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FORCE GENERATION BY MICROTUBULE ENDBINDING 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 EndTracking 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 EndTracking Motors ................. ...............29................ 1.4 Thermodynamic Driving Force ................. ...............30................ 1.5 Sum m ary .................. ...............3.. 1.............. 1.6 Outline of Dissertation ................ ...............31........... ... 2 MICROTUBULE ENDTRACKING MODEL ................ ...............37................ 2.1 EB 1 EndTracking 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 ENDTRACKING MODEL WITH MONOVALENT EBl..............__53 3.1 NonTethered Protofilament Growth ................ ........... ............... 53. .... 3.1.1 Thermodynamics of EB ltubulin interactions ................. ................ ...._.54 3.1.2 Kinetics of EB ltubulin 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 NonTethered ProtoHilaments ................. ...............64........... .... 3.3.2 Tethered ProtoHilaments ................. ...............65......_.._ .... 4 PROTOFILAMENT ENDTRACKING MODEL WITH DIVALENT EBl1................... .....73 4.1 NonTethered ProtoHilament Growth ................ ...............73................ 4.1.1 Kinetics of EB lTubulin Interactions ................ ...............73........... .. 4. 1.2 EB1 Occupational Probability Model .............. ...... ...............75 4.1.3 Average Fraction of EBlibound Subunits at Equilibrium ................... .............77 4. 1.4 Average Fraction of EBlibound 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 EBlibound subunits at equilibrium ................... .82 4.1.6.3 Average fraction of EBlibound subunits during protoHilament grow th ........._.._..... ......... ._ ...............82..... 4.2 Tethered ProtoHilament Growth Model ...._.._.._ ........._.._......_ ...........8 4.2.1 Kinetics of EB lTubulin Interactions ...._.._.._ ........... ......_.._.........8 4.2.2 ProtoHilament EndTracking Model .............. ...............87.... 4.2.3 Parameter Estimations ........._.._.... ...............87.._.._........ 4.2.4 Results ........._..._... ...............87.._.._.. ...... 4.3 Sum m ary .............. .. ............. ..........9 4.3.1 NonTethered 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 EndTracking Motors in Biology ..................._ ..............124 6.2 Microtubule EndTracking Model .............. ...............126.... 6.3 ProtoHilament EndTracking 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 NonTethered 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 13Protofilament Microtubule Model ................. ............. ............... 145 ... B.2 Protofilament Growth Model with Monovalent EB 1 .............. ......... .. ..............147 B.2.1 Occupational Probability of Monovalent EB 1 on a NonTethered 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 NonTethered Protofil1am ent ............... .... ............__.. ........_ __. .. .. .. ........5 B.3.2 Average Fraction of divalent EBlibound Protomers on Side of Protofil1am ent ............... .... ....__ ......__ ................5 B.3.3 Average Fraction of EBlibound 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 11 Thermodynamic equations characterizing the multiple steps for GDP to GTP conversion ........... ..... .._ ...............35... 12 Equilibrium constants used in energy equations ......___. .... ... .._ .. ......_.......3 41 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 412. .............109 LIST OF FIGURES Figure page 11 Microtubule structure................ ..............3 12 Chromosomal binding site of microtubules ......... ........ ................. ...............33 13 EB 1 binding to microtubule lattice. .............. ...............34.... 14 Concentration of EB 1 along length of microtubule. ...._._._._ .... ... .___ ........._.......34 21 Model for microtubule force generation by EB 1 endtracking motor. .............. .... ...........47 22 Reaction mechanisms of EB 1 endtracking motor. .......____ ..... ._ .............. ..48 23 Force dependence on EB 1 binding and equilibrium surface position. ............. .............49 24 Microtubule elongation in the absence of external force ................. ........................49 25 Distribution of protoHilament lengths for microtubule endtracking model .................. ....50 26 Effect of applied force on MT elongation rate............... ...............5...1 27 Thermodynamic versus simulated stall forces ................. ...............52............... 31 Schematic of nontethered, monovalent EB 1 endtracking motor mechanisms. ..............67 32 Various pathways of nontethered monovalent EB 1 binding to protoHilament. ................67 33 Choosing an optimal K~value for monovalent EBl1.....__.__. .... ... ._ ........._._......68 34 EB1 density profie on a nontethered microtubule protoHilament with monovalent E B 1. ............. ...............69..... 35 Effect of K; on profie of monovalent EB 1 occupational probability.. ............_ .............70 36 Schematic of tethered, monovalent EB 1 endtracking motor mechanisms.. ...................70 37 Force effects on a tethered protoHilament with monovalent EBl. ............. ...............71 38 Divalent EB 1 represented as divalent endtracking motor. ..........__.... ..._ ............72 41 Mechanisms of a nontethered, divalent endtracking motor.. ..........._._ ........._.. ......95 42 Mechanisms of equilibrium, side binding of EB 1 to protoHilament.. ................ ...............97 43 Choosing an optimal K~value for divalent EBl1.....__.....___ ..............__.......9 44 Effect of ko,, on optimalK; ........._._.._......_.. ...............98... 45 EB 1 equilibrium binding............... ...............99 46 Occupational probability of EB 1 along length of protoHilament.. ................ .................100 47 Time averaged EBlibound tubulin fraction at equilibrium ................. .....................101 48 Time averaged fraction of EBlibound subunits during protoHilament growth. ...............102 49 Mechanisms of tethered, protoHilament endtracking model with divalent EBl1.............103 410 Mechanisms of tubulin addition to linking proteinbound protoHilament. ................... ....105 411 Forcevelocity profies for tethered protofilaments bound to divalent EB 1 endtracking motors ................. ...............106........ ...... 412 Stall forces versus affinity modulation factor at various KT values ........._..... ..............109 413 Effect of f KT, and F on pathways taken ................. ...._ ...............110 414 Percent of time protoHilament bound and unbound to motile surface ................... ...........11 1 415 State of the terminal subunit (Sl) when f1 and f1000 ........._.__..... ..._._............1 12 416 Fraction of S1 subunits bound and unbound from motile surface .........._.... ..............113 417 Average state of unbound linking protein. ......___ ... ....._ ....___ ...........1 51 EM image of a ciliary plug at the end of a ciliary microtubule ................ ................. .120 52 Schematic of ciliary plug inserted into the lumen of a cilia/flagella microtubule ...........121 53 Mechanism of the ciliary/flagellar endtracking motor ........... ..... .___ ..............121 54 Force effects on ciliary microtubules............... .............12 55 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 filamentbound EB 1 CT: Effective concentration of tracking unit at protofilament plusend d:. Size of tubulin protomer dbE : State of tubulin protomer attached to the subunit on the minusside of a double bound EB 1 dimer dbE : State of tubulin protomer attached to the subunit on the plusside 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 plusend 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 trackbound protofilament (Kl' E k r/kf') K3: Equilibrium dissociation constant for EB 1 subunit binding to trackbound filamentTGDP (K3 k side'/k+side') K: Ratio of forward and reverse rate of EB 1 subunit binding to protofilament plusend (K k +/k) K': Equilibrium dissociation constant for EB 1 subunit binding to protofilament plusend (K' k /k+) Kd: Equilibrium dissociation constant for EB 1 (or TE) binding to filament bound TGDP (Kd kside/konside) Kd*: Equilibrium dissociation constant for EB 1 (or TE) in solution binding to TGTP at protofilament plusend Kp,: Equilibrium dissociation constant for reversible phosphate binding to T protomers KT: Equilibrium dissociation constant for track binding to solutionphase 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 filamentbound EB 1 binding to TGTP at protofilament plusend k side: Forward rate constant for subunit of filamentbound EB 1 binding to filamentbound TGDP k_: Reverse rate constant for EB 1 (or TE) in solution binding to TGTP at protofilament plusend k side: Reverse rate constant for EB 1 (or TE) binding to filamentbound TGDP ksT: Thermal energy (Boltzmann constant, k, x absolute temperature, T) k: Forward and reverse rate constants for tubulin in solution binding to protofilament plusend ke,o: Initial forward rate constant for tubulin in solution binding to protofilament plusend kfE: Forward rate constant for EBlibound tubulin in solution binding to TGTP at protofilament plusend kMT: Kinetochorebound 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 TGTP at protofilament plusend ko;2side: Forward rate constant for EB 1 (or TE) binding to filamentbound TGDP k,: Reverse rate constant for tubulin in solution binding to protofilament plusend k,E: Reverse rate constant for EBlibound tubulin in solution binding to TGTP at protofilament plusend kT: Forward rate constant for track binding to solutionphase EB 1 (or TE or TTE) kT : Reverse rate constant for track binding to solutionphase 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 EBlibound protomers on protofilament P,: Phosphate p: Probability of protomer bound to EB 1 subunit spend: Probability of EB 1 binding to the protofilament plusend peq: Equilibrium probability of protomer bound to EB 1 subunit pside: Equilibrium probability of EB 1 binding to filamentbound TGDP q : Probability of protomer in state dbE+ q : Probability of protomer in state dbE q: Probability of protomer attached to doublebound EB 1 subunit qeq: Equilibrium probability of protomer attached to doublebound EB 1 subunit Sl: State of terminal protomer in protofilament S2: State of penultimate protomer in protofilament TAC: TipAttachment Complex model Tb: Tubulin protomer [Tb]: Tubulin protomer concentration [Tb],: Critical tubulin concentration for free MT plusend [Tb]Ec: Critical TE concentration for free MT plusend TE: EBlbound tubulin protomer TGDP: GDPbound tubulin protomer TGDP concentration Critical tubulin concentration for TGDP at MT plusend GTPbound tubulin protomer Track bound to protofilamentbound EB 1 Track bound to protofilamentbound TE Track bound to protofilamentbound 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 filamentbound 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 protofilamentbound EB 1 subunit Transition state distance [TGDP] : [TGDP](),: TGTP : Tk2: Tk3: Tk4: Tk: TkE: TkTE: TkTTE: Tm: TTE: t: u: ueq: v: vr: w: weq: x: z: ze: A: AG: Net free energy change of the tubulin cycle AGo: AGrnloss : AG(+add: AGexchange : AGhydrolysis: AGPIrelease : 3: r: r: p : z Initial free energy change of the tubulin cycle Free energy of TGDP dissociation from MT minusend Free energy of TGTP addition to MT plusend Free energy of GDP/GTP exchange in solution Free energy of MTbound GTP hydrolysis Free energy of MTbound 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 ENDBINDING 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 microtubulebased 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 EndBinding 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 endtracking proteins is not fully understood, but they likely modulate microtubulemotile 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 forcedependent 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 endtracking motors that have a higher affinity for guanosine triphosphatebound tubulin than guanosine diphosphate bound tubulin, thereby allowing them to convert the chemical energy of microtubulefilament hydrolysis to mechanical work. These microtubule endtracking 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 solutionphase tubulin. These models explain the observed exponential profile of EB 1 on untethered filaments and predict that affinitymodulated endtracking 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). Endtracking proteins (a.k.a. tiptracking proteins), such as endbinding protein 1 (EBl) and adenomatous polyposis coli (APC), have previously been shown to bind specifically to the polymerizing microtubule plusend 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 endtracking proteins in force generation at these sites. A few models demonstrate how endbinding proteins may be involved in microtubule forcegeneration, suggesting that endtracking proteins bind weakly to the microtubule plusend 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 endbinding proteins and a motile surface. The obj ective of my thesis research was to help elucidate the role of microtubule elongation mediated by endbinding proteins in force generation. Our models explain and characterize the interaction of endbinding proteins with growing microtubule ends and their role in force generation Understanding the functions of microtubules and endtracking 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 endtracking protein APC not only plays a potentially key role in microtubulechromosome 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 8nm a/Ptubulin heterodimers; where the Psubunit can bind to either a guanosine triphosphate (GTP) or guanosine diphosphate (GDP) nucleotide (Farr et al., 1990). Tubulin bound to GTP assembles headtotail to form the 13 asymmetric, linear protofilaments of a microtubule (Figure 11) (Chretien et al., 1995; Chretien and Fuller, 2000). Each protofilament has the same polarity, with a Ptubulin at one end (minus or slowgrowing end), and an atubulin at the other (plus or fastgrowing 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), GTPtubulin protomers add to the plusend of a MT, the subunits then hydrolyze their bound GTP and subsequently release the hydrolyzed phosphate. During depolymerization (MT shortening), GDPtubulin subunits are released from the MT minusends 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 plusend during mitosis (Piehl and Cassimeris, 2003). Assuming irreversible elongation at the MT plusend in vivo, this value can be used to estimate the plusend protofilament effective onrate constant of tubulin ( f) association, rd [Tbh] (11) where d is the length of a protomer and [Tb] the intracellular GTPtubulin 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 offrate (k,) of 10.6 s^l, assuming [Tb]e = ky / k, (12) This calculated offrate is used to determine the reversible, elongation rate of the microtubule plusend (85 nm/s) by v, = (kf [Tbj kr)d. (13) 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 plusends without the involvement of endtracking 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 plusend 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 12) (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 plusend 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 GTPrich MT plusends (Severin et al., 1997). A complex of proteins is required for kinetochore capture by kMTs, but their interaction have not been elucidated (MimoriKiyosue and Tsukita, 2003). If one of the kinetochoreassociated proteins could recognize and track the GTPrich 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 kinetochorekMT attachment during mitosis have also been shown to bind to the plusends of MTs (endtracking proteins), suggesting their likely role in such a mechanism. Of particular interest here are the plusend 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 EndTracking Proteins Several MT endtracking proteins are thought to facilitate force generation by microtubule polymerization (Schuyler and Pellman, 2001). Endtracking proteins localize to the MT plusend, 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 endtracking 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, 30kDa leucine zipper protein (MimoriKiyosue 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 GTPrich tubulin protomers (polymerizing unit) at the polymerizing plusends of microtubules. EB 1 quickly disappears from the plus ends of depolymerizing GDPrich MT' s, indicating that the higher EB 1 density at plus ends requires polymerization and/or a GTPrich 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) (MimoriKiyosue and Tsukita, 2003). This hypothesis is supported by demonstrating that EBlinull Drosophila cells cause mitotic defects including mispositioning of kinetochores during congression (Rogers et al., 2002). Specific localization of EB 1 to GTPrich MT plusends is not understood, but may be the result of direct binding of EB 1 to the GTPstabilized conformation of the MT plus end, copolymerization with tubulin in solution, or recruitment by other proteins, and dissociation from GTPbound MT subunits (Figure 13). 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 14. 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 (MimoriKiyosue 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 TGTP did not bind to an EB 1 construct. Contrary to his finding, Juwana et al. (1999) demonstrated that recombinant EB 1 coprecipitates with purified bovine tubulin an immunoaffinity assay, despite the 100time lower concentration of EB1 than TGTP (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 tubulinaffinity 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 plusends 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 colocalize and interact with both EB 1 and polymerizing microtubule plusends at the kinetochore and at the cell cortex (Juwana et al., 1999). Like EB 1, APC falls off the microtubule upon plusend depolymerization. The Cterminal domain of APC (CAPC) 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 CAPC, 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 microtubulebinding domain nor the EB 1 binding domain of APC can be compromised to obtain proper chromosomal segregation. In the absence of the EBlibinding domain, APC localizes nonspecifically 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 kinetochoreassociated proteins (pl50Glued, CLIP170, and CLASPs) also have direct interactions with EB 1, have the ability to bind to the MT plusend, and are located at the kinetochoreMT interface (Folker et al., 2005; Hayashi et al., 2005; MimoriKiyosue et al., 2005). Therefore, these components may also be involved in activation of EB1 at the MT tip and/or linking the EBlibound MT plusend 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, membranebound capping structures (or MT "plugs") are persistently associated with the plusends of polymerizing 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 microtubule, and (b) platelike 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 crossreactivity (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 colocalizes with the plusends 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 endtracking properties of EB 1 are likely to play a role in MT elongationdependent force generation. 1.3 Force Generation Models Although much progress was made identifying microtubuleassociated proteins and their locations, how MT elongation is coupled to force generation has not been determined. Various forcegenerating models have been considered, including force from microtubule polymerization, force from motorprotein 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), (14) where N, = 13 is the number of protofilaments, kBT = 4.14 pNnm is the thermal energy (Boltzmann constant x absolute temperature), and d= 8 nm is longitudinal dimerrepeat distance. Under typical intracellular tubulin (Tb) concentrations of 1015 CIM and a plusend critical concentration [Tb], = 5 CIM (Walker et al., 1988), then Fsran = ~57 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 antipole 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 randomwalk approach, where the free energy source is the binding of GTPtubulin protomers to MT ends (Hill, 1985). The Tip Attachment Complex model (TAC) incorporates the idea of a "sleeve" in order to model forcegeneration 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 weakbinding 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 endtracking 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 motorinduced forcegeneration model. One such model is known as the "reverse PacMan" mechanism (Maddox et al., 2003), where plusend directed motors move kinetochores antipole ward during plusend kMT polymerization (Inoue and Salmon, 1995). The plusend directed motor protein, CENPE, 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 antipoleward kinetochore motility in the cell; there are other kinetochoreassociated motor proteins (i.e., MCAK) of unknown function. One recent forcegeneration model, the "slipclutch" model, integrates both the reverse PacMan and lateral TAC mechanisms. This model represents the polymerization state of the kinetochore by a "slipclutch" 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 EndTracking Motors Dickinson & Purich (2002) first proposed a model for actinbased motility whereby end tracking proteins tethered elongating filaments to motile obj ects and facilitated force generation. In this mechanochemical model for actin subunit addition, surfacebound endtracking proteins bind preferentially to newly added ATPbound terminal subunits on each subfilament and release from ADPbound 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 hydrolysisdriven processive tracking on the filament end gives the endbinding protein the characteristics of a molecular motor. We later proposed that the interaction of microtubule endtracking 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 GTPdriven 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 15). 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, (16) where AGo 11k,T is the standardstate free energy change for tubulin in vivo (Howard, 2001). The free energy changes for the individual assembly steps are listed in Table 11, where Kp, is the equilibrium dissociation constant of reversible phosphate binding to GDPtubulin protomers and Kx is the equilibrium constant for the GTP/GDP exchange reaction. Based on literature values (Table 12), the free energy from the combined filamentbound hydrolysis and phosphaterelease 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 plusends (~5.8 kBT). Hence, considerably greater forces can be expected by exploiting the ability of endtrackers like EB 1 that bind preferentially to TGTP protomers, thereby providing a pathway for harnessing the energy released by MTbound 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 endtracking proteins, EB 1 and APC. EB 1 is known to specifically localize to the GTPrich 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 forcegeneration mechanisms involving endbinding proteins and MTs have been proposed including TAC and models involving ATPdriven MT motors kinesis and dynein, which move on MT sides. This thesis explores the hypothesis that endtracking motor facilitate plusend 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 endtracking proteins binding specifically to the NTPbound monomers on the filaments, a feature correlates well with the properties of the MTs and their corresponding endtracking proteins. 1.6 Outline of Dissertation The layout of this dissertation is as follows. Chapter 2 describes a preliminary mechanochemical MT endtracking 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 endtracking 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. Endtracking models based on interactions of monovalent or divalent solutionphase 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 endtracking 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 endtracking from a rigid "plug," reflecting ciliary /flagellar growth. Finally, Chapter 6 summarizes the work completed and suggests future directions. GTPbound tubulin (~TGTP) GDPbound tubulin (TlG DP) } Protofilament S() end GTP GDP (+) end Figure 11. Microtubule structure. Tubulin bound to GTP polymerize into 13protofilament polymers: microtubules. Because tubulin is a heterodimer, the microtubule has a structural polarity with a plus and minus end. During MT polymerization, TGTP binds to the MT plusend, which induces hydrolysis of the penultimate tubulin subunit causing filamentbound GTP to be converted to GDP. TGDP dissociates from the minus end. 8 Figure 12. 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 2328, p. 1094). New York, New York.] ks' 1 k+ i' EB1 OR Figure 13. EB 1 binding to microtubule lattice. EB 1 has equal association and dissociation rates on GDPbound 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 lmicrotubule 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 TGTP SO TGDP Reaction Coordinate Figure 15. 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 GTPGDP exchange in solution. Table 11. Thermodynamic equations characterizing the multiple steps for GDP to GTP conversion lG  22 k,"T L, Hydrolysis + Pi Release III~ GTPDepolymerization ~OO 90 TGDP \yExchange Definition Equation AG(+)add = k,Tln([Tbj/[Tbje) AGP,reles = kg Tln([ ]/K, Acc3;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 TGTP to MT plusend Phosphate (Pi) release Loss of TGDP from MT minusend GDP/GTP Exchange Hydrolysis of MTbound GTP in terms of free energy of other steps Polymerization Table 12. 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 TGTP addition to MT plusend 0.03 C1M (Howard, 2001) [TGDP]()c TGDP addition to MT minusend 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 ENDTRACKING MODEL This chapter describes a preliminary model that simulates the growth of a 13protofilament microtubule (MT) in the presence of surfacetethered EB 1 endtracking motors. While this model does not account for EB 1 binding from solution, 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 EB 1 endtracking motor is that it captures filamentbound 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 plusend 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 Gaussianbased 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 EndTracking Motors The preferred binding of EB 1 to MT plusends is reminiscent of the interaction between endtracking proteins and actin in the actoclampin endtracking motor model and suggests that endbinding proteins may behave as endtracking motors. To explore this possibility, we model EB1 as a protein tethered to a motile surface on one end and interacting with the MT plusend through its MTbinding domain on the other end. There are two key features of a MT endtracking motor: affinitymodulated 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 endtracking motors to maintain a strong interaction with the protomiament even when other endtracking 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 MTbinding protein. 2.2 Microtubule Growth Model Our preliminary MT endtracking model illustrated in Figure 21 simulates the growth of a 13protoHilament microtubule bound to surfacetethered EB 1 motors and analyzes the force effects on the growth of the microtubule. By capturing part of the filamentbound 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 22 and include several possible endtracking "stepping motor" pathways for two EB 1 dimeric subunits (referred to hereafter as EB 1 "heads") operating at the plusend of each MT protofilaments. Considering Stage A, where only one EB 1 head is bound to terminal GTPbound 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 TGTP 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 TGTP 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" EBl1MT 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 endtracking cycle, the protofilament remains associated with the motile obj ect (i.e., the motor is processive). (Longterm 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 hydrolysisinduced affinity modulation, the principle of detailed balance would fix the relation among the various equilibrium dissociation constants in Reactions 13 shown in Figure 22, 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 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 F,,,, = (N k,T/d)1n([TbjKK/K, )=(Nk, T/d)(In(hl[Tbj/Tbj)+In~f)). (21) 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 GTPhydrolysisdriven 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 ~7pN stall force predicted for a Brownian Ratchet driven solely by the free energy of protomer addition (c.f., Eq. 14). While Eq. 21 provides a thermodynamic limit, MT growth by the endtracking 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 EBlibound 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 coiledcoil region and its two MTbinding 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 (22) 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) = (23) where ks is Boltzmann's constant, and Tis the absolute temperature. The positiondependent 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 (24) 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 TGDP subunit from solution. Because binding sites are at discrete positions spaced by distance d = 8 nm, then : = nd in Eq. 24. 