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Simulating Temperature Jumps for Protein Folding Studies

Permanent Link: http://ufdc.ufl.edu/UFE0021355/00001

Material Information

Title: Simulating Temperature Jumps for Protein Folding Studies
Physical Description: 1 online resource (104 p.)
Language: english
Creator: Kim, Seonah
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: biochemistry, cd, computational, folding, jump, kinetics, nmr, protein, shift, temperature
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Protein folding is described as a dynamic process of an ensemble of molecules reaching well-defined three dimensional structures to achieve biological activity from linear amino acids sequences. Many human diseases result from protein misfolding or aggregation. Enormous effort has been made both experimentally and theoretically for nearly 40 years to explain the basic principle and mechanism of protein folding and unfolding. Nonetheless, many of them are still unknown or incompletely understood, mainly due to the complexity of the systems and the fast folding time scale. Experimental and theoretical approaches are complementary with each other for the protein folding studies and hence, combination of the two is required to have better understanding. One of the most popular experimental methods for the protein folding studies is laser-induced temperature-jump (T-jump), because it has nanosecond resolution. In the first project, the T-jump on the polyalanine peptides (Ala20) was simulated as a proof-of-principle system to mimic the experimental measurements. Replica exchange molecular dynamics (REMD) were performed to obtain equilibrated ensembles as a proper conformational sampling, which was combined with multiplexed molecular dynamics to extract kinetic properties in line with experiments. In the second project, the same methodology used in the first project was applied to real proteins. Effect of frictional coefficient in the solvent model was approximated using Langevin dynamics. Computationall results on the two related 14-residue peptides were chosen and compared with experimental results. A ratio of relaxation time of the two peptides was determined by calculated Circular Dichroism (CD) spectra by a factor of ~1.2, while the experimental results were ~1.1.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Seonah Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Roitberg, Adrian E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021355:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021355/00001

Material Information

Title: Simulating Temperature Jumps for Protein Folding Studies
Physical Description: 1 online resource (104 p.)
Language: english
Creator: Kim, Seonah
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: biochemistry, cd, computational, folding, jump, kinetics, nmr, protein, shift, temperature
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Protein folding is described as a dynamic process of an ensemble of molecules reaching well-defined three dimensional structures to achieve biological activity from linear amino acids sequences. Many human diseases result from protein misfolding or aggregation. Enormous effort has been made both experimentally and theoretically for nearly 40 years to explain the basic principle and mechanism of protein folding and unfolding. Nonetheless, many of them are still unknown or incompletely understood, mainly due to the complexity of the systems and the fast folding time scale. Experimental and theoretical approaches are complementary with each other for the protein folding studies and hence, combination of the two is required to have better understanding. One of the most popular experimental methods for the protein folding studies is laser-induced temperature-jump (T-jump), because it has nanosecond resolution. In the first project, the T-jump on the polyalanine peptides (Ala20) was simulated as a proof-of-principle system to mimic the experimental measurements. Replica exchange molecular dynamics (REMD) were performed to obtain equilibrated ensembles as a proper conformational sampling, which was combined with multiplexed molecular dynamics to extract kinetic properties in line with experiments. In the second project, the same methodology used in the first project was applied to real proteins. Effect of frictional coefficient in the solvent model was approximated using Langevin dynamics. Computationall results on the two related 14-residue peptides were chosen and compared with experimental results. A ratio of relaxation time of the two peptides was determined by calculated Circular Dichroism (CD) spectra by a factor of ~1.2, while the experimental results were ~1.1.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Seonah Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Roitberg, Adrian E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021355:00001


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SIMULATING TEMPERATURE JUMPS FOR PROTEIN FOLDING STUDIES


By

SEONAH KIM















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 Seonah Kim




























To my loving husband Jiho and my family









ACKNOWLEDGMENTS

At the completion of this work I take great pleasure in acknowledging the people who have

supported me over the last couple of years. I gratefully thank and acknowledge my advisor, Prof.

Adrian Roitberg, for the opportunity to work with him. His guidance, humor, understanding, and

encouragement leaded me to overcome all the obstacles I had felt in studying. My personal goal

would not have been realized without the unconditional support from Prof. Roitberg. Those

words are not enough to thank my advisor Prof. Roitberg.

I would like to thank for kind advice and suggestions of my committee members, Prof.

Alexander Angerhofer, and Prof. Gail Fanucci as well, especially Prof. Ronald Castellano who

accepted to be my committee at the last moment. I am also greatly obligated to Prof. Steve

Hagen to give helpful discussions and suggestions through all my research.

I am grateful to Prof. Carlos Simmerling, Prof. Jeff Krause and Prof. Valeria Kleiman for

kindly providing invaluable advice within and beyond science, and to Prof. Thomas Cheatham

for hosting while I was in Utah. I gratefully acknowledge Georgios Leonis' and Julio Palma' s

help with their kind, helpful advice, supporting in and out of science through all my Ph. D. study

here in Gainesville. I also thank to all of my Quantum Theory Proj ect (QTP) buddies (Josh,

Andrew, Tom, Kelly, Joey, Lena and Martin) who share lots of party. I am so lucky to have all of

my buddies here in qtp and enjoyed every moment. I will miss them all. It has been great

pleasure to work with my group members (Dr. Gustavo Seabra, Ozlem Demir, Christina Crecca,

Hui Xiong, Dan Sindhikara, Yilin MEng, Lena Dolghih, and Mehrnoosh Arrar).

Above all I thank my husband, Jiho, for his love, support, encouragement, endless

corrections and personal sacrifice without which I would not have been able to accomplish this

work. Finally, I am forever indebted to my parents and brother, Jihwan, for their love,

understanding, prayer and encouragement when it was most required.












TABLE OF CONTENTS


page

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


LIST OF TABLES ............_...... ._ ...............7....


LIST OF FIGURES .............. ...............8.....


LI ST OF AB BREVIAT IONS ............_ ..... ..__ ............... 11..


AB S TRAC T ............._. .......... ..............._ 12...


CHAPTER


1 INTRODUCTION ................. ...............14.......... ......


1.1 Prologue ................. ...............14................
1.2 Structural Biology............... ...............17
1.3 Protein Folding ................... .. ........ ...........2
1.4 From the Experiment to the Simulation ................. ...............21...........
1.5 Temperature Jump Experiments ................. ...............22........ ...
1.6 Circular Dichroism .............. ...............24....
1.7 NMR Spectroscopy ................. ...............26........... ...
1.8 Overview of Research Proj ects .................. .... .... ......... ........ ........... ....2
1.8.1 First Proj ect: Simulating Temperature Jumps for Protein Folding ................... .....27
1.8.2 Second Proj ect: Folding Kinetics by Temperature-Jump Simulations of Two
Related 14-residue Peptides ................. ...............27........... ....

2 THEORY AND METHODS .............. ...............29....


2.1 Conformational Sampling............... ...............29
2.2 Force Field ................... ..... ......... .......... ...............32
2.3 Generalized Born (GB) Solvation Model ...._._._.. ........... ......... ..........3
2.4 Langevin Dynamics ......._.. .......... ......... .........___........3
2.5 Computation of Circular Dichroism (CD) ................. ...............36........... ..
2.6 Computation of NMR Chemical Shifts ............... ...............38....
2.6.1 Calculation of Proton ( H) Chemical Shift............... ...............38.
2.6.2 Calculation of 15N and 13C Chemical Shifts ......____ ..... ... .__ .............3


3 SIMULATING TEMPERATURE JUMP S FOR PROTEIN FOLDING. .............. ..... ..........42


3 .1 Introducti on ............ ..... .._ ...............42..
3.2 Methods ........................ ...............4
3.2.1 Simulation Details .............. ... ........ ..............4
3.2.2 Calculation of NMR Chemical Shifts............... ...............47.
3.2.3 Calculation of CD Spectra ............ ..... ._ ...............48












3.3 Results and Discussions................... ............4
3.3.1 Calculation of Chemical Shifts............... ...............52.
3.3.2 Calculation of Circular Dichroism .............. ...............56....

3.3.3 Optimum Number of Unfolded States? ................. ...............62.............
3.4 Conclusion ................ ...............64................


4 FOLDING KINETICS BY TEMPERATURE-JUMP SIMULATIONS OF TWO
RELATED 14-RESIDUE PEPTIDES............... ...............67


4. 1 Introducti on ................. ...............67........... ...
4.2 M ethods .............. ...............70...
4.3 Results and Discussion .............. ...............72....

4.3.1 Cluster Analysis............... ...............73
4.3.2 Helicity ................ .. .. ...............76
4.3.3 Calculation of CD Spectra ................. ...............76........... ..
4.3.4 Effects of Frictional Coefficients .............. ...............82....
4.4 Conclusion ................ ...............87.................


5 CONCLUSIONS .............. ...............89....


LIST OF REFERENCES ................. ...............91................


BIOGRAPHICAL SKETCH ................. ...............104......... ......










LIST OF TABLES


Table page

3-1 Comparison of calculated chemical shifts for residue 6 (A6) in polyalanine peptide
and experimental values for alanine residue from references............... ...............5

4-1 Relaxation and folding times of peptide 1 and peptide 2 from experimental and
computational d ata. .............. ...............8 2....

4-2 Folding times of the two peptides at different collision frequencies with Langevin
dynam ics. ............. ...............83.....










LIST OF FIGURES


Figure page

1-1 The general formula of an amino acid is showing a central carbon atom (C") is
attached to an amino group (NH2), a carbonyl group (COOH), a hydrogen atom, and
a side chain (R). ............. ...............14.....

1-2 The twenty different amino acids found in proteins. Side chains are shown in blue. .......15

1-3 The optical isomers of alanine, L and D forms. ......___ ... ...__... .....__... .......1

1-4 Part of the polypeptide chain shows to illustrate rigid peptide bond between C' (from
carbonyl group) and N (from amino group), two degrees of freedom,

from rotations around N-C" and C"-C' bonds, respectively ................. .....................16

1-5 Ramachandran plot for the 20-residue Trp-cage protein (PDB ID: 1L2Y). The dots
are created by each residue from the Trp-cage protein. The Trp-cage protein is one of
smallest folding protein-like molecule. The structure is showed in Figure 1-6 C.............18

1-6 The diagram of protein structures. A) Primary structure, B) Secondary structure
using an a-helix and P-sheet (PDB ID: MBH12), C) Tertiary structure (Trp-cage
protein), and D) Quaternary structure showed using hemoglobin complex (1GZX) ........19

1-7 The general scheme of Temperature-jump (T-jump) relaxation kinetics. ........................23

1-8 The standard curves for CD spectra of poly-L-lysine in different secondary structure
conformations taken from Campbell & Dwek, 1984............... ...............25..

2-1 The sketch of Replica-exchange method (REM) simulation in amber molecular
dynamics packages............... ...............3

2-2 Outline of 15N and 13C chemical shift calculation in SHIFTS program. ...........................41

3-1 The diagram of the T-jump setup. A) Experimental T-jump scheme, B)
Computational T-jump setup. ............. ...............47.....

3-2 Cluster analysis of polyalanine. A) All conformations from REMD at 151K are
superimposed on the reference structure of the largest cluster, B) a-helix reference
structure from the cluster analysis. ............. ...............49.....

3-3 Histograms from the C"-RMSD. A) Probability density of C"-RMSD at 181K (blue
line) and 214K (red line), B) A(probability at 214K probability at 181K) vs. C"-
RM SD ............. ...............50.....

3-4 Fractions of a-helical folded state (F), and unfolded state (U) as a function of
temperature using probability (C"-RMSD) ................. ...............51................










3-5 Averaged chemical shifts (proton, 13C, and 15N) for polyalanine peptide (ACE-
(ALA)20-NME) as a function of residue number, for all sixteen temperatures. ................53

3-6 1H-5N HSQC spectra of polyalanine peptide. A) At 181K and B) At 300K. ................ ..56

3-7 Two-dimensional 13 a 13Cp CTOsspeak shapes for polyalanine peptide, residue 5, 11
and 16 (AS, All, and Al6) from several temperatures. Color scheme are from the
maximum signal intensity (red) to the minimum signal intensity (blue)...........................57

3-8 Results of Circular dichroism. A) Calculated circular dichroism spectra of ACE-
(ALA)20-NME from 153K to 542K, B) Average of CD222 () Of each
simulated polyalanines. Error bars represent standard deviation............. ..............__ .59

3-9 Two circular dichroism (CD) spectra, pure a-helix and a-helix after minimization,
are compared to see different minimum wavelengths. ............. ...............60.....

3-10 Average of CD222 () Of T-jump simulation data (red: folded (F), and green:
unfolded (U) states) are fitted using a single exponential equation ................. ...............60

3-11 Folding (green) and unfolding (red) fractions from T-jump simulation data are
calculated and fitted using same h in Figure 3-10. ............. ...............61.....

3-12 Reference structures from cluster analysis. A) All conformations from REMD at
151K are superimposed on the two reference structures, B) a-helix and C) Coiled-
coil a-helix reference structures ................. ...............63................

3-13 C"-RMSD (residue 2-19) computed using different initial states from 1Cps MD
simulations. A) a-helix, B) Coiled-coil, and C) Unfolded initial states from C"-
RMSD1. D) a-helix, E) Coiled-coil, and F) Unfolded initial states from C"-RMSD2......64

3-14 C"-RMSD2 (with respect to coiled-coil a-helix) plots from 1Cps MD simulation using
coiled-coil a-helix initial state at 214K............... ...............66..

3-15 C"-RMSD (residue 2-19) relation plot using C"-RMSD1 (with respect to a-helix
reference structure) as x-axis and C"-RMSD2 (with respect to coiled-coil a-helix
reference structure) as y-axis, respectively, at 214K. ............. ...............66.....

4-1 Calculated helicity. A) Peptide 1 versus residue number, B) Peptide 2 versus residue
number based on DSSP method, respectively. ............. ...............73.....

4-2 Reference structures from cluster analysis of peptide 1 and peptide 2. A) All
conformations from REMD at 150K and are superimposed on the two reference
structure s, B) Two repre sentative structure s ....__. ................. ... ............. ..74

4-3 Populations of the representative clusters. A) Peptide 1 as a function of time from
20-500ns REMD, B) Peptide 2 as a function of time from 20-500ns REMD. At
200ns, simulations of both peptides are converged and stabilized ................. ...............75









4-4 The resulting curve for the two parameter calculations ([6,.]222 and x) from the plot
[6]222 VeTSUS 1/n from the Equation 4-1. The unit of molar ellipticity is deg-cm2
-dmol-l of both peptides............... ...............77

4-5 Fractional helicity (fH) Of peptide 1 (blue) and peptide 2 (red) as a function of
tem perature. ............. ...............77.....

4-6 Calculated circular dichrioism (CD) spectra. A) Peptide 1 from 150-726K, B)
Peptide 2 from 150-834K. The unit of molar ellipticity is deg-cm2 dmol-l of both
peptides. ............. ...............79.....

4-7 Average of mean residue ellipticities at 222ns () Of Simulated peptide 1
(blue) and peptide 2 (red) are shown as a function of temperature. The unit of molar
ellipticity is deg-cm2 dmol-l of both peptides. ............. ...............80.....

4-8 Average of CD222 () Of temperature jump (T-jump) simulation data
(collision frequency y = 1.0ps ). A) Peptide 1: black, B) Peptide 2: red with ftimng
curves, respectively. The unit of molar ellipticity is deg-cm2 dmol-l of both peptides.....81

4 -9 Comparisons of different collision frequencies, y=1.0, 5.0, 10.0, and 20.0ps- A)
Peptidel, B) Peptide 2 using average of CD222 () Of T-jump simulation data,
respectively. The unit of molar ellipticity is deg-cm2 dmol-l of both peptides. ...............84

4-10 Comparisons of the folding times at different collision frequencies of two peptides
with error bars and linear fits (dotted lines). The folding times and associate errors
were calculated from fitting curves of average of CD222 () fTOm Figure 4-9.......85

4-11 Friction dependence of peptide 1 (marked as 1) and peptide 2 (marked as 2) by T-
jump simulations. A) Y-intercept, B) Slope of folding time (z) obtained from linear
fits in Figure 4-10............... ...............86..









LIST OF ABBREVIATIONS

ACE: Acetyl beginning group

CD: Circular Dichroism

DSSP: Definition of the Secondary Structure of Proteins

GB: Generalized Born

HSQC: Hetero-nuclear Single Quantum Coherence

IR: Infrared

MD: Molecular Dynamics

NME: N-methylamine ending group

NMR: Nuclear Magnetic Resonance

NOEs: Nuclear Overhauser Effects

REMD: Replica Exchange Molecular Dynamics

T-jump: Temperature-jump









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

SIMULATING TEMPERATURE JUMPS FOR PROTEIN FOLDING STUDIES

By

Seonah Kim

December 2007

Chair: Adrian E. Roitberg
Major: Chemistry

Protein folding is described as a dynamic process of an ensemble of molecules reaching

well-defined three dimensional structures to achieve biological activity from linear amino acids

sequences. Many human diseases result from protein misfolding or aggregation. Enormous effort

has been made both experimentally and theoretically for nearly 40 years to explain the basic

principle and mechanism of protein folding and unfolding. Nonetheless, many of them are still

unknown or incompletely understood, mainly due to the complexity of the systems and the fast

folding time scale. Experimental and theoretical approaches are complementary with each other

for the protein folding studies and hence, combination of the two is required to have better

understanding.

One of the most popular experimental methods for the protein folding studies is laser-

induced temperature-jump (T-jump), because it has nanosecond resolution. In the first proj ect,

the T-jump on the polyalanine peptides (Ala20) WAS Simulated as a proof-of-principle system to

mimic the experimental measurements. Replica exchange molecular dynamics (REMD) were

performed to obtain equilibrated ensembles as a proper conformational sampling, which was

combined with multiplexed molecular dynamics to extract kinetic properties in line with

experiments.










In the second proj ect, the same methodology used in the first proj ect was applied to real

proteins. Effect of frictional coefficient in the solvent model was approximated using Langevin

dynamics. Computationall results on the two related 14-residue peptides were chosen and

compared with experimental results. A ratio of relaxation time of the two peptides was

determined by calculated Circular Dichroism (CD) spectra by a factor of ~1.2, while the

experimental results were ~1.1.









CHAPTER 1
INTTRODUCTION

1.1 Prologue

Proteins and nucleic acid are a starting point of life science, as they have a role in all living

processes. Protein folding studies are of great interest especially for many human diseases

associated with protein misfolding, such as cystic fibrosis, Alzheimer's, Parkinson' s disease and

Mad cow disease.'

Proteins are built by various combinations of commonly twenty amino acids. All amino

acids contain an amino group (NH2), a carboxyl group (COOH), and a distinctive R group

connected to a central carbon atom (C") (Finure 1-1).
ammno
group
NH2 carbon alpha atom (C")


side chain group H, i



CO OH
carb oxyl
group
Figure 1-1. The general formula of an amino acid is showing a central carbon atom (C") is
attached to an amino group (NH2), a carbonyl group (COOH), a hydrogen atom, and a
side chain (R).

The twenty amino acids found in proteins are shown in Figure 1-2. In general, four

different groups are connected to the central C" atom, making it a chiral center, except glycine,

where two H atoms link to Ca. Chiral molecules can have optical isomers, L- and D- forms

(Figure 1-3). Most of the amino acids in nature exist in L-form. Circular dichroism (CD)

spectroscopy, discussed in section 1.4,6 is a useful and critical tool to study chiral interactions.

A protein is made of combination of amino acids joined via a peptide bond, where the











Nonpolar amino acids


Ala, Alanine (A)

CH3
+H3N--C--COO


Val, Valine (V)

H3C\ .CH3
CH

+H3N--C-COO-


Leu, Leucine (L)

H3C\ .CH3
CH

CH2
+H3N-C-COO-


Ile, Isoleucine (I)

CH3
H2C\H/CH3

+H3N-C-COO


Pro, Proline (P)

H2
/C,
H2C CH2
HN--CH

+H3N--C COO


Gly, Glycine (G)



+H3N--C COO~








His, Histidine (H)



k~NH+

CH2
+H3N--C--COO


Cys, Cysteine (C)


CH2
+H3N-C-COO







Arg, Arginine (R)
NH 2

C=NH2
NH

CH2
CH2
CH2

~H3N-C;-COO


Phe, Phenylalanine (F) Met, Methionine (M) Trp, Tryptophan (W)
CH3 N


/ ~CH 2

CH2 CH2 CH2
+H3N-C~-COO~ H3C-C-COO- H3N-C-COO
H H H
Charged polar amino acids

Asp, Aspartic acid (D) Glu, Glutamic acid (E) Lys, Lysine (K)
.~~ OONH3


CH2 CH2
| CH2
'H3N-C-COO I
| CH2

+H3N C COO- H
CH2

H ~H3N C-COO


Uncharged polar amino acids

Asn, Asparagine (N) Gin, Glutamine (Q)

O /NH2 O C/NH2

CH2 CH2
+H3N-C-COO- CH2
H 'H3N-C-COO


Tyr, Tyrosine (Y)
OH


Thr, Threonine (T)

HO\ CH3
CH

+H3N C COO


Ser, Serine (S)

OH
CH2

+H3N C--COO-


Figure 1-2. The twenty different amino acids found in proteins. Side chains are shown in blue.





























C terminal


N terminal


Figure 1-4. Part of the polypeptide chain shows to illustrate rigid peptide bond between C' (from
carbonyl group) and N (from amino group), two degrees of freedom, cp and uy angels
from rotations around N-C" and C"-C' bonds, respectively.

carboxyl carbon atom (C-terminus) of one amino acid is bonded to the nitrogen atom (N-

terminus) in the amino group of the next amino acid (Figure 1-4). The connection of multiple

peptide bonds generates a backbone (or main chain) of a protein. The peptide group is generally

rigid enough to remain planar and in transrt~t~rt~t~rt~t~rt~ configuration. The amino acid sequence which is

linked by a peptide bond is called a primary structure.

Backbone rotations can occur around either the N-C" or the C"-C bonds; defined as phi

(cP) and psi (uy) angles, respectively. Different sets of values for phi and psi angles can denote

different protein (or peptide) conformation. Thus, the backbone conformations can be described


NH + NH3+
3 -


HC MH H C



CO O- CO O-


L-alanine D-alanine
Figure 1-3. The optical isomers of alanine, L and D forms.









by specifying these two angles. In Figurel-5, the pair of angles (cp and uy) are plotted in a

Ramachandran plot.' As shown in the figure, three main regions are allowed with respect to cp

and uy angles; one is the right-handed a-helix around (p=-570, uy=-470 (denoted aR), anOther is

the p sheet (parallel and antiparallel) around (p=-1250, uy=+1250 (denoted P) and the other is the

left-handed a-helix around (p=+570, uy=+470 (denoted aL). A left-handed polyproline II helix

(PPII) which is often observed in proline-rich sequences and sterically forced conformation for

polyproline, is also found in proteins around (p=-750, uy=+1450 (denoted PPII) and is an

important conformation in protein-protein interfaces.8,9

The three-dimensional form of local geometric arrangements within the peptide backbone

is called secondary structure. For example, a-helices in proteins appear when the cp and uy angles

are approximately -570 and -470, respectively and when hydrogen bond are formed between the

carbonyl oxygen of the ith residue and the amido portion of the i + 4th residue.

The tertiary structure is the combination of secondary structural units joined by a loop (or

turn). Furthermore, the quaternary structure can be defined when a protein involves more than

one polypeptide chain. The diagram of hierarchy of protein structures Figure 1-6 shows this

relationship.

1.2 Structural Biology

The subj ect of structural biology in proteins is the link between one-dimensional amino

acid sequences to their three-dimensional structure. Structural biology is the study of molecular

shape of biological macromolecules-proteins and nucleic acids- and their interactions. Studies of

structural biology began with the exploration of biological materials by using early microscopy

over 100 years ago. Thanks to development of new and powerful methods, such as X-ray

crystallography and nuclear magnetic resonance (NMR), it became one of the most important










+180


9 51 (10) 0











-180
-180 0 +180
phi ((p)
Figure 1-5. Ramachandran plot for the 20-residue Trp-cage protein (PDB ID: 1L2Y). The dots
are created by each residue from the Trp-cage protein. The Trp-cage protein is one of
smallest folding protein-like molecule. The structure is showed in Figure 1-6 C.

subj ects in molecular biology.10 There are two main experimental tools in structural biology; X-

ray crystallography and NMR. X-ray crystallography is the first discovered and the most

dominant tool, being widely used for structure determination of macromolecules. As of May

2007, 85% (37,101 out of total 43,633 structures) of released structures in Protein Data Bank

were obtained based on X-ray crystallographic results. However, its application is not universal

because it can be used only when high quality of crystal is available. Moreover, it provides

averaged atomic positions, with atomic displacement parameters (or B-factors) used to infer the

internal motions of proteins."
























a-helix


Figure 1-6. The diagram of protein structures. A) Primary structure, B) Secondary structure using
an a-helix and P-sheet (PDB ID: MBH12), C) Tertiary structure (Trp-cage protein),
and D) Quaternary structure showed using hemoglobin complex (1GZX)


A


B









The NMR technique has been also commonly used. It is, however, limited to smaller

molecular systems, but sensitive enough to recognize mobile regions of macromolecules in

aqueous solutions. In addition to effort on these technologies, theoretical work, especially

computational studies, are routinely used to support and complement experimental data.

1.3 Protein Folding

The dynamical process where a protein forms its well-defined three dimensional structures

to achieve biological activity is called "Protein Folding". For proteins that do fold, they usually

do into one specific unique state in just a few seconds (or less) from any starting conformation.

This state is defined as the "native state". Slight changes, such as pH, or temperature, can convert

biologically active protein molecules in native state (folded) to biologically inactive denatured

state (unfolded). Moreover, many human diseases result from protein misfolding or aggregation.

Therefore, protein folding research is the one of the most important subj ects in biology.

Although enormous efforts were made from experimental and theoretical studies for nearly 40

years, the pathways and mechanisms of protein folding have not been yet fully understood due to

the complexity of the systems and the fast time scale of folding.12-14

In the early 1960's, experimentally, Anfinsen et al. 15 studied the refolding of the

denatured bovine pancreatic ribonuclease (RNase). RNase as non-functional protein,

immediately returns to its native conformation (folding process) from a randomly coiled

structure, upon removal of the denaturant (8M urea), which helps to restore its enzymatic

activity. This behavior established the thermodynamicc hypothesis", which is directly related to

the tertiary structure of a protein and more importantly, there is a thermodynamically the most

stable minimum on the free energy profile, i.e., the free energy is being lowest in the native than

in the unfolded state.16 Thus, Anfinsen's observation opened a new era of protein folding

research.









