J.R.C. van der Maarel BIOPOLYMER PHYSICSpodgornik/download/Introduction.pdf4.4 Entangled polymer...

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INTRODUCTION TOBIOPOLYMERPHYSICS

INTRO

DU

CTION

TO BIO

POLYM

ER PHYSICS

This book provides an ideal introduction to

the physics of biopolymers. The structure,

dynamics, and properties of biopolymers

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

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.

INTRODUCTION TOBIOPOLYMERPHYSICS

ISBN-13 978-981-277-603-7ISBN-10 981-277-603-6

Johan R. C. van der Maarel

,!7IJ8B2-hhgadh!

J.R.C. van der Maarel

J.R.C. van der Maarel

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. Nucleic acids are

Chapter I: Biopolymers 2

involved with the storage of the genetic code (DNA) and the translation of the genetic information into protein products (RNA). Proteins catalyze biochemical reactions (enzymes), have structural or mechanical functions or are important in cell signalling and immune responses. The structural components of plants are primarily composed of the polysaccharide cellulose. Bacteria excrete polysaccharides for adhesion to surfaces and to avoid dehydration. Examples of these polysaccharides are dextran, xanthan and pullulan, which have found wide-spread applications in pharmacy, biotechnology and the food industry. The classification according to the functioning of the biopolymers is also not unique. An important exception is the ribosome; an organelle on which proteins are assembled. A ribosome contains 65% RNA and 35% protein. It can be considered an enzyme, but its active site is made of RNA. However, the functioning and purpose of biopolymers in the machinery of life is beyond the scope of this book. Here, we intend to explore the extent to which their properties can be understood in terms of concepts from physics and mathematics.

Like every polymer, biopolymers are strings or sequences of monomeric units or monomers for short. In many cases these strings are linear, but sometimes they are closed and circular, branched or even cross-linked. In the latter case, we are dealing with a gel. In this book, we will primarily focus on linear polymers, but we will also discuss star-branched polymers, spherical polymer brushes and closed circular, supercoiled DNA. The structure of any biopolymer is determined by the nature of the building blocks (i.e. the monomeric units) in combination with environmental conditions such as the temperature, the solvent (water) and the presence of salts and/or other molecular components. The monomeric units of nucleic acids, proteins and polysaccharides are largely different and will be discussed in the next section. A unique feature of biopolymers is that most of them are essentially heteropolymers, because they may contain a variety in monomeric units. The biological relevance of a biopolymer is ultimately based on the sequence of the monomers, i.e. the primary structure. In the case of DNA, the primary structure is the sequence of bases attached to the sugar rings, which determines the genetic code. For proteins, it is the amino acid sequence, which eventually determines, together with environmental conditions, their 3–dimensional shapes and biological functions. The properties of polysaccharides are also largely determined by the nature of the monomeric


units, more specifically in the way they are connected. A fundamental characteristic of biopolymers is the formation of

hierarchical structures at successive length scales. Starting from the primary structure, the monomeric units are organized in a certain local molecular conformation. This local conformation is commonly referred to as the secondary structure. Examples of secondary structures are the famous double-helical arrangement of the two opposing strands in the DNA molecule (the duplex) and α−helixes and β− sheets formed by the polypeptide chains in proteins. At a larger distance scale, a biopolymer can adopt a defined 3–dimensional conformation: the so-called tertiary structure. This is particularly relevant for proteins, which largely owe their biological functioning to their 3–dimensional structure, but also nucleic acids and polysaccharides have tertiary structures. An example of a nucleic acid with a tertiary structure is transfer RNA, which has an L–shaped 3–dimensional structure that allows them to fit into the active site of the ribosome (it transfers a specific amino acid residue to a growing polypeptide chain). Eventually, biopolymers can form even larger complexes among themselves and with other macromolecular components in the cell and organism.

Biopolymers have emergent properties associated with their hierarchical structures. Here, the meaning of emergence is that the biopolymers have properties that cannot be attributed to the individual building blocks. For instance, the nucleic acid bases are just molecular components made of carbon, nitrogen and oxygen. It is their specific sequence in a strand of the DNA or RNA molecule that carries the genetic code. This property cannot be attributed to the individual bases, but it has emerged from the assembling of the bases into the nucleic acid. It is also obvious that the activity of a protein is an emerging property of the hierarchical assembling of the amino acids. Here, it is even possible to replace a selected and limited number of amino acids by other amino acids without losing the biological function of the protein. Emergence is a general phenomenon associated with the assembling of building blocks into larger scale structures, both in civil engineering and in biology.

Throughout this book, we will almost exclusively deal with systems in thermodynamic equilibrium. Although this is of interest in its own right, it is only fair to say that the study of systems in thermodynamic equilibrium has a limited relevance for our understanding of life. It is commonly believed that


spontaneous assembling processes, driven by the minimization of the system’s free energy (self-assembly), are important in biology. However, one should bear in mind that life exists by the virtue of the dissipation of energy, mainly through the hydrolysis of adenosine triphosphate (ATP). By definition, a living organism is in a non-equilibrium state and it is not always possible to generalize the concepts obtained for equilibrium conditions. Understanding life on the basis of non-equilibrium, dissipative processes is clearly a challenge for the future.

1.2 Primary structures

1.2.1 Nucleic acid primary structure

There are two types of nucleic acids, ribonucleic acid (RNA) and deoxyribonucleic acid (DNA). As shown in Fig. 1.1, each molecule is a

O

O

PO O

O

O

OH

CH2 Base

3’

5’

5’ end

3’ end O

PO O

O

O

OH

CH2 Base

3’

5’

O

O

PO O

O

OCH2 Base

3’

5’

5’ end

3’ end O

PO O

O

OCH2 Base

3’

5’

RNA DNA

Figure 1.1 Chemical structures of ribonucleic acid (RNA, left) and deoxyribo-nucleic acid (DNA, right). The phosphate groups and the five carbon sugar rings are shown in detail. The bases are shown schematically, but their chemical structures are depicted in Fig. 1.2.


polymeric chain, in which the units are covalently linked by the phosphates. The monomeric units are the nucleotides. Each nucleotide is built around a five-carbon sugar; ribose in RNA and 2’–deoxyribose in DNA. In Fig. 1.1 the five carbon atoms of the sugar are counted from the one to which the base is attached at the right, down through the ring and then up to the fifth carbon at the upper left side. Besides a difference in bases, which will be discussed shortly, the chemical difference between RNA and DNA lies in the replacement of a hydroxyl group by a hydrogen atom at the 2’ position in DNA. The nucleotides are linked through the formation of a phosphodiester between the 5’ carbon of one nucleotide and the 3’ carbon of the next nucleotide. In this way, long nucleic acid chains sometimes contain millions of units which are attached to each other. It is important to realize that the string of nucleotides has a direction from the 3’ to the 5’ end. The phosphate group is a strong acid with a apK of around one. RNA and DNA are thus strong acids and under physiological conditions every phosphate moiety carries a negative charge. DNA and RNA are so-called polyelectrolytes and the presence of charge results in specific properties, such as an electrostatic contribution to the bending rigidity of the molecule. This and other effects of the presence of charge will be detailed in Chapter 3.

The backbone of the nucleic acid molecule is a repetitive structure and by itself it cannot store information. It is clear that the information storage

Sugar

Sugar

SugarSugar

N

N

N

NH

N

N

NH

O

NH

N

N

NH

O

HN

NO

O

HN

NO

O

N

N

Sugar

2

2

Guanine (G)DNA/RNA

Adenine (A)DNA/RNA

2

Thymine (T)DNA

Cytosine (C)DNA/RNA

Uracil (U)RNA

Figure 1.2 Pyrimidine (cytosine, C; thymine, T; uracil, U) and purine (adenine, A; guanine, G) bases in DNA and RNA.


capacity is derived from the sequence of bases, each of which is attached to the 1’ carbon of the sugar ring. There are two types of bases: the purines and pyrimidines. In the case of DNA, there are two purines, adenine (A) and guanine (G) and two pyrimidines, cytosine (C) and thymine (T). In the case of RNA, uracil (U) replaces thymine (see Fig. 1.2). DNA and RNA also contain a small fraction of chemically modified bases; some of these can induce alternate secondary structures, as will be discussed in Chapter 5. Note that the bases do not carry charge, but they can form hydrogen bonds.

1.2.2 Protein primary structure

All proteins are polymers and their monomeric units are α− amino acids. The amino group is attached to the α− carbon, i.e. the carbon next to the carboxyl group. Under physiological conditions, the amino acid is in its zwitterionic form; the amino group has picked up a proton and has become positively charged and the carboxyl group has dissociated a proton and is negatively charged. Besides the amino group, a hydrogen atom and a side group are also attached to the α− carbon of every amino acid. The amino acids are distinguished by their different side groups. Twenty chemically different amino acids are incorporated in proteins; their structures are shown in Fig. 1.3. In the simplest case, glycine, the side group is just a hydrogen atom. The amino acids can be grouped according to the physical-chemical properties of the side group: aliphatic, hydroxyl or sulphur containing, cyclic (proline), aromatic, basic or acidic. It is clear that the higher order secondary and tertiary structures of proteins are intimately related to these properties, together with environmental factors such as the solvent quality.

With the exception of glycine, there are always four different chemical groups attached to the α− carbon of every amino acid. Accordingly, amino acids are chiral and each one can occur in two different stereoisomers: the D– and L–forms. The L–form of alanine is displayed in Fig. 1.4; it has the amino, hydrogen, carboxyl and methyl groups arranged in a clockwise manner, when the α− carbon is viewed from the top with the amino and carboxyl groups pointing downwards and the hydrogen and methyl group pointing upwards. All amino acids incorporated by organisms into proteins are of the L–form. The chirality of the amino acids has an important consequence for the


secondary structure. For instance, owing to the steric interactions among the side groups, only right-handed α−helixes are possible. Left-handed helixes can be obtained by using synthetic amino acids in the D–form.

Amino acids can be covalently linked by the formation of a peptide bond between the α− carboxyl group and the α− amino group. This is illustrated in Fig. 1.4 for the link between alanine and glycine in order to form

C COOH N

H

3-

H

C COOH N

H

3- C COOH N

H

3- C COOH N

H

3- C COOH N

H

3-

C COOH N

H

3- C COOH N

H

3- C COOH N

H

3- C COOH N

H

3- C COOH N

H

3-

CH CH CH3 2

CH3CH3CH

CH3CH3

+ + + +

Glycine Alanine Valine Leucine Isoleucine

Aliphatic

OHNH

C COOH N

H

3- C COOH N

H

3- C COOH N

H

3- C COOH N

H

3- C COOH N

H

3-

HCOH CH2

CH3

H C2

CH2

+ + + + +

Serine Cysteine Threonine Methionine Proline

CH2

OH

CH2

SH CH2

CH3

S

CH

C COOH N

H

3-

CHCH3

CH2

+

CH3

2

Hydroxyl or Sulphur Cyclic

Phenylalanine Tyrosine Tryptophan Histidine Lysine Arginine

2CH 2CH 2CH

HNN H

+

+ + + + + +2CH 2

CH2

N H3

CH

CH2

CH2

+

2

CH2

C 2

CH

CH2

NH

+N H

2NH

Aromatic Basic

C COOH N

H

3- C COOH N

H

3- C COOH N

H

3- C COOH N

H

3-

2

C

NH2O

+ + + +

Aspartic acid Glutamic acid Asparagine Glutamine

Acidic

CH2

-

2

CH

COO-

2

CH

2CH

CH

C

NH2O

2CH

COO

Figure 1.3 The twenty standard α− amino acids found in proteins. Note that they have been arranged according to the properties of the side group. In organisms, more different amino acids are present, but those are not incorporated in proteins.


glycylalanine. The carbonyl C=O and the N–H bonds must remain in the same plane with only a little twisting around the C–N bond possible, because of the electron resonance structure of the peptide bond. Furthermore, due to the steric interaction between the side groups, the trans form is the favoured configuration. In this way, many amino acids can be linked to form a polypeptide. All proteins are polypeptides of a defined sequence determined by the genetic code. This sequence of amino acids is the primary structure of the protein, upon which all higher levels of organisation are based. As in the case of nucleic acids, the string of amino acids has a direction. At one side there is the amino N–terminus and at the other side the carboxyl C–terminus. Note that the polypeptide backbone is not charged besides the end groups. The charges of a protein are located in the side groups. Since a particular side group can be neutral or charged either positively or negatively, the net charge of a protein depends on the amino acid composition as well as the pH of the supporting medium. Under physiological conditions the protein is usually close to the iso-electric point, so that the positive and negative charges cancel out and the net charge is almost zero.

C

C

C

N

O

O

-+

C

C

O

O

-+

N

CC C

O

O

N

-O

C

CN+

Glycine Alanine

Glycylalanine

+O

C-terminusN-terminus

Figure 1.4 Formation of the peptide bond. Here, glycine and alanine are linked to form the dipeptide glycylalanine by the removal of a water molecule. Note that the peptide bond is planar. Redrawn from Ref. [1].


1.2.3 Polysaccharide primary structures

Polysaccharides are polymers of monosaccharides linked with glycosidic bonds. The monomers are cyclic structures, mostly containing 5 (pentoses) or 6 (hexoses) carbon atoms. An example of a pentose is ribose, which is one of the building blocks of nucleic acids. Many polysaccharides are made of hexoses, such as sucrose and galactose. Amylose and cellulose are linear chains of α−D–glucose and β− D–glucose, respectively (see Fig. 1.5). The glycosidic bond is formed between a hydroxyl group on one carbohydrate unit with a hydroxyl group on another unit. In the case of amylose and cellulose the links are formed between the first and fourth carbon atom. It is customary to indicate these bonds by the numbers of the linked atoms and the stereoisomer of the unit, so amylose and cellulose are linked by ,1 4α− and

,1 4β− glycosidic bonds, respectively. The primary structure of a polysaccharide can be more complicated. For instance, pullulan is a linear

OH

O

CH OH2

OHO

OH

O

CH OH2

OHO

OH

O

CH OH2

OH O

OH

O

CH OH2

OH O

Cellulose

O

O

Amylose

OH

O

CH OH2

OH

OHOH

OH

O

CH OH2

OH

OH

OH

α-D-glucose β-D-glucose

1 1

2 23 3

4 4

5 5

6 6

Figure 1.5 Amylose and cellulose are linear polysaccharides made by connecting α− D–glucose and β − D–glucose, through ,1 4α− and

,1 4β − glycosidic bonds, respectively and the removal of a water molecule.


polymer of maltotriose units. Three glucose units in maltotriose are connected by an ,1 4α− bond; whereas consecutive maltotriose units are connected by

,1 6α− bonds (see Fig. 1.6). Dextran is a branched polysaccharide made of many glucose molecules joined into chains of varying lengths. The linear chain sections are linked by ,1 6α− bonds between glucose units, while branches begin from ,1 3α− linkages (and in some cases, ,1 2α− and ,1 4α− linkages as well). Polysaccharides are never as complex as proteins or nucleic acids; they usually contain no more than two kinds of residues. Furthermore, polysaccharide chains have a random degree of polymerisation, in contrast with proteins and nucleic acids which are almost always of a defined length.

OH

O

CH OH2

OHO

OH

O

CH OH2

OHO

OH

O

CH 2

OHO

OH

OH

O

CH 2

OHOH O

OH

O

CH 2

OH OOH

O

CH 2

OHOH O

OH

O

CH 2

OHOH O

Pullulan

Dextran

Figure 1.6 Primary structures of pullulan and dextran. Pullulan is a linear polymer with a repeating maltotriose unit of three glucose units. Dextran has a branched structure with on average about 100 monomers between the branch points.


In their basic form, polysaccharides are uncharged. However, they are often functionalized with carboxyl groups, phosphate groups and/or sulphuric ester groups. The monomeric units contain many hydroxyl groups, which can engage in intra- and inter-molecular formation of hydrogen bonds. This hydrogen bonding keeps the chains together and contributes to the high tensile strength of the polymeric material. In this context, it is interesting to note that other forms of functionalization also occur. For instance, chitin, which is a major component of the exoskeletons of crustaceans and insects, can be described as cellulose with one hydroxyl group on carbon 2 of each glucose unit substituted by an acetylated amino (acetylamine) group. This substitution allows for increased hydrogen bonding, which gives the matrix formed by the polymer increased strength. Some polysaccharides such as cellulose are insoluble in water, whereas for others (e.g. dextran or pullulan) water is a moderate to excellent solvent.

