); % Random numbers for first set (between plotting)
diffs=randn(1,dnplot)*sqrt(2*Df/nf~dt); % dw for first set
jsim=1; % iteration index within set
zeq=0; kf=zeros(1,nf) ;
for i=1:nt
unbind=(runb(jsim,:)<(kf~dt)); % Identify those that unbind
if sum(unbind>=1) |i==1
z0f=z0f+unbind~d; % shift those that rebind Equilibrium position
SF=1;
kappai=kappa*((zeq)<=z0f)+kappa2*((zeq)>zf; % Vector of filament stiffness
while SFA2>1e-10;
zeq = kappai~z0f'/sum(kappai) ;
kappai=kappa*((zeq)<=z0f)+kappa2*((zeq)>z~) % Vector of filament stiffnesses
F=-kappai.*(zeq-z~f) ; % Vector of forces
SF=sum(F) ;











end
stiffness=sum(kappa*((zeq)<=z0f)+kappa2*(zq>") % total stiffness
fvar=kT/stiffness; % Position variance
Pr = d*F/kT; % Dimensionless force
tau = (exp(Pr)-1-Pr) ./Pr.A2; % Dimensionless Mean Time to shift
T = tau~dA2/Df; % Mean Time to shift (s)
kf =1./(Tmin+T) ; % Shift probability per unit time (s-1)
end
z=zeq+diffs(1,jsim)*sqrt(fvar); % Noisy position
zp(i) = z; % store position


jsim=jsim+1;
if i==nplot
tp=(1:i)*dt; th(ih)=t; % store Time
zh(ih)=zp(i); % store position
zhn(ih)=zp(i)+pn~randn(1,1); % Noisy position
tplot=th(max(ih-nbp,1) :ih) ;
zplot=zh(max(ih-nbp,1) :ih) ;
znplot=zhn(max(ih-nbp,1) :ih) ;
SUBPLOT(2,1,1), plot(tplot,znplot,'r',tplot,zplot,'b'); % Plot recent trajectory
tmin=th(max(ih-nbp,1)) ; tmax = max([th .1]);
zmin=zh(max(ih-nbp,1))-3; zmax = max(zh)+5;
axis([tmin tmax zmin zmax]); % AXes
zrng=(zmin:5.4:zmax) ;
nlin=1ength(zrng) ;
tlin=[ones(nlin,1)*tmin ones(nlin,1)*tmax];
zlin=[zrng' zrng'];
line(tlin', zlin') ;
SUBPLOT(2,1,2), hist(z~f-zeq,-5:10:max(z0f-zeq)+5); % Histogram of filament lengths
drawnow;
nplot=nplot+dnplot; % Update next iteration to plot
ih=ih+1; % Update plot index
runb=rand(dnplot,nf); % Generate random numbers for next set
diffs=randn(1,dnplot) ; % "
jsim=1; % Reset set index
end
t=t+dt; % Update time
end










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BIOGRAPHICAL SKETCH

Luz Elena Caro was born and raised in Delaware, and graduated from Middletown High

School in Middletown, DE. She attended the University of Delaware and obtained her B.ChE. in

Chemical Engineering. During her time at the university, Luz Elena completed two summer

internships at Merck & Co., Inc. After graduation, she interned at General Mills for a summer

before j oining the chemical engineering department at the University of Florida for her graduate

degree. Upn receiving her doctoral degree, Luz Elena will join the drug metabolism department

at Merck & Co., Inc. in West Point, PA as a senior research pharmacokineticist.





PAGE 1

1 FORCE GENERATION BY MICROT UBULE END-BINDING PROTEINS By LUZ ELENA CARO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

PAGE 2

2 2007 Luz Elena Caro

PAGE 3

3 To my supportive parents, Ruby and Armando Caro, encouraging siblings, Maritza, Mando, and Cesar, and my eternal best friend, Antonio.

PAGE 4

4 ACKNOWLEDGMENTS I acknowledge the support of m y advisor, Dr. Richard B. Dickinson, whose patience, guidance, and motivation provided the necessary t ools for my successful and rewarding graduate experience. The helpful comments and construc tive criticisms from my committee members were greatly appreciated. I thank Dr. Anuj Chauhan for th e continued encouragement and support throughout my professional career. The in sightful career advice and assistance provided by Dr. Jennifer Curtis was greatly appreciated. I recognize Dr. Anthony Ladd for introducing me to the exciting research at the University of Florida. The assistance provided by the Chemical Engineering faculty and staff was invaluable for my experience at the University of Florida. I thank Dr. Daniel Purich for his biochemistry and professional advice which helped me to develop my research skills. I am grateful for the expertise of the members in the biochemistry group (under advisement of Dr. Daniel Purich) wh ich helped me gain the proper biochemistry understanding needed for my graduate research; I thank Dr. William Zeile, Dr. Joseph Phillips, Dr. Fangliang Zhang. I thank my group members who helped me lear n experimental techni ques and exchanged ideas pertaining to research; my graduate expe rience was enriched by their companionship and support: Kimberly Interliggi, Colin Sturm, Ga urav Misra, Jeff Sharp, Huilian Ma, Adam Feinburgh.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES.........................................................................................................................9 LIST OF TERMS...........................................................................................................................11 ABSTRACT...................................................................................................................................17 CHAP TER 1 INTRODUCTION..................................................................................................................19 1.1 Microtubules .................................................................................................................20 1.2 End-Tracking Proteins .................................................................................................. 23 1.2.1 EB1 ................................................................................................................... 23 1.2.2 Adenom atous Polyposis Coli (APC)................................................................. 25 1.2.3 Ciliary and Flagellar M ovement....................................................................... 26 1.3 Force Generation Models ..............................................................................................27 1.3.1 Brownian Ratchet Models ................................................................................. 27 1.3.2 Sleeve Model .....................................................................................................28 1.3.3 Kinetochore Motors ..........................................................................................29 1.3.4 Filam ent End-Tracking Motors......................................................................... 29 1.4 Therm odynamic Driving Force..................................................................................... 30 1.5 Summ ary.......................................................................................................................31 1.6 Outline of Dissertation .................................................................................................. 31 2 MICROTUBULE END-TRACKING MODEL.....................................................................37 2.1 EB1 End-Tracking Motors ............................................................................................37 2.2 Microtubule Growth Model ..........................................................................................38 2.2.1 Param eter Estimations....................................................................................... 42 2.2.2 Elongation Rate in the Absence of External Force ........................................... 43 2.2.3 Force effects on elongation rate ........................................................................ 44 2.3 Summ ary.......................................................................................................................45 3 PROTOFILAMENT ENDTRACKING MODEL W ITH MONOVALENT EB1................ 53 3.1 Non-Tethered Protofilam ent Growth............................................................................ 53 3.1.1 Therm odynamics of EB1tubulin interactions..................................................54 3.1.2 Kinetics of EB1-tubu lin interactions .................................................................55 3.1.3 Param eter Estimations....................................................................................... 56 3.1.4 Results ............................................................................................................... 58

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6 3.2 Tethered P rotofilament Growth....................................................................................58 3.2.2 Model ................................................................................................................59 3.2.3 Param eter Estimations....................................................................................... 61 3.2.4 Results ............................................................................................................... 63 3.3 Summ ary.......................................................................................................................64 3.3.1 Non-Tethered Protofilam ents............................................................................ 64 3.3.2 Tethered P rotofilaments.................................................................................... 65 4 PROTOFILAMENT ENDTRACKING MODEL W ITH DIVALENT EB1........................ 73 4.1 Non-Tethered Protofilam ent Growth............................................................................ 73 4.1.1 Kinetics of EB1-Tubulin Interactions ............................................................... 73 4.1.2 EB1 Occupational Probability Model ...............................................................75 4.1.3 Average Fraction of EB1-bound Subunits at Equilibrium ................................ 77 4.1.4 Average Fraction of EB1-bound subun its during protofilam ent growth.......... 77 4.1.5 Param eter Estimations....................................................................................... 78 4.1.6 Results ............................................................................................................... 81 4.1.6.1 Occupational probability .................................................................... 81 4.1.6.2 Average fraction of EB1-bound subunits at equilibrium ....................82 4.1.6.3 Average fraction of EB1-bound subunits during protofilam ent growth.................................................................................................82 4.2 Tethered P rotofilament Growth Model......................................................................... 83 4.2.1 Kinetics of EB1-Tubulin Interactions ............................................................... 84 4.2.2 Protofilament End-Tracking Model.................................................................. 87 4.2.3 Param eter Estimations....................................................................................... 87 4.2.4 Results ............................................................................................................... 87 4.3 Summ ary.......................................................................................................................91 4.3.1 Non-Tethered Protofilam ents............................................................................ 92 4.3.2 Tethered P rotofilaments.................................................................................... 93 5 CILIARY PLUG MODEL...................................................................................................115 5.1 Model ..........................................................................................................................115 5.2 Parameter Estimations................................................................................................. 118 5.3 Results ........................................................................................................................ .119 5.4 Summ ary.....................................................................................................................120 6 DISCUSSION.......................................................................................................................124 6.1 Possible Roles of End-Track ing Motors in Biology ................................................... 124 6.2 Microtubule End-Tracking Model ..............................................................................126 6.3 Protofilam ent End-Tracking Models.......................................................................... 127 6.4 Future W ork................................................................................................................128 APPENDIX A PARAMETER ESTIMATIONS.......................................................................................... 130

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7 A.1 Concentrations of EB1 Species in Solution ................................................................130 A.2 Occupation Probability of Monovale nt EB1 Binding to Non-Tethered Protofilam ent...............................................................................................................131 A.3 Occupation Probability of Monovalent EB 1 Binding to Tethered Protofilam ent...... 132 A.4 Occupation Probability of Divalent EB 1 Binding to Tethered Protofilam ent............ 137 B MATLAB CODES............................................................................................................... 145 B.1 13-Protofilam ent Microtubule Model.........................................................................145 B.2 Protofilam ent Growth Model with Monovalent EB1................................................. 147 B.2.1 Occupational Probability of M onovalent EB1 on a Non-Tethered Protofilam ent................................................................................................... 147 B.2.2 Occupational Probability of Monovalent EB1 on a Tethered Protofilam ent.. 149 B.3 Protofilam ent Growth Model with Divalent EB1....................................................... 152 B.3.1 Occupational Probability of Divalent EB1 on a Non-Tethered Protofilam ent................................................................................................... 152 B.3.2 Average Fraction of divalent EB1-bound Protomers on Side of Protofilam ent................................................................................................... 155 B.3.3 Average Fraction of EB1-bound protom ers during protofilam ent growth..... 157 B.3.4 Tethered P rotofilament Growth with Divalent EB1....................................... 160 B.4 Ciliary Plug Model ...................................................................................................... 170 LIST OF REFERENCES.............................................................................................................172 BIOGRAPHICAL SKETCH.......................................................................................................178

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8 LIST OF TABLES Table page 1-1 Thermodynamic equations characterizing the multiple steps for GDP to GTP conversion ..................................................................................................................... .....35 1-2 Equilibrium constants used in energy equations................................................................ 36 4-1 Protofilament stall forc es at varying values of KT and affinity modulation factors. Stall forces (in units of pN) correspond to the data represented in Figure 4-12.............. 109

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9 LIST OF FIGURES Figure page 1-1 Microtubule structure...................................................................................................... ...33 1-2 Chromosomal binding site of m icrotubules....................................................................... 33 1-3 EB1 binding to microtubule lattice.................................................................................... 34 1-4 Concentration of EB1 along length of m icrotubule........................................................... 34 2-1 Model for microtubule force gene ration by EB1 end-tracking m otor............................... 47 2-2 Reaction mechanisms of EB1 end-tracking motor............................................................ 48 2-3 Force dependence on EB1 binding and equilibrium surface position............................... 49 2-4 Microtubule elongation in the absence of external force...................................................49 2-5 Distribution of protofilament lengt hs for m icrotubule end-tracking model...................... 50 2-6 Effect of applied force on MT elongation rate................................................................... 51 2-7 Thermodynamic versus simulated stall forces................................................................... 52 3-1 Schematic of non-tethered, monovalent EB1 end-tracking motor mechanisms................ 67 3-2 Various pathways of non-tethered m onovalent EB1 binding to protofilament................. 67 3-3 Choosing an optimal K1value for monovalent EB1........................................................... 68 3-4 EB1 density profile on a non-tethered m icrotubule protof ilament with monovalent EB1....................................................................................................................................69 3-5 Effect of K1 on profile of monovalent EB 1 occupational probability............................... 70 3-6 Schematic of tethered, monovalent EB1 end-tracking motor mechanisms....................... 70 3-7 Force effects on a tethered pr otofilam ent with monovalent EB1...................................... 71 3-8 Divalent EB1 represented as divalent end-tracking m otor................................................ 72 4-1 Mechanisms of a non-tethered, divalent end-tracking motor............................................ 95 4-2 Mechanisms of equilibrium, side binding of EB1 to protofilament.................................. 97 4-3 Choosing an optimal K1value for divalent EB1................................................................. 97

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10 4-4 Effect of kon on optimalK1.................................................................................................98 4-5 EB1 equilibrium binding.................................................................................................... 99 4-6 Occupational probab ility of EB1 along length of protofilam ent.....................................100 4-7 Time averaged EB1-bound tubulin fraction at equilibrium............................................. 101 4-8 Time averaged fraction of EB1-bound subunits during protofilam ent growth................ 102 4-9 Mechanisms of tethered, protofilament end-tracking m odel w ith divalent EB1............. 103 4-10 Mechanisms of tubulin addition to linking protein-bound protofilam ent........................105 4-11 Force-velocity profile s for tethered protofilam ents bound to divalent EB1 end-tracking motors......................................................................................................... 106 4-12 Stall forces versus affinity m odulation factor at various KT values.................................109 4-13 Effect of f KT, and F on pathways taken.........................................................................110 4-14 Percent of time protofilament bound and unbound to motile surface.............................. 111 4-15 State of the terminal subunit (S1) when f =1 and f =1000 .................................................112 4-16 Fraction of S1 subunits bound and unbound from motile surface................................... 113 4-17 Average state of unbound linking protein........................................................................ 114 5-1 EM image of a ciliary plug at the en d of a ciliary microtubule....................................... 120 5-2 Schematic of ciliary plug inserted into the lum en of a cilia /flagella microtubule........... 121 5-3 Mechanism of the ciliary/ flagellar end-tracking motor ................................................... 121 5-4 Force effects on ciliary m icrotubules............................................................................... 122 5-5 Ciliary plug movement................................................................................................... 123

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11 LIST OF TERMS ADP: Adenosine diphosphate APC: Adenomatous Polyposis Coli ATP: Adenosine triphosphate a: Width of protofilament Ceff: Effective concentration of a free subunit of filament-bound EB1 CT: Effective concentration of track ing unit at protofilament plus-end d: Size of tubulin protomer dbE-: State of tubulin protomer attached to the subunit on the minus-side of a doublebound EB1 dimer dbE+: State of tubulin protomer attached to the subunit on th e plus-side of a doublebound EB1 dimer [E]0: Total intracellular EB1 concentration [E]: Concentration of EB1 in solution EB1: End Binding Protein 1 Esp: Hookean Spring energy Df : Protofilament diffusivity dt : Time steps taken in simulation F : Force applied to microtubule plus-end Fstall: Stall force maximum achievable force f : Energy captured from hydrolysis that is used for affinity modulation GDP: Guanosine diphosphate nucleotide GTP: Guanosine triphosphate nucleotide [GDP]: Concentration of guanosine diphosphate nucleotide [GTP]: Concentration of gua nosine triphosphate nucleotide K1: Equilibrium dissociation constant for tubulin in solution binding to EB1

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12 (K1 k1/k1) K1 : Equilibrium dissociation constant for tubulin addition to track-bound protofilament (K1 k r/kf) K3: Equilibrium dissociation constant for EB1 subunit binding to track-bound filament-T-GDP (K3 k side/k+side) K : Ratio of forward and reverse rate of EB1 subunit binding to protofilament plus-end (K k +/k-) K : Equilibrium dissociation consta nt for EB1 subunit binding to protofilament plus-end (K k -/k+) Kd: Equilibrium dissociation constant fo r EB1 (or TE) binding to filamentbound T-GDP (Kd k-side/konside) Kd*: Equilibrium dissociation constant fo r EB1 (or TE) in solution binding to T-GTP at protofilament plus-end Kpi: Equilibrium dissociation constant for reversible phosphate binding to T protomers KT: Equilibrium dissociation constant for track binding to solution-phase EB1 (or TE or TTE) Kx: Equilibrium dissociation constant for the GTP/GDP exchange reaction k1: Forward rate constant for tubulin in solution binding to EB1 k-1: Reverse rate constant for tubulin in solution binding to EB1 k+: Forward rate constant for subunit of filament-bound EB1 binding to T-GTP at protofilament plus-end k+ side: Forward rate constant for subun it of filament-bound EB1 binding to filament-bound T-GDP k-: Reverse rate constant for EB1 (or TE) in solution binding to T-GTP at

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13 protofilament plus-end kside: Reverse rate constant for EB1 (or TE) binding to filament-bound T-GDP kBT : Thermal energy (Boltzmann constant, k, absolute temperature, T) kf: Forward and reverse rate constants for tubulin in solution binding to protofilament plus-end kf,0: Initial forward rate constant fo r tubulin in solution binding to protofilament plus-end kf E: Forward rate constant for EB1-bound t ubulin in solution binding to T-GTP at protofilament plus-end kMT: Kinetochore-bound microtubule kobs: Observed decay constant of EB1 on MT koff: Dissociation rate constant fo r EB1 dimer from protofilament kon: Forward rate constant for EB1 (or TE) in solution binding to T-GTP at protofilament plus-end kon side: Forward rate constant for EB1 (or TE) binding to filament-bound T-GDP kr: Reverse rate constant for tubulin in solution binding to protofilament plus-end kr E: Reverse rate constant for EB1-bound t ubulin in solution binding to T-GTP at protofilament plus-end kT: Forward rate constant for track bind ing to solution-phase EB1 (or TE or TTE) kT -: Reverse rate constant for track binding to solution-phase EB1 (or TE or TTE) L : Length of protofilament in ciliary plug LLF: Lock, Load, and Fire model

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14 MT: Microtubule N : Total number of protomers in a protofilament Np: Number of protofilaments tethered to motile object n: Position of tubulin protomer bound to track ns : Number of protomers between EB1 subunit at equilibrium position and final binding position Peq: Equilibrium fraction of EB1-bound protomers on protofilament Pi: Phosphate p: Probability of protomer bound to EB1 subunit pend: Probability of EB1 binding to the protofilament plus-end peq: Equilibrium probability of protomer bound to EB1 subunit pside: Equilibrium probability of EB1 binding to filament-bound T-GDP q+: Probability of protomer in state dbE+ q-: Probability of protomer in state dbEq: Probability of protomer atta ched to double-bound EB1 subunit qeq: Equilibrium probability of protomer attached to double-bound EB1 subunit S1: State of terminal protomer in protofilament S2: State of penultimate protomer in protofilament TAC: Tip-Attachment Complex model Tb: Tubulin protomer [Tb]: Tubulin protomer concentration [Tb]c: Critical tubulin concentra tion for free MT plus-end [Tb]E c: Critical TE concentration for free MT plus-end TE: EB1-bound tubulin protomer T-GDP: GDP-bound tubulin protomer

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15 [T-GDP]: T-GDP concentration [T-GDP](-)c: Critical tubulin concentrati on for T-GDP at MT plus-end T-GTP: GTP-bound tubulin protomer Tk2: Track bound to protofilament-bound EB1 Tk3: Track bound to protofilament-bound TE Tk4: Track bound to protofilament-bound dbE+ Tk: Track (tethering protein bound to motile surface) Tk-E: Track bound to EB1 in solution Tk-TE: Track bound to TE in solution Tk-TTE: Track bound to TTE in solution Tm: Time required for tubulin additi on and filament-bound GTP hydrolysis TTE: EB1 bound to two tubulin protomers t : Total simulation time u: Probability of protomer being unbound from EB1 ueq: Equilibrium probability of pr otomer being unbound from EB1 v : Irreversible velocity vr: Reversible velocity vmax: Maximum expected velocity w : Probability of protomer bound to TE weq: Equilibrium probability of protomer bound to TE x : Protofilament end position z : Equilibrium surface position ze: Equilibrium position of protofilament-bound EB1 subunit Transition state distance G Net free energy change of the tubulin cycle

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16 G0 Initial free energy change of the tubulin cycle G(-)loss Free energy of T-GDP dissociation from MT minus-end G(+)add Free energy of T-GTP ad dition to MT plus-end Gexchange Free energy of GDP/GTP exchange in solution Ghydrolysis Free energy of MT-bound GTP hydrolysis GPi-release Free energy of MT-bound phosphate (Pi) release Viscous drag coefficient Hookean spring constant Viscosity Stiffness of MT protofilament under compression 2 Surface density of EB1 on motile object Time required for ciliary plug to shift and rebind

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17 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FORCE GENERATION BY MICROT UBULE END-BINDING PROTEINS By Luz Elena Caro December 2007 Chair: Richard B. Dickinson Major: Chemical Engineering Microtubules are cytoskeletal filaments esse ntial for multiple cell functions, including motility of microorganisms and cell division. Of pa rticular interest is how these biological polymers generate the forces required for move ment of chromosomes during mitosis and for formation of cilia and flagella. Defective microt ubule-based force generati on can lead to various pathological complications; therefore, an understanding of microtubul e force generation is important for cancer research and biotechnology. The mechanism by which elongating microtubul es generate force is unknown. Several proteins, including End-Binding Protein 1 (EB1) and adenomatous polyposis coli (APC), specifically localize to microtubule elongating ends where the microtubule is tightly bound to a motile object and generating force. The role of these end-tracking proteins is not fully understood, but they likely modul ate microtubule-motile surface interactions, and may aid in force production. The objective of my research is to elucidate the role of polymerizing microtubules and endbinding proteins, specifically EB1, in force-dependent processes by formulating a model that explains their interaction and role in fo rce generation. The commonly assumed Brownian Ratchet model describing the forces caused by elongating microtubules cannot easily explain

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18 force generation during rapid elongation and stro ng attachment of the microtubule to the motile object. I propose a novel mechanism in which EB1 proteins behave as end-tracking motors that have a higher affinity for guanosine triphos phate-bound tubulin than guanosine diphosphatebound tubulin, thereby allowing them to convert the chemical energy of microtubule-filament hydrolysis to mechanical work. These microtubul e end-tracking motors are predicted to provide the required forces for cell motility and persis tent attachment between the motile surface and polymerizing microtubules. I have developed mechanochemical models that characterize th e kinetics of these molecular motors based on experimentally de termined binding parameters and thermodynamic constraints. These models account for the as sociation of EB1 to te thered and untethered elongating microtubule ends, in the absence or presence of force, and with or without EB1 binding to solution-phase tubulin. These models explain the observed expone ntial profile of EB1 on untethered filaments and predict that affinity -modulated end-tracking motors should achieve higher stall forces than with the Brownian Ratc het system, while maintaining a strong, persistent attachment to the motile object.

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19 CHAPTER 1 INTRODUCTION Forces produced by m icrotubule polymerizati on are required for chromosomal movement during mitosis and ciliary/flagellar formation (Dentler and Rosenbaum, 1977; Inoue and Salmon, 1995; Dogterom and Yurke, 1997). End-tracking proteins (a.k.a. tip -tracking proteins), such as end-binding protein 1 (EB1) and adenomatous pol yposis coli (APC), ha ve previously been shown to bind specifically to the polymerizing microtubule plus-end where the microtubule is tightly bound, at the kinetochore an d at the tips of growing cilia/fl agella (Allen and Borisy, 1974; Dentler, 1981; Severin et al., 1997), suggesting a possible role for end-tracking proteins in force generation at these sites. A few models dem onstrate how end-binding pr oteins may be involved in microtubule force-generation, suggesting that endtracking proteins bind weakly to the microtubule plus-end and serve as a linker between the MT and a motile surface (e.g., kinetochore) (Hill, 1985; Inoue and Salmon, 1995; Rieder and Salm on, 1998; Maddox et al., 2003; Maiato et al., 2004). However, these models cannot explain the st rong attachment of the microtubule to a motile object during elongati on, nor the energetics a nd mechanism of the interaction between the end-binding proteins and a motile surface. The objective of my thesis research was to help elucidate the role of microtubule el ongation mediated by end-binding proteins in force generation. Our models explai n and characterize the in teraction of end-binding proteins with growing mi crotubule ends and their role in force generation Understanding the functions of microtubules and end-tracking proteins in cellular motility and cell proliferation is of great importance to the medical field, part icularly in the area of cancer research. For example, the end-tracking protei n APC not only plays a po tentially key role in microtubule-chromosome attachment during cell division, but it also s uppresses excessive cell production that could lead to col on cancer. Cells with a specific mutation in APC, which prevent

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20 its binding to microtubules and EB 1, lead to aneuploi d progency and an absence of APCs tumor suppression function (Fodde et al ., 2001b; Kaplan et al., 2001). By providing insight into the potential function of these proteins and the inte raction among them is just one example of how our research can provide a signi ficant impact in cell biology. 1.1 Microtubules Microtubules (MTs) are versatile polym ers that occur in nearly every eukaryotic cell. They provide form and support in cells, aid in mitosis, guide transport of organelles, and enable cell motility (Olmsted and Borisy, 1973; Yaha ra and Edelman, 1975; Dentler and Rosenbaum, 1977). Microtubules are hollow, tubula r structures composed of 8-nm -tubulin heterodimers; where the subunit can bind to either a guanosine tr iphosphate (GTP) or guanosine diphosphate (GDP) nucleotide (Farr et al., 1990 ). Tubulin bound to GTP assemb les head-to-tail to form the 13 asymmetric, linear protofilame nts of a microtubule (Figure 11) (Chretien et al., 1995; Chretien and Fuller, 2000). Each protof ilament has the same polarity, with a -tubulin at one end (minus or slow-growing end), and an -tubulin at the other (plus or fast-growing end) (Chretien et al., 1995; Chretien and Fuller, 20 00). The structural polarity of the microtubules is important in their growth and ability to participate in many cellular functions. During microtubule polymerization (MT growth), GTP-tubulin protomer s add to the plus-end of a MT, the subunits then hydrolyze their bound GTP and subsequen tly release the hydrolyzed phosphate. During depolymerization (MT shortening) GDP-tubulin subunits are released from the MT minus-ends at a very rapid rate (D esai and Mitchison, 1997). The elongation velocity of a mi crotubule during polymerization, v is reported as 167 nm/s for the free, microtubule plus-end during mito sis (Piehl and Cassimeris, 2003). Assuming

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21 irreversible elongation at the MT plus-end in vivo this value can be used to estimate the plus-end protofilament effective on-ra te constant of tubulin (kf) association, Tbd v kf (1-1) where d is the length of a protomer and [Tb] the intracellular GTP-tubulin concentration (~10 M; (Mitchison and Kirschner, 1987); yielding kf = 2 M-1-s-1. The critical concentration for tubulin polymerization in vitro is [Tb]c = 5 M, which can be used to calculate an effective tubulin off-rate ( kr) of 10.6 s-1, assuming frckkTb /] [ (1-2) This calculated off-rate is used to determine the reversible, elongation rate of the microtubule plus-end (85 nm/s) by dkTbkvr fr. (1-3) For the purposes of model comparisons in subse quent chapters, these ra te constants for binding and dissociation of tubulin are assumed, and v and vr are taken as nominal velocities of irreversible and reversible elong ation, respectively, of MT plus -ends without the involvement of end-tracking proteins. Microtubule polymerizati on/depolymerization provides the forces required for cilia and flagella assembly as well as chromosomal alignment during mitosis. During microtubule elongation in both processes, the plus-end of the MT remains attached to the motile object (i.e., the ciliary/flagellar assembly plug or kinetochore, respectively) (Allen and Borisy, 1974; Dentler, 1981; Severin et al ., 1997). Microtubule assembly is known to play a key role throughout mitosis, the process of division a nd separation of the two identical daughter chromosomes (Inoue, 1981; Salmon, 1989; Rieder and Alexander, 1990) In an early stage of

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22 mitosis, replicated chromosomes (sister chromatids) are attached to each other at the centromere, which also serves as the binding site for the proteinaceous kine tochore structure (Figure 1-2) (Lodish et al., 1995). The outer plate of kinetochores contains proteins th at bind to kinetochore microtubules (kMTs). Other types of microtubules are also involved in mitosis, including astral and polar MTs. However, kMTs are of particular interest because of their role in properly arranging cellular chromosomes by generating force at the kinetochore (Lodish et al., 1995). The six phases of mitosis include prophase prometaphase, metaphase, anaphase, and telophase. During prometaphase, kM Ts of different length emanat e away from opposite poles of the cell, and bind their plus-end to the kinetochores of chromosomes. By rapid addition and loss of tubulin protomers at the ki netochore, the kMTs oscillate back and forth (directional instability), generating the force required to ba lance the length of kMTs on opposite ends of each chromosome (Skibbens et al., 1993). These oscillations eventually results in the alignment of the chromosomes at the spindle equator (congressi on). In metaphase, kMTs from opposite poles experience a net polymerization at the kinetochore and net depolymerization at the poles (MT flux) (Maddox et al., 2003), exerting tension on each of the chromosomes (Inoue, 1982). As a result, the chromosomes mainta in their alignment along the e quatorial plane. The spindle checkpoint then ensures chromosome s are properly attached to the spindle before anaphase onset by releasing an inhibitory signal that delays anaphase if kineto chores are unattached (Rieder et al., 1994; 1995). The kMTs proceed to depolymeri ze while still attached to the kinetochores during anaphase (Coue et al., 1991), pulling the si ster chromatids apart and moving them toward opposite poles for cellular division (cyt okinesis). Kinetochores stabilize microt ubules against disassembly by attaching specifically to elongating GTP-rich MT plus-ends (Severin et al., 1997). A comple x of proteins is required for

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23 kinetochore capture by kMTs, but their interaction have not been elucidated (Mimori-Kiyosue and Tsukita, 2003). If one of the kinetochore-a ssociated proteins coul d recognize and track the GTP-rich end, this protein could potentially pr ovide a mechanism that would couple kinetochore movement to force generated by MT polymerizat ion during mitosis. Se veral proteins that localize at the kinetochore-kMT at tachment during mitosis have al so been shown to bind to the plus-ends of MTs (end-tracking proteins), suggesti ng their likely role in such a mechanism. Of particular interest here are th e plus-end tracking prot eins EB1 and adenomatous polyposis coli. Although the interaction among thes e two proteins and the protei n/microtubule relationship is still unclear, a significant amount of recent research demonstrat es their potential role in kinetochore motility and kMT attachment, as discussed presently. 1.2 End-Tracking Proteins Several MT end-tracking proteins are thought to facilitate fo rce generation by microtubule polymerization (Schuyler and Pell man, 2001). End-tracking proteins localize to the MT plus-end, and when fluorescently labeled, they mark the translating polymerizing ends of MTs. Recent studies demonstrate the ability of a variety of proteins to tr ack the ends of growing MTs, including CLIPs, CLASPs, p150glued, APC, EB1, etc. It is sugges ted these end-tracking proteins aid in control of MT dynamics and in attach ment of MTs to a motile surface (i.e., the cell membrane or kinetochore) in several organisms, such as fungi and humans. 1.2.1 EB1 Of particular interest here is the EB1 prot ein, because it was f ound to localize at points where polymerizing MTs generate force (mitosis, cell growth, flag ellar movement, etc). EB1s specific localization sugges ts that EB1 has a role in modula ting the attachment of the MTs to motile surfaces and regulating MT dynamics at the attachment site to generate the forces during these cellular processes.

