Modeling the Physical Behavior of Helical Polymers

MODELING THE PHYSICAL BEHAVIOR OF HELICAL POLYMERS

A Dissertation

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulﬁllment

of the Requirements for the Degree

Doctor of Philosophy

Vikas Varshney

August, 2006 MODELING THE PHYSICAL BEHAVIOR OF HELICAL POLYMERS

Vikas Varshney

Dissertation

Approved: Accepted:

Advisor Department Chair Dr. Gustavo A. Carri Dr. Mark D. Foster

Committee Member Dean of the College Dr. Ali Dhinojwala Dr. Frank N. Kelley

Committee Member Dean of the Graduate School Dr. Wayne L. Mattice Dr. George R. Newkome

Committee Member Date Dr. Alexei P. Sokolov

Committee Member Dr. Jutta Luettmer-Strathmann

ii ABSTRACT

The main objective of this dissertation is to develop a minimal model for helical conformations present in semi-ﬂexible polymers and to study their consequences on the physical behavior of such polymers subjected to various external stimuli. The polymer chain is modeled with the Freely Rotating Chain model while the helical structure is realized using the geometrical concept of torsion of a curve. Thereafter, the changes in the conformations of polymer chain under various kinds of external stimuli such as long-range interactions and external mechanical forces are investigated using Monte Carlo simulations based on the Wang-Landau sampling algorithm.

The modeling approach is shown to successfully capture the phyiscal properties of the helix-coil transition in a semi-flexible polymer chain as a function of temperature and provides an opportunity to explore real-space characteristics of the transition, in addition to its thermodynamic properties. Furthermore, the model successfully captures the characteristic cooperativity of the transition and is shown to be in accordance with many experimental observations and theoretical predictions for helical polymers such as homopolypeptides. iii The minimal model is extended further to include long range attractive interactions among beads which lead to the collapse of the polymer chain. However, such collapse interferes with the presence of helical structure due to torsional energy, leading to a rich phase behavior of chain configurations where both interactions are significant. This phase behavior for various chain lengths is investigated in terms of temperature and strength of attractive interactions and the results suggest that the emergence of many stable chain configurations is the consequence of the coupling between the helix-coil and coil-globule transitions.

The original model is also extended to include the effect of an external mechanical force to study the force-elongation characteristics of helical, semi-flexible polymers. The analysis suggests a non-monotonic behavior of helix-coil transition temperature as a function of force which is attributed to the change in the nature of the helix-coil transition which becomes a helix-extended coil transition for strong forces. Moreover, force-elongation curves at constant temperature display three different behaviors depending on the temperature of the system. At temperatures below or slightly above the helix-coil transition temperature, the force-elongation curve shows one or two coexistence regions, respectively. In these regions, helical sequences and random coil domains coexist while the strength of the force and temperature determine the fraction of the polymer adopting each conformation. At high temperatures, our model recovers the elastic behavior of a random coil.

iv ACKNOWLEDGEMENTS

I would like to express my thankfulness towards my advisor, Dr. Gustavo A. Carri, without whose excellent guidance, continuous support and encouragement, I could not think of writing this dissertation. Dr. Carri has truly been an inspiration for me and provided me the wonderful opportunity of working and learning from him throughout the course of my studies at The University of Akron.

I would also like to express my gratitude towards my committee members:

Dr. Ali Dhinojwala, Dr. Alexei Sokolov, Dr. Wayne Mattice and Dr. Jutta Luettmer-

Strathmann for their continuous guidance along with their helpful, valuable discus- sions and suggestions during the course of my studies. I would also like to express my special thanks to Dr. Dhinojwala for providing me an excellent opportunity to work under his guidance. In addition, I would also like to thank National Science

Foundation, Ohio Board of Regents, Petroleum Research Fund and The University of Akron for their ﬁnancial support.

I would also like to express my appreciation to my group members (both Dr.

Carri and Dr. Dhinojwala’s group) especially Dr. Dirama, my housemates: Shishir,

Ani and Rajesh, and my friend Sudheer (Duke University) for being always with

v me in the good as well as bad times during my stay in Akron. I would also like to thank Dr. Rangwalla for providing the dissertation writing format in form of LATEX template.

Last but not least, I would like to express my whole-hearted gratitude towards my parents Shri Madan Gopal Varshney and Smt. Veena Varshney along with my brother Akash for being extremely supportive throughout my life; without their unconditional love and guidance I would not be standing on my feet.

vi TABLE OF CONTENTS

Page

LISTOFFIGURES ...... ix

CHAPTER

I. INTRODUCTION ...... 1

II. SIMULATIONMODELANDPROTOCOL ...... 8

2.1 A brief overview of models for the simulation of helical polymers . 8

2.2 Thebasicmodel...... 10

2.3 SimulationMethodology ...... 15

2.4 Extension of the basic model to study the collapse of a helical polymer ...... 25 2.5 Extension of the basic model to study the eﬀect of an external mechanical force on a helical polymer ...... 31 III. THE HELIX-COIL TRANSITION ...... 36

3.1 Introduction ...... 36

3.2 EstimationoftheDensityofStates ...... 52

3.3 ConﬁgurationalProperties ...... 55

3.4 ConformationalProperties ...... 60

3.5 ThermodynamicProperties ...... 69

vii 3.6 Eﬀect of interfacial penalty ...... 74

3.7 Conclusions ...... 77

IV. COLLAPSE OF A HELICAL POLYMER: INTERPLAY BE- TWEEN SECONDARY AND TERTIARY INTERACTIONS . . . . 80 4.1 Introduction ...... 80

4.2 DensityofStates...... 84

4.3 StateDiagrams ...... 85

4.4 Chain length eﬀects on the conﬁgurational, conformational and thermodynamicproperties ...... 92 4.5 Conclusions ...... 108

V. STRETCHING A HELICAL POLYMER: INTERPLAY BE- TWEEN THE HELIX-COIL TRANSITION AND AN EXTER- NAL MECHANICAL FORCE ...... 110 5.1 Introduction ...... 110

5.2 ConstantForceScenario ...... 117

5.3 ConstantTemperatureScenario ...... 121

5.4 Conclusions ...... 137

VI. CONCLUSIONS ...... 140

BIBLIOGRAPHY ...... 142

viii LIST OF FIGURES

Figure Page

1.1 Schematic of a helical curve carved through a cylinder...... 1

1.2 Schematic depicting various simulation methodologies with the respective length and time scales where they are applicable [6]...... 4 2.1 Schematic of our coarse-grained model...... 11

2.2 A schematic of the energy landscape as a function of a coordinate representing the configurations of the chain...... 17 2.3 Bin distribution for the configurations as a function of the parameters NH and NS. N corresponds to the number of beads in the polymerchain...... 20 2.4 Bin distribution for the configurations generated during the pre-runs as a function of the parameters NH and λ. N corresponds to the numberofbeadsinthepolymerchain...... 27 2.5 Applicationofforce...... 32

2.6 Bin distribution for the conﬁgurations as a function of the parameters NH and Rx...... 33 3.1 Schematic of the helix-coil transition [32]...... 37

3.2 The speciﬁc rotation of poly(γ-benzyl-L-glutamate)( ) and poly(β- benzyl-L-aspartate) ( )at546nm[34]...... ◦ 38 •

ix 3.3 Molecular weight dependence of the helix-coil transition shown as speciﬁc rotation at wavelength of 436 nm as a function of temperature for poly(ǫ-carbobenzoxy L-lysine)(PBCL) in m-cresol. Nh¯ represents the degree of polymerization [37]...... 40 3.4 Structure of an α-helix...... 43

3.5 Fraction of hydrogen bonds, θ, as a function of the statistical weight, s, for various values of the initiation parameter σ [28]...... 46 3.6 A typical ﬂat energy histogram obtained from the Wang-Landau sampling algorithm for N=60beads...... 53

3.7 Logarithm of the density of states, gtrue(NH ,NS) after rescaling as obtained from the Wang-Landau sampling algorithm for N=60 beads. . 54

3.8 Plot of the mean square radius of gyration, R2 , as a function of h gi temperature, T , for diﬀerent chain lengths and interfacial penalty η=0. The parameters are ( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N•=60...... 56 ∗ 2 3.9 Plot of the mean square radius of gyration, Rg , as a function of the number of beads, N. The parameters are:h ( i) low temperature • with torsion τhelix=0.87 (+90 dihedral angle), ()low temperature with torsion τhelix=0.253 (+16.7 dihedral angle) and (N) High temperature with torsion τhelix=0.87...... 57 3.10 Plot of the radial distribution function, ρ, for the chain ends as a function of the end-to-end distance, R, and temperature, T , for N=50beads...... 58

3.11 Plot of the mean square end to end distance, R2 , as a function of temperature, T , for a chain with N =60 beads.h i Continuous line (our simulation), ( )(Nagai’smodel)...... 59 • 3.12 Plot of the fraction of helical beads (order parameter, θ ) as a function of temperature, T , for diﬀerent chain lengths and interfacialh i penalty η=0. The parameters are ( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N=60...... • 61 ∗ 3.13 Plot of the fraction of helical beads (order parameter) as a function of temperature, T , for a chain with N =50 beads. Continuous line (our simulation), ( )(Bloomﬁeld)[64]...... 63 •

x 3.14 Plot of the fraction of helical beads (order parameter) as a function of temperature, T , for a chain with N =60 beads. Continuous line (our simulation), ( ) (Nagai’s model) [31]...... 64 •

3.15 Plot of the number of helical domains, Nhd , as a function of temperature, T , for different chain lengths andh interfaciali penalty η=0. The parameters are: ( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N=60.• ...... 67 ∗ 3.16 Plot of the average length of helical domain as a function of temperature, T , for different chain lengths and interfacial penalty η=0. The parameters are ( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N=60.• ...... 68 ∗ 3.17 Plot of the entropy, S, as a function of temperature, T , for different chain lengths and interfacial penalty η=0. The parameters are ( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N=60.• . 69 ∗ 3.18 Plot of the maximum entropy, S, as a function of chain length for hightemperatures...... 71 3.19 Plot of the internal energy, U, as a function of temperature, T , for different chain lengths and interfacial penalty η=0. The parameters are: ( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N•=60...... 71 ∗ 3.20 Plot of the minimum internal energy U as a function of chain length forlowtemperatures...... 72 3.21 Plot of the heat capacity, Cv, as a function of temperature, T , for different chain lengths and interfacial penalty η=0. The parameters are ( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N•=60...... 73 ∗ 3.22 Plot of the maximum height of the heat capacity as a function of chainlength...... 74

2 3.23 Plot of mean square radius of gyration, Rg , as a function of temperature, T , for a chain with N=60 beadsh fori diﬀerent values of the interfacial penalty, η. The parameters are: ( ) η=0, () η=200, () η=600 and (N) η=1000...... • 75

xi 3.24 Plot of the order parameter, θ , as a function of temperature, T , for a chain with N=60 beadsh fori diﬀerent values of the interfacial penalty, η. The parameters are: ( ) η=0, () η=200, () η=600 and (N) η=1000...... • 76

3.25 Plot of the number of helical domains, Nhd , as a function of temperature, T , for a chain with N=60 beadsh fori diﬀerent values of the interfacial penalty, η. The parameters are: ( ) η=0, () η=200, () η=600 and (N) η=1000...... • 76 3.26 Plot of the heat capacity, Cv, as a function of temperature, T , for a chain with N=60 beads for diﬀerent values of the interfacial penalty, η. The parameters are: ( ) η=0, () η=200, () η = 600 and (N) η =1000...... • 77

4.1 Logarithm of the density of states, gtrue(NH ,λ), after rescaling as obtained from the Wang-Landau sampling algorithm for N=60 beads. . 85

4.2 State diagram for a chain with 10 beads. Helical beads (dark segments),coilbeads(lightsegments)...... 87

4.3 State diagram for a chain with 30 beads. Helical beads (dark segments),coilbeads(lightsegments)...... 88

4.4 State diagram for a chain with 45 beads. Helical beads (dark segments),coilbeads(lightsegments)...... 89

4.5 State diagram for a chain with 60 beads. Helical beads (dark segments),coilbeads(lightsegments)...... 90

2 4.6 Plot of the mean square radius of gyration, Rg , as a function of temperature, T , for various chain lengths at ǫh= 0.i The parameters are: ( ) N=10 beads, ( ) N=30 beads, () N=45 beads and () N=60beads.∗ ...... • 93

4.7 Plot of the order parameter, θ , as a function of temperature, T , for various chain lengths at ǫh=i 0. The parameters are: ( ) N=10 ∗ beads, ( ) N=30 beads, () N=45 beads and () N=60 beads. . . . . 94 •

4.8 Plot of the number of helical domains, Nhd , as a function of temperature, T , for various chain lengthsh at ǫi= 0. The parameters are: ( ) N=10 beads, ( ) N=30 beads, () N=45 beads and () N =60beads...... ∗ • 94

xii 2 4.9 Plot of the mean square radius of gyration, Rg , as a function of epsilon, ǫ, for various chain lengths in the highh temperaturei regime (500K). The parameters are: () N=30 beads, () N=45 beads and ( ) N=60beads...... 97 • 4.10 Plot of chain entropy as a function of epsilon, ǫ, for various chain lengths in the high temperature regime (500 K). The parameters are: ( ) N=30 beads, () N=45 beads and () N=60 beads...... 98 • 2 4.11 Plot of the mean square radius of gyration, Rg , as a function of epsilon, ǫ, for various chain lengths in the lowh temperaturei regime. The parameters are: ( ) N=10 beads, ( ) N=30 beads, () N=45 beads and () N=60beads.∗ ...... • 100

4.12 Plot of the order parameter, θ , as a function of epsilon, ǫ, for various chain lengths in the lowh temperaturei regime. The parameters are: ( ) N=10 beads, ( ) N=30 beads, () N=45 beads and () ∗ • N=60beads...... 100

4.13 Plot of the number of helical domains, Nhd , as a function of epsilon, ǫ, for various chain lengths in the lowh temperaturei regime. The parameters are: ( ) N=10 beads, ( ) N=30 beads, () N=45 beads and () N=60beads.∗ ...... • 102 4.14 Plot of chain entropy as a function of epsilon, ǫ, for various chain lengths in the low temperature regime. The parameters are: ( ) N=30 beads, () N=45 beads and () N=60beads...... • . . . . 107

5.1 Logarithm of the density of states, gtrue(NH ,Rx), after rescaling as obtained from the Wang-Landau sampling algorithm for N=30 beads. . 116

5.2 Plot of the mean square end-to-end distance, R2 , as a function of temperature for diﬀerent values of the appliedh i force. The parameters are: ( ) F =0, () F =100, () F =200, (N) F =500, (H) F =1000, (+) F•=2000, ( ) F =4000, and (◭) F =8000...... 118 ∗ 5.3 Plot of the transition temperature, T ∗, as a function of the applied force F ...... 119

5.4 Plot of the mean end-to-end distance, Rx , as a function of the applied force, F , for diﬀerent temperatures.h i The parameters are: ( ) T =200 K, () T =315 K and () T =500K...... 122 • 5.5 Force curve of native Xanthan in PBS [110]...... 125

xiii 5.6 Force vs. extension profiles for EMBL3 λ DNA in PBS with two different pulling velocities. (o) 1µm/s and (+) 10µm/s. X-axis represent extension and Y -axis represent Force (pN) [107]...... 126 5.7 Stretching of a cross-linked DNA in 150 mM NaCl, 10 mM tris, 1mM EDTA, pH 8.0 buffer (black), NA2.5EDTA buffer (pH 8) with Na+ concentrations of 5 mM (red), 2.5 mM (green), and 0.625 mM (blue)[119]...... 127 5.8 Force as a function of extension per base pair for the single-stranded, (thin solid line) and double-stranded (dashed-line) DNA. The bold solid line is the DNA stretching curve assuming force-induced melting. fcr is the crossover force at which both ssDNA and dsDNA have equal extension and fov is the force where the over-stretching transitionoccurs[120]...... 128

5.9 Plot of the mean square end-to-end distance, R2 , as a function of the applied force, F , for diﬀerent temperatures.h i The parameters are: ( ) T =200 K, () T =315 K and () T =500K...... 129 • 5.10 Plot of the order parameter (fraction of beads in the helical state) as a function of the applied force, F , for diﬀerent temperatures. The parameters are: ( ) T =200 K, () T =315 K and () T =500 K. . . . . 131 • 5.11 Force-elongation curve at 315 K. Continuous curve (Buhot and Halperin), △ (oursimulation)...... 132 5.12 Order parameter as a function of the applied force, F , at 315 K. Continuous curve (Buhot and Halperin after multiplication by 0.83), △ (oursimulation)...... 133 5.13 Occurrence of plateaus in the extension-force curves signaled by the crossing of free energy curves corresponding to a polypeptide chain in a pure coil form (solid line) and in a pure helical form (dashed line)[113]...... 134 5.14 Schematic of various equilibrium conformations. The arrows represent an increase of the applied force. The dashed line, continuous line and dotted line arrows represent conformational changes at low temperatures (T =200 K), intermediate temperatures (T =315 K) and high temperatures (T =500 K), respectively...... 137

xiv CHAPTER I

INTRODUCTION

In the Oxford dictionary, the term ‘HELIX’ is deﬁned as ‘a shape like a spiral or a line curved around a CYLINDER or CONE’ [1]. This shape or curve is represented in

Figure 1.1. In macromolecular systems, this interesting geometric feature exists on the length scale of Angstroms (10−10m) and nanometers. The origin of this helicity in such systems is predominantly attributed to their chemical structure and surroundings.

Figure 1.1: Schematic of a helical curve carved through a cylinder.

While in synthetic macromolecules like isotactic polypropylene, the helix is a consequence of steric hinderance between side-groups and torsional considerations [2], it is the intra-molecular hydrogen bond formation between nearby repeat units which dominates the existence of helicity in biopolymers like proteins [3]. Another example

1 is DNA (deoxyribonucleic acid) which is a biopolymer that carries all of our genetic information; in this case the helical structure arises from inter-molecular hydrogen bonds between its two strands [4].

Helical polymers have a large impact on our day to day lives. From the biological perspective, proteins and DNA are an integral part of almost every biological activity. It is well known that proteins function eﬀectively only in their folded or native states in which the α-helical structure is one of its major components. Similarly,

DNA is known to reside in cells in its most common helical B-form. The sequences in DNA contain our genetic instructions which specify our biological development [4].

Also, synthetic helical polymers such as PTFE, PEG and isotactic polypropylene have their practical importance in one form or another as products that we use in our daily lives. The presence of helical structures in so many systems has motivated the research community to study helical polymers, both synthetic and biological, from various perspectives for past ﬁve decades; a whole gamut of properties including physical characteristics, conformational changes, interactions among themselves and with the surroundings, their response to various external stimuli from the molecular to the macroscopic level, etc. have been studied.

One of the most intensively investigated class of helical polymers is known as homopolypeptides. Homopolypeptides are macromolecules which have a repeat unit of a single amino acid and present a special case of proteins; hence, providing an opportunity to understand the complex behavior of such systems by studying their

2 simpler analogues. The helical conformation of these macromolecules is stable only under certain conditions and gets disrupted when these conditions are changed or perturbed. This conformational change in homopolypeptides or in helical polymers in general is known as the helix-coil transition. Since the pioneering work on the helix-coil transition in homopolypeptides and proteins carried out in the 1950s and

60s, as beautifully compiled by Poland and Scheraga [5], a number of experimental studies have been done, not only on homopolypeptides, but also on other polymers capable of adopting the helical conformation such as chiral polyisocyanates, PTFE,

PEG, iPP, etc.