In 1968, Cyrus Levinthal showed that it is impossible that the protein folds into its native

state by sampling all possible conformations." For example, if a 150-residue protein molecule

has only three stable conformations for each amino-acid (the allowed regions are a, P, and L

from the Ramachandran plot), then 315o=1068 pOssible conformations exist. In order to obtain the

native structure from a random search, it will take ~1048 years, substantially longer than the age

of the universe. This argument is popularly known as the "Levinthal's paradox". Therefore,

Levinthal conjectured that proteins must fold to their native, stable conformation by using well-

determined "folding pathways". These endeavors to find the correct pathways are continuing to

these days with increasing success.

1.4 From the Experiment to the Simulation

The present study aims towards simulating experimentally observed processes, by

computing thermodynamic and kinetic properties. Advances in computational power and speed

have opened the way to investigations of new fields and new possibilities. Karplus and his

coworkers first introduced molecular dynamics (MD) simulation of a biological macromolecule

in 1977.' The MD simulation can generate the configurations of the system based on Newton' s

law of motion and hence, can provide ultimate details of individual atomic motions as a function

of time. Hidden details of interest (including folding pathways) can be revealed by an MD

simulation. Thus, the simulation can play an important role in interpreting experimental

observations.

Experimental outcome is frequently compared with data calculated from MD simulations

in order to validate the methodology and estimate systematic errors. The simulation can be

carried out under conditions that are difficult or impossible to achieve in experiments, for

example, under very high temperature or pressure. Therefore, combination of experimental and

simulational information is of relevance in complementing and validating both approaches.









1.5 Temperature Jump Experiments

A maj or shortcoming of most experimental methods in protein folding is their limited time

resolution while key events might occur much faster than that. Relaxation methods that can

probe very fast time scales are of great interest in the studies of fast reactions.19,20 A powerful

relaxation method for the study of protein folding is temperature-jump (T-jump). The T-jump

experiments were originally developed in the 1950s with the application of resistive heating. It

was recently revisited and applied to protein folding dynamics with the use of modern laser

heating.21 Laser T-jump can reduce the dead time to nanosecond or picosecond scale and probe

the earliest folding events. Laser T-jump is also advantageous with regard to the fast rise time

coupled with a small amount of sample due to small heating volume. Fast T-jump experiments

are, hence, adequate for studies of kinetics and later combined with theoretical results for a

detailed description of biological systems.

A scheme of experimental T-jump relaxation kinetics is shown in Figure 1-7. The sample

is initially in an equilibrium state at the initial temperature, Tinitial. At t=0, a pulsed pump laser

increases the temperature of the solution until the final temperature (Trinal) is reached within a

nanosecond or shorter timescale.22 The final temperature should be sufficiently different from the

initial temperature to perturb the equilibrium significantly. Temperature change (AT) is generally

set up between 100C and 200C in an aqueous sample of protein. The unfolding process is

initiated rapidly, and the subsequent relaxation kinetics is monitored and recorded at a time

resolution of ~1ns 22,23 by using time-resolved infrared (IR) spectroscopy,19,24 UV Circular

Dichroism (CD),20,25 Or Trp-fluorescence,4,26,27 until the system reaches a new equilibrium. The

resulting kinetic spectra are then fitted to a proper relaxation curve; for example, single (for two-

state mechanism) or double (for three-state mechanism) exponential model.










initial equilibrium
at Tinitial


new e uilibrium


Signal UYIIIUI
at Tainal









t=0


t 10-s s
Figure 1-7. The general scheme of Temperature-jump (T-jump) relaxation kinetics.28

In a two-state folding mechanism, the process assumes only two states; the folded (F) and

unfolded (U) ensembles with kF and ker, the folding and unfolding rates, respectively.


Unfolded (U) 4 FFolded (F) (1-11


The folding rate equations are derived as,

d[F]
= kF [U] ke,[F] (1-2)

d[U]
= k,[F]-kF[U] (1-3)
where [U] and [F] are the concentrations of unfolded and folded states, respectively. If we

assume an initial concentration of 1, then this makes [U] = 1 [F] Thus, the time dependent

populations can then be directly solved leading to:29

F(t) = Czet + C, (1-4)
U(t) = 1- F(t) (1-5)
The population relaxation is dominated by a single rate constant,

Ai= k, +kF =kF (1+ K') =1/ rm,==1/ z, +1/ rF (1-6)









In this equation, ZF is the folding time (and z, is the unfolding time). The values of the two

constants C1 and C2 are COmpletely determined from the initial and final equilibrium


concentrations and are then not adjustable parameters. The equilibrium constant K (- ) can be


written as a ratio of the forward and backward rate constants and also determined by the

equilibrium concentrations of the folded and unfolded states. Therefore, we could derive the

folding (zF) and unfolding times (zu) by combining the relaxation results (Equation 1-6) and

equilibrium constant.

1.6 Circular Dichroism

Circular Dichroism (CD) spectroscopy is a widely used technique for the study of

secondary structures of polypeptides and proteins.30-34 CD measures the differential absorption

of choromophore -containing chiral molecules- between left circularly polarized light (LCPL)

and right circularly polarized light (RCPL), CD=Abs(LPCL)-Abs(RPCL), which arises from

structural asymmetry chirality. This, therefore, provides information about both conformation

and its change. It also valuable for analysis of macromolecules, globular structures, and drug

complexes in the field of biological, biochemical, chemical, and pharmaceutical sciences,

because it can give more detailed data than most absorption or fluorescence spectroscopy

techniques. 35,36

The CD data are reported in units of ab sorbance (Ab s(LPCL)-Ab s(RPCL)) or ellipticity

(6). Molar ellipticity (6) is usually utilized when CD involves molar concentration, and is defined

as:

[B]= 1000/Cl (1-7)
where C is the molar concentration (mol/L) and I is cell pathlength (cm) of the sample. The unit

of molar ellipticity is typically reported as deg-cm2 -dmol-l or deg-M^ml









CD data contain secondary structural information in terms of amide transitions in the

backbone chain. For example, a-helical content consists of a positive band at 190nm and two

negative bands at 208nm and 222nm,37 whereas P-sheet contains two opposite signs at 215nm

(minimum) and 198nm (maximum).38 In particular, a strong negative band with molar ellipticity

at 222nm is a key indication of helix formation of protein and peptides, because it is strongly

affected by structural changes between folding and unfolding processes. Helical content can be

monitored effectively in CD spectra and be used to understand protein folding and unfolding

procedure.39 Figure 1-8 shows the standard curves for CD spectra of poly-L-lysine in different

secondary structure conformations (random coil, P-sheet and a-helix) from Campbell et at. 40










E4-




P random coil




be~ta sheet


-4-


180 200 220 240
wavelength (nm)
Figure 1-8. The standard curves for CD spectra of poly-L-lysine in different secondary structure
conformations taken from Campbell & Dwek, 1984.40









1.7 NMR Spectroscopy

The spectroscopic measurements are extended to study more completely structural

properties of proteins and peptides including the kinetic and thermodynamic results on protein

folding.41 Nuclear magnetic resonance (NMR) spectroscopy is also an important tool in these

studies along with CD spectra. The most accessible quantities in NMR spectroscopy are

chemical shifts,42 nuclear Overhauser effects (NOEs), and scalar coupling constants.

Allerhand et al. showed significant differences between C" chemical shifts in random coil

and helical polypeptides.43 These observations showed that the chemical shift might be applied to

probe secondary structures of proteins. The chemical shifts of 1H, a-13C, and carbonyl-13C r

conformation-dependent, as shown by empirical 44,45 and ab initio studies.46 Therefore, they can

be used for determination of backbone conformations. Later, Dalgarno and his colleagues

defined the 'secondary structure shift' (sometimes called conformationall' or 'conformation-

dependent' shift), A6ss, as:47
AS = 3(1-8)

where Srcoil refers to the standard chemical shift measured from random coil. The relative

secondary structure shift (A6ss) could be correlated with the intensity of the helical CD signal

(6222). For example, upfield shifts for 1H, 1N and downfield shifts for 13C are observed for helix

formation.42

Since one-dimensional (lD) NMR spectra are too complex to interpret due to severe

overlapping signals, two-dimensional (2D) experiments are more popular in the studies of

protein folding at individual residues. The hetero-nuclear single quantum coherence (HSQC)

experiment is also frequently used in the field of protein NMR. The spectra are two-dimensional

between 1H and hetero nuclei (13C Of 15N). They contain thermodynamic information such that

the spectrum is well dispersed and all individual peaks are distinguishable when the protein is









folded. In this dissertation, we focused on conformational variations by direct use of chemical

shift and HSQC experiments.48

1.8 Overview of Research Projects

The main goal of this work is to simulate the experimental T-jump setup for the study of

protein folding. Computational methods used include replica exchange molecular dynamics

(REMD), calculated CD spectra, structural cluster analysis, and computational NMR chemical

shifts. Two proj ects have been performed: the first one is the T-jump simulation of Alanine20,

and the second project is the same type of study of two related 14-residue peptides.

1.8.1 First Project: Simulating Temperature Jumps for Protein Folding

My first project presents a new computational methodology aimed to calculate

thermodynamic and kinetic properties of peptide folding, and designed to mimic the way

experimental measurements of these properties are made. Particularly, I focus on T-jump

simulations of folding rates, and show how a combination of REMD followed by multiplexed

molecular dynamics starting from structures taken from the REMD runs can be used to extract

properties in line with experiments. A model system, Alanine20, WAS studied in this proj ect as a

proof of principle and description of the methodology.

1.8.2 Second Project: Folding Kinetics by Temperature-Jump Simulations of Two Related
14-residue Peptides

As follow-up of Project 1, REMD simulations of two closely related 14-residue peptides

were performed to obtain equilibrated ensembles. Snapshots from this ensemble were used as

initial structures for the T-jump simulations. These two 14-residue peptides are very similar but

have different experimental folding rates by a factor of ~2.9.49 They were selected by the

experimental group to compare end-capping effects which can stabilize a-helix. The folding

kinetics of the two 14-residue peptides is studied by using T-jump simulations, and their results









are analyzed using calculated CD spectra to obtain the folding and unfolding rate. For these two

systems, the relaxation time and folding/unfolding rate constants are calculated and compared

with experimental data.









CHAPTER 2
THEORY AND METHODS

In this chapter we will present some general discussion of theory and methods used in this

dissertation. In particular, we will address issues of sampling, force fields, Langevin dynamics,

generalized Born (GB) solvation model, and computation of circular dichroism (CD) and NMR

chemical shifts.

2.1 Conformational Sampling

The MD simulations of biomolecules are still immature due to both inaccuracy of the force

fields and inadequate conformational sampling associated with the number of degrees of freedom

of system. The energy surface of biological systems is generally rough and rugged, such that it

contains many local energy minima, which are isolated by high, insurmountable energy

barriers.so"si MD simulations may often get trapped in a local minimum and never reach the

global minimum. One way to overcome this sampling problem is to perform simulations in

generalized ensembles, where the construction of the ensemble is weighted by a non-Boltzmann

probability weight factor. Therefore, the resulting distribution guarantees a random walk in

energy space, producing much better sampling in the conformational space. The results need to

be properly re-weighted to give any thermodynamic quantity as a function of temperature.52-54

Many methods based on generalized ensemble algorithms have been introduced to

overcome sampling problems of biological molecules; the multicanonical algorithm (MUCA,55,56

also referred as entropic sampling" or adaptive umbrella sampling'"), simulated tempering

(ST),59 1/k-sampling,60 Tsallis statistics 61 with simulated tempering,62 replica-exchange method

(REM),63,64 and replica-exchange multicanonical algorithm (REMUCA).65 The replica- exchange

method (REM) (or parallel tempering 64,66) iS one of most widely used algorithms in a

generalized-ensemble.









Replica Exchange Method (REM). In REM, the standard Boltzmann weight factor can

be used. A number of non-interacting copies (replicas) can be simulated independently and

simultaneously at different temperatures by the conventional MD or Monte-Carlo (MC) methods.

Conformations are exchanged between different temperature replicas every few steps with a

specified transition probability that is defined by the Metropolis criterion. This exchange process

enforces random walks in temperature space, which in turn leads to random walks in potential

energy space. Consequently, REM has been widely applied to protein and peptide folding

research.50,64,67,68 This dissertation makes use of the REM algorithm, modified to be combined

with molecular dynamics, the so-called the replica exchange molecular dynamics (REMD).

In REM,SO an artificial system composed ofM non-interacting copies (or replicas) is

considered at M different temperatures, Tm (m=1, 2,..., M). The state of this generalized

ensemble is defined as X = (x,["'mi,..., xf" 1) with xj,', (p '1, q '1),,,where p '1 q '1 represent

moment and coordinates, respectively, for replica i at temperature m. Since the replicas are non-

interacting, the weight factor (W) for the state X is then given by the product of Boltzmann

factors for each replica or temperature, as shown in Equation 2-1:


WRA (X) = expI -= I9,,,~ 1 1 (2-1)


where p = (kB is the Boltzmann constant) and the Hamiltonian H(p '1, q 'l) is the sum of
k,T

kinetic and potential energies.

Now, one can attempt to exchange temperatures between the ith and jth replicas at

temperatures Tm and Tn, respectively. The new state of the system becomes:

X =(..., x[I', [1,, ),,.) X =(..., x,1 ..., xI,',...) (2-2)
The detailed balance condition needs to be applied to converge to an equilibrium ensemble:









WRAI (X~w(X 4 X' ) = WRAI (X' )w(X' 4 X) (2-3)
where w(X 4 X ) is the transition probability from state X to X' and WRAI (X) is the weight

factor of the state X. From Equation 2-1, 2-2, and 2-3, the exchange probability (P) is obtained:


P=w(X X ) W (X )
w(X 4 X) Wa (X)
-0,EeE, -fEbt., 40 r~,,a ,
=e e
e-P,E, e-PJEJ (2-4)
See




Exchange attempted (and rej ected)





T4 replica 4



T3 _tllllll replica 3



T2 r -' replica 2



T1 rl 1111 1 _' replica 1





Exchange accepted tm

Figure 2-1. The sketch of Replica-exchange method (REM) simulation in amber molecular
dynamics packages.









where A = (p, /7,)(E, E, ) and E is the potential energy of each replica. One can now obtain

the acceptance probability of replica exchange (P~accept)) by using a Metropolis criterion (or

Monte Carlo procedure):69


PJ(accept) = rmm, e "" ,= min l, e" (2-5)
The general simulation of the REM performs the following steps:

1. Each replica is simulated-based on canonical ensemble-in parallel and independently for
a certain number of MD steps.

2. Periodically, replicas with adj acent temperatures are swapped with acceptance
probability, P~accept) from Equation 2-5.

3. Repeat the process.

In step 2, the exchanges are only allowed between adjacent replicas in temperature, because the

acceptance ratio of the exchange decreases exponentially with increasing difference between the

two temperatures.63 Figure 2-1 shows a sketch of a REM simulation, describing the mechanism

of replica exchange (or rej section) between different temperatures.


2.2 Force Field

The force field representing the collection of molecular interactions represents the behavior

of all atoms and bonds with specific fitting parameters. Many different simulation packages have

been developed over the years; for example, AMBER,70,71 GROMACS,72 OPLS,73 and

CHARMM.74 Names of these packages generally imply the empirical force fields. In the

framework of this dissertation, the AMBER force field, the most commonly used for

biomolecular systems, was applied. Its potential energy function (U(R)) with a corresponding set

of empirical parameters is shown below:"










U (R = K, r -req 2 onds~
bonds
+ [ KH(B- 0q)2 Gu~e/BS
angles

+ C (1 + cos[nq5 7) dihzedrals (2-6)
dthedrals 2
atoms A B
+ van der Waals


+ [ electrostatic

where req and Beq are equilibration structural parameters. Kr, Ks, and V, are force constants, n is

multiplicity, and y is the phase angle for the torsional angle parameters. In addition, A, B, and q

are parameters related to the non-bonded potentials. Balance in parameterization can result in

reasonably good compromise between accuracy and computational efficiency, thus, reproducing

simulation results close to experimental ones.

For the non-bonded part, the van der Waals parameters are usually determined by

thermodynamic properties of various pure liquids.76,77 The electrostatic parameters are calibrated

using a restrained electrostatic potential fit (RESP) model.78,79 The parameters for the first three

internal terms (bond, angle, and dihedral) of Equation 2-6, come from a combination of

experimental data and high-level ab initio calculations.

Numerous IVD simulations have been run for proteins and nucleic acids under these force

fields and compared with experimental structures over the decades. Nevertheless, older A1VBER

type force fields present many deficiencies, such as the well-known over-stabilization of a-

helices. In order to avoid this tendency and thus to get better backbone dihedral angles, Hornak

et al. introduced new parameters, considering the energies of multiple conformations from high

level ab initio calculations.so Since there are many approximations within force fields,









optimization of force fields remains to be improved along with advance in experimental

technology.

2.3 Generalized Born (GB) Solvation Model

Simulations with an explicit treatment of solvent provide much improved accuracy, but is

computationally expensive for larger molecules, for example, proteins or nucleic acids.81-83

Alternatively, implicit solvation models have proven to be valuable tools for computational

efficiency and also relative simplicity. In particular, the Generalized Born (GB) model is one of

the most popular implicit solvation models. The solvent, such as water, in the GB model is

treated as an infinite continuum medium with the corresponding dielectric properties.81,84 Thus,

the GB model calculates approximate values of the solute-solvent electrostatic free energies of

solvation (Gol) and gives rapid estimates of Gol to save computation times in calculations.8ls

The electrostatic contribution of the solvation free energy in the reaction field, Greac, is

approximated by a system of simple ionic particles with radius a and charge q:84


Apot =~ 4 rea =~ 1E (2-7)
where E is the dielectric constant and this result is the well-known Born formula.86 If the simple

ion is expanded to a "molecule" consisting of spheres of radii (al, at, ..., an;) and charges (ql, qz,

..., qN) with the separation distance rt, between charges q, and q,, the polarization free energy

(Generalized Born (GB) equation) has been approximated as follows:s


B, 2\ e]i fGB 2s
where

fGB~~r i r 050,D=r,/4, )(2-9)
The fGB is defined as a function of an "effective Born radius"(a,) when the distance rzi 0 ,

while fGB- rJ as r,, -0.8









In the original model, the effective Born radius (ai) was computed by a numerical

integration procedure,81 but more recently "pairwise" approximations in which ai is estimated

via a summation over atom pairs, has been proposed by several groups.82,87,88 Therefore,

effective Born radius (ai) can be derived as:

a,' = p, '-fg(r:, ,rpp, pi) (2-10)

where p, is an intrinsic radius for atom i, and g() is a positive function which depends on the

positions and sizes of the atoms and also has scaling factor for an empirical correction.83,88,89

While the GB approximation is highly efficient for larger molecules, it is not so well

balanced between protein-protein and protein-solvent interactions, compared to explicit solvent

model. For example, over-stabilization of salt bridges has been frequently observed in GB

model, causing a significant conformational difference from explicit solvent model.90-93 In Order

to quantify the potential overstabilization of ion pairs for both models, Geney et al. performed

Potential of Mean Force (PMF) method of salt bridge formation and found an excessive strength

of salt bridges in GB.94

The lack of solvent friction in GB can accelerate conformational transition rates, resulting

in faster conformational sampling and at the same time, correctly predicting the native

conformations.95-98 Langevin Dynamics is one of the commonly employed methods to overcome

the frictional and high velocity collision problem.

2.4 Langevin Dynamics

Langevin dynamics complements Newton's second law to account for omitted (solvent)

degrees of freedom. The Langevin equation includes a frictional term in the form of a stochastic

differential equation and thus, it attempts to mimic the viscous aspect of a solvent. However, it is

an incomplete implicit solvent model, since it ignores electrostatic or hydrophobic effect. Those









effects are included via the implicit solvation models, such as GB, described above. Furthermore,

Langevin dynamics controls the temperature as a thermostat, thus approximating the canonical

ensemble.

The Langevin equation for motion of a particle i can be written using a stochastic

differential equation as:

d2X (t 2C t
m 2 () -7 mi + R, (t) (2-1 1)
dt2 dt
where m, and x, are the mass and position of particle i in the simulation, respectively. F, is an

interaction force between a particle of interest and other particles and R(t) is the force on the

particle due to random fluctuation by interaction with solvent molecules.99 The collision

frequency (y) is derived from the friction coefficient (5) by y=(/m (m is the mass of the particle)

and sometimes referred to as the friction coefficient in the literature. R(t) is a white thermal noise

that obeys the fluctuation-dissipation theorem at temperature T,100

~R(t)R(t ) = 27kTm3(t- t ) (2-12)
where kB is the Boltzmann constant.

2.5 Computation of Circular Dichroism (CD)

The CD spectra generally provides a direct measure of chiralityy" of the molecular

structure, since the magnitude and sign of the CD spectrum depend on the geometrical variables

and electronic structure of a molecule. The protein can be characterized as a collection of

independent chromophores. An individual chromophore that is sensitive to secondary structural

conformation and its interactions between the transitions on chromophores are the basis of

calculations of CD. The matrix method is most commonly used to compute the CD spectra of

proteins and peptides, where the excited states of each chromophore are subj ect to quantum

chemical treatment, considering interactions between the chromophores based on classical

physicS.101,102 The rotational strength, which gives the intensity of a CD band, can be









theoretically defined in terms of the imaginary part of the scalar product of the electric (CL) and

magnetic (m) transition dipole moments of an electronic transition, using the Resenfeld equation

and measures transitions of excited state.103 For an electronic transition from ground (0) to

excited (i) states (0 i), the rotational strength can be calculated according to:104

Ro, Im7, pe?,1*tW,#n;<,1(2-13)
where Im represents imaginary part, ry, and ry, are the ground and excited state wave function,

respectively. pe, and ps, are the electronic and magnetic transition dipole moment, respectively.

Since an electronic excitation occurs only within a group, rather than between groups, a

protein can be considered as a set of M non-interacting chromophoric groups in the matrix

method. The excited-state wave function of the whole system (77 is expressed as a linear

combination of the basis functions (Ora) for each chromophoric group with the ni excitation:




Each basis function is a product of M monomrer wave functions, such that:

to(~J =- 4,.0 a, (2-15)
where
molecule is not symmetrical, the CD spectrum can be obtained from the sum of these rotational

strengths derived as a nonzero value from each transition.

A Hamiltonian matrix of a protein is composed of the excitation energy of a single

chromophore (forming the diagonal elements) and the interaction between different

chromophoric groups (forming the off-diagonal elements). The off-diagonal elements can be

simplified by charge distribution of electronic interactions.'os Thus:

Ir ;kiC zqni'qklnl Iym~kln (2-16)
where m2 and n correspond to the point charge of the transitionJ j on chromophore i) and

transition 1(ton chromophore k), respectively, and r represents the distance between the point









charges. As an example, the Hamiltonian matrix for the amide electronic transitions between nxn

(at 220nm) and xx~n (at 193nm) is shown below:




H = """ "" (2-17)



The diagonalization of the matrix H using a unitary transformation provides the eigenvalues and

eigenvectors corresponding to all transitions of the protein. The eigenvalue gives information

about the excitation energy and the eigenvector describes the mixing of localized transitions. The

rotational strength (Equation 2-13) of each excited state can now be derived from the eigenvector

and be used to calculate the CD. In our work we will use the programs design by Sreerama and

Woody, where instead of performing a quantum chemical calculation they employed parameter

sets consisting of a combination of experimental data and theoretical parameterS.105,106

2.6 Computation of NMR Chemical Shifts

The NMR chemical shifts are affected by the environment. Calculating the shift is,

therefore, important for interpretation of structural information on macromolecules. Empirical

methods,44,48,107-111 Semi-empinical models 112,113 and ab initio quantum approaches 114-117 have

been tried to calculate the chemical shift.

2.6.1 Calculation of Proton ( H) Chemical Shift

An equation of the proton chemical shift is generally described in terms of various

contributions, as below:ll

AS = store ;0 = s, os, ang ~ HB 6e 6s,,de nsc, (2-18)
where 3;, = the random coil chemical shift value of an amino acid residue, St,, = the backbone

torsional contribution, 3,,,, = the ring current contribution, 6HB = the contribution arising from









hydrogen bond, 3, = the electric field or local charge contribution, 3side = the side chain

torsional contribution, and 3mase = other chemical shift contributions including solvent,

temperature, motional averaging, and covalent bond geometry. The empirical model (Equation 2-

18) was developed and parameterized to experimental shift data through literature analyses.

107,114,115,119,120 However, it simply represents rough and empirical knowledge of chemical shift

propensity, rather than unique (or complete) and quantitative description of proton chemical

shifts ( H).112

Quantum chemical shift calculations were performed to improve the accuracy of the

previous empirical models, considering ring current,121 electrostatic effects, structural

dependence of magnetic anisotropy, and close contact contributions."' As a result, a new

empirical model developed via a combination of the empirical formula with the quantum

calculation, was introduced with an improved prediction of proton shifts."l

2.6.2 Calculation of 1sN and 1C Chemical Shifts

Xu et al. predicted 15N, 13 u, 13Cp, and 13 carbonyll C) chemical shift in proteins, using a

mix of quantum chemistry and a database of experiments.122 Figure 2-2 presents an outline of

15N and 13C Shift prediction algorithm, where database of density-functional derived shifts in the

program SHIFTS (version 4.1) was used. This database is also used for chemical shift

calculations in this dissertation.122 The SHIFTS program first takes a protein structure in

Brookhaven (PDB) format and calculates the structural parameter of all the amino acids within a

given protein, such as backbone conformation, side-chain orientation, and hydrogen bonding

geometry.

The density-functional database was built based on the calculated chemical shift patterns of

1335 peptide sequences which are derived from 20 proteins. The calculated results identified









various significant potential contributions to the shift: the backbone cp and uy torsion angles of the

three consecutive residues (preceding (i 1), self (i), and following residues (i + 1)), side-chain

orientations of two consecutive residues and hydrogen bonding. Therefore, the total contribution

is given by the sum of the individual ones:

A(c) = C A(k, c)
k (2-19)
where k denotes one of the contributions, and c is either helix or sheet structure. The predicted

chemical shift, 6pred(C), iS then derived as:

3 pred (C=nFR)~C (2-20)
3RM. () is a reference chemical shift for an amino acid, where c = a for helix and c = P for sheet.

3RM ) is ideally determined by DFT calculation using the standard structure parameters from

the literature.122 Finally, the process is followed by side-chain orientation refinement based on

experimental shifts for an improved prediction.



