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 ~100nm motile object, this time would be ~10100 Cls, much faster than the cycle time for protomer addition. Therefore, we assume the instantaneous position of the motile obj ect remains in mechanical quasiequilibrium 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 25; the position of bound EB 1 heads in each stage can be determined from Figure 23. 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 (25) Solving Equation 25 for ze thus allows p(z) and the resulting transition probabilities for transition between states (kAt) 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 24, 25, 26, and 27, 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 22. The intracellular tubulin concentration [Tb] was assumed to be 10 C1M (Mitchison and Kirschner, 1987). We use the value of the plusend critical concentration [Tb], = 5 C1M estimated by Walker et al. (1988) from the ratio of on and offrate constants for elongation (8.9 CIM s^ and 44 s^l, respectively). The macroscopic onrate 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 onrate 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 offrates 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 zposition 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 MTbound TGDP 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 offrate constant koyS= 0.24 s^l for an EB 1 dimer from MT GDPsubunits was calculated from the measured velocity and the exponential decaylength 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 offrate 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 zze = d/2 from Equation 24. The primary simulation parameter was the total simulation time, t, which was set at 4 seconds. For f1 to f10,000, 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 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 p (Appendix B.1 contains the MATLABe code), the resulting MT position increases linearly with time. The tubulin onrate was chosen to yield the experimentally determined velocity of 167 nm/s for microtubules during mitosis (Piehl and Cassimeris, 2003). Figure 25 is a representation of the protofilament lengths and average equilibrium surface position corresponding to (t = 4s, v=165 nm/s). As seen in Figure 25, the maximum difference between the shortest and longest filaments is four subunits. This small difference reflects how the endtracking model can also ensure high fidelity: the protofilaments do not advance too far past one another during polymerization. This diagram shows that the endtracking 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 26 shows that the speed of MT polymerization decreased with increasing external load for all values of f calculated. When the endtracking protein was not affinitymodulated (f= 1), the velocity decreased linearly with increasing external force. As fwas increased, the endtracking 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 GTPbound filament, and promoted the forward MT assembly process. Although large tensile forces should dissociate the filament endtracking 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. 25) 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 forcelimitations 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 11). 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 25, and are compared to these simulated stall forces in Figure 27. The simulated and calculated results are comparable when the EB 1 motor has little affinity modulation (from f1 to f10); for ~f1, 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 endtracking motors are not affinity modulated (at f1) 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 forcedependent exponential equation (Figure 25). 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 13protofilament MT bound to surfacetethered EB 1 motors and serves to demonstrate the principles of force generation by processive MT endtracking motors. The key features of these endtracking motors are (1) their ability to capture filamentbound GTP hydrolysis and convert them to mechanical work (2) their dimeric structure which allow them to maintain persistent attachment of the MT plusend 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 TGTP versus TGDP 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 solutionphase EBl1. The proposed cofactor assisted endtracking model not only addresses the importance of a cofactor 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 TGTP Newvly Hydrolyzed ~TGDP TGDP EB1 dimner Figure 21. Model for microtubule force generation by EB 1 endtracking motor. Model that represents the distal attachment of tubulin protomers at the MT plusend. A uniform density of EB 1 dimers on the motile obj ect links the MT protofilaments to the surface. 1~ k S Newly Hydrolyzed TGDP k, av Z' EB1 dimer k side' St C (4) rStag eD Figure 22. Reaction mechanisms of EB 1 endtracking motor. Mechanism of the EB 1 endtracking motor on the plusend of one of the MT protofilaments (from upper left): One EB 1 head is initially bound to the terminal GTPtubulin 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 23. 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 24. Microtubule elongation in the absence of external force. (A) The position of a 13protofilament MT tethered to a surface by EB 1 endtracking 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 n1 Stage D n1 n2 S85 Ecr 82 819 80 HProtomers Avg z 1 2 3 4 5 6 7 8 9 10 11 1213 Protofilament Number Figure 25. Distribution of protofilament lengths for microtubule endtracking 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 26. 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 threeparameter 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 27. Thermodynamic versus simulated stall forces. The thermodynamic stall force was calculated for the various MT endtracking motor represented in Figure 4 by using equation 25. Comparison of the calculated and simulated stall forces is shown. When the hydrolysis affects the microtubule dynamics very little (f1 to f5) the model provides a good prediction for the EB 1 endtracking 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 ENDTRACKING MODEL WITH MONOVALENT EB 1 The microtubule endtracking model developed in Chapter 2 neglected solutionphase End Binding protein 1 (EBl1) and binding to microtubules and tubulin protomers. While not essential for endtracking, binding of EB 1 from solution is evident in the exponential profiles of bound EB 1 at elongating free plusends and the apparent equilibrium density of EB 1 along the length of the microtubule (MT) (Chapter 1 and Figure 13). To account for binding solutionphase 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 endtracking motor, with preferential binding to TGTP over TGDP, 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 GTPhy droly si sdriven 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 forcevelocity relationship of this the endtracking model is then compared to those of the simple Brownian Ratchet mechanism. 3.1 NonTethered Protofilament Growth We first consider growth of a single microtubule protofilament in the presence of a solutionphase 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 endtracking motor similar to that described in Chapter 2. The various reactions considered in the free MT model are shown in Figure 31. 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 ltubulin 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 nonterminal subunits. 3.1.1 Thermodynamics of EB1tubulin interactions As described in Chapter 1, free energy of the direct binding pathway is given simply by [Tb] AG(+)add = k,Tln~ [Tb], (31) 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 plusend and facilitates the net reaction of monomer addition. The principle of detailed balance requires that Eq. 32 holds whether E binds first to Tb in solution or E binds to the terminal subunit following monomer addition. Eq. (32) can be rewritten [Tb] AG(+)add = k,Tln k, T n f (33) [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 GDPtubulin 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 (34) and [E]o [Tb] /K, [ TE ] ,[Tb] [MT] K, Kd (35) 3.1.2 Kinetics of EB1tubulin interactions We have assumed that the affinity modulation factor must be greater than 1 for EB 1 to track on the plusends 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 32: EB 1 binds directly to the GTPrich protofilament plusend, EB 1 associates with TGDP 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 plusend, pend, and the equilibrium probability of EB 1 binding protofilament sides, peq. The value of peq WAS obtained by the steadystate 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] 1p) dt (36) The equations specific for solving the probability of EB 1 binding to the plusend and side of the protofilament are given by = k [E~c k p, kTbjk,E TE] p,_z p,) + kiEP + ki.cjp X,~ p) (37) dp3 = k [E],c, kp, p,[Tbj+kiE TE]c, +[T ktczp, [TbjeKdk,Ep102, (38) dt K, f respectively, where c, 1p, (see full derivation in Appendix A.2). Here, the index i represents the subunit on the protofilament numbered from the plusend. These two differential equations were numerically integrated (fourthorder RungeKutta 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 plusend 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. 11 and Eq. 12, respectively, assuming a maximum velocity v=170 nm/s (Piehl and Cassimeris, 2003). The dissociation constant for EB 1 to the GDPbound 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 plusend 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 offrates for EB 1 on both the sides and plusend of the protofilament are a fraction,J; less than their onrates. This onrate 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 steadystate, Eq. 39) and spend (from Eq. 38). pe (39) 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 steadystate. This procedure was repeated for various values of f and the resulting K; values are shown in Figure 33. 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 TGTP than TGDP (i.e., fmust be greater than two) in order for EB 1 to accumulate at the plusends of protofilaments as seen experimentally. The optimal value of K; (for greater than two) increases with increasing values of f At larger values off(f5 and f10), 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 plusend 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 33. 3.1.4 Results Figure 34 shows the steadystate EB 1 binding density profies for f=1 and f =1000. When ~f1 the steady state occupational probability is uniform along the length of the protoHilament. The slightly higher EB 1 density at the plusend reflects some benefit of copolymerization with tubulin. However, EB 1 is predicted to have a much larger density at the plusend when fis large. The EB 1 density decreases exponentially along the length of the protoHilament, consistent with experimental observations (Figure 14). This finding supports our assertation EB 1 must have a significantly higher affinity for GTPbound tubulin in order to track on the GTPrich protofilament plusends. The effect of K; on the EB 1, monovalent occupational probability profile is demonstrated in Figure 35. The model was simulated at K; values from 0.01 CIM to 1 CIM at f1000. 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 plusend 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 f1 in Figure 34. This behavior is expected because at larger values of K;, EB 1 has a large offrate 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 nontethered model (Figure 36). 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 plusend. 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 EBlibinds 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 nontethered monovalent endtracking 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 TkE 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=1wv[q (310) Similar to the derivation of Equations 37 and 38, the transition probabilities between states can be obtained from reaction rate constants for each pathway (Figure 36). The resulting differential equations for the probabilities of EB 1 and TkE binding to the protofilament (in terms of the kinetic rates) are given by Equations 311 and 312, respectively, where u, =1 q, p,, kT is onrate of the linking protein binding to solutionphase 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 1p,) jk,.EP1+k,u, +k,'yEl 1 Pr+1 (311) 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 (312) The differential equations for the probability of the track binding to either TE (313) or EB 1 (314) 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 (313) = k,.[E]yk, vk,[Tb]v +k,~l w koCs,2vu + k,29, (314) These ordinary differential equations were solved using a fourthorder RungeKutta 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 steadystate net rate of the tubulin addition and dissociation pathways: V =d [Tb]e (315) +kfE TE~]+ CI l)eF Tb]cE end +end ; There are two ways tubulin can add to the protofilament plusend, directly with an onrate ofkf or copolymerizing with EB 1 with an onrate 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 (eFd) and the dependence of the forward rate on the critical concentration when the protofilament plusend 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 TkTE addition, and the dependence of the forward rate on the critical concentration [Tb]cE when the protofilament plusend is bound to EBli. The probabilities, qend and pend, WeTO SOlVed from Equations 311 and 312 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, nontethered case. The kinetic rate constants were calculated from detailed balance. The onrates for an EB 1 subunit (or head) to the protofilament side (konside) and to the protofilament plusend (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 (316) Rearranging this equation gives kosd = k o'(317) onon =[E] +Kd where Kd side/konside. The offrate constant for EB 1 from the GDPbound tubulin subunits, k side, iS calculated from Kd. The offrate of EB 1 from the protofilament plusend, k_, is the equal to the offrate of EB 1 from divided by a factor of f The linking protein was assumed to be a flexible, springlike 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 3D normal distribution on a halfsphere. The normal Gaussian distribution of the spring is given by Equation 318, 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 p2 in the EB 1 effective concentration calculated in Chapter 2. 3.2.4 Results Figure 37A shows the predicted density of EB 1 along the length of the protofilament (zero represents the plusend). The protein species considered are the nontethered 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 plusend 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 plusend, and EB 1 is expected to have preferential binding to the protofilament plusend 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 onrate of linking protein to EB 1 at this location is much greater than its dissociation rate. The nontethered EB 1 experiences a small peak in probability near the protofilament plusend, likely because it was initially GTPbound, and eventually dissociates from TGDP. The forcevelocity profile for these mechanisms is shown in Figure 37B, 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 endtracking 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 endtracking 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 38), 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 endtracking 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 additiondriven (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 endtracking motor, with preferential binding to TGTP over TGDP, 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 NonTethered 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 TGTP in solution, this model accounts for several reaction pathways to allow EB 1 to bind with tubulin in solution as well as filamentbound tubulin. The dissociation constant for EB 1 and free TGTP was taken as that need to provide a 4.2 ratio of EBlibound subunits at the protofilament plusend versus protofilament sides, which would correlate well with experimental results. Large affinity modulation factors resulted in an equilibrium value for the tubulinEB 1 dissociation constant, and are therefore optimal for simulation purposes. Regardless of the value for other key kinetic rates (i.e., onrate of tubulin, kf, or onrate of EB 1 on protofilament sides, k ), it is required for EB1 to have a larger affinity for TGTP rather TGDP (f >1) in order to achieve the 4.2 ratio. This result supports the assertation that EB 1 has an affinitymodulated interaction with tubulin, which is not accounted for in the Brownian ratchet mechanisms, but is the key characteristic of the endtracking 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 ltubulin dissociation constant was used to calculate the binding probability of EB 1 to the plusend and sides of a microtubule protofilament. The results of the model demonstrate that the mechanism with no affinity modulation results in a nearuniform EB 1 density along the entire length of the protofilament. However, large affinity modulation results in a greater EB 1 binding at the protofilament plusend that decays along the length of the protofilament, a prediction which agrees to experimental results showing the decay of fluorescent EB 1 on a nontethered microtubule. To simplify this complex model, several assumptions were made. First, tubulin addition induces filamentbound hydrolysis at the protofilament plusend. The affinity modulation is assumed to affect only the offrates of the protein interactions and not the onrates. 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 cofactor protein in endtracking 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 nontethered, monovalent endtracking 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 nontethered) demonstrates that at large affinity modulation factors there is a high occupation of total EB 1 at the protofilament plusend, 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 freegrowing protofilaments. More importantly was the effect of force on the endtracking model. This model demonstrates the potential of the monovalent endtracking 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 springlike 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 endtracking complex and the protofilament that would affect the occupational probability ofEB1 and the forcevelocity profile. The subsequent tethered protofilament growth model with divalent endtracking EB 1 motors will account for the force effects on the linking protein. Stabilization .K* Release Figure 31. Schematic of nontethered, monovalent EB 1 endtracking 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 ltubulin complex ([Tb]cE). GTP hydrolysis on the penultimate subunit occurs upon tubulin addition to the protofilamnent plusend. Kd i s the EB1I dissociation from the protofilament plusend and Kd* is the dissociation constant for EB 1 from TGDP, where f Kd/Kd  TE Figure 32. Various pathways of nontethered monovalent EB 1 binding to protofilament. EB 1 can bind directly to the GTPrich protofilament plusend, or EB 1 can associate with TGDP on the side of the protofilament, or EB 1 can copolymerize with tubulin in solution. *) TGTP O TGDP 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 TubulinEB1 Eq. Dissoc. Constant, K1 (prM) Figure 33. 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 tiptoside 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 34. EB 1 density profile on a nontethered 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 plusend, which decreases along the length of the protofilament. 0  0 100 200 300 Tubulin Subunit Figure 35. 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 f1000 is shown. When K; is 0.01 CIM, EB 1 has a high occupational probability at the plusend, 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 f1 in Figure 34. r TGTP O *T"GDP Figure 36. Schematic of tethered, monovalent EB 1 endtracking motor mechanisms. Tubulin protomers can add directly to filament ends, or they can first bind to surfacetethered EB 1 in solution then add as an EB ltubulin complex. GTP hydrolysis on the penultimate subunit occurs upon tubulin addition to the protofilament plusend. 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 + EndTracking 1 1,5 2 2.5 i 0a,5 Figure 37. Force effects on a tethered protofilament with monovalent EBli. A) Occupational probability of EB 1 along length of protoHilament (zero represents plusend) at f 1000. Two protein species considered: untethered EB 1 and tethered EB 1 on protoHilament. Occupation of EB 1 for each species decreases along length of protoHilament. B) ForceVelocity profie. Maximum achievable force for endtracking 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) MTbinding domains "/~ SGTP tubulin O GDP tubulin ~hEB1 Figure 38. Divalent EB 1 represented as divalent endtracking motor. A) Depiction of EB 1 structure characterized from crystal structures. The Cterminus is represented by (C) and the Nterminus is represented by (N). Reprinted by permission from Macmillan Publishers Ltd: [Nature] (Honnappa et al., 2005) copyright (2005). B) Schematic of endtracking motor complex comparable to crystal structure of EBli. CHAPTER 4 PROTOFILAMENT ENDTRACKING MODEL WITH DIVALENT EB 1 The protofilament endtracking 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 endtracking motors. Similar to Chapter 3, we first model the growth of an untethered protofilament in the presence of solutionphase EB 1 endtracking 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 tiptoside 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 forcedependent velocity was compared to that of the Brownian Ratchet mechanism. 4.1 NonTethered Protofilament Growth As in the models from previous chapters, EB1 is assumed to preferentially binds to TGTP rather than TGDP. 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 EB1Tubulin Interactions The reactions considered in this protofilament model are shown in Figures 41 and 42. 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 solutionphase 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 noncooperative; 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(41) 2[E][Tb] [ TE ] = K, (42) [TTE] = [E[2b K, (43) Figure 41A 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 ltubulin complex (kfE). Since unbound EB1 can bind to Tb in two identical ways, the onrate 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 plusend (Figure 41B). One pathway involves direct binding of the EB 1 head to the terminal, GTPbound subunit (k ) after tubulin addition (kf). The other pathway allows the EB 1 head to bind to solutionphase Tb (k;) and facilitate tubulin addition by shuttling it to the protofilament plusend (kfE "''). The value for kfE "'' accounts for the onrate of TE and the local, effective concentration of the unbound EB 1 head (CeyS), which is represented by Equation 44. kE'"= kEC'# (44) 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 (doublebound to EB 1 on plusend) and the penultimate subunit is in state dbE (doublebound to EB 1 on minusend). Figure 41C shows how TE in solution can bind the protoHilament: binding directly to the protoHilament plusend (kon) after tubulin addition (kf), or by first binding to tubulin in solution (k;) then adding as an EB ltubulin complex (2kfE). Figure 42 shows the various mechanisms by which EB 1 can bind to the GDPrich protofilament side. As shown in the Eigure, free EB 1 can bind directly from solution to a TGDP 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 onrate of kogside (pathway C). 4.1.2 EB1 Occupational Probability Model A probabilistic model similar to the tethered, divalent endtracking model (Chapter 3) was developed to simulate the pathways shown in Figure 41. 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 45. 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 46 through 410 (Appendix A.3), where R+ and R_ are defined by Equations 411 and 412. 2ko,,[E]u, k p, + k, w, kl [Tb]pl + k qll k plul + k ~+q k plul ' (46) + R (p,_z pi ) +R (pl , kw, w +R w,_, w, )+R (w + ks,,[TE]u, kw, + k, [Tb]p, (47) q, k ~ ', qk ,q' +k ~p u +k pu ,u +R qy~,_ q, )+R qI 1 (48) = 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 (49) (410) 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 , (411) (412) At equilibrium, these probabilities reduce to Equations 413 through 416, where qeq and Kk side/k side 2ko,, [E ] Peq k usq w pe eg K " 4eq 4eq (413) (414) f eq = 2 K p g2< < (415) 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(416) 8 ko, [E ] k Sl" The results of these equations were used to analyze the distribution of the divalent, EB 1 endtracking motors on the nontethered protofilament, and determine the equilibrium EB 1 concentration along the protofilament, Peq. PeI', P, 'e, e, / 2 (417) 4.1.3 Average Fraction of EB1bound Subunits at Equilibrium A stochastic model was developed that determines the average binding fraction of EB 1 along a nongrowing 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 plusend, 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 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 EBlibinding state of each subunit was used to determine the fraction of EBlibound subunits in the protofilament; this fraction was averaged over time for a total simulation time of 40 seconds. 4.1.4 Average Fraction of EB1bound subunits during protofilament growth A stochastic model was also used to calculate the timeaveraged fraction of EBlibound subunits during protofilament growth. The pathways considered in this model are those shown in Figure 42., as well as the association and dissociation of tubulin from the protofilament plusend (Figure 41). 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 EBlibinding state of each subunit was determined for each time step and the fraction of EBlibound 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 41 and 42 have not yet been determined experimentally, including the dissociation constant for Tb and EB 1 in solution (K;) the EB 1 onrate for protofilamentbound TGTP (kon), the EB 1 offrate from protofilamentbound TGDP (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 filamentbound subunits attached to EB 1 is given by Equation 418 (see appendix A.3 for full derivation). [E]o [E] 1 p (418) [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 419, where ul/z ueq ([E][E]o/2). K (419) def kside 2kside kszd ksde 1/ 2 ,,1/2* Under this constraint, um/ is given by Equation 420, 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 =(420) 2[E]o 4K2 C, Assuming [Tb]=0, [E]o=0.27 CIM, [E]= [E]o/2, and Cey153 C1M, Equations 419 and 420 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 421i, where kofyis the known offrate 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 (421) [Tb] To determined the optimal value of K; (that provides a 4.2 tiptoside 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 43 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 TGTP than TGDP (i.e., f must be greater than one) in order for EB 1 to accumulate at the plusends 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 plusend). 