The occurrence of the helix-coil transition in such polymers is just a starting point to appreciate how this transition can inﬂuence their other physical properties.

Such properties include their mechanical stability in terms of the force-elongation behavior, shearing response, swelling behavior in case they are cross-linked, their use as the thermoplastics elastomer when they are synthesized in alterative isotactic-atactic block fashion, their response to solvent and pH, etc. Many of such studies on the helix- coil transition and related physical properties will be discussed in section 3.1 from both, experimental and theoretical, perspectives. However, it is pertinent to mention here that while the experimental studies provide an understanding of the macroscopic changes occurring during conformational changes, molecular theories serve as mathematical foundations which capture the experimentally observed behaviors and provide

3 microscopic insights into the mechanisms responsible for the observed experimental behaviors.

A third perspective, known as computer simulations or computer modeling, has often been employed to bridge between experimental observations and theoretical predictions by actually simulating the system with various models.

Figure 1.2: Schematic depicting various simulation methodologies with the respective length and time scales where they are applicable [6].

Simulations are further classiﬁed into various categories with respect to the study of length and time scale of the problem under investigation. These classi-

ﬁcations are schematically shown in ﬁgure 1.2 [6]. As far as the molecular level understanding of polymers is concerned approaches such as Quantum Mechanics,

Molecular Mechanics, and Monte Carlo Simulations are used. Quantum mechanical calculations are ﬁrst principle calculations based on electron probability distributions

4 obtained from Schrodinger’s equation [7]. These calculations are used to investigate small systems and to obtain detailed information of the local interactions between the atoms of interest. Molecular mechanics or molecular dynamics (MD) simulations are focused on the time evolution of the system in the picosecond to nanosecond time scale. This is a deterministic approach where the time evolution of the system is explored using classical Newton’s equations of motion. In MD simulations, these equations determine each newly generated configuration from the previous configuration by analyzing coordinates, velocities and forces on each atom of the system using various proposed algorithms. Once the system equilibrates, time averages of various properties of interest are calculated and analyzed [8]. Monte Carlo (MC) simulations, on the other hand, explore the potential energy landscape by stochastically generating random configurations of the system [8]. The motivation here is to capture the most probable configuration of the system under specified conditions and to observe how this most probable configuration changes as the condition are changed. There is no equation of motion in these simulations since each new configuration is generated stochastically by perturbing a small part of the system. The exploration of the potential energy surface proceeds by accepting such new configurations according to a defined acceptance criterion. Several Monte Carlo algorithms have been proposed in the literature with numerous ways to explore the potential energy surface efficiently.

In MC simulations, after generating a suﬃciently large number of new conﬁgurations,

5 also called Monte Carlo moves, equilibrium properties of the system are evaluated as ensemble averages, unlike in MD simulations where they are time averaged.

Coming back to our problem of helical polymers, any simulation study of their physical behaviors should comprise the characterization of their two possible states, the helical form and the random coil state, which are separated by a helix-coil transition. One of the most important features of this transition is its characteristic time scale. For example, it is well known that in the case of biopolymers like proteins which carry α-helical structures, this folding-unfolding transition occurs in the timescale of millisecond to seconds. The occurrence of helix-coil transition on such timescales narrows down the opportunity of exploring the physical behavior of helical polymers using computer simulations by removing the possibility of using molecular dynamics simulations. As discussed previously, since Monte Carlo simulations are time independent; they provide an opportunity to investigate the helix-coil transition and study its physical characteristics.

In order to understand the physical behavior of helical polymers, we have

ﬁrst addressed their most important characteristic: the helix-coil transition. Later on, we have extended our studies to investigate a few of the most important, fundamental problems of helical polymers where the helix-coil transition has signiﬁcant consequences. The goals of our studies are described below:

1. Building a simple and novel model to realize the helix-coil transition.

6 2. Incorporating the eﬀect of solvent in the basic model to study the collapse of helical polymers and hence, understand the possible coupling between the helix-coil and coil-globule transitions.

3. Incorporating the eﬀect of an external mechanical force in the basic model to study the force-elongation behavior of helical polymers and hence, study the competition between the helix-coil and helix-extended coil (stretched) transitions.

The basic outline of the dissertation is described as follows. Chapter 2 de- scribes our simulation protocol which includes the basic model, simulation methodology and the extension of the model to address aforementioned problems. Chap- ters 3, 4, and 5 focus on the motivation, results and discussion of the helix-coil transition, the effect of solvent and the effect of an external mechanical force, respectively. Chapter 6 concludes the dissertation with final conclusions and future outlook to address other related problems important from the perspective of the helix-coil transition.

7 CHAPTER II

SIMULATION MODEL AND PROTOCOL

2.1 A brief overview of models for the simulation of helical polymers

Over the years, helical polymers have been modeled using both atomistic and sim- pliﬁed, i.e. coarse-grained, models. From an atomistic perspective, issues such as adsorption, conformational transition, etc. have been studied for both proteins [9–12] and synthetic polymers [13,14]. In these studies, the molecule is build with atomistic resolution using its constitutive atoms which interact via electrostatic and van der

Waals interactions and moreover, are subjected to various conditions. The major shortcoming of such atomistic descriptions lies in their inherent diﬃculty to study large systems or high chain length for suﬃciently long times.

A number of coarse-grained and geometric models have been proposed to study helical polymers. In this regard, Baumgarten [15] proposed a simple theoretical model of helices with reversible helical sense by considering two nearest neighbor interactions along the chain. Similarly, Muthukumar et al. [16] studied the disen- tangling of helical DNAs by mimicking them as a pair of twisted chains on a lattice along one of its principle directions. In another model, Roas [17] modeled the helix by an electron density function in 3-dimensions to study the degree of chirality in

8 helical structures. Michels and coworkers [18] studied the helix-coil transition in self assembled polymers in which their helical structure originates from the association of discotic plates or bifunctional monomers. Yamamoto et al. [19] took a step forward to address the quite complex problem of crystallization using molecular dynamics simulations where they modiﬁed the torsional potential of polyethylene to generate helical structures which could mimic crystallization in helical polymers such as iPP

(isotactic polypropylene).

The foregoing discussion about the modeling of helical structures points to the fact that in order to address quite complex problems such as crystallization, it is more useful to use coarse-grained descriptions of the helix, yet keeping the important ingredients for helicity. In addition, it would not be incorrect to state that incorporating helicity in a ﬂexible/semiﬂexible polymer chain further adds a new dimension to the problem of interest. As mentioned before, it is also important to note that dealing with the problems at the atomistic level increases the number of degrees of freedom of the system which becomes a major concern since they increase tremendously with system size.

We have circumvented this problem by developing a minimal, coarse-grained model to study the physical behavior of helical polymers, focusing on homopolypeptides. Later on, further additions have been done to address certain speciﬁc problems as mentioned in Chapter 1. The coarse-graining process decreases the total number of possible states by removing states that diﬀer by properties with characteristic

9 length scales of the order of angstroms which are irrelevant for the large length scale behavior.

In the next few sections, our basic model along with our simulation methodology are discussed. As previously mentioned, this minimal model was constructed to capture the most important properties of helical polymers, i.e., helix-coil transition.

The basic description is followed by its speciﬁc extensions to study other physical problems such as the collapse of a helical polymer and its stretching behavior.

2.2 The basic model

Our basic model involves a single polymer chain represented as a sequence of beads through the Freely Rotating Chain Model [20]. The distance between beads was set to 1.53 in arbitrary units which also corresponds to the bead diameter. The bond length was kept ﬁxed during the course of the simulation. In addition, the bond angle was also kept ﬁxed at 109.3 degrees throughout the simulations. An illustration of our model and its ability to represent homopolypeptides is shown in Figure 2.1.

As previously mentioned in section 2.1, the initial motivation behind the development of the model was to capture the most important property of helical polymers, i.e., the helix-coil transition. Hence, the model should be built around the ideas of traditional helix-coil transition theories [5]. One experimentally observed characteristic of the helix-coil transition is its cooperative nature. The origin of this cooperativity emerges from the diﬀerent propensities of helix nucleation and helix-

10 Figure 2.1: Schematic of our coarse-grained model.

propagation along the chain and will be discussed in detail in section 3.1.3. However, in order to study the helix-coil transition along with its cooperativity, we ﬁrst needed to incorporate helicity into our model. Moreover, we also wanted our basic model to be as simple as possible so that it could be possible to extend it further to study other complex problems where the helix-coil transition has important consequences.

In order to achieve helicity in our model in a very simple and eﬃcient manner, we employed the concept of torsion.

If one imagines the bead representation of the polymer chain as the discrete version of a continuous, thread-like representation of the chain, then the torsion of the curve is a well-deﬁned mathematical quantity in both, continuous and discrete, representations of the polymer. Moreover, in the particular case of the discrete model, the torsion is related to two consecutive dihedral angles of the polymer. Equation 2.1

11 deﬁnes the torsion τ(x) of a curve parameterized by the vectorial ﬁeld r(x) which represent the polymer chain. x is the arc length parameter that can take any value in the interval [0,L], L being the contour length of the polymer chain.

(r′(x), r′′(x), r′′′(x)) τ(x)= (2.1) [(r′(x), r′′(x))] 2 | | where r′(x), r′′(x), r′′′(x) are the first, second and third order derivatives of r(x), respectively. The square brackets and parenthesis indicate vectorial and scalar triple product, (i.e. (A,B,C)=A.(BXC)), respectively. This definition of torsion is also valid for the discrete representation of the polymer chain; the only difference is that the derivatives of the field must be approximated using finite differences. For example, the first order derivative of the field on the ith bead is

r(i + 1) r(i 1) r′(i) − − (2.2) ≈ 2lk

where lk is the bond (Kuhn) length. Similar expressions are also available for the second and third order derivatives [21]. The dependence on the third order derivative implies that the torsion on the ith bead depends on the spatial locations of the beads i 2, i 1, i, i +1 and i + 2. Consequently, the torsion can be calculated for beads − − 3 to N 2 where N is the total number of beads in the polymer chain. − It is well known that a perfect helix is completely deﬁned by its two geometrical properties, namely; the pitch and the radius of the helix. This information can be translated into other mathematical variables known as curvature and torsion of the helix using standard formulas [22]. The curvature determines the bond angles

12 which are ﬁxed in our model. On the other hand, given the values of bond lengths and angles, the torsion is completely deﬁned by two consecutive dihedral angles.

A perfect helical curve has a fixed value of torsion. If we try to superimpose the beads that represent our polymer chain onto this imaginary helical curve, each bead of the chain should have a constant value of torsion. We called this value the torsion of the perfect helix, τhelix. The value τhelix was chosen to be 0.87 and corresponds to two consecutive dihedral angles of +90 degrees. In our simulations, we generated millions of configurations and for each generated configuration, we evaluated torsion on each bead. We identified the beads of the polymer chain as either helical beads or coil (not helical) beads based on following geometric criterion: ‘a bead has a helical conformation if the value of its torsion differs from the torsion of the perfect helix, τhelix, by less than a certain cutoff value, τcutoff ’. The cutoff value

τcutoff was set to 0.001. If this criterion was satisﬁed, then this helical bead carried a negative enthalpy, called C, which stabilized the helical conformation. Otherwise, the bead was assumed to be in the random coil state which was taken as the reference state of the system. The enthalpic parameter C was related to the standard parameter s of helix-coil transition theory as follows

s = exp( ∆F/k T ) (2.3) − B where ∆F = C T ∆S. Here, C provides the enthalpic contribution that arises − from the formation of a hydrogen bond. Furthermore, ∆S is the decrease in the entropy of the residue incorporated into a helical sequence due to the formation of a

13 new hydrogen bond. The origin of ∆S in our model arises from the Freely Rotating

Chain model and the constraints in the dihedral angles. In addition, both parameters

C and ∆S are negative. It can easily be deduced from the equation 2.3 that ∆F = 0 when T = Tm(= C/∆S). This temperature is known as the melting temperature.

In our case, for values of T < Tm, ∆F is negative, i.e., s > 1. In this regime, the formation of the helical state is favored. On the other hand, when T is higher than Tm,

∆F is positive, i.e., s < 1, suggesting that the preferred state at high temperatures is the coil state.

Another important parameter used in helix-coil transition theories is σ, which is a Boltzmann weight, and is related to the initiation of a new helix. It is associated with the entropic penalty for the formation of interfaces between random coil and helical domains. The fact that the polymer sacrifices additional configurational entropy for the nucleation of the helix makes the initiation an improbable event when compared to the propagation of an existing helix. This concept is further discussed in section 3.1.3. Consequently, σ is always less than unity; generally it is of the order of 10−2 to 10−4. In our model, we incorporated this initiation event as an energetic penalty for the formation of interfaces and was denoted as η which was positive by definition, therefore, it destabilized the formation of interfaces. The parameter η together with the Freely Rotating Chain model and the constraints on the dihedral angles determined the final value of the parameter σ of the traditional helix-coil transition theories. For simplicity, the parameters C and η were assumed to be constant.

14 2.3 Simulation Methodology

The initial configuration of the polymer chain was generated randomly as discussed by Mattice et. al. [20]. The first bead was placed at the origin and was not allowed to move during the course of the simulation. The initial position of the second bead was along the X-axis and at distance equal to 1.53 from the origin. The third bead was located on the X Y plane so that the bond angle was 109.3 degrees and, − moreover, the second bond had a positive projection onto the Y-axis. The positions of all other beads were computed using random dihedral angles with respect to the previous three beads. These angles were taken from a prescribed finite set of sixty four possible values, φ = 90+(mπ/32) where φ represented the dihedral angle and m varied from 0 to 63. This was done for the purpose of having a finite estimate for the density of states as explained in section 2.3.3. Pivot moves were used to change the configuration of the chain. In the pivot move, the ith bead was selected randomly and the rest of the polymer (beads i +1 to n) was rotated around the bond between beads i 1 and i by a randomly chosen angle. − The aforementioned system was simulated using Monte Carlo Simulations. In the next sub-section, a few important algorithms of Monte Carlo simulations along with their their advantages and disadvantages are discussed, concluding with the

Wang-Landau (WL) algorithm which was employed in our simulations. This discussion will be followed by the energetics of the model and a detailed description of the implementation of WL algorithm.

15 2.3.1 A Brief Overview of Monte Carlo Methods [23]

The most simple method for doing Monte Carlo simulations is known as ‘Simple Sam- pling’. Under such sampling, random numbers are generated uniformly in a defined interval and are distributed according to some pre-defined criteria. For example, by using such a simple strategy, one can deal with mathematical problems such as evaluation of multi-dimensional definite integrals which are intractable with analytical techniques. Another interesting problem with such simple sampling is to estimate the value of π numerically.

However, in order to study physical phenomena such as phase transitions,

‘Importance Sampling’ Monte Carlo techniques are employed. The main motivation behind such technique is to sample the most important phase space of the system more often, i.e., giving it importance. In importance sampling, the algorithm employed is known as the Metropolis algorithm and serves as the most fundamental algorithm to sample the phase space of a system. Due to its fundamental nature, this algorithm is discussed below.

The basic idea behind this algorithm can be put in words as ‘the product of the probability of ﬁnding the system in state n with its transition rate to state m, Wn→m is equal to the product of the probability of ﬁnding the system in state m with its transition rate to state n, Wm→n’. This expression is known as ‘detailed balance’ and can be written mathematically as shown in equation 2.4. Eventually, this expression gives rise to the acceptance probability criterion as shown in equation 2.5

16 Figure 2.2: A schematic of the energy landscape as a function of a coordinate representing the conﬁgurations of the chain.

Pn(t)Wn→m = Pm(t)Wm→n (2.4)

W → = exp( ∆E/k T ) (2.5) n m − B

The recipe of the Metropolis algorithm can be written as follows:

1. Choose an initial state.

2. Propose a new state.

3. Calculate the energy change ∆E between the current and the proposed states.

4. Generate a random number r such that 0

5. If r < exp( ∆E/k T ), accept the move. − B

17 6. Get another new state and follow 3.

After a set number of states have been tried, the properties of the system are determined and added to the statistical average which is being kept. The behavior is also pictorially shown in figure 2.2. It is clear from the figure that at sufficiently low temperatures, the system is quite susceptible of being trapped in a local energy minimum. This may sometimes lead to wrong understanding of the problem of interest.

One of the methods to circumvent this problem is known as ‘Parallel Tempering’.

In parallel tempering methods, many replicas of same system are run independently under various conditions, for example at various temperatures. In addition to normal

Metropolis criterion for each box, the state of two nearby boxes are also switched according to deﬁned acceptance criterion after certain number of Monte Carlo steps which helps in avoiding problems related to trapping in energy minima.

The methods discussed above only provide information of the system under the condition explored, for example, ﬁxed temperature or pressure. There are other types of Monte Carlo algorithms which can be used to predict the behavior of the system at a temperature other than that at which the simulation was performed.

These methods are known as ‘Reweighting Methods’. Under such methods comes sampling algorithms such as Umbrella sampling, Histogram (single, multi and broad) sampling and Multicanonical sampling. These methods are motivated to capture the probability distribution of the energy for unexplored conditions from the probability distribution obtained for the simulated conditions. The reweighting procedure may

18 be done either after the simulation is done or it may become an integral part of the simulation process itself. Among these methods, the Multicanonical algorithm serves as an eﬀective sampling scheme to avoid any trapping in energy minima without performing many simulations and hence has been employed in situations where the energy surface is quite complex [24].

Another method, known as the ‘Random Walk’ algorithm which has elements of the Multicanonical algorithm was proposed recently by Wang and Landau [25]. The similarity between both methods arise from the fact that both algorithms explore the energy landscape with the probability P (E) 1/g(E), g(E) being density of states ∝ for energy E. However, the temperature independent nature of the random walk algorithm along with its very simple implementation have made this algorithm an excellent starting point for the exploration of the energy landscape avoiding getting trapped in local energy minima. This algorithm has already been used extensively in a broad variety of systems and has been proven to be very eﬃcient in studying the folding behavior of polypeptides (please refer to section 4.1).

2.3.2 Energetic considerations of the model

In section 2.2, we briefly discussed two energetic parameters, C and η, to describe helical strength and interfacial penalty, respectively. It should be noted that these parameters are configuration independent. For the particular problem of interest, we also noted that each configuration generated during the simulation consists of two configuration-dependent parameters, complementary to the energetic parameters,

19 namely the number of beads in the helical state, NH and the number of interfaces between helical and coil domains, NS. The energy of the system is written as follows

E = C N + η N (2.6) × H × S

Figure 2.3: Bin distribution for the conﬁgurations as a function of the parameters

NH and NS. N corresponds to the number of beads in the polymer chain.

It was mentioned previously that parameters C and η were kept constant during the course of the simulation. Under such circumstances, the energy E is a function of NH and NS, and can be written as E(NH ,NS). This classification is schematically shown in Figure 2.3. In this figure, each generated configuration was

20 classiﬁed in terms of NH and NS. While shaded and black regions correspond to

‘feasible bins’, no conﬁguration is possible in unshaded regions. For example, if NH and NS are equal to 0, then bin number 1 (in black) is a feasible bin which represents an all-coil state. Similarly, black bin 2 represents a perfect helical conformation with

N = N 4 and N = 0. Along the same lines, black bin 3 corresponds to an H − S arbitrary conﬁguration in which we have 5 beads in the helical state and there are

3 interfaces between helical and coil domains. So each generated conﬁguration has a certain energy E(NH ,NS) given by equation 2.6 that was assigned to one of the shaded bins. Our ﬁnal goal was to estimate the total number of possible states in each of these shaded bins. The total number of states in each bin is commonly known as the density of states g(E). In this particular problem, the density of states g(NH ,NS) was estimated by the Wang-Landau algorithm.