Structure (X-ray, NMR, or modeling)


Calculate the structure parameters
for all amino acids in the protein


Calculate individual shift correction from DFT
A(k) = S(k) 6i, (k) Database


Calculate total contributions





Predict chemical shift

s6 pred ~ REF C ()


side-chain orientation refinement



output

Figure 2-2. Outline of 15N and 13C chemical shift calculation in SHIFTS program.122









CHAPTER 3
SIMULATING TEMPERATURE JUMPS FOR PROTEIN FOLDING

3.1 Introduction

Understanding the structure, kinetics and thermodynamics of protein folding is one of the

unsolved problems in biology. Many human diseases like Alzheimer's and mad cow disease 5

are directly associated with protein misfolding, unfolding and aggregation. When studying

protein and peptide folding, there are three main questions of interest. First, given a primary

sequence, is there a unique 3-D structure under physiological conditions and if so, what is that

structure? Second, how does the peptide fold into its native structure? Third, how fast does it

fold? 123-128 These questions are routinely answered in the laboratory using a mix of structural,

thermodynamic and kinetic methods.24,129 Both experimental and theoretical approaches to the

protein folding problem have been used to address these questions.25,28 It is basically impossible

to probe all possible protein conformations experimentally because data, when available, is

averaged over time and over many molecules (with the notable exception of single molecule

experiments that still average over time).24,13 Alternatively, molecular dynamics simulations can

be used to provide a detailed description of the system.131-134 In this chapter I focus on the speed

of protein folding, but it is clear that underlying methodology, structure and thermodynamics are

also available.

In recent years, the use of multiplexed molecular dynamics runs to study folding has

become commonplace. The availability of very large numbers of processors for short times has

permitted the existence of pioneering efforts such as Folding@Home.4,135-137 This technique has

been able to reproduce experimental folding rates for a number of systems.4,138-140 In its most

used, basic form, it starts from a very large number of initial conformations, and runs molecular

dynamics for each of them for a pre-defined amount of time (a small number of nanoseconds).









The initial coordinates are usually chosen either fully extended or taken from a high temperature

run. The procedure then monitors the evolution of each sample independently and, after the

predefined amount of time has passed, simply counts the number of runs that produced a folded

structure.138,141,142 Under the assumption of a single exponential decay, one can extrapolate to

long times and extract a rate constant for the process. In a single exponential decay assumption

(over all time scales) with a time constant z, and for M independent very short MD runs of time t,

one expects around tM/z runs to have folded.137 For typical values of t=50ns, z=10 Cps and

M=10,000 trajectories, one expects only 50 of the runs to have folded. These represent, by

definition, the fastest component of the folding. The basic extrapolation relies on an assumption

of a single exponential process over all time scales, even the very short ones. In other words, it is

assumed that the fastest folders truly follow the same pathways as the overall ensemble.

However, some of the assumptions in the Folding@Home style methods become invalid

under certain conditions. Several groups showed that an extrapolation of very short time decays

to asymptotic exponential behavior might be unreliable.143,144 Paci et at. 145 TepOrted that the

fastest folding events do not agree with a corresponding ensemble behavior obtained by the

distributed computing for a three-stranded antiparallel P-sheet peptide. This is in some sense

obvious: given a system with many free energy minima and a number of barriers (whose

ensemble relaxation should be described in terms of a master equation description), one should

not expect a single exponential decay to hold at ALL time scales.146 This has been best expressed

by Daggett and Fersht in ref.:147 "It is our opinion that, at the molecular level, intermediates are

always present. In other words, true two-state folding with only the denatured and native states

occupying free energy minima is implausible."









Moreover, Pande and his colleagues Start from a fully extended conformation for their

initial states. While they see evidence of very fast (tens of nanoseconds) relaxation to a

collapsed, unstructured state in times substantially shorter than their simulation times, it is clear

that this is a choice of unfolded ensemble that cannot be compared with typical kinetic

measurements (such as temperature jumps) which usually measure relaxation times between two

closely related equilibrium states.

Since in Folding@Home a folding event is only counted against a pre-determined

coordinate set (the folded state), it is possible that a number of the simulations have ended in a

different configuration. If this new state has lower (free) energy than the initial choice, then it

should be properly called the folded state. The solution to this problem is to run every simulation

long enough for a substantial percent of the ensemble to fold into the same state, which then is

defined, a posteriori, as the folded state of the system.

Recently, Pande et al. and Levy et. al,140,148-150 introduced a new method, namely,

Markovian state models (MSMs), to predict protein folding rate constants. The MSMs can

calculate both folding probability (Pfold) of all the configurations in a system and mean first

passage time (MFPT) from the unfolded state to the folded state.140

In the present report, the folding kinetics of a polyalanine peptide is described and

discussed based on MD simulation results. One of the most used experimental approaches to

protein folding kinetics is laser-induced temperature-jump (T-jump) spectroscopy.26,27,151 T-jump

can raise the temperature extremely fast (nanosecond scale) and record the relaxation to

equilibrium with a time resolution of ~1ns.22,23 Since the usual distributed computing procedure

(Folding@Home) is clearly not what is done experimentally, where, in the case of T-jump,

relaxation of the ensemble from one equilibrium distribution to another one is realized, it is









important to ask if a protocol that closely resembles an experimental T-jump could be designed

and explored. This chapter will show that this is indeed possible, focusing on a model system

that should be considered as a proof of principle calculation.

In the computational T-jump which is first introduced in this chapter, the temperature

will be increased suddenly, creating a system no longer at equilibrium and having to relax to a

new state of equilibrium. In this method, the conformational space of a polyalanine peptide is

pre-equilibrated at different temperatures using replica exchange molecular dynamics (REMD).

In this way, we will have a-priori knowledge of what the ensemble looks like at the initial time,

before the T-jump (equilibrium at T1ow), and as time approaches infinity (equilibrium at Thigh).

Then we will monitor a number of traj ectories started from structures taken from the T1ow

ensemble, run them at Thigh for ai pre-defined time, and then, at each time-slice, compute any

property we desire. In a similar way with experiment, this data will be analyzed after watching

the relaxation data, before deciding on what kinetic scheme will fit the data best. Additionally,

we can then go into the ensemble, and ask detailed structural questions about rates and pathways.

3.2 Methods

3.2.1 Simulation Details

The initial structure of the alanine polypeptide, ACE-(ALA)20-NME (ACE is acetyl

beginning group and NME is N-methylamine ending group), was built in an extended

conformation with the AMBER 8.0 simulation package" using recently published AMBER

ff99SB force Hield,76,152,153 which has been shown to provide improved agreement with

experiment.154,155 A cutoff of 16A+ was used to compute long-range interactions, and the

Hawkins, Cramer, Truhlar 88,89 pairwise generalized Born (GB) implicit solvent model, with

parameters from Tsui and Case,s was then applied to mimic the effects of water solvation.81'156

The system was initially subjected to 2,000 steps of minimization and the resulting









conformations were used as initial seeds for the REMD simulation.

The REMD method used the multisander implementation in the AMBER 9.0 molecular

dynamics program. Sixteen replicas were simulated for 200ns each (total of 3.2Cls) at

exponentially spaced temperatures, from 153K to 542K (153, 166, 181, 197, 214, 233, 253, 276,

300, 326, 355, 386, 420, 457, 498, and 542K). These temperatures resulted in an average

exchange rate of 15% between adj acent replicas. The SHAKE algorithm was used to constrain

the lengths of all bonds involving hydrogen and a 2-fs time step was used for every replica.

Exchanges between replicas was attempted every 10ps, resulting in 20,000 attempted exchanges

at each temperature. Conformations were recorded every 2ps from the simulation of each replica.

The first 10ns of the simulation were discarded and the latter 190ns were saved for further

calculations, giving a total of 19,000 configurations for analysis.

Conformations obtained from the REMD calculation were used as the initial structures for

the T-jump experiment. 362 starting configurations were selected, equispaced in time, collected

from the equilibrated simulations at 181K. Each member of this ensemble was then instantly and

independently heated up to the Thigh of 214K (the details of the T choices are presented later on)

and MD was run for each member for 35ns to simulate the relaxation after the T-jump. All MD

parameters are same as those used in REMD. Figure 3-1 shows a sketch of both the experimental

T-jump scheme and computational T-jump setup to illustrate how these two methods are similar

to each other.

In order to properly identify the structural elements of the ensemble, we performed a

cluster analysis with moil-view 15 using backbone RMSD for residues 2 to 19 as a similarity

criterion with average linkage.159 The clusters were defined using a bottom-up approach with a

similarity cutoff of 2A+, for C"-RMSD. The representative structures from the REMD ensemble










at 153K were used to determine the composition of the folded ensemble.

3.2.2 Calculation of NMR Chemical Shifts

NMR Chemical shifts were computed in order to predict structural information. SHIFTS

(version 4. 1) program (by David Case group) was employed to estimate proton amidee proton,

and H,), as well as 1N, 13 u, 13Cp, and 13C' (frOm CO bonds) chemical shift of the polyalanine

peptide.112,121,122 SHIFTS program recognizes a protein structure in Brookhaven (PDB) format. It

computes proton chemical shifts using empirical equations and 15N, 13 u, 13Cp, and 13C' chemical



Cell

Laser 2 Y


Detector
PC
Laser 1







362 equilibrated configurations from REMD @ T,(181 K)
T-jump at t= 0
.......181 214 K

at tl
- -- at t2
- --- at t3

MD @ Tf
for 35 ns*


********~a 3 n
(214K*
Figue 31. he dagrm o theT-jmp etup A)ExprimetalT-jmp shem, *
ComptatonalT-jmp stu*









shifts from a database based on density functional calculations.

2D NMR spectra obtained using HSQC experiment and 13C,/13Cp CTOsspeaks relationship

enable us to see structural distributions of folded and unfolded states.

3.2.3 Calculation of CD Spectra

Computations on CD spectra for estimating secondary structure and calculating folding

properties were performed by Sreerama and Woody.105,106 The matrix method 101 (in origin-

independent formulation of rotational strength 102) was employed with a transition parameter set

consisting of a combination of experimental data and theoretical parameters: the authors used

experimental datal60 for two amide xx*n transitions and parameters from intermediate neglect of

differential overlap/spectroscopic (INDO/S) wavefunctionsl61 for the nx*" transition. The

rotational strength was computed in order to generate CD spectra through Gaussian band. The

bandwidths assigned for the nx*" transition, and for two xx*n transitions were 10.5nm, 11.3nm,

and 7.2nm, respectively.39

3.3 Results and Discussions

The conformations at the lowest temperature (153K) from REMD were sorted into

clusters.15 ClUStering of the ensemble at 153K shows only two substantially populated clusters.

The largest cluster contains 76% of the structures, whereas the next largest cluster is 9%

populated, with no other cluster having a population higher than 6%. Figure 3-2 A shows the

ensemble at 153K superimposed on the representative structure of the largest populated cluster.

Figure 3-2 B shows the structure for that cluster, forming a well defined a-helix. This is then the

reference structure for future analysis. For the ensembles at 181 and 214K, the C"-RMSD

(residues 2-19) are computed, versus the representative structure of most populated cluster, and

obtain the histograms of Figure 3-3 A. There is a clear separation between the features in these

plots, which allows us to define two different types of states, which are named folded (F) and









unfolded (U). Structures are defined as folded (F) (a-helical) if their C"-RMSD is within 2.0A+ of

the reference structure, and labeled as unfolded (U) otherwise. Based on this clear structural

separation, the process is treated as a two-state equation between the folded and unfolded

ensembles.


kF

kg


Unfolded (U)


Folded (F)


(3-1)


The folding rate equations for each species are written as,

d[F]
= kF[U] -k,[F]
d[U]
= k,[F]- kF[U]






dt "


(3-2)

(3-3)


A B

Figure 3-2. Cluster analysis of polyalanine. A) All conformations from REMD at 151K are
superimposed on the reference structure of the largest cluster, B) a-helix reference
structure from the cluster analysis.











0.2

A L T =181K
T= 14K

0.15-







S0.05-




0.0


0 2 4 6 8 10
RMSD (angstrom)












-0.











0 2 4 6 8 10

RMSD angstromm)
Figure 3-3. Histograms from the C"-RMSD. A) Probability density of C"-RMSD at 181K (blue
line) and 214K (red line), B) A(probability at 214K probability at 181K) vs. C"-
RMSD.










In the theoretical work concentrations were replaced by populations, which is equivalent of

having a total initial concentration set to unity. This then makes (U) = 1 (F). The time

dependent populations can then be directly solved leading to:29

F(t) = Czet + C, (3 -4)
U(t) = 1- F(t) (3-5)
The population relaxation is dominated by a single rate constant,

Ai = k, + kF /z, rela U F1 z. In this equation, ZF is the folding time (and z, is the


unfolding time). The values of the two constants C1 and C2 are COmpletely determined from the

initial and final concentrations and are then not adjustable parameters. By recalling that the

equilibrium constant K can be written as a ratio of the forward and backward rate constants, the

two can be then separately determined. The population of two F and U ensembles versus

temperature is shown in Figure 3-4. They were computed from the C"-RMSD histograms

(referenced to the F structure) using a 2.0A+ cutoff. At low temperature the system shows mostly


1|1| I'ldI'l( II
folded (F)
0.8-



0.16--



0.4-



0.2-



160 180 200 220 240 260 280 300 320 340

Temperature (K)
Figure 3-4. Fractions of a-helical folded state (F), and unfolded state (U) as a function of
temperature using probability (C"-RMSD).









a-helical folded states but the populations of unfolded states rise as temperature increases

providing a reasonable melting curve. A clear melting temperature was found at 190K. The

low melting temperature compared with regular experiments can be assigned directly to issues

with the force field and solvation model.1,94 It does not however subtract from the main point of

this chapter which aims to present and test a new method. Based on the melting curve, T-jump

simulations were performed with the temperature jump bracketing the melting temperature, from

181K to 214K. By having the complete description of the temperature dependent ensembles we

can have complete knowledge of what both the initial t=0 (T=181K) and t=oo (T=214K)

ensembles look like. In Figure 3-3 A, we show the RMSD probability at both initial and final

temperatures for the T-jump. In Figure 3-3 B, the probability difference (214K 181K) validates

choices of temperatures, by showing a significant change in (F) and (U). These figures

completely determine the values of C1 and C2 in Equation 3-4.

3.3.1 Calculation of Chemical Shifts

Calculation of chemical shift was performed on the traj ectories obtained from 10-200ns

REMD simulations. All the chemical shifts ( H,, 1Hp, 1H (along the N-H bond), 13Cu, 13CP 13 ,

(along the carbonyl bond), 15N) were computed using SHIFTS program (section 3.2.2). The

spectra measured at 16 different temperatures were then averaged for each residue (total 1,900

structures). The averaged chemical shifts are plotted as a function of residue number (Figure 3-

5). As shown in Figure 3-5, the highest temperature is placed at the bottom and all the averaged

chemical shifts decrease with increasing temperature except 13Cp and 1H, chemical shifts. We

can expect that our prediction of 13Cp and 1H, chemical shifts compared with experimental data

(Table 3-1) shows prediction errors from side chain orientation.

Table 3-1 presents chemical shifts values obtained using REMD simulations at the lowest

(153K) and the highest temperature (514K) for a residue 6 (A6) in polyalanine peptide. For






































-







T~emperatures
He

~lllll lll l ,l, r~r llll



"1











Temperatures
Hp


Temperatures


sa I 1 I [ I I I

,Co







| 1 i l l i s l ~,,,,1,, i l l


Temnerat~lren


Tenineratulren


i7,ca,


,,,,~


"i


p
p ,,
m
177 5


I L ~I LI


I





I


Temperatures


Residue


Figure 3-5. Averaged chemical shifts (proton, 13C, and 15N) for polyalanine peptide (ACE-
(ALA)20-NME) as a function of residue number, for all sixteen temperatures.


C
he ,
e




3,8



h
I3.7

36

p
s

m


C
h
e


1


"2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Residue









Table 3-1. Comparison of calculated chemical shifts for residue 6 (A6) in polyalanine peptide
and experimental values for alanine residue from referenceS 4,411
Helixa 153Kb Random coiled 514Kd 6f 6RCe s1-s,
13C, 54.7 54.51 52.4 52.47 2.3 2.04
13C' 179.6 179.34 177.6 177.20 2.0 2.14
13Cp 19.74 18.48 19.26 20.06 0.48 -1.58
'5N 117.94 122.5 115.28 2.66
a-'H 3.91 4.33 3.98 -0.07
N-'H 7.68 8.15 7.55 0.13
SAll data are given in ppm.
ac" experimentally measured chemical shift values from references.
b, d COmputationally measured average chemical shift values from 10-200ns REMD simulations.
e chemical shift deviations, 6, (helix) SRC (random coil), from references.
chemical shift deviations, Si (low temperature) Sh (high temperature), from 10-200ns REMD
simulations.

comparison, it also provides the chemical shift values of both helix and random coil for a alanine

residue, measured experimentally by Wishart et at. 42,111 and Spera et al.44 As mentioned in the

previous section (cluster analysis), the conformation at the lowest temperature (153K) is mainly

well-defined a-helix (76% populated). The calculated chemical shifts (13C) aIt 153K (Table 3-1)

agree well with the helix chemical shifts values from the experiment. In contrast, the calculated

values at 514K are comparable with the random coil values from the experiment. The calculation

results make sense, disordered conformations of the peptide (random coil) are expected at the

higher temperature and more structured conformations (folded or packed conformation) at the

lower temperature. The results also indicate that with increasing temperature, there is an increase

of the random coil population (unfolded state) accompanied by a decrease of a-helical structure

(folded state (F)). Therefore, one can detect formation of the secondary structure by monitoring a

change in the chemical shift (aS).

The calculated 13 u, and 13C' chemical shift are in good agreement with experimental

results. However, the calculated 13Cp chemical shift differs from experimental one, due to errors

associated with fitting of reference shifts in the database of the SHIFTS program.112,121,122









Wishart et al. commented that error in 13C and 15N chemical shift data is relatively larger than

that in the proton chemical shift.42

The 2D spectrum, for example HSQC, can usually provide more complete structural

details, compared to 1D one. Figure 3-6 shows 1H-5N HSQC spectrum of a polyalanine peptide

at 181K and 300K. At higher temperature (300K, Figure 3-6 B), the spectrum is poorly resolved

with slow fluctuation, due to the disordered state. This indicates unfolded states are dominated.

On the contrary, the spectrum at lower temperature (181K, Figure 3-6 A) is well resolved and

dispersed, as a result of folded states. Figure 3-6 included only the cross peaks providing the

structural information. However, the kinetic analysis can be performed with additional intensity

information.

The 13C chemical shift for C, and Cp has been often used to determine backbone

conformation of proteins and peptides, as found in many of previous NMR studies, such as

empirical 44,45 and ab initiO 46 methods. This is because the C, and Cp chemical shifts are mostly

determined by the backbone cp and uy torsional angle.41 In this studies, 13Cd13Cp CTOsspeaks in

the folded state are relatively sharp (2-4 ppm full width), narrow, and well resolved. The

crosspeaks in the unfolded state are, however, broad, spreading toward upright. Figure 3-7 shows

two-dimensional calculated 13Cd13Cp CTOsspeaks for polyalanine peptides (residue 5 (AS), 11

(All), and 16 (Al6)) obtained at 153, 181, 214, and 300K. The spectra measured at different

temperatures were then averaged over the ensemble (total 1,900 structures). Three labeled

residues are all similar in the general trend on temperature. The crosspeak at the lowest

temperature (153K) is relatively ordered, sharp, and narrow, indicating significantly folded (or

helical) conformation. As temperature increases, the crosspeak profile becomes broader toward

upright, showing that conformational distribution is temperature dependent.

































































~18


These results are in accord with the HSQC results that covered in Figure 3-6.

3.3.2 Calculation of Circular Dichroism

For each member of the ensemble and at each temperature in the REMD simulation, CD spectra

were computed using the technique described in the methods section. The spectra for


115


115.5


116
15N
(ppm)
116,5


117


117.5


118


A





-






O O
O
OO
O
O OO
Or


1HI- (ppm)


B


O
O

O
Og O
O
O
O


15N
(ppm)
116.5


117


117.5


1H (ppm)

Figure 3-6. 1H-5N HSQC spectra of polyalanine peptide. A) At 181K and B) At 300K.










153K 18 1K 2 14K 300K iI












"Cscheical s ift(pm
Figure' 3-.T odmninl1 3C Tspa hpsfrplylnn etdrsde5 1 b
and` 16 (AS All an Al)fo eea emeaue.Clrsceeaefo h
maiu ina nest (e)t h minmu sina inenit (le .
each~~~~. teprtr eete vrgdoe ebr fteesml ttl190srcue) h
data~ is shw nFgr 38Afrteaerg pcr e sus tepraue Inareetwth
RMSD~r daa h oettmeauesetu eeblsta fa -eiwiea eprtr


inrass tesytmbeoesdsodrd.Te ode ttehs w mnm in~ the CD\ spectrm
on a 08m(xx*aid rastin)ad nthr t22n (-* rastinalngte aboy
bod) ExeietlC pctafra -ei hosadee iiumfr22mta o
208nm162- ~ Th caclae loca miiuma 28mdosno g eewtthex ri nalests
indiatin ths rniin ob eysniie 00 -eia oyaaieA E(l)0

NME wee biltusig TNKE moecuar ynaicsproram163164and200stes o









minimizations were performed to investigate the difference. Minimum wavelengths are

compared between two CD spectra (one is before minimization, i.e. pure 100% a-helix, and the

other is after minimization) in Figure 3-9. The 100% a-helix has a minimum at 222nm while the

structure after minimization has a minimum at 208nm.

This discrepancy is assigned to problems with the CD calculations parameters, and been in

the process of investigating this. Figure 3-8 B presents the average value of CD at 222nm

() VeTSus temperature. This is the type of signal one can follow when performing the T-

jump experiment.

T-jump simulations were performed and the resulting time traces were fitted to calculate

the folding and unfolding rates with initial and final temperatures determined by referring to the

melting curve in Figure 3-4. 362 structures from the ensemble at 181K were used as starting

points for molecular dynamic runs at 214K, hence simulating a T-jump. The temperature reaches

the new value in a time scale of the order of the thermostat coupling (0.1Ipsl time constant used)

and should be thought as having a very short dead-time. These simulations were run for a total of

35ns each (total of 12.67Cps). At each time slice, a CD spectrum was computed for each

independent MD run, and the average performed. The folded and unfolded states were added and

then VeTSus time is plotted for each state in Figure 3-10. The folded and unfolded states

were defined in Figure 3-4 using C"-RMSD probability. When fitting this kinetic trace, the

power of this method can be clearly seen. This trace is bounded by the fact that we know, from

the equilibrium runs, what the system looks like at t=0 and at t=oo. This leaves a single parameter

to be determined via fitting, assuming a two-state system. The calculated relaxation data are well

fitted using a single exponential equation, as shown in Figure 3-10, indicating that the suggested

reversible two-state kinetic mechanism (Equation 3-1) during the cluster analysis is valid. From












A 60[000


40000




20000




0




-20000a




-40000


Wavelength (nm)


~i -10000
E
rYrr
E
o
a,


a
,cL
-20000
uu


-30000 '
100 200 300 400
Temperature :KI

Figure 3-8. Results of Circular dichroism. A) Calculated circular dichroism spectra of ACE
(ALA)20-NME from 153K to 542K, B) Average of CD222 () Of each
simulated polyalanines. Error bars represent standard deviation.












80000


600000


40000


20000


0


-20000 -
-40000
200 250

Wavelength (nm)

Figure 3-9. Two circular dichroism (CD) spectra, pure a-helix and a-helix after minimization,
are compared to see different minimum wavelengths.


10+06


-2e+06




-3e+06


-4e+c06


time (ns)
Figure 3-10. Average of CD222 () Of T-jump simulation data (red: folded (F), and green:
unfolded (U) states) are fitted using a single exponential equation.











this fit the value of Ai = (k, +kF) in Equation 3-4 is determined to be ps and
36339

Treax,=36.34ns. This relaxation time was used from CD to predict the time evolution of the F and

U populations. Figure 3-11 has the simulated average populations versus time, and the

populations predicted from Equation 3-4 also. The curves in Figure 3-11 are not fitted, but are

instead simulated using the fit from Figure 3-10. The agreement between the simulated and raw

population data is excellent. Using our knowledge of the limiting populations at the initial and

final temperatures, equilibrium constants can be obtained. We can then separately determine the

folding and unfolding rate constants at the final temperature.

The computed values are kF=2. 11 x 10-5ps l, and ker=6.44 x 10-6ps l, corresponding to a

folding time zF=47.5ns and an unfolding time Tri=155ns. Williams et al. have simulated

0.7







0.6


~i0.5--
unfolded
u- ~rI folded



0.4 --





0.3 I
0 10 20 30

tirne (ns)

Figure 3-11. Folding (green) and unfolding (red) fractions from T-jump simulation data are
calculated and fitted using same h in Figure 3-10.









21-residue alanine-based peptide (Fs21 peptide, -A5-(A3 A)3A-) and estimated folding time, 16

to 180ns using T-jump experiment from 9.3 to 27.40C.19

In the present model we assume that the implicit solvent is a correct representation of

structure and thermodynamics. However, it does not properly represent the friction of water

solvent.95 The time scales seen in this chapter are then much faster than they would be otherwise.

The following chapter describes our continuing line of work on systems where experimental data

is available and using Langevin dynamics (which incorporates friction).

3.3.3 Optimum Number of Unfolded States?

Two clusters were found in the cluster analysis earlier. The most populated cluster is a-

helix (Figure 3-2 B) that occupies roughly 76% of the population. It is defined as a folded state

in the two-state mechanism (a-helix and unfolded state). The second largest cluster is 9%

populated coiled-coil a-helix (the structure is shown in Figure 3-12). If the coiled-coil a-helix is

added as the second folded state, three-state kinetic mechanism (a-helix, coiled-coil a-helix, and

unfolded state) can be suggested.

Three different states (a-helix, coiled-coil a-helix and unfolded states) are chosen for

initial MD simulation run for 1Cps at 214K in order to check how many different states exist. The

simulation time (1Lps) is chosen because it is estimated to be close to the folding time of small

proteins (11-21 residues peptides) as the lower limit.165 This simulation possibly shows all the

intermediates involved in the protein folding process.14 The C"-RMSD (residues 2 to 19) was

computed from the 1 Cps trajectory with respect to a-helical, coiled-coil reference structures

(Figure 3-13). These RMSD plot reflects the number of different states. By comparing a-helix

reference (Figures 3-13 A, B, and C) and coiled-coil a-helix reference structure (Figures 3-13 D,

E, and F), simulated structures are classified as follows: a-helix when C"-RMSD1 < 2.0A+ and

C"-RMSD2 > 5.0A+ and coiled-coil a-helix when C"-RMSD1 > 5.0A+ and C"-RMSD2




























A B C
Figure 3-12. Reference structures from cluster analysis. A) All conformations from REMD at
151K are superimposed on the two reference structures, B) a-helix and C) Coiled-coil
a-helix reference structures.