1.3 Secondary structures

1.3.1 Secondary structures of nucleic acids

The bases of DNA and RNA can form base-pairs stabilized with hydrogen bonds. As shown in Fig. 1.7, adenine can form two hydrogen bonds with thymine, whereas guanine can form three hydrogen bonds with cytosine. With these pairing arrangements between the purines and pyrimidines, the distances between the 1’ carbons of the attached sugars are the same (1.08 nm). In this way, two opposing single strands with a complementary base sequence can form a double helix, which is regular in diameter. This would not be possible if purines pair with purines and/or pyrimidines with pyrimidines. Besides hydrogen bonding, the double helix is stabilized by dispersion forces resulting from correlated electron charge fluctuations in the stack of base-pairs.

RNA is usually single-stranded, but most RNA molecules can form hair-pin structures by base-pairing of self-complementary regions within the same molecule. Single-stranded DNA with self-complementary base sequences can also fold back on itself and form single-chain stacked base structures. At elevated temperature and/or in the presence denaturing agents, the single-


stranded DNA molecule will take a random coil configuration. However, the canonical form of DNA is the double helix made of two complementary strands in an anti-parallel direction (the duplex). In the double helix, each strand can serve as a template for a complementary strand of DNA in the case of replication or for a complementary strand of messenger RNA in the case of the transcription of the genome for the synthesis of protein products.

The bases from the two opposing DNA strands in the duplex are stacked in the interior of the helix, whereas the two anti-parallel sugar-phosphate backbones are extended along the outside. The helix has a major and minor groove. Three secondary structures of the double-stranded DNA molecule have been identified: the A–, B– and Z–forms. The average values of the most important structural parameters are collected in Table I.I and space-filling representations are shown in Fig. 1.8. The main distinguishing features of these different secondary structures of DNA are2 • The A– and B–forms are right-handed and can be found in any sequence.

B is the dominant form under physiological conditions. The A–form is

H

H

H

H

Sugar

Sugar

N

N

N

N

N

N

N

O

N

N

N

N

O

Sugar

N

N

O

O

N

N

Sugar

Guanine Cytosine

HH

H

Adenine Thymine

H

Figure 1.7 Base-pairing of thymine with adenine and cytosine with guanine.


found at low hydration levels, such as in spun fibres. The Z–form is left-handed and occurs in alternating purine-pyrimidine sequences, particularly guanine-cytosine (GC).

• The double-stranded duplex in the A–form is thick and compressed along the helix; in the Z–form it is elongated and thin whereas in the B–form it is intermediate.

• There are 10, 11 and 12 base-pairs (bp) per turn in B–, A– and Z–DNA, respectively. The corresponding pitches are 3.2, 3.4 and 4.5 nm.

B-form A-form Z-form

Figure 1.8 Space-filling representations of double-stranded DNA in the B–, A– and Z–form.

Table I.I Structural properties of DNA in the A–, B– and Z–form.2

Geometrical attribute A–form B–form Z–form Helix sense right-handed right-handed left-handed

Repeating unit (bp) 1 1 2 Rotation/bp 32.7° 36.0° 60°/2

Mean bp/turn 11 10.0 12 Inclination of bp to axis +12° 2.4° -6.2° Rise/bp along axis (nm) 0.29 0.34 0.37 Pitch/turn of helix (nm) 3.2 3.4 4.5

Diameter (nm) 2.6 2.0 1.8 Minor groove depth (nm) 0.28 0.75 0.9 Minor groove width (nm) 1.10 0.57 0.4 Major groove depth (nm) 1.35 0.85 - Major groove width (nm) 0.27 1.17 -


• B–DNA has a wide major groove and a narrow minor groove, both of which are of similar depth. The A–form has a narrow, deep major groove and a wide, shallow minor groove. Z–DNA has a deep and narrow minor groove and no major groove.

• In B–DNA the base-pairs are almost perpendicular to the helix axis; those in A– and Z–DNA are inclined at larger angles.

We will almost exclusively deal with DNA in the regular B–form. Owing to the presence of the hydroxyl group at the 2’ position of the

ribose sugar, base-paired RNA adopts the A–form geometry.

1.3.2 Secondary structures of proteins

Proteins show a wide variety of secondary structures. These structures satisfy a number of criteria: the peptide bond is planar because little twisting is possible about the C–N bond; the steric interaction between the side groups of the amino acids is minimal and the structure is stabilized by hydrogen bonding between the oxygen of the carbonyl C=O groups and the hydrogen of the amide C–N groups. Two major secondary structures, which satisfy these criteria, are the α−helix and β− sheet (see Figs. 1.9 and 1.10, respectively). Note that these two structures are by no means the only ones. There exists other well defined, but less abundant secondary structures, such as the 103 helix and some specific sharp turn loop sequences. Furthermore, there are also significant parts of the polypeptide chain which cannot be classified as one of these secondary structures. The latter parts have an irregular structure, but they are not random coils.

The structure of the α−helix is shown in Fig. 1.9. There is an almost linear hydrogen bond formed between every carbonyl oxygen with an amide

Table I.II Geometrical attributes of polypeptide secondary structures.1

Structure Residues/ Turn

Rise/Residue (nm)

Pitch (nm)

α− helix 3.6 0.15 0.54 103 − helix 3.0 0.20 0.60

Parallel β − sheet 2.0 0.32 0.64 Anti-parallel β − sheet 2.0 0.34 0.68


hydrogen on the fourth residue up the chain (separated by two residues). The hydrogen bonds are almost parallel to the helix axis. There is little or no steric interaction among the side groups, because they are pointing outwards away from the central axis of the helix. The α−helix has 3.6 residues per turn, which results in a rise of 0.15 nm per residue and a pitch of 0.54 nm per turn. In the less abundant 103 − helix, there is a hydrogen bond between the carbonyl oxygen and the amide hydrogen of the third residue up the chain. Accordingly, the 103 − helix is less compressed in the longitudinal direction with 3.0 residues per turn and a rise of 0.20 nm per residue.

In the β− sheet, each residue is flipped by 180 degrees with respect to its preceding one and the polypeptide chain (β− strand) is folded in a zigzag fashion. As illustrated in Fig. 1.10, the linear hydrogen bonds are now formed

RC

N

C

O

O

O

O

O

O

O

O

C

C

C

C

C

C

C

C

N

N

N

N

N

C

C

C

C

C

C

C

N

N

CN

N

OC

N

R

R

R

R

R

R

R

R

R

R

C

C

O

C

C

C

C

C

C

N

N

N

N

C

C

C

C

C

C

N

N

CN

N

C

N

N

C

C

N

C

NC

CCCC

CCC

NC

C NNCCC

N

CC

CCCCCCCC

C

CCC

CC

NC

C

N

CCN

CN

CNN

CN

CC

C CCCNN

C

C

NC

NC

N

NC

NC

CCC

N

CCN

C

C

C

N

N

C

N

CC

CCCC C

N

C

NCC

C

C

CN

CO

COOON

CCCCO

CN

CN

CCCCCCCCCCC

CC

C

NC

N

Figure 1.9 Left: Irregular α− helical secondary structure of polypeptides. The hydrogen bonds between the carbonyl oxygen and the amide hydrogen are within a single polypeptide chain and almost parallel to the helix axis. The side groups point outwards. Right: Schematic helical ribbon representation showing the atoms of the backbone atoms only.


between adjacent chains almost perpendicular to the strand axis. Due to the consecutive flipping of the residues by 180 degrees, the side groups alternately point upwards and downwards away from the sheet. The β− sheet can be formed in two ways: parallel and anti-parallel. In the parallel configuration, the β− strands are all running in the same direction from the N– to the C–terminus. In the anti-parallel configuration, adjacent strands are running in opposite directions (as in Fig. 1.10). In the β− strand there are just two residues per turn, but the rise per residue differs between the parallel and anti-parallel configuration: 0.32 and 0.34 nm, respectively. The geometrical attributes of a number of secondary protein structures are collected in Table I.II.

C

N

N

N

NN

N

N

NN

N

N

N

N

C

C

C

CC

C

C

C

C

C

CC

CC

C

CC

CC

C C

C

C

C

C

C

C C

CNN

O

O

OO

O

O

OO

O

C

N

R

R

R

R

RR

R

R R

RR

R

R

R

N

N

NN

N

N

NN

N

N

N

N

C

C

C

CC

C

C

C

C

C

CC

CC

C

CC

CC

C C

O

O

O

O

C

C

R

C

C

C

C C

CNN

O

C

C

N

NC

C

C

C

C

N

C

C

C

C

C

N

N

C

C

CC

C

C

C

C

C

CC

C

N

C

N

C

C

N

C

C

C

N

C

NN

C

CC

N

C

C

N

C

C

C

C

C

C

N

N

C

C

C

N

C

C

CC

N

C

N

C

C

NN

C

C

C

C

C

C

NN

C

C

C

C

N

C

C

N

C

C

N

C

N

C

CC

C

N

C

C

C

C

C

Figure 1.10 Left: Irregular anti-parallel β− sheet secondary structure of polypeptides. The hydrogen bonds between the carbonyl oxygen and the amide hydrogen are between adjacent chains and almost perpendicular to the chain axis. The side groups alternately point upwards and downwards along the chain. Right: Schematic representation showing the atoms of the backbone atoms only and the coarse-grained arrows which show the directions from the N– to the C–terminus.


In proteins, the secondary structures may be deformed by the presence of the side groups. The α−helix and β− strand structures are often depicted by the coarse-grained helical ribbon and deformed arrow shapes as shown in Figs. 1.9 and 1.10, respectively. The arrow heads at the ends of the β− strands point in the direction from the N– to the C–terminus.

1.3.3 Secondary structures of polysaccharides

Polysaccharides with a complex and/or branched primary structure, such as pullulan and dextran respectively, take a random coil conformation when dissolved in a suitable solvent (water). If the primary structure is simple and regular, polysaccharides may exhibit a regular secondary structure. In amylose, the regular orientation of successive glucose residues results in a right-handed helix with six residues per turn. Cellulose can exist as fully extended chains with each residue flipped by 180 degrees with respect to its neighbour in the chain. The cellulose chains form ribbons that are packed side-by-side with hydrogen bonds within and between them; a structure which is reminiscent of the β− sheet. Xanthan is a linear polysaccharide with a repeating unit made of 5 sugar units. To every repeating unit of the main chain a small side-chain is attached consisting of three modified sugar units. Two of these xanthan chains are thought to form a double helix, which gives the molecule a high bending rigidity and accounts for its surprisingly high solution viscosity.

1.4 Tertiary structure and stabilizing interactions

Naked double-stranded DNA, that is DNA not complexed with proteins, behaves as a charged polymer and takes a random coil conformation in water or an aqueous buffer. However, the biological relevance of naked DNA is limited. Inside the capsid of certain bacteriophages, double-stranded DNA is compacted and essentially protein-free, except for the proteins which make up the structure of the capsid itself. In the nucleoid region of bacterial cells, the genome is thought to be compacted by specific interactions with proteins as well as by osmotic, depletion effects exerted by non-binding proteins dispersed in the cytoplasm (the latter effects will be discussed in Chapter 6). In eukaryotic cells, DNA is wrapped around histone proteins and looks like


beads on a string when observed with an electron microscope. A section of 146 base-pairs of DNA with a contour length of around 50 nm is wrapped in 1.65 left-handed turns around the histone octamer, which is composed of four identical pairs of histone proteins. This assembly of DNA and protein is called the nucleosome core particle. The nucleosome core particles are connected with sections of 50 base-pairs of ‘linker’ DNA, together with another histone protein, so that the total repeating unit of the beads on the string is around 200 base-pairs. The core particles are stacked into a higher order structure called chromatin, which is organized in a hierarchical manner up to the level of the chromosome. Besides the structure of the nucleosome core particle, the structure of chromatin is largely unknown.

A special category of double-stranded DNA is plasmid. Plasmids are separated from chromosomal DNA and they usually occur in bacteria. Their size varies from around two to more than 400 kilo base-pairs. Plasmids are widely used as cloning vectors in genetic engineering, because they easily transfer from one bacterial cell to another and it is easy to insert DNA fragments at their restriction sites. Plasmids are often, if not always, circular, but the strands of the duplex are usually twisted a couple of times about their long axes before they are closed in order to form the ring. As a result of this topological constraint and the fact that the double-stranded DNA molecule can support twist, the plasmid molecule takes a 3–dimensional, supercoiled configuration. Supercoiling is not exclusive to plasmids; it also occurs in sections of chromosomal DNA as a result of complexation of protein on DNA. We will discuss supercoiling and supercoiling-induced transitions in the secondary structure of double-stranded DNA in Chapter 5.

Unlike DNA, RNA is usually single-stranded and has a much shorter chain of nucleotides. Single RNA strands often have self-complementary bases, which allow them to take a tertiary conformation by intra-molecular base-pairing and the formation of hair-pin structures. The tertiary conformation is stabilized by hydrogen bonding through the hydroxyl group at the 2’ position of the ribose ring. The additional hydroxyl group also results in the A–form of the RNA double helix with a narrow, deep major groove and a wide, shallow minor groove. RNA molecules can also be packed into larger structures and/or form complexes with proteins. The ribosome is an example of the latter category.

Proteins have very rich tertiary structures, on which their biological


functions are based. They can be grouped according to their tertiary structures into two broad categories: the fibrous and globular proteins. The fibrous proteins are elongated and are usually of regular secondary structure. They are often structural elements in the cell and organism. The secondary structures of fibrous proteins can be, among others, α−helix (α−keratin) and β− sheet (silk fibroin). An interesting example is elastin, which has elastic properties because it contains cross-linked random coils. The random coils allow for the elastic deformation of the fibre without breaking the polypeptide bonds, like the cross-linked polymers in a natural or synthetic rubber. Globular proteins are compact and more or less spherical. The latter proteins often contain defined domains in which one can recognize structural elements such as bundles of α−helixes and assemblies of β− strands in the form of twisted sheets and barrels.

The folding of the protein from a random coil state with an astronomical number of molecular configurations into its native state with a small number of possible configurations is accompanied by a tremendous loss in configurational entropy. In order to render the native state thermodynamically stable, this loss in entropy should be compensated by stabilizing interactions within the polypeptide sequence and/or an increase in another form of entropy. As we will see shortly, both effects are involved in the folding process. In the folded state, the most important stabilizing interactions are: • Charge interactions. Many amino acids contain side groups which are

either positively or negatively charged under physiological conditions close to the iso-electric point. The electrostatic attractive forces stabilize the native state. Far from the iso-electric point, the protein acquires a net positive or negative charge (depending on acidic or basic conditions) and the mutual repulsion among these charges will contribute to the instability of the folded structure and might eventually result in denaturation.

• Hydrogen bonding. Many side groups contain functional groups which can be involved in the formation of hydrogen bonds with other side groups and if available with the carbonyl oxygen and amide hydrogen on the polypeptide backbone. Although a single hydrogen bond is relatively weak, the sheer number of them can add a significant contribution to the stabilization of the folded state.


• Van der Waals interaction. The interior of globular proteins is closely packed with many uncharged side groups. The weak attraction resulting from dipole and induced dipole interactions between these side groups adds up and results in a significant stabilizing force.

• Disulfide bonding. If the protein is meant to function in an external, oxidizing environment, as opposed to the reducing environment inside most cells, significant stabilization of the folded structure can come from the formation of disulfide bonds between cysteine residues.

• Hydrophobic interaction. Despite the fact that the aforementioned interactions stabilize the native state to a significant extent, the main contribution to the stability of the protein comes from the hydrophobic effect. If the hydrophobic side groups are buried in the interior of the globular protein, water molecules that were first restricted in their translational and rotational motions due to the interaction with the protein are released. This release of hydration water molecules results in an increase of the entropy of the whole system including protein and solvent, which partially offsets the tremendous loss in configurational entropy associated with the folding process.

Relatively small proteins fold spontaneously into their 3–dimensional, native tertiary structures. For longer polypeptide sequences, the folding process may be assisted with helper proteins called chaperones, thereby avoiding misfolded states and possibly amorphous aggregation. We will further discuss the scientifically challenging folding problem in Sec. 5.4. Finally, one should bear in mind that many, if not all proteins are multi-unit assemblies and that they form higher order complexes with other biopolymers, such as DNA and RNA in the machinery of life.

1.5 Questions

1. What are the differences between DNA and RNA from a primary structural point of view?

2. Describe the difference in molecular structure of amylose and cellulose. 3. Give a reason why water is a good solvent for dextran and not a good


solvent for cellulose. 4. Why does dextran not have a regular secondary structure as is found for

amylose? 5. Why is the right-handed α−helix much more abundant than the left-

handed α−helix in polypeptides of biological origin? Under which condition would the left-handed helix be more abundant?

6. Describe the differences between the parallel and the anti-parallel

β− sheets of polypeptides. Why do they have a slightly different rise per residue?

7. Give a reason why a single-stranded DNA molecule does not take such an

intricate tertiary structure as can be found in transfer RNA. 8. What happens to the 3–dimensional tertiary structure of a closed circular

and supercoiled DNA molecule when one of the strands of the duplex is cleaved by an enzyme or accidentally cut (nicked)?

