J.R.C. van der Maarel J.R.C. van der Maarel INTRODUCTION TO INTRODUCTION TO INTRODUCTION TO TO INTRODUCTION Johan R. C. van der Maarel BIOPOLYMER BIOPOLYMER PHYSICS his book provides an ideal introduction to Tthe physics of biopolymers. The structure, dynamics, and properties of biopolymers PHYSICS subjected to various forms of confinement are covered, and special attention is paid to the effect of charge and electrostatic screening (polyelectrolyte effect). By focusing on the development of physical intuition rather than PHYSICS BIOPOLYMER mathematical rigor, readers will be better prepared to address complicated, real issues in the life sciences or related fields such as material or food sciences. The book is designed to serve as a bridge between undergraduate textbooks in physical (bio)chemistry and the professional literature, and is thus especially suitable for advanced undergraduate or postgraduate students and professionals who have already acquired basic knowledge of physics, thermodynamics, and molecular biology. ISBN-13 978-981-277-603-7 ISBN-10 981-277-603-6 World Scientific World Scientific www.worldscientific.com ,!7IJ8B2-hhgadh! 6644 hc Copyright by Johan R. C. van der Maarel All rights reserved TO MY CELESTIAL DANCERS ANNE AND LIEVE AND TO PASCALE FOR HER FORBEARANCE vii PREFACE This book is an introduction to the physics of biopolymers. After a brief overview of the basic properties, we will focus on the structure and dynamics of biopolymers subjected to various forms of confinement. Examples are biopolymers in nano-channels, exposed to external forces, grafted at an interface to form a brush or under crowded conditions at high concentrations in the semi-dilute regime. Special attention will be paid to the effect of charge and electrostatic screening (polyelectrolyte effect). Along the way, we will also discuss higher order secondary and tertiary structures and their transitions. Finally, we will consider the properties of biopolymers in congested and crowded states, which bear resemblance to the situation in living cells and organisms. The book is primarily aimed at the development of physical intuition rather than mathematical rigor in order to prepare the reader to address complicated, real issues in the life sciences or other related fields such as material or food sciences. Most, if not all of the material has been treated with the simplest approach, without losing scientific significance. The mathematics is not too complicated and can be handled by anyone who has received a basic training in calculus. The book is intended to serve as a bridge between undergraduate textbooks in the area of physical (bio) chemistry and professional literature. Accordingly, it is targeted at the advanced undergraduate or postgraduate student as well as the professional, who has already acquired a basic knowledge of physics, thermodynamics and molecular biology. The book is based on my lecture notes for a course on biopolymer physics for fourth year students, which I teach at my home institution. Surely, the quantity of the material exceeds the amount which can be taught in a single viii term and the lecturer might want to make a selection. For instance, one can drop the section on polyelectrolyte brushes or one can skip one of the more specialized topics, such as the compaction of the genome in the capsid of bacteriophages. I plan to post the answers to the questions, small computer script files and other relevant updates (including corrections) on my research group’s website: http://www.physics.nus.edu.sg/~bcf/. It is a pleasure to thank all those people who have contributed, either directly or indirectly, to the writing of this book. First, there are my former teachers and colleagues who have diligently explained to me the older and therefore perhaps less known literature on polymers and polyelectrolyes. Then, of course, I owe thanks to my former and present students. They have pointed out many mistakes in my lecture notes on which this book is based and they have forced me to explain the material in as transparent a way as possible. Special thanks are due to Claire Lesieur for informing me about the status of our understanding of protein folding. I thank Rudi Podgornik for enlightening discussions about the Poisson–Boltzmann equation for polyelectrolytes in the presence of salt. Furthermore, I am grateful to Daniel Blackwood for proof-reading the manuscript. It goes without saying that the responsibility for any possible remaining errors and/or inconsistencies lies entirely with the author. Finally, I thank Pascale, Anne and Lieve for their patience and I apologize for the many hours I took from our precious family time. Singapore, July 2007. ix CONTENTS CHAPTER 1 BIOPOLYMERS 1 1.1 Introduction 1 1.2 Primary structures 4 1.2.1 Nucleic acid primary