Thermodynamics and statistical mechanics of polymers and proteins

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Thermodynamics and Statistical Mechanics of Polymers and Proteins

Michael Bachmann

Dedicated to my family

Contents

Outline 1 Introduction to Biopolymers: Proteins 1.1 The Trinity of Amino Acid Sequence, Structure, and Function . . 1.1.1 Ribosomal Synthesis of Proteins . . . . . . . . . . . . . . 1.1.2 From Sequence to Function: The Protein Folding Process 1.2 Molecular Modeling . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Covalent Bonds . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Effective Noncovalent Interactions and Nanoscopic Modeling: Towards a Semiclassical All-Atom Representation 1.3 All-Atom Peptide Modeling . . . . . . . . . . . . . . . . . . . . 1.3.1 “Generic” Semiclassical All-Atom Peptide Modeling . . . 1.3.2 Simplified All-Atom Modeling with Reduced Parameter Sets 1.4 The Mesoscopic Perspective . . . . . . . . . . . . . . . . . . . . 1.4.1 Why Coarse-Graining...? . . . . . . . . . . . . . . . . . . 1.4.2 The Origin of the Hydrophobic Force . . . . . . . . . . . 1.4.3 Coarse-Grained Hydrophobic-Polar Modeling . . . . . . . 2 Statistical Mechanics: A Modern Review 2.1 The Theory of Everything . . . . . . . . . . . . . . . . . . . . . 2.2 Thermodynamics and Statistical Mechanics . . . . . . . . . . . . 2.2.1 The Thermodynamic Limit . . . . . . . . . . . . . . . . 2.2.2 Thermodynamics of the Closed System: The Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Thermodynamic Equilibrium and the Statistical Nature of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Thermal Fluctuations: The Statistical Path Integral . . . . . . . 2.4 Phase and Pseudophase Transitions . . . . . . . . . . . . . . . . 2.5 Relevant Degrees of Freedom . . . . . . . . . . . . . . . . . . . 2.5.1 Coarse-Grained Modeling on Mesoscopic Scales . . . . . . 2.5.2 Macroscopic Relevant Degrees of Freedom: The FreeEnergy Landscape . . . . . . . . . . . . . . . . . . . . .

vii 1 1 4 6 7 7 9 11 11 13 14 16 17 18 23 23 25 25 27 29 37 40 42 43 44

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Contents

2.6 Kinetic Free-Energy Barrier and the Transition State . . . . . . . 3 The Complexity of Minimalistic Lattice Models Folding 3.1 Evolutionary Aspects . . . . . . . . . . . . . . . . . 3.2 On Self-Avoiding Walks and Contact Matrices . . . 3.3 Exact Statistical Analysis of Designing Sequences . . 3.4 Exact Density of States and Thermodynamics . . .

46

for Protein . . . .

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4 Monte Carlo and Chain Growth Methods for Molecular Simulations 4.1 Conventional Markov-Chain Monte Carlo Sampling . . . . . . . . 4.1.1 Ergodicity and Consequences of Finite Time Series . . . . 4.1.2 Master Equation . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Selection and Acceptance Probabilities . . . . . . . . . . 4.1.4 Simple Sampling . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Metropolis Sampling . . . . . . . . . . . . . . . . . . . . 4.2 Generalized-Ensemble Monte Carlo Methods . . . . . . . . . . . 4.2.1 Replica-Exchange Monte Carlo Method (Parallel Tempering) 4.2.2 Multicanonical Sampling . . . . . . . . . . . . . . . . . . 4.2.3 Wang-Landau Method . . . . . . . . . . . . . . . . . . . 4.3 Lattice Polymers: Monte Carlo Sampling vs. Rosenbluth Chain Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Pruned-Enriched Rosenbluth Method: Go with the Winners . . . 4.5 Canonical Chain Growth with PERM . . . . . . . . . . . . . . . 4.6 Multicanonical Chain-Growth Algorithm . . . . . . . . . . . . . . 4.6.1 General Aspects of Multicanonical Sampling . . . . . . . 4.6.2 Multicanonical Sampling of Rosenbluth-Weighted Chains . 4.6.3 Iterative Determination of the Density of States . . . . . 4.7 Validation and Performance . . . . . . . . . . . . . . . . . . . . 4.7.1 Comparison with Results from Exact Enumeration . . . . 4.7.2 Multiple Histogram Reweighting . . . . . . . . . . . . . . 5 Freezing and Collapse of Flexible Lattice Polymers 5.1 Conformational Transitions of Flexible Homopolymers . . . . . . 5.2 Modeling and Simulation of the Simplest Model for Flexible Polymers: Interacting Self-Avoiding Walks . . . . . . . . . . . . . . . 5.3 Energetic Fluctuations of Finite-Length Polymers . . . . . . . . . 5.3.1 The Expected Peak Structure of the Specific Heat . . . . 5.3.2 Simple-Cubic Lattice Polymers . . . . . . . . . . . . . .

51 51 52 55 61 65 65 65 67 69 70 71 77 77 77 77 77 80 81 84 84 85 86 89 89 90 93 93 94 95 96 96

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5.3.3 Polymers on the Face-Centered Cubic Lattice . . . . . . . 100 5.4 The Θ Transition Revisited . . . . . . . . . . . . . . . . . . . . 103 5.5 Freezing and Collapse in the Thermodynamic Limit . . . . . . . . 106 6 Crystallization of Elastic Polymers 6.1 Relevance of Surface Effects . . . . . . . . . . . . . . . . . . . . 6.2 Lennard-Jones Clusters . . . . . . . . . . . . . . . . . . . . . . . 6.3 Perfect Icosahedra . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Liquid-Solid Transitions of Elastic Flexible Polymers . . . . . . . 6.4.1 Finitely Extensible Nonlinear Elastic Lennard-Jones Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Classification of Geometries . . . . . . . . . . . . . . . . 6.4.3 Ground States . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Thermodynamics of Complete Icosahedra . . . . . . . . . 6.4.5 Liquid-Solid Transitions of Elastic Polymers . . . . . . . . 6.4.6 Long-Range Effects . . . . . . . . . . . . . . . . . . . .

109 109 110 111 113

7 Folding Properties of Hydrophobic-Polar Lattice Proteins 7.1 Lattice Model for Parallel β Helix with 42 Monomers . . . 7.2 Ten Designed 48-mers . . . . . . . . . . . . . . . . . . . 7.3 Beyond 100 Monomers ... . . . . . . . . . . . . . . . . . 7.4 Protein Folding as a Finite-Size Effect . . . . . . . . . . .

127 127 131 135 136

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113 114 116 118 120 124

8 Thermodynamic Properties of Mesoscopic Off-Lattice Heteropolymer Models 139 8.1 Simulations of a Hydrophobic-Polar Off-Lattice Model for Proteins 139 8.2 Similarity Measure and Order Parameter . . . . . . . . . . . . . 141 8.2.1 Root Mean Square Deviation . . . . . . . . . . . . . . . 141 8.2.2 Angular Overlap Order Parameter . . . . . . . . . . . . . 141 8.3 Search for Global Energy Minima . . . . . . . . . . . . . . . . . 144 8.4 Comparative Analysis of Thermodynamic Properties . . . . . . . 147 9 Characteristic Glassy Folding Channels and Kinetics of Two-State Folding 153 9.1 Tertiary Protein Folding from a Mesoscopic Perspective . . . . . 153 9.2 Identification of Characteristic Folding Channels . . . . . . . . . 154 9.3 G¯o Kinetics of Two-State Folding . . . . . . . . . . . . . . . . . 158 9.3.1 The Mesoscopic G¯o Model . . . . . . . . . . . . . . . . . 159 9.3.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 161 9.3.3 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 165

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Contents

9.3.4 Mesoscopic Heteropolymers vs. Real Proteins . . . . . . . 170 9.4 Microcanonical Effects and Definition of Temperature . . . . . . 170 10 Generic Geometries of Strings with Constraints 10.1 The Intrinsic Nature of Secondary Structures . . . . . . 10.2 Polymers with Thickness Constraint . . . . . . . . . . . 10.2.1 Global Radius of Curvature . . . . . . . . . . . 10.2.2 Thickness-Dependent Ground-State Properties . 10.2.3 Structural Phase Diagram of Tubelike Polymers 10.3 Secondary-Structure Phases of a Hydrophobic-Polar Heteropolymer Model . . . . . . . . . . . . . . . . . .

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175 175 176 176 178 180

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11 Statistical Analyses of Aggregation Processes 11.1 Pseudophase Separation in the Nucleation of Polymers . . . . . . 11.2 Mesoscopic Hydrophobic-Polar Aggregation Model . . . . . . . . 11.3 Order Parameter of Aggregation and Fluctuations . . . . . . . . 11.4 Statistics of the Two-Chain Heteropolymer System in Three Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Multicanonical Results . . . . . . . . . . . . . . . . . . . 11.4.2 Canonical Perspective . . . . . . . . . . . . . . . . . . . 11.4.3 Microcanonical Interpretation – The Backbending Effect . 11.5 Aggregation Transition in Larger Heteropolymer Systems . . . . .

185 185 186 188 189 189 192 194 199

12 Hierarchical Nature of Phase Transitions 205 12.1 Aggregation of Semiflexible Polymers . . . . . . . . . . . . . . . 205 12.2 Structural Transitions of Semiflexible Polymers with Different Bending Rigidities . . . . . . . . . . . . . . . . . . . . 206 12.3 Hierarchies of Subphase Transitions . . . . . . . . . . . . . . . . 210 13 Adsorption of Polymers at Solid Substrates 13.1 Structure Formation at Hybrid Interfaces of Soft and Solid Matter 13.2 Minimalistic Modeling and Simulation of Hybrid Interfaces . . . . 13.3 Contact-Density Chain-Growth Algorithm . . . . . . . . . . . . . 13.4 Pseudophase Diagram of a Flexible Polymer near an Attractive Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Solubility-Temperature Pseudophase Diagram . . . . . . . 13.4.2 Contact-Number Fluctuations . . . . . . . . . . . . . . . 13.4.3 Anisotropic Behavior of Gyration Tensor Components . . 13.5 The Whole Picture: The Free-Energy Landscape . . . . . . . . . 13.6 Continuum Model of Adsorption . . . . . . . . . . . . . . . . . .

213 213 214 217 218 218 220 222 223 228

Contents

13.6.1 Off-Lattice Modeling . . . . . . . . . . . . . . . . . . . . 13.6.2 Suitable Energetic and Structural Quantities for Phase Characterization . . . . . . . . . . . . . . . . . . . . . . 13.6.3 Comparative Discussion of Structural Fluctuations . . . . 13.6.4 Adsorption Parameters . . . . . . . . . . . . . . . . . . . 13.6.5 The Pseudophase Diagram of the Hybrid System in Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Comparison with Lattice Results . . . . . . . . . . . . . . . . . .

v

229 230 230 234 235 238

14 Hybrid Protein–Substrate Interfaces 14.1 Steps Towards Bionanotechnology . . . . . . . . . . . . . . . . . 14.2 Specific Peptide Adsorption at Different Substrates . . . . . . . . 14.2.1 Hybrid Lattice Model . . . . . . . . . . . . . . . . . . . 14.2.2 Substrate-Specific Conformational Adsorption Behavior in Dependence of Temperature and Solubility . . . . . . . . 14.3 Selected Semiconductor-Binding Synthetic Peptides . . . . . . . 14.4 Simulation of Semiconductor-Binding Peptides in Solution . . . . 14.4.1 Peptide Model and Simulation Details . . . . . . . . . . . 14.4.2 Temperature Dependence of Energetic Fluctuations and Secondary-Structure Contents . . . . . . . . . . . . . . . 14.4.3 Characterization of Secondary Structures at Room Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Modeling a Hybrid Peptide-Silicon Interface . . . . . . . . . . . . 14.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 Si(100) and the Role of Water . . . . . . . . . . . . . . 14.5.3 The Hybrid Model . . . . . . . . . . . . . . . . . . . . . 14.6 Sequence-Specific Peptide Adsorption at (100) Silicon Surfaces . 14.6.1 Thermal Fluctuations and Deformations upon Binding . . 14.6.2 Secondary-Structure Contents of the Peptides . . . . . . 14.6.3 Order Parameter of Adsorption and Nature of the Adsorption Transition . . . . . . . . . . . . . . . . . . . . . . .

241 241 243 243

15 Summary

273

Bibliography

279

Index

291

244 250 252 252 253 257 261 261 261 263 265 265 267 269

Outline

More than 30 million people, infected by the human immunodeficiency virus (HIV), suffer from the acquired immune deficiency syndrome (AIDS);1 2 billion humans carry the hepatitis B virus (HBV) within themselves and in more than 350 million cases the liver disease caused by the HBV is chronical and, therefore, currently incurable.2 These are only two examples of worldwide epidemics due to virus infections. Viruses typically consist of a compactly folded (often nicely crystallized) nucleic acid (single- or double-stranded RNA or DNA) encapsulated by a protein hull. Proteins in the hull are responsible for the fusion of the virus with a host cell. Virus replication, by DNA and RNA polymerase in the cell nucleus and protein synthesis in the ribosome, is only possible in a host cell. Since regular cell processes are disturbed by the virus infection, serious damages or even the destruction of the fine-tuned functional network within a biological organism can be the consequence. Another class of diseases is due to structural changes of proteins mediated by other molecules, so-called prions. As there is a strong causal connection between the three-dimensional structure of a protein and its biological function, refolding can cause the loss of functionality. A possible consequence is the death of cells. Examples for prion diseases in the brain are bovine spongiform encephalopathy (BSE)3 and its human form Creutzfeld-Jakob disease (CJD)4. A further source for damaging cellular networks is protein misfolding followed by amyloid aggregation. In the case of Alzheimer’s disease (AD), which is a neurodegenerative disease and the most common type of dementia, amyloid beta (Aβ) peptides in sufficiently high concentration experience structural changes and tend to form aggregates. Following the amyloid hypothesis, it is believed that these aggregates (which can also take fibrillar forms) are neurotoxic, i.e., they are able to fuse into cell membranes of neurons and open calcium ion channels. It is known that extracellular Ca2+ ions intruding into a neuron can promote its degeneration. About 24 million, mainly elderly people are currently affected.5 1

UNAIDS/WHO AIDS Epidemic Update: December 2007. WHO Fact Sheet No. 204 (2000). 3 The economic impact of BSE is disastrous. Following USA Today from August 4, 2006, US beef exports declined from $3.8 billion in 2003, before the first mad cow was detected in the USA, to $1.4 billion in 2005. 4 The WHO Fact Sheet No. 180 (2002) reports a rate of one per million people. 5 Alzheimer’s Disease International, Global Perspective 16(3), 1 (2006). 2

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Outline

This exemplified collection of diseases manifests the extraordinary importance of protein research. It comprises the understanding of their synthesis based on the genetic code (gene expression), the characteristics and specificity of folding and aggregation processes, as well as the unraveling of the dynamic and kinetic aspects of biological processes, where in almost all cases functional proteins are involved. Of course, treating patients suffering from these diseases is a medical problem, but revealing the nature of mechanisms behind the functioning of living organisms is an interdisciplinary task for the whole ensemble of natural sciences. This is also the case when it comes to nanotechnological applications. Science and technology meet and partly fuse at smallest length scales. In particular, learning from biological systems – from complex networks of biological functionality or from structural properties of single molecules – has become more and more important and relevant for special-purpose applications on micrometer or even nanometer scales. This also includes the interaction of such soft materials with solid matter, where systematic research is just in the beginning. Examples for such hybrid systems with enormous potential for future applications are, among many others, biosensors in form of adhesion-specific nanoarrays for the identification of proteins in solution and also nanoelectronic circuits on polymer basis. On the experimental side, the progress in the development of high-resolution equipment and new experimental techniques not only allows for detecting of what is happening on atomic scales, but also enables local manipulation of molecules which is essential for the design of specific applications. On the other hand, computational capacities have reached a level that make it now possible to study macromolecular systems also more systematically in simulations of suitable models by means of sophisticated numerical methods. At this point, the challenge for theoretical physics, in particular, is twofold: Firstly, the modeling and analysis of specific molecular structures at atomic scales and secondly, the generalization of the conformational (pseudophase) transitions accompanying structure formations processes within a mesoscopic frame. Both approaches facilitate the systematic understanding of molecular processes that is typically difficult to achieve in experiments. Protein folding, peptide aggregation, polymer collapse, crystallization, and adsorption of polymers and proteins to nanoparticles and solid substrates have an essential feature in common: In all these processes, structure formation is guided by a collective, cooperative behavior of the molecular subunits lining up to build chainlike macromolecules. In this process, polymers and proteins experience conformational transitions related to thermodynamic phase transitions. For chains of finite length, an important difference of crossovers between conformational (pseudo)phases is, however, that these transitions are typically rather smooth pro-

Outline

ix

cesses, i.e., thermodynamic activity is not necessarily signalized by strong entropic or energetic fluctuations. The interest in properties of finite-length polymers and proteins has grown rapidly within the past few years, not only because of the technological advances on the nanometer scale, but in particular due to the fact that the thermodynamics of small-scale systems is the key to the understanding of the biomechanical principles of signal exchange and transport processes being relevant for life, as, for example, receptor–ligand binding between proteins or molecular flow through nanopores. Here, we will disucss results from studies of thermodynamic properties of conformational transitions for single- and multiple-chain protein systems are presented and discussed, with particular interest devoted to folding, aggregation, and adsorption processes to solid substrates. By means of sophisticated Monte Carlo (MC) computer simulations [1], we investigate folding, cluster formation, and adsorption models on different scales, from the simplest mesoscopic lattice representations to complex force-field based microscopic models. From these studies on different levels of detail, we expect to reveal pseudouniversal aspects common to structure-formation transitions of different systems [2]. These approaches serve, for example, to identify suitable effective degrees of freedom, i.e, kinds of order parameters, that allow for the description of transition paths in the typically rugged free-energy landscape parametrized by only very few parameters. The quantitative studies on the basis of new approaches for hybrid protein-substrate models are of importance for the non-empirical designing of future nanotechnological applications, e.g., specific biosensors in biomedicine and organic optoelectronic devices. In the first chapter of this book, we begin with an introduction into the molecular structure and the modeling of biomolecules, exemplified for proteins. Fundamental aspects of thermodynamics and statistical mechanics, with emphasis on finite-size effects, are reviewed in Chapter 2. In Chapter 3, properties of the complete sequence and conformation space are systematically analyzed for short lattice proteins by exact enumeration of a minimalistic hydrophobic-polar heteropolymer model. Computer simulations of larger systems require efficient algorithms. A novel method for lattice polymers and proteins that combines multicanonical sampling and sophisticated chain-growth strategies is introduced in Chapter 4. As a first application of chain-growth methods, the study of homopolymer freezing and collapse transitions on regular lattices is the subject of Chapter 5. In this regard, the influence of surface and finite-size effects upon crystallization of elastic polymers is addressed in detail in Chapter 6. Returning to proteins, folding properties of comparatively long hydrophobic-polar lattice proteins are investigated in Chapter 7. In such studies, lattice effects are inevitable,

x

Outline

but undesired side-effects. Therefore, general thermodynamic folding properties are also analyzed for mesoscopic off-lattice heteropolymer models in Chapter 8 for several sequences of hydrophobic and polar monomers. Highlights of Chapter 9 are the identification and classification of realistic, sequence-dependent folding characteristics of coarse-grained heteropolymers and the analysis of two-state folding kinetics for a heteropolymer which also exhibits significant features of natural two-state folders. Generic local geometries like secondary structures are discussed in Chapter 10 by introducing tubelike polymers, i.e., stringlike objects with constraints. The extension of coarse-grained modeling to multiple-chain systems is described in Chapter 11, where also analyses of aggregation transitions of short heteropolymers in different statistical ensembles are presented. In Chapter 12, we unravel the hierarchical nature of phase transitions by discussing the exemplified aggregation transition of homopolymers. Pseudophase diagrams of adsorption processes of lattice and off-lattice homopolymers to solid substrates are discussed in detail in Chapter 13. Substrate-specific binding of heteropolymers to solid substrates is investigated in Chapter 14. This chapter also includes an atomistic modeling approach for hybrid interfaces and the specification to a realistic system of synthetic peptides near a silicon substrate. Eventually, the most essential points are summarized in Chapter 15. I would like to thank all who have actively and passively contributed to this work: Wolfhard Janke, head of the Computational Quantum Field Theory Group at the Institute of Theoretical Physics of the University of Leipzig; the former and present members of the Soft Matter Systems Research Group, including Thomas Vogel, Stefan Schnabel, Reinhard Schiemann, Anna Kallias, Jakob Schluttig, Christoph Junghans, and Monika M¨oddel. Furthermore, I am indebted to Anders Irb¨ack, Simon Mitternacht, Thomas Neuhaus, Tarık C ¸ elik, G¨okhan G¨oko˘glu, Handan Arkın, Karsten Goede, Marius Grundmann, Annette Beck-Sickinger, Kai Holland-Nell, Peter Grassberger, Hsiao-Ping Hsu, Kurt Binder, David P. Landau, Wolfgang Paul, Federica Rampf, Klaus Kroy, and Ulrich Behn.

1 Introduction to Biopolymers: Proteins

1.1 The Trinity of Amino Acid Sequence, Structure, and Function Proteins are highly specialized macromolecules performing essential functions in a biological system, such as controlling transport processes, stabilization of the cell structure, and enzymatic catalyzation of chemical reactions; others act as molecular motors in the complex machinery of molecular synthetization processes. Chemically, proteins are built up of sequences of amino acid residues linked by peptide bonds. The polypeptide chain consists of a linear backbone with the amino acid specific side chains attached to it. The atomic composition of the protein backbone is shown in Fig. 1.1. Typical proteins consist of up to N = 50, . . . , 3000 residues. The 20 different types of amino acids occurring in bioproteins are shown in Fig. 1.2. The side chains of these amino acids govern the principal specificity of each amino acid in protein folding processes and differ in chemical and physical properties under the influence of the surrounding solvent. Solubility in an aqueous environment is dependent of the occurrence of polar groups in the side chain, such as, e.g., the hydroxylic groups in serine and threonine. Hydrophobic side chains are insoluble and little reactive in a polar environment. Typical large and strongly hydrophobic side chains such as, for example, phenylalanine or tryptophan, possess aromatic rings. Others, like alanine or leucine with methine (-CH), methylene (-CH2), or methyl (-CH3) groups in the side chain, are aliphatic, i.e., these side chains only contain hydrogen and carbon atoms. Primarily only arginine and lysine (positively charged) and aspartic and glutamic acid (negatively charged) contribute explicitly to the total charge of a protein in a neutral environment. In addition, histidine is typically positively charged in a slightly acidic environment, but neutral in neutral solution. The frequency and sequential arrangement of hydrophobic, polar, and charged amino acid residues in the amino acid sequence (also called the primary struc-

2

1. Introduction to Biopolymers: Proteins

resi NH+ 3

Oi−1

Hαi

C′i−1 Cαi−1 Ni Hαi−1

C′i

Cαi

Cαi+1 Ni+1

Hi resi−1

COO−

Hi+1

Oi

Hαi+1

resi+1

Figure 1.1: Atomic composition of the protein backbone. Amino acids are connected by the peptide bond between C′i−1 and Ni . Side chains or amino acid residues (“res”) are usually connected to the backbone by a bond with the Cαi atom (except proline which has a second covalent bond to its backbone nitrogen).

ture) of a protein is mainly responsible for the formation of a stable and unique native conformation. Protein conformations or segments of it are typically classified on different length scales. Parts of protein conformations that form local symmetric substructures are called secondary structures. These include helices, sheets (or strands), and turns. Substructures of this type are common to all linelike objects and can be generally considered as the underlying geometry of linear polymers. The formation of secondary structures is not necessarily connected with the formation of hydrogen bonds, but these structures are essentially stabilized by hydrogen bonds. The whole conformation of a single protein, including secondary structures, is its tertiary structure. The tertiary fold typically consists of a very compact core of hydrophobic residues which is screened from the aqueous environment by a shell of polar amino acids being in direct contact with the solvent. This assembly of separate polar and hydrophobic parts is characteristic for proteins – it reduces entropy and thus ensures stability. Eventually, in large proteins or protein compounds, different hydrophobic domains can form widely independently. The global shape of the macromolecule is then classified as its quaternary structure [3–5]. The understanding of general aspects of the folding of proteins into native conformations is particularly essential, as the shape of the stable three-dimensional geometrical structure of a protein often determines the biological function of a protein – or its malfunction, if the protein has misfolded, refolded, or denatured. The spectrum of protein functions in a biological organism is manifold. Proteins are involved in almost all cell processes. Ion channels and nanopores formed by membrane proteins control ion and water flow into and out of the cell [6]. Figure 1.3 shows the atomic structure of the membrane protein aquaporine, a pore

1.1 The Trinity of Amino Acid Sequence, Structure, and Function + Nζ Oδ

1

Nδ

−

2 Oδ

2

Cǫ Cγ

Cδ Cγ

Aspartic Acid Asp, D

Cγ

1

2

Cδ

1

Cβ

Cγ

Glutamine Gln, Q

2

Sγ

Oγ

Cβ

Cβ

Cβ

Cβ

Leucine Leu, L

Isoleucine Ile, I Cζ

Serine Ser, S

2

Cη

2 Cε

Cδ

2

2

Cζ

1

Nε 2

Hα

Cδ

3 Cε Cγ

1

Cβ

Glycine Gly, G

Tryptophan Trp, W + Nη

Sδ

Cγ

2

3

Cδ

1 Cε

Cδ

1

Cβ

Cγ

Proline Pro, P

Oγ Cγ

1

2

Cβ

Phenylalanine Phe, F

Threonine Thr, T

2

Oη

1

Cǫ

Cζ

2

Nǫ

2

Nδ

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Cǫ

Cζ

Cδ

1

Cδ Cγ Cβ

Methionine Met, M

2

Cβ

1

Nǫ Cβ

Cδ

Nη

Cysteine Cys, C

Cε

Cζ

Cβ

Alanine Ala, A

Cγ

Oǫ

Glutamic Acid Glu, E

Cδ

Cγ

Valine Val, V

Cε

Cδ Cγ

Cβ

Lysine Lys, K

1 Cγ

Cβ

−

Cδ

Cβ

Asparagine Asn, N

Cδ

2

2

Nǫ

1

Oǫ

2

Cγ

Cβ

Cβ

Cγ

1

Oǫ

1 Oδ

Cγ

3

Arginine Arg, R

Cδ

2

Cγ

Cǫ

Cδ Cβ

Histidine His, H

1

2

Cγ Cβ

Tyrosine Tyr, Y

Figure 1.2: The side chains of the 20 amino acids lining up bioproteins. Side chains are bound to the backbone Cα atom, except proline that has an extra bond to the backbone nitrogen N. Heavy atoms have been assigned standard labels [3]. There is no side chain for glycine, the free 2 Cα bond is saturated by the hydrogen Hα .

4

1. Introduction to Biopolymers: Proteins

Figure 1.3: Membrane protein aquaporine (from Ref. [7]).

embedded in the cell membrane that is permeable for water molecules. The efficiency is extreme. Up to 3 000 000 000 water molecules can rush through this pore per second [8,9]. Other cell proteins like actin, for example, are responsible for the mechanical stability of the cell backbone. Actin polymerizes to filaments and these filaments can form stable networks. Beside cell stability, these networks also enable transport processes along these “tracks”, e.g., vesicle transport mediated by myosin proteins. The interplay between actin and myosin is also important for the ability of muscle tissue to contract. Biochemical reactions are catalyzed by enzymes. General structural protein stability but also the ability to locally unfold and refold just allow for receptor-ligand binding processes necessary for enzymatic activity. Large proteins or protein compounds form complex nanoscale machines which act as “molecular motors”, as, for example, in the DNA/RNA polymerases and ATP synthase.

1.1.1 Ribosomal Synthesis of Proteins The information about the amino acid composition of proteins is encoded in the DNA. In the polymerase II process, the genetic code is transcribed to the singlestranded messengerRNA (mRNA) sequence. The translation of the mRNA base code into amino acid sequences, i.e., the gene expression, is mediated by the ribosome. The ribosome itself is a macromolecular compound of large, multiple-

1.1 The Trinity of Amino Acid Sequence, Structure, and Function

5

Figure 1.4: The sequence of amino acids building up proteins is, based on the genetic DNA expression, synthesized in the ribosome [10].

domain proteins. A schematic snapshot of the ribosomal protein synthesis is shown in Fig. 1.4. Always three successive bases along the mRNA strand encode a single amino acid. The size of such a codon is intuitively clear: Since there are four different bases building up the RNA code and the complementary base necessary to form a base pair is unique, a single-base codon could encode only 4 amino acids and two bases 42 = 16 amino acid residues. Since 20 amino acids were identified in typical bioproteins, three bases are required to form a codon. The 43 = 64 possibilities not only allow multiple, redundant codes for amino acids (which is a first kind of genetic error correction), but also enable the definition of start and stop codons which are necessary to separate the codes for the different proteins within the linear RNA sequence [10,11]. After the ribosome has read a codon off the mRNA strand, a transferRNA (tRNA) molecule connects to the mRNA codon. A tRNA molecule mainly contains a three-base section, the anticodon, which is complementary to a specific codon at the mRNA, and the associated amino acid (aa) residue. Thus the tRNA’s serve as translators between the base codons and the amino acids. The ribosome separates the amino acid from the tRNA and attaches it to the already synthesized part of the protein sequence. This process continues until a stop codon is reached. Eventually, the protein is released into the aqueous solvent within the cell. It is widely believed that in this moment the protein is still unstructured and the formation of the functional structure is a spontaneous folding process.

1. Introduction to Biopolymers: Proteins

Free Energy

6

Order Parameter, Reaction Coordinate, Overlap, ...

Figure 1.5: Sketch of a free-energy landscape for a protein under folding conditions as a function of a single cooperativity parameter which is often referred to as a reaction coordinate in analogy to chemical reaction kinetics, as an order parameter by considering the folding process as an analog of a thermodynamic phase transition, or as a kind of overlap parameter similar to what is often used in descriptions of metastable systems such as spin glasses.

1.1.2 From Sequence to Function: The Protein Folding Process Anfinsen’s refolding experiments [12] showed that the native conformation is not a result of the synthetization process in the ribosome. Rather, it is a dynamical process which is strongly dependent on intrinsic properties such as the amino acid sequence, but it is also influenced by the solution properties and temperature of the surrounding solvent. Typical folding times are of the order of milliseconds to seconds. One of the most substantial problems in the understanding of protein folding is the “strategy” the protein follows in finding the unique conformation (geometrical structure), the so-called native fold. Thermodynamically, this native conformation represents the state of minimal free energy. Therefore, protein folding is not simply an energetic minimization process – it is also affected by entropic forces. This means that the folding trajectory is a stochastic process from a mostly random initial structure towards the global free-energy minimum, thereby circumventing free-energy barriers. The free-energy landscape of a protein, considered as a function of the protein’s degrees of freedom is extremely rugged and complex and it seems paradox that the protein is able to find the “needle in a haystick” by a stochastic search process within a relatively short time period. Protein folding is, however, also a process of high cooperativity, i.e., structure formation requires a collective arrangement of at least a subset of the degrees of freedom. For this reason, it is expected that a single or a few cooperativity parameter(s) – comparable to order parameters in thermodynamic phase transitions – allow(s) for the discrimination between dominating macrostates, i.e., the structural “phases”. A strongly simplified sketch of such a free-energy landscape as a function of a single cooperativity parameter is shown in Fig. 1.5. For sta-

1.2 Molecular Modeling

7

bility reasons, the single funnel-like global free-energy valley is sufficiently deep to prevent thermal unfolding. Local folding processes can cause weakly stable or metastable conformations that slow down the folding process. Thus, the folding channel is not necessarily smooth and local free-energy minima can be present in the funnel. The assumption of the existence of a reduced set of relevant collective degrees of freedom thus enables a generalized view of folding processes as structural or conformational transitions and their classification. Indeed, there has been an enormous progress in this direction within the past years, and candidates with comparatively simple folding trajectories were identified. This regards, e.g., single-exponential or downhill folding, where no barriers hinder and slow down the folding process. Another prominent example is two-state folding, a “firstorder-like”, “discontinuous” transition with a single barrier. More complex are “folding-through-intermediates” events with more than one free-energy barrier or even metastability, where the native fold is degenerate and, therefore, the formation of different structures is (almost) likely probable. The latter case is obviously important for proteins involved in mechanical or motoric processes, where local refolding can be necessary for fulfilling a specific biological function.

1.2 Molecular Modeling Structure formation at the atomic level is, in principle, a traditional quantumchemical many-body problem. Amino acids occurring in bioproteins contain between 7 (glycine) and 24 (tryptophan) atoms.1 Thus, typical bioproteins consist of hundreds to ten thousands of atoms. In general, the structural properties of macromolecules depend on two classes of chemical bonds: covalent and noncovalent bonds. Covalent bonds are based on common electron pairs shared between atoms and stabilize the chemical composition of the molecule. On the other hand, noncovalent bonds2 are based on much weaker effective interactions due to screening, polarization effects, or dipole moments, partly induced by the surrounding polar solvent. These interactions are responsible for the three-dimensional structure of macromolecules in solvent.

1.2.1 Covalent Bonds The formation of a covalent bond between atoms is a pure quantum-mechanical effect and due to an effective pairwise attraction between electrons in outer, unsaturated shells of two atoms. This spin-dependent exchange interaction over1 2

Numbers of atoms refer to uncharged amino acids within a polypeptide chain. Sometimes also denoted as nonbonded interactions.

8

1. Introduction to Biopolymers: Proteins

χ2 χ1 φ

ψ

NH+ 3

COO− ω

Phe

Figure 1.6: Definition of the backbone dihedral angles φ, ψ, and ω. Exemplified for phenylalanine, also the only two side-chain degrees of freedom χ1 and χ2 are denoted. The convention is that the torsional angles can have values between −180◦ and +180◦ , counted from the N-terminus − (NH+ 3 ) to the C-terminus (COO ) according to the right-hand rule and in the side chains starting α from the C atom.

compensates the electrostatic repulsion between the electrons and results in an electron pair which is shared by the atoms involved. Covalent bonds are very stable and a thermal decomposition, e.g., at room temperature, is extremely unlikely. The dissolution energy of biochemically relevant covalent bonds lies between 50 kcal/mol (disulfide bridges S–S) and 170 kcal/mol (C=O double bonds). For comparison, the thermal energy at room temperature Tr = 300 K is RTr ≈ 0.6 kcal/mol.3 This energy is also not sufficient to excite vibrations of covalent bonds at room temperature. Therefore, the effective bond lengths, i.e., the distances between the atom cores are rigid. Furthermore, the bond angles between two successive covalent bonds are relatively rigid. Anyhow, weak vibrational fluctuations are thermally excitable, but the typical fluctuation widths are with up to 5◦ comparatively small. In proteins, covalent bonds obviously stabilize the sequence of the amino acids linked by covalent peptide bonds, i.e., its primary structure. Although also torsional degrees of freedom are affected by covalent bonds, a subset of torsional angles is widely flexible: the so-called dihedral torsion angles. Figure 1.6 shows a conformation of a small peptide, with the endgroups − NH+ 3 and COO and the amino acid phenylalanine (Phe) highlighted. The dihedral angles in the backbone are typically denoted φ (torsional angle between atoms C′i−1, Ni , Cαi , and C′i of the (i − 1)th and ith amino acid), ψ (between The gas constant in molar units is R = NA kB ≈ 1.99 × 10−3 kcal/K mol, where NA ≈ 6.02 × 1023 mol−1 is the Avogadro constant and kB ≈ 3.30 × 10−27 kcal/K is the Boltzmann constant. 3

1.2 Molecular Modeling

9

Ni , Cαi , C′i , Ni+1). The angles φ and ψ are comparatively flexible in the interval −180◦ < φ, ψ ≤ 180◦, because the torsional barriers imposed by the electronic properties of the covalent bonds are rather weak. An exception is proline, where the particular geometry of the side chain restricts φ to a value close to -75◦. The angle ω is associated with the torsion of the peptide bond Cαi , C′i , Ni+1, and Cαi+1. However, the sp2 hybridizations of the C′ and N valence electrons and a p-electron uninvolved in the hybridizations that forms an electron cloud surrounding the C′ -N peptide bond entail a large torsional barrier. Thus, ω ≈ 180◦, and Cαi , C′i, Ni+1, and Cαi+1 form an almost planar trans conformation. Proline is special as it is bound to three massive radicals instead of the usual two. For this reason, there is also a non-negligible amount (about 10%) of proline involving peptide bonds in cis conformation (ω ≈ 0◦) [4]. Phenylalanine possesses two torsional side-chain angles, χ1 and χ2 , that can thermally be activated under physiological conditions. Depending on the type of amino acid and its atomic composition, the number of torsional side-chain angles varies. Thus, the three-dimensional geometric structure of proteins is little dependent of covalent bonds, and it is rather due to the much weaker effects between nonbonded atoms. Nonetheless, the rigidity of covalent bond lengths and bond angles affects the process of structure formation (e.g., the folding of a protein into the native state with lowest free energy). Generally, the steric constraints promote frustration and metastability and thus the existence of stable native conformations is a very particular property of the comparatively small number of functional bioproteins selected by evolution. However, the majority of all possible amino acid sequences suffers from degeneration effects, e.g., non-functional metastability, and also from only weakly stable conformations under physiological conditions. For this reason, such sequences play only a very minor role in biological systems (molecular motors, for example, require only small activation barriers for motion, but refolding regards often only small parts of the structure).

1.2.2 Effective Noncovalent Interactions and Nanoscopic Modeling: Towards a Semiclassical All-Atom Representation Noncovalent inter-atomic interactions are induced by van der Waals forces, electrostatic potentials between partial charges of atoms, torsional potentials dipole– dipole arrangements (hydrogen bond formation) and effective inter-monomeric forces, as, for example, caused by the hydrophobic effect, which is due to the interaction of the protein with the surrounding solvent. The standard approaches to calculate ground-state energies, transition probabilities, and quantum-mechanical expectation values for small molecules are quantum chemistry (QC) and densityfunctional theory (DFT), often in combination of analytical and numerical meth-

10

1. Introduction to Biopolymers: Proteins

ods. The bottleneck of these methods is that the electron distributions of heterogeneous macromolecules such as proteins are so complex that these approaches typically fail in predicting a precise energy spectrum of the whole system. The knowledge of the energy spectrum is quite important for the most relevant questions in protein-folding studies: The topology of the free-energy landscape as a function of a small set of relevant “folding coordinates” (similar to reaction coordinates in physical chemistry or order parameters in terminology of statistical physics), its barriers, and thus the possible folding trajectories under physiological conditions4. Thus, protein folding is a thermodynamic process in the presence of kinetic barriers and is often considered as a “conformational transition”, i.e., it exhibits features of thermodynamic phase transitions, although prerequisites of real phase transitions – for example the existence of a thermodynamic limit – are not satisfied. Conformational transitions are thus rather crossovers between different classes of structures (e.g., random coils, helices, and sheets). It should be noted, however, that for small molecules and short segments of large molecules, as, for example, isolated amino acids, QC and DFT are quite useful and results from these calculations enter partly into force fields in semiclassical models. Since a macromolecular quantum-chemical analysis is virtually impossible, the most promising approach employed in the past is investigating dynamics and statistics of semiclassical models. These models are usually based on an energy function consisting of effective potentials for noncovalent interactions. “Effective” here means that quantum-mechanical effects as well as the influence of the surrounding solvent enter into parameters describing individual properties of amino acids, their side chains, but also properties of individual atoms regarding their actual position in the molecule. Due to the large number of constraints (atomic van der Waals radii, fixed covalent bond lengths, energetic torsional barriers), the complexity of this parameter field is enormous and the modeling of the noncovalent interactions is a substantial problem. The parametrization of the associated potentials is frequently based on structural data from NMR or Xray experiments, physico-chemical estimations of enthalpies, bioinformatical data base analyses, but also on quantum-chemical and other theoretical approaches. The main difficulty and major source of error is that in all cases the results for a specific system or a subset of proteins stored in a data base are assumed to be reliable for all proteins. This finally sets up what is called the “force field” and which in certain models comprises of the order of O(103) parameters.

4

We refer to “physiological conditions” as the region of the space of external parameters temperature and pH value, where the aqueous solvent surrounding the protein is fluid and neutral under normal pressure.

1.3 All-Atom Peptide Modeling

11

1.3 All-Atom Peptide Modeling Although the nature of nonbonded atomic and molecular interactions is known in principle, semiclassical models of proteins differ not only in the parametrization of the potentials, but also in the form of the effective energetic contributions. In the following, two models are described as representatives of a large zoo of models which also claim to be “all-atom” representations of proteins. The problem which is common to many of the all-atom approaches is that these models are often gauged against a subset of the protein data bank (pdb) [7]. Within this subset, these models can be applied with some success to structure predictions. In structural predictions for sequences not in the gauge set, these models often fail. A frequent feature of such models is, for example, the overweighting of a certain type of secondary structures, either helices or strands. Another general problem is the enormous complexity of these models and the huge parameter sets, resulting in slow dynamics in computer simulations. This is particularly apparent in molecular dynamics simulations [13] of protein folding ranging from extremely (CPU) time consuming to simply impossible. Reasonable results can currently only be obtained in computer simulations of small peptides – but the understanding of their folding and aggregation behaviors could actually be the key to deeper insights into generic aspects of structure formation of large bioproteins.

1.3.1 “Generic” Semiclassical All-Atom Peptide Modeling In standard peptide models, all atoms and covalent bonds, bond and torsion angles of the molecule are assigned individual parameters mimicking quantum-mechanical effects. Examples of such parameters are van der Waals radii and partial charges of the atoms, lengths of covalent bonds between atoms, angles between successive covalent bonds, torsional angles and torsional barriers. Atomic parameters typically depend on the position of the respective atom in the amino acid and the chemical composition of the amino acid residue, even different parametrizations of the same type of atom are distinguished. The SMMP (Simple Molecular Mechanics for Proteins) [14] implementation of the ECEPP/3 (Empirical Conformational Energies for Proteins and Polypeptides) force field [15] knows, for example, seven different parametrizations of hydrogen, depending on the chemical group it belongs to (e.g., if it is part of an aliphatic, aromatic, hydroxylic or carboxylic, amide or amine group, bound to sulfur or to Cδ of proline). In these models, each atom i, located at the position ri , carries a partial charge qi . Covalent bonds between atoms, according to the chemical structure of the amino acids, are considered rigid, i.e., bond lengths are kept constant, as well as bond angles between covalent bonds, and certain rigid torsion angles. In some

12

1. Introduction to Biopolymers: Proteins

models, this constraint is weakened by allowing the bonds and bond angles to fluctuate slightly. Distances between nonbonded atoms i and j are defined as rij = |ri − rj | and measured in ˚ A in the following. The set of degrees of freedom covers all dihedral torsion angles ξ = {ξα } of αth residue’s backbone (φα , ψα , ωα ) (1) (2) and side chain (χ = χα , χα , . . .). The model incorporates electrostatic Coulomb interactions between the partial atomic charges (all energies in kcal/mol), X qi qj EC (ξ) = 332 , (1.1) εr (ξ) ij i,j effective atomic dipole-dipole interaction modeled via Lennard-Jones potentials5, ! X Aij Bij , (1.2) ELJ (ξ) = − 12 6 r r (ξ) (ξ) ij ij i,j O-H and N-H hydrogen-bond formation, EHB (ξ) =

X i,j

Dij Cij − 12 (ξ) 10 (ξ) rij rij

!

,

and considers dihedral torsional barriers (if any): X Etor(ξ) = Ul (1 ± cos(nl ξl )) .

(1.3)

(1.4)

l

The total energy of a conformation, whose structure is completely defined by the set of dihedral angles ξ, is E0(ξ) = EC (ξ) + ELJ (ξ) + EHB (ξ) + Etor(ξ).

(1.5)

The hundreds of parameters qi , Aij , Bij , Cij , Dij , Ul , and nl are listed in force fields, for example, in the already mentioned ECEPP/3 force field [15], one of the most commonly used all-atom force fields in the past. In typical simulations, the dielectric constant is set to ε = 2, which is the vacuum value. For simulations in implicit solvent (explicit solvent requires additional force fields for the solvent, e.g., water), the model is extended by the solvation-energy contribution, which is given by [17] X Esolv (ξ) = σiAi (ξ), (1.6) i

5

Here we use the standard Lennard-Jones parametrization as provided by the ECEPP/3 force field [15]. It should be noted that there exist more refined effective pair potentials for unbonded atomic interactions as given, e.g., in Ref. [16].

1.3 All-Atom Peptide Modeling

13

where Ai is the solvent-accessible surface area of the ith atom for a given conformation and σi is the solvation parameter for the ith atom. The values for σi depend on the type of the ith atom and are parametrized in a separate force field. A frequently used suggestion is, for example, given in Ref. [17]. The total potential energy of the molecule then reads Etot(ξ) = E0(ξ) + Esolv(ξ).

(1.7)

1.3.2 Simplified All-Atom Modeling with Reduced Parameter Sets An alternative, computationally much more efficient approach for modeling peptides is based on effective interactions with drastically reduced parameter sets. One such model that still contains all atoms of the peptide chain, including H atoms, but no explicit water molecules, has recently been designed by Irb¨ack et al. [18,19]. It assumes fixed bond angles, bond lengths, and peptide torsion angles (ω = 180◦), so that each amino acid has the Ramachandran angles φ, ψ and a number of side-chain torsion angles as its degrees of freedom. The energy function consists of four terms, E = Eev + Eloc + Ehb + Ehp .

(1.8)

The first term, Eev , represents excluded volume effects and is of the form X λij (σi + σj ) 12 Eev = κev , (1.9) r ij i<j where the sum is over all atom pairs. The parameters σi are atomic radii and λij is a scale factor, which is 1.0 for pairs connected by three covalent bonds and 0.75 otherwise. The second term represents an interaction between neighboring NH and CO partial charges along the backbone. It is given by X X qi qj , (1.10) Eloc = κloc rij I

i=N,H ∈I j=C,O ∈I

where the outer sum is over all amino acids and the qi ’s are partial charges. The H bond contribution, Ehb , consists of two parts: backbone–backbone bonds and backbone–side-chain bonds, X X (1) (2) Ehb = εhb u(rij )v(αij , βij ) + εhb u(rij )v(αij , βij ) , (1.11) bb-bb

bb-sc

14

1. Introduction to Biopolymers: Proteins

where rij denotes the OH distance, αij the NHO angle and βij the HOC angle. The function u(r) is given by σ 10 σ 12 hb hb −6 (1.12) u(r) = 5 r r and the angular dependence is ( (cos α cos β)1/2 if α, β > 90◦, v(α, β) = (1.13) 0 otherwise. The last energy term, Ehp , represents an effective hydrophobic attraction and has the form X Ehp = − MIJ CIJ , (1.14) I<J

where the sum is over all pairs of nonpolar amino acids. The parameters MIJ (≥ 0) are constants that determine the strength of attraction between amino acids I and J. CIJ is a geometric factor and a measure of the degree of contact between two side chains. It is defined as   X X 1 2 2   CIJ = f (min rij )+ f (min rij ) , (1.15) j∈AJ i∈AI NI + NJ i∈AI

j∈AJ

where AI denotes a predefined set of NI side-chain atoms for residue I. The function f (x) is given by f (x) = 1 if x < A, f (x) = 0 if x > B, and f (x) = (B − x)/(B − A) if A < x < B [A = (3.5 ˚ A)2 and B = (4.5 ˚ A)2]. The number of parameters needed in this model is about one order less compared to full-detail models based, for example, on the ECEPP force field as described in the previous section. Despite its simplicity and computational efficiency, it is still demanding to perform systematic, comparative studies of folding characteristics among different peptides. But, the introduction of the effective hydrophobic interaction (1.14) in this model shows the way how simplified protein models on a coarse-grained level would have to be designed. This is performed in the following section – driven to extremes.

1.4 The Mesoscopic Perspective Another, completely different vista is opened by employing minimalistic, coarsegrained protein models. Coarse-graining of models, where relevant length scales are increased by reducing the number of microscopic degrees of freedom, has proven to be very successful in polymer science. Although specificity is much

1.4 The Mesoscopic Perspective

15

more sensitive for proteins, since details (charges, polarity, etc.) and differences of the amino acid side chains can have strong influences on the fold, mesoscopic approaches are also of essential importance for the basic understanding of conformational transitions affecting the folding process. It is also the only possible approach for systematic analyses such as the evolutionarily significant question why only a few sequences in nature are “designing” and thus relevant for selective functions. On the other hand, what is the reason why proteins prefer a comparative small set of target structures, i.e., what explains the preference of designing sequences to fold into the same three-dimensional structure? Many of these questions are still widely unanswered yet. Actually, the complexity of these questions requires a huge number of comparative studies of complete classes of peptide sequences and structures that cannot be achieved by means of computer simulations of microscopic models. Currently only two approaches are promising. One is the bioinformatics approach of designing and scoring sequences and structures (and also possible combinations of receptors and ligands in aggregates), often based on data base scanning according to certain criteria. Another, more physically motivated approach makes use of coarse-grained models, where only a few specific properties of the monomers enter into the models. Frequently, only two types of amino acids are distinguished: hydrophobic (H) and polar (P) residues, giving the class of corresponding models the name “hydrophobic-polar” (HP) models (see Fig. 1.7). In the simplest case, the HP peptide chain is a linear, self-avoiding chain of H and P residues on a regular lattice [20,21]. Such models allow a comprising analysis of both, the conformation and sequence space, e.g., by exactly enumerating all combinatorial possibilities. Other important aspects in lattice model studies are the identification of lowest-energy conformations of comparatively long sequences and the characterization of the folding thermodynamics. Since lattice models suffer from undesired effects of the underlying lattice symmetries, simple hydrophobic-polar off-lattice models were defined. One such model is the AB model, where, for historical reasons, A symbolizes hydrophobic and B polar regions of the protein, whose conformations are modeled by polymer chains in continuum space governed by effective bending energy and van der Waals interactions [22]. These models allow for the analysis of different mutated sequences with respect to their folding characteristics. Here, the idea is that the folding transition is a kind of pseudophase transition which can in principle be described by one or a few order-like parameters. Depending on the sequence, the folding process can be highly cooperative (single-exponential), less cooperative depending on the height of a free-energy barrier (two-state folding), or even frustrating due to the existence of different barriers in a metastable regime (crystal

16

1. Introduction to Biopolymers: Proteins H P P P

H

Figure 1.7: Coarse-graining peptides in a “united atom” approach. Each amino acid is contracted to a single “Cα ” interaction point. The effective distance between adjacent, bonded interaction sites is about 3.8 ˚ A. In the coarse-grained hydrophobic-polar models considered here, the interaction sites have no steric extension. The excluded volume is modeled via type-specific Lennard-Jones pair potentials. In hydrophobic-polar (HP) peptide models, only hydrophobic (H) and polar (P) amino acid residues are distinguished.

or glassy phases). These characteristics known from functional proteins can be recovered in the AB model, which is computationally much less demanding than all-atom formulations and thus enables throughout theoretical analyses. It is a common feature of such coarse-grained models that these enable a broader view on the general problem of protein folding, but for precise, specific predictions, their applicability is limited. In analogy to magnetic systems, they are rather comparable with the Ising model for ferromagnets or the EdwardsAnderson-Ising model for spin glasses. It should also be remarked that, due to their nontrivial simplicity, coarse-grained models are also a perfect testing ground for newly developed algorithms.

1.4.1 Why Coarse-Graining...? Functional proteins in a biological organism are typically characterized by a unique three-dimensional molecular structure, which makes the protein selective for individual functions. In most cases, the free-energy landscape is believed to exhibit a rough shape with a large number of local minima and, for functional proteins, a deep, funnel-like global minimum. This assumed complexity is the reason, why it is difficult to understand how the random-coil conformation of covalently bonded amino acids spontaneously folds into a well-defined stable “native” conformation. Furthermore, it is expected that there are only a small number of folding

1.4 The Mesoscopic Perspective

17

Figure 1.8: (a) The C-peptide of ribonuclease A consists of 13 amino acids and is a typical α-helix former. (b) 7-residue segments of the Aβ peptide, associated with Alzheimer’s disease, tend to form planar shapes, so-called β-strands.

paths from any unfolded conformation to this final fold. Protein folding follows a strict hierarchy at different length scales. The primary structure is provided by the ribosome. Since subsequent amino acids are uniformly linked by a covalent peptide bond independent of the geometrical structure of the protein, the typical length scale of the primary structure is a single amino acid. The next level are secondary structures like α-helices, β-sheets, and turns. These substructures which are probably simply intrinsic geometries of any linelike object, are stabilized by backbone hydrogen bonding which typically involves segments of several subsequent amino acids. Therefore, the scale of secondary structures is determined by the typical segment sizes, which are of the order of ten amino acids. Consequently, secondary-structure formation is often the first step in protein folding. This is followed by the formation of global, single-domain tertiary structures. In fact, this process is what renders protein folding special. The main driving force for the folding of a complex domain, i.e., of up to hundreds of amino acids, is an effective cooperative interaction between many amino acid side chains and which is strongly influenced by the solubility properties (in particular its polarization) of the aqueous solvent the protein resides in. Roughly, amino acid side chains can be classified as polar, hydrophobic, and neutral. While polar residues favor contact with polar water molecules, hydrophobic acids avoid contact with water which results in an effective attraction between hydrophobic side chains. In consequence, this attractive force leads to a formation of a highly compact hydrophobic core, which is screened from the solvent by a shell of polar amino acids. For very large proteins, the final stage in the folding process is the quaternary structure, where the size of a protein domain is the typical length scale.

1.4.2 The Origin of the Hydrophobic Force In proteins, the size of individual secondary-structure segments like α-helices and β-strands (for examples see Fig. 1.8) is typically rather small. The reason is that

18

1. Introduction to Biopolymers: Proteins

Figure 1.9: (a) Tyrosine (Tyr) is an amino acid with an OH group in the side chain. Thus, it is hydrophilic as the OH dipole can form a hydrogen bond with a polar solvent molecule (water). (b) Phenylalanine (Phe), on the other hand, is a typical example for a hydrophobic amino acid. The CH2 -C6 H5 side chain does not contain a polar group. Phe in the surface-accessible protein shell would disturb the hydrogen-bond network of the solvent which is energetically disfavored.

proteins are “interacting polymers”, i.e., the amino acids interact with each other and form a globular or tertiary shape. This is due to the fact that amino acids noticeably differ only in their side chains. Adjacent amino acid backbones are connected via the peptide bond and electric dipoles formed by backbone atoms are typically involved in hydrogen-bond formation. Backbone-backbone interaction provides the symmetry of secondary structures. However, the interaction between the non-bonded side chains is non-uniform and strongly dependent of the side chain type. Roughly, two significantly different classes of side chains occur: hydrophilic ones that favor contact with a surrounding polar solvent like water and hydrophobic side chains which are non-polar, thus disfavoring contact with water molecules (for representatives of the two classes see Fig. 1.9). Therefore, the effective force that leads to the formation of a compact hydrophobic core surrounded by a screening shell of polar amino acids is called hydrophobic force. For spontaneously folding single-domain proteins it is the essential driving force in the tertiary folding process.

1.4.3 Coarse-Grained Hydrophobic-Polar Modeling The formation of tertiary hydrophobic-core structures is a complex process. Although atomic details, e.g., van der Waals volume exclusion separating side chains in linear and ring structures, polarizability, and partial charges, noticeably influence the folding process and the native fold, it should be possible to understand certain aspects of the folding characteristics, at least qualitatively, by means of coarse-grained models which are based on a few effective parameters. Minimalistic hydrophobic-polar lattice and off-lattice heteropolymer models, suitable for addressing these questions, are introduced in the following.

1.4 The Mesoscopic Perspective

19

Hydrophobic-Polar Lattice Proteins

The simplest model for a qualitative description of protein folding is the lattice hydrophobic-polar (HP) model [20]. In this model, the continuous conformational space is reduced to discrete regular lattices and conformations of proteins are modeled as self-avoiding walks restricted to the lattice. Assuming that the hydrophobic interaction is the most essential force towards the native fold, sequences of HP proteins consist of only two types of monomers (or classes of amino acids): Amino acids with high hydrophobicity are treated as hydrophobic monomers (H), while the class of polar (or hydrophilic) residues is represented by polar monomers (P ). In order to achieve the formation of a hydrophobic core surrounded by a shell of polar monomers, the interaction between hydrophobic monomers is attractive and short-range. In the standard formulation of the model [20], all other interactions are neglected. Variants of the HP model also take into account (weaker) interactions between H and P monomers as well as between polar monomers [21]. Although the HP model is extremely simple, it has been proven that identifying native conformations is an NP-complete problem in two and three dimensions [23]. Therefore, sophisticated algorithms were developed to find lowest-energy states for chains of up to 136 monomers. The methods applied are based on very different algorithms, ranging from exact enumeration in two dimensions [24,25] and three dimensions on cuboid (compact) lattices [21,26–28], and hydrophobiccore construction methods [29,30] over genetic algorithms [31–35], Monte Carlo simulations with different types of move sets [36–39], and generalized ensemble approaches [40] to Rosenbluth chain-growth methods [41] of the ’Go with the Winners’ type [42–48]. With some of these algorithms, thermodynamic quantities of lattice heteropolymers were studied as well [27,40,44,47–49]. In the HP model, a monomer of an HP sequence σ = (σ1, σ2, . . . , σN ) is characterized by its residual type (σi = P for polar and σi = H for hydrophobic residues), the position 1 ≤ i ≤ N within the chain of length N , and the spatial position xi to be measured in units of the lattice spacing. A conformation is then symbolized by the vector of the coordinates of successive monomers, X = (x1, x2, . . . , xN ). The distance between the ith and the jth monomer is denoted by rij = |xi − xj |. The bond length between adjacent monomers in the chain is identical with the spacing of the used regular lattice with coordination number q. These covalent bonds are thus not stretchable. A monomer and its nonbonded nearest neighbors may form so-called contacts. Therefore, the maximum number of contacts of a monomer within the chain is (q −2) and (q −1) for the monomers at the ends of the chain. To account for the excluded volume, lattice proteins are self-avoiding, i.e., two monomers cannot occupy the same lattice site. The total

20

1. Introduction to Biopolymers: Proteins

energy for an HP protein reads in energy units ε0 (we set ε0 = 1 in the following) X EHP = ε0 Cij Uσi σj , (1.16) hi,j>i+1i

where Cij = (1 − δi+1 j )∆(xij − 1) with 1, z = 0, (1.17) ∆(z) = 0, z 6= 0 is a symmetric N × N matrix called contact map and uHH uHP Uσi σj = (1.18) uHP uP P is the 2 × 2 interaction matrix. Its elements uσi σj correspond to the energy of HH, HP , and P P contacts. For labeling purposes we shall adopt the convention that σi = 0 = ˆ P and σi = 1 = ˆ H. In the simplest formulation [20], only the attractive hydrophobic interaction is nonzero, HP uHP uHP (1.19) HH = −1, HP = uP P = 0 (HP model). Therefore, UσHP = −δσi H δσj H . This parametrization, which we will traditionally i σj call the HP model in the following, has been extensively used to identify ground states of HP sequences, some of which are believed to show up qualitative properties comparable with realistic proteins whose 20-letter sequence was transcribed into the 2-letter code of the HP model [29,31,50–52]. This simple form of the standard HP model suffers, however, from the fact that the lowest-energy states are usually highly degenerate and therefore the number of designing sequences (i.e., sequences with unique ground state – up to the usual translational, rotational, and reflection symmetries) is very small, at least on the three-dimensional simple cubic (sc) lattice. Incorporating additional inter-residue interactions, symmetries are broken, degeneracies are smaller, and the number of designing sequences increases [27,28]. Based on the Miyazawa-Jernigan matrix [53] of inter-residue contact energies between real amino acids, an additional attractive nonzero energy contribution for contacts between H and P monomers is more realistic [21] and the the elements of the interaction matrix (1.18) are set to uMHP HH = −1,

uMHP HP = −1/2.3 ≈ −0.435,

uMHP P P = 0 (MHP model), (1.20)

corresponding to Ref. [21]. The factor 2.3 is a result of an analysis for the inter-residue energies of contacts between hydrophobic amino acids and contacts between hydrophobic and polar residues [53] which motivated the relation 2uHP > uP P + uHH [21]. In the following we call this variant the MHP model (mixed HP model).

1.4 The Mesoscopic Perspective

21

Going Off-Lattice: Heteropolymer Modeling in the Continuum

The lattice models discussed in the previous section suffer from the fact that the results for the finite-length heteropolymers typically depend on the underlying lattice type. It is difficult to separate realistic effects from artifacts induced by the use of a certain lattice structure. This problem can be avoided, in principle, by studying off-lattice heteropolymers, where the degrees of freedom are continuous. On the other hand, this advantage is partly counter-balanced by the increasing computational efforts for sampling the relevant regions of the conformational state space. In consequence, a precise analysis of statistical properties of off-lattice heteropolymers by means of sophisticated Monte Carlo methods can reliably be performed only for chains much shorter than those considered in the lattice studies. In the following, we focus on hydrophobic-polar heteropolymers described by the so-called AB model [22], where A monomers are hydrophobic and residues of type B are polar (or hydrophilic). We denote the spatial position of the ith monomer in a heteropolymer consisting of N residues by xi , i = 1, . . . , N , and the vector connecting nonadjacent monomers i and j by rij = xi − xj . For covalent bond vectors, we set |bi | ≡ |ri i+1| = 1. The bending angle between monomers k, k + 1, and k + 2 is ϑk (0 ≤ ϑk ≤ π) and σi = A, B symbolizes the type of the monomer. In the AB model [22], the energy of a conformation is given by ! N N −2 N −2 X X X 1 C(σi, σj ) 1 (1 − cos ϑk ) + 4 EAB = − , (1.21) 12 6 4 r r ij ij i=1 j=i+2 k=1

where the first term is the bending energy and the sum runs over the (N − 2) bending angles of successive bond vectors. The second term partially competes with the bending barrier by a potential of Lennard-Jones type. It depends on the distance between monomers being nonadjacent along the chain and accounts for the influence of the AB sequence on the energy. The long-range behavior is attractive for pairs of like monomers and repulsive for AB pairs of monomers:  σi, σj = A,  +1, σi, σj = B, C(σi, σj ) = +1/2, (1.22)  −1/2, σi 6= σj . The AB model is a Cα type model in that each residue is represented by only a single interaction site (the “Cα atom”). Thus, the natural dihedral torsional degrees of freedom of realistic protein backbones are replaced by virtual bond and torsion angles. The large torsional barrier of the peptide bond between neighboring amino acids is in the AB model effectively taken into account by introducing the bending energy.

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1. Introduction to Biopolymers: Proteins

Although this coarse-grained picture will obviously not be sufficient to reproduce microscopic properties of specific realistic proteins, it qualitatively exhibits, however, sequence-dependent features known from nature, as, for example, tertiary folding pathways known from two-state folding, folding through intermediates, and metastability [54,55]. The discussion of the capability of mesoscopic models in polymeric structure formation processes will be a central aspect throughout the following chapters.

2 Statistical Mechanics: A Modern Review

2.1 The Theory of Everything In the past century theoretical physicists developed two “fundamental” theories: the theory of gravity based on the concept of general relativity for stellar systems on large scales and quantum mechanics developed to explain physical effects on small, i.e., atomic scales. Quantum mechanics is inevitably connected to Heisenberg’s uncertainty principle and is thus, by definition, a statistical theory. A quantum system is considered to be in a state |ψ(t)i, and its time evolution is described by the Schr¨odinger equation i~

∂ ˆ |ψ(t)i = H|ψ(t)i, ∂t

(2.1)

ˆ = E ˆ1 is the system-specific Hamilton operator the energy E of the where H ˆ and thus E being system is associated to; ~ is Planck’s constant. For H time-independent, the formal solution of Eq. (2.1) reads |ψ(t)i = exp[−iE(t − t0 )/~]|ψ0i with |ψ0 i ≡ |ψ(t0 )i. Re-insertion into Eq. (2.1) yields ˆ 0 i = E|ψ0 i, H|ψ

(2.2)

known as the stationary Schr¨odinger equation. This theory would be completely deterministic, if the stationary state |ψ0 i would be known or could uniquely be identified or prepared by any measurement which in quantum mechanics ˆ 0 i, where is represented by applying a Hermitean operator to the state: A|ψ {|an i | n discrete} may be the orthonormalized, discrete set of eigenstates of ˆ n i = an |an i with the real eigenvalue an . Introducing the dual Aˆ such that A|a state vector han |, orthonormalization requires han |am i = δnm . The probability interpretation can be generalized to any point in time of the dynamics. The unique preparation of a certain quantum P state is, however, impossible. ˆ Expanding |ψ0i in eigenstates of A, |ψ0 i = n cn |an i, and thus cn = han |ψ0 i

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which is a complex number. If |ψ0 i is normalized, the real number |cn |2 = han |ψ0 ihan |ψ0 i⋆ is interpreted as the probability that the system is in the nth ˆ Since the system must be in any eigenstate |an i after the measurement of A. P eigenstate of Aˆ after the measurement, n |cn |2 = 1. An average or expectation P 2 ˆ ˆ = value is obtained as hAi n an |cn | = hψ0 |A|ψ0 i. The uncertainty principle causes that our information, even after “preparation” of a system state, must remain incomplete. This is because two Hermitean operaˆ 6= B ˆ Aˆ and thus the result of the simultanetors do not necessarily commute AˆB ous measurement depends on the order of operations. The most prominent example is the position-momentum uncertainty of a free particle of mass m, xˆpˆ 6= pˆxˆ: Knowing the initial coordinates exactly, no information about the momentum can be gained, i.e., according to Eq. (2.1) the information about the particle position is completely lost in the next moment, |x(t + ∆t)i = exp(−iE∆t/~)|x(t)i, provided E = p2/2m 6= 0.1 The consequence of these general considerations is that only statistical information is naturally provided by quantum mechanics. Quantum states of small systems fluctuate, only the statistical information, represented by expectation values and probability densities, is relevant. This also means that a single experiment is useless: Even possessing exact knowledge about the full spectrum of quantum levels the electron in a hydrogen atom can occupy does not help at all to “guess” the electron state of a single H atom in a single experiment. Only repeating an experiment many times or considering many atoms simultaneously in an experiment will provide the necessary information to obtain the spectrum that resembles the theoretically predicted one. And even this will not be perfect, spectral lines have a broadness and the spectrum is not discrete, it rather resembles a distribution with peaks at the right positions. This is due to the fact that the measurement is performed with a macroscopic equipment which interacts with the system. Also, the system is never isolated. Even in vacuum, particle and field fluctuations occur which influence “the system” – a lesson learned from quantum field theory. Thus what is really measured in experiments of systems on atomic scales is not the pure quantum mechanics of the system itself. Rather, the results reflect the complex and cooperative interaction of many agents. Cooperativity of many particles not only stabilizes our (necessarily restricted) information about the quantum system, it also stabilizes the matter. Macroscopic systems like solids are stable because of the collective behavior of the huge number of small quantum systems interacting with each other. On the other hand, the destabilization of a 1

If E = 0, p = 0, i.e., the momentum is sharply known. In this case, the particle position x is completely unknown. It is impossible to construct a state that violates the uncertainty relation. The quantum state with the smallest uncertainty ∆x∆p = ~/2 is the Gaussian wavepacket.

2.2 Thermodynamics and Statistical Mechanics

25

macroscopic system, such as the melting of a solid, again requires a cooperative effect that leads to a macroscopic phase transition. A laser only works because of the massive spontaneous, coherent and timely emission of radiation. In conclusion, although the quantum nature of small systems is apparent, it only promotes the formation of large stable structures, if many particles cooperatively interact with each other. However, the amount of data that would be required to characterize this macrostate on a quantum level can neither be calculated because of its giant extent nor measured because of principle and experimental uncertainties. Thus, is it really necessary to go down to the quantum level to identify the system parameters which somehow contain the condensed information of the collective behavior of the individual quantum states? Up to here, we have only talked about quantum fluctuations, but at nonzero temperatures, also thermal fluctuations are relevant. Thus, obviously, a theory that allows for the explanation of macroscopic phases and the transitions between these, i.e., a physical “theory of everything” [56], must be of statistical nature.