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 tiptoside binding ratios for these conditions are shown in Figure 44. When f1 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 f50, 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 onrate of EB1 and tubulin (k;) was assumed to be a typical value for proteinprotein binding interactions, 10 IM s^l. Consequently, the value of ky was determined from the optimal K; value (K;=kl/k;). The onrate constant for TE and TTE to the protofilament plus end, kfE, was assumed to be equal to the onrate of tubulin addition, k. The offrate constant of TE and TTE to the protofilament plusend, k; was calculated based on detailed balance, and is represented by Equation 422. kEE K, ko; (422) 4.1.6 Results 4.1.6.1 Occupational probability The ordinary differential equations that define the probabilities of EB 1 binding (Eq 46 to Eq. 410) were solved to determine the expected equilibrium fraction of EBlibound protomers in the protofilament (Figure 45). 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 46 to 410 were numerically integrated and solved at a set value ofK=37 and K;=0.65 CIM s^ using a fourthorder RungeKutta method in Matlab. The occupational probability of EB1 along the length of the protofilament for/1~ and f 1000 is shown in Figure 46. 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 GDPbound subunits, and this is the equilibrium EB 1 binding fraction determined earlier (Figure 45). 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 TGTP versus TGDP when#f1000. This decay profile is comparable with the experimental results shown in Figure 14. The ratio of the occupational probability at the plus and minus end of the protofilament for f1000 is about 4.3, which is similar to the tiptoside ratio of 4.2 observed in experiments. For both f1 and f1000. 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 EB1bound subunits at equilibrium The time averaged fraction of EBlibound protomers at f1 and f1000 and optimal values for K; (0.65 CIM) and kon (5 1M s^l) are displayed in Figure 47. 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 filamentbound TGDP, the time averaged fraction along the length of the protofilament is similar for f1 and f1000. 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 offrates and not the onrates on EB 1, TE, or TTE to the protofilament plusend, 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 f1 and ~f1000. 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 EB1bound subunits during protofilament growth Figure 48 displays the average fraction of EBlibound 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 f1 is overall slightly smaller than that of f1000, but both results show a larger fraction of bound EB 1 at the plusend. This result is expected since when/f1000 the rate of EB 1 addition at the protoHilament plusend is increased. When/f1000, there is a sharp decrease in the EBlibound 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 GDPbound tubulin subunits and hence significantly reduced offrates from the protomiament plusend (k,E and k_). Conversely, when f 1 the EBlbound fraction of subunits along the side of the protomiament fluctuates around the equilibrium value; since EB 1 has no preferential binding to GTP or GDPbound 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 tetheredprotofilament model discussed in section 3.2, considers the growth of a single microtubule protofilament in the presence of solutionphase, divalent EBli. A flexible binding protein provides as a link between EB 1 and a motile surface, allowing EB 1 to behave as an endtracking motor. The various reaction mechanisms considered for this tethered, protofilament model are those previously shown in Figures 41 and 42, and the binding pathways involving the surfacebound tethering protein (Figures 49 and 410). 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 (423) 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. 317 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 EB1Tubulin Interactions Figures 49 A and B shows the two methods by which EB 1 can bind to the surfacebound tethering protein, hereafter abbreviated Tk. Tk can associate with EB 1 in solution (pathway A) or with protofilamentbound 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 offrates. In pathway B, Tk binds to EB 1 on the protofilament (this could also be filamentbound TE or doubly bound EBli). The forward kinetic rate of this reaction, kT ', is represented by Equation 424, and is proportional to the forward rate of Tk binding to EB 1, kT, the effective concentration of Tk at the protofilament plusend, CT, and the effects of the transition state and spring energies. k,'= k,Czer(n1)dA/2kBTe ((n1)d)'2 2kB (424) The er(n1)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 plusend and increases toward the minusend 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, surfacetethered EB 1 protein and its equilibrium binding position on the protofilament is n1. Hence, the displacement distance of the spring is given by (n1) 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 49 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' (n1dA 2kB' (425) The mechanisms by which surfacetethered EB 1 can bind to the protomiament are shown in Figures 49 C and D. Surfacebound TE can attach to the protomiament with a forward rate of kon and a reverse rate of k_ '(pathway C), given by Equations 426 and 427, respectively. kon '= konfr es (n)dA 2k^BTe 11)d ) 2kB' (426) k '= kes (n1)dA, 2kB' (427) The contribution of the transition state and spring energies are equal to that in equations 424 and 425. If the surface is initially tethered to unbound EB 1 (pathway D), the onrate 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 onrate of EB 1 in solution to a protomiamentbound 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 minusend. These two potential pathways are shown in Figure 49 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 minusdirection (pathway F), the kinetic rates will be affected by the transition and spring energies. These rate equations are described in Equations 428 and 429. k side'= k side v(n 1 I 2)d) 2?kB' (428) k side = k sid~e (n1)dA, 2kB (429) Figure 410 shows the various ways tubulin can add to the protofilament plusend that involve the linking protein. Tubulin can add directly to the plusend of the surfacetethered protofilament (mechanism A), tubulin can be transferred to the protofilament plusend by surfacetethered TE (TkTE) or TTE (TkTTE) as seen in pathways B and C, respectively, or the protofilamenttethered TE can shuttle tubulin to the protofilament end (mechanism D). Only the onrates, not the reverse rates, for these reactions are affected by the interaction with the tethering protein. For all four pathways (AD), when there is an applied force against the surface (in the opposite direction of protofilament growth), F, the tubulin onrates are reduced by a factor ofeFd/kBT For direct tubulin addition in pathway A, the forward rate, kf' is proportional to the onrate of tubulin addition, kf, and the effect of force, as shown in Equation 430. 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 (430) Tubulin transferred to the protofilament end by TkTE or TkTTE (pathways B and C) are both proportional to the onrate of TE (or TTE) to the protofilament plus end, kfE, as seen in Equation 431. There is also an effect from CT and any load applied to the motile surface. The onrate of tubulin in these pathways is twice that of kfE because of EBli's dimeric structure. kf' kf F/k (431) Tubulin shuttled to the protofilament plusend by filamentbound EB 1 (pathway D) has a corresponding onrate of kfE "'. This forward rate (Equation 432) 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 3D normal distribution on a halfsphere (see section 4.2.3). kiE"= k," iE y((1d'z,2)d 2kB d kB' (432) 4.2.2 Protofilament EndTracking Model Considering the above reactions, stochastic models were performed to analyze the behavior of divalent, EB1 endtracking 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 onrate of Tk and EB 1 binding (kr) was estimated as 5 C1M s^ and the offrate, kT was calculated from the value of KT provided (Krk//kT). The value for v used to calculate the tubulin onrate from Equation 11 was 170 nm/s (Piehl and Cassimeris, 2003), which was assumed to be the irreversible elongation at the protofilament plusend. 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 surfacetethered 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 plusend (17addd) by the total simulation time, t. This model also provided the state of the terminal subunit, position of the linking protein, the timeaveraged fluorescence along the protofilament, and time spent in each pathway. Figure 411 shows the forcevelocity profiles for a polymerizing protofilament with surfacetethered EB 1 endtracking motors. To analyze the effect of the affinity modulation factor n the velocity profile, several affinity modulation factors were considered (Figures 411 AE). Regardless of the value of f the velocity decreased as the force increased because the force is opposite the direction of growth. For f1 and f10, 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 13), 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 plusend. 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 f1) to~1.7 pN (at f1000), as seen in Figure 411 F. Figure 412 summarizes the effect of KT and fon the stall force, with the corresponding data in Table 41. 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 f1, there is no affinity modulation and EB 1 binds to TGTP and TGDP 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 f1 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 endtracking 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 endtracking mechanism mediates tubulin addition and to understand the effects of f and KT on the velocity profiles (Figures 4 11 and 412), the frequency of the different pathways possible for association or dissociation of tubulin were measured and the resulting percentages are displayed in Figure 413. 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 plusend. 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 411 D). An increase in KT from 0.1 to 10 CIM when f1000 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 41 1 D). The same result was found when f1; at F=0 pN, the percentage spent in the forward pathway at Kr10 CIM (78%) was significantly greater than at K0. 1 CIM (32%), and resulted in a higher initial velocity at Kr10 CIM (Figure 411 A). When comparing the two affinity modulation factors (at Kr10 CM and F=2. 1 pN) the time spent associating tubulin at f1 was 93%, which was higher than when f1000 (85%). This result explains why lower values off resulted in negative velocities at large forces (Figure 411 F). The percent of time the protofilament spent bound and unbound to the motile surface is shown in Figure 414. The unbound percentage 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 433 shows that when the linking protein binds to EB 1 on the terminal subunit (n=1), the onrate is proportional to kTCT. But when n is greater than one, the onrate, 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 filamentbound 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 415, 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 f1, the terminal subunit was in state Tk3 or Tk4 a significant fraction of time; when ~f1000, 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 416. The most significant result is that when f1000, the unbound fraction decreased with increasing force, but when f1 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 f1, the frequency EBlibound tubulin addition decreased (Figure 413). However, with large affinity modulation (f1000), 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 417). 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 f1000, most of the linking protein are mostly bound to TE. When f1 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 nontethered, divalent EB 1 endtracing motors that processively linking protein the plusends 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 endtracking model has an advantage over the monovalent endtricking model and the Brownian ratchet mechanisms by maintaining a high EB 1 concentration at the protofilament plusend and allowing rapid MT polymerization 4.3.1 NonTethered 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 plusend or side) from solution or by copolymerization with EB 1, protofilamentbound tubulin can be in various EBlibinding 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 ltubulin 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 endtracking model over the Brownian ratchet mechanism to preferentially bind to the protofilament plusend 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 plusend 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 GDPbound subunits) and prevented the protofilament from growing. The results of this model show that the affinity modulation of EB 1 does not affect this sidebinding 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 EBlibound fraction of subunits shows a slightly larger fraction of EB 1 binding at the plusend when the affinity modulation factor is greater. 4.3.2 Tethered Protofilaments The tethered protofilament endtracking model simulates EB 1 endtracking motors operating on a growing protofilament plusend, and introduces a cofactor protein that tethers EB1 to a motile surface. Unlike the monovalent endtracking 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 plusend. This model also accounts for any transition state effects on the onand offrates due to binding between surfacetethered EB1 and the protofilament. The forcevelocity 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 endtracking 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 endtracking 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 endtracking 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 viceversa. 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 endtracking model is able to maintain a persistent attachment of the protofilament plusend (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%) untethered. This result suggests that the EB 1 endtracking motors are able to maintain persistent attachment of the protofilament end to the motile surface, translating its filamentbound hydrolysis energy to mechanical work and allowing the protofilament to grow even under large loads. Figure 41. Mechanisms of a nontethered, divalent endtracking motor. A) Topleft: EB 1 and TGTP free in solution. EB 1 binds to the protofilament in two ways: after tubulin addition (clockwise) or copolymerizing with tubulin (counterclockwise). Clockwise: TGTP adds to the protofilament end (kf) and induces hydrolysis of the penultimate tubulin subunit; EB 1 binds to TGTP at the protofilament end (2kon). Counterclockwise: EB1 and TGTP bind in solution (2kl); Together, EB1 and TGTP add to the protofilament end (k E). B) Toplet EB1 intite bound: to the ' GTPrich protofilament plusend. Free tubulin in solution binds to the protofilament in two ways: directly from solution (clockwise) or facilitated by the EB 1 motor (counterclockwi se). Clockwi se: Tubulin in solution adds to the protofilament end (kf), which induces hydrolysis of the EBlbound, penultimate tubulin subunit. The unbound EB 1 head binds to the GTPbound protofilament end (k,). Counterclockwise: The free EB 1 head binds to tubulin in solution (k;) and shuttles the protomer to the protofilament end (kyE '). C) TopLeft: TE and TGTP free in solution. TE binds to the protofilament in two ways: after tubulin addition (clockwise) or copolymerize as TTE (counterclockwise). Clockwise: TGTP binds to protofilament end (kf), inducing hydrolysis of the penultimate subunit. TE binds to the TGTP protofilament end (kon). Counterclockwise: TE binds to TGTP 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 ) TGDP < * 0'0 2k k _, G) 2kOn k Etc~l=! ~f~iX_7L? kk g 00 SCOCC) A EB1 TGTP A 2konslde a k s'de sdE Ssiddee konslde k s'de Figure 42. Mechanisms of equilibrium, side binding of EB1 to protofilament. Offrates of EB 1 binding to protofilamentbound 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 TubulinEB1 Eq. Dissoc. Constant, Ky (pM) Figure 43. 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 tiptoside 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 ef1. kon 0.1 af1,kon =1 Ti a f1 koin=10D 0" 1 00 *S, ko n 0 .1 w~ f5 0, k2n=10 m 10.   Rati o =42 01 0.01 0.1 1 10 TubulinEB1 Eq~.Diss~c. Constant, K,{prM) Figure 44. 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 TubulinEB1I Eq. Dissoc. Constant, K (ptM) Figure 45. 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 46. Occupational probability of EB1 along length of protofilament. Zero on the xaxis 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 47. Time averaged EBlibound tubulin fraction at equilibrium. Zero on the xaxis represents the protofilament plus (growing) end. Results for both f1 and f1000 shown. Values for other variables: K;=0.65 C1M, kon=5 IM s^l. Equilibrium EBlbound 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 48. Time averaged fraction of EBlibound subunits during protofilament growth. Zero on the xaxis 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 EBlbound fraction represented by solid line at 0.024. Simulation time used was 80 seconds and N=200. Figure 49. Mechanisms of tethered, protofilament endtracking model with divalent EB l. A) Pathway for tethering protein to bind to EB 1 (or TE) in solution. B) Tethering protein binds to protofilamentbound EB 1 (or TE). Energy is exerted by the spring, which is accounted for in the on and offrates (kT and kr'). C) Surfacetethered TE binds to protofilament plusend with on rate of kon'. D) Surface tethered EB 1 has twice the onrate 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 minusdirection 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 minusdirection exerts a force on the spring, which is accounted for in the on and off rates (kside' and kside ') kT kT~ nll n Xn+lXn+2 kT' k,. kan' k. 2kon k.' k side k side' O TGDP ~W) OCOO 0000 IrvL~ OOCO OCOO A EB1 TGTP k; k, kE 2hF' c kF" kE r O TGDP Figure 410. Mechanisms oftubulin addition to linking proteinbound protofilament. A) Tubulin can add to the plusend of a surfacetethered protofilament. B, C) Tubulin bound to TkE or TkTE attaches to the protofilament end. Tubulin bound to TkTE has two configurations with which it can bind. D) Tubulin addition is facilitated by the filamentbound EB 1 endtracking motor. The forces exerted on the spring are accounted for in the forward rate constants for each mechanism. HIY. 000 B ~000 rZEB1 TGTP Figure 411. Forcevelocity profiles for tethered protofilaments bound to divalent EB 1 endtracking motors. The effect of force and velocity of both the Brownian Ratchet and EndTracking 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) Forcevelocity 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 KeSpM . 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 412. Stall forces versus affinity modulation factor at various KT values. Resulting stall forces for data in Figures 319 AE. Thermodynamnic represents the thermodynamic values at the various values of the affinity modulation factor, based on ksT=4. 14 pNnm, d = 8 nm, [Tb] = 10 CIM, and [Tb], = 5 CIM. Data for this figure can be found in Table 41. Table 41. 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 pNnm, 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 413. 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 414. 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 413. 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 415. State of the terminal subunit (Sl) when f1 and f1000. 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: 1unbound, 2bound to E, 3bound to TE, 4bound to dbE Tk2bound to E tethered to linking protein, TK3bound to TE tethered to linking protein, Tk4bound 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 416. Fraction of S1 subunits bound and unbound from motile surface. Based on data presented in Figure 414, subunits in states 14 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 STkE OTkTE OTkTTE 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 417. Average state of unbound linking protein. The fraction of time the linking protein spent in each of its unbound states is shown: Tk unbound, TkE linking protein bound to EB1, TkTE linking protein bound to TE, TkTTE 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 plusends of ciliary MTs are attached to the cell membrane by MTcapping structures, which are located at the site of tubulin addition (Figure 51, (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 endtracking motor, similarly to the endtracking motors in cell division and cellular growth. This chapter discussed the EB 1 endtracking model developed for the ciliary plug. Essentially, the plug is the endtracking motor, which is behaves similar to the Lock, Load and Fire Mechanism (Dickinson and Purich, 2002). The advancement of the plug at the microtubule plusend occurs in three steps: tubulin addition, filamentbound 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 endtracking motor. 5.1 Model The physical characteristics of the plug make this model complex, but for simplification, the EB 1 endtracking 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 52), 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 endtracking motor are represented in Figure 53. The first step in this endtracking 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 TGDP, the motor rebinds to the GTPrich, filament plusend 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 51), 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 ratelimited by Tm (confirmed below), in which case Tm can again be estimated from v;;; (167 nm/s), i.e., T, = =0.05s (51) 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. 52) (Gardiner, 1986): D, dr dr2 F;+D = 1 kT dx' f dx'? (52) 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 53 and 54) This equation was used to generate forcevelocity 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 endtracking motor attached was also simulated. The filament end position, x, was governed by Equation 57. x acTI (57) 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 52 was solved and is represented in Equation 55. d2 ksT dF k,T k,T z(F) exD i i! DdF; k,T dF; dF; (53) (54) With r as a function of force, the equation that governs the velocity becomes a function of force, and is represented by Equation 56. 5( 5) d T, + r(F) v(F) (56) 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 14protofilmaent 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 51. The viscosity of the fluid used to calculate the drag coefficient was assumed to be that of water, 109 pNs/nm2 (Boal, 2002). Assuming each of the protofilaments to be a semiflexible 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 58 (Howard, 2001). k,T i2 L4 (58) The persistence length is represented by Equation 59, where B is the bending modulus of the filament (B = 1.2 x 1026 Nm2, (Mickey and Howard, 1995)). The resulting value for the compression stiffness is 1.1 pN/nm. S= B /kg T (59) 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 (510) The resulting stretch stiffness for a protofilament was determined to be 370 pN/nm. 5.3 Results Figure 54 shows the force effects on ciliary microtubules. In Figure 54A, 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 filamentbound GTP hydrolysis remains constant and is forceindependent, 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 54B. 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 55A, where the xaxis 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 endtracking motor for each protofilament must fill with tubulin before the ciliary plug can advance. The histogram in Figure 55B 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 55A. 5.4 Summary The ciliary plug was simulated as an EB 1 endtracking motor similar to the endtracking motors described in the Lock, Load and Fire Mechanism (Dickinson and Purich, 2002). The primary steps of this model are tubulin addition, filamentbound GTP hydrolysis and the shifting and rebinding of the ciliary plug (e.g., EBl) to the filament end. By analyzing the forcevelocity 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 endtracking motor to maintain fidelity of the microtubule by allowing the plug to advance only once all protofilaments are the same length. Figure 51. 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: 22592269. Copyright 1988 The Rockefeller University Press] ~ Independent Protofilaments Figure 52. 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 plusend. Red represents the GTPbound tubulin subunits. T~ubulin TGTP TG DP EndTracking M oto r/ E B Figure 53. Mechanism of the ciliary/flagellar endtracking motor. In the first step, tubulin adds to the MT plusend into the endtracking 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, 105 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 54. 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 55. 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 endtracking 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 EndTracking Motors in Biology The role of nucleotide hydrolysis in cytoskeletal molecular motor action is wellestablished 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 endtracking motors, we previously described how an microtubule filament endtracking motors can exploit nucleotide hydrolysis to generate significantly greater force than that predicted by a freefilament 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 plusend elongation to examine whether a hypothetical endtracking motor consisting of affinitymodulated interactions of EB 1 at MT and protofilament ends can propel obj ects (e.g., MTattached kinetochores or MTattached 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 endtracking 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; MimoriKiyosue et al., 2005). Kinetochores also stabilize MTs against disassembly by preferentially attaching to GTPcontaining Psubunits of tubulin subunits situated at or near MT plusends, and this property is likely to be the consequence of EBl1's ability to attach to polymerizing GTPrich MT subunits and to dissociate from GDPcontaining subunits, thereby providing a thermodynamic driving force for localization at or near the MT plusend. Capture of EBlirich MTs by kinetochores may allow those EB 1 molecules combining with APC to selfassemble into an endtracking motor unit that links force generation to MT polymerization and hydrolysis of MTbound GTP. It is known that in the absence of the EBl/APC complex, chromosomes fail to align at the metaphaseplate, 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 endtracking motor there and playing a role in elongation dependent force generation. In fact, it has previously been suggested that the pluglike structures found at the plusends of MTs in regenerating Chlamydmona~s flagella appear to be "MT assembly machines". The analogous geometry of the ciliary/flagellar plug and the tubuleattachment complex in the kinetochore would allow plug and kinetochorebound EB 1 to interact with their MT partners as an endtracking motor. This proposal does not preclude the action of other ATP hydrolysisdependent motors. For example, although the kinesinlike protein NOD lacks residues known to be critical for kinesin function, microtubule binding activates NOD's ATPase activity some 2000fold, 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. EndBinding Protein 1 (EBl1) has previously been shown to bind specifically to the polymerizing microtubule plusend where the microtubule is tightly bound, suggesting a possible role in force generation at these sites. We propose that endbinding proteins specificallyy EBl) behave as molecular motors that modulate the interaction between MTs and the motile obj ect, and generate the forces required for MTbased motility. Although the importance of EB 1 and its potential to behave as and endtracking motors has been discussed thoroughly in this research study, the models developed can be used to understand force generating mechanisms involving other endtracking proteins and their potential to act as motors (e.g., CLASPS, Clip170). 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 endtracking motor. 