< 2.0A+, where C"-RMSD1 was calculated from a-helix reference and C"-RMSD2 from coiled-

coil a-helix reference structure. Interestingly, several transitions, from coiled-coil a-helix to a-

helix folded state based in the rmsd range (0-2, 2-3, 3-5, and above SA+), are found in Figure 3-13

E. The fraction of four states was obtained by applying both C"-RMSD1 and C"-RMSD2, as

shown in Figure 3-14. In order to show relationship between two folded states, C"-RMSD1 as an

x-axis is plotted against C"-RMSD2 as an y-axis C"-RMSD (Figure 3-15), where four different

states are also found. Based on the results, the following kinetic mechanism can be suggested.

a-helix Unfolded Unfolded Coiled-coil (3-6)
k, (T T,~ k (T T,} k, (rrI

In Equation 3-6, two different unfolded states are separated by two representative

structures (U1 and U2). This implies that more representative structures can be defined without

limit. Based on the present experiments, the number of unfolded states is counted, introducing

the T-jump methodology. Optimal folding mechanism was not discussed in this chapter and

should be dealt in the near future.








(P A


;:L n I
e



ni iii~
1P~


i C


..D E F






Figure 3-13. C"-RMSD (residue 2-19) computed using different initial states from 1Cps M/D
simulations. A) a-helix, B) Coiled-coil, and C) Unfolded initial states from C"-
RMSD1. D) a-helix, E) Coiled-coil, and F) Unfolded initial states from C"-RMSD2.
3.4 Conclusion
This chapter presents the first attempt to using simulation conditions and predicting
observables as close as possible to experiment. As far as we know, this is the first time a
simulation of T-jump is used to calculate the folding rate constants.

By using a mix of very efficient sampling techniques (REMD) to properly populate
ensembles and regular MD simulations to study the non-equilibrium relaxation of the

populations, extracting kinetic and thermodynamic data are enabled. A set of folding/unfolding
time (rF=47.5ns and zU =155ns) was calculated by data fitting of calculated CD spectra.
The chemical shifts were also calculated and it showed systematic change in
conformational distributions depending on the temperatures. Finally, the number of folded and
unfolded states was determined and discussed, based on 1Cps long MD simulations.
In the future, implicit and explicit solvent models need to be compared and influence of
friction remains to be determined. As an extension of the present work, the following chapter


B










describes results obtained on real proteins using Langevin dynamics which considers frictional


effect.

































20000 40000 60000 80000


le+05


Time (ps)
Figure 3-14. C"-RMSD2 (with respect to coiled-coil a-helix) plots from 1Cps MD simulation
using coiled-coil a-helix initial state at 214K.


2 4 6 8


RMSD1 (alpha-he-lix)

Figure 3-15. C"-RMSD (residue 2-19) relation plot using C"-RMSD1 (with respect to a-helix
reference structure) as x-axis and C"-RMSD2 (with respect to coiled-coil a-helix
reference structure) as y-axis, respectively, at 214K.









CHAPTER 4
FOLDINTG KINETICS BY TEMPERATURE-JUMP SIMULATIONS OF TWO RELATED 14-
RESIDUE PEPTIDES

4.1 Introduction

Protein folding is described as a process of an ensemble of molecules reaching their

biologically active three dimensional structures from a linear chain of amino acids.99 Both

experimental and theoretical studies for over 40 years have converged to show and predict the

basic principle and particular mechanism of folding and unfolding of proteins.146 Their

mechanism and kinetics are still unknown or incompletely understood and therefore, are being

actively investigated in the molecular biology area. Particularly, simulations can describe the

folding process microscopically and its atomic details which are unavailable in experiments.

However, conventional molecular dynamics is unable to yield complete conformational

space sampling due to its high energy barriers and deep local minima in most of the systems.

Thus, the development of efficient algorithms becomes a point at issue. One of the oldest

methods is umbrella sampling technique that is combined with molecular dynamics simulations.

166 Brooks and coworkers 167 utilized this method extensively to understand thermodynamics and

kinetics of folding for biomolecular (or biological) systems. It was very powerful for the

investigation of the folding free energy landscape in several chosen reaction coordinates.

However, it was limited to small proteins (or peptides),168 and was not capable of determining

the biasing potential.

Replica exchange molecular dynamics (REMD),64 which adapts a random walk in potential

energy space, was introduced to overcome those problems. This technique shows not only

enhanced sampling but also increased speed of equilibration by treating the temperature as a

control parameter.









Direct experimental measurement of folding rates becomes possible by using the

temperature jump (T-jump) experiments in the nanosecond to microsecond time scale. The T-

jump rapidly changes the position of the equilibrium between folded and unfolded states by the

temperature change, and hence, reaction kinetics to the new equilibrium can be monitored 169

Moreover, the laser-induced T-jump technique was extended to far-infrared absorption (5.88 -

6.67 Cpm), which can observe protein secondary structure.29 Several groups applied this method

to helix-coil transition 19,170 and folding transition of hairpin structure.171'172

Combination of computational and experimental methods in studying protein folding is

advantageous.25 For example, Folding@Home typically runs a very large number of initial

conformations for a few nanoseconds, either fully extended or taken from a high temperature

run. Under the assumption of a single exponential decay that is based on two-state folding

kinetics, a folding time of 10 Cps is calculated when 50 out of 10,000 traj ectories have folded for a

50ns simulation. In order to verify the dynamics, the simulations were compared with T-jump

spectroscopic results.25,173

However, several groups pointed out that two-state model might not be applicable under

certain conditions. Daggett and Fersht 147 emphasize the problem in terms of intermediate states

that always exist between a denatured and native state. Sabelko et al. found non-exponential

folding kinetics of two proteins (yeast phosphoglycerate kinase (PGK) and a ubiquitin mutant)

from a nanosecond T-jump.173 F. Gai and coworkers also reported non-exponential behavior for

helix-coil transition kinetics.13 Therefore, the most important issue at this point is how we

perform our simulation in the conditions close to real experiments without any kinetic

assumption, while connecting theory and experiment.

In Chapter 3, we introduced Temperature-jump (T-jump) simulations of the polyalanine









peptides (Ala20) aS a proof-of-principle system.174 Since alanine-20 has high hydrophobicity and

insolubility in water," no experimental data have been reported. In this chapter, we apply the

same techniques (T-jump simulations) to model two related 14-residue peptides. Wang et at. 49

have previously studied the helix-coil kinetics of these two peptides using time-resolved infrared

(IR) spectroscopy, coupled with laser-induced T-jump technique. They reported that the one is a

general polyalanines derivative of Baldwin-type peptide,176 and the other contains helix stabilizer

and end-capping groups, showing difference in helicity and stability regarding the folding

kinetics. In this chapter, we simulated T-jump and calculated folding and unfolding rates of the

same peptides that Wang et al. 49 applied, using calculated circular dichroism (CD) spectra. The

results were then compared with the experimental data. Most importantly, the present work was

performed in a very similar way to the experiments, such that they were compared with each

other, minimizing assumptions. Initially, REMD simulations were performed to obtain

equilibrated ensembles to overcome energy barriers and accelerate the convergence. Snapshots

from these ensembles were used as initial structures for T-jump simulations.

The Effect of Frictional Coefficient. To further validate our T-jump methodology, we

focus on the influence of friction and random forces-introduced by the solvent-on the protein

folding kinetics. Kramers 1 proposed in its simplest form that the reaction rate (k) in the high

friction (y) limit should be proportional to the inverse of the friction of the solvent, k ac 1 / 7.

This inverse dependence of rates of protein folding on viscosity, rl (or, equivalently, the friction,

y) has been frequently reported experimentally.2395178,179

The effect of frictional coefficient through Langevin dynamics was studied for the folding

kinetics of two related 14-residue peptides. Different values of friction and random forces were

applied and compared with experimental results from Wang et al 49









4.2 Methods

Alanine-based peptides were originally designed by Marqusee and Baldwin 176 for ot-helix

formation. Two related 14-residue peptides, Ac-YGAKAAAAKAAAAG-NH2 (peptide 1), and

Ac-YGSPEAAAKAAAA-r-NH2 (peptide 2, where r represents D-Arg), were derived and tested

by F. Gai group.49 The initial structures for two related 14-residue peptides were constructed in

fully extended conformations with the AMBER 9.0 molecular simulation package.lso The

AMBER ff99SB force field 76,152,153 was used for both peptides. Modified generalized Born (GB)

implicit solvent model by A. Onufriev, D. Bashford and D. A. Case (GBOBC 181 was applied. The

system was initially subj ected to 500 steps of minimization, and then equilibrated by using

Langevin dynamics with collision frequency y=1.0psl for both peptides to account for frictional

effects. The resulting conformations were used as initial seeds for the REMD simulation.

All REMD simulations reported here were carried out using the multisander

implementation in the AMBER 9.0 simulation package. Sixteen replicas for peptide I were

simulated for 500ns each (total of 8Cps) at exponentially distributed temperatures, from 150K to

726K (150, 167, 185, 206, 228, 254, 282, 313, 348, 386, 429, 477, 529, 588, 653, and 726K).

Eighteen replicas for peptide 2 were simulated for 500ns each (total of 9Cps) from 150K to 834K

(150, 166, 184, 203, 225, 248, 275, 304, 336, 372, 411, 455, 503, 557, 616, 681, 754, and 834K).

The average exchange rate between adj acent replicas was 15%. The SHAKE algorithm was used

to constrain all the bond lengths involving hydrogen atoms, "' which allows an integration time

step of 2fs for each replica. Replica exchanges were attempted at every 10ps, resulting in 50,000

attempted exchanges at each temperature. Conformations from the simulation of each replica

were recorded every 2ps interval. The first 20ns of the simulation were discarded and the latter

480ns were saved for further calculation, giving a total of 48,000 configurations for both

peptides.









To determine the secondary structure, the program designed by Kabsch and Sander,

Definition of the Secondary Structure of Proteins (DS SP),182 was used for both peptides.

Percentage of a-helix (Helicity) was calculated for every residue and every replica, using this

program. We also performed a cluster analysis with moil-view 15 based on backbone RMSD for

residues 4-9 for peptide 1 and residues 5-10 for peptide 2 with a cutoff of 2A+.

Conformations obtained from the REMD calculation were used as the initial structures for

the T-jump experiment. We selected 1,200 starting configurations collected from the equilibrated

simulations for peptide 1 and peptide 2 at 282K and 304K, respectively. They are equispaced in

time (200-500ns REMD). Each member of this ensemble was then instantly and independently

heated up to the Thigh (the details of the temperature choices are discussed later) and MD was run

for 5ns to simulate the relaxation after the T-jump. For both of the peptides, all MD parameters

are the same as those used in REMD.

Several collision frequencies (y=1.0, 5.0, 10.0, and 20.0ps- ) were used for both peptides to

investigate the frictional effect on the rate of the protein folding.

In order to estimate secondary structure and folding properties, computations on CD

spectra were performed, in the same manner used by Sreerama and Woody (chapter 3).105 The

matrix method 101 was employed with a transition parameter set consisting of a combination of

experimental data and theoretical parameters: experimental data 160 for two amide xx*n transitions

and parameters from intermediate neglect of differential overlap/spectroscopic (INTDO/S)

wavefunctionS 161 for the nx*" transition. The rotational strength was computed to generate CD

spectra through Gaussian band. The bandwidths assigned for the nx*" transition, and for two xx*~n

transitions were 10.5nm, 11.3nm, and 7.2nm, respectively.39









4.3 Results and Discussion

The two related 14-residue peptides in this study were selected to prove a new

methodology (described in the chapter 3). Marqusee et at. 176 designed, synthesized, and tested

alanine-based peptides as helix-forming peptides. In their work, the poly-Ala helix containing (i

+ 4) Glu-...Lys+ salt bridges showed optimal behavior (-80% helicity). D-Arg was also chosen

because it is the most efficient a-helical C-capping residue.183 According to Huang et al., a

tripeptide, Ser-Pro-Glu, was selected as helix-stabilizing N-terminal sequence that occurs most

frequently at the N-terminus of helices in the WHATIF database of 1705 helices and also might

stabilize the helix by electrostatic interactions.130 Therefore, peptide 2 was built based on

possible a-helix-stabilizing effects, while peptide I was considered as a regular poly-Ala peptide

found in the experimental work.

We determined the secondary structure between two related 14-residue peptides to verify

our new methodology on the study of properties of peptides. Figure 4-1 shows plots of the

helicity of the secondary structure versus residue number to present the structural differences.

The overall helicity is expected to be low, since these two peptides are only 14-residue long.

However, significant differences are found between the two peptides in Figure 4-1. A high

helicity region in peptide 2 is seen between residue 5 (Proline) and 10 (Lysine). In contrast,

peptide 1 shows a low and broad distribution. The difference can be explained by a salt-bridge

interaction of two charged residues, Glu-(residue 6) and Lys (residue 10). When the two end

terminals (N-terminal and C-terminal) in peptide 2 are compared, C-terminal end with a D-Arg

residue shows 20-40% helicity, but N-terminal with an acetyl group shows almost zero helicity.

Simulations results are comparable with experimental ones in a sense that peptide 2 is more

helical than peptide 1.















IIII K




282 K


. ~275 K


0 5 10 15 20 0 5 10 15 20
residue number residue number
Figure 4-1. Calculated helicity. A) Peptide 1 versus residue number, B) Peptide 2 versus residue
number based on DSSP method, respectively.

4.3.1 Cluster Analysis

The conformations at the lowest temperature (150K for both peptides) from REMD were

sorted into clusters based on potential helical regions (residue 4 to 9 of peptide 1 and residue 5 to

10 of peptide 2) from DSSP method. The largest cluster in peptide 1 contains 71% of the

structures and the next largest cluster is 13% populated. In peptide 2, the largest cluster is 78%

and the next largest is 14% populated. No other clusters from both peptides have a population

higher than 10%. Figure 4-2 A shows all conformations superimposed on the representative








structures of the largest populated clusters for both peptides (red and yellow for peptide 1 and 2,

respectively) at 150K. Two representative structures from clusters are shown in Figure 4-2 B.

In Figure 4-3 the fractional sizes of each cluster in both peptides are plotted as a function

of time to evaluate convergence of the REMD simulations, where the first 20ns of the

simulations were discarded and the latter 480ns were used for calculation. Both peptides

converge to their final populations after approximately 200ns. The biggest cluster of both

peptides converges to a population of ~0.7-0.8 (70-80%). Therefore, all initial conformations

were taken from 200-500ns REMD for future analysis.


1 1


e a-heix O Extacende-beta
Figure 4-2. Reference structures from cluster analysis of peptide 1 and peptide 2. A) All
conformations from REMD at 150K and are superimposed on the two reference
structures, B) Two representative structures.


pep1


pep2
















Scluster5








CL 0.4--




0.2-




100 200 300 400 500
time ins)








cluster
~-- cluster
ul 0.8~ 1 cluster3
.P -F"~c'I clusterfj
rm c~luste~r7-

S0.4--




0.2-




100 200 300 400 500

time (ns)
Figure 4-3. Populations of the representative clusters. A) Peptide 1 as a function of time from 20-
500ns REMD, B) Peptide 2 as a function of time from 20-500ns REMD. At 200ns,
simulations of both peptides are converged and stabilized.









4.3.2 Helicity

Theoretical fractional helicity (fH) Of a peptide can be calculated using the mean residue

ellipticity at 222nm, [8 233,,491184,185


ft, (4 -1)


where [6, 222 is the mean residue ellipticity of an ideal peptide with 100% helicity, n is htheth

length of the potential helical region, and x is an empirical correction. In order to assign those

parameters, 14 polyalanines (ACE-(ALA)n-NME, n=5-14, 16, 18, 20, 22) with 100% helix were

built using HyperChem software 186 and subj ected to 5,000 steps of initial minimization. The CD

spectra ([8]322) were computed and plotted ([8]?32 versus 1/n) from the Equation 4-1 (Figure 4-

4). Values of x and [8, 1222 are taken to be 2.38 and -3 1403 deg cm2 dmof l, respectively, based

on the plot and fit in Figure 4-4. These two values are used for further calculations. Figure 4-5

shows the fractional helicity of peptide 1 and peptide2 from 200 500ns REMD as a function of

temperature (Equation 4-1).

Wang et at. 49 TepOrted that the helicity of peptide 1 and peptide 2 at 1 10C was

approximately 11% and 29% (by experiment), respectively. Marqusee et at. 176 Showed that the

16- or 18-residue Ala-based peptides contain approximately 25-50% helicity. Both theoretical

and experimental results clearly indicate that the helicity of peptide 2 is larger than that of

peptide 1.

4.3.3 Calculation of CD Spectra

For each member of the ensemble and at each temperature in the REMD simulation, CD

spectra were computed as mentioned in the methods section 4.2. The spectra obtained for each

peptide at a certain temperature were then averaged over members of the ensemble (49,000


































0.05 0.1 0 15


Figure 4-4. The resulting curve for the two parameter calculations ([6,.]222 and x) from the plot
[6]222 VeTSUS 1/n from the Equation 4-1. The unit of molar ellipticity is deg-cm2 dmol'
1 of both peptides.


0.3





,0.2





0.1





0


300 350 400 450 500 550
Temperature (K)


Figure 4-5. Fractional helicity (fH) of peptide 1 (blue) and peptide 2 (red) as a function of
temperature.









structures each). The resulting average spectra as a function of wavelength are shown in Figure

4-6. The CD spectra for an a-helix show two minima, one at 222nm (n-~n* transition along the

carbonyl bond) and the other at 208nm (n-~n* amide transition).162 In the plot, the molar

ellipticities at 222nm (6222) are linearly related to the helical minimum.184 Figure 4-7 shows

average values of CD at 222nm () for peptide 1 and peptide 2, as a function of

temperature. The observation showing higher helical population in peptide 2 than peptide 1, are

in a good agreement with the experimental results by Wang et at. 49, in the similar temperature

region. This type of signal is useful for the T-jump experiment.

T-jump simulations were performed and the resulting time traces were fitted to calculate

the folding and unfolding rates. This system appears to reach equilibrium after ~ 200ns of

REMD from the cluster analysis in Figure 4-3. Therefore, 1,200 initial configurations were

selected from 200 500ns REMD. The T-jump was then simulated from 313 to 348K for peptide

1 and 336 to 372K for peptide 2. Temperature change is achieved within a time scale of the order

of the thermostat coupling (0.1Ips time constant used), having a very short dead-time. The

simulations were run for a total of 5ns each (total of 6 Cps of each peptide). At each time slice, a

CD spectrum was computed for each independent MD run, and then they are averaged. The

resulting are plotted against time and fitted (Figure 4-8). As we mentioned in the

Chapter 3, the trace is bounded from the equilibrium runs, we know what the system looks like at

t=0 and at t=oo. This leaves a single parameter (h) to be determined via fitting from the Equation

3-4. The calculated relaxation data are fitted well using a single exponential function, indicating

occurrence of reversible two-state folding. Details of the two-state mechanism are covered in

Chapter 3.






























Temperatures

-40000II
210 220 23020

Wavelength (nm)




20000-












-20000


Temperatures


-410000
210 220 230 240

Wavelength (nm)
Figure 4-6. Calculated circular dichrioism (CD) spectra. A) Peptide 1 from 150-726K, B) Peptide
2 from 150-834K(. The unit of molar ellipticity is deg-cm2 dmol' of both peptides.


















-,5000~ I I-






-10000--






-15000
300 350 400 450 500 550
Temperature (K)

Figure 4-7. Average of mean residue ellipticities at 222ns () Of Simulated peptide 1
(blue) and peptide 2 (red) are shown as a function of temperature. The unit of molar
ellipticity is deg-cm2 dmor~ of both peptides.

In a two-state model, the relaxation time (Trelax) is obtained from the sum of the folding

rate (kf) and unfolding rate (k ),

3A = kf + k, = kr (1 + K) = 1/Trelax = 1/zf + 1/Tra (4-2)
In this equation, zf is the folding time, while Tr, is the unfolding time. The equilibrium constant K


(- ) is calculated by fractional helicity (fH) and is a ratio of the forward and backward rate


constants. When only two states, folded (F) and unfolded (U) states, exists and fractional helicity

of peptide 1 and peptide 2 are 12% at 348K and 15% at 372K, respectively, the K values of

peptide 1 (K1) and peptide 2 (K2) are calculated with the final temperatures of T-jump simulation

as the target temperatures, according to,











-2000


-2500-



-3000-







-4000



-4500--



-5000 I
U 1 2 3 4 ti
Time (ns)


-2000 II


I- peptide 2
-2500-B



-3000--











-4500



-5000 I
D 1 2 3 4 ti
Time (ns)


Figure 4-8. Average of CD222 ( frequency y = 1.0ps ). A) Peptide 1: black, B) Peptide 2: red with fitting curves,
respectively. The unit of molar ellipticity is deg-cm2 dmol-' of both peptides.










[U] k [U] k
K, -" 7.3 and K2 5.7 (4-3)
[F] k, [F] k,
The relaxation time and folding time obtained from experimental and computational results

are compared in the Table 4-1. The computed z values are 0.20ns for peptide 1 and 0. 17ns for

peptide 2, which correspond to a folding time zr-1.65ns of peptide 1 and zr-1.16ns of peptide 2,

respectively. Thus, the relaxation time of two peptides differ by a factor of ~1.2, which is the

ratio [(0.20ns) /(0. 17n2s)l. Similarly,, difference in experimental results was by a factor of ~1.1,

which is the ratio [(222n~s) /(204n2s) .49 This result validates our methodology for the kinetic

studies. In addition, ratio of folding time between two peptides is ~1.4 from the computation and

~2.9 from the experiments.49 The results show that peptide 2 with higher helicity folds faster

than peptide 1, indicating correlation between overall helix stability and folding time.

Table 4-1. Relaxation and folding times of peptide 1 and peptide 2 from experimental and
computational data.
Relaxation and folding times (ns)

z (exp.*) Z (com.**) zf (exp.*) Zf (com.**)
Peptide 1 222 0.20 ~ 2000 1.65
Peptide 2 204 0.17 ~ 700 1.16
* Experimental data obtained at 110C (284K) by Wang et al. 49
**" Computational data obtained by calculated CD () Of T-jump simulations (peptidel1 at 348K
and peptide 2 at 372K) in Figure 4-7.

The time scales seen in theoretical data are much shorter than in experimental data. This

mainly is due to the implicit solvent model used for the calculation, which accelerates the folding

and unfolding times. The solvent effect on the folding time including different frictional

coefficients will be discussed in the next section.

4.3.4 Effects of Frictional Coefficients

In order to generate data closer to the explicit solvent environment and also to show

independence of friction versus folding time, we studied the effect of frictional coefficient. To

investigate the frictional effect on the rate of the protein folding, we performed the T-jump










simulations (details in section 4.3.3) using Langevin dynamics with several collision frequencies

(y=1.0, 5.0, 10.0, and 20.0ps- ), for both peptides. Figure 4-9 presents the simulation results in

the plot of average values of CD at 222nm () VeTSus time. As we already mentioned in

Chapter 3 and section 4.3.3, all traces for both peptides were fitted to only a single parameter (h)

with the same values of Ct and C2 fTOm Equation 3-4 since we have complete knowledge of what

the system looks like at t=0 and t=oo. The calculated relaxation data in Figure 4-9 are then fitted

using a single exponential function, resulting in folding times (zf) from Equation 4-2. The folding

times are plotted as a function of collision frequency (y) in Figure 4-10 and the values of folding

time for two peptides are compared in the Table 4-2.

Table 4-2. Folding times of the two peptides at different collision frequencies with Langevin
dynamics.
Folding times (zf, ns)
Collision frequency, y (ps-') Peptide 1 Peptide 2
1.0 1.65 1.16
5.0 4.45 4.11
10.0 8.94 5.94
20.0 16.7 11.2

The folding time with higher friction is much longer than lower friction for both peptides.

The folding time is approximately linear to the collision frequency (Figure 4-10). However, it

does not seem to be closely fitted to the Kramers' relation kf '(= r,) ac 7. The Kramers'

model is not satisfactory, as its y-intercept is Eixed to be zero.23,177 Therefore, the results in

Figure 4-10 were further analyzed using the following linear model (a modified simple Kramers'

model, Equation 4-4) with a and b as variables which was suggested by Qiu et al.;23

r, =a +by (4-4)


SKramer's equation in its simplest form could be described as, k = (ma bi~ / 27ry) exp(-AE / RT) where ma~
and ab~ are the curvature of the potential energy surface at the bottom and top of the barrier, respectively, E is the
size of the energy barrier to conformational change, R is the gas constant, T is the temperature, and y is the friction
of the solvent.











-2000


-2500 -1~
-- gamma= 10.0
gamma = 20 0
-3000 -









-4500--


-5000-


-55000I|||
D 1 2 3 4 5
Time (ns)

-2000 I I
B gamma = 1.0
gamma =5.0
-2500--
gamma= 10.0
----- gamma =20.0
-3000--


S-3500-






-5 .000 -r




-5000 |

0 1 2 3 4 H
Time (ns)

Figure 4 -9. Comparisons of different collision frequencies, y=1.0, 5.0, 10.0, and 20.0ps- A)
Peptidel, B) Peptide 2 using average of CD222 () Of T-jump simulation data,
respectively. The unit of molar ellipticity is deg-cm2 dmol-l of both peptides.









The plots were well fitted with R2 = 0.999 for peptide 1 and R2 = 0.99 for peptide 2 (values of a

and b are given in Figure 4-11). This equation describes that the Kramers' relationship zf ac 7 is

preserved except the positive value of y-intercept (a) whereas a = 0 in simple Kramers' model.

Ansari et al. ls7 also suggested the relaxation rate equation using a modified Kramers'

model according to;2


k-' ac cr + r,


(4-5)


15 E


10~


0 5 10 15 20
collision frequencies
Figure 4-10. Comparisons of the folding times at different collision frequencies of two peptides
with error bars and linear fits (dotted lines). The folding times and associate errors
were calculated from fitting curves of average of CD222 () fTOm Figure 4-9.


2 The rate equation can be written as k = C- exp(-Eo / RT) where R is the gas constant, T is the

temperature, Eo is the average height of the potential energy barrier, C is adjustable parameter, a is the protein
friction, and rl is the solvent friction.


















0.5 --











0.



1 2

















0.5 -







1 2

Figure 4-11i. Friction dependence of peptide 1 (marked as 1) and peptide 2 (marked as 2) by T-
jump simulations. A) Y-intercept, B) Slope of folding time (z) obtained from linear
fits in Figure 4-10.









where o is the protein friction (or internal friction) and rl, is the solvent friction. Ansari et al. ls

mentioned that the folding rate can be determined as the sum of the solvent friction (rls) and the

protein friction (or internal friction, o) from Equation 4-5. Related with Ansari model,m" we

could suggest that y-intercept (a) is correlated with internal friction.