2.2 Thermodynamics and Statistical Mechanics 2.2.1 The Thermodynamic Limit One of the fundamental and universal conclusions from thermodynamics and its underlying basis, statistical mechanics, is the existence of competing principles of order and disorder. While on the one hand a physical system tends to reduce its energy E which typically leads to the formation of stable, symmetric structures with large-scale order (e.g., solids), the thermal environment of the system provokes distortations or fluctuations that can result in a less ordered state (as in liquids) or, in the extreme case, force complete disorder (gas phase). This tendency to increase the accessible phase space volume, i.e., allowing the system to change its configuration by thermal fluctuations, is closely connected to the entropy S of the system. Lighted from this point of view, it is not surprising that the empirical laws of thermodynamics rule these two quantities: The First Law states the conservation of (internal) system energy and the Second Law associates the direction of the evolution of processes with the system entropy which includes the important case of an isolated system that maximizes its entropy in equilibrium. Eventually, the Third Law declares the entropy to be constant at absolute zero temperature, independently of the control parameters of the system. At absolute zero, the system is in its energetic ground state Emin and since ground states are nondegenerate in natural systems, the entropy is not only minimal at T = 0, it is exactly zero. There is an additional law, the Zeroth Law, which propagates the definition of

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a measure for the thermal state of the system in equilibrium: the temperature. It is only consequent to define it in a way that relates to each other the competing effects of order and disorder: −1 ∂S(E, V, N ) T (E) = , (2.3) ∂E N,V where the subscript at the parentheses denotes the conserved quantities which here are the volume V and the particle number N . Doubtlessly being one of the most powerful theories – it is gererally applicable to all natural systems – it suffers from several conceptual problems. The physical basis of the axioms may be intuitively apparent, its quantitative description is, however, purely empirical and is at this level not based on a fundamental theory. This particularly regards the precise definition of the entropy. Thermodynamics cannot explain the diversity of structures belonging to the same thermodynamic phase, fluctuations are not considered. A macrostate is paramtrized by only a small number of macroscopic parameters: energy, entropy, temperature, pressure, volume, and chemical potential. Due to its long history and its empirical nature, the interpretation of thermodynamics was always closely connected to the available experimental methods and the systems that could be investigated. Thermodynamic systems were typically considered as “large”, at least large enough to neglect the individual microstates of each of the particles and the interactions among them. However, all these contributions enable the necessary cooperativity to form an ordered or disordered macrostate. In other words, thermodynamics makes heavy use of cooperativity without allowing for the introduction of this concept. The microscopic basis of thermodynamics is provided by the statistical mechanics. Interestingly, for a long time, it has not been considered as a stand-alone theory with all of its consequences, but rather as the microscopic theory behind thermodynamics. Thus, systems remained “large”, only the discussion of the thermodynamic limit (N → ∞, V → ∞, but N/V = const.) seemed to be of interest, which is justified for very large macroscopic systems, of course. However, finite-size effects, which by definition vanish in the thermodynamic limit, were not taken seriously for a long time. The identification of smallness effects in experiments has been and is still difficult. On the theoretical side, the situation was comparable. The idea of universality initiated the introduction of statistical field theory and renormalization group theory which provide information only in the somewhat diffuse “scalefree” regime near macroscopic phase transitions. To satisfy this huge interest in macroscopic effects, also computer simulations were performed. Since this is only possible for

2.2 Thermodynamics and Statistical Mechanics

27

finite systems, the hierarchical finite-size scaling theory was developed to get rid of the finite-size effects. This works fine for phase transitions where the entropy is still dominant in both phases which is the case in second-order or continuous transitions. If the entropy changes rapidly during the transition, as in first-order or dicontinuous phase transitons, and becomes small in the low-temperature phase, energetic effects dominate this type of transition. An example is crystallization which does not allow for a standard finite-scaling approach. Nucleation processes are governed by the competition of particles at the surface to the environment and in the bulk, and this renders each finite system unique. Systems which gained much attention more recently, are the biomolecules. These are rather small polymers and because of its functionalized chemical composition they sustain their size for all time. A finite-size scaling approach does not make sense at all. The folding of proteins, for example, is only due to finiteness effects under thermodynamic conditions. It is, therefore, necessary to develop or adapt concepts for understanding cooperative effects also for systems on rather small and mesoscopic length scales (nm to µm) [57].

2.2.2 Thermodynamics of the Closed System: The Canonical Ensemble For the molecular systems considered throughout this book, it is sufficient to concentrate on the situation, where the (small) system of interest is coupled to a “large” heatbath which provides the constant canonical temperature T . We further assume that the system-accessible volume V is constant and also is the size of the system, parametrized by the particle number N . This is the standard canonical (or NVT) ensemble. Thermodynamically, one speaks of a closed system (since dV = 0 and dN = 0) which together with the heatbath forms a thermally isolated system. The total energy Etot cannot change by definition and thus dEtot = dE + dEbath + dEint = 0. The energy is not necessarily an extensive variable as long as the interaction between the system and the heatbath, Eint , cannot be neglected. Extensitivity would mean that the energy of the combined system is simply be the sum of the energy of the system (E) and of the heatbath (Ebath). Typically, the coupling between the system and the heatbath is considered to be “sufficiently weak” to justify the assumption of extensitivity of the energy. Later on, in the discussion of aggregation processes of small systems, we will see that the energy is nonextensive if surface effects at the boundary of coupled small systems are relevant. In equilibrium, heatbath and system decouple entropically, such that dStot = dS + dSbath = 0. The canonical temperature T is for the finite system not necessarily identical to the caloric or microcanonical temperature Tmicro(E) defined in Eq (2.3). One

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must keep this in mind to better understand, why it is difficult to clearly identify transition temperatures in the subsequent discussions of structural transitions experienced by small molecular systems. The reason is that for constant heatbath temperature, the system energy E cannot be constant, since energy from the heatbath is pumped into the system or energy is dissipated from the system and absorbed by the heatbath. These energy transfers are mediated by the system– heatbath coupling degrees of freedom. It is, therefore, more useful to introduce an average energy, called the internal energy U . In the beginning of this section, the entropy S = S(E, V, N ) was introduced in dependence of the natural variables E, V , and N (all other thermodynamic quantities are dependent on the natural variables). Replacing E by U and inverting the relation for S, we obtain for the internal energy U = U (S, V, N ), or in differential form, ∂U ∂U ∂U dU (S, V, N ) = dS + dV + dN. (2.4) ∂S N,V ∂V S,N ∂N S,V

In analogy to Eq. (2.3), it is suitable to express the canonical temperature by the derivative of U with respect to S, −1 ∂U ; (2.5) T = ∂S N,V

T dS is associated with the heat exchange in the system. Since −pdV is the mechanical work necessary to change the volume, p = −(∂U/∂V )S,N corresponds to the pressure. The chemical potential is defined as µ = (∂U/∂N )S,V . Substituting these expressions, one recovers the differential form of the First Law: dU (S, V, N ) = T dS − pdV + µdN.

(2.6)

F (T, V, N ) = U (S(T, V, N ), V, N ) − T S(T, V, N ),

(2.7)

Thus, in the canonical ensemble dU = T dS, i.e., the internal energy can only be changed by heat transfer. Since the entropy is difficult to use as a control parameter in an experiment and it is also not a variable being representative for the ensemble (as are T , V , and N ), it would be useful to replace S by T . Since in Eq. (2.6) S is an independent variable and T not, we have to perform a Legendre transformation from the internal energy U (S, V, N ) to a thermodynamic potential called “free energy” by or differentially dF = −SdT − pdV − µdN . In the canonical ensemble, where dF = 0 in equilibrium, the free energy is the central thermodynamic potential, i.e., at the phase transition between two phases A and B, FA = FB , which defines the transition point.

2.2 Thermodynamics and Statistical Mechanics

29

2.2.3 Thermodynamic Equilibrium and the Statistical Nature of Entropy The introduction of the entropy in the context of thermodynamics was necessary to account for the tendency of a thermal environment to “disorder” a system. However, everything is based on the empirical rule given by the Second Law stating that T dS ≥ dU + pdV − µdN . In a thermally isolated system, where dU = 0, dV = 0, and dN = 0, the entropy increases until it has reached a maximum value which defines the thermal equilibrium state, where dS = 0. Hence, in the closed (or canonical) system, dS ≥ dU/T and thus (∂S/∂U )N,V ≥ T −1. In thermal equilibrium, the equality corresponds to the expression used to introduce the temperature in Eq. (2.5), and T is a unique measure associated to the equilibrium state. A prominent consequence of this relation is the heat transfer when thermally coupling two isolated systems I and II which shall be in equilibrium at temperatures TI and TII , respectively. After the coupling, the systems shall be closed systems, i.e., heat exchange is possible, but no particle transfer and no volume change. The combined system remains isolated, which entails dUtot = 0. This corresponds to the canonical situation of coupling a system to a heatbath: We expect the hotter system to transfer energy to the colder system until the equilibrium state of the total system is reached, where dStot = 0. The differential total entropy is given by 1 1 dStot = dSI + dSII = dUI + dUII . (2.8) TI TII Since the systems are coupled in a way that does not allow for the interaction of the systems, dUint = 0, and thus dUtot = dUI + dUII = 0 which implies dUII = −dUI . Inserting this into Eq. (2.8) yields 1 1 dStot = − dUI , (2.9) TI TII which gives insight into the direction of heat transfer. As long as, for example, TI > TII , the internal energy of system I decreases, dUI < 0 (and dUII > 0), since the Second Law requires dStot ≥ 0 for the combined, isolated system. In the irreversible nonequilibrium process of heat exchange, the temperatures in systems I and II change with time until the equilibrium is reached, where the temperatures are in both systems identical and the entropy indeed becomes maximal, dStot = 0. The somehow curious conclusion from these considerations is that with the entropy a fundamental quantity is introduced into thermodynamics which, however, is difficult to imagine and hardly accessible in direct experimental measurements. Therefore, the origin of this quantity cannot be of macroscopic nature and must

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be provided by a microscopic theory. The main assumptions postulated for this quantity are that it represents the thermal disorder of a system, it is extensive, and it takes its maximum value in the equilibrium state of the system. In the above discussed example of the two coupled systems, we have learned that energy is transfered from the hotter to the colder system. In consequence, the originally colder system takes up heat and the internal energy increases; but also the entropy of this system must increase.2 The increases of internal energy, entropy, and temperature combined with the empirical observation that symmetries or structures in the system become smaller or “melt” lead to the conclusion that the system state is getting more disordered under these conditions. It is, therefore, plausible to associate the entropy to the number of microstates which are accessible to the system. A microstate is a “snapshot” of the individual classical or quantum-mechanical states of the individual particles in the whole system. Such an indiviual state is in atomic systems the quantum state of the nth particle |an i; on larger scales, it can be sufficient to characterize it classically by the mechanical particle state given by position and momentum, sn = (xn, pn ). The microstates of an N -particle system are then given by the state vectors |a1 a2 . . . an . . . aN i or (s1 , s2, . . . , sn, . . . , sN ), respectively. Typically, an equilibrium ensemble of microstates dominates at a given temperature and represents the macrostate. Let us denote the number of all possible microstates in a system by W. Returning to the example of the two coupled systems, the respective numbers of microstates are WI and WII before the systems are coupled. Since each microstate in system I can be coupled with WII microstates in system II (or vice versa), the total number of microstates in the coupled system is multiplicative: Wtot = WI WII . Since the entropy shall be an extensive quantity, Boltzmann defined it as the logarithm of W: S = kB ln W, (2.10) where kB is the Boltzmann constant and is introduced into this relation to give S a physical dimension. This famous formula connects macroscopic thermodynamics with microscopic statistics. Since in the canonical ensemble a macrostate is expressed by T , V , and N , also W shall depend on these quantities. However, since the actual choice of a microstate does not only have an a priori probability ∼ W −1, it will also depend on the energy of the microstate and its energetic degeneracy. It is, therefore, desireable to generalize Eq. (2.10). For simplicity and without loss of generality, we concentrate on a discrete system with microstates 2

The entropy of the originally hotter system actually decreases, but the total entropy increases. The total entropy would remain unchanged only in the trivial case that the temperatures of the separate systems were already identical before the coupling. A decrease of the total entropy in an isolated system is impossible; otherwise the Second Law would be violated.

2.2 Thermodynamics and Statistical Mechanics

31

i that have P energies Ei. A microstate has a normalized probability of realization pi with i pi = 1. For a given temperature, we have to take into account that the microstates that form the ensemble which represents the macrostate of the system at this temperature possibly possess different energies. A microstate-based entropy definition would not be sufficiently general. Therefore, we average over all possible microstates and weight their contributions to the entropy with the probability of their realization, pi . Then, the expression (2.10) is replaced by S = −kB hln pi,

(2.11)

P with the definition of the statistical average hOi = i Oi pi . This statistical definition of the entropy and the principle of maximum entropy in equilibrium allows to determine the probability pi that a certain microstate is realized. Let us introduce the functional X 1 Φ[p] = S[p] + λm Cm [p], (2.12) kB m P where S[p]/kB = − i pi ln pi is the entropy functional and Cm[p] is a constraint as, e.g., the normalization of the probability is. The constraints are coupled into the functional by constant Lagrange multipliers λm , such that λm = ∂Φ[p]/∂Cm[p]. The maximization of the entropy under constraints is then performed by setting the variation of the functional (2.12) to zero: δΦ[p] = 0.

(2.13)

Since the microstate probabilities will depend on the constraints introduced, different thermodynamic situations correspond to different statistical “ensembles” of microstates. The most prominent ensembles of microstates are discussed in the following. Microcanonical Ensemble

The most obvious constraint is the P normalization of the probability pi which can simply be written as C1 [p] = i pi − 1. Inserting this into Eq. (2.12) and performing the variation (2.13) leads to X δpi (− ln pi − 1 + λ1 ) = 0. (2.14) i

Since arbitrary independent variations δpi can be performed for different microstates i, the left hand side only vanishes if the expression in the parantheses is zero. Hence, pi = exp(λ1 − 1) = const., i.e., all microstates have

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the of realization. Since the normalization constraint requires P same probability P i 1 = 1, it is useful to define the total number of available i pi = exp(λ1 − 1) microstates P which is called the partition sum and reads in this ensemble trivially Zmicro = i 1 = exp(1 − λ1 ). Thus, the probability for a microstate is a uniform distribution: pi =

1

Zmicro

= const.

(2.15)

For a physical system it is always convenient to group microstates with respect to their energies Ei, as the energy is a natural quantity that represents the state of a system. However, it is necessary to assume that energetic states can be and are typically degenerate, i.e., different microstates i and j can possess the same energy Ei = Ej . However, the degeneracy or the number of microstates gE P for a given energy itself will vary with the energy such that gE = i δE Ei is an energetic distribution. It is straightforward to extend this relation to systems with continuous state and energy space, where the number of states gE is replaced by the density of states g(E). In this light, the very general microstate probability (2.15) is not very helpful for the ensemble of all microstates. However, since we can assume that in a sufficiently small energy interval E − ∆E < E < E + ∆E (0 < ∆E ≪ |E|), the number of states will not noticeably change with E. Then, the assumption that all microstates in this microcanonical ensemble possess almost the same energy E ≈ const. is justified and the probability of realization for all microstates in the given energy shell is identical, as Eq. (2.15) states. Assuming for simplicity E = const, the partition sum P ensemble identical with the Pis in the microcanonical number of states Zmicro = k gEk δE Ek = gE = i δE Ei , where k numbers the energy levels and i labels the microstates as before. Thermodynamically, the microcanonical ensemble corresponds to a large isolated system, i.e., to a system with almost no energetic fluctuations in which case indeed U ≈ E ≈ const. Since in equlibrium S has reached its maximum value, this system possesses in equilibrium a caloric temperature Tmicro = (∂SE /∂E)−1. Since pi ≡ pE = gE−1 (under the idealized assumption E = const), we obtain SE = kB ln gE ,

(2.16)

i.e., the temperature Tmicro is statistically related to the change of the number of states with energy. Because of these properties, a microcanonical ensemble of microstates ideally represents a heatbath with fixed temperature Tmicro.

2.2 Thermodynamics and Statistical Mechanics

33

Canonical Ensemble

Embedding a (smaller) closed system into a heatbath, it will by heat transfer finally assume in equlibrium the heatbath temperature. This does not mean that in equilibrium the energy exchange between heatbath and system stops. Via the heatbath coupling, the system can gain energy from the heatbath by fluctuations and lose energy to it by dissipation. Thus, the system energy Ei of a current system microstate i will change if by the interaction with the heatbath a new microstate j is formed which possesses a different energy Ej 6= Ei. Whereas the energetic fluctuations and thus E can in principle take any value, the internal energy of the system, U , is constant in thermal equilibrium at a given constant temperature T , because dU = T dS and S takes its maximum value in equilibrium.3 Hence, it is useful to introduce the statistical average hEi and to relate it to the internal energy: U = hEi = const. For the derivation of the microstate probability pi from the principle of maximum entropy, it is therefore necessaryP to introduce into the functional (2.12) a second constraint C2 [p] = hEi − U = i pi Ei − U in addition to the normalization constraint C1 . Variation of the functional (2.12) on equal footing as before and setting the variation to zero leads to pi = exp(1 − λ1 − λ2 Ei). The Lagrange multiplier λ1 is determined from the normalization condition for pi and we define P from it the partition sum of the canonical ensemble as Zcan = exp(λ1 −1) = i exp(−λ2 Ei). Thus, for the microstate probability, pi =

1 −λ2 Ei . e Zcan

(2.17)

Since λ2 and Zcan are constants with respect to the energy, the canonical microstate probability pi is an exponentially decaying function of the microstate energy, in contrast to the uniformly distributed microstate propability in the microcanonical ensemble. The internal or mean energy is given by U = hEi = P i Ei exp(−λ2 Ei )/Zcan . In order to determine λ2 , we insert expression (2.17) into the entropy functional which yields S = kB ln Zcan + λ2 kB U. (2.18) Multiplying by T and comparing this result with Eq. (2.7), we can easily identify the terms from our statistical consideration with the macroscopic quantities from thermodynamics. Thus, the free energy of the system is related to the partition 3

The canonical temperature T of the closed system is identical with the microcanonical heatbath temperature, heatbath i.e., T = Tmicro . However, this does not mean that also the canonical and microcanonical system temperatures system coincide; actually Tmicro = T is only valid in the thermodynamic limit, where the energetic fluctuations of infinitely large systems vanish.

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sum via F (T, V, N ) = −kB T ln Zcan (T, V, N )

(2.19)

and the Lagrange multiplier λ2 corresponds to the inverse thermal energy, which is typically called β = 1/kB T . Finally, the canonical microstate probability can be written as 1 −Ei /kB T e . (2.20) pi = Zcan P and the partition sum is Zcan = i exp(−βEi). Although pi is the most general probability in the canonical ensemble, it is often useful to introduce the canonical energy distribution by grouping all microstates with the same energy. Since for a given temperature the partition sum is a constant independently of the grouping of microstates, we can rewrite it as X X X XX X −βE −βEi −βEi δE Ei = gE e−βE , = e = δE Ei e Zcan = e i

E

i

E

i

E

(2.21) where gE is again the number (or in continuum the density) of states. It is therefore convenient to introduce the canonical energy distribution by pcan,E =

1 gE e−βE , Zcan

(2.22)

or for a system with continuous (as in a classical system) or quasi-continuous (as in a quantum system with many close energy levels like in a solid) energy space, −1 −βE pcan (E)dE = Zcan g(E)e dE, in which case the partition sum is expressed by R the integral Zcan = dEg(E)e−βE and is called partition function. The internal or mean energy can now be written in the form X 1 U (T, V, N ) = hEi(T, V, N ) = gE (V, N )e−β(T )E (2.23) Zcan (T, V, N ) E

∂ ln Zcan (T, V, N ) (2.24) ∂T and obviously depends on the temperature T . In order to verify the expression (2.5) for the canonical temperature as introduced in the thermodynamical formalism, we invert the relation (2.23) to get T = T (U, V, N ). Then, the entropy (2.18) can also be written as a function in these variables and reads = kB T 2

S(U, V, N ) = kB ln Zcan (T (U, V, N ), V, N ) +

U T (U, V, N )

(2.25)

and the derivative with respect to U reproduces indeed the expected result (2.5). However, since U is constant in the canonical ensemble whereas E is not, T and

2.2 Thermodynamics and Statistical Mechanics

35

0.014 0.012

pcan (E)

0.010

360K

T = 280K

0.008

340K 300K

0.006

321.5K

0.004 0.002 0.000 200

220

240

260 280 E [kcal/mol]

300

320

Figure 2.1: XXXXXXXXXXXXXXXXXXXXXXXXX Canonical probability distribution for a system of four Aβ16-22 peptide segments (KLVFFAE, cf. Fig. 1.2 for the letter code) plotted for different temperatures. The width of the distribution is getting large close to the temperature T ≈ 321.5K, where the aggregation transiton of these peptides occurs. The aggregate consists of four planar β sheets, see Fig. 1.8(b).

Tmicro need not to be identical. The energy distribution (2.22) is exponentially supressed for high energy values which contribute only significantly at high temperatures. At low temperatures, pcan,E is dominated by microstates with small energies. Thus, if the system is not in a macrostate close to a phase transition with phase coexistence, pcan,E will typically possess a single peak with a maximum at a certain energy. This peak becomes sharper with increasing system size and in the thermodynamic limit microstates with the peak energy Emax dominate the whole ensemble. Then, hEi ≈ Emax , such that ∂S/∂U ≈ ∂S/∂Emax, i.e., microcanonical and canonical temperatures coincide. Thus, in the thermodynamic limit, the canonical and the microcanonical ensemble yield identical quantitative informations about the thermodynamic behavior of the system. The Grand Canonical Ensemble

Another frequently required demand is the fluctuation of the number of particles N in the system and to keep the chemical potential µ constant. This is the physical situation which is statistically provided by the grand canonical ensemble. We have already introduced the constraints with respect to the normalization of the microstate probability and the constant mean value of energy. A system that exchanges particles with its environment can only be in equilibrium with the surrounding “particle bath”, if the average number of sys-

36

2. Statistical Mechanics: A Modern Review

tem particles is constant. P This requires the introduction of a third constraint C3 [p] = hN i − N = i Ni pi − N , where Ni is the number of particles in the i microstate. This constraint is coupled with the Lagrange multiplier λ3 into the functional (2.12). Variation of the functional Φ[p] under all these constraints yields pi = exp(1 − λ1 − λ2 Ei − λ3Ni ) = exp(−λ2Ei − λ3 Ni )/Zgrand and T S = kB T ln Zgrand + λ2 kB T U + λ3 kB T N . In order to connect this relation to thermodynamics and for the identification of the Lagrange multipliers, we introduce in analogy to the free energy of closed systems the grand canonical potential Ω(T, V, µ) as the Legendre transformation of the free energy with respect to N and µ: Ω(T, V, µ) = F (T, V, N (T, V, µ)) − µN (T, V, µ) = U (T, V, µ) − T S(T, V, µ) − µN (T, V, µ). Comparison with the result from the variation gives Ω(T, V, µ) = −kB T ln Zgrand (T, V, µ), λ2 = 1/kB T = β, and λ3 = −µ/kB T = −µβ. Hence, the microstate probability in the grand canonical ensemble reads 1 −β(Ei −µNi) e (2.26) pi = Zgrand P with the partition sum Zgrand = i exp[−β(Ei − µNi )]. The entropy is finally given by S(T, V, µ) = kB ln Zgrand (T, V, µ) +

1 [U (T, V, µ) − µN (T, V, µ)]. T

(2.27)

Substituting T by T (U, V, µ) entails S(U, V, µ) = kB ln Zgrand (T (U, V, µ), V, µ) 1 [U − µN (T (U, V, µ), V, µ)]. + T (U, V, µ)

(2.28)

The grand canonical temperature can thus be expressed via T ≡ Tgrand = (∂S/∂U )−1 V,µ.

(2.29)

The derivation of the ensembles presented in this section is completely general and also valid for quantum systems. However, it is not complete as quantum statistics of quantum particles such as photons, electrons, protons, etc., requires the additional consideration of fermionic and bosonic particle symmetries, respectively, in dependence of their particle spin. In all applications presented in this book, we only consider classical or semiclassical systems, where the quantum effects are “hidden” in the parametrization of the effective potentials used in the models. The assumption is that mesoscopic systems such as macromolecules and molecular aggregates behave sufficiently cooperative to investigate the net effect only, but not the individual quantum-mechanical contributions. This is an idealization,

2.3 Thermal Fluctuations: The Statistical Path Integral

37

but it is based on the fact that for a correct physical description of macroscopic systems, a precise quantum-mechanical treatment is not necessarily required. This brings us back to the fundamental “theory of everything”. The demand for it is a profound philosophical problem and it is under violent debate. For the understanding of the dynamics of a macroscopic system, Newton’s equations of motion are completely sufficient. On the other hand, the question, why this macroscopic system is stable, requires the consideration of the fundamental principles valid at very short length scales, i.e., in the quantum regime. However, from our current understanding of physics, quantum physics in its current form is not the desired fundamental theory, as it does not allow for the quantization of gravity which is the dominant force in interactions of astronomically large systems, where Einstein’s field equations apply. Interestingly, the classical physics at intermediate length and energy scales is a limiting case of both, quantum physics and gravitation. It is the cooperative behavior of many particles or sections of a macroscopic system that governs its physical properties and, therefore, since cooperativity and statistics are inevitably connected, statistical mechanics is probably the most fundamental theoretical concept in the natural sciences.

2.3 Thermal Fluctuations: The Statistical Path Integral In the thermodynamical description, the capability of a system to react to excitations or changes of environmental parameters such as temperature or pressure is expressed by so-called response quantities like the specific heat or the compressibility, respectively. The specific heat, for example, quantifies the capacity of heat storage by the system. For a system with constant volume such as in the canonical ensemble, it is defined as the amount of heat exchange T dS while the temperature is changed, CV (T ) = T (∂S/∂T )N,V = (∂U/∂T )N,V . From the latter, the statistical expression 1 2 1

dhEi(T ) 2 2 = hE i(T ) − hEi (T ) = [hEi(T ) − E] CV (T ) = dT kB T 2 kB T 2 (2.30) can easily be derived. Thus, the specific heat corresponds to the fluctuations of energy and as such to the width of the energetic distribution pcan,E (T ). The larger the energetic fluctuation width, the larger is also the number of energetic states that can be thermally excited and, therefore, the larger is the specific heat. The fluctuation formula (2.30) can be generalized and the fluctuation of any quantity O can be defined via the temperature derivative of its mean value 1 dhOi = (hOEi − hOihEi) , dT kB T 2

(2.31)

38

2. Statistical Mechanics: A Modern Review

which is particularly interesting, if O can be considered as a suitable order parameter that allows for the quantitative separation of phases. The temperature, where the fluctuations become maximal, is an estimate for the transition temperature. However, for finite systems, different fluctuation quantities typically signal the same transition at noticeably different fluctuation peak temperatures. Only in the thermodynamic limit, these peak temperatures converge towards a single phase transition temperature. In the discussion of the Rcanonical ensemble, we have already introduced the partition function Zcan = dE g(E) exp(−βE) for a system with continuous energy space. This is the typical situation for a system with continuous mechanical phase-space degrees of freedom, position and momentum. Let’s denote by X = (x1 , x2, . . . , xn , . . . , xN ) the 3N -dimensional vector of the three-dimensional coordinates xn and by P = (p1, p2, . . . , pn , . . . , pN ) the 3N -dimensional vector of the momenta pn . Then, the canonical partition function, which accounts for the thermally weighted fluctuations of the phase-space variables, is written as Z Zcan = DP DX exp [−βH(P, X)] , (2.32)

where H(P, X) = E is the Hamilton function which corresponds to the total system energy E and Z N Z Z Y d3 p n d3 x n . (2.33) DP DX ≡ CN 3 (2π~) n=1

The prefactor CN = 1/N ! is the Boltzmann correction which takes into account the trivial multi-counting of microstates generated by permuting identical particles – provided the particles can be exchanged at all. For a single molecular chain, where monomers are bonded and cannot change their positions within the chain, no Boltzmann correction is needed. The factor 1/(2π~) in the integral measure sets the scale of the infinitesimal phase-space volume dpn dxn for each phase-space component. The “volume” element has the physical dimension of an action (measured in Js) and represents the smallest phase-space volume in which a single particle state can reside. This is a consequence of Heisenberg’s uncertainty principle. If the Hamilton function is of the standard form N X p2n H(P, X) = + V (X), 2m n=1

(2.34)

where the first term is the total kinetic energy (m is the particle mass, which shall be identical for all particles, for simplicity) and the second term is the potential

2.3 Thermal Fluctuations: The Statistical Path Integral

39

energy N X

N 1 X V2 (|xn − xm |), V1(xn ) + V (X) = 2 n,m=1 n=1

(2.35)

n6=m

representing the respective coupling to an external field and the interactions between the particles as the sum over all pair potentials which typically only depend on the particle-particle distance. Then the momentum integrals factorize and can exactly be solved for each of the components, yielding in total 1/λ3N th , where p λth = 2π~2 β/m (2.36) is called the thermal wavelength and sets the temperature-dependent length scale of the thermal fluctuations. The resulting expression for the partition sum reads Z Zcan = DX exp [−βV (X)] , (2.37)

where the integral measure is redefined via Z N Z Y d3 x n . DX ≡ CN 3 λ th n=1

(2.38)

Expression (2.37) is called the statistical path integral in analogy to the quantummechanical Feynman path integral introduced as an alternative way of quantization, where all possible particle paths, which in contrast to classical Newtonian mechanics may vary due to quantum fluctuations, are integrated over [58,59]. Mean values of quantities O that can be parametrized with respect to the momenta and coordinates are expressed as Z 1 hO(P, X)i = DP DX O(P, X) exp [−βH(P, X)] . (2.39) Zcan A prominent example is the total system energy E = H(P, X). Its mean value can with Eq. (2.39) be written as hEi = −

∂ ln Zcan , ∂β

(2.40)

in agreement with Eq. (2.24). It should be noted that the contribution for the kinetic energy can easily be separated. Since N Z N Z N ∂ Y d3 x n 1 X 2 1 Y −βE(X) 3 −βp2n /2m − e d pn e hp i = 2m n=1 n Zcan n=1 (2π~)3 ∂β n=1

40

2. Statistical Mechanics: A Modern Review

=

3 N kB T, 2

(2.41)

the mean total energy is 3 hEi = N kB T + hV (X)i. 2

(2.42)

For practical purposes as, for example, in Monte Carlo simulations, it is therefore completely sufficient to calculate the mean potential energy hV (X)i. Thus, in the canonical ensemble, mean energy differences are identical to mean potential energy differences. The contribution of the kinetic energy fluctuations to the specific heat is thus also constant: CV (T ) =

∂hEi(T ) 3 1 2 2 hV (X)i(T ) − hV (X)i (T ) . = N kB + dT 2 kB T 2

(2.43)

Not surprisingly, the kinetic energy contributions in Eqs. (2.42) and (2.43) are identical to the thermodynamic results for the ideal gas, the idealized model for noninteracting particles [V (X) = 0]. In the applications presented in the following chapters, we will omit these contributions, but we have to keep in mind that they must be added to make any statistical analysis of a Hamiltonian system quantitatively correct.

2.4 Phase and Pseudophase Transitions The set of microstates that dominates under given external conditions, like the equilibrium temperature, forms the macrostate of the system. If in a certain range of the external parameters the corresponding macrostates exhibit significant similarities, they are said to belong to the same phase. A system experiences a phase transition, if a small change of an external parameter leads to a dramatic change of the macrostate properties making it belonging to another phase. The quantitative analysis of phase transitions makes explicit use of the thermodynamic limit and is thus not directly transferrable to small systems. Usually, two types of phase transitions are distinguished. As an example, let us consider temperature-driven phase transitions. Discontinuous or first-order transitions refer to the discontinuity of the entropy as a function of temperature: ∂F = −S(T, V, N ). (2.44) ∂T N,V First-order transitions are characterized by the coexistence of two phases I and II at the transition temperature T0. The difference of the respective entropies near

2.4 Phase and Pseudophase Transitions

41

the transition point, multiplied by the transition temperature, defines the latent heat: ∆Qlat = lim T0[SII (T0 + τ ) − SI (T0 − τ )]. (2.45) τ →0

A phase transition is of first order if ∆Qlat > 0. Consequently, a transition is continuous or of second order, if ∆Qlat = 0. In this case, the entropy is continuous at the transition temperature. However, the second derivative of F with respect to T , 2 ∂ F ∂S 1 CV (T ), (2.46) = − = − ∂T 2 N,V ∂T N,V T is discontinuous and thus also the specific heat at the “critical” transition temperature TC . Defining the dimensionless parameter τ = (T − TC )/TC , the specific heat follows near the transition point a power law, CV (τ ) ∼ |τ |−α , where α is the critical exponent associated to the specific heat. Other fluctuation quantities such as the compressibility of a gas or the susceptibility of a magnetic system – both response quantities to external fields pressure and magnetic field, respectively – and correlation lengths also exhibit power law monotony near the transition temperature, but possess different characteristic critical exponents (γ for compressibility and susceptibility, ν for the correlation length). For the identification of phases and the type of phase transitions (e.g., the transition between the ferro- and the paramagnetic phase in a magnetic system), it is often very useful to define an order parameter = 0, T < TC , O (2.47) 6= 0, T > TC ,

which thus ideally allows for the unique separation of phases. Near the transition temperature, the order parameter also follows a power law in the ordered phase: O ∼ (−τ )β with the critical exponent β (not to be confused with the thermal energy). For a magnetic system, it is convenient to choose the mean value of the spontaneous magnetization as the order parameter, since it is zero in the disordered phase (paramagnetism) and nonzero in the ordered phase (ferromagnetism). In other systems, the definition of a suitable order parameter is much less obvious. One of the most striking advances in physics was the discovery of universality: Physically completely different systems that share the same values of critical exponents belong to the same universality class. Thus, physical systems can be classified with respect to their transition behavior. These powerful fundamental concepts, provided only by statistical physics, are idealized in a way that they can in this form only be applied to “large” systems,

42

2. Statistical Mechanics: A Modern Review

where the thermodynamic limit conditions are satisfied. Unfortunately, the transfer to systems on mesoscopic scales like the molecular systems discussed in this book, is not straightforward. The “collapse” of the fluctuation and correlation quantities at a single transition temperature TC does not occur for finite systems. Furthermore, a suitably defined parameter that could serve as an order parameter, will not behave exactly in the way that expression (2.47) suggests. Finally, most relevantly, fluctuations of thermodynamic quantities do not exhibit powerlaw behavior near the transition points. Thus, a classification of different finite systems by means of sets of critical exponents is impossible. Nonetheless, as we will show in the subsequent chapters of the book, different conformational transitions can exhibit similarities and a classification of molecular systems is highly desireable. This regards, in particular, the classification of proteins with respect to their folding behavior. Another point is the difficult classification of the transitions into first- and second-order transitions. Actually, even from “real” phase transitions which are of second order in the thermodynamic limit, it is known that they can exhibit first-order features in the finite system (dichotomic transitions). In order to discriminate among transitions of finite systems and systems in the thermodynamic limit, we will denote noticeable structural changes as pseudophase transitions. The associated dominating macrostates shall be denoted pseudophases rather than phases. It is necessary to further extend and to generalize the theory of phase transitions to make it applicable to finite systems as well. Thus, it is useful to return to the initial idea of cooperativity which is the basis of both, phase and pseudophase transitions.

2.5 Relevant Degrees of Freedom When a system behaves cooperatively, its degrees of freedom do not act independently. This is particularly apparent near transitions from ordered to disordered phases. In the disordered phase, the correlation length is very small, i.e., a local system change is not necessarily felt by distant parts of the system. A collective response to the excitation does not occur. If the system is in a macrostate near the transition point, it can happen that a system change due to a small change of the environmental parameters initiates a spontaneous ordering effect. An example is the behavior of a flexible polymer at the so-called Θ transition, where it “collapses” from large, extended random-coil structures to compact, densely packed globular conformations. A suitable model to describe the behavior of polymers in the random-coil phase are random-walk or self-avoiding random walk models (if volume exclusion effects must be considered). The ener-

2.5 Relevant Degrees of Freedom

43

getic interactions between the monomers are completely irrelevant; only “hard” constraints (connectivity, volume exclusion, chain length) do matter. In the globular phase, however, the attractive van der Waals forces between many atoms of the monomers, that stabilize the globular shapes, are necessarily to be taken into account in order to describe the formation of compact structures correctly. Since this structure formation process is highly cooperative, i.e., many atoms are involved, it is sufficient to introduce a general “hydrophobic” interaction between abstractified “united atoms” which comprise subsets of atoms. Thus, the effect can be qualitatively and, for large systems, even quantitatively be described by a simplified, coarse-grained model that in addition to the system constraints only includes an effective interaction among nonbonded monomers. Compared to an atomic model, the number of degrees of freedom being necessary for the understanding of the thermodynamics of the collapsed phase, can be drastically reduced. This is not only very helpful for the efficiency of computer simulations of such systems, it also helps understanding structure formation processes as effects of cooperativity. One of the most contemporary challenges in this field of research is to find the minimal set of degrees of freedom that allows for the description of structural transitions. On a very abstract level, this could be an order parameter or, in analogy to chemical reations, a reaction coordinate. The reason, why this set of degrees of freedom should be small is the reconstruction of the free-energy landscape in an intuitive way, i.e., to express the free energy of the system not only as a function of the external macroscopic thermodynamic parameters such as the temperature, but also as a function of the set of relevant degrees of freedom. As such, “paths” from a macrostate A to a state B (for example from an unfolded protein structure to the native state) can be parametrized and different paths can be assigned statistical weights. This helps to reconstruct the most likely paths the system can follow under thermal conditions.

2.5.1 Coarse-Grained Modeling on Mesoscopic Scales In Fig. 1.7, the general idea of coarse-graining an atomic model for a protein by introducing “united atoms” has already been depicted. The reduced set of coordinates of the monomers (i.e., the new interaction sites) represents the coarsegrained degrees of freedom. If it is justified that the main properties of structural behavior in a cooperative structure formation process can be described by these new monomer positions only, these can be considered as the relevant effective degrees of freedom. Other monomer properties, such as hydrophobicity or charges are then also encoded in effective parameters. In the following, we will develop the formalism to correctly derive the effective model which only depends on the

44

2. Statistical Mechanics: A Modern Review

relevant degrees of freedom. Let us represent the set of relevant mesoscopic degrees of freedom by the L-dimensional vector Q(X) = (q1(X), q2(X), . . . , ql (X), . . . , qL(X)).

(2.48)

Then, the path integral (2.37) can be decomposed into the new and the old coordinates by writing Z Z L Y L Zcan = λth DQ DX [δ(ql − qls (X))] e−βV (X) , (2.49) l=1

where qls (X)) represents the mapping from the X into the reduced Q space; δ(x) is the Dirac δ distribution. Thus, the partition function can now be expressed as a path integral in the Q space, Z ˜ Zcan = DQe−β V (Q) , (2.50) R Q R with DQ ≡ CL Ll=1 dql /λth . The coarse-grained model of the original system is therefore governed by the effective potential res (Q), V˜ (Q) = −kB T ln Zcan

where we have introduced the restricted partition function Z L Y L res [δ(ql − qls (X))] e−βV (X) . Zcan(Q) = λth DX

(2.51)

(2.52)

l=1

The coarse-grained potential (2.51) is the most general variant of an effective potential that would even allows for a correct quantitative description of the thermodynamic system behavior. However, typically, coarse-grained models are still drastically simplified such that models like the HP and the AB models for proteins (see Section 1.4) only represent approximate and sometimes crude versions of coarse-grained models. This is further reduction of complexity is justified as long as the main features of system relevant thermodynamic processes are reproduced, at least qualitatively.

2.5.2 Macroscopic Relevant Degrees of Freedom: The Free-Energy Landscape Comparing the formal expressions of Eqs. (2.19) and (2.51), we can immediately identify the effective potential as the free energy of the system parametrized

2.5 Relevant Degrees of Freedom

45

as a function of the relevant degrees of freedom. It therefore represents the free-energy landscape of the system under given external parameters. However, although the complexity of the original system has already been reduced, the dimension L of the Q space often remains large. For this reason, it does not help the intuition to “visualize” the macrostate of a system by means of a highdimensional free-energy landscape. Rather it appears necessary to introduce only very few components ql . For this, we free ourselves from the concept of coarsegraining the true degrees of freedom and introduce macroscopic parameters Q similar to what is generally done in thermodynamics. The few parameters take over the role of order parameters and shall as such represent the macrostate of the system. A simple and popular example from polymer science is the end-toend distance of a polymer. It is very large in the disordered, random-coil phase and very small in the phase of collapsed conformations. It therefore exhibits a sharp change while passing the collapse transition point and is perfectly suited to discriminate between these two phases and to identify the transition point. However, it is much less useful to signal the transition from the globular to the crystalline phase. This is a disadvantage of order parameters. In contrast to the very general coarse-grained model (2.51) based on mesoscopic degrees of freedom, the introduction of macroscopic relevant degrees of freedom is far less general and often only useful to discriminate parts of a phase diagram. With the vector (2.48) now representing the set of macroscopic relevant degrees of freedom, the statistical expectation value of each component is in the canonical ensemble given by Z 1 DX ql (X)e−βV (X) (2.53) hql (X)i = Zcan and the fluctuation width about this average value is obtained by the derivative with respect to the temperature dhql i 1 = (hql Ei − hql ihEi) , (2.54) dT kB T 2 in accordance with Eqs. (2.39) and (2.31), respectively. The probability (density) to find a system state that is represented by a certain vector of relevant degrees of freedom Q is given by res Zcan (Q) , (2.55) p(Q) = Zcan where the restricted partition function is given by Eq. (2.52). According to Eq. (2.19), we can relate this restricted partition function to the free energy of the system in this macrostate, res F res (T, V, N ; Q) = −kB T ln Zcan (T, V, N ; Q).

(2.56)

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2. Statistical Mechanics: A Modern Review

Figure 2.2: Sketch of a free-energy landscape parametrized by the single relevant degree of freedom q at the externally fixed temperature T . The two minima of F (T ; q) correspond to equilibrium states A and B (e.g., folded and unfolded macrostates of a protein).

This function represents for given system parameters T , V , and N the freeenergy landscape in dependence of the components of the vector of relevant degrees of freedom Q. Minima in this landscape correspond to locally stable (metastable) equilibrium system states. Peaks in this landscape represent freeenergy barriers. A structural transition requires the system to circumvent the barrier or to overcome it by a fluctuation with thermal energy that exceeds the barrier height.

2.6 Kinetic Free-Energy Barrier and the Transition State Figure 2.2 show a typical sketch of a free-energy landscape F (T ; q) at fixed temperature T . It is parametrized by the single relevant degree of freedom q. The minima FA = F (T ; qA) and FB = F (T ; qB) in the free-energy landscape correspond to the equilibrium states A and B, respectively. We assume that q is getting larger the more ordered the state is. For a protein with two-state folding characteristics, A is associated to the unfolded state (i.e., random, unstructured conformations), whereas B represents the ensemble of native-like conformations. Since B is the global minimum, the probability for the system to reside in B is larger than in A, i.e., the protein is likely to be folded. In this example, q could be defined as the number of “native contacts” nnative which is the number of correctly arranged monomer-monomer positions. This number obviously increases if the molecular conformations are getting closer to the native (functional) structure. The minima are separated by the maximum at the transition state, Fts = F (T ; qts). If q is indeed the relevant degree of freedom and the system resides in the local minimum at A, the free-energy difference ∆F ‡ = Fts − FA is the kinetic barrier the system has to climb to reach the transition state. Only after reaching

2.6 Kinetic Free-Energy Barrier and the Transition State

47

the ensemble of states belonging to the thermodynamically unstable transition macrostate, the system is with a certain probability p able to enter the ordered stable equilibrium state at B. With the probability 1−p it can also happen that the system returns to A again. However, since in the example shown in Fig. 2.2 FB < FA , the system will macroscopically reside in the thermodynamic equilibrium phase B, i.e., at the temperature T < TAB the population of microstates associated to phase B is the largest. If the temperature is increased such that T > TAB , A would become the dominant phase since FA < FB then. Thus, TAB is suitably defined as the transition temperature, where FA = FB and, therefore, both phases coexist. Since qA 6= qB at TAB , this transition is discontinuous. The energy difference UA − UB = TAB ∆S = ∆Qlat > 0 is the latent heat and corresponds to the energy required in the transition towards A to break the contacts that stabilized the ordered state in phase B. The average time required to reach the transition state depends on the barrier height ∆F ‡. This can be shown in a simple Markoffian-type transition state theory, where memory effects are neglected. Then, the time-dependent change of the probability p(Q) to find the system in a state Q is given by the “gain” of states originating in a single timestep ∆t from all other possible states Q′ and the “loss” from Q to Q′ . The so-called master equation reads Z Z ∆p(Q, t) ′ ′ ′ = DQ p(Q , t)T (Q → Q, ∆t) − DQ′ p(Q, t)T (Q → Q′ , ∆t), ∆t (2.57) ′ where T (Q → Q ) is the transition probability to reach the system state Q′ in a timestep ∆t if the system has originally been in the state Q before. In equilibrium, the probability distribution p(Q′, t) is constant in time, ∆p(Q, t)/∆t = 0. Since all states Q′ can be considered to be independent of each other, this leads to the detailed-balance equilibrium condition p(Q)T (Q → Q′ ) = p(Q′)T (Q′ → Q).

(2.58)

Inserting expression (2.55) for the probability density and Eq. (2.56) to relate the restricted partition function to the Q-dependent restricted free energy yields the transition rate T (Q → Q′ ) ′ kQ→Q = = e−∆FQ→Q′ /kB T , (2.59) ′ T (Q → Q) with ∆FQ→Q′ = F res (T ; Q′) − F res (T ; Q). The rate kQ→Q′ is smaller than unity if F res (T ; Q′) > F res (T ; Q), as expected. The transition rate can be related to the average time it needs to reach Q′ : −1 ∆FQ→Q′ /kB T τQ→Q′ ∼ kQ→Q . ′ = e

(2.60)

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2. Statistical Mechanics: A Modern Review

The transition time thus grows at constant temperature exponentially with the free-energy difference. Carrying over these results to the exemplified free-energy landscape shown in Fig. 2.2, we find that the time to reach the transition state from macrostate A is τA→ts ∼ e∆F

‡

/kB T

(2.61)

and grows exponentially with the barrier height. On the other hand, the time τts→B needed to reach the ordered state B from the transition state, decays exponentially with the free-energy difference |F res (T ; qB )−F res (T ; qts)|. Note that in this theory the total average time to reach the ordered state B from the disordered state A by passing the transition state is τA→B = τA→ts + τts→B . This exemplified behavior generally applies to systems of rather little cooperativity, where many local conformational changes are required to perform a macroscopic change of the system state. “Climbing the hill” towards the transition state is thus a necessity to reach another (meta)stable macrostate. In other words: To reach B from A in a single fluctuation without the intermediate residence in the transition state is extremely unlikely. Thus, since such a transition is a time-dependent process influenced and activated by thermal fluctuations, it is frequently called a kinetic transition with ∆F ‡ being the kinetic barrier. In systems of high cooperativity, the passage of a transition state is not necessary (it actually does not exist) and the transition can in principle occur within a single timestep. In this context, “cooperativity” does not necessarily mean that different parts of the system depend on and need to interact with each other to change the macrostate (or the phase). Rather, local parts of the system can react individually in the same way upon a weak change of the environmental conditions. In the freezing transition of water, nucleation cores form independently and attract other molecules in the local environment of each nucleus to join. This leads to macroscopic crystalline structures which finally bind to each other in order to reduce instabilities due to surface effects. However, the individual growth of the nucleation centers also causes dislocations that typically appear at the boundaries of these crystalline substructures. A molecular folding process is much more complex and generally requires interactive cooperativity, i.e., the formation of a unique functional protein fold can take comparatively long (up to the order of seconds) and include weakly stable intermediate states. The collective folding is also necessary to avoid dislocations. Hence, the free-energy contour depicted in Fig. 2.2 will only apply to a subclass of proteins, so-called two-state folders. The transition state theory and the assumption of “complex and rugged” free-

2.6 Kinetic Free-Energy Barrier and the Transition State

49

energy landscapes are still under debate. One reason is the inherent difficulty to identify the “true” relevant degrees of freedom Q which are typically highly system-specific. The problem is that the kinetic barrier ∆F ‡ frequently depends on the choice of Q which makes an experimental verification of a theoretically proposed free-energy landscape more difficult. Nonetheless, the free-energy landscape concept is helpful in understanding details of phase transitions (such as, e.g., the occurrence of barriers that slow down the transition process) and to quantify these. Throughout this book we will frequently discuss conformational transitions by means of interpretations of corresponding free-energy landscapes.

3 The Complexity of Minimalistic Lattice Models for Protein Folding

3.1 Evolutionary Aspects The number of different functional proteins encoded in the human DNA is of order 100 000 – an extremely small number compared to the total number of possibilities: Recalling that 20 amino acids line up natural proteins and typical proteins consist of N ∼ O(102 − 103) amino acid residues, the number of possible primary structures 20N lies somewhere far, far above 20100 ∼ 10130. Assuming all proteins were of size N = 100 and a single folding event would take 1 ms, a sequential enumeration process would need about 10119 years to generate structures of all sequences, irrespective of the decision about their “fitness”, i.e., the functionality and ability to efficiently cooperate with other proteins in a biological system. Of course, one might argue that the evolution is a highly parallelized process which drastically increases the generation rate. So, we can ask the question, how many processes can maximally run in parallel. The universe contains of the order of 1080 protons. Assuming that an average amino acid consists of at least 50 protons, a chain with N = 100 amino acids has of the order O(103) protons, i.e., 1077 sequences could be generated in each millisecond (forgetting for the moment that some proton-containing machinery is necessary for the generation process and only a small fraction of protons is assembled in earth-bound organic matter). The age of our universe is about 1010 years (we also forget that the Earth is even about one order of magnitude younger) or 1021 ms. Hence, about 1098 sequences could have been tested to date, if our drastic simplifications were right. But even this yet much too optimistic estimate is still noticeably smaller than the above mentioned reference number of 10130 possible sequences for a 100-mer. At least two conclusions can be drawn from this crude analysis. One thing is that the evolutionary process of generating and selecting sequences is ongoing as

52

3. The Complexity of Minimalistic Lattice Models for Protein Folding

it is likely that only a small fraction of functional proteins has been identified yet by nature. On the other hand, the existence of complex biological systems, where hundreds of thousands different types of macromolecules interact efficiently, can only be explained by means of efficient evolutionary strategies of adaptation to environmental conditions on Earth which dramatically changed through billions of years. Furthermore, the development from primitive to complex biological systems leads to the conclusion that within the evolutionary process of protein design, particular patterns in the genetic code have survived over generations, while others were improved (or deselected) by recombinations, selections, and mutations. But the sequence question is only one side. Another regards the geometric structures of proteins which are directly connected to biological functionalities. The conformational similarity among human functional proteins is also quite surprising; only of the order of 1 000 significantly different “folds” were identified [21]. Since the conformation space is infinitely large because of the continuous degrees of freedom and the sequence space is also giant, the protein folding problem is typically attacked from two sides: the direct folding problem, where the amino acid sequence is given and the associated native, functional conformation has to be identified, and the inverse folding problem, where one is interested in all sequences that fold into a given target conformation. With these two approaches, it is, however, virtually impossible to unravel evolutionary factors that led to the set of present functional proteins. Only for small, discrete protein models, a complete and exact statistical analysis of the entire sequence and conformations space is possible. Such an analysis [27,28] will be performed in the following by employing hydrophobic-polar (HP) lattice models [20,21], as introduced in Section 1.4.3.

3.2 On Self-Avoiding Walks and Contact Matrices Flexible lattice polymers are typically modeled by self-avoiding walks (SAW). The total number of conformations for a chain with N monomers is not known exactly. For N → ∞ it is widely believed that in leading order the scaling law [60,61] Cn = AµnC nγ−1

(3.1)

holds, where n = N −1 is the number of self-avoiding steps. In this expression, µC is the effective coordination number of the lattice, γ is a universal exponent, and A is a non-universal amplitude. In Table 3.1, the exactly enumerated number for selfavoiding conformations is listed for chains with up to N = n + 1 = 19 monomers. Based on these data we estimate for the simple-cubic lattice µC ≈ 4.684 and γ ≈ 1.16 [27] by extrapolating the results obtained with the ratio method [60,62]. These results are in good agreement with previous enumeration results [63–65],

3.2 On Self-Avoiding Walks and Contact Matrices

53

Table 3.1: Number of conformations CN and contact matrices MN for chains with N monomers (or, equivalently, self-avoiding walks with n = N − 1 steps). N 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1 n C MN 6 N 3 25 2 4 121 3 5 589 9 6 2 821 20 7 13 565 66 8 64 661 188 9 308 981 699 10 1 468 313 2 180 11 6 989 025 8 738 12 33 140 457 29 779 13 157 329 085 121 872 14 744 818 613 434 313 15 3 529 191 009 1 806 495 16 16 686 979 329 6 601 370 17 78 955 042 017 27 519 000 18 372 953 947 349 102 111 542

1 C /MN 6 N

13 40 65 141 206 344 442 674 800 1 113 1 290 1 715 1 954 2 528 2 869 3 652

Monte Carlo methods [66] and field-theoretic estimates [67] for γ. We are not going to go to extremes in extending the numbers of walks CN in Table 3.1 which have already been enumerated up to n = 26 steps (and hence C27 ≈ 5.49 × 1017 self-avoiding conformations with N = 27 monomers) [64]. Rather, we scan the combined space of HP sequences and conformations which contains for chains of N = 19 monomers 219C19 ≈ 1.17 × 1018 possible combinations. Therefore, the computational efforts in our study are comparably demanding. In models with the general form (1.16), where the calculation of the energy reduces to the summation over contacts (i.e., pairs of monomers being nearest neighbors on the lattice but nonadjacent along the chain) of a given conformation, the number of conformations that must necessarily be enumerated can drastically be decreased by considering only classes of conformations, so-called contact sets [25,68]. A contact set is uniquely characterized by a corresponding contact map (or contact matrix), but a single conformation is not. Thus, for determining energetic quantities of different sequences, it is sufficient to carry out enumerations over contact sets. In a first step, however, the contact sets and their degeneracy, i.e., the number of conformations belonging to each set, must be determined and stored. Then, the loop over all nonredundant sequences is performed for all contact sets instead of conformations. We have developed a parallelized exact enumeration program that efficiently scans the whole space of

54

3. The Complexity of Minimalistic Lattice Models for Protein Folding

1012 1010 108 1 Cn , 6 106 Mn

1 Cn 6

104

Mn

102 100

(a) 2

4

6

8

10 n

12

14

16

18

rnC

5 4.68 4.38 4 rnC ,

rnM

rnM 3 2

(b) 1

3

5

7

9

11 n

13

15

17

19

Figure 3.1: (a) Dependence of the numbers of self-avoiding walks Cn and contact matrices Mn on the number of steps n = N − 1. (b) Ratios of numbers of self-avoiding walks rnC = Cn /Cn−1 and contact matrices rnM = Mn /Mn−1 . The dotted lines indicate the values the respective series C M converge to, r∞ = µC ≈ 4.68 and r∞ = µM ≈ 4.38, respectively.

contact sets and nonredundant sequences [28]. In Table 3.1, the resulting numbers of contact sets MN are summarized and, although also growing exponentially [see Figs. 3.1(a) and (b)], the gain of efficiency by enumerating contact sets, is documented by the ratio between CN and MN in the last column. Assuming that the number of contact sets Mn follows a scaling law similar to Eq. (3.1), we estimated the effective coordination number to be approximately µM ≈ 4.38. Unfortunately, the ratios of numbers of contact sets for even and odd numbers of walks oscillate much stronger than for the number of conformations, as is shown in Fig. 3.1(b). This renders an accurate scaling analysis (in particular for the exponent γ) based on the data for the relatively

3.3 Exact Statistical Analysis of Designing Sequences

55

Table 3.2: Number of designing sequences SN (only relevant sequences, see text) in the HP and MHP models. N HP SN MHP SN

4 5 3 0 7 0

6 7 0 0 0 6

8 9 10 11 12 13 2 0 0 0 2 0 13 0 11 8 124 14

14 15 16 17 18 19 1 1 1 8 29 47 66 97 486 2196 9491 4885

small number of steps much more difficult than for self-avoiding walks.

3.3 Exact Statistical Analysis of Designing Sequences In this section, we analyze the complete sets SN of designing sequences for HP proteins of given numbers of residues N ≤ 19. A sequence σ is called designing, if there is only one conformation associated with the native ground state, not counting rotation, translation, and reflection symmetries that altogether contribute on a simple cubic lattice a symmetry factor 6 for linear, 24 for planar, and 48 for conformations spreading into all three spatial directions. In Table 3.2 we have listed the numbers of designing sequences SN we found for the two models. In contrast to previous investigations of HP proteins on the square lattice [25], the number of designing sequences obtained with the pure HP model is extremely small on the simple cubic lattice. This does not allow for a reasonable statistical study of general properties of designing sequences, at least for very short chains. The situation is much better using the more adequate MHP model. The first quantity under consideration is the hydrophobicity of a sequence σ, i.e., the number of hydrophobic monomers NH , normalized with respect to the total number of residues: N NH 1 X m(σ) = σi . (3.2) = N N i=1

The average hydrophobicity over a set of designing sequences of given length N is then defined by 1 X hmiN = m(σ). (3.3) SN σ∈SN

The hydrophobicity distribution for all sequences is not binomial since in our analysis we have distinguished only sequences that we call relevant, i.e., two sequences that are symmetric under reversal of their residues are identified and enter only once into the statistics. Therefore we consider, for example, only 10 relevant sequences with length N = 4 instead of 24 = 16. Taking into account all 2N sequences would obviously lead to a binomial distribution for NH , since

56

3. The Complexity of Minimalistic Lattice Models for Protein Folding

0.30 0.25 0.20 hN 0.15 0.10 0.05

(a)

0.00 0.0

0.1

0.2

0.3

0.4

0.5 m

0.6

0.7

0.8

0.9

1.0

0.30 0.25 0.20 bN 0.15 0.10 0.05 0.00

(b) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 N

Figure 3.2: (a) Distribution of hydrophobicity hN of all designing sequences with N = 18 monomers (solid line) compared with the distribution of hydrophobicity of all sequences of this length (dashed line) for the MHP model. (b) Widths of the hydrophobicity distribution of the designing sequences, bN , depending on the chain length N (solid line) compared with the widths of the hydrophobicity distribution of all sequences (dashed line) for the MHP model.

there are then exactly N (3.4) NH sequences with NH hydrophobic monomers. In Fig. 3.2(a), the distribution of hydrophobicity is plotted for the designing sequences with N = 18 monomers in the MHP model and, for comparison, for all sequences with N = 18. For this example, we see that the width of the hydrophobicity distribution for the designing sequences, which has its peak at hmiMHP ≈ 0.537 > 0.5, is smaller than that of the distribution over all sequences. 18

3.3 Exact Statistical Analysis of Designing Sequences

57

Table 3.3: Number of designable conformations DN in both models. N HP DN MHP DN

4 5 1 0 1 0

6 7 0 0 0 2

8 9 10 2 0 0 2 0 5

11 12 13 14 15 16 17 18 19 0 2 0 1 1 1 8 28 42 6 30 8 31 58 258 708 1447 1623

In order to gain more insight how the hydrophobicity distributions differ, we have compared the widths of both distributions in their dependence on the chain length N ≤ 19. This is shown in Fig. 3.2(b). It seems that for N → ∞ the widths of the hydrophobicity distributions for the designing sequences asymptotically approach the curve of the widths of the hydrophobicity distributions of all sequences. After having discussed sequential properties of designing sequences, we now analyze the properties of their unique ground-state structures, the native conformations. From Table 3.3 we read off that the number of different native conformations DN is usually much smaller than the number of designing sequences, i.e., several designing sequences share the same ground-state conformation. The number of designing sequences that fold into a certain given target conformation X(0) (or conformations being trivially symmetric to this by translations, rotations, and reflections) is called designability [69]: X (0) (0) ∆ Xgs(σ) − X , (3.5) FN (X ) = σ∈SN

where Xgs(σ) is the native (ground-state) conformation of the designing sequence σ. The function ∆(Z) is the generalization of Eq. (1.17) to 3N -dimensional vectors. It is unity for Z = 0 and zero otherwise. The designability is plotted in Fig. 3.3 for all native conformations that HP proteins with N = 17, 18, and 19 monomers can form in the MHP model. In this figure, the abscissa is the rank of the conformations, ordered according to their designability. The conformation with the lowest rank is therefore the most designable structure and we see that a majority of the designing sequences folds into a few number of highly designable conformations, while only a small number of designing sequences possesses a native conformation with low designability (note that the plot is logarithmic). Similar results were found, for example in Ref. [70], where the designability of compact conformations on cuboid lattices was investigated in detail. The left picture in Fig. 3.4 shows the conformation with the lowest rank (or highest designability) with N = 18 monomers. From our analysis we see that this characteristic distribution of the designing sequences is not restricted to cuboid lattices only. This result is less trivial than one may think at first sight. As we will show later on in the discussion

58

3. The Complexity of Minimalistic Lattice Models for Protein Folding

100

N = 17 N = 18 N = 19

FN 10

1 1

10

100

1000

rank

Figure 3.3: Designability FN of native conformations in the MHP model for N = 17, 18, and 19. The abscissa is the rank obtained by ordering all designable conformations according to their designability.

of the radius of gyration, native conformations are very compact, but only very few conformations are maximally compact (at least for N ≤ 19). For longer sequences similar results were found in Ref. [48]. Highly designable conformations are of great interest, since it is expected that they form a frame making them stable against mutations and thermodynamic fluctuations. Such fundamental structures are also relevant in nature, where in particular secondary structures (helices, sheets, hairpins) supply proteins with a stable backbone [70]. Conformational properties of polymers are usually studied in terms of the squared end-to-end distance Re2 = (xN − x1 )2

(3.6)

Figure 3.4: Structure (N = 18) with the highest designability of all native conformations (left) and with minimal radius of gyration (right).

3.3 Exact Statistical Analysis of Designing Sequences

59

and the squared radius of gyration Rg2

N 1 X ¯ )2 , (xi − x = N i=1

(3.7)

P ¯ = i xi /N is the center of mass of the polymer. In polymer physics where x both quantities are usually referred to as measures for the compactness of a conformation. A typical conformation with minimal radius of gyration for a chain with N = 18 monomers is shown in the right picture of Fig. 3.4. In Fig. 3.5(a) we compare the N -dependence of the averages of the native conformations found in the MHP model and all possible self-avoiding walks. The same quantities for the squared radius of gyration are shown in Fig. 3.5(b). The averages were obtained by calculating 1 X 2 2 SAW hRe,g i = Re,g (X), (3.8) CN X∈CN 1 X 2 2 MHP Re,g (Xgs(σ)), (3.9) hRe,g i = SN σ∈SN

where CN is the set of all self-avoiding conformations on a sc lattice. Figure 3.5(a) 2 SAW i ∼ n2ν with ν ≈ 0.59 shows for the mixed HP model that, compared to hRe,g (see Ref. [71] for a recent summary of estimates for ν), the average end-to-end 2 MHP distance hRe,g i of the native conformations only is much smaller. For even number of monomers, the ends of a HP protein can form contacts with each other 2 MHP on the sc lattice. Accordingly, the values of hRe,g i are smaller for N being even and the even-odd oscillations are very pronounced. The widths (or standard deviations) bRe2 of the distributions of the squared end-to-end distances are also very small. Even for heteropolymers with N = 19 monomers in total, there are virtually no native conformations, where the distance between the ends is larger than 3 lattice sites. We have checked this for the standard HP model, too, and found the same effect. Since the number of native conformations is very small in this model, we have not included these results in the figure. Depicting the average squared radius of gyration hRg2 i and the widths of the corresponding distribution of the radius of gyration in Fig. 3.5(b) for all self-avoiding conformations as well as for the native ones, we see that these results confirm the above remarks. As the average end-to-end distances of native conformations are much smaller than those for the bulk of all conformations, we observe the same trend for the and mean squared radii of gyration hRg2 iMHP and hRg2 iSAW and the widths bRMHP 2 g bRSAW as well. In particular, the width bRMHP is so small, that virtually all native 2 2 g g

60

3. The Complexity of Minimalistic Lattice Models for Protein Folding

12 (a)

10

hRe2 iSAW

8 hRe2 i, bR2e

bRSAW 2 e

6 4 hRe2 iMHP

2

bRMHP 2 e 0

3

1.6

5

7

9

11 N

13

15

17

19

(b)

1.4

hRg2 iSAW

1.2

Rg2 min

hRg2 iMHP

hRg2 i, 1 bR2 0.8

bRSAW 2 g

g

0.6 0.4

bRMHP 2 g

0.2 0

3

5

7

9

11 N

13

15

17

19

Figure 3.5: (a) Average squared end-to-end distances hRe2 i of native conformations in the MHP model compared with those of all self-avoiding walks (SAW). We have also inserted the widths bR2e of the corresponding distributions of end-to-end distances. (b) The same for the average squared radius of gyration hRg2 i. Since the radius of gyration is an appropriate measure for the compactness of a conformation, we have also plotted Rg2 min for the conformations with the minimal radius of gyration (or, equivalently, maximal compactness).

conformations possess the same radius of gyration. For this reason, we have also searched for the conformations having the smallest radius of gyration Rg2 min (these conformations are not necessarily native as we will see!) and inserted these values into this figure, too. We observe that these values differ only slightly from hRg2 iMHP . Thus we conclude that native conformations are very compact, but not necessarily maximally compact. This property has already been utilized in enumerations being performed a priori on compact lattices, [21,26,70] where, however, the proteins are confined by hand to live in small cuboids (e.g., of

3.4 Exact Density of States and Thermodynamics

100 10−1 10 hR2g

native

−2

10−3

100 10−1 10−2 10−3 10−4 10−5 0.0

61

native all

0.1

0.2

0.3

0.4

0.5

10−4 10−5 all

10−6 10−7 0.055 0.0579 0.060

0.065 Rg2 /Rg2 max

0.070

0.075

P Figure 3.6: Distribution hR2g (normalized to hR2g = 1) of squared radii of gyration (normalized with respect to the maximal radius of gyration Rg2 max = (N 2 − 1)/12 of a completely stretched conformation) of native conformations with N = 18 in the MHP model, compared with the histogram for all self-avoiding conformations. The vertical line refers to the minimal radius of gyration (Rg2 min /Rg2 max = 0.0579 for N = 18) and an associated structure is shown on the right-hand side of Fig. 3.4. The inset shows the distribution up to Rg2 /Rg2 max = 0.5.

size 3×3×3 or 4×3×3). Our results on the general sc lattice confirm that this assumption is justified to a great extent. Nevertheless, the slight deviation from the minimal radius of gyration native conformations exhibit is a remarkable result as it concerns about 90% of the whole set of native conformations! This can be seen in Fig. 3.6, where we have plotted the distribution of the squared radii of gyration for all self-avoiding conformations with N = 18 and the native states in the MHP model. All native conformations have a very small radius of gyration but only a few of them share the smallest possible value. A structure with the smallest radius of gyration is shown on the right-hand side of Fig. 3.4. It obviously differs from the most-designable conformation drawn on the left of the same figure.

3.4 Exact Density of States and Thermodynamics Returning to the simpler HP model (1.19), we now discuss thermodynamical properties of designing and nondesigning sequences. In Ref. [72], we conjectured for exemplified sequences of comparable 14-mers, one of them being designing, that designing sequences in the HP model seem to show up a much more pronounced low-temperature peak in the specific heat than the non-designing examples. This peak may be interpreted as kind of a conformational transition between structures with compact hydrophobic cores (ground states) and states where the

62

3. The Complexity of Minimalistic Lattice Models for Protein Folding

whole conformation is highly compact (globules) [47,48]. Another peak in the specific heat at higher temperatures, which is exhibited by all lattice proteins, is an indication for the usual globule–coil transition between compact and untangled conformations. We will return to this point again when we discuss conformational transitions in more detail later on. In order to study energetic thermodynamic quantities such as mean energy and specific heat we determine from the enumerated conformations for a given sequence the density of states of P g(E) that conveniently allows the calculation k the sum Z(T ) = E g(E) exp(−E/kB T ) and the moments hE iT = P partition k E E g(E) exp(−E/kB T )/Z, where the subscript T indicates the difference of calculating thermal mean values based on the Boltzmann probability from averages previously introduced in this section. Then, the specific heat as a function of temperature is given by the fluctuation formula CV (T ) = (hE 2iT − hEi2T )/kB T 2. In the HP model with pure hydrophobic interaction, the density of states shows up a monotonic growth with increasing energy, at least for the short chains in our study. For a reasonable comparison of the behavior of designing and nondesigning sequences, we here focus on 18-mers having the same hydrophobicity (mH = 8) and ground-state energy Emin = −9. There are in total 527 sequences with these properties, only two of which are designing. The densities of states for the two designing sequences and an example of a non-designing sequence are plotted in Fig. 3.7. We have already divided out a global symmetry factor 6 (number of possible directions for the link connecting the first two monomers) that all conformations on a sc lattice have in common. Since the ground-state conformations of the designing sequences spread into all three dimensions, an additional symmetry factor 4×2 = 8 (4 for rotations around the first bond, 2 for a remaining independent reflection) makes a number of conformations obsolete and the ground-state degeneracy of the designing sequences is indeed unity. Obviously this is not the case for the sequences we identified as non-designing. In fact, the uniqueness of the ground states of designing sequences is a remarkable property as there are not less than ∼ 1010 possible conformations of HP lattice proteins with 18 monomers. As we also see in Fig. 3.7, the ratio of the density of the first excited state (E = −8) for the designing and the non-designing sequences is smaller than for the ground state. This means that, at least for these short chains, the low-temperature behavior of the HP proteins in this model strongly depends on the degeneracy of the ground state. Furthermore, we expect that the low-temperature behavior of both designing sequences is very similar as their low-energy densities hardly differ. We have investigated this, once more for the 18mers with the properties described above, by considering the mean energy hEiT and the specific heat CV (T ). The results are shown in Figs. 3.8(a) and 3.8(b),

3.4 Exact Density of States and Thermodynamics

63

1010 108 1 106 g(E) 6 104 HPPPHPHPHHPPPHPHPH (d) HPHPHPPPHHPPPHPHPH (d) HPPHPHPPHPPHPHPPHH (n)

102 100 -10

-9

-8

-7

-6

-5 E

-4

-3

-2

-1

0

Figure 3.7: Density of states g(E) for two designing sequences (d) with N = 18, mH = 8, and Emin = −9 in the HP model. We have divided out the symmetry factor 6 that is common to all conformations. Three-dimensional conformations have an additional symmetry factor 8, such that the states with minimal energy for these two curves are indeed unique and the sequences are designing. For comparison we have also plotted g(E) for one exemplified non-designing sequence (n) out of 525 having the same properties as quoted above, but different sequences. The groundstate degeneracy for this example is g0 = g(Emin ) = 6 × 1840 (including all symmetries).

respectively. The two solid curves belong to the two designing sequences and the dashed lines are the minimum/maximum bounds of the respective quantities for the non-designing sequences. As a main result we find that designing and nondesigning sequences behave indeed differently for very low temperatures. There is a characteristic, pronounced low-temperature peak in the specific heat that can be interpreted as kind of transition between low-energy states with hydrophobic core and very compact globules. This confirms a similar observation for the 14-mers studied in Ref. [72]. The upper bound of the specific heats for non-designing sequences in Fig. 3.8(b) exposes two peaks. By analyzing our data for all 525 non-designing sequences we found that there are two groups: some of them experience two conformational transitions, while others do not show a characteristic low-temperature behavior. Thus, the only appearance of these two peaks is not a unique, characteristic property of designing sequences. In order to quantify this observation, we have studied all relevant 32 896 sequences with 16 monomers. Only one of these sequences is designing (HP2 HP2 HP HP H2 P HP H, with minimum energy Emin = −9), but in total there are 593 sequences, i.e., 1.8% of the relevant sequences, corresponding to curves of specific heats with two local maxima. It should be noted that the degeneracies of the ground states associated with

64

3. The Complexity of Minimalistic Lattice Models for Protein Folding

-2

-4

hEiT -6 -8 (a) -10 0.0

0.1

0.2

0.3 T

0.4

0.5

0.6

30 25 20 CV (T ) 15 10 5 0 0.0

(b) 0.1

0.2

0.3 T

0.4

0.5

0.6

Figure 3.8: (a) Mean energy hEiT and (b) specific heat CV for the two designing sequences with N = 18, mH = 8, and Emin = −9 (solid lines) in the HP model, whose densities of states were plotted in Fig. 3.7. The curves of the same quantities for the 525 non-designing sequences are completely included within the respective areas between the dashed lines. The low-temperature peak of the specific heat (near T = 0.14) is most pronounced for the two designing sequences which behave similarly for low temperatures.

these sequences are comparatively small. Combined exact enumeration studies of conformation and sequence space for lattice peptides noticeably longer than 19 monomers are currently computationally out of reach which is due to the exponential growth of the state space. Therefore, for longer sequences, primarily the direct folding problem is studied using computer simulation methods, i.e., lowestenergy conformations and thermodynamic folding properties are identified and analyzed for a given HP sequence. In the following, a novel computer simulation method is introduced that allows for an efficient sampling of the conformation space for long HP sequences in the direct folding problem.