6.2 Microtubule EndTracking Model We developed and analyzed a preliminary EB 1 filament endtracking model for MTs to determine the advantages of the mechanochemical process over the monomerdriven ratchet mechani sms. The two important properties of this model are (a) maintenance of a tight, persistent (processive) attachment at elongating MT plusends by means of EBlI's multivalent affinitymodulated 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 highfidelity pathway for assembling tethered MT's. For simplicity, our model only considers simple reactions of the EB 1 filament endtracking 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 EBliassociated 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 EBlprotofilament interaction. We previously suggested that EB 1 may be a polymerization cofactor acting together with APC to endtrack MT protofilaments (see MechanismC in (Dickinson et al., 2004). However, in view of the recent finding that EB 1 is a stable, twoheaded dimer (Honnappa et al., 2005), we now explain how such multivalency would allow EB 1 alone to operate as the endtracking motor (like MechanismA 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 forcedependence of elongation arising from the dependence of probability of the flexible EB 1 head binding at a specific MT lattice position. With hydrolysisdriven affinitymodulation 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 hydrolysismediated motors on an MT's thirteen protofilaments can yield kinetic stall forces of approximately 30 pN. This value is considerably larger than the ~7pN 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 EndTracking Models The microtubule endtracking model developed neglected solutionphase End Binding protein 1 (EBl1) and binding to microtubules and tubulin protomers. To account for binding solutionphase EB1, we developed simplified models that simulated the growth of a single protofilament in the presence of EB1 endtracking motors. The properties of all protofilament the endtracking models were compared to those of the simple Brownian Ratchet mechanism. Two of the models consider only freeprotofilament 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 GTPdriven affinity modulated binding of the EB 1 endtracking proteins is required in order to provide a 4.2 tiptoside 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 forcevelocity profiles show that the divalent, EB 1 endtracking model provides an great advantage over the monovalent end tracking model as well as a Brownian Ratchet mechanism. The divalent endtracking motors are able to provide processive endtracking 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 13protofilament, microtubule endtracking model should be considered. It is suggested to develop a stochastic model that includes all mechanisms discussed in the tetheredprotofilament model with divalent EB l. For simplifications it could be assumed that all protofilament behave independently, but whose individual EB 1 endtracking 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 endtracking 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 endtracking 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 plusend. 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 EBlicoated 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 velocityforce 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] (A1) E +M2T ++MT E Kd= [E][MT]/[MTE] (A2) The total concentration of EB1 is represented in all states is [E]o = [E] + [TE] + [MTE] (A3) Substituting A1 and A2 into A3 gives Equation A4. [E]o = [E] + [E][Tb]/K;+ [E][MT]/Kd (A4) such that solving for [E] yields. [E], [E] = o (A5) ,[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 noncooperative. Here, the relevant reaction and equilibrium equations are E + Tb t, TE K; =2 [E][Tb]/[TE] (A6) Tb + TE t, TTE K; = [Tb][TE]/(2 [TTE]) (A7) E +M2T ++MT E Kd= [E][MT]/[MTE] (A8) [E]o = [E] + [TE] + [TTE] + [MTE] (A9) Combining A6 and A7 yields [TE][Tb~] [E][Tb]2 [ TTE ] =  2K, $2(A10) Combining Eqs. A9 and A10 yields [E]o = [E] + 2[E] [Tb]/K;+[E][Tb]2/K;2 + [E][MT]/Kei (A11) hence [E], [E]= (A12) K, KdK from which [TE] and [TTE] can be calculated using Eqs. A6 and A7. A.2 Occupation Probability of Monovalent EB1 Binding to NonTethered 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 (A13) The dimensionless relationships in Equations A14 to A17 can be substituted into Equation A 13: k ,l[E]I 95 a '(A14) 'k,[T] a, (A15) Sk,[T] k,E[TE ] f= (A16) k,[T] T,, ,E1+kcT K )11T, (A17) 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 (A18) and plusend (A19), where u1p,. dp' =,c, a, p, +(1+ f)(p,_z p,)+T,' (p,,1 p) (A18) dzd dp, =Td 1~' 1 191 1 1Tr,01lp 192 Tr110 (A19) 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 TkE 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 A20. y = 1w v q (A20) The differential equations for the probabilities of EB 1 and TkE 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 )+ (A21) 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, 241 (A22) 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]yk,v vk,[T]v +k, w ko,,2Ced,i2vuz + k ;I9, (A23) (A24) To dedimensionalize time in these differential equations, the variable Tr~ was introduced, which is defined by Equation A25. 1k, E 1 + k,u, Tl k, [T](A25) 1c k, [T] k, 1 c1 = and TI,_,' [T] K, dEquation A25 gets reduced to A26. K,f f Setting Tr o T, Toouz Tnpl (A26) Dividing Equations A21 to A24 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,111 *ITr,01I 1 r, 1P+1 (A27) 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 11,~ 1 1 1 ,11 71 k, [T] k, [T] (A28) dwo k, [ T E] k, K, k, k, K, kE f1 dz k, [T] k, [T] k, k, [T] k, [T (A29)4 dv k, [E] k, K, k, k, K k, k Kd,; y v v + w1 vu +C q, dz k, [T] k, [T] k, k, [T] k, [T] k, [T] (0 To evaluate the probabilities ofE or TkTE binding to the terminal subunit in the protofilament, these probabilities were rewritten 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] (A31) kt~ K,; 1 kE [T E ] (1p,) + Tu, +T,~, q pp, T,,p(1 p k, [T] k, [T] dq k, Ct k, C s, kfE qCf ,1 yp, + vu1 + w(1q4 ) dz k, [T] k, [T] k, [T] + D1 Il P 9 (A32) 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 A27 to A730, the equations were de dimensionalized by using the following dimensionless parameters: k, K, k, k, [T] kf k, kE Sk, k, cp = eA(ld0 0.2 The resulting dedimensionalized 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) (A33) [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] [TE] C + (1+ 3 + 3 "f. aw)(q,_z q,) (A34) [T] [T] To K To To K [T] K,f [T] []~ dw [ T E ] K, K, ef,1 =7 ~~ y !wq ' w+ ,,q,p (A35) 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 (A36) C~Cf, PLU C aK [T] ,, [T] dp [ E] Kd qif ,1 qC dz [T] [T] [T] [T] (A3 7) [T] [nliiT] [T] [T] K,f f1 dq, C,, C,, C,, =Y "fg~yp,+p efglvu1+S ef~w(1ql) dz [T] [T] [T] YK, Kd/ f [TE] 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("dzm)/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 (A39) + (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 (A40) + (1+S+70.)219) Sy p+u+S q q1 92 at 7~+ g 700wS q (A41) dv . = TE/ P yv 1+ rfw a 70,pvu + a eq, dz f (A42) 1>1 1>1 dp a nu, a p1 yX8,pyp, p, S00 o*PI dz f (A43) +Y~pq, +3 (p) u S ep d = yXB,p yp, + a *X0, vu, + S00fow(1 q, ) dz (A44) fy +a* 9,++ ,+pz S ,q These differential equations (A39 to A44) 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~= 1y~ q 92 P,n,~ (A45) = 2kon [E]ul k p, + k, wl k, [T]p, + k q,_z k pu, u (46 dt (A47) dq, = k 9 s9+ su + tu (A48) dq, = k,9 1 k P z, k tu (A49) +R (q1 q, +R q 9 = k p, + kw, + k ql + k ql k P,zu k ~pzuA0 2kon [E]u, kon[T E]u, + R (u, u,)+R (u,, u,) R = kl[T] +k, E([T EF] +[T ET])+ kE efn 1 (A51) R_= kiu + k~u,E +I1 1 1 (A52) 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 ET])pi +/(a ,u + k,E 1' p2 k,E1 2)"i~?kt,l i (A53) dq, = k 2q k q, + k, p2 1, + k+2p 12 [T]+kE([T "E+ [TE T])b +kE;l eff y 1 k (A55) + u," + k, E(/ 1 1 2) du, = k_,p, +k ,w, +k_,,q,' k ~p2u 12kon[E]u, kon[T E]u, + ki[T](1 un) (kE([T E] +kfE[T E T]) 1, k~u(1u)+kE 1 1 2(A.56) Since an EB 1 bound to the protofilament plusend 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, (A57) 0 = kon[T' E]ueq k~weq + k, [T']peq~ kl weq (A58) 0 = kqeq + 2k peq eq~ (A59) 0 = k peq + k weq + k qeq 2k peq1, e 2kon[E]ueqo kon[T E]mue (A60) And the following holds true: ueq 1 eq eq eq", (A61) Solving Equation A59 gives: k _qeq = 2k peq elq (A62) Substituting this relationship into the other three equilibrium equations results in: 0 =2kon~[E~]ueq k +k, [T])peq +k weq (A63) 0 = k peq+k weq (2kon [E]+ kon [T~ E eql (A64) 0 = kon[TE]ueq k_ +k1 eq +k, [T]peq (A65) From these three relationships, the equations for weq, and peq WeTO SOlVed for: kon [E] (k +k '[T]) w, = 2 uq + ne k, k, (A66) I k +k +[T]}I Sk +k e (A67) At equilibrium, the following relationship is true: [T][E] [T E]= 2 K, (A68) Plugging this into Equation A67, gives: + 1 +, IT 2kon [E]k k Peq eq k k +k[T (A69) The terms in backets on, the top qand bottom arest idntcl thereforlwgeiat A69reucs o 2kon[E],[] ~ 2k,2n~~ [Tb]] eq K eq k (A72) 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 (A74) To determine the value ofKl, we first assume [Tb]=0, which reduces Eq. A75 to: "e, ([T] = 0) S2ko, [E] 2ko, [E] k 2k[E k kkk k, 2ko,, [E ] k k (A75) The fraction of filamentbound protomers attached to EB 1 at equilibrium defined by total amount of EB 1 minus the amount of EB 1 in solution: (A76) This is also equivalent to: [M~T] or p = E We, / 2+ p,,) =;t~k 2ko,, (1 Solving for p gives: p) (A77) 1 2k ~lin~ (A78) 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 A78 gives: 1 K d,eff +1 [M~T] or (A79) (A80) When half of the protofilament is saturated (p=1/2), Kd,ef is given by Equation A81 and ul/ ueq ([E][E]o/2). K d,ef f k (A81) Under this constraint, um/ is given by Equation A82, 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 (A82) Since k =kon*Cgt and k side"_konside CyrifOr the protofilament plus end, Eqluation A83 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) (A83) Substituting the definition of K, where K k./k gives: +1 + K +1 +8K E 4K fs (A84) Assuming [Tb]=0, [E]o=0.27C1M, [E]= [E]o/2, and C,f153 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, (A85) Determination of k: At steadystate, the measured offrate 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' (A86) Rearranging Eq. A86 gives Equation A87. 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 (A88) APPENDIX B MATLAB CODES B.1 13Protofilament Microtubule Model This stochastic program simulates a 13protofilament 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 13protofilament 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 (pNnm) 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 TGDP's (SA1) kD=0.5; % Dissociation rate constant for both EB1 arms on taxolstabilized MT wal(uM) L=10; % (nm) rho=1/(LA2) ; % (nmA2) %conversion=1.66e6; % conversion from uM to nmA3 %ks ol1=kof f/ (kD*/ 3 ; ksol=5e7; % ksol: onrate for both EB1 arms on MT in solution (nmA3/s) klon=8.9/13; % kl+: kon for tubulin dimer on MT (1/(uMs)) 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 TGTP/T GDP (1/s) (1.6*10A3) kZ of f=k 30f f/ f; % k2: koff for 1 EB1 arm between T GTP/TGDP(1/s) (0.02) dt=4.8e6; % 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 TGTP/TGDP % State 2: One EB1 arm bound between terminal 2 TGDP's % state 3: one EB1 arm bound between terminal 2 TGDP's, One arm btwn terminal TGTP/TGDP % state % state % state 4: one EB1 arm bound between terminal 2 TGDP's, One arm btwn lagging 2 TGDP's 5: one EB1 arm bound between lagging 2 TGDP's 6: one EB1 arm bound between lagging 2 TGDP's, One arm btwn terminal TGTP/TGDP 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)' % 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+(ps2).*(n1)+(ps==3).*(+)+p=4(nn+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 xaxis % Label yaxis % 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 xaxis ylabel('Filament End Position (nm) ') % Label yaxis B.2 Protofilament Growth Model with Monovalent EB1 B.2.1 Occupational Probability of Monovalent EB1 on a NonTethered Protofilament This probabilistic model simulates the free growth of a single protofilament in the presence of monovalent, EB 1 endtracking 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 onrate for tubulin % kr = 3.38; % s^2 kf = v/d/Tb; % uM^1s^1 onrate for tubulin taken assuming irreversible % elongation at observed elongation speed TC = 5; % uM plusend critical concentration kr = kf*Tc; % s^1 offrate, 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 tipto nsa omidl 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 EB1Tb concentration klus_side=kobs/(E+Kd); % uM^1 s^1 onrate constant for EB1 to MT side  kinus_side=kplus_side*Kd; % s^1 offrate 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 onrate constant for EB1 to terminal subunit % assumed same as on side kminus=enhqkminus_side/f; % s^1 offrate 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:n1).^2Q~.*(./iEB)2).*x((:1*ldlt/igB^2' % av=ay.*exp((0:n1)*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=1p; 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:n1; /Ti)= gy(i) .*u(i)av(i) .*p(i)+(1+b)*(p~)p(i)pi)Tiv(~+)pi % /In)=0 ; /In)= /In1) ; B.2.2 Occupational Probability of Monovalent EB1 on a Tethered Protofilament This probabilistic model simulates the growth of a surfacetethered protofilament in the presence of monovalent, EB 1 endtracking 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; % pNnm 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 onrate for tubulin taken assuming irreversible elongation at observed elongation speed TC = 5; % uM plusend critical concentration kr = kf*Tc; % s^1 offrate, 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 tiptomiddle 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 EB1Tb concentration kplus_side=kobs/(E+Kd); % uM^1 s^1 onrate constant for EB1 to MT side  kminus_side=kplus_side*Kd; % s^1 offrate 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 onrate constant for EB1 to terminal subunit  assumed same as on side kminus=kminus_side/f; % % s^1 offrate 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 3D normal distribution on halfsphere 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*j1) ; pend=xout(nt,2*j1); 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 = (im1)*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*Tkr) Attachprob = 1(1sum(qend))^13 function ff=bfrate(t,x,pars) n=(length(x)2)/2; j=1:n; p=x(2*]1) ; q=x(2*]) ; u=1pq; v=x(2*n+1) ; w=x(2*n+2) ; y=1vwsum(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(im1))*d^2/kT) = Fd +kappad~i meani = 0; if sum(q)>0 meani = sum(q.*j ')/sum(q); end FT = (meani1)*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(jim)*kappadqdeld)'; theta=exp(kappad/2*(jim).^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+line2deltaqpsiqmu/xi/fqpl qlp(1*p2) linel = (chiv(1)*(gamma~y~p(1)lpa~svulph)sv) (1)chv*h*w11) 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:n1; linel = alpha.*(epsqu(i)muqp(i))gammaqchiv(i). *y*~)gmabt~h~)*~) line2 = (1+delta*(eps/xi+chiv0*w))*(p(i1)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)qiq); line3 = psi*(deltaqmu/xi/f*(p(1)phi(1~ql)*q(1+u1*qi1q); fq(i)=linel+line2+line3; ff(1)=fp(1) ; ff(2)=fq(1) ; ff(2*i1)=fp~i); ff(2*i) = fq(i); linel = gamma*(epsqybetaqv)+eta*(xiwwvv)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/xiqybeta~w)+eta*(vx~wxiwela(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(n1)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)qnqn) fq(n)=linel+line2; ff(2*n1)=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 NonTethered Protofilament This probabilistic model simulates the free growth of a single protofilament in the presence of divalent, EB 1 endtracking 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  freeended MT's % Simulates freeended 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; % uMA1sA1 onrate for tubulin % kr = 3.38; % sA2 kf = v/d/Tb; % uMA1sA1 onrate for tubulin taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plusend critical concentration kr = kf*TC; % SA1 offrate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % sA1 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 nmA3  based on 3D normal distribution on half sphere Ceff = ceff/(6.0Z~22e23)e27/1000*1e6; % nm3 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; % onrate for EB1 subunit and Tb (uMA1*sA1), value for typical proteinprotein binding klm = kl*Kdl; % offrate for EB1 subunit and Tb (sA1) %% Equlibria E = EB1/((1+(Tb/Kdl))A2); % [EBl], Concentration of EB1 dimer in sol'n TE = 2*E*Tb/Kdl; % [EB1Tb], concentration of EB1 dimer bound to 1 tubulin protomer TTE = Tb*TE/(2*Kdl); % [EB1TbA2], 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 halflife 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; % uM1s1 fon = kon/kon_side; %% accelerated onrate at end kplus = kon*Ceff; foff = fon/f; kmi nus=fon/f~kmi nus_si de ; %% correspondi ng change offrate at end %% other parameters kfE = kf % onrate constant of TE and TTE to MT end uM1s1; krE = kfE*TC/Kdl~kminus/kon % offrate 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*j1)=p_eq; %initial conditions x0(3*j)=q_eq/2; x0(4)=0; x0(3*j2)=pi_eq; [tout, xout]=ode23s(@(t,x0) dfrate(t,x0,pars),tspan,x0); nt=1ength(tout); nmid=round(nt/2); pimid=xout(nmid,3*j2) ; pmid=xout(nmid,3*j1) ; qpmid=xout(nmid,3*j) ; piend=xout(nt,3*j2) ; pend=xout(nt,3*j1) ; qpend=xout(nt,3*j) ; qmend=[0 qpend(1:n1)]; qmmid=[0 qpmid(1:n1)]; 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*j2) ; p=x(3*j1) ; qp=x(3*j) ; qm=[0 qp(1:n1) ']'; u=1pqpqmpid; 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:n1; tmpl=2*konv(i)*E.*u(i)kmy(i).*p(i)+klm"pdilT~~)kyi.q~)kyi1 *~)*~+) tmp2=kmy(i+1).*qm(i+1)kpy(i1).*p(i).*ui+R*p1pi)Rmpi1pi; 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*bpil~i~)R*pdi1i~)+m(i~+) pid(i)) ; tmpl=2*kon*E~u(1)kminus~p(1)+klm~pid(1)k*bp1+mnssd~m2pu~iep1*() tmp2=kfE*TE*(1p(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*(1pid(1))(k"bkf*"TbkE*TE*pi(1) kfE*Ceff~pid(1)+krE~q qp)(1r~)+(krEu1+r~()*pd2r~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*j2)=fpi ; f(3*j1)=fp; f(3*j)=fqp; B.3.2 Average Fraction of divalent EB1bound Protomers on Side of Protofilament This stochastic model simulates the sidebinding of divalent EB 1 on a nongrowing 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 freeended 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 plusend critical concentration EB1 = .27; % uM EB1 concentration d = 8; % nm; % subunit length V = 170; % nm/s; % elongation speed kf = v/d/Tb; % uMA1sA1 onrate for tubulin taken assuming irreversible elongation at observed elongation speed kr = kf*TC; % SA1 offrate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % sA1 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 nmA3  based on 3D normal distribution on halfsphere Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm3 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; % onrate for EB1 subunit and Tb (uMA1*sA1), value for typical proteinprotein binding klm = kl*Kdl; % offrate for EB1 subunit and Tb (sA1) %% Equlibria E = EB1/((1+(Tb/Kdl))A2); % [EBl], Concentration of EB1 dimer in sol'n TE = 2*E*Tb/Kdl; % [EB1Tb], concentration of EB1 dimer bound to 1 tubulin protomer TTE = Tb*TE/(2*Kdl) ; % [EB1TbA2], 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 halflife 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; % uM1s1 fon = kon/kon_side; %% accelerated onrate at end kplus = kon*Ceff; foff = fon/f; kmi nus=fon/f~kmi nus_si de ; %% correspondi ng change offrate at end %% other parameters kfE = 10 % onrate constant of TE and TTE to MT end uM1s1; krE = kfE*TC/Kdl~kminus/kon % offrate constant of TE or TTE % Initial conditions N=40; % 50number 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; % 200run 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,:) bind plus side tstS=(((s==2)&[S(2:N)==1 0]) & rnsidel(it,:)<(kplus_side~dt + kplus_side~dt+kminus_side~dt)) &~(tst3tst4); % bind minus side tst6=((s==2)& rnsidel(it,:)<(kl*T~dt + 2*kplus_side~dt+kminus_side~dt)) &~(tst3tst4tst5); % bind T tst7=(S==3)& rnsidel(it,:) otst=(tstl+tst2+tst3+tst4+tstS+tst6+tst7+tt+s9tt0 if sum(otst)>0 ntav=itprevit; 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(ifnd41)=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(ifnd91)=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 EB1bound 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 freeended 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 yaxis. % 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; % uMA1sA1 onrate for tubulin taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plusend critical concentration kr = kf*TC; % SA1 offrate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % sA1 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 nmA3  based on 3D normal distribution on halfsphere Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm3 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; % onrate for EB1 subunit and T (uMA1*SA1), Value for typical proteinprotein binding klm = kl*Kdl; % offrate for EB1 subunit and T (SA1) %% Equlibria E = EB1/((1+(T/Kdl))A2); % [EBl], Concentration of EB1 dimer in sol'n TE = 2*E*T/Kdl; % [EB1T], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = T*TE/(2*Kdl); % [EB1TA2], 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 halflife 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; % uM1s1 fon = kon/kon_side; %% accelerated onrate at end kplus = kon*Ceff; foff = fon/f; kmi nus=fon/f~kmi nus_si de ; %% correspondi ng change offrate at end %% other parameters kfE = kf; % onrate constant of TE and TTE to MT end uM1s1; %kfE = le8; krE = kfE*TC/Kdl~kminus/kon; % offrate 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,Nl1]; kminusv=[kminus kminus_side~ones(1,N1)]; kplusv=[kplus kplus_side~ones(1,N1)]; 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,:) side tstS=(((s==2)&[S(2:N)==1 0]) & rnsidel(it,:)<(kplusv~dt + kplusv~dt+kminusv~dt)) &~(tst3tst4); % bind minus side tst6=((s==2)& rnsidel(it,:)<(klv*T~dt + 2*kplusv~dt+kminusv~dt)) &~(tst3tst4tst5); % bind T tst7=(s==3)& rnsidel(it,:) otst=(tstl+tst2+tst3+tst4+tstS+tst6+tst7+s8t9ttl) % Anything happen? if sum(otst)>0 Sold=s; % store old ntav=itprevit; % 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 timeaveraged 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(ifnd41)=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(ifnd91)=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) 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 &~(tallta2ta3); if tallta2ta3ta4 ntav=itprevit; 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:N1) ; 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 ta5ta6ta7ta8 Sold=s; ntav=itprevit; FL=(S==2)+(S==3)+.5*(S==5)+.5*(S==4); FLav=(previt*FLav+ntav*FL)/(ntav+previt); previt=it; nadd=nadd1; S(1:N1)=S(2:N) ; if s(N1)==4 s(N)=5; else S(N) = S(N1) ; 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 endtracking 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 yaxis. % 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; % uMA1sA1 onrate for tubulin taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plusend critical concentration kr = kf*TC; % SA1 offrate, assuming Tc=kr/kf; thalf=2.6; % s kobs=log(2)/thalf; % sA1 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 nmA3  based on 3D normal distribution on halfsphere Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm3 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; % onrate for EB1 subunit and T (uMA1*SA1), Value for typical proteinprotein binding klm = kl*Kdl; % offrate for EB1 subunit and T (SA1) %% Equlibria E = EB1/((1+(T/Kdl))A2); % [EBl], Concentration of EB1 dimer in sol'n TE = 2*E*T/Kdl; % [EB1T], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = T*TE/(2*Kdl); % [EB1TA2], 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 halflife kplus_side = K~kminus_side; %% by definition kon_side = kplus_side/Ceff; %f= 1; % affinity modulation factor ffvec(i run) ; %% Mixed model  chose kon, calculated koff from f kon = 5; % uM1s1 fon = kon/kon_side; %% accelerated onrate at end kplus = kon*Ceff; foff = fon/f; kmi nus=fon/f~kmi nus_si de ; %% correspondi ng change offrate at end %% Other parameters kfE = kf; % onrate constant of TE and TTE to MT end uM1s1; krE = kfE*TC/Kdl~kminus/kon; % offrate 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,Nl1]; kminusv=[kminus kminus_side~ones(1,N1)]; kplusv=[kplus kplus_side~ones(1,N1)]; 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; %pNnm 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 Trackerbound 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 MTbound 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,:) side tstS=(((s==2)&[S(2:N)==1 0]) & rnsidel(it,:)<(kplusv~dt + kplusv~dt+kminusv~dt)) &~(tst3tst4); % bind minus side tst6=((s==2)& rnsidel(it,:)<(klv*T~dt + 2*kplusv~dt+kminusv~dt)) &~(tst3tst4tst5); % bind T to E tst7=(s==3)& rnsidel(it,:) otst=(tstl+tst2+tst3+tst4+tstS+tst6+tst7+s8t9ttl) % Anything happen? if sum(otst)>0 Sold=s; % store old ntav=itprevit; % number of additional steps in average FL=(abs(s)==2)+(ass3+5(abs(s)==5)+.5*(abs(s)==4) % EB1 fluorsescence FLav=(previt*FLav+ntav*FL)/(ntav+previt); % Update timeaveraged 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(ifnd41)=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(ifnd91)=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) 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 &~(tallta2ta3) ; if tallta2ta3ta4 ntav=itprevit; FL=(abs(s)==2)+(ass3+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:N1) ; 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 ta5ta6ta7ta8 Sold=s; ntav=itprevit; FL=(abs(s)==2)+(ass3+5(abs(s)==5)+.5*(abs(s)==4) FLav=(previt*FLav+ntav*FL)/(ntav+previt); previt=it; nadd=nadd1; S(1:N1)=S(2:N) ; if s(N1)==4 s(N)=5; else S(N) = S(N1) ; 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) 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)) &~ (tT1tT2); % release EB1 %% Track = 2 Tracking unit bound w/ TE tT4 = (Track==2)& rndT(it) totC = cumsum((S==1).*konTy~dt); tT6 = ((Track==2)& rndT(it)<(totC(N)+klm~dt+kl*T~dt)) &~ (tT4tT5); % Binds free MT site tT7 = ((Track==2) & rndT(it)<(krT~dt+totc(N)+klm~dt+kl*T~dt))& (tT4tT5tT6) ; % release TE tT8 = ((Track =2) & rndT(it)<(kfEp*Ffac~dt+krT~dt+totc(N)+kl mdklTt)& (tT4tT5tT6tT7); % 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) &~ (tT9tT10); % 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(Track1) = 4; end if Track1==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:N1); 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:N1); 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) 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) &~(tallta2ta3); % add from TEbound MT if tallta2ta3ta4 ntav=itprevit; FL=(abs(s)==2)+(ass3+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:N1) ; 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 ta5ta6ta7ta8 Sold=s; ntav=itprevit; FL=(abs(s)==2)+(ass3+5(abs(s)==5)+.5*(abs(s)==4) FLav=(previt*FLav+ntav*FL)/(ntav+previt); previt=it; nadd=nadd1; if Track == if ta6 Track = 2; elseif ta7 Track = 3; end else Track=Track1; end S(1:N1)=S(2:N) ; if s(N1)==4 s(N)=5; elseif s(N1)==4 s(N)=5; else S(N) = S(N1) ; 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 unitE detachment kminusy =[kminus kminus_side.*ones(1,N1)]; ffac2 = exp(gamT*(Track1)*d~delta/kT); ffac3 = exp(gamT*(Track.5)*dA2/kT); 5% Doubly bound EB1 tst1 = (s(Track) = 5)& rndT(it) 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 plusside EB1 head (doubly bound) 5% bound TE 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) &~ (tst2tst2a) ; % dissociate T 5% bound E tst3 = (S(Track) == 2)& rndT(it) detach Tracker tst3b = ((s(Track) == 2)& rndT(it)<(kl*TE~dt+krT~ ffac2*dt+kminusv( Trc)fa2d) &~ (tst3tst3a) ; % add T tst3c = ((S(Track) == 2)& S(Track+1)==1 & rndT(it)<(kplus_side~ffac3*ffac2*dt+kl *TE~tkTfa2d~mns(rc)fa2d) &~ (tst3tst3altst3b); % bind second head in minusdirection if Track>1 tst3d = ((s(Track1)==1 & s(Track) == 2)& rndT(it)<(kplusv(Track 1)*dt+kplus_side~ffac3*ffac2*dt+kl *TE~d~r~fc*tkiuvTak*fc*dt) &~ (tst3tst3altst3bltst3c); % bind second head in plusdirection else tst3d=0; end if tst1 Track=Track1; 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(Track1)=4; Track = 0; if Track ;old1==1; s1(4,irun)=s1(4,irun)+1; end elseif tstlb S(Track) = 2; S(Track1) = ; if Track1==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(Track1) = 4; if Track_old1==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*(irun1)+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 13protofilament 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 MTbased 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 (pNnm) nf = 13; % No. filaments kappa = 0.15; % Filament compression stiffness (pN/nm) Df = 4e6; % Filament diffusivity (nmA2/s) deltaf = kT/Df; % Filament Drag (pNs/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>1e10; zeq = kappai~z0f'/sum(kappai) ; kappai=kappa*((zeq)<=z0f)+kappa2*((zeq)>z~) % Vector of filament stiffnesses F=kappai.