The simple Kramers' model (r af c 7 ) is well associated with the analysis of the protein

folding kinetics when folding is relatively slow (1/ks ~ms).23 However, our model shows better

agreement with Equation 4-4, instead of simple Kramer model, and thus internal friction can

influence the very fast folding reactions since the values of a and b in Figure 4-11 show fastest

(nanoseconds) timescales.

Zagrovic and Pande 95 Simulated TrpCage (TC~b) using the distributed computing

technique and analyzed the dependence of solvent viscosity. According to their results, if the

protein initially collapses into a random conformation and this continues until the protein folds

(unfolded random folded), the first step is mainly controlled by the solvent friction and the

second step is controlled only by the internal friction. Thus, internal friction would play a maj or

role for the folding rate when the second step becomes the rate-limiting step.

The time scale of present folding (Figure 4-10) is much shorter than that from the

experiments (microsecond scale).23 This is due to the difference in the composition of the solvent

between the experiment and simulation. The simulated results, however, show similar ratio of

folding times between two peptides to the experimental results.

4.4 Conclusion

We studied folding kinetics of two related 14-residue peptides by using T-jump

simulations that was recently introduced as a new computational methodology. Helicity and

folding kinetics of two alanine-based peptides were investigated and compared with










experimental data. Very efficient sampling techniques, such as REMD, are used to populate

equilibrium ensembles. Multiplexed MD simulations were then run to obtain kinetic information,

particularly the non-equilibrium relaxation of the populations. We found that peptide 2 having

more helicity, folds faster than peptide 1. The ratio of relaxation time of two peptides differed by

a factor of ~1.2, while the corresponding experimental results were ~1.1. Therefore, our new

methodology seems in good agreement with experimental data.

The effect of friction on the protein folding was also studied using Langevin dynamics. We

performed data fitting, using the modified Kramers' linear model, on simulated results from

different frictional coefficients. The observed nanosecond time scale of folding for both peptides

indicates that internal friction can influence the very fast folding reactions. The composition of

solvent made a significant effect on the folding kinetics, such that the nanosecond time scale in

the simulation was obtained for the microsecond time scale in the experiments.









CHAPTER 5
CONCLUSIONS

Simulations opened the way to investigations of new fields and possibilities since Karplus

et al. first introduced MD simulation of biomolecular systems.' MD simulations can provide

ultimate details of individual atomic motions including pathways while experimental results

usually show only averaged structure. However, approximations of molecular interactions are

one of the well known drawbacks of MD simulations. Therefore, combining experimental and

computational results address in complementing and overcoming limitations of both approaches.

Moreover, understanding the structure, kinetics and thermodynamics of protein folding

remains one of the unsolved problems for both computational and experimental biophysists.

Laser-induced T-jump is one of the most popular experimental methods in protein folding

research since the timescale of T-jump extends from nanoseconds to milliseconds, which is an

appropriate range for studies of folding kinetics.

In the first proj ect, we introduced the computational T-jump: the temperature increases

suddenly, creating the relaxation of the ensemble from one equilibrium distribution (at low

temperature) to another one (at high temperature) by using proper conformational sampling

method, which is REMD. Our method is designed and explored closely resembling to an

experimental T-jump while the usual distributed computing procedure uses multiplexed MD runs

to study folding. The alanine polypeptide (ACE-(ALA)20-NME) was used as our model system

in this proj ect as a proof of principle and description of the methodology. A set of

folding/unfolding time (zF=47.5ns and zu=155ns) was calculated from fitting the results of CD

spectra and the changes of conformational distributions were shown through computation of

NMR chemical shifts.










The second proj ect extended the method to real proteins to get consistent results using

Langevin dynamics which includes frictional effects and random forces. In this proj ect two

related 14-residue peptides were chosen and compared with the experimental data. The ratio of

relaxation times of the two peptides is by a factor of ~1.2, while the corresponding experimental

result was ~1.1.









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BIOGRAPHICAL SKETCH

Seonah Kim, was born in Daegu, South Korea. She went to college at Yeoungnam

University, maj oring in industrial chemistry and graduated with a bachelor' s degree. In

August 1999, she started graduate school at the University of Houston, in the Computer Science

Department and earned her master's degree. She moved to Gainesville, Florida, in July 2003.

There she entered the University of Florida's Ph.D. program in chemistry, specializing in

computational chemistry.





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1 SIMULATING TEMPERATURE JUMPS FO R PROTEIN FOLDING STUDIES By SEONAH KIM 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

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2 2007 Seonah Kim

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3 To my loving husband Jiho and my family

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4 ACKNOWLEDGMENTS At the completion of this work I take great pleasure in acknowledging the people who have supported me over the last couple of years. I grat efully thank and acknowledge my advisor, Prof. Adrian Roitberg, for the opportunity to work wi th him. His guidance, humor, understanding, and encouragement leaded me to overcome all the obs tacles I had felt in studying. My personal goal would not have been realized without the unconditional support from Prof. Roitberg. Those words are not enough to thank my advisor Prof. Roitberg. I would like to thank for kind advice and s uggestions of my committee members, Prof. Alexander Angerhofer, and Prof. Gail Fanucci as well, especially Prof. Ronald Castellano who accepted to be my committee at the last moment. I am also greatly obligated to Prof. Steve Hagen to give helpful discussions an d suggestions through all my research. I am grateful to Prof. Carlos Simmerling, Pr of. Jeff Krause and Prof Valeria Kleiman for kindly providing invaluable advice within and beyond science, and to Prof. Thomas Cheatham for hosting while I was in Utah. I gratefully ac knowledge Georgios Leonis and Julio Palmas help with their kind, helpful a dvice, supporting in and out of sc ience through all my Ph. D. study here in Gainesville. I also thank to all of my Quantum Theory Project (QTP) buddies (Josh, Andrew, Tom, Kelly, Joey, Lena and Martin) who shar e lots of party. I am so lucky to have all of my buddies here in qtp and enjoyed every moment I will miss them all. It has been great pleasure to work with my group members (Dr. Gu stavo Seabra, Ozlem Demir, Christina Crecca, Hui Xiong, Dan Sindhikara, Yilin MEng, Le na Dolghih, and Mehrnoosh Arrar). Above all I thank my husband, Jiho, for his love, support, encouragement, endless corrections and personal sacrifice without which I would not have been able to accomplish this work. Finally, I am forever indebted to my parents and brother, Jihwan, for their love, understanding, prayer and encourag ement when it was most required.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 LIST OF ABBREVIATIONS........................................................................................................11 ABSTRACT....................................................................................................................... ............12 CHAPTER 1 INTRODUCTION..................................................................................................................14 1.1 Prologue................................................................................................................... .........14 1.2 Structural Biology......................................................................................................... ....17 1.3 Protein Folding............................................................................................................ .....20 1.4 From the Experiment to the Simulation............................................................................21 1.5 Temperature Jump Experiments.......................................................................................22 1.6 Circular Dichroism......................................................................................................... ..24 1.7 NMR Spectroscopy...........................................................................................................26 1.8 Overview of Research Projects.........................................................................................27 1.8.1 First Project: Simulating Temperature Jumps for Protein Folding........................27 1.8.2 Second Project: Folding Kinetics by Te mperature-Jump Simulations of Two Related 14-residue Peptides.........................................................................................27 2 THEORY AND METHODS..................................................................................................29 2.1 Conformational Sampling.................................................................................................29 2.2 Force Field................................................................................................................ ........32 2.3 Generalized Born (GB) Solvation Model.........................................................................34 2.4 Langevin Dynamics..........................................................................................................35 2.5 Computation of Circ ular Dichroism (CD)........................................................................36 2.6 Computation of NMR Chemical Shifts............................................................................38 2.6.1 Calculation of Proton (1H) Chemical Shift.............................................................38 2.6.2 Calculation of 15N and 13C Chemical Shifts...........................................................39 3 SIMULATING TEMPERATURE JUM PS FOR PROTEIN FOLDING...............................42 3.1 Introduction............................................................................................................... ........42 3.2 Methods.................................................................................................................... ........45 3.2.1 Simulation Details..................................................................................................45 3.2.2 Calculation of NMR Chemical Shifts.....................................................................47 3.2.3 Calculation of CD Spectra......................................................................................48

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6 3.3 Results and Discussions....................................................................................................48 3.3.1 Calculation of Chemical Shifts...............................................................................52 3.3.2 Calculation of Circular Dichroism.........................................................................56 3.3.3 Optimum Number of Unfolded States?..................................................................62 3.4 Conclusion................................................................................................................. .......64 4 FOLDING KINETICS BY TEMPERATU RE-JUMP SIMULATIONS OF TWO RELATED 14-RESIDUE PEPTIDES....................................................................................67 4.1 Introduction............................................................................................................... ........67 4.2 Methods.................................................................................................................... ........70 4.3 Results and Discussion.....................................................................................................72 4.3.1 Cluster Analysis......................................................................................................73 4.3.2 Helicity................................................................................................................. ..76 4.3.3 Calculation of CD Spectra......................................................................................76 4.3.4 Effects of Frictional Coefficients...........................................................................82 4.4 Conclusion................................................................................................................. .......87 5 CONCLUSIONS....................................................................................................................89 LIST OF REFERENCES............................................................................................................. ..91 BIOGRAPHICAL SKETCH.......................................................................................................104

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7 LIST OF TABLES Table page 3-1 Comparison of calculated chemical shifts for residue 6 (A6) in polyalanine peptide and experimental values for al anine residue from references............................................54 4-1 Relaxation and folding times of peptid e 1 and peptide 2 from experimental and computational data.............................................................................................................82 4-2 Folding times of the two peptides at different collision frequencies with Langevin dynamics....................................................................................................................... .....83

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8 LIST OF FIGURES Figure page 1-1 The general formula of an amino acid is showing a central carbon atom (C) is attached to an amino group (NH2), a carbonyl group (COOH) a hydrogen atom, and a side chain (R)............................................................................................................... ...14 1-2 The twenty different amino acids found in proteins. Side chains are shown in blue........15 1-3 The optical isomers of alanine, L and D forms..................................................................16 1-4 Part of the polypeptide chain shows to illustrate rigid peptid e bond between C (from carbonyl group) and N (from amino group), two degrees of freedom, and angels from rotations around NC and CC bonds, respectively.............................................16 1-5 Ramachandran plot for the 20-residue Trp-cage protein (PDB ID: 1L2Y). The dots are created by each residue from the Trp-cage protein. The Trp-cage protein is one of smallest folding protein-like molecule. Th e structure is showed in Figure 1-6 C.............18 1-6 The diagram of protein structures. A) Primary structure, B) Secondary structure using an -helix and -sheet (PDB ID: MBH12), C) Tertiary structure (Trp-cage protein), and D) Quaternary structure showed using hemoglobin complex (1GZX)........19 1-7 The general scheme of Temperatur e-jump (T-jump) relaxation kinetics.........................23 1-8 The standard curves for CD spectra of pol y-L-lysine in different secondary structure conformations taken from Campbell & Dwek, 1984.........................................................25 2-1 The sketch of Replica-exchange me thod (REM) simulation in amber molecular dynamics packages.............................................................................................................31 2-2 Outline of 15N and 13C chemical shift calculation in SHIFTS program............................41 3-1 The diagram of the T-jump setup. A) Experimental T-jump scheme, B) Computational T-jump setup.............................................................................................47 3-2 Cluster analysis of polyalanine. A) All conformations from REMD at 151K are superimposed on the reference struct ure of the largest cluster, B) -helix reference structure from the cluster analysis.....................................................................................49 3-3 Histograms from the C-RMSD. A) Probability density of C-RMSD at 181K (blue line) and 214K (red line), B) (probability at 214K probability at 181K) vs. CRMSD........................................................................................................................... .....50 3-4 Fractions of -helical folded state (F), and unf olded state (U) as a function of temperature using probability (C-RMSD)........................................................................51

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9 3-5 Averaged chemical shifts (proton, 13C, and 15N) for polyalanine peptide (ACE (ALA)20NME) as a function of residue numbe r, for all sixteen temperatures.................53 3-6 1H15N HSQC spectra of polyalanine pep tide. A) At 181K and B) At 300K...................56 3-7 Two-dimensional 13C/13C crosspeak shapes for polyala nine peptide, residue 5, 11 and 16 (A5, A11, and A16) from several te mperatures. Color scheme are from the maximum signal intensity (red) to th e minimum signal intensity (blue)...........................57 3-8 Results of Circular dichroism. A) Calc ulated circular dichroism spectra of ACE (ALA)20NME from 153K to 542K, B) Average of CD222 () of each simulated polyalanines. Error bars represent standard deviation.......................................59 3-9 Two circular dichroism (CD) spectra, pure -helix and -helix after minimization, are compared to see differ ent minimum wavelengths.......................................................60 3-10 Average of CD222 () of T-jump simulation data (red: folded (F), and green: unfolded (U) states) are fitted usi ng a single exponential equation...................................60 3-11 Folding (green) and unfolding (red) fract ions from T-jump simulation data are calculated and fitted using same in Figure 3-10.............................................................61 3-12 Reference structures from cluster anal ysis. A) All conformations from REMD at 151K are superimposed on the two reference structures, B) -helix and C) Coiledcoil -helix reference structures.........................................................................................63 3-13 C-RMSD (residue 2-19) computed using different initial states from 1 s MD simulations. A) -helix, B) Coiled-coil, and C) Unfolded initial states from CRMSD1. D) -helix, E) Coiled-coil, and F) Un folded initial states from C-RMSD2......64 3-14 C-RMSD2 (with respect to coiled-coil -helix) plots from 1 s MD simulation using coiled-coil -helix initial state at 214K..............................................................................66 3-15 C-RMSD (residue 2-19) relation plot using C-RMSD1 (with respect to -helix reference structure) as x-axis and C-RMSD2 (with respect to coiled-coil -helix reference structure) as y-ax is, respectively, at 214K.........................................................66 4-1 Calculated helicity. A) Peptide 1 versus residue number, B) Peptide 2 versus residue number based on DSSP method, respectively...................................................................73 4-2 Reference structures from cluster anal ysis of peptide 1 and peptide 2. A) All conformations from REMD at 150K and are superimposed on the two reference structures, B) Two repres entative structures......................................................................74 4-3 Populations of the representative clusters A) Peptide 1 as a function of time from 20-500ns REMD, B) Peptide 2 as a func tion of time from 20-500ns REMD. At 200ns, simulations of both peptides are converged and stabilized....................................75

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10 4-4 The resulting curve for th e two parameter calculations ([ ]222 and x) from the plot [ ]222 versus 1/n from the Equation 4-1. Th e unit of molar ellipticity is deg cm2 dmol-1 of both peptides......................................................................................................77 4-5 Fractional helicity ( fH) of peptide 1 (blue) and pep tide 2 (red) as a function of temperature.................................................................................................................... ....77 4-6 Calculated circular dichrioism (C D) spectra. A) Pep tide 1 from 150-726K, B) Peptide 2 from 150-834K. The unit of molar ellipticity is deg cm2 dmol-1 of both peptides....................................................................................................................... .......79 4-7 Average of mean residue ellipticities at 222ns () of simulated peptide 1 (blue) and peptide 2 (red) are shown as a function of temperature. The unit of molar ellipticity is deg cm2 dmol-1 of both peptides...................................................................80 4-8 Average of CD222 () of temperature jump (T-jump) simulation data (collision frequency = 1.0ps -1). A) Peptide 1: black, B) Peptide 2: red with fitting curves, respectively. The unit of molar ellipticity is deg cm2 dmol-1 of both peptides.....81 4 -9 Comparisons of different collision frequencies, =1.0, 5.0, 10.0, and 20.0ps-1. A) Peptide1, B) Peptide 2 using average of CD222 () of T-jump simulation data, respectively. The unit of molar ellipticity is deg cm2 dmol-1 of both peptides.................84 4-10 Comparisons of the folding times at diffe rent collision frequencies of two peptides with error bars and linear fits (dotted lin es). The folding times and associate errors were calculated from fitting curves of average of CD222 () from Figure 4-9.......85 4-11 Friction dependence of peptide 1 (marked as 1 ) and peptide 2 (marked as 2 ) by Tjump simulations. A) Y-intercept, B) Slope of folding time ( ) obtained from linear fits in Figure 4-10............................................................................................................ ...86

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11 LIST OF ABBREVIATIONS ACE: Acetyl beginning group CD: Circular Dichroism DSSP: Definition of the Secondary Structure of Proteins GB: Generalized Born HSQC: Hetero-nuclear Single Quantum Coherence IR: Infrared MD: Molecular Dynamics NME: N-methylamine ending group NMR: Nuclear Magnetic Resonance NOEs: Nuclear Overhauser Effects REMD: Replica Exchange Molecular Dynamics T-jump: Temperature-jump

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12 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 SIMULATING TEMPERATURE JUMPS FO R PROTEIN FOLDING STUDIES By Seonah Kim December 2007 Chair: Adrian E. Roitberg Major: Chemistry Protein folding is described as a dynamic pro cess of an ensemble of molecules reaching well-defined three dimensional structures to achi eve biological activity from linear amino acids sequences. Many human diseases result from protei n misfolding or aggregation. Enormous effort has been made both experimentally and theoretic ally for nearly 40 years to explain the basic principle and mechanism of protein folding and unfolding. Nonetheless, ma ny of them are still unknown or incompletely understood, mainly due to the complexity of the systems and the fast folding time scale. Experimental and theoretical approaches are complementary with each other for the protein folding studies and hence, comb ination of the two is required to have better understanding. One of the most popular experimental methods for the protein folding studies is laserinduced temperature-jump (T-jump), because it has nanosecond resolution. In the first project, the T-jump on the polyalanine peptides (Ala20) was simulated as a proofof-principle system to mimic the experimental measurements. Repli ca exchange molecular dynamics (REMD) were performed to obtain equilibrated ensembles as a proper conformational sampling, which was combined with multiplexed molecular dynamics to extract kinetic properties in line with experiments.

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13 In the second project, the same methodology used in the first project was applied to real proteins. Effect of frictional coefficient in the solvent model was approximated using Langevin dynamics. Computationall results on the two rela ted 14-residue peptides were chosen and compared with experimental results. A ratio of relaxation time of the two peptides was determined by calculated Circular Dichroism (C D) spectra by a factor of ~1.2, while the experimental results were ~1.1.

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14 CHAPTER 1 INTRODUCTION 1.1 Prologue Proteins and nucleic acid are a st arting point of life scie nce, as they have a role in all living processes. Protein folding studi es are of great interest espe cially for many human diseases associated with protein misfolding, such as cyst ic fibrosis, Alzheimers, Parkinsons disease and Mad cow disease.1-5 Proteins are built by various combinations of commonly twenty amino acids. All amino acids contain an amino group (NH2), a carboxyl group (COOH), a nd a distinctive R group connected to a central carbon atom (C) (Figure 1-1). Figure 1-1. The general formula of an ami no acid is showing a central carbon atom (C) is attached to an amino group (NH2), a carbonyl group (COOH) a hydrogen atom, and a side chain (R). The twenty amino acids found in proteins ar e shown in Figure 1-2. In general, four different groups are connected to the central C atom, making it a chiral center, except glycine, where two H atoms link to C. Chiral molecules can have optical isomers, Land Dforms (Figure 1-3). Most of the amino acids in nature exist in L-form. Circular dichroism (CD) spectroscopy, discusse d in section 1.4,6 is a useful and critical tool to study chiral interactions. A protein is made of combination of ami no acids joined via a peptide bond, where the side chain group carboxyl group carbon_alpha atom (C) CH H3C NH2 COOH amino group

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15 Nonpolaraminoacids+H3NC H COOCH3 Ala,Alanine(A)H3C CH CH3 C +H3N COOH Val,Valine(V)H3C CH CH3 CH2 C +H3N COOH Leu,Leucine(L)CH3H2C H C CH3 C +H3N COOH Ile,Isoleucine(I)H2C H2C HN CH CH2 Pro,Proline(P)C COOH +H3N CH2 C +H3N COOH Phe,Phenylalanine(F)CH3S CH2 CH2 C COOH3C H Met,Methionine(M) H N CH2 C COOH +H3N Trp,Tryptophan(W)SH CH2 C H +H3N COOCys,Cysteine(C)+H3NC COOH H Gly,Glycine(G)ChargedpolaraminoacidsO C OCH2 C COOH +H3N Asp,Asparticacid(D)O C OCH2 CH2 C +H3N COOH Glu,Glutamicacid(E)NH3 +CH2 CH2 CH2 CH2 C +H3N COOH Lys,Lysine(K)NH2C NH CH2 CH2 CH2 C NH2 + COO+H3N H Arg,Arginine(R)NH+H N CH2 C COOH +H3N His,Histidine(H)UnchargedpolaraminoacidsO C NH2 CH2 C COO+H3N H Asn,Asparagine(N)CH2C COO+H3N H CH2 C NH2 O Gln,Glutamine(Q)OH CH2 C COO+H3N H Ser,Serine(S)CCOOH +H3N CH HO CH3 Thr,Threonine(T) CH2 C COOH +H3N Tyr,Tyrosine(Y)OH Figure 1-2. The twenty different amino acids found in proteins. Side chains are shown in blue.

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16 Figure 1-3. The optical isomers of alanine, L and D forms. Figure 1-4. Part of the polypeptide chain shows to illustrate rigid pept ide bond between C (from carbonyl group) and N (from amino group), two degrees of freedom, and angels from rotations around NC and CC bonds, respectively. carboxyl carbon atom (C-terminus) of one amino acid is bonded to the nitrogen atom (Nterminus) in the amino group of the next ami no acid (Figure 1-4). The connection of multiple peptide bonds generates a backbone (or main chai n) of a protein. The pe ptide group is generally rigid enough to remain planar and in trans configuration. The amino acid sequence which is linked by a peptide bond is ca lled a primary structure. Backbone rotations can occur around either the NC or the CC bonds; defined as phi ( ) and psi ( ) angles, respectively. Different sets of values for phi and psi angles can denote different protein (or peptide) conformation. Thus the backbone conformations can be described CH H3C NH3 + COOC H CH3 NH3 + COOL-alanine D-alanine peptide bond C terminal N terminal

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17 by specifying these two angles. In Figure1-5, the pair of angles ( and ) are plotted in a Ramachandran plot.7 As shown in the figure, three main regions are allowed with respect to and angles; one is the right-handed -helix around =, = (denoted R), another is the sheet (parallel and antiparallel) around =, =+125 (denoted ) and the other is the left-handed -helix around =+57, =+47 (denoted L). A left-handed polyproline II helix (PPII) which is often observed in proline-rich se quences and sterically forced conformation for polyproline, is also f ound in proteins around =, =+145 (denoted PPII) and is an important conformation in protein-protein interfaces.8,9 The three-dimensional form of local geometri c arrangements within the peptide backbone is called secondary structure. For example, -helices in proteins appear when the and angles are approximately and respectively a nd when hydrogen bond are formed between the carbonyl oxygen of the ith residue and the amido portion of the i + 4th residue. The tertiary structure is the co mbination of secondary structur al units joined by a loop (or turn). Furthermore, the quaternary structure can be defined when a protein involves more than one polypeptide chain. The diagram of hierarchy of protein structures Figure 1-6 shows this relationship. 1.2 Structural Biology The subject of structural bi ology in proteins is the link between one-dimensional amino acid sequences to their three-dime nsional structure. Structural biology is the study of molecular shape of biological macromoleculesproteins and nuc leic acids and their in teractions. Studies of structural biology began with the exploration of biological materials by using early microscopy over 100 years ago. Thanks to development of new and powerful methods, such as X-ray crystallography and nuclear magnetic resonance (NMR), it became one of the most important

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18 Figure 1-5. Ramachandran plot for the 20-residu e Trp-cage protein (PDB ID: 1L2Y). The dots are created by each residue from the Trp-cage protein. The Trp-cage protein is one of smallest folding protein-like molecule. Th e structure is showed in Figure 1-6 C. subjects in molecular biology.10 There are two main experimental tools in struct ural biology; Xray crystallography and NMR. X -ray crystallography is the first discovered and the most dominant tool, being widely used for structur e determination of macromolecules. As of May 2007, 85% (37,101 out of total 43,633 structures) of released structures in Protein Data Bank were obtained based on X-ray crys tallographic results. However, it s application is not universal because it can be used only when high quality of crystal is available. Moreover, it provides averaged atomic positions, with atomic displacement parameters (or B-factors) used to infer the internal motions of proteins.11 -180 -180 +180 +180 0 0 p hi ( ) p si ( ) R L PPII

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19 A B -helix -sheet C D Figure 1-6. The diagram of protein structures. A) Primary structure, B) Secondary structure using an -helix and -sheet (PDB ID: MBH12), C) Tertia ry structure (Trp-cage protein), and D) Quaternary structure show ed using hemoglobin complex (1GZX)

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20 The NMR technique has been also commonly used. It is, however, limited to smaller molecular systems, but sensitive enough to reco gnize mobile regions of macromolecules in aqueous solutions. In addition to effort on thes e technologies, theoretic al work, especially computational studies, are routinely used to support and complement experimental data. 1.3 Protein Folding The dynamical process where a protein forms its well-defined three dimensional structures to achieve biological activity is called Protein Folding. For proteins that do fold, they usually do into one specific unique state in just a few seconds (or less ) from any starting conformation. This state is defined as the nativ e state. Slight changes, such as pH, or temperature, can convert biologically active protein molecules in native state (folded) to biologically inactive denatured state (unfolded). Moreover, many human diseases result from protein mi sfolding or aggregation. Therefore, protein folding research is the one of the most important subjects in biology. Although enormous efforts were made from experi mental and theoretical studies for nearly 40 years, the pathways and mechanisms of protein folding have not been yet fully understood due to the complexity of the systems and the fast time scale of folding.12-14 In the early 1960s, experimentally, Anfinsen et al. 15 studied the refolding of the denatured bovine pancreatic ribonuclease (RNase). RNase as non-functional protein, immediately returns to its native conformati on (folding process) from a randomly coiled structure, upon removal of the denaturant (8 M urea), which helps to restore its enzymatic activity. This behavior establis hed the thermodynamic hypothesis, which is directly related to the tertiary structure of a protein and more im portantly, there is a thermodynamically the most stable minimum on the free energy pr ofile, i.e., the free energy is be ing lowest in the native than in the unfolded state.16 Thus, Anfinsens observation opene d a new era of protein folding research.