9. Why is purine or pyrimidine base-pairing not suitable for the formation

of a double helix of two opposing strands of nucleic acid? 10. A protein made of 101 residues in its random coil state can exist in 3 to

the power 100 conformations, if each link between residues has three equally probable configurations (see Sec. 5.4). a. Estimate the change in configurational entropy if the protein folds

into a native structure with only one conformation. b. Suppose that the protein folds into a single α−helix. Calculate the

stabilization energy pertaining to the formation of the intra-molecular hydrogen bonds between carbonyl oxygen and amide hydrogen. Assume that each hydrogen bond contributes 5 kJ/mol to the stabilization energy.

c. Is the α−helix stable at 298 K?


CHAPTER 2

POLYMER CONFORMATION

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.

2.1 The ideal chain

Every polymer is a sequence of units called monomers. We will first discuss the simplest model: the ideal chain (see Fig. 2.1). Let there be a total of 1N + monomers per chain and each monomer has a centre of mass position vector iR . In the ideal chain model, the step vector of a link between subsequent

identical monomers 1i i il R R −= − describes a random walk with step length il l= through space (there are N links). In this ideal chain, the orientation

of a specific link is uncorrelated with the orientation of the other links. Furthermore, there are no interactions between segments which are not directly linked with each other (no long-range volume interactions). Since every link has a random orientation, irrespective of the orientation of the other links, this model is also referred to as the random flight chain.

Chapter 2: Polymer Conformation 24

The total length of the chain with N links measured along the contour is given by the sum of the step lengths

iL N l Nl= = (2.1)

and the end point vector h between the first and last monomer (with index 0 and N , respectively) reads

0 1

NN ii

h R R l=

= − =∑ (2.2)

To gauge the physical extent of the chain, it is useful to calculate the mean square end-to-end point distance

2h h h= i (2.3)

where the brackets denote an average over all possible chain configurations and the dot represents the in-product of the two end-to-end point vectors. With Eq. (2.2) and the identities

2i il l l=i (2.4)

cos2i j ijl l l θ=i (2.5)

the mean square end-to-end point distance can be expressed as

cos

21 1 1

1 12 21 2 1 2

2 2

N N Ni j i ii j i

N N N Ni j iji j i j

i j i j

h h h l l l l

l l Nl l θ

= = =

− −

= = = =< <

= = = +

= +

∑ ∑ ∑

∑ ∑ ∑ ∑

i i i

i (2.6)

where the factor of two comes from the restriction in the double summation to indexes i j< . For a random flight chain, the steps are uncorrelated, so that

0i j i jl l l l= =i i and cos 0ijθ = . Accordingly, the mean square end-

h

R

NR

0

l R R 1i i i

Figure 2.1 An ideal random flight chain.


to-end point distance takes the simple form

2 2h Nl= (2.7)

The typical radius of the random flight chain is on the same order of magnitude as the square root of the mean square end-to-end distance

1 22 1 2R h N l (2.8)

The coil radius is proportional to the square root of the number of links. Note that the contour length of the chain is proportional to the number of links, so that for a long chain the characteristic size of the coil is much shorter than the chain length measured along the contour.

The second moment of the mass distribution (radius of gyration gR ) can be derived in a similar way and takes the form

12 2 22 1 2

1 16

N Ng iji j

i j

R h NlN

−

= =<

= =∑ ∑ (2.9)

with 2ijh being the mean square distance between segments i and j .

Due to the random nature of the ideal chain, the end-to-end distance h h= obeys Gaussian statistics with probability

( ) exp3 2 2

2 2

3 32 2

Nh

P hNl Nlπ

⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟= −⎜⎟⎜ ⎟⎟⎜ ⎜ ⎟⎜⎝ ⎠ ⎝ ⎠ (2.10)

According to Boltzmann’s law, the entropy of the chain is proportional to the logarithm of the probability and takes the form

ln2

0 2

32

Nh

S k P S kNl

= = − (2.11)

In the ideal chain, the monomeric units have no interactions and hence zero internal energy ( 0U = ). Accordingly, the configurational free energy of a chain with end-to-end point distance h only comes from the entropy and reads

( )2

0 2

32h

F h U TS F kTNl

= − = + (2.12)

The latter expression will be used many times, for instance for the calculation of the response of the chain subjected to long-range excluded volume interactions among the segments.


2.2 The Kuhn chain

In the ideal chain model, orientation correlation between segments is neglected and the chain takes a random flight through 3D space. We will now consider orientation correlation between segments which are not too far separated from each other along the contour; the Kuhn chain. Irrespective of orientation correlation, the mean square end-to-end distance of a chain reads [Eq. (2.6)]

cos12 21 2

2N N

iji ji j

h l N θ−

= =<

⎡ ⎤⎢ ⎥= +⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∑ (2.13)

We will now investigate how short-range orientation correlation between the segments, as expressed by the double summation over the angular correlation in Eq. (2.13), modifies the characteristic size of a polymer chain.

The Kuhn chain is based on two assumptions: (a) Short-range orientation correlation along the contour. The segments are assumed to lose their orientation correlation over a short distance along the contour, separated by, say, s monomers, i.e. cos 0ijθ = for j i s− > . Accordingly, the only terms contributing to the double summation in Eq. (2.13) come from pairs with i j i s< < + :

cos cos1 1

1 2 1 2

N N N Nij iji j i j

i j i j i s

θ θ− −

= = = =< < < +

∑ ∑ ∑ ∑ (2.14)

(b) The chain is homogeneous and end effects are neglected. Under the latter assumptions, the summation over the index j in the range i j i s< < + will give an identical result for every values of i . Since there are a total of N values of i and we neglect end point effects, we can set i to the first link ( 1i = ) and replace the first summation by a multiplication with the number of links N

cos cos111 2 2

N N sij ji j j

i j i s

Nθ θ−

= = =< < +

∑ ∑ ∑ (2.15)

For a Kuhn chain including short-range orientation correlation, with Eq. (2.15) the mean square end-to-end distance Eq. (2.13) can be written as

2 2h Nl σ= (2.16)

with


cos 121 2

sjj

σ θ=

= + ∑ (2.17)

Accordingly, including short-range orientation correlation modifies the mean square end-to-end distance by a multiplicative factor σ . An important observation is that the mean square end-to-end distance remains proportional to the number of links N , irrespective of the inclusion of short-range orientation correlation.

The effect of the short-range orientation correlation between the segments can be eliminated by renormalization of the step length l . For this purpose, we replace the sequence of N segments of length l by a sequence of kN N σ= Kuhn segments with Kuhn length kl lσ= , so that the total

contour length of the chain is preserved k kL Nl N l= = (2.18)

With the renormalized step length, the mean square end-to-end distance Eq. (2.16) reads

2 2 2k kh Nl N lσ= = (2.19)

With the renormalized step length, the expression for the end-to-end distance Eq. (2.19) is the same as the one for the ideal chain Eq. (2.7). Accordingly, the step length has been renormalized in such a way that there is no short-range orientation correlation between the Kuhn segments.

2.3 The worm-like chain

In the ideal and Kuhn chain models, the most important property of a polymer chain, that is its flexibility, is concentrated at the connection points of the segments. However, very often a polymer, such as DNA, can be described by a thin, elastic filament obeying Hooke’s elasticity law under small deformations. Such model of a polymer chain is called the persistent or worm-like chain model (Fig. 2.2). In the worm-like chain model, we take the limit to a continuous chain by letting the step length l go to zero and by letting the number of segments N go to infinity, in such a way that the contour length L Nl= is constant.

Consider an elastic filament of length s with a constant curvature sθ (radius of curvature cR s θ= , see Fig. 2.3). According to Hooke’s law, the


elastic bending energy is an extensive property proportional to the length of the filament s and the square of the curvature

21

2 bU ssθ

κ⎛ ⎞⎟⎜Δ = ⎟⎜ ⎟⎜⎝ ⎠

(2.20)

with bκ being a characteristic bending rigidity constant. The mean square bending angle can be calculated by taking the thermal average

( )

( )

exp

exp

22 2 2

b

U kT d skT

U kT d

θ θθ

κθ

−Δ= =

−Δ

∫∫

(2.21)

where the factor of two accounts for the fact that the bending of the filament can occur in two directions independently (in and out of plane).

The directional correlation of the worm-like chain ( )cos sθ , with ( )sθ being the angle enclosed by the orientation vectors separated by a distance s along the contour, is exponential

( )cos expp

ssL

θ⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎟⎜⎝ ⎠

(2.22)

with the persistence length pL . The persistence length is the typical length scale over which the orientation correlation is lost. To relate the persistence length to the bending rigidity, we consider a short filament near the rod limit of length ps L . For such small length s , we can expand the cosine term in the directional correlation Eq. (2.22) up to and including the second order in the bending angle ( )sθ

Figure 2.2 A worm-like chain.


( ) ( )cos 211

2s sθ θ= − + (2.23)

Likewise, we can expand the exponential up to and including the first order in the length s

exp 1p p

s sL L

⎛ ⎞⎟⎜− = − +⎟⎜ ⎟⎟⎜⎝ ⎠ (2.24)

and

( )2 2p

ss

Lθ = (2.25)

We have already calculated the mean square bending angle with the help of Hooke’s law and from a comparison of Eqs. (2.21) and (2.25) it follows that

bpL kT

κ= (2.26)

Accordingly, the persistence length is the bending rigidity in units kT . Furthermore, it can easily be shown that a short filament with a contour length ps L has a thermally averaged bending energy U kTΔ = in accordance with the equipartition theorem. In the case of DNA, the persistence length is around 50 nm, which corresponds to 150 base-pairs.

For the worm-like chain in the continuum approximation, the end-to-end vector h can be written as

( )0

Lh l s ds= ∫ (2.27)

s

( )sθ

cR s θ=

( )sθ

Figure 2.3 An elastic filament of length s and curvature sθ .


where ( )l s r s= ∂ ∂ is the tangent unit vector of chain direction at a distance s from the start of the chain measured along the contour (Fig. 2.4). In analogy with Eq. (2.3), the mean square end-to-end distance of the worm-like chain can be obtained from

( ) ( )

( )

' '

cos , '

2

0 0

0 02

L L

L L s

h ds ds l s l s

ds dt t t s sθ−

= =

= −

∫ ∫

∫ ∫

i (2.28)

With the directional correlation Eq. (2.22), the double integral is readily calculated, so that the mean square end-to-end point distance takes the form

exp2 22 1pp p

L Lh L

L L

⎡ ⎛ ⎞⎤⎟⎜⎢ ⎥= − + − ⎟⎜ ⎟⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦ (2.29)

Two limiting situations are of special interest:

,2 2ph L L L= (rod) (2.30)

,2 2 p ph LL L L= (coil) (2.31)

In the case of pL L the chain is a rigid rod, whereas for pL L the mean square end-to-end point distance of a Gaussian coil is recovered. For the Kuhn chain, we obtained 2 2

k k kh N l Ll= = [Eq. (2.19)]. From a comparison with the relevant expression of the worm-like chain Eq. (2.31), it follows that 2k pl L= (2.32)

Since by definition there is no orientation correlation between Kuhn segments, the orientation correlation is lost over a distance of twice the persistence length.

The radius of gyration of the worm-like chain can be calculated according to

s

s’ ( )l s

( ) ' l s

Figure 2.4 The tangent vector along the space curve.


( )' '2 22 0

1 L L

gs

R ds ds h s sL

= −∫ ∫ (2.33)

where ( )'2h s s− denotes the mean square distance between points s and 's on the space curve. Note that the coordinate 's runs from s to L to avoid

double counting. Expression Eq. (2.29) for the mean square end-to-end distance should also be valid for any sub-section of the chain, so that we can write

' '' exp2

22 0

21

L Lpg

s p p

L s s s sR ds ds

L LL

⎡ ⎛ ⎞⎤− − ⎟⎜⎢ ⎥= − + − ⎟⎜ ⎟⎟⎜⎢ ⎥⎝ ⎠⎣ ⎦∫ ∫ (2.34)

The double integral is readily done and we obtain the Benoit–Doty equation for the radius of gyration of a worm-like chain

exp2 3 2 2 32

3 6 6 13p

g p p pp

L LR L L L LL L

LL

⎡ ⎛ ⎛ ⎞⎞⎤⎟⎟⎜ ⎜⎢ ⎥= − + − − − ⎟⎟⎜ ⎜ ⎟⎟⎟⎟⎜ ⎜⎢ ⎥⎝ ⎝ ⎠⎠⎣ ⎦ (2.35)

which has the limiting forms

,2 2112g pR L L L= (rod) (2.36)

,2 13g p pR LL L L= (coil) (2.37)

Besides the numerical constants, the limiting forms of the radius of gyration are the same as the one for the end-to-end point distance.

The ideal, Kuhn and worm-like chain models have in common that they predict that the characteristic size of the coil scales with the square root of the contour length. Accordingly, orientation correlation between segments separated along the contour by a relatively small number of units has no qualitative influence on the properties of the chain at a larger distance scale. At a local distance scale, however, the short-range orientation correlation results in an increase in statistical step length and a decrease in local flexibility of the chain. These models ignore interactions between segments which are separated over large distances along the contour, but close spatially due to the coiling of the chain. As we will see in the next section, these volume interactions result in swelling of the chain with a change in the chain length dependence of the characteristic size of the coil.


2.4 Excluded volume interactions

We now consider the interaction between two segments which are close together spatially, but separated over a large distance as measured along the contour (see Fig. 2.5). For such an interaction pair, there can be a repulsive energy of 1 2 kT B , where B is an excluded volume parameter depending on thermodynamic properties (B has the dimension of a volume and is on the order of 3l ). Let c be the monomer concentration; its average value inside the coil domain is determined by the size of the coil R , i.e.

3

Nc

R (2.38)

The repulsive energy per unit volume is proportional to the density of the contact pairs, that is to the concentration squared, 2c . The free energy of repulsion per unit volume now takes the form

212repF kT B c (per unit volume) (2.39)

An essential assumption is to replace the average over the square of the monomer concentration inside the chain by the square of the average concentration

2

226

Nc c

R= (2.40)

This is typical of a mean field approach; fluctuations and correlations between the monomers are ignored. The total free energy of repulsion now follows from integration of the free energy per unit volume Eq. (2.39) over the chain volume 3R

( ),

2

3rep coilBN

F R kTR

(2.41)

where constants on the order of unity have been dropped. The repulsive interactions tend to swell the chain, because for larger R the free energy decreases.

The swelling of the chain by the repulsive interactions is counterbalanced by the elastic force due to the restriction in configurational degrees of freedom of the monomers with increasing radius. For the elastic free energy, we can take the ideal chain result [Eq. (2.12)]


( )2

2elasR

F R kTNl

(2.42)

and the equilibrium size of the chain can be derived from minimization of the total free energy with respect to the radius R

( ) ( ),2 3 2 2

0rep coil elas BN R R NlF FR R

∂ +∂ += =

∂ ∂ (2.43)

Omitting the numerical factors, we find the Flory radius of the swollen chain

( )1 53 3 5FR B l lN (2.44)

(note that 3B l is dimensionless and of order unity). More precise calculations give a Flory exponent 0.588 for the dependence of the radius on N , which is remarkably close to 3 5 .

In general, we can say that the characteristic size of a polymer chain scales with the number of links according to a power law

R lN ν (2.45)

where the exponent ν depends on solvent conditions (Table II.I). In a good solvent, the polymer chain is swollen and ν takes the Flory value 3 5ν = . In a poor solvent, the monomers collapse and they form a compact globule. For a closely packed three-dimensional globule with N links in a volume proportional to 3R , the exponent takes the value 1 3 . Finally, in a so called theta solvent attractive and repulsive forces between the monomers are balanced and the chain exhibits ideal behaviour with exponent 1 2 .

So far, we have discussed the properties of a single, isolated chain. We will now focus on the conformation of polymer chains in a confinement such as within a narrow or wide tube or if we expose the chain to traction. These

Figure 2.5 An interaction pair, spatially close but far away along the contour.


analyses will also serve to introduce scaling concepts, which will also be of importance if we want to understand the behaviour of more dense systems with overlapping chains in the semi-dilute regime.

2.5 Confinement in a tube; introduction to scaling

Scaling in polymer physics is based on the existence of a certain, unique length scale within which the chain is unperturbed by external factors. In order to demonstrate this concept, let us first consider a polymer chain confined in a tube with a diameter far exceeding the persistence length

pD L (see Fig. 2.6). Furthermore, we ignore interactions between the monomers and the wall of the tube; all derived properties of the confinement are related to the restriction in configurational degrees of freedom and screening of excluded volume interactions.