*(zeqz~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)1Pr) ./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 (s1) 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(ihnbp,1) :ih) ; zplot=zh(max(ihnbp,1) :ih) ; znplot=zhn(max(ihnbp,1) :ih) ; SUBPLOT(2,1,1), plot(tplot,znplot,'r',tplot,zplot,'b'); % Plot recent trajectory tmin=th(max(ihnbp,1)) ; tmax = max([th .1]); zmin=zh(max(ihnbp,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~fzeq,5:10:max(z0fzeq)+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 LIST OF REFERENCES Abrieu, A., J. 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M~olBiol Cell. 12:430816. Tirnauer, J. S., S. Grego, E. D. Salmon and T. J. Mitchison. 2002b. EBlmicrotubule interactions in Xenopus egg extracts: role of EB 1 in microtubule stabilization and mechanisms of targeting to microtubules. Mol1Biol Cell. 10:361426. Walker, R. A., E. T. O'Brien, N. K. Pryer, M. F. Soboeiro, W. A. Voter, H. P. Erickson and E. D. Salmon. 1988. Dynamic instability of individual microtubules analyzed by video light microscopy: rate constants and transition frequencies. JCell Biol. 4: 143748. Yahara, I. and G. M. Edelman. 1975. Modulation of lymphocyte receptor mobility by locally bound concanavalin A. Proc NatlAcad Sci USA. 4:157983. Zeeberg, B. and M. Caplow. 1979. Determination of free and bound microtubular protein and guanine nucleotide under equilibrium conditions. Biochemistry. 18:38806. 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 ENDBINDING 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. PAGE 5 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 EndTracking 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 EndTracking Motors......................................................................... 29 1.4 Therm odynamic Driving Force..................................................................................... 30 1.5 Summ ary.......................................................................................................................31 1.6 Outline of Dissertation .................................................................................................. 31 2 MICROTUBULE ENDTRACKING MODEL.....................................................................37 2.1 EB1 EndTracking 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 NonTethered Protofilam ent Growth............................................................................ 53 3.1.1 Therm odynamics of EB1tubulin interactions..................................................54 3.1.2 Kinetics of EB1tubu lin interactions .................................................................55 3.1.3 Param eter Estimations....................................................................................... 56 3.1.4 Results ............................................................................................................... 58 PAGE 6 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 NonTethered Protofilam ents............................................................................ 64 3.3.2 Tethered P rotofilaments.................................................................................... 65 4 PROTOFILAMENT ENDTRACKING MODEL W ITH DIVALENT EB1........................ 73 4.1 NonTethered Protofilam ent Growth............................................................................ 73 4.1.1 Kinetics of EB1Tubulin Interactions ............................................................... 73 4.1.2 EB1 Occupational Probability Model ...............................................................75 4.1.3 Average Fraction of EB1bound Subunits at Equilibrium ................................ 77 4.1.4 Average Fraction of EB1bound 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 EB1bound subunits at equilibrium ....................82 4.1.6.3 Average fraction of EB1bound subunits during protofilam ent growth.................................................................................................82 4.2 Tethered P rotofilament Growth Model......................................................................... 83 4.2.1 Kinetics of EB1Tubulin Interactions ............................................................... 84 4.2.2 Protofilament EndTracking Model.................................................................. 87 4.2.3 Param eter Estimations....................................................................................... 87 4.2.4 Results ............................................................................................................... 87 4.3 Summ ary.......................................................................................................................91 4.3.1 NonTethered 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 EndTrack ing Motors in Biology ................................................... 124 6.2 Microtubule EndTracking Model ..............................................................................126 6.3 Protofilam ent EndTracking Models.......................................................................... 127 6.4 Future W ork................................................................................................................128 APPENDIX A PARAMETER ESTIMATIONS.......................................................................................... 130 PAGE 7 7 A.1 Concentrations of EB1 Species in Solution ................................................................130 A.2 Occupation Probability of Monovale nt EB1 Binding to NonTethered 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 13Protofilam 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 NonTethered 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 NonTethered Protofilam ent................................................................................................... 152 B.3.2 Average Fraction of divalent EB1bound Protomers on Side of Protofilam ent................................................................................................... 155 B.3.3 Average Fraction of EB1bound 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 PAGE 8 8 LIST OF TABLES Table page 11 Thermodynamic equations characterizing the multiple steps for GDP to GTP conversion ..................................................................................................................... .....35 12 Equilibrium constants used in energy equations................................................................ 36 41 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 412.............. 109 PAGE 9 9 LIST OF FIGURES Figure page 11 Microtubule structure...................................................................................................... ...33 12 Chromosomal binding site of m icrotubules....................................................................... 33 13 EB1 binding to microtubule lattice.................................................................................... 34 14 Concentration of EB1 along length of m icrotubule........................................................... 34 21 Model for microtubule force gene ration by EB1 endtracking m otor............................... 47 22 Reaction mechanisms of EB1 endtracking motor............................................................ 48 23 Force dependence on EB1 binding and equilibrium surface position............................... 49 24 Microtubule elongation in the absence of external force...................................................49 25 Distribution of protofilament lengt hs for m icrotubule endtracking model...................... 50 26 Effect of applied force on MT elongation rate................................................................... 51 27 Thermodynamic versus simulated stall forces................................................................... 52 31 Schematic of nontethered, monovalent EB1 endtracking motor mechanisms................ 67 32 Various pathways of nontethered m onovalent EB1 binding to protofilament................. 67 33 Choosing an optimal K1value for monovalent EB1........................................................... 68 34 EB1 density profile on a nontethered m icrotubule protof ilament with monovalent EB1....................................................................................................................................69 35 Effect of K1 on profile of monovalent EB 1 occupational probability............................... 70 36 Schematic of tethered, monovalent EB1 endtracking motor mechanisms....................... 70 37 Force effects on a tethered pr otofilam ent with monovalent EB1...................................... 71 38 Divalent EB1 represented as divalent endtracking m otor................................................ 72 41 Mechanisms of a nontethered, divalent endtracking motor............................................ 95 42 Mechanisms of equilibrium, side binding of EB1 to protofilament.................................. 97 43 Choosing an optimal K1value for divalent EB1................................................................. 97 PAGE 10 10 44 Effect of kon on optimalK1.................................................................................................98 45 EB1 equilibrium binding.................................................................................................... 99 46 Occupational probab ility of EB1 along length of protofilam ent.....................................100 47 Time averaged EB1bound tubulin fraction at equilibrium............................................. 101 48 Time averaged fraction of EB1bound subunits during protofilam ent growth................ 102 49 Mechanisms of tethered, protofilament endtracking m odel w ith divalent EB1............. 103 410 Mechanisms of tubulin addition to linking proteinbound protofilam ent........................105 411 Forcevelocity profile s for tethered protofilam ents bound to divalent EB1 endtracking motors......................................................................................................... 106 412 Stall forces versus affinity m odulation factor at various KT values.................................109 413 Effect of f KT, and F on pathways taken.........................................................................110 414 Percent of time protofilament bound and unbound to motile surface.............................. 111 415 State of the terminal subunit (S1) when f =1 and f =1000 .................................................112 416 Fraction of S1 subunits bound and unbound from motile surface................................... 113 417 Average state of unbound linking protein........................................................................ 114 51 EM image of a ciliary plug at the en d of a ciliary microtubule....................................... 120 52 Schematic of ciliary plug inserted into the lum en of a cilia /flagella microtubule........... 121 53 Mechanism of the ciliary/ flagellar endtracking motor ................................................... 121 54 Force effects on ciliary m icrotubules............................................................................... 122 55 Ciliary plug movement................................................................................................... 123 PAGE 11 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 filamentbound EB1 CT: Effective concentration of track ing unit at protofilament plusend d: Size of tubulin protomer dbE: State of tubulin protomer attached to the subunit on the minusside of a doublebound EB1 dimer dbE+: State of tubulin protomer attached to the subunit on th e plusside 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 plusend 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 PAGE 12 12 (K1 k1/k1) K1 : Equilibrium dissociation constant for tubulin addition to trackbound protofilament (K1 k r/kf) K3: Equilibrium dissociation constant for EB1 subunit binding to trackbound filamentTGDP (K3 k side/k+side) K : Ratio of forward and reverse rate of EB1 subunit binding to protofilament plusend (K k +/k) K : Equilibrium dissociation consta nt for EB1 subunit binding to protofilament plusend (K k /k+) Kd: Equilibrium dissociation constant fo r EB1 (or TE) binding to filamentbound TGDP (Kd kside/konside) Kd*: Equilibrium dissociation constant fo r EB1 (or TE) in solution binding to TGTP at protofilament plusend Kpi: Equilibrium dissociation constant for reversible phosphate binding to T protomers KT: Equilibrium dissociation constant for track binding to solutionphase 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 k1: Reverse rate constant for tubulin in solution binding to EB1 k+: Forward rate constant for subunit of filamentbound EB1 binding to TGTP at protofilament plusend k+ side: Forward rate constant for subun it of filamentbound EB1 binding to filamentbound TGDP k: Reverse rate constant for EB1 (or TE) in solution binding to TGTP at PAGE 13 13 protofilament plusend kside: Reverse rate constant for EB1 (or TE) binding to filamentbound TGDP kBT : Thermal energy (Boltzmann constant, k, absolute temperature, T) kf: Forward and reverse rate constants for tubulin in solution binding to protofilament plusend kf,0: Initial forward rate constant fo r tubulin in solution binding to protofilament plusend kf E: Forward rate constant for EB1bound t ubulin in solution binding to TGTP at protofilament plusend kMT: Kinetochorebound 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 TGTP at protofilament plusend kon side: Forward rate constant for EB1 (or TE) binding to filamentbound TGDP kr: Reverse rate constant for tubulin in solution binding to protofilament plusend kr E: Reverse rate constant for EB1bound t ubulin in solution binding to TGTP at protofilament plusend kT: Forward rate constant for track bind ing to solutionphase EB1 (or TE or TTE) kT : Reverse rate constant for track binding to solutionphase EB1 (or TE or TTE) L : Length of protofilament in ciliary plug LLF: Lock, Load, and Fire model PAGE 14 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 EB1bound protomers on protofilament Pi: Phosphate p: Probability of protomer bound to EB1 subunit pend: Probability of EB1 binding to the protofilament plusend peq: Equilibrium probability of protomer bound to EB1 subunit pside: Equilibrium probability of EB1 binding to filamentbound TGDP q+: Probability of protomer in state dbE+ q: Probability of protomer in state dbEq: Probability of protomer atta ched to doublebound EB1 subunit qeq: Equilibrium probability of protomer attached to doublebound EB1 subunit S1: State of terminal protomer in protofilament S2: State of penultimate protomer in protofilament TAC: TipAttachment Complex model Tb: Tubulin protomer [Tb]: Tubulin protomer concentration [Tb]c: Critical tubulin concentra tion for free MT plusend [Tb]E c: Critical TE concentration for free MT plusend TE: EB1bound tubulin protomer TGDP: GDPbound tubulin protomer PAGE 15 15 [TGDP]: TGDP concentration [TGDP]()c: Critical tubulin concentrati on for TGDP at MT plusend TGTP: GTPbound tubulin protomer Tk2: Track bound to protofilamentbound EB1 Tk3: Track bound to protofilamentbound TE Tk4: Track bound to protofilamentbound dbE+ Tk: Track (tethering protein bound to motile surface) TkE: Track bound to EB1 in solution TkTE: Track bound to TE in solution TkTTE: Track bound to TTE in solution Tm: Time required for tubulin additi on and filamentbound 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 protofilamentbound EB1 subunit Transition state distance G Net free energy change of the tubulin cycle PAGE 16 16 G0 Initial free energy change of the tubulin cycle G()loss Free energy of TGDP dissociation from MT minusend G(+)add Free energy of TGTP ad dition to MT plusend Gexchange Free energy of GDP/GTP exchange in solution Ghydrolysis Free energy of MTbound GTP hydrolysis GPirelease Free energy of MTbound 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 PAGE 17 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 ENDBINDING 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 ubulebased 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 EndBinding 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 endtracking proteins is not fully understood, but they likely modul ate microtubulemotile 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 forcedependent 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 PAGE 18 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 endtracking motors that have a higher affinity for guanosine triphos phatebound tubulin than guanosine diphosphatebound tubulin, thereby allowing them to convert the chemical energy of microtubulefilament hydrolysis to mechanical work. These microtubul e endtracking 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 solutionphase tubulin. These models explain the observed expone ntial profile of EB1 on untethered filaments and predict that affinity modulated endtracking motors should achieve higher stall forces than with the Brownian Ratc het system, while maintaining a strong, persistent attachment to the motile object. PAGE 19 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). Endtracking proteins (a.k.a. tip tracking proteins), such as endbinding protein 1 (EB1) and adenomatous pol yposis coli (APC), ha ve previously been shown to bind specifically to the polymerizing microtubule plusend 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 endtracking proteins in force generation at these sites. A few models dem onstrate how endbinding pr oteins may be involved in microtubule forcegeneration, suggesting that endtracking proteins bind weakly to the microtubule plusend 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 endbinding proteins and a motile surface. The objective of my thesis research was to help elucidate the role of microtubule el ongation mediated by endbinding proteins in force generation. Our models explai n and characterize the in teraction of endbinding proteins with growing mi crotubule ends and their role in force generation Understanding the functions of microtubules and endtracking 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 endtracking protei n APC not only plays a po tentially key role in microtubulechromosome 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 PAGE 20 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 8nm 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 headtotail 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 slowgrowing end), and an tubulin at the other (plus or fastgrowing 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), GTPtubulin protomer s add to the plusend of a MT, the subunits then hydrolyze their bound GTP and subsequen tly release the hydrolyzed phosphate. During depolymerization (MT shortening) GDPtubulin subunits are released from the MT minusends 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 plusend during mito sis (Piehl and Cassimeris, 2003). Assuming PAGE 21 21 irreversible elongation at the MT plusend in vivo this value can be used to estimate the plusend protofilament effective onra te constant of tubulin (kf) association, Tbd v kf (11) where d is the length of a protomer and [Tb] the intracellular GTPtubulin concentration (~10 M; (Mitchison and Kirschner, 1987); yielding kf = 2 M1s1. The critical concentration for tubulin polymerization in vitro is [Tb]c = 5 M, which can be used to calculate an effective tubulin offrate ( kr) of 10.6 s1, assuming frckkTb /] [ (12) This calculated offrate is used to determine the reversible, elongation rate of the microtubule plusend (85 nm/s) by dkTbkvr fr. (13) 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 endtracking 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 plusend 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 PAGE 22 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 12) (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 plusend 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 GTPrich MT plusends (Severin et al., 1997). A comple x of proteins is required for PAGE 23 23 kinetochore capture by kMTs, but their interaction have not been elucidated (MimoriKiyosue and Tsukita, 2003). If one of the kinetochorea ssociated proteins coul d recognize and track the GTPrich 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 kinetochorekMT at tachment during mitosis have al so been shown to bind to the plusends of MTs (endtracking proteins), suggesti ng their likely role in such a mechanism. Of particular interest here are th e plusend 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 EndTracking Proteins Several MT endtracking proteins are thought to facilitate fo rce generation by microtubule polymerization (Schuyler and Pell man, 2001). Endtracking proteins localize to the MT plusend, 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 endtracking 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. PAGE 24 24 EB1 is a dimeric, 30kDa leucine zipper protein (MimoriKiyosue 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 plusends of mi crotubules. EB1 quickly disapp ears from the plus ends of depolymerizing GDPrich MTs, indicating that the higher EB1 de nsity at plus ends requires polymerization and/or a GTPrich 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) (MimoriKiyosue and Tsukita, 2003). This hypothesis is supported by dem onstrating that EB1null Drosophila cells cause mitotic defects including mispositioning of kinetochores dur ing congression (Rogers et al., 2002). Specific localization of EB1 to GT Prich MT plusends is not understood, but may be the result of direct binding of EB1 to the GTPstabilized conformation of the MT plus end, copolymerization with tubulin in solution, or recr uitment by other proteins, and dissociation from GTPbound MT subunits (Figure 13). 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 14. 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 (MimoriKiyosue 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, PAGE 25 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 TGTP did not bind to an EB1 construct. Contrary to his finding, Juwana et al. (1999) demonstrated that recombinant EB1 coprecipitates with purified bovine tubulin an immunoaffinity assay, despite the 100time lower concentration of EB1 than TGTP (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 tubulinaffinity 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 plusends 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 colocalize and interact with both EB1 and pol ymerizing microtubule plusends at the kinetochore and at the ce ll cortex (Juwana et al., 1999). Like EB1, APC falls off the microtubule upon plusend depolymerization. The Cterminal do main of APC (CAPC) 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 CAPC, 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 EB1binding domain, APC localizes nonspecifi 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 kinetochoreasso ciated proteins (p150Glued, CLIP170, and CLASPs) also have direct interactions with EB1, have the ability to bind to the MT plusend, and are located at the kinetochoreMT interface (Folke r et al., 2005; Hayashi et al., 2005; MimoriKiyosue et al., 2005). Therefore, these components may also be involved in activation of EB1 at the MT tip and/or linking the EB1bound MT pl usend 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, membranebound capping struct ures (or MT plugs) are persistently associated with the plusends 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 crossreactivity (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 colocalizes with the plusends 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 endtracking properties of EB1 ar e likely to play a role in MT elongationdepe ndent force generation. 1.3 Force Generation Models Although much progress was made identifying mi crotubuleassociated proteins and their locations, how MT elongation is coupled to force generation has not been determined. Various forcegenerating models have been considered, including force from microtubule polymerization, force from moto rprotein 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, (14) PAGE 28 28 where 13 pN is the number of protofilaments, TkB = 4.14 pNnm is the thermal energy (Boltzmann constant absolute temperature), and d = 8 nm is longitudinal dimerrepeat distance. Under typical intracellular tubulin (Tb) concentrations of 1015 M and a plusend critical concentration [Tb]c = 5 M (Walker et al., 1988), then Fstall = ~57 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 antipole 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 randomwalk approach, where the freeenergy source is the binding of GTPtubulin protomers to MT ends (Hill, 1985). The Tip Attachment Complex model (TAC) incor porates the idea of a sleeve in order to model forcegeneration 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 weakbinding 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 endtracking 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 motori nduced forcegeneration model. One such model is known as the reverse PacMan mechanism (Maddox et al., 2003), where plusend directed motors move kinetochores antipole ward during plusend kMT polymerization (Inoue and Salmon, 1995). The plusend directed motor protein, CENPE, 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 antipoleward kinetochore motility in the cell; there are other kinetochoreassociated mo tor proteins (i.e., MCAK) of unknown function. One recent forcegeneration model, the slipclutch model, integrates both the reverse PacMan and lateralTAC mechanisms. This model represen ts the polymerization st ate of the kinetochore by a slipclutch 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 EndTracking Motors Dickinson & Purich (2002) first proposed a model for actinbased motility whereby endtracking proteins tethered elongating filaments to motile objects and facili tated force generation. In this mechanochemical model for actin s ubunit addition, surfacebound endtracking proteins bind preferentially to newly added ATPbound termin al subunits on each subfilament and release PAGE 30 30 from ADPbound 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 hydrolysisdriven processive tracking on the filament end gives the endbinding protein the characteristics of a molecular motor. We later proposed that the interaction of microtubule endtracking 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 GTPdriven 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 15). The net free energy of this cycle is the sum of these individual free energies exchange loss releaseP hydrolysis F addGG G GGGi (15) this is equal to the net free energy of GTP hydrolysis: i BPGDPGTPTkGG ln0, (16) where TkGB110 is the standardstate fr ee energy change for tubulin in vivo (Howard, 2001). The free energy changes for the individual asse mbly steps are listed in Table 11, where KPi is the equilibrium dissociation constant of reversible phosphate binding to GDPtubulin protomers and KX is the equilibrium constant for the GTP/GDP exchange reaction. Based on literature values (Table 12), the free energy from the combin ed filamentbound hydrolysis and phosphaterelease 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 plusends (~5.8 kBT ). Hence, considerably greater forces can be expected by exploiting the ability of endtrackers like EB 1 that bind preferentially to TGTP protomers, thereby providing a pathway fo r harnessing the energy released by MTbound 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 ndtracking proteins, EB1 and APC. EB1 is known to spec ifically localize to the GTPri 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 forcegeneration mechanisms invol ving endbinding proteins and MTs have been proposed including TAC and mode ls involving ATPdriven MT motors kinesis and dynein, which move on MT sides. This thesis explores the hypothesis that endtracking motor facilitate plusend 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 endtracking proteins binding specifically to the NTPbound monomers on the filaments, a feature correlates well with the properties of the MTs a nd their correspond ing endtracking proteins. 1.6 Outline of Dissertation The layout of this dissertation is as fo llows. Chapter 2 describes a preliminary mechanochemical MT endtracking 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 endtracking 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. Endtracking 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 endtracking 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 endtracking from a rigid plug, reflecting ciliary /flagellar gr owth. Finally, Chapter 6 summarizes the work completed and suggests future directions. PAGE 33 33 Figure 11. Microtubule structure. Tubulin bound to GTP polymerize into 13protofilament polymers: microtubules. Because tubulin is a heterodimer, the microtubule has a structural polarity with a plus and mi nus end. During MT polymerization, TGTP binds to the MT plusend, which induces hydrolysis of the penultimate tubulin subunit causing filamentbound GTP to be converted to GDP. TGDP dissociates from the minus end. Figure 12. 