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21 In 1968, Cyrus Levinthal showed that it is impossi ble that the protein folds into its native state by sampling all pos sible conformations.17 For example, if a 150-residue protein molecule has only three stable conformations for each amino-acid (the allowed regions are and L from the Ramachandran plot), then 3150 1068 possible conformations exis t. In order to obtain the native structure from a random search, it will take ~1048 years, substantially longer than the age of the universe. This argument is popularly known as the Levinthals paradox. Therefore, Levinthal conjectured that protei ns must fold to their native, stable conformation by using welldetermined folding pathways. These endeavors to find the correct path ways are continuing to these days with increasing success. 1.4 From the Experiment to the Simulation The present study aims towards simulating experimentally observed processes, by computing thermodynamic and kinetic properties. Advances in computational power and speed have opened the way to investigations of new fields and new possibilities. Karplus and his coworkers first introduce d molecular dynamics (MD) simulation of a biological macromolecule in 1977.18 The MD simulation can generate the confi gurations of the system based on Newtons law of motion and hence, can provide ultimate details of individual atomic motions as a function of time. Hidden details of interest (includi ng folding pathways) can be revealed by an MD simulation. Thus, the simulation can play an im portant role in interpreting experimental observations. Experimental outcome is frequently compared with data calculated from MD simulations in order to validate the methodology and estimate systematic errors. The simulation can be carried out under conditions that are difficult or impossible to achieve in experiments, for example, under very high temperature or pressure Therefore, combination of experimental and simulational information is of relevance in complementing and validating both approaches.

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22 1.5 Temperature Jump Experiments A major shortcoming of most experimental me thods in protein folding is their limited time resolution while key events might occur much fa ster than that. Relaxation methods that can probe very fast time scales are of great interest in the studie s of fast reactions.19,20 A powerful relaxation method for the study of protein foldi ng is temperature-jump (T-jump). The T-jump experiments were originally developed in the 1950s with the application of resistive heating. It was recently revisited and applied to protein fo lding dynamics with the use of modern laser heating.21 Laser T-jump can reduce the dead time to nanosecond or picosecond scale and probe the earliest folding events. Laser T-jump is also advantageous with regard to the fast rise time coupled with a small amount of sample due to small heating volume. Fast T-jump experiments are, hence, adequate for studies of kinetics and later combined with theoretical results for a detailed description of biological systems. A scheme of experimental T-jump relaxation kinetics is shown in Figure 1-7. The sample is initially in an equilibrium st ate at the initial temperature, Tinitial. At t=0, a pulsed pump laser increases the temperature of the solution until the final temperature (Tfinal) is reached within a nanosecond or shorter timescale.22 The final temperature should be sufficiently different from the initial temperature to perturb the equilib rium significantly. Te mperature change ( T) is generally set up between 10C and 20C in an aqueous sa mple of protein. The unfolding process is initiated rapidly, and the subse quent relaxation kinetics is mon itored and recorded at a time resolution of ~1ns 22,23 by using time-resolved infrared (IR) spectroscopy,19,24 UV Circular Dichroism (CD),20,25 or Trp-fluorescence,4,26,27 until the system reaches a new equilibrium. The resulting kinetic spectra are then fitted to a prope r relaxation curve; for example, single (for twostate mechanism) or double (for threestate mechanism) exponential model.

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23 Figure 1-7. The general scheme of Temperat ure-jump (T-jump) re laxation kinetics.28 In a two-state folding mechanism, the proce ss assumes only two states; the folded (F) and unfolded (U) ensembles with kF and kU, the folding and unfolding rates, respectively. The folding rate equations are derived as, [] [][]FUdF kUkF dt (1-2) [] [][]UFdU kFkU dt (1-3) where ] [ U and ] [ F are the concentrations of unfolded a nd folded states, respectively. If we assume an initial concentration of 1, then this makes ] [ 1 ] [ F U Thus, the time dependent populations can then be dire ctly solved leading to:29 12()tFtCeC (1-4) ) ( 1 ) ( t F t U (1-5) The population relaxation is dominated by a single rate constant, F U relax F F UK k k k / 1 / 1 / 1 ) 1 (1 (1-6) t = 0 initial equilibrium at Tinitial new equilibrium at Tfinal t t 10-8 s Signal kU kF Unfolded (U) Folded (F) ( 1-1 )

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24 In this equation, F is the folding time (and U is the unfolding time). The values of the two constants C1 and C2 are completely determined from the initial and final equilibrium concentrations and are then not adjustab le parameters. The equilibrium constant K (= F Uk k) can be written as a ratio of the forward and backward rate constants and also determined by the equilibrium concentrations of the folded and unf olded states. Therefore, we could derive the folding ( F) and unfolding times ( U) by combining the relaxation results (Equation 1-6) and equilibrium constant. 1.6 Circular Dichroism Circular Dichroism (CD) spectroscopy is a widely used technique for the study of secondary structures of polypeptides and proteins.30-34 CD measures the differential absorption of choromophore containing chiral molecules between left circ ularly polarized light (LCPL) and right circularly polarized light (RCPL), CD=Abs(LPCL)A bs(RPCL), which arises from structural asymmetry chirality. This, therefor e, provides information about both conformation and its change. It also valuable for analysis of macromolecules, globular structures, and drug complexes in the field of biological, bioc hemical, chemical, and pharmaceutical sciences, because it can give more detailed data than most absorption or fluorescence spectroscopy techniques. 35,36 The CD data are reported in units of abso rbance (Abs(LPCL)Abs(R PCL)) or ellipticity ( ). Molar ellipticity ( ) is usually utilized when CD involves molar concentration, and is defined as: Cl 100 (1-7) where C is the molar concentration (mol/L) and l is cell pathlength (cm) of the sample. The unit of molar ellipticity is typically reported as deg cm2 dmol-1 or deg M-1 m-1.

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25 CD data contain secondary st ructural information in terms of amide transitions in the backbone chain. For example, -helical content consists of a positive band at 190nm and two negative bands at 208nm and 222nm,37 whereas -sheet contains two opposite signs at 215nm (minimum) and 198nm (maximum).38 In particular, a strong negativ e band with molar ellipticity at 222nm is a key indication of helix formation of protein and peptides, because it is strongly affected by structural changes between folding and unfolding processes. Helical content can be monitored effectively in CD spectra and be us ed to understand protei n folding and unfolding procedure.39 Figure 1-8 shows the standard curves for CD spectra of poly-L-lysine in different secondary structure conformations (random coil, -sheet and -helix) from Campbell et al. 40 Figure 1-8. The standard curves for CD spectra of poly-L-lysine in different secondary structure conformations taken from Campbell & Dwek, 1984.40

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26 1.7 NMR Spectroscopy The spectroscopic measurements are extende d to study more completely structural properties of proteins and peptides including th e kinetic and thermodynamic results on protein folding.41 Nuclear magnetic resonance (NMR) spectrosc opy is also an important tool in these studies along with CD spectra. The most accessible quantities in NMR spectroscopy are chemical shifts,42 nuclear Overhauser effects (NOEs) and scalar coupling constants. Allerhand et al. showed significant differences between C chemical shifts in random coil and helical polypeptides.43 These observations showed that the chemical shift might be applied to probe secondary structures of pr oteins. The chemical shifts of 1H, -13C, and carbonyl-13C are conformation-dependent, as shown by empirical 44,45 and ab initio studies.46 Therefore, they can be used for determination of backbone confor mations. Later, Dalgarno and his colleagues defined the secondary structure shift (some times called conformational or conformationdependent shift), ss, as:47 coil r obs ss (1-8) where r.coil refers to the standard chemical shif t measured from random coil. The relative secondary structure shift ( ss) could be correlated with the in tensity of the helical CD signal ( 222). For example, upfield shifts for 1H, 15N and downfield shifts for 13C are observed for helix formation.42 Since one-dimensional (1D) NMR spectra are too complex to interpret due to severe overlapping signals, two-dimensional (2D) expe riments are more popular in the studies of protein folding at individual residues. The hetero-nuclear si ngle quantum coherence (HSQC) experiment is also frequently us ed in the field of pr otein NMR. The spectra are two-dimensional between 1H and hetero nuclei (13C or 15N). They contain thermodynamic information such that the spectrum is well dispersed a nd all individual peaks are distinguishable when the protein is

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27 folded. In this dissertation, we focused on conf ormational variations by direct use of chemical shift and HSQC experiments.48 1.8 Overview of Research Projects The main goal of this work is to simulate the experimental T-jump setup for the study of protein folding. Computational methods used include replica excha nge molecular dynamics (REMD), calculated CD spectra, structural clus ter analysis, and computational NMR chemical shifts. Two projects have been performed: the first one is the T-jump simulation of Alanine20, and the second project is the same type of study of two related 14-residue peptides. 1.8.1 First Project: Simulating Temp erature Jumps for Protein Folding My first project presents a new comp utational methodology aimed to calculate thermodynamic and kinetic properties of peptid e folding, and designed to mimic the way experimental measurements of these propertie s are made. Particularly, I focus on T-jump simulations of folding rates, and show how a combination of REMD followed by multiplexed molecular dynamics starting from structures take n from the REMD runs can be used to extract properties in line with experiments. A model system, Alanine20, was studied in this project as a proof of principle and desc ription of the methodology. 1.8.2 Second Project: Folding Kinetics by Temperature-Jump Simulations of Two Related 14-residue Peptides As follow-up of Project 1, REMD simulations of two closely relate d 14-residue peptides were performed to obtain equilibrated ensembles. Snapshots from this ensemble were used as initial structures for the T-jump simulations. Th ese two 14-residue peptides are very similar but have different experimental fold ing rates by a factor of ~2.9.49 They were selected by the experimental group to compare endcapping effects which can stabilize -helix. The folding kinetics of the two 14-residue pe ptides is studied by using T-jump simulations, and their results

PAGE 28

28 are analyzed using calculated CD spectra to obt ain the folding and unfolding rate. For these two systems, the relaxation time and folding/unfoldi ng rate constants are calculated and compared with experimental data.

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29 CHAPTER 2 THEORY AND METHODS In this chapter we will present some general di scussion of theory and methods used in this dissertation. In particular, we will address issu es of sampling, force fields, Langevin dynamics, generalized Born (GB) solvation model, and co mputation of circular dichroism (CD) and NMR chemical shifts. 2.1 Conformational Sampling The MD simulations of biomolecules are still i mmature due to both inaccuracy of the force fields and inadequate conformati onal sampling associated with th e number of degrees of freedom of system. The energy surface of biological system s is generally rough and rugged, such that it contains many local energy minima, which ar e isolated by high, insurmountable energy barriers.50,51 MD simulations may often get trapped in a local minimum and never reach the global minimum. One way to overcome this samp ling problem is to perform simulations in generalized ensembles, where th e construction of the ensemble is weighted by a non-Boltzmann probability weight factor. Therefore, the resu lting distribution guarantees a random walk in energy space, producing much better sampling in th e conformational space. The results need to be properly re-weighted to give any therm odynamic quantity as a function of temperature.52-54 Many methods based on generalized ensemble algorithms have been introduced to overcome sampling problems of biological mo lecules; the multicanonical algorithm (MUCA,55,56 also referred as entropic sampling57 or adaptive umbrella sampling58), simulated tempering (ST),59 1/k-sampling,60 Tsallis statistics 61 with simulated tempering,62 replica-exchange method (REM),63,64 and replica-exchange multi canonical algorithm (REMUCA).65 The replicaexchange method (REM) (or parallel tempering 64,66) is one of most widely used algorithms in a generalized-ensemble.

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30 Replica Exchange Method (REM). In REM, the standard Boltzmann weight factor can be used. A number of non-interacting copies (replicas) can be simula ted independently and simultaneously at different temperatures by the conventional MD or Mont e-Carlo (MC) methods. Conformations are exchanged betw een different temperature replicas every few steps with a specified transition probability that is defined by the Metropolis cr iterion. This exchange process enforces random walks in temperature space, which in turn leads to random walks in potential energy space. Consequently, REM has been wide ly applied to protein and peptide folding research.50,64,67,68 This dissertation makes use of the REM algorithm, modified to be combined with molecular dynamics, the so-called the repl ica exchange molecular dynamics (REMD). In REM,50 an artificial system composed of M non-interacting copies (or replicas) is considered at M different temperatures, Tm (m=1, 2,, M). The state of this generalized ensemble is defined as ) ,..., ()] ( [ )] 1 ( [ 1 M i M ix x Xwith m i i i mq p x) (] [ ] [ ] [where] [ ip, ] [ iqrepresent momenta and coordinates, respectively, for replica i at temperature m. Since the replicas are noninteracting, the weight factor (W) for the state X is then gi ven by the product of Boltzmann factors for each replica or temper ature, as shown in Equation 2-1: M i i i i m REMq p H X W1 ] [ ] [ ) () ( exp ) ( (2-1) where T kB1 ( kB is the Boltzmann constant) and the Hamiltonian ) (] [ ] [i iq p H is the sum of kinetic and potential energies. Now, one can attempt to exchange temperatures between the i th and j th replicas at temperatures Tm and Tn, respectively. The new state of the system becomes: ,...) ,..., (..., ,...) ,..., (...,' '] [ ] [ ] [ ] [i n j m j n i mx x X x x X (2-2) The detailed balance condition n eeds to be applied to converge to an equilibrium ensemble:

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31 ) ( ) ( ) ( ) (' 'X X w X W X X w X WREM REM (2-3) where ) ('X X w is the transition probability from state X to X and ) ( X WREM is the weight factor of the state X. From Equation 2-1, 2-2, and 2-3, the exchange probability ( P ) is obtained: e e e e e e e X W X W X X w X X w Pj i j i j j i i j i i j j j i i j i i jE E E E E E E E E E REM REM ) )( ( ') ( ) ( ) ( ) ( (2-4) Figure 2-1. The sketch of Replica-exchange method (REM) simulation in amber molecular dynamics packages. Exchange attempted (and rejected) Exchange accepted T time T1 T2 T3 T4 replica 1 replica 2 replica 3 replica 4

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32 where ) )( (j i j iE E and E is the potential energy of each replica. One can now obtain the acceptance probability of replica exchange ( P(accept) ) by using a Metropolis criterion (or Monte Carlo procedure):69 e e accept Pj iE E j i, 1 min 1 min ) () )( ( (2-5) The general simulation of the RE M performs the following steps: 1. Each replica is simulated-based on canonical ensemble-in parallel a nd independently for a certain number of MD steps. 2. Periodically, replicas with adjacent te mperatures are swapped with acceptance probability, P(accept) from Equation 2-5. 3. Repeat the process. In step 2, the exchanges are only allowed betwee n adjacent replicas in temperature, because the acceptance ratio of the exchange decreases expone ntially with increasing difference between the two temperatures.63 Figure 2-1 shows a sketch of a RE M simulation, describing the mechanism of replica exchange (or rejection) between different temperatures. 2.2 Force Field The force field representing the collection of mo lecular interactions represents the behavior of all atoms and bonds with specific fitting para meters. Many different si mulation packages have been developed over the y ears; for example, AMBER,70,71 GROMACS,72 OPLS,73 and CHARMM.74 Names of these packages generally im ply the empirical force fields. In the framework of this dissertation, the AMBER force field, the most commonly used for biomolecular systems, was applie d. Its potential energy function ( U(R) ) with a corresponding set of empirical parameters is shown below:75

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33 tic electrosta R q q Waals der van R B R A s dihedral n V angles K bonds r r K R Uatoms j i ij j i atoms j i ij ij ij ij dihedrals n angles eq bonds eq r 6 12 2 2]) cos[ 1 ( 2 ) ( ) ( ) ( (2-6) where req and eq are equilibration structural parameters. Kr, K, and Vn are force constants, n is multiplicity, and is the phase angle for the torsi onal angle parameters. In addition, A B and q are parameters related to the non-bonded potentia ls. Balance in parameterization can result in reasonably good compromise between accuracy a nd computational efficiency, thus, reproducing simulation results close to experimental ones. For the non-bonded part, the van der Waals parameters are usually determined by thermodynamic properties of various pure liquids.76,77 The electrostatic parameters are calibrated using a restrained electrostatic potential fit (RESP) model.78,79 The parameters for the first three internal terms (bond, angle, and dihedral) of Equation 2-6, come from a combination of experimental data and high-level ab initio calculations. Numerous MD simulations have been run for proteins and nucleic acids under these force fields and compared with experimental structur es over the decades. Ne vertheless, older AMBER type force fields present ma ny deficiencies, such as the well-known over-sta bilization of helices. In order to avoid this tendency and thus to get better backbone di hedral angles, Hornak et al. introduced new parameters, considering the energies of multiple conformations from high level ab initio calculations.80 Since there are many approxima tions within force fields,

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34 optimization of force fields remains to be improved along with advance in experimental technology. 2.3 Generalized Born (GB) Solvation Model Simulations with an explicit treatment of solvent provide much improved accuracy, but is computationally expensive for larger molecule s, for example, protei ns or nucleic acids.81-83 Alternatively, implicit solvation models have proven to be valuable tools for computational efficiency and also relative simplicity. In partic ular, the Generalized Born (GB) model is one of the most popular implicit solvati on models. The solvent, such as water, in the GB model is treated as an infinite cont inuum medium with the corres ponding dielectric properties.81,84 Thus, the GB model calculates approxima te values of the solute-solvent electrostatic free energies of solvation (Gpol) and gives rapid estimates of Gpol to save computation times in calculations.81,85 The electrostatic contribution of the solv ation free energy in the reaction field, reac, is approximated by a system of simple ionic particles with radius and charge q :84 1 1 2 2 12 q q Greac pol (2-7) where is the dielectric constant and this result is the well-known Born formula.86 If the simple ion is expanded to a molecule consisting of spheres of radii ( 1, 2, N) and charges ( q1, q2, qN) with the separation distance rij between charges qi and qj, the polarization free energy (Generalized Born (GB) equation) has been approximated as follows:81 N i N i j GB j i polf q q G 1 1 2 1 (2-8) where ) 4 /( ) ( ) (2 2 5 0 5 0 2 2 ij ij j i ij D ij ij GBr D e r f (2-9) TheGBf is defined as a function of an effective Born radius( i) when the distance rij 0 while GBf rij as rij .87

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35 In the original model, the effective Born radius ( i) was computed by a numerical integration procedure,81 but more recen tly pairwise approximations in which i is estimated via a summation over atom pairs, has been proposed by several groups.82,87,88 Therefore, effective Born radius ( i) can be derived as: i j j i j i i ir r g ) (1 1 (2-10) where i is an intrinsic radius for atom i and g() is a positive function which depends on the positions and sizes of the atoms and also has scaling factor for an empirical correction.83,88,89 While the GB approximation is highly efficien t for larger molecules, it is not so well balanced between protein-protein and protein-solvent interactions, compared to explicit solvent model. For example, over-stabiliz ation of salt bridges has been frequently observed in GB model, causing a significant conformationa l difference from explicit solvent model.90-93 In order to quantify the potential overstabilizati on of ion pairs for both models, Geney et al. performed Potential of Mean Force (PMF) method of salt br idge formation and found an excessive strength of salt bridges in GB.94 The lack of solvent friction in GB can acceler ate conformational transition rates, resulting in faster conformational sampling and at the same time, correctly predicting the native conformations.95-98 Langevin Dynamics is one of the commonly employed methods to overcome the frictional and high velocity collision problem. 2.4 Langevin Dynamics Langevin dynamics complements Newtons s econd law to account for omitted (solvent) degrees of freedom. The Langevin eq uation includes a frictional term in the form of a stochastic differential equation and thus, it attempts to mimic the viscous aspect of a solvent. However, it is an incomplete implicit solvent model, since it ignores electrostatic or hydrophobic effect. Those

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36 effects are included via the implicit solvation m odels, such as GB, described above. Furthermore, Langevin dynamics controls the temperature as a thermostat, thus approximating the canonical ensemble. The Langevin equation for motion of a particle i can be written using a stochastic differential equation as: ) ( ) ( ) ( ) (2 2t R m dt t dx t x F dt t x d mi i i i i i i i (2-11) where mi and xi are the mass and position of particle i in the simulation, respectively. Fi is an interaction force between a particle of interest and other particles and R(t) is the force on the particle due to random fluctuation by interaction with solvent molecules.99 The collision frequency ( ) is derived from the friction coefficient ( ) by = /m (m is the mass of the particle) and sometimes referred to as the friction coefficient in the literature. R(t) is a white thermal noise that obeys the fluctuation-dissip ation theorem at temperature T,100 ) ( 2 ) ( ) (' 't t Tm k t R t RB (2-12) where kB is the Boltzmann constant. 2.5 Computation of Circ ular Dichroism (CD) The CD spectra generally provides a direct measure of chirality of the molecular structure, since the magnitude and sign of the CD spectrum depend on the geometrical variables and electronic structure of a molecule. The pr otein can be characterized as a collection of independent chromophores. An individual chromop hore that is sensitive to secondary structural conformation and its interactions between the transitions on chromophores are the basis of calculations of CD. The matrix method is most commonly used to compute the CD spectra of proteins and peptides, where the excited stat es of each chromophore ar e subject to quantum chemical treatment, considering interactions between the chromophor es based on classical physics.101,102 The rotational strength, which gives the intensity of a CD band, can be

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37 theoretically defined in terms of the imaginary part of the sc alar product of the electric ( ) and magnetic (m) transition dipole moments of an electronic transition, using the Resenfeld equation and measures transitions of excited state.103 For an electronic transition from ground (0) to excited (i) states (0 i), the rotational strength ca n be calculated according to:104 0 0| | | | Im m i i e oiR (2-13) where Im represents imaginary part, 0 and i are the ground and excite d state wave function, respectively. e and m are the electronic and magnetic transition dipole moment, respectively. Since an electronic excitation occurs only w ithin a group, rather than between groups, a protein can be considered as a set of M noninteracting chromophoric groups in the matrix method. The excited-state wave function of the whole system ( T) is expressed as a linear combination of the basis functions ( ia) for each chromophoric group with the ni excitation: M i n a ia ia Tic (2-14) Each basis function is a product of M monomer wave functions, such that: 0 0 10... ... ...M j ia ia (2-15) where ia corresponds to the wave function of chromophore i for ( 0 a ) excitation. When the molecule is not symmetrical, the CD spectrum can be obtained from the sum of these rotational strengths derived as a nonzero value from each transition. A Hamiltonian matrix of a protein is com posed of the excitation energy of a single chromophore (forming the diagonal elemen ts) and the interaction between different chromophoric groups (forming the off-diagonal elements). The off-diagonal elements can be simplified by charge distributi on of electronic interactions.105 Thus: mn k ijm k ijm kl ijr q q Vln ln ,/ (2-16) where m and n correspond to the point charge of the transition j (on chromophore i ) and transition l (on chromophore k ), respectively, and r represents the distance between the point

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38 charges. As an example, the Hamiltonian matrix for the amide electronic transitions between n (at 220nm) and (at 193nm) is shown below: 2 22 12 12 22 2 21 12 12 21 1 11 12 12 11 1* * * * * * * E V V V V E V V V V E V V V V E Hn n n n n n n n n n n n n n (2-17) The diagonalization of the matrix H using a unit ary transformation provides the eigenvalues and eigenvectors corresponding to all transitions of the protein. The eigenvalue gives information about the excitation energy and th e eigenvector describes the mixing of localized transitions. The rotational strength (Equation 2-13 ) of each excited state can now be derived from the eigenvector and be used to calculate the CD. In our work we will use the programs design by Sreerama and Woody, where instead of performing a quantum chemical calculation they employed parameter sets consisting of a combination of experi mental data and theoretical parameters.105,106 2.6 Computation of NMR Chemical Shifts The NMR chemical shifts are affected by the environment. Calculating the shift is, therefore, important for interpretation of stru ctural information on macromolecules. Empirical methods,44,48,107-111 semi-empirical models 112,113 and ab initio quantum approaches 114-117 have been tried to calculate the chemical shift. 2.6.1 Calculation of Proton (1H) Chemical Shift An equation of the proton chemical shift is generally described in terms of various contributions, as below:118 misc side e HB ring tor rc total (2-18) where rc = the random coil chemical shift value of an amino acid residue, tor = the backbone torsional contribution, ring = the ring current contribution, HB = the contribution arising from

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39 hydrogen bond, e = the electric field or local charge contribution, side = the side chain torsional contribution, and misc = other chemical shift contributions including solvent, temperature, motional averaging, and covalent bond geometry. The empirical model (Equation 218) was developed and parameterized to experimental shift data through literature analyses. 107,114,115,119,120 However, it simply represents rough a nd empirical knowledge of chemical shift propensity, rather than unique (or complete) an d quantitative descrip tion of proton chemical shifts (1H).112 Quantum chemical shift calculations were pe rformed to improve the accuracy of the previous empirical models, considering ring current,121 electrostatic effects, structural dependence of magnetic anisotropy, and close contact contributions.115 As a result, a new empirical model developed via a combination of the empirical formula with the quantum calculation, was introduced with an im proved prediction of proton shifts.115 2.6.2 Calculation of 15N and 13C Chemical Shifts Xu et al. predicted 15N, 13C, 13C, and 13C (carbonyl C) chemical shift in proteins, using a mix of quantum chemistry and a database of experiments.122 Figure 2-2 presents an outline of 15N and 13C shift prediction algorithm, where database of density-functional derived shifts in the program SHIFTS (version 4.1) was used. This database is also used for chemical shift calculations in this dissertation.122 The SHIFTS program first ta kes a protein structure in Brookhaven (PDB) format and calcul ates the structural parameter of all the amino acids within a given protein, such as backbone conformation, side-chain orientation, and hydrogen bonding geometry. The density-functional database was built based on the calculated chemical shift patterns of 1335 peptide sequences which are derived from 20 proteins. The calculated results identified

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40 various significant pote ntial contributions to the shift: the backbone and torsion angles of the three consecutive residues (precedin g (i 1), self (i), and following residues (i + 1)), side-chain orientations of two consecutive residues and hy drogen bonding. Therefore, the total contribution is given by the sum of the individual ones: kc k c ) ( ) ( (2-19) where k denotes one of the contributions, and c is either helix or sheet structure. The predicted chemical shift, pred(c), is then derived as: ) ( ) ( ) ( c c cREF pred (2-20) ) ( cREF is a reference chemical shift for an amino acid, where c = for helix and c = for sheet. ) ( cREF is ideally determined by DFT calculation using the standard structure parameters from the literature.122 Finally, the process is followed by side -chain orientation refinement based on experimental shifts for an improved prediction.