2.3.3 The Wang-Landau Algorithm

In the previous sub-section, the energy of the conﬁguration was described in terms of two independent, but conﬁguration-dependent variables, NH and NS. In order to study the system of interest, we extended the original Wang-Landau algorithm to 2-dimensions as discussed by de Pablo et al. [26] and implemented it to estimate g(NH ,NS).

At the beginning of the simulation, the density of states g(NH ,NS), which was unknown a priori, was initialized to unity for all possible values of NH and NS.

The random walk was then started by changing the conﬁguration of the polymer.

21 The transition probability for switching the polymer conﬁguration from (NHi,NSi) to (NHf ,NSf ) was deﬁned by

g(NHi,NSi) P rob(NHi,NSi NHf ,NSf )= min 1, (2.7) → g(NHf ,NSf )!

Each time a move was accepted, the density of states of the new configuration was updated by multiplying the existing value by a modification factor f, i.e., g(N ,N ) g(N ,N ) x f. However, if the move was rejected, then the density Hf Sf → Hf Sf of states of the old configuration was updated, g(N ,N ) g(N ,N ) x f. This Hi Si → Hi Si modification of the density of states allowed the random walk to explore NH and NS spaces quickly and efficiently. The starting value of f was taken to be e = 2.71828, as recommended by Wang and Landau. After a move was completed, the corresponding histogram H(NH ,NS) was updated along with the modification of the density of states. Once the histogram was ‘flat’ within some tolerance, the value of f was modified using the recommendation of Wang and Landau: fnew = √fcurrent. At this point, the histogram was reset to zero and the above procedure was started again with the updated modification factor. This procedure was repeated until the value of f was very close to 1. We stopped our simulations when f 1 became smaller than − 10−7.

Once the relative density of states is known, they were further rescaled to get the true density of states g(NH ,NS) as follows

g(NH ,NS) gtrue(NH ,NS)= XTotconf (2.8) g(NH ,NS) NH ,NS X

22 where T otconf is total number of possible conﬁgurations that the chain could take which, for our model, was 64N−3. Using the density of states, various conformational, conﬁgurational and thermodynamic quantities were calculated using standard for- mulae from statistical mechanics. The mathematical expressions for the canonical partition function, Helmholtz free energy, internal energy, entropy and heat capacity are listed below

Z(T,η)= g (N ,N ) e−β(NH ×C+NS ×η) (2.9) true H S × NH ,NS X

F (T,η)= T ln g (N ,N ) e−β(NH ×C+NS ×η) (2.10) − true H S × NH ,NS X (N C + N η) g (N ,N ).e−β(NH ×C+NS ×η) H × S × × true H S NH ,NS U(T,η)= X (2.11) g (N ,N ) e−β(NH ×C+NS ×η) true H S × NH ,NS X U(T,C,η) F (T,C,η) S(T,η)= − (2.12) T

U 2 U 2 Cv(T,η)= h iT −h iT (2.13) T 2 where F (T,η), U(T,η), S(T,η) and Cv(T,η) are in units of Boltzmann constant, kB. It should be noted here that properties such as free energy F (T,η), and entropy

S(T,η), which directly depend upon the lnZ are aﬀected by choice of dihedral angles,

φ, used in the simulations. The choice of φ ultimately decides the estimation of density of states and hence aﬀect those properties directly which are proportional to density of states or partition function.

23 Apart from these thermodynamic quantities, the ensemble average of other quantities of interest were calculated using following equation.

A(N ,N ) g (N ,N ) e−β(NH ×C+NS ×η) H S × true H S × NH ,NS A(T,η) = X (2.14) h i g (N ,N ) e−β(NH ×C+NS ×η) true H S × NH ,NS X

For the evaluation of any desired property A(NH ,NS) as mentioned in previous equation, production run for approximately 5 million accepted moves were performed for each chain length using Wang-Landau criterion. Thereafter, each con-

ﬁgurational and conformational property was then ensemble averaged for each pair of feasible NH and NS before incorporating into equation 2.14. Few of such properties include mean square end-to-end distance R2 , mean square radius of gyration Rg2 , h i h i average number of helical domains N , etc. h hdi

In the remaining two sections, the extension of our minimal model is discussed to address some fundamental problems where helix-coil transition has important consequences. Here, it is pertinent to mention that for all subsequent studies, the effect of the number of interfaces between helical and coil domains, NS, was not incorporated, explicitly. It will become clear in section 3.4 that we were able to realize helix-coil transition irrespective of NS. For all subsequent studies, NS have been replaced by new different parameter. The definition of each parameter will become clear while discussing each study separately.

24 2.4 Extension of the basic model to study the collapse of a helical polymer

In order to study the collapse of helical polymers, non-local attractive interactions were taken into account in addition to local torsional energetics. The modeling of non-local interactions and the modiﬁed simulation methodology are discussed below.

2.4.1 Modeling non-local interactions

The eﬀect of long-range, inter-bead interactions was described in an implicit manner through a modiﬁed Lennard-Jones potential of the form

12 6 σb σb 4.ǫ. if rij σb  rij − rij ≥ V = ! (2.15) ij    if r < σ ∞ ij b    where rij is the distance between beads i and j, ǫ is the strength of the interaction and σ is the bead diameter. The potential was summed over all pairs (1 2 and 1 3 b − − interactions were not included) to obtain the total interaction energy for a particular configuration of the chain. The excluded volume interactions were considered through hard core potentials as shown in equation 2.15, i.e. all the configurations with any rij less than σ were discarded. Hence, it is safe to assume that only negative contributions were considered while calculating the total interaction energy, hence, allowing us to concentrate only on attractive interactions as per equation 2.15. Those configurations with any rij less than σb were discarded due to hard core repulsion.

25 We note that the quantity Vij/ǫ depends only on the configuration of i j>i X X the chain and is independent of ǫ. Thus, we defined a compactness parameter λ as follows, σ 6 σ 12 λ = V /ǫ = 4 b b (2.16) − ij × r − r i j>i ij ij ! X X By definition large values of λ corresponded to more compact configurations and consequently, λ is capable of detecting collapsed conformations of the chain.

2.4.2 Simulation Methodology

Let us consider an arbitrary configuration of the polymer chain which was generated during the course of the simulation. This configuration contained two relevant parameters of interest: the number of beads in the helical state, NH , and the compactness parameter, λ, that quantifies the global packing of the chain. The former is a measure of local interactions while the latter is a consequence of the non-local or tertiary interactions. Then, the energy for this configuration is written as

E = C N ǫ λ (2.17) × H − ×

It is clear from equation 2.17 that the energy of a configuration depends on NH and λ, and can be written as E = E(NH ,λ). As mentioned previously the parameter C quantifies the stabilization of the helix while ǫ determines the strength of the van der Waals interactions, as defined in equation 2.15. Each energy state was associated with a number of possible configurational states; thus, the total number of configurational states per unit energy, defined as the density of states (DOS),

26 g(E) could be written as a function of NH and λ. We denoted it as g(NH ,λ). This distribution of the DOS on NH and λ is illustrated in figure 2.4. In this schematic, which consists of many small squares, the horizontal axis corresponds to the number of beads in the helical state, NH , which increases stepwise, while the vertical axis corresponds to the total attractive interaction for each configuration as quantified by

λ, which is a continuous variable. We discretized the vertical axis by binning it in intervals of width equal to α. The values of α were taken as 1, 1, 4 and 5 for the study of chain lengths of 10, 30, 45 and 60, respectively. This coarse-graining of α had to be done due to the computational limitations in the case of large chain lengths.

Figure 2.4: Bin distribution for the conﬁgurations generated during the pre-runs as a function of the parameters NH and λ. N corresponds to the number of beads in the polymer chain.

27 Similar to figure 2.3, this figure was also classified into shaded ‘feasible’ and unshaded ‘non-feasible’ bins. For the determination of feasible and unfeasible bins for this particular problem, pre-simulations were run for approximately 20 million accepted moves in order to explore the energy space in terms of NH and λ using bin widths α as described in the previous paragraph. Once the energy space was explored, the system was simulated using standard Wang-Landau implementation. For chain lengths of 10, 30, 45 and 60, the maximum attractive interaction was found to be approximately 12, 72, 104 and 140. Hence, the total number of bins in λ dimension to be used were 12, 72, 26 and 28 for corresponding chain lengths, respectivelys.

The recognition of such feasible bins is shown in figure 2.4. In this figure, while black bin 1 corresponds to an all-helical state which can have only one value of λ (since the perfect helix has only one possible configuration), black bins 2 and 3 correspond to all-coil configurations with relatively different packing configurations.

Similarly any general conﬁguration can be put in any of the shaded bins in ﬁgure 2.4.

For example, if the configuration of the chain with length equal to 45 beads has four helical beads and λ = 17.57, then this configuration would be assigned to the black bin 4. We estimated the density of states for our model to address this problem using the Wang-Landau sampling algorithm as discussed in section 2.3.3, the only difference being that interface parameter NS and interfacial penalty η were replaced by the compactness parameter λ and the van der waals interaction parameter ǫ,

28 respectively. For this study, the acceptance criterion has the form

g(NHi,λi) P rob(NHi,λi NHf ,λf )= min 1, (2.18) → g(NHf ,λf )!

Here, it is worth mentioning that for chain lengths equal to 45 and 60 beads, the simulations were performed using the parallel implementation of the Wang-

Landau algorithm as suggested by Schulz and collaborators [27] which is described below.

First of all, the bin diagram shown in figure 2.4 was divided into 8 and 24 horizontal windows of energy for chain lengths of 45 and 60 beads, respectively, in such a way that each window contained a certain range of the number of helical beads and the whole range for the compactness parameter λ. Moreover, parts of consecutive windows were overlapped with each other. After that, each window was simulated using the Wang-Landau algorithm independently as suggested by Schulz [27] considering boundary effects to estimate density of states, (DOS). When the density of states for each window was estimated, the overlapping portion of consecutive windows was used for overall match up and rescaling of the global density of states for the whole range of NH and compactness parameter λ. It should be noted here that the overlapping of the windows was performed after the final stage of the simulation, i.e., when the value of convergence parameter f became close to 1.00.

Once the DOS was estimated from the Wang-Landau algorithm, it was further rescaled to get the true density of states, gtrue(NH ,λ), as discussed in equation 2.8. It should be noted here that equation 2.8 was applied over only bins which were visited

29 during the coarse of the simulation. However, it is possible that the new bins could be explored if the simulations were to run 10 times longer. Similar to section 2.3.3, once the DOS, gtrue(NH ,λ) was known, various ensemble averages of thermodynamic quantities were calculated as follows:

Z(T, ǫ)= g (N ,λ) e−β(NH ×C−λ×ǫ) (2.19) true H × NXH ,λ

F (T, ǫ)= T ln g (N ,λ) e−β(NH ×C−λ×ǫ) (2.20) − true H × NXH ,λ (N C λ ǫ) g (N ,λ).e−β(NH ×C−λ×ǫ) H × − × × true H NH ,λ U(T, ǫ)= X (2.21) g (N ,λ) e−β(NH ×C−λ×ǫ) true H × NXH ,λ U(T,C,λ) F (T,C,λ) S(T, ǫ)= − (2.22) T

U 2 U 2 Cv(T, ǫ)= h iT −h iT (2.23) T 2

AA(N ,λ) g (N ,λ) e−β(NH ×C−λ×ǫ) H × true H × NH ,λ AA(T, ǫ) = X (2.24) h i g (N ,λ) e−β(NH ×C−λ×ǫ) true H × NXH ,λ where Z(T, ǫ), F (T, ǫ), U(T, ǫ), S(T, ǫ) and Cv(T, ǫ) are the partition function, Helmholtz free energy, internal energy, entropy and heat capacity, respectively. Moreover, eq. 2.24 was used to calculate various conﬁgurational and conformational quantities

AA(T, ǫ) of interest. For this problem, it should be noted that the configurations of h i the chain in terms of NH and λ were stored during the simulations with the motivation to capture as many different configurations of the system as possible. Moreover,

30 the quantity AA(NH ,λ) used in equation 2.24 are essentially ensemble averages for each feasible pair of NH and λ.

2.5 Extension of the basic model to study the eﬀect of an external mechanical force

on a helical polymer

The modeling of the external mechanical force and the simulation methodology employed are described henceforth.

2.5.1 Modeling Force

The application of force on the polymer chain is schematically shown in ﬁgure 2.5.

As shown in this figure, one end of the polymer chain was fixed at origin while the polymer chain was stretched by applying force at its other end. We set the force to point in the positive X-direction and investigated the equilibrium conformations of the polymer chain in terms of configurational, conformational and thermodynamic properties.

2.5.2 Simulation Methodology

Let us consider an arbitrary conﬁguration of the polymer chain which was generated through the course of the simulation. For this case, each conﬁguration had two relevant parameters: the number of beads in the helical state, NH , and the projection of the end-to-end vector along the direction of the applied mechanical force,

Rx. Note that since the force was modeled to act in the positive X-direction, only

31 Figure 2.5: Application of force.

the X-component of end-to-end vector was relevant. If we imagine that a random conﬁguration of the chain was somehow generated by the application of a certain force acting along positive X-direction, then the energy for this conﬁguration could be written as follows

E = C N F R (2.25) × H − | | × x which implies that the contribution of the force term is zero when F =0or Rx = 0. In this case, the energy only depends on NH . Similarly, if NH = 0 for any conﬁguration, then the energy depends only on Rx.

It is clear from eq. 2.25 that, for a constant value of force F , the energy E of a conﬁguration depends on NH and Rx, and can be written as E = E(NH ,Rx).

Hence, the density of states (DOS) for this energy, g(E), is also a function of NH and Rx, and can be denoted as g(NH ,Rx). Furthermore, provided that the force acts

32 along the positive x-direction, we can safely assume that g(N ,R ) = g(N , R ) H x H − x due to the symmetric nature of the system with respect to Y Z plane. For this − reason, we only considered the modulus of Rx when estimating the DOS. However, we used both g(N ,R ) and g(N , R ) separately when calculating various equilibrium H x H − x properties, as explained below.

Figure 2.6: Bin distribution for the conﬁgurations as a function of the parameters

NH and Rx.

The dependence of the DOS on NH and Rx is illustrated more clearly in

ﬁgure 2.6. In this schematic, the horizontal axis corresponds to number of beads in the helical state, NH , which increases discretely, while the vertical axis is continuous because Rx is a real variable. So, we discretized the vertical axis by binning it in

33 intervals of width equal to 0.5 in the same arbitrary units of the bond length. which resulted in total number of bins in Rx dimension to be twice of extended chain length.

Therefore, any configuration generated by the simulation was allocated in one of the squares shown in figure 2.6. For example, if the configuration of the chain has four helical beads and Rx = 2.34, then this configuration would be assigned to the black shaded bin.

The system was simulated using the Wang-Landau algorithm as mentioned in section 2.3.3 with only one modiﬁcation in the acceptance criterion. For this study, the acceptance criterion was

g(NHi,Rxi) P rob(NHi,Rxi NHf ,Rxf )= min 1, (2.26) → g(NHf ,Rxf )!

Once the density of states g(NH ,Rx) was estimated, it was further trans- formed into the true density of states as mentioned in equation 2.8 as discussed in section 2.4.2. Thereafter, various thermodynamic and conﬁgurational properties were evaluated as follows:

Z(T,F )= g (N ,R ) (e1 + e2) (2.27) true H x × NH ,Rx X A(T,F )= T ln g (N ,R ) (e1 + e2) (2.28) − true H x × NH ,Rx X g (N ,R ) (N C R F ) e1 +(N C + R F ) e2 true H x × H × − x × × H × x × × NH ,Rx U(T,F )= X g (N ,R ) (e1 + e2) true H x × NH ,Rx X (2.29) U(T,C,F ) A(T,C,F ) S(T,F )= − (2.30) T

34 U 2 U 2 Cv(T,F )= h iT −h iT (2.31) T 2 g (N ,R ) AA(N ,R ) e1 + AA(N , R ) e2 true H x × H x × H − x × NH ,Rx AA(T,F ) = X h i g (N ,R ) (e1 + e2) true H x × NH ,Rx X (2.32) where e1 = e−β(NH ×C−Rx×F ) and e2 = e−β(NH ×C+Rx×F )

As described previously, Z(T,F ), A(T,F ), U(T,F ), S(T,F ) and Cv(T,F ) are the partition function, Hemholtz free energy, internal energy, entropy and heat capacity, respectively; and were described in units of Boltzmann constant, kB. For the sake of completeness, it should be noted that eq. 2.32 could be used to calculate various conﬁgurational and conformational quantities AA(T,F ) of interest. It should h i be noted that for the evaluation of any desired property A(NH ,Rx) as mentioned in previous equation, production run for approximately 5 to 10 million accepted moves were performed for each chain length using Wang-Landau criterion. Thereafter, each conﬁgurational and conformational property was then ensemble averaged for each pair of feasible NH and Rx before incorporating into previous equation.

35 CHAPTER III

THE HELIX-COIL TRANSITION

3.1 Introduction

The terms ‘helix’ and ‘random coil’ are two of the most important conformational states in biopolymers. If the free energy of the system is controlled by entropy, the macromolecule exists in the random coil state; whereas the helical conformation is favored whenever the enthaplic contributions dominate the free energy. These enthalpic contributions arise from the intra-chain hydrogen bonded interactions which stabilize the helical conformation of the macromolecule. This transition from one state to another has important consequences on the conformation of the chain leading to the

‘helix-coil’ transition [5]. Since its discovery in late 50’s [28] and early 60s [29–31], the research community has studied the helix-coil transition and its consequences due to its presence in biopolymers; this transition has been an active area of research since then. Figure 3.1 shows the schematic of the conformational change during the helix-coil transition. Recently, Teramoto has reviewed this transition [32].

3.1.1 Helix-Coil Transition: Experimental Perspective

Experimentally, both the helical and coil conformations of various polypeptides i.e., poly (L-amino acid)s have been studied through X-ray diﬀraction, light scattering

36 Figure 3.1: Schematic of the helix-coil transition [32].

and hydrodynamic studies [33]. However, the transition region, connecting both conformations is often characterized in terms of their absorption difference of left- and right-handed circularly polarized light through circular dichroism measurements as a function of variables such as temperature, solvent quality and others. Some of the earliest experimental findings are shown in figures 3.2 and 3.3 where the optical activity of a few polypeptides is plotted as the function of solvent quality and temperature, respectively.