Within a length scale on the order of the diameter of the tube D , the monomers are unaffected by the confinement, that is they behave as if the tube is not there. We can accordingly define domains of diameter D , in which the chain is unperturbed and in which the monomers are distributed as in an isolated chain without confinement. These domains are commonly referred to as blobs. Let there be g links inside each blob (g N< ) of diameter D . By analogy of Eq. (2.45),

D lg ν (2.46)

and

( )1g D l ν (2.47)

The chain inside the tube can be considered a linear sequence of N g blobs; it is a renormalized chain with N g segments of unit length D . The extension of the chain in the longitudinal direction induced by the confinement is given by

Table II.I Exponent ν in R lN ν under various conditions.

Condition Exponent ν Good solvent, swollen chain ν = 3/5

Theta solvent, ideal chain ν = 1/2 Poor solvent, globule ν = 1/3


( )( )1R N g D Nl D l ν ν−= (2.48)

Inside the blobs, the chain can be swollen due to excluded volume interactions, so that with 3 5ν = the extension takes the form

( ) 2 3R Nl D l − (2.49)

The elongation inside the tube is proportional to the length of the polymer and increases with decreasing diameter according to 2 3D− . The monomer concentration inside the tube is then given by

( )4 332

Nc l l DD R

− (2.50)

and does not depend on the number of links N (or the length of the chain). We can also derive an expression for the free energy of confinement based

on scaling arguments. First of all, the free energy of confinement should be an extensive property, that is it should be proportional to the length of the polymer and hence the number of links confF N∼ (2.51)

(recall that ~R N ). Furthermore, the dimension of the free energy should be that of the thermal energy. We can write confF kT ϕ (2.52)

where ϕ should be dimensionless and proportional to N in order to satisfy Eq. (2.51). Now, we have to realize that there are two relevant length scales to describe our system; the Flory radius FR and the diameter D . The function ϕ should depend on FR and D , but in such a way that it is dimensionless.

D

Blobs

g links

Figure 2.6 A polymer in a tube. The polymer is coiled, because pD L .


An obvious choice is a power law

( )nFR Dϕ = (2.53)

in which the exponent n can be determined from the requirement that ϕ should be proportional to N . With Eq. (2.45) and 3 5ν = we obtain

( ) ( )3 5 nnconf FF kT R D kT lN D (2.54)

and 5 3n = . As a result, the scaling law of the free energy of confinement of a flexible polymer inside a tube reads

( )5 3confF kT N l D , pD L (2.55)

In the next section, we will consider the opposite case, in which the polymer is confined in a narrow channel with a width much smaller than the persistence length.

2.6 Deflection in a narrow tube

If the diameter of the tube is much smaller than the persistence length pD L , the polymer will undulate inside the channel and will only bend

when it bounces off the wall (Fig. 2.7). Following Odijk, we define a characteristic deflection length λ and deflection angle θ .5 The deflection length should be a function of the two relevant length scales of the system: D and pL . For small θ , the deflection length and angle are related through

Dθ

λ (2.56)

Furthermore, recall that for an elastic worm-like filament with a length s the mean square bending angle is given by

λ

θ D

Figure 2.7 A worm-like chain in a narrow tube.


( )2 2p

ss

Lθ = (2.57)

[Eq. (2.25)]. Hence, for a deflection length segment with s λ= , one obtains

2

2 2p

DLλ

θλ

⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠ (2.58)

from which it follows that the deflection length scales as

2 3 1 3pD Lλ (2.59)

The extension of the chain with contour length L in the narrow tube of diameter D is given by

cos 211

2R L Lθ θ

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠ (2.60)

and the relative decrease in length with respect to the fully stretched configuration takes the form

( )2 3212 p

L RD L

Lθ

−, pD L (2.61)

100

10−1

100

40030D (nm)

DN

A e

xten

sion

Figure 2.8 Extension versus channel diameter as measured with fluorescence microscopy of λ−phage DNA confined in nano-channels. The extension is normalized to the dye adjusted contour length of 18.6 micrometres. The solid line represents a power-law fit with slope –0.85 ± 0.05. The dashed line is the prediction of the extension for a worm-like chain in a narrow tube. Redrawn from Ref. [6].


With increasing diameter and decreasing persistence length, the chain becomes more wrinkled inside the tube.

The free energy of confinement of a chain in a narrow tube can also be derived using scaling arguments. Again, the free energy should have the dimension of the thermal energy kT . Furthermore, the free energy should be extensive in the contour length of the polymer. The other relevant length scale is the deflection length λ . The free energy of confinement is proportional to the number of deflection length segments, or in other words the number of wall induced bends, inside the tube

1 3 2 31confp

FL D

LkT λ− − , pD L (2.62)

Note that this free energy of confinement is of entropic origin; it results from fluctuations of the worm-like chain about the classical path induced by deflections from the wall. The scaling relations for the extension of fluorescent labelled DNA molecules in nano-channels have been verified by Reisner et al. and are shown in Fig. 2.8.6

2.7 Stars and radial brushes

We will now consider a different kind of confinement; we connect a number of chains to each other at one point and form a polymer star or, alternatively, we anchor them at a spherical core particle and form a spherical brush (see Fig. 2.9). The radial monomer density in a star or spherical brush can be neatly solved using scaling concepts in the Daoud-Cotton expanding blob model.7 A particular test chain of the star is confined in a conical volume by the presence of the other chains. This is in contrast with the previously treated situations, where the polymer was confined by a hard wall. In the Daoud-Cotton model, the polymer chains form radial strings of blobs of expanding size ( )rξ with increasing distance away from the centre of the star. An important notion is that the blobs are space-filling, so that ( )r rξ (2.63)

Furthermore, the blob size ( )rξ is related to the number of links inside the blob ( )g r through the Flory relation

( )r lg νξ (2.64)


and, inversely

( ) ( ) ( )1 1g r l r lν νξ (2.65)

Note that both g and ξ depend on the distance r away from the centre of the star. The monomer density is given by the number of links inside the blob divided by its volume

( ) ( )1 33 3 1r g g r l r ννρ ξ −− (2.66)

and

( ) 1r rρ −∼ , 1 2ν = (2.67)

( ) 4 3r rρ −∼ , 3 5ν = (2.68)

in theta and good solvents, respectively. Due to the fact that the blob size increases, the segment density decreases with increasing distance away from the centre of the star according to a power law with a negative exponent on the order of unity.

2.8 Chains under traction

We will now consider a polymer chain exposed to a tension force. With the advent of the optical tweezers technique, this has become quite a popular activity in order to investigate the mechanical properties of biopolymers such as DNA. We will start our discussion with an ideal chain under small tension.

r

Figure 2.9 Space-filling radial strings of blobs in a spherical brush.


Then we will move on to the worm-like chain and a swollen chain under both small and large tension forces.

2.8.1 An ideal chain under small tension

We consider an ideal chain exposed to a small stretching force f . Let the end-to-end distance of the chain under the force be fh . At equilibrium, the stretching force is balanced by the elastic force exerted by the chain. The elastic force is the derivative of the configurational free energy evaluated at the imposed end-to-end distance fh h=

f

elas

h h

Ff

h =

∂=

∂ (2.69)

With the ideal chain free energy being

( )2

2elash

F h kTNl

(2.70)

[Eq. (2.12)] we obtain for the extension

2 2

0f

Nl Rh f f

kT kT (2.71)

with 0R the unperturbed, ideal chain radius. The extension is linear in the force f , so that Hooke’s law is obeyed. Furthermore, the extension is proportional to the number of links, which shows that the force is transmitted through the links. It should also be noted that the elastic force is completely of entropic origin due to the restriction of the configurational degrees of freedom of monomers upon stretching of the chain.

2.8.2 Worm-like chain

For the worm-like chain model, no closed analytical solution for the force-extension relation is available. However, the normalized force pfL kT versus relative extension fh L has been solved numerically by Fixman and Kovac and is displayed in Fig. 2.10.8

The corresponding expression for the linear response of the ideal chain, Eq. (2.71), is also displayed. A comparison with the full numerical solution shows that the latter approximation is valid for fh L smaller than, say,


0.25. For the analysis of experimental data, it is useful to have an interpolation formula valid for any value of the extension. Marko and Siggia proposed9

( )2

1 144 1

f

pf

hkTfL Lh L

⎡ ⎤⎢ ⎥

= − +⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

(2.72)

As can be seen in Fig. 2.10, this equation reproduces the general behaviour well, but it deviates from the numerical result by as much as 10% for

.0 5fh L = . Interpolation formulae such as Eq. (2.72) have been used to extract the persistence length of DNA from experimental force versus extension data in optical tweezers setups.10,11 The results agree with a DNA persistence length around 50 nm, despite the fact that the worm-like chain model in its present form does not include volume interactions (see Fig. 2.11).

0 0.2 0.4 0.6 0.8 110

−1

100

101

102

<h>f /L

fLp/k

T

Figure 2.10 Normalized force versus extension as obtained from the numerical solution (solid line), the limiting law for small tension (dashed line) and Marko and Siggia’s interpolation formula (dashed-dotted line).9


2.8.3 Swollen chain

In order to derive the extension of a swollen chain we will use a scaling argument. The chain under tension f can be considered a chain of blobs; every blob has a size ξ and an elastic stretching energy f kTξ = (Fig. 2.12). The other relevant length scale is the Flory radius FR . The end-to-end distance under the force f is expected to be proportional to the size of the chain without the applied force Ffh R ϕ (2.73)

where ϕ is an unknown dimensionless function of the two relevant length scales FR and ξ . Again, we will use a power law

( )nFRϕ ξ= (2.74)

so that

Figure 2.11 Force versus extension of DNA measured with an optical tweezers set up. The solid line is a fit to the numerical solution of the worm-like chain model. The inset is a blow up of the small force region. Reprinted with permission from Ref. [12]. Copyright Biophysical Society (1999).


( ) 1n

n nF F Ff

fh R R R

kTξ + ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(2.75)

For a small force, the extension should obey Hooke’s law and be linear in f , which implies that 1n = . The expression for the extension of a swollen chain under a small tension is thus similar to the one pertaining to the ideal chain Eq. (2.71), but with the unperturbed radius 0R replaced by the Flory radius FR

2 6 5 2F

fR N l

h f fkT kT

, FfR kT (2.76)

An interesting result is that for the swollen chain the extension is not linear in the number of links, because the force can now also be transmitted by segment contact pairs through volume interactions.

For large tension, the extension should be proportional to the number of blobs and hence to the number of links N . With 3 5

FR N l , the extension Eq. (2.75) reads

( )1 3 51n

nnf

fh l N

kT++ ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(2.77)

so that n should take the value 2 3 for a linear dependence of the extension in N . For a swollen chain under large tension, we accordingly obtain

2 3

ffl

h NlkT

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠ , fl kT (2.78)

Note that due to the large tension the extension is now non-linear in the force.

Equation (2.78) can be derived in an alternative manner. One can argue that within the blobs the chain is unperturbed by the stretching force, because

f-f

Blob

ξ

Figure 2.12 A chain of blobs under a large tension.


the elastic stretching energy is less than kT . On one hand, inside the blob the chain is swollen and the blob size is related to the number of links g inside the blob according to

3 5lgξ (2.79)

On the other hand the blob size should be inversely proportional to the stretching force

kTf

ξ = (2.80)

so that

5 3kT

gfl

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠ (2.81)

For large tension, the chain can be considered as a linear sequence of N g blobs of size ξ and so the total length of the chain of blobs along the stretching force is given by

2 3

ffl

h N g NlkT

ξ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

, fl kT (2.82)

The projection of the string of blobs onto a plane normal to the force is an ideal string (the blobs do not exert an excluded volume effect, see below) and its lateral dimension is given by

( )1 6

1 2 1 2 kTR N g N l

flξ⊥

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠ (2.83)

Besides elongation in the longitudinal direction, the chain also shrinks in the lateral direction under the strong pulling force.

Now that we know the elongation of the swollen chain under a small and large tensional force, we can derive the corresponding elastic, configurational free energies by integration of the force over the extension. For a small tensional force, the relation between the extension and force is given by Eq. (2.76) and we obtain

( )2

elasF

hF h kT

R⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(swollen chain, FfR kT ) (2.84)

We will use the elastic free energy of the mildly perturbed chain to derive the Rouse–Zimm relaxation time in Chapter 4 on polymer dynamics. The


configurational free energy of the swollen chain is similar to the one pertaining to the ideal chain, but with the unperturbed radius 0R replaced by the Flory radius FR . In the case of a large tensional force, we obtain in a similar way to Eq. (2.78)

( )5 2

elasF

hF h kT

R⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(swollen chain, fl kT ) (2.85)

This form of the free energy can be used to derive the physical extent of an osmotic polyelectrolyte star or spherical brush, where the arms are significantly stretched due to the osmotic pressure of the small ions trapped in the brush.

2.9 From the dilute to the semi-dilute regime

If we dissolve a polymer in a suitable solvent and if the concentration is sufficiently low so that the average inter coil distance far exceeds the size of the coil given by FR , we obtain a dilute solution of coils with coil concentration c N (c is the monomer concentration). The equation of state (osmotic pressure versus monomer concentration) of such a dilute solution is proportional to the coil concentration and given by van’t Hoff’s equation

ckT NΠ

= (2.86)

With increasing concentration the coils are accommodated in a progressively smaller volume and at a certain critical concentration *c they will start to interpenetrate (see Fig. 2.13). We can find *c by realizing that at the overlap concentration the average inter-coil distance should match the characteristic coil size

*

1 3

FN

Rc⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(2.87)

so that

* 3 4 5c l N− − (2.88)

The overlap concentration is strongly dependent on the step length (persistence length) and polymer length. Above *c the chains interpenetrate,


individual coils are no longer discernible and the so-called semi-dilute regime is formed.

The thermodynamics and chain statistics in the semi-dilute regime can be analyzed with scaling concepts. The system is characterized by the extent to which it has penetrated the semi-dilute regime *c c . For the equation of state, we propose a scaling relation

ckT N

ϕΠ

= (2.89)

with ϕ an unknown, dimensionless function depending on *c c . For * 1c c , one obviously has 1ϕ = in order to recover van’t Hoff’s law Eq.

(2.86). In the semi-dilute regime ( * 1c c ), where the chains are interpenetrated, the osmotic pressure is expected to be independent on the chain length or the number of links per chain N . We again propose a power law

*

ncc

ϕ⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠

, *

1cc

(2.90)

so that

*

nc ckT N c

⎛ ⎞Π ⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠ (2.91)

With the expression for the overlap concentration Eq. (2.88), the osmotic pressure is given by

( )4 5 11 3 nn nc l NkT

−+Π (2.92)

c <c* c =c* c >c* Figure 2.13 From the dilute to the semi-dilute regime.


For 5 4n = the dependence of N is seen to vanish and the equation of state takes the form

( )9 43 3l clkT

−Π (2.93)

The scaling of the critical overlap concentration and osmotic pressure in the semi-dilute regime has been experimentally observed for polymers in a good solvent (Fig. 2.14).

The scaling of the equation of state of the semi-dilute polymer solution suggests the existence of a unique length scale within which the chain is unperturbed by the presence of the other chains (see Fig. 2.15). Within a certain correlation length or blob size ξ , the monomers are unaffected by the other chains, that is they behave as if the other chains are not there. We can find how the blob size depends on the monomer concentration and length of the links using a similar scaling approach as for the equation of state.

2.0

1.0

0.0-1.6 0.0 1.6

log(c/c*)

log(Π

M/c

RT

)

Figure 2.14 Reduced osmotic pressure of poly(α− methylstyrene) in toluene against the scaled concentration *c c . The molecular weight ranges from 20.4× 104 to 7.5× 106. The slope of the full line is the renormalization group theory result with slope 1.32. Redrawn from Ref. [13].


In the dilute regime, where the chains do not interpenetrate, the monomers in a particular coil are unaffected by the other coils within a characteristic distance scale FR . Accordingly, let us propose a scaling relation FRξ ϕ (2.94)

where 1ϕ = for * 1c c . In the semi-dilute regime, again we will use a power law ( )* mc cϕ = , so that

*

m

Fc

Rc

ξ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

(2.95)

With Eq. (2.88), the scaling law for the correlation length takes the form

( )4 3 53 1 mm mc l Nξ ++ (2.96)

Now, we have to realize that once the chains are interpenetrated the blob size should be independent on N . For 3 4m =− the dependence of N is seen to vanish and thus the correlation length decreases with increasing concentration according to

( ) 3 43l clξ− (2.97)

Inside the blobs, the chain is swollen and the blob size is related to the number of links g according to the Flory relation

3 5lgξ (2.98)

With the correlation length Eq. (2.97), the number of links per blob takes the

Blob ξ

Figure 2.15 The renormalized chain of blobs in the semi-dilute regime.


form

( ) ( ) 5 45 3 3g l clξ− (2.99)

Alternatively, one can derive the number of links per blob from the fact that the blobs are space-filling 3g c ξ .