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 2328, p. 1094). New York, New York.] PAGE 34 34 Figure 13. EB1 binding to microt ubule lattice. EB1 has equal asso ciation and dissociation rates on GDPbound microtubule latti ce. EB1 may bind directly to the microtubule plusend or copolymerize with t ubulin in solution first. Figure 14. 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. EB1mi 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) .] PAGE 35 35 Figure 15. 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 GTPGDP exchange in solution. Table 11. Thermodynamic equations characte rizing the multiple steps for GDP to GTP conversion Definition Equation Addition of TGTP to MT plusend c B addTbTbTkG /ln)( Phosphate ( Pi ) release pi B releasePKPTkGi/ln Loss of TGDP from MT minusend c B lossGDPTGDPTTkG)( )(/ ln GDP/GTP Exchange X B B exchangeKTk GTPGDPTGDPTbTkG ln / ln Hydrolysis of MTbound GTP in terms of free energy of other steps c c XP B hydrolysis FTb GDPT KKTkG Gi)( 0ln PAGE 36 36 Table 12. 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 TGTP addition to MT plusend 0.03 M (Howard, 2001) [TGDP]()c TGDP addition to MT minusend 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 ENDTRACKING MODEL This chapter describ es a preliminary model that simulates the growth of a 13protofilament microtubule (MT) in the presen ce of surfacetethered EB1 endtracking 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 endtracking motor is that it captures filamentbound 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 plusend 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 Gaussianbased 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 EndTracking Motors The preferred binding of EB1 to MT plusends is reminiscent of the interaction between endtracking proteins and actin in the actoclamp in endtracking motor m odel and suggests that endbinding proteins may behave as endtracking 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 plusend through its MTbinding domain on the other end. There are two key features of a MT endtracking motor: affinitymodulated 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 endtracking motors to maintain a strong inter action with the protofilament even when other endtracking 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 MTbinding protein. 2.2 Microtubule Growth Model Our preliminary MT endtracking model illustrated in Figure 21 simulates the growth of a 13protofilament microtubule bound to surfacetethered EB1 motors and analyzes the force effects on the growth of the microtubule. By capturing part of the filamentbound 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 22 and include several possible endtracking stepping motor pathways for two EB1 dimeric subunits (referred to hereafter as EB1 heads) operating at the plusend of each MT protofilaments. Considering Stage A, where only one EB1 head is bound to terminal GTPbound 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 TGTP 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 TGTP 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 EB1MT 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 endtracking cycle, the protofilament remains associated with the motile object (i.e., the motor is processive). (Longterm 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 hydrolysisinduced affinity modulation, the principle of detailed balance would fix the relation among the various equilibrium dissociation constants in Reactions 13 shown in Figure 22, 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 (21) 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 GTPhydrolysisdriven 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 ~7pN stall force predicted for a Brownian Ratchet driven solely by the free energy of protom er addition (c.f., Eq. 14). While Eq. 21 provides a thermodynamic limit, MT growth by the endtracking 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 EB1bound 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 coiledcoi l region and its two MTbinding 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 (22) 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 (23) where kB is Boltzmanns constant, and T is the absolute temperature. The positiondependent binding rate constant k+( z) (s1) of the EB1 head to th e MT lattice at distance z is taken as )()( zCkzkeffon or: /2 )(2/)( 22Tk e kzkB Tkzz onBe (24) where is the EB1 spacing distance, k1 is the forward association rate constant ( M1s1 or nm3/s) for EB1 binding to a TGDP subunit from solu tion. Because binding s ites are at discrete positions spaced by distance d = 8 nm, then z = nd in Eq. 24. 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 ~100nm motile object, this time would be ~10100 s, much faster than the cycle time for protomer addition. Therefore, we assume the instantaneous position of the motile object remains in mechanical quasiequilibrium 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 25; the position of bound EB1 h eads in each stage can be determined from Figure 23. ))32(())12(())1(()(D state C state B state A state e i e i e i eizdn zdn zdn zdnF (25) Solving Equation 25 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 24, 25, 26, and 27, 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 22. The intracellular tubulin concentration [Tb] was assumed to be 10 M (Mitchison and Kirschner, 1987). We use the value of the plusend critical concentration [Tb]c = 5 M estimated by Walker et al. (1988) from th e ratio of onand offrate constants for elongation (8.9 M1s1 and 44 s1, respectively). The macroscopic onrate constant (8.9 M1s1) 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 onrate constant kf= 8.9/13 M1s1 or 0.68 M1s1 and kr = 44/13 s1 or 3.4 s1. 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 zpositi 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 MTbound TGDP subunit, kon = 25 M1s1 = 57 nm3/s, was assumed by taking a typical association rate constant for protein binding in solution (Eig en and Hammes, 1963). The offrate constant koff = 0.24 s1 for an EB1 dimer from MT GDPsubunits wa s calculated from the measured velocity and the exponential decayleng 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 offrate 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 24. 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 onrate was chosen to yield the experimentally determined velo city of 167 nm/s for microtubules during mitosis (Piehl and Cassimeris, 2003). Figure 25 is a representation of the protofilament lengths and average equilibrium surface position corresponding to (st 4 v =165 nm/s). As seen in Figure 25, the maximum difference between the shortest and lo ngest filaments is four subunits. This small difference reflects how the endtracking 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 endtracking 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 26 shows that the speed of MT polymerization decreased with increasing extern al load for all values of f calculated. When the endtracking protein wa s not affinitymodulated ( f = 1), the velocity decreased linearly with increasing external force. As f was increased, the endtracking 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 GTPbound filament, and promoted the forw ard MT assembly process. Although large tensile forces should dissociate the filament endtracking 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. 25) 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 forcelimitations on the reaction kinetics. By increasing f the forward PAGE 45 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 11). 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 25, and are compared to these simulated stall forces in Figure 27. 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 endtracking 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 forcedependen t exponential equation (Figure 25). 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 13protofilament MT bound to surfacetethered EB1 motors and serves to dem onstrate the principles of force generation by processive MT endtracking motors. The key feat ures of these endtracking motors are (1) their PAGE 46 46 ability to capture filamentbound GTP hydrolysis and convert them to mechanical work (2) their dimeric structure which allow them to maintain persistent attachment of the MT plusend 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 TGTP versus TGDP 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 solutionphase EB1. The propos ed cofactor assisted endtracking model not only addresses the importance of a cofactor 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. PAGE 47 47 Figure 21. Model for microtubule force genera tion by EB1 endtracking motor. Model that represents the distal attachme nt of tubulin protomers at the MT plusend. A uniform density of EB1 dimers on the motile object links the MT protofilaments to the surface. PAGE 48 48 Figure 22. Reaction mechanisms of EB1 endtracking motor. Mechanism of the EB1 endtracking motor on the plusend of one of the MT protofilaments (from upper left): One EB1 head is initially bound to th e terminal GTPtubulin 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. PAGE 49 49 Figure 23. 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 24. Microtubule elongation in the absence of external force. (A) The position of a 13protofilament MT tethered to a surface by EB1 endtracking 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. PAGE 50 50 Figure 25. Distribution of protofilament lengths for microt ubule endtracking 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. PAGE 51 51 Figure 26. 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 threeparameter 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. PAGE 52 52 Figure 27. Thermodynamic versus simulated st all forces. The thermodynamic stall force was calculated for the various MT endtracki ng motor represented in Figure 4 by using equation 25. 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 endtracking 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. PAGE 53 53 CHAPTER 3 PROTOFILAMENT ENDTRACKING MODEL W ITH MONOVALENT EB1 The microtubule endtracking model developed in Chapter 2 neglected solutionphase End Binding protein 1 (EB1) and binding to microtubules and tubulin prot omers. While not essential for endtracking, binding of EB1 from soluti on is evident in the e xponential prof iles of bound EB1 at elongating free plusends and the apparent equilibrium density of EB1 along the length of the microtubule (MT) (Chapter 1 and Figure 13). To account for binding solutionphase 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 endtracking motor, with preferential binding to TGTP over TGDP, 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 tiptoside ratio of EB1 density requires GTPhydrolysisdriven 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 forcevelocity relations hip of this the endtracking model is then compared to those of the simple Brownian Ratchet mechanism. 3.1 NonTethered Protofilament Growth We first consider growth of a single micr otubule protofilament in the presence of a solutionphase monovalent EB1 and then show in Section 3.2 how linking the growing tip to a PAGE 54 54 surface containing a flexible binding protein for EB 1 forms an endtracking motor similar to that described in Chapter 2. The various reactions considered in the free MT model are shown in Figure 31. 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 EB1tubulin 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 EB1tubulin interactions As described in Chapter 1, free energy of the direct binding pathway is given simply by c B addTkG ]Tb[ ]Tb[ ln)( (31) 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 )( (32) 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. 32 holds whether E binds first to Tb in solution or E binds to the terminal subunit following monomer addition. Eq. (32) can be rewritten fTk TkGB B addln [Tb] Tb][ lnc )( (33) 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 GDPtubulin with equal affinity. PAGE 55 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 (34) and dK MT K Tb KTbE TE ][][ 1 /][ ][1 1 0 (35) 3.1.2 Kinetics of EB1tubulin interactions We have assumed that the affinity modulation factor must be greater than 1 for EB1 to track on the plusends 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 32: EB1 binds directly to the GTPrich protofilament plusend, EB1 associates with TGDP 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 plusend, pend, and the equilibrium probability of EB1 binding protofilament sides, peq. The value of peq was obtained by the steadystate 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 (36) PAGE 56 56 The equations specific for solving the probability of EB 1 binding to the plusend and side of the protofilament are given by iir E rii E f fiii i i ippckpkppTEkTbkpkcEk dt dp 111 1 (37) fK cpkKTb pckTcTEkTbppkcEk dt dpE fd c f c E f 1 21 21 1 111,111, 1][ ][, (38) 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 plusend. These two differential equations were numerically integrated (fourthorder R ungeKutta 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 plusend 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. 12, respectively, assuming a maximum velocity v =170 nm/s (Piehl and Cassimeris, 2003). Th e dissociation constant for EB1 to the GDPbound side ( Kd) of protofilaments was taken as 0.5 M, based on an in vitro study on EB1 PAGE 57 57 MT binding interactions (Tirnaue r et al., 2002b). The rate equati ons for the on rates of EB1 to the MT plusend 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 offrates for EB1 on both the sides and pl usend of the protofila ment are a fraction, f less than their onrates. This onrate 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 steadystate, Eq. 39) and pend (from Eq. 38). E K pd eq 1 1 (39) 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 steadystate. This procedure was repeated for various values of f and the resulting K1 values are shown in Figure 33. 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 TGTP than TGDP (i.e., f must be greater than two) in order for EB1 to accumulate at the plusends 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 plusend 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 33. 3.1.4 Results Figure 34 shows the steadystate 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 plusend reflects some be nefit of copolymerization with tubulin. However, EB1 is predicted to have a much larger density at the plusend when f is large. The EB1 density decreases exponentially along the length of the protofilament, consistent with experimental observations (Figure 14). This fi nding supports our assertation EB1 must have a significantly higher affinity for GTPbound tubul in in order to track on the GTPrich protofilament plusends. The effect of K1 on the EB1, monovalent o ccupational probability pr ofile is demonstrated in Figure 35. 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 plusend 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 34. This behavior is expected because at larger values of K1, EB1 has a large offrate 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., PAGE 59 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 nontethered m odel (Figure 36). 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 plusend. 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 EB1binds 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 nontethered monovalent endtr 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: PAGE 60 60 pi = probability of EB1 bound to the protofilament qi = probability of TkE 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 (310) Similar to the derivation of Equations 37 and 38, the transition probabilities between states can be obtained from reaction rate consta nts for each pathway (Figure 36). The resulting differential equations for the probabilities of EB1 and TkE binding to the protofilament (in terms of the kinetic rates) are given by Equations 311 and 312, respectively, where ui =1qipi, kT is onrate of the linking protein binding to solutionphase 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 ,,) ][][( ][ (311) ii E rr E riiieff E f E f f iiieff iiTiieffiT iqqqkukpkqqwCkTEkTbk qkvuCkqkypCk dt dq 11 11 1 ,,) ][][( (312) The differential equations for the probability of the track binding to e ither TE (313) or EB1 (314) 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 (313) PAGE 61 61 ii i iieffion i T TqkvuCkwkvTbkvkyEk dt dv, ,, 1 1][ ][ (314) These ordinary differential equations were solv ed using a fourthorder RungeKutta 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 steadystate 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 (315) There are two ways tubulin can add to the protof ilament plusend, directly with an onrate of kf or copolymerizing with EB1 with an onrate 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 (eFd) and the dependence of the forward rate on the critical con centration when the protofilament plusend 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 TkTE addition, and the dependence of the forward ra te on the critical concentration [Tb]c E when the protofilament plusend is bound to EB1. The probabilities, qend and pend, were solved from Equations 311 and 312 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, nontethered case. The kinetic rate constants were cal culated from detailed balance. The onrates for an EB1 subunit (or head) to the protofilament side ( kon side) and to the PAGE 62 62 protofilament plusend ( kon) were calculated based on the obser ved decay rate of an EB1 dimer from the MT side, koff = 0.11 s1. 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 (316) Rearranging this equation gives d off on side onKE k kk (317) where Kd kside/ kon side. The offrate constant for EB 1 from the GDPbound tubulin subunits, kside, is calculated from Kd. The offrate of EB1 from the protofilament plusend, k, is the equal to the offrate of EB1 from divided by a factor of f The linking protein was assumed to be a flex ible, springlike 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 3D normal distribution on a halfsphere. The normal Gaussian distribution of the spri ng is given by Equa tion 318, where is the spring constant and is equal to kBT /2, s B TATk d C 1 2 exp 2 12 (318) 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. PAGE 63 63 3.2.4 Results Figure 37A shows the predicted density of EB1 along the length of the protofilament (zero represents the plusend). The protein species consider ed are the nontethered 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 plusend 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 plusend, and EB1 is expected to have preferential binding to the protofilament plusend 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 onrate of linking protein to EB1 at this location is much greater than its dissociation rate. Th e nontethered EB1 experiences a small peak in probability near the protofilament plusend, likely because it was initially GTPbound, and eventually dissociates from TGDP. The forcevelocity profile for these mechanis ms is shown in Figure 37B, 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 endtracking 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 endtracking 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 38), 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 endtracking 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 additiondriven (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 endtracking motor, with preferential binding to TGTP over TGDP, 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 NonTethered 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 TGTP in solution, this model accounts for se veral reaction pathways to allo w EB1 to bind with tubulin in solution as well as filamentbound tubulin. The dissociation constant for EB1 and free TGTP was taken as that need to provide a 4.2 ratio of EB1bound subunits at the protofilament plusend versus protofilament sides, which would correlat e well with experimental results. Large affinity modulation factors resulted in an equilibrium value for the tubulinEB1 dissociation constant, and are therefore optimal for simulation purposes. Regardless of the value for other key kinetic rates (i.e., onrate of tubulin, kf, or onrate of EB1 on protofilament sides, k+), it is required for EB1 to have a larger affinity for TGTP rather TGDP ( f >1) in order to achieve the 4.2 ratio. PAGE 65 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 endtracking 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 EB1tubulin dissociation constant was used to calculate the binding probability of EB1 to the pluse 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 nearuniform 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 plusend that decays along the length of the protofilament, a prediction which agrees to experimental results showing the decay of fluorescent EB1 on a nonteth ered microtubule. To simplify this complex model, several assu mptions were made. First, tubulin addition induces filamentbound hydrolysis at the protofilament plusend. The affinity modulation is assumed to affect only the offrates of the protei n interactions and not the onrates. 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 cofactor protein in endtracking 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 nontet hered, monovalent endtracking PAGE 66 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 nontethered) demonstrates that at large affinity modulation factors th ere is a high occupation of total EB1 at the protofilament plusend, 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 freegrowing protofilaments. More importantly was the effect of force on the endtracking model. This model demonstrates the potential of the monovalent endtracking 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 springlike 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 endtracking complex and the protofilament that would affect the occupational probability of EB1 and the forcevelocity profile. The subsequent tethered protofilament growth model with divalent e ndtracking EB1 motors will account for the force effects on the linking protein. PAGE 67 67 Figure 31. Schematic of nontethered, monova lent EB1 endtracking 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 EB1tubulin complex ([Tb]c E). GTP hydrolysis on the penultimate subunit occurs upon tubulin addition to the protofilament plusend. Kd is the EB1 dissociation from the protofilament plusend and Kd is the dissociation constant for EB1 from TGDP, where f = Kd/Kd *. Figure 32. Various pathways of nontethered monovalent EB 1 binding to protofilament. EB1 can bind directly to the GTP rich protofilament plusend, or EB1 can associate with TGDP on the side of the protofilament, or EB1 can copolymerize with tubulin in solution. PAGE 68 68 Figure 33. 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 M1s1. The value of K1 required to achieve a tiptoside 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. PAGE 69 69 Figure 34. EB1 density profile on a nontethered 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 plusend, which decreases along the length of the protofilament. PAGE 70 70 Figure 35. 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 plusend, 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 34. Figure 36. Schematic of tethered, monovalent EB1 endtracking motor mechanisms. Tubulin protomers can add directly to filament ends or they can first bind to surfacetethered EB1 in solution then add as an EB1tubulin complex. GTP hydrolysis on the penultimate subunit occurs upon tubulin additi on to the protofilament plusend. EB1 is allowed to dissociate from surface linki ng protein (pathway not shown here). PAGE 71 71 Figure 37. Force effects on a tethered protof ilament with monovalent EB1. A) Occupational probability of EB1 along lengt h of protofilament (zero represents plusend) at f =1000. Two protein species considered: untethered EB1 and tethered EB1 on protofilament. Occupation of EB1 for each sp ecies decreases along length of protofilament. B) ForceVelocity profile. Maximum achiev able force for endtracking model at f =1000 (~1.2 pN) exceeds that of Brownian Ratchet model (0.4 pN) PAGE 72 72 Figure 38. Divalent EB1 represented as divalent endtracking motor. A) Depiction of EB1 structure characterized from crystal structur es. The Cterminus is represented by (C) and the Nterminus is represented by (N). Reprinted by permission from Macmillan Publishers Ltd: [ Nature ] (Honnappa et al., 2005) copyrig ht (2005). B) Schem atic of endtracking motor complex comparable to crystal structure of EB1. PAGE 73 73 CHAPTER 4 PROTOFILAMENT ENDTRACKING MODEL W ITH DIVALENT EB1 The protofilament endtracking 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 endtracking motors. Similar to Chapter 3, we first model the growth of an untethered protofilament in the presence of solutionphas e EB1 endtracking 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 tiptoside 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 forcedependent velocity was compared to that of the Brownian Ratchet mechanism. 4.1 NonTethered Protofilament Growth As in the models from previous chapters, EB1 is assumed to preferentially binds to TGTP rather than TGDP. 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 EB1Tubulin Interactions The reactions considered in this protofilament model are shown in Figures 41 and 42. 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 solutionphase 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 noncooperative; this pr operty allows EB1 to bind to PAGE 74 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 (41) 1]][[2 ][ K TbE TE (42) 2 1 2K TbE TTE (43) Figure 41A 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 EB1tubulin complex ( kf E). Since unbound EB1 can bind to Tb in two identical ways, the onrate 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 plusend (Figure 41B). One pathway involves direct binding of the EB1 head to the terminal, GTPbound subunit ( k+) after tubulin addition ( kf). The other pathway allows the EB1 head to bind to solutionphase Tb ( k1) and facilitate tubuli n addition by shuttling it to the protofilament plusend ( kf E ). The value for kf E accounts for the onrate of TE and the local, effective concentration of the unbound EB1 head ( Ceff), which is represented by Equation 44. eff E f E fCkk ''' (44) 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+ (doublebound to EB1 on plusend) and the penultimate subunit is in state dbE(doublebound to EB1 on minusend). Figure 41C shows how TE in PAGE 75 75 solution can bind the protofilament: binding di rectly to the protofilament plusend ( kon) after tubulin addition ( kf), or by first binding to tubulin in solution ( k1) then adding as an EB1tubulin complex (2 kf E). Figure 42 shows the various mechanisms by which EB1 can bind to the GDPrich protofilament side. As shown in the figure, fr ee EB1 can bind directly from solution to a TGDP 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 onrate of kon side (pathway C). 4.1.2 EB1 Occupational Probability Model A probabilistic model similar to the tethered, divalent endtracking m odel (Chapter 3) was developed to simulate the pathways shown in Figure 41. 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 45. iiii iwpqqu 1 (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), PAGE 76 76 and are represented by Equations 46 through 410 (Appendix A.3), where R+ and Rare defined by Equations 411 and 412. ii ii ii ii ii ii i i ion ippRppR upkqkupkqkpTbkwkpkuEk dt dp 1 1 1 11,1 1 1 1][ ][2 (46) ii ii i i i i on iwwRwwRwkpTbkwkuTEk dt dw 1 1 1 1][ ][ (47) ii ii iiiiiiii ii iqqRqqRupkupkqkqk dt dq1 1 11. 