It is known from X-ray diﬀraction studies that poly(γ-benzyl-L-glutamate) exists in an α-helix (discussed in next section) form in its solid state [35]. However, in solution its conformational behavior is quite sensitive to the solvent used. For example, when this polypeptide is dissolved in strongly interacting solvents, such as dichloro-acetic acid (DCA), the helical structure breaks down completely [34], while

37 Figure 3.2: The speciﬁc rotation of poly(γ-benzyl-L-glutamate)( ) and poly(β- ◦ benzyl-L-aspartate) ( ) at 546 nm [34]. •

38 this structure remains intact in non-interacting solvents like chloroform [36]. In this regard, ﬁgure 3.2 shows the conformational changes in poly(γ-benzyl-L-glutamate) in

DCA-chloroform mixtures characterized through the specific rotation measurements at the wavelength of 546 nm. It is clear from the figure that around 68% volume fraction of DCA, there is a sharp change in specific rotation behavior of poly(γ-benzyl-

L-glutamate) which is associated with drastic conformational changes accompanying the helix to coil transition as the volume fraction of DCA is increased.

As far as the temperature dependence of the transition is concerned, it varies from system to system. In general, the α-helix is disrupted as the temperature is increased. However, there are certain polypeptides which show opposite trends too, also known as inverse transition. One of such behavior is shown in figure 3.3 where the specific rotation of poly(ǫ-carbobenzoxy L-lysine) in m-cresol is plotted as a function of temperature for various molecular weights [37]. The inverse nature of this transition arises from the fact that not only the intra-chain hydrogen bonds dictate the total enthaplic contribution but also the hydrogen bonds between amino-acid residues and solvent molecules are important. Moreover, the entropy of the solvent also becomes a relevant variable in addition to the chain entropy. In addition to the inverse transition behavior, the figure also shows that the transition becomes sharper as the molecular weight is increased.

Apart from the dependence on solvent and temperature, the helix-coil transition has also been studied as a function of other external stimuli. For example,

39 Figure 3.3: Molecular weight dependence of the helix-coil transition shown as spe- ciﬁc rotation at wavelength of 436 nm as a function of temperature for poly(ǫ- carbobenzoxy L-lysine)(PBCL) in m-cresol. Nh¯ represents the degree of polymerization [37].

40 Granick et al. [38] studied the stabilization of the α-helical conformation of poly(L- glutamic acid) in the adsorbed state on charged polymer brushes. Other surface studies include synthesis and characterization of surface grafted poly(L-glutamic acid) and poly(L-lysine) by Chang et al. which adopts the α-helical conformation [39,40].

They investigated the pH dependence of the helix-coil transition and showed that the transition was accompanied by a change in thickness and refractive index of the ﬁlm.

They also suggested that, as compared to solvated free chains, the grafted chains favor the helical conformation as a function of pH. The effect of surfactant and ions was also explored. Along the same lines, Burkett et al. [41] also discussed observed differences between the preference of the α-helical conformation in free and adsorbed polypeptides and attributed it to increased electrostatic interactions. McHugh et al. [42] studied the effect of simple shear on the conformations of poly(L-lysine).

They observed a reversible helix-stretched coil transition at 87.5% methanol-water composition above the critical shear rate and described it in terms of an activated jump process. Zozulya et al. [43] studied the eﬀect of Na+ and Mg2+ ions on the conformational transition of poly(dA)-poly(dT), also sometimes known as DNA polymers. Furthermore, Green et al. [44] discussed the eﬀect of non-racemic chiral solvent such as chlorinated hydrocarbons on the formation of helical structures in mixtures of racemic poly(n-hexyl isocynate).

It is quite clear that most studies of the helix-coil transition are focussed on homopolypeptides as they are the simplest cases of poly(aminoacids)/proteins.

41 Among these studies, the α-helical conformation of polypeptides emerges to be the most crucial one and is worth discussing due to its widely recognized importance.

3.1.2 The Structure of the α-helix

The α-helix model was first described by Pauling, Corey and Branson in 1951 [45], and was supported by various experiments done on proteins thereafter. The first convinc- ing proof of its existence came with the first protein crystal structure of myoglobin, in which most secondary structure was found to be helical. Subsequent experiments ver- ified its existence in almost all globular proteins. Now, it is believed that the α-helix is the most abundant secondary structure in proteins, with 30% of residues found in this state. A schematic diagram of a poly amino acid in the α-helical conformation is shown in figure 3.4 [46].

In an α-helix, the linear translation is a rise of 0.54 nm and circular rotation is 3.6 residues per turn of the helix. This combination of translation and rotation puts the amide groups of residue i and i + 4 close enough so that a strong hydrogen bond is formed between carbonyl O of one residue and amide H of other. The helix is stabilized by such i, i + 4 hydrogen bonds. Here, the essential feature of the α-helix model is that, if there are j residues in helical conformations, only j-2 hydrogen bonds are formed [46]. Some other not so common helices found in protein structures are

310-helix and π-helix which have diﬀerent values for linear translation and residues per turn than α-helix.

42 Figure 3.4: Structure of an α-helix.

43 The conformation of a protein or polypeptide can be described by the sequence of backbone dihedral angle pairs φ and ϕ. Out of various possible values of φ and ϕ, most combinations are sterically excluded leaving only the broad β range and narrower α region in the φ-ϕ plane. One of the main reasons why the α-helix is so stable is that a succession of the sterically allowed angles, in the narrower α region, naturally position the backbone NH and CO groups towards each other for strong hydrogen bond formation.

3.1.3 Helix-Coil Transition: Theoretical Perspective

The original helix-coil transition theories, proposed in 60’s were based on the 1- dimensional Ising model [5]. It has been found that linear, helical macromolecules are well represented by the linear Ising model and hence, the helix-coil transition of biopolymers/homopolypeptides can be well mapped onto it. Moreover, it is well known that the linear Ising model cannot have any thermodynamic transitions. Since helix-coil transition theories are built on the basis of the linear Ising model, it is im- plicitly assumed that the helix-coil transition is not a true thermodynamic transition.

Instead, it is a conformational or order-disorder transition from an all helical conformation to a random coil conformation. In order to get the essence of how the helix-coil transition is actually treated theoretically, two of the most important theoretical treatments are discussed in next sub-sections.

If we consider the structure of α-helix in a polypeptide chain as discussed in the previous section, there are 3 kinds of important states possible, namely

44 a) a coil state, b) a helical state that contributes a hydrogen bond and c) a helical state that does not contribute a hydrogen bond.

In each of the two theories to be discussed next, there is a statistical weight assigned to each of above 3 states. A statistical weight is a measure of the probability of the occurrence of a certain state. In general, it can be written as eκ where κ is a real number. If this number is positive, the statistical weight is larger than one, i.e. that state is favorable. On the other hand, if κ is negative, the state becomes unfavorable. In both theories the coil state (a) has been taken as the reference state.

However, the theoretical interpretation of the other 2 states (b and c) is discussed below. It should be noted that, although the theories are not discussed in great detail below, it is worth mentioning that these theories are essentially matrix treatments of statistical weights used to compute the properties of interest.

3.1.3.1 The Zimm-Bragg Theory

The Zimm-Bragg (ZB) model [28] counts the number of hydrogen bonds in each conformation of the polymer chain. In this model, the units are considered as peptide groups and are classiﬁed on the basis of whether their NH group participate in hydrogen bonds within the helix. As mentioned above, the coil state is taken as reference.

The terminal hydrogen-bonded units have a statistical weight of σ1/2, and internal hydrogen bonded units carry a statistical weight of s. s is the propagation parameter and σ is the initiation parameter. Here, it is worth mentioning that these two param-

45 eters deﬁne the crux of Zimm-Braggs theory. The most important diﬀerence between both parameters is the fact that the initiation parameter σ is always less than one while the paramater s can be higher or lower than one depending upon temperature.

Mathematically, the statistical weight s is deﬁned as

E S s = exp − Hbd (3.1) RT − R

Figure 3.5: Fraction of hydrogen bonds, θ, as a function of the statistical weight, s, for various values of the initiation parameter σ [28].

The most fundamental feature of the thermodynamics of the helix-coil transition is that the initiation of a new helix is much more diﬃcult than the propagation

46 of an existing helix. The main reason for this is the following. For the formation of the first hydrogen bond, the chain must sacrifice the conformational entropy of three consecutive residues. Once the initiation has occured, the addition of another residue to the helix propagates the structure. This is the entropy of only one residue. This fact makes the initiation a less probable event than the propagation of an existing helix. Because of this reason, the value of σ is taken to be less than one in the ZB model. The parameter σ is also known as the cooperativity parameter. The lower the value of sigma is, the more cooperative the transition is. The cooperativity of the transition is identified by the sharpness of the transition. This behavior is shown in figure 3.5 where the residual helical content is plotted as a function of parameter s for various values of σ. The complete helix-coil equilibrium is governed by the statistical weight for all possible conformations. The probability of each conformation is calculated by the statistical weight of that conformation divided by the partition function (sum of all statistical weights) of the system. Moreover, various equilibrium properties such as the mean helical length, the mean number of hydrogen bonds, the fraction of residues in helical state, etc. can be calculated by applying an appropriate mathematical treatment to the partition function.

3.1.3.2 Lifson-Roig Theory

In the Lifson-Roig (LR) model [29], each residue is either assigned the conformation of helix (h) or coil (c). These assignments depend upon the helical dihedral angles pair (φ,ϕ) of that particular residue. Hence, a peptide of N residues can be written

47 as a string of N ‘h’s or ‘c’s, adding up to 2N total conformations. A particular residue is assigned a statistical weight depending upon its conformation and surrounding residues’ conformations. A residue in an h conformation with an h on either side has a statistical weight of w. As mentioned before, coil residues are used as reference and are assigned statistical weight of u. The two h residues at the extremes of a helical domain are assigned weights of v. The statistical mechanical treatment of the system is similar to that of ZB model and all the equilibrium properties can be evaluated in a similar fashion.

It is worth mentioning that the LR model is easier to handle conceptually for heteropolymers, especially, since the w and v parameters are assigned to individual residues. In this case, the substitution of one amino acid at a certain position changes the w and v values at that position. On the other hand, in the ZB model, the initiation parameter σ is associated with several residues. For example, in case of heteropolymers, the initiation probability would vary along the chain due to the diﬀerent relative sequencing of amino-acids. Moreover, s is associated with a peptide group rather than a residue. Another important diﬀerence is that the ZB model statistical weights for all ‘chc’ and ‘chhc’ sequences are assigned zero while they are all considered in LR model. This zeroing of ‘chc’ and ‘chhc’ sequences in the ZB model excludes a large number of conformations that contain a residue with helical conformation (φ,ϕ) but without a hydrogen bond. The parameters of the ZB and LR

48 models are related according to following equations:

w s = (3.2) u v2 σ = (3.3) u2

Another important point to notice here is that there are some conformations of partly helical polymers that are not allowed due to steric eﬀects. These steric eﬀects are not included in the helix-coil transition theories as they assign the same weight (unity) for all coil residues. The inclusion of steric hindrances (exclusion) in the theories would lead to the statistical weight of the coil residue to vary from unity and could be lower than unity in many cases.

Apart from the discussed theoretical treatments of the helix-coil transition, other important theories proposed in the 60s are worth mentioning like the one by

Gibbs and Di Marzio [47], Zimm and Rice [30] and Nagai [31]. Particularly, Nagai combined the tranditional matrix method and the Freely Jointed Chain model to calculate real space properties such as the mean square end-to-end distance. Recently

Halperin [48] and Pincus [49] employed Nagai’s model to address the force-elongation behavior of homopolypeptides which will be discussed in great detail in Chapter 5.

Other theoretical treatments have extended the matrix treatments of the helix-coil transition to address specific problems. For example, Pincus et al. [50] discussed the effect of solvent composition using a simple model that improves the hydrogen bonding ability of the solvent and makes it less cooperative without affecting transition temperature. Tanaka [51] proposed a statistical mechanical treatment to study fun-

49 damental properties of helices induced on a polymer chain to hydrogen bonding of chiral and achiral molecules on the side chain. Alternative approaches to the helix- coil transition have also been developed. For example, Muthukumar [52] developed a statistical mechanics approach based on a novel ﬁeld-theoretic method to study the helix-coil transition, among other physical phenomena; and Paoletti et al. [53] developed a fully thermodynamic approach to describe counter-ion condensation on a biopolyelectrolyte capable of a conformational transition between two diﬀerent conformations. Doig [46] has also reviewed recent advances in helix-coil transition theories focussing on side chain interactions, helix-dipoles, 310 helix formation, helix dipoles, etc.

3.1.4 Helix-Coil Transition: Simulation Perspective

The helix-coil transition in polypeptides has also attracted a lot of attention from both molecular dynamics [11, 13, 14, 54] and Monte Carlo simulations perspective [55–57].

Particularly in Monte Carlo simulations, with the increase in computational speed and the development of very eﬃcient algorithms over the last decade such as the Mul- ticanonical algorithm and the Wang-Landau algorithm, researchers have been able to run atomistic and coarse-grained Monte Carlo simulations of polymers with secondary structures. As discussed in section 2.3.1, these approaches are very powerful because they carry out a random walk on the energy surface avoiding trapping in local energy minima and, in principle, are capable of capturing the thermodynamics of the diﬀerent systems accurately. As a consequence of this, the aforementioned

50 algorithms have proven to be very eﬃcient in studies of folding single peptides into

α-helical structures. For example, Hansmann and coworkers have investigated the folding behavior of metenkephalin [24] and poly(L-analine) [10] using Multicanonical algorithm, while de Pablo has used the Wang-Landau algorithm to study polypeptide folding on a lattice [58] as well as united atom models [59].

The only disadvantage with current random walk algorithms is that the computational time needed to generate the ﬂat energy histogram required by the algorithms increases rapidly with the number of states, putting many systems of current experimental and theoretical interest out of the reach of atomistic computer simulations. The increase in the number of states with increasing number of atoms implies that if we want to address the general behavior of many-chain systems of current interest like hydrogels of diblock copolypeptides [60] and networks of helix-forming polymers [61], or the general behavior of charged or chiral many body-systems involv- ing only one helical chain like a single helical polyelectrolyte with its counter-ions [53] and salt ions, or the induction of helical structures on a polymer due to hydrogen bonding of chiral and achiral molecules [51], we must coarse-grain the description of the polymer as much as possible without losing the helical characteristics of the chain. The coarse-graining process decreases the total number of possible states by removing states that diﬀer by properties with characteristic length scales of the order of angstroms which are irrelevant for the large length scale behavior. Therefore, it allows us to use the algorithms mentioned previously to study systems with more

51 than one polymer or systems for which an explicit (coarse-grained) description of the counter ions or solvent molecules is essential.

Motivated by the capability of such algorithms, we have proposed a minimal model (as discussed in Chapter 2) to realize the helix-coil transition. Our model is a real-space realization of the traditional ideas of helix-coil transition theory that bridge between the Ising-like concepts used in the traditional matrix treatments of the helix-coil transition and the real-space, helical structure of the chain.

The discussion of this chapter begins with section 3.2 which brieﬂy reviews the estimated density of states. Afterward, the observed helix-coil transition is discussed in sections 3.3, 3.4 and 3.5 as a function of temperature from the perspective of conﬁgurational, conformational and thermodynamic properties. In these sections, special attention has been paid in the comparison of our results with other simulation, experimental and theoretical studies. Then, section 3.6 discusses the consequences and importance of the interfacial penalty on the helix-coil transition. Finally, the chapter is concluded with the summary of our results and proposing a broader set of problems where our model could be useful.

3.2 Estimation of the Density of States

One of the most important and necessary conditions of the Wang-Landau algorithm is the ﬂattening of the energy histogram at every step of the Wang-Landau iteration algorithm (please refer section 2.3.3) within certain tolerance. In our simulations,

52 we used a tolerance of 20%, as suggested by Wang and Landau [25]. The energy histogram is plotted in figure 3.6 as a function of the number of beads in the helical state, NH , and the number of interfaces, NS, for a chain length of 60 beads (last iteration). The figure shows the flatness of the histogram and, in addition, shows the convergence of the density of states.

Figure 3.6: A typical ﬂat energy histogram obtained from the Wang-Landau sampling algorithm for N=60 beads.

Figure 3.7 shows the rescaled density of states as a function of NH and NS for a chain length of 60 beads. Firstly, the ﬁgure supports the fact that the random coil conformation (NH = 0, NS = 0) has the highest number of states. Secondly, the

53 Figure 3.7: Logarithm of the density of states, gtrue(NH ,NS) after rescaling as obtained from the Wang-Landau sampling algorithm for N=60 beads.

54 ﬁgure also supports the fact that the perfect helix (NH = 56, NS = 0) has the lowest density of states (ln(DOS) very close to zero). Thirdly, the ﬁgure is also in agreement with the fact that the logarithm of the density of states decreases with increasing

NH , as each bead added to the helix reduces the number of possible conﬁgurations.

Furthermore, for a particular NH , the DOS again decreases as the number of interfaces increases.

3.3 Conﬁgurational Properties

We start analyzing the consequences of our minimal model by considering the con-

figurational properties of the polymer. For the purpose of clarity, the interfacial parameter, η, has been set to 0 in sections 3.3, 3.4 and 3.5. However, the consequences of the interfacial penalty are discussed in section 3.6. Figure 3.8 shows the behavior of the mean square radius of gyration, R2 , as a function of temperature, h gi T for six different chain lengths. This figure clearly shows that at low temperatures the model predicts an extended (rod like) configuration for the polymer chain while, at high temperatures, the polymer decreases its size which supports the existence of random coil conformation.

One way to confirm the presence of these two conformations is to analyze the dependence of the chain dimensions, i.e. R2 , as a function of chain length N h gi at low and high temperatures. This dependence has been plotted in figure 3.9 on a double logarithmic plot. This figure shows that R2 scales as N 2 at low temperatures h gi

55 400

300

2 200

100

0 200 250 300 350 400 T[ o K]

Figure 3.8: Plot of the mean square radius of gyration, R2 , as a function of tem- h gi perature, T , for diﬀerent chain lengths and interfacial penalty η=0. The parameters are ( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N=60. • ∗

indicating a rod-like, extended structure and as N 1.26 at high temperatures indicating a random coil conformation with excluded volume interactions. The small deviation of the exponent from 1.17 at high temperatures indicates that the polymer chain may still be short, i.e., the chain length is not large enough to be in the Self-Avoiding

Walk (SAW) regime. The ﬁgure also shows that for smaller chain length, the chain dimensions are larger than those of random coil in contrast to what is known for poly(amino-acids) [20] when the τhelix=0.87 is employed. We have attributed this behavior to the value of torsion employed. In order to support our argument, we also run a few simulations with τhelix=0.253 which corresponds to helices where the

56 Figure 3.9: Plot of the mean square radius of gyration, R2 , as a function of h gi the number of beads, N. The parameters are: ( ) low temperature with torsion •

τhelix=0.87 (+90 dihedral angle), ()low temperature with torsion τhelix=0.253 (+16.7 dihedral angle) and (N) High temperature with torsion τhelix=0.87.

57 pitch of helix is equal to bead diameter. The behavior of such chains is also shown in ﬁgure 3.9. It is clear that, at low temperatures a perfect helical chain does have smaller chain dimensions than its random coil counterpart. Furthermore, we found that when N = 92, both chain dimensions are equal.

Figure 3.10: Plot of the radial distribution function, ρ, for the chain ends as a function of the end-to-end distance, R, and temperature, T , for N=50 beads.