Above the overlap concentration, we can describe the polymer solution as a system of closely packed blobs of decreasing size (~ 3 4c− ) and decreasing number of links per blob (~ 5 4c− ) with increasing monomer concentration (c ). The osmotic pressure is simply proportional to the blob concentration

3

1kT ξΠ (2.100)

and the free energy is extensive in the number of blobs and proportional to the osmotic pressure (pure substance)

( )9 433 3

F V Vcl

kT lξ= (2.101)

with the total volume V .

2.10 Chain statistics in the semi-dilute regime

In the semi-dilute regime, the size of the coils is expected to decrease below the Flory radius FR , due to the increasing interactions among the coils. However, since the coils interpenetrate, the extent to which the coils shrink is not clear beforehand. In order to derive the characteristic size of the chains in the semi-dilute regime, we will use a scaling argument together with the analogy of the concentrated polymer solution and a melt (that is a polymer system without solvent).3 In the semi-dilute regime, the chains are considered as sequences of N g blobs of size ξ . In other words, the chains are renormalized and the blobs are the basic units rather than the individual monomers. The blobs are closely packed, so that the solution can be viewed as a melt of renormalized chains; the characteristic distance between the renormalized chains, that is the mesh size, equals the correlation length ξ .

Let us now consider a grey test chain of blobs in a melt of identical black chains (Fig. 2.16). Since the blobs are closely packed and space-filling, the blob concentration 3

blobc c g ξ−= = is constant. A blob in the grey chain


experiences repulsive interactions from blobs in the same grey chain as well as from those in the other black chains. The total repulsive energy is thus given by the sum tot grey blackF F F= + (2.102)

and the force by the spatial derivative

greytot blackFF Fr r r

∂∂ ∂= +

∂ ∂ ∂ (2.103)

The repulsive energy is proportional to the blob concentration, so that the force is proportional to

greyblob blackcc cr r r

∂∂ ∂= +

∂ ∂ ∂ (2.104)

In the melt of renormalized chains, fluctuations in the blob density are vanishingly small and 0blobc r∂ ∂ = . Accordingly, the force exerted by the black blobs is cancelled out by the one originating from the grey blobs and the statistics of the chain of blobs remains ideal.

cblob

3ξ −

r

Figure 2.16 Melt formed by the chains of blobs in the semi-dilute regime. The total blob concentration is constant, so that the force exerted by the blobs in the black chains is cancelled out by the force originating from the blobs in the grey chain as indicated by the arrows. Redrawn from Ref. [3].


An alternative way to understand that excluded volume interactions are screened beyond the correlation length is to recognize the fact that segments pertaining to different blobs cannot form contact pairs.

Therefore, for an ideal chain of N g blobs of size ξ in the semi-dilute regime, the dimension is given by the ideal chain result

( )1 2R N g ξ (2.105)

With the correlation length Eq. (2.97) and the number of segments per blob Eq. (2.99), the radius of the chain in the semi-dilute regime takes the form

( ) 1 81 2 3R N l cl− (2.106)

The ideal behaviour is expressed by the square root dependence of the chain size on the number of links. Furthermore, the weak decrease in size with concentration to the power 1 8− indicates the interpenetration of the coils until eventually a mesh of chains of segments is formed.

2.11 Questions

1. Derive the expression for the radius of gyration of an ideal chain. 2. The size of a polymer chain R scales with the number of links N with

step length l according to a power law R N lν . Derive the value of the exponent ν for

a. A chain under theta conditions. b. A swollen chain in a good solvent. c. A compact, closely packed globule. 3. A polymer chain with persistence length pL is confined in a tube of

diameter D , so that pL D . a. Derive the elongation of the polymer along the tube. b. Derive the scaling relation of the free energy of confinement in

terms of the number of links N , D and pL .


4. A polymer chain with persistence length pL is confined in a narrow tube of diameter D , so that pL D

a. Derive the scaling law for the deflection length and the elongation

of the polymer along the tube. b. Derive the scaling relation of the free energy of confinement in

terms of the number of links N , D and pL . 5. A polymer star contains many polymer arms extending outwards from the

centre of the star. Calculate the radial density of the monomers away from the centre of the star.

6. Derive the extension of a polymer chain exposed to a stretching force for

the following cases: a. An ideal chain in the linear regime (small force). b. As in a), but for a swollen chain in a good solvent. c. A swollen chain under large tension. 7. Derive a scaling law for the elastic free energy of a swollen chain under

large tension in a good solvent. 8. The elastic properties of DNA can be measured in a single molecule

stretching experiment. a. Explain why the relative contribution of the long-range excluded

volume effect to the stretching decreases with increasing stretching force.

b. Alternatively, DNA can be stretched by confinement in a nano- channel with a diameter D . Derive the relationship between D and f and calculate which value of D corresponds to a stretching force .0 02f = pN.

9. With increasing monomer concentration, a polymer in a good solvent

exhibits a transition from the dilute to the semi-dilute regime. Derive scaling relations for:


a. The overlap concentration *c b. The equation of state (osmotic pressure versus monomer

concentration) of the semi-dilute regime. c. The correlation length (blob size) in the semi-dilute regime. d. The size of the chain in the semi-dilute regime.

e. Repeat a) through d) but this time for a polymer in a theta solvent. 10. Explain why the ‘chain of blobs’ in the semi-dilute regime behaves like an

ideal chain. 11. Consider neutral polymer stars dispersed in a good solvent. Each star has

aN arms and each arm comprises N links.

a. Derive the expression for the monomer density ( )rρ at a distance r away from the centre of the star. Take the number of arms per star aN explicitly into account.

b. Derive the expression for the size of the neutral star. How does the size depend on the number of arms?

c. Derive the expression for the overlap concentration and the osmotic pressure of a semi-dilute solution of neutral polymer stars. How does the pressure compare to that of a semi-dilute solution of linear polymers?


CHAPTER 3

POLYELECTROLYTES

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.

3.1 Counterion condensation

Many biopolymers carry a large number of charges. For instance, the spine axis projected distance between the negatively charged nucleotides in a double-stranded DNA molecule in the B–form is .0 171A= nm (i.e. half the base-pair spacing). This projected distance between the charges is significantly shorter than the Bjerrum length Bl . The Bjerrum length is the distance over which two interacting elementary charges have an electrostatic energy kT , i.e.

2 4Bl e kTπε= (see Appendix A). At 298 K in water, the Bjerrum length takes the value 0.71 nm, so that the electrostatic energy pertaining to a neighbouring pair of phosphate charges along the backbone of DNA amounts

Chapter 3: Polyelectrolytes 56

to .4 2Bl A= times the thermal energy kT . As a result of this high charge density, small counterions will gather (condense) upon the DNA chain and the effective distance between the charges is effectively increased to Bl . The counterions form a double layer around the polymer and the electrostatic energy is renormalized to kT .

In an early treatment of the double layer by Oosawa, the charged polymer (polyion) is described as a straight line-charge on a local distance scale.14 Furthermore, all ions come from the dissociation of the polyelectrolyte; there is no additional low molecular weight salt in the solution. With a distance between the charges A , the bare linear charge density is given by

0eA

ρ = (3.1)

The polyion is then placed into two coaxial cylinders; an inner region I of radius 1r and an outer region II of radius 2r (see Fig. 3.1). Let the concentrations in region I and II be 1c and 2c , respectively (total counterion concentration c ). If β is the fraction of counterions in region II (free counterions) and if ϕ is the volume fraction of region I, then we can write

1 1cc

βϕ−

= , 2

1cc

βϕ

=−

(3.2)

Now, we assume that the potential in the radial direction away from the polyion is a step function, with 1ψ and 2ψ the values of the potential in regions I and II, respectively. Of course, in reality the potential should decrease monotonously with increasing distance in the radial direction away

1y 1y

II

ψ

1r 2r

2ψ

1ψ

I

Figure 3.1 Oosawa’s two state condensation model.


from the polyion. A more realistic picture of the double layer will be discussed below. For now, we will assume that the counterions are distributed over the two regions in a step-like fashion according to Boltzmann’s law

exp1 2e

c ckTδψ⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠

(3.3)

with 1 2δψ ψ ψ= − . With Eq. (3.2), we can write a relation

exp11

ekT

β β δψϕ ϕ

⎛ ⎞− ⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠− (3.4)

between the fraction of free counterions in region II and the volume fraction of region I. By taking the logarithm, we obtain

ln ln11

ekT

β β δψϕ ϕ

⎛ ⎞ ⎛ ⎞− ⎟ ⎟⎜ ⎜= −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠− (3.5)

For points outside a charged cylinder of infinite length, the electrostatic potential is given by

( ) ln2

r rρ

ψπε

= (3.6)

with ρ being the charge per unit length. In our situation, there is a fraction 1 β− of the oppositely charged counterions condensed in region I, so that the net charge per unit length of the inner cylinder amounts to

0eA

ρ β βρ= = (3.7)

Furthermore, it is assumed that the counterions in region II do not contribute to the screening. With Eqs. (3.6) and (3.7), the difference in potential between region I and II takes the form

( )ln1 2 1 22e

r rA

βδψ ψ ψ

πε= − = (3.8)

The ratio of the cylinder radii can be expressed as the volume fraction of region I: ( )21 2r rϕ = , so that the potential difference reads

ln4eA

βδψ ϕ

πε= (3.9)

For a dilute polymer with 1ϕ , the relation between the fraction of free counterions and the volume fraction of region I [Eq. (3.5)] takes the form


( )( )ln ln11

β βϕ

βλ−

=−

( 1ϕ ) (3.10)

with the charge density parameter

BlA

λ = (3.11)

The fraction of free counterions β , which is equivalent to the renormalization of the charge density according to Eq. (3.7), can be found from the behaviour of Eq. (3.10) for a vanishingly small volume fraction

1ϕ (i.e. for lnϕ →−∞ ). Unfortunately, Eq. (3.10) has no analytical solution, but two regions of the structural charge density parameter λ can easily and clearly be discerned (see Fig. 3.2).

(a) Low charge density 1λ < . For low charge density with 1λ < , we have 1 0βλ− > and ( )( ) ( )ln 1 1β β βλ− − goes to minus infinity if and only if

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 50

0. 2

0. 4

0. 6

0. 8

1

λ

(e/l

)

ρ

β = 1/λβB

Figure 3.2 Fraction of free counterions (top) and effective linear charge density (bottom) versus the charge density parameter Bl A of a polyion in the limit of infinite dilution.


1β = (recall that 0 1β≤ ≤ and 0λ > ). In this region of λ , the bare distance between charges exceeds the Bjerrum length BA l> , the counterion fraction in region I is zero (all counterions are free) and the effective charge density of the polyion equals the bare charge density

0eA

ρ ρ= = ( )BA l> (3.12)

This is consistent with our preliminary conjecture that for a distance between the charges A exceeding the Bjerrum length Bl , the corresponding electrostatic energy is less than kT and there is no need for counterion condensation. It is convenient to define the effective number of charges per unit length as eff eν ρ= . In the absence of counterion condensation, one obtains the trivial result

1eff Aν = ( )BA l> (3.13)

(b) High charge density 1λ > . In the high charge region 1λ > , Eq. (3.10) shows a divergence to minus infinity for 1β λ= . Hence, the linear charge density is renormalized

0

B

el

ρρ

λ= = ( )BA l< (3.14)

due to the condensation of a fraction 1 1 λ− of the counterions on the chain. The electrostatic energy is accordingly reduced to kT . In the condensed layer, the effective distance between net charges equals the Bjerrum length Bl , so that

1eff

Blν = ( )BA l< (3.15)

Counterion condensation can be verified experimentally by the measurement of the osmotic pressure. If a solution with volume V contains PN polyions and if each polyion dissociates Z counterions, the ideal osmotic

pressure is given by van’t Hoff’s equation

P Pideal

N ZNkT

V+

Π = (3.16)

However, for all but very short polyelectrolytes 1Z and the osmotic pressure is dominated by the counterion contribution


Pideal

ZNkT kT cV

Π = (3.17)

with the monomer concentration Pc ZN V= (we assume that every monomer carries a charge). One can always define an osmotic coefficient Φ , being the ratio of the real and the ideal osmotic pressures, so that

idealΦ = Π Π . In the Oosawa condensation model, only free counterions are thought to contribute to the osmotic pressure and

( )( )

1

1B

effB

A lA

A lν

λ⎧ >⎪⎪Φ = ⎨⎪ <⎪⎩

∼ (3.18)

In the case of DNA at 298 K, one has .4 2λ = and 24% of the counterions are free. Accordingly, Oosawa’s model predicts .0 24Φ = , which is in good agreement with experimental results for DNA at fairly high concentration (see Fig. 3.3).15 At lower polyelectrolyte concentrations, the experimental values of the osmotic coefficient approach ( )1 2λ rather than 1 λ . Surely, the several assumptions in Oosawa’s condensation model are not fulfilled in a real polyelectrolyte solution.

The model for the double layer clearly needs improvement with respect to the neglect of electrostatic screening in the double layer and the corresponding unrealistic step-like radial dependence of the electrostatic

0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

c (M)

Π/Π

idea

l

Figure 3.3 The osmotic coefficient as a function of the counterion concentration of salt-free DNA solutions (0.1 M corresponds to 33 g of DNA/L). Redrawn from Ref. [15].


potential. Furthermore, the fraction of condensed counterions has been derived for a line charge of vanishing diameter, whereas a real biopolymer has a finite cross-section. Finally, a biopolymer solution under physiological conditions will certainly contain simple salts besides counterions. These features are included, at least to some extent, in Manning’s treatment of the problem in the Debye–Hückel approximation.

3.2 The electrostatic potential

A polyelectrolyte is often immersed in a salt solution with a salt concentration in excess of the counterion concentration. In Manning’s treatment, it is assumed that a certain fraction of the counterions is condensed on the chain, so that the electrostatic energy of a test charge at any point in the solution is less than kT .16 In this case, the charges interact through a screened Coulomb potential with a Debye screening length 1κ− determined by the salt concentration sρ according to 2 8 B slκ π ρ= (see Appendix A). Furthermore, as illustrated in Fig. 3.4, we assume that the polyelectrolyte is locally rod-like. At a distance r in the radial direction away from the spine axis of the rod-like polymer section, the infinitesimal potential element ( ),d r zψ originating from an infinitesimal charge element dzρ at position z is given by

( )( )exp ',

'1

4r

d r z dzrκ

ψ ρπε

−= (3.19)

with

( )'1 22 2r r z= + (3.20)

In order to derive the potential, we have to integrate z along the polymer axis. If the length L of the rod-like section is much larger than the screening length, so that 1Lκ , finite length effects can be neglected and the integration limits can be set to ±∞ . Accordingly, we obtain

( )( )( )

( )

exp1 22 2

1 22 24

r zr dz

r z

κρψ

πε∞

−∞

− +=

+∫ (3.21)

This expression can be simplified by using the substitution ( )1 22 21u z r= + , so that


( )( )

( )( )

exp01 21 22 21

rur du K r

u

κρ ρψ κ

πε πε∞ −

= =−

∫ (3.22)

with 0K being the zero order modified Bessel function. With eff eν ρ= , the reduced potential e kTφ ψ= takes the form ( ) ( )02 eff Br l K rφ ν κ= (3.23)

This potential can also be derived by solving the Poisson–Boltzmann equation in the Debye–Hückel approximation.

We can find the effective number of charges per unit length effν using the condensation concept. Following Oosawa, we assume that a fraction 1 β− of the counterions are condensed on the chain, so that the charge per unit length amounts to 0βρ . As we have seen in the previous section, for a line charge we obtain a charge density of 0ρ βρ= . However, a real biopolymer has a finite cross-section and it is reasonable to model the polyion as a cylinder of radius a . The electric field at the surface of a cylinder of radius a and charge per unit length 0βρ reads

( ) 0

2E r a

aβρπε

= =− (3.24)

The electric field is also given by the derivative of potential Eq. (3.22)

r

r’

z

Figure 3.4 A rod-like polyion segment with the radial coordinate r away from the spine axis.


evaluated at r a= , so that

( )( ) 12r a

dE r a K a

drψ ρκ

κπε=

= = =− (3.25)

with 1K being the first order modified Bessel function. Accordingly, in the Manning condensation model, the charge density reads

( )0

1a K aβρ

ρκ κ

= (3.26)

and the effective number of charges per unit length eff eν ρ= is given by

( )1

1eff a K a A

βν

κ κ= (3.27)

Again, we consider the two domains of the charge density parameter λ . For 1λ < , no counterion condensation occurs, 1β = and effν takes the form

( )1

1eff A a K aν

κ κ= ( BA l> ) (3.28)

In the condensation range with 1λ > , one has BA lβ = and

( )1

1eff

Bl a K aν

κ κ= ( BA l< ) (3.29)

Note that the effective number of charges per unit length depends on the Debye length (and thus salt concentration through 2 8 B slκ π ρ= ) and the polymer radius through aκ .