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24 EB1 is a dimeric, 30-kDa leucine zipper protein (Mimori-Kiyosue et al., 2000) with two MT binding domains. EB1 binds to microtubules throughout the ce ll cycle. During mitosis and cell growth, EB1 specifically localizes to the GTP -rich tubulin protomers (polymerizing unit) at the polymerizing plus-ends of mi crotubules. EB1 quickly disapp ears from the plus ends of depolymerizing GDP-rich MTs, indicating that the higher EB1 de nsity at plus ends requires polymerization and/or a GTP-rich MT end. This association/disso ciation behavior suggests that EB1 has a role in targeting th e MTs to a motile surface and/or regulating MT dynamics at the attachment site (cell membrane or kinetochore) (Mimori-Kiyosue and Tsukita, 2003). This hypothesis is supported by dem onstrating that EB1-null Drosophila cells cause mitotic defects including mis-positioning of kinetochores dur ing congression (Rogers et al., 2002). Specific localization of EB1 to GT P-rich MT plus-ends is not understood, but may be the result of direct binding of EB1 to the GTP-stabilized conformation of the MT plus end, co-polymerization with tubulin in solution, or recr uitment by other proteins, and dissociation from GTP-bound MT subunits (Figure 1-3). A study performed by Tirnauer et al. (2002b) provided important quantitative data that can be used to evaluate and provide parameters for m odels of EB1 interactions with MTs. As shown in Figure 1-4. They observed an exponentially decreasing density of EB1 from the MT tips to a uniform density on the MT sides, with 4.2x grea ter EB1 density relative to the sides. They measured the equilibrium dissociation consta nt of EB1 to MT sides in vitro to be Kd = 0.5 M. Equilibrium binding EB1 from the cytoplasm to MT sides also explains the faint uniform fluorescence of EB1 along the side of polymerizing MTs in vivo (Mimori-Kiyosue et al., 2000). The above results suggested that EB1 may either bind near plus ends with higher affinity than MT sides, or it could copolym erize with tubulin at plus ends be fore release at filament sides,

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25 which would require association between EB1 and t ubulin in solution. Ho wever, direct binding between EB1 and tubulin protomers in solution is uncertain. Vincent Ga ches (2005) group used sucrose gradient centrifugation to find that bovine brain T-GTP did not bind to an EB1 construct. Contrary to his finding, Juwana et al. (1999) demonstrated that recombinant EB1 co-precipitates with purified bovine tubulin an immunoaffinity assay, despite the 100-time lower concentration of EB1 than T-GTP (Tirnauer et al., 2002a; Tirn auer et al., 2002b). Ho wever, other studies report no interaction between purified EB1 and tubulin. For example, Ligon et al. (2006) showed that full length human EB1 did not bind to a tubulin-affinity matrix. Nevertheless, lack of binding in vitro does not rule out EB1s inte raction with tubulin protomers in vivo, which may require cytoplasmic components or conditions not present in these in vitro experiments. Consistent with this possibi lity, EB1 and tubulin alone in vitro do not promote microtubule polymerization (Nakamura et al., 2001; Tirnauer et al., 2002b; Ligon et al., 2003). 1.2.2 Adenomatous Polyposis Coli (APC) EB1 may be recruited to the MT plus-ends by other proteins such as APC. APC is a dimeric tumor suppressor protein th at plays an important protein ro le in preventing colon cancer. APC is known to co-localize and interact with both EB1 and pol ymerizing microtubule plus-ends at the kinetochore and at the ce ll cortex (Juwana et al., 1999). Like EB1, APC falls off the microtubule upon plus-end depolymerization. The C-terminal do main of APC (C-APC) is responsible for its associa tion with EB1 and microtubules (Bu and Su, 2003), which is diminished upon phosphorylation of APC. In the absence of C-APC, there is an ineffective connection between kMTs and the kinetochore (F odde et al., 2001a; Kaplan et al., 2001; Green and Kaplan, 2003), about 75% of cells exhibit failed chromosome congression (Green and Kaplan, 2003), and chromosome segregation is de fective (which may be responsible for colon cancer) (Fodde et al., 2001a; Kapl an et al., 2001; Green and Kaplan 2003). Studies also indicate

PAGE 26

26 that neither the microtubulebinding domain nor the EB1 binding domain of APC can be compromised to obtain proper chromosomal segreg ation. In the absence of the EB1-binding domain, APC localizes non-specifi cally to MTs (Askham et al., 2000), and in the presence of only the EB1 binding domain, APC distributes throughout the entire cell without binding to microtubules or kinetochores (Green and Kaplan, 2003). These observations suggest that APC may modulate plus end attachment of EB1 to kMTs, help kMTs target the kinetochore, and (in association with EB1) aid in regulating kMT polymerization during mitosis. Other kinetochore-asso ciated proteins (p150Glued, CLIP-170, and CLASPs) also have direct interactions with EB1, have the ability to bind to the MT plus-end, and are located at the kinetochore-MT interface (Folke r et al., 2005; Hayashi et al., 2005; Mimori-Kiyosue et al., 2005). Therefore, these components may also be involved in activation of EB1 at the MT tip and/or linking the EB1-bound MT pl us-end to the kinetochore. 1.2.3 Ciliary and Flagellar Movement Another example of force gene ration mediated by polymerizing microtubules that remain attached to the motile object is c iliary/flagellar formation and regeneration. During formation of these organelles, membrane-bound capping struct ures (or MT plugs) are persistently associated with the plus-ends of polymeriz ing MTs during MT assembly and disassembly (Suprenant and Dentler, 1988). These capping structures consist of (a) a pluglike unit that inserts into the lumen of the micr otubule, and (b) platelike structure that joins the plug to the membrane. Interestingly, components of the capping structure have been found to resemble proteins within the kinetochore, as indicated by their antigenic cross-reactivity (Miller et al., 1990), and these findings suggest th at the kinetochore and ciliar y/flagellar capping structures may interact with polymerizi ng microtubules in a similar ma nner. In this regard, EB1 co-localizes with the plus-ends of microtubules within cilia/flagella as well as those attached to

PAGE 27

27 kinetochores (Pedersen et al., 2003; Schroder et al., 2007; Sloboda and Howard, 2007). Depleted or mutated EB1 microtubule ends significantly reduces the efficien cy of primary cilia assembly in fibroblasts (Schroder et al., 2007). Because the sites of EB1 lo calization are involved in force generation in the above organelles the MT end-tracking properties of EB1 ar e likely to play a role in MT elongation-depe ndent force generation. 1.3 Force Generation Models Although much progress was made identifying mi crotubule-associated proteins and their locations, how MT elongation is coupled to force generation has not been determined. Various force-generating models have been considered, including force from microtubule polymerization, force from moto r-protein activity, and force from affinity modulation (Mitchison and Salmon, 2001). 1.3.1 Brownian Ratchet Models It is commonly assumed that the Brownian Ra tchet model describes the protrusive forces caused by elongating microtubules (Peskin et al., 1993; Mogiln er and Oster, 1996). The thermodynamic driving force in this model is the free energy change of pr otomer addition to free protofilament ends (Hill, 1981; Ther iot, 2000). An essential feature of this model is that thermal fluctuations open a gap between the free protofila ment plus end and the motile surface to allow addition of each new protomer. Because the pr otofilaments must freely fluctuate from the surface, the Brownian ratchet m echanism therefore cannot easily explain force generation during rapid elongation and strong attachment of the elongating microtubule end to the motile object. The thermodynamic stall force associated with the Brownian Ratchet model is limited by the free energy of protomer addition, and is given by c p stalldTkNFTbTblnB, (1-4)

PAGE 28

28 where 13 pN is the number of protofilaments, TkB = 4.14 pN-nm is the thermal energy (Boltzmann constant absolute temperature), and d = 8 nm is longitudinal dimer-repeat distance. Under typical intracellular tubulin (Tb) concentrations of 10-15 M and a plus-end critical concentration [Tb]c = 5 M (Walker et al., 1988), then Fstall = ~5-7 pN, or ~ 0.5 pN per protofilament. 1.3.2 Sleeve Model The Hill sleeve model couples polymerization of MTs with the force generated at the anti-pole ward moving kinetochore in cells. Th e model assumes MTs are inserted into a sleeve and tubulin dimers are added to the growing MT through the center of the sleev e. Movement of a MT through the sleeve as it grow s is accounted for by a random-walk approach, where the freeenergy source is the binding of GTP-tubulin protomers to MT ends (Hill, 1985). The Tip Attachment Complex model (TAC) incor porates the idea of a sleeve in order to model force-generation by MT polymerization in the presence of linker proteins. In TAC models, the tip of the microtubule inserts into a sleeve containing li nker proteins that bind weakly to the subunits at/near th e ends of MTs, and are assume d to grow freely by means of a Brownian ratchet mechanism (Inoue and Salmon, 1995). The weak-binding properties of the TAC linker proteins are assumed to allow the TAC to advance with the growing MT tip without hindering elongation. Therefore, the assumed bo nds between the TAC lin ker proteins and MT have the seemingly contradictory properties of being strong enough to sust ain attachment of the motile object, while at the same time being w eak enough for their rapid unbinding/rebinding to permit unhindered elongation. In contrast, the models we are proposing suggest that linker proteins behave as end-tracking motors that have unique binding prot eins allowing them to

PAGE 29

29 maintain a strong, persistent attachment betw een the protofilament and a motile surface during MT elongation. 1.3.3 Kinetochore Motors Several researchers have proposed a motor-i nduced force-generation model. One such model is known as the reverse Pac-Man mechanism (Maddox et al., 2003), where plus-end directed motors move kinetochores anti-pole ward during plus-end kMT polymerization (Inoue and Salmon, 1995). The plus-end directed motor protein, CENP-E, was assumed to play this role because of its localization to the kinetochor e and its role in sensing kMT attachment at the kinetochore (Abrieu et al., 2000). However, rece nt experimental evidence shows that the CENPE protein is not required for chromosome congression (McEwen et al., 2001). This result does not dismiss the possibility that MT motors contri bute to anti-poleward kinetochore motility in the cell; there are other kinetochore-associated mo tor proteins (i.e., MCAK) of unknown function. One recent force-generation model, the slip-clutch model, integrates both the reverse Pac-Man and lateralTAC mechanisms. This model represen ts the polymerization st ate of the kinetochore by a slip-clutch mechanism involving molecular moto rs and linker protei ns that are attached to the kinetochore and bind along the wall of MTs. The energetics of such a mechanism have not yet been analyzed, but it suggests that the proteins involved provide force at the kinetochore, and prevent strong forces from pulling MT plus ends out of their kinetochore attachment sites (Maddox et al., 2003). 1.3.4 Filament End-Tracking Motors Dickinson & Purich (2002) first proposed a model for actin-based motility whereby endtracking proteins tethered elongating filaments to motile objects and facili tated force generation. In this mechanochemical model for actin s ubunit addition, surface-bound end-tracking proteins bind preferentially to newly added ATP-bound termin al subunits on each subfilament and release

PAGE 30

30 from ADP-bound penultimate subunits. This cycle f acilitates force genera tion of persistently tethered filaments by capturing the free energy of ATP hydrolysis in the monomer addition cycle. The ATP hydrolysis-driven processive tracking on the filament end gives the end-binding protein the characteristics of a molecular motor. We later proposed that the interaction of microtubule end-tracking proteins with terminal GTP subunits could similarly explain force generation and persistent attachment of MTs at motile objects (Dickinson et al., 2004). The models presented in this thesis are quanti tative extensions of that initial model. 1.4 Thermodynamic Driving Force The thermodynamic advantage of GTP-driven affi nity modulated interactions can be seen by accounting for the free energy requirements of th e tubulin polymerization cycle. The net free energy of the tubulin cycle (G ) is partitioned among the five key steps of the tubulin cycle: tubulin addition (polymerization), filament GTP hydrol ysis, phosphate (Pi) release, depolymerization, and GDP/GTP exchange in solu tion (Figure 1-5). The net free energy of this cycle is the sum of these individual free energies exchange loss releaseP hydrolysis F addGG G GGGi (1-5) this is equal to the net free energy of GTP hydrolysis: i BPGDPGTPTkGG ln0, (1-6) where TkGB110 is the standard-state fr ee energy change for tubulin in vivo (Howard, 2001). The free energy changes for the individual asse mbly steps are listed in Table 1-1, where KPi is the equilibrium dissociation constant of reversible phosphate binding to GDP-tubulin protomers and KX is the equilibrium constant for the GTP/GDP exchange reaction. Based on literature values (Table 1-2), the free energy from the combin ed filament-bound hydrolysis and phosphate-release steps account for ~11 kBT of energy, which is nearly half of the total energy of the tubulin cycle

PAGE 31

31 (G 22 kBT; (Howard, 2001), and is sign ificantly greater than th e free energy of monomer addition at the MT plus-ends (~5.8 kBT ). Hence, considerably greater forces can be expected by exploiting the ability of end-trackers like EB 1 that bind preferentially to T-GTP protomers, thereby providing a pathway fo r harnessing the energy released by MT-bound GTP hydrolysis to facilitate protomer addition and resultant force generation. 1.5 Summary Microtubule polymers play an essential role in force generated during cell division, ciliary movement, and many other cell processes. The pola rity of MTs is key features that allow them to provide guided transport and to target specif ic proteins, such as e nd-tracking proteins, EB1 and APC. EB1 is known to spec ifically localize to the GTP-ri ch end of MTs when MTs are polymerizing at the leading edge of growing cells and when MTs are polymerizing at the kinetochore during mitosis. These properties sugg est a critical role of EB1 force generation by MTs. Prior force-generation mechanisms invol ving end-binding proteins and MTs have been proposed including TAC and mode ls involving ATP-driven MT motors kinesis and dynein, which move on MT sides. This thesis explores the hypothesis that end-tracking motor facilitate plus-end attachment and force generation, by harnessing the energy nuc leotide triphosphate (NTP) hydrolysis and converting it to mechanical work. The key feature of this model is that the end-tracking proteins binding specifically to the NTP-bound monomers on the filaments, a feature correlates well with the properties of the MTs a nd their correspond ing end-tracking proteins. 1.6 Outline of Dissertation The layout of this dissertation is as fo llows. Chapter 2 describes a preliminary mechanochemical MT end-tracking model which was first developed to demonstrate how endtracking proteins on a motile objec t (e.g., kinetochore) can facil itate MT attachment, elongation

PAGE 32

32 and force generation. This model demonstrates th e principles of filament end-tracking and force generation and assumes EB1 is immobilized at the motile object, but it does not account for the interaction of EB1 from solution with MTs. End-tracking models base d on interactions of monovalent or divalent solutionphase EB1 with MT protofilamen ts are modeled in Chapters 3 and 4, respectively. Chapter 3 first treats the simpler case of monovalent EB1 to illustrate how the exponential EB1 density on MT tips results from affinity m odulated interactions and how simply allowing EB1 to bind reversible to flex ible proteins (e.g., APC) in the kinetochore comprises an end-tracking motors. Chapter 4 then addresses the more realistic (and complex) case of divalent EB1, which makes similar pred ictions at the monovalent case, but predicts enhanced processivity due to EB1 s divalent interactions with the MT lattice. Both Chapters 3 and 4 discuss the growth of a single protofilament allowing EB1 binding, the probabilistic model used to determine optimal kinetic parameters, and stochastic simulations of protofilament growth against a load. Chapter 5 explores an MT m odel with EB1 end-tracking from a rigid plug, reflecting ciliary /flagellar gr owth. Finally, Chapter 6 summarizes the work completed and suggests future directions.

PAGE 33

33 Figure 1-1. Microtubule structure. Tubulin bound to GTP polymerize into 13-protofilament polymers: microtubules. Because tubulin is a heterodimer, the microtubule has a structural polarity with a plus and mi nus end. During MT polymerization, T-GTP binds to the MT plus-end, which induces hydrolysis of the penultimate tubulin subunit causing filament-bound GTP to be converted to GDP. T-GDP dissociates from the minus end. Figure 1-2. Chromosomal bindi ng site of microtubules. Two sister chromatids bind at the centromere to form a chromosome. Ki netochore microtubules bind to the chromosome in the kinetochore at the cen tromere [Reprinted with permission from Lodish, H. 1995. Molecular Cell Biology (Figure 23-28, p. 1094). New York, New York.]

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34 Figure 1-3. EB1 binding to microt ubule lattice. EB1 has equal asso ciation and dissociation rates on GDP-bound microtubule latti ce. EB1 may bind directly to the microtubule plusend or copolymerize with t ubulin in solution first. Figure 1-4. Concentration of EB1 along length of microtubule. By m easuring the fluorescence intensity of EB1 on the microt ubule, there is an experimentally determined decay of EB1 concentration along the protofilamen t. [Reprinted with permission from Tirnauer, J. 2002. EB1-mi crotubule interactions in Xenopus egg extracts: role of EB1 in microtubule stabilization and mechan isms of targeting to microtubules. Molecular Biology of the Cell. (Pg. 3622, Figure 4) .]

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35 Figure 1-5. Thermodynamics of GDP to GTP tubu lin exchange cycle. The free energy change of the tubulin cycle, G, is -22 kBT which is partitioned among the various steps: polymerization, hydrolysis and phosphate re lease, depolymerization, and GTP-GDP exchange in solution. Table 1-1. Thermodynamic equations characte rizing the multiple steps for GDP to GTP conversion Definition Equation Addition of T-GTP to MT plus-end c B addTbTbTkG /ln)( Phosphate ( Pi ) release pi B releasePKPTkGi/ln Loss of T-GDP from MT minus-end c B lossGDPTGDPTTkG)( )(/ ln GDP/GTP Exchange X B B exchangeKTk GTPGDPTGDPTbTkG ln / ln Hydrolysis of MT-bound GTP in terms of free energy of other steps c c XP B hydrolysis FTb GDPT KKTkG Gi)( 0ln

PAGE 36

36 Table 1-2. Equilibrium constants used in energy equations Symbol Reaction Value Reference Kx GTP/GDP exchange 3.00 (Zeeberg and Caplow, 1979) Kp Pi binding to filaments 25.00 mM (Carlier et al., 1988) [Tb]c T-GTP addition to MT plus-end 0.03 M (Howard, 2001) [T-GDP](-)c T-GDP addition to MT minus-end 90.00 M (Howard, 2001) *Calculated from the ratio of measured equilibrium dissociation constants of nucleotide binding to the protomer, i.e., Kx = KGDP/ KGTP

PAGE 37

37 CHAPTER 2 MICROTUBULE END-TRACKING MODEL This chapter describ es a preliminary model that simulates the growth of a 13-protofilament microtubule (MT) in the presen ce of surface-tethered EB1 end-tracking motors. While this model does not account for EB1 binding from solu tion, it does illustrate the principle of MT endtracking and force generation on a motile object. As described in the previous chapter, the key feature of the EB1 end-tracking motor is that it captures filament-bound GTP hydrolysis energy and converts it to mechanical work. In th e model presented here, EB1s dimeric structure allows it to maintain persistent attachment of the MT plus-end and the motile surface (i.e., a processive motor) and it is expected to allow fo r larger stall forces than the Brownian Ratchet Model. EB1 is modeled as a Hookean spring whose binding to the MT depends on its Gaussian-based probability density, which is a function of EB1s equilibrium and binding positions. An external load applied to the mo tile surface affects the pr obability of EB1 binding and the velocity and maximum achievable force of the microtubule. The velocity as a function of applied force and the resulting stall forces are simulated and analyzed. 2.1 EB1 End-Tracking Motors The preferred binding of EB1 to MT plus-ends is reminiscent of the interaction between end-tracking proteins and actin in the actoclamp in end-tracking motor m odel and suggests that end-binding proteins may behave as end-tracking motors. To expl ore this possibility, we model EB1 as a protein tethered to a motile surface on one end and interacting with the MT plus-end through its MT-binding domain on the other end. There are two key features of a MT end-tracking motor: affinity-modulated interactio n driven by hydrolysis of GTP on the filament end, and multiple or multivalent interactions with the filament end to maintain its possession to the motile surface. EB1 is assumed to bind prefer entially to filament GTP subunits and release

PAGE 38

38 from GDP subunits, thereby capturing some of th e available hydrolysis energy, stabilizing GTPbound terminal subunits, and increasing the net free energy of protomer addition. Because EB1 dimers are multivalent and multiple EB1 molecu les can interact with each MT end, the end-tracking motors to maintain a strong inter action with the protofilament even when other end-tracking units release, thereby allowing th e motor can advance pr ocessively along the polymerizing MT end. This processive action is driven by GTP hydrolysis and is the primary characteristic of other molecular motors, such as kinesin, except in this case hydrolysis occurs on the MT rather than on the MT-binding protein. 2.2 Microtubule Growth Model Our preliminary MT end-tracking model illustrated in Figure 2-1 simulates the growth of a 13-protofilament microtubule bound to surface-tethered EB1 motors and analyzes the force effects on the growth of the microtubule. By capturing part of the filament-bound GTP hydrolysis energy and converting it to mechanical wo rk, the resulting stall force is expected to exceed that of the Brownian Ratchet Mechanism, which is driven solely by free energy of monomer addition. The model assumes that EB1 is tethered to the motile object and does not bind to tubulin protomers from solution, although solution phase EB1 exists in the cytoplasm and likely interacts with tubulin in solution (see Chap ter 1). These complications are addressed in the subsequent chapters. The key reactions for the present model are shown in Figure 2-2 and include several possible end-tracking stepping motor pathways for two EB1 dimeric subunits (referred to hereafter as EB1 heads) operating at the plus-end of each MT protofilaments. Considering Stage A, where only one EB1 head is bound to terminal GTP-bound subunit, as the beginning of the cycle, monomers can add direc tly from solution (Reaction 1), which triggers hydrolysis on the now penultimate subunit, resulting in Stage B. The second EB1 head then binds the new terminal subunit (Reaction 2, resu lting in Stage C). The first EB1 head then

PAGE 39

39 releases from the penultimate subun it (Reaction 3) to rest ore Stage A, with the net effect of the cycle of having added one protomer. We also allow for binding of the second EB1 head in the wrong direction (Reaction 5) or T-GTP addition when both heads are bound (Reaction 4), either of which results resulting in Stage D. Note that of the two EB1 heads remains associated with terminal T-GTP until hydrolysis of its GTP is induced when a new tubulin protomer adds to the protofilament end and/or when the second EB 1 head binds the newly added Tb protomer. Hydrolysis weakens the older EB1-MT bond, ther eby releasing that EB1 head to bind to the next added Tb protomer in the cycle. Because at least one EB1 head should be bound at any time during the end-tracking cycle, the protofilament remains associated with the motile object (i.e., the motor is processive). (Long-term processivi ty may not be essentia l when there is a high density of EB1 molecules on the surface near the protofilament; even if both heads are released, other EB1 molecules would quickly capture the protofilament end.) In the absence of hydrolysis-induced affinity modulation, the principle of detailed balance would fix the relation among the various equilibrium dissociation constants in Reactions 1-3 shown in Figure 2-2, such that 3 21Tb K KKc where frkkK /1, kkK /2 and kkKside/3are the equilibrium dissociati on constants for Reactions 1, 2, and 3, respectively. However, affinity modulation is assumed to increase sidek thereby increasing K3 by a factor f such that fK KKc/]Tb[3 21 The value of f reflects the portion of th e GTP hydrolysis energy that can be transduced into work in each endtracking cycle. Because hydrolysis and protomer addition are the two sources of energy used for force generation in this mechanism, the thermodynamic stall force is characterized by fTbTbdTkNKKKTbdTkNFc B B stallln ln ln13 (2-1)

PAGE 40

40 The first term on the right hand represents th e contribution of tubulin addition without GTP hydrolysis (same as that of a free MT in the Brownian Ratchet model). The second term corresponds to the benefit of ha ving GTP-hydrolysis-driven affinity modulation. For example, f = 1000 corresponds to ~7 kBT additional energy captu red per cycle, putting Fstall at ~54 pN, a value that is much higher than the ~7-pN stall force predicted for a Brownian Ratchet driven solely by the free energy of protom er addition (c.f., Eq. 1-4). While Eq. 2-1 provides a thermodynamic limit, MT growth by the end-tracking cycle may kinetically stall at a lower force, whose value can be determined by stochastic simulation. To simulate the elongation of the 13 protofilaments of an EB1-bound microtubul e, we made several simplifying assumptions about the binding properties of EB1 to the protofilament lattice. We assume that only one EB1 dimer operates processively on each of the 13 protofilaments at any one time. Any lateral effects among adjacent protof ilaments on their elongation are neglected. Because the EB1 dimer has flexible segments between its coiled-coi l region and its two MT-binding heads (Honnappa et al., 2005), we modeled each EB 1 head as a Hookean springs with spring constant The contribution of the spring energy, Es given by 22 e szz E (2-2) where z and ze are the instantaneous and equilibrium positions, respectively, of EB1s MT binding domain. All EB1 molecules bound to the motile object are assumed to have the same equilibrium position, hence ze determines the position of the translating motile object relative to the MT (assumed fixed in space). Assuming EB1 is present at the motile object with a mean lateral spacing, the effective local concentration of EB1 at the motile object is

PAGE 41

41 thus)()(2zpzCeff where p(z) is the Boltzmanns distribution of the EB1 binding position, i.e., /2 )(2/)(2Tk e zpB TkzzBe (2-3) where kB is Boltzmanns constant, and T is the absolute temperature. The position-dependent binding rate constant k+( z) (s-1) of the EB1 head to th e MT lattice at distance z is taken as )()( zCkzkeffon or: /2 )(2/)( 22Tk e kzkB Tkzz onBe (2-4) where is the EB1 spacing distance, k1 is the forward association rate constant ( M-1s-1 or nm3/s) for EB1 binding to a T-GDP subunit from solu tion. Because binding s ites are at discrete positions spaced by distance d = 8 nm, then z = nd in Eq. 2-4. While it is possible that a st ressed bond may have an increas ed or decreased dissociation rate (i.e., slip bond or catch bond, respectivel y) under several piconewtons of force (Bell, 1978; Dembo, 1994); (Dembo et al., 1988), we assume the simplest case where the EB1 bonds are neither catch nor slip under th e forces involved here, and for ce is not assumed to not affect the dissociation rate constants of EB1 releasing from MT sides. The characteristic time for forces to relax between transitions is ~ /13 where is the viscous drag coefficient (drag force/velocity) of the motile object propelled by the MT. For a ~100-nm motile object, this time would be ~10-100 s, much faster than the cycle time for protomer addition. Therefore, we assume the instantaneous position of the motile object remains in mechanical quasi-equilibrium with the external load, F such that its position ze is determined by the balance of spring forces due to the bound EB1 heads. The equation for the external load is

PAGE 42

42 given by Equation 2-5; the position of bound EB1 h eads in each stage can be determined from Figure 2-3. ))32(())12(())1(()(D state C state B state A state e i e i e i eizdn zdn zdn zdnF (2-5) Solving Equation 2-5 for ze thus allows p(z) and the resulting tran sition probabilities for transition between states (k t ) and a time step of t to be calculated for each EB1 head at each time point in the simulation. In the simulation re sults shown in Figures 2-4, 2-5, 2-6, and 2-7, t was taken to 2 s. This time increment was chosen to be ten percent of the inverse of the largest kinetic constant to ensure that the ki netics of all reactions was accounted for. 2.2.1 Parameter Estimations The key parameters in this model include [Tb], [Tb]c, and the kinetic rate constants shown in Figure 2-2. The intracellular tubulin concentration [Tb] was assumed to be 10 M (Mitchison and Kirschner, 1987). We use the value of the plus-end critical concentration [Tb]c = 5 M estimated by Walker et al. (1988) from th e ratio of onand off-rate constants for elongation (8.9 M-1s-1 and 44 s-1, respectively). The macroscopic on-rate constant (8.9 M-1s-1) from MT elongation rate measurements reflects th e collective assembly of the 13 protofilaments on the MT tip; however, the growth rule for indivi dual protofilaments is uncertain. We therefore made the simplest assumption that each protof ilaments operates independently and elongates reversibly with on-rate constant kf= 8.9/13 M-1s-1 or 0.68 M-1s-1 and kr = 44/13 s-1 or 3.4 s-1. The MT reversible elongation speed used in the model was assumed to that determined by Piehl and Cassemeris (167 nm/s) and not the velocity calc ulated by the on and off -rates from Walker et al. The spring constant of an EB1 head was estimated as = kBT/2 where nm is the estimated standard deviation in the z-positi on of an EB1 head based on EM micrographs