To elaborate on this conformational structure a little further, we plotted the radial distribution function (RDF) which is the probability of ﬁnding the polymer with an end-to-end distance equal to R, for the chain ends as a function of the end- to-end distance R and temperature, T . Figure 3.10 shows this plot for a polymer with

58 fifty beads. At high temperatures, the radial distribution function shows a shallow peak for small values of the end-to-end distance, R. This is the typical behavior of polymers with low degrees of stiffness (random coils). However, as the temperature is reduced, the model predicts a shift in the peak to larger values of the end-to-end distance. This figure is very similar to the RDF for worm-like polymers as discussed by Frey et al. [62] where the temperature is substituted by the persistence length of the polymer chain. Using the similarities with the worm-like chain model we can say that the polymer chain becomes stiffer as the temperature is reduced supporting the helical structure of the chain.

4000

3000

2 2000

1000

0 260 280 300 320 340 T[ o K]

Figure 3.11: Plot of the mean square end to end distance, R2 , as a function of h i temperature, T , for a chain with N =60 beads. Continuous line (our simulation), ( ) • (Nagai’s model).

59 Furthermore, for the purpose of making our discussion of the conﬁgurational properties more balanced and objective, it is appropriate to compare the results with the ones obtained using one of the well-established theories of helix-coil transition.

In this regard, ﬁgure 3.11 shows a quantitative comparison between our result for

R2 and the one obtained from Nagai’s model [31] for a chain with sixty beads using h i the equations 91 and 112 from the reference. The parameters ﬁtted were l1 and l0 which correspond to the increase step length when a segment becomes a part of helix and random coil ,respectively; and α, the helix nucleation parameter. The values of the parameters of Nagai’s model are l1=3, l0=2.5, N=60 and α=0.25. The value of the enthalpy used in the expression for the parameter σ, (analogous to parameter s in Zimm-Braggs theory) was -7.5 Kcal.mol−1 and the value of the entropy was -12 calmol−1K−1. As seen in the ﬁgure, there is good quantitative agreement between both models.

3.4 Conformational Properties

The behavior of the fraction of the polymer in the helical conformation (order parameter) as a function of temperature, T is shown in ﬁgure 3.12 for all six chain lengths studied. It is clear from the ﬁgure that the order parameter is close to one at low temperatures, thus predicting helix formation and it approaches zero at high temperatures where no helices are present. One important observation is the sigmoidal shape of the curves. This shape is a direct consequence of the cooperativity of the

60 transition and is captured correctly by the model. Furthermore, this result agrees on a qualitative level with predictions arising from other computer simulation studies on atomistic models [24, 63], theories [28, 29, 47] and experimental observations [32].

Another important feature of this result is its dependence on chain length. As the chain length increases, the transition temperature increases until it reaches a limiting value. This behavior is in good agreement with the predictions from standard helix-coil transition theories and is explained below.

0.8

0.6

0.4

Fraction of Helical Beads 0.2

0 200 250 300 350 400 T[ o K]

Figure 3.12: Plot of the fraction of helical beads (order parameter, θ ) as a func- h i tion of temperature, T , for diﬀerent chain lengths and interfacial penalty η=0. The parameters are ( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) • ∗ N=60.

61 It is known that at the temperature where a system undergoes a discontinuous transition between two states, the free energy of both states is the same. In general, the diﬀerence in the free energy can be written as ∆F =∆H T ∆S, where ∆H and − ∆S are the change in enthalpy and entropy, respectively. It means that the occurrence of both states is equally probable. For our model it is ∆F =(N 4).C T.(N 3).∆S − − − where ∆S is the change in entropy when a bead changes its conformation from the coil to the helical state. In such case, the transition temperature can be deﬁned as

(N 4).C T ∗ = − (3.4) (N 3).∆S − In equation 3.4, the factor of N-4 arises from the fact that the chain can only have N-4 beads in the helical state for a N bead system, while the total number of dihedral angles, N-3, deﬁnes the total loss in entropy. It is clear from the equation that the transition temperature increases with an increase in chain length, but it saturates when the chain length becomes suﬃciently high. For our model, putting the values of ∆H and ∆S as -1300 and -ln(64) gives a transition temperature for

∗ inﬁnite chain length, T∞, equal to 313K.

For a discontinuous transition, at transition temperature, both states coexist. For a moment if we assume that in order to satisfy the criteria of free energy diﬀerence by helix-coil transition, the perfect helical state and perfect random coil state should co-exist. However, when characterized in terms of helical content, the ensemble averaging of helicity of both states leads to the value of 0.5 at T . The ∗ value of 0.5 corresponds to s = 1 in the high chain length limit suggesting that the

62 free energy diﬀerence is indeed zero at the mid point of the transition. In simpler words, at transition temperature, each bead could exist in either helical or coil state with equal probability.

0.8

0.6

0.4

0.2 Fraction of Helical Beads

0 260 280 300 320 340 o T [ K]

Figure 3.13: Plot of the fraction of helical beads (order parameter) as a function of temperature, T , for a chain with N =50 beads. Continuous line (our simulation), ( ) • (Bloomﬁeld) [64].

Figures 3.13 and 3.14 show a comparison of the fraction of the polymer in the helical conformation predicted by our simulation study with the results obtained using two theoretical approaches. First, we compare our result with the one obtained by Bloomﬁeld [64] in Figure 3.13. In this case the polymer has ﬁfty beads. The parameter η is set to zero. In addition, the functional dependence of the parameter s (in

63 1

0.8

0.6

0.4

0.2 Fraction of Helical Beads

0 260 280 300 320 340 T[ o K]

Figure 3.14: Plot of the fraction of helical beads (order parameter) as a function of temperature, T , for a chain with N =60 beads. Continuous line (our simulation), ( ) • (Nagai’s model) [31].

Bloomﬁeld’s notation) on temperature was assumed to be of the form of equation 3.5 and was ﬁtted with equation 3.6 from reference [64].

1300 s(T )= exp +∆S (3.5) T s 2σ (1 s) θ = X 1+ − − (3.6) h i 1+ s + (1 s)2 + 4σs (1 s)2 + 4σs − − where θ is the order parameter.p The optimal valuesp of the nucleation parameter σ h i (in Bloomﬁeld’s notation) and ∆S are 0.02 and -4.163, respectively. It is important to point out that the value of σ(= 0.02) is less than one. This suggests that the cooperativity of the transition is included in our model to some extent even without incorporating the interfacial penalty explicitly (η=0). This is the natural consequence

64 of the restrictions introduced by the FRC model and the constraints on the dihedral angles together with the criterion used to determine the helical conformation. This conclusion is easy to understand if we follow the subsequent line of reasoning.

As discussed in section 2.2, the torsion of a bead depends upon two consecutive dihedral angles along the chain. So, when the helix is nucleated, i.e. a certain bead adopts the helical state, the change in entropy associated with it corresponds to the restriction of two dihedral angles. In other words, the system loses entropy equal to 2 ln(64). However, for the next bead to adopt the helical state, the system only ∗ has to lose an amount of entropy equal to ln(64) since one of the dihedral anlges is already in place to favor the helical conformation. This is also known as the propagation of the helix. It is this extra decrease in entropy (ln(64)) during the nucleation of the helix which contributes to the inherent cooperativity in our model.

Before proceeding further, let me discuss some consequences of using φ = 64 on the transition features. As mentioned before, the conformational entropy per bond is quite higher than that of continuous freely rotating chain model (ln(64) vs. ln(2π)). The main reason for using 64 possible dihedral angles was to capture the cooperative nature of the transition to a signiﬁcant amount, inherently. Such values of φ resulted in the cooperativity of the order of 10−2 in our model for helix-coil transition. As a consequence of assigning such a high conformational entropy, the

ﬁtting of the model with that of Nagai [31] resulted in extremely high values of ∆H

(about a order higher in comparison to what has been found for homopolypeptides).

65 Moreover, such a high value for ∆H did affect the behavior of saturation of transition temperature as a function of chain length. In homopolypeptides, such a saturation is known to occur around N ∼ 1000. This saturation point depends upon ∆H. The higher the value of ∆H, the shorter the chain length required in order to attain the saturation point. Since, the fitted value of our ∆H was about a order of magnitude higher than those for homopolypeptides, the saturation did occur around N = 60 for our model (see figure 3.12). In order to get a better fit of chain length dependence of experimental data, a lower value of φ should be used. Such a prediction of φ could be determined by performing a model molecular dynamics simulation with a purely freely rotating chain with no restriction on φ, assigning a helical state only when a particular dihedral angle visits a certain defined range of possible dihedral angles.

The results predicted by such a MD simulation should be compared with those of

MC simulations with diﬀerent values of possible dihedral angles in order to get the better estimate of φ. However, it should be noted here that such a reduction shall decrease the cooperativity, σ of the model.

Results for the order parameter have also been compared with the predictions by Nagai’s model [31] and is shown in figure 3.14. In order to be consistent with the comparison presented in figure 3.11, the same values for the parameters were used. It should be noted that these parameters were optimized to provide a good quantitative agreement between both results ( R2 and θ ). Thus, we regard the comparison h i h i showed in figure 3.14 to be quite good since there are no adjustable parameters. It

66 should also be observed that the transition starts at the same temperature (when approached from high temperatures). But, the width of the transition is not the same for both models. This is a consequence of the parameters used which can be optimized to obtain a very good quantitative agreement for the fraction of the polymer in the helical conformation. This would decrease the quality of the agreement shown in ﬁgure 3.11. However, a compromise could be reached such that the descriptions of both quantities are acceptable. In addition, it is also important to note that Nagai’s model does not include excluded volume interactions and uses the Freely Jointed (not

Rotating) Chain model. These differences might also contribute to the quantitative differences observed in figures 3.11 and 3.14.

1 Number of Helical Domains

0 200 250 300 350 400 T[ o K]

Figure 3.15: Plot of the number of helical domains, N , as a function of tempera- h hdi ture, T , for diﬀerent chain lengths and interfacial penalty η=0. The parameters are:

( ) N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N=60. • ∗

67 60

10 Average Length of a Helical Domain 0 200 250 300 350 400 T[ o K]

Figure 3.16: Plot of the average length of helical domain as a function of temperature,

T , for diﬀerent chain lengths and interfacial penalty η=0. The parameters are ( ) • N=10, () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N=60. ∗

Figures 3.15 and 3.16 provide further insight into the mechanism behind the helix-coil transition. For example, figure 3.15 shows the average number of helical domains, N , as a function of temperature for chains of different lengths. It can h hdi be noted that for short chains (N=10), N is always less than one while, for long h hdi chains, it is larger than one near the transition temperature. These results are in good qualitative agreement with the original results of Lifson and Roig [29] and indicate that short chains undergo the helix-coil transition by unwinding from the ends while long chains break into multiple helical domains near the transition temperature before undergoing the transition. Further insight is provided by figure 3.16 where we plot the average length of a helical domain as a function of temperature. This figure clearly

68 shows that, on average, the length of a helical domain decreases in a monotonic manner when the temperature is increased and is supported by helix-coil transition theories [29].

3.5 Thermodynamic Properties

250

200

150

Entropy 100

0 200 250 300 350 400 o T[ K]

Figure 3.17: Plot of the entropy, S, as a function of temperature, T , for diﬀerent chain lengths and interfacial penalty η=0. The parameters are ( ) N=10, () N=20, • () N=30, (N) N=40, (H) N=50 and ( ) N=60. ∗

We start the thermodynamic analysis of the model with the evaluation of the entropy of the system. Figure 3.17 shows the entropy, S as a function of temperature,

T for chains of diﬀerent lengths. The ﬁrst observation is that the entropy increases

69 with increasing temperature as expected from the second law of thermodynamics.

Furthermore, the entropy approaches zero when the temperature approaches absolute zero in agreement with the third law of thermodynamics and, hence, is independent of the chain length. This is the natural consequence of the formation of helices at low temperatures. Once the complete helix is formed, the total number of degrees of freedom is ﬁve, two rotational, which are irrelevant due to the isotropic symmetry of the problem and three translational, which are removed from the simulation by

fixing the position of the first bead to the origin of the coordinate system. Thus, all curves should collapse onto a single curve at low temperatures, as predicted by our model. Furthermore, figure 3.18 shows the dependence of the entropy on chain length in the high temperature region. It should be observed that our model predicts a linear dependence of the entropy on the number of beads as it should be because entropy is an extensive thermodynamic variable.

It should be noted here that as mentioned before, our model attributes a higher conformational entropy per bond than that of pure freely rotating chain model.

Such a diﬀerence is reﬂected in properties which are directly proportional to the lnZ,

Z being the partition function. Such properties include properties like entropy, free energy, etc. Hence, in our model also, the entropy of the random coil is substantially higher than that of freely rotating chain.

Figures 3.19 and 3.20 show the same plots for the internal energy, U. Fig- ure 3.19 shows the dependence on temperature for diﬀerent chain lengths. The inter-

70 300

250

200

150 Entropy 100

0 0 10 20 30 40 50 60 Number of Beads

Figure 3.18: Plot of the maximum entropy, S, as a function of chain length for high temperatures.

-20000

-40000 Internal Energy -60000

-80000 200 250 300 350 400 o T[ K]

Figure 3.19: Plot of the internal energy, U, as a function of temperature, T , for diﬀerent chain lengths and interfacial penalty η=0. The parameters are: ( ) N=10, • () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N=60. ∗

71 nal energy increases as the temperature is raised and reaches the asymptotic value of zero at high temperatures. This is a consequence of using the random coil conformation as our reference state. It is interesting to note that the predicted functional dependence on temperature also agrees on a qualitative level with recent predictions arising from the Zimm-Bragg theory which were used to describe experimental data by Linder et. al. [65]. Figure 3.20 shows the predicted dependence of the internal energy on chain length for low temperatures. As in the case of the entropy, the internal energy is a linear function of the number of beads in agreement with the fact that it is an extensive thermodynamic variable.

-20000

-40000 Internal Energy -60000

-80000 0 10 20 30 40 50 60 Number of Beads

Figure 3.20: Plot of the minimum internal energy U as a function of chain length for low temperatures.

72 2000

1500

Cv 1000

500

0 200 250 300 350 400 T[ o K]

Figure 3.21: Plot of the heat capacity, Cv, as a function of temperature, T , for diﬀerent chain lengths and interfacial penalty η=0. The parameters are ( ) N=10, • () N=20, () N=30, (N) N=40, (H) N=50 and ( ) N=60. ∗

Figure 3.21 shows the heat capacity, Cv, as a function of temperature for polymers with various chain lengths. This ﬁgure shows that an increase in the length of the polymer increases the height of the peak and shifts the transition temperature toward higher values until it saturates at temperatures close to 313K for large values of

N. This behavior has already been discussed previously in section 3.4. In addition, the width of the transition at half height decreases monotonically with increasing chain length until it reaches a limiting non-zero value, governed by the nucleation parameter. Figure 3.22 shows the height of the peak as a function of the number of beads. The relationship is linear in agreement with the fact that the heat capacity is

73 an extensive thermodynamic variable. In addition, the behavior of heat capacity as shown in ﬁgure 3.21 also agrees with atomistic simulations of polyalanine [63] as well as lattice models of proteins [58].

2000

1500

1000 Maximum Height 500

0 0 10 20 30 40 50 60 Number of Beads

Figure 3.22: Plot of the maximum height of the heat capacity as a function of chain length.

3.6 Eﬀect of interfacial penalty

In this section, the effect of the interfacial penalty on the helix-coil transition has been studied. For this purpose, we kept the length of the chain fixed (N=60) and varied the interfacial penalty parameter η. Figures 3.23, 3.24, 3.25 and 3.26 show four different properties: R2 , the fraction of helical beads, the number of helical h gi

74 domains and the heat capacity as a function of temperature for diﬀerent values of the parameter η.

300 > 2 g 200

100

200 250 300 350 400 T [ oK]

Figure 3.23: Plot of mean square radius of gyration, R2 , as a function of temper- h gi ature, T , for a chain with N=60 beads for diﬀerent values of the interfacial penalty,

η. The parameters are: ( ) η=0, () η=200, () η=600 and (N) η=1000. •

The main conclusion from all the ﬁgures is that the higher the interfacial penalty, the narrower the transition region. In other words, as the interfacial penalty increases, the helix-coil transition approaches the all-to-none transition originally proposed by Schellman [5]. This is the natural consequence of hindering those conformations with many helical domains which are very expensive in energy. Therefore, the chain prefers to go from the perfect helix to the random coil by avoiding those conformations which require the formation of interfaces. This behavior is most clear

75 1

0.8

0.6

0.4

Fraction of Helical Beads 0.2

0 200 250 300 350 400 T [ oK]

Figure 3.24: Plot of the order parameter, θ , as a function of temperature, T , h i for a chain with N=60 beads for diﬀerent values of the interfacial penalty, η. The parameters are: ( ) η=0, () η=200, () η=600 and (N) η=1000. •

1 Number of Helical Domains

0 200 250 300 350 400 T [ oK]

Figure 3.25: Plot of the number of helical domains, N , as a function of temper- h hdi ature, T , for a chain with N=60 beads for diﬀerent values of the interfacial penalty,

η. The parameters are: ( ) η=0, () η=200, () η=600 and (N) η=1000. •

76 12000

10000

8000 C v 6000

4000

2000

0 200 250 300 350 400 T [ oK]

Figure 3.26: Plot of the heat capacity, Cv, as a function of temperature, T , for a chain with N=60 beads for diﬀerent values of the interfacial penalty, η. The parameters are: ( ) η=0, () η=200, () η = 600 and (N) η = 1000. •

in ﬁgure 3.25 which shows that at suﬃciently high interfacial penalties, the unwrap- ping of the helical structure occurs from the ends, thus minimizing the number of interfaces.

3.7 Conclusions

In this chapter, a novel approach to the simulation of the helix-coil transition of polymers has been explored. The approach was built on the traditional concepts of helix-coil transition theory and added the geometrical property of torsion of a curve. This new parameter allowed us to bridge the gap between the conformational and conﬁgurational properties of the polymer chain. Therefore, real-space properties

77 like the radial distribution function were calculated in addition to the traditional conformational properties like the fraction of the polymer in the helical conformation.

The cooperative nature of the helix-coil transition was captured by employing the concept of torsion which was used to determine the conformational state of each bead.

It was shown that the use of torsion as a criterion to determine the conformational state of each bead led to results that agree with predictions arising from other computational, theoretical as well as experimental studies on a qualitative and quantitative level. Therefore, this model might be used to study systems where the helical structure of the polymer is important but the atomistic details are not. One example of this is the case of networks of helical polymers where the atomistic details are not so relevant for the mechanical and thermodynamic properties but, the helical structure of the polymer must be described correctly. Other problems where our minimal model may be useful are the conformational changes induced in helical polymers due to external forces or solvent quality.

Another area where our model might have interesting consequences is the one of polyelectrolytes and polyampholytes. In these ﬁelds, the possibility of helix formation will add another degree of freedom to these already very challenging systems.

Therefore, we foresee an enrichment of the physics of these systems. For example, it would be very interesting to study how helix formation gets coupled and modiﬁes the diﬀerent scaling regimes predicted for polyelectrolyte systems or, how helix formation

78 couples to the counter-ion condensation mechanism. In the next two chapters, we address some of these problems which include the eﬀect of external mechanical force and the eﬀect of solvent quality (implicit solvent model).