4 Monte Carlo and Chain Growth Methods for Molecular Simulations

4.1 Conventional Markov-Chain Monte Carlo Sampling 4.1.1 Ergodicity and Consequences of Finite Time Series The general idea behind all Monte Carlo methodologies is an efficient stochastic sampling of the configurational or conformational phase space or parts of it with the objective to obtain reasonable approximations for statistical quantities such as expectation values, probabilities, fluctuations, correlation functions, densities of states, etc. A given system conformation (e.g., the geometric structure of a molecule) X is locally or globally modified to yield a conformation X′. This update or “move” is then accepted with the transition probability T (X → X′). Frequently used updates for polymer models are, for example, random translational changes of single monomer positions, bond angle modifications, or rotations about covalent bond axes. More global updates consist of combined local updates, which can be necessary to satisfy constraints such as fixed bond lengths or simply to improve efficiency. It is, however, a necessary condition for correct statistical sampling that Monte Carlo moves are ergodic, i.e., the chosen set of moves must in principle guarantee to reach any conformation out of any other conformation. Since this is often hard to prove and an insufficient choice of move sets can result in systematic errors, great care must be dedicated to choose appropriate moves or sets of moves. Since molecular models often contain constraints, the construction of global moves can be demanding. Therefore, reasonable and efficient moves have to be chosen in correspondence to the model of a system to be simulated. A Monte Carlo update corresponds to the discrete “time step” ∆t in the simulation process. In order to reduce correlations, typically a number of updates is performed between measurements of a quantity O. This series of updates is

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4. Monte Carlo and Chain Growth Methods for Molecular Simulations

called a “sweep” and the “time” passed in a sweep is τ = N ∆t if the sweep consists of N updates. Thus, if M sweeps are performed, the discrete “time series” of O(t) is expressed by the vector (O(T0 + τ ), O(T0 + 2τ ), . . . , O(T0 + mτ ), . . . , O(T0 + Mτ )) and represents the Monte Carlo trajectory. The period of equilibration T0 sets the starting point of the measurement. For convenience, we use the abbreviation Om ≡ O(T0 + mτ ) in the following. According to the ergodic theory, averaging a quantity over an infinitely long time series is identical to perform the statistical ensemble average: Z 1 T lim dt O(t) ≡ hOi. (4.1) T →∞ T 0 This is the formal basis for Monte Carlo sampling. However, only finite time series can be simulated on a computer. For a finite number of sweeps M in a sample k, P (k) (k) O ≈ the relation (4.1) can only be satisfied approximately, M −1 M O = m m=1 (k) hOi. Note that the mean value O will depend on the sample k, meaning that it (k ′ ) (k) is likely that another sample k ′ will yield a different value O 6= O . In order to define a reasonable estimate for a statistical error, it is necessary to start from the assumption that we have generated an infinite number of independent samples k. (k) In this case the distribution of the estimates O is Gaussian, according to the central limit theorem. The exact average of the estimates is then given by hOi. The statistical error of O is thus suitably defined as the standard deviation of the Gaussian: v u rD q M X M E u 1 X 2 2 t 2 εO = O − hOi = hO i − hOi = Amn σO2 m , (4.2) 2 M m=1 n=1 where

Amn =

hOm On i − hOm ihOn i 2 i − hO i2 hOm m

(4.3)

2 i − hOm i2 is the variance of the is the autocorrelation function and σO2 m = hOm distribution of individual data Om . If the Monte Carlo updates in each sample are performed completely randomly without memory, i.e., a new conformation is created independently of the one in the step before (which is a possible but typically very inefficient strategy), two measured values Om and On are uncorrelated, if m 6= n. Then, the autocorrelation function simplifies to Amn = δmn and the statistical error satisfies the celebrated relation σO (4.4) εO = √ m . M

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Since the exact distribution of Om values and the “true” expectation value hOi are unchanged in the simulation (but unfortunately unknown), the standard √ 1deviation σOm is constant, too. Thus, the statistical error decreases with 1/ M . In practice, most of the efficient Monte Carlo techniques generate correlated data, in which case we have to fall back to the more general formula (4.2). It can conveniently be rewritten as p εO = σOm / Meff (4.5)

with the effective statistics Meff = M/τac ≤ M, where τac corresponds to the autocorrelation time. This means, the statistics is effectively reduced by the number of sweeps until the correlations have decayed.2 Since it takes at least the time τac to generate statistically independent conformations, a sweep can simply contain as many updates as necessary to satisfy τ ≈ τac without losing effective statistics. In this case M converges to Meff as τac ≈ 1, since τac is always measured in units of τ , and the data entering into the effective statistics are virtually uncorrelated. This is also the general idea behind advanced error estimation methods such as binning and jackknife analyses [73,74]. It is not a necessary condition; more sweeps with less updates in each sweep, i.e., periods between measurements shorter than τac only yield redundant statistical information. This is not even wrong, but computationally inefficient as it does not improve the statistical error (4.5).

4.1.2 Master Equation Beside ergodicity, another demand for correct statistical sampling is to ensure that the probability distribution p(X) associated to the desired statistical ensemble is independent of time. This can only be achieved in the simulation, if the relevant part of the phase space is sampled sufficiently efficient to allow for quick convergence towards a stable estimate for p(X). Monte Carlo simulations follow a Markoffian dynamics, i.e., the update of a given conformation X to a new one X′ is not influenced by the history that lead to X, i.e., the dynamics does not possess an explicit memory. As we have already done in the description of the transition state theory in Section 2.6, we shall use the master equation (2.57) to describe such a process. Here, we use summations instead of integrations to 1

2 2 For the actual calculation, it is a problem that σO is unknown. However, what can be estimated is σ ˜O = m m 2

2 2 O2 − O and for its expected value we thus obtain h˜ σO i = σO (1 − 1/M ). The 1/M correction is the systematic m m error due to the finiteness of the time series, called bias. The bias-corrected relation for the statistical error reads qP 2 finally εO = [M (M − 1)]−1/2 m (Om − O) [74]. 2 For a detailed discussion of the autocorrelation function and the calculation of the autocorrelation time, see, e.g., Ref. [74].

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emphasize the discreteness of the Markov process in the simulation: ∆p(X, t) X [p(X′)T (X′ → X; ∆t) − p(X)T (X → X′ ; ∆t)], = ∆t ′

(4.6)

X

where T (X → X′ ; ∆t) is the transition probability from X to X′ in a single update (or time step P ∆t). Due ′to particle conservation, it satisfies the normalization condition X′ T (X → X ; ∆t) = 1, i.e., whatever update we perform, we must end up with a state X′ which is an element of the conformational space. The condition ∆p(X, t)/∆t = 0 ensures that the ensemble is in a stationary state if the right-hand side of Eq. (4.6) vanishes. Since the stationarity condition also allows solutions where the distribution function p(X) dynamically changes on cycles which, however, is not the physical situation in a statistical equilibrium ensemble, we demand more rigorously that the expression in the brackets vanishes. This is called the detailed balance condition. Consequently, the ratio of transition rates is given by T (X → X′; ∆t) p(X′) = . (4.7) T (X′ → X; ∆t) p(X) From this relation, it follows that it is obviously a good idea to construct an efficient Markov chain Monte Carlo algorithm, i.e., to choose appropriate acceptance probabilites for the Monte Carlo updates to yield the correct transition probability T (X → X′; ∆t), by taking into account the basic microstate probabilities of the statistical ensemble to be simulated. Markov Monte Carlo algorithms of the canonical ensemble, for example, have to satify T (X → X′ ; ∆t) = e−β∆E , ′ T (X → X; ∆t)

(4.8)

where ∆E = E(X′) − E(X) is the energy difference between the new and the old state. Thus, the transition rate to reach a state X′, which is energetically favored compared to the inital state X, grows exponenially with ∆E < 0. “Climbing the hill” towards a state with higher energy (∆E > 0) is, on the other hand, exponentially suppressed. This is in correspondence with the interpretation of the Markov transition state theory discussed in Section 2.6. Hence, it is possible to study the kinetic behavior (identification of free-energy barriers, measuring the height of barriers, estimating transition rates, etc.) of a series of processes in equilibrium – for example the folding and unfolding behavior of a protein – by means of Monte Carlo simulations. To quantify the dynamics of a process, i.e., the explicit time dependence is, however, less sensical as the conformational change in a single time step depends on the move set and does not follow a

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physical, e.g., Newtonian, dynamics.3

4.1.3 Selection and Acceptance Probabilities In order to correctly satisfy the detailed balance condition (4.7) in a Monte Carlo simulation, we have to take into account that each Monte Carlo step consists of two parts. First, a Monte Carlo update of the current state is suggested and second, we have to decide whether or not to accept it according to our sampling strategy. In fact, both steps are independent of each other in the sense that each possible update can be combined with any sampling method. Therefore, it is useful to split the transition probability T (X → X′ ; ∆t) into a part that depends on the selection probability s(X → X′) for a desired update from X to X′ and the acceptance probability a(X → X′) for this update: T (X → X′ ; ∆t) = s(X → X′ )a(X → X′).

(4.9)

The acceptance probability is typically used in the form a(X → X′ ) = min (1, σ(X, X′)w(X → X′)) ,

(4.10)

with the ratio of microstate probabilities p(X′) w(X → X ) = p(X) ′

(4.11)

and the ratio of forward and backward selection probabilities s(X′ → X) . σ(X, X ) = s(X → X′ ) ′

(4.12)

The expression (4.10) for the acceptance probability naturally fulfils the detailedbalance condition (4.7). The selection ratio σ(X, X′) is unity, if the forward and backward selection probabilities are identical. This is typically the case for “simple” local Monte Carlo updates. If, for example, the update is a translation of a coordinate, x′ = x + ∆x, where ∆x ∈ [−x0, +x0] is chosen from a uniform random distribution, the forward selection for a translation by ∆x is equally probable to the backward move, i.e., to translate the particle by −∆x. This is also valid 3

The natural way to study the time dependence of Newtonian mechanics is typically based on molecular dynamics methods which, however, suffer from severe problems to ensure the correct statistical sampling at finite temperatures. The system needs to be coupled to the heatbath by means of a thermostat [13]. The correct implementation and parametrization of this coupling is highly nontrivial and can cause systematic errors [75]. From a more formal point of view, it is even questionable what “dynamics” shall mean in a thermal system, where even under the same thermodynamic conditions trajectories run typically differently, due to the “random” thermal fluctuations caused by interactions with the huge number [O(1023 ) per mol] of mechanically untraceable heatbath particles.

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for rotations about bonds in a molecular system such as rotations about dihedral angles in a protein. If selection probabilities for forward and backward moves differ, the selection rate is not unity. This is often the case in complex, global updates which comprise several steps. Then, the determination of the correct selection probabilities can be difficult and the selection rate has typically to be estimated in test runs first. To this class of updates belong the biased Gaussian steps [76], where a series of torsional updates of a few sequential protein backbone dihedral angles are performed in order to ensure that the update does not drastically change the protein conformation. Note that the overall efficiency of a Monte Carlo simulation depends on both, a model-specific choice of a suitable set of moves and an efficient microstate sampling strategy based on w(X → X′).

4.1.4 Simple Sampling The choice of the microstate probabilities p(X) is not necessarily coupled to a certain physical statistical ensemble. Thus, the simplest choice is a uniform probability p(X) = 1 independently of ensemble-specific microstate properties. Thus also w(X → X′) = 1 and if Monte Carlo updates used satisfy σ(X, X′) = 1, the acceptance probability is trivially also unity, a(X → X′ ) = 1, i.e., all generated Monte Carlo updates are accepted, independently of the type of the update. Thus, updates of system degrees of freedom can be performed randomly, where the random numbers are chosen from a uniform distribution. This method is called simple sampling. However, its applicability is quite limited. Consider, for example, the estimation of the density of states for a discrete system with this method. After having performed aP series of M updates, we will have obtained an energetic M −1 histogram h(E) = M m=1 δEm ,E which represents an estimate for the density of states. expectation value of the energy can be estimated by E = PM The canonical P −1 −Em /kB T M = E Eh(E)e−E/kB T . If the microstates are generated m=1 Em e randomly from a uniform distribution, it is obvious that we will sample the states X with an energy E(X) in accordance with their system-specific frequency or degeneracy. High-frequency states thermodynamically dominate in the purely disordered phase. However, near phase transitions towards more ordered phases, the density of states drops rapidly – typically by many orders of magnitude. The degeneracies of the lowest-energy states representing the most ordered states are so small that the thermodynamically most interesting transition region spans even in rather small systems often hundreds to thousands orders of magnitude.4 4

In order to get an impression of the large numbers consider the 2D Ising model with 50 × 50 = 2500 spins. The total number of spin configurations is 22500 ≈ 10752 . The maximally disordered energetic state has also a degeneracy of this order. Since the ground-state degeneracy is 2 (all spins up or all down), i.e., it is of the order

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To bridge a region of 100 orders of magnitude by simple sampling would roughly mean to perform about 10100 updates in order to find a single ordered state. Assuming that a simple single update would require only a few CPU operations, it will at least take 1 ns on standard CPU cores. Even under this optimistic assumption, it would take more than 1083 years to perform 10100 updates on a single core! Thus, for studies of phase transitions, simple sampling is of little use.

4.1.5 Metropolis Sampling From the dominance of a certain restricted space of microstates in ordered phases, it is obviously a good idea to primarily concentrate in a simulation on a precise sampling of the microstates that form the macrostate under given external parameters such as, for example, the temperature. The canonical probability distribution functions, like the exemplified curves plotted in Fig. 2.1, clearly show that within the certain stable phases, only a limited energetic space of microstates is noticeably populated, whereas the probability densities drop off rapidly in the tails. Thus, an efficient sampling of this state space should yield the relevant information within comparatively short Markov chain Monte Carlo runs. This strategy is called importance sampling. The standard importance sampling variant is the Metropolis method [78], where the algorithmic microstate probability p(X) is identified with the canonical microstate probability p(X) ∼ e−βE(X) at the given temperature T (β = 1/kB T ). Thus, the acceptance probability (4.10) is governed by the ratio of the canonical thermal weights of the microstates: ′

w(X → X′) = e−β[E(X )−E(X)] .

(4.13)

According to Eq. (4.10), a Monte Carlo update from X to X′ with σ(X, X′) = 1 is accepted, if the energy of the new microstate is lower than before, E(X′) < E(X). If this update would provoke an increase of energy, E(X′) > E(X), the conformational change is accepted only with the probability e−β∆E , where ∆E = E(X′) − E(X). Technically, a random number r ∈ [0, 1) from a uniform distribution is drawn; if r ≤ e−β∆E , the move is still accepted, whereas it is rejected otherwise. Thus, the acceptance probability is exponentially suppressed with ∆E and the Metropolis simulation yields, at least in principle, a time series which is inherently correctly sampled in accordance with the canonical statistics. The arithmetic mean value of a quantity O over the finite Metropolis time PMseries is already −1 an estimate for the canonical expectation value: O = M m=1 Om ≈ hOi. In the hypothetic case of an infinitely long simulation (M → ∞), this relation of 100 , the density of states of this rather small system covers more than 750 orders of magnitude.

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is an exact equality, i.e., the deviation is due to the finiteness of the time series only. However, it is just this restriction to a finite amount of data which limits the quality of Metropolis data. Because of the canonical sampling, reasonable statistics is only obtained in the energetic region which is most dominant for a given temperature, whereas in the tails of the canonical distributions such as those shown in Fig. 2.1 the statistics is rather poor. Thus, there are three physically particularly interesting cases where Metropolis sampling as standalone method is little efficient. First, for low temperatures, where lowest-energy states dominate, the widths of the canonical distributions are extremely small and since β ∼ 1/T is very large, energetic “uphill” updates are strongly suppressed by the Boltzmann weight e−β∆E → 0. That means, once caught in a low-energy state, the simulation freezes and it remains trapped in a low-energy state for a long period. Second, p near a second-order phase transition, the width or standard deviation σE = hE 2 i − hEi2 of the canonical energy distribution function gets very large at the critical temperature TC , as it corresponds to the maximum (or, in the thermodynamic limit, the divergence) of the specific CV = σE2 /kB T 2 . Thus, a large energetic space must precisely be sampled (“critical fluctuations”) which requires high statistics. Since in Metropolis dynamics, “uphill moves” with ∆E > 0 are only accepted with a reasonable rate, if at the transition point the ratio ∆E/kB TC > 0 is not too large, it can take a long time to reach a high-energy state if starting from the low-energy end. Since near TC the correlation length diverges like ξ ∼ |τ |−ν [with τ = (T − TC )/TC ] and the correlation time in the Monte Carlo dynamics behaves like tcorr ∼ |τ |−νz , the dynamic exponent z allows to distinguish the efficiencies of different algorithms. The larger the value of z, the less effecient is the method. Unfortunately, the standard Metropolis method turns out to be one of the least efficient methods in sampling critical properties of systems exhibiting a second-order phase transition. The third reason is that the Metropolis method does also perform poorly at first-order phase transitions. In this case, the canonical distribution function is bimodal, i.e., it exhibits two separate peaks with a highly suppressed energetic region inbetween, since two phases coexist. For the reasons already outlined, the Metropolis method cannot energetically “jump” from the low- to the high-energy phase; it rather would have to explore the valley step by step. Since the energetic region between the phases is entropically suppressed – the number of possible states the system can assume is simply too small – it is thus quite unlikely that this “diffusion process” will lead the system into the high-energy phase, or it will at least take extremely long. However, apart from lowest-energy and phase transition regions, the Metropo-

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lis method can successfully be employed, often in combination with reweighting techniques. Single-Histogram Reweighting

A standard Metropolis simulation is performed at a given temperature, say T0 . However, it is often desirable to get also quantitative informations about the changes of the thermodynamic behavior at nearby temperatures. Since Metropolis sampling is not a priori restricted to a limited phase space, at least in principle, it is indeed theoretically possible to reweight Metropolis data obtained for a given temperature T0 = 1/kB β0 to a different one, T = 1/kB β. The idea is to “divide out” the Boltzmann factor e−β0 E in the estimates for any quantity at the simulation temperature and to multiply it by e−βE :

−(β−β )E PM 0 Oe Om e−(β−β0 )Em T0 m=1 , (4.14) hOiT = −(β−β )E ≈ OT = PM −(β−β0 )Em 0 e e T0 m=1

where we have again considered that the MC time series of length M is finite. In practice, the applicability of this simple reweighting method is rather limited in case the data series was generated in a single Metropolis run, since the error in the tails of the simulated canonical histograms rapidly increases with the distance from the peak. By reweighting, one of the noisy tails will gain the more statistical weight the larger the difference between the temperatures T0 and T is. In combination with the generalized-ensemble methods to be discussed later in this chapter, however, single-histogram reweighting is the only way of extracting the canonical statistics off the simulated histograms and works perfectly. Multiple-Histogram Reweighting

From each Metropolis run, an estimate for the density of states g(E) can easily be calculated.PSince the histogram measured in a simulation at temperature T , hT (E) = M m=1 δE Em , is an estimate for the canonial distribution function pcan,T (E) ∼ g(E)e−βE , the estimate for the density of states is obtained by reweighting, g(E) = hT (E)eβE . However, since in a “real” Metropolis run at the single temperature T accurate data can only be obtained in a certain energy interval which depends on T , the estimate g(E) is restricted to this typically rather narrow energy interval and does by far not cover the whole energetic region reasonably well. Thus, the question is whether the combination of Metropolis data obtained in simulations at different temperatures, can yield an improved estimate g(E). This is indeed possible by means of the multiple-histogram reweighting method [79],

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sometimes also called “weighted histogram analysis method” (WHAM) [80]. Even though the general idea is simple, the actual implementation is not trivial. The reason is that conventional Monte Carlo simulation techniques such as theP Metropolis method cannot yield absolute estimates for the partition sum ZT = E g(E)e−βE , i.e., estimates for the density of states at different energies gi (E) and gj (E ′) can only be related to each other if obtained in the same run, i.e., i = j, but not if performed under different conditions. This is not a problem for the estimation of mean values or normalized distribution functions at fixed temperatures as long as the Metropolis data obtained in the respective temperature threads are used, but interpolation to temperatures where no data were explicitly generated, is virtually impossible. Also the multiple-histogram reweighting method does not solve the problem of getting absolute quantities, but at least a “reference partition function” is introduced, which the estimates of the density of states obtained in runs at different simulation temperatures can be related to. Thus, interpolating the data between different temperatures becomes possible. Basically, the idea is to perform a weighted average of the histograms hi (E), measured in Monte Carlo simulations for different temperatures, i.e., at βi (where i = 1, 2, . . . , I indexes the simulation thread), in order to obtain an estimator for the density of states by combining the histograms in an optimal way: P gi (E)wi(E) gˆ(E) = iP . (4.15) i wi (E)

The exact density of states is given by g(E) = pcan,T (E)ZT eβE and since the normalized histogram hi (E)/Mi obtained in the ith simulation thread is an estimator for the canonical distribution function pcan,Ti (E), the density of states is in this thread estimated by hi (E) βiE Zi e , (4.16) gi (E) = Mi where Zi is the unknown partition function at the ith temperature. Since in Metropolis simulations the best-sampled energy region depends on the simulation temperature, the number of histogram entries for a given energy will differ from thread to thread. Thus, the data of the thread with high statistics at E should in this interpolation scheme get more weight than histograms with less entries at E. Therefore, the weight shall be controlled by the errors of the individual histograms. A possibility to determine a set of optimal weights is to reduce the deviation of the estimate gˆ(E) for the density of states from the unknown exact distribution hgi(E), where the symbol h. . .i is used to refer to this quantity as the true distribution which would have been hypothetically obtained in an infinite number of threads (it should not be confused with a statistical ensemble

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average). As usual, the “best” estimate is the one that minimizes the variance g − hgi)2 i. Inserting the relation (4.15) and minimizing with respect to σgˆ2 = h(ˆ the weights wi yields a solution wi =

1 , σg2i

(4.17)

where σg2i = h(gi − hgi i)2 i is the exact variance of gi in the ith thread. Because of Eq. (4.16) and the fact that Zi is an energy-independent constant in the ith thread, we can now concentrate on the discussion of the error of the ith histogram, since σg2i = σh2i Zi2e2βi E /Mi2 . The variance σh2i is also an unknown quantity and, in principle, an estimator for this variance would be needed. This would yield an expression that includes the autocorrelation time [79,80] – similar to the discussion below Eq. (4.5). However, to correctly keep track of the correlations in histogram reweighting is difficult and thus also the estimation of error propagation is nontrivial. Therefore, we follow the standard argument based on the assumption of uncorrelated Monte Carlo dynamics (which is typically not perfectly true, of course). The consequence of this idealization will be that the weights (4.17) are not necessarily optimal anymore (the applicability of the method itself is not dependent of the choice of wi, but the error of the final histogram will depend on the weights). In order to determine σh2i for uncorrelated data, we only need to calculate the probability P (hi ) that in the ith thread a state with energy E (for simplicity we assume that the problem is discrete) is hit hi times in Mi trials, where each hit occurs with the probability phit . This leads to the binomial distribution with the hit average hhi i = Mi phit. In the limit of small hit probabilities (a reasonable assumption in general if the number of energy bins is large, and, in particular, for the tails of the histogram) the binomial turns into the Poissonian distribution P (hi ) → hhi ihi e−hhii /hi! with identical variance and expectation value, σh2i = hhi i. Insertion into Eq. (4.17) yields the weights wi (E) =

Mi2 . hhi i(E)Zie2βiE

(4.18)

Since hhi i(E) is exact, the exact density of states can also be written as g(E) =

hhi i(E) βi E Zi e Mi

(4.19)

which is valid for all threads, i.e., the left-hand side is independent of i. This enables us to replace hhi i everywhere. Inserting expression (4.18) into Eq. (4.15)

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and utilizing the relation (4.19) to replace hhi i, we finally end up with the estimator for the density of states in the form PI hi (E) gˆ(E) = PI i=1 −1 , (4.20) −βi E e M Z i i i=1

where the unknown partition sum is given by X Zi = gˆ(E)e−βiE ,

(4.21)

E

i.e., the set of equations (4.20) and (4.21) must be solved iteratively.5 One (0) initializes the recursion with guessed values Zi for all threads, calculates the (0) (1) first estimate gˆ(1) (E) using Zi , re-inserts this into Eq. (4.21) to obtain Zi , and continues until the recursion process has converged close enough to a fixed point. There is a technical aspect that should be taken into account in an actual calculation. Since the density of states can even for small systems cover many orders of magnitude and also the Boltzmann factor can become very large, the application of the recursion relations (4.20) and (4.21) often results in overflow errors since the floating-point data types cannot handle these numbers. At this point, it is helpful to change to a logarithmic representation which however, makes it necessary to think about adding up large numbers in logarithmic form. Consider the special but important case of two positive real numbers a ≥ 0 and 0 ≤ b ≤ a which are too large to be stored such that we wish to use their logarithmic representations alog = log a and blog = log b instead. Since the result of the addition, c = a + b, will also be too large, we introduce clog = log c as well. The summation is then performed by writing c = eclog = ealog + eblog . Since a ≥ b (and thus also alog ≥ blog), it is useful to separate a, and to rewrite the sum as eclog = ealog (1 + eblog −alog ). Taking the logarithms yields the desired result, where only the logarithmic representations are needed to perform the summation: clog = alog + log(1 + x), where x = b/a = eblog−alog ∈ [0, 1]. The upper limit x = 1 is obviously associated to a = b, whereas the lower limit x = 0 matters if a ≥ 0, b = 0.6 Since the logarithm of the density of states is proportional to 5

Note that for a system with continuous energy space which is partitioned into bins of width ∆E in the simulation, the right-hand side of Eq. (4.21) must still be multiplied by ∆E. 6 At the lower limit, there is a numerical problem, if blog − alog ≪ 0 (or x = b/a ≪ 1) is so small that the minimum allowed floating-point number is underflown by x. This typically occurs if a and b differ by many tens to thousands orders of magnitude (depending on the floating-point number precision). In this case, the difference between c and a cannot be resolved, as the error in clog = alog + O(x) is smaller than the numerical resolution; in which case we simply set clog = alog . If this is not acceptable and a higher resolution is really needed, non-standard concepts of handling numbers with arbitrary precision could be an alternative.

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the microcanonical entropy, S(E) ∼ log g(E), the logarithmic representation has even an important physical meaning.

4.2 Generalized-Ensemble Monte Carlo Methods The Metropolis method is the simplest method and for this reason it is a good starting point for the simulation of a complex system. However, it is also one of the least efficient methods and thus one will often end up with the question of how to improve the efficiency of the sampling. The little efficiency of the Metropolis method in simulations at the most interesting temperatures lies in its construction as it is based on the importance sampling with respect to the thermodynamically most relevant phase space region. One of the most frequently used “tricks” is to employ a modified statistical ensemble within the simulation run and to reweight to the canonical ensemble after the simulation. Thus, the simulation is performed in a typically artificial generalized ensemble.

4.2.1 Replica-Exchange Monte Carlo Method (Parallel Tempering) 4.2.2 Multicanonical Sampling 4.2.3 Wang-Landau Method

4.3 Lattice Polymers: Monte Carlo Sampling vs. Rosenbluth Chain Growth Computer simulations of long lattice peptides are particularly demanding. The reason is that the native fold, i.e., the ground-state or lowest-energy conformation itself, plays an essential role in protein science and that it is, in the discrete lattice representation, non- or low-degenerate. As mentioned earlier, lattice polymers are modeled by self-avoiding walks. This takes into account the finite volume and the uniqueness of the monomers. A lattice site can hence be occupied by a single monomer only. This has the consequence that the number of very dense conformations of a polymer is by orders of magnitude lower than that of random coil states. In Monte Carlo simulations, particular attention must therefore be dedicated to efficient update procedures which also allow the sampling of dense conformations. In polymer simulations, so-called move sets were applied with some success to study the behavior near the Θ point, which denotes the phase transition, where polymers subject to an attractive interaction collapse from random coils to compact conformations. Move sets being widely used usually consist of semilocal transformations that change the position of a single monomer and a single bond

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vector (end flips), a single monomer position but two bonds (corner flips), two positions and three bonds (crankshaft) or moves with more changes [36–39], and nonlocal pivot rotations, where the ith monomer serves as pivot point and one of the two partial chains connected with it is rotated about any axis through the pivot [77]. An example of such a moveset is shown in Fig. 4.1. In a conventional Metropolis Markov chain Monte Carlo method [78], the structural update from a given conformation X = (x1, x2, . . . , xN ), where the xi ’s are the lattice monomer positions of a chain of length N , to a new conformation X′ is accepted with the probability n o ′ −β[E(X′ )−E(X)] w(X → X ) = min 1, e , (4.22) where β = 1/kB T is the thermal energy. On regular lattices with small coordination numbers, such as the widely considered simple-cubic lattice, the updating of conformations employing move sets becomes inefficient, however, the more dense the conformation is. At low temperatures, the acceptance rate of locally changing a dense conformation decreases drastically and the simulation threatens to get stuck in a specific conformation or to oscillate between two states. Since the search for ground states is an essential aspect of studying lattice proteins, the application of move sets is not very useful, at least for chains of reasonable length. A more promising alternative is the completely different approach based on chain growth. The polymer grows by attaching the nth monomer at a randomly chosen nearest-neighbor site of the (n − 1)th monomer. The growth is stopped, if the total length N of the chain is reached or the randomly selected continuation of the chain is already occupied. In both cases, the next chain is started to grow from the first monomer. This simple chain growth is also not yet very efficient, since the number of discarded chains grows exponentially with the chain length. The performance can be improved with the Rosenbluth chain growth method [41], where first the free nearest neighbors of the (n − 1)th monomer are determined and then the new monomer is placed to one of the unoccupied sites. Since the probability for the next monomer to be set varies with the number of free neighbors, this implies a bias given by !−1 n Y ml pn = , (4.23) l=2

where ml is the number of free neighbors to place the lth monomer. The bias is corrected by assigning a Rosenbluth weight factor WnR = p−1 n

(4.24)

4.3 Lattice Polymers: Monte Carlo Sampling vs. Rosenbluth Chain Growth

end flip

79

corner flip

i

crankshaft move

pivot move

Figure 4.1: Example for a move set consisting of end and corner flips, crankshaft moves and pivot updates on a square lattice.

to each chain that has been generated by this procedure. An illustrative example for the bias in the Rosenbluth chain-growth method is shown in Fig. 4.2. The two depicted linear chains are grown on a square lattice from both ends (labeled by “1”). According to Rosenbluth sampling, the chain is continued if the number of free neighbor sites is m ≥ 1. Since the number of free nearest-neighbor places varies, different probabilities for the continuation of the chain occur. Since both conformations are identical, the probability of creation should be the same. This requires the introduction of the correction weights. Although this biased growth is more efficient than simple sampling, this method suffers from attrition too: If all nearest neighbors are occupied, i.e., the chain was running into a “dead end” (attrition point), the complete chain has to be discarded and the growth process has to be started anew. In order to improve the efficiency of Rosenbluth chain growth considerably, a strategy to increase the number of successfully created chains is useful: ’Go with the Winners’.

80

4. Monte Carlo and Chain Growth Methods for Molecular Simulations 1/4

1/2

1/3

1

1/3

1/3

1/3

1

1/3

1/4

Figure 4.2: Square-lattice example for the bias implied by Rosenbluth sampling. Both walks shown are grown from the monomer labeled “1”. Although the shapes are identical, they are created with different probabilities (left: p = 1/108, right: p = 1/72).

4.4 Pruned-Enriched Rosenbluth Method: Go with the Winners Combining the Rosenbluth chain-growth method with population control, as it is done in PERM (Pruned-Enriched Rosenbluth Method) [43–45], leads to a further considerable improvement of the efficiency by increasing the number of successfully generated chains. This method renders particularly useful for studying the Θ point of polymers, since then the Rosenbluth weights of the statistically relevant chains approximately cancel against their Boltzmann probability. The (a-thermal) Rosenbluth weight factor WnR is therefore replaced by WnPERM

=

n Y l=2

ml e−(El −El−1 )/kB T ,

2≤n≤N

(E1 = 0,

W1PERM = 1),

where T is the temperature and El is the energy of the partial chain Xl = (x1 , . . . , xl ) created with Rosenbluth chain growth. In PERM, population control works as follows. If a chain has reached length n, its weight WnPERM is calculated and compared with suitably chosen upper and lower threshold values, Wn> and Wn< , respectively. For WnPERM > Wn> , identical copies are created which grow then independently. The weight is equally divided among them. If WnPERM < Wn<, the chain is pruned with some probability, say 1/2, and in case of survival, its weight is doubled. For a value of the weight lying between the thresholds, the chain is simply continued without enriching or pruning the sample. The upper and lower thresholds Wn> and Wn< are empirically parametrized. Although their values do not influence the validity of the method, a careful choice can drastically improve the efficiency of the method (the “worst” case is Wn> = ∞ and Wn< = 0, in which case PERM is simply identical with Rosenbluth sampling). An efficient way of parametrization is dynamical adaption of the values [43–48] with respect to the actual number of generated chains cn with length n and their estimated

4.5 Canonical Chain Growth with PERM

81

7

2 0.5ln hRgyr,ee i

6 2 hRgyr i

5 4 3

2 hRee i

2 1 0

2

3

4

5

6

7

8

9

10

11

ln N 2 2 Figure 4.3: Scaling of mean square radius of gyration hRgyr i and end-to-end distance hRee i for self-avoiding walks. Data points refer to results from PERM runs for N = 16, 32, . . . , 32 768 steps. Lines manifest the respective power-law behaviors.

partition sum Zn =

1 X PERM Wn (t), c1 t

(4.25)

where c1 is the number of growth starts (also called “tours”) and t counts the generated conformations with n monomers. Useful choices of the threshold values are c2n > Wn = C1 Zn 2 , Wn< = C2 Wn>, (4.26) c1 where C1 , C2 ≤ 1 are constants. For the first tour, Wn> = ∞ and Wn< = 0, i.e., no pruning and enriching. Results of a simple athermal application of PERM to self-avoiding walks on a simple-cubic lattice are plotted in Fig. 4.3, where the scaling behavior 2 2 hRgyr,ee i ∼ N 2ν of the mean square radius of gyration hRgyr i and end-to-end 2 distance hRee i with the number of steps N is shown. Data were obtained for chains of N = 16, 32, . . . , 32 768 steps. For both quantities, the slope of the lines in the logarithmic plot is ν = 0.59, which is close to the precisely known critical exponent ν = 0.588 . . . [81].

4.5 Canonical Chain Growth with PERM In recently proposed new PERM variants nPERMss is [new PERM with simple/importance sampling (ss/is)], a considerable improvement was achieved by creating different copies, i.e., the chains are identical in (n − 1) monomers but

82

4. Monte Carlo and Chain Growth Methods for Molecular Simulations

have different continuations, instead of completely identical ones, since identical partial chains usually show up a similar evolution. Because of the different continuations, the weights of the copies can differ. Therefore it is not possible to decide about the number of copies on the basis of a joint weight. The suggestion is to calculate first a predicted weight which is then compared with the upper threshold Wn> in order to determine the number of clones. Another improvement of PERM being followed up since first applications to lattice proteins is that the threshold values Wn> and Wn< are no longer constants, but are dynamically adapted with regard to the present estimate for the partition sum and to the number of successfully created chains with length n. The partition sum is proportional to the sum over the weight factors of all conformations of chains with length n, created with a Rosenbluth chain-growth method like, for instance, nPERMss is : Zn =

1 X nPERMssis Wn (Xn,t). c1 t

(4.27)

Here, Xn,t denotes the tth generated conformation of length n. The proportionality constant is the inverse of the number of chain growth starts c1 . Note that due to this normalization it is possible to estimate the degeneracy of the energy states. This is in striking contrast to importance sampling Monte Carlo methods, where the overall constant on the r.h.s. of Eq. (4.27) cannot be determined and hence only relative degeneracies can be estimated. Since nPERMss and nPERMis, respectively, are possible fundamental ingredients for our algorithm, it is useful to recall in some detail how these chain-growth algorithms work. The main difference in comparison with the original PERM is that, if the sample of chains of length n − 1 shall be enriched, the continuations to an unoccupied nearest-neighbor site have to be different, i.e., the weights of these chains with length n can differ. Therefore it is impossible to calculate a uniform weight like WnPERM as given in Eq. (4.25) before deciding whether to enrich, to prune, or simply to continue the current chain of length n − 1. As proposed in Ref. [46], it is therefore useful to control the population on the basis of a predicted weight Wnpred which is introduced as Wnpred

=

nPERMss Wn−1 is

mn X

ss

is , χnPERM α

(4.28)

α=1

where mn denotes the number of free neighboring sites to continue with the nPERMss is nth monomer. The “importances” χα differ for nPERMss and nPERMis. Due to its characterization as a simple sampling algorithm (nPERMss), where all continuations are equally probable, and as a method with importance sampling

4.5 Canonical Chain Growth with PERM

83

(nPERMis), the importances may be defined as (α) 1 −β(En −En−1 ) nPERMis (α) e . χnPERMss = 1, χ = m + α α n 2

(4.29)

(α)

The expression for nPERMis involves the energy En of the choice α ∈ [1, mn] (α) for placing the nth monomer and the number of free neighbors mn of this choice which is identical with mn+1 , provided the αth continuation was indeed selected for placing the nth monomer. Since χnPERMis contains informations beyond the α nth continuation of the chain, nPERMis controls the further growth better than nPERMss. The predicted weight for the nth monomer is now used to decide how the growth of the chain is continued. If the predicted weight is bigger than the current threshold, Wnpred > Wn>, and mn > 1, the sample of chains is enriched and the number of copies k is determined according to the empirical rule k = min[mn , int(Wnpred/Wn>)]. Thus, 2 ≤ k ≤ mn different continuations will be followed up. Using nPERMss, the k continuations are chosen randomly with equal probability among the mn possibilities, while for nPERMis the probability of selecting a certain k-tuple A = {α1 , . . . , αk } of different continuations is given by P χα α∈A pA = P P . (4.30) χα A α∈A

Considering the probabilities pA as partial intervalsP of certain length, arranging them successively in the total interval [0, 1] (since A pA = 1), and drawing a random number r ∈ [0, 1), one selects the tuple whose interval contains r. This tuple of different sites is then chosen to continue the chain. The corresponding weights are [46]: nPERMss is Wn,α j

=

nPERMss Wn−1 is

m n e−β mn k pA k

(αj )

En

−En−1

,

(4.31)

where j ∈ {1, . . . , k} is the index of the αj th continuation within the tuple A. nPERMss In the special case of simple sampling this expression reduces to Wn,α = j (α )

nPERMss Wn−1 mn exp[−β(En j − En−1)]/k. If the predicted weight is less than the lower threshold, Wnpred < Wn<, however, the growth of this chain is stopped with probability 1/2. In this case, one traces the chain back to the last branching point, where the growth can be continued again, or, if there are no branching points, a new tour is started. If the chain survives, the continuation of the chain follows the same procedure as described above, but now with k = 1, where Eq. (4.30)

84

4. Monte Carlo and Chain Growth Methods for Molecular Simulations

P simplifies to pA = χα / α χα since A = {α}. In this case the weight of the chain is taken twice. For Wn< ≤ Wnpred ≤ Wn> , the chain is continued without enriching or pruning (once more with k = 1). The first tour, where the nth monomer is attached for the first time, is started with bounds set to Wn> = ∞ and Wn< = 0, thus avoiding enrichment and pruning. For the following tours, we use (4.26). The constant C1 ≤ 1 controls the number of successfully generated chains per tour. For the lower bound a useful choice is C2 = 0.2 [46].

4.6 Multicanonical Chain-Growth Algorithm The efficiency of PERM depends on the simulation temperature. Therefore, a precise estimation of the density of states requires separate simulations at different temperatures. Then, the density of states can be constructed by means of the multiple-histogram reweighting method [79]. Although being a powerful method, it is difficult to keep track of the statistical errors involved in the individual histograms obtained in the simulations. An alternative approach, in which the density of states g(E) is obtained within a single simulation without the necessity of a subsequent multi-histogram reweighting, is the combination of PERM with multicanonical sampling, the socalled multicanonical chain-growth method [47,48], which will be described in the following. It should be noted that there is another powerful, related method, called flat-histogram PERM [82], which is based on a different ansatz, but it also aims at an improved sampling of “rare” conformations, compared to PERM.

4.6.1 General Aspects of Multicanonical Sampling The general idea of multicanonical sampling [83–85] is to simulate the thermodynamic behavior of the system in a generalized (multicanonical) ensemble, where the energetic macrostates are distributed uniformly, pmuca(E) = const, which implies the introduction of multicanonical weight factors Wmuca (E). In typical multicanonical Monte Carlo simulations, the dynamics is therefore governed by a random walk in energy space. Hence, the sampling of entropically rare events is, in principle, as frequent as the sampling of highly degenerate energetic states. The acceptance probability for a new system configuration X′ with energy E(X′) is wmuca (X → X′) = min[1, exp{S(E(X)) − S(E(X′))}], where S(E(X)) = − ln Wmuca (E(X)) is related to the microcanonical entropy. The canonical energy distribution pcan (E) ∼ g(E) exp(−E/kB T )

(4.32)

4.6 Multicanonical Chain-Growth Algorithm

85

for a given temperature T is related to the multicanonical histogram via −1 pcan (E) ∼ Wmuca (E)pmuca(E)e−E/kB T ,

(4.33)

which implies that the multicanonical weights are proportional to the inverse density of states, Wmuca (E) ∼ g −1(E). Since g(E) is unknown, the determination of the weights Wmuca (E) is not straightforward and must be performed in the first stage of the simulation in an iterative procedure [84].

4.6.2 Multicanonical Sampling of Rosenbluth-Weighted Chains For a given temperature, the nPERMss is algorithms yield accurate canonical distributions over some orders of magnitude. In order to construct the entire density of states, standard reweighting procedures may be applied, requiring simulations for different temperatures [79]. The low-temperature distributions are, however, very sensitive against fluctuations of weights which inevitably occur because the number of energetic states is low, but the weights are high. Thus, it is difficult to obtain a correct distribution of energetic states, since this requires a reasonable number of hits of low-energy states. Therefore we assign the chains an additional weight, the multicanonical weight factor Wnflat, chosen such that all possible energetic states of a chain of length n possess almost equal probability of realization. The first advantage is that states having a small Boltzmann probability compared to others are hit more frequently. Second, the multicanonical weights introduced in that manner are proportional to the inverse canonical distribution at temperature T , Wnflat(E) ∼ 1/pcan,T (E), with respect to the inverse density of states n Wnflat (E) ∼ gn−1(E)

(4.34)

for T → ∞. Thus, only one simulation is required and a multi-histogram reweighting is not necessary. The multicanonical weight factors are unknown in the beginning and have to be determined iteratively. Before we discuss the technical aspects regarding this novel method, we first explain it more formally. The energy-dependent multicanonical weights (4.34) are trivially introduced into the partition sum as suitable “decomposition of unity” in the following way: −1 1 X nPERMssis Zn = Wn (Xn,t)Wnflat(E(Xn,t)) Wnflat(E(Xn,t)) . (4.35) c1 t

Since we are going to simulate at infinite temperature, we express with (4.34) the partition sum which then coincides with the total number of all possible conformations as 1 X gn (E(Xn,t))Wn(Xn,t) (4.36) Zn = c1 t

86

4. Monte Carlo and Chain Growth Methods for Molecular Simulations

with the combined weight ss

Wn (Xn,t) = WnnPERMis (Xn,t)Wnflat(E(Xn,t)).

(4.37)

Taking this as the probability for generating chains of length n, pn ∼ Wn , leads to the desired flat distribution Hn (E), from which the density of states is obtained by Hn (E) gn (E) ∼ flat . (4.38) Wn (E) The canonical distribution at any temperature T is calculated by simply reweighting the density of states to this temperature, pcan,T (E) ∼ gn (E) exp (−E/kB T ). n

4.6.3 Iterative Determination of the Density of States In the following, we will describe our procedure for the iterative determination of the multicanonical weights, from which we obtain an estimate for the density of states. Since there are no informations about an appropriate choice for the multicanonical weights in the beginning, we set them in the zeroth iteration for flat,(0) all chains 2 ≤ n ≤ N and energies E equal to unity, Wn (E) = 1, and the (0) histograms to be flattened are initialized with Hn (E) = 0. These assumptions render the zeroth iteration a pure nPERMss is run. Since we set β = 1/kB T = 0 (and kB = 1) from the beginning, the accumulated histogram of all generated chains of length n, X nPERMss (0) is Hn (E) = Wn,t δEt E , (4.39) t

is a first estimate of the density of states. In order to obtain a flat histogram in the next iteration, the multicanonical weights flat,(0)

Wnflat,(1) (E)

=

Wn

(E)

(0) Hn (E) (1)

∀ n, E

(4.40)

are updated and the histogram is reset, Hn (E) = 0. The first and all following iterations are multicanonical chain growth runs and proceed along similar lines as described above, with some modifications. The prediction for the new weight follows again (4.28), but the importances χisα (4.29) are in the ith iteration introduced as flat,(i) (α) 1 Wn (En ) χis,(i) = mn(α) + . (4.41) α flat,(i) 2 W (E ) n−1 n−1

4.6 Multicanonical Chain-Growth Algorithm

87

ss,(i)

In the simple sampling case, we still have χα = 1. If the sample is enriched (Wnpred > Wn> ) the weight (4.31) of a chain with length n choosing the αj th continuation is now replaced by flat,(i)

ss,is Wn,α j

=

ss,is Wn−1

(α )

mn Wn (En j ) , flat,(i) mn W (E ) n−1 k pA n−1 k

(4.42)

where in the simple sampling case (ss) pA and the binomial factor again cancel each other. If Wn< ≤ Wnpred ≤ Wn>, an nth possible continuation is chosen (selected as described for the enrichment case, but with k = 1) and the weight is as in Eq. (4.42). Assuming that Wnpred < Wn< and that the chain has survived pruning (as usual with probability 1/2), we proceed as in the latter case and the chain is assigned twice that weight. The upper threshold value is now determined in analogy to Eq. (4.26) via Wn>

=

where Znflat =

2 flat cn C1 Zn 2 , c1

1 X ss,is Wn,t c1 t

(4.43)

(4.44)

is the estimated partition sum according to the new distribution provided by the weights (4.42) for chains with n monomers. Whenever a new iteration is started, Znflat, cn , Wn< are reset to zero, and Wn> to infinity (i.e. to the upper limit of the data type used to store this quantity). If a chain of length n with the energy E was created, the histogram is increased by its weight: X ss,is Hn(i) (E) = Wn,t δEt E . (4.45) t

From iteration to iteration, this histogram approaches the desired flat distribution Hn (E) and after the final iteration i = I, the density of states is estimated by (I)

gn(I) (E)

=

Hn (E) flat,(I)

Wn

(E)

,

2 ≤ n ≤ N,

(4.46)

in analogy to Eq. (4.38). Figure 4.4 shows how the estimate for the density of states of a lattice protein with 42 monomers evolves with increasing number of iterations. The 0th iteration is the initial pure nPERMis run at β = 0. This does not render, however, a proper

88

4. Monte Carlo and Chain Growth Methods for Molecular Simulations 0 0th

-5 1st

-10 -15

g E)

log10 (

-20

6th 9th

-25 -30 -35

-30

-25

-20

E

-15

-10

-5

0

(i)

Figure 4.4: Estimates for the density of states g42 (E) for an exemplified heteropolymer with 42 monomers after several recursion levels. Since the curves would fall on top of each other, we have added, for better distinction, a suitable offset to the curves of the 1st, 6th, and 9th run. The estimate of the 0th run is normalized to unity.

image of the abilities of nPERMis which works much better at finite temperatures. Iterations 1 to 8 are used to determine the multicanonical weights over the entire energy space E ∈ [−34, 0]. Then, the 9th iteration is the measuring run which gives a very accurate estimate for the density of states covering about 25 orders of magnitude. In standard multicanonical chain-growth simulations for HP lattice proteins with up to 100 monomers, it is sufficient to perform about 30 iterations. The runs 0 to I − 1 are usually terminated after 105 –106 chains of total length N are generated, while in the measuring run (i = I) usually 107 –109 conformations are enough to obtain reasonable statistics. The parameter C1 in Eq. (4.43) that controls the pruning/enrichment statistics and thus how many chains of complete length N are generated per tour, is often set to C1 = 0.01, such that on average 10 complete chains were successfully constructed within each tour [47,48]. With this choice, the probability for pruning the current chain or enriching the sample is about 20%. In almost all started tours c1 at least one chain achieves then its complete length. Thus the ratio between successfully finished tours and started tours is very close to unity, assuring that our algorithm performs with quite good efficiency. Unlike typical applications of multicanonical or flat histogram algorithms in importance sampling schemes, where all energetic states become equally probable such that the dynamics of the simulation corresponds to a random walk in energy space, the distribution to be flattened in our method is the histogram that

4.7 Validation and Performance

89

Table 4.1: Sequences, hydrophobicity nH , and global minimum energy Emin with degeneracy g0ex (without rotations, reflections, and translations) of the exactly enumerated 14-mers used for validation of our algorithm. The last column contains the predictions for the ground-state degeneracy obtained with our method. No. 14.1 14.2 14.3 14.4

sequence HP HP H2P HP H2P2 H H2 P2 HP HP H2P HP H H2 P HP HP2HP HP H2 H2 P HP2HP HP H2P H

nH 8 8 8 8

Emin −8 −8 −8 −8

g0ex 1 2 2 4

g0 0.98± 0.03 2.00± 0.07 2.00± 0.06 3.99± 0.13

accumulates the weights of the conformations. Hence, if the histogram is flat, a small number of high-weighted conformations with low energy E has the same probability as a large number of appropriate conformations with energy E ′ > E carrying usually lower weights. Therefore the number of actual low-energy hits remains lower than the number of hits of states with high energy. In order to accumulate enough statistics in the low-energy region, the comparative large number of generated conformations in the measuring run is required.

4.7 Validation and Performance Before we discuss the physical results obtained with the multicanonical chaingrowth algorithm, we first remark on tests validating the method. We compared the specific heat for very short chains with data from exact enumeration and found that our method reproduces the exact results with high accuracy. For a chain with 42 monomers, where exact results are not available, we performed a multi-histogram reweighting [79] from canonical distributions at different temperatures obtained from original nPERMis runs. Here, it turned out that our method shows up a considerably higher performance (higher accuracy in spite of lower statistics at comparable CPU times). We also compared with implementations based on sophisticated importance sampling Monte Carlo schemes, e.g., we have also performed multicanonical sampling [83–85] and Wang-Landau simulations [86] in combination with conformational updates different from chain growth (e.g. move sets as described in Section 4.3). For the present applications, however, all of these attempts proved to be less efficient than our multicanonical chain-growth method [47,48].

4.7.1 Comparison with Results from Exact Enumeration As a first validation of our method, we apply it to a set of 14-mers with some interesting properties (see Table 4.1) regarding the relation between their ground-

90

4. Monte Carlo and Chain Growth Methods for Molecular Simulations -2.0

-3.0

-4.0 log10 ε(T ) -5.0 14.1

14.2

-6.0 14.3 -7.0

0.1

0.2

0.3

0.4

14.4

0.5

0.6

0.7

0.8

0.9

1.0

T

Figure 4.5: Logarithmic plot of the relative errors of our estimates for the specific heats of the 14-mers given in Table 4.1.

state degeneracy and the strength of the low-temperature conformational transition between lowest-energy states and compact globules [47,48,72]. In finitesize systems, (pseudo)transitions are usually identified through structural peculiarities (maxima for strong transitions or “shoulders” for weak transitions) in the temperature-dependent behavior of fluctuations of thermodynamic quantities. Usually, it is hard to obtain a quite accurate estimation of the fluctuations in the low-temperature region. Thus, it is a good test of our method to calculate fluctuating quantities for the 14-mers listed in Table 4.1 and to compare with results that are still available by exactly enumerating all possible 943 974 510 conformations (except translations) [72]. Therefore, we determined with our method the densities of states for these 14-mers and calculated the fluctuations of the energy around the mean value in order to obtain the specific heat. We generated 109 chains and the results for the specific heats turned out to be highly accurate. This is demonstrated in Fig. 4.5, where we have plotted for the exemplified 14-mers the relative errors ε(T ) = |CVex (T ) − CV (T )|/CVex(T ) of our estimates CV (T ) compared with the specific heats CVex(T ) obtained by the exact enumeration procedure. We see that, except for very low temperatures, the relative error is uniformly smaller than 10−3.

4.7.2 Multiple Histogram Reweighting The calculation of the density of states by means of canonical stochastic algorithms cannot be achieved by simply reweighting one canonical histogram, obtained for a given temperature, to as many as necessary distributions to cover the

4.7 Validation and Performance

91

whole temperature region, since the overlap between the sampled distribution and most of the reweighted histograms is too small [87]. As this simple reweighting only works in a certain region around the temperature the simulation was performed, a multiple application of the reweighting procedure at different sampling temperatures is necessary [79]. For the sequence of a 42-mer to be studied in detail in Chapter 7, we performed the multiple reweighting of 5 overlapping histograms obtained by separate nPERMis runs at temperatures 0.3, 0.5, 0.8, 1.5, and 3.0 in order to estimate the density of states. The histograms as well as the resulting density of states are shown in Fig. 4.6, where we have also plotted the density of states being obtained by means of our multicanonical sampling algorithm. Each of the histograms contains statistics of 8×107 chains. This number was adequately chosen such that the density of states from histogram reweighting matches within the error bars of the density of states obtained with our algorithm that also inherently supplies us with the absolute density of states. Note that these absolute values cannot be obtained by means of the multiple-histogram reweighting procedure, where the normalization is initially arbitrary. In this example, the density of states was obtained by accumulating statistics of 5×107 chains. This means that 8 times more chains were necessary to approximately achieve the accuracy with the multiple histogram reweighting method. The iterative period for the determination of the multicanonical weights is no drawback, as it takes in our implementation only 10% compared to the production run. Therefore we conclude that our dynamical method is more efficient and also more elegant than a static reweighting scheme, where also a reliable estimation of statistical errors is cumbersome.

92

4. Monte Carlo and Chain Growth Methods for Molecular Simulations

70 60 50 log10 g(E), 40 log10 H T (E)

T = 0.3

log10 H T (E)

0.5 0.8

30

1.5

3.0

20 log10 g(E) 10 0 -35

-30

-25

-20

-15

-10

-5

0

E

Figure 4.6: Histograms H T (E) obtained by single nPERMis runs for 5 different temperatures T = 0.3, 0.5, 0.8, 1.5, and 3.0 (dashed lines). The resulting density of states g(E) obtained by multiple histogram reweighting (long dashed line) lies within the error bars of the density of states calculated by means of our method (solid line).

5 Freezing and Collapse of Flexible Lattice Polymers

5.1 Conformational Transitions of Flexible Homopolymers As a first application of the sophisticated chain-growth methods introduced in the previous chapter, we analyze the crystallization and collapse transition of a simple model for flexible polymer chains on simple cubic and face-centered cubic lattices [88]. The analysis of conformational transitions a single polymer in solvent can experience is surprisingly difficult. In good solvent (or high temperatures), solvent molecules occupy binding sites of the polymer and, therefore, the probability of noncovalent bonds between attractive segments of the polymer is small. The dominating structures in this phase are dissolved or random coils. Approaching the critical point at the Θ temperature, the polymer collapses and in a cooperative arrangement of the monomers, globular conformations are favorably formed. At the Θ point, which has already been studied over many decades, the infinitely long polymer behaves like a Gaussian chain, i.e., the effective repulsion due to the volume exclusion constraint is exactly balanced by the attractive monomermonomer interaction. Below the Θ temperature, the polymer enters the globular phase, where the influence of the solvent is small. Globules are very compact conformations, but there is little internal structure, i.e., the globular phase is still entropy-dominated. For this reason, a further transition towards low-degenerate energetic states is expected to happen: the freezing or crystallization of the polymer. Since this transition can be considered as a liquid-solid phase separation process, it is expected to be of first order, in contrast to the Θ transition, which exhibits characteristics of a second-order phase transition [89,90]. The complexity of this problem appears in the quantitative description of these processes. From the analysis of the corresponding field theory [91] it is known

94

5. Freezing and Collapse of Flexible Lattice Polymers

that for the Θ transition the upper critical dimension is dc = 3, i.e., multiplicative and additive logarithmic corrections to the Gaussian scaling are expected and, indeed, predicted by field theory [92–95]. However, until now neither experiments nor computer simulations could convincingly provide evidence for these logarithmic corrections. This not only regards analyses of different single-polymer models [43,96–101], but also the related problem of critical mixing and unmixing in polymer solutions [102–106]. In a remarkable recent study of a bond-fluctuation polymer model, it was shown that, depending on the intramolecular interaction range, collapse and freezing transition can fall together in the thermodynamic limit [99,100]. This surprising phenomenon is, however, not general. For an off-lattice bead-spring polymer with FENE (finitely extensible nonlinear elastic) bond potential and intra-monomer Lennard–Jones interaction, for example, it is expected that both transitions remain well separated in the limit of infinitely long chains [101]. Here, we investigate collapse and freezing of a single homopolymer restricted to simple cubic (sc) and face-centered cubic (fcc) lattices. We primarily focus on the freezing transition, where comparatively little is known as most of the analytical and computational studies in the past were dedicated to the controversially discussed collapse transition; see, e.g., Refs. [43,96,103–105,107–113]. A precise statistical analysis of the conformational space relevant in this low-temperature transition regime is difficult as it is widely dominated by highly compact low-energy conformations which are entropically suppressed. Most promising for these studies appear the chain-growth methods based on PERM discussed in the previous chapter, which, in their original formulation [43], are particularly useful for the sampling in the Θ regime. For the analysis of the freezing transition, the generalized multicanonical [47,48] or flat-histogram variants [82] are more efficient. The precision of these algorithms when applied to lattice polymers, is manifested by unraveling even finite-length effects induced by symmetries of the underlying lattice, as will be seen in the following.

5.2 Modeling and Simulation of the Simplest Model for Flexible Polymers: Interacting Self-Avoiding Walks Segments of polymers do not intersect, i.e., a lattice site can only be occupied by a single monomer. Thus, polymer conformations on lattices are typically modeled by self-avoiding walks which we have already investigated in Section 3.2. Since monomers of flexible polymers typically experience mutual attractive forces, we here employ the interacting self-avoiding walk (ISAW) model for lattice polymers. In order to mimic the “poor solvent” behavior in the energetic regime, i.e., at

5.3 Energetic Fluctuations of Finite-Length Polymers

95

low temperatures, nearest-neighbor contacts of nonadjacent monomers reduce the energy. Thus, the most compact conformations possess the lowest energy. Formally, the total energy of a conformation X = (x1, x2, . . . , xN ) of a chain with N beads is simply given as E(X) = −ε0 nNN(X),

(5.1)

where ε0 is an unimportant energy scale (which is set ε0 ≡ 1 in the following) and nNN (X) is the number of nearest-neighbor contacts between nonbonded monomers. Since in this model all nearest-neighbor contacts are attractive, a homopolymer can be considered as a “homogeneous heteropolymer” with hydrophobic monomers only. Not surprisingly, Eq. (5.1) is thus a special case of the HP model (1.16). For the simulation of this model in the Θ regime, employing the nPERMss (new PERM with simple sampling) variant [46] is very efficient and polymer chains with lengths of up to 32 000 (sc) and 4 000 (fcc) monomers, respectively, can be analyzed with reliable statistics. For the analysis of the conformational behavior below the Θ point, the more sophisticated multicanonical [47,48] or flat-histogram techniques [82] are more efficient as these methods increase, in particular, the sampling of entropically suppressed (“rare”) conformations, which are, for example, essential for the study of the freezing transition. Due to the much higher demands in this regime, maximum chain lengths, for which precise results are reliably be obtained, are N = 125 (sc) and N = 56 (fcc), respectively.

5.3 Energetic Fluctuations of Finite-Length Polymers It was recently found √ for a bond-fluctuation model with inter-monomeric interaction radius r = 6 that in the infinite chain-length limit collapse and freezing are indistinguishable phase transitions appearing at the same temperature (the Θ temperature TΘ) [99,100]. In a bead-spring FENE model analysis [101], this phenomenon could not be observed: Both transitions exist in the thermodynamic limit and the crossover peaks in the specific heat remain well-separated. The same observation was made independently in Ref. [100] for the bond-fluctuation model with increased interaction range. In the following, we perform for the lattice polymer model (5.1) a detailed analysis of these transitions on regular sc and fcc lattices and discuss the expected behavior in the thermodynamic limit.

96

5. Freezing and Collapse of Flexible Lattice Polymers

5.3.1 The Expected Peak Structure of the Specific Heat Statistical fluctuations of the energy, as expressed by the specific heat, can signalize thermodynamic activity. Peaks of the specific heat as a function of temperature are indicators for transitions or crossovers between physically different macrostates of the system. In the thermodynamic limit, the collective activity, which influences typically most of the system particles, corresponds to thermodynamic phase transitions. For a flexible polymer, three main phases are expected: The random-coil phase for temperatures T > TΘ, where conformations are unstructured and dissolved; the globular phase in the temperature interval Tm < T < TΘ (Tm: melting temperature) with condensed, but unstructured (“liquid”) conformations dominating; and for T < Tm the “solid” phase characterized by locally crystalline or amorphous metastable structures. In computer simulations, only polymers of finite length are accessible and, therefore, the specific heat possesses typically a less pronounced peak structure, as finite-length effects can induce additional signals of structural activity and shift the transition temperatures. These effects, which are typically connected with surface-reducing monomer rearrangements, are even amplified by steric constraints in lattice models as used in our study. Although these pseudotransitions are undesired in the analysis of the thermodynamic transitions, their importance in realistic systems is currently increasing with the high-resolution equipment available in experiment and technology. The miniaturization of electronic circuits on polymer basis and possible nanosensory applications in biomedicine will, therefore, require a more emphasized analysis of the finite-length effects in the future.

5.3.2 Simple-Cubic Lattice Polymers Figure 5.1 shows typical examples of specific heats for very short chains on the sc lattice and documents the difficulty of identifying the phase structure of flexible homopolymers. The 27-mer exhibits only a single dominating peak – which is actually only an sc lattice effect. The reason is that the ground states are cubic (3×3×3) and the energy gap towards the first excited states is ∆E = 2.1 Actually, also the most pronounced peaks for N = 48 (4×4×3) and N = 64 (4×4×4) are due to the excitation of perfectly cuboid and cubic ground states, respectively. The first significant onset of the collapse transition is seen for the 48-mer close to T ≈ 1.4. A clear discrimination between the excitation and the melting transition is virtually impossible. In these examples, solely for N = 64 1

This gap is an artifact of the simple-cubic lattice and the resulting excitation transition is a non-cooperative effect: Consider a cuboid conformation with a chain end located in one of the corners. It forms two energetic contacts with the nearest neighbors nonadjacent within the chain. Performing a local off-cube pivot-rotation of the bond the chain end is connected with, these two contacts are lost and none new is formed.

5.3 Energetic Fluctuations of Finite-Length Polymers

97

0.9

N N N N N

0.8

CV (T )/N

0.7 0.6

= 27 = 34 = 48 = 53 = 64

0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

T Figure 5.1: Examples of specific-heat curves (per monomer) for a few exemplified short homopolymers on the sc lattice. Absolute errors (not shown) are smaller than 0.03 in the vicinity of the low-temperature peaks and smaller than 10−5 in the onset of the Θ-transition region near T ≈ 1.5.

three separate peaks are present. The plots in Figs. 5.2(a)–5.2(c) show representative conformations in the different pseudophases of the 64-mer. Due to the energy gap, the excitations of the cubic ground state with energy E = −81 (not shown) to conformations with E = −79 [Fig. 5.2(a)] result in a pseudotransition which is represented by the first specific-heat peak in Fig. 5.1. The second less-pronounced peak in Fig. 5.1 around T ≈ 0.6 − 0.7 signalizes the melting into globular structures, whereas at still higher temperatures T ≈ 1.5 the wellknown collapse peak indicates the dissolution into the random-coil phase. The distribution of the maximum values of the specific heat CVmax with respect to the maximum temperatures TCVmax is shown in Fig. 5.3. Not surprisingly, the peaks belonging to the excitation and freezing transitions (+) appear to be irregularly “scattered” in the low-temperature interval 0 < TCVmax < 0.8. The height of the peaks indicating the collapse transition of the finite-length polymers (⊙) is, on the other hand, monotonously increasing with the collapse-peak temperature. Figures 5.4(a) and 5.4(b), showing the respective chain-length dependence of the maximum temperatures and maximum specific-heat values, reveal a more systematic picture. At least from the results for the short chains shown, general scaling properties for the freezing transition cannot be read off at all. The reason is that the low-temperature behavior of these short chains is widely governed by lattice effects. This is clearly seen by the “sawtooth” segments. Whenever the sc chain possesses a “magic” length Nc such that the ground state is cubic or cuboid (i.e., Nc ∈ Nc = {8, 12, 18, 27, 36, 48, 64, 80, 100, 125, . . . }), the energy gap ∆E = 2 between the ground-state conformation and the first excited state

98

5. Freezing and Collapse of Flexible Lattice Polymers (a)

(b)

E = −79

E = −60

(c)

E=0

Figure 5.2: Representative conformations of a 64-mer in the different pseudophases: (a) Excitation from the perfect 4×4×4 cubic ground state (not shown, E = −81) to the first excited crystal state, (b) transition towards globular states, and (c) dissolution into random-coil conformations. 0.9 0.8

CVmax/N

0.7 0.6 0.5 0.4 0.3 0.2 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

TCVmax Figure 5.3: Map of specific-heat maxima for several chain lengths taken from the interval N ∈ [8, 125]. Circles (⊙) symbolize the peaks (if any) identified as signals of the collapse (TCVmax > 1). The low-temperature peaks (+) belong to the excitation/freezing transitions (TCVmax < 0.8). The group of points in the lower left corner corresponds to polymers with Nc + 1 monomers, where Nc denotes the “magic” lengths allowing for cubic or cuboid ground-state conformations (see Fig. 5.4 and text).