PAGE 43

43 (Honnappa et al., 2005). The spacing = 7.5 nm was chosen assuming EB1 dimers are closely packed on the motile object. The association rate constant for an EB1 dimer on a MT-bound T-GDP subunit, kon = 25 M-1s-1 = 57 nm3/s, was assumed by taking a typical association rate constant for protein binding in solution (Eig en and Hammes, 1963). The off-rate constant koff = 0.24 s-1 for an EB1 dimer from MT GDP-subunits wa s calculated from the measured velocity and the exponential decay-leng th of EB1 dissociating from the wall of a polymerizing MT (Tirnauer et al., 2002a). However, this value re flects the probability of both EB1 heads being released simultaneously, which is assumed to be proportional to the off-rate of one EB1 head, kside, multiplied by the probability of the other head being dissociated, which is K3/(1+ K3), such that koff = kside [ K3/(1+ K3)], where kkKside 3and k+ is calculated at zze = d/2 from Equation 2-4. The primary simulation parameter was the total simulation time, t which was set at 4 seconds. For f =1 to f =10,000, data points for pNF 20 were obtained using a simulation time of 24 seconds to allow sufficient time for the microtubule to equilibrate. 2.2.2 Elongation Rate in the Absence of External Force A typical simulated trajectory for a surfacetethered, polymerizing microtubule in the absence of external load is given in Figure 24. Assuming an affinity modulation of 1000, and choosing optimal values for kon and (Appendix B.1 contains the MATLAB code), the resulting MT position increases linearly with time. The tubulin on-rate was chosen to yield the experimentally determined velo city of 167 nm/s for microtubules during mitosis (Piehl and Cassimeris, 2003). Figure 2-5 is a representation of the protofilament lengths and average equilibrium surface position corresponding to (st 4 v =165 nm/s). As seen in Figure 2-5, the maximum difference between the shortest and lo ngest filaments is four subunits. This small difference reflects how the end-tracking model can also ensure high fidel ity: the protofilaments

PAGE 44

44 do not advance too far past one another during po lymerization. This diagram shows that the end-tracking motors also maintain the average eq uilibrium position near the filament ends. The equilibrium surface position, z, is not located at the average filament end position since it is dependent on the individual springs binding location. 2.2.3 Force effects on elongation rate To analyze the effect of applied force on the polymerization rate of EB1 tethered microtubules, F was varied over a range of -4 pN to 34 pN. Figure 2-6 shows that the speed of MT polymerization decreased with increasing extern al load for all values of f calculated. When the end-tracking protein wa s not affinity-modulated ( f = 1), the velocity decreased linearly with increasing external force. As f was increased, the end-tracking motor was able to capture some of the filament hydrolysis energy to elongate mo re rapidly under significant forces, with the velocity depending approximately exponentially on the compressive force. Negative (tensile) forces applied to the surface increased the polymerization rate of growing MTs slightly until the maximum rate was reached. Moreover, tensile fo rces increased the probability of EB1 binding to the GTP-bound filament, and promoted the forw ard MT assembly process. Although large tensile forces should dissociate the filament end-tracking motors from the MT and thereby detach the MT from motile objec t, the possible of complete di ssociation of the EB1 molecule was allowed in our simulations. As the modulation factor f increased from 1 to 10, the dependence of velocity on the force resulted in a faster elongati on (Fig. 2-5) and a greater maximum achievable force. Once the modulation factor became greater than 10, there was no significant effect of f on the polymerization rate, and the micr otubule achieved similar stall fo rces. These observations can be explained by the force-limitations on the reaction kinetics. By increasing f the forward

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45 reaction in step 2 is favored, increas ing the rate of polymerization. Once f becomes greater than 10, the forward reaction in both steps 2 and 5 be come essentially irreversible (Equation 1-1). Further increasing the modulation factor has minimal effect on the rate of reaction, MT polymerization, and stall force. The kinetic stall force for each simulation was ta ken as the force at which the speed of the MT is less than 0.1% of the velocity when there is no force ( F =0). The thermodynamic stall forces predicted for the microt ubules at various values of f were calculated from Equation 2-5, and are compared to these simulated stall forces in Figure 2-7. The simulated and calculated results are comparable when the EB1 mo tor has little affinity modulation (from f =1 to f =10); for f =1, the thermodynamic and simulated stall force is approxim ately 7 pN. However, as f increases, the simulated stall force deviates from the expected thermodynamic limit. This phenomenon can be explained by the kinetic and thermodynamic properties. When the end-tracking motors are not affinity modulated (at f =1) the critical tubulin concentration for MT assembly remains relatively large, and the velocity is thermodynami cally limited; once the thermodynamic stall force is achieved, the MT will experience negative velocities, or net depolymerization. At la rger modulation factors ( f > 1), the effective crit ical concentration is reduced and the MT dynamics ar e kinetically, rather than thermodynamically, limited, and the velocity can be approximated by a force-dependen t exponential equation (Figure 2-5). That is, for large values of f MTs are predicted to kinetically st all at much lower forces than the thermodynamic stall force. 2.3 Summary This preliminary model simulates the gr owth of a 13-protofilament MT bound to surface-tethered EB1 motors and serves to dem onstrate the principles of force generation by processive MT end-tracking motors. The key feat ures of these end-tracking motors are (1) their

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46 ability to capture filament-bound GTP hydrolysis and convert them to mechanical work (2) their dimeric structure which allow them to maintain persistent attachment of the MT plus-end to the motile surface. EB1 was modeled as a Hookean spring whose association rate with the MT is governed by the probability density function of the spring and varies depending on an external applied force. The dissociation rate of EB1 from the MT was determined by its affinity to T-GTP versus T-GDP subunits, or affinity modulation factor, f The resulting velocity as a function of applied force was dete rmined at varying values of f The model demonstrates EB1s ability to maintain fidelity of the MT, with a maximum difference in protofilament length of four subunits. In addition, an increasing affinity modu lation of EB1 results in an increase in stall force, with a maximum stall force that is significantly greater than that predicted by the Brownian Ratchet mechanism. The primary limitation of this model is that EB 1 does not bind to tubulin in solution, nor does it account for solution-phase EB1. The propos ed co-factor assisted end-tracking model not only addresses the importance of a co-factor such as APC, whic h could be critical in the monomer addition step, but also the issue of solution phase EB1 and tubulin binding. This solution binding may be essential, and is addressed further in Chapters 3 and 4.

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47 Figure 2-1. Model for microtubule force genera tion by EB1 end-tracking motor. Model that represents the distal attachme nt of tubulin protomers at the MT plus-end. A uniform density of EB1 dimers on the motile object links the MT protofilaments to the surface.

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48 Figure 2-2. Reaction mechanisms of EB1 end-tracking motor. Mechanism of the EB1 end-tracking motor on the plus-end of one of the MT protofilaments (from upper left): One EB1 head is initially bound to th e terminal GTP-tubulin subunit. Step 1: A tubulin protomer binds to the filament e nd from solution. Step 2: The second EB1 head binds to the newly added terminal subunit. The complex can now follow two different pathways, 3 or 4. Step 3: Binding of second EB1 head to MT end induced hydrolysis of penultimate subunit and attenuates affinity of EB1 bound to the penultimate subunit; this EB1 head is released from the MT and the protofilament is returned to its original state. Step 4: A tubulin protomer adds to the filament end from solution, inducing hydrolysis of the pe nultimate subunit. Step 5: Affinity of distal EB1 to hydrolyzed subunit is attenuated and is released from the MT.

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49 Figure 2-3. Force dependence on EB1 binding and equilibrium surface position. Binding position of EB1 dimers in each stage, where n represents the position of the bound tubulin subunit along the protofilament. Figure 2-4. Microtubule elongation in the absence of external force. (A) The position of a 13-protofilament MT tethered to a surface by EB1 end-tracking motors is plotted as a function of time. No external forces are applied to the surface, the modulation factor is set to 1000, and optimal values of kon, and L are used (See Appendix B.1). The average velocity of 172 nm/s is n ear the set value of 167 nm/s.

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50 Figure 2-5. Distribution of protofilament lengths for microt ubule end-tracking model. Filament lengths for microtubule de scribed in Figure 1 (st 4 ,snmv /660 ). The maximum difference between the shortest and longest filaments is four subunits. The solid blue line represents the average equilibrium position of the microtubule.

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51 Figure 2-6. Effect of applied fo rce on MT elongation rate. The de pendence of velocity on force is presented for models with various modulation factors: 1, 5, 10, 100, 1,000, and 10,000. For f =1, the velocity decreases linearly with force, shown by the fitted line. For f =5 to f =10,000, the velocity decreases expone ntially with increasing force. The data is fitted to a three-parameter exponen tial equation represented by the solid line. The stall force for each simulation is estimated as the force at which the velocity is less than 0.1% of the velocity when F =0. The simulation time was set at 4s. For f =5 to f =10,000, data points for pNF 20 were obtained using a simulation time of 24 seconds to allow sufficient time for the microtubule to equilibrate.

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52 Figure 2-7. Thermodynamic versus simulated st all forces. The thermodynamic stall force was calculated for the various MT end-tracki ng motor represented in Figure 4 by using equation 2-5. Comparison of the calculated and simulated stall forces is shown. When the hydrolysis affects the mi crotubule dynamics very little ( f =1 to f =5) the model provides a good prediction for the EB1 end-tracking model. At higher f values, the data deviates from thermodynami c predictions. The simulated stall force only slightly increases once f becomes greater than 100. The solid lines represent logarithmic fit to each of the data presented.

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53 CHAPTER 3 PROTOFILAMENT END-TRACKING MODEL W ITH MONOVALENT EB1 The microtubule end-tracking model developed in Chapter 2 neglected solution-phase End Binding protein 1 (EB1) and binding to microtubules and tubulin prot omers. While not essential for end-tracking, binding of EB1 from soluti on is evident in the e xponential prof iles of bound EB1 at elongating free plus-ends and the apparent equilibrium density of EB1 along the length of the microtubule (MT) (Chapter 1 and Figure 1-3). To account for binding solution-phase EB1, we first developed a simplified model that simula tes the growth of a singl e protofilament in the presence of a monovalent EB1 protein... A model for more complex and realistic case of dimeric EB1 binding is presented in the next chapter. In the previous chapter, it was assumed that the EB1 protein behave s as an end-tracking motor, with preferential binding to T-GTP over T-GDP, and an affinity modulation factor greater than 1. In this chapter, the a ssertion that EB1 has a higher affinity for GTP subunits is supported by showing that the observed 4.2 tip-to-side ratio of EB1 density requires GTP-hydrolysis-driven affinity modulated binding. We do so by first modeling free filament growth with EB1 binding, but without attachment of EB1 a motile object, and comparing the predicted EB1 density along the length of the MT to the experimental result s. We then allow EB1 to interact with a linker protein at the motile object (e.g., Adenomatous Po lyposis Coli, APC) and predict the resulting MT dynamics and force generation. The force-velocity relations hip of this the end-tracking model is then compared to those of the simple Brownian Ratchet mechanism. 3.1 Non-Tethered Protofilament Growth We first consider growth of a single micr otubule protofilament in the presence of a solution-phase monovalent EB1 and then show in Section 3.2 how linking the growing tip to a

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54 surface containing a flexible binding protein for EB 1 forms an end-tracking motor similar to that described in Chapter 2. The various reactions considered in the free MT model are shown in Figure 3-1. Tubulin protomers (Tb) can add directly to filament ends (equilibrium dissociation constant [Tb]c), or they can first bind to EB1 (E) in solution ( K1) then add as an EB1-tubulin complex ([Tb]c E) In either pathway, tubulin addition is assumed to be followed by prompt GTP hydrolysis on the penultimate subunit. Because EB1 is assumed to have a lower affinity for GDP subunits (equilibrium dissociation constant Kd > Kd *), the energy provided by GTP hydrolysis later releases the EB1 from nonterminal subunits. 3.1.1 Thermodynamics of EB1-tubulin interactions As described in Chapter 1, free energy of the direct binding pathway is given simply by c B addTkG ]Tb[ ]Tb[ ln)( (3-1) but the free energy of the net reaction i nvolving binding and release of EB1 is d B d B B addK E Tk K E Tk TkG ][ ln ][ ln [Tb] Tb][ ln* c )( (3-2) hence more negative than that of monomer addi tion. In this way, EB1 binding temporarily stabilizes the protofilament plus -end and facilitates the net re action of monomer addition. The principle of detailed balance requ ires that Eq. 3-2 holds whether E binds first to Tb in solution or E binds to the terminal subunit following monomer addition. Eq. (3-2) can be re-written fTk TkGB B addln [Tb] Tb][ lnc )( (3-3) where f = Kd/Kd is the affinity modulation factor. Like in the previous chapter, at f = 1 would represent the case in which EB1 binds to both GTPas well as GDP-tubulin with equal affinity.

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55 A value greater than one means signifies that th e affinity of EB1 to GTP is greater than its affinity to GDP. To determine the solution phase concentration of E and TE, we assume [E] is determined by equilibrium binding with Tb and to sides of the MTs within the cell (at subun it concentration [MT]). As derived in the Appendix (A.1), this assumption yields dK MT K Tb E E ][][ 1 ][1 0 (3-4) and dK MT K Tb KTbE TE ][][ 1 /][ ][1 1 0 (3-5) 3.1.2 Kinetics of EB1-tubulin interactions We have assumed that the affinity modulation factor must be greater than 1 for EB1 to track on the plus-ends of protofilaments (i.e., EB 1 must have a higher a ffinity for GTP rather than GDP). To test this assertion, we de veloped a probabilistic m odel accounting various EB1 binding pathways shown in Figure 3-2: EB1 binds directly to the GTP-rich protofilament plus-end, EB1 associates with T-GDP on the si de of the protofilament, or EB1 copolymerize with tubulin in solution. This model was used to predict the probability of EB1 binding to the protofilament plus-end, pend, and the equilibrium probability of EB1 binding protofilament sides, peq. The value of peq was obtained by the steady-state of a differential equation describing the probability of EB1 binding to an MT side (far fro m the plus end) as a function of concentrations and reactions rates, which is given by )1(][][ ][ pEkTEkpkpkpTbk dt dpon E f side E r f (3-6)

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56 The equations specific for solving the probability of EB 1 binding to the plus-end and side of the protofilament are given by iir E rii E f fiii i i ippckpkppTEkTbkpkcEk dt dp 111 1 (3-7) fK cpkKTb pckTcTEkTbppkcEk dt dpE fd c f c E f 1 21 21 1 111,111, 1][ ][, (3-8) respectively, where ci 1pi (see full derivation in A ppendix A.2). Here, the index i represents the subunit on the protofilament numbered from th e plus-end. These two differential equations were numerically integrated (fourth-order R unge-Kutta method; Appendix B.3 contains the Matlab code) under set parameters in order to calculate the occupational probability ( pi) of EB1 along the length of a free protofilament (equivalent to the EB1 binding density). 3.1.3 Parameter Estimations The key parameters in this model, which include [Tb], [Tb]c, [MT], [E]0, f and the kinetic rate constants, were obtained from literatur e or calculated based on known values. The intracellular tubulin and microtubule concentrat ion, [Tb] and [MT], was assumed to be 10 M (Mitchison and Kirschner, 1987), the value of the plus-end critical concentration [Tb]c = 5 M was estimated by Walker et al. (Walker et al., 19 88), and the total intracellular concentration of EB1 was estimated as 0.27 M (Tirnauer et al., 2002b). Unless otherwise indicated, the value f= 103 was chosen for the affinity m odulation factor, which reflects 7 kBT of the available GTP hydrolysis energy captured for affinity modulatio n. (As shown below, many predicted properties because asymptotically independent of f >> 1 ). The rate constants, kf and kr were assumed to be the same for free MT elongation as from Eq. 11 and Eq. 1-2, respectively, assuming a maximum velocity v =170 nm/s (Piehl and Cassimeris, 2003). Th e dissociation constant for EB1 to the GDP-bound side ( Kd) of protofilaments was taken as 0.5 M, based on an in vitro study on EB1-

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57 MT binding interactions (Tirnaue r et al., 2002b). The rate equati ons for the on rates of EB1 to the MT plus-end and sides are assumed to be eq ual and are based on the observed decay rate constant of EB1 form MT sides determined in a study by Tirnauer et al (T irnauer et al., 2002b). The off-rates for EB1 on both the sides and pl us-end of the protofila ment are a fraction, f less than their on-rates. This on-rate constant of TE binding to plus ends is assu med equivalent to that for Tb, (i.e., kf E= kf). Experimental data has not validated a dissoci ation constant for the binding of EB1 and tubulin in solution, K1, so its optimal value was determined from peq (at steady-state, Eq. 3-9) and pend (from Eq. 3-8). E K pd eq 1 1 (3-9) Assuming a value for f pend was calculated for va rious values of K1. The ratio of pend to peq was determined at each chosen K1 value to determine which K1 resulted a pend : peq ratio of 4.2 at steady-state. This procedure was repeated for various values of f and the resulting K1 values are shown in Figure 3-3. When f is equal to one or two, the EB1 binding ratio remains below the expected 4.2 value at all values of K1. This result suggests that EB 1 must have a greater affinity for T-GTP than T-GDP (i.e., f must be greater than two) in order for EB1 to accumulate at the plus-ends of protofilaments as seen e xperimentally. The optimal value of K1 (for f greater than two) increases with increasing values of f At larger values of f ( f =5 and f =10), the optimal value for K1 is approximately 0.21 M. Increasing f past 10 does not provide any additional effect on K1. By increasing f the binding reaction of EB1 (to either the plus-end or the side of the protofilament) is favored. Once f becomes greater than ~10, thes e forward reactions essentially become irreversible and the probability of EB 1 binding to the protofilament end is no longer

PAGE 58

58 dependent on K1. Thus, further increasing f has minimal effect on the net rate of tubulin addition. Assuming an affinity modulati on factor of 1000, the optimal value for K1 (0.21 M) was determined from the results obtain in Figure 3-3. 3.1.4 Results Figure 3-4 shows the steady-state EB1 binding density profiles for f=1 and f =1000. When f =1 the steady state occupational probability is uniform along the length of the protofilament. The slightly higher EB1 density at the plus-end reflects some be nefit of copolymerization with tubulin. However, EB1 is predicted to have a much larger density at the plus-end when f is large. The EB1 density decreases exponentially along the length of the protofilament, consistent with experimental observations (Figure 1-4). This fi nding supports our assertation EB1 must have a significantly higher affinity for GTP-bound tubul in in order to track on the GTP-rich protofilament plus-ends. The effect of K1 on the EB1, monovalent o ccupational probability pr ofile is demonstrated in Figure 3-5. The model was simulated at K1 values from 0.01 M to 1 M at f =1000. When K1 is small (e.g., 0.01 M), EB1 preferentially binds to tubulin protomers in solution, therefore EB1 has a high occupational probability at the plus-end of a protofilament, which decreases along the length of the protofilament. This decay profile flattens out as K1 increases; at K1 =1 M the profile is similar to the profile of f =1 in Figure 3-4. This behavior is expected because at larger values of K1, EB1 has a large off-rate from tubulin protomers in solution; therefore it can bind along the entire length of the protofilament. 3.2 Tethered Protofilament Growth Similar to the above model of untethered protof ilament, this model simulates the growth of microtubules that bind to monova lent EB1 motors, but also in troduces a linking protein (e.g.,

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59 APC) that tethers the protofilament via EB1 to a motile surface. Here, we assume reversible binding of the linking protein on the motile object to EB1 from solution or on the MT lattice. Otherwise, the assumptions and parameter values from the previous model were applied in this model. This model has the similar pathways as seen in the non-tethered m odel (Figure 3-6). In Mechanism A, EB1 binds directly to the protofilament; in M echanism B, EB1 copolymerizes with tubulin, and Mechanism C (not shown) is a combination of A and B, but it also allows EB1 to dissociate from the tethering protein. The te rminal subunit of a protofilament is assumed not to dissociate when bound to an EB1 molecule. Consider the initial configur ation of each cycle as the st ate with the EB1 motor bound at the protofilament plus-end. When tubulin a dds to the protofilament, hydrolysis of the penultimate subunit that is bound to the motor is induced, and the motors reduced affinity for the protofilament causes EB1 to dissociate from either the protofilament (A and B) or the tethering protein (C). In mechanism A, the moto r can directly rebind to the protofilament plusend; whereas the motor in B has to copolymerize w ith tubulin in solution first, and the motor in C has to wait until EB1-binds to the protofilament before either of the two motors can attach to the protofilament plus end. In each of th e mechanisms, once the motor rebinds, the surface advances. These motors can continue to act processively on the end of the microtubule to generate force and propel the surface forward. 3.2.2 Model To simulate this monovalent EB1 molecular motor, a probabilistic model similar to the non-tethered monovalent end-tr acking model was derived to simulate the EB1 fluorescence along a protofilament based on the probability of EB1 and the tethering protein (Tk) making transitions between different bi nding states. The relevant prob abilities considered were:

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60 pi = probability of EB1 bound to the protofilament qi = probability of Tk-E bound to the protofilament w = probability of Tk bound to TE in solution v = probability of Tk bound to E in solution y = probability of Tk being unbound The probability of Tk being unbound, y, is represented by: iqvwy 1 (3-10) Similar to the derivation of Equations 3-7 and 3-8, the transition probabilities between states can be obtained from reaction rate consta nts for each pathway (Figure 3-6). The resulting differential equations for the probabilities of EB1 and Tk-E binding to the protofilament (in terms of the kinetic rates) are given by Equations 3-11 and 3-12, respectively, where ui =1qipi, kT is on-rate of the linking protein binding to solution-phase EB1, and Ceff is the effective local concentration of the linking protei n near the protofilament. ii E rr E riiieff E f E f f iiTiieffiTi ion ippqkukpkppwCkTEkTbk qkypCkpkuEk dt dp 11 11 1 ,,) ][][( ][ (3-11) ii E rr E riiieff E f E f f iiieff iiTiieffiT iqqqkukpkqqwCkTEkTbk qkvuCkqkypCk dt dq 11 11 1 ,,) ][][( (3-12) The differential equations for the probability of the track binding to e ither TE (3-13) or EB1 (3-14) in solution were also determined by th e reaction rates and corr esponding probability for that reaction. 1 1, 1 1][ ][ qkwCkwkvTbkwkyTEk dt dwE r eff E f T T (3-13)

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61 ii i iieffion i T TqkvuCkwkvTbkvkyEk dt dv, ,, 1 1][ ][ (3-14) These ordinary differential equations were solv ed using a fourth-order Runge-Kutta method in Matlab in order to determine the occupati onal probability of EB 1 along the length of a protofilament, as well as the effect of force on the velocity of the filament. The velocity of the protofilament is obtained to the steady-state ne t rate of the tubulin addition and dissociation pathways: end end E c Fd T E f end end c Fd fpqTbewCTEk pq Tb Tb eTbk V 1 (3-15) There are two ways tubulin can add to the protof ilament plus-end, directly with an on-rate of kf or copolymerizing with EB1 with an on-rate of kf E, hence, there are two rates of tubulin addition included in the equation. Assu ming direct tubulin addition, th e first term of the equation accounts for the effect of applied force on direct tubulin addition (e-Fd) and the dependence of the forward rate on the critical con centration when the protofilament plus-end is not bound to EB1. The second term represents the case when tubulin copolymerizes with EB1. This part of the equation accounts for: the effect of applied for ce on both direct TE addition and Tk-TE addition, and the dependence of the forward ra te on the critical concentration [Tb]c E when the protofilament plus-end is bound to EB1. The probabilities, qend and pend, were solved from Equations 3-11 and 3-12 for the protofilament plus end. 3.2.3 Parameter Estimations The protein concentrations used for the simulatio ns in this section are the same as those in the monovalent, non-tethered case. The kinetic rate constants were cal culated from detailed balance. The on-rates for an EB1 subunit (or head) to the protofilament side ( kon side) and to the

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62 protofilament plus-end ( kon) were calculated based on the obser ved decay rate of an EB1 dimer from the MT side, koff = 0.11 s-1. EB1 in solution can bind one of its heads to the side of a protofilament with a rate of kon side[E], and dissociates from the protofilament with a rate of kside. At equilibrium, the decay rate of an EB1 dimer from the MT side, koff is equal to the sum of these two terms: side side on offkEkk (3-16) Rearranging this equation gives d off on side onKE k kk (3-17) where Kd kside/ kon side. The off-rate constant for EB 1 from the GDP-bound tubulin subunits, kside, is calculated from Kd. The off-rate of EB1 from the protofilament plus-end, k-, is the equal to the off-rate of EB1 from divided by a factor of f The linking protein was assumed to be a flex ible, spring-like tethering region with position fluctuations () of 10 nm. The resulting effective concen tration of the linking protein near the protofilament is estimated like Ceff, for a 3-D normal distribution on a half-sphere. The normal Gaussian distribution of the spri ng is given by Equa tion 3-18, where is the spring constant and is equal to kBT /2, s B TATk d C 1 2 exp 2 12 (3-18) The surface area of the binding location, As, is estimated as half a sphere (22) since the linking protein can only bind to the one half of the micr otubule at a time. This value is analogous to -2 in the EB1 effective concentrati on calculated in Chapter 2.

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63 3.2.4 Results Figure 3-7A shows the predicted density of EB1 along the length of the protofilament (zero represents the plus-end). The protein species consider ed are the non-tethered EB1 protein in solution, the EB1 tethered to the protofilament and the sum of the two species. The density of the tethered EB1 species shows a high concentration of EB1 at the protofilament plus-end which decreases along the length of the protofilament. Th is decay behavior is expected; it requires more energy for the linking protein (spring) to maintain attachment at distances from the protofilament plus-end, and EB1 is expected to have preferential binding to the protofilament plus-end due to its affinity modulation. The unattached EB1 protein does not s eem to bind significantly at the protofilament plus end, most lik ely because the on-rate of linking protein to EB1 at this location is much greater than its dissociation rate. Th e non-tethered EB1 experiences a small peak in probability near the protofilament plus-end, likely because it was initially GTP-bound, and eventually dissociates from T-GDP. The force-velocity profile for these mechanis ms is shown in Figure 3-7B, which compares a protofilament whose driving fo rce is the monovalent EB1 motor to where the driving force is solely free monomer addition. At an affinity modulation factor of 1000, the end-tracking model provides a higher maximum achievable force (~1. 2 pN) demonstrating its advantage over the thermal ratchet model, whose stall force is 0.4 pN However the advantage is modest because of the monovalent nature of this end-tracking motor requires it to detach from the protofilament during the cycle, thereby still permit tubulin dissociation, which is energetically favored while EB1 is unbound. However, it is known that EB1 is actually a dimer (Figure 3-8), with two MT binding domains, which may facilitate processiv ity by allowing one EB1 head to remain bound while the other head releases. EB1 may theref ore behave as a divale nt motor, which would provide the end-tracking model with the advant age to allow rapid MT polymerization while

PAGE 64

64 maintaining a persistent attachment between the MT and the motile surface. This idea is explored in Chapter 4. 3.3 Summary This chapter described two models that simulate the growth of a single protofilament in the presence of a monovalent EB1 protein to dete rmine the advantages of the mechanochemical process over a simple monomer addition-driven (Brownian ratchet) mechanism. They key characteristic of these models is that they account for the reaction between solution phase EB1 and tubulin protomers. In the pr evious chapter, it was assumed that the EB1 protein behaves as an end-tracking motor, with preferential binding to T-GTP over T-GDP, and an affinity modulation factor greater than one In this chapter, this assu mption is supported by our finding that affinity modulation is necessary to achieve the observed high density of EB1 at filament ends relative to filament sides. 3.3.1 Non-Tethered Protofilaments The first model presented eliminates any fo rce effects by allowing free filament growth and EB1 binding, and assumes that neither the EB1 nor the protofilament are tethered to a motile surface. Although experimental results are not conc lusive as to whether EB1 binds to T-GTP in solution, this model accounts for se veral reaction pathways to allo w EB1 to bind with tubulin in solution as well as filament-bound tubulin. The dissociation constant for EB1 and free T-GTP was taken as that need to provide a 4.2 ratio of EB1-bound subunits at the protofilament plus-end versus protofilament sides, which would correlat e well with experimental results. Large affinity modulation factors resulted in an equilibrium value for the tubulin-EB1 dissociation constant, and are therefore optimal for simulation purposes. Regardless of the value for other key kinetic rates (i.e., on-rate of tubulin, kf, or on-rate of EB1 on protofilament sides, k+), it is required for EB1 to have a larger affinity for T-GTP rather T-GDP ( f >1) in order to achieve the 4.2 ratio.