1, .11. (48) ii ii iiiiii iiii iqqRqqRupkupkqkqk dt dq1 1 11, 1,11, (49) ii ii i onioniiiiiii ii i iuuRuuR uTEkuEkupkupkqkqkwkpk dt du 1 1 1, 1,][][2 (410) 1])[]([][ wCkTTETEkTbkReff E f E f f (411) 111 1qpwkukRE rr (412) At equilibrium, these probabilities redu ce to Equations 413 through 416, where qeq qeq + + qeq and K k+ side/ kside. eq on equ k Ek p][2 (413) eq eqp K Tb w1 (414) eqeq equpKq 2 (415) PAGE 77 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 (416) The results of these equations were used to analyze the distribution of the divalent, EB1 endtracking motors on the nontethered protofilament, and determine the equilibrium EB1 concentration along the protofilament, Peq. 2/eqeqeq eqqwpP (417) 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 nongrowing 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 pluse 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 EB1binding state of each subunit was used to determine the fraction of EB1bound subunits in the protofilament; this fraction was averaged over time for a total simulation time of 40 seconds. 4.1.4 Average Fraction of EB1bound subunits during protofilament growth A stochastic model was also used to calcu late the timeaveraged fraction of EB1bound subunits during protofilament growth. The pathways considered in this model are those shown in Figure 42., as well as the association and dissociation of tubulin from the protofilament plusend (Figure 41). The stochastic model used to simulate these pathways is very similar to PAGE 78 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 EB1binding state of each subunit was determined for each time step and the fraction of EB1bound 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 41 and 42 ha ve not yet been determined experimentally, including the dissociation constant for Tb and EB1 in solution ( K1) the EB1 onrate for protofilamentbound TGTP ( kon), the EB1 offrate from protofilamentbound TGDP ( 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 filamentbound subunits attached to EB1 is given by Equation 418 (see appendix A.3 for full derivation). 1 ][ 1 ][ ][][, 0 0 tot effdMT K E EE (418) 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 419, 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 (419) Under this constraint, u1/2 is given by Equation 420, where k+ side= kon sideCeff for the protofilament plus end, and kside= k+ side/ K. PAGE 79 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 ][ (420) Assuming [Tb]=0, [E]0=0.27 M, [E]=[E]0/2, and Ceff=153 M, Equations 419 and 420 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 421, where koff is the known offrate of dimeric EB1 from a protofilament ( koff =0.26 s1, (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 (421) To determined the optimal value of K1 (that provides a 4.2 tiptoside 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 43 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 TGTP than TGDP (i.e., f must be greater than one) in order for EB1 to accumulate at the plusends 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 PAGE 80 80 these rates (EB1 binding to the protofilament plusend). 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 M1s1, 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 M1s1 to 10 M1s1. The resulting EB1 tiptoside binding ratios for these conditions are shown in Figure 44. 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 M1s1 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 onrate of EB1 and tubulin ( k1) was assumed to be a typical value for proteinprotein binding interactions, 10 M1s1. Consequently, the value of k1 was determined from the optimal K1 value ( K1= k1 / k1). The onrate constant for TE and TTE to the protofilament plus end, kf E, was assumed to be equal to the onrate of tubulin addition, kf. The offrate constant of TE and TTE to the protofilament plusend, kr E, was calculated based on detailed balance, and is represented by Equation 422. on c E f E rkkK Tk k 1 (422) PAGE 81 81 4.1.6 Results 4.1.6.1 Occupational probability The ordinary differential equations that define the probabilities of EB1 binding (Eq 46 to Eq. 410) were solved to determine the expect ed equilibrium fraction of EB1bound protomers in the protofilament (Figure 45). This fraction was evaluated by determining Peq from Equation 417 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 46 to 410 were numerically integrated and solved at a set value of K =37 and K1=0.65 M1s1 using a fourthorder RungeKutta method in Matlab. The occupational probability of EB1 along the length of the protofilament for f =1 and f =1000 is shown in Figure 46. 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 GDPbound subunits, and this is the equilibrium EB1 binding fraction determ ined earlier (Figure 45). 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 TGTP versus TGDP when f =1000. This decay profile is comparable with the experimental results show n in Figure 14. The ratio of the occupational PAGE 82 82 probability at the plus and minus end of the protofilament for f =1000 is about 4.3, which is similar to the tiptoside 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 EB1bound subunits at equilibrium The time averaged fraction of EB1bound protomers at f =1 and f =1000 and optimal values for K1 (0.65 M) and kon (5 M1s1) are displayed in Figure 47. 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 filamentbound TGDP, 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 offrates and not the onrates on EB1, TE, or TTE to the protof ilament plusend, 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 EB1bound su bunits during protofilament growth Figure 48 displays the average fraction of EB1bound 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 M1s1 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 plusend. This PAGE 83 83 result is expected since when f =1000 the rate of EB1 addition at the protofilament plusend is increased. When f =1000, there is a sharp decrease in the EB1bound 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 GDPbound tubulin subunits and hence significantly reduced offrates from the protofilament plusend ( kr E and k). Conversely, when f =1 the EB1bound 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 theredprotofilament model disc ussed in section 3.2, considers the growth of a single microtubule protofilament in the presence of solutionphase, divalent EB1. A flexible binding protein provides as a link between EB1 and a motile surface, allowing EB1 to behave as an endtracking motor. The various reaction mechanisms considered for this tethered, protofilament model are those previously s hown in Figures 41 and 42, and the binding pathways involving the surfacebound tethering protein (Figures 49 and 410). 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 (423) 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. 317 in section 3.2.3. The assumptions and parameter values used are the same as those in section 4.1. PAGE 84 84 4.2.1 Kinetics of EB1Tubulin Interactions Figures 49 A and B shows the two methods by which EB1 can bind to the surfacebound tethering protein, hereafter abbreviated Tk. Tk can associate with EB1 in solution (pathway A) or with protofilamentbound 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 offrates. In pathway B, Tk binds to EB1 on the protof ilament (this could also be filamentbound TE or doubly bound EB1). The forward kinetic rate of this reaction, kT is represented by Equation 424, and is proportional to the forw ard rate of Tk binding to EB1, kT, the effective concentration of Tk at the protofilament plusend, CT, and the effects of the transition state and spring energies. TkdnTkdn TTTB BeeCkk21 212' (424) 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 plusend and increases toward the minusend 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, surfacetethered EB1 protein and its equilibrium binding position on the protofilament is n1. Hence, the displacement distance of the spring is given by ( n1 ) 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 49 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: PAGE 85 85 Tkdn TTBekk21' (425) The mechanisms by which surfacetethered EB 1 can bind to the protofilament are shown in Figures 49 C and D. Surfacebound TE can attach to the protofilament w ith a forward rate of kon and a reverse rate of k (pathway C), given by Equations 426 and 427, respectively. TkdnTkdn TononB BeeCkk21 212' (426) TkdnBekk21' (427) The contribution of the transition st ate and spring energies are equa l to that in equations 424 and 425. If the surface is initially tethered to unbound EB1 (pathway D), the onrate 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 onra te of EB1 in solution to a protofilamentbound 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 minusend. These two potential pathways are shown in Figure 49 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 minusdirection (pathway F), the kine tic rates will be affect ed by the transition and spring energies. These rate equations ar e described in Equations 428 and 429. TkdnTkdn side sideB Beekk221 212' (428) Tkdn side sideBekk21' (429) PAGE 86 86 Figure 410 shows the various wa ys tubulin can add to the protofilament plusend that involve the linking protein. Tubulin can add dire ctly to the plusend of the surfacetethered protofilament (mechanism A), tubulin can be transferred to the protofilament plusend by surfacetethered TE (TkTE) or TTE (TkTTE) as seen in pathways B and C, respectively, or the protofilamenttethered TE can shut tle tubulin to the protofilament end (mechanism D). Only the onrates, not the reverse rates, for these reactions are affected by the interaction with the tethering protein. For all four path ways (AD), when there is an applied force against the surface (in the opposite direction of protofilament growth), F the tubulin onrates are reduced by a factor ofTkFdBe/ For direct tubulin addition in pathway A, the forward rate, kf is proportional to the onrate of tubulin addition, kf, and the effect of force, as shown in Equation 430. 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' (430) Tubulin transferred to the pr otofilament end by TkTE or TkTTE (pathways B and C) are both proportional to the onrate of TE (o r TTE) to the protofilament plus end, kf E, as seen in Equation 431. There is also an effect from CT and any load applied to the motile surface. The onrate of tubulin in these pathways is twice that of kf E because of EB1s dimeric structure. kTFd T E f E feCkk/' (431) Tubulin shuttled to the protofilament pluse nd by filamentbound EB1 (pathway D) has a corresponding onrate of kf E. This forward rate (Equation 432) 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 3D nor mal distribution on a halfsphere (see section 4.2.3). PAGE 87 87 TkFdTkd eff E f E fB Be eCkk/ 2212'' (432) 4.2.2 Protofilament EndTracking Model Considering the above reactions, st ochastic models were performed to analyze the behavior of divalent, EB1 endtracking 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 onrate of Tk and EB1 binding (kT) was estimated as 5 M1s1 and the offrate, kT was calculated from the value of KT provided ( KT=kT / kT). The value for v used to calculate the tubulin onrate from Equa tion 11 was 170 nm/s (Piehl and Cassimeris, 2003), which was assumed to be the irreversible elongation at the protofilament plusend. 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 surfacetethered 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 PAGE 88 88 plusend ( naddd) by the total simulation time, t This model also provided the state of the terminal subunit, position of the linking pr otein, the timeaveraged fluorescence along the protofilament, and time spent in each pathway. Figure 411 shows the forcevelocity prof iles for a polymerizing protofilament with surfacetethered EB1 endtracking motors. To analyze the effect of the affinity modulation factor n the velocity profile, several affinity modulation factors were considered (Figures 411 AE). 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 13), 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 plusend. 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 M1s1, the stall force PAGE 89 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 411 F. Figure 412 summarizes the effect of KT and f on the stall force, with the corresponding data in Table 41. 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 TGTP a nd TGDP 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 endtracking 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 endtracking mechanism mediates tubulin addition and to understand the effects of f and KT on the velocity profiles (Figur es 411 and 412), 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 413. For an affinity modulation factor of PAGE 90 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 plusend. 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 411 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 411 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 411 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 411 F). The percent of time the protofilament spen t bound and unbound to the motile surface is shown in Figure 414. 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 433 shows that when the linking protein binds to EB1 on the terminal subunit (n =1), the onrate is proportional to kTCT. But when n is greater than one, the onrate, 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 filamentbound 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 415, where states Tk2, Tk3, and Tk4 represent states in which the PAGE 91 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 416. 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 EB1bound tubulin addition decreased (Figure 413). 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 417). 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 PAGE 92 92 nontethered, divalent EB1 endtracing motors that processively linking pr otein the plusends 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 endtracking model has an advantage over the monovale nt endtricking model and the Brownian ratchet mechanisms by maintaining a high EB1 concentration at the protofilament plusend and allowing ra pid MT polymerization 4.3.1 NonTethered 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 plusend or side) from solution or by copolymerization with EB1, pr otofilamentbound tubulin can be in various EB1binding 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. EB1tubulin 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 endtracking model over the Brownian ratchet mechanism to preferentially bind to the protofilament plusend 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 plusend 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 GDPbound subunits) and prevented the protofilament from growing. The results of this model PAGE 93 93 show that the affinity modulation of EB1 does not affect this sidebinding 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 EB1bound fraction of subunits shows a s lightly larger fraction of EB1 binding at the plusend when the affinity m odulation factor is greater. 4.3.2 Tethered Protofilaments The tethered protofilament endtracking model simulates EB1 endtracking motors operating on a growing protofilament plusend, a nd introduces a cofactor protein that tethers EB1 to a motile surface. Unlik e the monovalent endtracking 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 plusend. This model also accounts for any transition state effects on the onand o ffrates due to binding be tween surfacetethered EB1 and the protofilament. The forcevelocity 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 endtr 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 endtracking 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 PAGE 94 94 endtracking 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 viceversa. 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 endtracking model is able to maintain a persistent attachment of the protofilament plusend (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%) untethered. This result suggests that the EB1 endtracking motors are able to maintain persistent attachment of the protofila ment end to the motile surface, translating its filamentbound hydrolysis energy to mechanical work and allowi ng the protofilament to grow even under large loads. PAGE 95 95 Figure 41. Mechanisms of a nontethered, divalent endtracking motor. A) Topleft: EB1 and TGTP free in solution. EB1 binds to the protofilament in two ways: after tubulin addition (clockwise) or copolymerizing with tubulin (counter clockwise). Clockwise: TGTP adds to the protofilament end ( kf) and induces hydrolysis of the penultimate tubulin subunit; EB1 binds to TGTP at the protofilament end (2 kon). Counterclockwise: EB1 and TGTP bind in solution (2 k1); Together, EB1 and TGTP add to the protofilament end (kf E). B) Topleft: EB1 initiates bound to the GTPrich protofilament plusend. Free tubulin in solution binds to the protofilament in two ways: directly from solution (clockwise) or facilitated by the EB1 motor (counterclockwise). Clockw ise: Tubulin in solution adds to the protofilament end ( kf), which induces hydrolysis of the EB1boun d, penultimate tubulin subunit. The unbound EB1 head binds to the GTPbound protofilament end ( k+). Counterclockwise: The free EB1 head binds to tubulin in solution ( k1) and shuttles the protomer to the protofilament end ( kf E ). C) TopLeft: TE and TGTP free in solution. TE binds to the protofilament in two ways: after tubulin addition (clockwise) or copolymerize as TTE (counte rclockwise). Clockwise: TGTP binds to protofilament end ( kf), inducing hydrolysis of the penultimate subunit. TE binds to the TGTP protofilament end ( kon). Counterclockwise: TE binds to TGTP in solution ( k1). TTE binds to the protofilament end ( kf E) and induces hydrolysis of the penultimate tubulin subunit. PAGE 96 96 PAGE 97 97 Figure 42. Mechanisms of equilibrium, side binding of EB1 to protofilament. Offrates of EB1 binding to protofilamentbound GDP affected by affinity modulation factor. Tubulin addition and dissociation pathway neglected for this equilibrium mechanism. Figure 43. 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 M1s1. The value of K1 required to achieve a tiptoside 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. PAGE 98 98 Figure 44. 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 M1s1, 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. PAGE 99 99 Figure 45. EB1 equilibrium binding. Data shown in figure is for a kon value of 5 M1s1 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. PAGE 100 100 Figure 46. Occupational probabili ty of EB1 along length of protofilament. Zero on the xaxis represents the protofilament plus (gro wing) end. Simulation time used was 40 seconds. Values for other variables: K1=0.65 M, kon=5 M1s1. 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. PAGE 101 101 Figure 47. Time averaged EB1bound tubulin fraction at equilibrium. Zero on the xaxis 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 M1s1. Equilibrium EB1bound fraction represented by solid line at 0.024. Simulation time used was 80 seconds and N =200 PAGE 102 102 Figure 48. Time averaged fraction of EB1boun d subunits during protofilament growth. Zero on the xaxis 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 M1s1. Equilibrium EB1bound fraction represented by solid line at 0.024. Simulation time used was 80 seconds and N =200. PAGE 103 103 Figure 49. Mechanisms of tethered, protofilame nt endtracking model w ith divalent EB1. A) Pathway for tethering protein to bind to EB1 (or TE) in solution. B) Tethering protein binds to protofilamentbound EB1 (o r TE). Energy is exerted by the spring, which is accounted for in the onand offrates ( kT and kT ). C) Surfacetethered TE binds to protofilament plusend with on rate of kon D) Surface tethered EB1 has twice the onrate 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 minusdirection exerts no force on th e spring, and has an on and off rate of k+ side and kside, respectively. EB1 walking in the minusdirection exerts a force on the spring, which is accounted for in the on and off rates ( k+ side and kside ) PAGE 104 104 PAGE 105 105 Figure 410. Mechanisms of tubulin addition to linking proteinbound protofilament. A) Tubulin can add to the plusend of a surfacetether ed protofilament. B, C) Tubulin bound to TkE or TkTE attaches to the protofilame nt end. Tubulin bound to TkTE has two configurations with which it can bind. D) Tubulin addition is facilitated by the filamentbound EB1 endtracking motor. The forces exerted on the spring are accounted for in the forward rate constants for each mechanism. PAGE 106 106 Figure 411. Forcevelocity profiles for tethered protofilaments bound to divalent EB1 endtracking motors. The effect of force and velocity of both the Brownian Ratchet and EndTracking models are shown. Simulation time used was 40 seconds. Values for other variables: k1=10 M1s1, K1=0. kon=5 M1s1. 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) Forcevelocity profiles shown for varying values of f (1,10,100,1000,10000) when KT =10 M. PAGE 107 107 PAGE 108 108 PAGE 109 109 Figure 412. Stall forces versus affi nity modulation factor at various KT values. Resulting stall forces for data in Figures 319 AE. Thermodynamic represents the thermodynamic values at the various values of th e affinity modulation factor, based on kBT =4.14 pNnm, d = 8 nm, [Tb] = 10 M, and [Tb]c = 5 M. Data for this figure can be found in Table 41. Table 41. 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 412. 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 pNnm, d = 8 nm, [ Tb ] = 10 M, and [Tb]c = 5 M. PAGE 110 110 Figure 413. 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. PAGE 111 111 Figure 414. 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 413. 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. PAGE 112 112 Figure 415. 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: 1unbound, 2bound to E, 3bound to TE, 4bound to dbE+, Tk2bound to E tethered to linking protein, TK3bound to TE tethered to linking protein, Tk4bound to dbE+ tethered to linking protein. PAGE 113 113 Figure 416. Fraction of S1 subunits bound and unbound from motile surface. Based on data presented in Figure 414, subunits in states 14 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. PAGE 114 114 Figure 417. 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, TkE linking protein bound to EB1, TkTE linking protein b ound to TE, TkTTE linking protein bound to TTE. Each of the eight combinations of variables (f KT, F ) was considered. PAGE 115 115 CHAPTER 5 CILIARY PLUG MODEL Cilium is a motile organelle made up of an array of MTs. The plusends of ciliary MTs are attached to the cell membrane by MTcapping struct ures, which are located at the site of tubulin addition (Figure 51, (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 endtracking motor, similarly to the endtracking motors in cell division and cellular growth. This chapter discussed the EB1 endtrackin g model developed for the ciliary plug. Essentially, the plug is the endtracking 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 plusend occurs in three steps: tubulin addition, filamentbound 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 endtracking motor. 5.1 Model The physical characteristics of the plug make this model complex, but for simplification, the EB1 endtracking 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 PAGE 116 116 distance L (See Figure 52), 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 endtracking motor are represented in Fi gure 53. The first step in this endtracking 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 TGDP, the motor rebinds to the GTPrich, filament plusend 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 51), 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 ratelimited by Tm (confirmed below), in which case Tm can again be estimated from vmax (167 nm/s), i.e., s v d Tm05.0max (51) 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. 52) (Gardiner, 1986): 1 '2 2 dx d D dx d F Tk Df B f (52) 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 53 and 54) PAGE 117 117 Tk DB f (53) aL L /log 4 (54) 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 52 was solved and is represented in Equation 55. dF Tk dF Tk Tk dF dF Tk D d FB B B B f x2 2 2exp )(1 (55) With as a function of force, the equation that govern s the velocity becomes a function of force, and is represented by Equation 56. )( )( FT d Fvm (56) 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 endtracking motor attached was also simulated. The filament end position, x was governed by Equation 57. TkxB (57) 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 PAGE 118 118 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 14protofilmaent 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 51. The viscosity of the fluid used to calculate the dr ag coefficient was assumed to be that of water, 109 pNs/nm2 (Boal, 2002). Assuming each of the protofilaments to be a semiflexible 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 58 (Howard, 2001). 4 2L TkB (58) PAGE 119 119 The persistence length is repr esented by Equation 59, where B is the bending modulus of the filament ( B = 1.2 x 1026 Nm2, (Mickey and Howard, 1995)). The resulting value for the compression stiffness is 1.1 pN/nm. TkBB/ (59) 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 (510) The resulting stretch stiffness for a protofilament was determined to be 370 pN/nm. 5.3 Results Figure 54 shows the force effects on ciliary microtubules. In Figure 54A, 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 filamentbound GTP hydrolysis remains constant and is forceindependent, 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 54B. 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 55A, 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 PAGE 120 120 usually about 8nm, which is the size of the tubuli n dimer. The reasoning fo r the step size is that the endtracking motor for each protofilament must fill with tubulin before the ciliary plug can advance. The histogram in Figure 55B 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 55A. 5.4 Summary The ciliary plug was simulated as an EB1 e ndtracking motor similar to the endtracking motors described in the Lock, Load and Fire Mechanism (Dickinson and Purich, 2002). The primary steps of this model are tubulin addition, filamentbound GTP hydrolysis and the shifting and rebinding of the ciliary plug (e.g., EB1) to the filament end. By analyzing the forcevelocity 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 endtracking motor to maintain fidelity of the microtubule by allowing the plug to advance only once all protofilaments are the same length. Figure 51. 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: 22592269. Copyright 1988 The Rockefeller University Press] PAGE 121 121 Figure 52. 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 plusend. Red represents the GTPbound tubulin subunits. Figure 53. Mechanism of the ciliary/flagellar endtracking motor. In the first step, tubulin adds to the MT plusend into the endtracking co mplex. This binding induces hydrolysis attenuating the affinity of the complex to the protofilament. The surface advances to the filament end. PAGE 122 122 Figure 54. 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. PAGE 123 123 Figure 55. 