79 CHAPTER IV

COLLAPSE OF A HELICAL POLYMER: INTERPLAY BETWEEN

SECONDARY AND TERTIARY INTERACTIONS

4.1 Introduction

Understanding the molecular mechanisms that lead to the folding or collapse of macromolecules as a consequence of changes in environmental conditions such as a decrease in temperature, solvent quality and other physical variables is a first-class challenge in polymer physics [66, 67]. Perhaps, the most illustrative example of this problem can be found in the coil-globule transition, i.e. the collapse transition, typ- ically observed in synthetic polymers as well as biopolymers. Due to the relevance of this transition to various macromolecular systems like polymer solutions, network collapse, protein folding, DNA packaging and others, extensive experimental [68–74], theoretical [75–77] as well as computer simulation studies [78–82] have been carried out to fully understand this transition. While a significant amount of information about the conformation and configuration of the chains has been gained through experimental techniques such as Scanning Force Microscopy (SFM) [70], Dynamic

Light Scattering (DLS) [71], Fluorescence measurements [72] and others; theoretical and computer simulation studies have complemented the experimental observations

80 and led to a deeper understanding of the thermodynamics of the coil-globule transition.

It is well known that some macromolecules can also adopt equilibrium conformations other than the random coil and globular states. One of such conformations is the helical conformation which is commonly observed in synthetic as well as biological polymers. If helical and globular conformations can form simultaneously, then a coupling and competition between the helix-coil and coil-globule transitions can be expected which may result in a novel and rich physical behavior of the macromolecule.

Biopolymers such as proteins and DNAs belong to this category and put forward a more realistic picture of such conformational changes only present in macromolecules.

Interestingly, the possibility of this coupled behavior between the coil-globule and helix-coil transitions has already been suggested by Di Marzio [83, 84] as one particular case of a more general classiﬁcation of possible couplings of phase transitions in macromolecular systems. Recently, Matsuyama [85] has studied one of such couplings between the behavior of a polymer threading a membrane and the coil-globule transition.

The understanding of the helix-coil transition is important for the understanding of the conformational changes in biopolymers. It has been reviewed by

Poland and Scheraga [5]. Indeed, the helix-coil transition is a consequence of the competition between enthalpically favored intra-molecular, short-range hydrogen bonds between nearby residues and the overall entropy of the chain. In addition, long-

81 range interactions such as electrostatic interactions, van der Waals interactions and long-range (along the polymer backbone) hydrogen bonds also do play a very sig- niﬁcant role in determining the overall shape of the macromolecule. Interestingly, in proteins [3], it is the delicate balance between the aforementioned short-range and long-range interactions which determine its 3-dimensional native state. Thus, helical polymers provide us with an excellent opportunity to explore the conformational space of macromolecules where both short-range and long-range interactions are important. These systems will display the coupling between the helix-coil and coil-globule transitions where short-range interactions stabilize the helical state while long-range interactions control the overall 3-dimensional shape and entropy favors the random coil state.

Due to the hierarchal structure of proteins, it is quite a challenge at present to explore the above transitions and their possible coupling using deterministic approaches such as molecular dynamics simulations due to computational limitations.

However, with the advent of algorithms capable of escaping energy minima, such as the Multicanonical [10, 24] and Wang-Landau algorithm [25], it possible to address the coupling of transitions in macromolecular systems using stochastic approaches like Monte Carlo simulations. As an example, before the advent of the Wang-Landau algorithm, Shakhnovich et. al. [86] studied the eﬀect of relative strength of secondary vs. tertiary interactions on the folding of macromolecules capable of forming helices

82 using standard Metropolis Monte Carlo. However, in order to carry out the study, they had to run 300 diﬀerent simulations approximately.

The Wang-Landau algorithm along with various modiﬁcations have been used to study conformational changes in macromolecules. For example, de Pablo and coworkers implemented the Wang-Landau algorithm to study polypeptide folding on a lattice [58] as well as united atom models [59]. Vorontsov-Velyaminovall et al. [87,88] studied ring polymer chains on both lattice and continuous models via the Wang-

Landau algorithm. Abrams, et al. [89] investigated the thermodynamics of the coil- globule transition in confined geometries of a spring-bead heteropolymers comprising of hydrophobic and hydrophilic beads through off-lattice Wang-Landau Monte Carlo simulations. Parker et al. [90] combined the Metropolis Monte Carlo algorithm with the Wang-Landau algorithm (density guided importance sampling) to study protein folding through reduced model (RAFT model). Carbajal-Tinoco et al. [91] presented a model of protein folding based on effective interactions between peptide amino-acid residues and investigated the thermodynamics of formation of secondary structures such as α-helices and β-sheets. Cavalli et al. [92] compared the sequence based transferable energy functions (TEF) and structure based G¯olike energy function for the reversible folding of 20-residue peptide. Muller and coworkers [81] studied the collapse of a single semi-flexible macromolecule through expanded ensemble Monte

Carlo simulation and discussed the stability regions of coil, globule, and rod-like conformations. Paul and coworkers [82] also addressed the phase behavior of lattice

83 ﬂexible polymer through the Wang-Landau sampling algorithm, suggesting a two- step collapse through a coil-globular transition followed by crystallization at lower temperature.

In this chapter, we have extended our minimal model to incorporate long- range interactions (tertiary interactions) in addition to modeling hydrogen bonding and addressed the possible interference of short-range, secondary and long-range, tertiary interactions on the conformational changes of a helical homopolypeptide.

The simulations were motivated to understand the possible coupling between the helix-coil and coil-globule transitions for homopolypeptides. The discussion focusses on the understanding of the observed conformational changes from the perspective of state diagrams, conﬁgurational, conformational and thermodynamic properties in diﬀerent regimes alongwith their chain length dependence.

4.2 Density of States

As mentioned in section 2.4, the simulations were done for chain lengths of 10, 30,

45 and 60 beads. The section also discussed the detailed simulation procedure to estimate density of states gtrue(NH ,λ). The logarithm of the rescaled density of states gtrue(NH ,λ) is presented in figure 4.1 for a chain length of 60 beads. It is quite clear from the figure that, irrespective of compactness parameter λ, the number of possible configurations increases exponentially with the decrease in NH . Moreover, the maximum value of the compactness parameter also increases with decrease in

84 Figure 4.1: Logarithm of the density of states, gtrue(NH ,λ), after rescaling as obtained from the Wang-Landau sampling algorithm for N=60 beads.

NH . However, it is surprising to note that the maximum does not occur at NH = 0.

This behavior gives rise to very interesting features in the state diagrams which are discussed below.

4.3 State Diagrams

State diagrams in macromolecular systems describe various possible conformational, stable states as a function of two varibles. Figures 4.2, 4.3, 4.4 and 4.5 show the state diagrams for chain lengths equal to 10, 30, 45 and 60 beads, respectively. All

ﬁgures show the existence of stable conformational states of the polymer chain in

85 various regions of the state diagrams. In all cases, the dark and light segments represent helical and coil beads, respectively. The boundaries shown in the state diagrams describe the transitions from one conﬁgurational state to another one and were obtained from the peak in the heat capacity, Cv (as function of ǫ and T ). The

figures also show many boundaries (small peaks in Cv) ending before merging with the other boundaries (large peaks). This occurs when a small peak approaches a large one; it becomes shoulder or is completely absorbed by the large peak. Hence, such peaks could not be resolved numerically. Interestingly, all the state diagrams depict similar configurations of the chain in the limiting regimes; i.e., a helical configuration for small values of ǫ and at low T , a random-coil configuration for small values of ǫ and at high T , an amorphous globular configuration for large values of ǫ and at high

T , and a globular configuration with residual secondary structure for large values of ǫ and at low T which results from the competition between long-range attractive interactions among the beads and the favored secondary structure due to stabilization of helical beads. The latter configuration is not observed in figure 4.2 due to the short length of the polymer.

The aforementioned configurations at low T and high ǫ mimic the globular configurations of some proteins where residual secondary structure persists even in globular state [93]. In such cases (Figures 4.3, 4.4 and 4.5), we also observe the emergence of a few more regimes in the low-temperature limit as ǫ increases. These regions of the state diagram do not correspond to one specific structure of the polymer chain

86 Figure 4.2: State diagram for a chain with 10 beads. Helical beads (dark segments), coil beads (light segments).

87 Figure 4.3: State diagram for a chain with 30 beads. Helical beads (dark segments), coil beads (light segments).

88 Figure 4.4: State diagram for a chain with 45 beads. Helical beads (dark segments), coil beads (light segments).

89 Figure 4.5: State diagram for a chain with 60 beads. Helical beads (dark segments), coil beads (light segments).

90 but rather to an ensemble of conﬁgurations with similar characteristics. Snapshots of the polymer taken from the simulations are shown in the ﬁgures.

Let us now consider each region separately. Firstly, in the low ǫ limit and at low temperatures, the chain is stable in the helical configuration which undergoes the traditional helix-coil transition [5] upon an increase in temperature. Secondly, in the high T limit, the chain undergoes the standard coil-globule transition with an increase in ǫ. Thirdly, in the low T limit, we observe a cascade of continuous conformational transitions starting from a perfect helical state for small values of ǫ and ending in a collapsed globular state with residual secondary structure; except in the case of very short chains (figure 4.2). However, when comparing the various state diagrams, we found that the number of these low-temperature transitions increases with increasing chain length. For example, we find that for 30, 45 and 60 beads, we have 3, 5 and 6 continuous transitions, respectively. These transitions give rise to various configurations which are present in all the state diagrams. The emergence of these transitions is a consequence of the competition between inter-bead attractive interactions, which bring the beads closer to each other and the torsional energy, which stabilizes the helical configuration of the chain. This competition results in the breaking of the helical structure with increasing ǫ such that, once the structure is broken, it adopts a configuration that minimizes the interaction energy. This is achieved by folding the helical strands into a parallel arrangement such that they are close to each other. For example, during the first transition a perfect helical structure

91 is converted into a hairpin-like structure in all the state diagrams. The further analysis of all the regimes and various transitions is described in the next sections in terms of various conﬁgurational, conformational and thermodynamic properties.

4.4 Chain length eﬀects on the conﬁgurational, conformational and thermodynamic

properties

4.4.1 Low ǫ regime

All the state diagrams show the presence of the standard helix-coil transition for low values of ǫ. Since we have discussed this transition in great detail in chapter 3, we only present a very brief description of the chain length dependence of the various properties. The mean square radius of gyration, R2 , is plotted as a function of T in h gi figure 4.6 for all four chain lengths. It is clear from the figure that for all cases, R2 h gi decreases with increasing temperature indicating the presence of a transition where the chain changes its conformation from an extended one to a more compact one. In addition, figure 4.7 illustrates the cooperative nature of this transition through the typical sigmoid-like shape of the order parameter (i.e. helical content) as a function of T . The transition occurs at T ∗ which is determined by the parameter C (see chapter 2) of our model. At low temperatures, the helical content is one implying an all-helical configuration but, as the temperature increases, the helical content decreases to zero indicating the melting of the helical strands. The cooperative nature

92 of the order parameter has already been compared with helix-coil transition theories in Chapter 3.

300 >

2 200

100

0 200 250 300 350 400 Temperature (T)

Figure 4.6: Plot of the mean square radius of gyration, R2 , as a function of h gi temperature, T , for various chain lengths at ǫ = 0. The parameters are: ( ) N=10 ∗ beads, ( ) N=30 beads, () N=45 beads and () N=60 beads. •

Further insight regarding the breaking mechanism of the helical structure with increasing temperature can be obtained from ﬁgure 4.8 where number of helical domains, N , is plotted as a function of T . The peak around T ∗ indicates that h hdi the helical structure breaks into more than one helical strand during the transition in agreement with helix-coil transition theories [5]. Moreover, N increases with h hdi increasing chain length indicating that the longer the chain, the larger the number of

93 1

0.8 θ>

0.6

0.4

Order Parameter < 0.2

0 0 100 200 300 400 500 Temperature (T)

Figure 4.7: Plot of the order parameter, θ , as a function of temperature, T , for h i various chain lengths at ǫ = 0. The parameters are: ( ) N=10 beads, ( ) N=30 ∗ • beads, () N=45 beads and () N=60 beads.

2.5

1.5

0.5 Number of Helical Domains

200 250 300 350 Temperature (T)

Figure 4.8: Plot of the number of helical domains, N , as a function of temper- h hdi ature, T , for various chain lengths at ǫ = 0. The parameters are: ( ) N=10 beads, ∗ ( ) N=30 beads, () N=45 beads and () N = 60 beads. •

94 intermediate helical strands during the transition. However, for short chains N h hdi decreases monotonically which indicates that the chain unwinds from the ends during the transition.

4.4.2 High T regime

As mentioned previously, the coil-globule transition is observed in the high temperature regime as a function of ǫ. In this temperature regime the helical conformation is unstable; thus, it is absent and no interference with the coil-globule transition occurs. Therefore, at low values of ǫ, the chain adopts the random-coil conﬁguration due to the dominance of entropy. However, as the magnitude of ǫ is increased, the chain starts to collapse leading to the coil-globule transition. This is conﬁrmed by

figure 4.9 where the normalized chain dimensions, as quantified by the mean square radius of gyration divided by the number of beads N, R2 /N, is plotted as a function h gi of ǫ for chain lengths equal to 30, 45 and 60 beads, respectively. First of all, the figure clearly shows the decrease in chain dimensions with increasing ǫ indicating the collapse of the polymer chain which reaches a plateau for large values of ǫ. Secondly, we observe a change in the trend of R2 /N for various chain lengths. Interestingly, the h gi crossover occurs around ǫ 200 where the normalized chain dimensions are same for ∼ all chain lengths. This suggests that the attractive interactions at ǫ 200 are enough ∼ to overcome the effect of excluded volume interactions due to hard core repulsions.

At lower values of ǫ, the excluded volume interactions dominate while attractive in-

95 teractions overwhelm the former interactions at higher values of ǫ leading to collapse of the chain.

We have observed that the scaling law R2 /N N α is valid over the entire h gi ∝ range of values of ǫ. The exponent α was found to be positive for small values of ǫ, approximately zero for ǫ 200 and and negative for values of ǫ above 200. Specifically, ∼ the values of α are 0.28, -0.01 and -0.21 at ǫ = 0, 200 and 800, respectively. The difference in rescaling exponents α is attributed to the fact that the simulated chain lengths are still short. From figure 4.9, it is also clear that the transition becomes sharper with increasing chain lengths. Moreover, it also shows that the collapse of the chain shifts towards lower values of ǫ as the chain length increases. This is the direct consequence of the inter-bead attractive interactions increasing proportionally to DP 2 (degree of polymerization) while the chain entropy increases only linearly with

DP . Thus, the longer the chain, the lower the value of ǫ needed for the transition to occur.

Figure 4.10 shows the conﬁgurational entropy of the chain as a function of

ǫ for various chain lengths. It is clear from the figure that the entropy decreases with increasing ǫ, as expected. Moreover, for large values of ǫ the entropy levels off. Similar to the previous figure, the transition becomes sharper with increasing chain length. This increase in the sharpness of the transition implies its increasing cooperative nature. The cooperative nature of the coil-globule transition has also been discussed by Kuznetsov, et. al. [66]. Figure 4.10 also shows that the transition

96 Figure 4.9: Plot of the mean square radius of gyration, R2 , as a function of epsilon, h gi ǫ, for various chain lengths in the high temperature regime (500K). The parameters are: () N=30 beads, () N=45 beads and ( ) N=60 beads. •

shifts towards lower values of ǫ with an increase in chain length. Again, such a shift can be explained by the scaling arguments presented in the previous paragraph. It is interesting to note that the transition seems to occur at diﬀerent values of ǫ when

ﬁgures 4.9 and 4.10 are compared. Further analysis shows that the decrease in the entropy of the chain with increasing ǫ is negligible until the value of ǫ is large enough to completely overwhelm the eﬀect of excluded volume interactions. This implies that the entropy of the chain remains approximately constant until the chain has collapsed,

ǫ 200 in ﬁgure 4.10. Only when the chain is almost collapsed the entropy of the ∼ chain decreases with increasing ǫ.

97 300

250

200

150 Entropy 100

0 0 200 400 600 800 1000 epsilon (ε)

Figure 4.10: Plot of chain entropy as a function of epsilon, ǫ, for various chain lengths in the high temperature regime (500 K). The parameters are: ( ) N=30 beads, () • N=45 beads and () N=60 beads.

Finally, it should be noted that helical properties like the helical content and number of helical domains do not provide any additional insight into the coil-globule transition as these properties are all very close to zero in the high temperature regime.

4.4.3 Low T regime

As discussed previously, in the low temperature regime, we observe a series of continuous conformational transitions which involve the collapse of the helical structure with increasing ǫ. The sharpness of these transitions is a direct result of the competition between various stable states that are present at low temperatures. In this

98 section, we analyze these transitions using various conﬁgurational, conformational and thermodynamic properties along with their chain length dependences.

We start with the analysis of the mean-square radius of gyration, R2 , as h gi a function of ǫ in figure 4.11 for chains with 10, 30, 45 and 60 beads. As expected, the figure shows a decrease in R2 with increasing ǫ indicating the collapse of the h gi chain into more compact configurations. While for DP = 10 only one sharp decrease is observed, the case is different for DP = 30, 45 and 60. For these chain lengths a series of sharp, but continuous decreases in R2 are observed at specific values of ǫ. h gi These changes clearly indicate a substantial change in the dimensions of the chain which implies the collapse of the helical structure upon an increase in ǫ. Moreover, these changes indicate that the molecular mechanism behind the collapse of the chain is related to the breaking of the helical structure.

To validate this argument, we plot the order parameter, θ , in ﬁgure 4.12 h i as a function of ǫ. It is clear from the ﬁgure that for small values of ǫ, the chain is in the helical state. However, as ǫ increases, θ decreases in a step-like manner at h i the same values of ǫ as in case of R2 , hence supporting our argument about the h gi mechanism responsible for the collapse of the chain. As expected, we observe only a single transition for DP = 10 and a cascade of step-like transitions for DP = 30,

45 and 60. Moreover, for latter cases, we observe residual secondary structure in the collapsed globular conﬁguration of the chain which mimics the collapse states of certain proteins [93].

99 400

300 > 2 200

100

0 0 200 400 600 800 1000 epsilon (ε)

Figure 4.11: Plot of the mean square radius of gyration, R2 , as a function of epsilon, h gi ǫ, for various chain lengths in the low temperature regime. The parameters are: ( ) ∗ N=10 beads, ( ) N=30 beads, () N=45 beads and () N=60 beads. •

0.8 θ>

0.6

0.4

Order Paramter < 0.2

0 0 200 400 600 800 1000 epsilon (ε)

Figure 4.12: Plot of the order parameter, θ , as a function of epsilon, ǫ, for various h i chain lengths in the low temperature regime. The parameters are: ( ) N=10 beads, ∗ ( ) N=30 beads, () N=45 beads and () N=60 beads. •

100 The decrease of the order parameter is due to the breaking of the helical structure caused by the inter-bead, long-range attractive interactions and was observed for all chain lengths studied (10, 30, 45 and 60 beads). This mechanism is similar to the one present in the standard helix-coil transition of long enough chains where the polymer undergoes the transition by breaking the all-helical conformation into shorter helical strands as the temperature is increased. However, the helix- coil transition does not consider long-range interactions and the helical strands do not aggregate like in this case. Since the underlying mechanisms for breaking the helical structure are similar, cooperativity is expected to occur in the various, low- temperature transitions. This observation agrees with the experiments of Moore and collaborators [94, 95] who studied the helix-coil transition of phenylene-ethynylene oligomers at constant temperature as a function of solvent quality and observed the typical sigmoid-type shape of order parameter which indicates cooperativity. How- ever, they observed only one transition which is due to the low DP used in their studies. Interestingly, as mentioned before, we also observe only a single transition when we studied chains with 10 beads.