structures 4 1.2.2 Protein primary structures 6 1.2.3 Polysaccharide primary structures 9 1.3 Secondary structures 11 1.3.1 Secondary structures of nucleic acids 11 1.3.2 Secondary structures of proteins 14 1.3.3 Secondary structures of polysaccharides 17 1.4 Tertiary structure and stabilizing interactions 17 1.5 Questions 20 CHAPTER 2 POLYMER CONFORMATION 23 2.1 The ideal chain 23 2.2 The Kuhn chain 26 2.3 The worm-like chain 27 2.4 Excluded volume interactions 32 2.5 Confinement in a tube; introduction to scaling 34 2.6 Deflection in a narrow tube 36 2.7 Stars and radial brushes 38 2.8 Chains under traction 39 2.8.1 An ideal chain under small tension 40 2.8.2 Worm-like chain 40 2.8.3 Swollen chain 42 2.9 From the dilute to the semi-dilute regime 45 2.10 Chain statistics in the semi-dilute regime 49 2.11 Questions 51 x CHAPTER 3 POLYELECTROLYTES 55 3.1 Counterion condensation 55 3.2 The electrostatic potential 61 3.3 The non-linear Poisson–Boltzmann equation 66 3.3.1 Polyelectrolytes in excess salt 66 3.3.2 Charge distribution in the cell model 69 3.4 The electrostatic persistence length 76 3.5 Electrostatic excluded volume 80 3.6 Flexible chains and electrostatic blobs 87 3.7 Spherical polyelectrolyte brushes 89 3.7.1 Spherical polyelectrolyte brush without salt 89 3.7.2 Salted spherical polyelectrolyte brush 94 3.8 Polyelectrolytes in the semi-dilute regime 99 3.8.1 Salt-free polyelectrolytes; a hierarchy of blobs 99 3.8.2 Salted polyelectrolytes 101 3.9 Questions 103 CHAPTER 4 POLYMER DYNAMICS 105 4.1 Single chain dynamics 105 4.2 Pulling a chain into a hole 111 4.3 Dynamics of non-entangled chains in the semi-dilute regime 114 4.4 Entangled polymer dynamics; reptation 117 4.5 Dynamic scaling of polyelectrolytes 121 4.5.1 Polyelectrolytes without salt 121 4.5.2 Salted polyelectrolytes 124 4.5.3 Comparison with experimental results 126 4.6 Gel electrophoresis 130 4.7 Questions 134 CHAPTER 5 HIGHER ORDER STRUCTURES AND THEIR TRANSITIONS 137 5.1 Supercoiled DNA 137 5.1.1 Topology 138 5.1.2 Molecular free energy 142 5.1.3 Long-range structure and branching 151 5.2 Alternate secondary DNA structures 155 5.2.1 B–Z transition 155 5.2.2 Cruciforms 159 5.3 Helix-coil transition 161 5.4 Protein folding 167 5.5 Questions 171 xi CHAPTER 6 MESOSCOPIC STRUCTURES 175 6.1 Lyotropic liquid crystals 175 6.1.1 Virial theory 177 6.1.2 Liquid crystalline orientation order 182 6.1.3 Isotropic-anisotropic phase coexistence 185 6.2 Hexagonal packing of DNA 190 6.2.1 Undulation enhanced electrostatic interaction 191 6.2.1 Melting of the hexagonal phase 196 6.2.2 DNA equation of state 198 6.3 Bacteriophage DNA packaging 201 6.4 Crowding and entropy driven interactions (depletion) 208 6.4.1 Entropic colloidal interactions in solutions of macromolecules 210 6.4.2 Phase separation of small particles in a polymer solution 215 6.5 Questions 220 APPENDIX A: POISSON–BOLTZMANN THEORY FOR A MONOVALENT SALT 223 APPENDIX B: SUMMARY OF SCALING LAWS 227 APPENDIX C: LIST OF IMPORTANT SYMBOLS 229 RECOMMENDED READING 233 REFERENCES 235 INDEX 243 Introduction to Biopolymer Physics 1 CHAPTER 1 BIOPOLYMERS In this chapter, the basic properties of biopolymers will be briefly discussed. We will group them according to nucleic acids, proteins and polysaccharides and we will summarize their main biological functions. Biopolymers have the unique feature that they exhibit a hierarchy in their molecular structures. Associated with these structures, their biological functions emerge almost naturally. In the latter context, think about the importance of the double- helical structure of DNA for the replication process. It is important to realize that these biological functions are based on the way the building blocks (nucleotides, amino acids, carbohydrates, etc.) are assembled. We will subsequently present the primary, secondary and some tertiary structures of nucleic acids, proteins and polysaccharides and show how they are stabilized by interactions. However, a detailed discussion of the chemical composition of the various biopolymers and their biological functions is beyond the scope of this book and for this purpose the reader is referred to the dedicated literature (see, for instance, the textbooks of Mathews, van Holde and Ahern and Bloomfield, Crothers and Tinoco).1,2 1.1 Introduction Biopolymers or biomacromolecules can be roughly classified according to three different categories: nucleic acids, proteins and polysaccharides (carbohydrates). It should be born in mind that this classification is not strict and that there are important exceptions. An example is glycoprotein, which is a combination of protein and carbohydrate and plays a role in, among others, immune cell recognition and tissue adhesion. The biological functions of nucleic acids, proteins and polysaccharides are also different.