5.3 Energetic Fluctuations of Finite-Length Polymers

99

2 1.8

(a)

TCVmax (N )

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 10

20

30

40

50

60

70

80

90

100 110 120

N 0.9

(b)

CVmax(N )/N

0.8

0.7

0.6

0.5

0.4

0.3

0.2 10

20

30

40

50

60

70

80

90

100 110 120

N Figure 5.4: (a) Collapse (⊙) and crystallization/excitation (+) peak temperatures of the specific heat for all chain lengths in the interval N ∈ [8, 125], (b) values of the specific-heat maxima in the same interval. Error bars for the collapse transition data (not shown) are much smaller than the symbol size. Θ peaks appear starting from N = 41. For the sake of clarity, not all intermediate Θ data points are shown (only for N = 41, 45, 50, . . .).

entails a virtual energetic barrier which results in an excitation transition. Since entropy is less relevant in this regime, this energetic effect is not “averaged out” and, therefore, causes a pronounced peak in the specific heat [see Fig. 5.4(b)] at comparatively low temperatures [Fig. 5.4(a)]. This peculiar sc lattice effect vanishes widely by increasing the length by unity, i.e., for chain lengths Nc + 1. In this case, the excitation peak either vanishes or remains as a remnant of less thermodynamic significance. The latter appears particularly in those cases, where N = Nc + 1 with Nc = L3 (with L being any positive integer) is a chain length allowing for perfectly cubic ground states. Increasing the polymer length further, the freezing peak dominates at low temperatures. Its peak increases with the chain length, whereas the peak temperature decreases. Actually, with

100

5. Freezing and Collapse of Flexible Lattice Polymers

increasing chain length, the character of the transition converts from freezing to excitation, i.e., the entropic freedom that still accompanies the melting/freezing process decreases with increasing chain length. In other words, cooperativity is lost: only a small fraction of monomers – residing in the surface hull – is entropically sufficiently flexible to compete the energetic gain of highly compact conformations. This flexibility is reduced the more, the closer the chain length N approaches a number in the “magic” set Nc . If the next length belonging to Nc is reached, the next discontinuity in the monotonic behavior occurs. Since noticeable “jumps” are only present for chain lengths whose ground states are close to cubes (Nc = L3 ) or cuboids with Nc = L2 (L±1), the length of the branches in between 2/3 scales with ∆Nc ∼ L2 ∼ Nc . Therefore, only for very long chains on the sc lattice, for which, however, a precise analysis of the low-temperature behavior is extremely difficult, a reasonable scaling analysis for TCVmax (N ) and CVmax(N ) could be performed.

5.3.3 Polymers on the Face-Centered Cubic Lattice The general behavior of polymers on the fcc lattice is comparable to what we found for the sc polymers. The main difference is that excitations play only a minor role, and the freezing transition dominates the conformational behavior of the fcc polymers at low temperatures. Nonetheless, finite-length effects are still apparent as can be seen in the chain-length dependence of the peak temperatures and peak values of the specific heats plotted in Fig. 5.5(a) and Fig. 5.5(b), respectively. Figure 5.5(a) shows that the locations of the freezing and collapse transitions clearly deviate with increasing chain lengths and we hence can conclude that also for fcc polymers there is no obvious indication that freezing and collapse could fall together in the thermodynamic limit. Similar to the sc polymers, the finite-length effects at very low temperatures are apparently caused by the usual compromise between maximum compactness, i.e., maximum number of energetic (nearest-neighbor) contacts, and steric constraints of the underlying rigid lattice. The effects are smaller than in the case of the sc lattice, as there are no obvious “magic” topologies in the fcc case. Groundstate conformations for a few small polymers on the fcc lattice are shown in Fig. 5.6. The general tendency is that the lowest-energy conformations consist of layers of net planes with (111) orientation, i.e., the layers themselves possess √ triangular pattern with side lengths equal to the fcc nearest-neighbor distance 2 (in units of the lattice constant). This is not surprising, as these conformations are tightly packed which ensures a maximum number of nearest-neighbor contacts and, therefore, lowest conformational energy. An obvious example is the groundstate conformation of the 13-mer as shown in Fig. 5.6(a) which corresponds to the

5.3 Energetic Fluctuations of Finite-Length Polymers

101

3.2 2.9

(a)

TCVmax (N )

2.6 2.3 2 1.7 1.4 1.1 0.8 0.5 0.2 10

15

20

25

30

35

40

45

50

55

N (b)

CVmax(N )/N

0.7

0.6

0.5

0.4

0.3

0.2 10

15

20

25

30

35

40

45

50

55

N Figure 5.5: Peak temperatures (a) and peak values (b) of the specific heat for all chain lengths N = 8, . . . , 56 of polymers on the fcc lattice. Circles (⊙) symbolize the collapse peaks and low-temperature peaks (+) signalize the excitation/freezing transitions. The error bars for the collapse transition are typically much smaller than the symbol size. Only for the freezing transition of longer chains, the statistical uncertainties are a little bit larger and visible in the plots. Θ peaks appear starting from N = 19. For clarity, Θ data points are only shown for N = 19, 25, 30, . . ..

intuitive guess for the most closely packed structure on an fcc lattice: a monomer with its 12 nearest neighbors (“3–7–3” layer structure). A simple contact counting yields 36 nearest-neighbor contacts which, by subtracting the N −1 = 12 covalent (nonenergetic) bonds, is equivalent to an energy E = −24. However, this lowestenergy conformation is degenerate. There is another conformation (not shown) consisting of only two “layers”, one containing 6 (a triangle) and the other 7 (a hexagon) monomers (“6–7” structure), with the same number of contacts. A special case is the 18-mer. As Fig. 5.6(b) shows, its ground state is formed by a complete triangle with 6 monomers, a hexagon in the intermediate layer with 7 monomers, and an incomplete triangle (possessing a “hole” at a corner) with 5

102

5. Freezing and Collapse of Flexible Lattice Polymers

(a)

(b)

N = 13 E0 = −24 (d)

N = 27 E0 = −67

(c)

N = 18 E0 = −39

N = 19 E0 = −42 (e)

N = 30 E0 = −77

Figure 5.6: Ground-state conformations and energies of the (a) 13-, (b) 18-, (c) 19-, (d) 27-, and (e) 30-mer on the fcc-lattice (bonds not shown).

monomers (“6–7–5” structure). Although this imperfection seems to destroy all rotational symmetries, it is compensated by an additional symmetry: Exchanging any of the triangle corners with the hole does not change the conformation at all! Thus, the seeming imperfection has a similar effect as the energetic excitation and causes a trivial entropic transition. This explains, at least partly, why the 18-mer exclusively exhibits an additional peak in the specific heat at very low temperatures [see Fig. 5.5(a)]. A similar reasoning presumably also applies to the anomalous low-temperature peaks of the 32-, 46-, and 56-mers, but for these larger ground-state conformations it does not make much sense to go into such intricate details. The expectation that the 19-mer, which can form a perfect shape without any “holes” (“6–7–6” structure), is a prototype of peculiar behavior is, however, wrong. This is due to the existence of degenerate less symmetric ground-state conformations [as the exemplified conformation in Fig. 5.6(c)]. The described geometric peculiarities are, however, only properties of very short chains. One of the largest of the “small” chains that still possesses a nonspherical ground state, is the 27-mer with the ground-state conformation shown in Fig. 5.6(d). For larger systems, the relative importance of the interior monomers will increase, because of the larger number of possible contacts. This requires the number of surface monomers to be as small as possible which results in compact, sphere-like shapes. A representative example is the 30-mer shown in Fig. 5.6(e).

5.4 The Θ Transition Revisited

103

Table 5.1: TΘ values on the sc and fcc lattice from literature. lattice type sc

fcc

3.64 3.713 3.650 3.716 3.60 3.62 3.717 3.717 3.745 3.71 8.06 8.20 8.264

TΘ ... ± ± ± ± ± ± ±

4.13 0.007 0.08 0.007 0.05 0.08 0.003 0.002

± ... ±

0.01 9.43 0.02

model single chain single chain single chain single chain single chain single chain single chain polymer solution lattice theory polymer solution single chain single chain lattice theory

Ref. [107] [108] [110] [96] [97] [98] [43] [103] [112] [104,105] [107] [109,111] [112]

5.4 The Θ Transition Revisited The scaling behavior of several quantities at and close to the Θ point in three dimensions has been the subject of a large number of field-theoretic and computational studies [43,96,103–105,107–113]. Nonetheless, the somewhat annoying result is that the nature of this phase transition is not yet completely understood. The associated tricritical limn→0 O(n) field theory has an upper critical dimension dc = 3, but the predicted logarithmic corrections [92–94] could not yet be clearly confirmed from the numerical data produced so far. In our study of freezing and collapse on regular lattices, we here mainly focus on the critical temperature TΘ for polymers on the sc and on the fcc lattice. The sc value of TΘ has already been precisely estimated in several studies, but only a few values are known for the fcc case. Some previous estimates in the literature are compiled in Table 5.1. As our main interest is dedicated to the expected difference of the collapse and freezing temperatures, we will focus here on the scaling behavior of the finite-size deviation of the maximum specific-heat temperature of a finite-length polymer from the Θ temperature, Tc (N ) − TΘ, as it has also been studied for the bond-fluctuation model [99,100] and the off-lattice FENE polymer [101], as well as for polymer solution models [102,104,105]. In the latter case, Flory–Huggins mean-field theory [114] suggests 1 1 1 1 − ∼√ + , (5.2) Tcrit (N ) TΘ N 2N where Tcrit (N ) is the critical temperature of a solution of chains of finite length

0.48

0.28

0.45

0.26

0.42

0.24

0.39

0.22

0.36

0.2

0.33

0.18

0.3

0.16 s

0.27

1/T f

(N )

5. Freezing and Collapse of Flexible Lattice Polymers

1/T s (N )

104

0.14

f

0

0.015

0.03

0.045

0.06

√ 1/ N

0.075

0.09

0.105

Figure 5.7: Inverse collapse temperatures for several chain lengths on sc (N ≤ 32 000) and fcc lattices (N ≤ 4 000). Drawn lines are fits according to Eq. (5.5).

N and TΘ = limN →∞ Tcrit (N ) is the collapse transition temperature. In this case, field theory [92] predicts a multiplicative logarithmic correction of the form Tcrit (N ) − TΘ ∼ N −1/2[ln N ]−3/11. Logarithmic corrections to the mean-field theory of single chains are known, for example, for the finite-chain Boyle temperature TB (N ), where the second virial coefficient vanishes. The scaling of the deviation of TB (N ) from TΘ reads [96]: TB (N ) − TΘ ∼ √

1 . N (ln N )7/11

(5.3)

In Ref. [101], it is claimed that, for their data obtained from simulations with the FENE potential, this expression can also be used as a fit ansatz for Tc (N ) − TΘ. However, also the mean-field-motivated fit without explicit logarithmic corrections, a2 a1 Tc (N ) − TΘ = √ + , (5.4) N N has been found to be consistent with the off-lattice data [101], and also with the results obtained by means of the bond-fluctuation model of single chains with up to 512 monomers [99,100]. Up to corrections of order N −3/2, Eq. (5.4) is equivalent to 1 a ˜1 a ˜2 1 − =√ + , (5.5) Tc (N ) TΘ N N which was found to be consistent with numerical data obtained in grand canonical analyses of lattice homopolymers and the bond-fluctuation model [102,104,105].

5.4 The Θ Transition Revisited

105

The situation remains diffuse as there is still no striking evidence for the predicted logarithmic corrections (i.e., for the field-theoretical tricritical interpretation of the Θ point) from experimental or numerical data. Using our data from independent long-chain nPERMss [46] chain-growth simulations (sc: Nmax = 32 000, fcc: Nmax = 4 000) in the vicinity of the collapse transition, we have performed a scaling analysis of the N -dependent collapse transition temperatures Tc (N ), identified as the collapse peak temperatures of the individual specificheat curves, and estimated from it the N → ∞ limit TΘ. For the single-chain system, field theory [93] predicts the specific heat to scale at the Θ point like CV (T = TΘ)/N ∼ (ln N )3/11. Short-chain simulations [110] did not reveal a logarithmic behavior at all, whereas for long chains a scaling closer to ln N was read off [43]. The situation is similar for structural quantities such as the end-to-end distance and the gyration radius. Figure 5.7 shows our data points of the inverse collapse temperature Tc−1 from the simulations on the sc (left scale) and on the fcc lattice (right scale), plotted against N −1/2. Error bars for the individual data points in Fig. 5.7 were obtained by jackknife error estimation [73] from several independent simulation runs. Also shown are respective fits according to the ansatz (5.5). Optimal fit parameters using the data in the intervals 200 ≤ N ≤ 32 000 (sc) and 100 ≤ N ≤ 4 000 (fcc) were found to ˜2 ≈ 8.0 (sc) and TΘfcc = 8.18(2), a˜1 ≈ 1.0, be TΘsc = 3.72(1), a˜1 ≈ 2.5, and a and a ˜2 ≈ 5.5 (fcc). In addition, also other fit functions motivated by field theory and mean-field-like approaches, corresponding to Eqs. (5.2)–(5.5), each of which also with different fit ranges, have been evaluated by means of χ2 tests [88]. From these results, it can be concluded that the two-parameter mean-field-like fits (5.4) and (5.5) as well as the single-parameter fit according to (5.2) are consistent with the data. Surprisingly poor, on the other hand, is the goodness of the fit against the logarithmic scaling (5.3). Even more astonishing is, however, the good coincidence with a logarithmic fit of the “wrong” form N −1/2(ln N )7/11 with the data. Summarizing these results, if logarithmic corrections as predicted by tricritical field theory are present at all, even chain lengths N = 32 000 on an sc lattice are too small to observe deviations from the mean-field picture. At least, the goodness of the logarithmic fit with the “wrong” exponent +7/11 could lead to the speculative conclusion that for N ≤ 32 000 multiplicative and additive logarithmic corrections to scaling are hidden in the fit parameters of the “mean-field-like fits”. The subleading additive corrections are expected to be of the form ln(ln N )/ ln2 N [95]. They thus not only disappear very slowly – they are also even of the same size as the leading scaling behavior, which makes it extremely unlikely to observe the logarithmic corrections in computational studies at all [95]. Similar additive logarithmic scaling is also known, for example,

106

5. Freezing and Collapse of Flexible Lattice Polymers

from studies of the two-dimensional XY spin model [115]. The estimated sc Θ temperatures from the good fits are in perfect agreement with the most reliable estimates from literature. In the fcc case, only the fit function (5.5) is independent of the data sets used and, therefore, consistent with the data obtained for all chain lengths. However, the noticeable improvement of the goodness for the fits to (5.2), (5.4), and the “wrong” N −1/2(ln N )7/11 form by excluding the very short chains from the data sets considered, leads to the conclusion that even chains with N = 4 000 monomers on the fcc lattice are also too short to find evidence for the logarithmic corrections to mean-field scaling. Our best estimates for the fcc Θ temperature agree nicely with the results from Refs. [109,111].

5.5 Freezing and Collapse in the Thermodynamic Limit From the results for the freezing and the collapse transition, we conclude that both transitions remain well separated also in the extrapolation towards the thermodynamic limit. This is the expected behavior as it is a consequence of the extremely short range of attraction in the nearest-neighbor lattice models used. Considering a more general square-well contact potential between nonbonded monomers in our parametrization,   ∞ r ≤ 1, v(r) = −1 1 < r ≤ λ, (5.6)  0 λ < r,

the attractive interaction range is simply R = λ−1. In our single-chain study of sc and fcc lattice models, we have λ → 1 and thus R → 0. Since this R value is well below a crossover threshold known for colloids interacting via Lennard–Jones-like (1) and Yukawa potentials, where different solid phases can coexist, Rc ≈ 0.01 [116– 118], we interpret our low-temperature transition as the restructuring or “freezing” of compact globular shapes into the (widely amorphous) polymer crystals. Following Ref. [119], there is also another phase boundary, namely between stable and metastable colloidal vapor-liquid (or coil-globule) transitions, in the (2) range 0.13 < Rc < 0.15. Other theoretical and experimental approaches yield (1) (2) slightly larger values, Rc ≈ 0.25 [116,120–122]. Below Rc , the liquid (globule) phase is only metastable. The specific bond-fluctuation model used in Ref. [99] corresponds to R = 0.225, i.e., it lies in the crossover regime between the stable and metastable liquid phase [100]. Consequently, the crystallization and collapse transition merge in the infinite-chain limit and a stable liquid phase was only found in a subsequent study of a bond-fluctuation model with larger interaction range [100].

5.5 Freezing and Collapse in the Thermodynamic Limit

107

Qualitatively, analogous to the behavior of colloids, our considerations would explain the separate stable crystal, globule, and random-coil (pseudo)phases that we have clearly identified in our lattice polymer study. Since the range of interactions seems to play a crucial, quantitative role, it is an interesting, still widely open question to what extent the colloidal picture in the compact crystalline and globular phases is systematically modified for polymers with different nonbonded interaction ranges, where steric constraints (through covalent bonds) are a priori not negligible.

6 Crystallization of Elastic Polymers

6.1 Relevance of Surface Effects A central result of the discussion in the last chapter was the strong influence of finite-size effects on the freezing behavior of flexible polymers constrained to regular lattices. Thus, (unphysical) lattice effects interfere with (physical) finitesize effects and the question remains how polymer crystals of small size could look like. Since all effects in the freezing regime are sensitive to system or model details, this question cannot be answered in general. Nonetheless, it is obvious that the surface exposed to a different environment, e.g., a solvent, is relevant for the formation of the whole crystalline or amorphous structure. This is true for any physical system. If a system tries to avoid contact with the environment (a polymer in bad solvent or a set of mutually attracting particles in vacuum), it will form a shape with a minimal surface. A system which can be considered as a continuum like a water droplet, will preferrably form a spherical shape. But what if the system is “small” and discrete? Small crystals consisting of a few hundred cold atoms such as argon [123] but also as different systems as spherical virus hulls enclosing the coaxially wound genetic material [124,125] exhibit an icosahedral or icosahedral-like shape. But why is just the icosahedral assembly naturally favored? The capsid of spherical viruses is formed by protein assemblies, the protomers, and the highly symmetric morphological arrangement of the protomers in icosahedral capsids reduces the number of genes that are necessary to encode the capsid proteins. Furthermore, the formation of crystalline facets decreases the surface energy, which is particularly relevant for small atomic clusters. Another point is that the arrangement of a finite number of constituents (atoms, proteins, monomers) on the facets of an icosahedron optimizes the interior space filling and thus reduces the total energy of the system. This is also reason why it can be expected that elastic polymers favor icosahedral shapes as well [126,127].

110

6. Crystallization of Elastic Polymers

Figure 6.1: (a) Anti-Mackay and (b) Mackay growth overlayer on the facet of an icosahedron [129].

6.2 Lennard-Jones Clusters In linear elastic polymers, bonds that connect monomers are not stiff. Within a certain range, bonded monomers can adapt their distance to external perturbations without much energetic effort. One can imagine the bond as a rather floppy spring. For this reason, crystalline structures of elastic polymers with nonbonded monomers interacting with each other, will exhibit similarities compared to atomic or Lennard-Jones clusters, provided the energy and length scales of bonded and nonbonded interactions enable this. It is, therefore, instructive to review the size-dependent features of small Lennard-Jones clusters. Many-particle systems governed by van der Waals forces are typically described by Lennard-Jones (LJ) pair potentials VLJ (rij ) = 4ǫ[(σ/rij )12 − (σ/rij )6],

(6.1)

where rij is the distance between two atoms located at ri and rj (i, j = 1, . . . , N ), respectively. An important example are atomic clusters whose structural properties have been subject of numerous studies, mainly focusing on the identification of ground-states and their classification. It has been estimated [128] that icosahedral-like LJ clusters are favored for systems with N < 1690 atoms. Larger systems prefer decahedral structures until for N > 213000 face-centered cubic (fcc) crystals dominate. In the small-cluster regime, energetically optimal icosahedral-like structures form by atomic assembly in overlayers on facets of an icosahedral core. There are two generic scenarios (see Fig. 6.1): Either hexagonal closest packing (hcp) is energetically preferred (anti-Mackay growth), or the atoms in the surface layer continue the fcc-shaped tetrahedral segment of the interior icosahedron (Mackay growth) [129]. The surprisingly strong dependence of structural liquid-solid transitions on the system size has its origin in the different structure optimization strategies. “Magic” system sizes allow for the formation of most stable complete icosahedra (N = 13, 55, 147, 309, 561, 923, . . .) [130]. Except for a few exceptional cases – for 13 ≤ N ≤ 147 these are N = 38 (fcc truncated octahedron), 75-77 (Marks decahedra), 98 (Leary tetrahedron), and

6.3 Perfect Icosahedra

111

102-104 (Marks decahedra) [128] – LJ clusters typically possess an icosahedral core and overlayers are of Mackay (N = 31-54, 82-84, 86-97, 99-101, 105-146, . . .) or anti-Mackay (N = 14-30, 56-81, 85, . . .) type [131]. Although it seems that there are strong analogies in liquid-solid transitions of LJ clusters and classes of flexible polymers, comparatively few systematic attempts were undertaken to relate for finite systems structural properties of LJ clusters and frozen polymers [126,127,132–135]. In contrast, the analogy of the generic phase diagram for colloids and polymers has been addressed in numerous studies (for a recent overview, see, e.g., Ref. [136]).

6.3 Perfect Icosahedra Since we will consider in the following polymer chains with lengths in the icosahedral regime, it is instructive to review properties of perfect icosahedra. The surface of the icosahedron consists of 20 triangular faces, where two of each share an edge and five a common vertex, i.e., in total there are 30 edges and 12 vertices. Thus, the smallest number of LJ particles being necessary to form a stable, perfectly icosahedral shape is N1 = 13 (12 on the surface, one in the center). The next larger icosahedral structure possesses two overlayers which are of Mackay type [see Fig. 6.1(b)]. In this case, the top layer is composed of 60 identical triangles sharing 42 vertices. Therefore, N2 = 55 particles are needed to construct an icosahedron with two layers surrounding the central particle. The numbers 13 and 55 are “magic numbers” because only systems with precisely these sizes can form perfectly icosahedral shapes. Putting the third complete layer on top would require N3 = 147 particles, which is the next magic number in this hierarchy. This can be continued ad infinitum. Exemplified magic numbers of particles Nn , the number of particles in the interior bulk (i.e., without the top layer) Nnbulk , and in the top layer Nnsurf are listed in Table 6.1 for icosahedra with up to 6 overlayers. Interpreting the process of structure formation of a small system of LJ particles as a conformational transition, it is particularly interesting to consider the structural behavior at the surface as it seems that it is for these system sizes more relevant than the bulk effects. In the thermodynamic limit, of course, the phase transition will be mainly driven by the crystal formation in the bulk (fcc structure in the case of LJ particles). However, the systems we are going to study in this chapter are so small that the general aspects of nucleation transitions for very large systems are not valid anymore. The crystallization of small systems depends extremely on the precise system size – a lesson that has already been taught in Sec 5.3 – and this is due to surface effects.

112

6. Crystallization of Elastic Polymers

Table 6.1: Exemplified numbers of particles in the interior bulk (Nnbulk ) and in the surface layer (N surf ) for icosahedral shapes with n ≤ 6 Mackay layers surrounding the central particle; Nn = Nnbulk + N surf is the total “magic” number of particles. Also given is the surface-to-volume ratio rsv = Nnsurf /Nnbulk . n 1 2 3 4 5 6

Nnbulk 1 13 55 147 309 561

Nnsurf 12 42 92 162 252 362

Nn 13 55 147 309 561 923

rsv 12.00 3.23 1.67 1.10 0.82 0.65

For this reason it is instructive to study the surface-to-volume ratio of these icosahedral structures. As can be read off from the numbers listed in Table 6.1, the surface-to-volume ratio rsv = Nnsurf /Nnbulk not surprisingly decreases with the number of particles Nn , but this happens slowly. The magic numbers listed in Table 6.1 follow a clear hierarchy. The number of particles in the surface layer can be derived from the Mackay overlayer growth principle as depicted in Fig. 6.1(b). Since the order of the layer n corresponds to (n + 1) particles lininig up at each edge, the new top layer contains (n − 1)(n − 2)/2 particles in the center of each facet, surrounded by (n − 1) non-vertex particles in the edges and the vertices, the total number of particles in the surface layer is: 1 Nnsurf = 20 (n − 1)(n − 2) + 30(n − 1) + 12 = 10n2 + 2. (6.2) 2 The number of particles in the bulk is easier to calculate by the recursive relation bulk surf bulk Nnbulk = Nn−1 + Nn−1 = Nn−1 + 10n2 − 20n + 12,

(6.3)

as the growth of the bulk volume with n is associated with adding the number of surface particles of the former surface layer of order (n − 1) to the previous volume. The ansatz Nnbulk = an3 + bn2 + cn2 + d is suitable to solve the difference equation originating from Eq. (6.3). This yields 10 3 11 n − 5n2 + n − 1, (6.4) 3 3 where we have made use of the numbers listed in Table 6.1 to determine the parameters. Finally, the magic numbers are obtained from Nn = Nnsurf + Nnbulk and can be expressed as Nnbulk =

Nn =

11 10 3 n + 5n2 + n + 1. 3 3

(6.5)

6.4 Liquid-Solid Transitions of Elastic Flexible Polymers

113

In the limit of large systems (n → ∞), the surface-to-volume ratio thus vanishes like rsv ∼ n−1, (6.6) or, with Eq. (6.5), rsv ∼ Nn−1/3.

(6.7)

This rather slow decrease gives a strong indication for the expectation that even the crystallization process of comparatively large systems with hundreds to thousands of particles will significantly be influenced by strong surface effects. This is not only true for magic LJ clusters; it is a general result, which is also valid for systems that do not form perfectly icosahedral shapes. As we will show in the following systematic analysis for elastic and flexible LJ polymers, size-dependent pecularities are relevant for the individual crystallization processes of these systems, too.

6.4 Liquid-Solid Transitions of Elastic Flexible Polymers 6.4.1 Finitely Extensible Nonlinear Elastic Lennard-Jones Polymers The crystallization of polymers can nicely be studied by means of a model that enables the formation of icosahedral or icosahedral-like global energy minimum shapes which are virtually identical with LJ clusters of the same size. This means that bonds between monomers must be highly elastic such that the energetic excitation barriers for changes of the bond length within a certain range are extremely small. This is realized in a frequently used model for flexible polymers, where covalent bonds are modeled by a so-called finitely extensible nonlinear elastic (FENE) anharmonic potential [137,138]. Similar to the LJ rare-gas model, all monomers also interact via a pairwise LJ potential. Thus the polymer energy is given by N −1 N X 1 X mod VFENE (rii+1), (6.8) V (rij ) + E= 2 i,j=1 LJ i=1 i6=j

with the FENE potential between adjacent monomers VFENE (rii+1) = −

K 2 R ln{1 − [(rii+1 − r0)/R]2}. 2

(6.9)

The potential possesses a minimum coinciding with r0 and diverges for r → r0 ±R, where R parametrizes the elasticity range of the bond. In the following, we consider only R = 0.3 and the spring constant K = 40. The LJ potential is

114

6. Crystallization of Elastic Polymers

Figure 6.2: Possible shell conformations (rc = 2.5σ) of a monomer possessing 12 (a)-(d) or 11 (e) neighbors: (a) icosahedron, (b) elongated pentagonal pyramid [140], (c) cuboctahedron (fcc), (d) triangular orthobicupola (hcp) [140], (e) incomplete icosahedron. Sticks illustrate shell contacts, not bonds.

truncated and shifted and reads mod VLJ (rij ) = VLJ (min(rij , rc)) − VLJ (rc),

(6.10)

with VLJ (rij ) from Eq. (6.1). As usual, rij denotes the distance between the ith and jth monomer and rc is the cutoff distance. The energy scale ǫ is set to 1 and σ = 2−1/6r0 with the minimum-potential distance r0 = 0.7. Later on, we will discuss the influence of the cutoff and results for cutoff distances rc = 2.5σ, 5σ, ∞ are then compared. Monte Carlo simulations of this model, where the unraveling of the details of the liquid-solid transition requires high accuracy, are best performed by employing sophisticated generalized-ensemble simulations with an appropriate set of updates [126,127].

6.4.2 Classification of Geometries Structural changes between different geometries appear to be a characteristic feature of the behavior of LJ clusters at low temperatures. Since we expect a similar behavior also for elastic polymers, it is interesting to classify the geometrical structures in order to specify conformational transitions. While many studies refer to bond orientation parameters [139] that are calculated via spherical harmonics, we restrict ourselves here to informations provided by the contact map which can be updated instantly during the simulation without further computational expenses. We consider two monomers as being in contact if their distance is smaller than a threshold rcontact. As a consequence of this definition also bonded monomers do not need to be necessarily in contact. The total number of monomer contacts is not an appropriate measure for classification since in the interior of a frozen polymer every monomer has usually exactly 12 neighbors. Instead, the contacts between that 12 neighbors reflect their arrangement which corresponds to the local geometry. In Fig. 6.2(a)-(d), different conformations of basic cells composed of a monomer and its 12 neighbors

6.4 Liquid-Solid Transitions of Elastic Flexible Polymers

115

are shown. Counting the contacts between the neighbors is a simple but efficient way to characterize different types: Only the icosahedral cell [Fig. 6.2(a)] reveals 30 shell contacts corresponding to the 30 edges of an icosahedron. It always appears in the center of an icosahedral conformation. Consequently, if there is no such basic element, the global geometry cannot be icosahedral! On the other hand, icosahedral cells are also formed by a sufficiently large anti-Mackay overlayer at the corners of the icosahedral core. If the number of outer monomers is too small, one might find instead the defected icosahedral cell [Fig. 6.2(e)], where the central monomer and its 11 neighbors form 25 shell contacts. The total number of both structures nic is a suitable “order” parameter which allows a classification of the global geometry at low temperatures, given roughly by   = 0, nonicosahedral = 1, icosahedral + Mackay, nic (6.11)  ≥ 2, icosahedral + anti-Mackay.

More precisely, if nic = 0, the polymer forms a nonicosahedral structure, e.g., it is decahedral or fcc-like; nic = 1 indicates icosahedral geometry with Mackay overlayer or a complete icosahedron which might possess a few monomers bound in anti-Mackay type. Finally, for nic ≥ 2, the monomers form an icosahedral core with a considerably extended anti-Mackay overlayer. The probabilities pnic (T ) for the different values of nic as a function of temperature provide the necessary information to reveal structural transitions. Figure 6.2(b) shows the elongated pentagonal pyramid with 25 shell contacts which is the basic module of five-fold symmetry axes in icosahedra and decahedra. It also occurs along the edges of an icosahedral core, which is covered by an anti-Mackay overlayer. Hence, it appears in icosahedral conformations of polymers with N ≥ 31 monomers and in decahedral structures. Besides, it is formed at the edges of the central tetrahedron in conformations with a tetrahedral symmetry. An example is the ground state of the LJ cluster with N = 98 or the low-energy minima for N = 159 and N = 234. In consequence, the total number of elongated pentagonal pyramids nepp can be used to distinguish decahedral and tetrahedral structures. Figures 6.2(c) and (d) show conformations which hardly differ because both of them possess 24 neighbor-neighbor contacts and occur in almost all geometries considered here. Only the truncated fcc-octahedron (i.e., the ground state of the 38mer) does not exhibit triangular orthobicupolae [Fig. 6.2(d)]. Cuboctahedra [Fig. 6.2(c)] are related to fcc- and triangular orthobicupolae [Fig. 6.2(d)] to hcp-packing. For the classification of the formation of crystalline structures, we will in the following mainly focus on the analysis of the number of complete and defected icosahedral cells, nic . Sophisticated, e.g.,

116

6. Crystallization of Elastic Polymers

1

g(E)

10−500 10−1000 10−1500 N = 13 N = 55 N = 147 N = 309

10−2000 10−2500 -6

-5

-4

-3

-2

-1

0

E/N

Figure 6.3: Density of states for polymers forming complete icosahedra.

multicanonical, simulations with a suitable set of Monte Carlo updates are needed to obtain the expectation values and distributions of these indicators with high accuracy [126,127]. For the discussion of the thermodynamics of crystallization by means of energetic fluctuations, a precise estimate of the density of states g(E) is necessary as the specific heat is derived from it. As usual, peaks in the specific heat signalize transitions between phases accompanied by strong energetic fluctuations. The density of states as obtained from multicanonical simulations [127] is plotted in Fig. 6.3 for several magic chain lengths and covers hundreds to thousands orders of magnitude, reflecting the power of the employed simulation method.

6.4.3 Ground States Global energy minimum conformations of the elastic polymers are virtually identical to ground-state configurations of LJ clusters, i.e., for almost all system sizes, the ground state is icosahedral or icosahedral-like (Fig. 6.4). At small temperatures, the covalent, elastic bonds cause only small deviations from the corresponding atomic cluster, since the minimum of the FENE potential is close to the equilibrium distance. The bonds between adjacent monomers arrange themselves in a way that minimizes the bond potential. As a consequence, bonds between different shells of an icosahedral core are rare since the corresponding monomer distances are smaller than the equilibrium distance and would entail higher bond energies. For longer chains, the central icosahedral cell is strongly compressed

6.4 Liquid-Solid Transitions of Elastic Flexible Polymers

117

Figure 6.4: Putative icosahedral or icosahedral-like ground-state state conformations for different system sizes.

and one end of the chain is usually located in the center of the cell, thus avoiding the inclusion of the most inappropriate distance (from the central monomer to one of its neighbors) in the chain twice. A similar effect occurs in decahedral conformations. Bonds between monomers on the central axis are favorable since their length is close to the optimum. This forces the polymer chain to adapt this axis at low temperatures [Figs. 6.5(a),(c)]. Perfect icosahedra are found as expected for system sizes N = 13, 55, 147, 309. If the system size exceeds these “magic” sizes, the polymer builds an icosahedral core with an anti-Mackay overlayer which grows with the chain length. At some point (N > 30 and N > 80), the overlayer adopts the structure of the core by changing to Mackay type. Further increase of the number of monomers completes the outer shell and leads to the next perfect icosahedral shape. A few polymers of certain sizes possess ground-state conformations that correspond to different nonicosahedral geometries. One finds a truncated octahedron for N = 38 and a decahedral conformation for N = 75 − 77 [Fig. 6.5(a)]. Some deviations are caused by the cutoff of the LJ potential, so the chains with N = 81, 85, 87, 98, 102 monomers possess lowest-energy structures that do not

118

6. Crystallization of Elastic Polymers

Figure 6.5: Nonicosahedral ground-state conformations: (a) N = 75, rc = 2.5σ decahedral, (b) N = 98, rc = 5σ tetrahedral, (c) N = 102, rc = 5σ decahedral.

correspond to the respective clusters unless an untruncated LJ potential is applied. Using the cutoff rc = 2.5σ, icosahedral ground states with Mackay overlayer for N = 81, 85, 98, 102 are found, and for N = 87 a conformation with two merged icosahedral cores (Fig. 6.6).

6.4.4 Thermodynamics of Complete Icosahedra Chains that form complete icosahedra exhibit a very clear and uniform thermodynamic behavior. Two separate conformational transitions occur and are indicated by peaks in the specific CV (T ) heat and in the fluctuations of the radius of gyration dhrgyri/dT . As shown in Figs. 6.7(a) and (b), the icosahedra melt in the interval 0.3 < T < 0.5 and a liquidlike regime is reached where the polymer arranges still in a globular shape but exhibits no distinct structure. Hence, the icosahedral order parameter hnic i ≈ 1 changes to hnic i ≈ 0. The corresponding peak in the normalized specific heat increases rapidly with system size and allows in principle a precise determination of the melting temperature. There are some characteristic features that can be read off from Figs. 6.7(a) and (b): First, the transitions are particularly strong for chains with “magic” length, which possess perfectly icosahedral ground-state morphology (e.g., N = 13, 55, 147, 309). A second type of liquid-solid transition consists of two steps, at higher temperatures the formation of an icosahedral core with anti-Mackay overlayer that transforms at lower temperatures by monomer rearrangement at the

Figure 6.6: The minimum-energy structure for N = 87, rc = 2.5σ is formed by two entangled icosahedral cores. The ends of the chain coincide with the respective centers.

6.4 Liquid-Solid Transitions of Elastic Flexible Polymers 100

(a)

90 80

N = 13 N = 55 N = 147 N = 309

18 14

70

12 C(T )/N

C(T )/N

N N N N N N N

16

60 50

10 8

119

= 31 = 38 = 45 = 55 = 60 = 70 = 80

6 40

4

30

2

20

0 0

0.1

0.2

10

T

0.3

0.4

0.5

0 0.2

0.4

1.6

(b)

0.8

1 T

1.2

1.4

1.6

1.8

2

N = 13 N = 31 N = 55 N = 60 N = 70 N = 80 N = 147 N = 309

1.4 1.2

d(hrgyr i/N 1/3 )/dT

0.6

1 0.8 0.6 0.4 0.2 0 -0.2 0.5

1

1.5

2

2.5

T

Figure 6.7: (a) Specific heats (in the inset shifted by a constant value for the sake of clarity) and (b) fluctuations of the mean radius of gyration dhrgyr i/dT as functions of the temperature for chains of different length.

surface into an energetically more favored Mackay layer (“solid-solid” transition). This is the preferred scenario for most of the chains with lengths in the intervals 31 ≤ N ≤ 54 or 81 ≤ N ≤ 146 that make the occupation of edge positions in the outer shell unavoidable. In most of the remaining cases, typically anti-Mackay layers form. However, all solid-solid or liquid-solid transitions considered here must not be understood as thermodynamic transitions in a strict sense since all investigated systems are small and dominated by finite-size effects. For longer chains

120

6. Crystallization of Elastic Polymers

one would expect, in analogy to the thermodynamic behavior of LJ clusters, the crossover from icosahedral ground states to decahedral (N & 1500) and later to fcc-like [128] structures (N & 200000) which exhibit a different crystallization behavior. Therefore the extrapolation towards the thermodynamic limit by means of finite-size scaling is not an appropriate choice. To conclude, although the elastic polymers are entropically restricted by the covalent bonds, we see that the general, qualitative behavior in the freezing regime exhibits noticeable similarities compared to LJ cluster formation. However, it is worth noting that for N = 309 the results differs considerably from the pure LJ cluster in which case two clearly separated peaks were found [130]. Increasing T further leads to the collapse transition with transition temperatures in the interval 1.0 ≤ T ≤ 2.0. For higher temperatures, the chains arrange randomly in extended conformations. This cross-over is hardly signaled in the specific heat curves, as it is suppressed by finite-size effects. However, geometric quantities like the radius of gyration and its fluctuation [Fig. 6.7(b)] give some insight. It is obvious that the solid-liquid transition remains well separated from the coil-globule collapse, moreover the intermediate temperature interval increases within the investigated chain length interval. The solid phase is dominated by extremely stable icosahedra which, however, depending on the temperature, exhibit surface defects. Although the mobility of a great majority of the monomers is strongly restricted, there are still changes which can be observed. Firstly, the linkage of the monomers of conformations representative for the ensemble in this regime can still vary strongly. Only at extremely low temperatures the lowest-energy conformation gains thermodynamic relevance. Starting from the center, the number of bonds connecting different shells is reduced. At T = 0 the first three shells are connected only by a single bond, such that for N = 13, 55, 147 one end of the chain is the central monomer and the other is at the surface. For N = 309 bonds between the corners of the third and the fourth shell exist also at T = 0, since the length of this bond is roughly equal to alternative bonds within the third shell, which include the monomers at the corners. Secondly the entire icosahedron undergoes a compactification with decreasing temperature.

6.4.5 Liquid-Solid Transitions of Elastic Polymers In the following, we systematically analyze the behavior of small elastic polymers in the liquid-solid transition regime. Particular emphasis will be dedicated to the chain-length dependence of geometric changes in the conformations of the polymers while passing the transition line. The specific-heat curves for elastic polymers with different chain lengths has

6.4 Liquid-Solid Transitions of Elastic Flexible Polymers

p

1 0.8 0.6 0.4 0.2 0

121

N = 31

1 0.8 0.6 0.4 0.2 0

N = 38

1 0.8 0.6 0.4 0.2 0

nic = 0 nic = 1 nic > 1 0

0.1

N = 55

0.2

0.3

0.4

0.5

T

Figure 6.8: Temperature dependence of the probability of icosahedral (Mackay: nic = 1, antiMackay: nic > 1) and nonicosahedral (nic = 0) structures for exemplified polymers with lengths N = 31, 38, 55. In the left panel, the corresponding lowest-energy morphologies are shown.

already been plotted in Fig. 6.7(a). The ground state of the short 13mer is the energetically stable icosahedron whose almost perfect symmetry is only slightly disturbed by the FENE bonds. The melting transition is indicated by the peak at T = 0.33. With increasing system size the low-temperature conformations are less symmetric and the peak becomes broader. For N > 30, the crossover from Mackay-like ground states to anti-Mackay conformations takes place, the corresponding peak increases with growing N and shifts from T ≈ 0.04 to higher temperatures. For the next “magic” polymer with N = 55, there is only one melting transition left and anti-Mackay-like structures are strongly suppressed. As an effect of the bond elasticity, the melting transitions occur at slightly higher temperatures than in the case of pure LJ clusters [131]. From the structural point of view, it is instructive to study the temperature dependence of the probability pnic that the “order” parameter, as in defined in Eq. (6.11), is nic ≤ 1 or nic ≥ 2. Figure 6.8 shows the respective populations pnic for the structural morphologies of the chains with N = 31, 38, 55, parametrized by nic , as a function of temperature. For N = 31, one finds that liquid structures with nic = 0 dominate above T = 0.5, i.e., no icosahedral cells are present.

6. Crystallization of Elastic Polymers

C(T )/N

8 6 4 2 0

C(T )/N nic ≤ 1 nic ≥ 2

8 6 4 2 0 8 6 4 2 0 0

0.1

0.2

0.3

N = 81

1 0.8 0.6 0.4 0.2 0

N = 82

1 0.8 0.6 0.4 0.2 0

N = 85

1 0.8 0.6 0.4 0.2 0

0.4

pnic

122

0.5

T Figure 6.9: Whereas the Mackay–anti-Mackay crossover is not recognizable in the specific heat curves, the probability of occurrence of specified numbers of icosahedral cells, pnic , reveals the transition temperature (rc = 2.5σ).

Decreasing the temperature and passing T ≈ 0.4, nucleation begins and the populations of structures with icosahedral cells (nic = 1 and nic > 1) increase. As has already been mentioned, the associated energetic fluctuations are also signaled in the specific heat [see inset of Fig. 6.7(a)]. Cooling further, always nic > 1, showing that the remaining monomers build up an anti-Mackay overlayer and create additional icosahedra. Very close to the temperature of the “solid-solid” transition near T ≈ 0.04, where also the specific heat exhibits a significant peak, the transition from anti-Mackay to Mackay overlayer occurs and the ensemble is dominated by frozen structures containing a single icosahedral cell surrounded by an incomplete Mackay overlayer. The exceptional case of the 38mer shows a significantly different behavior. A single icosahedral core forms and in the interval 0.08 < T < 0.19 icosahedra with Mackay overlayer are dominant. Although the energetic fluctuations are weak, near T ≈ 0.08, a surprisingly strong structural crossover to nonicosahedral structures occurs: the formation of a maximally compact fcc truncated octahedron.

6.4 Liquid-Solid Transitions of Elastic Flexible Polymers

C(T )/N nic ≤ 1 nic ≥ 2

N = 90

1 0.8 0.6 0.4 0.2 0

8 6 4 2 0

1 0.8 0.6 N = 100 0.4 0.2 0

8 6 4 2 0

1 0.8 N = 110 0.6 0.4 0.2 0 0.4 0.5

0

0.1

0.2

0.3

pnic

C(T )/N

8 6 4 2 0

123

T Figure 6.10: Already for N ≥ 110 the entire solid phase is dominated by Mackay (nic = 1) conformations (rc = 2.5σ).

The “magic” 55mer exhibits a very pronounced transition from unstructured globules to icosahedral conformations with complete Mackay overlayer at a comparatively high temperature (T ≈ 0.33). Below this temperature, the groundstate structure has already formed and is sufficiently robust to resist the thermal fluctuations. The behavior of chains containing between 55 and 147 monomers again generally corresponds to that of LJ clusters [131] with the differences that the melting transition occurs at higher temperatures and that the Mackay–anti-Mackay transition temperature increases much faster with system size. Besides there are a few chains with ground states of types different to LJ clusters which is induced by the truncation of the LJ potential as mentioned above. Anti-Mackay ground states are found up to system sizes of N = 80. In contrast to the clusters, N = 81 and N = 85 possess a global-energy minimum of Mackay type. Hence a solid-solid transition is encountered also in these cases. However, no peaks indicating the solid-solid transition are seen in the specific heat for N = 81, 82, 85, and the crossover to anti-Mackay conformations can only be identified in changes of nic .

124

6. Crystallization of Elastic Polymers

In Fig. 6.9, the probability pnic is plotted for these cases. While one would expect higher transition temperatures for growing system size, this prediction fails in the case of N = 85, where the anti-Mackay energy minimum is almost as deep as the ground state due to an optimal arrangement of the outer monomers. Nevertheless, for N ≥ 90, the transition shifts rapidly to higher temperatures (Fig. 6.9) and manifests in the specific heat as well. Whereas in the case of pure LJ clusters, the two peaks remain separated up to sizes of 130 atoms [131], we observe both transitions merging already for N ≈ 100. In contrast to the polymers, the Mackay–anti-Mackay transition temperature of clusters even decreases near N = 120, presumably because the anti-Mackay overlayer is almost complete, leading to a spherical and therefore stable conformation. This indicates that in this polymer model anti-Mackay structures loose weight compared to atomic clusters. It is also worth noting that for N = 75 one finds a crossover between decahedral and icosahedral conformations which is indicated by a small peak at T ≈ 0.8.

6.4.6 Long-Range Effects At this point we cannot yet judge whether the differences to the behavior of atomic clusters at medium temperatures are caused by the truncation of the LJ potential or by the polymer topology, i.e., by the bonds connecting to beads. To answer this question, we will discuss the change in long-range ordering when modifying the cutoff rc . First, let us consider the polymer with N = 85 monomers employing the original LJ potential, where rc = ∞. The corresponding LJ cluster is the largest with an anti-Mackay ground state in the interval 55 < N < 147 [129]. Without cutoff, any deviation of polymer crystalization compared to the cluster behavior is caused by the bond potential only. In contrast to the previous discussion with truncated potential (rc = 2.5σ), we retain the anti-Mackay ground state where the 30 outer monomers completely cover 10 faces of the icosahedral core and build an energetically favored structure [Fig. 6.11(a)]. One might notice that there are no bonds connecting monomers on different faces directly since the bond length would be too far from the potential minimum. This means that for very low energies only a few bond configurations are allowed and that the anti-Mackay state is much less metastable than the Mackay state [Fig. 6.11(b)] for which many more bond configurations with low energies are possible. This leads to an entropic dominance of the latter in the temperature interval 0.002 < T < 0.08 as visible in Fig. 6.11, a solid-solid transition that has not been reported for atomic clusters. This transitions is also signaled by the specific heat at T ≈ 0.002, since at this temperature the energetically favored anti-Mackay state prevails. The transition back to anti-Mackay conformations extends over a wider temperature

6.4 Liquid-Solid Transitions of Elastic Flexible Polymers

125 1

2.5

2.3 2.2 2.1 2

0.5

0.5 0 0

1.9

pnic

2.4

C(T )/N

1

9 8 7 6 5 4 3 2 1 0 0.1

0.2

0.3

0.4

0.5

C(T )/N nic ≤ 1 nic ≥ 2

1.8 1.7 1.6 0.001

0.002

0.003

0.004

0.005

0.006

0 0.007

T

Figure 6.11: (Left) Conformations of minimal energy for N = 85 with (a) anti-Mackay and (b) Mackay overlayer. (Right) The polymer with N = 85 and untruncated LJ potential (rc = ∞) changes from anti-Mackay (T < 0.002) to Mackay conformations (0.002 < T < 0.1) and back (0.1 < T < 0.34).

interval around T ≈ 0.08 and cannot be localized in the specific heat. Again, the change from a LJ cluster to a polymer leads to a greater prominence of Mackay conformations in this regime. In order to study the influence of the cutoff on the thermodynamics at higher temperatures we now consider the doubled cutoff, rc = 5σ ≈ 3.12, and compare the exemplified system with N = 102. For the corresponding cluster, two separate transitions were observed at medium temperatures which stands in contrast to the polymer with original cutoff (rc = 2.5σ), where both transitions merge. Applying the enlarged cutoff we observe only a slight shift in the peak positions whereas the major differences compared to the behavior of the LJ cluster as demonstrated in Ref. [131] persist (Figs. 6.12). It is still impossible to distinguish both conformational transitions clearly by means of the specific heat, i.e., the temperature domain where anti-Mackay like conformations prevail is very small. One may conclude that this difference to the cluster behavior (i.e., the suppression of anti-Mackay states) is mainly an effect of the bonds. The second interesting result are the different ground states. As in the case of unbounded clusters with untruncated LJ potential, the ground-state conformation is decahedral for N = 102 [Fig. 6.5(c)]. We can also identify the solid-solid transition towards Mackay structures: The decahedral-icosahedral crossover occurs at the temperature T ≈ 0.02 (see Fig. 6.12), with a relatively prominent signal in the specific heat. To conclude, there is some evidence that during the change from atomic LJ

126

6. Crystallization of Elastic Polymers

1

8

C(T )/N nic = 0 nic = 1 nic ≥ 2

6

0.6

rc = 2.5σ

0.4

2

0.2

0

0 pnic

C(T )/N

4

0.8

1

8

1.7 1.6 1.5

6 4

1

2

rc = 5σ

0 0.02

0.8 0.6 0.4 0.2

0.04

0

0 0

0.1

0.2

0.3

0.4

0.5

T

Figure 6.12: The solid-solid transition to the decahedral ground state for the 102mer is induced by long-range effects.

clusters to polymers, Mackay-like structures are more favored independently of a truncation of the LJ potential. This results in a shift of the Mackay–antiMackay (or Mackay overlayer melting-) transition to higher temperatures and the formation of a Mackay dominated temperature interval, as it was discussed for the 85mer with untruncated LJ potential (see also the discussion in Refs. [126,127]). After the general discussion of geometric properties and conformational transitions of linear polymers, we now return to the much more specific proteinlike heteropolymers, whose folding behavior is strongly influenced by disorder effects induced by the sequence of heterogeneous monomers.

7 Folding Properties of Hydrophobic-Polar Lattice Proteins

7.1 Lattice Model for Parallel β Helix with 42 Monomers As an example for gaining deeper insights into conformational transitions accompanying the tertiary folding behavior of proteins, multicanonical chain-growth simulations were performed for a 42-mer with the sequence P H2 P HP H2 P HP HP2 H3 P HP H2 P HP H3 P2 HP HP H2 P HP H2 P that forms a parallel helix in the ground state [47,48]. Originally, it was designed to serve as a lattice model of the parallel β helix of pectate lyase C [141]. But there are additional properties that make it an interesting and challenging system. The ground-state energy is known to be Emin = −34. In the multicanonical chain-growth simulations [47,48], the ground-state degeneracy was estimated to be g0 = 3.9 ± 0.4, which is in perfect agreement with the known value g0ex = 4 (except translational, rotational, and reflection symmetries) [30]. As we will see, there are two conformational transitions. At low temperatures fluctuations of energetic and structural quantities signalize a (pseudo)transition between the lowest-energy states possessing compact hydrophobic cores and the regime of the globule conformations. This transition is in addition to the usual one between globules and random coils. The average structural properties at finite temperatures can be best characterized by the mean end-to-end distance hRee i(T ) and the mean radius of gyration hRgyr i(T ). As these quantities carry shape informations, their calculation is not exclusively based on the density of states gN (E(XN,t). Therefore expectation values of such quantities O are obtained from the time series of the measuring run of the multicanonical chain-growth simulation at infinite temperature by using the general formula 1 X hOi(T ) = O(XN,t )WN (XN,t)gN (E(XN,t))e−βE(XN,t) , (7.1) ZN t

128

7. Folding Properties of Hydrophobic-Polar Lattice Proteins 0.17 0.15 0.13 hRee i(T ) , 0.11 N

hRee i(T )/N

hRgyr i(T ) 0.09 N

hRgyr i(T )/N

0.07 0.05 0.03 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

T

Figure 7.1: Mean end-to-end distance hRee i and mean radius of gyration hRgyr i of the 42-mer.

with the estimate for the partition sum X ZN = WN (XN,t )gN (E(XN,t))e−βE(XN,t) .

(7.2)

t

The simulation results for hRee i(T ) and hRgyr i(T ) of the 42-mer are shown in Fig. 7.1. Statistical errors were estimated as usual by using the jackknife binning method [73]. The pronounced minimum in the end-to-end distance can be interpreted as an indication of the transition between the lowest-energy states and globules: The small number of ground states have similar and highly symmetric shapes (due to the reflection symmetry of the sequence) but the ends of the chain are polar and therefore they are not required to reside close to each other. Increasing the temperature allows the protein to fold into conformations different from the ground states and contacts between the ends become more likely. Therefore, the mean end-to-end distance decreases and the protein has entered the globule “phase”. Further increasing the temperature leads then to a disentangling of the globules and random coil conformations with larger end-to-end distances dominate. In Fig. 7.2, we have plotted the specific heat and the derivatives of the mean end-to-end distance and of the mean radius of gyration with respect to the temperature, 1 d (7.3) hRee i(T ) = 2 (hERee i − hEihRee i) , dT T d 1 hRgyr i(T ) = 2 (hERgyr i − hEihRgyr i) . (7.4) dT T Two temperature regions of conformational activity (shaded in gray), where the curves of the fluctuating quantities exhibit extremal points, can clearly be sepa-

7.1 Lattice Model for Parallel β Helix with 42 Monomers

129

0.20 CV (T )/N

1.00

0.15 0.10

0.50 CV (T ) N

0.05 d hRgyr i(T ) dT N

0.00

-0.05 d hRgyr i(T ) dT N -0.10

-0.50 -1.00

-0.15

d hRee i(T ) dT N

-1.50 0.0

d hRee i(T ) , 0.00 dT N

0.1

0.2

0.3

0.4

0.5

-0.20 0.6

0.7

0.8

0.9

-0.25 1.0

T

Figure 7.2: Specific heat CV and derivatives w.r.t. temperature of mean end-to-end distance hRee i and radius of gyration hRgyr i as functions of temperature for the 42-mer. The ground-state – (1) (2) globule transition occurs between T0 ≈ 0.24 and T0 ≈ 0.28, while the globule – random coil (1) (2) transition takes place between T1 ≈ 0.53 and T1 ≈ 0.70 (shaded areas).

rated. We estimate the temperature region of the ground-state – globule tran(1) (2) sition to be within T0 ≈ 0.24 and T0 ≈ 0.28. The globule – random coil (1) (2) transition takes place between T1 ≈ 0.53 and T1 ≈ 0.70. For high temperatures, random conformations are favored. In consequence, in the corresponding, rather entropy-dominated ensemble, the high-degenerate high-energy structures govern the thermodynamic behavior of the macrostates. A typical representative is shown as an inset in the high-temperature pseudophase in Fig. 7.2. Annealing the system (or, equivalently, decreasing the solvent quality), the heteropolymer experiences a conformational transition towards globular macrostates. A characteristic feature of these intermediary “molten” globules is the compactness of the dominating conformations as expressed by a small gyration radius. Nonetheless, the conformations do not exhibit a noticeable internal longrange symmetry and behave rather like a fluid. Local conformational changes are not hindered by strong free-energy barriers. The situation changes by entering the low-temperature (or poor-solvent) conformational phase. In this region, energy dominates over entropy and the effectively attractive hydrophobic force favors the formation of a maximally compact core of hydrophobic monomers. Polar residues are expelled to the surface of the globule and form a shell that screens the core from the (fictitious) aqueous environment. The existence of the hydrophobic-core collapse renders the folding behavior of a heteropolymer different from crystallization or amorphous transitions of ho-

130

7. Folding Properties of Hydrophobic-Polar Lattice Proteins

(a)

0.4

0.3

an;T P42 (E ) 0.2

T

= 0:30

0.1 T

0.0 -34

-33

-32

= 0:24

-31

-30

-29

-28

-27

-26

-25

E

0.14 (b)

0.12 0.10

T = 0.5

T = 1.0

0.08 can,T (E) P42 0.06 0.04 0.02 0.00 -32

-28

-24

-20

-16

-12

-8

-4

0

E

Figure 7.3: Canonical distributions for the 42-mer at temperatures (a) T = 0.24, 0.25, . . . , 0.30 (1) (2) close to the ground-state – globule transition region between T0 ≈ 0.24 and T0 ≈ 0.28, (b) T = 0.50, 0.55, . . . , 1.0. The high-temperature peak of the specific heat in Fig. 7.2 is near (1) (2) T1 ≈ 0.53, but at T1 ≈ 0.73 the distribution has the largest width [48]. Near this temperature, the mean radius of gyration and the mean end-to-end distance (see Figs. 7.1 and 7.2) have their biggest slope.

mopolymers. The reason is the disorder induced by the sequence of different monomer types. The hydrophobic-core formation is the main cooperative conformational transition which accompanies the tertiary folding process of a singledomain protein. In Fig. 7.3 we have plotted the canonical distributions pcan,T 42 (E) for different temperatures in the vicinity of the two transitions. From Fig. 7.3(a) we read off that the distributions possess two peaks at temperatures within that region where the ground-state – globule transition takes place. This is interpreted as indication of a “first-order-like” transition, i.e., both types of macrostates coexist in this

7.2 Ten Designed 48-mers

131

Table 7.1: Ground-state energies Emin and degeneracies g0 as estimated with the multicanonical chain-growth method [47,48] for ten HP sequences with 48 monomers. For comparison, we have < also quoted the lower bounds on native degeneracies gCHCC obtained by means of the CHCC (constraint-based hydrophobic core construction) method [29] as given in Ref. [50]. In both cases the constant factor 48 from rotational and reflection symmetries of conformations spreading into all three spatial directions was divided out. No. 48.1 48.2 48.3 48.4 48.5 48.6 48.7 48.8 48.9 48.10

sequence Emin HPH2 P2 H4 PH3 P2 H2 P2 HPH3 PHPH2 P2 H2 P3 HP8 H2 −32 H4 PH2 PH5 P2 HP2 H2 P2 HP6 HP2 HP3 HP2 H2 P2 H3 PH −34 PHPH2 PH6 P2 HPHP2 HPH2 PHPHP3 HP2 H2 P2 H2 P2 HPHP2 HP −34 PHPH2 P2 HPH3 P2 H2 PH2 P3 H5 P2 HPH2 PHPHP4 HP2 HPHP −33 P2 HP3 HPH4 P2 H4 PH2 PH3 P2 HPHPHP2 HP6 H2 PH2 PH −32 H3 P3 H2 PHPH2 PH2 PH2 PHP7 HPHP2 HP3 HP2 H6 PH −32 PHP4 HPH3 PHPH4 PH2 PH2 P3 HPHP3 H3 P2 H2 P2 H2 P3 H −32 PH2 PH3 PH4 P2 H3 P6 HPH2 P2 H2 PHP3 H2 PHPHPH2 P3 −31 PHPHP4 HPHPHP2 HPH6 P2 H3 PHP2 HPH2 P2 HPH3 P4 H −34 PH2 P6 H2 P3 H3 PHP2 HPH2 P2 HP2 HP2 H2 P2 H7 P2 H2 −33

< g0 (×103 ) gCHCC (×103 ) 5226 ± 812 1500 17 ± 8 14 6.6 ± 2.8 5.0 60 ± 13 62 1200 ± 332 54 96 ± 19 52 58 ± 21 59 22201 ± 6594 306 1.4 ± 0.5 1.0 187 ± 87 188

temperature region [40]. The behavior in the vicinity of the globule – random coil transition is less spectacular as can be seen in Fig. 7.3(b), and since the energy distribution shows up one peak only, this transition could be denoted as being “second-order-like”. The width of the distributions grows with increasing temperature until it has reached its maximum value which is located near T ≈ 0.7. For higher temperatures, the distributions become narrower again [48]. Since finite-size scaling is impossible because of the non-continuable sequences of different types of monomers, “transitions” between classes of protein shapes are, of course, to be distinguished from phase transitions in the strict thermodynamic sense. In conclusion, conformational transitions for polymers of finite size, such as proteins, are usually weak and therefore difficult to identify. Thus, these considerations are, of course, of limited thermodynamic significance. From a technical point of view, however, it is of some importance as Markovian Monte Carlo algorithms can show up problems with sampling the entire energy space, as the probability in the gap between the two peaks can be suppressed by many orders of magnitude (what is obviously not the case in our example of the 42mer) and tunnelings are extremely rare. Just for such situations, flat histogram algorithms have primarily been developed [83–85].

7.2 Ten Designed 48-mers We have also analyzed the ten designed sequences with 48 monomers given in Ref. [50]. The ratio between the numbers of hydrophobic and polar residues is one half for these HP proteins, i.e., the hydrophobicity is nH = 24. In Table 7.1

132

7. Folding Properties of Hydrophobic-Polar Lattice Proteins 35

2.0 log10 H(E)

30

1.5

25 log10 g(E)

1.0 log10 H(E)

20

0.5

15

0.0 log10 g(E)

48.1 48.5 48.6 48.7

10

-0.5

5 -30

-25

-20

-15

-10

-5

0

-1.0

E

Figure 7.4: Logarithmic plots of the densities of states g(E) and “flat” histograms H(E) for the sequences 48.1, 48.5, 48.6, and 48.7 from Table 7.1 that have the same lowest energy Emin = −32. The normalization of the histograms of these examples was chosen such that they coincide at maximum energy, log10 H(Emax = 0) = 1.

we have listed the sequences and ground-state properties. The minimum energies we found in multicanonical chain-growth simulations [48] coincide with the values given in Refs. [46] and [50]. Figure 7.4 shows the densities of states for selected 48-mers and the multicanonical histograms of the production run. Note that for Rosenbluth chain-growth methods (a-thermal or at β = 0) the histogram for chains of length N is obtained by accumulating their individual Rosenbluth weights WNR , which explains the poorer performance near the minimum energy, where a small number of states enters with big weights. This differs from the usual procedure in algorithms with importance sampling, where the counter of an energy bin being hit by an appropriate state is incremented by unity. In Fig. 7.5 we have plotted the mean energy, free energy, and entropy as functions of temperature for these lattice proteins. These results were obtained by means of the density of states as sampled with our algorithm. For T → 0 the curves for both hEi and F merge into the four different values of Emin (= −34, −33, −32, −31) while the entropy exhibits the ground-state degeneracies, S → ln g0 . Our estimates for the degeneracies g0 of the ground-state energies, and < for comparison, the lower bounds gCHCC given in Ref. [29], are listed in Table 7.1. The lower bounds were obtained with the constrained-based hydrophobic core construction (CHCC) method [29]. Our values lie indeed above these lower bounds or include it within the range of statistical errors. Notice that for the sequences 48.1, 48.5, and 48.8, our estimates for the ground-state degeneracy are much < higher than the bounds gCHCC . In these cases the smallest frame containing the

7.2 Ten Designed 48-mers

133

-0.5

-0.6 hE i(T )

N F (T ) N

1.2

1.0 hE i(T )=N 0.8

, -0.7

0.6

F (T )=N -0.8

S (T ) N

0.4

S (T )=N -0.9 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.2 0.40

T

Figure 7.5: Mean energy hEi(T ), Helmholtz free energy F (T ), and entropy S(T ) for 48-mers with the sequences given in Table 7.1.

entire hydrophobic core is rather large (cube containing 4 × 3 × 3 = 36 monomers with surface area A = 32 [bond length]2) such that enumeration of this frame is cumbersome. For 48.5 and 48.8, we further found ground-state conformations lying in less compact frames (48.5: A = 32, 40, 42, 48, 52, 54 [bond length]2, 48.8: A = 32, 40, 42 [bond length]2) and those conformations would require still more effort to be identified with the CHCC algorithm, which was designed to locate global energy minima and therefore starts the search beginning from the most compact hydrophobic frames. The ground-state energies of these examples are rather high (Emin = −31 for 48.8, and Emin = −32 for 48.1 and 48.5) and therefore a higher degeneracy seems to be natural. This is, however, only true, if there does not exist a conformational barrier that separates the compact H-core low-energy states from the general compact globules. Comparing the groundstate degeneracies and the low-temperature behavior of the specific heats for the sequences 48.1, 48.5, 48.6, and 48.7 (all of them having global energy minima with Emin = −32) as shown in Figs. 7.4 and 7.6, respectively, we observe that 48.6 and 48.7 with rather low ground-state degeneracy actually possess a pronounced low-temperature peak in the specific heat, while the higher-degenerate proteins 48.1 and 48.5 only show up a weak indication of a structural transition at low temperatures. The HP proteins 48.2, 48.3, and 48.9, which have the lowest minimum energy Emin = −34 among the examples in Table 7.1, have also the lowest ground-state degeneracies. These three candidates seem indeed to exhibit a rather strong ground-state – globule transition, as can be read off from the associated specific heats in Fig. 7.6. We have again measured the mean end-to-end distances and mean radii of

134

7. Folding Properties of Hydrophobic-Polar Lattice Proteins

0.17

1.2 1.0

hRee i(T )/N

48.1

0.15

CV (T )/N

hRee i(T ) , 0.11 N

CV (T ) 0.6 N

0.09 hRgyr i(T ) N

0.4

0.07

0.2 0.0 0.0

1.0

0.13

0.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

48.2

0.15 CV (T )/N 0.13

0.8

0.11

CV (T ) 0.6 N

0.03 1.0

0.0 0.0

0.17 48.3

0.1

0.2

0.3

0.4

1.0

CV (T )/N hRee i(T )/N

0.13 0.11

CV (T ) 0.6 N

hRee i(T ) , N

0.09 hRgyr i(T ) N

0.4

0.07

0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.03 1.0

hRee i(T )/N

48.5

CV (T ) 0.6 N

0.15

CV (T )/N

0.15 CV (T )/N 0.13 0.11

CV (T ) 0.6 N

hRee i(T ) , 0.11 N

0.07

hRgyr i(T )/N

0.2

0.3

0.4

0.4

0.5

0.6

0.7

0.8

0.9

0.03 1.0

0.11

0.2

0.07 hRgyr i(T )/N

0.3

0.4

0.5

0.6

hRee i(T ) , N

0.09 hRgyr i(T ) N

hRee i(T )/N

0.2

0.7

0.8

0.9

0.05 0.03 1.0

0.13 CV (T )/N 0.11

48.9

0.15 CV (T )/N

0.13

0.8

hRee i(T ) , 0.11 N

CV (T ) 0.6 N

0.09 hRgyr i(T ) N hRee i(T )/N

0.2 0.0 0.0

0.07 0.05

hRgyr i(T )/N 0.1

0.2

0.3

0.4

0.5 T

0.09 hRgyr i(T ) N

hRee i(T )/N

0.4

hRee i(T ) , N

0.07 hRgyr i(T )/N 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.05 0.03 1.0 0.17

48.8

0.15 0.13

0.8 CV (T )/N

CV (T ) 0.6 N 0.4

0.11

0.0 0.0

0.09 hRgyr i(T ) N

hRee i(T )/N

0.2

0.07 hRgyr i(T )/N

0.1

0.2

0.3

0.4

0.5

0.6

hRee i(T ) , N

0.7

0.8

0.9

0.05 0.03 1.0

T 0.17

0.4

0.03 1.0

0.15

T 1.2 1.0

0.9

CV (T ) 0.6 N

1.0

0.13

CV (T )/N

0.1

0.8

1.2

0.15

CV (T ) 0.6 N

0.0 0.0

0.7

0.17

0.8

0.0 0.0

0.17 48.7

0.4

0.6

T

1.2

0.8

0.05

48.6

T

1.0

0.5

0.2

0.05

0.3

0.07

hRgyr i(T )/N 0.1

hRee i(T ) , N

0.09 hRgyr i(T ) N

hRee i(T )/N

0.4

1.0

0.09 hRgyr i(T ) N

0.4

0.2

0.03 1.0 0.17

1.2

0.13

0.1

0.9

T

0.8

0.0 0.0

0.8

0.8

0.0 0.0

0.17

0.2

0.7

48.4

T 1.2 1.0

0.6

0.2

0.05

hRgyr i(T )/N 0.1

0.5

1.2

0.15

0.8

0.0 0.0

0.05

T

1.2 1.0

0.07 hRgyr i(T )/N

T

hRee i(T ) , N

0.09 hRgyr i(T ) N

hRee i(T )/N

0.4 0.2

0.05

hRgyr i(T )/N

0.17

1.2

0.6

0.7

0.8

0.9

0.03 1.0

0.17

1.2 1.0

48.10

0.15

CV (T )/N

0.13

0.8

0.11

CV (T ) 0.6 N

0.2 0.0 0.0

0.09 hRgyr i(T ) N

hRee i(T )/N

0.4

0.07

hRgyr i(T )/N 0.1

0.2

0.3

0.4

0.5

0.6

0.7

hRee i(T ) , N

0.8

0.9

0.05 0.03 1.0

T

Figure 7.6: Heat capacities CV (T ), mean end-to-end distances hRee i(T ), and mean radii of gyration hRgyr i(T ) of the ten designed 48-mers.

7.3 Beyond 100 Monomers ...

135

gyration which are also plotted as functions of temperature into Fig. 7.6. Both quantities usually serve to interpret the conformational compactness of polymers. For HP proteins, the end-to-end distance is strongly influenced, however, by the types of monomers attached to the ends of the chain. It is easily seen from the figures that the 48-mers with sequences starting and ending with a hydrophobic residue (48.1, 48.2, and 48.6) have a smaller mean end-to-end distance at low temperatures than the other examples from Table 7.1. The reason is that the ends can form hydrophobic contacts and therefore a reduction of the energy can be achieved. Thus, in these cases contacts between ends are usually favorable and the mean end-to-end distance is close to the mean radius of gyration. Interestingly, there exists indeed a crossover region, where hRee i < hRgyr i. Comparing with the behavior of the specific heat, this interval is close to the region, where the phase dominated by low-energy states crosses over to the globule-favored phase. The hydrophobic contact between the ends is strong enough to resist the thermal fluctuations in that temperature interval. The reason is that, once such a hydrophobic contact between the ends is established, usually other in-chain hydrophobic monomers are attracted and form a hydrophobic core surrounding the end-to-end contact. Thus, before the contact between the ends is broken, an increase of the temperature first leads to a melting of the surrounding contacts. The entropic freedom to form new conformations is large since the low-energy states are all relatively high degenerate and do not possess symmetries requiring an appropriate amount of heat to be broken. For sequences possessing mixed or purely polar ends, the mean end-to-end distance and mean radius of gyration differ much stronger, as there is no energetic reason, why the ends should be nearest neighbors. In conclusion, we see that for longer chains the strength of the low-temperature transition not only depends on low ground-state degeneracies as it does for short chains [72]. Rather, the influence of the higher-excited states cannot be neglected. A striking example is sequence 48.4 with rather low ground-state degeneracy, but only weak signals for a low-temperature transition.