Figure 4.13 provides further insight into the behavior of the system at low temperatures; we plot the number of helical strands, N , as a function of ǫ for h hdi various chain lengths. Apart from the case of chain length equal to 10, the ﬁgure shows that for all other chain lengths, N ﬁrst increases and then decreases before h hdi reaching a constant value for large values of ǫ. In order to understand this inter-

101 7

1 Number of Helical Domains

0 0 200 400 600 800 1000 epsilon (ε)

Figure 4.13: Plot of the number of helical domains, N , as a function of epsilon, h hdi ǫ, for various chain lengths in the low temperature regime. The parameters are: ( ) ∗ N=10 beads, ( ) N=30 beads, () N=45 beads and () N=60 beads. •

esting, non-monotonic behavior we notice that, although N increases due to the h hdi breaking of helical structure, the total helical content, i.e., order parameter decreases monotonically. This leads to the conclusion that there should be a maximum in the dependence of N on ǫ after which N should begin to decrease with increasing h hdi h hdi ǫ. Moreover, the maximum in N depends on the length of the chain. In our case, h hdi this happens when N is equal to 3, 5 and 6 for chain lengths of 30, 45 and 60 h hdi beads, respectively. However, N does not approach zero on further increase of h hdi ǫ which implies that some residual secondary structure remains even for large values of ǫ where the chain becomes a collapsed globule. This residual structure can

102 be melted with an increase in temperature. This behavior can also be observed in

ﬁgures 4.2, 4.3, 4.4 and 4.5 (high ǫ regime) for chain lengths equal to 30, 45 and 60 beads, respectively.

Let us now discuss the fundamental differences in the behavior of N in h hdi the low T and low ǫ regimes for long chains. Previously, in subsection 4.4.1, it was said that in the low ǫ regime, N first increases with increasing temperature in h hdi continuous fashion before decaying down to zero. During this transition no specific configurations of the chain could be identified. This was due to the fact that when the temperature is increased and the helix breaks into shorter helical strands, it does so in such a way that the shorter helical strands can have any orientation in space and, moreover, can have any length within the limitation imposed by the total length of the chain. This generates a very large number of possible configurations with similar values of N . This is a consequence of the small value of ǫ where long h hdi range interactions are weak. What is observed is an ensemble average of the number of helical domains as a function of temperature. However, in the low T limit, the situation is different. As ǫ increases, the long range interactions become stronger and start to control the behavior of the chain. When they are strong enough to break the helix, these long range attractions are the ones that determine the set of configurations of the chain that are the most stable ones. These configurations are the ones where there are two helical strands parallel to each other and in spatial

103 contact. Thus, the diﬀerence in the molecular mechanisms active in both regimes is due to the presence of the long range interactions.

Figures 4.11, 4.12 and 4.13 clearly show that an increase in the length of the chain enriches the physics of the state diagram with novel, stable conformations of the chain. Indeed, a number of new conformations with more helical strands appear between the all-helical and the collapsed globular conformations. Therefore, more and sharper transitions are observed in the low temperature regime. However, due to

finite size effects, they are not transitions in the thermodynamic sense as discussed by Stukan and co-workers [96]. Moreover, the dependence on chain length follows the opposite trend reported for semi-flexible chains. The decrease in R2 , θ along with h gi h i the changes in N as described in figures 4.11, 4.12 and 4.13, respectively support h hdi the proposed mechanism for the breaking of the helical structure which is illustrated in figures 4.3, 4.4 and 4.5 with several snapshots of the polymer chain.

During the first step-like transition, the all-helical conformation breaks into two smaller, but relatively long helical strands which may have any relative orientation in space. However, the most probable configuration is the one where they are aligned parallel to each other and are located side by side, as shown in figures 4.3, 4.4 and 4.5.

This relative alignment is favored by the increase in the number of contacts between the beads of both helical strands which decreases the LJ interaction energy and the helical conformation of both strands which decreases the torsional energy. This ﬁrst transition was observed for all chain lengths studied (30, 45 and 60) making it a

104 characteristic feature of the model. Furthermore, we would like to emphasize here that the first transition shifts to lower values of ǫ as the chain length increases, as depicted in figure 4.12. This shift results from a similar argument to the one discussed previously for the coil-globule transition, but with a small modification.

On one hand, the LJ attractive interaction among beads increases as DP 2 while, on the other hand, the helical interactions increase linearly with DP . Hence, the

LJ interactions overwhelm the enthalpy gain due to the formation of longer or more helices as the chain length increases. Thus, smaller values of ǫ are required to break various structures. For example, for 30, 45 and 60-bead chains at 200K the all-helical conformation breaks around ǫ 300, 225 and 175, respectively. ∼

A further increase in ǫ leads to a transition to a conﬁguration with three helical strands parallel to each other. Interestingly, this transition was also observed for all the chain lengths studied. Further increases in ǫ take the system into new regimes with more helical strands parallel to each other. Also, the same energetic argument used to rationalize the formation of the ﬁrst state is also applicable to the other states until the LJ attractive interactions overwhelm the formation of helices and control the behavior of the chain.

The length of the chain has a fundamental effect on the number and characteristics of each of these new, low-temperature states. An increase in the length of the chain increases the number of the low-temperature, intermediate states. For example, in the case of 60 beads in figure 4.5, we observe configurational states with 4 and 6

105 helical strands parallel to each other. However, the parallel alignment of the helices is disturbed by the large number of coil beads at higher values of ǫ. Here, we detect a transition between two conﬁgurational states, both with six helical strands, where the main diﬀerence is the degree of alignment of the strands resulting from higher value of ǫ and less helical beads. A similar behavior was also observed for chains with

45 beads in figure 4.4. In some cases, we also observe that N does not have an h hdi integral value. This kind of behavior suggests that in such cases, there exist two or more stable states with similar free energies, each having the same order parameter but different number of helical strands and all of them contribute to the partition function significantly. One of such cases was observed in figure 4.4 for 45 beads in

ǫ 300 and T 170 250 range, where we observe N 4.3. In all cases, we ∼ ∼ − h hdi ∼ observe a final, stable collapsed globular configuration with residual secondary structure where the number of helices (local residual secondary structure) increases with chain length, i.e., 2, 4 and 5 for chain lengths of 30, 45 and 60 beads, respectively. In addition, the degree of alignment seems to change with changes in chain length. For example, for the 30-bead case the residual helical strands were found to be parallel to each other unlike the 60-bead case where the five residual strands were observed to be in a non-parallel configuration.

The picture that has emerged from our study is also supported by thermodynamic properties such as entropy which is plotted in ﬁgure 4.14 for chain lengths equal to 10, 30, 45 and 60 in the low-temperature regime. For small values of ǫ, we

106 200

150

100 Entropy

0 0 200 400 600 800 1000 epsilon (ε)

Figure 4.14: Plot of chain entropy as a function of epsilon, ǫ, for various chain lengths in the low temperature regime. The parameters are: ( ) N=30 beads, () • N=45 beads and () N=60 beads.

observe that the entropy is close to zero for all the chain lengths studied. This is a consequence of the all-helical conﬁguration of the chain. However, as the various transitions start to occur, we observe a sharp increase in entropy with increasing ǫ.

Note that all the transitions in the entropy occur at same values of ǫ where the other conﬁgurational and conformational properties showed sharp changes. The increase in entropy with each transition suggests that there are many conformations of the chain in each low-temperature regime, similar to the case of the collapsed globule at high temperatures. This provides further proof that the low-temperature regimes can

107 not be associated with speciﬁc structures of the chain but rather with ensembles of various conformations of the chain.

4.5 Conclusions

In this chapter, we have successfully shown that the overall 3-dimensional structure of a helical polymer is the consequence of a delicate balance between short range secondary and long range tertiary interactions. In our simulations, this interplay has been shown to occur at low temperatures where both interactions are significant. The relative strength of tertiary interactions has been shown to have a significant effect on the folding or collapsed behavior of a helical macromolecule, even for the case of homopolypeptides. Moreover, this effect increases with an increase in chain length as supported by a larger number of low temperatures transitions at higher chain lengths.

For weak tertiary interactions and at high temperatures, our model predicted the independent occurrence of the helix-coil and coil-globule transitions, respectively.

However, as the temperature was reduced and ǫ was increased, these transitions were coupled and interfered with each other. The result of this coupling was the emergence of a plethora of novel conformational states which were stable at low temperatures. These states were found to be ensemble averages of conformations with similar characteristics; for example, all the conformations in one state had the same number of helical strands. Moreover, they were separated by transition lines where

108 the polymer underwent a substantial change in its conﬁgurational, conformational and thermodynamic properties.

The analysis of many properties led to the conclusion that as ǫ was increased, the helical strands broke into shorter ones which adopted a parallel conﬁguration to minimize the Lennard-Jones interactions. With further increase in ǫ, as helical strands got shorter, the parallel orientation of such strands also got perturbed until the globular state with residual secondary structure was reached. Further increase of the strength of the tertiary interactions did not lead to any new structure of the chain.

Finally, the effect of the geometrical parameters such as bond lengths, bong angles, and others has not been fully explored. Changes in the values of these parameters would definitely affect the distribution of the various possible states in the state diagram, especially in the low T regime. However, we expect that the overall features of the transitions along with the rich phase behavior of state diagrams would still persist independently of the choice of the values of the parameters used.

109 CHAPTER V

STRETCHING A HELICAL POLYMER: INTERPLAY BETWEEN THE

HELIX-COIL TRANSITION AND AN EXTERNAL MECHANICAL FORCE

5.1 Introduction

It is a well-established fact that novel conformations in biological as well as synthetic macromolecules can be induced by the application of a mechanical force [97–101].

These conformations are diﬀerent from the ones generated by more traditional methods like chemical or thermal denaturation and are inaccessible to conventional methods of measurement like X-ray crystallography and NMR [102]. Thus, the application of a mechanical force and the measurement of the resulting strain, generally referred to as Single Molecule Force Spectroscopy (SMFS) have provided a novel perspective on the structure of macromolecules and the determinants of their mechanical stability. Moreover, it has been suggested that the understanding of such force-elongation characteristics of basic structural elements such as α-helices is very important for the design of protein and polymer based micro-machineries [103].

The understanding of mechanical properties of single macromolecules was started by Bustamante, et al. [101] who studied the elasticity of single stranded DNA using magnetic beads. Since then, various novel techniques such as Atomic Force

110 Microscopy and optical tweezers have been used routinely to investigate the elastic characteristics of synthetic [103–105] as well as biological macromolecules, such as DNA [98, 106, 107] and RNA [108], polysaccharides [109, 110], proteins [111, 112] and others. For example, Chu and coworkers [106] studied the relaxation of a single

DNA molecule stretched by elongational ﬂow. Bustamante et al. studied the overstretching of double stranded DNA (B-form) [107] using laser tweezers and observed a reversible folding/stretching transition as a function of force. In contrast, Marko and coworkers [98] observed diﬀerent structural transitions in twisted and stretched

DNA molecule depending upon whether the stretched molecule was allowed to relax or held fixed and characterized various DNA conformations (B, S, P, Z form) based on these transitions. Reif and coworkers have studied the mechanical behavior of polysaccharides such as Dextran [109] and Xanthan [110], as a function of applied force. In case of Dextran, they suggested different regimes of chain conformations governed by entropy, twist in bond angles and extension in bond angles as the force increases. They also identified clear differences in native and denatured polysaccharides, for example in Xanthan, in terms of force-elongation characteristics. Reif and coworkers also studied proteins such as Titin [111], a muscle protein used for structural integrity using AFM techniques. They observed a saw-tooth pattern in applied force with periodicity of 25-28 nm as a function of extension and suggested the unfolding of individual immunoglobin titin segments. Bustamante and coworkers [112] have also studied the stretching behavior of Titin as a function of force using laser

111 tweezers and concluded that it behaves like a highly non-linear entropic spring based on its reversible folding/unfolding characteristics at low and high forces, respectively.

SMFS has been used to study synthetic polymers as well. In this regard, Rief et al. [104] suggested the presence of a helical structure in synthetic polymers such as poly(ethylene-glycol) in water based on single molecule force measurement by AFM and discussed various elastic responses in PEG as a function of force. Along the same lines, Zhang and co-workers [105] also discussed a multiple stranded supras- tructure based on hydrogen bond analysis and suggested that such polymers behave as ideal entropic springs upon the application of force. However, the results by Ikai and coworkers [103] on poly-L-glutamic acid suggested that its helical conformation could be stretched almost fully with continuous increase in force.

The abundant experimental results obtained for synthetic as well as bio- macromolecules using SMFS pose an enormous challenge on the theoretical community who has to develop analytical models to capture the observed experimental behaviors. Developing such models has proven to be a formidable task due to the high complexity of the aforementioned biological macromolecules. Indeed, these systems are highly complex due to many reasons like molecular-level structural properties which include chemical structure and tacticity, and energetic considerations like electrostatic interactions, hydrogen bonding capabilities, preferred dihedral angles, van der Waals interactions, etc. The complexity of these systems at the molecular level results in a hierarchy of structures, proteins (with their primary, secondary and ter-

112 tiary structures) being one of the most common examples. However, as discussed in chapter 1, homopolypeptides such as poly(analine), poly-L-glutamic acid, etc. are much simpler systems to study and have been modeled theoretically by various research groups in recent years in terms of their force-elongation behavior. For example,

Pincus and collaborators [49] studied the eﬀect of an external applied force on the helix-coil transition of a single homopolypeptide chain modeled as a Freely Jointed

Chain and predicted an asymmetric sigmoidal shape with a pseudo-plateau in its stress-strain plot. Halperin and coworkers [48, 113] addressed the extension of he- licogenic polypeptides and rod-coil block copolymers. It has been found that the application of a mechanical force on these polypeptides at temperatures just above their melting temperatures, T ∗, results in a force-elongation behavior that displays two plateaus associated with the co-existence of helical and coil domains. For low to moderate values of the applied force the ﬁrst plateau corresponds to the helix- formation induced by chain extension which is accompanied by the corresponding loss in conﬁgurational entropy. On the other hand, for large forces, the plateau corresponds to the extension-induced melting of the helical structure which occurs when the end-to-end distance exceeds the length of the helix. Kessler and Rabin [114] have also studied the force-elongation behavior of a helical spring and found that under some conditions when the force is increased, the end-to-end distance displays upward jumps (stretching instabilities). They showed that each jump was associated with the

113 disappearance of helical turns and that the number of jumps depends on the length of the spring.

The subject of stretching bio-macromolecules and synthetic polymers has also been investigated through various complementary computational approaches like molecular dynamics and Monte Carlo simulations. The advantages of these methods are clear; as discussed in chapter 1, these approaches provide a molecular-level understanding of the conformations, conﬁgurations and interactions of the macromolecule. For example, Olson et al. [115] have reviewed the deformations in DNA from the perspective of simulations. Okazaki et al. [116] observed a non-monotonic force-elongation behavior by performing molecular dynamics simulations using an

AFM cantilever. In their study, the initial increase in the force was attributed to

Hooke’s law while the subsequent decrease was assigned to the rupture of the α- helical structure. Grunze et al. [117] investigated the equilibrium conformations of poly(ethylene oxide) as a function of extension using ab-initio calculations of potential energy surfaces and demonstrated the occurrence of transitions from the amorphous to the helical conformers in the low force regimes and the stretching of the helical to planar all-trans conformations in high force regime. Recently, de Pablo and collaborators [118] have also studied reversible mechanical unfolding of protein oligomers using

Monte Carlo simulations through the Wang-Landau algorithm by constructing the potential of mean force associated with the unfolding behavior. In their study they calculated the density of states in the framework of an expanded ensemble, which

114 they refer as the EXEDOS (expanded ensemble density of states) method, to analyze the force elongation behavior in term of potential of mean force as the function of extension.

In this chapter, we have explored how a helical semiflexible polymer described by our model responds when a mechanical force is applied to it. The discussion has been restricted to the ‘quasi-static force’ [48] where the rate of equilibration of the configurations of the chain is much faster than the rate of change of the force. Fol- lowing the above line of thought, the force-elongation behavior of helical semiflexible polymers has been computed using Monte Carlo simulations and the equilibrium properties of the chain have been rationalized in the various regimes in terms of its configurational, conformational and thermodynamic properties. The appropriate extension of our model has been described in section 2.5. For this study, chain lengths of 10, 20 and 30 beads were simulated to understand the effect of the external mechanical force. However, only the results for a chain length of 30 beads have been presented. Here, it is worth mentioning that the results for 10 and 20 beads also follow the same trend.

As far as the ﬂow of this chapter is concerned, section 5.2 discusses the tran- sitional conformational changes in the chain as a function of temperature at constant values of force while section 5.3 deals with consequences of applied force at constant temperature. In addition to qualitative comparisons with various experimental ﬁnd-

115 ings, the simulation results have also been compared with the theoretical predictions of Bahut and Halperin for the extension of polypeptides quantitatively.

Figure 5.1: Logarithm of the density of states, gtrue(NH ,Rx), after rescaling as obtained from the Wang-Landau sampling algorithm for N=30 beads.

As mentioned in section 2.5, the density of states were estimated using Monte

Carlo simulations based on the Wang-Landau algorithm and is shown in ﬁgure 5.1 where the logarithm of the density of states is plotted as a function of projection of chain end-to-end distance, Rx and the number of beads in the helical state, NH . The

ﬁgure clearly suggests that the number of states where the chain is in the coil conformation are much higher than those that correspond to the perfect helical structure.

The part of graph where the logarithm of the DOS is zero corresponds to unfeasible conformations, i.e. the simulation did not ﬁnd any conﬁgurations. With this estimate

116 of the density of states, various properties on interest have been evaluated and are discussed below.

5.2 Constant Force Scenario

Let us start by presenting the results for the case of a thirty-bead chain under a constant mechanical force, F . Figure 5.2 shows the mean square end-to-end distance,

R2 , as a function of temperature, T in the range 100-500 K for forces with increasing h i strengths. It should be noted that the unit of force F is Kelvin per unit length. At

F = 0, the polymer chain undergoes the traditional helix-coil transition exactly as discussed in section 3.3. In this case, the equilibrium conformation at low temperature corresponds to an all-helix conformation while, at high temperatures, the chain adopts the random coil conformation. However, as the strength of the force increases, some new features start to appear. It is clear from the ﬁgure that at any temperature,

2 2 R 6 R . This result is the direct consequence of the fact that the h iForce=0 ≥ h iForce=0 force is stretching the polymer chain in its direction and, hence, increasing its mean square dimensions. Figure 5.2 also depicts that for weak forces, the chain dimension decreases with increasing temperature while it increases for strong forces. We refer to the decrease in R2 with increasing temperature as the ‘helix-coil’ transition whereas h i we call the increase in R2 with increasing temperature the ‘helix-extended coil’ or h i ‘helix-stretch’ transition. It is also observed that at suﬃciently high strengths of the force, no transition is observed in the investigated temperature range. This is a result

117 of the high strength of the applied force that destroys the helical structure of the chain even at low temperatures where the chain displays an extended coil conformation.

Figure 5.2: Plot of the mean square end-to-end distance, R2 , as a function of h i temperature for diﬀerent values of the applied force. The parameters are: ( ) F =0, • () F =100, () F =200, (N) F =500, (H) F =1000, (+) F =2000, ( ) F =4000, and ∗ (◭) F =8000.