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65 This result supports the a ssertation that EB1 has an affinity -modulated interaction with tubulin, which is not accounted for in the Brownian ratche t mechanisms, but is the key characteristic of the end-tracking model. This model predicts the density of EB1 bound along the length of a protofilament, and compares the results from affinity modulation to a mechanism with no affinity modulation. The optimal, equilibrium EB1-tubulin dissociation constant was used to calculate the binding probability of EB1 to the plus-e nd and sides of a microtubule prot ofilament. The results of the model demonstrate that the mechanism with no a ffinity modulation results in a near-uniform EB1 density along the entir e length of the protofilament. Ho wever, large affinity modulation results in a greater EB1 binding at the protofilame nt plus-end that decays along the length of the protofilament, a prediction which agrees to experimental results showing the decay of fluorescent EB1 on a non-teth ered microtubule. To simplify this complex model, several assu mptions were made. First, tubulin addition induces filament-bound hydrolysis at the protofilament plus-end. The affinity modulation is assumed to affect only the off-rates of the protei n interactions and not the on-rates. Because EB1 stabilizes the protofilament end, it is assumed that the terminal subunit of a protofilament cannot dissociate if bound to an EB1 molecule. All protein concentr ations are considered to be constant. 3.3.2 Tethered Protofilaments We have previously proposed a potential role of EB1 acting as a co-factor protein in end-tracking mechanisms (Dickinson et al., 2004). Consistent with this proposition, the second model allows EB1 binding to be translated to MT force generation by introducing a linking protein that attaches the monova lent EB1 protein to a motile surface. To simulate this monovalent EB1 molecular motor, a model similar to the non-tet hered, monovalent end-tracking

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66 model was used. The reaction mechanisms c onsidered were the same with exception of association and dissociation of EB 1 from the surface linking protein. As a result, these motors act processively on the end of th e microtubule to generate force a nd propel the surface forward. The occupational probability of all EB1 speci es (tethered and non-tethered) demonstrates that at large affinity modulation factors th ere is a high occupation of total EB1 at the protofilament plus-end, which decays along the length of the protofilament. This decayed concentration of EB1 along the protofilament is comparable to the decay profile shown by Tirnauer et al. (Tirnauer et al., 2002b) for EB1 on free-growing protofilaments. More importantly was the effect of force on the endtracking model. This model demonstrates the potential of the monovalent end-tracking motor to provide a higher maximum achievable force (~1.2 pN) than the thermal ratchet model (0.4 pN). However, the advantage is not that significant because it was assumed that EB1 is a monovalent protein instead of its true configuration as a diva lent protein. For simplification, this modeling approach neglected the potential energy exerted by compression and extension of the spring-like linking protein, particularly wh en a load is applied to the motile surface. When a load is introduced, there is an associated change in the kinetic reactions between the end-tracking complex and the protofilament that would affect the occupational probability of EB1 and the force-velocity profile. The subsequent tethered protofilament growth model with divalent e nd-tracking EB1 motors will account for the force effects on the linking protein.

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67 Figure 3-1. Schematic of non-tethered, monova lent EB1 end-tracking motor mechanisms. Tubulin protomers (Tb) can add directly to filament ends with an equilibrium dissociation constant [Tb]c, or they can first bind to EB1 (E) in solution (with dissociation constant K1) then add as an EB1-tubulin complex ([Tb]c E). GTP hydrolysis on the penultimate subunit occurs upon tubulin addition to the protofilament plus-end. Kd is the EB1 dissociation from the protofilament plus-end and Kd is the dissociation constant for EB1 from T-GDP, where f = Kd/Kd *. Figure 3-2. Various pathways of non-tethered monovalent EB 1 binding to protofilament. EB1 can bind directly to the GTP -rich protofilament plus-end, or EB1 can associate with T-GDP on the side of the protofilament, or EB1 can copolymerize with tubulin in solution.

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68 Figure 3-3. Choosing an optimal K1value for monovalent EB1. The experimentally determined ratio of EB1 binding to the tip versus the side of a protofilament (4.2) is represented by the dotted line. Each curve represents a di fferent affinity modul ation factor value, f and the data points correspond to the EB 1 binding ratio at various values of K1 and a kon of 2.1 M-1s-1. The value of K1 required to achieve a tip-to-side binding ratio of 4.2 for f < 50 increases with increasing f The optimal value of K1 chosen was 0.21 M where f >10. The simulation time was 1000 seconds.

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69 Figure 3-4. EB1 density profile on a non-tethered microtubule pr otofilament with monovalent EB1. Considering the various mechanisms of EB1 binding, the occupation probability for both f =1 and f =1000 are shown. At f =1 the steady state occupational probability is uniform along the length of the protofilament. When f =1000, EB1 has a high occupational probability at the plus-end, which decreases along the length of the protofilament.

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70 Figure 3-5. Effect of K1 on profile of monovalent EB1 occupational probability. Considering the various mechanisms of EB1 binding, the occupation probability for K1 from 0.01 to 1 M at f =1000 is shown. When K1 is 0.01 M, EB1 has a high occupational probability at the plus-end, which decreases along the length of the protofilament. This decay profile flattens out as K1 increases; at K1 =1 M the profile is similar to the profile of f =1 in Figure 3-4. Figure 3-6. Schematic of tethered, monovalent EB1 end-tracking motor mechanisms. Tubulin protomers can add directly to filament ends or they can first bind to surface-tethered EB1 in solution then add as an EB1-tubulin complex. GTP hydrolysis on the penultimate subunit occurs upon tubulin additi on to the protofilament plus-end. EB1 is allowed to dissociate from surface linki ng protein (pathway not shown here).

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71 Figure 3-7. Force effects on a tethered protof ilament with monovalent EB1. A) Occupational probability of EB1 along lengt h of protofilament (zero represents plus-end) at f =1000. Two protein species considered: un-tethered EB1 and tethered EB1 on protofilament. Occupation of EB1 for each sp ecies decreases along length of protofilament. B) Force-Velocity profile. Maximum achiev able force for end-tracking model at f =1000 (~1.2 pN) exceeds that of Brownian Ratchet model (0.4 pN)

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72 Figure 3-8. Divalent EB1 represented as divalent end-tracking motor. A) Depiction of EB1 structure characterized from crystal structur es. The C-terminus is represented by (C) and the N-terminus is represented by (N). Reprinted by permission from Macmillan Publishers Ltd: [ Nature ] (Honnappa et al., 2005) copyrig ht (2005). B) Schem atic of end-tracking motor complex comparable to crystal structure of EB1.

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73 CHAPTER 4 PROTOFILAMENT END-TRACKING MODEL W ITH DIVALENT EB1 The protofilament end-tracking model describe d in Chapter 3 is a simplified model that does not account for the divalent structure of EB1. This chapter discusses the more realistic models developed that simulate the growth of a protofilament in the presence of divalent EB1 end-tracking motors. Similar to Chapter 3, we first model the growth of an untethered protofilament in the presence of solution-phas e EB1 end-tracking motors. The model accounts for the solution binding of tubulin and EB1, and predicts the EB1 density along a polymerizing protofilament, with a 4.2 tip-to-side ratio of EB1. The second model allows EB1 to bind to a motile surface via a tethering protein. The resu lting protofilament dynamics were analyzed, and the force-dependent velocity was compared to that of the Brownian Ratchet mechanism. 4.1 Non-Tethered Protofilament Growth As in the models from previous chapters, EB1 is assumed to preferentially binds to T-GTP rather than T-GDP. For simplicity, we assume that if one subunit (head) is bound to the protofilament, the remaining unbound head can only bind to an adjacent tubulin subunit. 4.1.1 Kinetics of EB1-Tubulin Interactions The reactions considered in this protofilament model are shown in Figures 4-1 and 4-2. We assume for all pathways that GTP hydrolysis of the penultimate subunit occurs immediately after tubulin addition (O'Brien et al., 1987; Schilstra et al., 1987; Stewart et al., 1990) and that solution-phase EB1 can exist in three forms: unbound (E), bound to one tubulin protomer (TE), or bound to two tubulin protomer s (TTE). To determine the concentrations of these three species, we determined [E] by assuming equilibri um binding with Tb and microtubule sides, and equilibrium binding of TE and Tb. EB1 is a homodimer, so it is assumed that both tubulin binding domains (heads) are iden tical and non-cooperative; this pr operty allows EB1 to bind to

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74 its first or second tubulin protomer with an equilibrium dissociation constant, K1. As derived in Appendix A.1, these assumptions results in the following concentrations: dK MT K Tb E E ][][ 1 ][ ][2 1 0 (4-1) 1]][[2 ][ K TbE TE (4-2) 2 1 2K TbE TTE (4-3) Figure 4-1A shows the two methods in wh ich unbound EB1 in solution can bind to a protofilament, by adding dire ctly to filament ends ( kon) after tubulin addition ( kf), or by first binding to tubulin in solution ( k1) then adding as an EB1-tubulin complex ( kf E). Since unbound EB1 can bind to Tb in two identical ways, the on-rate for these bindi ng steps is doubled (2 k1 or 2kon). Once EB1 is bound, there are two pathways that result in attachment of EB1s second head to the protofilament plus-end (Figure 4-1B). One pathway involves direct binding of the EB1 head to the terminal, GTP-bound subunit ( k+) after tubulin addition ( kf). The other pathway allows the EB1 head to bind to solution-phase Tb ( k1) and facilitate tubuli n addition by shuttling it to the protofilament plus-end ( kf E ). The value for kf E accounts for the on-rate of TE and the local, effective concentration of the unbound EB1 head ( Ceff), which is represented by Equation 4-4. eff E f E fCkk ''' (4-4) The terminal two subunits at this stage are both bound to the same EB1 protein. The state of the terminal subunit is referred to as dbE+ (double-bound to EB1 on plus-end) and the penultimate subunit is in state dbE(double-bound to EB1 on minus-end). Figure 4-1C shows how TE in

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75 solution can bind the protofilament: binding di rectly to the protofilament plus-end ( kon) after tubulin addition ( kf), or by first binding to tubulin in solution ( k1) then adding as an EB1-tubulin complex (2 kf E). Figure 4-2 shows the various mechanisms by which EB1 can bind to the GDP-rich protofilament side. As shown in the figure, fr ee EB1 can bind directly from solution to a T-GDP subunit (pathway A), and subse quently bind its unbound head the neighboring subunit (pathway E). Additionally, TE can bind to the side of a protofilament with an on-rate of kon side (pathway C). 4.1.2 EB1 Occupational Probability Model A probabilistic model similar to the tethered, divalent end-tracking m odel (Chapter 3) was developed to simulate the pathways shown in Figure 4-1. This model determines the EB1 density along a protofilament ba sed on the probability of each t ubulin subunit being in a specific EB1 binding state. The relevant pr obabilities considered were: pi : probability of subunit i bound to EB1 head (o ther head unbound) wi : probability of subunit i bound to TE qi +: probability of subunit i in state dbE+ qi : probability of subunit i in state dbEui : probability of subunit i being unbound The probability of the subunit being unbound, ui, is represented by Equation 4-5. iiii iwpqqu 1 (4-5) The probability of Tb being in any one of these binding states is based on its reaction for that pathway, the probability of the reaction, and the corresponding protein concentrations. The probabilities are defined by a set of ordinary differential equations (in terms of the kinetic rates),

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76 and are represented by Equations 46 through 4-10 (Appendix A.3), where R+ and Rare defined by Equations 4-11 and 4-12. ii ii ii ii ii ii i i ion ippRppR upkqkupkqkpTbkwkpkuEk dt dp 1 1 1 11,1 1 1 1][ ][2 (4-6) ii ii i i i i on iwwRwwRwkpTbkwkuTEk dt dw 1 1 1 1][ ][ (4-7) ii ii iiiiiiii ii iqqRqqRupkupkqkqk dt dq1 1 11. 1, .11. (4-8) ii ii iiiiii iiii iqqRqqRupkupkqkqk dt dq1 1 11, 1,11, (4-9) ii ii i onioniiiiiii ii i iuuRuuR uTEkuEkupkupkqkqkwkpk dt du 1 1 1, 1,][][2 (4-10) 1])[]([][ wCkTTETEkTbkReff E f E f f (4-11) 111 1qpwkukRE rr (4-12) At equilibrium, these probabilities redu ce to Equations 4-13 through 4-16, where qeq qeq + + qeq and K k+ side/ kside. eq on equ k Ek p][2 (4-13) eq eqp K Tb w1 (4-14) eqeq equpKq 2 (4-15)

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77 side side on side side on side side on side side on eqk Ek K k Ek K k Ek K T k Ek K T u ][ 8 ][ 161 ][2 1 ][ 1 ][2 1 ][2 1 1 (4-16) The results of these equations were used to analyze the distribution of the divalent, EB1 end-tracking motors on the non-tethered protofilament, and determine the equilibrium EB1 concentration along the protofilament, Peq. 2/eqeqeq eqqwpP (4-17) 4.1.3 Average Fraction of EB1bound Subunits at Equilibrium A stochastic model was developed that determines the average binding fraction of EB1 along a non-growing protofilament in order to test the previous probabilistic model and compare the results. The pathways considered for this m odel are those where EB1 bi nds to the side of the protofilament and not the plus-e nd, which are shown in Figure 42. To model these pathways, the state of each tubulin subunit was analyzed. During the simulation, the state of each subunit in the protofilament was initially unbound from EB1. (The Matlab code can be found in Appendix B.3.2). The transition probability in time t for each pathway reaction was analyzed; if that reaction occurred, then the state of the tubulin subunit would change to its new state. The EB1-binding state of each subunit was used to determine the fraction of EB1-bound subunits in the protofilament; this fraction was averaged over time for a total simulation time of 40 seconds. 4.1.4 Average Fraction of EB1-bound subunits during protofilament growth A stochastic model was also used to calcu late the time-averaged fraction of EB1-bound subunits during protofilament growth. The pathways considered in this model are those shown in Figure 4-2., as well as the association and dissociation of tubulin from the protofilament plus-end (Figure 4-1). The stochastic model used to simulate these pathways is very similar to

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78 the previous model; it utili zes the same kinetic parameters, a nd the state of the tubulin subunits was determined from reaction rate for each path way. (Appendix B.3.3 contains the Matlab code for this model.) The EB1-binding state of each subunit was determined for each time step and the fraction of EB1-bound subunits in the protofilament was averaged over time for a total simulation time of 40 seconds. 4.1.5 Parameter Estimations Several key kinetic parameters listed in Figures 4-1 and 4-2 ha ve not yet been determined experimentally, including the dissociation constant for Tb and EB1 in solution ( K1) the EB1 on-rate for protofilament-bound T-GTP ( kon), the EB1 off-rate from protofilament-bound T-GDP ( kside) and the value of K To solve for these parameters, it was first assumed the protofilament was at equilibrium (i.e., it does not polymerize). At equilibrium, the fraction of filament-bound subunits attached to EB1 is given by Equation 4-18 (see appendix A.3 for full derivation). 1 ][ 1 ][ ][][, 0 0 tot effdMT K E EE (4-18) When half of the protofilament is saturated wi th EB1, the effective equilibrium dissociation constant of EB1 and the protofilament ( Kd,eff) is given by Equation 4-19, where u1/2 ueq ([E]-[E]0/2). 2/1 2/1 ,2 1 1 u k k k k Kside side on side side effd (4-19) Under this constraint, u1/2 is given by Equation 4-20, where k+ side= kon sideCeff for the protofilament plus end, and kside= k+ side/ K.

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79 eff eff eff effC E K C E K C E K C E K u0 2 0 2 2 0 0 2/1][ 4 ][ 81 ][ 1 ][ (4-20) Assuming [Tb]=0, [E]0=0.27 M, [E]=[E]0/2, and Ceff=153 M, Equations 4-19 and 4-20 and the experimentally determined value for Kd,eff of 0.44 M (Tirnauer et al., 2002b) where used to determine the value of K as 37. With this known value of K kside can be represented as a function of K1 from Equation 4-21, where koff is the known off-rate of dimeric EB1 from a protofilament ( koff =0.26 s-1, (Tirnauer et al., 2002b )). Appendix A.3 contains the derivations for these equations and parameter calculations. 1 1][ 1 1 K Tb uK kKkeq off side (4-21) To determined the optimal value of K1 (that provides a 4.2 tip-to-side EB1 binding ratio), the probabilistic model disc ussed in section 4.1.2 was simulated under different values of f and K1 (Appendix B.3.1 contains the Matlab code). Figure 4-3 shows the results of these simulations. When f is equal to one, the EB1 bindi ng ratio remains below the expected 4.2 value at all values of K1. This result suggests that EB1 must have a greater affinity for T-GTP than T-GDP (i.e., f must be greater than one) in order for EB1 to accumulate at the plus-ends of protofilaments as seen experimentally. The optimal value of K1 for f greater than one increases with increasing values of f At larger values of f (50 and 500), the optimal value for K1 is approximately 0.65 M. Increasing f past 50 does not provide any additional effect on K1. The reasoning behind the trends in these results lies in the reaction rates. Both kr E and kare inversely proportional to f so an increase in f causes the forward reactions to be fa vored in the mechanisms corresponding to

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80 these rates (EB1 binding to the protofilament plus-end). But, as f approaches infinity, it reaches a point in which the kr E and kbecome zero and no increase in f will favor the forward reaction further. The optimal values of K1 and f chosen for all simulations used in this chapter were: K1 = 0.65 M and f 50. The value of kon used to determine K1 in the previous analysis was estimated as 1 M-1s-1, but this value has not been experimentally determined. To ensure that the value of kon chosen does not affect the binding ratio of EB1 (or K1), we analyzed the effect of kon on K1. The model from section 4.1.2 was simulated for affinity modu lation factors from 1 and 50, and the value of kon was varied from 0.1 M-1s-1 to 10 M-1s-1. The resulting EB1 tip-to-side binding ratios for these conditions are shown in Figure 4-4. When f =1 and K1 is less than 0.1 M, there is no affect of kon on the binding ratio, and when K1 is greater than 0.1 M, there is a minimal effect of kon. In either case, the binding ratio still fails to obtai n the optimal value of 4.2. The optimal value of K1 (0.65 M) is not affected by the value of kon for f =50, therefore an average value of 5 M-1s-1 for kon was chosen to be used for all further simulations. The kinetic parameters used in this chapter that were also used in Chapter 3 were determined the same way. Additionally, the on-rate of EB1 and tubulin ( k1) was assumed to be a typical value for protein-protein binding interactions, 10 M-1s-1. Consequently, the value of k1 was determined from the optimal K1 value ( K1= k1 -/ k1). The on-rate constant for TE and TTE to the protofilament plus end, kf E, was assumed to be equal to the on-rate of tubulin addition, kf. The off-rate constant of TE and TTE to the protofilament plus-end, kr E, was calculated based on detailed balance, and is represented by Equation 4-22. on c E f E rkkK Tk k 1 (4-22)

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81 4.1.6 Results 4.1.6.1 Occupational probability The ordinary differential equations that define the probabilities of EB1 binding (Eq 4-6 to Eq. 4-10) were solved to determine the expect ed equilibrium fraction of EB1-bound protomers in the protofilament (Figure 4-5). This fraction was evaluated by determining Peq from Equation 4-17 at various values of K1. The percent of the protofila ment bound to EB1 increases with increasing K1. At large values of K1 ( K1 > 10), the EB1 binding fraction reaches an equilibrium, with approximately 40% of the protofilament bound. The value of K1 used for simulations (0.65 M) corresponds to about 2.5% of EB1 bound to the protofilament at equilibrium. The data for this plot was recreated for f =1000; since the equilibrium bindi ng probability of EB1 is not dependent on the affinity modulation fact or, the results were the same as for f =1 (data not shown). The set of ordinary differential equations in Equations 4-6 to 4-10 were numerically integrated and solved at a set value of K =37 and K1=0.65 M-1s-1 using a fourth-order Runge-Kutta method in Matlab. The occupational probability of EB1 along the length of the protofilament for f =1 and f =1000 is shown in Figure 4-6. When f =1, the EB1 density is nearly constant along the length of the protofilament at 2.5%. This behavior is expected since at f =1 EB1 does not have preferentia l binding to GTP or GDP-bound subunits, and this is the equilibrium EB1 binding fraction determ ined earlier (Figure 4-5). At f =1000, the probability at the protofilament end is 0.107, which decreases along the length of the prot ofilament to a value of 0.025. It is expected that there is a higher occupational probability at the protofilament plus end since EB1 has a higher affinity for T-GTP versus T-GDP when f =1000. This decay profile is comparable with the experimental results show n in Figure 1-4. The ratio of the occupational

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82 probability at the plus and minus end of the protofilament for f =1000 is about 4.3, which is similar to the tip-to-side ratio of 4.2 observed in experiments. For both f =1 and f =1000. The occupational probability at the first tubulin subunit is significantly less than that of the rest of the filament because the value of qi is zero for the first subunit. 4.1.6.2 Average fraction of EB1-bound subunits at equilibrium The time averaged fraction of EB1-bound protomers at f =1 and f =1000 and optimal values for K1 (0.65 M) and kon (5 M-1s-1) are displayed in Figure 4-7. The calculated equilibrium fraction of 0.024 from the probabilistic model is also shown on the plot for comparison with results from the stochastic model. Since th is model only considers the binding of EB1 to filament-bound T-GDP, the time averaged fracti on along the length of the protofilament is similar for f =1 and f =1000. In both cases, the fraction fluc tuates around the equilibrium value of 0.024. Since it is assumed that the affinity modu lation factor only affect s the off-rates and not the on-rates on EB1, TE, or TTE to the protof ilament plus-end, the only rates affected by f are kand kr E. These rates do not correspond to any of the pathways considered for this model; therefore, it is expected that there be a si milarity between the tw o curves generated at f =1 and f =1000. The data for this plot was recreated for a simulation time of 20 seconds, which resulted in no noticeable difference in th e plots (data not shown). 4.1.6.3 Average fraction of EB1-bound su bunits during protofilament growth Figure 4-8 displays the average fraction of EB1-bound subunits on a protofilament when polymerization is allowed to occur, at affinity modulation factors f = 1 and f = 1000. The equilibrium percentage of EB1 bound (2.4%) for K1=0.65 M and kon=5 M-1s-1 is shown for comparison. The results show that the fraction corresponding to f =1 is overall slightly smaller than that of f =1000, but both results show a larger fr action of bound EB1 at the plus-end. This

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83 result is expected since when f =1000 the rate of EB1 addition at the protofilament plus-end is increased. When f =1000, there is a sharp decrease in the EB1-bound fraction along the length of the protofilament. This behavior is due to the large affinity modul ation factor, which results in a lower affinity for EB1 to GDP-bound tubulin subunits and hence significantly reduced off-rates from the protofilament plus-end ( kr E and k-). Conversely, when f =1 the EB1-bound fraction of subunits along the side of the protofilament fluc tuates around the equilibrium value; since EB1 has no preferential binding to GTPor GDPbound subunits, there are significant amounts of EB1 bound along the side of the protofilament. 4.2 Tethered Protofilament Growth Model This model, similar to the te thered-protofilament model disc ussed in section 3.2, considers the growth of a single microtubule protofilament in the presence of solution-phase, divalent EB1. A flexible binding protein provides as a link between EB1 and a motile surface, allowing EB1 to behave as an end-tracking motor. The various reaction mechanisms considered for this tethered, protofilament model are those previously s hown in Figures 4-1 and 4-2, and the binding pathways involving the surface-bound tethering protein (Figures 4-9 and 4-10). The tethering protein was modeled as a Hookean spring whic h exerts energy on the motile surface under a load. The spring is defined by its spring constant, which is given by 2 TkB (4-23) The thermal energy is given by kBT and represents the variance in its position fluctuations. The effective concentration of the linking protein near the protofilament is obtained from normal Gaussian distribution of the spring given by Eq. 3-17 in section 3.2.3. The assumptions and parameter values used are the same as those in section 4.1.

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84 4.2.1 Kinetics of EB1-Tubulin Interactions Figures 4-9 A and B shows the two methods by which EB1 can bind to the surface-bound tethering protein, hereafter abbreviated Tk. Tk can associate with EB1 in solution (pathway A) or with protofilament-bound EB1 (pathway B). Although pathway A only shows the reaction between free EB1 and Tk, this reaction can occur with TE or TTE under the same onand off-rates. In pathway B, Tk binds to EB1 on the protof ilament (this could also be filament-bound TE or doubly bound EB1). The forward kinetic rate of this reaction, kT is represented by Equation 4-24, and is proportional to the forw ard rate of Tk binding to EB1, kT, the effective concentration of Tk at the protofilament plus-end, CT, and the effects of the transition state and spring energies. TkdnTkdn TTTB BeeCkk21 212' (4-24) The TkdnBe21 term represents the contribution of the tr ansition state effects from force, where is the transition state distance. Th e subunit position on th e protofilament ( n) is equal to one at the plus-end and increases toward the minus-end of the filament. The TkdnBe212term corresponds to the effect of stretc hing the tethering protein (or spring) from its initial position to its binding position on the protofilament. Since the tether ing proteins unbound, equilibrium position is one, the number of subunits between an unbound, surface-tethered EB1 protein and its equilibrium binding position on the protofilament is n-1. Hence, the displacement distance of the spring is given by ( n-1 ) d where d is the length of a subunit (8 nm). The effects of the transition state energy is associated with a bond unde r tension; therefore it also affects the reverse rate constants. Since the dissociation pathway in 4-9 B allows the EB1 spring to return to its equilibrium position, the only energy associated with the reverse rate, kT is that of the transition state:

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85 Tkdn TTBekk21' (4-25) The mechanisms by which surface-tethered EB 1 can bind to the protofilament are shown in Figures 4-9 C and D. Surface-bound TE can attach to the protofilament w ith a forward rate of kon and a reverse rate of k- (pathway C), given by Equations 4-26 and 4-27, respectively. TkdnTkdn TononB BeeCkk21 212' (4-26) TkdnBekk21' (4-27) The contribution of the transition st ate and spring energies are equa l to that in equations 4-24 and 4-25. If the surface is initially tethered to unbound EB1 (pathway D), the on-rate of EB1 to the filament is twice that of kon since EB1 is a homodimer that can bind with either one of its heads equally. For either pathway, C or D, the on-ra te of EB1 in solution to a protofilament-bound subunit changes depending on whether EB1 bi nds to the terminal tubulin subunit ( kon) or a subunit on the side of the filament ( kon side). EB1 bound to the protofilament by only one of it s heads has the potential to walk along the protofilament toward the plusor minus-end. These two potential pathways are shown in Figure 4-9 E and F. If the EB1 motor walks in the plus direction (pathway E), no energy is exerted on the spring and the rates of reaction ar e those for a single EB1 head binding to the protofilament subunit. However, these rates will depend on whether the EB1 head binds to the terminal subunit ( k+ and k-) or to the side of the filament ( k+ side and kside). If the EB1 motor walks in the minus-direction (pathway F), the kine tic rates will be affect ed by the transition and spring energies. These rate equations ar e described in Equations 4-28 and 4-29. TkdnTkdn side sideB Beekk221 212' (4-28) Tkdn side sideBekk21' (4-29)

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86 Figure 4-10 shows the various wa ys tubulin can add to the protofilament plus-end that involve the linking protein. Tubulin can add dire ctly to the plus-end of the surface-tethered protofilament (mechanism A), tubulin can be transferred to the protofilament plus-end by surface-tethered TE (Tk-TE) or TTE (Tk-TTE) as seen in pathways B and C, respectively, or the protofilament-tethered TE can shut tle tubulin to the protofilament end (mechanism D). Only the on-rates, not the reverse rates, for these reactions are affected by the interaction with the tethering protein. For all four path ways (A-D), when there is an applied force against the surface (in the opposite direction of protofilament growth), F the tubulin on-rates are reduced by a factor ofTkFdBe/ For direct tubulin addition in pathway A, the forward rate, kf is proportional to the on-rate of tubulin addition, kf, and the effect of force, as shown in Equation 4-30. There is also an effect of the spring energy due to the insertion of tubulin and extension of the spring. TkFdTkdn ffB Be ekk/ 212' (4-30) Tubulin transferred to the pr otofilament end by Tk-TE or Tk-TTE (pathways B and C) are both proportional to the on-rate of TE (o r TTE) to the protofilament plus end, kf E, as seen in Equation 4-31. There is also an effect from CT and any load applied to the motile surface. The on-rate of tubulin in these pathways is twice that of kf E because of EB1s dimeric structure. kTFd T E f E feCkk/' (4-31) Tubulin shuttled to the protofilament plus-e nd by filament-bound EB1 (pathway D) has a corresponding on-rate of kf E. This forward rate (Equation 4-32) has an effect from applied force, spring energy, and from the local effective concentration of the EB1 head, Ceff. This local concentration is estimated based on the 3-D nor mal distribution on a half-sphere (see section 4.2.3).