7.3 Beyond 100 Monomers ... It has already been mentioned that it is currently only possible to study long proteins by means of strongly simplified models. One such example which is often used as a benchmark in efficiency tests of new algorithms, is the HP transcription of the 103-mer cytochrome c [51] with 37 hydrophobic and 66 polar monomers1. 1

The HP sequence is P2 H2 P5 H2 P2 H2 P HP2 HP7 HP3 H2 P H2 P6 HP2 HP HP2 HP5 H3 P4 H2 P H2 P5 H2 P4 H4 P HP8 H5 P2 HP2 [51].

136

7. Folding Properties of Hydrophobic-Polar Lattice Proteins

Figure 7.7: Low-energy conformation of the 103-mer with E = −56 (i.e., there are 56 pair contacts among hydrophobic monomers in the standard HP model) found in multicanonical chaingrowth simulations [48]. Red spheres correspond to hydrophobic monomers and light spheres mark polar residues.

As the conformation with the maximally compact hydrophobic core is in the focus of interest, many attempts to identify the lowest-energy conformation have been performed by employing several minimization strategies. Early estimates for the “ground state” energy range from Emin = −49 employing a contactinteraction Monte Carlo method [52] to more recent values Emin = −54 found with nPERMis [46] and, with an additional bias suppressing contacts between H and P monomers, even Emin = −55 [46]. Without any bias and special parameter adjustments, the multicanonical chain-growth algorithm decreased the value to Emin = −56 (a representative conformation is shown in Fig. 7.7) [48]. The density of states, estimated in these simulations, covers more than 50 orders of magnitude. The degeneracy of conformations with Emin = −56 was determined to be of order 1016 such that it was likely that there exist still one or more even lower-lying energetic states [48]. Actually, recently a conformation was found by means of a fragment-regrowth Monte Carlo method [142] that possesses the smaller energy Emin = −57. Eventually, based on a newly developed constraintbased hydrophobic-core optimization method [143], the exact ground-state energy was determined as Emin = −58 [144].

7.4 Protein Folding as a Finite-Size Effect Understanding protein folding by means of equilibrium statistical mechanics and thermodynamics is a difficult task. A single folding event of a protein cannot occur “in equilibrium” with its environment. But protein folding is often considered as a folding/unfolding process with folding and unfolding rates which are constant in a stationary state that defines the “chemical equilibrium”. Thus, the statistical

7.4 Protein Folding as a Finite-Size Effect

137

properties of an infinitely long sequence of folding/unfolding cycles under constant external conditions (which are mediated by the surrounding solvent) can then also be understood – at least in parts – from a thermodynamical point of view. In particular, folding and unfolding of a protein are conformational transitions and one is tempted to simply take over the conceptual philosophy behind thermodynamic phase transitions, in particular known from “freezing/melting” and “condensation/evaporation” transitions of gases. But, this approach is necessarily wrong. Thermodynamic phase transitions occur only in the thermodynamic limit, i.e., in infinitely large systems. A protein is, however, a heteropolymer uniquely defined by its finite amino acid sequence, which is actually comparatively short and cannot be made longer without changing its specific properties. This is different for polymerized molecules (“homopolymers”), where the infinite-length chain limit can be defined, in principle. The intensely studied collapse or Θ transition between the random-coil and the globular phase is such a phase transition in the truest sense, where a finite-size scaling towards the infinitely long chain is feasible [88]. In this case, also a classification of the phase transitions into continuous transitions (where the latent heat vanishes and fluctuations exhibit power-law behavior close to the critical point) and discontinuous transitions (with nonvanishing latent heat) is possible. For proteins (or heteropolymers with a “disordered” sequence), a finite-size scaling is useless and so a classification of conformational transitions. Nonetheless, cooperative conformational changes are often referred to as “folding”, “hydrophobic-collapse”, “hydrophobic-core formation”, or “glassy” transitions. All these transitions are defined on the basis of certain parameters, also called “order parameters” or “reaction coordinates”, but should not be confused with thermodynamic phase transitions. The onset of finite-system transitions is also less spectacular: Their identification on the basis of peaks and “shoulders” in fluctuations of energetic and structural quantities and interpretation in terms of “order parameters” is a rather intricate procedure. Since the different fluctuations do not “collapse” for finite systems, a unique transition temperature can often not be defined. Despite a surprisingly high cooperativity, collective changes of protein conformations are not happening in a single step. As found in studies of lattice models like the examples presented in this chapter, transition regions separate the “pseudophases”, where random coils, maximally compact globules, or states with compact hydrophobic core dominate. Although the lattice models are very useful in unraveling generic folding characteristics, they suffer from lattice artifacts, which are, however, less relevant for long chains. In order to obtain a more precise and thus finer resolved image of folding characteristics, it is necessary to “get rid of the lattice” and to allow the

138

7. Folding Properties of Hydrophobic-Polar Lattice Proteins

coarse-grained protein to fold into the three-dimensional continuum.

8 Thermodynamic Properties of Mesoscopic Off-Lattice Heteropolymer Models

8.1 Simulations of a Hydrophobic-Polar Off-Lattice Model for Proteins A manifest off-lattice generalization of the HP model is the AB model (1.21), where the hydrophobic monomers are labeled by A and the polar or hydrophilic ones by B [22]. The contact interaction is replaced by a distance-dependent Lennard-Jones type of potential accounting for short-range excluded volume repulsion and long-range interaction, the latter being attractive for AA and BB pairs and repulsive for AB pairs of monomers. An additional interaction accounts for the bending energy of any pair of successive bonds. This model was first applied in two dimensions [22] and generalized to three-dimensional AB proteins [145], partially with modifications taking implicitly into account additional torsional energy contributions of each bond [146,147]. Performing multicanonical simulations of this model and analyzing the results for several AB heteropolymers [148], the comparison of the respective folding properties allows for a classification of characteristic structure formation behaviors which are found in similar form for “real” proteins. Before these properties are discussed in detail, we investigate general thermodynamic and lowest-energy conformation properties of the peptides studied. In order to locate global energy minima of a complex system, it is often useful to apply specially biased algorithms that only serve this purpose. We used the energy landscape paving (ELP) minimizer [149] to find global energy minima of the sequences under consideration. The ELP minimization is a Monte Carlo optimization method, where the energy landscape is locally flattened. This means that if a state X with energy E(X) is hit, the energy is increased by a “penalty” which itself depends on the histogram of any suitably chosen order parameter.

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8. Thermodynamic Properties of Mesoscopic Off-Lattice Heteropolymer Models

Figure 8.1: Spherical update of the bond vector between the ith and (i + 1)th monomer.

The simplest choice is the energy distribution H(E) such that the energy change in one update step is Et+1 = Et +f (H(Et)). Thus, the Boltzmann probability for a Metropolis update becomes a function of “time” t: p(Et) = exp(−Et/kB T ), where kB T is the thermal energy at the temperature T . The advantage of this method is that local energy minima are filled up and the likelihood of touching again recently visited regions decreases. This method has successfully proved to be applicable to find global energy minima in rough energy landscapes of proteins [149,150] and AB heteropolymers [148]. Of course, as a consequence of the bias, stochastic methods along the line of ELP violate detailed balance and it is therefore inappropriate to apply those for uncovering thermodynamic properties of a statistical system. The general procedure of performing multicanonical simulations has already been described in Section 4.6.1 and thus we concentrate in the following on ergodic update mechanisms required in these simulations. For off-lattice heteropolymers governed by the AB model (1.21), the bond length between adjacent monomers in the chain is fixed. In real proteins this Cαn –Cαn+1 distance is about 3.8 ˚ A. This “virtual peptide bond” is considered as one of the length scales in the mesoscopic AB model and is set to unity here. Therefore, the (i + 1)th monomer lies on the surface of a sphere with radius unity around the ith monomer. Thus, spherical coordinates are the natural choice for calculating the new position of the (i + 1)th monomer on this sphere. For the reason of efficiency, we do not select any point on the sphere but restrict the choice to a spherical cap with maximum opening angle 2θmax (the dark area in Fig. 8.1). Thus, to change the position of the (i +1)th monomer to (i +1)′, we select the angles θ and ϕ randomly from the respective intervals cos θmax ≤ cos θ ≤ 1 and 0 ≤ ϕ ≤ 2π, which ensure a uniform distribution of the (i + 1)th monomer positions on the associated spherical cap. After updating the position of the (i + 1)th monomer, the following monomers in the chain are simply translated according to the corresponding bond vectors which remain unchanged in this type of update. Only the bond vector between the ith and the (i + 1)th monomers is rotated, all others keep their direction.

8.2 Similarity Measure and Order Parameter

141

This is similar to single spin updates in local-update Monte Carlo simulations of the classical Heisenberg model with the difference that in addition to local energy changes long-range interactions of the monomers, changing their relative position to each other, have to be computed anew after the update. For simulations in the state space of dense conformations it is recommendable to choose a rather small opening angle, e.g., cos θmax = 0.99, in order to be able to sample also very narrow and deep valleys in the landscape of angles. In the following, we investigate the capability of the multicanonical method to identify lowest-energy conformations by using spherical-cap updates. These structures are then compared with “putative” ground-state conformations identified by means of the minimization procedure ELP. For the comparison, we will use two measures of structural similarities, root mean square deviation and the angular overlap parameter. Both will be introduced in the next section.

8.2 Similarity Measure and Order Parameter 8.2.1 Root Mean Square Deviation In order to check the structural similarities of two conformations X = (x1, x2, . . . , xN ) and X′ = (x′ 1 , x′2 , . . . , x′ N ), the standard measure is the root mean square deviation (rmsd) of the respective pairs, v u N u1 X t rmsd = min |˜ xi − x˜′ i |2 . (8.1) N i=1

˜ i = xi − x0 and P the positions with respect to the Here, x x˜′i = x′ i − x′ 0 denote P N ′ ′ centers of masses x0 = j=1 xj /N and x 0 = N j=1 x j /N of the ith monomer in the two conformations. Obviously, the rmsd is zero for exactly coinciding conformations and the larger the value the worse the coincidence. The minimization of the sum in Eq. (8.1) is performed with respect to a global relative rotation of the two conformations in order to find the best match. For the explicit calculation the exact quaternion-based optimization procedure described in Ref. [151] might be used.

8.2.2 Angular Overlap Order Parameter The folding process of proteins is necessarily accompanied by cooperative conformational changes. Although not phase transitions in the strict sense, it should be expected that one or a few parameters can be defined that enable the description of the structural ordering process. The number of degrees of freedom

142

8. Thermodynamic Properties of Mesoscopic Off-Lattice Heteropolymer Models

in most all-atom models is given by the dihedral torsional backbone and sidechain angles. In coarse-grained Cα models as the AB model, the original dihedral angles are replaced by a set of virtual torsional and bond angles. In fact, the number of degrees of freedom is not necessarily reduced in simplified off-lattice models. Therefore, the complexity of the space of degrees of freedom is comparable with more realistic models, and it is also a challenge to identify a suitable order parameter for the folding in such minimalistic heteropolymer models. On the other hand, the computational simplicity of these models allows for a more systematic and efficient analysis of the heteropolymer folding process. In Fig. 8.2, we show the probability distributions pang(Θ, Φ) of all successive pairs of virtual bond angles Θi = π − ϑi and torsion angles1 Φi for the exemplified AB sequence A4 B2A4 BA2BA3 B2A at several temperatures. This plot can be considered as the AB analog of the Ramachandran map for real proteins. Although this representation is not appropriate to describe the folding process, which will be rather complicated for this example as described later on, a few interesting features can already be read off from this figure. At the temperature T = 0.3, we observe two domains in this landscape, i.e., a structural pre-ordering has already taken place. The distribution is noticeably peaked for bond angles around 90◦ and torsion angles close to 0◦, i.e., almost perfectly planar cis conformations are favored in the ensemble as well as segments with bond angles between 60◦ and 70◦ for a broad distribution of torsion angles mainly between 40◦ and 100◦. The reason for the large width of the torsion-angle distribution in this region is that the temperature is still to high for fine-structuring within the conformations. Explicit torsional barriers might stabilize these segments even at this temperature but are disregarded in the model. Decreasing the temperature down to T = 0.1, we see that the landscape of this accumulated distribution of the degrees of freedom becomes very complex, and the peaks are much sharper. In fact, close to T ≈ 0.1, we observe a conformational transition towards the formation of the ground states. Actually, the complexity of this landscape can be understood better when considering the folding channels in the following, where we will see that this heteropolymer exhibits metastability and therefore rather glassy behavior. A remarkable aspect is the formation of the peaks in the bond-angle distribution at low temperatures close to 60◦, 90◦, and 120◦, as these angles are typical base angles in face-centered cubic crystals. As this concerns only segments of the conformations, the conformational transition is actually not a crystallization. Concluding, distributions of degrees of freedom 1

For a polymer with N monomers, there are in this model (N − 2) bond and (N − 3) torsion angles. In order to form successive pairs of these angles, we have left out the last bond angle (counted from the first monomer in the sequence) in Fig. 8.2.

8.2 Similarity Measure and Order Parameter

143

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pang (Φ, Θ)

T = 0.2

pang (Φ, Θ)

T = 0.3

60

30

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60

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180 180 150 120 90 Φ

60

30

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150 Θ

180 180 150 120 90 Φ

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180

Figure 8.2: Exemplified bond and torsion angle distributions of sequence A4 B2 A4 BA2 BA3 B2 A at different temperatures. The distributions of the torsion angles are reflection-symmetric and therefore only the positive intervals are shown.

are not quite useful to describe the folding process. For this reason it is necessary to define a suitable effective system parameter [152,153]. A useful choice will be discussed in the following. In analogy to studies of the specific folding behavior in all-atom protein models [154,155], a generalized variant of the overlap order parameter was introduced in Refs. [54,55,148]. The idea is to define a simple and computationally low-cost measure for the similarity of two conformations, where the differences of the angular degrees of freedom are calculated. We define the overlap parameter as follows: Q(X, X′) = 1 − d(X, X′). (8.2)

With Nb = N − 2 and Nt = N − 3 being the respective numbers of bond angles Θi and torsional angles Φi , the angular deviation between the conformations is calculated according to !# "N Nt b X X 1 d(X, X′) = drt (Φi , Φ′i) , (8.3) db (Θi , Θ′i) + min r=± π(Nb + Nt) i=1 i=1

where

db (Θi, Θ′i) = |Θi − Θ′i |, ′ ′ ′ d± t (Φi, Φi ) = min (|Φi ± Φi |, 2π − |Φi ± Φi |) .

(8.4)

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8. Thermodynamic Properties of Mesoscopic Off-Lattice Heteropolymer Models

Table 8.1: Fibonacci sequences [22] used for our analyses of ground-state properties. label F1 F2 F3 F4

N 13 21 34 55

sequence AB2 AB2 ABAB2 AB BABAB2 ABAB2 AB2 ABAB2 AB AB2 AB2 ABAB2 AB2 ABAB2 ABAB2 AB2 ABAB2 AB BABAB2 ABAB2 AB2 ABAB2 ABAB2 AB2 ABAB2 AB2 ABAB2 ABAB2 AB2 ABAB2 AB

Here we have taken into account that the AB model is invariant under the reflection symmetry Φi → −Φi . Thus, it is not useful to distinguish between reflection-symmetric conformations and therefore only the larger overlap is considered. Since −π ≤ Φi ≤ π and 0 ≤ Θi ≤ π, the overlap is unity, if all angles of the conformations X and X′ coincide, else 0 ≤ Q < 1. It should be noted that the average overlap of a random conformation with the corresponding reference state is for the sequences considered close to hQi ≈ 0.66. As a rule of thumb, it can be concluded that values Q < 0.8 indicate weak or no significant similarity of a given structure with the reference conformation.

8.3 Search for Global Energy Minima The multicanonical algorithm primarily serves to study thermodynamic properties, e.g., the “phase” behavior of the AB heteropolymers. Before discussing these results, however, it is necessary to analyze the capability of the multicanonical method to find lowest-energy conformations, in particular the native fold, because the identification of these structures is not only interesting as a by-product of the simulation. Rather, since they dominate the low-temperature behavior, it is necessary that they are generated frequently in the multicanonical sampling. In Table 8.1 we list the Fibonacci sequences [22] that have already been studTable 8.2: Estimates for AB model global energy minima of the Fibonacci sequences listed in Table 8.1 [148], obtained with multicanonical (MUCA) sampling and ELP minimization [149] compared with values quoted in Ref. [145] employing off-lattice PERM and after subsequent conjugate-gradient (CG) minimization. MUCA ELP PERM CG label Emin [148] Emin [148] Emin [145] Emin [145] F1 −4.967 −4.967 −3.973 −4.962 F2 −12.296 −12.316 −7.686 −11.524 F3 −25.321 −25.476 −12.860 −21.568 F4 −41.502 −42.428 −20.107 −32.884

8.3 Search for Global Energy Minima

145

Table 8.3: Root mean square deviations rmsd and overlap Q of lowest-energy conformations found in the multicanonical simulations and compared with structures obtained by ELP optimization for the Fibonacci sequences of length N given in Table 8.1. N F1 F2 F3 F4

rmsd 0.015 0.025 0.162 2.271

Q 0.994 0.992 0.979 0.766

ied in Ref. [145] by means of an off-lattice variant of the improved version [46] of the celebrated chain-growth algorithm with population control, PERM [43], described in detail in Chapter 4 for lattice heteropolymers. In Ref. [145], first estimates for the putative ground-state energies of the Fibonacci sequences with 13 to 55 monomers were given for the AB model in three dimensions. We compare these results with the respective lowest energies found in multicanonical simulations [148] and with the results obtained with the biased minimization algorithm ELP. It turns out that the ground-state energies found by multicanonical sampling agree well with what comes out by the ELP minimizer, cf. Table 8.2. Note that the multicanonical algorithm is not tuned to give good results in the lowenergy sector only. For all sequences studied, the same algorithm also yielded the thermodynamic results to be discussed in the following section. Another interesting result is that our estimates for the ground-state energies in Table 8.2 lie significantly below the energies quoted in Ref. [145], obtained with the offlattice PERM variant, and our values are even lower than the energies obtained by PERM and subsequent conjugate-gradient minimization in the attraction basin. Not unexpectedly, this is particularly pronounced for the longest chain considered. In Table 8.3, we list the values of rmsd and Q for the lowest-energy conformations found for the Fibonacci sequences of Table 8.1. Since the AB model is energetically invariant under reflection symmetry, the landscape of the free energy as function of the bond and torsion angles is trivially symmetric with respect to the torsional degrees of freedom. Consequently, unless exceptional cases, there is thus a trivial twofold energetic degeneracy of the global energy minimum, but the respective rmsd and overlap parameter are different for the associated conformations. The values quoted were obtained by comparing the lowest-energy conformation found with both reference conformations and quoting the value indicating better coincidence (i.e., lower rmsd and higher overlap). This obvious ambiguity can be circumvented by adding a symmetry-breaking term to the models that disfavors, e.g., left-handed helicity [156].

146

8. Thermodynamic Properties of Mesoscopic Off-Lattice Heteropolymer Models

Figure 8.3: Side (left) and top view (right) of the putative global energy minimum conformation of the 55-mer (F4) found with the ELP minimization algorithm (red spheres: hydrophobic monomers – A, light spheres: hydrophilic – B).

We see that for the shortest sequences (with 13 and 21 monomers) the coincidence of the lowest-energy structures found is extremely good and we are pretty sure that we have found the respective basins containing the ground states. In the case of the 34-mer both structures coincide still very well and it seems that the attraction valley towards the ground state was found within the multicanonical simulation. The situation is probably more complex for the 55-mer. As the parameters tell us, the lowest-energy conformations identified by ELP minimization and multicanonical simulation show significant structural differences. It is likely that both conformations do not belong to the same attraction basin and it is a future task to reveal whether this is a first indication of metastability. Answering this question is strongly related with the problem of identifying the folding path or an appropriate parameterization of the free-energy landscape. Due to hidden barriers it is practically impossible that ordinary stand-alone multicanonical sampling will be able to achieve this for such relatively long sequences. Note that, in our multicanonical simulation for the 55-mer, we precisely sampled the density of states over 120 orders of magnitude, i.e., the probability for finding randomly the lowest-energy conformation (that we identified with multicanonical sampling) in the conformational space is even less than 10−120! In Fig. 8.3 we show two views of the global energy minimum conformation with energy Emin ≈ −42.4 for the 55-mer found by applying the ELP minimizer. The hydrophobic core is tube-like and the chain forms a helical structure (which is here an intrinsic property of the model and is not due to hydrogen-bonding obviously being not supplied by the model).

8.4 Comparative Analysis of Thermodynamic Properties

147

Table 8.4: Sequences used in the study of thermodynamic properties of heteropolymers as introduced in Ref. [146] and the purely hydrophobic homopolymer A20 . The number of hydrophobic monomers is denoted by #A. No. 20.1 20.2 20.3 20.4 20.5 20.6 20.7

sequence BA6 BA4 BA2 BA2 B2 BA2 BA4 BABA2 BA5 B A4 B2 A4 BA2 BA3 B2 A A4 BA2 BABA2 B2 A3 BA2 BA2 B2 A3 B3 ABABA2 BAB A3 B2 AB2 ABAB2 ABABABA A20

#A 14 14 14 14 10 10 20

8.4 Comparative Analysis of Thermodynamic Properties Now, we are focusing on thermodynamic properties of heteropolymers, in particular on conformational transitions heteropolymers pass from random coils to native conformations with compact hydrophobic core. We investigate six exemplified heteropolymers with 20 monomers as introduced in Ref. [146]. The associated sequences are listed in Table 8.4. The hydrophobicity (= #A monomers in the sequence) is identical (=14) for the first four sequences 20.1–20.4, while sequences 20.5 and 20.6 possess only 10 hydrophobic residues. Sequence 20.7 is the homopolymer A20 consisting of hydrophobic residues only. As described in Section 8.3, the multicanonical method is capable to find even the lowest-energy states without any biasing or quenching. This is evident from the results listed in Table 8.5, where multicanonical estimates of the Table 8.5: Minimal energies and temperatures of the maximum specific heats as obtained by multicanonical sampling for the six heteropolymers 20.1–20.6 listed in Table 8.4. The global maximum of the respective specific heats is indicated by a star (⋆). The specific heat of sequence (2) (1) 20.6 possesses only one maximum at TC ≈ 0.35. The value given for TC belongs to the pronounced turning point. For comparison, we have also given the globally minimal energies found from minimization with ELP as well as the respective rmsd and the structural overlap parameter Q of the corresponding minimum energy conformations [148]. No. 20.1 20.2 20.3 20.4 20.5 20.6

MUCA Emin −33.766 −33.920 −33.582 −34.496 −19.647 −19.322

ELP Emin −33.810 −33.926 −33.578 −34.498 −19.653 −19.326

rmsd 0.048 0.015 0.025 0.030 0.017 0.047

Q 0.954 0.992 0.990 0.985 0.988 0.989

(1)

TC 0.27(3)⋆ 0.26(4)⋆ 0.25(3)⋆ 0.26(3) 0.15(2) 0.15(2)

(2)

TC 0.61(5) 0.69(4) 0.69(3) 0.66(2)⋆ 0.41(1)⋆ 0.35(1)⋆

180

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8. Thermodynamic Properties of Mesoscopic Off-Lattice Heteropolymer Models

hmuca (E)

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0 10 20 30 40 50 60 E

Figure 8.4: Multicanonical histogram hmuca (E) and logarithm of the density of states g(E) for sequence 20.1 [55].

global energy minima are once more compared with optimized structures obtained with the ELP minimization method. The values obtained with multicanonical sampling agree pretty well with those from ELP minimization. The respective structural coincidences are confirmed by the values for the rmsd and the overlap also being given in the table. In order to identify conformational transitions, we calculated the specific P heat CV (T ) = (hE 2i − hEi2 )/kB T 2 with P hE k i = E g(E)E k exp(−E/kB T )/ E g(E) exp(−E/kB T ) from the density of states g(E). We sampled conformations with energy values lying in the interval [−60.0, 50.0], discretized in bins of size 0.01, and required the multicanonical histogram to be flat for at least 70% of the core of this interval, i.e., within the energy range [−43.5, 33.5]. Within and partly beyond this region, we achieved almost perfect flatness, i.e., the ratios between the mean and maximal histogram value, Hmean /Hmax , as well as the ratio between minimum and mean, Hmin/Hmean , exceeded 0.9. As a consequence of this high-accurate sampling of the energy space this enabled us to calculate the density of states very precisely over about 70 orders of magnitude. The energy scale is, of course, bounded from below by the ground-state energy Emin, and that we closely approximated this value can be seen by the strong decrease of the logarithm of the density of states for the lowest energies in the exemplified plots of the density of states and the multicanonical (“flat”) histogram for sequence 20.1 in Fig. 8.4. This strong decrease of the density-of-states curves near the ground-state energy is common to all short heteropolymer sequences studied. It reflects the isolated character of the ground state within the energy landscape. We used the density of states to calculate the specific heats of the 20-mers in

8.4 Comparative Analysis of Thermodynamic Properties 5.0

20.1 20.2 20.3 20.4 20.5 20.6 20.7

4.5 4.0 3.5 CV (T ) N

149

3.0 2.5 2.0 1.5 1.0 0.5 0.0

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0.2

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Figure 8.5: Specific heats of the 20-mers listed in Table 8.4.

both models. The results are shown in Fig. 8.5. A first observation is that the specific heats show up two distinct peaks with the low-temperature peak located (1) (2) at TC and the high-temperature peak at TC compiled in Table 8.5. The sequences considered here are very short and the native fold contains a single hydrophobic core. Interpreting the curves for the specific heats in Fig. 8.5 in terms of conformational transitions, we conclude that the heteropolymers tend to (2) (1) form, within the temperature region TC < T < TC , intermediate states (often also called traps) comparable with globules in the collapsed phase of polymers. For sequences 20.5 and, in particular, 20.6 the smaller number of hydrophobic (2) (1) monomers causes a much sharper transition at TC than at TC (where, in fact, the specific heat of sequence 20.6 possesses only a turning point). The pro(2) nounced transition near TC is connected with a dramatic change of the radius of gyration, as can be seen later in Fig. 8.7, indicating the collapse from stretched to highly compact conformations with decreasing temperature. The conformations (2) dominant for high temperatures T > TC are random coils, while for temper(1) atures T < TC primarily conformations with compact hydrophobic core are favored. In-between, there is the intermediary globular “phase”. As it has already been outlined, these “phases” are not phases in the strict thermodynamic sense, since for heteropolymers of the type we used in this study (this means we are not focused on sequences that have special symmetries, as for example diblock copolymers AnBm ), a thermodynamic limit is in principle nonsensical. Therefore, conformational transitions of heteropolymers are not true phase transitions. As

150

8. Thermodynamic Properties of Mesoscopic Off-Lattice Heteropolymer Models 0.5

2.5 (a)

CV (T ) N

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0.2 5 dhR i gyr N dT 0.1

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Figure 8.6: Fluctuations of energy (specific heat), radius of gyration and end-to-end distance for sequence 20.3.

a consequence, fluctuating quantities, for example the derivatives with respect to the temperature of the mean radius of gyration, dhRgyr i/dT , and the mean end-to-end distance, dhRee i/dT , do not indicate conformational activity at the same temperatures, as well as when compared with the specific heat. Similar to the lattice analyses of heteropolymer folding, conformational transitions of heteropolymers happen within a certain interval of temperatures, not at a fixed critical temperature. This is a typical finite-size effect and, for this rea(1) (2) son, the peak temperatures TC and TC defined above for the specific heat are only representatives for the entire intervals. In order to make this more explicit, we consider sequence 20.3 in more detail. In Fig. 8.6, we compare the energetic fluctuations (in form of the specific heat CV ) with the respective fluctuations of radius of gyration and end-to-end distance, dhRee,gyr i/dT = (hRee,gyr Ei−hRee,gyr ihEi)/kB T 2 . Obviously, the temperatures with maximal fluctuations are not identical and the shaded areas are spanned over the temperature intervals, where strongest activity is expected. We observe for this example that two such centers of activity can be separated linked by an intermediary interval of globular traps. In fact, there is a minimum of the specific heat at TCmin between (1) (2) (2) TC and TC , but the height of the barrier at TC is rather small and the globules are not very stable. It is also interesting to compare the thermodynamic behavior of the heteropolymers with the homopolymer consisting of 20 A-type monomers, A20. This is the consequent off-lattice generalization of self-avoiding interacting walks on the lattice (ISAW) that have been extensively studied over the past decades. In contrast to heteropolymers, homopolymers show up a characteristic second-order phase transition between random coil conformations (“good solvent”) and com-

8.4 Comparative Analysis of Thermodynamic Properties

0.14

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Figure 8.7: Mean radius of gyration hRgyr i as a function of the temperature T for the sequences 20.1–20.4 (solid curves), 20.5, 20.6 (long dashes), and for the homopolymer A20 (short-dashed curve).

pact globules (“poor solvent”), the Θ transition [91] (cf. also the analysis of the collapse of lattice homopolymers in Section 5.4 and Ref. [88]). The specific heat of this short homopolymer, also plotted in Fig. 8.5, shows that the collapse from random coils (high temperatures) to globular conformations (low temperatures) happens, roughly, in one step. There is only one energetic barrier as indicated by the single peak of the specific heats. The onset of the separate collapse transition can only be guessed as a the specific heat exhibits only a smart shoulder near T ≈ 0.9. In the crystallization studies of flexible polymers on regular lattices in Chapter 5, we have shown that both transitions (crystallization and collapse) are well separated in the thermodynamic limit [88]. This depends, however, on the interaction range and in bond-fluctuations studies it was shown that for models with a certain interaction thresholds, both transitions fall together in the thermodynamic limit [99,100]. In Fig. 8.7, we have plotted the mean radii of gyration as a function of the temperature for the sequences from Table 8.4 in comparison with the homopolymer. For all temperatures in the interval plotted, the homopolymer obviously takes more compact conformations than the heteropolymers, since its mean radius of gyration is always smaller. This different behavior is an indication for a rearrangement of the monomers that is particular for heteropolymers: the formation of the hydrophobic core surrounded by the hydrophilic monomers. Since the homopolymer trivially also takes in the ground state a hydrophobic core conformation (since it only consists of hydrophobic monomers), which is obviously more compact than the complete conformations of the heteropolymers, we conclude that hydrophobic monomers weaken the compactness of low-temperature conformations. Thus, ho-

152

8. Thermodynamic Properties of Mesoscopic Off-Lattice Heteropolymer Models

mopolymers and heteropolymers show a different “phase” behavior in the dense phase. Homopolymers fold into globular conformations which are hydrophobic cores with maximum number of hydrophobic contacts. Heteropolymers also form very compact hydrophobic cores which are, of course, smaller than that of the homopolymer due to the smaller number of hydrophobic monomers in the sequence. In total, however, heteropolymers are less compact than homopolymers because the hydrophilic monomers are pushed off the core and arrange themselves in a shell around the hydrophobic core. We also see in Fig. 8.7 a clear tendency that the mean radius of gyration and thus the compactness strongly depends on the hydrophobicity of the sequence, i.e., the number of hydrophobic monomers. The curves for sequences 20.5 and 20.6 (long-dashed curves) with 10 A’s in the sequence can clearly be separated from the other heteropolymers in the study (with 14 hydrophobic monomers) and the homopolymer. This supports the assumption that for heteropolymers the formation of a hydrophobic core is more favorable than the folding into an entire maximally compact conformation. The tertiary formation of the hydrophobic core will be discussed in detail in the following chapter.

9 Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

9.1 Tertiary Protein Folding from a Mesoscopic Perspective Folding of linear chains of amino acids, i.e., bioproteins and synthetic peptides, is, for single-domain macromolecules, accompanied by the formation of secondary structures (helices, sheets, turns) and the tertiary hydrophobic-core collapse. While secondary structures are typically localized and thus limited to segments of the peptide, the effective hydrophobic interaction between nonbonded, nonpolar amino acid side chains results in a global, cooperative arrangement favoring folds with compact hydrophobic core and a surrounding polar shell that screens the core from the polar solvent. Systematic analyses for unraveling general folding principles are extremely difficult in microscopic all-atom approaches, since the folding process is strongly dependent on the “disordered” sequence of amino acids and the native-fold formation is inevitably connected with, at least, significant parts of the sequence. Moreover, for most proteins, the folding process is relatively slow (microseconds to seconds), which is due to a complex, rugged shape of the free-energy landscape [157–159] with “hidden” barriers, depending on sequence properties. Although there is no obvious system parameter that allows for a general description of the accompanying conformational transitions in folding processes (as, for example, the reaction coordinate in chemical reactions), it is known that there are only a few classes of characteristic folding behaviors, mainly single-exponential folding, two-state folding, folding through intermediates, and glass-like folding into metastable conformations [152–155,160–162]. Thus, if a classification of folding characteristics is useful at all, strongly simplified models should reveal statistical [54,55] and kinetic [163] pseudouniversal properties. The reason why it appears useful to use a simplified, mesoscopic model like the AB model is two-fold: Firstly, it is believed that tertiary folding

154

9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

is mainly based on effective hydrophobic interactions such that atomic details play a minor role. Secondly, systematic comparative folding studies for mutated or permuted sequences are computationally extremely demanding at the atomic level and are to date virtually impossible for realistic proteins. We will show in the following that by employing the AB heteropolymer model [22] and monitoring the simple angular “order” parameter introduced in Section 8.2.2, it is indeed possible to identify different complex folding characteristics. This is comparable to studies of phase transitions based on effective order parameters in other disordered systems such as, e.g., spin glasses, where simplified models are successfully employed [164]. The individual folding trajectories will be characterized by a similarity parameter which is related to the replica overlap parameter used in spin-glass analyses. This is useful as the amino acid sequence induces intrinsic disorder and frustration into the system and therefore a peptide behaves similar to a spin system with a quenched disorder configuration of couplings.

9.2 Identification of Characteristic Folding Channels For the qualitative discussion of the folding behavior it is useful to consider the histogram of energy E and angular overlap parameter Q, as defined in Section 8.2.2, obtained from multicanonical simulations, X Hmuca(E, Q) = δE,E(Xt ) δQ,Q(Xt ,X(0) ) , (9.1) t

where the sum runs over all Monte Carlo sweeps t. In Figs. 9.1(a)–9.1(c), the multicanonical histograms Hmuca (E, Q) are plotted for three of the sequences listed in Table 8.4: 20.1, 20.3, and 20.4. Ideally, multicanonical sampling yields a constant energy distribution hmuca (E) =

Z1

dQ Hmuca(E, Q) = const.

(9.2)

0

In consequence, the distribution Hmuca (E, Q) can suitably be used to identify the folding channels, independently of temperature. This is more difficult with temperature-dependent canonical distributions P can (E, Q), which can, of course, be obtained from Hmuca (E, Q) by a simple reweighting procedure, P can (E, Q) ∼ Hmuca (E, Q)g(E) exp(−E/kB T ). Nonetheless, it should be noted that, since there is a unique one-to-one correspondence between the average energy hEi and temperature T , regions of changes in the monotonic behavior of Hmuca (E, Q) can also be assigned a temperature, where a conformational transition occurs.

9.2 Identification of Characteristic Folding Channels

155

(a) -26

N

0.6 0.7 0.8

Q

-27 F (Q)

D

0.5

T = 0.4

-26.5

log10 Hmuca (E, Q) 3 2 1 0

0.9 1.0

-28

-29

-30

-31

-32

-33

-34

T = 0.2

D

-27.5

-35

T = 0.1

-28

N

E

T = 0.05

-28.5 0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Q

(b) -29 -29.5

log10 Hmuca (E, Q)

T = 0.2

D

I

4 2 0 0.5

D

0.6 0.7

0.8

Q

0.9 1.0

-28

-29

-30

-31

-32

-33

-34

F (Q)

-30

N

T = 0.1

I

-30.5

-35

T = 0.05 -31

E

T = 0.01

-31.5

N 0.7

0.75

0.8

0.85

0.9

0.95

1

Q

(c) -29

M2

log10 Hmuca (E, Q)

-29.5

4 2 0

-30

D

0.6 0.7 Q

0.8 0.9 1.0

-30

-31

-32

-33

-34

F (Q)

M1

0.5

T = 0.2

D

T = 0.1 -30.5

-35

T = 0.05

-31

M2

E

-31.5 0.7

0.75

T = 0.01

0.8

0.85

M1 0.9

0.95

1

Q

Figure 9.1: Multicanonical histograms Hmuca (E, Q) of energy E and angular overlap parameter Q and free-energy landscapes F (Q) at different temperatures for the three sequences (see Table 8.4) (a) 20.1, (b) 20.4, and (c) 20.3. The reference folds reside at Q = 1 and E = Emin . Pseudophases are symbolized by D (denatured states), N (native folds), I (intermediates), and M (metastable states). Representative conformations in intermediate and folded phases are also shown [54,55].

156

9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

Interpreting the ridges of the probability distributions in the left-hand panel of Fig. 9.1 as folding channels, it can clearly be seen that the heteropolymers exhibit noticeable differences in the folding behavior towards the native conformations (N). Considering natural proteins it would not be surprising that different sequences of amino acids cause in many cases not only different native folds but also vary in their folding behavior. Here we are considering, however, a highly minimalistic heteropolymer model and hitherto it was not clear whether it would be possible to separate characteristic folding channels in this simple model, but as Fig. 9.1 demonstrates, in fact, it is. For sequence 20.1, we identify in Fig. 9.1(a) a typical two-state characteristics. Approaching from high energies (or high temperatures), the conformations in the ensemble D have an angular overlap Q ≈ 0.7, which means that there is no significant similarity with the reference structure, i.e., the ensemble D consists mainly of unfolded peptides. For energies E < −30 a second branch opens. This channel (N) leads to the native conformation (for which Q = 1 and Emin ≈ −33.8). The constant-energy distribution, where the main and native-fold channels D and N coexist, exhibits two peaks noticeably separated by a well. Therefore, the conformational transition between the channels looks first-order-like, which is typical for two-state folding. The main channel D contains the ensemble of unfolded conformations, whereas the native-fold channel N represents the folded states. The two-state behavior is confirmed by analyzing the temperature dependence of the minima in the free-energy landscape. The free energy as a function of the “order” parameter Q at fixed temperature can be suitably defined as: F (Q) = −kB T ln p(Q).

(9.3)

DX δ(Q0 − Q(X, X(0))) e−E(X)/kB T

(9.4)

In this expression, p(Q0) =

Z

is related to the probability of finding a conformation with a given value of Q in the canonical ensemble at temperature T . The formal integration runs over all possible conformations X. In the right-hand panel of Fig. 9.1(a), the free-energy landscape at various temperatures is shown for sequence 20.1. At comparatively high temperatures (T = 0.4), only the unfolded states (Q ≈ 0.71) in the main folding channel D dominate. Decreasing the temperature, the second (native-fold) channel N begins to form (Q ≈ 0.9), but the global free-energy minimum is still associated with the main channel. Near T ≈ 0.1, both free-energy minima have approximately the same value, the folding transition occurs. The discontinuous character of this conformational transition is manifest by the existence of the freeenergy barrier between the two macrostates. For even smaller temperatures, the

9.2 Identification of Characteristic Folding Channels

157

native-fold-like conformations (Q > 0.95) dominate and fold smoothly towards the Q = 1 reference conformation, which is the lowest-energy conformation found in the simulation. A significantly different folding behavior is noticed for the heteropolymer with sequence 20.4. The corresponding multicanonical histogram is shown in Fig. 9.1(b) and represents a folding event through an intermediate macrostate. The main channel D bifurcates and a side channel I branches off continuously. This branching is followed by the formation of a third channel N, which ends in the native fold. The characteristics of folding-through-intermediates is also confirmed by the free-energy landscapes as shown for this sequence in Fig. 9.1(b) at different temperatures. Approaching from high energies, the ensemble of denatured conformations D (Q ≈ 0.76) is dominant. Close to the transition temperature T ≈ 0.05, the intermediary phase I is reached. The overlap of these intermediary conformations with the native fold is about Q ≈ 0.9. Decreasing the temperature further below the native-folding threshold close to T = 0.01, the hydrophobiccore formation is finished and stable native-fold-like conformations with Q > 0.97 dominate (N). The most extreme behavior of the three exemplified sequences is exhibited by the heteropolymer 20.3. The main channel D does not decay in favor of a nativefold channel. In fact, we observe both, the formation of two separate native-fold channels M1 and M2 . Channel M1 advances towards the Q = 1 fold and M2 ends up in a completely different conformation with approximately the same energy (E ≈ −33.512). The spatial structures of these two conformations are noticeably different and their mutual overlap is correspondingly very small, Q ≈ 0.746. It should also be noted that the lowest-energy conformations in the main channel D have only slightly larger energies than the two native folds. Thus, the folding of this heteropolymer is accompanied by a very complex folding characteristics. In fact, this multiple-peak distribution near minimum energies is a strong indication for metastability. A native fold in the natural sense does not exist, the Q = 1 conformation is only a reference state but the folding towards this structure is not distinguished as it is in the folding characteristics of sequences 20.1 and 20.4. This explains also, why the bond- and torsion-angle distribution in Fig. 8.2 possesses so many spikes: it represents rather the ensemble of amorphous conformations than a distinct footprint of a distinguished native fold. The amorphous folding behavior is also seen in the free-energy landscapes in Fig. 9.1(c). Above the folding transitions (T = 0.2) the typical sequence-independent denatured conformations with hQi ≈ 0.77 dominate (D). Then, in the annealing process, several channels are formed and coexist. The two most prominent channels (to which the lowestenergy conformations belong that we found in the simulations) eventually lead for

158

9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

T ≈ 0.01 to ensembles of macrostates with Q > 0.97 (M1), and conformations with Q < 0.75 (M2). The lowest-energy conformation found in this regime is structurally different but energetically degenerate compared with the reference conformation. After having shown that it is indeed possible to classify protein folding characteristics employing a mesoscopic model, we will consider now kinetic properties of a particularly interesting folding behavior: two-state folding.

9.3 G¯ o Kinetics of Two-State Folding Spontaneous protein folding is a dynamic process, which starts after the generation of the DNA-encoded amino acid sequence in the ribosome and it is in many cases finished, when the functional conformation, the native fold, is formed. As this process takes microseconds to seconds, a dynamical computational analysis of an appropriate microscopic model, which could lead to a better understanding of the conformational transitions accompanying folding [165], is extremely demanding. Since protein folding is a thermodynamic process at finite temperature, a certain folding trajectory in the free-energy landscape is influenced by Brownian collisions with surrounding solvent molecules. Therefore, it is more favorable to study the kinetics of the folding process by averaging over an appropriate ensemble of trajectories. A significant problem is that the complexity of detailed semiclassical microscopic models based on force-fields and solvent parameter sets (or explicit solvent) rules out molecular dynamics (MD) in many cases and, therefore, Markovian Monte Carlo (MC) dynamics is a frequently used method for such kinetic studies.1 It is obvious, however, that the time scale provided by MC is not directly comparable with the time scale of the folding process. It is widely believed that the folding path of a protein is strongly correlated with contact ordering [166], i.e., the order of the successive contact formation between residues and, therefore, long-range correlations and memory effects can significantly influence the kinetics. A few years ago, experimental evidence was found that classes of proteins show particular simple folding characteristics, single exponential and two-state folding [167,168]. In the two-state folding process, which we will focus on in the following example, the peptide is either in an unfolded, denatured state or it possesses a native-like, folded structure. In contrast to the barrier-free single1

Another reason why MD is typically outperformed by MC if thermodynamics becomes relevant is the fact that it is difficult to obtain the correct statistical distributions with MD. The proper usage of thermostats in MD remains a notorious problem [75].

9.3 G¯ o Kinetics of Two-State Folding

159 10

5

4

cV (AB)

9

AB

8

G¯oL

3.5

7

3

6

2.5

5

2

4

1.5

3

1

2

0.5

1

0 0.1

cV (G¯oL)

4.5

0 0.2

0.3

0.4

0.5

0.6

T

Figure 9.2: Specific heats for the peptide with sequence 20.6 as obtained with the AB model [148] and the gauged G¯oL model as defined in Eq. (9.5) [163].

exponential folding, there exists an unstable transition state to be passed in the two-state folding process. Due to the comparatively simple folding characteristics, strongly simplified, effective models were established. Knowledge-based models of G¯o-like type [169–172] were investigated in numerous recent studies [153,173– 181]. In G¯o-like models the native fold must be known and is taken as input for the energy function. The energy of an actual conformation depends on its structural deviation from the native fold (e.g., by counting the number of already established native contacts). By definition, the energy is minimal, if conformation and native fold are identical in all degrees of freedom involved in the model. The simplicity of the model entails reduced computational complexity and also MD simulations, e.g., based on Langevin dynamics [179], can successfully be performed. Here, we follow a different approach. We also study a G¯o-like model, but it is based on a minimalistic coarse-grained hydrophobic-polar representation of the heteropolymer [163].

9.3.1 The Mesoscopic G¯ o Model In the following, we consider the exemplified AB sequence 20.6 (see Table 8.4). For our comparative model study, we employ the physically motivated AB model [22] and a knowledge-based model of G¯o type [169,170,174], which is referred to as the G¯oL (G¯o-like) model throughout the following [163]. In the AB model, the heteropolymer with sequence 20.6 experiences a hydrophobic collapse transition which is signalized by the peak in the specific-heat

160

9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

Figure 9.3: Putative global-energy minimum of sequence 20.6 in the AB model with EAB ≈ −19.3. Red monomers are hydrophobic (A) and light residues polar (B).

curve plotted in Fig. 9.2. The transition between random coils and native-like hydrophobic-core conformations is a (pseudo)phase separation process and the folding transition of two-state (folded/unfolded) type. In particular, for such systems it is known from model studies of realistic amino acid sequences that knowledge-based G¯o type models reveal kinetic aspects of folding and unfolding processes reasonably well [174,179]. For performing kinetic studies of the peptide with sequence 20.6 in the simplified hydrophobic-polar approach as well, we use the (putative) global-energy minimum, identified in energy-landscape paving (ELP) minimizations [149] of the AB model [148], as input for the definition of a hydrophobic-polar G¯oL model. The (putative) native conformation X(0) is shown in Fig. 9.3 and its energy is EAB ≈ −19.3 in the units of the AB model (1.21). In G¯oL models, the “energy” of a given conformation is related to its similarity with the ground state. This means, “energy” in the G¯oL model plays rather the role of a similarity or “order” parameter and is, therefore, not a potential energy in the usual physical sense (as there is no physical force associated with it). Denoting the (N − 2) bending (0) angles of the global-energy minimum conformation by ϑk , its (N − 3) torsional (0) (0) angles as ϕl , and the inter-monomer distances by rij , we define the G¯oL model according to the representation in Ref. [174] as: EG¯oL (X)/ε = Kϑ

N −2 X k=1

ϑk −

(0) ϑk

2

+

−3 XN X

n=1,3 l=1

Kϕ(n)

n

h io (0) 1 − cos n ϕl − ϕl

 " " (0) #10 non-native # (0) 12 native X 1 X rij r + 5 ij −6 + 12 . r r r ij ij ij i<j−1 i<j−1

(9.5)

9.3 G¯ o Kinetics of Two-State Folding

161

The last two sums run over all pairs of nonbonded monomers. In the case the (0) pair (i, j) forms a native contact, i.e., rij < rcut and rij < rcut , the monomers experience a short-range 10-12 Lennard-Jones attraction, while for non-native contacts an overall repulsive 1/r12 contribution is taken into account. The con(1) (3) stants Kϑ = 20, Kϕ = 0.5, and Kϕ = 0.25 weigh the relative strengths of the angular energy contributions. The values were adjusted to have a reasonable coincidence of the peak temperature of the specific heat compared to the results obtained with the AB model [148] (see Fig. 9.2). The free overall energy scale ε was chosen such that EAB (X(0) ) = EG¯oL(X(0) ) = −ε ntot, where ntot is the total number of native contacts. The definition of a native contact requires the introduction of a cutoff radius, which we chose as rcut = 1.14. This entails ntot = 20 for the native conformation X(0) , and therefore ε ≈ 0.966. The choice of the cutoff radius is not very sensitive with respect to the results, provided it is not too small. For the thermodynamic analyses, we performed parallel tempering simulations [182–184] and for the kinetic studies, a standard implementation of the Metropolis Monte Carlo method was used [163]. Chain updates were performed employing the spherical-cap updates as described and parametrized in Section 8.1 and in Ref. [148]. A sweep consists of (N − 1) sequential bond vector update trials.

9.3.2 Thermodynamics For the study of the folding transition, the introduction of an effective parameter is useful which uniquely describes the macrostate of the ensemble of heteropolymer conformations. A widely used measure is the contact number q(X) which is for a given conformation X simply defined as the fraction of the already formed native contacts n(X) in conformation X and the total number of native contacts ntot in the final fold, i.e., q(X) = n(X)/ntot. Then, the statistical ensemble average of this quantity hq(X)i at a given temperature characterizes its macrostate. Roughly, for a two-state folder, if hq(X)i > 0.5, native-like conformations are dominating the statistical ensemble. If less than half the total number of contacts is formed, the heteropolymer tends to reside in the pseudophase of denatured conformations. Note that folding transitions are not sharp phase transitions in the thermodynamic sense as the heteropolymer sequence is of finite length and cannot be extended. Another suitable parameter is, once more, the angular overlap parameter Q(X) introduced in Section 8.2.2. The advantage is that it is particularly useful for classifying intermediate or metastable structures with stable, but non-native contacts. In Fig. 9.4, the fluctuations of both parameters, i.e., dhqi/dT and dhQi/dT , re-

162

9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding 0 -0.1

0.1 dhqi/dT

dhQi/dT or dhqi/dT

-0.2 dhQi/dT

-0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 0.25

0.3

0.35

0.4

0.45

0.5

0.55

T

Figure 9.4: Fluctuations of the similarity parameter q and the angular overlap parameter Q as functions of the temperature. For comparison, the specific heat (dotted line) is also plotted into this figure.

spectively, are plotted as functions of the temperature for the G¯oL model of sequence 20.6. We clearly see that the temperature region of conformational activity as signalized by these two “order” parameters coincides with the thermally active region indicated by the peak of the specific heat, which is also shown for comparison. The folding temperature, i.e., the temperature of maximum activity, is Tf ≈ 0.36. The classification of the heteropolymer with sequence 20.6 as two-state folder arises from the analysis of the free-energy landscape. We assume that q is a suitable parameter that describes the macrostate of the system adequately. Considering this parameter as a constraint, we can formally average out the conformational degrees of freedom and the probability for a conformation in a macrostate with contact parameter q ′ reads Z 1 p(q ′) = hδ(q ′ − q(X))i = DX δ(q ′ − q(X)) e−EG¯oL(X)/kB T , (9.6) Z

where Z denotes QN −1 partition function. The integral measure is QNthe 3unrestricted simply DX = i=1[d ri] i=1 [δ(rii+1 − 1)]. Expression (9.6) can be used to define a free energy as function of the q parameter by F (q) = −kB T ln p(q).

(9.7)

Since the value of q is a qualitative measure for the macrostate the system resides in, the minimum of the free energy at a given temperature T is related to the

9.3 G¯ o Kinetics of Two-State Folding

163

F (q) = −kB T ln p(q)

1.5 1.4

T = 0.340

1.3

T = 0.355

1.2

T = 0.365

1.1 1 0.9 0.8 0.7 0.2

0.3

0.4

0.5 0.6 0.7 q = n/ntot

0.8

0.9

1

Figure 9.5: Free-energy landscape F (q) of 20.6 for different temperatures close to the transition point.

dominant macrostate in the canonical ensemble at this temperature. Actually, as can be read off from Fig. 9.5, the folding transition of the heteropolymer with sequence 20.6 is a phase-separation process, i.e., at the transition point close to Tf ≈ 0.36, the folded and the denatured pseudophase coexist and the transition state barrier possesses a local maximum close to q ≈ 0.5, as expected for a typical two-state folding characteristics. A useful measure of structure formation is the radial distribution function and its dependence on temperature. Due to translational invariance in the threedimensional space, three degrees of freedom contribute only a volume factor to the partition function. Therefore, we utilize this by fixing the position of the first monomer, r1 = 0. Thus, we measure radial distances r of the other monomers with respect to the first one and we define the radial distribution function as + * N X 1 δ (|ri| − r) , (9.8) G(r) = 4πr2 i=3

where i = 2 is excluded as, by definition, the virtual covalent bonds are rigid and, R∞ 2 therefore, r12 = 1. Actually, from our definition, 4π 0 dr r G(r) = N − 2. In Fig. 9.6, the radial distribution functions of S are shown for (a) the lowest-energy conformation and the ensembles at different temperatures in the (b) AB model and (c) G¯oL model. Although the global-energy minimum conformation shown in Fig. 9.3 does obviously not form a regular crystal structure, the Lennard-Jones like interactions in combination with the rigid virtual bonds induce preferable sub-

164

9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

5 (a)

G(r) · 4πr2

4 3 2 1 2

G(r)/108 · r2

(b)

1

6 5 4 3 2 1

G(r)/108 · r2

(c)

1

1.5

2

2.5 r

3

3.5

4

Figure 9.6: Radial distribution functions of 20.6 (a) for the lowest-energy conformation (see Fig. 9.3), (b) at 16 temperatures in the interval T ∈ [0.1, 1.5] (curves from top to bottom) employing the AB model, and (c) at 18 temperatures in the interval T ∈ [0.1, 1.5] for the G¯oL model.

structures, favoring, e.g., bond angles of 60◦, 90◦, and 120◦ [55]. Therefore, the peaks of the radial distances from the first monomer for the global energy minimum conformation in Fig. 9.6(a) can be partly explained by these local segments. The first peak at r = 21/6 ≈ 1.12 is related to the minimum potential distance between two nonbonded A monomers (and the reference monomer at r1 = 0 is of type A). Also the location of other peaks can be deduced from Fig. 9.3 by similar geometric arguments.

9.3 G¯ o Kinetics of Two-State Folding

165

The temperature dependence of the peak evolution is shown for the AB model in Fig. 9.6(b) and for the G¯oL model in Fig. 9.6(c). As a first result, we see that there is nice coincidence not only in the location of the peaks, but also in the fact that their is no indication of intermediary, weakly stable conformations. Actually, the folding characteristics is similar in both models, at least from the thermodynamic point of view. At high temperatures, random-coil conformations dominate: There is no significant structuring in the radial distribution function. Passing the folding transition temperature, local, planar structures form first, before the tertiary, three-dimensional ordering towards the native conformation occurs.

9.3.3 Kinetics The advantage of the simple G¯oL model for two-state folding is that it enables likewise kinetic studies of folding and unfolding events. In fact, this is the main purpose of this knowledge-based model, because kinetics studies of physically motivated models are typically computationally extremely demanding. This is unfortunately also the case for folding studies employing the AB model. A striking example is shown in Fig. 9.7 where for a folding event the gray-scale coded average probability of native-contact formation is plotted for the original AB model [Fig. 9.7(a)] and the G¯oL model [Fig. 9.7(b)]. The average was taken over a “time” window [tMC − ∆t, tMC + ∆t] with ∆t = 1 000 MC steps. The darker a bar, the higher the probability that the associated contact is formed. As the simulation was carried out in the folding regime, the probability of native-bond formation increases with the number of total MC steps. It is, however, obvious that the folding is much slower using the AB model, whereas in the G¯oL model most of the native contacts have already been formed after 80 000 MC steps. The native conformation of the considered sequence possesses partly kind of zigzag structures, or “turns” (see Fig. 9.3). The folding of these segments is particularly simple and the probability of the formation of the associated native contacts of monomers i with monomers i + 2 or i + 3 increases much faster than for the other contacts. But even in this case, the G¯oL kinetics is unbeatable. This example exhibits the dilemma of physics-based models in studying kinetic aspects of structure formation by means of computer simulations. It is a notoriously difficult problem, because the time scale of molecular dynamics is typically too small to see folding events, but also Markov chain Monte Carlo dynamics of physical models is typically too slow. Just for kinetic aspects, where absolute time scales are widely irrelevant, knowledge-based G¯oL models are an alternative that allow for increasing the sampling efficiency, at least for systems with simple folding characteristics (as, for example, two-state folding). Actually, for

166

9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

(a)

Native Contacts

20000

40000

60000

18:20 16:18 14:18 14:16 9:20 9:11 6:20 6:9 3:20 3:18 3:16 3:6 2:20 2:18 2:16 2:14 2:11 2:9 2:6 1:20 1:18 1:14 1:11 1:9

80000 18:20 16:18 14:18 14:16 9:20 9:11 6:20 6:9 3:20 3:18 3:16 3:6 2:20 2:18 2:16 2:14 2:11 2:9 2:6 1:20 1:18 1:14 1:11 1:9

20000

40000 MC Steps

60000

80000

20000

40000

60000

80000

Native Contacts

(b) 18:20 16:18 14:18 14:16 9:20 9:11 6:20 6:9 3:20 3:18 3:16 3:6 2:20 2:18 2:16 2:14 2:11 2:9 2:6 1:20 1:18 1:14 1:11 1:9

18:20 16:18 14:18 14:16 9:20 9:11 6:20 6:9 3:20 3:18 3:16 3:6 2:20 2:18 2:16 2:14 2:11 2:9 2:6 1:20 1:18 1:14 1:11 1:9 20000

40000 MC Steps

60000

80000

Figure 9.7: Gray-scale coded averages of the probabilities for native-contact formation of sequence 20.6 as a function of Monte Carlo time for the (a) AB and the (b) G¯oL model, averaged over 1 000 folding events. The probability is calculated as a combined temporal average over 2 000 MC steps in the ensemble of the 1 000 folding events. The darker the bars are, the higher is the probability that the associated contact has formed. The labels of the pair contacts refer to the numbering in the native conformation shown in Fig. 9.3.

such models an adequate statistical sampling of ensembles close to the transition state can be achieved by sophisticated Monte Carlo methods where efficiency is typically gained by simulating generalized ensembles at the expense of an artificial

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167

Figure 9.8: Single folding event for the heteropolymer with sequence 20.6 in the G¯oL model at T = 0.3 < Tf ≈ 0.36. The description of the gray scale code is given in the text.

dynamics. Therefore, it is widely believed that free-energy-driven dynamics, as it is relevant for protein folding, can be provided reasonably only by BoltzmannMarkov chains of conformational updates. One possibility to achieve this is the application of a conventional Metropolis Monte Carlo method. Although the overall time scale is left open, this method allows for comparative folding (unfolding) studies at various temperatures. The kinetics of the folding (unfolding) transitions is thereby obtained by averaging over sufficiently many folding (unfolding) trajectories. In Fig. 9.8, snapshots of a single G¯oL folding event of the heteropolymer with sequence 20.6 at T = 0.3 are shown at different times tMC . The gray scale of the monomers encodes the variance of the monomer positions, σr2i = r2i −ri 2 , averaged over the MC time interval [tMC − 1 000, tMC + 1 000]. The first monomer is fixed and, therefore, not moving. The higher the mobility of a monomer in this time interval, the brighter is its color. For σr2i < 0.1, the monomer is rendered in black, and for σr2i > 1 in white. The gray scales are linearly interpolated in-between these

9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

q = n/ntot

168

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Q

0.9

0.8

5 E

0 -5 -10 -15

0

8000

16000

24000 32000 40000 Monte Carlo Steps

48000

56000

Figure 9.9: Temporal averages of the native-contact and the overlap similarity parameters q and Q, respectively, and the energy for the folding event shown in Fig. 9.8. The temporal averages are calculated every 4000 MC steps over the time interval [tMC − 1 000, tMC + 1 000].

boundaries. Although there are periods of relaxation and local unfolding, a stable intermediate conformation is not present and the folding process is a relatively “smooth” process. This is also confirmed by the more quantitative analysis of the same folding event in Fig. 9.9, where the temporal averages of the similarity parameters q and Q, and of the energy E are shown. Since the temperature lies sufficiently far below the folding temperature (Tf ≈ 0.36), the free-energy landscape does not exhibit substantial barriers which hinder the folding process. Nonetheless, the chevron plot shown in Fig. 9.10 exhibits a rollover which means that the folding characteristics is not perfectly of two-state type, in which case the folding (unfolding) branches would be almost linear [179]. In this plot, the temperature dependence of the mean first passage time τMFP is presented. We define τMFP as the average number of MC steps that are necessary to form at least 13 native contacts in the folding simulations, starting from a random

9.3 G¯ o Kinetics of Two-State Folding

169

-8.0

− ln τMFP

-9.0 -10.0 -11.0 -12.0 -13.0 -3.6

-3.4

-3.2

-3.0

-2.8 −1/T

-2.6

-2.4

-2.2

-2.0

Figure 9.10: Chevron plot of the mean-first passage times from folding (•) and unfolding (◦) events at different temperatures. The hypothetic intersection point corresponds to the transition state.

conformation. In the unfolding simulations, we start from the native state and τMFP is the number of MC steps required to reach a conformation with less than 7 native contacts, i.e., 13 native contacts are broken. In all simulations performed at different temperatures, τMFP is averaged from the first passage times of a few hundred respective folding and unfolding trajectories. Assuming a linear dependence at least in the transition state region, τMFP is directly related f,u to exponential folding and unfolding rates kf,u ≈ 1/τMFP ∼ exp(−εf,u/kB T ), respectively, where the constants εf,u determine the kinetic folding (unfolding) propensities. The dashed lines in Fig. 9.10 are tangents to the logarithmic folding and unfolding curves at the transition state. The slopes are the folding (unfolding) propensities and have in our case values of εf ≈ −1.32 and εu ≈ 5.0. In this variant of the chevron plot, which is similar to the presentations discussed in Refs. [179,185], the temperature T mimics the effect of the denaturant concentration that is in experimental studies the more generic external control parameter. The hypothetic intersection point of the folding and unfolding branches defines the transition state. The transition state temperature estimated from this analysis coincides very nicely with the folding temperature Tf ≈ 0.36 as identified in our discussion of the thermodynamic properties of the system. This result also demonstrates that the description of the folding and unfolding transitions from the kinetic point of view is not only qualitatively, but even quantitatively consistent with the thermodynamic results from the canonical-ensemble analysis.

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9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

9.3.4 Mesoscopic Heteropolymers vs. Real Proteins It is not the purpose of this coarse-grained analysis to explain the two-state folding characteristics of a specific, real protein. Rather, we have shown that – although also only exemplified for a single sequence – it is actually useful to study thermodynamic and kinetic properties of mesoscopic peptide models without introducing atomic details. The main focus of such models is pointed towards general features of the folding transition (measured in terms of “order” parameters being specific for the corresponding transition, such as, e.g., the contact and overlap parameters investigated in this analysis) that are common to a number of proteins behaving qualitatively similarly. It is then furthermore assumed that these proteins can be grouped into classes of certain folding characteristics. The sequence 20.6 considered here is not obtained from a one-to-one hydrophobic-polar transcription of a real amino acid sequence. Such a mapping is not particularly useful. Rather, the heteropolymer 20.6 is considered as a representative that exhibits two-state folding characteristics in the coarse-grained model described here. This implies, in general, that the classification of peptide folding behaviors is not necessarily connected with detailed atomic correlations and particular contact-ordering. It is rather an intrinsic property of protein-like heteropolymers and can thus already be discovered by employing models on mesoscopic scales [54,55,186]. Several proteins are known to be two-state folders and their folding transitions exhibit the features we have also seen in our present coarse-grained model study. A famous example is chymotrypsin inhibitor 2 (CI2) [167], one of the first proteins, where two-state folding characteristics has experimentally been identified. Clear signals of a first-order-like folding-unfolding transition were also seen in computational G¯o model analyses of that peptide [174,179]. It is clear, however, that a precise characterization of the transition state ensemble, which is required for a better understanding of the folding (or unfolding) process of a specific peptide (e.g., secondary-structure formation or disruption in CI2 [178]), is not possible. In the models used in our study, for example, only tertiary folding aspects based on hydrophobic-core formation are considered as being relevant. Nonetheless, as shown in our investigation of sequence 20.6, macroscopic quantities or cooperativity parameters manifest a qualitatively similar behavior of the heteropolymer considered here, compared to real two-state folders.

9.4 Microcanonical Effects and Definition of Temperature The folded structure of a functional bioprotein is thermodynamically stable under physiological conditions, i.e., thermal fluctuations do not lead to significant globular conformational changes. To force tertiary unfolding requires an activation

9.4 Microcanonical Effects and Definition of Temperature

171

energy that is much larger than the energy of the thermal fluctuations. This activation barrier can be drastically reduced by the influence of other proteins like prions. The Creutzfeld-Jakob disease is an example for the disastrous consequences prion-mediated degeneration of proteins can cause in the brain. The folded structure and the statistical ensemble of native-like structures, which are morphologically identical to the native fold, form a macrostate. It represents a conformational phase which is energy-dominated. Functionality of the native structure is only assured if entropic effects are of little relevance. A significant change of the environmental conditions such as temperature, pH value, or, following the above mentioned example, the prion concentration, can destabilize the folded phase. Entropy becomes relevant, the entropic contribution to the free energy starts dominating over energy. Consequently, the hydrophobic core decays. This does not necessarily lead to a globular unfolding of the protein. A rather compact intermediate conformational phase can be stable [27]. However, further imbalancing the conditions will finally lead to the phase of randomly unstructured coils. The latter transition is often called “folding/unfolding transition”, whereas the hydrophobic core formation is referred to as “glassy transition”, as unresolvable competing energetic effects may result in frustration. The primary structure, i.e., the sequence of different amino acids lining up in proteins, is already sort of quenched disorder. Simple examples for these transitions accompanying peptide folding processes have already been discussed in the analyses of Figs. 7.2 and 8.6 for an exemplified lattice and an off-lattice heteropolymer, respectively. Obviously, the peak temperatures of the specific heat CV and the fluctuations of end-to-end distance and radius of gyration do not perfectly coincide, although all fluctuating quantities clearly signalize the transitions. This is a typical indication for the finiteness of the system. There are no transition points in protein structure-formation processes, but rather transition regions (shaded areas in Fig. 7.2 and 8.6). This separates conformational transitions of finite-length polymers (pseudophase transitions) from thermodynamic phase transitions being considered in the thermodynamic limit. The smallness of such systems can cause surprising side-effects in nucleation processes which protein folding belongs to. Since the formation of the solventaccessible hydrophilic surface and the bulky hydrophobic core is crucial for the whole tertiary folding process, the competition between surface and volume effects significantly influences the thermodynamics of nucleation. For this reason, it is not obvious at all, which statistical ensemble represents the appropriate frame for the thermodynamic analysis of folding processes. This is even more intricate as one may think. It is, for example, quite common to interpret phase transitions by

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9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

means of fluctuating quantities calculated within the canonical formalism. Transition points are characterized by divergences in the fluctuations (second-order phase transitions) or entropy discontinuities (first-order transitions), occurring at unique transition temperatures. This standard analysis is based on the assumption that the temperature is a well-defined quantity, as it seems to be an easily accessible control parameter in experiments. This assumption is true for very large systems (N → ∞) with vanishing surface/volume ratio in equilibrium, where surface fluctuations are irrelevant. The microcanonical Hertz entropy S(E) = ln G(E) RE (with kB = 1 in our units), where G(E) = Emin dE ′ g(E ′) is the integrated density of states, is a concave function and thus the microcanonical temperature, defined by the mapping F : E 7→ T =: T (E) = [∂S(E)/∂E]−1, never decreases with increasing energy E. A discrimination of the parameter “temperature” in the canonical ensemble and the microcanonical (caloric) temperature T (E) is not necessary, as energetic fluctuations vanish and thus the canonical and the microcanonical ensemble are equivalent in the thermodynamic limit. But what if surface fluctuations are non-negligible? In this case, the canonical temperature can be a badly defined control parameter for studies of nucleation transitions with phase separation.2 This becomes apparent in the following microcanonical folding analysis of the hydrophobic-polar heteropolymer sequence 20.6 (cf. Table 8.4), employing the AB model. From multicanonical computer simulations, an accurate estimate of the density of states g(E) can be obtained. For this particular heteropolymer, it turns out that the entropy S(E) exhibits a convex region, i.e., a tangent with two touching points, at Efold and Eunf > Efold , can be constructed. This socalled Gibbs hull is then parametrized by H(E) = S(Efold) + E/Tfold, where Tfold = [∂H(E)/∂E]−1 = [(∂S(E)/∂E)Efold,Eunf ]−1 is the microcanonically defined folding temperature, which is here Tfold ≈ 0.36. As shown in Fig. 9.11, the difference S(E) − H(E) has two zeros at Efold and Eunf , and a noticeable well in-between with the local minimum at Esep . At this point the deviation S(Esep) − H(Esep) is called surface entropy as the convexity of the entropy in this region is caused by surface effects. However, the most striking feature in Fig. 9.11 is the qualitative change of the microcanonical temperature T (E) in the transition region: Approaching from small energies (folded phase), the curve passes the folding temperature Tfold and follows the overheating branch. It then decreases with increasing energy (passing again Tfold ) before following the undercooling branch in reverse direction, crossing Tfold for the third time. In the unfolded phase, temperature and energy increase 2

Folding or “nucleation” processes of proteins are strongly dependent on the sequence of amino acids. Thus, folding is no generic phase transition and terms like “nucleation” should be used with some care.