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 endtracking 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. PAGE 124 124 CHAPTER 6 DISCUSSION 6.1 Possible Roles of EndTracking Motors in Biology The role of nucleotide hydrolysis in cytoskel etal molecular motor action is wellestablished 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 endtracking motors, we previously described ho w an microtubule filament endtracking motors can exploit nucleotide hydrolysis to generate significantly greater force than that predicted by a freefilament 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 plusend elongation to examine whether a hypothetical endtracking motor consis ting of affinitymodulated in teractions of EB1 at MT and protofilament ends can propel objects (e.g., MTa ttached kinetochores or MTattached 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 endtracking 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; MimoriKiyosue et al., 2005). Kinetochores also stabilize MTs against disassembly by preferentially attaching to GTPcontaining subunits of tubulin subunits situated at or near MT plusends, and this prop erty is likely to be the consequence of EB1s ability to attach to polymerizing GTPrich MT subunits and to dissociate from GDPcontaining subunits, thereby providing a thermodynamic drivin g force for localization at or near the MT PAGE 125 125 plusend. Capture of EB1rich MTs by kinetoch ores may allow those EB1 molecules combining with APC to selfassemble into an endtracki ng motor unit that links force generation to MT polymerization and hydrolysis of MTbound GTP. It is known that in the absence of the EB1/APC complex, chromosomes fail to align at the metaphaseplate, 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 endtrack ing motor there and playing a role in elongationdependent force generation. In fact, it has previous ly been suggested that the pluglike structures found at the plusends of MTs in regenerating Chlamydimonas flagella appear to be MT assembly machines. The analogous geomet ry of the ciliary/flagellar plug and the tubuleattachment complex in the kinetochore would allow plugand kinetochorebound EB1 to interact with their MT partners as an endtracking motor. This proposal does not preclude the action of other ATP hydrolysisdependent moto rs. For example, although the kinesinlike protein NOD lacks residues known to be critical for kinesin function, microtubule binding activates NODs ATPase activity some 2000fo 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. EndBinding Protein 1 (EB1) has previously been sh own to bind specifically to the polymerizing microtubule plusend where the mi crotubule is tightly bound, sugges ting a possible role in force generation at these sites. We propose that e ndbinding proteins (specifically EB1) behave as PAGE 126 126 molecular motors that modulate the interaction between MTs and the motile object, and generate the forces required for MTbased motility. Although the importance of EB1 and its potentia l to behave as and endtracking motors has been discussed thoroughly in this research study, the models developed can be used to understand force generating mechanisms invol ving other endtracking proteins and their potential to act as motors (e.g., CLASPS, C lip170). 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 endtracking motor. 6.2 Microtubule EndTracking Model We developed and analyzed a preliminary EB 1 filament endtracking model for MTs to determine the advantages of the mechanochemical process over the monomerdriven ratchet mechanisms. The two important properties of this model are (a) maintenance of a tight, persistent (processive) attachment at elongat ing MT plusends by means of EB1s multivalent affinitymodulated interactions; and (b) a mech anism for the assembly of the MT end by EB1 dimers bound on the motile object, thus affording a highfidelity pathway for assembling tethered MTs. For simplicity, our model only considers simple reactions of the EB1 filament endtracking 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 EB1asso 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 EB1protofilament interaction. We previously suggested that EB1 may be a polyme rization cofactor acting together with APC to PAGE 127 127 endtrack MT protofilaments (see MechanismC in (Dickinson et al., 2004). However, in view of the recent finding that EB1 is a stable, twoheaded dimer (Honnappa et al., 2005), we now explain how such multivalency would allow EB1 al one to operate as the endtracking motor (like MechanismA 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 forcedependence of elongation arising from the dependence of probability of the flexible EB1 head binding at a specific MT lattice position. With hydrolysisdriven affinitymodulation 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 hydrolysismediated motors on an MTs thirteen protofilaments can yield kinetic stall forces of approximately 30 pN. This value is considerably larger than the ~7pN 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 EndTracking Models The microtubule endtracking model devel oped neglected solutionphase End Binding protein 1 (EB1) and binding to microtubules and tubulin protomers. To account for binding solutionphase EB1, we developed simplified mo dels that simulated th e growth of a single protofilament in the presence of EB1 endtracking motors. The properties of all protofilament the endtracking models were compared to those of the simple Brownian Ratchet mechanism. Two of the models consider only freeprotofilament 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 PAGE 128 128 confirm the assumption that GTPdriven affinity modulated binding of the EB1 endtracking proteins is required in order to provide a 4.2 tiptoside 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 forcevelocity profiles show that the divalent, EB1 endtracking model provides an great advantage over the monovalent endtracking model as well as a Brownian Ratchet mech anism. The divalent endtracking motors are able to provide processive endtracking 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 13protofilament, mi crotubule endtracking model should be considered. It is suggested to develop a stochastic model that includes all mechanisms discussed in the tetheredprotofilament model with divalent EB1. For simplifications it could be assumed that all protofilament behave independently, but whose individual EB1 endtracking 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 endtracking 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 endtracking 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 PAGE 129 129 associates with the MT plusend. 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 EB1coated 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 velocityforce relationships coul d also be determined. This technique would provide more accurate stall force estimations for compar ison with the simulated results. PAGE 130 130 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] (A1) EMTMTEdK Kd = [E][MT]/[MTE] (A2) The total concentration of EB1 is represented in all states is [E]0 = [E] + [TE] + [MTE] (A3) Substituting A1 and A2 into A3 gives Equation A4. [E]0 = [E] + [E][Tb]/ K1+ [E][MT]/ Kd (A4) such that solving for [E] yields. dK MT K Tb E E ][][ 1 ][ ][1 0 (A5) 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 noncooperative. Here, the relevant reaction and equilibrium equations are E + Tb TE K1 =2 [E][Tb]/[TE] (A6) Tb + TE TTE K1 = [Tb][TE]/(2 [TTE]) (A7) EMTMTEdK Kd = [E][MT]/[MTE] (A8) [E]0 = [E] + [TE] + [TTE] + [MTE] (A9) PAGE 131 131 Combining A6 and A7 yields 2 1 2 1]][[ 2 ]][[ ][ K TbE K TbTE TTE (A10) Combining Eqs. A9 and A10 yields [E]0 = [E] + 2[E][Tb]/ K1+[E][Tb]2/ K1 2 + [E][MT]/Kd (A11) hence dK MT K Tb E E ][][ 1 ][ ][2 1 0 (A12) from which [TE] and [TTE] can be calculated using Eqs. A6 and A7. A.2 Occupation Probability of Monovalent EB1 Binding to NonTethered 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[ (A13) The dimensionless relationships in Equations A14 to A17 can be substituted into Equation A13: ][ ][,Tk Ekf ii i (A14) ][,Tk kf i i (A15) ][ ][ Tk ETkf E f (A16) 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 (A17) PAGE 132 132 The resulting equations represent the differentia l equation for the probability of EB1 binding to the protofilament side (A18) and plusend (A19), where u 1pi. iirii iiii d ippTpp pc d dp 1 1 1)1( (A18) 21 1 1,21 1 0,111111 1cpTpcTcppc dt dpr r d (A19) 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 TkE 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 A20. iqvwy1 (A20) The differential equations for the probabilities of EB1 and TkE binding to the protofilament are ii E rr E riiieff E f E f f iiTiieffiTi ion ippqkukpkppwCkTEkTbk qkypCkpkuEk dt dp 11 11 1 ,,) ][][( ][ (A21) PAGE 133 133 ii E r r E riiieff E f E f f iiieffoniiTiieffiT iqqqkukpkqqwCkTEkTk qkvuCkqkypCk dt dq 11 1 1 1 ,,) ][][( (A22) 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 (A23) ii i iieffion i T TqkvuCkwkvTkvkyEk dt dv, ,, 1 1][ ][ (A24) To dedimensionalize time in these differential equations, the variable Tr 1 was introduced, which is defined by Equation A25. ][1 1 1Tk ukpk Tf r E r r (A25) Setting ][1 0,T T Tc r and fK K k k T T Td f E f c r 1 1 1,][ Equation A25 gets reduced to A26. 1 1 1,1 1 0, 1pTuTTr rr (A26) Dividing Equations A21 to A24 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( ][ ][][][ ][ (A27) PAGE 134 134 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( ][][][][ (A28) 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 (A29) 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 (A30) To evaluate the probabilities of E or TkTE binding to the termin al subunit in the protofilament, these probabilities were rewritten 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 (A31) )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 (A32) To solve for the probabilities ( pi, qi, w v ) from Equations A27 to A730, 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 PAGE 135 135 The resulting dedimensionalized 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( ][ ][][][ ][ (A33) 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( ][][][][ (A34) 11 1 1,1 1, 1][][][][ ][ qTw T C w T K vw T K y T ET dt dwr eff t (A35) 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 (A36) )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 (A37) )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 (A38) PAGE 136 136 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( (A39) ii ii ii iii ii iii iqqq f up f qqw qvu qyp d dq 111 11 1 00) 1( (A40) 11 00q f w wvwy dt dw (A41) ii i iii i iq vu q f vu wvvy d dv 1 1 11 11 (A42) )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 (A43) PAGE 137 137 )1( 1 )1(211 21 1 1 1 1 1 00 111 111 1qq f qp f uq f qw vu yp d dq (A44) These differential equations (A39 to A44) 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 (A45) ii ii ii ii ii i i i i ion ippRppRupkqk upkqkpTkwkpkuEk dt dp 1 1 1 11, 1 1 1 1][ ][2 (A46) ii ii i i ii on iwwRwwRwkpTkwkuETk dt dw 1 1 1 1][ ][ (A47) PAGE 138 138 i i i i iiiiii ii ii iqqRqqR upkupkqkqk dt dq1 1 11. 1, .11. (A48) i i i i iiiiii iiii iqqRqqR upkupkqkqk dt dq1 1 11, 1,11, (A49) ii ii i onion iiiiiii iii iuuRuuRuETkuEk upkupkqkqkwkpk dt du 1 1 1,1,][][2 (A50) 1])[]([][wCkTETETkTkReff E f E f f (A51) 111 1qpwkukRE rr (A52) 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 (A53) )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 (A54) 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 (A55) PAGE 139 139 211 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 (A56) 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 (A57) eq eq eq eq onwkpTkwkuETk 1 1][ ][0 (A58) eqeq equpkqk 2 0 (A59) eq oneq on eqeq eq eq equETkuEkupkqkwkpk][][2 2 0 (A60) And the following holds true: eqeqeq eqwpqu 1 (A61) Solving Equation A59 gives: eqeq equpkqk 2 (A62) Substituting this relationship into the other three equilibrium equations results in: eq eq eq onwkpTkkuEk 1 1][ ][20 (A63) eq on on eq equETkEkwkpk][][2 0 (A64) eq eq eq onpTkwkkuETk][ ][01 1 (A65) From these three relationships, the equations for weq, and peq were solved for: PAGE 140 140 eq eq on eqp k Tkk u k Ek w 1 1 1][ ][ 2 (A66) eq on on equ k Tkk kk ETk k k Ek p 1 1 1][ ][1][2 (A67) At equilibrium, the followi ng relationship is true: 1]][[ 2][ K ET ET (A68) Plugging this into Equation A67, gives: eq on equ k Tkk T k k k k k Ek p 1 1 1 1 1][ 1 ][1 ][2 (A69) The terms in brackets on the top and bottom ar e identical, therefore it A69 reduces to: eq on equ k Ek p ][2 (A70) Consequently, weq and qeq reduce to Equation A71. eq eqp K Tb w1 (A71) eqeq equp k k q 2 (A72) Substituting A70, 71, 72 into Equation A61 results in the following equation: 2 1][2 2 ][ 1 ][2 110eq on eq onu k Ek k k u K T k Ek (A73) PAGE 141 141 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 (A74) To determine the value of K1, we first assume [Tb]=0, which reduces Eq. A75 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 (A75) The fraction of filamentbound 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 (A76) 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 (A77) Solving for gives: 1 2 1 ][ 1 1 eq on eq totu k k u k k MT (A78) The effective equilibrium dissociation c onstant of EB1 and the protofilament ( Kd,eff) is defined as: PAGE 142 142 eq on eq effdu k k u k k K 2 1 1, (A79) Simplifying Equation A78 gives: 1 ][ 1, tot effdMT K (A80) When half of the protofilament is saturated (=1/2), Kd,eff is given by Equation A81 and u1/2 ueq ([E][E]0/2). 2/1 2/1 ,2 1 1 u k k k k Kon effd (A81) Under this constraint, u1/2 is given by Equation A82, 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/]([ (A82) Since k+=kon* Ceff and k+ side=kon side* Ceff for the protofilament plus end, Equation A83 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/]([ (A83) Substituting the definition of K where kkK/gives: PAGE 143 143 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 (A84) 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 (A85) Determination of k: At steadystate, the measured offrate offk = 0.26 s1 is related to the kby: 1][ 1 1 2/ )( K T Ku k qp pk keq eqeqeq eqeq off (A86) Rearranging Eq. A86 gives Equation A87. 1][ 1 1 K T Ku kkeq off (A87) PAGE 144 144 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 (A88) PAGE 145 145 APPENDIX B MATLAB CODES B.1 13Protofilament Microtubule Model This stochastic program simulates a 13protofilament 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 13protofilament 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 (pNnm) 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 TGDP's (s^1) kD=0.5; % Dissociation rate constant for both EB1 arms on taxolstabilized 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: onrate for both EB1 arms on MT in solution (nm^3/s) k1on=8.9/13; % k1+: kon for tubulin dimer on MT (1/(uMs)) 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 TGTP/TGDP (1/s) (1.6*10^3) k2off=k3off/f; % k2: koff for 1 EB1 arm between TGTP/TGDP(1/s) (0.02) dt=4.8e6; % 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 TGTP/TGDP % State 2: One EB1 arm bound between terminal 2 TGDP's % State 3: One EB1 arm bound between terminal 2 TGDP's, One arm btwn terminal TGTP/TGDP PAGE 146 146 % State 4: One EB1 arm bound between terminal 2 TGDP's, One arm btwn lagging 2 TGDP's % State 5: One EB1 arm bound between lagging 2 TGDP's % State 6: One EB1 arm bound between lagging 2 TGDP's, One arm btwn terminal TGTP/TGDP 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)' PAGE 147 147 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 xaxis ylabel( 'Filament End Position (nm)' ) % Label yaxis B.2 Protofilament Growth Model with Monovalent EB1 B.2.1 Occupational Probability of Monovalent EB1 on a NonTethered Protofilament This probabilistic model simulates the free grow th of a single protofilament in the presence of monovalent, EB1 endtracking 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 onrate for tubulin % kr = 3.38; % s^2 kf = V/d/Tb; % uM^1s^1 onrate for tubulin taken assuming irreversible % elongation at observed elongation speed Tc = 5; % uM plusend critical concentration kr = kf*Tc; % s^1 offrate, 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 tiptomiddle 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 EB1Tb concentration kplus_side=kobs/(E+Kd); % uM^1 s^1 onrate constant for EB1 to MT side kminus_side=kplus_side*Kd; % s^1 offrate constant for EB1 from MT side enh = kf/kplus_side % End binding Rate enhancement factor % Roughly estimated parameters PAGE 148 148 f = 1000; % affinity modulation factor sigma = 10 ; % nm stdev of EB1 position fluctuations kplus=enh*kplus_side; % uM^1 s^1 onrate constant for EB1 to terminal subunit % assumed same as on side kminus=enh*kminus_side/f; % s^1 offrate 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:n1).^2.*(d./sigEB1)^2)'.*exp((0:n1)*d*delta/sigEB1^2)'; % av=av.*exp((0:n1)*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=1p; 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:n1; f(i)= gv(i).*u(i)av(i).*p(i)+(1+b)*(p(i1)p(i))+Trinv*(p(i+1)p(i)); %f(n)=0; f(n)=f(n1); PAGE 149 149 B.2.2 Occupational Probability of Monoval ent EB1 on a Tethered Protofilament This probabilistic model simulates the growth of a surfacetethered protofilament in the presence of monovalent, EB1 endtracking 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; % pNnm 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 onrate for tubulin taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plusend critical concentration kr = kf*Tc; % s^1 offrate, 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 tiptomiddle 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 EB1Tb concentration kplus_side=kobs/(E+Kd); % uM^1 s^1 onrate constant for EB1 to MT side kminus_side=kplus_side*Kd; % s^1 offrate 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 onrate constant for EB1 to terminal subunit assumed same as on side kminus=kminus_side/f; % % s^1 offrate 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 3D normal distribution on halfsphere PAGE 150 150 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*j1); pend=xout(nt,2*j1); 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 = (im1)*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*Tkr) PAGE 151 151 Attachprob = 1(1sum(qend))^13 function ff=bfrate(t,x,pars) n=(length(x)2)/2; j=1:n; p=x(2*j1); q=x(2*j); u=1pq; v=x(2*n+1); w=x(2*n+2); y=1vwsum(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(im1))*d^2/kT) = Fd +kappad*i meani = 0; if sum(q)>0 meani = sum(q.*j')/sum(q); end FT = (meani1)*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(jim)*kappad*deld)'; theta=exp(kappad/2*(jim).^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+line2delta*psi*mu/xi/f*p(1)*(1p(2)); line1 = (chiv(1)*(gamma*y*p(1)alpha_s*v*u(1))+chiv0*phi0*w*(1q(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)*(1q(1)); fq(1) =line1+line2+line3; i=2:n1; PAGE 152 152 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(i1)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(i1)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*i1)=fp(i); ff(2*i) = fq(i); line1 = gamma*(eps*ybeta*v)+eta*(xi*wv)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*ybeta*w)+eta*(vxi*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(n1)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(n1)q(n)); fq(n)=line1+line2; ff(2*n1)=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 NonTethered Protofilament This probabilistic model simulates the free grow th of a single protofilament in the presence of divalent, EB1 endtracking 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 freeended MT's % Simulates freeended 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 PAGE 153 153 % 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 onrate for tubulin % kr = 3.38; % s^2 kf = V/d/Tb; % uM^1s^1 onrate for tubulin taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plusend critical concentration kr = kf*Tc; % s^1 offrate, 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 3D normal distribution on halfsphere Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm3 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; % Onrate for EB1 subunit and Tb (uM^1*s^1), Value for typical proteinprotein binding k1m = k1*Kd1; % Offrate 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; % [EB1Tb], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = Tb*TE/(2*Kd1); % [EB1Tb^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 halflife 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; % uM1s1 fon = kon/kon_side; %%% accelerated onrate at end kplus = kon*Ceff; foff = fon/f; kminus=fon/f*kminus_side; %%% corresponding change offrate at end %%% Other parameters kfE = kf % onrate constant of TE and TTE to MT end uM1s1; krE = kfE*Tc/Kd1*kminus/kon % offrate 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*j1)=p_eq; %initial conditions x0(3*j)=q_eq/2; x0(4)=0; x0(3*j2)=pi_eq; [tout, xout]=ode23s(@(t,x0) dfrate(t,x0,pars),tspan,x0); PAGE 154 154 nt=length(tout); nmid=round(nt/2); pimid=xout(nmid,3*j2); pmid=xout(nmid,3*j1); qpmid=xout(nmid,3*j); piend=xout(nt,3*j2); pend=xout(nt,3*j1); qpend=xout(nt,3*j); qmend=[0 qpend(1:n1)]; qmmid=[0 qpmid(1:n1)]; 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*j2); p=x(3*j1); qp=x(3*j); qm=[0 qp(1:n1)']'; u=1pqpqmpid; 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:n1; tmp1=2*konv(i)*E.*u(i)kmv(i).*p(i)+k1m*pid(i)k1*Tb*p(i)+kmv(i).*qp(i1)kpv(i+1).*p(i).*u(i+1); tmp2=kmv(i+1).*qm(i+1)kpv(i1).*p(i).*u(i1)+Rp*(p(i1)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(i1)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(i1)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*(1p(1))(kf*Tb+2*kfE*TTE)*p(1)+(kr*u(1)+krE*(pid(1)+qp(1)))*p(2)krE*p(1)*(1p(1)); PAGE 155 155 fp(1)=tmp1+tmp2; tmp1 = kon*TE*u(1)kminus*pid(1)+k1*Tb*p(1)k1m*pid(1); tmp2 = 2*kfE*TTE*(1pid(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)*(1pid(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*j2)=fpi; f(3*j1)=fp; f(3*j)=fqp; B.3.2 Average Fraction of divalent EB1bound Protomers on Side of Protofilament This stochastic model simulates the side binding of divalent EB1 on a nongrowing 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 freeended 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 plusend 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 onrate for tubulin taken assuming irreversible elongation at observed elongation speed kr = kf*Tc; % s^1 offrate, 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 3D normal distribution on halfsphere PAGE 156 156 Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm3 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; % Onrate for EB1 subunit and Tb (uM^1*s^1), Value for typical proteinprotein binding k1m = k1*Kd1; % Offrate 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; % [EB1Tb], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = Tb*TE/(2*Kd1); % [EB1Tb^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 halflife 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; % uM1s1 fon = kon/kon_side; %%% accelerated onrate at end kplus = kon*Ceff; foff = fon/f; kminus=fon/f*kminus_side; %%% corresponding change offrate at end %%% Other parameters kfE = 10 % onrate constant of TE and TTE to MT end uM1s1; krE = kfE*Tc/Kd1*kminus/kon % offrate constant of TE or TTE % Initial conditions N=40; % 50number 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; % 200run 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,:) PAGE 157 157 tst5=(((S==2)&[S(2:N)==1 0]) & rnside1(it,:)<(kplus_side*dt + kplus_side*dt+kminus_side*dt)) &~(tst3tst4); % bind minus side tst6=((S==2)& rnside1(it,:)<(k1*T*dt + 2*kplus_side*dt+kminus_side*dt)) &~(tst3tst4tst5); % bind T tst7=(S==3)& rnside1(it,:) PAGE 158 158 T = 10; % uM tubulin dimer concentration V = 170; % nm/s; % elongation speed d = 8; % nm; % subunit length kf = V/d/T; % uM^1s^1 onrate for tubulin taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plusend critical concentration kr = kf*Tc; % s^1 offrate, 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 3D normal distribution on halfsphere Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm3 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; % Onrate for EB1 subunit and T (uM^1*s^1), Value for typical proteinprotein binding k1m = k1*Kd1; % Offrate 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; % [EB1T], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = T*TE/(2*Kd1); % [EB1T^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 halflife 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; % uM1s1 fon = kon/kon_side; %%% accelerated onrate at end kplus = kon*Ceff; foff = fon/f; kminus=fon/f*kminus_side; %%% corresponding change offrate at end %%% Other parameters kfE = kf; % onrate constant of TE and TTE to MT end uM1s1; %kfE = 1e8; krE = kfE*Tc/Kd1*kminus/kon; % offrate 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 PAGE 159 159 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,N1)]; kminusv=[kminus kminus_side*ones(1,N1)]; kplusv=[kplus kplus_side*ones(1,N1)]; 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,:) PAGE 160 160 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:N1); 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 ta5ta6ta7ta8 Sold=S; ntav=itprevit; FL=(S==2)+(S==3)+.5*(S==5)+.5*(S==4); FLav=(previt*FLav+ntav*FL)/(ntav+previt); previt=it; nadd=nadd1; S(1:N1)=S(2:N); if S(N1)==4 S(N)=5; else S(N) = S(N1); 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 endtracking 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 PAGE 161 161 % 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 yaxis. % 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 onrate for tubulin taken assuming irreversible elongation at observed elongation speed Tc = 5; % uM plusend critical concentration kr = kf*Tc; % s^1 offrate, 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 3D normal distribution on halfsphere Ceff = ceff/(6.022e23)*1e27/1000*1e6; % nm3 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; % Onrate for EB1 subunit and T (uM^1*s^1), Value for typical proteinprotein binding k1m = k1*Kd1; % Offrate 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; % [EB1T], Concentration of EB1 dimer bound to 1 tubulin protomer TTE = T*TE/(2*Kd1); % [EB1T^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 halflife 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; % uM1s1 fon = kon/kon_side; %%% accelerated onrate at end kplus = kon*Ceff; foff = fon/f; kminus=fon/f*kminus_side; %%% corresponding change offrate at end PAGE 162 162 %%% Other parameters kfE = kf; % onrate constant of TE and TTE to MT end uM1s1; krE = kfE*Tc/Kd1*kminus/kon; % offrate 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,N1)]; kminusv=[kminus kminus_side*ones(1,N1)]; kplusv=[kplus kplus_side*ones(1,N1)]; 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; %pNnm 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 Trackerbound 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 MTbound 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,:) PAGE 163 163 if sum(otst)>0 Sold=S; % Store old ntav=itprevit; % 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 timeaveraged 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(ifnd41)=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(ifnd91)=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) PAGE 164 164 ta8 = roff(it)<(S(1)==4)*krE*dt; if ta5ta6ta7ta8 Sold=S; ntav=itprevit; 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=nadd1; S(1:N1)=S(2:N); if S(N1)==4 S(N)=5; else S(N) = S(N1); 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) PAGE 165 165 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(Track1) = 4; end if Track1==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:N1); 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:N1); 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; PAGE 166 166 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) PAGE 167 167 Track = 3; end else Track=Track1; end S(1:N1)=S(2:N); if S(N1)==4 S(N)=5; elseif S(N1)==4 S(N)=5; else S(N) = S(N1); 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 unitE detachment kminusv =[kminus kminus_side.*ones(1,N1)]; ffac2 = exp(gamT*(Track1)*d*delta/kT); ffac3 = exp(gamT*(Track.5)*d^2/kT); %%% Doubly bound EB1 tst1 = (S(Track) == 5)& rndT(it) PAGE 168 168 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)) &~ (tst2tst2a); % dissociate T %%% bound E tst3 = (S(Track) == 2)& rndT(it) PAGE 169 169 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(Track1) = 4; if Track_old1==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*(irun1)+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; PAGE 170 170 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 13protofilament 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 MTbased 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 (pNnm) nf = 13; % No. filaments kappa = 0.15; % Filament compression stiffness (pN/nm) Df = 4e6; % Filament diffusivity (nm^2/s) deltaf = kT/Df; % Filament Drag (pNs/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>1e10; zeq = kappai*z0f'/sum(kappai); kappai=kappa*((zeq)<=z0f)+kappa2*((zeq)>z0f); % Vector of filament stiffnesses F=kappai.*(zeqz0f); % Vector of forces sF=sum(F); PAGE 171 171 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)1Pr)./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 (s1) 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(ihnbp,1):ih); zplot=zh(max(ihnbp,1):ih); znplot=zhn(max(ihnbp,1):ih); SUBPLOT(2,1,1), plot(tplot,znplot, 'r' ,tplot,zplot, 'b' ); % Plot recent trajectory tmin=th(max(ihnbp,1)); tmax = max([th .1]); zmin=zh(max(ihnbp,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(z0fzeq,5:10:max(z0fzeq)+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 PAGE 172 172 LIST OF REFERENCES Abrieu, A., J. 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PAGE 178 178 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. 