Introduction to Biopolymer Physics

The following sections are included:

• PREFACE

• CONTENTS

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

In this chapter we will review some concepts in polymer physics as far as the conformational, static properties are concerned. We will closely follow the textbooks of de Gennes, Grosberg and Khokhlov.^3,4 The discussed polymers are homogeneous with every monomer having the unique function of serving as a link in a long chain. Here, we will not consider variation in secondary structure along the chain and/or specific tertiary structures. First we will discuss the properties of single chain molecules and how their size depends on the molecular weight and the interactions among the segments. Then we will move on to chains in various forms of spatial confinement and/or subjected to external perturbations such as a pulling force. Finally, we will discuss the properties of dense polymer solutions, where the chains significantly interpenetrate in the semi-dilute regime.

Most biopolymers can be ionized and carry electric charges. In the case of nucleic acids, every phosphate moiety connecting the nucleotides is negatively charged. The acidic or basic side groups of amino acids can be positively or negatively charged depending on the pH of the supporting medium. Of course, on a macroscopic scale electro-neutrality is preserved, so that the biopolymer charge should be compensated by the charge of small counterions dispersed in the medium. In addition to these counterions, the system might also include a low molecular weight salt. In this chapter we will review the basic properties of biopolymers related to the presence of charge. First we will discuss counterion condensation and the formation of the double layer around the polyion in terms of a critical linear charge density on the polymer. Then we will move on to electrostatic effects on the chain flexibility and volume interactions. Finally, we will discuss the properties of spherical polyelectrolyte brushes and polyelectrolytes in the semi-dilute regime.

In this chapter, we will discuss the dynamics of polymer chains in solution. We will focus on the global dynamics, involving the motion of the chain as a whole. First, we will consider dilute solutions, in which the chains do not overlap. This will allow us to introduce two important models, depending on the draining conditions of the coil. We will also derive expressions for some molecular transport properties, that is the polymer self-diffusion coefficient and the viscosity. As another example of a dynamic property, we will present an analysis of how long it takes to pull a polymer inside a hole. This issue is important for the understanding of certain micro-fluidic processes involving the translocation of a polymer inside a nano-channel. Then we will move on to a discussion of the dynamics in the semi-dilute regime, where the dynamics are strongly affected by the chain overlap and the possible formation of transient, topological constraints. We will also discuss the dynamics in saltfree and salted polyelectrolyte solutions. Finally, we will present the principle of gel electrophoresis in order to separate biomacromolecules on the basis of their molecular weight.

A unique property of biopolymers is that they can exhibit higher order secondary or tertiary structures. For instance, DNA is often, if not always, in a supercoiled conformation, in which the molecule is interwound and forms a higher order helix. Proteins fold into well defined molecular structures and their functioning is related to their specific conformation. Transitions between different molecular structures can be triggered by e.g. ligand binding and chemical modification, as well as environmental factors such as ionic strength and temperature. First we will discuss the medium to long-range structure of the supercoiled DNA molecule and how this is related to the topological constraint. Then we will move on to the energetics of supercoiling-induced, cooperative transitions, that is the B‒Z transition and the extrusion of a cruciform in an inverted repeat, palindromic base-pair sequence. For a further discussion of the concept of cooperativity, we will present the statistical-mechanical formalism of the helix to coil transition. Finally, we will discuss some aspects of protein folding and how we can describe and visualize the folding process in a meaningful manner.

In vivo, biopolymers are often, if not always, under crowded conditions and/or condensed into compact structures. Of course, many of the biological phenomena are related to the machinery of life, which involves structure and dynamics at the atomic level. However, it is of interest to see to what extent the properties of dense biopolymer systems can be described on the mesoscopic distance scale, that is on a scale of typically a few to tens of nanometres, with physical laws based on statistical properties such as entropy, density and charge. First we will focus on the formation of lyotropic liquid crystals of (locally) rod-like polymers, exclusively based on the competition of orientation entropy and the restriction in translational degrees of freedom. Then we will move on to the high density, hexagonal phase of DNA, for which we will derive the equation of state and a stability criterion based on a fluctuation enhanced electrostatic theory. With the equation of state, we will subsequently analyze the packing and the packaging force of the genome inside the capsid of bacteriophages. We will conclude with a description of entropy driven interactions and phase separation in ternary systems of large colloidal or small (nano) particles, an inert osmotic agent and a common solvent.

The following sections are included:

• APPENDIX A: POISSON–BOLTZMANN THEORY FOR A MONOVALENT SALT

• APPENDIX B: SUMMARY OF SCALING LAWS

• APPENDIX C: LIST OF IMPORTANT SYMBOLS

• Recommended reading

• References

• INDEX

• ABOUT THE AUTHOR