9.4 Microcanonical Effects and Definition of Temperature

173

0.50

0.50

S=H -0.50 S(E) − H(E)

0.40

-1.00

Tfold 0.35

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S(E) − H(E)

T (E)

0.45

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T (E) 0.30

-2.50 0.25 -12

-10

-8 -6 Efold

-4

-2

0

Esep

Eunf

2

-3.00 E

Figure 9.11: Folding transition as a phase-separation process for sequence 20.6 in the AB model. In the transition region, the caloric temperature T (E) of the protein decreases with increasing total energy. Folded and unfolded conformations coexist at the folding transition temperature, where S(E) = H(E), corresponding to Tfold ≈ 0.36. The energetic transition region is bounded by the energies Efold ≈ −6.3 and Eunf ≈ 0.15. Folding and unfolding regions are separated at Esep ≈ −2.0.

as expected. The unusual backbending of the caloric temperature curve within the transition region is not an artifact of the theory. It is a physical effect and has been confirmed in sodium cluster formation experiments [187], where a similar behavior was observed.3 In Fig. 9.12, results from the canonical calculations (mean energy hEi and specific heat per monomer cV ) are shown as functions of the temperature. The specific heat exhibits a clear peak near T = 0.35 which is close to the folding temperature Tfold , as defined before in the microcanonical analysis. The loss of information by the canonical averaging process is apparent by comparing hEi and the inverse, non-unique mapping F −1 of microcanonical temperature and energy. The temperature decrease in the transition region from the folded to the unfolded structures is unseen in the plot of hEi. Eventually, as we had already mentioned, there is also no unique canonical folding temperature signaled by peaks of fluctuating quantities; there are rather transition regions. Therefore, for small systems, the definition of transitions based on the canonical temperature is indeed little useful and the interpretation 3

It is sometimes argued that proteins fold in solvent, where the solvent serves as heat bath. This would provide a fixed canonical temperature such that the canonical interpretation is sufficient to understand the transition. However, the solvent-protein interaction is actually implicitly contained in the heteropolymer model and, nonetheless, the microcanonical analysis reveals this effect which is simply “lost” by integrating out the energetic fluctuations in the canonical ensemble (see Fig. 9.12).

174

9. Characteristic Glassy Folding Channels and Kinetics of Two-State Folding

10

5.5 5.0 4.0

[F : E 7→ T ]−1

0

3.5 3.0

hEi(T )

2.5

-5

cV

hEi, E

4.5

cV (T )

5

2.0 1.5

-10

1.0 0.5

-15 0.20

0.25

0.30 0.35 Tfold

0.40 T

0.45

0.50

0.55

0.0 0.60

Figure 9.12: Comparison of canonical and microcanonical analysis for the folding transition of 20.6: The fluctuations of energy as represented by the specific heat per monomer, cV (T ), exhibit a sharp peak near T ≈ 0.35. The canonical mean energy hEi(T ) crosses over from the folded to the unfolded conformations. However, the canonical calculation averages out the overheating/undercooling branches and the backbending effect which are clearly signaled by the microcanonical analysis of the (inverse) mapping between temperature and energy. The microcanonically defined “folding temperature” Tfold is close to the peak temperature of the specific heat.

of these cooperative effects as thermodynamic phase transitions has noticeable limitations. This not only regards the impossibility to precisely identify definite transition points. It is even more fundamental to ask the question which of the typically used statistical ensembles provides the most comprehensive interpretation of finite-system structure formation processes. We have shown that the microcanonical analysis of folding thermodynamics is particularly advantageous, as, e.g., the remarkable temperature backbending effect is averaged out in the canonical ensembles, where the temperature is considered as an external control parameter which seems to be questionable for small systems. As we will see in Chapter 11, this also applies to molecular aggregation, where similar phenomena can occur. Up to now, we have mainly investigated tertiary conformational transitions such as polymer collapse and crystallization, and also protein folding, which all require a cooperative behavior of the monomers on a lengths scale that corresponds to the chain length. We will now turn to effects on smaller (but still mesoscopic) scales that entail the formation of local symmetries. This class of symmetries is known as the secondary structures.

10 Generic Geometries of Strings with Constraints

10.1 The Intrinsic Nature of Secondary Structures Resolving structural properties of single molecules is a fundamental issue as molecular functionality strongly depends on the capability of the molecules to form stable conformations. Experimentally, the identification of substructures is typically performed, for example, by means of single-molecule microscopy, X-ray analyses of polymer crystals, or NMR for polymers in solution. With these methods, structural details of specific molecules are identified, but these can frequently not be generalized systematically with respect to characteristic features being equally relevant for different polymers. Therefore, the identification of generic conformational properties of polymer classes is highly desirable. The to-date most promising approach to attack this problem is to analyze polymer conformations by means of comparative computer simulations of polymer models on mesoscopic scales, i.e., by introducing relevant cooperative degrees of freedom and additional constraints. In these modeling approaches – we have already made use of it in the previous chapters – the linear polymer is considered as a chain of beads and springs. Monomeric properties are accumulated in an effective, specifically parametrized single interaction point of dimension zero (“united atom approach”). Noncovalent van der Waals interactions between pairs of such interaction points are typically modeled by Lennard-Jones (LJ) potentials. In such models, only the repulsive short-range part of the LJ potentials keeps the monomers pairwisely apart. Although such models have proven to be quite useful in identifying universal aspects of global structure formation processes, these approaches are less useful in this form to describe local symmetric arrangements of segements of the chain. For the identification of underlying secondary structure segments like helices,

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strands, and turns, however, the modeling of volume exclusion by means of pure LJ pair potentials is not sufficient to form clearly distinct secondary structures enabling a classification scheme. Segments of such secondary structures were found, e.g., in dynamical LJ polymer studies of transient states occurring in the collapse process [188] or as ground states in models with stiffness [189], explicit hydrogen bonding [190,191], or explicit solvent particles [192,193]. It could also be shown that helical structures form by introducing anisotropic monomer-monomer potentials in conjunction with a wormlike backbone model [194] or by combining excluded volume and torsional interactions [195]. The formation of secondary structures requires cooperative behavior of adjacent monomers, i.e., in addition to pairwise repulsion, information about the relative position of the monomers to each other in the chain is necessary to effectively model the competition between noncovalent monomeric attraction and short-range repulsion due to volume exclusion effects [196]. The simplest way to achieve this in a general, mesoscopic model is to introduce a hard single-parameter thickness constraint and, thus, to consider a polymer chain rather as a topologically three-dimensional tubelike object than as a one-dimensional, linelike string of monomers [197,198]. This approach differs from frequently studied cylindrical tube models [199], in which case the tube thickness rather mimics volume exclusion but not cooperativity such that explicit modeling of hydrogen bonds is required to generate and stabilize secondary structures. In the following, we use the global radius of curvature as a quantity that effectively includes many-body interactions among the monomers. Defining a lower bound will allow for introducing a thickness constraint. Then, the polymer behaves like a tube. Variations of thickness and temperature will enable us to classify the geometric secondary phases classes of short polymers (or segments of larger polymers) can reside in [200–202].

10.2 Polymers with Thickness Constraint 10.2.1 Global Radius of Curvature A natural choice for parametrizing the thickness of a polymer conformation with N monomers, X = (x1 , . . . , xN ), is the global radius of curvature rgc [203]. It is defined as the radius rc of the smallest circle connecting any three different monomer positions xi , xj , xk (i, j, k = 1, . . . , N ): rgc(X) = min{rc (xi, xj , xk ) ∀i, j, k | i 6= j 6= k}.

(10.1)

10.2 Polymers with Thickness Constraint

177

2 1 rc (x2 , x10 , x9 ) 9 8

10

rc (x8 , x9 , x10 )

Figure 10.1: Two examples of circles with different radii of curvature. The small circle is defined by the three consecutive monomers 8,9, and 10 in which case its radius is called the local radius of curvature.

Denoting the distance between two points by rij = |xi − xj | and the area of the triangle, spanned by any three points, by A∆ (xi, xj , xk ), rc is given as rij rjk rik . (10.2) rc = 4A∆(xi , xj , xk ) This is illustrated in Fig. 10.1, where two exemplified circles with their associated radius of curvature are depicted. Eventually, with these definitions, the polymer tube X has the “thickness” (or diameter) d(X) = 2rgc(X) [203,204]. We consider linear, flexible polymers with stiff bonds of unit length (ri i+1 = 1) and pairwise interactions among nonbonded monomers are modeled by a standard LJ potential. Thus, the energy of a conformation X reads X E(X) = VLJ (rij ), (10.3) i,j>i+1

where as usual VLJ (rij ) = 4ε[(σ/rij )12 − (σ/rij )6]. By setting σ = 1, VLJ (rij ) min vanishes for rij = 1 and is minimal at rij = 21/6 ≈ 1.122. Since we are interested in classifying conformational pseudophases of polymers with respect to their thickness, it is useful to introduce the restricted conformational space Rρ = {X | rgc(X) > ρ} of all conformations X with a global radius of curvature larger than a thickness constraint ρ, which can be understood as an effective measure for the extension of the polymer side chains. Given ρ, obviously only conformations with rgc ≥ ρ can occur. The canonical partition function of the restricted conformational space thus reads Z Zρ = DX Θ(rgc(X) − ρ)e−E(X)/kB T , (10.4)

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where kB T is the thermal energy (we use units in which ε = kB = 1 in the following) and Θ(z) is the Heaviside function. In this thickness-restricted space, canonical statistical averages of any quantity O are then calculated via hOiρ = R −1 Zρ DX O(X)Θ(rgc(X) − ρ) exp[−E(X)/kB T ]. It must be emphasized that the detailed thermodynamic analysis aiming at the entire structural phase diagram requires precise datasets that can only be obtained by means of sophisticated generalized-ensemble methods such as parallel tempering [182–184], multicanonical sampling [83–85], and the Wang-Landau method [86]. In order to identify the main features, we shall focus in the following on the tube polymer with N = 9 monomers [200] as most of the results obtained for this system can be generalized and are thus also common to the longer chains [201,202]. It should also be noted at this place that this consideration does not make much sense for much longer chains as large secondary structures in single molecules (such as proteins) become unstable and a rearrangement of secondary-structure segments in favor of tertiary alignments occurs.

10.2.2 Thickness-Dependent Ground-State Properties Figure 10.2 shows the ground-state energy per monomer as a function of ρ. Also shown are lowest-energy conformations for exemplified values of ρ. For the linelike 9mer (i.e., ρ = 0) in our units Emin/N = −1.85. The thickness constraint min becomes relevant, if ρ is larger than half the characteristic length scale rij of the LJ potential: In the interval 2−5/6 ≈ 0.561 < ρ < ρα ≈ 0.686 conformations are pre-helical. The nonbonded interaction distance is still allowed to be so small that structures are deformed. Nonetheless, the onset of helix formation is clearly visible. Optimal space-filling helical symmetry is reached when approaching ρα , where the ground-state conformation takes the perfect α-helical shape (see inset of Fig. 10.2). All torsional angles are identical (near 41.6◦) and also all local radii are constant; the number of monomers per winding is 3.6. Note that for proteins, where the effective distance between two Cα atoms is about 3.8 ˚ A, ρα in our units ˚ corresponds indeed to a pitch of about 5.4 A as known from α-helices of proteins. Thus, an α-helix is a natural geometric shape for tubelike polymers. Hydrogen bonds stabilize these structures in nature – but are not a necessary prerequisite for forming such secondary structures. This is indeed a substantial result as it shows that helical secondary structures form a subset of dominant basic geometries of stringlike objects with thickness. In other words, we can expect that there is a helical phase which, in dependence on thickness and external parameters such as the temperature, is stable for a whole class of polymers with different molecular compositions. For larger values of ρ, helices unwind, i.e., the pitch gets larger and the number

10.2 Polymers with Thickness Constraint

179

0 -0.2 -0.4

Emin /N

-0.6

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-0.8 -1 -1.2

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1.5

1.6

1.7

ρ

ρ = 0.60

0.69

0.73

0.78

0.88

0.92

1.20

Figure 10.2: Ground-state energy per monomer Emin /N of tubelike polymers with nine monomers as a function of the global radius of curvature constraint ρ (solid line). For comparison, also the energy curve of the perfect α-helix is plotted (dashed line). The inset shows that for a small interval around ρ ≈ 0.686, the ground-state structure is perfectly α-helical. Also depicted are side and top views of putative ground-state conformations for various exemplified values of ρ. For the purpose of clarity, the conformations are not shown with their natural thickness.

of monomers per winding increases. However, helical structures still dominate the ground-state conformations. It should be noted that the simple polymer model is energetically invariant under helicity reversal, i.e., left-handed helices or segments are not explicitly disfavored and are, therefore, also equally present in the conformational space. In the interval ρα ≤ ρ . 0.92, there are also stable helical conformations in the vicinity of ρ ≈ 0.73 (winding number ≈ 4.5) and ρ ≈ 0.78 (winding number ≈ 5.0). Near ρ ≈ 0.92, the final helical state has been reached. The thickness has increased in such a way that the most compact conformation is a helix with a single winding. After that, a topological change occurs and the ground-state conformations are getting flatter. The helix finally opens up and planar conformations with similarities to β-hairpins become dominant. These structures are still stabilized by nonbonded LJ interactions between pairs of monomers. Increasing the thickness further leads to a breaking of these contacts and ringlike conformations become relevant [204]. For values

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rgc ≈ N/2π, ground-state conformations are almost perfect circles with radius rgc . The existence of ringlike conformations is a consequence of the long-range monomer-monomer attraction. Eventually, for ρ → ∞, the effective stiffness increases, also the end contacts disappear, and only thick rods are still present. After these preparatory considerations of ground-state properties, we are now going to discuss the thermodynamic behavior of the tube polymers.

10.2.3 Structural Phase Diagram of Tubelike Polymers Based on the peak structure of the specific heat as a function of temperature T and thickness constraint ρ, we are now going to identify the structure of the conformational ρ-T (pseudo)phase diagram. This requires a very precise statistical analysis. Even for such a small polymer with 9 monomers, hundreds of separate generalized-ensemble computer simulations had to be performed [200,202]. The phase-diagram topology turns out to be general and valid for larger polymers, too [202]. Since even the expected shifts of the transition lines due to finite-length corrections are very small, one has good reason to assume that the pseudophase diagram of the 9mer reflects the general phase structure of short tubelike polymers pretty well. This is partly due to the fact that the polymer thickness as defined via the global radius of curvature is a length-independent constraint and the chains are short enough to prevent the formation of tertiary structures (as, e.g., arrangements of different secondary-structure segments forming a tertiary domain). For longer chains, however, tertiary structures are definitely relevant. For the chain with N = 13 monomers, first indications of structure formation on globular length scales were found [202]. Figure. 10.3 shows the specific-heat landscape CV (ρ, T ) = (hE 2 iρ − hEi2ρ )/T 2 for a 9mer as obtained from reweighting the density of states for given thickness constraints ρ. This profile therefore represents the structure of the conformational phase diagram for a tubelike polymer in the perspective of secondary structures. Dark regions correspond to strong energetic fluctuations, i.e., the darker the region the larger is the specific-heat value. Data points (+) mark the peaks or ridges of the profile and indicate conformational activity and thus represent transitions between different conformational pseudophases. Error bars are not shown for clarity but are sufficiently small (for most data points smaller than symbol size), so that the identified phase boundaries are statistically significant. Guided by the analyses of the ground-state properties, one can identify four principal pseudophases1. In region α, helical conformations are the most relevant 1

We note that there are singular points in the parameter space corresponding √ to special geometric representations of secondary structures. For the chain with length N = 8 and rgc ≈ 1/ 2, for example, the degenerate ground-state conformation exhibits an almost perfect alignment of the chain along the edges of a cube.

10.2 Polymers with Thickness Constraint

181

CV (ρ, T )

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ρ

Figure 10.3: (a) Perspective and (b) projected view of the specific-heat profile CV (ρ, T ) for a 9mer which is interpreted as structural phase diagram of thermodynamically relevant tube polymer conformations in thickness–temperature parameter space. Dark regions and data points (+) indicate the ridges of the landscape and separate conformational phases. Helical or helixlike conformations dominate in region α, sheets in region β, rings in region γ, and stiff rods in pseudophase δ. Circles (⊙) indicate the locations where the exemplified conformations of Table 10.1 are relevant. The general structure of the phase diagram in the secondary-structure sector remains widely unchanged also for the longer polymers.

structures. In particular, the α-helix resides in this pseudophase. Characteristic for the transition from pseudophase α to β is the unwinding of the helical structures which are getting more planar. Thus, region β is dominated by simple sheet-like structures. Since the 9mer is rather short, the only sheet-like class of conformations is the hairpin. For longer chains, one also finds more complex sheets, e.g., lamellar structures [197,201,202]. A characteristic property of the hairpins is that these are still stabilized by nonbonded interactions. These break with larger thickness and higher temperature. Entering pseudophase γ, dominating structures possess ringlike shapes. Finally, region δ is the phase of random coils, which are getting stiffer for large thickness and eventually resembling rods. Represen-

182

10. Generic Geometries of Strings with Constraints

Table 10.1: Exemplified conformations being thermodynamically relevant in the respective pseudophases shown in Fig. 10.3, visualized in different representations. phase

type

α

helix

β

sheet

γ

ring

δ

rod

views of representative example

tative polymer conformations dominating the pseudophases in the regions α to δ are depicted in Table 10.1 in different representations.

10.3 Secondary-Structure Phases of a Hydrophobic-Polar Heteropolymer Model Proteins form the most prominent class of polymers, where different types of secondary structures occur, typically within the same molecule. It is therefore useful to extend the homopolymer model employed in the previous section by intoducing two types of monomers which can be hydrophobic or polar. For this purpose, we will now combine the already introduced and for the linelike heteropolymers in Chapter 8 extensively studied AB model with the thickness constraint. In order to get an impression of the effects caused by introducing a sequencedependent model, we here consider the exemplified Fibonacci sequence F1 (AB2 AB2ABAB2 AB). The ground-state properties have already been discussed in the linelike AB model in Section 8.3. The linelike case corresponds to the constraint ρ = 0 (no restriction of the conformational space). Figure 10.4 shows the structural phase diagram analogously to Fig. 10.3, as well as selected groundstate conformations. The general structure including the several separated struc-

10.3 Secondary-Structure Phases of a Hydrophobic-Polar

Heteropolymer Model

183

0.2

CV (ρ, T )

T

γ β

δ

0.1

α

δ

0

α

β

γ

1.2 1.1

0 0.6

0.7

0.8

0.9

1.1

ρ = 0.7

0.8

0.1

1.0 0.9

1.2

ρ

a)

c)

1.0

0.05

ρ

0.15

0.8

T

0.7

b) 4 3 2 1

4 3 2 1

9 10 11 12

9 10 11 12

0.9

1.0

1.1

1.2

Figure 10.4: Structural phase diagram for the N = 13 Fibonacci AB heteropolymer sequence F1 (see Table 8.1). (a) Top-view with marked peak positions of the specific heat for various parameters ρ, (b) qualitative view of the specific-heat landscape. Gray scales encode the value of the specific heat, (c) exemplified ground-state conformations shown from different viewpoints, A monomers are marked in red (dark gray), B monomers are white.

tural subphases is similar to that for the homopolymer. The most noteworthy section of the phase diagram is the very stable β-sheet region in the interval 0.90 ≤ ρ ≤ 1.01, as T → 0. The dominant conformations are indeed “planar” (see low-energy conformations depicted in Fig. 10.4). and the qualitative properties do not change in the entire region. A quantitatively remarkable fact is the variation of the intra-monomer distances. We note, that the interaction length between the opposite hydrophobic A monomers 1-12 (r1,12 = 1.13, see Fig. 10.4) for monomer numbering) and 4-9 (r4,9 = 1.15) in the sheet conformation does not change in the whole region of thickness constraint at all. On the other hand, the distances of B monomer pairs 2-11 and 3-10 increase (∆r2,11 = ∆r3,10 = 0.27) and decrease between the A monomers 1-4 and 9-12 (∆r1,4 = ∆r9,12 = −0.10; cp. the conformations at ρ = 0.9 and ρ = 1.0). The van-der-Waals attraction between the A monomers is thus responsible for the stabilization of the β-sheet structures. For smaller thickness constraints, structures with helical properties are found, which depends on the monomer sequence. We note here a very pronounced conformational transition from random coils to native conformations in the temperature interval 0.1 . T . 0.15. With increasing thickness the ground-state conformation becomes ringlike and finally switches to a stretched rod.

184

10. Generic Geometries of Strings with Constraints

To conclude, the tube constaint concept is well suited to mimic the manybody volume extension of polymers with extended side chains like, for example, the residual groups of amino acids in proteins. After the discussion of single-chain properties, we are now going to investigate thermodynamic properties of the formation of structures of several individual chains. In these systems, single-chain structuring processes will compete with cooperative effects among different chains, which finally leads to another type of structural transition, the aggregation or clustering of polymers.

11 Statistical Analyses of Aggregation Processes

11.1 Pseudophase Separation in the Nucleation of Polymers Beside receptor-ligand binding mechanisms, folding and aggregation of proteins belong to the biologically most relevant molecular structure formation processes. While the specific binding between receptors and ligands is not necessarily accompanied by global structural changes, protein folding and oligomerization of peptides are typically cooperative conformational transitions [205]. Proteins and their aggregates are comparatively small systems. A typical protein consists of a sequence of some hundred amino acids and aggregates are often formed by only a few peptides. A very prominent example is the extracellular aggregation of the Aβ peptide, which is associated with Alzheimer’s disease. Following the amyloid hypothesis, it is believed that these aggregates (which can also take fibrillar forms [206]) are neurotoxic, i.e., they are able to fuse into cell membranes of neurons and open calcium ion channels. It is known that extracellular Ca2+ ions intruding into a neuron can promote its degeneration [207–209]. Conformational transitions proteins experience during structuring and aggregation are not phase transitions in the strict thermodynamic sense and their statistical analysis is usually based on studies of signals exposed by energetic and structural fluctuations, as well as system-specific “order” parameters. In these studies, the temperature T is considered as an adjustable, external control parameter and, for the analysis of the pseudophase transitions, the peak structure of quantities such as the specific heat and the fluctuations of the gyration tensor components or “order” parameter as functions of the temperature are investigated. The natural ensemble for this kind of analysis is the canonical ensemble, where the possible states of the system with energies E are distributed according to the Boltzmann probability exp(−E/kB T ), where kB is the Boltzmann constant. However, phase

186

11. Statistical Analyses of Aggregation Processes

separation processes of small systems as, e.g., droplet condensation, are accompanied by surface effects at the interface between the pseudophases [210,211]. This is reflected by the behavior of the microcanonical entropy S(E), which exhibits a convex monotony in the transition region. Consequences are the backbending of the caloric temperature T (E) = (∂S/∂E)−1, i.e., the decrease of temperature with increasing system energy, and the negativity of the microcanonical specific heat CV (E) = (∂T (E)/∂E)−1 = −(∂S/∂E)2/(∂ 2S/∂E 2) [213]. The physical reason is that the free energy balance in phase equilibrium requires the minimization of the interfacial surface and, therefore, the loss of entropy [212–214]. A reduction of the entropy can, however, only be achieved by transferring energy into the system. It is a surprising fact that this so-called backbending effect is indeed observed in transitions with phase separation. Although this phenomenon has already been known for a long time from astrophysical systems [215], it has been widely ignored since then as somehow “exotic” effect. Recently, however, experimental evidence was found from melting studies of sodium clusters by photofragmentation [187]. Bimodality and negative specific heats are also known from nuclei fragmentation experiments and models [216,217], as well as from spin models on finite lattices which experience first-order transitions in the thermodynamic limit [218,219]. This phenomenon is also observed in a large number of other isolated finite model systems for evaporation and melting effects [220,221]. The following discussion of the aggregation behavior is based on multicanonical computer simulations of a mesoscopic hydrophobic-polar heteropolymer model for aggregation, which is based on the simple AB off-lattice model, originally introduced to study tertiary folding of proteins from a coarse-grained point of view [213,214].

11.2 Mesoscopic Hydrophobic-Polar Aggregation Model For studies of heteropolymer aggregation on mesoscopic scales, a novel model is employed that is based on the hydrophobic-polar single-chain AB model [22]. As for modeling heteropolymer folding, we assume here that the tertiary folding process of the individual chains is governed by hydrophobic-core formation in an aqueous environment. For systems of more than one chain, we further take into account that the interaction strengths between nonbonded residues are independent of the individual properties of the chains the residues belong to. Therefore, we use the same parameter sets as in the AB model for the pairwise interactions

11.2 Mesoscopic Hydrophobic-Polar Aggregation Model

187

between residues of different chains. Our aggregation model reads [213,214] E=

X µ

(µ)

EAB +

XX

Φ(riµjν ; σiµ , σjν ),

(11.1)

µ<ν iµ ,jν

where µ, ν label the M polymers interacting with each other, and iµ , jν index the Nµ,ν monomers of the respective µth and νth polymer. The intrinsic single-chain energy of the µth polymer is given by [cf. Eq. (1.21)] (µ)

EAB =

X 1X (1 − cos ϑiµ ) + Φ(riµjµ ; σiµ , σjµ ), 4 i j >i +1 µ

µ

(11.2)

µ

with 0 ≤ ϑiµ ≤ π denoting the bending angle between monomers iµ , iµ + 1, and iµ + 2. The nonbonded inter-residue pair potential i h −6 −12 Φ(riµjν ; σiµ , σjν ) = 4 riµ jν − C(σiµ , σjν )riµ jν

(11.3)

depends on the distance riµ jν between the residues, and on their type, σiµ = A, B. The long-range behavior is attractive for like pairs of residues [C(A, A) = 1, C(B, B) = 0.5] and repulsive otherwise [C(A, B) = C(B, A) = −0.5]. The lengths of all virtual peptide bonds are set to unity. Here, we report on results obtained from statistical mechanics studies of the aggregation processes of short polymers by means of multicanonical simulations. Our primary interest is dedicated to the heteropolymer with the Fibonacci sequence F1 (AB2AB2 ABAB2AB) from Table 8.1, whose single-chain properties have already been listed in Tables 8.2 and 8.3 (see also Ref. [148]). In the following, we are going to study the thermodynamics of systems with up to 4 chains of this sequence over the whole energy and temperature regime. In the simulations, conformational changes of the individual chains included spherical updates [148] (see Section 8.1) and semilocal crankshaft moves, i.e., rotations around the axis between the nth and (n + 2)th residue. A typical multicanonical run contained of the order of 1010 single updates. The polymer chains were embedded into a cubic box with edge lengths L and periodic boundary conditions were used. In our simulations, the edge lengths of the simulation box were chosen to be L = 40 which is sufficient to reduce undesired finite-size effects. For cross-checks we have also performed replica-exchange (parallel tempering) simulations [182–184]. Verifying lowest-energy conformations found in the multicanonical simulations, we have also performed optimization runs using the energy-landscape paving (ELP) method [149].

188

11. Statistical Analyses of Aggregation Processes

11.3 Order Parameter of Aggregation and Fluctuations In order to distinguish between the fragmented and the aggregated regime, we introduce the “order” parameter M 1 X 2 d (rCOM,µ , rCOM,ν ) , Γ = 2M 2 µ,ν=1 per 2

(11.4)

where the summations are taken over the minimum distances dper = (1) (2) (3) dper , dper , dper of the respective centers of mass of the chains (or their periodic continuations). The center of mass of the µth chain in a box with periodic bound PN ary conditions is defined as rCOM,µ = iµµ=1 dper riµ , r1µ + r1µ /Nµ , where r1µ is the coordinate vector of the first monomer and serves as a reference coordinate in a local coordinate system. Our aggregation parameter is to be considered as a qualitative measure; roughly, fragmentation corresponds to large values of Γ, aggregation requires the centers of masses to be in close distance in which case Γ is comparatively small. Despite its qualitative nature, it turns out to be a surprisingly manifest indicator for the aggregation transition and allows even a clear discrimination of different aggregation pathways, as will be seen later on. According to the Boltzmann distribution, we define canonical expectation values of any observable O by M Z Y 1 DXµ O({Xµ })e−E({Xµ})/kB T , (11.5) hOi(T ) = Zcan (T ) µ=1 where the canonical partition function Zcan is given by Zcan (T ) =

M Z Y

µ=1

DXµ e−E({Xµ })/kB T .

(11.6)

Formally, the integrations are performed over all possible conformations Xµ of the M chains. Similarly to the specific heat per monomer cV (T ) = dhEi/Ntot dT = PM (hE 2 i − hEi2 )/NtotkB T 2 (with Ntot = µ=1 Nµ ) which expresses the thermal fluctuations of energy, the temperature derivative of hΓi per monomer, dhΓi/Ntot dT = (hΓEi − hΓihEi)/Ntot kB T 2 , is a useful indicator for cooperative behavior of the multiple-chain system. Since the system size is small – the number of monomers Ntot as well as the number of chains M – aggregation

11.4 Statistics of the Two-Chain Heteropolymer System in Three Ensembles

189

transitions, if any, are expected to be signalized by the peak structure of the fluctuating quantities as functions of the temperature. This requires the temperature to be a unique external control parameter which is a natural choice in the canonical statistical ensemble. Furthermore, this is a typically easily adjustable and, therefore, convenient parameter in experiments. However, aggregation is a phase separation process and, since the system is small, there is no uniform mapping between temperature and energy [213,214]. For this reason, the total system energy is the more appropriate external parameter. Thus, the microcanonical interpretation will turn out to be the more favorable description, at least in the transition region. We will discuss this in detail in the following section.

11.4 Statistics of the Two-Chain Heteropolymer System in Three Ensembles For the qualitative description of the aggregation and the accompanied conformational cooperativity within the whole system, it is sufficient to consider a very small system which is computationally reliably tractable and thus yields precise results for all energies and temperatures. Our heteropolymer system consists of two identical chains with the amino acid composition F1 and will be denoted as 2×F1. In the following, we discuss the aggregation behavior of this system from the multicanonical, the canonical, and the microcanonical point of view.

11.4.1 Multicanonical Results In a multicanonical simulation, the phase space is sampled in such a way that the energy distribution gets as flat as possible. Thermodynamically, this means that the sampling of the phase space is performed for all temperatures within a single simulation [83,84,87]. The desired information for the thermodynamic behavior of the system at a certain temperature is then obtained by simply reweighting the multicanonical into the respective canonical distribution, according to Eq. (4.33). Since the multicanonical ensemblee contains all thermodynamic informations, including the conformational transitions, it is quite useful to measure within the simulation the multicanonical histogram (cp. also Section 9.2) X (11.7) Hmuca (E0, Γ0 ) = δE,E0 δΓ,Γ0 , t

where t labels the Monte Carlo “time” steps. More formally, this distribution can be expressed as a conformation-space integral Hmuca (E0, Γ0) ∝ hδ(E − E0)δ(Γ − Γ0 )imuca =

190

11. Statistical Analyses of Aggregation Processes

=

1

M Z Y

Zmuca µ=1

DXµ δ(E({Xµ}) − E0 )δ(Γ({Xµ}) − Γ0 )

× exp [−Hmuca (E({Xµ}))/kB T ] ∝ e−Fmuca (E0 ,Γ0 )/kB T

(11.8)

with the multicanonical energy Hmuca (E) = E − kB T ln Wmuca(E; T ) which is independent of temperature. The multicanonical partition function is also trivially a constant in temperature, M Z Y (11.9) DXµ e−Hmuca (E({Xµ }))/kB T = constT . Zmuca = µ=1

It is obvious that integrating Hmuca (E, Γ) over Γ recovers the uniform multicanonical energy distribution: Z∞ 0

dΓ Hmuca (E, Γ) ∼ hmuca (E).

(11.10)

The canonical distribution of energy and Γ parameter at temperature T can be retained, similar to inverting Eq. (4.33), by performing the simple reweighting −1 (E; T ), Hcan (E, Γ; T ) = Hmuca (E, Γ)Wmuca

(11.11)

which is, due to the restriction to a certain temperature, less favorable to gain an overall impression of the phase behavior (i.e., the transition pathway) of the system, compared to the multicanonical analog Hmuca(E, Γ). In Fig. 11.1(a), Hmuca (E, Γ) is shown for the two-peptide system 2×F1 as a color-coded projection onto the E-Γ plane, which is the direct output obtained in the multicanonical simulation. Qualitatively, we observe two separate main branches (which are “channels” in the corresponding free-energy landscape), between which a noticeable transition occurs. In the vicinity of the energy Esep ≈ −3.15, both channels overlap, i.e., the associated macrostates coexist. Since Γ is an effective measure for the spatial distance between the two peptides, it is obvious that conformations with separated or fragmented peptides belong to the dominating channel in the regime of high energies and large Γ values, whereas the aggregates are accumulated in the narrow low-energy and small-Γ channel. Thus, the main observation from the multicanonical, comprising point of view is that the aggregation transition is a phase-separation process which, even for this small system, already appears in a surprisingly clear fashion. The high precision of the multicanonical method allows us even to reveal further details in the lowest-energy aggregation regime, which is usually a notoriously

11.4 Statistics of the Two-Chain Heteropolymer System in Three Ensembles

191

(a) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

-20.0

-15.0

-10.0

-5.0

0.0

5.0

16.0 14.0 12.0 10.0 8.0 Γ 6.0 4.0 2.0 0.0 10.0

E

(b) intermediate log10 Hmuca (E, Γ)

capped entangled

2.5 2.0 1.5

-18.0

1.0

-18.2

0.5

-18.4

0.0 0.9

0.8

0.7

0.6

0.5

E

0.4

Γ

Figure 11.1: (a) Multicanonical histogram log10 Hmuca as a function of energy E and aggregation parameter Γ, (b) section of log10 Hmuca in the low-energy tail [214].

difficult sampling problem. Figure 11.1(b) shows that the tight aggregation channel splits into three separate, almost degenerate subchannels at lowest energies. From the analysis of the conformations in this region, we find that representative conformations with smallest Γ values, Γ ≈ 0.45, are typically entangled, while those with Γ ≈ 0.8 have a spherically-capped shape. This is the subchannel connected to the lowest-energy states. Examples are shown in Fig. 11.2. The also highly compact conformations belonging to the intermediate subphase do not exhibit such characteristic features and are rather globules without noticeable internal symmetries. In all cases, the aggregates contain a single compact core of hydrophobic residues. Thus, the aggregation is not a simple docking process of two prefolded peptides, but a complex cooperative folding-binding process. This is a consequence of the energetically favored hydrophobic inter-residue con-

192

11. Statistical Analyses of Aggregation Processes

entangled

capped

Figure 11.2: Representatives and schematic characteristics of entangled and spherically-capped conformations dominating the lowest-energy branches in the multicanonical histogram shown in Fig. 11.1(b). Red spheres correspond to hydrophobic (A), light ones to polar (B) residues.

tacts which, as the results show, overcompensate the entropic steric constraints. The story is, however, even more interesting, as also non-negligible surface effects come into play. After the following standard canonical analysis, this will be discussed in more detail in the subsequent microcanonical interpretation of our results.

11.4.2 Canonical Perspective Phase transitions are typically described in the canonical ensemble with the temperature kept fixed. This is also natural from an experimentalist’s point of view, since the temperature is a convenient external control parameter. The macrostates are weighted according to the Boltzmann distribution (4.32). A nice feature of the canonical ensemble is that the temperature dependence of fluctuations of thermodynamic quantities is usually a very useful indicator for phase or pseudophase transitions. This cooperative thermodynamic activity is usually signalized by peaks or, in the thermodynamic limit (if it exists), by divergences of these fluctuations. Even for small systems, peak temperatures can frequently be identified with transition temperatures. Although in these cases peak temperatures typically depend on the fluctuating quantities considered, in most cases associated pseudophase transitions are doubtlessly manifest. In our aggregation study of the 2×F1 system we obtain from the canonical analysis a surprisingly clear picture of the aggregation transition. Figure 11.3(a) shows the canonical mean energy hEi and the specific heat per monomer cV , plotted as functions of the temperature T . In Fig. 11.3(b), the temperature

0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 0.05

25.0 hEi(T )/Ntot

10.0 cV (T ) (a) 0.10

0.15

0.20 T

0.25

5.0 0.0 0.35

0.30

20.0 hΓi(T )/Ntot

0.4

15.0

0.3 10.0 0.2 0.1

dhΓi/Ntot dT

0.10

0.15

0.20 T

0.25

0.30

dhΓi/Ntot dT

hΓi(T )/Ntot

20.0 15.0

0.5

0.0 0.05

193

cV (T )

hEi(T )/Ntot

11.4 Statistics of the Two-Chain Heteropolymer System in Three Ensembles

5.0 (b) 0.0 0.35

Figure 11.3: (a) Mean energy hEi/Ntot and specific heat per monomer cV , and (b) hΓi/Ntot and dhΓi/Ntot dT as functions of the temperature.

dependence of the mean aggregation order parameter hΓi and the fluctuations of Γ are shown. The aggregation transition is signalized by very sharp peaks and from both figures we read off peak temperatures close to Tagg ≈ 0.20. The aggregation of the two peptides is a single-step process, in which the formation of the aggregate with a common compact hydrophobic core governs the folding behavior of the individual chains. Folding and binding are not separate processes. The dominance of the inter-chain binding interaction can also be seen by considering the lowest-energy conformation found in our simulations. The energy of this conformation, which is shown in Fig. 11.4, is Emin ≈ −18.4 in our energy units. The peptide-peptide binding energy [i.e., the second term in Eq. (11.1)] (1,2) is with EAB,min ≈ −11.4 much stronger than the intrinsic single-chain energies (2) (1) EAB,min ≈ −3.2 and EAB,min ≈ −3.8, respectively. The single-chain minimum

194

11. Statistical Analyses of Aggregation Processes

Figure 11.4: The minimum-energy 2×F1 complex with Emin ≈ −18.4 as found in the simulations is a capped aggregate [213,214]. single ≈ −5.0 [148] noticeably smaller. energy is with EAB,min The comparatively strong inter-chain interaction and the strength of the aggregation transition despite the smallness of the system lead to the conclusion that surface effects are of essential importance for the aggregation of the peptides. This is actually confirmed by a detailed microcanonical analysis which is performed in the next subsection.

11.4.3 Microcanonical Interpretation – The Backbending Effect In the microcanonical analysis, the system energy E is kept (almost) fixed and treated as an external control parameter. The system can only take macrostates with energies in the interval (E, E + ∆E) with ∆E being sufficiently small to satisfy ∆G(E) = g(E)∆E, where ∆G(E) is the phase space volume of this energetic shell. In the limit ∆E → 0, the total phase space volume up to the energy E can thus be expressed as Z E G(E) = dE ′ g(E ′ ). (11.12) Emin

Since g(E) is positive for all E, G(E) is a monotonically increasing function and this quantity is suitably related to the microcanonical entropy S(E) of the system. In the definition of Hertz, S(E) = kB ln G(E).

(11.13)

Alternatively, the entropy is often directly related to the density of states g(E) and defined as S(E) = kB ln g(E). (11.14) The density of states exhibits a decrease much faster than exponential towards the low-energy states. For this reason, the phase-space volume at energy E

11.4 Statistics of the Two-Chain Heteropolymer System in Three Ensembles 20.0

8.0

10.0

T

−1

(E)

7.0

0.0 -10.0 S(E)

−1 Tagg

-30.0

T>−1

-40.0 -60.0

HS (E)

-70.0 -15.0

3.0

S(E) Eagg

T −1 (E)

T<−1

-20.0

-50.0

195

E> Esep E

E< Efrag

2.0 5.0

Figure 11.5: Microcanonical Hertz entropy S(E) of the 2×F1 system, concave Gibbs hull HS (E), and inverse caloric temperature T −1 (E) as functions of energy. The phase separation regime ranges from Eagg to Efrag . Between T<−1 and T>−1 , the temperature is no suitable external control parameter and the canonical interpretation is not useful: The inverse caloric temperature T −1 (E) exhibits an obvious backbending in the transition region. Note the second, less-pronounced backbending in the energy range E< < E < Efrag .

is strongly dominated by the number of states in the energy shell ∆E. Thus G(E) ≈ ∆G(E) ∼ g(E) is directly related to the density of states. This virtual identity breaks down in the higher-energy region, where ln g(E) is getting flat – in our case far above the energetic regions being relevant for the discussion of the aggregation transition (i.e., for energies E ≫ Efrag , see Fig. 11.5). Actually, both definitions of the entropy lead to virtually identical results in the analysis of the aggregation transition [213,214]. The (reciprocal) slope of the microcanonical entropy fixes the temperature scale and the corresponding caloric temperature is then defined via T (E) = (∂S(E)/∂E)−1 for fixed volume V and particle number Ntot. As long as the mapping between the caloric temperature T and the system energy E is bijective, the canonical analysis of crossover and phase transitions is suitable since the temperature can be treated as external control parameter. For systems, where this condition is not satisfied, however, in a standard canonical analysis one may easily miss a physical effect accompanying condensation processes: Due to surface effects (the formation of the contact surface between the peptides requires a rearrangement of monomers in the surfaces of the individual peptides), additional energy does not necessarily lead to an increase of temperature of the condensate. Actually, the aggregate can even become colder. The supply of additional energy supports the fragmentation of parts of the aggregate, but this is overcompensated by cooperative processes of the particles aiming to reduce

196

11. Statistical Analyses of Aggregation Processes

the surface tension. Condensation processes are phase-separation processes and as such aggregated and fragmented phases coexist. Since in this phase-separation region T and E are not bijective, this phenomenon is called the “backbending effect”. The probably most important class of systems exhibiting this effect is characterized by their smallness and the capability to form aggregates, depending on the interaction range. The fact that this effect could be indirectly observed in sodium clustering experiments [187] gives rise to the hope that backbending could also be observed in aggregation processes of small peptides. Since the 2×F1 system apparently belongs to this class, the backbending effect is also observed in the aggregation/fragmentation transition of this system. This is shown in Fig. 11.5, where the microcanonical entropy S(E) is plotted as function of the system energy. The phase-separation region of aggregated and fragmented conformations lies between Eagg ≈ −8.85 and Efrag ≈ 1.05. Constructing the concave Gibbs hull HS (E) by linearly connecting S(Eagg) and S(Efrag ) (straight dashed line in Fig. 11.5), the entropic deviation due to surface effects is simply ∆S(E) = HS (E) − S(E). The deviation is maximal for E = Esep and ∆S(Esep) ≡ ∆Ssurf is the surface entropy. The Gibbs hull also defines the aggregation transition temperature −1 ∂HS (E) . (11.15) Tagg = ∂E For the 2×F1 system, we find Tagg ≈ 0.198, which is virtually identical with the peak temperatures of the fluctuating quantities discussed in Section 11.4.2. The inverse caloric temperature T −1(E) is also plotted into Fig. 11.5. For a fixed temperature in the interval T< < T < T> (T< ≈ 0.169 and T> ≈ 0.231), different energetic macrostates coexist. This is a consequence of the backbending effect. Within the backbending region, the temperature decreases with increasing −1 system energy. The horizontal line at Tagg ≈ 5.04 is the Maxwell construction, i.e., the slope of the Gibbs hull HS (E). Although the transition seems to have similarities with the van der Waals description of the condensation/evaporation transition of gases – the “overheating” of the aggregate between Tagg and T> (within the energy interval Eagg < E < E> ≈ −5.13) is as apparent as the “undercooling” of the fragments between T< and Tagg (in the energy interval Efrag > E > E< ≈ −1.13) – it is important to notice that in contrast to the van der Waals picture the backbending effect in-between is a real physical effect. Another essential result is that in the transition region the temperature is not a suitable external control parameter: The macrostate of the system cannot be adjusted by fixing the temperature. The better choice is the system energy which is unfortunately difficult to control in experiments. Another direct conse-

11.4 Statistics of the Two-Chain Heteropolymer System in Three Ensembles

197

400.0

CV (E)

200.0 0.0 -200.0 -400.0 -20.0

-15.0

Eagg E

Esep Efrag

5.0

Figure 11.6: Microcanonical specific heat CV (E) for the 2×F1 complex. Note the negativity in the backbending regions [213].

quence of the energetic ambiguity for a fixed temperature between T< and T> is that the canonical interpretation is not suitable for detecting the backbending phenomenon. The most remarkable result is the negativity of the specific heat of the system in the backbending region, as shown in Fig. 11.6. A negative specific heat in the phase separation regime is due to the nonextensitivity of the energy of the two subsystems resulting from the interaction between the polymers. “Heating” a large aggregate would lead to the stretching of monomer-monomer contact distances, i.e., the potential energy of an exemplified pair of monomers increases, while kinetic energy and, therefore, temperature remain widely constant. In a comparatively small aggregate, additional energy leads to cooperative rearrangements of monomers in the aggregate in order to reduce surface tension, i.e, the formation of molten globular aggregates is suppressed. In consequence, kinetic energy is transferred into potential energy and the temperature decreases. In this regime, the aggregate becomes colder, although the total energy increases [213]. The precise microcanonical analysis reveals also a further detail of the aggregation transition. Close to Epre ≈ −0.32, the T −1 curve in Fig. 11.5 exhibits another “backbending” which signalizes a second, but unstable transition of the same type. The associated transition temperature Tpre ≈ 0.18 is smaller than Tagg, but this transition occurs in the energetic region where fragmented states dominate. Thus this transition can be interpreted as the premelting of aggregates

198

11. Statistical Analyses of Aggregation Processes 4.0 Tpre ≈ 0.18

3.0 2.5 2.0 1.5 1.0 0.5 -15.0

Tagg ≈ 0.20 -10.0 Eagg

premolten fragments aggregates coexistence regime

aggregates

coexistence regime

log10 hcan (E)

3.5

-5.0 E

0.0 Epre Efrag

5.0

Figure 11.7: Logarithmic plots of the canonical energy histograms (not normalized) at T ≈ 0.18 and T ≈ 0.20, respectively.

by forming intermediate states. These intermediate structures are rather weakly stable: The population of the premolten aggregates never dominates. In particular, at Tpre , where premolten aggregates and fragments coexist, the population of compact aggregates is much larger. This can nicely be seen in the canonical energy histograms at these temperatures plotted in Fig. 11.7, where the second backbending is only signalized by a small cusp in the coexistence region. Since both transitions are phase-separation processes, structure formation is accompanied by releasing latent heat which can be defined as the energetic widths of the phase coexistence regimes, i.e., ∆Qagg = Efrag −Eagg = Tagg[S(Efrag )−S(Eagg)] ≈ 9.90 and ∆Qpre = Efrag − Epre = Tpre [S(Efrag) − S(Epre )] ≈ 1.37. Obviously, the energy required to melt the premolten aggregate is much smaller than to dissolve a compact (solid) aggregate. For the comparison of the surface entropies, we use the definition (11.14) of the entropy. In the case of the aggregation transition, the surface entropy is agg agg ∆Ssurf ≈ ∆Ssurf = HS (Esep) − S(Esep), where HS (E) ≈ HS (E) is the concave Gibbs hull of S(E). Since HS (Esep) = HS (Efrag) − (Efrag − Esep )/Tagg and HS (Efrag ) = S(Efrag), the surface entropy is agg = S(Efrag ) − S(Esep) − ∆Ssurf

1 Tagg

(Efrag − Esep ).

(11.16)

Yet utilizing that with Eq. (11.11) the canonical distribution hcan (E) = R dΓ Hcan (E, Γ; T ) at Tagg (shown in Fig. 11.7) is hcan (E) ∼ g(E) exp(−E/kB Tagg), the surface entropy can be written in the simple

11.5 Aggregation Transition in Larger Heteropolymer Systems

199

and computationally convenient form [218]: agg = kB ln ∆Ssurf

hcan (Efrag) . hcan (Esep)

(11.17)

A similar expression is valid for the coexistence of premolten and fragmented states at Tpre The corresponding canonical distribution is also shown in Fig. 11.7. Thus, we obtain (in units of kB ) for the surface entropy of the aggregation transition pre agg ≈ 0.04, confirming the weakness of ≈ 2.48 and for the premelting ∆Ssurf ∆Ssurf the interface between premolten aggregates and fragmented states.

11.5 Aggregation Transition in Larger Heteropolymer Systems The statements in the previous section for the 2×F1 system are also, in general, valid for larger systems. This is the result of computer simulations for systems consisting of three (in the following referred to as 3×F1) and four (4×F1) identical peptides with sequence F1. Although the formation of compact hydrophobic cores is more complex in larger compounds of our exemplified sequence F1, the aggregation transition is little influenced by this. This is nicely seen in Figs. 11.8(a) and 11.8(b), where the temperature dependence of the canonical expectation values of Γ and E, as well as for their fluctuations, are shown for the 3×F1 system. For comparison, also results for the 4×F1 system are plotted into the same figures. Note that for the 4×F1 system finite-size effects are larger since, for computational reasons, we have kept the edge length of the simulation box L = 40, which is smaller than the successive arrangement of four straight chains with 13 monomers. This influences primarily the entropy in the high-energy regime far above the aggregation transition energy. Nonetheless, in the canonical interpretation, it acts back on the transition as undesired states (chain ends overlapping due to the periodic boundary conditions) are (weakly) populated at the transition temperature, whereas others are suppressed. We have performed a detailed analysis of the box size dependence (results not shown) and found that the canonical transition temperature scales slightly, but noticeably with the box size. Thus, the results obtained by canonical statistics for the 4×F1 system should not quantitatively be compared to the canonical results for the 2×F1 and 3×F1 systems. As it has already been discussed for the 2×F1 system, there are also for the larger systems no obvious signals for separate aggregation and hydrophobic-core formation processes. Only weak activity in the energy fluctuations in the temperature region below the aggregation transition temperature indicates that local

200

11. Statistical Analyses of Aggregation Processes 0.5

45.0 40.0

0.3

4×F1 3×F1

30.0

-0.1

25.0

-0.3

20.0 15.0

-0.5

3×F1

-0.9 0.05

10.0

4×F1

-0.7 0.10

0.15

0.20 T

(a) 0.25

0.5 0.4

3×F1

0.3 0.2 4×F1

0.1 0.0 0.05

0.10

0.15

0.20 T

0.25

0.30

5.0 0.0 0.35

22.0 20.0 18.0 16.0 3×F1 14.0 12.0 10.0 4×F1 8.0 6.0 4.0 (b) 2.0 0.0 0.30 0.35

dhΓi(T )/Ntot dT

hΓi(T )/Ntot

35.0 cV (T )

hEi(T )/Ntot

0.1

Figure 11.8: (a) Mean energy hEi/Ntot and specific heat per monomer cV , (b) mean aggregation parameter hΓi/Ntot and its fluctuations dhΓi/Ntot dT as functions of the temperature for the 3×F1 and 4×F1 heteropolymer systems.

restructuring processes of little cooperativity (comparable with the discussion of the premolten aggregates in the discussion of the 2×F1 system) are still happening. The strength of the aggregation transition is also documented by the fact that the peak temperatures of energetic and aggregation parameter fluctuations are virtually identical for the 3×F1 system, i.e., the aggregation temperature is Tagg ≈ 0.21 (for 4×F1 Tagg ≈ 0.22). For homogeneous multiple-chain systems two variants of thermodynamic limits are of particular interest: (i) M → ∞, while Nµ = const, (ii) Nµ → ∞ with M = const; both limits considered for constant polymer density. Since for proteins the sequence of amino acids is fixed, in this case only (i) is relevant and it is future work to perform a scaling analysis for multiple-peptide systems in this limit. A particularly interesting question is to what extent remnants of the finite-system effects survive in the limit of an infinite number of chains, dependent of the peptide

11.5 Aggregation Transition in Larger Heteropolymer Systems 1.0

0.16

-9.0 -0.9

0.12 0.10 2

s(e)

-0.7

0.08

∆s(e)

0.06

3

0.04 4

-0.5

-0.3 e

-0.1

∆s(e)

s(e), hs (e)

hs (e)

-5.0 -7.0

0.14

2 3 4

-1.0 -3.0

201

0.02 0.1

0.00 0.3

Figure 11.9: Microcanonical entropies per monomer s(e), respective Gibbs constructions hs (e) (left-hand scale), and deviations ∆s(e) = hs (e) − s(e) (right-hand scale) for 2×F1 (labeled as 2), 3×F1 (3), and 4×F1 (4) as functions of the energy per monomer e.

density. Since we have focused our study on the precise analysis of systems of few peptides for all energies and temperatures, it is computationally inevitable to restrict oneself to small systems, for which a scaling analysis is not very useful. Nonetheless, it is instructive to devote a few interesting remarks to the comparison of, once more, microcanonical aspects of the aggregation transition in dependence of the system size. In Fig. 11.9, the microcanonical entropies per monomer s(e) = S(e)/Ntot (shifted by an unimportant constant for clearer visibility) and the corresponding Gibbs hulls hs (e) = HS (e)/Ntot are shown for 2×F1 (in the figure denoted by “2”), 3×F1 (“3”), and 4×F1 (“4”), respectively, as functions of the energy per monomer e = E/Ntot. Although the convex entropic “intruder” is apparent for larger systems as well, its relative strength decreases with increasing number of chains. The slopes of the respective Gibbs constructions determine the aggrega3×F1 4×F1 tion temperature (11.15) which are found to be Tagg ≈ 0.212 and Tagg ≈ 0.217 confirming the peak temperatures of the fluctuation quantities plotted in Fig. 11.8. The existence of the interfacial boundary entails a transition barrier whose strength is characterized by the surface entropy ∆Ssurf. In Fig. 11.9, the individual entropic deviations per monomer, ∆s(e) = ∆S(e)/Ntot are also shown and the maximum deviations, i.e., the surface entropies ∆Ssurf and relative surface entropies per monomer ∆ssurf = ∆Ssurf/Ntot are listed in Table 11.1. There is no apparent difference between the values of ∆Ssurf that would indicate a trend for a vanishing of the absolute surface barrier in larger systems. However, the

202

11. Statistical Analyses of Aggregation Processes

Table 11.1: Aggregation temperatures Tagg , surface entropies ∆Ssurf , relative surface entropies per monomer ∆ssurf , relative aggregation and fragmentation energies per monomer, eagg and efrag , respectively, latent heat per monomer ∆q, and phase-separation entropy per monomer ∆q/Tagg . All quantities for systems consisting of two, three, and four 13-mers with AB sequence F1. system 2×F1 3×F1 4×F1

Tagg ∆Ssurf 0.198 2.48 0.212 2.60 0.217 2.30

∆ssurf 0.10 0.07 0.04

eagg −0.34 −0.40 −0.43

efrag ∆q ∆q/Tagg 0.04 0.38 1.92 0.05 0.45 2.12 0.05 0.48 2.21

relative surface entropy ∆ssurf obviously decreases. Whether or not it vanishes in the thermodynamic limit cannot be decided from our results and is a study worth in its own right. It is also interesting that subleading effects increase and the double-well form found for 2×F1 changes by higher-order effects, and it seems that for larger systems the almost single-step aggregation of 2×F1 is replaced by a multiplestep process. Not surprisingly, the fragmented phase is hardly influenced by side effects and the rightmost minimum in Fig. 11.9 lies well at efrag = Efrag /Ntot ≈ 0.04 − 0.05. Since the Gibbs construction covers the whole convex region of s(e), the aggregation energy per monomer eagg = Eagg/Ntot corresponds to the leftmost minimum and its value changes noticeably with the number of chains. In consequence, the latent heat per monomer ∆q = ∆Q/Ntot = Tagg[S(Efrag ) − S(Eagg)]/Ntot that is required to fragment the aggregate increases from two to four chains in the system (see Table 11.1). Although the systems under consideration are too small to extrapolate phase transition properties in the thermodynamic limit, it is obvious that the aggregation-fragmentation transition exhibits strong similarities to condensation-evaporation transitions of colloidal systems. Given that, the entropic transition barrier ∆q/Tagg, which we see increasing with the number of chains (cf. the values in Table 11.1), would survive in the thermodynamic limit and the transition was first-order-like. More surprising would be, however, if the convex intruder would not disappear, i.e., if the absolute and relative surface entropies ∆Ssurf and ∆ssurf do not vanish. This is definitely a question of fundamental interest as the common claim is that pure surface effects typically exhibited only by “small” systems are irrelevant in the thermodynamic limit. This requires studies of much larger systems. It should clearly be noted, however, that protein aggregates forming themselves in biological systems often consist only of a few peptides and are definitely of small size and the surface effects are responsible for structure formation and are not unimportant side effects. One should keep in mind that standard thermodynamics and the thermodynamic limit are somewhat theoretical

11.5 Aggregation Transition in Larger Heteropolymer Systems

203

constructs valid only for very large systems. The increasing interest in physical properties of small systems, in particular in conformational transitions in molecular systems, requires in part a revision of dogmatic thermodynamic views. Indeed, by means of today’s chemo-analytical and experimental equipment, effects like those described in this chapter, should actually experimentally be verifiable as these are real physical effects. For studies of the condensation of atoms, where a similar behavior occurs, such experiments have actually already been performed [187].

12 Hierarchical Nature of Phase Transitions

12.1 Aggregation of Semiflexible Polymers In the following, as an example for a first-order phase transition, the aggregation of interacting semiflexible polymers is discussed by analyzing results from multicanonical computer simulations of a mesoscopic bead-stick model, where nonbonded monomers interact via Lennard-Jones potentials. As we will see, aggregation of semiflexible polymers turns out to be a process, in which the constituents experience strong structural fluctuations, similar to peptides in coupled foldingbinding cluster formation processes. In contrast to the proteinlike hydrophobicpolar heteropolymer model investigated in the previous chapter, aggregation and crystallization are separate processes for a homopolymer with the same small bending rigidity. Rather stiff semiflexible polymers form a liquid-crystal-like phase, as expected. In analogy to the heteropolymer case, the first-order-like aggregation transition of the complexes is accompanied by strong system-size dependent hierarchical surface effects. In consequence, polymer aggregation is also a phaseseparation process with entropy reduction [222]. Cluster formation and crystallization of polymers are processes which are interesting for technological applications, e.g., for the design of new materials with certain mechanical properties or nanoelectronic organic devices and polymeric solar cells. From a biophysical point of view, the understanding of oligomerization, but also the (de)fragmentation in semiflexible biopolymer systems like actin networks is of substantial relevance. This requires a systematic analysis of the basic properties of the polymeric cluster formation processes, in particular, for small polymer complexes on the nanoscale, where surface effects are competing noticeably with structure-formation processes in the interior of the aggregate. A further motivation for investigating the aggregation transition of semiflexible homopolymer chains derives from the intriguing results of the similar aggregation process for peptides [213,214] discussed in Chapter 11, which were modeled as

206

12. Hierarchical Nature of Phase Transitions

heteropolymers with a sequence of two types of monomers, hydrophobic (A) and hydrophilic ones (B). By specializing the previously employed heteropolymer model to the apparently simpler homopolymer case, we now aim by comparison at isolating those properties which were mainly driven by the sequence properties of heteropolymers. In fact, while in both cases the aggregation transition is a phaseseparation process, we will show below that for homopolymers the aggregation and crystallization (if any) are separate conformational transitions – unlike the heteropolymer aggregates where they were found to coincide [213,214]. Again, we will explain the physical origin causing these differences within the microcanonical formalism [210,215], which proves to be particularly suitable for this type of problem.

12.2 Structural Transitions of Semiflexible Polymers with Different Bending Rigidities We thus consider the same model as in Section 11.2, but here we assume that all monomers iµ = 1, . . . , N (µ) of the µth chain (µ = 1, . . . , M) at positions xiµ are hydrophobic (A). The bonds between adjacent monomers are taken to be rigid (bead-stick model) and pairwise interactions among nonbonded monomers are modeled by a Lennard-Jones potential ], − ri−6 VLJ (riµjν ) = 4[ri−12 µ jν µ jν

(12.1)

where riµ jν = |xiµ − xjν | is the distance between monomers iµ and jν of the µth and νth chain, respectively. Intra-chain (µ = ν) and inter-chain (µ 6= ν) contacts are not distinguished energetically. The semiflexibility of a chain is described by the bending energy X (µ) Ebend = κ (12.2) (1 − cos ϑiµ ) , iµ

where 0 ≤ ϑiµ ≤ π is the bending angle formed by the monomers iµ , iµ + 1, and iµ + 2. For the comparison with the heteropolymer aggregation, we consider a bending rigidity κ = 0.25, which is at the rather floppy end of semiflexibility. Thus, the single-chain energy reads X (µ) (µ) E = Ebend + VLJ (riµ jµ ) (12.3) jµ >iµ +1

and the total energy of the polymer system is given by X XX E= E (µ) + VLJ (riµjν ) . µ

µ<ν iµ ,jν

(12.4)

12.2 Structural Transitions of Semiflexible Polymers with Different Bending Rigidities 207

All chains are assumed to have the same degree of polymerisation, i.e., the same number of monomers, N (µ) = N , µ = 1, . . . , M. The following results were obtained in multicanonical computer simulations [83] for different system sizes. The multicanonical weights were calculated recursively [84,85] in 50 iterations with 5 × 106 × M 2 single updates (spherical pivot rotations [148], semilocal crankshaft moves [214]) each. The production run with fixed multicanonical weights included 5 × 108 × M 2 single updates. The generic result obtained from these simulations is a precise estimate for the density of states g(E) or microcanonical entropy S(E) = kB ln g(E), which the subsequent microcanonical analysis is based on. In Fig. 12.1(a), the specific-heat curve for a system of two identical semiflexible polymers (2 × A13) is compared with the energetic fluctuations of a single chain (1 × A13). The single chain exhibits a very weak coil-globule collapse transition (shoulder near T ≈ 0.88), whereas the crystallization near T ≈ 0.24 is a pronounced, separate process. The thermodynamic phase behavior of single semiflexible polymers in solvent has already been subject of numerous studies, with particular focus on stiffness and finite chain length effects [223–225], where it was shown that the globule-solid transition is more influenced by stiffness effects than the coil-globule transition. In particular, for longer chains, it was found that, depending on the stiffness, single collapsed semiflexible polymers form spherical, ellipsoidal, and disklike globules, as well as toroids [225]. The first result for the semiflexible multiple-chain system read off from Fig.12.1(a) is that aggregation and collapse are not separate processes (near T ≈ 0.97), similar to the corresponding heteropolymer system, cp. Fig. 11.3(a). At about T ≈ 0.24 (close to the single-chain freezing temperature), the multiple-chain homopolymer complex crystallizes in a separate process. This is in strong contrast to the heteropolymer systems, where aggregation, collapse, and crystallization (hydrophobic-core formation) is a single-step process [213,214]. In Fig. 12.1(a), we compare with a two-chain system with much stronger bending rigidity κ = 10. As expected, there is a single aggregation transition near the temperature, where the system of less stiff semiflexible polymers with κ = 0.25 collapses. However, a further crystallization process at lower temperatures does not occur: There is no globular (liquid) pseudophase of defragmented relatively stiff semiflexible polymers. In Fig. 12.1(b), the canonical expectation value of the aggregation parameter Γ as defined in Section 11.3, hΓi, and its fluctuation dhΓi/dT are plotted. The peak position of dhΓi/dT coincides nicely with the corresponding peak temperature of the specific heat and thus signals the aggregation transition. In Fig. 12.2, the aggregation parameter and its fluctuations are shown for differ-

208

12. Hierarchical Nature of Phase Transitions 5.0

cV (T )

4.0

2×A13, κ = 10 2×A13, κ = 0.25

(a)

3.0 2.0 1.0

1×A13

hΓi(T )/Ntot

0.5

(b)

hΓi(T )/Ntot

0.4 0.3 0.2 0.1 0.0 0.0

dhΓi(T )/dT Ntot 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 2.0

dhΓi(T )/dT Ntot

0.0

T

Figure 12.1: (a) Specific heat per monomer as a function of temperature for a single semiflexible homopolymer with 13 monomers (1×A13 , κ = 0.25) and for systems of two such chains (2×A13 ) with different bending rigidities. (b) Canonical expectation value hΓi and fluctuation dhΓi/dT of the aggregation parameter Γ for the two-chain system with κ = 0.25.

ent system sizes of up to four chains. The peak shifts towards higher temperatures and gets sharper with increasing number of chains, since intra-chain and interchain monomer-monomer contacts are not energetically distinguished. The hypothetic maximal total number of intrinsic contacts is nmax intra = M(N −2)(N −1)/2 ∼ 2 MN = N Ntot , and for the maximally possible number of inter-chain contacts 2 2 2 2 1 one has nmax inter = M(M − 1)N /2 ∼ M N = Ntot . For large M, the relative fraction rinter of the inter-chain contacts, nmax 1 2 1 1 rinter = max inter max = 1 − 1− 1− +O , (12.5) nintra + ninter M N N M 2N behaves like rinter ∼ 1 − M −1 , i.e., the relative influence of inter-chain contacts increases rapidly towards unity with the number of chains. In consequence, aggregation dominates over collapse of the individual chains. Even for the two-chain system 2 × A13 this estimate is reasonable: The energy of the lowest-energy conformation found numerically is E (min) ≈ −83.61 and the contribution of the inter(min) (min) (min) chain contacts is Einter ≈ −50.20 ≈ rinter E (min) with rinter ≈ 0.60. This coincides nicely with the corresponding value of the above contact ratio, rinter ≈ 0.56. In fact, considering energetic and structural fluctuations of the larger systems with 1

Due to excluded volume and optimal space-filling constraints, these numbers need to be scaled by about q/Ntot , where q is the effective coordination number (which is, e.g., q ≈ 10.0 for a perfect fcc crystal).

12.2 Structural Transitions of Semiflexible Polymers with Different Bending Rigidities 209

hΓi(T )/Ntot

0.6 0.5

2×A13

(a)

3×A13

0.4 0.3

4×A13

0.2 0.1

dhΓi(T )/dT Ntot

0.0 2.0

(b)

1.5

2×A13

1.0

3×A13

4×A13

0.5 0.0 0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

T

Figure 12.2: (a) Canonical expectation values and (b) fluctuations of the aggregation parameter Γ for multi-chain systems with κ = 0.25. In (a), also an exemplified globular 4 × A13 aggregate is depicted.

three and four chains, there are no indications for an additional collapse transition at temperatures higher than the aggregation transition. Below the aggregation temperature, the entropic loss of the individual chains is overcompensated by the energetic gain of forming a joint globular aggregate. However, the entropic change while passing the aggregation transition is noticeably smaller than what was found for heteropolymer systems, where no separate freezing transition occurs [214]. In the intermediate fluid “globular” pseudophase, the aggregate of the homopolymers thus behaves like a single chain of length MN . Consequently, reducing the temperature, the aggregate optimizes the monomer arrangements in order to maximize energetic contacts. Indicated by the peak in the specific heat near T ≈ 0.2 (see Fig. 12.1), the small globular aggregates freeze into spherical amorphous structures with a maximum number of inter-chain contacts. For rather stiff semiflexible polymers (as our example with κ = 10), however, a separate freezing transition does not occur and the peak of cV near T ≈ 0.9 indicates a single-step transition from rod-like coils to a liquid-crystal-like phase [225]. In the studies of heteropolymer aggregation in Section 11.4, we also observed a pronounced single transition of a different nature. In this case, the formation of a heteropolymer complex consisting of different chains with compact hydrophobic core was roughly a single-step process, because hydrophobic-core formation (“freezing”) favors conformations with very small entropy. For semiflexible homopolymer systems, we find in the rather floppy limit that the freezing

210

12. Hierarchical Nature of Phase Transitions

System 2×A13 3×A13 4×A13

Tagg ∆ssurf 0.973 0.024 1.118 0.016 1.172 0.009

eagg −1.566 −1.730 −1.892

efrag ∆q ∆q/Tagg −0.944 0.622 0.639 −0.831 0.899 0.804 −0.799 1.093 0.932

Table 12.1: Aggregation temperatures Tagg , relative surface entropies per monomer ∆ssurf , relative aggregation and fragmentation energies per monomer, eagg and efrag , respectively, latent heat per monomer ∆q, and phase-separation entropy per monomer ∆q/Tagg .

temperature (T ≈ 0.2, for single chains or globular aggregates with κ = 0.25) is almost identical with the aggregation temperature for heteropolymer systems (compare Figs. 11.3 and 12.1). This coincidence in the behavior of these different systems is due to the formation of a single, very compact hydrophobic domain (the monomers of the interacting homopolymers are as hydophobic as the A-type monomers of the heteropolymer) which maximizes the number of energetic contacts. Stiff homopolymers, on the other hand, cannot form a maximally compact hydrophobic core and hence do not crystallize in a separate transition.