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87 TkFdTkd eff E f E fB Be eCkk/ 2212'' (4-32) 4.2.2 Protofilament End-Tracking Model Considering the above reactions, st ochastic models were performed to analyze the behavior of divalent, EB1 end-tracking motors operating on a single, growing mi crotubule protofilament. The model used to simulate the various reaction pathways is very similar to the model in section 4.1.4, where the pathway taken by the EB1 motor was determined by the probability of the corresponding kinetic reaction occurring (Appe ndix B.3.4 contains Matlab code). 4.2.3 Parameter Estimations The kinetic parameters used in this chapter sect ion that were also used in section 4.1 were determined the same way. The on-rate of Tk and EB1 binding (kT) was estimated as 5 M-1s-1 and the off-rate, kT was calculated from the value of KT provided ( KT=kT -/ kT). The value for v used to calculate the tubulin on-rate from Equa tion 1-1 was 170 nm/s (Piehl and Cassimeris, 2003), which was assumed to be the irreversible elongation at the protofilament plus-end. CT was estimated as 100 M. It was assumed that the bond between EB1 and the protofilament is a slip bond (i.e., tension force on the motile surface would increase the dissociate rate of EB1 to the microtubule). The transition state distance fo r this slip bond was estimated as 20 percent of a typical bond length, or 1 nm. 4.2.4 Results In the presence of a force, F the surface-tethered protofilament polymerized in the direction of the surface. The effect of force on the velo city of the protofilament was analyzed. The values for F were varied, which consequently affected the kinetic rate equa tions and corresponding probability for the pathways that are dependent on force. The resultant protofilament velocity was determined by dividing the total length of tubulin dimers added to the protofilament

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88 plus-end ( naddd) by the total simulation time, t This model also provided the state of the terminal subunit, position of the linking pr otein, the time-averaged fluorescence along the protofilament, and time spent in each pathway. Figure 4-11 shows the force-velocity prof iles for a polymerizing protofilament with surface-tethered EB1 end-tracking motors. To analyze the effect of the affinity modulation factor n the velocity profile, several affinity modulation factors were considered (Figures 4-11 A-E). Regardless of the value of f the velocity decreased as th e force increased because the force is opposite the direction of growth. For f =1 and f =10, velocities at for ces greater than the stall force (force at which the velocity is zero) were negative; at larger values of f the velocity decayed slower and approached zero as the force increased. These figures also show the effect of KT on the velocity profile. Since KT is the dissociation constant for EB1 and Tk, it represen ts the strength of the interaction between the protofilament and the motile surface, and the protof ilament cannot attach to the surface if EB1 is not bound to Tk. For all values of f a KT value of 10 M resulted in a maximum velocity of approximately 80 nm/s. This value is similar to the expected reversible elongation speed of the protofilaments is 85 nm/s (Equation 1-3), based on the rates determined for tubulin polymerization and depolymerization. At decreasing values of KT, the velocity at F =0 decreased, which is possibly due to the tether between the protofilament and th e motile surface. At lower values of KT this interaction is less likely to dissociate, therefore more energy is required to insert a tubulin at the plus-end. At all affinity modulation factors, the value of KT did not affect the stall forces. However, it is expected that as KT increases, the protofilament will spen d less time attached to the surface and will not be able to generate significant forces against a load. At KT =10 M-1s-1, the stall force

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89 increased with increasing values of f, from 0.37 pN (at f =1) to ~1.7 pN (at f =1000), as seen in Figure 4-11 F. Figure 4-12 summarizes the effect of KT and f on the stall force, with the corresponding data in Table 4-1. The thermodynamic stall fo rces are shown for comp arison to the simulation results. It is clear from the diagram that KT has little effect on the stall forces. When f =1, there is no affinity modulation and EB1 binds to T-GTP a nd T-GDP with equal affinity, hence the model is comparable with the Brownian Ratchet Mechanism. Therefore, it is not surprising that the resulting maximum achievable force at f =1 correlates well with the thermodynamics values, and is equal to that of the Brownian Ratchet Model, 0.37 pN. For f >1, the stall forces were lower than the predicted thermodynamic values. At incr easing values of the affinity modulation factor, the simulated stall forces increasingly de viated from the thermodynamic values. The reason the reactions stalled at forces lower than the thermodynamic limit is that there are parallel pathways of tubulin addition/dissociation (i.e., the direct tubulin addition/dissociation pathway and the end-tracking pathway), and the net tubulin diss ociation is favored thermodynamically for the direct pathway and at higher forces. Increasing f past a value of 1000 did not provide any additional effect on the stall force. Both kr E and kare inversely proportional to f so an increase in f favors the forward reactions for the mechan isms corresponding to these rates. But, as f approaches infinity, it reaches a point in which the kr E and kbecome zero and no increase in f will favor the forward reaction further. To determine how this end-tracking mechanism mediates tubulin addition and to understand the effects of f and KT on the velocity profiles (Figur es 4-11 and 4-12), the frequency of the different pathways possible for associatio n or dissociation of tubulin were measured and the resulting percentages are disp layed in Figure 4-13. For an affinity modulation factor of

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90 1000, when the F =0 and KT =0.1 M, the protofilament spent 47% of its time in free tubulin association at the protofilament plus-end. But, when the force was increased to 2.1 pN (near the stall force), the percentage in the forward a nd reverse pathways were equal (50%), which explains the zero velocity at this fo rce (Figure 4-11 D). An increase in KT from 0.1 to 10 M when f =1000 and F =0, resulted in a larger percentage of time spent associating tubulin (47% versus 58%, respectively), which explains why th e initial velocity was slightly higher when KT =10 M (Figure 4-11 D). The same result was found when f =1; at F =0 pN, the percentage spent in the forward pathway at KT=10 M (78%) was significantly greater than at KT=0.1 M (32%), and resulted in a higher initial velocity at KT=10 M (Figure 4-11 A). When comparing the two affinity modulation factors (at KT=10 M and F =2.1 pN) the time spent associating tubulin at f =1 was 93%, which was higher than when f =1000 (85%). This result explains why lower values of f resulted in negative velocities at large forces (Figure 4-11 F). The percent of time the protofilament spen t bound and unbound to the motile surface is shown in Figure 4-14. The unbound percen tage increased with larger values of KT or F Also, when a protofilament was surfacetethered, it was usually bound at its terminal or penultimate subunit. The forward rate equation in Equation 4-33 shows that when the linking protein binds to EB1 on the terminal subunit (n =1), the on-rate is proportional to kTCT. But when n is greater than one, the on-rate, kT, is reduced to nearly zero. Therefore, no matter what the value of kT, the linking protein either binds to terminally bound EB1 or most likely it does not bind to any filament-bound EB1. The state of the terminal su bunit in the filament was determined for each simulation to analyze the effect of f and KT on the EB1 binding behavior. The fraction of time spent in each state is shown in Figure 4-15, where states Tk2, Tk3, and Tk4 represent states in which the

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91 linking protein is bound to EB1, TTE or dbE+ on the protofilament, respectively. For all variations of KT, F and f, most of the time the terminal subunit was in the unbound state (state 1). When f =1, the terminal subunit was in state Tk3 or Tk4 a significant fraction of time; when f =1000, S1 was in state 4 and Tk4 a large amount of time. The most significant difference in the state of the filament is when KT is 0.1 versus when KT is 10 (for both values of f ); the larger KT value resulted in more subunits being unbound from EB1. A graphical representa tion of the bound versus unbound fraction of terminal subunits for each combination of f KT, and F is shown in Figure 4-16. The most significant result is that when f =1000, the unbound fraction decreased with increasing force, but when f =1 the unbound fraction increased with increasing force. This result has a signific ant implication for the role of the motor. When the force was increased at f =1, the frequency EB1-bound tubulin addition decreased (Figure 4-13). However, with large affinity modulation (f =1000), the frequency of tubulin addition occurs increased. Therefore, at large forces, affinity modulation allows EB1 to facilitate tubulin addition and maintain a pers istent attachment to the motile surface. When the linking protein is unbound from the protofilament, the state of the linking protein varied depending on the force and the affinity m odulation factor (Figure 4-17). When the force was zero the state of the linking protein was mo stly either unbound or bound to E or TE, which makes it easier to bind to the protofilament. When the force increased to 2.1 pN and f =1000, most of the linking protein are mostly bound to TE. When f =1 and force is 2.1 pN, most of the linking proteins were unbound or bound to TTE, which makes it easier to bind to the protofilament. 4.3 Summary To account for the dimeric structure of EB1, this chapter discusses the models we have developed that simulate the grow th of one protofilament in the presence of either tethered or

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92 non-tethered, divalent EB1 end-tracing motors that processively linking pr otein the plus-ends of protofilaments. Because EB1 is divalent, even if one of its heads dissociates upon hydrolysis of its bound tubulin, the other EB1 head can remain bound to the protofilament. Hence, the divalent end-tracking model has an advantage over the monovale nt end-tricking model and the Brownian ratchet mechanisms by maintaining a high EB1 concentration at the protofilament plus-end and allowing ra pid MT polymerization 4.3.1 Non-Tethered Protofilaments This model assumes that EB1 is not tethered to a motile surface, but is allowed to bind to tubulin in solution. By allowing tubulin addition (to the protofilament plus-end or side) from solution or by copolymerization with EB1, pr otofilament-bound tubulin can be in various EB1-binding states. The probability of tubulin be ing in any one of these states was used to determine the optimal dissociation consta nt for EB1 and tubulin in solution ( K1) that would result in the 4.2 binding ratio. As in the monovalent case, the Brownian ra tchet mechanism was not able to obtain the expected 4.2 EB1 binding ratio at any value of K1. EB1-tubulin interactions with large affinity modulation resulted in an optimal value for K1 of 0.65 M, which was used to determine the occupational probability of EB1 alon g the length of the protofilament. The results of this analysis demonstrates the advantage of the end-tracking model over the Brownian ratchet mechanism to preferentially bind to the protofilament plus-end and provide a decay behavior as seen in experiments. In addition, the model is able to simulate the occupational probability providing a 4.2 ratio of EB1 bind ing at the plus-end versus the si de of the protofilament. We also created a model that analyzes the average, equilibrium fraction of EB1 bound to the protofilament. This model only allows EB1 to bind to the sides of a protofilament (rich in GDP-bound subunits) and prevented the protofilament from growing. The results of this model

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93 show that the affinity modulation of EB1 does not affect this side-binding behavior. The resulting fraction of subunits bound to EB1 was 2. 6%, which is close to the expected equilibrium value of 2.4%. The same analysis was performed for growing protofilaments. The resulting average EB1-bound fraction of subunits shows a s lightly larger fraction of EB1 binding at the plus-end when the affinity m odulation factor is greater. 4.3.2 Tethered Protofilaments The tethered protofilament end-tracking model simulates EB1 end-tracking motors operating on a growing protofilament plus-end, a nd introduces a co-factor protein that tethers EB1 to a motile surface. Unlik e the monovalent end-tracking model, this model allows association and dissociation of the tethering protein to the mo tile surface and of EB1 to the tethering protein. The tethering protein was mode led as a Hookean spring, which translates its potential energy from mechanical work at the protofilament plus-end. This model also accounts for any transition state effects on the on-and o ff-rates due to binding be tween surface-tethered EB1 and the protofilament. The force-velocity relationships developed from this model were compared to the Brownian ratchet mechanism. Under no affinity modulation, the model predicts values consistent with the thermodynamics values and comparable to the Brownian Ratchet mechanism, with a resulting stall of 0.37 pN. The end-tr acking model provides a stall force up to 5 times greater than that of the Brownian Ratchet m echanism. Depending on the affinity of the interaction between EB1 and tubulin, the result ing stall force in the end-tracking model can range from 0.72 pN to 1.95 pN. However, as affinity modulation increases, the resulting stall forces deviate from the stall forces pr edicted by thermodynamics because the net tubulin dissociation is favored thermodynamically fo r the direct pathway and at higher forces The effect of the dissociation rate of EB1 from the linking protein does not affect the stall force of the

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94 end-tracking model, but it does affect the maximum protofilament velocity. We show that an increase in KT results in an increase in the maximum velocity, and vice-versa. A KT value of 10 M allows the protofilament to grow at a rate of 80 nm/s which is comparable to the calculated value of 85 nm/s for reversible elongation. At large forces (2.1 pN), the end-tracking model is able to maintain a persistent attachment of the protofilament plus-end (s pecifically the terminal and penultimate subunits) to the motile surface (71% of time); whereas the protofilament in the Brownian ratchet model spends most of the time (36%) un-tethered. This result suggests that the EB1 end-tracking motors are able to maintain persistent attachment of the protofila ment end to the motile surface, translating its filament-bound hydrolysis energy to mechanical work and allowi ng the protofilament to grow even under large loads.

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95 Figure 4-1. Mechanisms of a non-tethered, divalent end-tracking motor. A) Top-left: EB1 and T-GTP free in solution. EB1 binds to the protofilament in two ways: after tubulin addition (clockwise) or copolymerizing with tubulin (counter -clockwise). Clockwise: T-GTP adds to the protofilament end ( kf) and induces hydrolysis of the penultimate tubulin subunit; EB1 binds to T-GTP at the protofilament end (2 kon). Counter-clockwise: EB1 and T-GTP bind in solution (2 k1); Together, EB1 and T-GTP add to the protofilament end (kf E). B) Top-left: EB1 initiates bound to the GTP-rich protofilament plus-end. Free tubulin in solution binds to the protofilament in two ways: directly from solution (clockwise) or facilitated by the EB1 motor (counter-clockwise). Clockw ise: Tubulin in solution adds to the protofilament end ( kf), which induces hydrolysis of the EB1-boun d, penultimate tubulin subunit. The unbound EB1 head binds to the GTP-bound protofilament end ( k+). Counter-clockwise: The free EB1 head binds to tubulin in solution ( k1) and shuttles the protomer to the protofilament end ( kf E ). C) Top-Left: TE and T-GTP free in solution. TE binds to the protofilament in two ways: after tubulin addition (clockwise) or copolymerize as TTE (counte r-clockwise). Clockwise: T-GTP binds to protofilament end ( kf), inducing hydrolysis of the penultimate subunit. TE binds to the T-GTP protofilament end ( kon). Counter-clockwise: TE binds to T-GTP in solution ( k1). TTE binds to the protofilament end ( kf E) and induces hydrolysis of the penultimate tubulin subunit.

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96

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97 Figure 4-2. Mechanisms of equilibrium, side binding of EB1 to protofilament. Off-rates of EB1 binding to protofilament-bound GDP affected by affinity modulation factor. Tubulin addition and dissociation pathway neglected for this equilibrium mechanism. Figure 4-3. Choosing an optimal K1value for divalent EB1. The experimentally determined ratio of EB1 binding to the tip versus the side of a protofilament (4.2) is represented by the dotted line. Each curve represents a diffe rent affinity modulation factor value, f and the data points correspond to the EB1 binding ratio at various values of K1 and a kon of 1 M-1s-1. The value of K1 required to achieve a tip-to-side binding ratio of 4.2 for 50 > f > 1 increases with increasing f The optimal value of K1 chosen was 0.65 M where f >10. The simulation time was 40 seconds.

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98 Figure 4-4. Effect of kon on optimal K1. The experimentally determined ratio of EB1 binding to the tip versus the side of a protofilament (4 .2) is represented by the dotted line. For affinity modulation factors of 1 and 50, the value of kon was varied from 0.1 to 10 M-1s-1, and K1 from 0.01 to 5 M. The simulation time was 40 seconds. The optimal value of K1 is not significantly a ffected by the value of kon, and the optimal K1 remains at 0.65 M when f =50.

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99 Figure 4-5. EB1 equilibrium binding. Data shown in figure is for a kon value of 5 M-1s-1 and a simulation time of 40 seconds. At equilibrium, the percent of the protofilament bound to EB1 increases with increasing K1. At large values of K1 ( K1>10), the protofilament reaches an equilibrium w ith approximately 40% of EB1 bound. The value of K1 used for simulations (0.65 M) corresponds to an expected 2.4% of EB1 bound to the protofilament.

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100 Figure 4-6. Occupational probabili ty of EB1 along length of protofilament. Zero on the x-axis represents the protofilament plus (gro wing) end. Simulation time used was 40 seconds. Values for other variables: K1=0.65 M, kon=5 M-1s-1. Probability of EB1 when f =1 is nearly constant along the le ngth of the protofilament. When f =1000, occupational probability at protofilament end (0.107) is ~4.2 times higher than f =1 (0.025); the probability decays along the length of the protofilament.

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101 Figure 4-7. Time averaged EB1-bound tubulin fraction at equilibrium. Zero on the x-axis represents the protofilament plus (growing) end. Results for both f =1 and f =1000 shown. Values for other variables: K1=0.65 M, kon=5 M-1s-1. Equilibrium EB1-bound fraction represented by solid line at 0.024. Simulation time used was 80 seconds and N =200

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102 Figure 4-8. Time averaged fraction of EB1-boun d subunits during protofilament growth. Zero on the x-axis represents the protofilament plus (growing) end. Results for both f =1 and f =1000 shown. Values for other variables: K1=0.65 M, kon=5 M-1s-1. Equilibrium EB1-bound fraction represented by solid line at 0.024. Simulation time used was 80 seconds and N =200.

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103 Figure 4-9. Mechanisms of tethered, protofilame nt end-tracking model w ith divalent EB1. A) Pathway for tethering protein to bind to EB1 (or TE) in solution. B) Tethering protein binds to protofilament-bound EB1 (o r TE). Energy is exerted by the spring, which is accounted for in the onand off-rates ( kT and kT- ). C) Surface-tethered TE binds to protofilament plus-end with on rate of kon D) Surface tethered EB1 has twice the on-rate due to EB1s dimeric st ructure. E) EB1 can walk along the protofilament in the plus direction (E) or minus direction (F). EB1 walking toward the minus-direction exerts no force on th e spring, and has an on and off rate of k+ side and kside, respectively. EB1 walking in the minus-direction exerts a force on the spring, which is accounted for in the on and off rates ( k+ side and kside )

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104

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105 Figure 4-10. Mechanisms of tubulin addition to linking protein-bound protofilament. A) Tubulin can add to the plus-end of a surface-tether ed protofilament. B, C) Tubulin bound to Tk-E or Tk-TE attaches to the protofilame nt end. Tubulin bound to Tk-TE has two configurations with which it can bind. D) Tubulin addition is facilitated by the filament-bound EB1 end-tracking motor. The forces exerted on the spring are accounted for in the forward rate constants for each mechanism.

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106 Figure 4-11. Force-velocity profiles for tethered protofilaments bound to divalent EB1 end-tracking motors. The effect of force and velocity of both the Brownian Ratchet and End-Tracking models are shown. Simulation time used was 40 seconds. Values for other variables: k1=10 M-1s-1, K1=0. kon=5 M-1s-1. KT values were varied (0.1, 1, 5, and 10 M) in A E to analyze the effects on the stall force. A) f =1 B) f =10 C) f =100 D) f =1000 E) f =10,000 F) Force-velocity profiles shown for varying values of f (1,10,100,1000,10000) when KT =10 M.

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107

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108

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109 Figure 4-12. Stall forces versus affi nity modulation factor at various KT values. Resulting stall forces for data in Figures 3-19 A-E. Thermodynamic represents the thermodynamic values at the various values of th e affinity modulation factor, based on kBT =4.14 pN-nm, d = 8 nm, [Tb] = 10 M, and [Tb]c = 5 M. Data for this figure can be found in Table 4-1. Table 4-1. Protofilament stall forces at varying values of KT and affinity modulation factors. Stall forces (in units of pN) correspond to the data represented in Figure 4-12. f Thermodynamic* KT = 0.1 M KT = 1 M KT = 5 M KT = 10 M 1 0.36 0.37 0.37 0.38 0.36 10 1.55 0.72 0.73 0.73 0.72 100 2.74 1.22 1.2 1.22 1.21 1,000 3.93 1.69 1.65 1.66 1.78 10,000 5.13 1.78 1.7 1.95 1.72 *Thermodynamic values show the expect ed thermodynamic stall forces when kBT = 4.14 pN-nm, d = 8 nm, [ Tb ] = 10 M, and [Tb]c = 5 M.

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110 Figure 4-13. Effect of f KT, and F on pathways taken. For each affinity modulation factor value (1 and 1000), forces of 0 and 2.13 pN were analyzed for both a KT value of 0.1 and 10 M. For each f KT, F combination, the percentage of time the protofilament advanced along a pathway that resulted in association or dissociation of a tubulin protomer is shown. Pathways occurring less than 5% of the time are not shown.

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111 Figure 4-14. Percent of time protofilament bound and unbound to motile surface. The percentages listed are based on the same combinations of f, KT, and F values considered in Figure 4-13. Protofilaments bound to the motile surface were consistently tethered at the terminal tubulin subunit (S1) or the second tubulin subunit (S2). Percentage of time bound protofilaments we re tethered to either S1 or S2 shown for each combination of f, KT, and F values.

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112 Figure 4-15. State of the terminal subunit (S1) when f =1 and f =1000. KT values of 0.1 and 10 M and F values of 0 and 2.1 pN were analyzed. The fraction of time the terminal tubulin subunit (S1) remained in each of each of the 7 different states is shown. The various states the protofilament subunits include: 1-unbound, 2-bound to E, 3-bound to TE, 4-bound to dbE+, Tk2-bound to E tethered to linking protein, TK3-bound to TE tethered to linking protein, Tk4-bound to dbE+ tethered to linking protein.

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113 Figure 4-16. Fraction of S1 subunits bound and unbound from motile surface. Based on data presented in Figure 4-14, subunits in states 1-4 were considered unbound and subunits in states Tk2,Tk3, Tk4 were cons idered bound. All eigh t different variable combinations of f KT, and F are presented.

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114 Figure 4-17. Average state of unbound linking protei n. The fraction of time the linking protein spent in each of its unbound states is shown: Tk unbound, Tk-E linking protein bound to EB1, Tk-TE linking protein b ound to TE, Tk-TTE linking protein bound to TTE. Each of the eight combinations of variables (f KT, F ) was considered.

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115 CHAPTER 5 CILIARY PLUG MODEL Cilium is a motile organelle made up of an array of MTs. The plus-ends of ciliary MTs are attached to the cell membrane by MT-capping struct ures, which are located at the site of tubulin addition (Figure 5-1, (Suprenant and Dentler, 1 988)). As the protofilamen ts polymerize, the cap remains tethered to the filament end and push es the cell membrane forward. As mentioned earlier, EB1 has also been localized at the plus ends of ciliary microtubules. EB1 tends to localize at sites of MT force generation, ther efore it was assumed that EB1 may be behaving end-tracking motor, similarly to the end-tracking motors in cell division and cellular growth. This chapter discussed the EB1 end-trackin g model developed for the ciliary plug. Essentially, the plug is the end-tracking motor, which is behaves similar to the Lock, Load and Fire Mechanism (Dickinson and Pu rich, 2002). The advancement of the plug at the microtubule plus-end occurs in three steps: tubulin addition, filament-bound GTP hydrolysis, and the shifting and rebinding of the ciliary plug (e.g., EB1) to the filament end. The key parameters used to simulate this model are the diffusivity of the mi crotubule in the medium, th e length of the ciliary plug, the expected microtubule velocity, and th e force applied against the plug. The forcevelocity relationship for the microtubule is anal yzed to determine the maximum achievable force the microtubule can withstand with the EB1 end-tracking motor. 5.1 Model The physical characteristics of the plug make this model complex, but for simplification, the EB1 end-tracking motor is represented as a plug with multiple tubulin binding sites. The plug is inserted into the microtubule; hence, it was assumed that the plug creates a region where the protofilaments are separated from one another. The length of this region is labeled by a

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116 distance L (See Figure 5-2), where the 13 protofilament s are assumed to be independent of one another, and each of the motors opera tes on a single protofilament The three steps of the end-tracking motor are represented in Fi gure 5-3. The first step in this end-tracking model is addition of tubulin to the filament end, which induces hydrolysis of the penultimate subunit. Because the EB1 plug has a low affinity for T-GDP, the motor rebinds to the GTP-rich, filament plus-end causing the plug to advance. This model is very similar to the Lock, Load and Fire Mechanism (LLF) proposed in 2002 (Dickinson and Purich, 2002). The total time to complete one cycle is Tm plus Tm is the time it takes for the filament to add a dimer and undergo hydrolysis (Equation 5-1), time required for the plug to shift and relock to the new dimer following hydrolysis, and d represents the length of a tubulin protomer. We anticipate that elongation of unload ed protofilaments is rate-limited by Tm (confirmed below), in which case Tm can again be estimated from vmax (167 nm/s), i.e., s v d Tm05.0max (5-1) Following the approach of Dickinson & Purich (2002), the mean shift time is taken as the time required for the protofilament end to diffuse a distance d and rebind and can be solved by the differential equation given by (E q. 5-2) (Gardiner, 1986): 1 '2 2 dx d D dx d F Tk Df B f (5-2) where x is the protofilam ent end position, F is the force subjected to the protofilament end, and the protofilaments fluctuate in position with a characteristic diffusivity, Df. This diffusivity is dependent on the drag coefficient, and hence becomes a function of the length of the independent protofilament (Equation 5-3 and 5-4)

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117 Tk DB f (5-3) aL L /log 4 (5-4) By varying the length of the protofilame nt, we can also analyze the dependence of and elongation rate on force. To de termine the force dependence of on force, the differential equation in 5-2 was solved and is represented in Equation 5-5. dF Tk dF Tk Tk dF dF Tk D d FB B B B f x2 2 2exp )(1 (5-5) With as a function of force, the equation that govern s the velocity becomes a function of force, and is represented by Equation 5-6. )( )( FT d Fvm (5-6) This equation was used to generate forcevelocity profiles for the ciliary plug model. Based on the compression stiffness of a protofilament, and the thermal energy, kB the stepwise motion of a microtubule with the ci liary end-tracking motor attached was also simulated. The filament end position, x was governed by Equation 5-7. TkxB (5-7) In the simulations for this model, the 13 protofilaments were initiated at different, random lengths. Prior to polymerization, the length of one of the protofilaments was set so that the initial, equilibrium force was balanced. For a pr otofilament to go through one cycle of tubulin addition, GTP hydrolysis, surface advancement, the probability of the cycle occurring was evaluated. This probability was determined based on the rate of the cycle reaction, 1/ Tm. If any

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118 of the protofilaments underwent shifting and rebinding, the new plug equilibrium position was determined by zeq=i x / i, where i is either the stretch of compression stiffness of each protofilament; the protofilament has a stretch stiffness if its length is less than the equilibrium value, and it is under compression when its leng th is greater than the equilibrium value. The resulting force on each filament is equal to its stiffness times the displacement of the protofilament from equilibrium, and the overall force on the ciliary plug is the sum of these individual forces. 5.2 Parameter Estimations The key parameters used for this model were eith er based on literature values or estimated. The width of the protomer, a was calculated as 5.15 nm from a=2R / N where R is the radius of the protofilament and is equal to 11.48 nm for a 14-protofilmaent mi crotubule (Mickey and Howard, 1995). The length of the ciliary plug ( L ), or region where protofilaments are assumed to be independent, is estimated to be 75 nm from the EM image of the ciliary plug in Figure 5-1. The viscosity of the fluid used to calculate the dr ag coefficient was assumed to be that of water, 10-9 pN-s/nm2 (Boal, 2002). Assuming each of the protofilaments to be a semi-flexible rod, their filament compression () and stretch stiffness () were calculated. The compressi on stiffness is defined by the persistence length of the filament (), the thermal energy, and the length of the filament, and is represented by Equation 5-8 (Howard, 2001). 4 2L TkB (5-8)

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119 The persistence length is repr esented by Equation 5-9, where B is the bending modulus of the filament ( B = 1.2 x 10-26 N-m2, (Mickey and Howard, 1995)). The resulting value for the compression stiffness is 1.1 pN/nm. TkBB/ (5-9) The stretch stiffness is proportional to Youngs modulus ( Y =1.9 GPa, (Howard, 2001)), the cross sectional area of the microtubule ( A =190 nm2, (Gittes et al., 1993)), a nd the length of the rod being stretched, L : LAY (5-10) The resulting stretch stiffness for a protofilament was determined to be 370 pN/nm. 5.3 Results Figure 5-4 shows the force effects on ciliary microtubules. In Figure 5-4A, the mean time to shift as a function of force is shown for va rious protofilament lengths. Regardless of the force, there is little effect of length on the cycl e time. The time required for tubulin addition and filament-bound GTP hydrolysis remains constant and is force-independent, so the cycle time of the filaments is initially governed by Tm. As the load on the filament increases, the model is governed by the time it takes for the plug to advance (). The effect of force and corresponding cycle time on the protofilament velocity is shown in Figure 5-4B. Again, the lengths of the protofilaments have little effect on the velocity of the microtubul e. As the cycle time increases with increasing forces, the velocity exponentia lly decays to its maximum achievable force, or stall force ( Fstall). The approximate stall for the microt ubules simulated is approximately 12 pN. The position versus time data is re presented by Figure 5-5A, where the x -axis is representative of the end position of the ciliary plug. This figure shows how the ciliary plug advances as a steady rate for a short time then ju mps to a new position. The size of this jump is

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120 usually about 8nm, which is the size of the tubuli n dimer. The reasoning fo r the step size is that the end-tracking motor for each protofilament must fill with tubulin before the ciliary plug can advance. The histogram in Figure 5-5B show s the number of protofilaments at each length greater than the equilibri um position. In this simulation, th e protofilament end positions relative to the equilibrium position range from -5 to 55 nm after the simulation time of 1.6 s from Figure 5-5A. 5.4 Summary The ciliary plug was simulated as an EB1 e nd-tracking motor similar to the end-tracking motors described in the Lock, Load and Fire Mechanism (Dickinson and Purich, 2002). The primary steps of this model are tubulin addition, filament-bound GTP hydrolysis and the shifting and rebinding of the ciliary plug (e.g., EB1) to the filament end. By analyzing the force-velocity profile of this mechanism, we found the stall force to be approximately 12 pN at various protofilament lengths, which is significantly great er than the stall force of 4.8 pN predicted by the Brownian ratchet mechanism. The results also shows the strong depend ence of the stall force on the time for shift/rebinding of the ciliary plug to the filament end. The velocity profile shows the ability of the end-tracking motor to maintain fidelity of the microtubule by allowing the plug to advance only once all protofilaments are the same length. Figure 5-1. EM image of a ciliary plug at the en d of a ciliary microtubule. The average length of the plug is approximately 75 nm. [Reproduced from The Journal of Cell Biology, 1988, 107: 2259-2269. Copyright 1988 The Rockefeller University Press]

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121 Figure 5-2. Schematic of ciliary plug inserted in to the lumen of a cilia/f lagella microtubule. The microtubules behave as independent prot ofilaments for a distance L from the MT plus-end. Red represents the GTP-bound tubulin subunits. Figure 5-3. Mechanism of the ciliary/flagellar endtracking motor. In the first step, tubulin adds to the MT plus-end into the end-tracking co mplex. This binding induces hydrolysis attenuating the affinity of the complex to the protofilament. The surface advances to the filament end.

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122 Figure 5-4. Force effects on ciliar y microtubules. Initially the filament is governed by the time it takes for monomer addition & hydrolysis; however as the load on the filament increases, the model is governed by the time it takes for the plug to advance.

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123 Figure 5-5. Ciliary plug mo vement. A) Position versus Time Based on the compression stiffness and the thermal energy kB the stepwise motion of a ciliary plug with the end-tracking motor attached is shown. This motion shows how the ciliary plug advances as a steady rate for a short ti me then jumps to a new position. B) The histogram shows the length of the protofilaments.