In addition to previous discussion, ﬁgure 5.2 also highlights another interesting feature related to the behavior of the transition temperature as a function of the applied force. It is observed that the transition temperature T ∗, ﬁrst increases and then begins to decrease with increasing strength of the applied force, F . This trend

118 in T ∗ is more clearly observed in ﬁgure 5.3 where the position of the peak of the heat capacity, which corresponds to T ∗, is plotted as a function of the applied force. Apart from other features which were discussed in the previous paragraph, a maximum in

T ∗ is observed for values of the force close to 1000. Figure 5.3 also shows a change in the behavior of T ∗ for values of the force close to 1000. This crossover indicates a change in the character of the transition from the ‘helix-coil’ to the ‘helix-extended coil’ transition. We have rationalized the behavior of T ∗ and the consequent change in the nature of the transition below.

Figure 5.3: Plot of the transition temperature, T ∗, as a function of the applied force

F .

119 When the system undergoes the ‘helix-coil’ transition at low strengths of the force, this transition is essentially temperature-driven, i.e., temperature is the governing factor for breaking the helical structure. So, weak forces favor the ‘formation of the helical conformation’ and, hence, shift the transition temperature to higher temperatures. The stabilization of the helical structure by the application of a weak force can be rationalized using basic concepts of statistical mechanics as follows. When the force is applied, the entropy of the chain is reduced and the probability of visiting states that correspond to stretched configurations of the chain increases. Among these states are those that include helical sequences which is justified by the higher values of R2 at low T as compared to high T . Thus, stretching the chain increases h i the probability of visiting states with helical domains and, hence, the formation of helical sequences is facilitated by the application of a weak force. On the other hand, the ‘helix-extended coil’ is a force-driven transition which happens for high strengths of the force, i.e. the force is the primary factor for the disintegration of the helical structure. During this transition, the melting of the helical domains is a consequence of the overstretching (with respect to the length of the helix) of the chain. With the application of a strong force in the low temperature regime, the probability of the beads to adopt a helical conformation decreases due to overstretching, this results in a relatively low number of helical beads and consequently, a reduction in the transition temperature. Finally, the transition from one kind of behavior to the other one occurs for strengths of the force near the maximum in figure 5.3 where the effect of force to

120 induce the ‘helix-coil’ transition cancels its eﬀect to induce the ‘helix-extended coil’ transition.

5.3 Constant Temperature Scenario

We now proceed to study how various properties of the chain depend on the applied force when the temperature is kept constant. It has been stated in the literature that during the formation and disintegration of the helical structure due to the application of force two co-existing phases occur in a certain temperature window [113]. One of those phases consists of helical sequences while the other consists of either a random coil or extended coil. This temperature window is governed by factors like the elastic free energy of the stretched coil, Ael, the Zimm-Bragg parameter s, and the ratio

Rhelical ν = Rstretched , where Rhelical and Rstretched correspond to the end-to-end distance of the perfect helix and fully stretched polymer, respectively. The lower limit of this temperature window is Tl where the helix-coil transition takes place in absence of force, i.e. where F is zero, while the upper limit is determined by Tc which is deﬁned as the temperature where the free energy Ael is equal to the free energy of a helix

Ah. Tc is approximately 375 K for our model. Furthermore, at higher temperatures, no co-existing regions are observed while at lower temperatures, there is single co- existing region corresponding to the transformation of helical domains into extended coil structures due to the application of force. Therefore, in this part of our study

121 we explore three temperatures corresponding to three diﬀerent regimes, i.e., T =200

K (T < Tl), T =315 K (Tl Tc).

Figure 5.4: Plot of the mean end-to-end distance, R , as a function of the applied h xi force, F , for diﬀerent temperatures. The parameters are: ( ) T =200 K, () T =315 • K and () T =500 K.

Figure 5.4 shows the mean end-to-end distance, R , as a function of the h xi force for the temperatures mentioned above. R and R are equal to zero for all h yi h zi values of temperature and force due to the symmetry of the problem. It can easily be seen from ﬁgure 5.4 that irrespective of the temperature R is always zero in h xi the absence of force, as expected. As soon as the force starts acting on the chain

122 in the positive X direction, the configurations of the chain shift towards positive values of R , as can be observed in the figure. At low temperatures (T =200 K), h xi figure 5.4 shows an initial steep increase in R for low values of force followed by a h xi plateau and an additional increase for higher values of force before the final plateau is reached. The initial increase in R should not be considered as a co-existence of h xi helical and coil domains. This increase simply reflects the alignment of the helical structure along the direction of the applied force. However, the second steep increase in R , which occurs when F 5000, is indeed a co-existence region due to the h xi ≈ breaking of the helical structure as suggested by theoretical work of Halperin [113].

At high temperatures, T =500 K, there is no such co-existence of helical and coils domains as expected because the helical sequences have been melted. Here, the initial increase of R followed by a plateau region simply reflect the elastic nature h xi of the random coil conformation of the chain at low values of force followed by the limited chain extensibility for stronger forces. Like in the low temperature case, intermediate temperatures (T =315 K) show two steep increases in R as a function h xi of force. The first one occurs when F 200 while the second one happens when ≈ F 3000. Interestingly, unlike in the low temperature scenario, here both increases ≈ in R correspond to coexistence of helical and coil domains. The first coexistence h xi region occurs for weak forces and corresponds to the formation of helical structure from coil domains while the other coexistence region corresponds to the disintegration of helical sequences into extended coils.

123 In order to make our study more balanced and objective, we also carried out a qualitative comparison of the force-elongation curve predicted by our model with a few experimental results as shown in ﬁgures 5.5, 5.6, 5.7 and 5.8 in which the force elongation behavior of Xanthan [110] and DNA (experimental [107, 119] and theoretical [120]) are shown.

In ﬁgure 5.5 we show the force-elongation curve of Xanthan which is an ex- tracellular bacterial polysaccharide that forms an ordered helical secondary structure in its native form stabilized by non-covalent bonds in solution. Though, upon an increase in temperature, Xanthan undergoes a temperature-induced order-disorder transition into a disordered single-strand state, it does show non-linear force elongation characteristics. Figure 5.5 shows two force-elongation curves of native Xanthan.

This ﬁgure shows a monotonic increase in force followed by a relative sharp transition to a plateau. Upon further elongation of the molecule, the force starts to increase again until failure occurs. It can be observed that this behavior is in good qualitative agreement with the force-elongation behavior predicted by our model and showed in

ﬁgure 5.4. It is interesting to note that another polysaccharide called Dextran [109] also shows this kind of force-elongation behavior. In this case, the macromolecule undergoes a transition between two conformations with diﬀerent segmental elasticities.

As far as a comparison with DNA studies is concerned, ﬁgures 5.6, 5.7 and 5.8 clearly show the characteristic plateau in the force-elongation curves for double stranded DNA in the helical form. In all ﬁgures, it is the plateau around 60pN

124 of force which corresponds to the melting of helical domains. It is quite fascinating to observe that our minimal model is able to capture the most signiﬁcant feature of force-elongation curves for molecules as complex as DNA even though it was tailored for helical homopolymers.

Figure 5.5: Force curve of native Xanthan in PBS [110].

The different initial slopes of R for low and intermediate temperatures h xi can be better understood by studying the mean square end-to-end distance, R2 , h i which we have plotted in figure 5.9. This figure clearly shows that irrespective of the strength of the force R2 is never zero. Furthermore, it is clear from the figure that h i

125 Figure 5.6: Force vs. extension proﬁles for EMBL3 λ DNA in PBS with two diﬀerent pulling velocities. (o) 1µm/s and (+) 10µm/s. X-axis represent extension and Y -axis represent Force (pN) [107].

126 Figure 5.7: Stretching of a cross-linked DNA in 150 mM NaCl, 10 mM tris, 1mM

+ EDTA, pH 8.0 buﬀer (black), NA2.5EDTA buﬀer (pH 8) with Na concentrations of

5 mM (red), 2.5 mM (green), and 0.625 mM (blue) [119].

127 Figure 5.8: Force as a function of extension per base pair for the single-stranded,

(thin solid line) and double-stranded (dashed-line) DNA. The bold solid line is the

DNA stretching curve assuming force-induced melting. fcr is the crossover force at which both ssDNA and dsDNA have equal extension and fov is the force where the over-stretching transition occurs [120].

128 Figure 5.9: Plot of the mean square end-to-end distance, R2 , as a function of the h i applied force, F , for diﬀerent temperatures. The parameters are: ( ) T =200 K, () • T =315 K and () T =500 K.

129 at low temperatures (T =200 K) there is no initial increase in R2 for low values of h i force. This directly supports our previous argument that the initial steep increase in R is simply due to the alignment of the helical structure. However, at T =315 h xi K, one can clearly observe an increase in R2 , similar to the one in R as seen h i h xi in ﬁgure 5.4, for low values of the force. This indicates that as the force increases, an increasing number of beads in the coil conformation become part of the existing helical sequences. Another important feature of ﬁgures 5.4 and 5.9 is the fact that the force required to break the helical structure decreases with increasing temperature

(second co-existence region). The breaking of helical structure is essentially a force driven transition, so the higher the temperature is, the more it favors the melting of helical structures and consequently, weaker forces are needed for the transition to occur, as mentioned previously. The remaining features in ﬁgure 5.9 are similar to the ones in ﬁgure 5.4 and can be rationalized using the similar arguments.

Figure 5.10 shows the fraction of helical beads (order parameter) as a function of force, F for three temperatures: 200 K, 315 K and 500 K. At high temperatures,

500 K, no co-existence is observed; the order parameter always remains equal to zero and reﬂects the extension of a random coil. Furthermore, it also corroborates that at high temperatures, the helical conformation is not a feasible equilibrium structure, irrespective of the force that is acting on the chain. On the other hand, the ﬁgure does show a co-existence regime at low temperatures, 200 K, where the sharp decay in the order parameter for high values of force clearly reveals the disintegration of helical

130 1

0.8

0.6

0.4 Order Parameter 0.2

0 0 2000 4000 6000 8000 Force Applied

Figure 5.10: Plot of the order parameter (fraction of beads in the helical state) as a function of the applied force, F , for diﬀerent temperatures. The parameters are: ( ) • T =200 K, () T =315 K and () T =500 K.

structures. Moreover, ﬁgure 5.10 also conﬁrms the formation and disintegration of helical structures as a function of force at intermediate temperatures, 315 K, as demonstrated by an initial increase in order parameter followed by a sharp decrease to zero at higher values of force. Therefore, it corroborates the existence of two co-existing regions of helical and coil domains.

We also compared our results with the theoretical predictions made by Buhot and Halperin [113]. Figure 5.11 shows a quantitative comparison of our result for 315

K with the one obtained using the theory of Buhot and Halperin. The parameters of the theory are: a=5.073, ν=0.961, nh=1134, P =7.23, σ=0.035 and ∆S=-4.17,

131 40

30 > x 20

0 0 2000 4000 6000 8000 10000 F

Figure 5.11: Force-elongation curve at 315 K. Continuous curve (Buhot and

Halperin), △ (our simulation).

where both a and P are expressed in the same arbitrary units used to measure the bond length in our simulation study, and ∆S is in units of Boltzmann constant. It can be observed that the quantitative agreement between the theoretical and simulation results is quite remarkable if we consider that the theory does not account for excluded volume interactions and uses the Freely Jointed Chain model while the simulation does include excluded volume interactions and uses the Freely Rotating

Chain model. This suggests that the force-elongation behavior of the helical polymer does not depend strongly on the local details of the model, as long as the polymer adopts a helical structure.

132 1

0.8

0.6

0.4 Order Parameter

0.2

0 0 2000 4000 6000 F

Figure 5.12: Order parameter as a function of the applied force, F , at 315 K. Con- tinuous curve (Buhot and Halperin after multiplication by 0.83), △ (our simulation).

Figure 5.12 shows the order parameter as a function of the applied force at

315 K using the values of the parameters of the theory mentioned in the previous paragraph. The results obtained from the theory of Buhot and Halperin had to be multiplied by an arbitrary prefactor equal to 0.83 to reach the agreement shown in the ﬁgure. After multiplication the quantitative agreement between simulation and theory is again remarkable. The origin of the prefactor could be attributed to the use of diﬀerent models to describe the polymer chain, Freely Rotating Chain against Freely Jointed Chain and the presence or not of excluded volume interactions.

However, it is surprising to see that these diﬀerences can be accounted for by a simple multiplicative prefactor in the order parameter.

133 Figure 5.13: Occurrence of plateaus in the extension-force curves signaled by the crossing of free energy curves corresponding to a polypeptide chain in a pure coil form (solid line) and in a pure helical form (dashed line) [113].

134 It is pertinent to mention here that the theoretical calculations of Halperin

[113] indicate the presence of three different temperature scenarios, as discussed previously in this section. In this regard, figure 5.13 shows the free energy as a function of extension. The dashed line corresponds to the free energy of a perfect helix, and the solid line corresponds to the free energy of the pure coil as suggested by its parabolic shape. It is clear that wherever the two graphs intersect, we should observe a transition, since at such points the free energy difference between the helical and the coil forms is zero. When T is low, the intersection only occurs on the vertical portion of the dashed line. This corresponds to the transition from the helical state to the stretched or extended-coil state. When T is high, no intersection is observed, and, hence, the system never adopts the helical form, we only observe the elastic nature of the chain as a function of distance. However, when T is between Tl and Tc (in the

∗ ﬁgure, T is same as Tl in our model), the intersection occurs on both horizontal and vertical segments of the dashed line. Here, the horizontal intersection corresponds to the formation of helical structure due the application of force while the vertical intersection corresponds to breaking of helical structure as mentioned before. Each time an intersection occurs, it gives rise to a plateau in the force-elongation curve which is attributed to the co-existence of both possible states.

Finally, it is important to emphasize two quite diﬀerent behaviors of the force-elongation curves in the various experimental and simulation results, namely: the ‘plateau-like’ and ‘sawtooth-like’ behavior. The two diﬀerent behaviors are con-

135 sequences of the governing parameter in the study, i.e. extension- or strain-controlled experiment and force-controlled experiment. In strain-controlled experiments, we observe the ‘sawtooth-like’ behavior of the force-elongation curve and is associated with the rupture/unfolding of individual helical domains/rigid segments of the protein/chain under study. Initially, the force rises with extension, but as soon as a certain segment of the chain/protein unfolds, the force decreases sharply by a significant amount because the unfolded segment can extend very easily up to a certain length. Later on, this cycle is repeated until the whole protein/chain is unfolded or stretched. Here, it is pertinent to mention that such ‘sawtooth’ proﬁle does not appear for molecules like dsDNA even for constant velocity experiments as shown in

ﬁgure 5.6 [119] due to the inter-strand hydrogen bonding nature of its base pairs in both of its strands. So, when the dsDNA is extended, the whole DNA extends until the rupture occurs which leads to the overstretching transition. However, in the force- controlled experiments, the origin of the plateau can be understood with following line of reasoning. In such experiments, the applied force is distributed throughout the length of the chain. So, when the force is large enough, it breaks the folded structure as a whole. Once, the structure breaks down, this force is large enough to stretch the chain fully, that is where the so called plateau is observed in the force-elongation plot. Once the chain is fully extended, the force increases sharply again due to the limited extension of the chain.

136 5.4 Conclusions

In this chapter, the effect of an external mechanical force on a semiflexible polymer capable of forming helices was explored using Monte Carlo simulations. Three possible transitions, namely; ‘helix-coil transition’, ‘helix-extended coil transition’ and ‘coil- extended coil transition’ were observed depending on the simulation conditions and are shown schematically in figure 5.14. In this figure, the tails of the arrows correspond to weak forces while the heads of the arrows indicate strong forces.

Figure 5.14: Schematic of various equilibrium conformations. The arrows represent an increase of the applied force. The dashed line, continuous line and dotted line arrows represent conformational changes at low temperatures (T =200 K), intermediate temperatures (T =315 K) and high temperatures (T =500 K), respectively.

137 In the constant force situation, the changes in the chain dimension were in- dicative of either a ‘helix-coil’ or ‘helix-extended coil’ transition. In addition, the non-monotonic behavior was discussed from the perspective of formation or disintegration of helices due to the application of an external force. However, in the constant temperature scenario it was observed that, depending on the temperature of the system, three possible situations are encountered, as schematically shown in ﬁgure 5.14.

If the temperature is below the transition temperature of helix-coil transition, then the initial application of force ﬁrst orients the helical structure parallel to the force and then it starts melting the helical structure which leads to a co-existence of helical sequences and random (extended) coil domains. A further increase of the applied force leads to a fully extended coil conformation. At temperatures slightly above the transition temperature, the elastic behavior of the chain results in two co-existence regions. Here, weak forces favor the formation of helical beads (order parameter), i.e. the application of force stabilizes the helical conformation. This leads to a co- existence between helical and random coil domains. A further increase of the applied force starts melting the helical structures and leads to a co-existence between helical sequences and extended coil conformations. At high temperatures we found that the elastic behavior of a random coil was recovered by our model.

We would like to ﬁnish the discussion of this chapter with the note that our minimal model did capture the most important characteristics of the force-elongation curves of macromolecules capable of forming helical structures. Furthermore, our

138 model was able to capture the interplay of two important parameters: force and temperature on the conformational changes of helical macromolecules.

139 CHAPTER VI

CONCLUSIONS

In this dissertation, a minimal model of a single semi-ﬂexible polymer has been presented. The model mimics homopolypeptides on a coarse-grained level and was motivated to study their conformational changes during the helix-coil transition. The model mimics the helix stabilization due to intra-molecular hydrogen bonds (short range interactions) in homopolypeptides by a simple, novel and geometric concept of torsion. This concept has shown to capture the cooperativity of the helix-coil transition successfully and is supported by comparisons with experimental observations and theoretical predictions. The cooperativity of the helix-coil transition was further investigated as a function of the stability of interfaces between coil and helical beads.

As the penalty for the formation of interfaces was increased, the transition became more cooperative eventually leading to an all-to-none transition.

The model was extended further to include long-range interactions to study the collapse of helical semi-flexible polymers. The long range interactions were modeled by Lennard-Jones interactions. The interference of both short- and long-range interactions resulted in a rich phase behavior of chain configurations in regions where both interactions were significant. The investigation of the effect of molecular weight

140 suggests that the relative contribution of short- and long-range interactions is critical in deﬁning the collapse behavior of the polymer, as shown by the increase in the richness of the phase behavior with increasing molecular weight.

The force-elongation behavior of a semi-flexible helical polymer was also studied by extending the basic model to incorporate an external mechanical force. The study presented one of the first simulation investigations which attempted to understand the behavior of the conformational changes occurring in a helical polymer when a wide range of forces were applied. Our simulation results were in agreement with theoretical predictions suggesting three different temperature regimes of different conformational changes. The study also confirmed the existence of a temperature window where two co-existing regions of ‘helix-coil’ and ‘helix-extended coil’ were observed as a function of force. This co-existence of two states emerges as a plateau in the force vs. elongation (steep rise in our simulations) curves as observed and predicted by a number of experimentalists and theorists.

It is pertinent to mention here that though it is not included this dissertation, our model has also been used to study mechanical and conformational properties of networks of helical polymers based on the Three-Chain model [121] in terms of stress- strain behavior [122] and its temperature dependence [123]. Furthermore, the eﬀect of counter-ion condensation on the polyelectrolytes capable of forming helices is under investigation currently.

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