Now we know the potential, we can calculate the electrostatic energy associated with the interaction between the charged polymer and the small ions. This electrostatic free energy allows us to derive the osmotic coefficient of a solution of polyelectrolytes of vanishingly small diameter (line charge) in the limit of infinite dilution. We follow the same line of reasoning as in the calculation of the activity of a simple electrolyte solution in the framework of the Debye–Hückel theory. In this procedure, the charge density is varied from zero to its final value ρ by writing d dρ ρ λ= with 0 1λ≤ ≤ . The chain generates a potential ( )ψ λ depending on the value of the charging parameter λ . Now, we consider a chain element with length ds and self electrostatic free energy ( )ds dψ λ ρ . We derive the electrostatic free energy by integration along the contour and by charging the chain from zero charge to its nominal charge density ρ . This will be done with and without taking into account the


interaction of the polymer with the small ions. The difference then gives the screening contribution to the electrostatic free energy.

First, we consider the reference state, i.e. a charged rod of length L and charge density effeρ ν= without electrostatic interaction with its ion atmosphere. The potential is given by Eq. (3.6) and the electrostatic free energy follows from the double integration

lim ln1 2

0 0 02

Leleceff B

r

Fds d l r

kTλ ν λ

→=− ∫ ∫ (3.30)

We have taken the limit 0r → because we want to derive the osmotic coeffcient of a line charge of vanishingly small diameter. Note that the logarithm diverges for 0r → , but, as we will see shortly, this presents no difficulty. Next, we consider the electrostatic free energy of a charged chain, including the electrostatic interaction with the ion atmosphere. In an excess of salt, the potential is given by Eq. (3.23). Since we are interested in the limit

0rκ → , we can use the asymptotic form ( ) ( )ln0K r rκ κ−∼ . We thus obtain for the electrostatic free energy

( )lim ln1

2

0 0 02

Leleceff B

r

Fds d l r

kTλ ν λ κ

→=− ∫ ∫ (3.31)

The excess electrostatic free energy (per chain) follows from the difference

ln2elec elec elec eff BF F F kT l Lν κΔ = − =− (3.32)

where we have used ( )[ ]lim ln ln ln0r

r rκ κ→

− = . Note that the divergence at 0r → has cancelled out in the difference.

The electrostatic contribution to the osmotic pressure of a solution containing PN charged polymers can be obtained from the derivative of the excess electrostatic free energy with respect to the volume V :

2 2elec P eleceff B

N Fc l A

kT kT Vν

Π ∂Δ=− =−

∂ (3.33)

Here, we have used L ZA= , ( )2V Vκ κ∂ ∂ =− and Pc ZN V= . According to Oosawa’s condensation concept, the total osmotic pressure

is given by the sum of the contributions of the salt and the free counterions (a small contribution from the polymer is neglected). Without considering screening effects, the osmotic coefficient is given by Eqs. (3.18) and the ideal counterion contribution reads effcA kTν . The contribution from the salt is 2 sρ , because the cations and anions contribute equally to the osmotic


pressure (possible non-ideality of the salt is neglected). We also need to add the negative contribution Eq. (3.33) coming from the electrostatic interaction of the charged polymer with the ion atmosphere (screening effect), so that the total osmotic pressure takes the form ( )1 2 2eff eff B skT cA lν ν ρΠ = − + (3.34)

The ideal osmotic pressure, i.e. the osmotic pressure of a hypothetical solution without electrostatic screening effects and without counterion condensation, but including the salt, is given by 2ideal skT c ρΠ = + (3.35)

From the ratio of the real and ideal osmotic pressures follows the osmotic coefficient

( )1 2 2

2eff eff B s

ideal s

cA lc

ν ν ρρ

− +ΠΦ = =

Π + (3.36)

For a line charge we have 1eff Aν = ( BA l> ) or 1eff Blν = ( BA l< ), so that we can write

( )

( )( )

1 2 21 2

1 2 21 2

sB

s

sB

s

cA l

c

cA l

c

λ ρρ

λ ρρ

⎧ − +⎪⎪ >⎪ +⎪⎪Φ = ⎨⎪ +⎪ <⎪⎪ +⎪⎩

(3.37)

In order to gauge the effect of counterion screening on the osmotic coefficient, we take the salt-free limit 0sρ → . Note that by taking this limit we implicitly assume that the model is still valid if the screening only comes from the counterions. In the case of a vanishingly small salt concentration ( 0sρ → ), the osmotic coefficient takes the Manning value

( )

( ) ( )1 2

1 2B

B

A l

A l

λλ

⎧ − >⎪⎪Φ = ⎨⎪ <⎪⎩ (3.38)

These expressions can be compared to Oosawa’s results for a polyelectrolyte without simple salt Eqs. (3.18). For low polyelectrolyte charge ( BA l> ), the osmotic coefficient is now seen to decrease with increasing charge. In the case of a highly charged polyelectrolyte ( BA l< ), Manning’s value is a factor of two lower than Oosawa’s prediction. The difference is clearly due to the inclusion of the effect of screening in the ion atmosphere. However, it should be noted that the Manning value refers to a polyelectrolyte of vanishingly


small diameter and at infinite dilution. For a real polyelectrolyte solution, a better estimation of the osmotic coefficient can be obtained by solving the non-linear Poisson–Boltzmann equation in the cell model (see below).

Although the derived expressions for the potential are relatively simple, the use of the screened Coulomb potential at any point (Debye–Hückel approximation) in combination with the condensation concept is problematic. First of all, the condensation framework does not tell us how and where the condensed ions are distributed with respect to the chain. Furthermore, in close proximity to highly charged chains, the electric field is expected to be strong and the use of a screened Coulomb potential is not justified. At larger distances away from the chain, the functional dependence of the derived potentials should be more truthful to reality. In this distance range, the corresponding electrostatic energies are on the order of or less than kT , which supports the Debye–Hückel approximation. In the next section, we will solve the full, non-linear Poisson–Boltzmann equation in order to obtain a better picture of the potential and more accurate values of the effective charge density parameter.

3.3 Non-linear Poisson–Boltzmann theory

3.3.1 Polyelectrolytes in excess salt

A more realistic picture of the potential can be obtained by solving the non-linear Poisson–Boltzmann equation in cylindrical geometry.17 Again, we assume that the polyion is bathed in a salt solution with a concentration far exceeding the counterion concentration (the salt-free case will be discussed in the next section). If the counterion contribution is neglected, the non-linear Poisson–Boltzmann equation for the reduced potential e kTφ ψ= takes the form

sinh2 21 d dr

r dr drφ

φ κ φ⎛ ⎞⎟⎜∇ = =⎟⎜ ⎟⎜⎝ ⎠

(3.39)

At the surface of the polyion, the derivative of the reduced potential should satisfy the boundary condition


2

r a

ddr aφ λ

=

−= (3.40)

with charge density parameter Bl Aλ = . This boundary condition results from the fact that the derivative of the potential at the surface should match the electric field [Eq. (3.24), but for the reduced potential and with 1β = ]. Now, let us define a distance *r at which the electrostatic energy has reduced to kT and ( )* 1rφ = . For larger distances *r r≥ the electrostatic energy is equal or smaller than kT and the Poisson–Boltzmann equation may be linearized. In the previous section, we have already found the solution ( ) ( )02 eff Br l K rφ ν κ= , *r r≥ (3.41)

The effective number of charges per unit length effν now follows from ( )* 1rφ = , so that

0 1 2 3 4 5 60

1

2

3

4

5

6

7

(nm)

φ

r

Figure 3.5 Reduced potential around DNA in 10 mM of monovalent salt. The solid line is the numerical solution to the non-linear Poisson–Boltzmann equation in the inner double layer range. The dashed line is the solution to the linearized Poisson–Boltzmann equation. The reduced potential drops to unity at *r = 3.9 nm.


( )*0

12effBl K r

νκ

= (3.42)

The distance *r and, accordingly, effν can be found by solving the non-linear Poisson–Boltzmann equation for the reduced potential ( )rφ in the range *r r< with a numerical method (for instance a fourth order Runge-Kutta). In this backtracking procedure, we start at *r r= with ( )* 1rφ = and */ r rd drφ = = ( ) ( )* *1 0K r K rκ κ κ− and evaluate the potential back to r a= with the help of Eq. (3.39). The distance *r is then optimized, in an iterative manner, so that the boundary condition Eq. (3.40) at the surface of the polyion is satisfied. As a result of this procedure, as illustrated in Fig. 3.5, we seamlessly match the analytic tail Eq. (3.41) [with effν following from Eq.

0 50 100 1500

1

2

3

4

5

6

7

8

ν eff

(nm−

1 )

ρ (mM)s

Figure 3.6 Effective number of charges per unit length of DNA versus salt concentration. The solid line follows from the numerical solution to the non-linear Poisson–Boltzmann equation, whereas the dashed line represents Manning’s condensation result.


(3.42)] for large distances with the numerical solution to the non-linear Poisson–Boltzmann equation in the inner double layer region.

The effective number of charges per unit length increases with increasing salt concentration and has been set out in Fig. 3.6 (for DNA with outer diameter 2 nm at 298 K). The numerical result from Eq. (3.42) is always significantly larger than the condensation value given by Eq. (3.29). Recall that the limiting value for effν in the absence of salt, as given by Oosawa’s treatment, equals 1 Bl , i.e. 1.41 nm-1.

If the contribution of the counterions to the screening cannot be neglected, the situation becomes problematic and an analytic expression for the long-range potential is not available. It is however always possible to solve the non-linear Poisson–Boltzmann equation in the cell model by numerical means. As will be shown in the next section, the salt-free case can be solved analytically. Despite the fact that in biophysical and biotechnological applications polyelectrolytes are almost never truly salt-free, the analytic theory provides valuable insight into the counterion condensation phenomenon and results in useful expressions for the osmotic pressure even in the presence of salt.

3.3.2 Charge distribution in the cell model

A proven concept for the description of the double layer is the cell model. In this model, the polyion is placed in a coaxial electroneutral cell with a radius determined by the monomer concentration 2 1R A cπ −= . Inside the cell, the Poisson–Boltzmann equation can be solved either numerically (in the presence of salt) or analytically if all ions originate from the polyelectrolyte.18,19,20 The potential should satisfy a number of boundary conditions. At the surface of the polyion, the derivative of the potential should match the electric field, so that / 2r ad dr aφ λ= =− [Eq. (3.40)]. At the cell boundary, right between the polyions, the electric field should be zero

/ 0r Rd drφ = = owing to the electroneutrality requirement. Furthermore, we can arbitrarily set the potential to zero at the cell boundary: ( ) 0Rφ = . In the presence of simple salt, the boundary conditions have to be supplemented with the condition that the integrated radial ion densities should match the overall salt and counterion concentrations. We will first discuss the analytic


solution of the non-linear Poisson–Boltzmann equation in the cell model for the case of a salt-free polyelectrolyte solution. Subsequently, we will consider a polyelectrolyte at finite concentration in the presence of salt.

In the absence of salt, the non-linear Poisson–Boltzmann equation for the reduced potential e kTφ ψ= takes the form

( )exp21 d dr

r dr drφ

κ φ⎛ ⎞⎟⎜ =− −⎟⎜ ⎟⎜⎝ ⎠

(3.43)

The screening is caused by (monovalent) counterions only and we define the screening length 2 4 B Rlκ π ρ= with Rρ being the concentration of the counterions at the cell boundary. Note that this definition of the screening length differs from the one for a simple salt solution, because the charges on the polymer are thought not to contribute to the screening. Since at the cell boundary the electric field vanishes, the radial counterion density is given by the profile ( ) ( )expc Rrρ ρ φ= − . In the range a r R< < , Eq. (3.43) has an analytical solution, which is given by18,20

( )[ ]ln sinh ln2 2

222

rB Ar

B

κφ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎣ ⎦

(3.44)

where A and B are integration constants. There are no closed analytical expressions available for the integration constants, so that the potential still has to be solved by numerical means. From the boundary conditions follow the transcendental equations ( )[ ]coth ln1 0B B AR+ = (3.45)

and

( )[ ]coth ln

211

BB B R a

λ−

=+

(3.46)

Note that the profile depends on the polymer concentration through the cell radius R . For a certain concentration and a certain value of the charge density parameter λ , the parameter B can be solved with Eq. (3.46). At finite concentrations and with increasing charge density parameter λ , the value of B decreases from unity to zero at 1λ ≤ . A further increase in λ results in a transition of the value of B from a real to an imaginary number. The transition is not sharp, but occurs in a range of λ less than one. Eventually, B levels off at a constant, imaginary value. Both the range in the transition from


real to imaginary values and the limiting high charge value depend on the polymer concentration. After the parameter B has been determined, solving the integration constant A and calculating the radial counterion profile with Eqs. (3.45) and (3.44), respectively, is quite straightforward.

Fig. 3.7 shows the radial density profiles for mono-, di- and trivalent counterions as obtained with Monte Carlo computer simulation as well as the non-linear Poisson–Boltzmann equation.21 With increasing valence deviations are observed, which are related to ion correlation effects. These correlation effects are not captured by the mean field Poisson–Boltzmann approach. For monovalent ions, Poisson–Boltzmann theory generally gives good results. In the case of multivalent ions, more sophisticated methods such as Monte Carlo or full atomic scale Molecular Dynamics computer simulations are clearly necessary.

1 2 3 4 5 6 7 8

10−7

10−6

10−5

10−4

10−3

10−2

r (nm)

ρ(r)

Figure 3.7 Radial density profiles for mono-, di- and trivalent counterions around DNA from top to bottom. The solid lines result from the solution to the non-linear Poisson–Boltzmann equation, whereas the dashed lines were obtained with a Monte Carlo computer simulation.21


The cylindrical Fourier (Hankel) transforms of the radial counterion density profiles

( ) ( ) ( )00

2cellr

c ca q dr rJ qr rπ ρ= ∫ (3.47)

were also measured with small angle neutron scattering (SANS) and contrast matching techniques. The results for monovalent tetramethylammonium and the biologically relevant divalent putrescine and trivalent spermidine polyamines are displayed in Fig. 3.8. With increasing valence, the transform shifts to higher values of momentum transfer, which indicates a stronger spatial confinement of the ions around the DNA due to increased DNA-counterion interaction. Also displayed in Fig. 3.8 are the corresponding transforms of the Monte Carlo profiles. Satisfactory agreement between the experimental data and the model calculations is observed.

Besides the electrostatic potential, the non-linear Poisson–Boltzmann gives information about colligative properties such as the osmotic pressure and Donnan salt exlusion. At the cell boundary, the electric field vanishes so that there is no electrical pressure. The overall constant equilibrium pressure of the solution equals the mechanical pressure at the cell boundary with the

10q (nm )-1

0

1

a c

2 3

0.5

- 0.5

Figure 3.8 Cylindrical Fourier transforms of the radial counterion density profiles for monovalent tetramethylammonium (green dots), divalent putrescine (blue dots) and trivalent spermidine (red dots) as measured by SANS. The lines are the theoretical results based on the profiles obtained with Monte Carlo computer simulation.21


local counterion concentration Rρ . If there is no salt, the osmotic pressure is thus simply RkT ρΠ = (3.48)

The concentration of counterions at the cell boundary (commonly referred to as the osmotically free counterions) can be expressed in terms of the integration constant B and takes the form20

21

2RBcρ

λ−

= (3.49)

Accordingly, we obtain for the osmotic coefficient

21

2R Bcρ

λ−

Φ = = (3.50)

Note that the osmotic coefficient is concentration dependent through a concentration dependence of the integration constant B . In the limit of infinite dilution and/or for a polyion of vanishingly small radius (R a → ∞ ), one has 1B λ= − for BA l> ( 1λ < ) and 1B = for BA l< ( 1λ > ). We thus recover from the solution of the non-linear Poisson–Boltzmann equation in the cell model for a polyelectrolyte without salt the limiting Manning values of the osmotic coefficient Eq. (3.38). Furthermore, the Poisson–Boltzmann treatment yields critical behaviour of the osmotic coefficient at 1λ = , but in the limit of infinite dilution of the polymer only.

In the presence of simple salt, the Poisson–Boltzmann equation can no longer be solved analytically. However, it is quite straightforward to do this by numerical means with, e.g. a Runge-Kutta procedure. The parameters, which need to be optimized in order to satisfy the boundary conditions, are the concentrations of the cations and anions at the cell boundary, Rρ

+ and Rρ− , respectively. In addition to the usual boundary conditions, the integrated radial ion densities should match the overall concentrations of the counterions and the salt. A general phenomenon is that the anions coming from the salt are expelled from the polyion and accumulate in the region with low potential at the cell boundary (provided the polyion is negatively charged). As we will see shortly, the cation concentration at the cell boundary is close to the sum of the osmotically free counterions (as derived by solving the non-linear Poisson–Boltzmann equation without salt) and the simple salt concentration.