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124 CHAPTER 6 DISCUSSION 6.1 Possible Roles of End-Tracking Motors in Biology The role of nucleotide hydrolysis in cytoskel etal molecular motor action is well-established for myosin, kinesin, and dynein. The affinity of a myosin head for the actin filament lattice is modulated by ATP hydrolysis, and dynein and kinesin act analogously in their binding and release from the microtubule la ttice. In defining a new class of cytoskeletal filament end-tracking motors, we previously described ho w an microtubule filament end-tracking motors can exploit nucleotide hydrolysis to generate significantly greater force than that predicted by a free-filament thermal ratchet (i.e., the elastic Brownian ratchet mechanism), and these ideas were generalized based on thermodynamic considerations (Dic kinson et al., 2004). In this report, we used known kinetic properties of EB1 binding an d MT plus-end elongation to examine whether a hypothetical end-tracking motor consis ting of affinity-modulated in teractions of EB1 at MT and protofilament ends can propel objects (e.g., MT-a ttached kinetochores or MT-attached ciliary plugs) at typically observed velocities while operating against appreciable loads. While there is no direct evid ence that force production by polymerizing MTs is governed by an end-tracking motor mechanism, several e xperimental observati ons suggest that the properties and interactions of EB1 are compatible with such a model. Kinetochores, for example, are known to selectively bind EB1 by means of AP C and/or other adapter proteins (Folker et al., 2005; Hayashi et al., 2005; Mimori-Kiyosue et al., 2005). Kinetochores also stabilize MTs against disassembly by preferentially attaching to GTP-containing -subunits of tubulin subunits situated at or near MT plus-ends, and this prop erty is likely to be the consequence of EB1s ability to attach to polymerizing GTP-rich MT subunits and to dissociate from GDP-containing subunits, thereby providing a thermodynamic drivin g force for localization at or near the MT

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125 plus-end. Capture of EB1-rich MTs by kinetoch ores may allow those EB1 molecules combining with APC to self-assemble into an end-tracki ng motor unit that links force generation to MT polymerization and hydrolysis of MT-bound GTP. It is known that in the absence of the EB1/APC complex, chromosomes fail to align at the metaphase-plate, presumably due to disrupted MT polymerization and kinetochore attachment. The distal tips of ciliary/flagellar MTs are likewise decorated with EB1 proteins during formation and regeneration, suggesting EB1 may serve a similar role in forming an end-track ing motor there and playing a role in elongationdependent force generation. In fact, it has previous ly been suggested that the plug-like structures found at the plus-ends of MTs in regenerating Chlamydimonas flagella appear to be MT assembly machines. The analogous geomet ry of the ciliary/flagellar plug and the tubule-attachment complex in the kinetochore would allow plugand kinetochore-bound EB1 to interact with their MT partners as an end-tracking motor. This proposal does not preclude the action of other ATP hydrolysis-dependent moto rs. For example, although the kinesin-like protein NOD lacks residues known to be critical for kinesin function, microtubule binding activates NODs ATPase activity some 2000-fo ld, a property that (Matthies et al., 2001) suggested may allow chromosomes to be transiently attached to MTs with out producing vectorial transport. The Brownian Ratchet mechanism proposed for force generation by MTs in TAC models (Inoue and Salmon, 1995)) does not allow a stro ng association between the filaments and the motile object, and cannot predict substantial force generation at low protomer concentrations. End-Binding Protein 1 (EB1) has previously been sh own to bind specifically to the polymerizing microtubule plus-end where the mi crotubule is tightly bound, sugges ting a possible role in force generation at these sites. We propose that e nd-binding proteins (specifically EB1) behave as

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126 molecular motors that modulate the interaction between MTs and the motile object, and generate the forces required for MT-based motility. Although the importance of EB1 and its potentia l to behave as and end-tracking motors has been discussed thoroughly in this research study, the models developed can be used to understand force generating mechanisms invol ving other end-tracking proteins and their potential to act as motors (e.g., CLASPS, C lip-170). Adenomatous Polyposis Coli (or APC), which has an important role in preventing colon can cer, is like EB1 in that it is found at the tips of microtubules where microtubules bind to the chromosome at the kinetochore. It therefore also has the potential to behave like an end-tracking motor. 6.2 Microtubule End-Tracking Model We developed and analyzed a preliminary EB 1 filament end-tracking model for MTs to determine the advantages of the mechanochemical process over the monomer-driven ratchet mechanisms. The two important properties of this model are (a) maintenance of a tight, persistent (processive) attachment at elongat ing MT plus-ends by means of EB1s multivalent affinity-modulated interactions; and (b) a mech anism for the assembly of the MT end by EB1 dimers bound on the motile object, thus affording a high-fidelity pathway for assembling tethered MTs. For simplicity, our model only considers simple reactions of the EB1 filament end-tracking motors. The details of the key assumptions applied to facilitate our analysis of this mechanism do not compromise key results of high force genera tion and processivity. For example, the effect of interactions among protofilaments on EB1-asso ciated MT assembly was neglected. Although we accounted for EB1 flexibility, we neglected any contribution of the flexibility of the protofilaments themselves in net compliance of the EB1-protofilament interaction. We previously suggested that EB1 may be a polyme rization cofactor acting together with APC to

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127 end-track MT protofilaments (see Mechanism-C in (Dickinson et al., 2004). However, in view of the recent finding that EB1 is a stable, two-headed dimer (Honnappa et al., 2005), we now explain how such multivalency would allow EB1 al one to operate as the end-tracking motor (like Mechanism-A in Dickinson et al. (2004)). Ei ther mechanism could capture energy from GTP hydrolysis and potentially translate it to mechanical work. We simulated the kinetics of the latter m echanism by characterizing each reaction step based on its corresponding kinetic rate constant, with force-dependence of elongation arising from the dependence of probability of the flexible EB1 head binding at a specific MT lattice position. With hydrolysis-driven affinity-modulation factor f greater than 10, our model recapitulates experimental, irreversible polymeriz ation rates for free MTs of 170 nm/s. In the presence of an opposing force, the collective action of hydrolysis-mediated motors on an MTs thirteen protofilaments can yield kinetic stall forces of approximately 30 pN. This value is considerably larger than the ~7-pN achievabl e maximum force provided by the energy of monomer addition alone (i.e., without the benef it of GTP hydrolysis) in a Brownian Ratchet mechanism. 6.3 Protofilament End-Tracking Models The microtubule end-tracking model devel oped neglected solution-phase End Binding protein 1 (EB1) and binding to microtubules and tubulin protomers. To account for binding solution-phase EB1, we developed simplified mo dels that simulated th e growth of a single protofilament in the presence of EB1 end-tracking motors. The properties of all protofilament the end-tracking models were compared to those of the simple Brownian Ratchet mechanism. Two of the models consider only free-protofilament growth operating with either monovalent or divalent EB1 proteins. The si mulations for both models include d a probabilistic analysis to determine the expected EB1 occupancy along th e length of the protof ilament. The results

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128 confirm the assumption that GTP-driven affinity modulated binding of the EB1 end-tracking proteins is required in order to provide a 4.2 tip-to-side binding ra tio as observed in experiments. We also developed two other protofilament mode ls that allow EB1 to interact with a linker protein on a motile object (e.g., Adenomatou s Polyposis Coli, APC), one model contained monovalent EB1 and the other had divalent EB1. By applying a load on the motile surface, we analyzed the resulting MT dynamics and force gene ration. The force-velocity profiles show that the divalent, EB1 end-tracking model provides an great advantage over the monovalent endtracking model as well as a Brownian Ratchet mech anism. The divalent end-tracking motors are able to provide processive end-tracking and pe rsistent attachment to the motile surface during protofilament polymerization. These divalent motor characteristics allow the protofilament to obtain much higher stall forces than predicted by the monovalent case or by a system with no affinity modulation (e.g., Brownian Ratchet model). 6.4 Future Work Further analysis of a 13-protofilament, mi crotubule end-tracking model should be considered. It is suggested to develop a stochastic model that includes all mechanisms discussed in the tethered-protofilament model with divalent EB1. For simplifications it could be assumed that all protofilament behave independently, but whose individual EB1 end-tracking motors each contribute to the equilibrium position of the motile surface, much like the ciliary plug model. Based on results from the protofilament models, it is expected that the MT end-tracking model will predict greater stall forces than that of the Brownian ratchet model at large affinity modulation values. Although much of the literature supports our proposed EB1 end-tracking mechanism, there remains definitive experimental literature that conf irms this model. Future studies could clarify some of the assumptions made, and help to better characterize the mechanism by which EB1

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129 associates with the MT plus-end. Of particular interest is whether EB1 together with growing MTs can generate the force predicted by our simulations while remaining persistently attached to the motile object. This hypothesis might be tested by adding EB1-coated beads to a solution of tubulin and MTs and with fluorescence microscopy determine if the MT binds to the beads and remains persistently attached as it polymerizes. Using optical trapping techniques, the velocity-force relationships coul d also be determined. This technique would provide more accurate stall force estimations for compar ison with the simulated results.

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130 APPENDIX A PARAMETER ESTIMATIONS A.1 Concentrations of EB1 Species in Solution Monovalent EB1: The reaction equations and co rresponding equilibrium equations considered for monovalent EB1 binding to tubulin protomers (Tb) and mi crotubule sides (MT) in solution are: TETbEK1 K1 = [E][Tb]/[TE] (A-1) EMTMTEdK Kd = [E][MT]/[MTE] (A-2) The total concentration of EB1 is represented in all states is [E]0 = [E] + [TE] + [MTE] (A-3) Substituting A-1 and A-2 into A-3 gives Equation A-4. [E]0 = [E] + [E][Tb]/ K1+ [E][MT]/ Kd (A-4) such that solving for [E] yields. dK MT K Tb E E ][][ 1 ][ ][1 0 (A-5) Divalent EB1: For divalent EB1, two tubulin protomers can bind to each EB1 molecule (E) to form TE or TTE. The two binding sites are assumed identical and non-cooperative. Here, the relevant reaction and equilibrium equations are E + Tb TE K1 =2 [E][Tb]/[TE] (A-6) Tb + TE TTE K1 = [Tb][TE]/(2 [TTE]) (A-7) EMTMTEdK Kd = [E][MT]/[MTE] (A-8) [E]0 = [E] + [TE] + [TTE] + [MTE] (A-9)

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131 Combining A-6 and A-7 yields 2 1 2 1]][[ 2 ]][[ ][ K TbE K TbTE TTE (A-10) Combining Eqs. A-9 and A-10 yields [E]0 = [E] + 2[E][Tb]/ K1+[E][Tb]2/ K1 2 + [E][MT]/Kd (A-11) hence dK MT K Tb E E ][][ 1 ][ ][2 1 0 (A-12) from which [TE] and [TTE] can be calculated using Eqs. A-6 and A-7. A.2 Occupation Probability of Monovalent EB1 Binding to Non-Tethered Protofilament The probability of tubulin being bound to EB1 is given by the following: ii E r riii E f fiiiipppkukppuTEkTbkpkuEBk d dp 11 1 1 1][][ ]1[ (A-13) The dimensionless relationships in Equations A14 to A-17 can be substituted into Equation A13: ][ ][,Tk Ekf ii i (A-14) ][,Tk kf i i (A-15) ][ ][ Tk ETkf E f (A-16) 1 1 1,1 1 0,1 1 1 11 1][][][ pTcTp fK K k k T T c T T Tk ckpk Tr r d f E f c c f r E r r (A-17)

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132 The resulting equations represent the differentia l equation for the probability of EB1 binding to the protofilament side (A-18) and plus-end (A-19), where u 1pi. iirii iiii d ippTpp pc d dp 1 1 1)1( (A-18) 21 1 1,21 1 0,111111 1cpTpcTcppc dt dpr r d (A-19) A.3 Occupation Probability of Monovalent EB1 Binding to Tethered Protofilament This model determines the EB1 fluorescence along the protofilament based on the probability of each tubulin protomer being in a specific EB1 binding state. The binding states considered were: pi = probability of EB1 bound to tubulin protomer in protofilament qi = probability of Tk-E bound to tubulin in protofilament w = probability of Tk bound to TE in solution v = probability of Tk bound to E in solution y = probability of Tk being unbound The probability of Tk being unbound, y, is represented by Equation A-20. iqvwy1 (A-20) The differential equations for the probabilities of EB1 and Tk-E binding to the protofilament are ii E rr E riiieff E f E f f iiTiieffiTi ion ippqkukpkppwCkTEkTbk qkypCkpkuEk dt dp 11 11 1 ,,) ][][( ][ (A-21)

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133 ii E r r E riiieff E f E f f iiieffoniiTiieffiT iqqqkukpkqqwCkTEkTk qkvuCkqkypCk dt dq 11 1 1 1 ,,) ][][( (A-22) where ui =1qipi,,. The differential equations for the prob ability of the track binding to either TE or EB1 are given below: 1 1, 1 1][ ][qkwCkwkvTkwkyTEk dt dwE r eff E f T T (A-23) ii i iieffion i T TqkvuCkwkvTkvkyEk dt dv, ,, 1 1][ ][ (A-24) To de-dimensionalize time in these differential equations, the variable Tr -1 was introduced, which is defined by Equation A-25. ][1 1 1Tk ukpk Tf r E r r (A-25) Setting ][1 0,T T Tc r and fK K k k T T Td f E f c r 1 1 1,][ Equation A-25 gets reduced to A-26. 1 1 1,1 1 0, 1pTuTTr rr (A-26) Dividing Equations A-21 to A-24 by kf [T], results in the following differential equations with dimensionless time: ii r r rii ieff f E f f E f i it f it i ieff f it i d f i f ippqTuTpTppw T C k k T ET k k q T K k k yp T C k k p T K k k u T E k k d dp 11 1 1,1 1 0,1 1 1, 1 ,, ,) ][][ ][ 1( ][ ][][][ ][ (A-27)

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134 ii r r rii ieff f E f f E f i d f i ieff f i it f it i ieff f it iqqqTuTpTqqw T C k k T ET k k q T K k k vu T C k k q T K k k yp T C k k d dq 11 1 1,1 1 0,1 1 1, 1 ,, ,) ][][ ][ 1( ][][][][ (A-28) 1 1 1, 1, 11 1][ ][ ][][ ][ qTw T C k k w T K k k v k k w T K k k y T ET k k d dwr eff f E f ff t f t f t (A-29) i i d f i i i ieff f i i f f t f t f tq T K k k vu T C k k w T K k k v k k v T K k k y T E k k d dv ][ ][ ][ ][][ ][,, 11 1 (A-30) To evaluate the probabilities of E or Tk-TE binding to the termin al subunit in the protofilament, these probabilities were re-written for the case when i 1: )1( )1( ][ ][ ][ ][ ][][][ ][21 1 1,21 1 1,1 1 0,1 1 1,1, 1 1, 11 1, 1, 1 ** 1 1ppTpqTuTp T ET k k q T K k k wp T C k k pyp T C k k p T K k k u T E k k d dpr r r f E f t f t eff f E f eff f t d f f (A-31) )1( ][ ][][][ ][ 1 ][][ )1( ][ ][ ][21 1 1, 21 1 1 1 ** 1,1, 1 1, 1 1, 1 1, 1, 1qq fK K T T k k qp fK K T T k k u T T q T ET k k T K k k T K k k qw T C k k vu T C k k yp T C k k d dqd c f E f dc f E f c f E f D f t f t eff f E f eff f eff f t (A-32) To solve for the probabilities ( pi, qi, w v ) from Equations A-27 to A7-30, the equations were dedimensionalized by using the follo wing dimensionless parameters: fk k ][ T Kt fk k1 f tk k f E fk k 2/mzid ie

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135 The resulting de-dimensionalized differential equations are: ii dc c dc ii eff ii t ii ieff i d i ippq fK K T T u T T p fK K T T ppw T C T ET q T K yp T C p T K u T E d dp 111 1 1 1 1 1 1 1, ,][][][ ) ][][ ][ 1( ][ ][][][ ][ (A-33) ii dc c dc ii eff ii d ii ieff ii it ii ieff iqqq fK K T T u T T p fK K T T qqw T C T ET q T K vu T C q T K yp T C d dq 111 1 1 1 1 11 1, ,][][][ ) ][][ ][ 1( ][][][][ (A-34) 11 1 1,1 1, 1][][][][ ][ qTw T C w T K vw T K y T ET dt dwr eff t (A-35) ii d i ii ieff i ii d i eff tq T K vu T C q fT K vu T C w T K vv T K y T E d dv ][ ][ ][][][][][ ][1 1 1 1, 1 (A-36) )1( ][][ )1( ][ ][ ][ ][ ][][ / ][ ][21 1 1,211 1 1 1 11 11 1, 111 1, 1 1 1ppTpq fK K T T u T T p T ET q T K wp T C pyp T C p T fK u T E d dpr dc c t eff eff d (A-37) )1( ][ ][][][ ][ 1 ][ / ][ )1( ][ ][ ][211 1 21 1 1 1 1 1 1 1 1, 11 1, 11 1, 1qq fK K T T qp fK K T T u T T q T ET T fK T K qw T C vu T C yp T C d dqdc dc c d t eff eff eff (A-38)

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136 For further simplification, more dimensionless para meters were substituted into these differential equations: fk k* ][ ][ T E ][1T K ][0,T Ceff ][ T Kd ][ T Tc 22/)( mzid ie Both iand represent the effect of force on the pr obabilities for binding, based on a normal Gaussian distribution with a variance, 2, and filament end position, zm. The rewritten differential equations are as follows: ii ii ii iii i i ippq f up f ppw qyp pu d dp 111 11 1 00) 1( (A-39) ii ii ii iii ii iii iqqq f up f qqw qvu qyp d dq 111 11 1 00) 1( (A-40) 11 00q f w wvwy dt dw (A-41) ii i iii i iq vu q f vu wvvy d dv 1 1 11 11 (A-42) )1( )1(2 1 211 1 1 11 100 1111 1 1 1pp f pq f up q wp pyp p f u d dp (A-43)

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137 )1( 1 )1(211 21 1 1 1 1 1 00 111 111 1qq f qp f uq f qw vu yp d dq (A-44) These differential equations (A-39 to A-44) were solved with given kinetic parameters in Matlab (Appendix B.2.2) to determine the probability of the various EB1 binding interactions with the protomers in a protofilament. A.4 Occupation Probability of Divalent EB1 Binding to Tethered Protofilament This model determines the EB1 fluorescence along the protofilament based on the probability of each tubulin protomer being in a specific EB1 binding state. The binding states considered were: pi : probability of protomer i bound to EB1 subunit (o ther subunit unbound) wi : probability of protomer i bound to TE qi +: probability of protomer i in state dbE+ qi : probability of protomer i in state dbEui : probability of protomer i being unbound The probability of the protomer being unbound, ui, is represented by iiii iwpqqu 1 (A-45) ii ii ii ii ii i i i i ion ippRppRupkqk upkqkpTkwkpkuEk dt dp 1 1 1 11, 1 1 1 1][ ][2 (A-46) ii ii i i ii on iwwRwwRwkpTkwkuETk dt dw 1 1 1 1][ ][ (A-47)

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138 i i i i iiiiii ii ii iqqRqqR upkupkqkqk dt dq1 1 11. 1, .11. (A-48) i i i i iiiiii iiii iqqRqqR upkupkqkqk dt dq1 1 11, 1,11, (A-49) ii ii i onion iiiiiii iii iuuRuuRuETkuEk upkupkqkqkwkpk dt du 1 1 1,1,][][2 (A-50) 1])[]([][wCkTETETkTkReff E f E f f (A-51) 111 1qpwkukRE rr (A-52) The probabilities for the terminal pr otomer subunit are listed below: )1( ])[][()1]([ ][ ][22 1 21 1 1 1 21 2 1 1111 1 1ppkpwkuk pTETkTkpETk upkqkpTkwkpkuEk dt dpE r E r r E f f E f on (A-53) )1( ])[][()1]([ ][ ][2 1 21 1 1 1 1 1 111 11 1 1 wkwpkukqkwCk wETkTkwTETk wkpTkwkuETk dt dwE r E r r E r eff E f E f f E f on (A-54) 211 1 1 1 1 212.121,11.22. 1])[]([][ qpwkuk qkwCkqTETETkTk upkupkqkqk dt dqE r r E r eff E f E E f f (A-55)

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139 211 21 1 1 1 1 121,11,11,11, 1)1( ])[]([)1]([ ][][2 upwkuuk uTETkETkuTk uETkuEkupkqkwkpk dt duE r r E f E f f on on (A-56) Since an EB1 bound to the protofilament plus -end cannot bind in the negative direction, 01dtdq At equilibrium (the protofila ment does not polymerize), when i =1, eq eqeqqqq these differential equations reduce to: eqeq eq eq eq eq eq onupkqkpTkwkpkuEk 2 ][ ][201 1 (A-57) eq eq eq eq onwkpTkwkuETk 1 1][ ][0 (A-58) eqeq equpkqk 2 0 (A-59) eq oneq on eqeq eq eq equETkuEkupkqkwkpk][][2 2 0 (A-60) And the following holds true: eqeqeq eqwpqu 1 (A-61) Solving Equation A-59 gives: eqeq equpkqk 2 (A-62) Substituting this relationship into the other three equilibrium equations results in: eq eq eq onwkpTkkuEk 1 1][ ][20 (A-63) eq on on eq equETkEkwkpk][][2 0 (A-64) eq eq eq onpTkwkkuETk][ ][01 1 (A-65) From these three relationships, the equations for weq, and peq were solved for:

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140 eq eq on eqp k Tkk u k Ek w 1 1 1][ ][ 2 (A-66) eq on on equ k Tkk kk ETk k k Ek p 1 1 1][ ][1][2 (A-67) At equilibrium, the followi ng relationship is true: 1]][[ 2][ K ET ET (A-68) Plugging this into Equation A-67, gives: eq on equ k Tkk T k k k k k Ek p 1 1 1 1 1][ 1 ][1 ][2 (A-69) The terms in brackets on the top and bottom ar e identical, therefore it A-69 reduces to: eq on equ k Ek p ][2 (A-70) Consequently, weq and qeq reduce to Equation A-71. eq eqp K Tb w1 (A-71) eqeq equp k k q 2 (A-72) Substituting A-70, 71, 72 into Equation A-61 results in the following equation: 2 1][2 2 ][ 1 ][2 110eq on eq onu k Ek k k u K T k Ek (A-73)

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141 Solving for ueq gives: k Ek k k k Ek k k k Ek K T k Ek K T uon on on on eq][2 4 ][2 81 ][2 1 ][ 1 ][2 1 ][2 1 1 (A-74) To determine the value of K1, we first assume [Tb]=0, which reduces Eq. A-75 to: k Ek k k k Ek k k k Ek k Ek Tuon on on on eq][2 4 ][2 81 ][2 1 ][2 )0]([2 (A-75) The fraction of filament-bound protomers attach ed to EB1 at equilibrium defined by total amount of EB1 minus the am ount of EB1 in solution: 0 0][ E EE (A-76) This is also equivalent to: eq on eq tot eq eq totu k k u k k MT pq E MT )1(2 1 ][ )2/( ][0 (A-77) Solving for gives: 1 2 1 ][ 1 1 eq on eq totu k k u k k MT (A-78) The effective equilibrium dissociation c onstant of EB1 and the protofilament ( Kd,eff) is defined as:

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142 eq on eq effdu k k u k k K 2 1 1, (A-79) Simplifying Equation A-78 gives: 1 ][ 1, tot effdMT K (A-80) When half of the protofilament is saturated (=1/2), Kd,eff is given by Equation A-81 and u1/2 ueq ([E]-[E]0/2). 2/1 2/1 ,2 1 1 u k k k k Kon effd (A-81) Under this constraint, u1/2 is given by Equation A-82, k ECk k k k ECk k k k Ek k Ek EEuueffon on effon on on on eq 0 0 2 0 0 0 2/14 81 1 )2/]([ (A-82) Since k+=kon* Ceff and k+ side=kon side* Ceff for the protofilament plus end, Equation A-83 can be rewritten as: eff eff eff eff eqCk Ek k k Ck Ek k k Ck Ek Ck Ek EEuu 0 0 2 0 0 0 2/14 81 1 )2/]([ (A-83) Substituting the definition of K where kkK/gives:

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143 eff eff eff effC E K C E K C E K C E K u0 2 0 2 2 0 0 2/14 81 1 (A-84) Assuming [ Tb ]=0, [ E ]0=0.27 M, [E]=[E]0/2, and Ceff=153 M and the experimentally determined value for Kd,eff of 0.44 M (Tirnauer et al., 2002b) where used to determine the value of K as 37. With kkK/, ueq can be calculated for a given 1][ K T : 2 1 0 2 2 1 0 2 2 1 0 1 1 01 ][ 8 1 ][ 161 1 ][ 2 1 ][ 1 1 ][ 2 K T C E K K T C E K K T C KE K T K T C KE ueff eff eff eff eq (A-85) Determination of k: At steady-state, the measured off-rate offk = 0.26 s-1 is related to the kby: 1][ 1 1 2/ )( K T Ku k qp pk keq eqeqeq eqeq off (A-86) Rearranging Eq. A-86 gives Equation A-87. 1][ 1 1 K T Ku kkeq off (A-87)

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144 The only free parameter is K1 for the probabilities and variables. To find the value of K1, the probabilistic model is used to determine the value that provides the following ratio 2/ )2/2/ (1 eqeqeq ii iiiqp qqp = 4.2 (A-88)

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145 APPENDIX B MATLAB CODES B.1 13-Protofilament Microtubule Model This stochastic program simulates a 13-protofilament MT polymerizing against a motile surface with a constant load. The values of f and sigma can be varied to determine the resulting velocity of the microtubule and each of the prot ofilament end positions. The kinetic parameters were estimated or used from literature values. % This program simulates a 13-protofilament MT polymerizing against a motile surface with a constant load. clear all ; hold off ; Tc = 5.3; % Tubulin critical concentration (free filament) (uM) Tb = 15; % Tubulin dimer concentration (uM) d=8; % Size of tubulin dimer (nm) nf=13; % Number of protofilaments kT = 4.14; % Thermal energy (pN-nm) Howard J 2001 sigma=10; % 5 (nm) kappa=kT/(sigma^2); % (pN/nm) f = 1000; % Affinity modulation (Kd reduced by a factor of f) v=167; % velocity of MT end growth (nm/s) Lhalf=700; % half length of MT (nm) koff=v/Lhalf; % off rate of both EB1 arms coming off 2 adjacent T-GDP's (s^-1) kD=0.5; % Dissociation rate constant for both EB1 arms on taxol-stabilized MT wal(uM) L=10; % (nm) rho=1/(L^2); % (nm^-2) %conversion=1.66e6; % conversion from uM to nm^3 %ksol=koff/(kD*f); ksol=5e7; % ksol: on-rate for both EB1 arms on MT in solution (nm^3/s) k1on=8.9/13; % k1+: kon for tubulin dimer on MT (1/(uM-s)) k1off=44/13; % k1-: koff for tubulin dimer on MT (1/s) k3on=((rho*ksol)/((sigma)*(2*pi)^0.5)); % k3+: kon for 1 EB1 arm on MT wall (1/s) (1.6*10^3) k3off=(1/2)*koff+(1/2)*((koff^2)+(4*k3on*koff))^0.5; % k3-: koff for 1 EB1 arm from MT wall (1/s) (19.6) k2on=k3on; % k2+: kon for 1 EB1 arm between T-GTP/TGDP (1/s) (1.6*10^3) k2off=k3off/f; % k2-: koff for 1 EB1 arm between TGTP/T-GDP(1/s) (0.02) dt=4.8e-6; % dt should be at least 0.1 x 1/fastest time constant (s) tim=1; % initialize time nt=round(tim/dt); % Number of time steps nshow0=1000; % Initial value for dummy index used to minimize number of 'n' and'z0's displayed nshow=nshow0; % Let nshow equal 100 for first iteration nplot=round(nt/nshow0); % Set number of time steps that will be stored t=(1:nplot)*nshow0*dt; % Calculate time from number of time steps taken (has 1000 elements) F=0; % Constant force applied to surface (pN) q=0; % Initial value for dummy variable used in "position" loop z0=zeros(1,nf); % Initial filament equilibrium position z=(1/nf)*sum(z0)-F/(nf*kappa); % Initial position of motile surface n=zeros(1,nf); % Initial number of tubulin dimers on protofilament ps=ones(1,nf); % Initial state of each protofilament position=zeros(1,nplot); % Initial vector of z for each time step % State 1: One EB1 arm bound between terminal T-GTP/T-GDP % State 2: One EB1 arm bound between terminal 2 T-GDP's % State 3: One EB1 arm bound between terminal 2 T-GDP's, One arm btwn terminal T-GTP/T-GDP

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146 % State 4: One EB1 arm bound between terminal 2 T-GDP's, One arm btwn lagging 2 T-GDP's % State 5: One EB1 arm bound between lagging 2 T-GDP's % State 6: One EB1 arm bound between lagging 2 T-GDP's, One arm btwn terminal T-GTP/T-GDP rn1=rand(nf,nshow0); % Generate a (nf x nt) matrix of random numbers (from 0 to 1) for loop. % Generating random numbers OUTSIDE of loop makes program faster to run rn2=rand(nf,nshow0); rn3=rand(nf,nshow0); rn4=rand(nf,nshow0); rn5=rand(nf,nshow0); rn6=rand(nf,nshow0); rn7=rand(nf,nshow0); rn8=rand(nf,nshow0); rn9=rand(nf,nshow0); rn10=rand(nf,nshow0); jshow=1; jstore=1; % Indices for storing, plotting data beta=d^2/2/sigma^2; % shortcut parameter in calculating fordependence. np=round(nt/nshow0); % number of plotted points zp = zeros(1,np); t=zp; % storage vectors for position plot for j=1:nt % from 1 to 100,000 itst = [rn1(:,jshow)'0)) + max(tnum).*(max(tnum)>0); % update ps based on transitions (note if one filament makes two transitions in one step (shouldn't happen often) only one leading to larger ps value is used. fz = (ps==1).*n+(ps==2).*(n-1)+(ps==3).*(n-1+n)+(ps==4).*(n-1+n-2); % Dimenionless forces (correct for multiple springs in states 3,4 ns = sum((ps==1)+(ps==2)+(ps==3)*2+(ps==4)*2); z=sum(fz)/ns F/d/ns/kappa; % New Equlibrium position (dimensionless) jshow=jshow+1; % update jshow -if j==nshow % when j is a multiple of 100*time step j/nt % display percent of the loop performed n; % display number of dimers added (vector) z0; % display equilibrium position for protofilament (vector) nshow=nshow+nshow0; % new value for loop %bar(n); % Bar plot of subfilament lengths %drawnow; rn1=rand(nf,nshow0); % regenerate numbers for running loop to make program faster to run rn2=rand(nf,nshow0); rn3=rand(nf,nshow0); rn4=rand(nf,nshow0); rn5=rand(nf,nshow0); rn6=rand(nf,nshow0); rn7=rand(nf,nshow0); rn8=rand(nf,nshow0); rn9=rand(nf,nshow0); rn10=rand(nf,nshow0); jshow=1; % reset jshow for new random numbers zp(jstore)=z; % Position of motile surface at each 'jstore' time step t(jstore)=j*dt; % store data for only those positions plotted jstore=jstore+1; % update index for storing data end end % end "j" loop position=z*d ksol sigma L dt subplot(2,1,1) plot(t,zp*d, 'r' ) % Plot Surface Position (z) vs. Time (t) xlabel( 'Time (sec)' ) % Label x-axis ylabel( 'Surface Position (nm)' ) % Label y-axis title( 'Surface Position vs Time' ) % Label title of plot axis([0 1 -2 300]) %axis([0 dt*nt min(zp)*d max(zp)*d]) % Set axis plotting range