12.3 Hierarchies of Subphase Transitions For a deeper understanding of the aggregation transition, we now analyze entropic effects accompanying the aggregation transition in the microcanonical ensemble for an isolated system of multiple polymer chains. The microcanonical entropies per monomer, s = S/Ntot , are shown in Fig. 12.3(a) as functions of the total energy per monomer, e = E/Ntot. The reciprocal caloric temperature, T −1(E) = ∂S/∂E, is plotted in Fig. 12.3(b). The plots reveal that increasing the energy entails a reduction of temperature in the transition region, i.e., the backbending effect which was already discussed in Section 11.4.3. In the study of the heteropolymer aggregates, we used the microcanonical view to identify the peptide aggregation as so-called “coupled binding-folding transition”, i.e., the individual heteropolymer chains refold during the binding process and the finally formed aggregate possesses a single compact hydrophobic domain. By closer inspection of Fig. 12.3(b), we find for the homopolymer system a hiercharchical substructure caused by these surface effects. The frequency of oscillations of the curves, increasing with system size, reveals that the aggregation transition is actually a composition of different subprocesses, each of which is an individual phase-separation process. The amplitude of these oscillations decreases with system size showing that these subprocesses comprise a smaller surface-

12.3 Hierarchies of Subphase Transitions

211

s(e), hs(e)

0.5 0.2

(a)

3×A13

2×A13

-0.1

4×A13

-0.4 -0.7 -1.0

T −1(e)

1.2

(b)

2×A13

1.1 1.0

3×A13

0.9 0.8

4×A13

0.7

2×A13

∆s(e)

(c) 0.02

3×A13 4×A13

0.01

0.00

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

e

Figure 12.3: (a) Microcanonical entropy per monomer s(e) and the Gibbs constructions hs (e) (dashed lines), (b) reciprocal caloric temperatures T −1 (e) and Maxwell constructions, and (c) relative surface entropies per monomer ∆ssurf (e).

entropic barrier [see Fig. 12.3(c)]. The 2 × A13 system exhibits a single backbending effect as only two chains aggregate. For the three-chain system 3 × A13, a different scenario is apparent. In the higher-energy regime, first two chains stick together, and the formation of the three-chain globule is a separate process at lower energies. This hierarchical procedure continues for larger systems as, e.g., for 4×A13. However, the impact of the individual backbending effects is getting weaker and, in the thermodynamic limit, these effects are expected to disappear asymptotically, whereas the firstorder character of this transition remains. The horizontal lines in Fig. 12.3(b) are the respective Maxwell constructions and define the aggregation temperatures Tagg. In the following, we denote the leftmost (rightmost) energy where T −1(e) = −1 Tagg as eagg (efrag ). In the entropy curves, the Maxwell constructions correspond to −1 concave hulls hs (e) with ∂hs(e)/∂e = Tagg (Gibbs constructions) in the transition regime [see Fig. 12.3(a)], where entropy is reduced due to surface effects. For a quantitative analysis, the deviation ∆s(e) = hs (e) − s(e) is plotted

212

12. Hierarchical Nature of Phase Transitions

in Fig. 12.3(c). Within the transition region (i.e., for eagg ≤ e ≤ efrag ), the peak height ∆ssurf = maxeagg≤e≤efrag ∆s(e) defines the surface entropy. The energetic width of the phase-coexistence regime is the latent heat per monomer, ∆q = efrag − eagg = Tagg[s(efrag ) − s(eagg)]). Thus the entropic phase separation barrier ∆q/Tagg should survive in the thermodynamic limit, if the aggregation of semiflexible polymers is a first-order-like phase-separation process with coexistence of aggregates and fragments. Values for these quantities are listed in Table 12.1 for the three polymer systems considered in this study. We see that with increasing system size the surface entropies ∆ssurf decrease and thus the influence of surface effects is getting weaker. On the other hand, the latent heat per monomer increases, supporting the first-order character of the aggregation transition. We have seen in this analysis that the aggregation of interacting semiflexible polymers is a first-order phase-separation process. It is accompanied by sequentially ordered, i.e., hierarchical, subphase transitions of aggregation. Without the microcanonical analysis, it would have been difficult to reveal this hierarchical character of the transition. This analysis indeed sheds new light on the general nature of phase transitions which appear to be composed of separate subprocesses. Thus, the cooperative interplay of small parts of the system that undergoes a phase transition is hierarchical and not a single-step process. This is probably a general feature of all phase transitions and thus of fundamental importance for understanding the onset of phase transitions in general.

13 Adsorption of Polymers at Solid Substrates

13.1 Structure Formation at Hybrid Interfaces of Soft and Solid Matter The requirement of higher integration scales in electronic circuits, the onset of nanosensory applications in biomedicine, but also the fascinating capabilities of modern experimental setup with its enormous potential in polymer and surface research led recently to an increasing interest at the hybrid interface of organic and inorganic matter [226–231]. This also includes numerous detailed studies, e.g., of polymer film wetting phenomena [232–234], pattern recognition [235,236], protein–ligand binding and docking [237–239], charged adsorbed polymers [240] as well as electrophoretic polymer deposition and growth at surfaces [241]. The recent developments in single molecule experiments at the nanometer scale, e.g., by means of atomic force microscopy (AFM) [242,243] and optical tweezers [244,245], allow now for a more detailed exploration of structural properties of polymers in the vicinity of adsorbing substrates [231]. The possibility to perform such studies is of essential biological and technological significance. From the biological point of view, the understanding of the binding and docking mechanisms of proteins at cell membranes is important for the reconstruction of biological cell processes. Similarly, specificity of peptides and binding affinity to selected substrates could be of great importance for future electronic nanoscale circuits and pattern recognition nanosensory devices [235]. In this chapter, theoretical approaches to the modeling of hybrid systems of soft and solid matter will be discussed. A comprising analysis of conformational transitions experienced by polymers in the binding process, of the phase-diagram structure, and of dominant polymer conformations in these phases can currently only be performed efficiently by means of computer simulations of hybrid models

214

13. Adsorption of Polymers at Solid Substrates

Figure 13.1: Cavity model used for studies of the adsorption of single polymers at solid substrates on the simple-cubic lattice. The lower of the two parallel surfaces is attractive to the polymer, the upper is steric only. The distance between the surfaces is zw lattice units.

on mesoscopic scales. The influence of adhesion and steric hindrance for polymers grafted to a flat substrate [246–252], conformational pseudophase transitions for nongrafted polymers and peptides in a cavity with attractive substrate [253– 257], adsorption dynamics and substructure formation of adhered polymers [258], and the shape response to pulling forces [259,260] or external fields [261] were subject of computer simulations and analytical approaches of different such models [262,263]. The question how a flexible substrate, e.g., a cell membrane, bends as a reaction of a grafted polymer, was, for example, addressed in Ref. [264]. Proteins exhibit a strong specificity as the affinity of peptides to adsorb at surfaces depends on the amino acid sequence, solvent properties, and substrate shape. This was experimentally and numerically studied, e.g., for peptide-metal [226,265] and peptide-semiconductor [227–229] interfaces. Binding/folding and docking properties of lattice heteropolymers at an adsorbing surface were also subject of numerical studies [239].

13.2 Minimalistic Modeling and Simulation of Hybrid Interfaces The simplest, but very instructive model of a hybrid system is shown in Fig. 13.1. As in the discussion of general heteropolymer folding properties, employing a minimalistic simple-cubic (sc) lattice model [47] allows a systematic analysis of the conformational phases experienced by a nongrafted polymer in a cavity with one adhesive surface. The polymer can move between the two infinitely extended parallel planar walls, separated by a distance zw expressed in lattice units. The substrate is short-range attractive to the monomers of the polymer chain, while

13.2 Minimalistic Modeling and Simulation of Hybrid Interfaces

215

the influence of the other wall is purely steric. Denoting the number of nearest-neighbor, but nonadjacent monomer-monomer contacts by nm and the number of nearest-neighbor monomer-substrate contacts by ns , the energy of the hybrid system can be expressed in the simplest model [247] as Es (ns, nm) = −εs ns − εm nm , (13.1) where εs and εm are the respective contact energy scales, which are left open in the following. For simplicity, we perform a simple rescaling and set εs = ε0 and εm = ε0s. Here we have introduced the overall energy scale ε0 and the dimensionless reciprocal solubility s that controls the quality of the implicit solvent surrounding the polymer (the larger the s, the worse the solvent). Since contacts with the substrate usually entail a reduction of monomer-monomer contacts, there are two competing forces (rated against each other by the energy scales) affecting the formation of intrinsic and surface contacts. Here, we mainly focus on the conformational transitions the polymer experiences under different environmental conditions [253,254]. Concretely, we are interested in the dependence of energetic and structural quantities on temperature T and reciprocal solubility s in equilibrium. The probability (per unit area) for a conformation with ns surface and nm monomer-monomer contacts at temperature T and reciprocal solubility s is given by 1 (13.2) pT,s(ns , nm) = gns nm eε0 (ns +snm)/kB T , Z where gns nm = δns 0 gnum +(1−δns0 )gnbs nm is the contact density and Z the partition sum. In this decomposition, gnum stands for the density of unbound conformations, whereas gnbs nm is the density of surface and intrinsic contacts of all conformations bound to the substrate. Obviously, the number of the conformations without contact to the attractive substrate, gnum , depends on the distance zw between the cavity walls. For a sufficiently large distance zw from the substrate the influence of the neutral surface on the unbound polymer is small. For zw → ∞, however, gnum formally diverges. Therefore, the nonadhesive, impenetrable steric wall is necessary for regularization. The partition Psum of the system as a function of the two parameters T and s is simply Z = ns ,nm gns nm exp{ε0(ns + snm )/kB T } and the statistical average of any function O(ns , nm ) is given by the formula X O(ns , nm )pT,s(ns, nm), (13.3) hOi(T, s) = ns ,nm

which is very convenient since it only requires to estimate the contact density gns nm in the simulation. Denoting contact correlation matrix elements as Mxy (T, s) =

216

13. Adsorption of Polymers at Solid Substrates

hxyic = hxyi − hxihyi with x, y = ns , nm, the specific heat can be written as 2 ε0 1 (1, s) M(T, s) . (13.4) CV (T, s) = kB s kB T All quantities depending only on the contact numbers nm and ns can therefore simply be calculated from the estimate of the contact density gns nm provided by our simulation method. Although the two contact parameters are sufficient to describe the macrostate of the system and their fluctuations characterize the main pseudophase transition lines, it is often useful to introduce also nonenergetic quantities such as the end-to-end distance and the gyration tensor for gaining more detailed structural information of the polymer. For our specific problem at hand it is particularly useful to study the structural anisotropy of the adsorbed polymer in the different phases. To this end, we define the general gyration tensor for a polymer chain of N beads with the components 2 Rij

N 1 X (n) (n) = x − xi xj − xj , N n=1 i

(13.5)

(n)

where xi , i = 1, 2, 3, is the ith Cartesian coordinate of the nth monomer and PN (n) xi = n=1 xi /N is the center of mass with respect to the ith coordinate. Anisotropy in the polymer fluctuations is connected with the system’s geometry and therefore it will be sufficient to study the components of the gyration tensor parallel (x, y components) and perpendicular (in z direction) to the planar walls, N 2 2 X 1 Rk2 = x(n) − x + y (n) − y N n=1

and 2 R⊥

N 2 1 X (n) z −z . = N n=1

(13.6)

(13.7)

2 The gyration radius is then simply the trace of the gyration tensor, Rgyr = P3 2 2 2 2 Tr R = i=1 Rii = Rk + R⊥ . The calculation of statistical averages for quantities R that are not necessarily functions of the contact numbers ns and nm cannot be performed P via Eq. (13.3). In this case only the more general relation hRi = X R(X) exp{−Es (X)/kB T }/Z holds, where the sum runs over all polymer P conformations X. Introducing the accumulated density ′ ′ Racc (ns , nm) = X R(X)δns (X) n′s δnm (X) n′m /gn′s n′m , where δij is the Kronecker

13.3 Contact-Density Chain-Growth Algorithm

217

symbol, the expectation value can be expressed, however, in a form similar to Eq. (13.3): X hRi = Racc (ns , nm)pT,s(ns, nm ). (13.8) ns ,nm

The quantity Racc (ns , nm) can easily be measured in simulations with the contact density chain-growth algorithm. In the following, natural units are used as usual, i.e., we set kB = ε0 ≡ 1.

13.3 Contact-Density Chain-Growth Algorithm Lattice models of hybrid systems with decoupled energy scales as in Eq. (13.1), are most efficiently simulated by means of an improved variant of the recently developed multicanonical chain-growth sampling [2,47,48] or the flat-histogram chain-growth method [82]. The main advantage of the improved methods is that these directly sample the contact density gns nm . The contact-density chain growth method [2] generalizes the ordinary multicanonical version [47,48] which samples the density of states, i.e., the number of states for given energy. Here, the two independent energy scales εm and εs or their ratio s are set, respectively, after the simulation. This allows to introduce the reciprocal solubility s as a second external environmental parameter in addition to the temperature T . The determination of the contact density gnm ns follows similar lines as the multicanonical chain-growth method for the estimation of the density of states. In fact, the only change in the algorithm described in Section 4.6 is that the multicanonical chain-growth weights, defined in Eq. (4.42), are replaced by

ss,is Wn,α j

(n,αj ) (n,αj ) ns , nm m ss,is n , = Wn−1 flat,(i) (n−1) (n−1) mn ns , nm k pA Wn−1 k flat,(i) Wn

flat,(i)

(n,α )

(n,α )

(13.9)

where the multicontact weights Wn (ns j , nm j ) ∼ 1/g (n,αj ) (n,αj ) have nm ,ns again to be determined recursively. The extension of this method incorporating more than two system parameters is straightforward, but the efficiency of flattening the high-dimensional histograms at all levels of the growth process decreases, whereas the storage requirements for these fields rapidly increase.

218

13. Adsorption of Polymers at Solid Substrates

13.4 Pseudophase Diagram of a Flexible Polymer near an Attractive Substrate For an exemplified study of a hybrid system in equilibrium we choose a polymer with 179 monomers. Since its length is a prime number, the polymer is unable to form perfect cuboid conformations on the sc lattice, as it is, e.g., the case for a 100-mer [253]. There, two low-temperature subphases, dominated by the same 4×5×5 cuboid, were found. In one subphase it had 20 surface contacts, while in the other the cuboid was simply rotated, entailing 25 surface contacts. This is a typical example, where the exact number of monomers in the linear chain is directly connected to the occurrence of such specific pseudophases which are not, of course, phases in the traditional view. Nonetheless, the enormous progress in high-resolution experimental structure analyses and in the technological equipment for precise polymer deposition, as well as the natural finite length of classes of polymers (e.g., peptides and proteins), explain the growing interest in pseudophases and the conformational transitions between them. Here, we mainly focus on the expected thermodynamic phase transitions [247,248] and low-temperature higher-order layering pseudophase transitions [250]. The following results were obtained from contact-density chain-growth simulations [254] of the 179-mer in a cavity with zw = 200 (see Fig. 13.1), choosing uniformly distributed starting points at random. In eight independent runs 1.6 × 109 polymer conformations were generated in total. The resulting contact density gns nm and accumulated densities like Racc (ns, nm) are independent of external parameters such as temperature T and reciprocal solubility s. Concrete values of statistical quantities for specific parameter settings are obtained by simple reweighting as in Eqs. (13.2) and (13.8).

13.4.1 Solubility-Temperature Pseudophase Diagram As it has been outlined earlier, finite-size systems usually exhibit a zoo of crossover- or pseudotransitions, most of which disappear in the thermodynamic limit. These peculiarities of finite systems are also relevant for the polymer adsorption problem considered here. In Fig. 13.2 we have plotted the projection of the specific heat profile onto the solubility-temperature plane as obtained from a simulation of a lattice homopolymer with 179 monomers in a cavity with zw = 200. The color code reflects the value of the specific heat and the brighter the shading, the larger the value of CV . Black and white lines emphasize the ridges of the profile. Since we consider the specific heat as appropriate to identify pseudophases, these ridges mark the pseudophase boundaries. As expected, the pseudophase diagram is divided into two main parts, the phases of adsorption and desorption.

13.4 Pseudophase Diagram of a Flexible Polymer near an Attractive Substrate

219

5.0 4.0 3.0 2.0 s 1.0 0.0 -1.0 -2.0 0.0

1.0

2.0

3.0

4.0

5.0

T

Figure 13.2: Solubility-temperature pseudophase diagram of a 179-mer [254]. The color encodes the specific heat as a function of reciprocal solubility s and temperature T – the brighter the larger its value. Drawn lines emphasize the ridges of the profile and indicate transitions between the different conformational phases. Black lines mark expected thermodynamical phase transitions, while white lines belong to pseudotransitions specific to finite-length polymers. Along the dashed black line coexisting desorbed and adsorbed conformations are equally probable.

The two desorbed pseudophases DC (desorbed-compact conformations) and DE (desorbed-expanded structures) are separated by the collapse transition line which corresponds to the Θ transition of the infinite-length polymer which is allowed to extend into the three spatial dimensions.1 The region of the adsorbed pseudophases is much more complex, and little is known about its details, since it is relevant at lower temperatures, where conventional Monte Carlo methods with pivot-like updates usually tend to fail. The presence of general phases of adsorbed-expanded (AE) and adsorbed-compact (AC1, AC2) conformations was postulated in adsorption studies of grafted polymers and the existence of an additional phase of surface-attached globules (AG) was assumed [247,248,250]. In a recent study [250], it was argued that the layering transition between AC1 and AC2 is a thermodynamic phase transition. Although the polymer in our study is still relatively small, we can clearly identify pseudophases in Fig. 13.2 which can be assigned these labels, too. Those regions are separated by the black lines indicating the transitions between them. We expect that these are transitions in the thermodynamic meaning; only the 1

In very poor solvent (i.e., for very large values of s), the DC phase may degenerate, at least for flexible finitelength polymers, into separate globular and crystalline subphases (see, e.g., Refs. [88,99,100]), but this effect is not in the focus of the present study of adsorption properties.

220

13. Adsorption of Polymers at Solid Substrates 800 600

210 hnm i

180 hn2s ic

CV , 400

hn2s ic ,

hn2m ic ,

hns nm ic

150 120 hns i,

hn2m ic

hns i

90 hnm i

200

-200 0.0

60

CV

0

hns nm ic 0.5

1.0

1.5

2.0 T

30 2.5

3.0

3.5

0 4.0

Figure 13.3: Expectation values, self- and cross-correlations of the contact numbers ns and nm as functions of the temperature T in comparison with the specific heat for a 179-mer in solvent with s = 1.

precise location of the transition lines will still change with increasing length of the polymer. Thus, this picture confirms the previously assumed phases and it provides evidence that the AG phase is indeed there. Furthermore, we have also highlighted by white lines transitions between pseudophases which will probably not survive in the thermodynamic limit. This concerns, e.g., the higher-order layering transitions among the compact pseudophases AC2a1,2-d. In the following sections we will analyze the properties of the pseudophases in more detail.

13.4.2 Contact-Number Fluctuations The contact numbers ns and nm can be considered as system parameters appropriately describing the state of the system and are therefore useful to identify the pseudophases. Peaks and dips in the external-parameter dependence of selfcorrelations hn2s ic , hn2m ic and cross-correlations hns nm ic indicate activity in the contact-number fluctuations and, analyzing the expectation values hns i and hnm i in these active regions of the external parameters T and s, allow for an interpretation of the respective conformational transitions between the pseudophases. In Fig. 13.3, we have plotted for the 179-mer these quantities and, for comparison, the specific heat as functions of the temperature T at a fixed solvent parameter s = 1. This example is quite illustrative as the system experiences several conformational transitions when increasing the temperature starting from T = 0 (see Fig. 13.2). At temperatures very close to T = 0 (pseudophase AC1) all 179 monomers have contact to the substrate and 153 monomer-monomer contacts are formed. This is the most compact contact set being possible for

13.4 Pseudophase Diagram of a Flexible Polymer near an Attractive Substrate

221

topologically two-dimensional, filmlike conformations. It should be noted, however, that approximately 2 × 1018 conformations (self-avoiding walks) belong to this contact set.2 This high degeneracy is an artifact of the minimalistic lattice polymer model used. It is remarkable that the conformations with the highest number of total contacts n = ns + nm are filmlike compact (n = 332). All other conformations we found possess less contacts, even the most compact contact set that dominates the five-layer pseudophase AC2a1 (ns = 36, nm = 263, i.e., n = 299). The reason is that for low temperatures, those macrostates are formed which are energetically favored. Entropy is not yet relevant – for the s = 1 example hns i drops to 149 only up to T ≈ 0.3. Increasing the temperature further, the situation dramatically changes, as can be seen in Fig. 13.3. In a highly cooperative process, the average number of intrinsic contacts hnm i significantly increases (to ≈ 208) at the expense of surface contacts (hns i drops to approximately 104). Consequently, the strong fluctuations hn2s,m ic signalize a conformational transition, and the anticorrelation indicated by hns nm ic confirms that surface contacts turn into intrinsic contacts, which indirectly leads to the conclusion that the filmlike structure is given up in favor of layered, spatially three-dimensional conformations. The system has entered subphase AGe which is the part of the phase AG, where two-layer conformations dominate. The subphase transition near T ≈ 0.7 from the two-layer (AGe) to the bulky regime of AG is due to the ongoing, rather unstructured expansion of the polymer into the z direction by forming so-called surface-attached globules [248]. This is accompanied by a further reduction of surface contacts, while the number of intrinsic contacts changes weakly. Approaching T ≈ 2.0, the situation is just vice versa. Intrinsic contacts dissolve and the system experiences a conformational phase transition from globular conformations in AG to random strands in AE. Crossing this transition line, the system enters the good-solvent regime. Eventually, close to T ≈ 2.8, the polymer unbinds off the substrate. A clear signal is observed in the fluctuations of ns , i.e., the number of average surface contacts rapidly decreases. The expanded polymer is “free” and the influence of both walls is effectively steric. This phase (AE) is closely related to the typical random-coil phase of entirely free and dissolved polymers in good solvent. This example shows that a study of the contact number fluctuations is indeed sufficient to qualitatively identify and describe the conformational transitions between the pseudophases of the hybrid system. For this reason, ns and nm are adequate system parameters, which play a similar role like order parameters in 2

In Chapter 4, it has already been stated that it is an enormous advantage of the simple-sampling algorithms based on Rosenbluth sampling [41] compared to importance-sampling methods that they allow for the approximation of the degeneracy (“density”) of states absolutely, i.e., free energy and entropy can explicitly be determined.

222

13. Adsorption of Polymers at Solid Substrates

thermodynamic phase transitions.

13.4.3 Anisotropic Behavior of Gyration Tensor Components One of the most interesting structural quantities in studies of polymer phase transitions is the gyration tensor (13.5). For the hybrid system considered here, it is expected that the respective components parallel (13.6) and perpendicular (13.7) to the substrate will behave differently when the polymer passes pseudotransition lines. In order to prove this anisotropy explicitly, we have plotted in Fig. 13.4 the expectation values, hRk,⊥ i, and the fluctuations of these two components, dhRk,⊥ i/dT , again for the polymer in solvent with s = 1. For interpreting the peaks of the fluctuations, we have also included once more the specific heat curve for comparison. The immediate observation is that the temperatures, where one or both gyration tensor components exhibit peaks, almost perfectly coincide with those of the specific heat. This is a strong confirmation for the phase diagram in Fig. 13.2 which is based on the specific heat. Obviously, even for the rather short polymer with 179 monomers, we encounter the onset of fluctuation collapse near the (pseudo)phase transitions [254]. This is very promising for future quantitative finite-size scaling analyses. At very low temperatures, i.e., in pseudophase AC1, we have argued in the previous section that the dominant polymer conformation is the most compact single-layer film. This is confirmed by the behavior of hRk i and hR⊥ i, the latter being zero in this phase. A simple argument that the structure is indeed maximally compact is as follows. It is well known that the most compact shape in the two-dimensional continuous space is the circle. For n monomers residing in it, n ≈ πr2 , where r is the (dimensionless) radius of this circle. The usual squared gyration radius is Z 1 1 2 2 circ (13.10) d2 r ′ r ′ = r 2 Rgyr (≈ Rk2 ) = 2 πr 2 r′ ≤r

p circ ≈ n/2π ≈ 5.34 for n = N = 179. Indeed, this is close and therefore Rgyr to the value Rk ≈ 5.46 of the ground-state conformation we identified in phase AC1. Note that the most compact shape in the simple lattice polymer model we used in our study is a square and not a discretized circle.3 Near T ≈ 0.3, the strong layering transition from AC1 to AGe is accompanied by an immediate decrease of hRk i, while hR⊥ i rapidly increases from zero to about 0.5 which is exactly the gyration radius (perpendicular to the layers) of a two-layer 3

This is also true for most compact three-dimensional structures which are cubelike. Therefore, globules in the lattice model are rather cubes than spheres.

13.5 The Whole Picture: The Free-Energy Landscape 4

5

3 CV /50,

2

4 √ hRk i/ 2

CV /50

3

dhR⊥ i/dT, 1 dhRk i/dT

2

0

dhR⊥ i/dT

-1 -2 0.0

223

dhRk i/dT

hR⊥ i, √ hRk i/ 2

1

hR⊥ i 0.5

1.0

1.5

2.0 T

2.5

3.0

3.5

0 4.0

Figure 13.4: Anisotropic behavior of gyration tensor components parallel and perpendicular to the substrate and their fluctuations as functions of the temperature T for a 179-mer at s = 1. For comparison, we have also plotted the associated specific-heat curve.

system, where both layers cover approximately the same area. Note that the single layers are still compact, but not maximally. Applying the same approximation as in Eq. (13.10), the planar gyration radius for each of the two layers is now (with circ n ≈ N/2) Rgyr ≈ 3.77, while we measured in this phase (AGe) Rk ≈ 4.05. This separates the subphase AGe from the other two-layer pseudophase AC2d in Fig. 13.2, where the dominating conformation has perfect two-layer (lattice) structure with Rk ≈ 3.85 (this is the same 2% difference between continuous and lattice calculation for perfect shapes as above). We will discuss the conformational peculiarities in the following in more detail. The subphase transition from AGe to AG near T ≈ 0.7 is accompanied by a further decrease of hRk i whereas hR⊥ i increases, i.e., the height of the surface-adsorbed globule increases at the expense of the width. This tendency is stopped when approaching the transition (T ≈ 2.0) from the globular regime AG to the phase of expanded, but still adsorbed conformations. While hR⊥ i remains widely constant (the fluctuation does not signalize any transition), the polymer strongly extends in the directions parallel to the substrate, as indicated by the peak of dhRk i/dT . After unbinding from the substrate, parallel and perpendicular gyration radii behave widely isotropically (hR⊥ i2 ≈ hRk i2 /2 ≈ hRgyr i2 /3) as the influence of the isotropy-disturbing walls is weak in this regime.

13.5 The Whole Picture: The Free-Energy Landscape It was shown in Section 13.4.2 that the contact numbers ns and nm are unique system parameters for the pseudophase identification of the hybrid system. We

224

13. Adsorption of Polymers at Solid Substrates

define the restricted partition sum for a macrostate with ns surface contacts and nm monomer-monomer contacts by X ′ ′ δn′s ns δn′m nm gn′s n′m e−Es (ns ,nm )/kB T = gns nm e−Es (ns ,nm )/kB T , ZT,s (ns, nm ) = n′s ,n′m

(13.11) such that Z = ns ,nm ZT,s (ns , nm ). Assuming as usual that the dominating macrostate is given by the minimum of the free energy as a function of appropriate system parameters, it is useful to define the specific contact free energy as a function of the contact numbers ns and nm , P

FT,s(ns , nm) = −kB T ln gns nm e−Es (ns ,nm )/kB T = Es (ns, nm ) − T S(ns , nm), (13.12) identifying kB ln gns nm ≡ S(ns, nm ) as a “microcontact” entropy. For given external parameters T and s, this relation can be used to determine the minimum of the contact free energy and therefore allows the identification of the dominant macrostate with respect to the contact numbers. In turn, this quantity allows for an alternative representation of the pseudophase diagram, complementary to the one shown in Fig. 13.2 in that it is related to the contact numbers ns and nm . This is done by determining for (in principle) all values of the external parameters T and s the minima of the contact free energy (13.12). Then the pair of values ns and nm of the minimum contact free-energy state are marked in an ns -nm phase diagram. This is shown in Fig. 13.5, where all free-energy minima of the 179-mer for the parameter set T ∈ [0, 10] and s ∈ [−2, 10] are included and, based on the arguments of the previous section, differently shaded according to the pseudophase they belong to. A nice aspect of this representation is that it allows for the differentiation of continuous and discontinuous pseudophase transitions. The first important observation is that the diagram is divided into two separate regions, the pseudophases of desorbed conformations (DC and DE) and the remaining different phases of adsorption. The “space” in-between is blank, i.e., none of these (possible) conformations was found to be a free-energy minimum conformation. This shows that transitions between the adsorbed and desorbed pseudophases are always first-order-like. It should be noted, that the regime of contact pairs (ns, nm) lying above the shown compact phases is forbidden, i.e., conformations with such contact numbers do not exist on the sc lattice. The second remarkable result is that the pseudophases DC, DE, AE, and AG are “bulky”, while all AC subphases are highly localized in the plot of the freeenergy minima. Comparing with Fig. 13.2, the conclusion is that conformations in the AC phases are energetically favored (more explicitly, for s/T > 0.8 in AC1 and s/T > 2.2 in the AC2 subphases), while the behavior in the other

13.5 The Whole Picture: The Free-Energy Landscape

225

280 240 200 160 nm

s=1

120 80 40 0 0

20

40

60

80

100

120

140

160

180

ns

Figure 13.5: Map of all minima of the contact free energy FT,s (ns , nm ) in the parameter intervals T ∈ [0, 10] and s ∈ [−2, 10] for the 179-mer. The solid line connects the free-energy minima taken by the polymer in solvent with s = 1 by increasing the temperature from T = 0 to T = 5 and thus symbolizes its “path” through the free-energy landscape. The solid line is only a guide to the eyes.

pseudophases is entropy-dominated: The number of conformations with similar contact numbers in the globular or expanded regime is higher than the rather exceptional conformations in the compact phases, i.e., for sufficiently small s/T ratios the entropic effect overcompensates the energetic contribution to the free energy. The subphases AC2a1,2-d are strongly localized, thornlike “peninsulas” standing out from the AG regime. The discrete number and their separation leads to the conclusion that they have related structures. Indeed, as can be seen in Table 13.1, where we have listed representative conformations for all pseudophases, the few conformations dominating these subphases exhibit compact layered structures. The most compact three-dimensional conformation with 263 monomer-monomer contacts and 36 surface contacts is favored in subphase AC2a1 and possesses five layers. Starting from this subphase and increasing the temperature, two things may happen. A rather small change is accompanied with the transition to AC2a2, where the number of intrinsic contacts is reduced but the global five-layer structure remains. On the other hand, passing the transition line towards AC2b, the monomers prefer to arrange in compact four-layer conformations. Advancing towards AC2d, the typical conformations reduce layer by layer in order to increase the number of surface contacts. In AC2d, there are still two layers lying almost perfectly on top of each other. This is similar in subphase AGe, where also two-layer but less compact conformations dominate. In pseudophase AC1

226

13. Adsorption of Polymers at Solid Substrates

Table 13.1: Representative minimum free-energy examples of conformations in the different pseudophases of a 179-mer in a cavity. The substrate is shaded in light gray.

pseudophase

example

ns nm

DC

0 219

DE

0

50

AE

135 33

AG

49 227

AC2a1

36 263

AC2a2

39 256

AC2b

46 257

AC2c

60 251

AC2d

90 231

AGe

103 207

AC1

179 153

only the film-like surface layer remains. The reason for the differentiation of the phases AC1 and AC2 of layered conformations is that the transition from singleto double-layer conformations is expected to be a real phase transition, while the transitions between the higher-layer AC2 subphases are assumed to disappear in

13.5 The Whole Picture: The Free-Energy Landscape

227

the thermodynamic limit [250]. As can be seen in Fig. 13.2, a transition between AC1 and the phase of adsorbed, expanded conformations, AE, is possible. Since these two phases are connected in Fig. 13.5, we expect that the transition in between is second-order-like. Indeed, this transition is strongly related with the two-dimensional Θ transition since, close to the transition line, all monomers form a planar (surface-)layer. Similarly, there is also a second-order-like transition line s0 (T ) between AG and AE which separates the regions of poor (AG: s > s0 ) and good (AE: s < s0 ) solvent. Also, the transition between the desorbed compact (DC) and expanded (DE) conformations is second-order-like: This transition is strongly related with the well-known Θ transition in three dimensions [91]. Eventually, the transitions from the layer-phases AC2a2, AC2b, AC2c, and AGe to the globular pseudophase AG as well as transitions between pseudophases dominated by the same layer type (i.e., between the two-layer subphases AC2d and AGe, and between the five-layer subphases AC2a1 and AC2a2) are expected to be continuous. On the other hand, the transitions among the energetically caused compact low-temperature pseudophases are rather first-order-like, due to their noticeable localization in the map of free-energy minima (Fig. 13.5). The possible transitions (see Fig. 13.2) are AC2a1,2–AC2b, AC2b–AC2c, and AC2c–AC2d, respectively. Even more interesting, however, are the transitions from the single-layer pseudophase AC1 to the double-layer subphases AC2d and AGe. In the previous sections we have already discussed this transition for the special choice s = 1, where near T ≈ 0.3 the fluctuations of the contact numbers and the components of the gyration tensor exhibit a strong activity. We have included into Fig. 13.5 the “path” of macrostates the system passes by increasing the temperature from T = 0 to T = 5. At T = 0 the system is in a film-like, single-layer state. Near T ≈ 0.3 it indeed suddenly rearranges into two layers and enters subphase AGe in a single step. In Fig. 13.6(a), we have plotted the probability distribution pT,s(ns , nm) for s = 1 and T = 0.34 and it can clearly be seen that two distinguished macrostates coexist [266,267]. Increasing the temperature further, the system undergoes the continuous transitions from AGe via AG until it unfolds when entering pseudophase AE. The system is still in contact with the substrate. Close to a temperature T ≈ 2.4, however, the unbinding of the polymer off the substrate happens (from AE to DE). Comparing Figs. 13.5 and 13.6(b), where the probability distribution at T = 2.44 is shown, we see also a clear indication for a discontinuous transition. Note that we consider here the transition state, where the two minima of the free energy coincide4 (see also the black dashed line in Fig. 13.2) and not the point, 4

Strictly speaking, the proper definition of the coexistence point is the temperature, where the weights under

228

13. Adsorption of Polymers at Solid Substrates

pT,s (ns , nm )

(a)

5 × 10−2 4 3 2 1 0

AGe AC1

80

100

120

ns

140

160

180

260 180 n m 100

pT,s (ns , nm ) 5 × 10−4 4 3 2 1 0

(b) DE

0

20

AE

40

ns

60

80

100 120

0

160 80 n m

Figure 13.6: Probability distributions pT,s (ns , nm ) for the 179-mer in solvent with s = 1 (a) near the layering transition from AC1 to AGe at T ≈ 0.34 and (b) near the adsorption-desorption transition from AE1 to DE at T ≈ 2.44. Both transitions are expected to be real phase transitions in the thermodynamic limit and look first-order-like.

where the width of the distribution, i.e., the specific heat, is maximal. Since the system is finite, the transition temperature (T ≈ 2.8), as signaled by the fluctuations studied in the previous sections, deviates slightly from the transitionstate temperature reported here.

13.6 Continuum Model of Adsorption In order to get rid of undesired lattice effects like the almost cuboid form of the most compact adsorbed conformations in the subphases of AC2 in the phase diagram of a polymer on a simple-cubic lattice (see Fig. 13.2), we now investigate the two peaks are equal, see, e.g., Refs. [266,267].

13.6 Continuum Model of Adsorption

229

the structure of conformational phases of a semiflexible off-lattice polymer near an attractive substrate [257]. In this polymer–substrate model, nonbonded pairs of monomers as well as monomers and the substrate interact via attractive van der Waals forces. We simply use the semiflexible homopolymer variant of the AB model [22] that we have already employed to understand protein folding properties from a mesoscopic perspective [27,28] in Chapters 8 and 9.

13.6.1 Off-Lattice Modeling As on the lattice, we assume that adjacent monomers are connected by rigid covalent bonds. Thus, the distance is fixed and set to unity. Bond and torsional angles are free to rotate. The energy function consists of three terms, E = Vbend + VLJ + Vsur ,

(13.13)

associated with the bending stiffness (Vbend ), monomer–monomer LennardJones interaction (VLJ ), and monomer–surface attraction (Vsur ). The Lennard-Jones of nonbonded monomers is of standard form, VLJ = PN −2 PN potential −6 −12 4 i=1 j=i+2(rij − rij ) where rij = |rj − ri | is the distance between the monomers i and j. The lowest-energy distance of the Lennard-Jones potential between two monomers is rmin = 21/6 ≈ 1.12 and P is hence slightly larger than the −2 unity bond length. The bending energy is Vbend = N i=1 [1 − cos(ϑi )]/4, where ϑi is the bending angle in the interval [0, π). The bending energy can be considered as a penalty for successive bonds deviating from a straight arrangement. For the interaction with the substrate we assume that each monomer interacts with each atom of the substrate via a Lennard-Jones potential. Considering for simplicity the attractive substrate as a continuum with atomic density n in the half-space z < 0, the surface interaction potential of the ith monomer in a distance zi from the surface can be written as h Z 2π Z ∞ Z zi i−3 i−6 h 2 2 vsur (zi) = 4n − z ′ + r2 z ′ + r2 dφ dz ′ drr 0 0 −∞ 2πn 2 −9 = (13.14) zi − zi−3 3 15 PN PN −3 −9 and thus Vsur = v (z ) = ǫ sur i s i=1 i=1 (2zi /15 − zi ), where ǫs = 2πn/3 defines the surface attraction strength. As such it weighs the energy scales of intrinsic monomer–monomer attraction and monomer–surface attraction. In order to prevent the molecule from escaping in the simulations of this model, the upper half-space z > 0 is regularized by a steric wall which is placed a distance z = Lbox away from the attractive surface. The following results for a

230

13. Adsorption of Polymers at Solid Substrates

chain with N = 20 monomers were obtained in multicanonical simulations for 51 different surface attraction strengths ǫs , ranging from ǫs = 0, . . . , 5, reweighted to temperatures T ∈ (0, 5]. Each simulation consisted of 108 sweeps [257].

13.6.2 Suitable Energetic and Structural Quantities for Phase Characterization In order to obtain as much information as possible about the canonical equilibrium behavior, we define the following suitable quantities O. Next to the canonical expectation values hOi, we also determine the fluctuations about these averages, as represented by the temperature derivative dhOi/dT = (hOEi − hOi hEi) /T 2 (kB = 1). Apart from energetic fluctuations such as the specific heat per monomer cV = d hEi /N dT and fluctuations of structural quantities like the parallel and perpendicular components of the gyration tensor, Rk and R⊥ , respectively [cf. Eqs. (13.6) and (13.7)], clear evidence that the polymer is on average freely moving in the box or very close to the surface can bePprovided by the average distance of the center-of-mass of the polymer, hzcm i = N i=1 hzi i/N , to the surface. Another useful quantity is the mean number of monomers docked to the surface. A single-layer structure is formed if all monomers are attached at the substrate; if none is attached, the polymer is quasifree (desorbed) in solvent. The surface potential is a continuous potential and in order to distinguish monomers docked to the substrate from those not being docked, it is reasonable to introduce a cutoff. A monomer i shall be defined as being “docked” if zi < zc ≡ 1.2. The corresponding measured quantity is the average ratio hns i of monomers docked to the surface and the P total number of monomers. This can be expressed as ns = Ns/N with Ns = N i=1 Θ(zc − zi ), where Θ(z) is the Heaviside step function.

13.6.3 Comparative Discussion of Structural Fluctuations The radius of gyration provides an excellent measure of the globular compactness of polymer conformations and its components parallel [see Figs. 13.7(a,b)] and perpendicular [Figs. 13.7(c,d)] to the surface, respectively, are helpful in indicating structural changes induced by the presence of an attractive substrate.

For example, for ǫs ≥ 3.4, hR⊥i vanishes at low temperatures, while Rk attains small values at lower attraction strengths ǫs . The vanishing of hR⊥ i corresponds to conformations, where the polymer is spread out flat on the surface without any extension into the third dimension. The associated pseudophases are called adsorbed compact (AC1) and adsorbed expanded (AE1) phase. The ‘1’ is

13.6 Continuum Model of Adsorption

(a)

(b)

231

(c)

(d)

2.6

2.4

1.4

1.2

2.2

ǫs = 4

2

ǫs = 5 1

hR⊥ i (T )

Rk (T )

ǫs = 3

1.8

ǫs = 3.4

ǫs = 3

ǫs = 0

0.8

ǫs = 1 0.6

ǫs = 2

1.6

ǫs = 3.4 ǫs = 4 ǫs = 5

0.4 1.4

ǫs = 2 ǫs = 1

1.2

0.2

ǫs = 0 0

1 0

0.5

1

1.5

2

2.5

T

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

T

Figure 13.7: (a) Parallel component of the radius of gyration Rk as a function of temperature

T and adsorption strength ǫs for a 20mer, (b) Rk for selected values of ǫs , (c) and (d) the same for the perpendicular component hR⊥ i.

tagged in order to distinguish these phases from topologically three-dimensional phases, such as, e.g., the AC2 subphases. The phases AC1 and AE1 are separated by the freezing transition such that polymer structures in AC1 are maximally compact at lower temperatures, while AE1 conformations are less compact and more flexible but still lie rather planar at the surface. In order to verify that conformations in AC1 are indeed maximally compact single layers, we argue as follows. The most compact shape in the two-dimensional

(2D) continuous space is the circular disk. Thus one can calculate Rk for a disk and compare it with the simulated value. Assuming N monomers to be distributed evenly in the disk, N ≈ πr2 , where r is the radius of the disk in units of the mean distance of neighboring monomers. The radius of gyration in the same units is thus given by Z N 1 1 2 2 . (13.15) d2 r′ r′2 = r2 ≈ Rgyr,disk = Rk,disk = 2 πr r′ ≤r 2 2π Since we have two types of mean distances between monomers in compact conformations dependent on whether they are adjacent p along the chain or

not,we expect for disk-like conformations on the surface: 20/2π ≈ 1.784 < Rk,20 <

232

13. Adsorption of Polymers at Solid Substrates

p 2.003 ≈ rmin 20/2π. The simulated value is Rk,20 ≈ 1.81 which nicely fits the estimate. The argument is similar for sphere-like three-dimensional (3D) compact conformations with N = 4πr3 /3. Corresponding conformations are found as free desorbed compact chains (DC), as well as adsorbed compact polymer conformations (AC2a) for weak surface attraction. In this case, the radius of gyration is given by 2/3 Z 3 3N 3 3 2 d3r′ r′2 = r2 ≈ . (13.16) Rgyr,sph = 3 4πr r′ ≤r 5 5 4π

The estimate 1.464 < Rgyr,20 < 1.684 slightly is slighly larger than the simulated value Rgyr,20 = 1.242 which can be explained by the fact that the mass of the polymer is not uniformly distributed in the sphere as it is assumed in the calculation. For a compact packing of discrete monomer positions, it is more realistic that the outer thin shell of the sphere does not contain any monomers. Performing the integration not from r′ = 0 to r′ = r, but only to r′ = r − ε, reduces the estimated radius of gyration significantly already for small ε due to the increased weight of the outer shells in higher dimensions. Taking this effect into account, the thus obtained values of hRgyri seem to be even more reasonable. The most pronounced transition is the strong layering transition at ǫs ≈ 3.4 that separates regions of planar conformations (AC1, AE1) plane from the region of stable double-layer structures (AC2b) and adsorbed globules (AG), below and above the freezing transition respectively. For high surface attraction strengths ǫs , it is energetically favorable to form as many surface contacts as possible. In the layering-transition region, a higher number of monomer–monomer contacts causes the double-layer structures to have just the same energy as single-layer structures. For lower ǫs -values, the double-layer structures possess the lowest energies. Hence, this transition is a sharp energetic transition. Although for the considered short chain no higher-layer structures exist, the components Rk,⊥ indicate some activity for low surface attraction strengths. For N = 20, ǫs ≈ 1.4 is the lowest attraction strength, where still stable doublelayer conformations are found below the freezing transition. What follows for lower ǫs values after a seemingly continuous transition is a low-temperature subphase of surface attached compact conformations, called AC2a. AC2a conformations occur, if the monomer–surface attraction is not strong enough to induce layering in compact attached structures. The characterization of structures in this subphase requires some care, as system-size effects are dominant. Although the surface attraction is sufficiently strong to enable polymer–substrate contacts, compact desorbed polymer conformations below the Θ-transition are not expected to change much. Thus, layering effects do not occur.

13.6 Continuum Model of Adsorption

(a)

(b)

233

(c)

(d)

0.8

ǫs = 3

0.7

0.6

ǫs = 5

ǫs = 2

ǫs = 4

0.5

ǫs = 1

0.6 0.4

d hR⊥ i /dT (T )

d Rk /dT (T )

0.5 0.4 0.3

ǫs = 1

ǫs = 0

0.2 0.1

ǫs = 2

ǫs = 4

0.3

0.2

ǫs = 5

0.1

ǫs = 0

ǫs = 3.4

0

0

-0.1

ǫs = 3

ǫs = 3.4

-0.2 0

0.5

-0.1 1

1.5

T

2

2.5

0

0.5

1

1.5

2

2.5

T

Figure 13.8: (a) d Rk /dT as a function of T and ǫs , (b) d Rk /dT for selected values of ǫs , (c) and (d) the same for d hR⊥ i /dT .

In the wetting transition, elastic polymers with stretchable bonds can form perfectly icosahedral morphologies. This would additionally stabilize the polymer conformation and is already known from studies of atomic clusters. The smallest icosahedron with characteristic fivefold symmetry is formed by 13 atoms. Larger perfect icosahedra require also a “magic” number (55,147,309,. . .) of atoms. Thus, it is plausible that the wetting transition is accompanied by strong finitesize effects and morphologies of adsorbed crystalline structures depend on the precise length of the polymer. Raising the temperature above the freezing temperature, polymers form adsorbed and still rather compact conformations which look like globular, unstructured drops on the surface (AG: surface-attached globules [248]). This phase was also found in the lattice-polymer adsorption study. At even higher temperatures, two scenarios can be distinguished in dependence of the relative strengths of monomer–monomer and monomer–substrate interactions. In the first case, the polymer first desorbs from the surface [from AG to the desorbed globular (DG) bulk phase] and disentangles at even higher temperatures [from DG to the desorbed expanded bulk phase (DE)]. In the latter case, the polymer expands while it is still located at the surface (from AG to AE2) and desorbs at higher temperatures (from AE2 to DE). Due to the higher relative number of monomer–monomer con-

234

13. Adsorption of Polymers at Solid Substrates

(a)

(c)

(b)

(d)

ǫs = 5

1

20

ǫs = 0

ǫs = 4 ǫs = 3.4

0.8

ǫs = 1

ǫs = 3 hns i (T )

hzcm i (T )

15

ǫs = 2 ǫs = 3

10

ǫs = 3.4

ǫs = 4

0.6

ǫs = 2

0.4

ǫs = 5

5

ǫs = 1 0.2

ǫs = 0 0

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

T

5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

T

Figure 13.9: (a) Mean center-of-mass coordinate hzcm i as a function of T and ǫs , (b) hzcm i for selected values of ǫs , (c) and (d) the same for the mean number of surface contacts hns i.

tacts in compact bulk conformations of longer chains, the Θ-temperature increases with N . The same holds true for the surface attraction strength ǫs associated with the layering transition.

13.6.4 Adsorption Parameters The adsorption transition can be discussed best when looking at the distance of the mean center-of-mass distance hzcm i of the polymer from the surface [Figs. 13.9(a) and (b)] and the mean number of surface contacts hns i [Figs. 13.9(c) and (d)]. As can be seen in Fig. 13.9(a), for large temperatures and small values of ǫs , the polymer can move freely within the simulation box and the influence of the substrate is purely steric. Thus the mean of the center-of-mass distance hzcm i of the polymer from the surface is just half the height of the simulation box. On the other hand, for large enough surface attraction strengths and low temperatures, the polymer favors surface contacts and the mean center-ofmass distance converges to hzcm i ≈ 0.858, corresponding to the minimum-energy distance of the surface attraction potential for single-layer structures (and correspondingly larger values for double-layer and globular structures).

13.6 Continuum Model of Adsorption

235

One clearly identifies a quite sharp adsorption transition that divides the projection of hzcm i in Fig. 13.9(a) into an adsorbed (bright) regime and a desorbed (dark) regime. This transition appears as a straight line in the phase diagram and is parametrized by ǫs ∝ T . Intuitively, this makes sense since at higher T the stronger Brownian fluctuation is more likely to overcome the surface attraction. For the detailed discussion of these adsorbed phases, let us concentrate ourselves on the mean number of surface contacts hns i shown in Figs. 13.9(c) and (d). Unlike in the simple-cubic lattice studies, where one finds hns i ≈ 1/l for an llayer structure [250], we find for double-layer structures hns i > 1/2. The reason is that most compact multi-layer structures are cuboids on the lattice, while in the continuum case, “layered” conformations correspond to semispherical shapes, where – for optimization of the surface of the compact shape – the surface layer contains more monomers than the upper layers. Since this only regards the outer part of the layers, the difference is more pronounced the shorter the chain is. Further layering transitions are not observed. When the double-layer structure gets unstable at lower values of ǫs , hns i starts to decrease again. The conformations in AC and AC2a thus do not exhibit a pronounced number of surface contacts, and hns i varies with ǫs . To conclude, the single- to double-layer “layering transition” is a topological transition from two-dimensional to three-dimensional polymer conformations adsorbed at the substrate. The solvent-exposed part of the adsorbed compact polymer structure, which is not in direct contact with the substrate, reduces under poor solvent conditions the contact surface to the solvent. Due to the larger number of degrees of freedom for the off-lattice polymer, layered structures are not favored in this case. Thus, higher-order layering transitions are not identified (which in part is also due to the short length of the chain), but are also not expected in pronounced form.

13.6.5 The Pseudophase Diagram of the Hybrid System in Continuum To summarize all the informations gained from the different observables, we construct the approximate boundaries of different regimes in the T -ǫs plane. The pseudophase diagram is displayed in Fig. 13.10 and the different pseudophases are denoted by the abbreviations introduced in the previous subsections. Transitions between conformational phases are indicated by stripes. It must be noted that their thickness is due to the the different locations of peaks when considering different fluctuating quantities. This is a typical feature of a finite system, where different indicators of transitions (e.g., transition temperatures) do not necessarily coincide. Since this uncertainty is of principle nature, it again questions the usefulness of a quantitative discussion of (pseudo)phase transitions of a finite system by a canonical-ensemble analysis (see also the discussion of nucleation transitions

236

13. Adsorption of Polymers at Solid Substrates

3

DE

2

AE2

1

DG DC 0

T

AG

AC2a 1

AE1 AC1

AC2b 2

ǫs

3

4

5

0

Figure 13.10: Structural phase diagram of an off-lattice homopolymer with 20 monomers interacting with a continuous attractive substrate. The diagram, which is parametrized by the surface attraction strength ǫs and the temperature T , was constructed by means of the informations gained from the quantities discussed before. Stripes separate the individual conformational phases. The thickness reflects the (principle!) uncertainty in the estimation of the location of the transition points and arises from the input of the different quantities investigated for a small system. The conformational phases are discussed in the text.

in Chapters 11 and 12 in this context). The transition lines in the pseudophase diagram (Fig. 13.10), which also still vary with chain length N , represent the best compromise of all canonical quantities analyzed separately. Only in the thermodynamic limit of infinitely long chains, most of the identified pseudophase transitions are expected to occur at sharp values of the parameters ǫs and T for all observables. Taking that into account, the pseudophase diagram gives a good qualitative overview of the behavior of polymers near attractive substrates in dependence of environmental parameters such as solvent quality and temperature. The locations of the phase boundaries should be considered as rough guidelines. For the exemplified 20mer, the following pseudophases can be associated to the different regions in Fig. 13.10. Exemplified conformations being representative for the different phases are shown in Fig. 13.11. Since in the hybrid model discussed here, where the chain is not grafted at the substrate, the polymer can completely desorb from the surface and thus the typical polymer bulk phases are present in the phase diagram, too. These phase are denoted as DE (desorbed expanded) which corresponds to the random-coil phase of the quasifree desorbed polymer, DG (desorbed globular) representing the globular phase of the desorbed chain, and DC (desorbed compact) for the maximally compact, spherically shaped crystalline structures that dominate this desorption phase below the freezing-

13.6 Continuum Model of Adsorption

pseudophase

237

typi al onguration

DE DG DC AE1 AE2 AC1 AG AC2a AC2b Figure 13.11: Representative examples of conformations for a 20mer in the different regions of the T -ǫs pseudophase diagram, shown in Fig. 13.10. DE, DG, and DC represent bulk “phases”, where the polymer is preferably desorbed. In regions AE1, AE2, AC1, AG, AC2a, and AC2b, conformations are favorably adsorbed.

transition temperature. In the adsorption part of the phase diagram, also expanded, compact and globular phase are found, which, however, differ from their desorption counterparts. Two phases of adsorbed expanded conformations are distinguished, AE1 (adsorbed expanded single layer) which labels the phase of expanded, rather planar but little compact random-coil conformations and AE2 (adsorbed expanded threedimensional conformations), associated with adsorbed, unstructured random-coillike expanded conformations with typically more than half of the monomers in contact with the attractive substrate. Highly compact structures are found in AC1 (adsorbed compact single layer), where adsorbed circularly compact filmlike (i.e., two-dimensional) conformations dominate, and in AC2a,b – two subphases of adsorbed compact three-dimensional conformations. AC2a corresponds to adsorbed compact, semi-spherically shaped crystalline conformations, whereas AC2b (adsorbed compact double layers) is the subphase of adsorbed, compact “doublelayer” conformations, where the occupation of the surface layer is slightly larger than that of the other layer. AC2a and AC2b are subphases in the regime of

238

13. Adsorption of Polymers at Solid Substrates

the phase diagram, where adsorbed compact and topologically three-dimensional conformations are dominant. Since pronounced layering transitions as observed in lattice-polymer models are not expected to be relevant in continuum models, the discrimination of AC2a and AC2b is likely to be irrelevant in the thermodynamic limit. However, AC2a,b differ qualitatively from the phase AC1 of topologically two-dimensional polymer films and one thus expects that the transition between filmlike (AC1) and semispherical conformations (AC2) is of thermodynamic relevance. Finally, there is the phase of adsorbed globular three-dimensional conformations, AG, where representative conformations are surface-attached globular conformations and look like internally unstructured drops on the surface. The famous wetting transiton corresponds to passing the transition lines DG → AG → AC2b → AC1 for fluid droplets, while compact polymers wet the surface in the direction DC → AC2a → AC2b → AC1. The latter process can also be interpreted as a melting transition which is simply induced by an increase of the surface attraction strength. In cases, where the maximally compact conformation is more stable due to the high symmetry of the intrinsic icosahedral structure (as, e.g., for a chain of length N = 13), an additional subphase exists: AC (adsorbed icosahedral compact conformations) which is similar to DC, but polymers are in touch with the surface under the given external parameters in this phase.

13.7 Comparison with Lattice Results Finally, we compare the phase diagrams obtained from the sc lattice and the off-lattice models (Figs. 13.2 and 13.10). The energy of the system is given by Eq. (14.1), which we rewrite lattice L L L L here as EL ns , nm = −ǫs ns − ǫLm nLm ; nLs is the number of nearest-neighbor monomer–substrate contacts, nLm the number of nearest-neighbor, but nonadjacent monomer–monomer contacts and ǫLs and ǫLm are the respective contact energy scales. The phase diagram shown in Fig. 13.2 is parametrized by temperature T and monomer–monomer interaction strength. The phase diagram corresponds to the specific-heat profile and surface–monomer attraction strength is fixed. But, from the known contact density (see Section 13.2), the specific-heat profile can also be calculated for fixed monomer–monomer interaction strength ǫLm = 1, while varying the surface attraction parameter ǫLs which corresponds to the off-lattice approach in the previous section. Denoting the energy and temperature units in the original lattice model by E ′ and T ′ , respectively, a simple rescaling yields EL E′ = ′ T T

⇔

nLs + snLm ǫLs nLs + nLm = T′ T

⇔

T′ T = , s

1 ǫLs = , (13.17) s

13.7 Comparison with Lattice Results

239

Figure 13.12: Pseudophase diagram of a lattice polymer with 179 monomers as in Fig. 13.2, but here parametrized by the surface attraction strength ǫLs and the temperature T . The color encodes the specific-heat profile; the darker the color, the larger its value.

where s = ǫLm /ǫLs is the ratio of energy scales of intrinsic and surface contacts as introduced in Section 13.2. Certain similarities between the phase diagrams obtained from the rescaled lattice approach, shown in Fig. 13.12, and that of the off-lattice model (Fig. 13.10) are obvious. For instance, the adsorption transition line is parametrized in both models by ǫs ∝ T . Different, however, is not only the slope that depends on the system’s geometry and energy scales. Also, while for the off-lattice model the extrapolation of the transition line seems to go through the origin ǫs = 0 and T = 0, there is an offset observed in the lattice-system analysis such that the extrapolated transition line roughly crosses ǫLs = 0.4 and T = 0. This might be due to the intrinsic cuboidal structure of the polymer conformations on the sc lattice that possess planar surfaces at low temperatures even in the bulk. Unlike for off-lattice models, where a compact polymer attains a spherical shape, such a cuboidal conformation is likely to dock at a substrate without substantial conformational rearrangements. Here lies an important difference between lattice and off-lattice models. The off-lattice model provides, for sufficiently small surface attraction strengths, a competition between most compact spherical conformations that do not possess planar regions on the polymer surface, and less compact conformations with planar regions that allow for more surface contacts but reduce the number of intrinsic contacts. This also explains, why the wetting transition is more difficult to observe in adsorption studies on regular lattices. On the other hand, AC2 conformations at

240

13. Adsorption of Polymers at Solid Substrates

low T and for ǫs between the adsorption and the single-double layering transitions can be observed in both models. Similarly in both models, there exists the AG pseudophase of surface-attached globules. While for the off-lattice system, apart from the wetting transition, there is only the transition from AC2a (semi-spherical shaped) to AC2b (double-layer structures), on the lattice AC2 comprises a zoo of higher-layer subphase transitions (see Table 13.1). Decreasing the surface attraction at low temperatures, layer after layer is added until the number of layers is the same as in the most compact conformation. A lattice polymer has no other choice than forming layers in this regime. The layering transition from AC1 to AC2 is very sharp in both models. Also the shape of the transition region from topologically two-dimensional adsorbed to three-dimensional adsorbed conformations looks very similar. Summarizing we conclude that, in particular, the high-temperature pseudophases DE, DC/DG, AG, AE, nicely correspond to each other in both models. Noticeable qualitative deviations occur, as expected, in those regions of the pseudophase diagram, where compact conformations are dominant and (unphysical) lattice effects are influential.

14 Hybrid Protein–Substrate Interfaces

14.1 Steps Towards Bionanotechnology The advancing progress in manipulating soft and solid materials at the nanometer scale opens up new vistas for potential bionanotechnological applications of hybrid organic-inorganic interfaces [231,268]. This includes, e.g., nanosensors being sensitive to specific biomolecules (“nanoarrays”), as well as organic electronic devices on polymer basis which have, for example, already been realized in organic light-emitting diodes [269]. An important development in this direction is the identification of proteins that can bind to specific compounds. Over the last decade, genetic engineering techniques have been successfully employed to find peptides with affinity for, e.g., metals [226,270], semiconductors [227–229] and carbon nanotubes [271]. However, the mechanisms by which peptides bind to these materials are not completely understood; it is, for example, unclear what role conformational changes play in the binding process. In these mainly experimental studies, it has also been shown that the binding of peptides on metal and semiconductor surfaces depends on the types of amino acids [272] and on the sequences of the residues in the peptide chain [226– 229]. These experiments reveal many different interesting and important problems, which are related to general aspects of the question why and how proteins fold. This regards, for example, the character of the adsorption process, i.e., whether the peptides simply dock to the substrate without noticeable structural changes or whether they perform conformational transitions while binding. Another point is how secondary structures of peptide folds in the bulk influence the binding behavior to substrates (see Fig. 14.1). In helical structures, for example, side chains are radially directed and – due to the helical symmetry – residues with a certain distance in the sequence arrange linearly. But, under certain conditions (e.g., in the presence of energetically attractive substrates) it could be more favorable, if the protein prefers to take rather flat conformations in order to increase

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Figure 14.1: Two synthetic peptides with sequence AQNPSDNNTHTH facing a bare (100) Si surface. Snapshot from a computer simulation performed with the BONSAI package [273].

the number of surface contacts. If native folds are resistant against refolding due to rather little reactive substrate surfaces, the adsorption propensity is either weak, or a perfect matching of protein structure and the crystal surface, e.g., with respect to polarization, is required. In this case, the peptide simply docks to the substrate. It is also feasible, however, that the peptide is unstructured under given conditions (temperature, pH value of the surrounding solvent) and the binding process to the substrate is accompanied by refolding processes, i.e., the adsorption process is a coupled binding-folding scenario [239,274,275]. Peptide-specific affinity to dock or refold when binding [276,277] is of great importance in pattern recognition processes [236,278–282] such as receptor-ligand binding [237,238]. From the recent experiments of the adsorption of short peptides at semiconductor substrates, it is known that different surface properties (materials such as Si or GaAs, crystal orientation, etc.) as well as different amino acid sequences strongly influence the binding properties of these peptides at the substrate [227– 229]. This specificity will be of particular importance for future sensory devices and pattern recognition [236] at the nanometer scale. The reasons for this binding specificity are far from being clear, and it is a major challenge from both the experimental and the theoretical point of view to understand the basic principles of substrate–peptide cooperativity. This problem can be seen as embedded into a class of similar studies, where the adsorption and docking behavior of polymers is essential, e.g., protein–ligand binding [237], prewetting and layering transitions in polymer solutions as well as dewetting of polymer films [232], molecular pattern formation, electrophoretic polymer deposition and growth [241].

14.2 Specific Peptide Adsorption at Different Substrates

243

In the following, we first discuss the substrate-specificity of heteropolymer adhesion by employing a simple hybrid lattice model [2,255,256,283]. After having gained this qualitative insight into the structural binding behavior, a realistic hybrid peptide–semiconductor system is investigated by means of a sophisticated all-atom modeling approach.

14.2 Specific Peptide Adsorption at Different Substrates 14.2.1 Hybrid Lattice Model In the following, we identify conformational transitions of a nongrafted hydrophobic–polar heteropolymer with 103 residues in the vicinity of a polar, a hydrophobic, and a uniformly attractive substrate. Introducing only two system parameters, the numbers of surface contacts and intrinsic hydrophobic contacts, respectively, we obtain surprisingly complex temperature and solvent dependent, substrate-specific pseudophase diagrams [255,256]. For the study of hybrid peptide-substrate models, we use again the HP transcription of the 103-residue protein cytochrome c, whose low-energy conformations and thermodynamic properties were extensively studied in the past [46,48,51,52] and in Section 7.3. The HP sequence contains 37 hydrophobic and 66 polar residues. A conformation with a highly compact hydrophobic-core, exhibiting 56 hydrophobic contacts, is shown in Fig. 7.7. This lattice peptide resides in a cavity with an attractive substrate (see Fig. 13.1). The distance between the attractive and the steric wall, zw , is chosen sufficiently large to keep the influence on the unbound heteropolymer small (in the actual example zw = 200). In order to study the specificity of residue binding, we distinguish three substrates with different affinities to attract the peptide monomers: (a) the type-independent attractive, (b) the hydrophobic, and (c) the polar substrate. The number of corresponding nearest-neighbor contacts between monomers and substrate shall be P denoted as nH+P , nH s s , and ns , respectively. In analogy to the polymer-substrate model (13.1), we express the energy of the hybrid peptide-substrate system simply by Es(ns , nHH) = −ε0(ns + snHH ), (14.1) where ns = nH+P , nPs , or nH s s depending on the substrate (we set ε0 = 1 in the following). nHH denotes the number of intrinsic nearest-neighbor contacts between hydrophobic monomers only. Nearest-neighbor pairs of polar monomers (P P ) and contacts between polar and hydrophobic residues (HP ) are considered to be nonenergetic in this model (as in the standard HP model). The solubility (or reciprocal solvent parameter) s is, as well as the temperature T , are external

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parameters. Contact-density chain-growth simulations allow a direct estimation of the degeneracy (or density) g(ns, nHH) of macrostates of the system with given contact numbers ns and nHH [255,256].

14.2.2 Substrate-Specific Conformational Adsorption Behavior in Dependence of Temperature and Solubility In Figs. 14.2(a)–14.2(c) the color-encoded profiles of the specific heats for the different substrates are shown (the brighter the larger the value of CV ). The ridges (for accentuation marked by white and gray lines) indicate boundaries of the pseudophases. The gray lines belong to the main transition lines, while the white lines separate pseudophases that strongly depend on specific properties of the heteropolymer, such as its exact number and sequence of hydrophobic and polar monomers. With its degeneracy g(ns , nHH), we define the contact free energy as FT,s(ns, nHH ) = Es (ns, nHH) − T ln g(ns , nHH) and the probability for a macrostate with ns substrate and nHH hydrophobic contacts as pT,s (ns, nHH) ∼ g(ns , nHH) exp(−Es /T ). Assuming that the minimum of the (0) (0) free-energy landscape FT,s(ns , nHH) → min for given external parameters s and (0) (0) T is related to the class of macrostates with ns surface and nHH hydrophobic contacts, this class dominates the phase the system resides in. For this reason, it is instructive to calculate all minima of the contact free energy and to determine the associated contact numbers in a wide range of values for the external parameters. The map of all possible free-energy minima in the range of external parameters T ∈ [0, 10] and s ∈ [−2, 10] is shown in Fig. 14.3 for the peptide in the vicinity of a substrate that is equally attractive for both hydrophobic and polar monomers. Solid lines visualize “paths” through the free-energy landscape when changing temperature under constant solvent (s = const) conditions. Let us follow the exemplified trajectory for s = 2.5. Starting at very low temperatures, we know from the pseudophase diagram in Fig. 14.2(a) that the system resides in pseudophase AC1. This means that the macrostate of the peptide is dominated by the class of compact, filmlike singlelayer conformations. The system obviously prefers surface contacts at the expense of hydrophobic contacts. Nonetheless, the formation of compact hydrophobic domains in the two-dimensional topology is energetically favored but maximal compactness is hindered by the steric influence of the substrate-binding polar residues. Increasing the temperature, the system experiences close to T ≈ 0.35 a sharp first-order-like conformational transition, and a second layer forms (AC2). This

14.2 Specific Peptide Adsorption at Different Substrates

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Figure 14.2: Specific-heat profiles as a function of temperature T and solubility parameter s of the 103-mer near three different substrates that are attractive for (a) all, (b) only hydrophobic, and (c) only polar monomers. White lines indicate the ridges of the profile. Gray lines mark the main “phase boundaries”. The dashed black line represents the first-order-like binding/unbinding transition state, where the contact free energy possesses two minima (the adsorbed and the desorbed state). In the left panel typical conformations dominating the associated AC phases of the different systems are shown.

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is a mainly entropy-driven transition as the extension into the third dimension perpendicular to the substrate surface increases the number of possible peptide conformations. Furthermore, the loss of energetically favored substrate contacts of polar monomers is partly compensated by the energetic gain due to the more compact hydrophobic domains. Increasing the temperature further, the density of the hydrophobic domains reduces and overall compact conformations dominate in the globular pseudophase AG. Reaching AE, the number of hydrophobic contacts decreases further, and also the total number of substrate-contacts. Extended, dissolved conformations dominate. The transitions from AC2 to AE via AG are comparatively “smooth”, i.e., no immediate changes in the contact numbers passing the transition lines are noticed. Therefore, these conformational transitions could be classified as second-order-like. The situation is different when approaching the unbinding transition line from AE close to T ≈ 2.14. This transition is accompanied by a dramatic loss of substrate contacts – the peptide desorbs from the substrate and behaves in pseudophase DE like a free peptide, i.e., the substrate and the opposite neutral wall regularize the translational degree of freedom perpendicular to the walls, but rotational symmetries are unbroken (at least for conformations not touching one of the walls). As the probability distribution in Fig. 14.3 shows, the unbinding transition is also first-order-like, i.e., close to the transition line, there is a coexistence of adsorbing and desorbing classes of conformations. Despite the surprisingly complex phase behavior there are main “phases” that can be distinguished in all three systems. These are separated in Figs. 14.2(a)– 14.2(c) by gray lines. Comparing the three systems we find that they all possess pseudophases, where adsorbed compact (AC), adsorbed expanded (AE), desorbed compact (DC), and desorbed expanded (DE) conformations dominate. “Compact” here means that the heteropolymer has formed a dense hydrophobic core, while expanded conformations are dissolved, random-coil like. The sequence and substrate specificity of heteropolymers generates, of course, a rich set of new interesting and selective phenomena not available for homopolymers. One example is the pseudophase of adsorbed globules (AG), which is noticeably present only in those systems, where all monomers are equally attractive to the substrate [Fig. 14.2(a)] and where polar monomers favor contact with the surface [Fig. 14.2(b)]. In this phase, the conformations are intermediates in the binding/unbinding region. This means that monomers currently desorbed from the substrate have not yet found their position within a compact conformation. Therefore, the hydrophobic core, which is smaller than in the respective adsorbed phase (i.e., at constant solubility s), appears as a loose cluster of hydrophobic monomers. In Figs. 14.4(a)–14.4(c), we have plotted, exemplified for s = 2, the sta-

14.2 Specific Peptide Adsorption at Different Substrates

247

Figure 14.3: Contact-number map of all free-energy minima for the 103-mer and substrate equally attractive to all monomers. Full circles correspond to minima of the contact free energy FT,s (nH+P , nHH ) in the parameter space T ∈ [0, 10], s ∈ [−2, 10]. Lines illustrate how the s contact free energy changes with the temperature at constant solvent parameter s. For the exemplified solvent with s = 2.5, the peptide experiences near T = 0.35 a sharp first-order-like layering transition between single- to double-layer conformations (AC1,2). Passing the regimes of adsorbed globules (AG) and expanded conformations (AE), the discontinuous binding/unbinding transition from AE to DE happens near T = 2.14. In the DE phase the ensemble is dominated by desorbed, expanded conformations. Representative conformations of the phases are shown next to the respective peaks of the probability distributions.