In analogy with the salt-free case and because of the electroneutrality of the cell, the osmotic pressure of the salted polyelectrolyte solution is proportional to the total small ion density at the cell boundary

RRkT ρ ρ+ −Π = + (3.51)

The ideal osmotic pressure is given by Eq. (3.35) (the contribution from the polymer is neglected), so that the osmotic coefficient takes the form

2RR

scρ ρ

ρ

+ −+Φ =

+ (3.52)

Numerical results for the osmotic pressure of solutions of double-stranded DNA are displayed in Fig. 3.9. The calculations were done for 0.001 and 0.1 mole of nucleotide/L and for monovalent salt concentrations ranging from 0.01 to 5 times the DNA concentration. With increasing salt concentration, the osmotic coefficient increases and approaches unity for 5s cρ ≥ . Furthermore, the osmotic coefficient increases with increasing DNA concentration, a phenomenon which is not captured by the Manning approach (the latter approach is valid in the limit of infinite dilution only).

10−2

10−1

1000

0.2

0.4

0.6

0.8

1

0.001 M

0.1 M

ρ /cs

Φ

Figure 3.9 Osmotic coefficients Φ of 0.001 and 0.1 mole of DNA nucleotides/L versus the ratio of the salt and DNA concentration s cρ . Solid curves: results from the numerical solution of the Poisson–Boltzmann equation in the cell model with DNA diameter 0D = 2 nm [Eq. (3.52)]. Dashed curves: approximation according to the additivity rule Eq. (3.53).


For a salted polyelectrolyte solution, the total osmotic pressure is often expressed as the sum of the osmotic pressure of the salt-free polyelectrolyte (with counterion concentration at the cell boundary Rρ ) and the one of the salt solution without polyelectrolyte. In other words, the osmotic coefficient is approximated by the additivity rule

22

R s

scρ ρ

ρ+

Φ =+

(3.53)

As can be seen in Fig. 3.9, this approximation works well for diluted DNA solutions with concentrations less than, say, 0.01 mole of nucleotides/L. At higher DNA concentration, significant deviations between the prediction of Eq. (3.53) and the numerical results are observed. Nevertheless, the additivity rule provides a good first order approximation, even if the polyelectrolyte concentration is fairly high.

At sufficiently low polyelectrolyte concentration, the anions from the salt are accumulated in the region of low potential at the cell boundary with a concentration R sρ ρ− . The cation concentration at the cell boundary is approximately R sRρ ρ ρ+ + ( 2R R sRρ ρ ρ ρ+ −+ + ). Since the system is in chemical equilibrium, the activity of the salt should be constant through the solution. In the presence of the charged polymer, the activity of the salt is thus given by

( )2s s R sa a a ρ ρ ρ+ −= + (3.54)

In practical situations, the polyelectrolyte solution is often separated from a salt reservoir by a semi-permeable membrane. The membrane allows the passage of water molecules and small ions, but is impermeable for the macromolecule. Later we will see examples in the context of polyelectrolyte brushes and the packaging of the genome inside the capsid of bacteriophages. If the salt concentration in the reservoir is 'sρ , the balance of the activities of the salt across the membrane requires

( )'2s s R sρ ρ ρ ρ= + (3.55)

or

( )'1 22 22 4s R s Rρ ρ ρ ρ= + − (3.56)

In the case of excess salt in the outer reservoir 's Rρ ρ , the latter equation


can be expanded up to and including the second order in 'R sρ ρ and we obtain the limiting form

''

12 4R R

s ss

ρ ρρ ρ

ρ⎛ ⎞⎟⎜ ⎟= − −⎜ ⎟⎜ ⎟⎜⎝ ⎠

(3.57)

The Donnan salt exclusion parameter is defined as the difference in salt concentration between the polyelectrolyte solution and the reservoir and normalized to the polymer charge concentration:

'

'1

2 4 2s s

scρ ρ

ρ⎛ ⎞− Φ Φ Φ⎟⎜ ⎟Γ = = −⎜ ⎟⎜ ⎟⎜⎝ ⎠

(3.58)

with Φ being the osmotic coefficient of the corresponding polyelectrolyte solution without salt. For a very dilute polyelectrolyte solution, we obtain

( )1 2 2λΓ − for BA l> and 4λΓ in the case of high charge BA l< .

3.4 The electrostatic persistence length

If the biopolymer carries charge, the chain statistics are modified by the short-range electrostatic repulsions between segments which are not too far separated along the contour as well as electrostatic excluded volume interactions between segments separated over a long distance along the contour but close together spatially. We will first discuss the effect of the electrostatic repulsion on the bending rigidity, which is the persistence length. In the next section, we will treat electrostatic excluded volume interactions.

In order to derive an expression for the electrostatic contribution to the persistence length, we first consider an uncharged elastic filament of length s and with a small, but constant curvature sθ (radius of curvature cR s θ= , see Fig. 3.10). According to Hooke’s law, the elastic bending energy is given by

2

00

12 pU kT L

sθ

Δ = (3.59)

with 0pL being the bare persistence length pertaining to a chain without

charge interactions (see previous chapter). For a charged filament, the electrostatic repulsion between the charges on the chain results in an increased bending energy 0 eU U UΔ =Δ +Δ (3.60)


For small curvature, the electrostatic bending energy eUΔ should also be proportional to the length of the filament s and the square of the curvature ( )2sθ . In analogy with Eq. (3.59), the electrostatic persistence length e

pL is defined by

21

2e

e pU kT Lsθ

Δ = (3.61)

The total bending energy with respect to the straight configuration is the sum of the bare and electrostatic contributions

( )2

012

ep pU kT L L

sθ

Δ = + (3.62)

so that the total persistence length is given by the sum of the bare and electrostatic persistence lengths

0 ep p pL L L= + (3.63)

In the case of DNA, the bare persistence length is around 50 nm. The electrostatic contribution should depend on the charge density of the polymer and the ionic strength of the supporting medium.

In order to find an expression for epL , we consider the electrostatic

bending energy of an elastic filament with charge density effeρ ν= and length s .22,23 We will treat the electrostatic interactions in the Debye–Hückel approximation, so that the charges on the chain interact through a screened Coulomb potential. Accordingly, the present approach is tailored for the situation in which the range of electrostatic interactions exceeds the typical

m

mnθ

cR s θ=

n

smnrmn

Figure 3.10 An elastic filament of length s and curvature sθ


dimension of the double layer, i.e. for intrinsically stiff biopolymers such as DNA. Let us consider two charge elements dnρ and dmρ at positions n and m respectively, as measured from the start of the chain and separated by a distance mns m n= − along the contour. If the charge elements interact through a screened Coulomb potential, the corresponding electrostatic energy with respect to the straight configuration is given by

( )[ ] [ ]exp exp,

2

4mn mn

emn mn

r sdU m n dndm

r sρ κ κπε

⎡ ⎤− −= −⎢ ⎥

⎢ ⎥⎣ ⎦ (3.64)

where mnr is the distance between positions n and m in the conformation with curvature sθ . With goniometry, the latter distance can be expressed in the curvature

sin sin2 22 2mn mn

mn cs s

r Rs

θ θθ

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= =⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠ (3.65)

In analogy with Eq. (3.61), the expression for the electrostatic energy Eq. (3.64) can be expanded up to and including the second order in the bending angle θ . After some algebra, we obtain

( ) [ ]( ), exp2 2

21

4 24e mn mn mndU m n dndm s s s

sρ θ

κ κπε

= − + (3.66)

The total electrostatic bending energy follows from integration of the positions n and m along the contour

[ ]( )exp ( ) ( ) ( )

2 2

2

0

4 24

1

e

s s

n

Us

dn dm m n m n m n

ρ θπε

κ κ

Δ = ×

− − + − −∫ ∫ (3.67)

where position m runs from n to s in order to avoid double counting. The double integration is readily done and the result is

( )2 2

4 24eU sh sρ θ

κπε

Δ = (3.68)

where the function ( )h x has the form

( ) ( )( )exp2 3 3 2 13 8 8 5h x x x x x x x− − − − −= − + − + + (3.69)

In practice, the length of the filament far exceeds the screening length 1κ− ( 1sκ ) and we can use the asymptotic form ( ) 23h x x−= . With effeρ ν= the total electrostatic bending energy then takes the form


2 2

2

12 4

eff Be

lU kT

sν θκ

Δ = (3.70)

and from a comparison with Eq. (3.61) it follows immediately what is the Odijk-Skolnick-Fixman expression for the electrostatic persistence length

2

24eff Be

pl

Lνκ

= (3.71)

Note that epL is proportional to the square of the effective number of charges

per unit length 2effν . It is also proportional to the square of the inverse

screening length 2κ− and hence to the inverse of the salt concentration 1sρ− .

100

102

40

50

60

70

80

90

100

ρ (mM)

L

(nm

)

101

s

p

Figure 3.11 Salt concentration dependence of the DNA persistence length. The solid curve is the theoretical result, whereas the symbols refer to experiments. Open symbols result from single molecule stretching experiments of λ − phage DNA,24 solid symbols were obtained from an analysis of light scattering data pertaining to T7–phage DNA.25


The salt concentration dependence of the DNA persistence length is depicted in Fig. 3.11. The solid curve is the theoretical result calculated for a bare persistence length of 50 nm and an effective number of charges per unit length resulting from the solution to the non-linear Poisson–Boltzmann equation with a bare diameter of 2 nm. Note that the persistence length increases significantly for an ionic strength below 10 mM. Also displayed are the results obtained from single molecule stretching experiments of λ−phage DNA by Baumann et al. and a light scattering experiment of T7–phage DNA by Sobel and Harpst.24,25 The light scattering data were corrected for excluded volume effects as described in the next section. The experimental results compare favourably with the theoretical prediction, although the prediction falls a bit below the data in the low ionic strength range. These deviations might be related to the use of the screened Coulomb potential, which underestimates the electrostatic repulsion at a short distance scale on the order of the double layer thickness.

3.5 Electrostatic excluded volume

We will now consider electrostatic interactions among segments which are close together spatially, but separated over a long distance along the contour. As we have seen in the previous chapter, long-range excluded volume interactions in a polymer chain result in swelling of the chain. Deviations from the unperturbed state with radius of gyration 2

0gR can be expressed by

2 2 20g S gR Rα= (3.72)

The swelling factor Sα is related to the excluded volume parameter

3 2 1 2

2

328 pp

Lz

LLβ

π

⎛ ⎞ ⎛ ⎞⎟⎜ ⎟⎜⎟= ⎜ ⎟⎜⎟ ⎟⎜ ⎟⎜⎟⎜ ⎝ ⎠⎝ ⎠ (3.73)

according to, for instance the Yamakawa–Tanaka approximation26

( ) .. . . 0 462 0 541 0 459 1 6 04S zα = + + (3.74)

The latter approximation has the correct limiting form for a small perturbation of the ideal chain behaviour expressed by relatively small values of the expansion parameter z . In the asymptotic limit for large z , that is for long chains, the modified Flory result


lim .5 1 276sz

zα→∞

= (3.75)

provides a fairly satisfactorily description of the expansion coefficient. For the calculation of the electrostatic excluded volume, we will follow a

procedure delineated by Fixman and Skolnick.27 The parameter z in the expression of the swelling parameter is proportional to the effective volume of exclusion that one Kuhn segment presents to another

[ ]exp1dr U kTβ = − −∫ (3.76)

where U is the interaction energy, r is the difference in the positions of the centres of mass and the brackets denote an average over the respective orientation of the segments. We assume that the interaction energy can be divided into two contributions 0 eU U U= + (3.77)

where 0U and eU are the hard-core and electrostatic interaction energies, respectively. The hard-core interaction energy is infinite when the segments overlap, otherwise it is zero. Accordingly, we can split the excluded volume

φ

2D

L Figure 3.12 A rod excludes a volume sin22DL φ to another rod (top and side view).


into a hard-core and an electrostatic contribution

0 eβ β β= + (3.78)

with

0overlapVdrβ = ∫ (3.79)

[ ]exp1non overlap

e eV

dr U kTβ−

= − −∫ (3.80)

Note that the hard-core contribution has to be calculated in the overlap region of the centre of mass coordinate r , whereas in the calculation of the electrostatic part r is confined to the region of space where the particles do not overlap.

We will now calculate the excluded volume of the two interacting Kuhn segments with the assumption that they can be modelled as rigid rods of length 2k pl L= and bare diameter 0D . We thus neglect the effects of chain flexibility. As can be seen in Fig. 3.12, the volume excluded by one Kuhn segment with respect to another segment skewed at an angle φ is given by

( ) sin20 02 kD lβ φ φ= (3.81)

Rr

s

φ

Figure 3.13 Two rods separated at a centre of mass distance R and skewed at an angle φ .


The angle φ has to be averaged over all orientations, so that the bare contribution takes the form

( ) ( )sin sin2 2 22 2

0 0 0 00 0

22k kd l D d l D

π π πβ φ φ β φ φ φ= = =∫ ∫ (3.82)

The electrostatic contribution to the excluded volume can be expressed in a similar way

( ) [ ]( )sin exp0

2 22

02 1e k e

Dl d dR U kT

πβ φ φ

∞= − −∫ ∫ (3.83)

where the factor of two comes from the negative range of the centre of mass separation R . In order to calculate the electrostatic contribution, we need to evaluate the electrostatic interaction energy eU of the two interacting rods separated at a centre of mass distance R and skewed at an angle φ (for the definition of the angles see Fig. 3.13).28

As in the case of our calculation of the persistence length, we will evaluate the electrostatic interaction energy in the Debye–Hückel approximation with a screened Coulomb potential. For screened electrostatics in excess salt, the reduced potential of a rod has the form ( ) ( )02 eff Br l K rφ ν κ= (3.84)

[Eq. (3.41), do not confuse the reduced potential ( )rφ and the skewed angle φ ]. Now, consider a charge element dsρ at position s on the other rod, separated at a distance r in the radial direction away from first rod, so that

( )sin 22 2r R s φ= + (3.85)

and the corresponding electrostatic energy (in units kT ) is given by

effds dseρφ ν φ= (3.86)

The total electrostatic interaction energy follows from the integration of s along the rod

( )2

20

22

k

k

leeff B

l

Ul ds K r

kTν κ

−= ∫ (3.87)

With coordinate transformations 'R Rκ= , ' sins sκ φ= and 'r rκ= , we can write

( )sin

sin' '

sin

2 2

02

2 k

k

leff Be

l

lUds K r

kT

φκ

φκ

νκ φ −

= ∫ (3.88)


We now consider segments with a length far exceeding the screening length, i.e. 1klκ . In this case, the effects of the finite length of the segment can be neglected and the integration limits can be extended to ±∞ . Furthermore, the argument of the Bessel function 'r is symmetric with respect to sign inversion of 's , so that the range of integration can be restricted to the positive domain

( )( )' ' 'sin

21 22 2

00

4 eff Be lUds K s R

kTνκ φ

∞= +∫ (3.89)

The integration is readily done using

( )( ) ( )exp1 22 2

00 2dxK x z z

π∞+ = −∫ (3.90)

and the result for the electrostatic energy takes the form

( )expsin

e w RUkT

κφ−

= (3.91)

with

2 12 eff Bw lπν κ−= (3.92)

Note that the electrostatic interaction drives the rods away from each other in a preferentially perpendicular configuration. Besides an estimation of the excluded volume effect in a polymer coil, Eq. (3.91) can also be used for the calculation of the second virial coefficient and critical boundaries pertaining to the formation of a liquid crystal (see Sec. 6.1).

The remaining integrations in the electrostatic excluded volume Eq. (3.83) cannot be done in closed analytical form. After some algebra, the expression can be rearranged

( )'2 12e kl R wβ κ−= ; ( )' exp 0w w Dκ= − (3.93)

where

( ) ( ) ( )( )sin

sin exp2 2 1

0 01

yR y d dx x x

π φφ φ −= − −∫ ∫ (3.94)

For large y , the latter function has the asymptotic form

( ) ( )ln / ln1 2 24

R y yπ

γ= + − + (3.95)

with Euler’s constant γ = 0.57721. The electrostatic contribution to the excluded volume is thus given by


( )ln ' / ln2 1 1 2 22e kl wπ

β κ γ−= + − + (3.96)

The total excluded volume is the sum of the electrostatic and hard core contributions. With Eqs. (3.78), (3.82) and (3.96), the total excluded volume reads

2

2 k effl Dπ

β = (3.97)

where we have defined an effective diameter of the polymer given by

( )ln ' / ln10 1 2 2effD D wκ γ−= + + − + (3.98)

Note that the effective diameter depends on the screening length 1κ− as well as the effective number of charges per unit length through the parameter 'w . The effective diameter of DNA decreases from around 50 nm in 1 mM monovalent salt to its bare value of 2 nm in very high salt concentrations exceeding 1 M (see Fig. 3.14).