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41 Figure 2-2. Outline of 15N and 13C chemical shift calculation in SHIFTS program.122 Structure (X-ray, NMR, or modeling) Calculate individual shift correction Calculate total contributions Predict chemical shift side-chain orientation refinement Calculate the structure parameters for all amino acids in the protein ) ( ) ( ) ( k k kref kc k c ) ( ) () ( ) ( ) ( c c cREF pred output from DFT Database

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42 CHAPTER 3 SIMULATING TEMPERATURE JU MPS FOR PROTEIN FOLDING 3.1 Introduction Understanding the structure, kinetics and ther modynamics of protein folding is one of the unsolved problems in biology. Many human di seases like Alzheimers and mad cow disease 1-5 are directly associated with protein misfol ding, unfolding and aggr egation. When studying protein and peptide folding, there are three main questions of interest. First, given a primary sequence, is there a unique 3-D structure under physiological condi tions and if so, what is that structure? Second, how does the peptide fold into its native structure? Third, how fast does it fold? 123-128 These questions are routinely answered in the laboratory using a mix of structural, thermodynamic and kinetic methods.24,129 Both experimental and theo retical approaches to the protein folding problem have been used to address these questions.25,28 It is basically impossible to probe all possible protein conformations expe rimentally because data, when available, is averaged over time and over many molecules (with the notable exception of single molecule experiments that still average over time).24,130 Alternatively, molecular dynamics simulations can be used to provide a detailed description of the system.131-134 In this chapter I focus on the speed of protein folding, but it is cl ear that underlying methodology, structure and thermodynamics are also available. In recent years, the use of multiplexed mo lecular dynamics runs to study folding has become commonplace. The availability of very la rge numbers of processors for short times has permitted the existence of pioneering efforts such as Folding@Home.4,135-137 This technique has been able to reproduce experimental folding rates for a number of systems.4,138-140 In its most used, basic form, it starts from a very large num ber of initial conformatio ns, and runs molecular dynamics for each of them for a pre-defined amo unt of time (a small number of nanoseconds).

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43 The initial coordinates are usually chosen either fully extended or taken from a high temperature run. The procedure then monitors the evolutio n of each sample independently and, after the predefined amount of time has passed, simply co unts the number of runs that produced a folded structure.138,141,142 Under the assumption of a single exponential decay, one can extrapolate to long times and extract a rate constant for the process. In a single exponential decay assumption (over all time scales) with a time constant and for M independent very short MD runs of time t, one expects around tM/ runs to have folded.137 For typical values of t=50ns, =10 s and M=10,000 trajectories, one expects only 50 of th e runs to have folded. These represent, by definition, the fastest component of the folding. The basic extrapolation relies on an assumption of a single exponential process over all time scales even the very short ones. In other words, it is assumed that the fastest folders truly follow the same pathways as the overall ensemble. However, some of the assumptions in the Folding@Home style methods become invalid under certain conditions. Several groups showed that an extrapolation of very short time decays to asymptotic exponential be havior might be unreliable.143,144 Paci et al. 145 reported that the fastest folding events do not agree with a co rresponding ensemble beha vior obtained by the distributed computing for a three-stranded antiparallel -sheet peptide. This is in some sense obvious: given a system with many free energy minima and a number of barriers (whose ensemble relaxation should be de scribed in terms of a master equation description), one should not expect a single exponential decay to hold at ALL time scales.146 This has been best expressed by Daggett and Fersht in ref.:147 It is our opinion that, at the molecular level, intermediates are always present. In other words, true two-state fo lding with only the denatured and native states occupying free energy minima is implausible.

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44 Moreover, Pande and his colleagues4 start from a fully extended conformation for their initial states. While they see evidence of ve ry fast (tens of nanoseconds) relaxation to a collapsed, unstructured state in times substantiall y shorter than their simulation times, it is clear that this is a choice of unfo lded ensemble that cannot be compared with typical kinetic measurements (such as temperature jumps) whic h usually measure relaxation times between two closely related equilibrium states. Since in Folding@Home a folding event is only counted against a pre-determined coordinate set (the folded state), it is possible that a number of the simulations have ended in a different configuration. If this new state has lowe r (free) energy than the initial choice, then it should be properly called the folded state. The so lution to this problem is to run every simulation long enough for a substantial percen t of the ensemble to fold into the same state, which then is defined, a posteriori, as the folded state of the system. Recently, Pande et al. and Levy et. al ,140,148-150 introduced a new method, namely, Markovian state models (MSMs), to predict pr otein folding rate constants. The MSMs can calculate both folding probability (Pfold) of all the configurations in a system and mean first passage time (MFPT) from the unfolded state to the folded state.140 In the present report, the folding kinetics of a polyalanine peptide is described and discussed based on MD simulation results. One of the most used experimental approaches to protein folding kinetics is laser-induce d temperature-jump (T-jump) spectroscopy.26,27,151 T-jump can raise the temperature extremely fast (nanos econd scale) and record the relaxation to equilibrium with a time resolution of ~1ns.22,23 Since the usual distribu ted computing procedure (Folding@Home) is clearly not what is done experimentally, where, in the case of T-jump, relaxation of the ensemble from one equilibrium distribution to another one is realized, it is

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45 important to ask if a protocol that closely rese mbles an experimental T-jump could be designed and explored. This chapter will show that this is indeed possible, focusing on a model system that should be considered as a proof of principle calculation. In the computational T-jump which is first in troduced in this chapter, the temperature will be increased suddenly, creating a system no longer at equilibrium and having to relax to a new state of equilibrium. In this method, the conformational space of a polyalanine peptide is pre-equilibrated at different temperatures using replica exch ange molecular dynamics (REMD). In this way, we will have a-priori knowledge of what the ensemble looks like at the initial time, before the T-jump (equilibrium at Tlow), and as time approaches infinity (equilibrium at Thigh). Then we will monitor a number of trajectories started from structures taken from the Tlow ensemble, run them at Thigh for a pre-defined time, and then, at each time-slice, compute any property we desire. In a similar way with experiment, this data will be analyzed after watching the relaxation data, before deciding on what kinetic scheme will fit the data best. Additionally, we can then go into the ensemble, and ask detaile d structural questions ab out rates and pathways. 3.2 Methods 3.2.1 Simulation Details The initial structure of the al anine polypeptide, ACE(ALA)20NME (ACE is acetyl beginning group and NME is N-methylamine ending group), was built in an extended conformation with the AMBER 8.0 simulation package75 using recently published AMBER ff99SB force field,76,152,153 which has been shown to provide improved agreement with experiment.154,155 A cutoff of 16 was used to comp ute long-range interactions, and the Hawkins, Cramer, Truhlar 88,89 pairwise generalized Born (GB) implicit solvent model, with parameters from Tsui and Case,85 was then applied to mimic the effects of water solvation.81,156 The system was initially subjected to 2,00 0 steps of minimization and the resulting

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46 conformations were used as initial seeds for the REMD simulation. The REMD method used the multisander im plementation in the AMBER 9.0 molecular dynamics program. Sixteen replicas were simu lated for 200ns each (total of 3.2s) at exponentially spaced temperatures, from 153K to 542K (153, 166, 181, 197, 214, 233, 253, 276, 300, 326, 355, 386, 420, 457, 498, and 542K). These temperatures resulted in an average exchange rate of 15% between adjacent replic as. The SHAKE algorithm was used to constrain the lengths of all bonds involving hydrogen157 and a 2-fs time step was used for every replica. Exchanges between replicas was attempted every 10ps, resulting in 20,000 attempted exchanges at each temperature. Conformations were recorded every 2ps from the simulation of each replica. The first 10ns of the simulation were discarde d and the latter 190ns were saved for further calculations, giving a total of 19, 000 configurations for analysis. Conformations obtained from the REMD calculati on were used as the initial structures for the T-jump experiment. 362 starting configurations were selected, equispaced in time, collected from the equilibrated simulations at 181K. Each member of this ensemble was then instantly and independently heated up to the Thigh of 214K (the details of the T choices are presented later on) and MD was run for each member for 35ns to simu late the relaxation after the T-jump. All MD parameters are same as those used in REMD. Figu re 3-1 shows a sketch of both the experimental T-jump scheme and computational T-jump setup to illustrate how these two methods are similar to each other. In order to properly identify the structural elements of the ensemble, we performed a cluster analysis with moil-view 158 using backbone RMSD for residues 2 to 19 as a similarity criterion with average linkage.159 The clusters were defined usi ng a bottom-up approach with a similarity cutoff of 2, for C-RMSD. The representative struct ures from the REMD ensemble

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47 at 153K were used to determine the composition of the folded ensemble. 3.2.2 Calculation of NMR Chemical Shifts NMR Chemical shifts were computed in order to predict structural information. SHIFTS (version 4.1) program (by David Case group) was employed to estimate proton (amide proton, and H), as well as 15N, 13C, 13C, and 13C' (from CO bonds) chemical shift of the polyalanine peptide.112,121,122 SHIFTS program recognizes a protein st ructure in Brookhaven (PDB) format. It computes proton chemical shifts using empirical equations and 15N, 13C, 13C, and 13C' chemical Figure 3-1. The diagram of the T-jump setu p. A) Experimental T-jump scheme, B) Computational T-jump setup. Laser 2 Cell Laser 1 Detector PC Tjump at t = 0 181 214 K 362 equilibrated configurations from REMD @ Ti(181 K) t MD @ Tf for 35 ns at t1 at t2 at t3 at t35 ns (214K) A B

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48 shifts from a database based on density functional calculations. 2D NMR spectra obtained us ing HSQC experiment and 13C/13C crosspeaks relationship enable us to see structural distributio ns of folded and unfolded states. 3.2.3 Calculation of CD Spectra Computations on CD spectra for estimating s econdary structure and calculating folding properties were performed by Sreerama and Woody.105,106 The matrix method 101 (in originindependent formulation of rotational strength 102) was employed with a transition parameter set consisting of a combination of experimental da ta and theoretical parameters: the authors used experimental data160 for two amide transitions and parameters from intermediate neglect of differential overlap/spectros copic (INDO/S) wavefunctions161 for the n transition. The rotational strength was computed in order to ge nerate CD spectra through Gaussian band. The bandwidths assigned for the n transition, and for two transitions were 10.5nm, 11.3nm, and 7.2nm, respectively.39 3.3 Results and Discussions The conformations at the lowest temperat ure (153K) from REMD were sorted into clusters.159 Clustering of the ensemble at 153K show s only two substantially populated clusters. The largest cluster contains 76% of the struct ures, whereas the next largest cluster is 9% populated, with no other cluster having a popula tion higher than 6%. Figure 3-2 A shows the ensemble at 153K superimposed on the representa tive structure of the larg est populated cluster. Figure 3-2 B shows the structure for that cluster, forming a well defined -helix. This is then the reference structure for futu re analysis. For the ensembles at 181 and 214K, the C-RMSD (residues 2-19) are computed, versus the representa tive structure of most populated cluster, and obtain the histograms of Figure 3-3 A. There is a clear separati on between the features in these plots, which allows us to define two different types of states, which are named folded (F) and

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49 unfolded (U). Structures are defined as folded (F) ( -helical) if their C-RMSD is within 2.0 of the reference structure, and labe led as unfolded (U) otherwise. Based on this clear structural separation, the process is treated as a twostate equation between th e folded and unfolded ensembles. The folding rate equations for each species are written as, [] [][]FUdF kUkF dt (3-2) [] [][]UFdU kFkU dt (3-3) Figure 3-2. Cluster analysis of polyalanine. A) All conformations from REMD at 151K are superimposed on the reference structure of the largest cluster, B) -helix reference structure from the cluster analysis. kU kF Unfolded (U) Folded (F) (3-1) A Lase B

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50 Figure 3-3. Histograms from the C-RMSD. A) Probability density of C-RMSD at 181K (blue line) and 214K (red line), B) (probability at 214K probability at 181K) vs. CRMSD. A B

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51 In the theoretical work concentrations were re placed by populations, which is equivalent of having a total initial concentration set to unit y. This then makes (U) = 1 (F). The time dependent populations can then be directly solved leading to:29 12()tFtCeC (3-4) ) ( 1 ) ( t F t U (3-5) The population relaxation is dominated by a single rate constant, 1/1/1/UFrelaxUFkk In this equation, F is the folding time (and U is the unfolding time). The values of the two constants C1 and C2 are completely determined from the initial and final concentrations and are then not adjustable parameters. By recalling that the equilibrium constant K can be written as a ratio of the forward and backward rate constants, the two can be then separately determined. The population of two F and U ensembles versus temperature is shown in Figure 3-4. They were computed from the C-RMSD histograms (referenced to the F structure) using a 2.0 cuto ff. At low temperature the system shows mostly Figure 3-4. Fractions of -helical folded state (F), and unf olded state (U) as a function of temperature using probability (C-RMSD).

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52 -helical folded states but the populations of unfolded states rise as temperature increases providing a reasonable melting curve. A clear melting temperature was found at 190K. The low melting temperature compared w ith regular experiments can be assigned directly to issues with the force field and solvation model.1,94 It does not however subtra ct from the main point of this chapter which aims to present and test a new method. Based on the melting curve, T-jump simulations were performed with the temperature jump bracketing the melting temperature, from 181K to 214K. By having the complete descript ion of the temperature dependent ensembles we can have complete knowledge of what both the initial t=0 (T=181K) and t= (T=214K) ensembles look like. In Figure 3-3 A, we show the RMSD probability at both initial and final temperatures for the T-jump. In Figure 3-3 B, the probability difference (214K 181K) validates choices of temperatures, by showing a signific ant change in (F) and (U). These figures completely determine the values of C1 and C2 in Equation 3-4. 3.3.1 Calculation of Chemical Shifts Calculation of chemical shift was performed on the trajectories obtained from 10-200ns REMD simulations. All the chemical shifts (1H, 1H, 1H (along the NH bond), 13C, 13C, 13C' (along the carbonyl bond), 15N) were computed using SHIFTS program (section 3.2.2). The spectra measured at 16 different temperatures were then averag ed for each residue (total 1,900 structures). The averaged chemical shifts are pl otted as a function of residue number (Figure 35). As shown in Figure 3-5, the highest temperature is placed at the bottom and all the averaged chemical shifts decrease with increasing temperature except 13C and 1H chemical shifts. We can expect that our prediction of 13C and 1H chemical shifts compared with experimental data (Table 3-1) shows prediction errors from side chain orientation. Table 3-1 presents chemical shifts values obtained using REMD simulations at the lowest (153K) and the highest temperatur e (514K) for a residue 6 (A6) in polyalanine peptide. For

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53 Figure 3-5. Averaged chemical shifts (proton, 13C, and 15N) for polyalanine peptide (ACE (ALA)20NME) as a function of residue number, for all sixteen temperatures. Temperatures Tem p eratures Temperatures Temperatures Temperatures Temperatures Temperatures

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54 Table 3-1. Comparison of calculated chemical sh ifts for residue 6 (A6) in polyalanine peptide and experimental values for al anine residue from references 42,44,111* Helixa 153Kb Random coilc514Kd f RCe l hf 13C 54.7 54.51 52.4 52.47 2.3 2.04 13C' 179.6 179.34 177.6 177.20 2.0 2.14 13C 19.74 18.48 19.26 20.06 0.48 -1.58 15N 117.94 122.5 115.28 2.66 1H 3.91 4.33 3.98 -0.07 N1H 7.68 8.15 7.55 0.13 All data are given in ppm. a, c experimentally measured chemical shift values from references. b, d computationally measured average chemical shift values from 10-200ns REMD simulations. e chemical shift deviations, f (helix) RC (random coil), from references. f chemical shift deviations, l (low temperature) h (high temperature), from 10-200ns REMD simulations. comparison, it also provides the chemical shift values of both helix and random coil for a alanine residue, measured experimentally by Wishart et al. 42,111 and Spera et al.44 As mentioned in the previous section (cluster analysis), the conforma tion at the lowest temperature (153K) is mainly well-defined -helix (76% populated). The calculated chemical shifts (13C) at 153K (Table 3-1) agree well with the helix chemical shifts values from the experiment. In contrast, the calculated values at 514K are comparable with the random coil values from the experiment. The calculation results make sense, disordered conformations of the peptide (random coil) are expected at the higher temperature and more structured conforma tions (folded or packed conformation) at the lower temperature. The results also indicate that w ith increasing temperature, there is an increase of the random coil population (unfolded state) accompanied by a decrease of -helical structure (folded state (F)). Therefore, one can detect form ation of the secondary structure by monitoring a change in the chemical shift ( ). The calculated 13C, and 13C' chemical shift are in good agreement with experimental results. However, the calculated 13C chemical shift differs from experimental one, due to errors associated with fitting of reference shifts in the database of the SHIFTS program.112,121,122

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55 Wishart et al. commented that error in 13C and 15N chemical shift data is relatively larger than that in the proton chemical shift.42 The 2D spectrum, for example HSQC, can us ually provide more complete structural details, compared to 1D one. Figure 3-6 shows 1H15N HSQC spectrum of a polyalanine peptide at 181K and 300K. At higher temp erature (300K, Figure 3-6 B), the spectrum is poorly resolved with slow fluctuation, due to the disordered stat e. This indicates unfolded states are dominated. On the contrary, the spectrum at lower temperature (181K, Figure 3-6 A) is well resolved and dispersed, as a result of folded states. Figur e 3-6 included only the cross peaks providing the structural information. However, the kinetic anal ysis can be performed w ith additional intensity information. The 13C chemical shift for C and C has been often used to determine backbone conformation of proteins and pe ptides, as found in many of previous NMR studies, such as empirical 44,45 and ab initio 46 methods. This is because the C and C chemical shifts are mostly determined by the backbone and torsional angle.41 In this studies, 13C/13C crosspeaks in the folded state are relatively sharp (2-4 ppm full width), narrow, and well resolved. The crosspeaks in the unfolded state are, however, broad, spreading toward upright. Figure 3-7 shows two-dimensional calculated 13C/13C crosspeaks for polyalanine pe ptides (residue 5 (A5), 11 (A11), and 16 (A16)) obtained at 153, 181, 214, and 300K. The spectra measured at different temperatures were then averaged over the ense mble (total 1,900 stru ctures). Three labeled residues are all similar in the general trend on temperature. The crosspeak at the lowest temperature (153K) is relatively ordered, sharp, and narrow, indicating significantly folded (or helical) conformation. As temper ature increases, the crosspeak pr ofile becomes broader toward upright, showing that conformational distribution is temperature dependent.

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56 These results are in accord with the HSQC results that covered in Figure 3-6. 3.3.2 Calculation of Circular Dichroism For each member of the ensemble and at each te mperature in the REMD simulation, CD spectra were computed using the technique describe d in the methods section. The spectra for Figure 3-6. 1H15N HSQC spectra of polyalanine pep tide. A) At 181K and B) At 300K. A B

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57 Figure 3-7. Two-dimensional 13C/13C crosspeak shapes for polyal anine peptide, residue 5, 11 and 16 (A5, A11, and A16) from several temperatures. Color scheme are from the maximum signal intensity (red) to th e minimum signal intensity (blue). each temperature were then averaged over members of the ensemble (total 1,900 structures). The data is shown in Figure 3-8 A for the average sp ectra versus temperature. In agreement with the RMSD data, the lowest temperature spectrum resembles that of an -helix, while as temperature increases, the system becomes disordered. The fo lded state has two minima in the CD spectrum, one at 208nm ( amide transition) and another at 222nm (n* transition along the carbonyl bond). Experimental CD spectra for an -helix shows a deeper minimum for 222nm than for 208nm.162 The calculated local minimum at 208nm does not agree with the experimental results, indicating those tran sitions to be very sensitive. 100% -helical poly-alanines, ACE(Ala)20 NME, were built using TINKER molecular dynamics program 163,164 and 200 steps of

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58 minimizations were performed to investigat e the difference. Minimum wavelengths are compared between two CD spectra (one is before minimization, i.e. pure 100% -helix, and the other is after minimization) in Figure 3-9. The 100% -helix has a minimum at 222nm while the structure after minimization has a minimum at 208nm. This discrepancy is assigned to problems with the CD calculations parameters, and been in the process of investigating this. Figure 3-8 B presents the average value of CD at 222nm () versus temperature. This is the type of signal one can follow when performing the Tjump experiment. T-jump simulations were performed and the re sulting time traces were fitted to calculate the folding and unfolding rates wi th initial and final temperatures determined by referring to the melting curve in Figure 3-4. 362 structures from the ensemble at 181K were used as starting points for molecular dynamic runs at 214K, hen ce simulating a T-jump. The temperature reaches the new value in a time scale of the order of the thermostat coupling (0.1ps-1 time constant used) and should be thought as having a very short dead -time. These simulations were run for a total of 35ns each (total of 12.67s). At each time slice, a CD spectrum was computed for each independent MD run, and the average performed. The folded and unfolded states were added and then versus time is plotted for each state in Figure 3-10. The folded and unfolded states were defined in Figure 3-4 using C-RMSD probability. When fitting this kinetic trace, the power of this method can be clearly seen. This trace is bounded by the fact that we know, from the equilibrium runs, what the sy stem looks like at t=0 and at t= This leaves a single parameter to be determined via fitting, assuming a two-state system. The calculated relaxation data are well fitted using a single exponential equation, as sh own in Figure 3-10, indicating that the suggested reversible two-state kinetic mechanism (Equation 3-1) during the cluster analysis is valid. From

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59 Figure 3-8. Results of Circular dichroism. A) Calculated circ ular dichroism spectra of ACE (ALA)20NME from 153K to 542K, B) Average of CD222 () of each simulated polyalanines. Error bars represent standard deviation. A B

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60 Figure 3-9. Two circular di chroism (CD) spectra, pure -helix and -helix after minimization, are compared to see different minimum wavelengths. Figure 3-10. Average of CD222 () of T-jump simulation data (red: folded (F), and green: unfolded (U) states) are fitted usin g a single exponential equation.

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61 this fit the value of ()UFkk in Equation 3-4 is determined to be 1 36339 1ps and relax=36.34ns. This relaxation time was used from CD to predict the time evolution of the F and U populations. Figure 3-11 has the simulated average populations versus time, and the populations predicted from Equation 3-4 also. The curves in Figure 3-11 ar e not fitted, but are instead simulated using the fit from Figure 3-10. The agreement between the simulated and raw population data is excellent. Using our knowledge of the limiting populations at the initial and final temperatures, equilibrium constants can be obtained. We can then separately determine the folding and unfolding rate constants at the final temperature. The computed values are kF=2.11 10-5ps-1, and kU=6.44 10-6ps-1, corresponding to a folding time F=47.5ns and an unfolding time U=155ns. Williams et al. have simulated Figure 3-11. Folding (green) and unfolding (red ) fractions from T-jump simulation data are calculated and fitted using same in Figure 3-10.

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62 21-residue alanine-based peptide (Fs21 peptide, A5(A3RA)3A) and estimated folding time, 16 to 180ns using T-jump experiment from 9.3 to 27.4C.19 In the present model we assume that the imp licit solvent is a correct representation of structure and thermodynamics. However, it does not properly represent the friction of water solvent.95 The time scales seen in this chapter are then much faster than they would be otherwise. The following chapter describes our continuing line of work on systems where experimental data is available and using Langevin dynam ics (which incorporates friction). 3.3.3 Optimum Number of Unfolded States? Two clusters were found in the cluster analys is earlier. The most populated cluster is helix (Figure 3-2 B) that occupies roughly 76% of the population. It is defined as a folded state in the two-state mechanism ( -helix and unfolded state). Th e second largest cluster is 9% populated coiled-coil -helix (the structure is shown in Figure 3-12). If the coiled-coil -helix is added as the second folded state, three-state kinetic mechanism ( -helix, coiled-coil -helix, and unfolded state) can be suggested. Three different states ( -helix, coiled-coil -helix and unfolded states) are chosen for initial MD simulation run for 1 s at 214K in order to check how many different states exist. The simulation time (1 s) is chosen because it is estimated to be close to the folding time of small proteins (11-21 residues peptides) as the lower limit.165 This simulation possibly shows all the intermediates involved in the protein folding process.14 The C-RMSD (residues 2 to 19) was computed from the 1 s trajectory with respect to -helical, coiled-coil reference structures (Figure 3-13). These RMSD plot reflects the number of different states. By comparing -helix reference (Figures 3-13 A, B, and C) and coiled-coil -helix reference structure (Figures 3-13 D, E, and F), simulated structures are classified as follows: -helix when C-RMSD1 < 2.0 and C-RMSD2 > 5.0 and coiled-coil -helix when C-RMSD1 > 5.0 and C-RMSD2

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63 Figure 3-12. Reference structures from cluster analysis. A) All conformations from REMD at 151K are superimposed on the two reference structures, B) -helix and C) Coiled-coil -helix reference structures. < 2.0, where C-RMSD1 was calculated from -helix reference and C-RMSD2 from coiledcoil -helix reference structure. Interestingly, several transitions, from coiled-coil -helix to helix folded state based in the rmsd range (0-2, 2-3, 3-5, and above 5), are found in Figure 3-13 E. The fraction of four states was obtained by applying both C-RMSD1 and C-RMSD2, as shown in Figure 3-14. In order to show relationship between two folded states, C-RMSD1 as an x-axis is plotted against C-RMSD2 as an y-axis C-RMSD (Figure 3-15), where four different states are also found. Based on the results, th e following kinetic mechanism can be suggested. In Equation 3-6, two different unfolded st ates are separated by two representative structures (U1 and U2). This implies that more representati ve structures can be defined without limit. Based on the present experiments, the numb er of unfolded states is counted, introducing the T-jump methodology. Optimal folding mechan ism was not discussed in this chapter and should be dealt in the near future. k -2 k 3 k 2 k 1 k 1 -helix Coiled-coil ( CC) k 3 Unfolded (U 1 ) Unfolded (U 2 ) (3-6) A B C

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64 Figure 3-13. C-RMSD (residue 2-19) computed using different initial states from 1 s MD simulations. A) -helix, B) Coiled-coil, and C) Unfolded initial states from CRMSD1. D) -helix, E) Coiled-coil, and F) Un folded initial states from C-RMSD2. 3.4 Conclusion This chapter presents the first attempt to using simulation conditions and predicting observables as close as possible to experiment. As far as we know, this is the first time a simulation of T-jump is used to calculate the folding rate constants. By using a mix of very efficient sampling techniques (REMD) to properly populate ensembles and regular MD simulations to study the non-equilibrium relaxation of the populations, extracting kinetic and thermodynamic data are enabled. A set of folding/unfolding time ( F= 47.5ns and U =155ns) was calculated by data fitting of calculated CD spectra. The chemical shifts were also calculated and it showed systematic change in conformational distributions depending on the temp eratures. Finally, the number of folded and unfolded states was determined and discussed, based on 1 s long MD simulations. In the future, implicit and explicit solvent models need to be compared and influence of friction remains to be determined. As an extension of the present work, the following chapter A B C D E F

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65 describes results obtained on real proteins usin g Langevin dynamics wh ich considers frictional effect.