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tistical averages of the contact numbers ns and nHH as well as their variances and covariances for the three systems. For comparison we have also included the specific heat, whose peaks correspond to the intersected transition lines of Figs. 14.2(a)–14.2(c) at s = 2. From Figs. 14.4(a) and 14.4(c) we read off that the transition from AC to AG near T ≈ 0.4 is mediated by fluctuations of the intrinsic hydrophobic contacts. The very dense hydrophobic domains in the AC subphases lose their compactness. This transition is absent in the hydrophobicsubstrate system [Fig. 14.4(b)]. The signal seen belongs to a hydrophobic layering AC subphase transition, which influences mainly the number of surface contacts nHs . The second peak of the specific heats belongs to the transition between adsorbed compact or globular (AC/AG) and expanded (AE) conformations. This behavior is similar in all three systems. Remarkably, it is accompanied by a strong anti-correlation between surface and intrinsic contact numbers, ns and nHH . Not surprisingly, the hydrophobic contact number nHH fluctuates stronger than the number of surface contacts, but apparently in a different way. Dense conformations with hydrophobic core (and therefore many hydrophobic contacts) possess a relatively small number of surface contacts. Vice versa, conformations with many surface contacts cannot form compact hydrophobic domains. Finally, the third specific heat peak marks the binding/unbinding transition, which is, as expected, due to a strong fluctuation of the surface contact number. The strongest difference between the three systems is their behavior in pseudophase AC, which is roughly parametrized by s > 5T . When hydrophobic and polar monomers are equally attracted by the substrate [Fig. 14.2(a)], we find three AC subphases in the parameter space plotted. In subphase AC1, film-like conformations dominate, i.e., all 103 monomers are in contact with the substrate. Due to the good solvent quality in this region, the formation of a hydrophobic core is less attractive than the maximal deposition of all monomers at the , nHH)min = (103, 32). In fact, instead of a surface, the ground state is (nH+P s single compact hydrophobic core there are nonconnected hydrophobic clusters. At least on the used simple cubic lattice and the chosen sequence, the formation of a single hydrophobic core is necessarily accompanied by an unbinding of certain polar monomers and, in consequence, an extension of the conformation into the third spatial dimension. In fact, this happens when entering , nHH)min = (64, 47)], where a single hydrophobic two-layer domain AC2 [(nH+P s has formed at the expense of losing surface contacts. In AC3, the heteropolymer has maximized the number of hydrophobic contacts. Solely, local arrangements of monomers on the surface of the very compact structure can lead to the still possible maximum number of substrate contacts. FT,s is minimal for , nHH)min = (40, 52). (nH+P s

14.2 Specific Peptide Adsorption at Different Substrates

249

200.0 hnH+P i s

150.0 CV

CV ,

6hn2HH ic

0.0 -50.0 0.0

hnHH i

0.5

1.0

1.5

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2.5

2 h(nH s ) ic ,

hnH s nHH ic , hn2HH ic

3.0

3.5

20.0

0.0 4.0 50.0

hnHH i

(b)

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H+P 60.0 hns i, hnHH i 40.0

6hn2HH ic

50.0

150.0

80.0

h(nH+P )2 ic /4 s

100.0 h(nH+P )2 ic /4, s hnH+P nHH ic , s

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)

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2 h(nH s ) ic

30.0

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0.0

20.0

hnH s i,

hnHH i

10.0 hnH s nHH ic

-50.0 0.0

0.5

1.0

1.5

0.0 2.5

2.0

T 150.0 120.0 CV ,

90.0

h(nPs )2 ic /2,

60.0

hnPs nHH ic , hn2HH ic

60.0

hnHH i

(c)

hnPs i

hn2HH ic

0.0 -30.0 0.0

40.0 30.0

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h(nPs )2 ic /2

50.0

20.0

1.0

hnHH i

10.0

hnPs nHH ic 0.5

hnPs i,

1.5 T

2.0

2.5

0.0 3.0

Figure 14.4: Temperature dependence of specific heat, correlation matrix components, and contact number expectation values of the 103-mer for surfaces attractive for (a) all, (b) only hydrophobic, and (c) only polar monomers at s = 2.

The behavior of the heteropolymer adsorbed at a surface that is only attractive to hydrophobic monomers [Fig. 14.2(b)] is apparently different in the AC phase, compared to the behavior near the type-independently attractive sub-

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strate. Since surface contacts of polar monomers are energetically not favored, the subphase structure is determined by the competition of two hydrophobic forces: substrate attraction and formation of intrinsic contacts. In AC1, the number of hydrophobic substrate contacts is maximal for the single hydrophobic layer, (nHH s , nHH )min = (37, 42). The single two-dimensional hydrophobic domain is also maximally compact, at the expense of displacing polar monomers into a second layer. In subphase AC2 intrinsic contacts are entropically broken with minimal free energy for 35 ≤ nHH ≤ 40, while nHH = 37 remains maxis mal. Another AC subphase, AC3, exhibits a hydrophobic layering transition at the expense of hydrophobic substrate contacts. Much more interesting is the subphase transition from AC1 to AC5. The number of hydrophobic substrate contacts nHH of the ground-state conformation dramatically decreases (from 37 s to 4) and the hydrophobic monomers collapse in a one-step process from the compact two-dimensional domain to the maximally compact three-dimensional hydrophobic core. The conformations are mushroom-like structures grafted at the substrate. AC4 is similar to AC5, with advancing desorption. Not less exciting is the subphase structure of the heteropolymer interacting with a polar substrate [Fig. 14.2(c)]. For small values of s and T , the behavior of the heteropolymer is dominated by the competition between polar monomers contacting the substrate and hydrophobic monomers favoring the formation of a hydrophobic core, which, however, also requires cooperativity of the polar monomers. In AC1, film-like conformations (nPs = 66, nHH = 31) with disconnected hydrophobic clusters dominate. Entering AC2, hydrophobic contacts are energetically favored and a second hydrophobic layer forms at the expense of a reduction of polar substrate contacts [(nPs , nHH)min = (61, 37)]. In AC3, the upper layer is mainly hydrophobic [(nPs , nHH)min = (53, 45)], while the poor quality of the solvent (s large) and the comparatively strong hydrophobic force let the conformation further collapse [AC4: (nPs , nHH)min = (42, 52)] and the steric cooperativity forces more polar monomers to break the contact to the surface and to form a shell surrounding the hydrophobic core [(nPs , nHH)min = (33, 54) in AC5]. After these general considerations on the adsorption behavior of heteropolymers on simple-cubic lattices, we will study specific adhesion properties of semiconductor-binding synthetic peptides in the following.

14.3 Selected Semiconductor-Binding Synthetic Peptides Here, we focus on the solution and adsorption behavior of synthetic 12-residue peptides, whose adhesion properties to surfaces of GaAs and Si crystals were

14.3 Selected Semiconductor-Binding Synthetic Peptides

251

studied in recent experiments [227–229]. It could be shown that the adsorption of peptides to semiconductor substrates strongly depends on intrinsic properties of the hybrid peptide-semiconductor system. Whaley et al. [227] investigated by means of the phage-display approach [284] a library of about 109 random peptides, each containing 12 amino acids, and extracted those few sequences that exhibited reasonably good adsorption to gallium arsenide (GaAs). One of these sequences, AQNPSDNNTHTH1 , was investigated in more detail in AFM experiments [228,229] with respect to different semiconductor substrates, crystal orientations, and sequence mutation and permutation. A particularly suitable quantity that can be measured in AFM experiments is the peptide adhesion coefficient (PAC), defined as the percentage of surface coverage, after drying and washing of the samples which were originally in contact with the peptide solution. This quantity was measured for different peptide-substrate combinations [228,229], and it was found to show a clear dependence on both peptide and substrate. In Table 14.1, the sequences used in the experiment and their PAC values for adsorption to GaAs(100) and Si(100) surfaces are listed. In these experiments, it was confirmed that the peptide with sequence S1 has a high binding affinity to GaAs(100) surfaces, but it was also shown that the binding to (100) silicon (Si) is extremely poor.2 Exchanging the basic histidines against the nonpolar alanines led to a worse binding to GaAs(100) (which can be considered as a polar substrate), but the adsorption strength to bare, nonoxidized Si(100) (which typically behaves “hydrophobic”) increased noticeably. Another remarkable result was that a random permutation in the order of the amino acids of the original sequence (to TNHDHSNAPTNQ) made the peptide equally attractive to the Si(100) and GaAs(100) substrates. Eventually it could be shown that the crystal orientation of the semiconductor surface also influences the binding behavior. The mutated sequences S1′ and S3′, also listed in Table 14.1, possess also surprising folding and adsorption properties that will be discussed subsequently in this chapter. These are results of computer simulations [285,286]; experimental data are not yet available for these sequences. How the binding occurs in these peptide-substrate systems is unclear. However, although the bound peptides were found to form clusters [229], it seems unlikely that the peptides aggregate before binding to the surface, because the hydrophobicity of the peptides studied is low and the peptide concentration was extremely low, in the nanomolar range. Measurements of circular dichroism (CD) 1

For the amino acid code, refer Fig. 1.2. Here and in the following, only substrates with (100) surfaces are considered. In the experimental studies, also peptide adhesion properties at crystals with other orientations were investigated [228,229]. 2

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Table 14.1: Sequences and their PAC values [229] for adsorption to GaAs(100) and Si(100) surfaces. For the peptides S1′ and S3′ , experimental PAC data are not yet available. label S1 S1′ S2 S3 S3′ S4

sequence AQNPSDNNTHTH AQNTSDNNPHTH AQNPSDNNTATA TNHDHSNAPTNQ TNHPHSNADTNQ AQAPSDAATHTH

PAC (GaAs) 25%

PAC (Si) 1%

14% 17%

3% 16%

21%

6%

spectra suggest that all four experimentally studied small peptides are, as expected, largely unstructured in solution [229], thus favoring coupled folding and binding over docking. Recent studies have found that the adhesion propensity of peptides to various surfaces can be in part explained in terms of adhesion properties of their constituent amino acids [230,287]. However, the amino acid composition alone cannot explain the PAC values obtained experimentally for the peptides studied here. In fact, some of these peptides share exactly the same amino acid composition, but still have quite different adhesion properties. In order to explain the adhesion properties, it might thus be necessary to take structural characteristics into account. However, as already indicated, the CD measurements did not reveal any clear structural differences between these peptides [229]. In the following, we first discuss the aqueous solution behavior of these peptides, and look for possible structural differences not seen in the CD analysis [285,288]. Simulating the actual binding of the peptides to the semiconductor substrates is more challenging due to uncertainties about the precise form of the peptide-surface interactions and their dependence on solvation effects. The modeling of the interaction of peptides with Si(100) surfaces [286] will be described in detail in the second part of this chapter.

14.4 Simulation of Semiconductor-Binding Peptides in Solution 14.4.1 Peptide Model and Simulation Details The peptide model used for this analysis is described in detail in Section 1.3.2. It contains all atoms of the peptide chain, including H atoms, but no explicit water molecules. Fixed bond angles, bond lengths and peptide torsion angles (180◦) are assumed, so that each amino acid has the Ramachandran angles φ, ψ and a

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number of side-chain torsion angles as its degrees of freedom [18,19]. In our simulations, thermodynamic quantities are obtained by canonical statistics. Each conformation X is uniquely defined by the set of degrees of freedom, i.e., the set of dihedral backbone and side-chain angles of each amino acid (1) j = 1, . . . , M: ξj = φj , ψj , χj , . . . (ωj = 180◦ is fixed in our model). The Boltzmann probability for a conformation X is p(X) = exp[−E(X)/RT ]/Z and the partition function is given by Z Z = DX e−E(X)/RT , (14.2) where DX is the formal integral measure for all possible conformations in the space of the degrees of freedom. The gas constant takes the value R ≈ 1.99 × 10−3 kcal/K mol. The statistical average of any quantity O is obtained by Z 1 hOi = DX O(X)e−E(X)/RT . (14.3) Z

In order to investigate the solution behavior of the peptides S1–S4, simulatedtempering [289,290] simulations with eight temperatures in the range 275– 369 K were performed, and some reference runs at a constant temperature of 1 000 K [285]. The conformational updates we use are rotations of single backbone and side-chain torsion angles, and a semilocal backbone update, biased Gaussian steps (BGS) [76], which updates seven or eight consecutive angles in a manner that keeps the rest of the molecule approximately fixed. In the simulatedtempering runs these updates are called in different proportions at different temperatures with more BGS at lower temperatures. At 299 K, the fractions of attempted single-angle backbone moves, side-chain moves and BGS are 0.29, 0.51 and 0.20, respectively. In the 1 000 K simulations the corresponding fractions are 0.245, 0.51 and 0.245. The simulations were carried out using the software package PROFASI [291]. Each simulation comprises 109 elementary MC steps. The results of these simulations were analyzed using multi-histogram techniques [79] and statistical uncertainties quoted are 1σ errors obtained by the jackknife method [73].

14.4.2 Temperature Dependence of Energetic Fluctuations and Secondary-Structure Contents Cooperative structural activity is typically signaled by a peak in the statistical fluctuations of system relevant quantities, such as the energy. Figure 14.5 shows how the specific heat CV (T ) = dhEi/dT varies with temperature for the sequences S1–S4. The qualitative behavior of the three sequences S1, S2, and S4

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140 120

S3

CV(T)/R

100 S1 80 60

S2

40

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20 0 260

280

300

320

340

360

T (K) Figure 14.5: Temperature dependence of the specific heat CV in units of the gas constant R for the sequences S1–S4.

is virtually identical. For all three sequences, the specific heat exhibits a broad peak with maximum around 280 K. In the temperature regime where these peaks occur, it turns out that the secondary-structure content of these three sequences changes relatively rapidly. As the temperature decreases, the α-helix content, hnα i, increases, whereas the β-strand content, hnβ i, decreases slightly, as can be seen from Fig. 14.6. These results indicate that the structures with lowest energy are α-helical for S1, S2 and S4. It should be noted, however, that the α-helix content remains small, < 0.25, all the way down to 273 K. Sequence S3 shows a markedly different behavior. The specific heat does not exhibit a maximum within the temperature range studied. It increases monotonically with decreasing temperature (see Fig. 14.5). Furthermore, the β-strand content remains larger than the α-helix content at low temperature for this sequence (see Fig. 14.6). The β-strand content does not decrease with decreasing temperature, and the α-helix content increases much less than for the other sequences. Figure 14.7 shows typical low-energy conformations for the four different sequences, as obtained by simulated annealing [292]. As one might expect from the temperature dependence of the α-helix and β-strand contents, the structure is α-helical for S1, S2 and S4. However, the α-helix does not span the entire chain, but rather the region between residues 3 to 12. That the beginning of the

14.4 Simulation of Semiconductor-Binding Peptides in Solution

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T (K) Figure 14.6: Temperature dependence of (a) the α-helix content hnα i and (b) the β-strand content hnβ i, for the sequences S1–S4. We define a residue as α-helical if its Ramachandran angles φ and ψ satisfy φ ∈ (−90◦ , −30◦ ) and ψ ∈ (−77◦ , −17◦ ), and hnα i denotes the fraction of the 10 inner residues that are α-helical. Similarly, hnβ i is the fraction of the 10 inner residues with Ramachandran angles satisfying φ ∈ (−150◦ , −90◦ ) and ψ ∈ (90◦ , 150◦ ).

sequence does not make α-helix structure is not unexpected, because there is a proline at position 4. The lowest-energy structure we find for S3 is a β-hairpin. Its turn is at residues 6 and 7. The second strand of the β-hairpin, spanning residues 8–12, is not perfect but broken in the vicinity of the proline at position 9.

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Figure 14.7: Typical low-energy conformations for (a) S1, (b) S2, (c) S3, and (d) S4. These structures were obtained as the lowest-energy structures in ten simulated annealing runs for each sequence, starting from random conformations. In each run, the temperature was decreased geometrically from 369 K to 0.7 K in 100 steps. At each temperature 100 000 elementary MC steps were performed.

It must be stressed that the states illustrated in Fig. 14.7 are only weakly populated at room temperature, as is evident from the secondary-structure contents shown in Fig. 14.6. These results are thus consistent with the CD analysis of the solution behavior of these peptides [229], at room temperature and pH 7.6, which suggests that they all are largely unstructured. Our conclusion that the α-helix content, at low temperature, is higher than the β-strand content for S1 and S2, is in agreement with a previous study of S1–S3 based on the ECEPP/3 force field [288], where, however, the native topology of S3 was found to be α-helical as well. Having studied the overall structure and the temperature dependence, we now turn to a more detailed structural description at T = 299 K, which is close to where the CD measurements were taken [229]. This discussion will mainly focus on S1 and S3, as the double mutant S2 and the triple mutant S4 show a behavior very similar to that of S1.

14.4 Simulation of Semiconductor-Binding Peptides in Solution

(a)

(b)

40 30 20

40 30 20

0

1

2

3

4

RMSD (Å)

5

6

(c)

50

E (kcal/mol)

50

E (kcal/mol)

E (kcal/mol)

50

257

40 30 20

0

1

2

3

4

RMSD (Å)

5

6

0

2

4

6

8

10

12

RMSD (Å)

Figure 14.8: Free energies F (∆, E) calculated as functions of RMSD, ∆, and energy, E, for S1 and S3 at T = 299 K. The reference structure is either an α-helix or a β-hairpin (see text). The contours are spaced at intervals of 1 RT . Contours more than 6 RT above the minimum free energy are not shown. The free energy F (∆, E) is defined by P (∆, E) ∝ exp(−F (∆, E)/RT ), where P (∆, E) is the joint probability distribution of ∆ and E at temperature T . (a) RMSD from the α-helix for S1 (calculated over residues 5–12). (b) RMSD from the α-helix for S3 (residues 1–8). (c) RMSD from the β-hairpin for S3 (all residues). Note that the scale of the abscissa is different in (c).

14.4.3 Characterization of Secondary Structures at Room Temperature To further elucidate the structure and free-energy landscape of these peptides, we analyze root-mean-square deviations (RMSD) from suitable reference structures (calculated over backbone atoms). We first consider an α-helical reference structure. The N-terminal part of S1 is rather flexible due to a proline at position 4. Similarly, the C-terminal part of S3 is flexible, due to a proline at position 9. To reduce noise, we omit these tails when calculating RMSD. The reference structure used is an α-helix with 8 residues. With RMSD calculated this way, we study the free energy F (∆, E) as a function of RMSD, ∆, and energy, E, at 299 K. Figures 14.8(a) and 14.8(b) show contour plots of F (∆, E) for S1 and S3. For both sequences, the free-energy minimum is at an RMSD of about 3.4 ˚ A, which is approximately the average value for random structures, as obtained from reference runs at 1 000 K. This finding supports the conclusion that S1 and S3 both are largely unstructured at 299 K. A clear local free-energy minimum corresponding to α-helix structure is missing for both sequences. For S1, there is, however, a valley from the global minimum in the direction of low RMSD and low energy, and there is a small but significant fraction of α-helical conformations with ∆ ∼ 1 ˚ A and relatively low energy. For S3, there is a valley in the same direction, but it is less pronounced, and conformations with a ∆ as small as 1 ˚ A are rare. There is also a second valley for S3, where the lowest populated energies are found. The appearance of this second valley, where ∆ > 3 ˚ A, is not unexpected, given

258

14. Hybrid Protein–Substrate Interfaces 0.3 S2 0.25

(a)

S1 S4

〈 χα(i)〉

0.2 S3' 0.15 0.1

S3 S1'

0.05 0 2

4

6

8

10

12

i (residue #) 0.35

(b)

0.3

〈 χβ(i)〉

0.25

S1' S3

0.2

S1 S2

0.15 S3' S4

0.1 0.05 0 2

4

6

8

10

12

i (residue #) Figure 14.9: Secondary-structure profiles for S1–S4, and S1′ and S3′ at T = 299 K. (a) The probability that residue i is in the α-helix state, hχα (i)i, against i. (b) The probability that residue i is in the β-strand state, hχβ (i)i, against i. The lines are only guides to the eye.

that the lowest-energy structure found for S3 is a β-hairpin [see Fig. 14.7(c)]. Figure 14.8(c) shows F (∆, E) for S3 when this β-hairpin is taken as the reference structure. A local minimum with ∆ ∼ 1 ˚ A and low energy can be found, but it is very weakly populated. The dominating global minimum corresponds to unstructured conformations. In fact, the average RMSD from the β-hairpin for random S3 conformations, as obtained from a control run at 1 000 K, is about 6˚ A, which is approximately where the global minimum is found at T = 299 K. Next we examine how the α-helix and β-strand contents (as defined in the caption of Fig. 14.6) vary along the chains. Let χα (i) = 1 if residue i is in the

14.4 Simulation of Semiconductor-Binding Peptides in Solution

259

α-helix state and χα (i) = 0 otherwise, so that hχα (i)i is the probability of finding residue i in the α-helix state, and let χβ (i) denote the corresponding function for the β-strand state. Figure 14.9 shows hχα (i)i and hχβ (i)i against i for S1–S4 at T = 299 K. The low-energy conformations of S1, S2, and S4, shown in Fig. 14.7, contain an α-helix starting near position 3 and ending at the C terminus. The α-helix probability profile in Fig. 14.9(a) reveals that the stability of this α-helix is not uniform along the chain; its N-terminal part is most stable, whereas the stability decreases significantly toward the C terminus. For S3, it can be seen from Fig. 14.9(b) that the values of hχβ (i)i are similar in the two regions that make the strands of the β-hairpin in Fig. 14.9(c). An exception is Pro9, for which hχβ (i)i is strictly zero (proline has a fixed φ = −65◦ in the model, which falls outside the φ interval in our β-strand definition). We also note that the two end residues tend to be unstructured for all four sequences, with relatively small values of both hχα (i)i and hχβ (i)i. From the single-residue probabilities hχα (i)i and hχβ (i)i, one cannot tell whether or not the formation of secondary structure is cooperative. To study that for S1, S2, and S4, we calculate the helix-helix correlation coefficient for neighboring residues at T = 299 K, as defined by (α)

(α) ri i+1

where

Ci i+1

, =q (α) (α) Cii Ci+1 i+1

(α)

Cij = hχα (i)χα (j)i − hχα (i)ihχα (j)i. (α)

(14.4)

(14.5)

For all three peptides, we find that the largest ri i+1 values occur in the region from i = 4 to i = 9 and are in the range 0.3–0.5. These values indicate that helix formation is a rather weakly cooperative process for these peptides. Consequently, the free-energy barrier to helix formation should be low, a conclusion that is in line (α) with the results shown in Fig. 14.8(a). For S3, ri i+1 is about 0.3 or smaller for (β) all i. The analogous strand-strand correlation coefficient ri i+1, defined in terms of χβ (i), is smaller than 0.25 for all i for all the four sequences. Another way of analyzing secondary-structure correlations is to look at the typical lengths of unbroken α-helix and β-strand segments. Specifically, we calculate the fraction of conformations, at fixed T , that have at least one unbroken α-helix (β-strand) stretch with 3 residues or more, which we denote by λα (λβ ). Table 14.2 shows λα and λβ for S1 and S3 at three different temperatures. For S1 at T = 299 K, we find that λα = 0.12. This result can be compared with what one would expect if the χα (i) were independent random variables with idependent individual distributions, given by Fig. 14.9(a). In this uncorrelated

260

14. Hybrid Protein–Substrate Interfaces

Table 14.2: The fraction λα (λβ ) of conformations that have at least one continuous α-helix (β-strand) segment of length 3 or more. T (K) λα λβ

S1 S3 275 299 369 275 299 369 0.21 0.12 0.03 0.06 0.04 0.01 0.03 0.03 0.04 0.04 0.04 0.03

case, it turns out that one would find λα = 0.04. This comparison shows that the correlations are significant but not very strong. For S3, we find that λβ = 0.04 at T = 299 K. A calculation analogous to that for S1, shows that λβ = 0.04 is precisely what one would expect in the absence of correlations. Hence, we find that secondary-structure correlations are very weak for S3. Finally, it is also instructive to identify the backbone H bonds that are most (1) likely to occur. We consider an H bond formed if its energy is < −ǫhb /3. For S1, we find that the bonds NH(Asp6)-CO(Asn3) and NH(Asn7)-CO(Asn3) occur in ≈ 38% and ≈ 34% of the conformations, respectively, at T = 299 K, whereas no other backbone H bond has a frequency of occurrence above 15%. These results confirm that the α-helix seen in low-energy conformations for S1 is most stable in its N-terminal part. Note also that in our simulations this helix often starts with a fork-like H bonding; the CO(Asn3) group acts as an acceptor for two bonds. For S3, there is only one backbone H bond that occurs in more than 15% of the conformations at T = 299 K, namely NH(Asn11)-CO(Ala8) with a frequency of occurrence of ≈ 21%. The paucity of H bonds underscores the notion that this peptide is highly flexible. Why do we find a different behavior for S3? A major reason is the different position of the proline; the proline residue with its special geometry is at position 9 in the sequence S3, but at position 4 in S1, S2, and S4. To gauge the importance of the proline location, we repeated the same calculations for a variant of S3, S3′, with Asp4 and Pro9 interchanged (see Table 14.1). We find that the behavior of S3′ closely resembles that of S1, S2 and S4. As an example, we show in Fig. 14.9 the α-helix and β-strand probability profiles for S3′ . The S3′ profiles are nearly identical to those for S1, S2, and S4. In the reshuffling of S1 to get S3, the change of proline position thus seems particularly important. We also studied the sequence obtained by interchanging Pro4 and Thr9 in S1, which we call S1′ (see Table 14.1). We find that this transposition of S1 leads to a behavior similar to that of S3, as is illustrated by Fig. 14.9, which confirms the importance of the position of the proline.

14.5 Modeling a Hybrid Peptide-Silicon Interface

261

14.5 Modeling a Hybrid Peptide-Silicon Interface 14.5.1 Introduction In the past few years, attention has been dedicated to adhesion and selfassembly of polymers, proteins, or protein-like synthetic peptides to crystalline and amorphous solid materials such as, e.g., metals [226,265,270,293–297], semiconductors [227–229,298], carbon [299,300], carbon nanotubes [271,301], mica [302,303], and silica [304–307]. Since a generic, computationally feasible model for hybrid interfaces of proteins and solid materials, in particular semiconductors, does not exist, we will focus here on silicon substrates. One reason is that the binding behavior of the peptides S1, S2, and S4 on one hand and S3 on the other is so different [228,229] (the corresponding peptide adhesion coefficients are listed in Table 14.1) that the study of (100) silicon is intriguing. Compared with the other material, where PAC values are quoted in Table 14.1, gallium-arsenide, a bare, not oxidized (100) silicon surface repels water molecules, i.e., it is effectively hydrophobic. For computational reasons it is highly desirable to generalize the implicit-solvent peptide model, employed in the study of the peptides in solution in the previous section, by incorporating the interaction with the substrate. A rather polar substrate such as, e.g., GaAs, appears attractive to water molecules. In such cases, the influence of single water molecules and even water layers upon the peptide binding propensity is expected to be strong and the additional simulation of explicit water molecules can hardly be avoided [296,297]. In the following, we investigate and compare silicon-binding properties of four of the synthetic peptides with 12 residues listed in Table 14.1, S1, S3, S1′, and S3′ [286]. Silicon (Si) is one of the technically most important semiconductors, as it serves, for example, as carrier substrate for almost all electronic circuits. For this reason, electronic properties of Si are well-investigated. Also surface properties of Si have been subject of numerous studies. This regards, for example, oxidation processes in air [308,309] and water [310,311], as well as the formation of hydride surface structures, frequently in connection with etching processes [298,312–315]. The binding characteristics to Si substrates of small organic compounds and, for example, their influence on surface re-structuring of the particularly reactive Si(100)-2×1 surface has also found broad attraction [298,316–322].

14.5.2 Si(100) and the Role of Water Since it is important for the modeling, we here review properties of silicon surfaces and their preparation in exeriments. In the experiments of peptide adhesion

262

14. Hybrid Protein–Substrate Interfaces

to semiconductor substrates [228,229], the Si(100) surfaces were cleaned in a solution of NH4F and HF. The adsorption experiments were then performed in distilled water. This standard procedure ensures that the Si surface is widely free of oxide which causes strong hydrophobic properties [308,310]. The initial Si–F bonds after etching are replaced by Si–H bonds in the rinsing process in de-ionized water. Although the oxidation proceeds also in water [310,311], there are clear indications (maximum water droplet contact angle after removing the samples off the peptide solution) that the hydrophobicity of the Si samples is widely sustained during the peptide adsorption process. It is also known that Si surfaces are comparatively rough after HF treatment [312]. This renders a detailed modeling intricate, even more as the reactivity of the surface is strongly influenced by roughness effects (such as, e.g., steps). Si(100)-2×1 surfaces are known to form Si–Si dimers on top of the surface [298] with highly reactive dangling bonds. From the considerations and the experimental preparations described above, it seems plausible that these bonds are mainly passivated by hydrogen, forming hydride layers [298,310,312]. It should be emphasized that under these conditions the surface structure of Si(100) is substantially different from oxidized Si(100) which is polar and in effect hydrophilic [308]. This can nicely be seen in Fig. 14.10, where for (a) S1 and (b) S3 the relative peptide surface coverage (peptide adhesion coefficient, PAC) is shown for bare and oxidized Si(100) and GaAs(100) samples after different washing cycles. These data were obtained from AFM experiments of the peptides S1 and S3 interacting with Si(100) and GaAs(100) substrates in solution [286]. The experimental setup is identical with that of previous studies [228,229]. The main result is that the binding of S1 to oxidized GaAs(100) and Si(100) surfaces is virtually independent of the substrate type which is widely screened by the top oxygen layer. The different adhesion propensities to the “bare” (hydrated) substrates lead to the conclusion that oxidation has not yet strongly progressed during the peptide adsorption process. These characteristic properties of HF treated Si(100) surfaces in de-ionized water effectively enter into the definition of our hybrid model which will serve as the basis of our analysis and interpretation of the specificity of peptide adhesion on these interfaces. From these considerations we conclude that the key role of water is the slowing down of the oxidation process of the Si(100) surface, but for the actual binding process its influence is rather small (up to screening effects). In particular, we do not expect that stable water layers form between adsorbate and substrate.

14.5 Modeling a Hybrid Peptide-Silicon Interface

263

Figure 14.10: Experimental PAC values for S1 at bare and oxidized Si(100) after several washing cycles. For comparison, also data for S1 adhesion to bare and oxidized GaAs(100) substrates are shown. Lines are only guides to the eye. From [286].

14.5.3 The Hybrid Model Our model contains all peptide atoms, while the substrate is simplified and consists only of atomic layers with surface specific atomic density. The substrate and its surface structure itself is fixed and thus its energy is not considered in the model. Therefore, the energy of a single peptide with conformation X (where dihedral backbone and side-chain angles are the degrees of freedom) and interacting with the substrate, whose surface structure is characterized by the Miller index (hkl), is generally written as Si(hkl)

E(X) = Epep (X) + Epep-sub(z).

(14.6)

Here, z = (z1, z2 , . . . , zN ) is the perpendicular distance vector of all N peptide atoms from the surface layer of the substrate. The effect of the surrounding solvent is implicitly contained in the force field parameters. For the peptide, we employ the simplified all-atom model [18,19,291] described in Section 1.3.2, which has already been used in the study of the peptides in solution in Section 14.4. The interaction of the peptide with the substrate is modeled in a simplified way, i.e., each peptide atom feels the mean field of the atomic substrate layers. The atomic density of these layers is dependent on the surface characteristics,

264

14. Hybrid Protein–Substrate Interfaces

i.e., it depends on the crystal orientation (the Miller index hkl) of the substrate at the surface. We make the following assumptions for setting up the model: (i) According to our considerations about bare Si(100) in de-ionized water, the Si(100) surface is considered to be hydrophobic. This has the effect that it is not favorable for water molecules to reside between the adsorbed peptide and the substrate. Furthermore, polarization effects between side chains and substrate are not expected. We expect that hydrophobic side chains (in the sequences only proline and alanine) feel an effective attraction by the hydrophobic substrate, as there is no competition with other hydrophobic side chains in the folding process. (ii) Since dangling bonds on the Si(100) surface are probably saturated by covalent bonds to hydrogen atoms (due to the HF etching process), we assume that covalent bonds between peptide and surface atoms are not formed. Thus, the surface is also considered to be uncharged [323]. (iii) Si dimers sticking off the substrate are not considered. This and the hydration effect are expected to weakly screen the peptide from the substrate. From these assumptions, we use a generic noncovalent Lennard-Jones approach for modeling the interaction between peptide atoms and surface layer [297,299,300], " 10 4 # N X 2 σi,Si σi,Si Si(hkl) 2 Epep-sub(z) = 2πρSi(hkl) εi,Si σi,Si , (14.7) − 5 z z i i i=1 where ρSi(hkl) is the atomic density of the Si(hkl) surface layer. Si has diamond structure and since we assume the surface to be ideally flat (i.e., neither relaxed, −2 reconstructed, nor rough), ρSi(100) ≈ 0.068 ˚ A . The noncovalent interaction between the peptide atoms and the Si substrate is parametrized by force-field √ parameters εi,Si = εi εSi and σi,Si = σi + σSi and thus depends on the energy depths εi and van der Waals radii σi of the individual atoms. The parameter values used in the simulations are listed in Table 14.3. The computer simulations of the peptide adhesion to Si(100) surfaces were performed employing the BONSAI (Bio-Organic Nucleation and Self-Assembly at Interfaces) package [273]. For statistical sampling, a multiple-thread variant of the multicanonical Monte Carlo method has been applied. In each run about 109 updates were performed. The adsorption simulations require a set of several conformational updates, such as forward and backward pivot rotations about single dihedral backbone and side-chain torsion angles, as well as rigid-body rotations and translations. In

14.6 Sequence-Specific Peptide Adsorption at (100) Silicon Surfaces

265

the simulations, a simulation box of dimension [40 ˚ A]3 with periodic boundary conditions parallel to the substrate was used. In perpendicular direction mobility is restricted by the Si substrate residing by definition at z = 0. The influence of the wall parallel to the substrate is simply steric, i.e., the atoms experience hard-wall repulsion at z = zmax = 40 ˚ A.

14.6 Sequence-Specific Peptide Adsorption at (100) Silicon Surfaces 14.6.1 Thermal Fluctuations and Deformations upon Binding The adsorption of the peptides at the semiconductor surface is a conformational pseudophase transition. Since the peptides are known to be highly flexible and – as we have seen in Sections 14.4.2 and 14.4.3 – widely unstructured in solution at room temperature [285], the adsorption process is accompanied by significant structural changes of the peptides. This can be seen in Fig. 14.11, where the specific heat CV (T ) = (hE 2i − hEi2 )/RT 2 is plotted for each of the peptides considered in this adsorption study. The peaks for S1 and S3′ and the increase towards lower temperatures for S3 and S1′ indicate energetic activity that signalizes the onset of a crossover between random-coil structures in solution and adsorbed conformations. Although the adsorbed conformations are also not expected to exhibit clear symmetries, the restriction to form comparatively flat structures (in order to maximize the number of substrate contacts) reduces entropic freedom. Thus, at room temperature, surface-attached peptides are expected to be compact without noticeable internal structure and the adsorption process is connected with peptide refolding. The deformation of the peptides due to adsorption is apparent from the analysis of the gyration tensor components perpendicular and parallel to the

Table 14.3: Atomic van der Waals radii σ and energy depths ε used in the simulations of peptide adsorption at a (100) silicon substrate. atom Si H C O N

σ [˚ A] 2.0 [324] 1.2 [325] 1.8 [324,325] 1.5 [324,325] 1.6 [324,325]

ε [kcal/mol] 0.05 [304] 0.04 [325] 0.05 [325] 0.08 [325] 0.09 [325]

266

14. Hybrid Protein–Substrate Interfaces 90 80

CV (T )/R

70

S1 S3’

60

S3

50

S1’

40 30 20 260

280

300

320 T [K]

340

360

380

Figure 14.11: Specific heats of the peptides S1, S3, S1′ , and S3′ near a bare (100) silicon substrate in units of R.

substrate. We define the gyration radius of the heavy atoms as N

2 Rgyr

h 1 X (xi − x0 )2 , = Nh i=1

(14.8)

where the sum runs over the Nh heavy (non-hydrogen) atoms of the peptide, xi P h is the position of the ith heavy atom, and x0 = N i=1 xi /Nh . For identifying asymmetries in the peptide conformations and orientations, we use the simple 2 2 + Rk2 , where R⊥ and Rk are the components perpendecomposition Rgyr = R⊥ dicular and parallel to the substrate, respectively. The temperature dependence of the thermal averages of the components is shown in Fig. 14.12 for the peptides in presence of the Si(100) substrate and, for comparison, in solution (without substrate). The “thickness” hR⊥ i of the structures in perpendicular direction is in the shown temperature interval much smaller than the average extension without substrate [Fig. 14.12(a)]. This is different for the component hRk i parallel to the substrate, i.e., the planar extension is larger than in the bulk case [Fig. 14.12(b)]. An interesting feature is that S1 and S3′ behave similarly, as well as S3 and S1′. This is the same grouping that has already been observed in the previous study of the peptides in the substrate-free case (see Fig. 14.9), where this behavior has been found to be due to the position of proline in the sequence [285]. Although the adsorption at the Si(100) substrate is a quite different process, we find here the same groups forming.

14.6 Sequence-Specific Peptide Adsorption at (100) Silicon Surfaces

267

6.0 5.5 without substrate

5.0 hR⊥ i [˚ A]

4.5

S3,S1’

4.0 S1,S3’

3.5 3.0

S1,S3’

2.5

S3,S1’

2.0 260

with Si(100) substrate (a)

280

300

320 T [K]

340

360

380

6.0 5.5

√ hRk i/ 2 [˚ A]

5.0 4.5 4.0

S3,S1’ with Si(100) substrate S1,S3’ S3,S1’

without substrate

S1,S3’

3.5 3.0 2.5 2.0 260

(b) 280

300

320 T [K]

340

360

380

Figure 14.12: Gyration tensor components of the peptides (a) perpendicular and (b) parallel to the substrate. For comparison, also results for the peptides in bare solution (i.e., without substrate) are shown.

14.6.2 Secondary-Structure Contents of the Peptides As expected, adsorbed peptides do not exhibit clear structures at room temperature. In Fig. 14.13, the respective α-helix and β-strand contents of the adsorbed conformations are shown. Although noticeable differences for the mentioned peptide groups are found, there is no significant population of α-helical or β-sheet structures at least at room temperature. Nonetheless, there is a tendency that residues of S1 and S3′ are rather in α and residues of S3 and S1′ in β state. The

268

14. Hybrid Protein–Substrate Interfaces

0.24 0.22

(a)

S1

0.20 0.18 hnα i

0.16

S3’

0.14 0.12

S1’

0.10 0.08

S3

0.06 260

280

300

320 T [K]

340

360

380

0.24 S1’

0.22

(b)

0.20 0.18 hnβ i

0.16 0.14

S3 S3’ S1

0.12 0.10 0.08 0.06 260

280

300

320 T [K]

340

360

380

Figure 14.13: (a) α-helix and (b) β-strand content of the adsorbed peptides as functions of temperature.

small secondary-structure contents are quite similar to what we have found for the peptides in solution (without substrate; see Fig. 14.6) [285], which were qualitatively consistent with analyses of CD spectra [229]. It is noticeable, however, that there is no obvious stabilization of secondary structures near the Si(100) substrate. In recent adsorption experiments of another synthetic peptide binding at silica nanoparticles, such an effect could actually be observed [307]. Figure 14.14 shows low-energy conformations and orientations of the peptides,

14.6 Sequence-Specific Peptide Adsorption at (100) Silicon Surfaces

(a)

269

(b)

4Pro

9Pro

Figure 14.14: Lowest-energy conformations for (a) S1 and (b) S3 near the Si(100) surface.

an α-helix for S1 [Fig. 14.14(a)] and a β-strand for S3 [Fig. 14.14(b)], identified in the simulations. The shown peptide structures have strong similarities with the low-energy conformations of the peptides in solution, cf. Fig. 14.7. Proline (at positions 4 in S1 and 9 in S3) lies in both lowest-energy conformations very close to the substrate and influences the orientation of the peptide on the substrate. Together with the mutual reversal of the structural properties after changing the proline positions this confirms our prediction that not only the folding in solution, but also the adsorption properties of the investigated sequences are strongly dependent on the respective proline positions in the sequences. In order to verify this, we have also performed simulations of a simple model where the Lennard-Jones-based peptide-substrate interaction (14.7) is replaced by an effective attractive hydrophobic energy term for the interaction of the only hydrophobic residues in the peptide sequences, proline and alanine, with the substrate. In fact, the results (not shown) exhibit the same tendency in the adsorption behavior as already described for the model (14.7), i.e., at room temperature S3 forms more surface contacts with a Si(100) substrate than S1. This consistency strengthens the conclusion that the influence of the proline position in the sequence is indeed essential for the adsorption of the considered peptides.

14.6.3 Order Parameter of Adsorption and Nature of the Adsorption Transition In order to quantify the degree of adsorption, we define the ratio of heavy (nonhydrogen) atoms located in a distance ≤ 5 ˚ A above the substrate, nh, and the total number of heavy atoms, Nh , as the adsorption parameter q = nh /Nh . Figure 14.15 shows the temperature dependence of the mean value of q for the four sequences studied. There is a noticeable difference in the number of contacts between heavy atoms and the surface for the sequences S1 and S3. Although a

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14. Hybrid Protein–Substrate Interfaces

0.22 0.20 S3’

S3, S1’

hqi

0.18 S1

0.16 0.14 0.12 0.10 0.08 260

280

300

320 T [K]

340

360

380

Figure 14.15: Temperature dependence of the mean “order” parameter hqi.

direct quantitative comparison with experimental data is not yet possible, the tendency that S3 forms at average more contacts than S1 at a Si(100) surface is completely consistent with the experimental results. Our earlier prediction that S1′ behaves similar like S3 and S3′ similar like S1 is also apparent from this plot. The experimental verification of this assumption is still pending. The adhesion of the peptides at the Si(100) substrate exhibits all features of a phase-separation process. Exemplified for peptide S3, Fig. 14.16 shows the plot of the canonical probability distribution pcan(E, q) ∼ hδ(E − E(X))δ(q − q(X))i at room temperature (T = 300 K). The peak at (E, q) ≈ (80.5 kcal/mol, 0.0) corresponds to conformations not being in contact with the substrate. It is sep-

pcan (E, q) 2.5

0.5

2.0

0.4

1.5

0.3 q

1.0 0.5 0.0 0.5

0.2

95

0.1

85 0.4

0.3 q

0.2

65 0.1

0.0

55

75 E [kcal/mol]

55

65

75

85

95

0.0

E [kcal/mol]

Figure 14.16: Perspective (left) and top (right) view of the unnormalized canonical probability distribution pcan (E, q) for S3 at T = 300 K.

14.6 Sequence-Specific Peptide Adsorption at (100) Silicon Surfaces

271

arated from another peak near (E, q) ≈ (74.5 kcal/mol, 0.17) and belongs to conformations with about 17% of the heavy atoms with distances ≤ 5 ˚ A from the substrate surface. That means, adsorbed and desorbed conformations coexist, the gap in-between separates the two pseudophases and causes a kinetic free-energy barrier with a height of about 1 kcal/mol. Thus, the adsorption transition is a first-order-like pseudophase transition. This is general property of small polymer systems and the overall behavior is quite similar to what has been discussed in Chapter 13 for simplified hybrid lattice models of polymers and peptides near attractive substrates [254–256].

15 Summary

The analyses in the previous chapters have shown that it is indeed possible to reveal characteristic features of structure formation processes of polymers, in particular proteins, by means of minimalistic coarse-grained models. This is essential, as a generalized view of conformational transitions occurring in folding, aggregation, and adsorption processes of molecular systems can only possess a solid basis, if a classification of features common to different systems enables the introduction of suitable models on mesoscopic scales. This regards, for example, the statistical analysis of the combined space of sequences and conformations for hydrophobic-polar lattice proteins, which is necessary for a better understanding of the evolutionary aspect of protein selection. To date, such an analysis is virtually impossible for “realistic” models. We have exactly enumerated all possible sequences and conformations of hydrophobic-polar proteins with up to 19 monomers on a simple-cubic lattice, extending earlier studies on compact cubic lattices. Amongst other aspects, we dedicated particular interest to general properties of native states (conformations with minimum energy), of designing sequences (non-degenerate native states), and the designability of native conformations (how many peptides with different sequences share the same native structure). Comparing two hydrophobic-polar models, we found that only a few sequences are “designing”, i.e., possess a non- or low-degenerate ground state. Furthermore, we found that there is only a small number of designable conformations, most of the proteins with different sequences fold into. Both results are in agreement with experimental findings from analyses of real proteins, confirming that simple protein models are indeed suitable to unravel certain general properties of proteins and protein folding, at least on a statistical level. Another primary topic of this work is the discussion of thermodynamic and kinetic properties of hydrophobic-polar heteropolymer folding, again having in mind that certain aspects are directly comparable to characteristic folding behaviors of

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real proteins. By means of the newly developed multicanonical chain-growth algorithm, we have investigated the statistical mechanics of the direct folding problem (i.e., with given hydrophobic-polar sequence) for several proteins on a simple cubic lattice in the canonical ensemble. Our method allowed us to analyze energetic and structural fluctuations for all temperatures, thus enabling us to identify conformational transitions accompanying the folding process. Since proteins are finite-length linear polymer chains with graven “disorder” due to the amino acid sequence, conformational transitions are not phase transitions in the strict thermodynamic sense. Attempting finite-size scaling analyses is useless since changing the chain length necessarily modifies the amino acid sequence and thus also the folding properties. Therefore, a thermodynamic limit does not exist for a single heteropolymer chain with disordered amino acid sequence. The only way out would be the averaging over all possible heteropolymer sequences in direct analogy to studies of spin glasses. Although this is indeed a quite interesting problem, one does not learn much about specific folding properties of single sequences, which was in the focus of our folding study of longer hydrophobic-polar sequences. Depending on the heteropolymer sequence, we confirmed in our studies that typically two general transitions occur in heteropolymer folding. One is the folding transition from random coils to compact globular conformations common to all heteropolymers (i.e., little sequence-specific), a finite-length analog to the collapse (or Θ) transition known from homopolymers, which is in the latter case a true phase transition. The stability of the globular or intermediary (pseudo)phase of heteropolymers depends, however, strongly on the heteropolymer sequence. The second general transition at lower temperature (or worse solvent quality) is sort of a glassy transition as it results in the formation of the native conformation(s) with small entropy. During this transition, the highly compact hydrophobic core is formed, surrounded by a shell of polar residues which screens the core from the solvent. The kinetics of this transition strongly depends on the heteropolymer sequence. Hydrophobic-core formation is typically a (“first-order-like”) phase separation process and the sharpness and height of the free-energy barrier separating the hydrophobic-core and globular (pseudo)phases are measures for the stability of the hydrophobic core. It is assumed that a large set of the comparatively few functional bioproteins in nature exhibits such a large barrier preventing unfolding into nonfunctional conformations. This is also one of the common arguments, why under physiological conditions only a very small number among the possible protein sequences can be functional at all. In order to better understand the transition into the native conformations and to get rid of lattice effects, we have performed sophisticated Markov chain Monte

275

Carlo simulations of a hydrophobic-polar off-lattice protein model on mesoscopic scales. Although the model is still extremely minimalistic, we have found quite surprising characteristic folding features comparable to those of real proteins. Analyzing transition channels and free-energy landscapes based on a suitably defined similarity or “order” parameter, we identified folding behaviors which are known from real proteins in a like manner: two-state folding with a single kinetic barrier and unique native state, folding towards the native fold through intermediates over different barriers, and metastability with different, almost degenerate, native states. For an exemplified two-state folder, we also performed a kinetic Monte Carlo analysis of a suitably adapted hydrophobic-polar and coarse-grained model of G¯o type, where the energy of the peptide is defined by the similarity to the native fold. This allows for efficient simulations of a large number of folding and unfolding events required for a kinetic analysis by averaging over folding and unfolding trajectories. Also in this study, we found strong analogies to known properties of natural two-state folders. These results confirm once more that the investigation of coarse-grained models, which are much simpler and thus computationally much less demanding than so-called “realistic” models, is very useful for revealing general or “pseudouniversal” aspects of tertiary protein folding. Coarse-grained models enable a broader view on the general problem of protein folding, but for precise, specific predictions, their applicability is limited. The investigation of folding details (e.g., the folding of rather local segments such as secondary structures) requires more specific models, typically incorporating all atoms in semiclassical approaches. In analogy to magnetic systems, protein-like models on mesoscopic scales are rather comparable with the Ising model for ferromagnets or the Edwards-Anderson-Ising model for spin glasses. It should also be remarked that, due to their nontrivial simplicity, coarse-grained models are also a perfect testing ground for newly developed algorithms. Gaining deeper insights into the spontaneous folding of single proteins is also essential for a better understanding of the aggregation of organic substances and the adsorption of organic to inorganic matter. Another major part of the results presented is devoted to studies in this field. Following the same strategy as in the analysis of heteropolymer folding, we have extended and generalized the coarse-grained hydrophobic-polar heteropolymer models for our studies of protein aggregation and adsorption of polymers and proteins to solid substrates. In our analysis of heteropolymer aggregation, we particularly focused on the interpretation of our results from multicanonical Markov chain Monte Carlo simulations in different statistical ensembles. In analogy to the problem of interpreting conformational transitions in protein folding processes in terms of statistical fluctuations,

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we paid particular attention to inevitable finite-size effects in protein aggregation. In many-body systems, two thermodynamic limits can be discriminated, in principle. In one case, the number of chains is kept constant, while the length of the individual chains is extended to infinity. This is obviously not applicable for proteins. Second, the length of the chains remains fixed, but the number of chains is increased to infinity. This is actually interesting for systems of many proteins. Here, we analyzed and compared small systems of different numbers of short peptides and investigated finite-size properties in multicanonical, canonical, and microcanonical ensembles. Each of these analyses has advantages – in the multicanonical case the aggregation channels towards the different aggregates can nicely be visualized and applying the canonical formalism reveals strong fluctuations in the vicinity of the aggregation transition which allow for a precise estimation of the aggregation transition temperature for the finite system, but also for a finite-size scaling analysis toward the system with infinitely many chains. But, analyzing the aggregation of a few peptides from the microcanonical perspective uncovers an underlying physical effect, the backbending effect, which is “averaged out” in the canonical analysis. “Backbending” means that in the transition region the caloric temperature decreases with increasing energy. This is due to surface effects, additional energy does not lead to an increase of the caloric temperature; rather it is used to rearrange monomers in order to reduce surface tension at the expense of entropy. In effect, the protein complex is getting colder. Increasing the number of peptides in the system, we could show, however, that the effect becomes less relevant, although the latent heat increases and thus the firstorder character of this phase-separation process is getting stronger. Nonetheless, in biological aggregation processes typically only a few proteins are involved and thus the effect should be apparent. The “physical reality” of this effect has already been confirmed in atomic cluster formation experiments. However, the experimental verification in polymeric systems is still pending. One of the essential questions in aggregation processes among polymers, but also in the interaction of soft and solid matter is, how the mutual influence induces conformational changes. Two potential scenarios leading to the formation of complexes are conceivable. If the external force is attractive, but weaker than the intrinsic, intermonomeric forces that form the polymer or protein conformation, the proximity of an attractive partner – another polymer or a solid substrate – is not sufficient to refold the polymer and the aggregation or adsorption is a simple docking process. Unless the match is perfect, the binding force that holds the compound together is rather weak. On the other hand, if the external force entails refolding of the polymer, it can better adapt to the target structure (e.g., a crystalline substrate), or, if both partners experience conformational changes,

277

a new, highly compact compound can form. In this so-called coupled foldingbinding process, the binding force is typically stronger than in the docking case. In our aggregation study of a few short peptides, we observed such a behavior. Whether docking or coupled binding-folding occurs in aggregation or adsorption processes is not only a question of the properties of the two constituents. The formation of compounds is also influenced by external parameters like temperature and solvent quality. In order to obtain a general view of the variety of pseudophase transitions in adsorption processes, we have investigated the full phase diagram of a single flexible polymer near an attractive solid, homogeneous, and flat substrate on a simple-cubic lattice. By means of chain-growth simulations, employing a generalized form of our multicanonical chain-growth algorithm, we obtained the complete temperature-solubility (pseudo)phase diagram for a polymer of finite length within a single simulation and identified transition lines which are expected to survive in the thermodynamic limit as well as subphases which are specific to the number of monomers in the polymer chain. Depending on the external parameters temperature and solubility, phases of dominant expanded random-coil-like, globular, and compact conformations occur in the adsorption regime. The region of compact conformations splits into two phases which are believed to be also separated in the thermodynamic limit: filmlike, topologically two-dimensional compact conformations with almost all monomers having contact to the substrate and highly compact, but layered three-dimensional conformations. Subphase transitions between conformations with a different number of compact layers are also present, but these sensitively depend on the number of monomers and will probably disappear in the thermodynamic limit. As a first step toward a better understanding of substrate-specificity and sequence-dependence in heteropolymer adhesion processes, we have analyzed the pseudophase diagram of an exemplified hydrophobic-polar heteropolymer near an attractive monomer type independent substrate, as well as substrates being specifically attractive to hydrophobic or polar residues only. Not unexpectedly, the different substrate properties influence location and character of the conformational pseudophases in the temperature-solubility pseudophase diagram. This substrate-specific adsorption behavior, which has also been observed in recent experiments, is possibly quite interesting for future bionanosensory or nanoelectronic applications. In order to understand details of the specific adsorption properties of the short synthetic peptides used in these experiments, we have also performed multicanonical computer simulations of these peptides in solution as well as in interaction with a Si surface oriented in (100) direction. We not only confirmed specific adsorption properties of the peptides to (100) silicon as observed in the experiments. We also found that the composition of the amino acid sequence is

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particularly important as steric constraints of amino acids like proline play not only an important role in secondary-structure formation in solvent, but also influence the binding process in a noticeable way. Summarizing, a variety of studies regarding structure formation processes of macromolecules was presented in this work. The focus was laid on conformational transitions accompanying folding and aggregation of protein-like heteropolymers, as well as the adsorption of polymers and proteins to solid substrates. For this reason, particular emphasis was placed on simplified lattice and off-lattice models on mesoscopic scales. This allowed a much broader and more general view upon the conformational mechanics of the systems studied. The results, which are in qualitative correspondence to known features of realistic systems, justify the assumption that a large class of conformational transitions of finite polymeric systems possesses pseudouniversal properties that can indeed be studied with minimalistic models. In many cases it is also illusionary to employ full-detail models as, for example, in analyses of the combined sequence and conformation space of proteins. Nonetheless, if one is interested in experimentally competitive, quantitative analyses of atomic details of structural changes, it is inevitable to make use of finer grained or even all-atom models. We have done so in our studies regarding aspects of solution behavior and adsorption propensity of short synthetic peptides that had recently been investigated experimentally. Our results were mainly obtained by means of sophisticated generalized-ensemble chain-growth and Markov chain Monte Carlo computer simulations, partly newly developed or generalized for these purposes. It is a non-negligible fact that even with today’s equipment computer simulations of polymers, in particular, proteins, are extremely demanding and efficient algorithms are required. Despite the enormous progress in protein research in the past few years, it will remain one of the biggest scientific future challenges to uncover the principal secrets of cooperative conformational activity in structure formation processes of proteins.

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289

Index

AIDS, vii amyloid β peptide, vii, 173 aggregation, vii hypothesis, vii, 173 aquaporine, 2 barrier, 7, 10, 15, 46, 47, 138, 156, 262 activation, 9, 159 bending, 21 conformational, 121 energetic, 86, 101, 139 entropic, 190, 199, 200 excitation, 101 free-energy, 6, 7, 15, 46, 117, 141, 144, 247, 259, 262 hidden, 134, 141 kinetic, 10, 46, 48, 49, 259, 263 phase separation, 200 surface, 189, 199 torsional, 9–12, 21, 130 transition, 189, 190 transition state, 151 bioprotein, 1, 3, 5, 7, 9, 11, 141, 158, 262 bovine spongiform encephalopathy, vii Boyle temperature, 92 canonical analysis, 157, 180, 183, 223, 264 ensemble, 27, 28, 30, 33, 34, 37, 38, 40, 45, 144, 151, 157, 160– 162, 173, 177, 180, 223, 262

temperature, 27, 28, 33–35, 160, 161, 187 chain growth contact-density, 205, 206, 231 multicanonical, 65, 71, 72, 74–77, 82, 83, 115, 119, 120, 124, 205, 262, 265 Rosenbluth, 19, 65–69, 120, 209 chymotrypsin inhibitor 2, 158 cooperativity, 6, 24, 37, 42, 43, 48, 86, 125, 164, 188, 238 conformational, 177 parameter, 6, 26, 158 steric, 238 substrate-peptide, 230 cytochrome c, 123, 231 dementia, vii detailed balance, 47, 128 disease Alzheimer’s, vii, 17, 173 Creutzfeld-Jakob, vii, 159 liver, vii neurodegenerative, vii, 173 prion, vii, 159 effect backbending, 162, 174, 182, 184, 198, 199, 264 bond elasticity, 109 bulk, 99 cooperative, 25, 27, 43, 162, 172 degeneration, 9 disorder, 114

292

energetic, 27, 86, 159 entropic, 159, 198, 213 excluded volume, 13, 42, 164 finite-size, ix, 26, 27, 97, 107, 108, 124, 138, 175, 187, 221, 264 finiteness, 27, 82, 84, 87, 195 hydration, 252 hydrophobic, 9 lattice, ix, 15, 84, 85, 97, 216, 228, 262 long range, 112, 146 macroscopic, 26 memory, 47, 146 microcanonical, 158 noncooperative, 84 polarization, 7, 252 quantum-mechanical, 7, 10, 11, 36 roughness, 250 screening, 7, 250 smallness, 26 solvation, 240 stiffness, 195 surface, ix, 27, 48, 97, 99, 101, 160, 174, 180, 182–184, 190, 193, 198–200, 264 energy-landscape paving method, 127, 129, 133–136, 148, 175 ensemble canonical, 27, 28, 30, 33, 34, 37, 38, 40, 45, 144, 151, 157, 160– 162, 173, 177, 180, 223, 262 generalized, 19, 72, 154 grand canonical, 35, 36 microcanonical, 31–33, 35, 160, 198, 264 multicanonical, 72, 177 entropy Hertz, 160, 182

Index

microcanonical, 72, 160, 174, 182–184, 189, 195, 198 enumeration, ix, 19, 53, 64, 77, 78 enzyme, 1, 4 finite-size deviation, 91 effect, ix, 26, 27, 97, 107, 108, 124, 138, 175, 187, 221, 264 scaling, 27, 108, 119, 125, 210, 262, 264 system, 77, 206 free energy, 6, 28, 33, 36, 43–45, 120, 144, 150, 174, 209, 212, 213, 215, 238, 245 barrier, 6, 7, 15, 46, 117, 141, 144, 247, 259, 262 contact, 212, 213, 233, 235 landscape, ix, 6, 10, 16, 43, 44, 46, 48, 49, 128, 133, 134, 141, 143–146, 150, 156, 178, 211, 232, 245, 263 gene expression, viii, 4 grand canonical ensemble, 35, 36 potential, 36 temperature, 36 hybrid interface, x, 201, 202, 229, 249 model, ix, 201, 224, 231, 250, 259 system, viii, 201–203, 205, 206, 209, 211, 223, 231, 239 hydrophobicity, 19, 43, 55, 57, 62, 119, 135, 140, 239, 250 interface hybrid, x, 201, 202, 229, 249 organic-inorganic, 229 peptide-semiconductor, 202

Index

peptide-silicon, 249 protein-substrate, 229 pseudophase, 174 kinetic barrier, 10, 46, 48, 49, 259, 263 transition, 48 kinetics, 141, 146, 153 chemical reaction, 6 folding, x, 141, 146, 148, 153, 155, 157, 261 G¯o, 146 transition, 262 unfolding, 155, 157 landscape, 130, 169 energy, 127, 136 free-energy, ix, 6, 10, 16, 43, 44, 46, 48, 49, 128, 133, 134, 141, 143–146, 150, 156, 178, 211, 232, 245, 263 of angles, 129 specific-heat, 168, 171 Lennard-Jones cluster, 98, 99, 101–104, 108, 109, 111–113 polymer, 101, 113 potential, 12, 16, 21, 82, 94, 98, 101, 105, 111–113, 127, 149, 151, 163, 165, 166, 193, 194, 217, 252 master equation, 47 method contact-density chain growth, 205, 206, 231 energy-landscape paving, 127, 129, 133–136, 148, 175 exact enumeration, ix, 19, 53, 64, 77, 78

293

flat-histogram, 72, 76, 82, 83, 119, 205 generalized-ensemble, 19, 166, 265, 266 molecular dynamics, 11, 146, 153 Monte Carlo, ix, 19, 21, 40, 53, 65, 70, 72, 77, 102, 104, 119, 124, 127, 129, 142, 146, 149, 154, 155, 207, 252, 263, 266 multicanonical, ix, 72, 76–78, 82, 83, 104, 127–129, 132–135, 142, 160, 166, 174, 175, 177, 178, 193, 195, 205, 218, 252, 263, 265 multicanonical chain growth, 65, 71, 72, 74–77, 82, 83, 115, 119, 120, 124, 205, 262, 265 parallel tempering, 149, 166, 175 pruned-enriched Rosenbluth, 67– 72, 74, 75, 78, 83, 92, 133 Rosenbluth chain growth, 19, 65– 69, 120, 209 Wang-Landau, 77, 166 microcanonical analysis, 160–162, 182, 185, 195, 200 ensemble, 31–33, 35, 160, 198, 264 entropy, 72, 160, 174, 182–184, 189, 195, 198 specific heat, 174, 185 temperature, 27, 32, 33, 35, 160– 162, 174, 183, 184, 198, 264 model AB, 15, 21, 44, 127, 130, 133, 147, 151, 160, 170, 174 adsorption, ix, 216 aggregation, 174, 175 all-atom, 9, 11, 13, 130, 131, 231,

294

251 bond fluctuation, 82, 83, 91, 94 coarse-grained, x, 14, 16, 18, 43– 45, 130, 158, 261 effective, 43, 147 FENE polymer, 83 flexible polymer, 81, 101 G¯o, 147, 151, 158 Heisenberg, 129 hybrid, ix, 201, 205, 224, 231, 250, 259 hydrophobic-polar, ix, 15, 18, 20, 44, 52, 124, 127, 170, 193, 261 Ising, 16 lattice, 15, 19, 51, 83, 94, 115, 125, 202, 209, 226, 231 mesoscopic, x, 22, 43, 127, 141, 146, 158, 164 microscopic, ix, 15, 146 mixed hydrophobic-polar, 20, 55 off-lattice, x, 15, 21, 127, 130, 174, 217 peptide, 11, 14, 52, 240 protein, 131, 158, 261 self-avoiding walk, 42, 65, 82 semiclassical, 10, 11, 146 spin glass, 16 tube, 164 multicanonical chain growth, 65, 71, 72, 74–77, 82, 83, 115, 119, 120, 124, 205, 262, 265 ensemble, 72, 177 histogram, 72, 120, 136, 142, 143, 145, 177, 178 method, ix, 72, 76–78, 82, 83, 104, 127–129, 132–135, 142, 160, 166, 174, 175, 177, 178, 193, 195, 205, 218, 252, 263,

Index

265 weight, 72–75, 79, 195 myosin, 4 nanoarray, viii, 229 nanoelectronics, viii, 193, 265 nanoparticle, viii, 256 nanopore, ix, 2 nanosensor, 84, 201, 229, 265 neuron, vii, 173 order parameter, ix, 6, 10, 15, 38, 41–43, 45, 125, 127, 129, 144, 148, 158, 173, 258, 263 adsorption, 209, 257 aggregation, 176, 181 angular overlap, 129, 131, 142 icosahedral, 103, 106, 109 parallel tempering method, 149, 175 pectate lyase C, 115 peptide adhesion coefficient, 239, 250 adsorption, 202, 230, 231, 249, 252, 253, 258 aggregation, viii, 182, 193, 264 amyloid β, vii, 173 bond, 1, 2, 8, 9, 17, 18, 21, 175 modeling, 11 synthetic, x, 141, 238, 249, 265, 266 polymer adsorption, viii, 201, 204, 220, 266 aggregation, 173, 193 collapse, viii, 81, 139, 162 crystal, 97, 163

166,

249, 239, 198,

128,

256,

206,

Index

crystallization, viii, 81, 97, 101, 112, 162, 193 elastic, 82, 91, 97, 101, 102, 104, 108, 221 flexible, 42, 52, 81, 82, 84, 97, 99, 101, 139, 165, 206, 265 freezing, 81, 97 lattice, 52, 65, 81–84, 87, 95, 139, 206, 209, 210, 226–228 Lennard-Jones, 101, 113 linear, 2, 114, 165, 262 nucleation, 173 semiflexible, 193–197, 200, 217 stiff, 197, 198 tube, 164–166, 168 polymerase, vii, 4 potential bond, 104, 112 chemical, 26, 28, 35 effective, 10, 12, 36, 44 electrostatic, 9 FENE, 82, 92, 101, 104 grand canonical, 36 Lennard-Jones, 12, 16, 21, 82, 94, 98, 101, 105, 111–113, 127, 149, 151, 163, 165, 166, 193, 194, 217, 252 square-well, 94 surface, 217, 218, 222 thermodynamic, 28 torsion, 9 Yukawa, 94 protein, vii, 1, 4, 8, 16, 46, 115, 127, 170, 261 adsorption, 229, 249, 263 aggregation, 173, 190, 263 backbone, 1, 2, 21 design, 52 folding, viii, 1, 2, 6, 9–11, 16, 17,

295

27, 51, 124, 129, 141, 146, 155, 159, 173, 217, 261, 263 functional, viii, 9, 16, 48, 51, 52, 158, 262 hull, vii hydrophobic-polar, 19, 51, 55, 59, 62, 115, 121, 127, 261 lattice, ix, 19, 51, 62, 66, 69, 75, 115, 120, 261 membrane, 2 misfolding, vii off-lattice, 127 synthesis, vii, 4 pruned-enriched Rosenbluth method, 67–72, 74, 75, 78, 83, 92, 133 flat-histogram, 72, 76, 82, 83, 205 receptor-ligand binding, ix, 4, 173, 230 ribonuclease A, 17 ribosome, vii, 4–6, 17, 146 scaling analysis, 55, 87, 92, 188, 189, 210, 262, 264 finite-size, 27, 108, 119, 125, 210, 262, 264 Gaussian, 82 law, 52, 54 logarithmic, 94 mean-field, 94 solvent, 1, 2, 5, 7, 9, 10, 12, 17, 81, 97, 125, 141, 195, 218, 230, 266 accessible surface, 13, 159 explicit, 12, 146 good, 81, 138, 209, 215, 236 implicit, 12, 203, 249, 251 parameter, 208, 231, 235 poor, 82, 117, 139, 207, 223, 238

296

quality, 117, 224, 236, 238, 262, 265 specific heat, 37, 40, 41, 61, 77, 83, 84, 86, 106, 116, 139, 162, 171, 180, 197, 204, 232, 241 landscape, 168, 171 microcanonical, 174, 185 negative, 174, 185 structure α-helix, 245 β-sheet, 171, 255 amorphous, 97, 197 anti-Mackay, 109, 112 compact, 225, 236 crystalline, 48, 97, 98, 103, 151, 221, 224 cuboidal, 227 decahedral, 98, 103 designable, 57 diamond, 252 face-centered cubic, 99, 108 filmlike, 209 functional, 46 helical, 134, 164, 166, 167, 169, 229 hierarchical, 198 icosahedral, 98–100, 226 Mackay, 113, 114 native, 46 nonicosahedral, 103, 110 prediction, 11 primary, 2, 8, 17, 51, 159 quaternary, 2, 17 secondary, 2, 11, 17, 18, 58, 141, 158, 163, 164, 166, 168, 170, 229, 241, 244, 247, 248, 255, 263, 266 sheetlike, 169 surface, 249–251

Index

target, 15, 264 tertiary, 2, 17, 18, 168 tetrahedral, 103 surface adhesive, 202 attraction strength, 217, 220, 222, 226, 227 attractive, 217 barrier, 189, 199 contacts, 203, 206, 209, 212, 220, 222, 223, 227, 230–232, 236, 257 coverage, 239, 250 crystal, 230 defects, 108 effects, ix, 27, 48, 97, 99, 101, 160, 174, 180, 182–184, 190, 193, 198–200, 264 energy, 97 entropy, 160, 184, 186, 187, 189, 190, 199, 200 fluctuations, 160 GaAs, 239, 240, 250 hydrophilic, 159 interfacial, 174 layer, 98, 100, 214, 215, 223, 225, 251, 252 metal, 229 minimal, 97 monomers, 91 neutral, 203 potential, 217, 218, 222 semiconductor, 229, 239, 253 Si, 239, 240, 249, 250, 252, 253, 258, 265 solvent-accessible, 13, 159 structure, 249–251 tension, 184, 185, 264 temperature

Index

Θ, 81, 83, 91, 93, 94, 222 aggregation, 184, 187–190, 197– 199, 264 Boyle, 92 canonical, 27, 28, 33–35, 160, 161, 187 critical, 41, 91, 92 folding, 150, 153, 156, 157, 160– 162 freezing, 91, 195, 198, 221, 225 grand canonical, 36 heatbath, 28, 33 melting, 84, 106 microcanonical, 27, 32, 33, 35, 160–162, 174, 183, 184, 198, 264 transition, 28, 38, 40–42, 84, 92, 108, 110, 111, 125, 145, 160, 180, 185, 187, 216, 223 translation, 4 van der Waals force, 9, 43, 98, 217 interaction, 15, 18, 163 radius, 10, 11, 252 virus, vii, 97 capsid, vii, 97 hepatitis B, vii human immunodeficiency, vii Wang-Landau method, 77, 166

297