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147 zproto=n*d; subplot(2,1,2) bar(zproto, 'b' ) hold on zi=[position,position,position,position,position,position,position,position,position,position,pos ition,position,position]; plot(zi, 'r' ) xlabel( 'Protofilament') % Label x-axis ylabel( 'Filament End Position (nm)' ) % Label y-axis B.2 Protofilament Growth Model with Monovalent EB1 B.2.1 Occupational Probability of Monovalent EB1 on a Non-Tethered Protofilament This probabilistic model simulates the free grow th of a single protofilament in the presence of monovalent, EB1 end-tracking motors. The valu e of the affinity modulation factor can be varied to determine the resulting EB1 density al ong a protofilament. The kinetic parameters were estimated or used from literature values. % Probabalistic model free MT's % Simulates free MT's in presence of EB1 % Monovlanet EB1 % Plots: Occupation Probability vs Subunit clear all ; n=400; % number of subunits to simulate tspan=[0 1000]; j=1:n; x0=zeros(n,1); % Parameters % Fixed parameters Tb = 10; % uM tubulin dimer concentration MT = 10; % uM microtubule concentration V = 170; % nm/s; % elongation speed d = 8; % nm; % subunit length % kf = 0.68; % uM^-1s^-1 on-rate for tubulin % kr = 3.38; % s^-2 kf = V/d/Tb; % uM^-1s^-1 on-rate for tubulin -taken assuming irreversible % elongation at observed elongation speed Tc = 5; % uM plus-end critical concentration kr = kf*Tc; % s^-1 off-rate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % s^-1 observed decay rate constant of EB1 from MT sides EB1tot = 0.27; % uM Total EB1 concentration K1 = .2; % uM Equilibrium dissocation constant of Tb for E in solution % determined be value need for 4.2:1 tip-to-middle concentration Kd = 0.5; %uM Equilibrium dissocation constant of E for MT sides E = EB1tot/(1+Tb/K1+MT/Kd); %uM Equilibrium value of EB1 concentration TE = Tb/K1*E; %uM Equilibrium value of EB1-Tb concentration kplus_side=kobs/(E+Kd); % uM^-1 s^-1 on-rate constant for EB1 to MT side -kminus_side=kplus_side*Kd; % s^-1 off-rate constant for EB1 from MT side enh = kf/kplus_side % End binding Rate enhancement factor % Roughly estimated parameters

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148 f = 1000; % affinity modulation factor sigma = 10 ; % nm stdev of EB1 position fluctuations kplus=enh*kplus_side; % uM^-1 s^-1 on-rate constant for EB1 to terminal subunit % assumed same as on side kminus=enh*kminus_side/f; % s^-1 off-rate constant for EB1 from MT tip kfE = kf; % binding of TE to end; % Dimensionless parameters a = kminus_side/(kf*Tb); as = kminus/(kf*Tb); g = kplus_side*E/(kf*Tb); gs = kplus*E/(kf*Tb); b=kfE*TE/kf/Tb; % Fed parameters kT=4.14; sigEB1 = 10; % nm stdev of EB1 linkage position amp=100; % EB1 local concentration increase factor delta=1; % nm characteristic interaction distance %stiffEB1=kT/sigEB1^2; % EB1 linkage stiffness av = a*ones(n,1); av(1)=as; gv = g*ones(n,1); gv(1)=gs; % gv = gv.*amp.*exp(-(0:n-1).^2.*(d./sigEB1)^2)'.*exp((0:n-1)*d*delta/sigEB1^2)'; % av=av.*exp((0:n-1)*d*delta/sigEB1^2)'; Trinv0=Tc/[Tb]; Trinv1=Tc/[Tb]*Kd/K1/f*kfE/kf; [tout, xout]=ode23s(@(t,x0) sfrate(t,x0,av,gv,b,Trinv0,Trinv1),tspan,x0); nt=length(tout); nmid=round(nt/2); pmid=xout(nmid,j); pend=xout(nt,j); fluor=pend; fluorm=pmid; plot(1:n,pmid,'k:' ,1:n,pend, 'k' ); xlabel( 'subunit' ) ylabel( 'occupation probability' ) function f=sfrate(t,x,av,gv,b,Trinv0,Trinv1) n=length(x); j=1:n; p=x(j); u=1-p; f=zeros(n,1); Trinv=Trinv0*u(1)+Trinv1*p(1); f(1) = gv(1)*u(1)-av(1)*p(1)-p(1)+b*u(1)+Trinv0*u(1)*p(2)-Trinv1*p(1)*u(2); i=2:n-1; f(i)= gv(i).*u(i)-av(i).*p(i)+(1+b)*(p(i-1)-p(i))+Trinv*(p(i+1)-p(i)); %f(n)=0; f(n)=f(n-1);

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149 B.2.2 Occupational Probability of Monoval ent EB1 on a Tethered Protofilament This probabilistic model simulates the growth of a surface-tethered protofilament in the presence of monovalent, EB1 end-tracking motors. The value of the affinity modulation factor can be varied to determine the resulting EB 1 density along a protofilament. The kinetic parameters were estimated or used from literature values. % brunode.m % Probabalistic model tethered MT's % Simulates tethered MT's in presence of EB1 % Monovlanet EB1 % Plots: Occupation Probability vs Subunit clear all; n=200; % number of subunits to simulate tspan=[0 1000]; j=1:n; x0=zeros(2*n+2,1); % Parameters % Fixed parameters kT = 4.1; % pN-nm thermal energy T = 10; % uM tubulin dimer concentration MT = 10; % uM microtubule concentration V = 170; % nm/s; % elongation speed d = 8; % nm; % subunit length kf = V/d/T; % uM^-1s^-1 on-rate for tubulin -taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plus-end critical concentration kr = kf*Tc; % s^-1 off-rate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % s^-1 observed decay rate constant of EB1 from MT sides EB1tot = 0.27; % uM Total EB1 concentration K1 = .16; % uM Equilibrium dissocation constant of T for E in solution -determined be value need for 4.2:1 tip-to-middle concentration Kd = 0.5; %uM Equilibrium dissocation constant of E for MT sides E = EB1tot/(1+T/K1+MT/Kd); %uM Equilibrium value of EB1 concentration TE = T/K1*E; %uM Equilibrium value of EB1-Tb concentration kplus_side=kobs/(E+Kd); % uM^-1 s^-1 on-rate constant for EB1 to MT side -kminus_side=kplus_side*Kd; % s^-1 off-rate constant for EB1 from MT side % Roughly estimated parameters f = 1; % affinity modulation factor TcE = Tc*Kd/K1/f; sigma = 5 ; % nm stdev of EB1 position fluctuations Delta = 0; % bond distance kplus=kplus_side; % uM^-1 s^-1 on-rate constant for EB1 to terminal subunit -assumed same as on side kminus=kminus_side/f; % % s^-1 off-rate constant for EB1 from MT tip kfE = kf; % binding of TE to end; cef f0 = 2/(2*pi)^(3/2)/sigma^3; % concentration in nm^-3 -based on 3-D normal distribution on half-sphere

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150 Ceff0 = ceff0/(6.022e23)*1e27/1000*1e6; % conc in uM: nm^-3 x (1 mol/ 6.022e23) x (1e27 nm^3/m^3) x (1 m^3/1000 L) x (10^6 uM/M) % Varied parameters Kt = 5; % uM Force =1*kT*log(T/Tc)/d; % load in pN positive if compressive, negative if tensile %% Dimensionless parameters alpha=kplus_side/kf; % alpha_s =kplus/kf; % gamma = 1; %kt/kf; delta = 1; %kfE/kf eta = 1; %k1/kf; chi = Ceff0/T; beta = Kt/T; mu = Kd/T; psi = Tc/T; xi = K1/T; epsilon=E/T; deld = Delta/d; kappa = kT/sigma^2; kappad= kappa*d^2/kT; Fd = Force*d/kT; pars = [alpha alpha_s gamma delta eta chi beta mu psi xi epsilon f kappad deld Fd]; alpha=pars(1) alpha_s =pars(2) gamma=pars(3) delta=pars(4) eta = pars(5) chi = pars(6) beta = pars(7) mu = pars(8) psi = pars(9) xi = pars(10) eps = pars (11) f= pars(12) kappad = pars(13) % kappad = kappa*d^2/kT; deld = pars(14) % deld = Delta/d; Fd = pars(15) % Fd = F*d/kT neg if under compression Fnet*d/kT = F*d/kT+kappa*i*d*(d/kT) = Fd +kappad*i [tout, xout]=ode23s(@(t,x0) bfrate(t,x0,pars),tspan,x0); nt=length(tout); nmid=round(nt/2); pmid=xout(nmid,2*j-1); pend=xout(nt,2*j-1); qmid= xout(nmid,2*j); qend=xout(nt,2*j); fluor=pend+qend; fluorm=pmid+qmid; plot(1:n,pmid,'k:' ,1:n,qmid, 'b:' ,1:n,pend, 'k' ,1:n,qend, 'b' ,1:n,fluor, 'g' ); xlabel( 'subunit' ) ylabel( 'occupation probability' ) w = xout(nt,2*n+2); v = xout(nt,2*n+1); im = sum(qend.*(1:length(qend)))/sum(qend) FT = (im-1)*kappad; Fnetd = Fd+FT; V1 = kf*T*(exp(-Fd)-Tc/T*(1-(qend(1)+pend(1)))) V2 = kfE*((TE+Ceff0*w)*exp(-Fd)-TcE*(qend(1)+pend(1))) V=V1+V2 relV=V/(kf*T-kr)

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151 Attachprob = 1-(1-sum(qend))^13 function ff=bfrate(t,x,pars) n=(length(x)-2)/2; j=1:n; p=x(2*j-1); q=x(2*j); u=1-p-q; v=x(2*n+1); w=x(2*n+2); y=1-v-w-sum(q); fp=zeros(n,1); fq=zeros(n,1); ff=zeros(2*n+2,1); alpha=pars(1) ; alpha_s =pars(2); gamma=pars(3); delta=pars(4); eta = pars(5) ; chi = pars(6); beta = pars(7); mu = pars(8); psi = pars(9); xi = pars(10); eps = pars (11); f= pars(12); kappad = pars(13); % kappad = kappa*d^2/kT; deld = pars(14); % deld = Delta/d; Fd = pars(15); % Fd = F*d/kT pos if under compression Fnet*d/kT = F*d/kT+kappa*(i-(im-1))*d^2/kT) = Fd +kappad*i meani = 0; if sum(q)>0 meani = sum(q.*j')/sum(q); end FT = (meani-1)*kappad; Fnetd = Fd+FT; %afac=exp(Fnetd); %psi = psi*afac; % alpha=alpha*exp(Fnetd); alpha_s=alpha_s*afac; gamma=gamma*afac; eta=eta*afac; im =1; %im = Fd/kappad+meani; % mean subunit position for unstressed trackers afac=exp(Fd); psi = psi*afac; alpha=alpha*afac; alpha_s=alpha_s*afac; gamma=gamma*afac; eta=eta*afac; phi=exp(abs(j-im)*kappad*deld)'; theta=exp(-kappad/2*(j-im).^2)'; phi0 = exp(abs(im)*kappad*deld); theta0 = exp(-kappad/2*(im).^2); chiv=chi*theta(j).*phi(j); chiv0 = chi*theta0*phi0; line1 = alpha_s*(eps*u(1)-mu/f*p(1))-gamma*chiv(1)*y*p(1) p(1)-delta*chiv0*w*p(1); line2 = gamma*beta*phi(1)*q(1)+delta*eps/xi*(1p(1))+psi*(u(1)+delta*mu/f/xi*phi(1)*q(1))*p(2); fp(1) = line1+line2-delta*psi*mu/xi/f*p(1)*(1-p(2)); line1 = (chiv(1)*(gamma*y*p(1)-alpha_s*v*u(1))+chiv0*phi0*w*(1-q(1))); line2 = -(gamma*beta*phi(1)+alpha_s*mu/f*phi(1)+1+delta*eps/xi)*q(1); line3 = psi*(u(1)+delta*mu/xi/f*p(1))*q(2)-delta*psi*mu/xi/f*phi(1)*q(1)*(1-q(1)); fq(1) =line1+line2+line3; i=2:n-1;

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152 line1 = alpha.*(eps*u(i)-mu*p(i))-gamma*chiv(i).*y.*p(i)+gamma*beta*phi(i).*q(i); line2 = (1+delta*(eps/xi+chiv0*w))*(p(i-1)-p(i)); line3 = psi*(delta*mu/xi/f*(p(1)-theta(1)*q(1))+u(1))*(p(i+1)-p(i)); fp(i) = line1+line2+line3; line1 = gamma*(chiv(i).*y.*p(i)-beta*phi(i).*q(i))+alpha*(chiv(i).*v.*u(i)mu*phi(i).*q(i)); line2 = (1+delta*eps/xi+delta*chiv0*w)*(q(i-1)-q(i)); line3 = psi*(delta*mu/xi/f*(p(1)-phi(1)*q(1))+u(1))*(q(i+1)-q(i)); fq(i)=line1+line2+line3; ff(1)=fp(1); ff(2)=fq(1); ff(2*i-1)=fp(i); ff(2*i) = fq(i); line1 = gamma*(eps*y-beta*v)+eta*(xi*w-v)-alpha_s*(chiv(1)*v*u(1)-mu/f*phi(1)*q(1)); line2 = alpha*sum(mu*phi(2:n).*q(2:n)-v*chiv(2:n).*u(2:n)); ff(2*n+1) = line1+line2; ff(2*n+2)= gamma*(eps/xi*y-beta*w)+eta*(v-xi*w)-delta*(chiv0*wpsi*mu/xi/f*phi(1)*q(1)); V1 = (1+delta*eps/xi+delta*chiv0*w); V2 = psi*(delta*mu/xi/f*(p(1)-phi(1)*q(1))+u(1)); line1 = alpha.*(eps*u(n)-mu*p(n))-gamma*chiv(n).*y.*p(n)+gamma*beta*phi(n).*q(n); line2 = (1+delta*(eps/xi+chiv0*w))*(p(n-1)-p(n)); fp(n) = line1+line2; line1 = gamma*(chiv(n).*y.*p(n)-beta*phi(n).*q(n))+alpha*(chiv(n).*v.*u(n)mu*phi(n).*q(n)); line2 = (1+delta*eps/xi+delta*chiv0*w)*(q(n-1)-q(n)); fq(n)=line1+line2; ff(2*n-1)=fp(n); ff(2*n)=fq(n); B.3 Protofilament Growth Model with Divalent EB1 B.3.1 Occupational Probability of Divalen t EB1 on a Non-Tethered Protofilament This probabilistic model simulates the free grow th of a single protofilament in the presence of divalent, EB1 end-tracking motors. The value of the affinity modulation factor can be varied to determine the result ing EB1 density along a protofilament. The kinetic parameters were estimated or used from literature values. % Probabilistic model -free-ended MT's % Simulates free-ended MT's in presence of EB1 % Divalent EB1 % Inputs: kon % Outputs: EB1 Tip: Side Binding Ratio % Plots: Occupation Probability vs Subunit n=400; % number of subunits to simulate tspan=[0 140]; j=1:n; x0=zeros(3*n,1); % Parameters

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153 % Fixed parameters Tb = 10; % uM tubulin dimer concentration MT = 10; % uM microtubule concentration V = 170; % nm/s; % elongation speed d = 8; % nm; % subunit length % kf = 0.68; % uM^-1s^-1 on-rate for tubulin % kr = 3.38; % s^-2 kf = V/d/Tb; % uM^-1s^-1 on-rate for tubulin -taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plus-end critical concentration kr = kf*Tc; % s^-1 off-rate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % s^-1 observed decay rate constant of EB1 from MT sides EB1 = 0.27; % uM Total EB1 concentration sigma = 10 ; % nm stdev of EB1 position fluctuations ceff = 2*exp(-(8/10)^2/2)/(2*pi)^(3/2)/sigma^3; % concentration in nm^-3 -based on 3-D normal distribution on halfsphere Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm-3 x (1 mol/ 6.022e23) x (1e27 nm^3/m^3) x (1 m^3/1000 L) x (10^6 uM/M) K = 37; % equals kplus_side/kminus_side, valued required for Kdeff = 0.44 %%%% Guessed parameters Kd1 = .65; % Kd, Dissociation constant for EB1 subunit and Tb, Kd1=k1m/k1 (uM) Value for typical monovalent protein %Kd1=Kd1vec(irun); k1 = 10; % On-rate for EB1 subunit and Tb (uM^-1*s^-1), Value for typical protein-protein binding k1m = k1*Kd1; % Off-rate for EB1 subunit and Tb (s^-1) %%% Equlibria E = EB1/((1+(Tb/Kd1))^2); % [EB1], Concentration of EB1 dimer in sol'n TE = 2*E*Tb/Kd1; % [EB1-Tb], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = Tb*TE/(2*Kd1); % [EB1-Tb^2], Concentration of EB1 dimer bound to 2 tubulin protomers a=4*K^2*E/Ceff; b=(1+Tb/Kd1)*2*K*E/Ceff+1; u_eq = (-b+sqrt(b^2+4*a))/2/a p_eq=2*K*E/Ceff*u_eq; q_eq = 2*K*p_eq*u_eq; pi_eq = Tb/Kd1*p_eq; fl_eq=p_eq+pi_eq+q_eq/2 %%% Equilibrium flourescence conce chec k = u_eq+p_eq+q_eq+pi_eq %%% should equal one %% Determine kminus_side, kplus_side, kon kminus_side = kobs*(1+K*u_eq/(1+Tb/Kd1)); %% Based on FRAP half-life kplus_side = K*kminus_side; %% by definition kon_side = kplus_side/Ceff; f = 1000 ; % affinity modulation factor %f=fvec(irun); %% Mixed model -chose kon, calculated koff from f kon = 5; % uM-1s-1 fon = kon/kon_side; %%% accelerated on-rate at end kplus = kon*Ceff; foff = fon/f; kminus=fon/f*kminus_side; %%% corresponding change off-rate at end %%% Other parameters kfE = kf % on-rate constant of TE and TTE to MT end uM-1s-1; krE = kfE*Tc/Kd1*kminus/kon % off-rate constant of TE or TTE %% Dimensionless parameters pars =[kminus_side kminus kplus_side kplus kf kr krE kfE k1 k1m Tb E TE TTE Ceff]; x0(3*j-1)=p_eq; %initial conditions x0(3*j)=q_eq/2; x0(4)=0; x0(3*j-2)=pi_eq; [tout, xout]=ode23s(@(t,x0) dfrate(t,x0,pars),tspan,x0);

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154 nt=length(tout); nmid=round(nt/2); pimid=xout(nmid,3*j-2); pmid=xout(nmid,3*j-1); qpmid=xout(nmid,3*j); piend=xout(nt,3*j-2); pend=xout(nt,3*j-1); qpend=xout(nt,3*j); qmend=[0 qpend(1:n-1)]; qmmid=[0 qpmid(1:n-1)]; fluor=pend+.5*qpend+.5*qmend+piend; fluorm=pmid+.5*qmmid+.5*qpmid+pimid; pp=polyfit(20:150, log(fluor(20:150)-fl_eq),1); fitf=exp(pp(2)+pp(1)*(1:n)); plot(1:n,fluor, 'go' ,1:n,fitf+fl_eq); xlabel( 'subunit' ) ylabel( 'occupation probability' ) tip_ratio=real(exp(pp(2))+fl_eq)/fl_eq fl_eq function f=dfrate(t,x,pars) %pars = [kminus_side kminus kplus_side kplus kf kr krE kfE k1 k1m Tb E TE TTE Ceff]; kminus_side=pars(1); kminus=pars(2); kplus_side=pars(3); kplus=pars(4); kf=pars(5); kr=pars(6); krE=pars(7); kfE=pars(8); k1=pars(9); k1m=pars(10); Tb=pars(11); E=pars(12); TE=pars(13); TTE=pars(14); Ceff=pars(15); kon_side=kplus_side/Ceff; kon=kplus/Ceff; n=length(x)/3; j=1:n; pid=x(3*j-2); p=x(3*j-1); qp=x(3*j); qm=[0 qp(1:n-1)']'; u=1-p-qp-qm-pid; konv = ones(n,1)*kon_side; konv(1)=kon; kpv = konv*Ceff; kmv=ones(n,1)*kminus_side; kmv(1)=kminus; fp=zeros(n,1); fqp=fp; fqm=fp; fpi=fp; f=zeros(3*n,1); Rp=kf*Tb+kfE*(TE+2*TTE)+kfE*Ceff*pid(1); Rm=kr*u(1)+krE*(pid(1)+p(1)+qp(1)); i=2:n-1; tmp1=2*konv(i)*E.*u(i)-kmv(i).*p(i)+k1m*pid(i)-k1*Tb*p(i)+kmv(i).*qp(i-1)-kpv(i+1).*p(i).*u(i+1); tmp2=kmv(i+1).*qm(i+1)-kpv(i-1).*p(i).*u(i-1)+Rp*(p(i-1)-p(i))+Rm*(p(i+1)-p(i)); fp(i) = tmp1+tmp2; fqp(i)=-kmv(i+1).*qm(i+1)-kmv(i).*qp(i)+kpv(i).*p(i+1).*u(i)+kpv(i+1).*p(i).*u(i+1)+Rp*(qp(i-1)qp(i))+Rm*(qp(i+1)-qp(i)); fpi(i)=konv(i)*TE.*u(i)-kmv(i).*pid(i)+k1*Tb*p(i)-k1m*pid(i)+Rp*(pid(i-1)-pid(i))+Rm*(pid(i+1)pid(i)); tmp1=2*kon*E*u(1)-kminus*p(1)+k1m*pid(1)-k1*Tb*p(1)+kminus_side*qm(2)-kplus_side*p(1)*u(2); tmp2=kfE*TE*(1-p(1))-(kf*Tb+2*kfE*TTE)*p(1)+(kr*u(1)+krE*(pid(1)+qp(1)))*p(2)-krE*p(1)*(1-p(1));

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155 fp(1)=tmp1+tmp2; tmp1 = kon*TE*u(1)-kminus*pid(1)+k1*Tb*p(1)-k1m*pid(1); tmp2 = 2*kfE*TTE*(1-pid(1))-(kf*Tb+kfE*TE)*pid(1)kfE*Ceff*pid(1)+krE*qp(1)+(kr*u(1)+krE*p(1))*pid(2)-krE*pid(1)*(1-pid(2)); fpi(1)=tmp1+tmp2; tmp1= -kminus_side*qm(2)-kminus*qp(1)+kplus*p(2)*u(1)+kplus_side*p(1)*u(2); tmp2= -(kf*Tb+kfE*TE+2*kfE*TTE)*qp(1)+kfE*Ceff*pid(1)krE*qp(1)+(kr*u(1)+krE*(pid(1)+p(1)))*qp(2); fqp(1) =tmp1+tmp2; fp(n)=0; fqp(n)=0; fpi(n)=0; f(3*j-2)=fpi; f(3*j-1)=fp; f(3*j)=fqp; B.3.2 Average Fraction of divalent EB1-bound Protomers on Side of Protofilament This stochastic model simulates the side -binding of divalent EB1 on a non-growing protofilament. The value of the affinity modulati on factor can be varied to determine the timeaveraged fluorescence of EB1 along the length of the protofilament and the state of the subunits in the protofilament. The kinetic parameters were estimated or used from literature values. % Probabalistic model % Simulates free-ended MT's in presence of EB1 % Divalent EB1 % Inputs: f, Kd1, kon % Outputs: Time Avg Fluorescence % Plots: Time Avg Fluorescence vs Subunit clear all ; n=400; % number of subunits to simulate 400 tspan=[0 40]; 40 j=1:n; x0=zeros(3*n,1); % Determine Parameters % Fixed parameters Tb = 10; % uM tubulin dimer concentration Tc = 5; % uM plus-end critical concentration EB1 = 0.27; % uM EB1 concentration d = 8; % nm; % subunit length V = 170; % nm/s; % elongation speed kf = V/d/Tb; % uM^-1s^-1 on-rate for tubulin -taken assuming irreversible elongation at observed elongation speed kr = kf*Tc; % s^-1 off-rate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % s^-1 decay rate constant of EB1 from MT sides sigma = 10 ; % nm stdev of EB1 position fluctuations ceff = 2*exp(-(8/10)^2/2)/(2*pi)^(3/2)/sigma^3; % concentration in nm^-3 -based on 3-D normal distribution on half-sphere

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156 Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm-3 x (1 mol/ 6.022e23) x (1e27 nm^3/m^3) x (1 m^3/1000 L) x (10^6 uM/M) K = 37; % equals kplus_side/kminus_side, valued required for Kdeff = 0.44 %%%% Guessed parameters Kd1 = 0.65; % Kd, Dissociation constant for EB1 subunit and Tb, Kd1=k1m/k1 (uM) Value for typical monovalent protein %Kd1=Kd1vec(irun); k1 = 10; % On-rate for EB1 subunit and Tb (uM^-1*s^-1), Value for typical protein-protein binding k1m = k1*Kd1; % Off-rate for EB1 subunit and Tb (s^-1) %%% Equlibria E = EB1/((1+(Tb/Kd1))^2); % [EB1], Concentration of EB1 dimer in sol'n TE = 2*E*Tb/Kd1; % [EB1-Tb], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = Tb*TE/(2*Kd1); % [EB1-Tb^2], Concentration of EB1 dimer bound to 2 tubulin protomers a=4*K^2*E/Ceff; b=(1+Tb/Kd1)*2*K*E/Ceff+1; u_eq = (-b+sqrt(b^2+4*a))/2/a p_eq=2*K*E/Ceff*u_eq; q_eq = 2*K*p_eq*u_eq; pi_eq = Tb/Kd1*p_eq; fl_eq=p_eq+pi_eq+q_eq/2 %%% Equilibrium flourescence conce check = u_eq+p_eq+q_eq+pi_eq %%% should equal one %% Determine kminus_side, kplus_side, kon kminus_side = kobs*(1+K*u_eq/(1+Tb/Kd1)); %% Based on FRAP half-life kplus_side = K*kminus_side; %% by definition kon_side = kplus_side/Ceff; f = 1000 ; % affinity modulation factor %f=fvec(irun); %% Mixed model -chose kon, calculated koff from f kon = 10; % uM-1s-1 fon = kon/kon_side; %%% accelerated on-rate at end kplus = kon*Ceff; foff = fon/f; kminus=fon/f*kminus_side; %%% corresponding change off-rate at end %%% Other parameters kfE = 10 % on-rate constant of TE and TTE to MT end uM-1s-1; krE = kfE*Tc/Kd1*kminus/kon % off-rate constant of TE or TTE % Initial conditions N=40; % 50-number of subunits to simulate S=ones(1,N); %S = 1 if unocupplied; 2 if bound to E, 3 if bound to TEE, 4 if bound to +side of doubly bound, 5 if bound to -side tim=10; % 200-run time (s) chartime=1/max([kfE*TTE kfE*TE kf*Tb kr krE kon*Tb k1*Tb k1 kplus kplus_side kminus kminus_side]); %Characteristic time dt = chartime/5; %simulation time increment nt=round(tim/dt); rnside1=rand(nt,N); radd = rand(nt,1); previt=0; FLav=0*S; for it=1:nt %% Side binding tst1=(S==1)& rnside1(it,:)<2*kon_side*E*dt; % binds E tst2= (S==1)& rnside1(it,:)<(kon_side*TE*dt+2*kon_side*E*dt) &~tst1; % or binds TE tst3=(S==2)& rnside1(it,:)
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157 tst5=(((S==2)&[S(2:N)==1 0]) & rnside1(it,:)<(kplus_side*dt + kplus_side*dt+kminus_side*dt)) &~(tst3|tst4); % bind minus side tst6=((S==2)& rnside1(it,:)<(k1*T*dt + 2*kplus_side*dt+kminus_side*dt)) &~(tst3|tst4|tst5); % bind T tst7=(S==3)& rnside1(it,:)0 ntav=it-previt; FL=(S==2)+(S==3)+.5*(S==5)+.5*(S==4); FLav=(previt*FLav+ntav*FL)/(ntav+previt); previt=it; ifnd1=find(tst1); S(ifnd1)=2; ifnd2=find(tst2); S(ifnd2)=3; ifnd3=find(tst3); S(ifnd3)=1; ifnd4=find(tst4); S(ifnd4)=5; S(ifnd4-1)=4; ifnd5=find(tst5); S(ifnd5)=4; S(ifnd5+1)=5; ifnd6=find(tst6); S(ifnd6)=3; ifnd7=find(tst7); S(ifnd7)=2; ifnd8=find(tst8); S(ifnd8)=1; S(ifnd8+1)=2; ifnd9=find(tst9); S(ifnd9)=1; S(ifnd9-1)=2; ifnd10=find(tst10); S(ifnd10)=1; it S; FLav plot(1:N,FLav,[1 N],[fl_eq fl_eq]); axis([0 N 0 1]); drawnow; end end B.3.3 Average Fraction of EB1-bound pr otomers during protofilament growth This stochastic model simulates the time averaged fluorescence of EB1 along a nontethered, single microtubule protofilament in th e presence of divalent EB1 during protofilament growth. The value of the affinity modulation factor can be varied to determine its affect on the EB1 fluorescence. The state of th e subunits in the prot ofilament can also be determined. % Simulates free-ended MT's in presence of EB1 % Stochastic model % Divalent EB1 % Inputs: f, Kd1, kon % Outputs: Velocity, State of Subunits, Time Avg Fluorescence % Plots: Time Avg Fluorescence vs Subunit clear all ; tic; N=200; % number of subunits to simulate 200 tim=40; % run time (s)40 axmax=.2; % max y-axis. % Determine Parameters % Fixed parameters