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66 Figure 3-14. C-RMSD2 (with respect to coiled-coil -helix) plots from 1 s MD simulation using coiled-coil -helix initial state at 214K. Figure 3-15. C-RMSD (residue 2-19) relation plot using C-RMSD1 (with respect to -helix reference structure) as x-axis and C-RMSD2 (with respect to coiled-coil -helix reference structure) as y-axis, respectively, at 214K. CC U 2 U 1 -helix -helix coiled-coil (CC) Unfolded (U1) Unfolded (U2)

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67 CHAPTER 4 FOLDING KINETICS BY TEM PERATURE-JUMP SIMULATIONS OF TWO RELATED 14RESIDUE PEPTIDES 4.1 Introduction Protein folding is described as a process of an ensemble of molecules reaching their biologically active three dimensional stru ctures from a linear chain of amino acids.99 Both experimental and theoretical studies for over 40 years have converged to show and predict the basic principle and particular mechanism of folding and unfolding of proteins.146 Their mechanism and kinetics are still unknown or in completely understood a nd therefore, are being actively investigated in the molecular biology ar ea. Particularly, simulations can describe the folding process microscopically and its atomic details which are unavailable in experiments. However, conventional molecular dynamics is unable to yield complete conformational space sampling due to its high energy barriers and deep local minima in most of the systems. Thus, the development of efficient algorithms becomes a point at issue. One of the oldest methods is umbrella sampling technique that is combined with molecular dynamics simulations. 166 Brooks and coworkers 167 utilized this method extensively to understand thermodynamics and kinetics of folding for biomolecular (or biolog ical) systems. It was very powerful for the investigation of the folding free energy lands cape in several chosen reaction coordinates. However, it was limited to small proteins (or peptides),168 and was not capable of determining the biasing potential. Replica exchange molecular dynamics (REMD),64 which adapts a random walk in potential energy space, was introduced to overcome thos e problems. This technique shows not only enhanced sampling but also increased speed of equilibration by treating the temperature as a control parameter.

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68 Direct experimental measurement of fo lding rates becomes possible by using the temperature jump (T-jump) experiments in the nanosecond to microsecond time scale. The Tjump rapidly changes the position of the equilibrium between fold ed and unfolded states by the temperature change, and hence, reaction kineti cs to the new equilibrium can be monitored 169. Moreover, the laser-induced T-jump technique wa s extended to far-infrared absorption (5.88 6.67 m), which can observe protein secondary structure.29 Several groups applied this method to helix-coil transition 19,170 and folding transition of hairpin structure.171,172 Combination of computational and experiment al methods in studying protein folding is advantageous.25 For example, Folding@Home typically runs a very large number of initial conformations for a few nanoseconds, either fu lly extended or taken from a high temperature run. Under the assumption of a single exponentia l decay that is based on two-state folding kinetics, a folding time of 10 s is calculated when 50 out of 10,000 trajectories have folded for a 50ns simulation. In order to verify the dynamics, the simulations were compared with T-jump spectroscopic results.25,173 However, several groups pointed out that tw o-state model might not be applicable under certain conditions. Daggett and Fersht 147 emphasize the problem in terms of intermediate states that always exist between a dena tured and native state. Sabelko et al. found non-exponential folding kinetics of two proteins (yeast phos phoglycerate kinase (PGK) and a ubiquitin mutant) from a nanosecond T-jump.173 F. Gai and coworkers also repo rted non-exponential behavior for helix-coil transition kinetics.130 Therefore, the most important issue at this point is how we perform our simulation in the conditions clos e to real experiment s without any kinetic assumption, while connecting theory and experiment. In Chapter 3, we introduced Temperature-ju mp (T-jump) simulations of the polyalanine

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69 peptides (Ala20) as a proof-of-principle system.174 Since alanine-20 has high hydrophobicity and insolubility in water,175 no experimental data have been repo rted. In this chapter, we apply the same techniques (T-jump simulations) to model two related 14-residue peptides. Wang et al. 49 have previously studied the helixcoil kinetics of these two peptid es using time-resolved infrared (IR) spectroscopy, coupled with laser-induced T-jump technique. They reported that the one is a general polyalanines derivative of Baldwin-type peptide,176 and the other contai ns helix stabilizer and end-capping groups, showing difference in helicity and stability regarding the folding kinetics. In this chapter, we simulated T-jump and calculated folding and unfolding rates of the same peptides that Wang et al. 49 applied, using calcu lated circular dichroism (CD) spectra. The results were then compared with the experiment al data. Most importantly, the present work was performed in a very similar way to the experiment s, such that they were compared with each other, minimizing assumptions. Initially, RE MD simulations were performed to obtain equilibrated ensembles to overcome energy barri ers and accelerate the convergence. Snapshots from these ensembles were used as initial structures for T-jump simulations. The Effect of Frictional Coefficient. To further validate our T-jump methodology, we focus on the influence of friction and random forcesintroduced by the solventon the protein folding kinetics. Kramers 177 proposed in its simplest form that the reaction rate ( k ) in the high friction ( ) limit should be proportional to the in verse of the friction of the solvent, / 1 k. This inverse dependence of rates of protein folding on viscosity, (or, equivalently, the friction, ) has been frequently reported experimentally.23,95,178,179 The effect of frictional coefficient through Langevin dynamics was studied for the folding kinetics of two related 14-residue peptides. Differe nt values of friction an d random forces were applied and compared with experimental results from Wang et al.49

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70 4.2 Methods Alanine-based peptides were originally designed by Marqusee and Baldwin 176 for -helix formation. Two related 14-residue peptides, Ac-Y GAKAAAAKAAAAG-NH2 (peptide 1), and Ac-YGSPEAAAKAAAA-r-NH2 (peptide 2, where r represents D-Arg), were derived and tested by F. Gai group.49 The initial structures for two related 14-residue peptides were constructed in fully extended conformations with the AMBER 9.0 molecular simulation package.180 The AMBER ff99SB force field 76,152,153 was used for both peptides. Mo dified generalized Born (GB) implicit solvent model by A. Onufriev, D. Bashford and D. A. Case (GBOBC)181 was applied. The system was initially subjected to 500 steps of minimization, and then equilibrated by using Langevin dynamics with collision frequency =1.0ps-1 for both peptides to account for frictional effects. The resulting conformations were used as initial seeds for the REMD simulation. All REMD simulations reported he re were carried out using the multisander implementation in the AMBER 9.0 simulation pack age. Sixteen replicas for peptide 1 were simulated for 500ns each (total of 8 s) at exponentially distributed temperatures, from 150K to 726K (150, 167, 185, 206, 228, 254, 282, 313, 348, 386, 429, 477, 529, 588, 653, and 726K). Eighteen replicas for peptide 2 were simulated for 500ns each (total of 9 s) from 150K to 834K (150, 166, 184, 203, 225, 248, 275, 304, 336, 372, 411, 455, 503, 557, 616, 681, 754, and 834K). The average exchange rate between adjacent repl icas was 15%. The SHAKE algorithm was used to constrain all the bond le ngths involving hydrogen atoms,157 which allows an integration time step of 2fs for each replica. Re plica exchanges were attempted at every 10ps, resulting in 50,000 attempted exchanges at each temperature. Conf ormations from the simulation of each replica were recorded every 2ps interval The first 20ns of the simulatio n were discarded and the latter 480ns were saved for further calculation, givi ng a total of 48,000 configurations for both peptides.

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71 To determine the secondary structure, th e program designed by Kabsch and Sander, Definition of the Secondary Structure of Proteins (DSSP),182 was used for both peptides. Percentage of -helix (Helicity) was calculated for ever y residue and every replica, using this program. We also performed a cl uster analysis with moil-view 158 based on backbone RMSD for residues 4-9 for peptide 1 and residues 5-10 for peptide 2 with a cutoff of 2. Conformations obtained from the REMD calculati on were used as the initial structures for the T-jump experiment. We selected 1,200 starting configurations collected from the equilibrated simulations for peptide 1 and peptide 2 at 282K and 304K, respectively. They are equispaced in time (200-500ns REMD). Each member of this ensemble was then instantly and independently heated up to the Thigh (the details of the temperature choi ces are discussed later) and MD was run for 5ns to simulate the relaxation after the T-jump For both of the peptides, all MD parameters are the same as those used in REMD. Several collision frequencies ( =1.0, 5.0, 10.0, and 20.0ps-1) were used for both peptides to investigate the frictional effect on the rate of the protein folding. In order to estimate secondary structure a nd folding properties, computations on CD spectra were performed, in the same manne r used by Sreerama and Woody (chapter 3).105 The matrix method 101 was employed with a transition parame ter set consisting of a combination of experimental data and theoretical parameters: experimental data 160 for two amide transitions and parameters from intermediate neglect of differential overlap/spectroscopic (INDO/S) wavefunctions 161 for the n transition. The rotational strength was computed to generate CD spectra through Gaussian band. The bandwidths assigned for the n transition, and for two transitions were 10.5nm, 11.3nm, and 7.2nm, respectively.39

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72 4.3 Results and Discussion The two related 14-residue peptides in this study were selected to prove a new methodology (described in the chapter 3). Marqusee et al. 176 designed, synthesized, and tested alanine-based peptides as helix-forming peptides. In their work, the poly-Ala helix containing (i + 4) Glu-Lys+ salt bridges showed optimal behavior ( 80% helicity). D-Arg was also chosen because it is the most efficient -helical C-capping residue.183 According to Huang et al. a tripeptide, Ser-Pro-Glu, was selected as helix-stabilizing N-terminal sequence that occurs most frequently at the N-terminus of helices in th e WHATIF database of 1705 helices and also might stabilize the helix by electrostatic interactions.130 Therefore, peptide 2 was built based on possible -helix-stabilizing effects, while peptide 1 wa s considered as a regular poly-Ala peptide found in the experimental work. We determined the secondary structure between two related 14-residue peptides to verify our new methodology on the study of properties of peptides. Figure 4-1 shows plots of the helicity of the secondary structure versus residu e number to present the structural differences. The overall helicity is expected to be low, si nce these two peptides ar e only 14-residue long. However, significant differences are found be tween the two peptides in Figure 4-1. A high helicity region in peptide 2 is seen between re sidue 5 (Proline) and 10 (Lysine). In contrast, peptide 1 shows a low and broad distribution. Th e difference can be explained by a salt-bridge interaction of two charged residues, Glu-(residue 6) and Lys+(residue 10). When the two end terminals (N-terminal and C-terminal) in peptide 2 are compared, C-terminal end with a D-Arg residue shows 20-40% helicity, but N-terminal with an acetyl group shows almost zero helicity. Simulations results are comparable with experime ntal ones in a sense that peptide 2 is more helical than peptide 1.

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73 Figure 4-1. Calculated helicity. A) Peptide 1 versus residue number, B) Peptide 2 versus residue number based on DSSP method, respectively. 4.3.1 Cluster Analysis The conformations at the lowest temperature (150K for both peptides) from REMD were sorted into clusters based on potential helical regi ons (residue 4 to 9 of pe ptide 1 and residue 5 to 10 of peptide 2) from DSSP method. The larges t cluster in peptide 1 contains 71% of the structures and the next largest cl uster is 13% populated. In peptid e 2, the largest cluster is 78% and the next largest is 14% populated. No other clusters from both peptides have a population higher than 10%. Figure 4-2 A sh ows all conformations superimposed on the representative A B

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74 structures of the largest populated clusters for bo th peptides (red and yellow for peptide 1 and 2, respectively) at 150K. Two representative struct ures from clusters are shown in Figure 4-2 B. In Figure 4-3 the fractional sizes of each cluste r in both peptides are plotted as a function of time to evaluate convergence of the RE MD simulations, where the first 20ns of the simulations were discarded and the latter 480ns were used fo r calculation. Both peptides converge to their final populations after appr oximately 200ns. The biggest cluster of both peptides converges to a population of ~0.7-0.8 (70-80%). Therefore, all initial conformations were taken from 200-500ns REMD for future analysis. Figure 4-2. Reference structures from cluster analysis of peptide 1 and peptide 2. A) All conformations from REMD at 150K and are superimposed on the two reference structures, B) Two representative structures. A B

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75 Figure 4-3. Populations of the repr esentative clusters. A) Peptide 1 as a function of time from 20500ns REMD, B) Peptide 2 as a function of time from 20-500ns REMD. At 200ns, simulations of both peptides are converged and stabilized. A B

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76 4.3.2 Helicity Theoretical fractional helicity ( fH) of a peptide can be calcula ted using the mean residue ellipticity at 222nm, 222 ,49,184,185 ) 1 ( ] [ ] [222 222n x fH (4-1) where 222 is the mean residue el lipticity of an ideal pe ptide with 100% helicity, n is the the length of the potential helical region, and x is an empirical correction. In order to assign those parameters, 14 polyalanines (ACE-(ALA)n-NME, n=5-14, 16, 18, 20, 22) with 100% helix were built using HyperChem software 186 and subjected to 5,000 steps of initial minimization. The CD spectra (222 ) were computed and plotted ( 222 versus 1/n) from the Equation 4-1 (Figure 44). Values of x and 222] [ are taken to be 2.38 and -31403 deg cm2 dmol-1, respectively, based on the plot and fit in Figure 4-4. These two va lues are used for further calculations. Figure 4-5 shows the fractional helicity of peptide 1 and pe ptide2 from 200 500ns REMD as a function of temperature (Equation 4-1). Wang et al. 49 reported that the helicity of peptide 1 and peptide 2 at 11C was approximately 11% and 29% (by experiment), respectively. Marqusee et al.176 showed that the 16or 18-residue Ala-based pept ides contain approximately 25-50% helicity. Both theoretical and experimental results clearly indicate that th e helicity of peptide 2 is larger than that of peptide 1. 4.3.3 Calculation of CD Spectra For each member of the ensemble and at each temperature in the REMD simulation, CD spectra were computed as mentioned in the me thods section 4.2. The spectra obtained for each peptide at a certain temperature were then av eraged over members of the ensemble (49,000

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77 Figure 4-4. The resulting curve fo r the two parameter calculations ([ ]222 and x) from the plot [ ]222 versus 1/n from the Equation 4-1. The unit of molar ellipticity is deg cm2 dmol1 of both peptides. Figure 4-5. Fractional helicity ( fH) of peptide 1 (blue) and peptide 2 (red) as a function of temperature. ) 38 2 1 ( 31403 ) 1 ( ] [ ] [222 222n n x

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78 structures each). The resulting average spectra as a function of wavelength are shown in Figure 4-6. The CD spectra for an -helix show two minima, one at 222nm (n* transition along the carbonyl bond) and the other at 208nm ( amide transition).162 In the plot, the molar ellipticities at 222nm ( 222) are linearly related to the helical minimum.184 Figure 4-7 shows average values of CD at 222nm () for peptide 1 and peptide 2, as a function of temperature. The observation showing higher helical population in peptide 2 than peptide 1, are in a good agreement with the experimental results by Wang et al. 49, in the similar temperature region. This type of signal is us eful for the T-jump experiment. T-jump simulations were performed and the re sulting time traces were fitted to calculate the folding and unfolding rates. This system appears to reach equilibrium after ~ 200ns of REMD from the cluster analysis in Figure 4-3. Therefore, 1,200 initia l configurations were selected from 200 500ns REMD. The T-jump wa s then simulated from 313 to 348K for peptide 1 and 336 to 372K for peptide 2. Temperature change is achieved within a time scale of the order of the thermostat coupling (0.1ps time constant used), having a very short dead-time. The simulations were run for a to tal of 5ns each (total of 6 s of each peptide). At each time slice, a CD spectrum was computed for each independent MD run, and then they are averaged. The resulting are plotted against time and fitted (Figure 4-8). As we mentioned in the Chapter 3, the trace is bounded from the equilibri um runs, we know what th e system looks like at t=0 and at t= This leaves a single parameter ( ) to be determined via fitting from the Equation 3-4. The calculated relaxation data are fitted well using a single exponential function, indicating occurrence of reversible two-state folding. Deta ils of the two-state mechanism are covered in Chapter 3.

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79 Figure 4-6. Calculated ci rcular dichrioism (CD) spectra. A) Peptide 1 from 150-726K, B) Peptide 2 from 150-834K. The unit of molar ellipticity is deg cm2 dmol-1 of both peptides. Tem p eratures Tem p eratures A B

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80 Figure 4-7. Average of mean resi due ellipticities at 222ns () of simulated peptide 1 (blue) and peptide 2 (red) are shown as a f unction of temperature. The unit of molar ellipticity is deg cm2 dmol-1 of both peptides. In a two-state model, the relaxation time ( relax) is obtained from the sum of the folding rate ( kf) and unfolding rate ( ku), = kf + ku = kf (1 + K) = 1/ relax = 1/ f + 1/ u (4-2) In this equation, f is the folding time, while u is the unfolding time. The equilibrium constant K (= f uk k) is calculated by fractional helicity ( fH) and is a ratio of the forward and backward rate constants. When only two states, folded (F) and un folded (U) states, exists and fractional helicity of peptide 1 and peptide 2 are 12% at 348K and 15% at 372K, respectively, the K values of peptide 1 (K1) and peptide 2 (K2) are calculated with the final temperatures of T-jump simulation as the target temperatures, according to,

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81 Figure 4-8. Average of CD222 () of temperature jump (T-jump) simulation data (collision frequency = 1.0ps -1). A) Peptide 1: black, B) Pep tide 2: red with fitting curves, respectively. The unit of molar ellipticity is deg cm2 dmol-1 of both peptides. A B

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82 3 7 ] [ ] [1 f uk k F U K and 7 5 ] [ ] [2 f uk k F U K (4-3) The relaxation time and folding time obtained from experimental and computational results are compared in the Table 4-1. The computed values are 0.20ns for peptide 1 and 0.17ns for peptide 2, which correspond to a folding time f=1.65ns of peptide 1 and f=1.16ns of peptide 2, respectively. Thus, the relaxation time of two peptides differ by a factor of ~1.2, which is the ratio ) 17 0 /( ) 20 0 ( ns ns. Similarly,, difference in experimental results was by a factor of ~1.1, which is the ratio ) 204 /( ) 222 ( ns ns.49 This result validates our methodology for the kinetic studies. In addition, ratio of folding time between two peptides is ~1.4 from the computation and ~2.9 from the experiments.49 The results show that peptide 2 with higher helicity folds faster than peptide 1, indicating correlation between overall helix stability and folding time. Table 4-1. Relaxation and folding times of pe ptide 1 and peptide 2 from experimental and computational data. Relaxation and folding times (ns) (exp.*) (com.**) f (exp.*) f (com.**) Peptide 1 222 0.20 ~ 2000 1.65 Peptide 2 204 0.17 ~ 700 1.16 Experimental data obtained at 11C (284K) by Wang et al. 49 ** Computational data obtained by calculated CD () of T-jump simulations (peptide1 at 348K and peptide 2 at 372K) in Figure 4-7. The time scales seen in theoretical data are mu ch shorter than in experimental data. This mainly is due to the implicit solvent model used for the calculation, which accelerates the folding and unfolding times. The solvent effect on the folding time including different frictional coefficients will be discussed in the next section. 4.3.4 Effects of Frictional Coefficients In order to generate data cl oser to the explicit solvent environment and also to show independence of friction versus folding time, we studied the effect of fr ictional coefficient. To investigate the frictional effect on the rate of the protein folding, we performed the T-jump

PAGE 83

83 simulations (details in section 4.3.3) using Langevin dynamics w ith several collision frequencies ( =1.0, 5.0, 10.0, and 20.0ps-1), for both peptides. Figure 4-9 pr esents the simulation results in the plot of average values of CD at 222nm () versus time. As we already mentioned in Chapter 3 and section 4.3.3, all traces for both peptides were fitted to only a single parameter ( ) with the same values of C1 and C2 from Equation 3-4 since we have complete knowledge of what the system looks like at t=0 and t= The calculated relaxation data in Figure 4-9 are then fitted using a single exponential function, resulting in folding times ( f) from Equation 4-2. The folding times are plotted as a function of collision frequency ( ) in Figure 4-10 and th e values of folding time for two peptides are compared in the Table 4-2. Table 4-2. Folding times of the two peptides at different collision frequencies with Langevin dynamics. Folding times ( f, ns) Collision frequency, (ps-1) Peptide 1 Peptide 2 1.0 1.65 1.16 5.0 4.45 4.11 10.0 8.94 5.94 20.0 16.7 11.2 The folding time with higher friction is much longer than lower friction for both peptides. The folding time is approximately linear to the collision frequency (Figure 4-10). However, it does not seem to be closely fitted to the Kramers relation1, ) (1 f fk The Kramers model is not satisfactory, as its y-intercept is fixed to be zero.23,177 Therefore, the results in Figure 4-10 were further analyzed using the follo wing linear model (a modified simple Kramers model, Equation 4-4) with a and b as variables which was suggested by Qiu et al. ;23 b af (4-4) 1 Kramers equation in its simplest form could be described as, ) / exp( ) 2 / ( RT E kb a where a and b are the curvature of the po tential energy surface at the bottom and top of the barrier, respectively, E is the size of the energy barrier to conformational change, R is the gas constant, T is the temperature, and is the friction of the solvent.

PAGE 84

84 Figure 4 -9. Comparisons of diffe rent collision frequencies, =1.0, 5.0, 10.0, and 20.0ps-1. A) Peptide1, B) Peptide 2 using average of CD222 () of T-jump simulation data, respectively. The unit of molar ellipticity is deg cm2 dmol-1 of both peptides. A B

PAGE 85

85 The plots were well fitted with R2 = 0.999 for peptide 1 and R2 = 0.99 for peptide 2 (values of a and b are given in Figure 4-11). This equation describes that the Kramers relationship f is preserved except the positiv e value of y-intercept ( a ) whereas a = 0 in simple Kramers model. Ansari et al. 187 also suggested the relaxation rate equation using a modified Kramers model according to;2 sk 1 (4-5) Figure 4-10. Comparisons of the folding times at different collision frequencies of two peptides with error bars and linear fits (dotted lines). The folding times and associate errors were calculated from fitting curves of average of CD222 () from Figure 4-9. 2 The rate equation can be written as ) / exp(0RT E C ks where R is the gas constant, T is the temperature, E0 is the average height of the potential en ergy barrier, C is adjustable parameter, is the protein friction, and s is the solvent friction.

PAGE 86

86 Figure 4-11. Friction dependence of peptide 1 (marked as 1) and peptide 2 (marked as 2) by Tjump simulations. A) Y-intercept, B) Slope of folding time ( ) obtained from linear fits in Figure 4-10. A B

PAGE 87

87 where is the protein friction (o r internal friction) and s is the solvent friction. Ansari et al. 187 mentioned that the folding rate can be dete rmined as the sum of the solvent friction (s) and the protein friction (or internal friction, ) from Equation 4-5. Related with Ansari model,187 we could suggest that y-intercept (a) is correlated with internal friction. The simple Kramers model ( f) is well associated with the analysis of the protein folding kinetics when folding is relatively slow (fk/ 1~ms).23 However, our model shows better agreement with Equation 4-4, instead of simple Kramer model, and thus internal friction can influence the very fast folding reactions since the values of a and b in Figure 4-11 show fastest (nanoseconds) timescales. Zagrovic and Pande 95 simulated TrpCage (TC5b) us ing the distributed computing technique and analyzed the dependence of solvent viscosity. According to their results, if the protein initially collapses into a random confor mation and this continues until the protein folds (unfolded random folded), the first step is mainly controlled by the solvent friction and the second step is controlled only by the internal friction. Thus, inte rnal friction would play a major role for the folding rate when the se cond step becomes the rate-limiting step. The time scale of present folding (Figure 410) is much shorter than that from the experiments (microsecond scale).23 This is due to the difference in the composition of the solvent between the experiment and simulation. The simu lated results, however, show similar ratio of folding times between two peptides to the experimental results. 4.4 Conclusion We studied folding kinetics of two rela ted 14-residue peptides by using T-jump simulations that was recently introduced as a new computational methodology. Helicity and folding kinetics of two alanine-based peptid es were investigated and compared with

PAGE 88

88 experimental data. Very effici ent sampling techniques, such as REMD, are used to populate equilibrium ensembles. Multiplexed MD simulations were then run to obtain kinetic information, particularly the non-equilibrium relaxation of the populations. We found that peptide 2 having more helicity, folds faster than peptide 1. The ratio of relaxation time of two peptides differed by a factor of ~1.2, while the corresponding experi mental results were ~1.1. Therefore, our new methodology seems in good agreement with experimental data. The effect of friction on the protein folding was also studied usi ng Langevin dynamics. We performed data fitting, using the modified Kr amers linear model, on simulated results from different frictional coefficients The observed nanosecond time scale of folding for both peptides indicates that internal friction can influence th e very fast folding reactions. The composition of solvent made a significant effect on the folding ki netics, such that the nanosecond time scale in the simulation was obtained for the micr osecond time scale in the experiments.

PAGE 89

89 CHAPTER 5 CONCLUSIONS Simulations opened the way to investigations of new fields and possibi lities since Karplus et al. first introduced MD simula tion of biomolecular systems.18 MD simulations can provide ultimate details of individual atomic motions including pathways while experimental results usually show only averaged structure. However, approximations of molecular interactions are one of the well known drawbacks of MD simulati ons. Therefore, combining experimental and computational results address in complementing and overcoming limitations of both approaches. Moreover, understanding the structure, kine tics and thermodynamics of protein folding remains one of the unsolved problems for both computational and expe rimental biophysists. Laser-induced T-jump is one of the most popul ar experimental methods in protein folding research since the timescale of T-jump extends from nanoseconds to milliseconds, which is an appropriate range for stud ies of folding kinetics. In the first project, we introduced the computational T-jump: the temperature increases suddenly, creating the relaxation of the ensemb le from one equilibrium distribution (at low temperature) to another one (at high temperat ure) by using proper conformational sampling method, which is REMD. Our method is design ed and explored closely resembling to an experimental T-jump while the usual distributed computing procedure uses multiplexed MD runs to study folding. The alanine polypeptide (ACE(ALA)20NME) was used as our model system in this project as a proof of principle and description of the methodology. A set of folding/unfolding time ( F=47.5ns and U=155ns) was calculated from fitting the results of CD spectra and the changes of conformational dist ributions were shown through computation of NMR chemical shifts.

PAGE 90

90 The second project extended the method to real proteins to get consistent results using Langevin dynamics which includes frictional eff ects and random forces. In this project two related 14-residue pe ptides were chosen and compared with the experimental data. The ratio of relaxation times of the two peptid es is by a factor of ~1.2, wh ile the corresponding experimental result was ~1.1.

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104 BIOGRAPHICAL SKETCH Seonah Kim, was born in Daegu, South Ko rea. She went to college at Yeoungnam University, majoring in industr ial chemistry and graduated w ith a bachelors degree. In August 1999, she started graduate school at the Un iversity of Houston, in the Computer Science Department and earned her master's degree. She moved to Gainesville, Florida, in July 2003. There she entered the University of Florida's Ph. D. program in chemistry, specializing in computational chemistry.