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158 T = 10; % uM tubulin dimer concentration V = 170; % nm/s; % elongation speed d = 8; % nm; % subunit length kf = V/d/T; % uM^-1s^-1 on-rate for tubulin -taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plus-end critical concentration kr = kf*Tc; % s^-1 off-rate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % s^-1 decay rate constant of EB1 from MT sides EB1 = 0.27; % uM EB1 concentration sigma = 10 ; % nm stdev of EB1 position fluctuations ceff = 2*exp(-(8/10)^2/2)/(2*pi)^(3/2)/sigma^3; % concentration in nm^-3 -based on 3-D normal distribution on half-sphere Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm-3 x (1 mol/ 6.022e23) x (1e27 nm^3/m^3) x (1 m^3/1000 L) x (10^6 uM/M) K = 37; % equals kplus_side/kminus_side, valued required for Kdeff = 0.44 %%%% Guessed parameters Kd1 = .65; % Kd, Dissociation constant for EB1 subunit and T, Kd1=k1m/k1 (uM) Value for typical monovalent protein %Kd1=Kd1vec(irun); k1 = 10; % On-rate for EB1 subunit and T (uM^-1*s^-1), Value for typical protein-protein binding k1m = k1*Kd1; % Off-rate for EB1 subunit and T (s^-1) %%% Equlibria E = EB1/((1+(T/Kd1))^2); % [EB1], Concentration of EB1 dimer in sol'n TE = 2*E*T/Kd1; % [EB1-T], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = T*TE/(2*Kd1); % [EB1-T^2], Concentration of EB1 dimer bound to 2 tubulin protomers a=4*K^2*E/Ceff; b=(1+T/Kd1)*2*K*E/Ceff+1; u_eq = (-b+sqrt(b^2+4*a))/2/a; p_eq=2*K*E/Ceff*u_eq; q_eq = 2*K*p_eq*u_eq; pi_eq = T/Kd1*p_eq; fl_eq=p_eq+pi_eq+q_eq/2; %%% Equilibrium flourescence conce check = u_eq+p_eq+q_eq+pi_eq; %%% should equal one %% D etermine kminus_side, kplus_side, kon kminus_side = kobs*(1+K*u_eq/(1+T/Kd1)); %% Based on FRAP half-life kplus_side = K*kminus_side; %% by definition kon_side = kplus_side/Ceff; f = 1 ; % affinity modulation factor %f=fvec(irun); %% Mixed model -chose kon, calculated koff from f kon = 5; % uM-1s-1 fon = kon/kon_side; %%% accelerated on-rate at end kplus = kon*Ceff; foff = fon/f; kminus=fon/f*kminus_side; %%% corresponding change off-rate at end %%% Other parameters kfE = kf; % on-rate constant of TE and TTE to MT end uM-1s-1; %kfE = 1e-8; krE = kfE*Tc/Kd1*kminus/kon; % off-rate constant of TE or TTE % Initial conditions S=ones(1,N); %S = 1 if unocupplied; 2 if bound to E, 3 if bound to TEE, 4 if bound to +side of doubly bound, 5 if bound to -side chartime=1/max([kfE*TTE kfE*TE kf*T kr krE kon*T k1*T k1 kplus kplus_side kminus kminus_side]); %Characteristic time dt = chartime/10; %simulation time increment

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159 nt=round(tim/dt); rnside1=rand(nt,N); radd = rand(nt,1); roff=rand(nt,1); previt=0; FLav=0*S; nadd=1; konv=[kon kon_side*ones(1,N-1)]; kminusv=[kminus kminus_side*ones(1,N-1)]; kplusv=[kplus kplus_side*ones(1,N-1)]; k1v=k1*ones(1,N); k1mv=k1m*ones(1,N); for it=1:nt %% Side binding tst1=(S==1)& rnside1(it,:)<2*konv*E*dt; % binds E tst2= (S==1)& rnside1(it,:)<(konv*TE*dt+2*konv*E*dt) &~tst1; % or binds TE tst3=(S==2)& rnside1(it,:)0 Sold=S; % Store old ntav=it-previt; % number of additional steps in average FL=(S==2)+(S==3)+.5*(S==5)+.5*(S==4); % EB1 fluorsescence FLav=(previt*FLav+ntav*FL)/(ntav+previt); % Update time-averaged fluorescence previt=it; % update ifnd1=find(tst1); S(ifnd1)=2; ifnd2=find(tst2); S(ifnd2)=3; ifnd3=find(tst3); S(ifnd3)=1; ifnd4=find(tst4); S(ifnd4)=5; S(ifnd4-1)=4; ifnd5=find(tst5); S(ifnd5)=4; S(ifnd5+1)=5; ifnd6=find(tst6); S(ifnd6)=3; ifnd7=find(tst7); S(ifnd7)=2; ifnd8=find(tst8); S(ifnd8)=1; S(ifnd8+1)=2; ifnd9=find(tst9); S(ifnd9)=1; S(ifnd9-1)=2; ifnd10=find(tst10); S(ifnd10)=1; S=S(1:N); FLav; plot(1:N,FLav,[1 N],[fl_eq fl_eq], 'r' ); axis([0 N 0 axmax]); drawnow; Veloc = nadd/it/dt*d; end % Tubulin addition ta1 = radd(it)
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160 FL=(S==2)+(S==3)+.5*(S==5)+.5*(S==4); FLav=(previt*FLav+ntav*FL)/(ntav+previt); previt=it; nadd=nadd+1; S(2:N)=S(1:N-1); if ta1 S(1)=1; elseif ta2 S(1)=2; elseif ta3; S(1)=3; elseif ta4 S(1)=4; S(2)=5; end end % Tubulin removal ta5 = roff(it)<(S(1)==1)*kr*dt; ta6 = roff(it)<(S(1)==2)*krE*dt; ta7 = roff(it)<(S(1)==3)*krE*dt; ta8 = roff(it)<(S(1)==4)*krE*dt; if ta5|ta6|ta7|ta8 Sold=S; ntav=it-previt; FL=(S==2)+(S==3)+.5*(S==5)+.5*(S==4); FLav=(previt*FLav+ntav*FL)/(ntav+previt); previt=it; nadd=nadd-1; S(1:N-1)=S(2:N); if S(N-1)==4 S(N)=5; else S(N) = S(N-1); end if ta8 S(1)=3; end end if S(1)==5 [ta1 ta2 ta3 ta4 ta5 ta6 ta7 ta8]; otst; pause end it/nt end % Velocity(irun)=Veloc; B.3.4 Tethered Protofilament Growth with Divalent EB1 This stochastic model simula tes the growth of a single mi crotubule protofilament in the presence of divalent, EB1 end-tracking motors and an applied force. The value of the affinity modulation factor, applied force, and KT can be varied to determine the resulting velocity. This model also provides the state of the terminal subunit, position of the tracking unit, the time average fluorescence along the protofilament, and time spent in each pathway. The kinetic parameters were estimated or used from literature values. % Stochastic model % Simulates tethered MT's in presence of EB1 % Divalent EB1 % Inputs: f, Kd1, kon

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161 % Outputs: Velocity, State of Subunit, Position of Track % Plots: Time Avg Fluorescence vs Subunit %clear all; tic rnside1=0; radd = 0; roff=0; rndT=0; tim=1; % run time (s) axmax=1; % max y-axis. % Determine Parameters % Fixed parameters T = 10 ; % uM tubulin dimer concentration V = 170; % nm/s; % elongation speed d = 8; % nm; % subunit length kf = V/d/T; % uM^-1s^-1 on-rate for tubulin -taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plus-end critical concentration kr = kf*Tc; % s^-1 off-rate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % s^-1 decay rate constant of EB1 from MT sides EB1 = 0.27; % uM EB1 concentration sigma = 10 ; % nm stdev of EB1 position fluctuations ceff = 2*exp(-(8/10)^2/2)/(2*pi)^(3/2)/sigma^3; % concentration in nm^-3 -based on 3-D normal distribution on half-sphere Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm-3 x (1 mol/ 6.022e23) x (1e27 nm^3/m^3) x (1 m^3/1000 L) x (10^6 uM/M) K = 37; % equals kplus_side/kminus_side, valued required for Kdeff = 0.44 Kd1 = .65; % Kd, Dissociation constant for EB1 subunit and T, Kd1=k1m/k1 (uM) Value for typical monovalent protein %Kd1=Kd1vec(irun); k1 = 10; % On-rate for EB1 subunit and T (uM^-1*s^-1), Value for typical protein-protein binding k1m = k1*Kd1; % Off-rate for EB1 subunit and T (s^-1) %%% Equlibria E = EB1/((1+(T/Kd1))^2); % [EB1], Concentration of EB1 dimer in sol'n TE = 2*E*T/Kd1; % [EB1-T], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = T*TE/(2*Kd1); % [EB1-T^2], Concentration of EB1 dimer bound to 2 tubulin protomers a=4*K^2*E/Ceff; b=(1+T/Kd1)*2*K*E/Ceff+1; u_eq = (-b+sqrt(b^2+4*a))/2/a; p_eq=2*K*E/Ceff*u_eq; q_eq = 2*K*p_eq*u_eq; pi_eq = T/Kd1*p_eq; fl_eq=p_eq+pi_eq+q_eq/2; %%% Equilibrium flourescence conc check = u_eq+p_eq+q_eq+pi_eq; %%% should equal one %% Determine kminus_side, kplus_side, kon kminus_side = kobs*(1+K*u_eq/(1+T/Kd1)); %% Based on FRAP half-life kplus_side = K*kminus_side; %% by definition kon_side = kplus_side/Ceff; %f = 1; % affinity modulation factor f=fvec(irun); %% Mixed model -chose kon, calculated koff from f kon = 5; % uM-1s-1 fon = kon/kon_side; %%% accelerated on-rate at end kplus = kon*Ceff; foff = fon/f; kminus=fon/f*kminus_side; %%% corresponding change off-rate at end

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162 %%% Other parameters kfE = kf; % on-rate constant of TE and TTE to MT end uM-1s-1; krE = kfE*Tc/Kd1*kminus/kon; % off-rate constant of TE or TTE % Initial conditions S=ones(1,N); %S = 1 if unocupplied; 2 if bound to E, 3 if bound to TEE, 4 if bound to +side of doubly bound, 5 if bound to -side previt=0; FLav=0*S; nadd=1; konv=[kon kon_side*ones(1,N-1)]; kminusv=[kminus kminus_side*ones(1,N-1)]; kplusv=[kplus kplus_side*ones(1,N-1)]; k1v=k1*ones(1,N); k1mv=k1m*ones(1,N); %KT =10;% 5; % eq. dissoc const. for tracker binding to EB1 KT=KTvec(irun); kfT = 5; % krT = KT*kfT; %%%%% Tracking unit parameters sigT = 10; % nm; tracking unit stdev kT=4.1; %pN-nm d=8; %nm spacing gamT = kT/sigT^2; % pN/nm Tracking unit stiffness CpT0 = 100; %% uM effective concentration of Tracking unit end MT end delta = 1; % nm -transition state distance %q=0; q=qvec(irun); Force = q*log(T/Tc)*kT/d; %pN Ffac = exp(-Force*d/kT); konTv = konv.*CpT0.*exp(-gamT*((1:N)-1).^2*d^2/2/kT+gamT*((1:N)-1)*delta*d/kT); % Effect of stretching on Tracker-bound Eb1 binding kfEp = kfE*CpT0; % forward rate for transfer of tubulin from tracking unit (based on detailed balance) kfTv = kfT.*CpT0.*exp(-gamT*((1:N)-1).^2*d^2/2/kT+gamT*((1:N)-1)*delta*d/kT); % Effect of stretching on Tracker binding to MT-bound EB1 Track = 0; Trackdist=0*(1:N); chartime=1/max([kfEp sum(konTv) sum(kfTv) kfT*TTE kfT*TE kfT*E kfE*TTE kfE*TE kf*T kr krE kon*T k1*T k1 kplus kplus_side kminus kminus_side]); %Characteristic time dt = chartime/20; %simulation time increment nt=round(tim/dt); rnside1=rand(nt,N); radd = rand(nt,1); roff=rand(nt,1); rndT=rand(nt,1); for it=1:nt; %% Start time loop %% Side binding tst1=(S==1)& rnside1(it,:)<2*konv*E*dt; % binds E tst2= (S==1)& rnside1(it,:)<(konv*TE*dt+2*konv*E*dt) &~tst1; % or binds TE tst3=(S==2)& rnside1(it,:)
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163 if sum(otst)>0 Sold=S; % Store old ntav=it-previt; % number of additional steps in average FL=(abs(S)==2)+(abs(S)==3)+.5*(abs(S)==5)+.5*(abs(S)==4); % EB1 fluorsescence FLav=(previt*FLav+ntav*FL)/(ntav+previt); % Update time-averaged fluorescence previt=it; % update ifnd1=find(tst1); S(ifnd1)=2; ifnd2=find(tst2); S(ifnd2)=3; ifnd3=find(tst3); S(ifnd3)=1; ifnd4=find(tst4); S(ifnd4)=5; S(ifnd4-1)=4; ifnd5=find(tst5); S(ifnd5)=4; S(ifnd5+1)=5; ifnd6=find(tst6); S(ifnd6)=3; ifnd7=find(tst7); S(ifnd7)=2; ifnd8=find(tst8); S(ifnd8)=1; S(ifnd8+1)=2; S=S(1:N); ifnd9=find(tst9); S(ifnd9)=1; S(ifnd9-1)=2; ifnd10=find(tst10); S(ifnd10)=1; if ifnd3>0 & ifnd3(1)==1 | ifnd8>0 & ifnd8(1)==1| ifnd10>0 & ifnd10(1)==1; S1(1,irun)=S1(1,irun)+1; elseif ifnd1>0 & ifnd1(1)==1| ifnd7>0 & ifnd7(1)==1| ifnd9>0 & ifnd9(1)==2; S1(2,irun)=S1(2,irun)+1; elseif ifnd2>0 & ifnd2(1)==1| ifnd6>0 & ifnd6(1)==1; S1(3,irun)=S1(3,irun)+1; elseif ifnd4>0 & ifnd4(1)==2| ifnd5>0 & ifnd5(1)==1; S1(4,irun)=S1(4,irun)+1; end end if Track<1 %%% start unbound tracker loop % Tubulin addition ta1 = radd(it)
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164 ta8 = roff(it)<(S(1)==4)*krE*dt; if ta5|ta6|ta7|ta8 Sold=S; ntav=it-previt; FL=(abs(S)==2)+(abs(S)==3)+.5*(abs(S)==5)+.5*(abs(S)==4); FLav=(previt*FLav+ntav*FL)/(ntav+previt); previt=it; nadd=nadd-1; S(1:N-1)=S(2:N); if S(N-1)==4 S(N)=5; else S(N) = S(N-1); end if ta5 count(5,irun)=count(5,irun)+1; elseif ta6 count(6,irun)=count(6,irun)+1; elseif ta7 count(7,irun)=count(7,irun)+1; elseif ta8 S(1)=3; count(8,irun)=count(8,irun)+1; end if S(1)==1; S1(1,irun)=S1(1,irun)+1; elseif S(1)==2; S1(2,irun)=S1(2,irun)+1; elseif S(1)==3; S1(3,irun)=S1(3,irun)+1; elseif S(1)==4; S1(4,irun)=S1(4,irun)+1; end end %%% Track =0 Tracking unit unbound tTa = (Track==0)& rndT(it)
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165 Track = -3; elseif tTd %% find which subunit bound ifnd=find(rndT(it)<(totA+kfT*TTE*dt+kfT*TE*dt+kfT*E*dt)); Track=min(ifnd); S(Track)=-2; if Track==1; S1(6,irun)=S1(6,irun)+1; end elseif tTe %% find which subunit bound ifnd=find(rndT(it)<(totA2+totA(N)+kfT*TTE*dt+kfT*TE*dt+kfT*E*dt)); Track=min(ifnd); S(Track)=-3; if Track==1; S1(7,irun)=S1(7,irun)+1; end elseif tTf %% find which subunit bound ifnd=find(rndT(it)<(totA3+totA2(N)+totA(N)+kfT*TTE*dt+kfT*TE*dt+kfT*E*dt)); Track=min(ifnd); if Track>1 S(Track)=-5; S(Track-1) = -4; end if Track-1==1; S1(8,irun)=S1(8,irun)+1; end elseif tT1 Track = -2; elseif tT2 %% find which subunit bound ifnd=find(rndT(it)<(totB+2*k1*T*dt)); Track=min(ifnd); S(Track) = -2; if Track==1; S1(6,irun)=S1(6,irun)+1; end elseif tT3 Track = 0; elseif tT4 Track = -3; elseif tT5 Track = -1; elseif tT6 % find which subunit bound ifnd=find(rndT(it)<(totC+k1m*dt+k1*T*dt)); Track=min(ifnd); S(Track) = -3; if Track==1; S1(7,irun)=S1(7,irun)+1; end elseif tT7 Track =0; elseif tT8 Track =1; S(2:N)=S(1:N-1); S(1)=-2; nadd=nadd+1; count(9,irun)=count(9,irun)+1; S1(6,irun)=S1(6,irun)+1; elseif tT9 Track = -2; elseif tT10 Track = 0; elseif tT11 Track =1; S(2:N)=S(1:N-1); S(1)=-3; nadd=nadd+1; count(10,irun)=count(10,irun)+1; S1(7,irun)=S1(7,irun)+1; end if Track==-3; T1(1,irun)=T1(1,irun)+1; elseif Track==-2; T1(2,irun)=T1(2,irun)+1; elseif Track==-1; T1(3,irun)=T1(3,irun)+1; elseif Track==0;

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166 T1(4,irun)=T1(4,irun)+1; else T1(Track,irun)=T1(Track,irun)+1; end elseif Track>0 Trackdist(Track)=Trackdist(Track)+1; ffac = exp(-gamT*(Track-.5)*d^2/kT); % rate factor due to stretching tracking unit upon addition % Tubulin addition ta1 = radd(it)
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167 Track = -3; end else Track=Track-1; end S(1:N-1)=S(2:N); if S(N-1)==4 S(N)=5; elseif S(N-1)==-4 S(N)=-5; else S(N) = S(N-1); end if ta5 count(15,irun)=count(15,irun)+1; elseif ta6 count(16,irun)=count(16,irun)+1; elseif ta7 count(17,irun)=count(17,irun)+1; elseif ta8 if S(1)==-5; S(1)=-3; else S(1)=3; end count(18,irun)=count(18,irun)+1; end if S(1)==1; S1(1,irun)=S1(1,irun)+1; elseif S(1)==2; S1(2,irun)=S1(2,irun)+1; elseif S(1)==3; S1(3,irun)=S1(3,irun)+1; elseif S(1)==4; S1(4,irun)=S1(4,irun)+1; elseif S(1)==-1; S1(5,irun)=S1(5,irun)+1; elseif S(1)==-2; S1(6,irun)=S1(6,irun)+1; elseif S(1)==-3; S1(7,irun)=S1(7,irun)+1; elseif S(1)==-4; S1(8,irun)=S1(8,irun)+1; end if Track==-3; T1(1,irun)=T1(1,irun)+1; elseif Track==-2; T1(2,irun)=T1(2,irun)+1; elseif Track==-1; T1(3,irun)=T1(3,irun)+1; elseif Track==0; T1(4,irun)=T1(4,irun)+1; else T1(Track,irun)=T1(Track,irun)+1; end end if Track>0 %%% Tracking unit-E detachment kminusv =[kminus kminus_side.*ones(1,N-1)]; ffac2 = exp(gamT*(Track-1)*d*delta/kT); ffac3 = exp(-gamT*(Track-.5)*d^2/kT); %%% Doubly bound EB1 tst1 = (S(Track) == -5)& rndT(it)1))*dt+krT*ffac2*dt+kminusv(Track)*ffac2*dt)) &~ (tst1|tst1a); % detachment of plus-side EB1 head (doubly bound) %%% bound T-E tst2 = (S(Track) == -3)& rndT(it)
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168 tst2a = ((S(Track) == -3)& rndT(it)<(krT*ffac2*dt+kminusv(Track)*ffac2*dt)) &~ tst2; % detach Track from TE tst2b = ((S(Track) == -3)& rndT(it)<(k1m*dt+krT*ffac2*dt+kminusv(Track)*ffac2*dt)) &~ (tst2|tst2a); % dissociate T %%% bound E tst3 = (S(Track) == -2)& rndT(it)1 tst3d = ((S(Track-1)==1 & S(Track) == -2)& rndT(it)<(kplusv(Track1)*dt+kplus_side*ffac3*ffac2*dt+k1*TE*dt+krT*ffac2*dt+kminusv(Track)*ffac2*dt)) &~ (tst3|tst3a|tst3b|tst3c); % bind second head in plus-direction else tst3d=0; end if tst1 Track=Track-1; S(Track) = -2; S(Track+1) = 1; if Track==1; S1(6,irun)=S1(6,irun)+1; end elseif tst1a Track_old=Track; S(Track) = 5; S(Track-1)=4; Track = 0; if Track_old-1==1; S1(4,irun)=S1(4,irun)+1; end elseif tst1b S(Track) = -2; S(Track-1) = 1; if Track-1==1; S1(1,irun)=S1(1,irun)+1; end elseif tst2 Track_old=Track; S(Track) = 1; Track = -2; if Track_old==1; S1(1,irun)=S1(1,irun)+1; end elseif tst2a Track_old=Track; S(Track) = 3; Track = 0; if Track_old==1; S1(3,irun)=S1(3,irun)+1; end e lseif tst2b S(Track) = -2; if Track==1; S1(6,irun)=S1(6,irun)+1; end elseif tst3 Track_old=Track; S(Track) = 1; Track = -1; if Track_old==1; S1(1,irun)=S1(1,irun)+1; end elseif tst3a Track_old=Track; S(Track) = 2; Track = 0; if Track_old==1; S1(2,irun)=S1(2,irun)+1; end elseif tst3b S(Track) = -3; if Track==1;

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169 S1(7,irun)=S1(7,irun)+1; end elseif tst3c Track_old=Track; S(Track+1) = -5; S(Track) = -4; Track=Track+1; if Track_old==1; S1(8,irun)=S1(8,irun)+1; end elseif tst3d Track_old=Track; S(Track) = -5; S(Track-1) = -4; if Track_old-1==1; S1(8,irun)=S1(8,irun)+1; end end if Track==-3; T1(1,irun)=T1(1,irun)+1; elseif Track==-2; T1(2,irun)=T1(2,irun)+1; elseif Track==-1; T1(3,irun)=T1(3,irun)+1; elseif Track==0; T1(4,irun)=T1(4,irun)+1; else T1(Track,irun)=T1(Track,irun)+1; end end % Ends "if Track>0" Loop end %Ends "if Track<1, elseif Track>0 Loop" irun percent=(nt*(irun-1)+it)/(nt*nrun) Veloc = nadd/it/dt*d; end % Ends "for it=1:nt" time loop kon_vect=kon*vector; Kd1v=Kd1*vector; timv=tim*vector; Nv=N*vector; F = (qvec*log(T/Tc)*kT/d); Velocity(irun)=Veloc; matrix=[kon_vect',Kd1v',KTvec',fvec',qvec',F',Velocity',timv',Nv']; xlswrite( 'sim_track_M.xls' matrix', 'matrix' ); % Save position & time data in Excel xlswrite( 'sim_track_M.xls' count, 'count' ); % Save position & time data in Excel xlswrite( 'sim_track_M.xls' S1, 'S' ); % Save The following is a macro that runs the stoc hastic model above at various values of f KT, and q % This macro runs track for multiple parameter sets % Inputs: f, KT, q % Outputs: Velocity, state of subunit, location of tracking unit clear all ; tic; fvec=[1000*ones(1,20)]; % f qvec=[0,0.25,0.5,1,2,3,4,5,6,7,0,0.25,0.5,1,2,3,4,5,6,7]; KTvec=[0.1*ones(1,10),10*ones(1,10)]; N=40; % number of subunits to simulate Velocity=0*fvec; nrun=length(fvec); vector=ones(1,nrun); count=zeros(18,nrun); S1=zeros(8,nrun); T1=zeros(N+4,nrun); for irun=1:nrun; track end Velocity(irun)=Veloc; kon_vect=kon*vector;

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170 Kd1v=Kd1*vector; timv=tim*vector; Nv=N*vector; F = (qvec*log(T/Tc)*kT/d); matrix=[kon_vect',Kd1v',KTvec',fvec',qvec',F',Velocity',timv',Nv']; xlswrite( 'sim_track_M.xls' matrix', 'matrix' ); % Save position & time data in Excel xlswrite( 'sim_track_M.xls' count, 'count' ); % Save position & time data in Excel xlswrite( 'sim_track_M.xls' S1, 'S' ); % Save state of subunit data in Excel S' Track Velocity time=toc/3600 B.4 Ciliary Plug Model This model simulates a 13-protofilament MT polymerizing in a ciliary plug against a motile surface with a constant load. The value of the applied force and protofilament length can be varied to determine the trajectory of the ci liary plug (position versus time) and the resulting velocity. The kinetic parameters were esti mated or used from literature values. % Simulates MT-based motility in ciliary plugs based on the LLF model % Trajectory between steps not simulated (fast version) clear all ; hold off ; % Filament Parameters kT=4.14; % Thermal energy (pN-nm) nf = 13; % No. filaments kappa = 0.15; % Filament compression stiffness (pN/nm) Df = 4e6; % Filament diffusivity (nm^2/s) deltaf = kT/Df; % Filament Drag (pN-s/nm) v=167; % Expected velocity (nm/s) d = 8 ; % Subunit length (nm) Tmin = d/v; % Mean time to load (s) Kappa2 = 60; % filament stretch stiffness (pN/nm) pn=1. ; % Positioning error (nm) %% Simulation setup z0f=rand(1,nf)*100; % random initial distributin of filament lengths z0f(1)=-kappa/kappa2*sum(z0f(2:nf)); % Set filament 1 position to balance forces dt=.005*Tmin; % Simulation time increment nt = 2^18; % total time steps zp = 0*(1:nt); t=0; z=0; % Initialize t=time; z= position of motile surface nplot = 2^4; dnplot=nplot; % Time steps between plotting ih=1; nh=nt/nplot; zh=zeros(1,nh); th=zh; zhn=zh; % Plotting storage vectors/variables nbp=10*round(Tmin/(dnplot*dt)) % Sets plotting range based on number of expecte steps runb=rand(dnplot,nf); % Random numbers for first set (between plotting) diffs=randn(1,dnplot)*sqrt(2*Df/nf*dt); % dW for first set jsim=1; % iteration index within set zeq=0; kf=zeros(1,nf); for i=1:nt unbind=(runb(jsim,:)<(kf*dt)); % Identify those that unbind if sum(unbind>=1) | i==1 z0f=z0f+unbind*d; % Shift those that rebind Equilibrium position sF=1; kappai=kappa*((zeq)<=z0f)+kappa2*((zeq)>z0f); % Vector of filament stiffness while sF^2>1e-10; zeq = kappai*z0f'/sum(kappai); kappai=kappa*((zeq)<=z0f)+kappa2*((zeq)>z0f); % Vector of filament stiffnesses F=-kappai.*(zeq-z0f); % Vector of forces sF=sum(F);

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171 end stiffness=sum(kappa*((zeq)<=z0f)+kappa2*((zeq)>z0f)); % total stiffness fvar=kT/stiffness; % Position variance Pr = d*F/kT; % Dimensionless force tau = (exp(Pr)-1-Pr)./Pr.^2; % Dimensionless Mean Time to Shift T = tau*d^2/Df; % Mean Time to Shift (s) kf =1./(Tmin+T) ; % Shift probability per unit time (s-1) end z=zeq+diffs(1,jsim)*sqrt(fvar); % Noisy position zp(i) = z; % Store position jsim=jsim+1; if i==nplot tp=(1:i)*dt; th(ih)=t; % Store Time zh(ih)=zp(i); % Store position zhn(ih)=zp(i)+pn*randn(1,1); % Noisy position tplot=th(max(ih-nbp,1):ih); zplot=zh(max(ih-nbp,1):ih); znplot=zhn(max(ih-nbp,1):ih); SUBPLOT(2,1,1), plot(tplot,znplot, 'r' ,tplot,zplot, 'b' ); % Plot recent trajectory tmin=th(max(ih-nbp,1)); tmax = max([th .1]); zmin=zh(max(ih-nbp,1))-3; zmax = max(zh)+5; axis([tmin tmax zmin zmax]); % Axes zrng=(zmin:5.4:zmax); nlin=length(zrng); tlin=[ones(nlin,1)*tmin ones(nlin,1)*tmax]; zlin=[zrng' zrng']; line(tlin', zlin'); SUBPLOT(2,1,2), hist(z0f-zeq,-5:10:max(z0f-zeq)+5); % Histogram of filament lengths drawnow; nplot=nplot+dnplot; % Update next iteration to plot ih=ih+1; % Update plot index runb=rand(dnplot,nf); % Generate random numbers for next set diffs=randn(1,dnplot); % jsim=1; % Reset set index end t=t+dt; % Update time end

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178 BIOGRAPHICAL SKETCH Luz Elena Caro was born and rais ed in Dela ware, and graduated from Middletown High School in Middletown, DE. She attended the Univer sity of Delaware and obtained her B.ChE. in Chemical Engineering. During her time at th e university, Luz Elena completed two summer internships at Merck & Co., Inc. After graduation, she interned at General Mills for a summer before joining the chemical engine ering department at the University of Florida for her graduate degree. Upn receiving her doctoral degree, Luz El ena will join the drug metabolism department at Merck & Co., Inc. in West Point, PA as a senior research pharmacokineticist.