The radii of gyration of T7–phage DNA (39936 base-pairs, contour length . 13 7L = micrometres) in monovalent salt have been measured by

10−3

10−2

10−1

100

0

10

20

30

40

50

60

ρs (M)

Def

f (nm

)

Figure 3.14 Effective diameter of DNA versus salt concentration calculatewith bare diameter 0D = 2 nm and the effective number of charges per unlength from the numerical solution to the non-linear Poisson–Boltzmanequation.


Sobel and Harpst25 by light scattering and are collected in Table III.I. As expected, the coil swells with decreasing ionic strength from about 500 nm at high salt concentration to 700 nm in 5 mM NaCl. The calculated effective diameters [Eq. (3.98)] and corresponding values of the swelling parameter [Eq. (3.74)] are also collected in Table III.I. Due to electrostatic effects on the persistence length and the excluded volume, the coil swells by almost 30% with respect to the unperturbed state. The persistence lengths were subsequently fitted with the help of the Benoit–Doty equation

exp2 3 2 2 320

3 6 6 13p

g p p pp

L LR L L L LL L

LL

⎡ ⎛ ⎛ ⎞⎞⎤⎟⎟⎜ ⎜⎢ ⎥= − + − − − ⎟⎟⎜ ⎜ ⎟⎟⎟⎟⎜ ⎜⎢ ⎥⎝ ⎝ ⎠⎠⎣ ⎦ (3.99)

and are also collected in Table III.I. As can be seen in Fig. 3.11, the values of the persistence length obtained from the scattering data agree with the ones obtained in a single molecule stretching experiment. This confirms our correction procedure for the electrostatic excluded volume effect.

The strong dependence of the effective diameter and the persistence length on the salt concentration has a significant effect on many biologically relevant properties of DNA. However, for monovalent salt concentrations exceeding, say, 10 mM, the polyelectrolyte effect becomes progressively less important and, in many aspects, neutral polymer behaviour is recovered.

Table III.I Ionic strength dependence of the radius of gyration25 1 22

gR , effective diameter effD , swelling factor Sα and persistence length pL of T7–phage DNA (contour length 13.66 micrometres) in excess simple salt.

sC (mM) 1 22

gR (nm) effD (nm) Sα pL (nm)

5 700 23 1.27 68 10 670 16 1.21 68 20 590 11 1.22 52

100 550 5.5 1.14 52 200 530 4.3 1.12 50 500 490 3.2 1.11 43

1,000 484 2.7 1.10 43


3.6 Flexible chains and electrostatic blobs

So far, we have considered polyelectrolytes with a rod-like conformation at a local distance scale. The locally rod-like conformation can be due to the strong electrostatic repulsion between the charges on the chain or due to the intrinsic rigidity related to the secondary structure (for instance the double-helical structure of double-stranded DNA). In the case of a weakly charged and flexible polyelectrolyte, the electrostatic forces might not be strong enough to stiffen the chain and the chain remains coiled at a local distance scale. At a larger distance scale, the chain will nevertheless be stretched out owing to the long-range Coulomb repulsion (see Fig. 3.15). As in the case of a polymer under a large tensional force, we can derive the physical extent of the intrinsically flexible, salt-free polyelectrolyte chain with a scaling argument.

We will assume that there is no or minimal screening of Coulomb interaction. Let a fraction qf of the monomers carry an elementary charge ( 1qf < ). The chain can be envisaged as a chain of electrostatic blobs; every blob has a size elecξ . If the electrostatic screening length is much larger than the blob size, the electrostatic repulsion energy of a pair of blobs is given by Coulomb’s law 2 2 2 4q elec elecf g e kTπεξ = (there are q elecf g charges per blob). On one hand, the blob size

2 2elec q elec Bf g lξ = (3.100)

is determined by the electrostatic energy. On the other hand, the blob size is related to the number of links elecg inside the blobs according to

elec eleclg νξ (3.101)

From the combination of Eqs. (3.100) and (3.101), we obtain

ξ e

Figure 3.15 Chain of electrostatic blobs.


( )1 1elecg B ν− , ( )1

elec l B ν νξ − (3.102)

where the constant B depends on the chain statistics inside the blobs

( )( ) ( ) ( ) ( )

( ) ( )

1 321 22

2 72

1 2

3 5

q Bq B

q B

f l lB f l l

f l l

ν ν ν

ν

− −⎧⎪ =⎪⎪= = ⎨⎪⎪ =⎪⎩

(3.103)

Due to the minimal screening conditions, the chain of electrostatic blobs will take an elongated, straight configuration with a renormalized contour length elec elecL N g N lBξ (3.104)

Note that, despite the overall extended configuration, within the blobs the chain is coiled. In the last section of this chapter, we will use the renormalized contour length for the derivation of a scaling law of the salt-free polyelectrolyte in the semi-dilute regime.

The effective number of charges per unit length of the chain of blobs (the linear charge density) is given by the number of charges per blob divided by the blob size

1 1 1q eleceff q elec B

elec

f gf g lν

ξ− − −= = (3.105)

With increasing charge fraction the blob size decreases, the chain stretches out and the effective number of charges per unit length increases until effν reaches Oosawa’s counterion condensation threshold 1

eff Blν −= . As can be seen in Eq. (3.105), in order to satsfy 1

eff Blν −= there must be one charge per electrostatic blob 1q elecf g = and the blob size must be equal to the Bjerrum length elec Blξ = . For a higher fraction of charged links, these critical values of the blob size and the effective number of charges per unit length are effectively preserved by the condensation of counterions inside the electrostatic blobs. Of course, when the blob size approaches the Bjerrum length, the chain becomes more and more locally stretched with a scaling exponent ν on the order of unity instead of the Gaussian value 1 2 . However, this has no consequence for the critical values of the blob size and effective charge density, because Eq. (3.105) holds irrespective of the chain statistics inside the blob.


3.7 Spherical polyelectrolyte brushes

We now consider the polyelectrolyte analogy of the radial polymer brush (Fig. 3.16). Polyelectrolyte brushes have found wide-spread technological applications from the stabilization of colloidal suspensions to the control of gelation, lubrication and flow behaviour. Besides these technological applications, polyelectrolyte brushes are thought to be a relevant model system for the external envelope of certain micro-organisms and to play a role in cell recognition and the lubrication and cushioning properties of the synovial fluid.29,30 As in the case of the neutral spherical brush, scaling concepts have proven to be extremely valuable for the understanding of the charge ordering.31,32 It is convenient to start the discussion with a polyelectrolyte brush without a simple salt (i.e. all small ions come from the dissociation of the polyelectrolyte). We will assume that the local chain statistics are Gaussian, but the calculations can also be done for a swollen chain. The presence of excluded volume interactions results in slightly different scaling laws, but no qualitatively different behaviour. We leave it to the reader to derive the equations for the spherical polyelctrolyte brush with 3 5ν = .

Figure 3.16 A schematic drawing of a spherical polyelectrolyte brush. The brush surrounds a core particle and the small ions are trapped inside the brush.


3.7.1 Spherical polyelectrolyte brush without salt

We consider a brush consisting of aN arms anchored at a spherical core particle; each arm contains N links. A fraction qf of the links is charged, so that the brush carries a total of a qN f N elementary charges. When the fraction qf of ionized groups is small, the electrostatic screening length is much larger

than the brush size; inside the brush there is no screening of the Coulomb interaction (unscreened brush). The a qN f N counterions coming from the dissociation of the polyelectrolyte are accordingly thought to be dispersed in the surrounding medium outside the brush. The radius R of the charged brush is determined by the balance of the energy of electrostatic repulsion given by Coulomb’s law

( )2B a qelec l N f NFkT R

(3.106)

and the elastic free energy of all aN arms of the brush

2

2elas

aF R

NkT Nl

(3.107)

For a swollen chain with 3 5ν = the corresponding elastic free energy is given by Eq. (2.85). From the equilibrium condition ( ) 0elas elecF F R∂ + ∂ = , we can immediately derive a scaling law for the characteristic size of the unscreened spherical brush

( )1 31 3 2 3a B qR N Nl l l f (3.108)

The arms of the brush are extended in a similar way as a linear polyelectrolyte chain [see Eq. (3.104) with 1 2ν = ], but in the brush there is an additional inter-arm electrostatic repulsion described by the factor 1 3

aN . With increasing charge fraction qf , the counterions are increasingly

confined inside the domain of the brush and the concomitant osmotic pressure becomes the main contribution to the stretching force. Eventually, almost all counterions are trapped inside the brush with a concentration proportional to 3

a qN f N R . The balance of the osmotic stretching force exerted by the trapped counterions

2

a qN f NRkT RΔΠ (3.109)

and the elastic force on the arms elasF R∂ ∂ results in a scaling relation of the


radial dimension of the osmotic brush

/1 2qR Nl f (3.110)

Now, the characteristic size scales with the square root of the charge fraction and does not depend on the number of arms [with 3 5ν = one obtains a similar relationship /2 5

qR Nl f ]. The transition between the unscreened and the osmotic brush can be found from the matching of the sizes of the unscreened [Eq. (3.108)] and osmotic [Eq. (3.110)] brushes and occurs at a critical charge fraction

( )* 2 2q B af l l N− − (3.111)

Therefore, the unscreened brush can only be observed for brushes with a very weak charge ( * 1qf ) and/or a relatively small number of arms. In practice, almost all polyelectrolyte brushes are in the osmotic regime. The experimental scaling of the size of the salt-free osmotic brush as a function of the charge fraction is displayed in Fig. 3.17.

As in the case of the neutral brush, the internal structure of the brush can be described by the blob model. The anchored chains form radial strings of

0 0.5 10

10

20

30

40

50

60

2/5

D (n

m)

fq Figure 3.17 Experimental brush diameter versus the charge fraction.32 The bottom and top dashed curves represent the core and fully expanded diameter, respectively. The solid line denotes the scaling result for an osmotic, salt-free polyelectrolyte brush in a good solvent ( ν = 3/5 ).


blobs of size ( )rξ with increasing distance away from the core. However, due to the electrostatic effects, the scaling of the blob-size and hence the radial monomer density will be different from the one in the neutral brush (see Fig. 3.18). Each blob contains ( )g r links, so that the radial monomer density is given by the number of monomers ( )aN g r in the shell of radius r and thickness ( )rξ

( ) 2aN grr

ρξ

(3.112)

Inside the shell, the charge density equals the counterion density ( )qf rρ , because the charge of the shell is completely screened and compensated by an equal amount of counterions (local electroneutrality condition).

The radial scaling of the blob size ( )rξ can be derived from the balance of the elastic stretching force and the osmotic pressure exerted by the trapped counterions. Consider the boundary between region I and II in the top panel of Fig. 3.18 (the top panel refers to the brush in the absence of salt). Across the boundary at position r, the osmotic force is proportional to the difference in counterion densities in the two adjacent shells

( )2

2q II I

rr f

kTρ ρ

ΔΠ− (3.113)

The corresponding elastic force is given by the difference in the elastic forces

2 2

a II a I

II I

N Ng l g lξ ξ

− (3.114)

pertaining to the blobs in region I and II, respectively. At equilibrium, the osmotic and elastic forces should balance, so one obtains

22a

qN

r fgl

ξρ (3.115)

With the general expression for the radial monomer density Eq. (3.112), the osmotic blob size takes the form

1 2qg l fξ (3.116)

Note that the expression of the osmotic blob size is the same as the one for the total size of the brush Eq. (3.110), but with N replaced by g . Inside the blob, the chain statistics are Gaussian with ( )2g lξ , so that

1 2ql fξ − (3.117)


and 1qg f − . Owing to the trapping of the counterions, the net charge per

blob is on the order of unity ( 1qg f ). Since the fraction of confined counterions does not vary along the radius, both the blob size ξ and the number of links per blob g are constant. The formation of radial strings of blobs of uniform size and hence uniform mass per unit length, results in a radial monomer density profile

( ) 1 1 2 2a qr N l f rρ − − − (3.118)

This expression can be derived with Eq. (3.112) together with 1qg f − and

1 2ql fξ − .

The scaling results of the salt-free, osmotic brush are markedly different from those of the neutral brush. The blobs of the osmotic brush are not space-filling, but their size is uniform and restricted by the condensation of the

(b)

I II

r

r

(a)

I II

Figure 3.18 Radial strings of blobs in an osmotic (a) salt-free and (b) a salted spherical brush grafted on a core particle.


counterions onto the arms. This results in a radial density scaling exponent of two instead of unity as derived for the neutral brush. Furthermore, as shown by Eq. (3.110), for high charges the arms become fully stretched and the osmotic brush takes a much more extended conformation than the neutral brush.

3.7.2 Salted spherical polyelectrolyte brush

The scaling laws for the osmotic brush were derived for a brush without salt. An additional screening of Coulomb interaction becomes important when the concentration sρ of salt exceeds the concentration of charged monomers qf ρ inside the brush. The stretching force is now proportional to the difference in osmotic pressure of coions and counterions in- and outside the brush (the counterions have an opposite charge and the coions have the same charge with respect to the polymer charge). Note that the counterions now come from the dissociation of the polyelectrolyte as well as from the salt; the coions originate exclusively from the salt. In the presence of salt, the osmotic pressure in the surrounding medium increases and the pressure difference with respect to the inside of the brush decreases. Accordingly, the osmotic brush will contract and the polyelectrolyte arms will become more and more coiled. However, as we will see below, we will not recover the intuitively expected neutral brush behaviour at very high salt concentration. As in the case of the salt-free osmotic brush, the salted brush can be treated by scaling arguments.

Let the brush be negatively charged, so that the counterions are cationic. Inside the brush, the counter- and coion densities are denoted by ρ+ and ρ− , respectively. The brush is electroneutral and the counterion density should equal the sum of the coion and the negatively charged monomer densities

qfρ ρ ρ+ −= + (3.119)

For now, we ignore any spatial dependence of the monomer density inside the brush. This should not be problematic as far as the derivation of a scaling law for the overall size of the brush is concerned (we ignore numerical factors on the order of unity anyway). Outside the brush in the bulk of the solution, the densities of the co- and counterions are equal and are determined by the salt concentration


b sbρ ρ ρ+ −= = (3.120)

At thermodynamic equilibrium, the chemical potentials of the electrolyte inside the brush and in the bulk of the solution are equal, which results in the Donnan rule for the distribution of the co- and counterions

2b sbρ ρ ρ ρ ρ+ − + −= = (3.121)

Eqs. (3.119) and (3.121) can be solved for the co- and counterion densities inside the brush

( )1 22 2 22 4q s qf fρ ρ ρ ρ± = + ± (3.122)

Owing to Donnan salt partitioning, the densities of the co- and counterions inside the brush are different and depend on the salt versus polymer charge concentration.

The osmotic stretching force is now proportional to the difference in co- and counterion densities in- and outside the brush and is given by

( )

( )( )

22

1 22 2 2 24 2

bb

q s s

RR

kT

R f

ρ ρ ρ ρ

ρ ρ ρ

+ − + −ΔΠ+ − −

+ − (3.123)

In the salt dominated regime, the salt concentration far exceeds the polymer charge concentration s qfρ ρ . Accordingly, the osmotic stretching force can be expanded up to and including the second order in q sf ρ ρ and takes the limiting form

2 2 22q

s

R fRkT

ρρ

ΔΠ ( s qfρ ρ ) (3.124)

(a factor of 4 has been dropped). As in our derivation of the size of the salt-free osmotic brush, for the polymer charge concentration inside the brush we take the average value 3

q a qf N f N Rρ . The osmotic stretching force is counterbalanced by the elastic force given by the derivative of the configurational free energy Eq. (3.107),

2

1 elasa

F RN

kT R Nl∂∂

(3.125)

so that

( )1 52 2 3 1 5a q sR N l f N ρ− (3.126)


The brush shows a moderate contraction with increasing salt concentration to the power 1 5− . The experimental scaling of the size of the osmotic brush as a function of the salt concentration is displayed in Fig. 3.19. Note the robustness of the outer brush diameter against the presence of salt.

In order to derive the radial scaling of the blob size and monomer density away from the centre of the core, we have to consider the differential osmotic force inside the salted brush. Consider the boundary between shells I and II in the bottom panel of Fig. 3.18. Across the boundary at position r , the osmotic force is proportional to the difference in co- and counterion densities in the two adjacent shells

( )2

2II III I

rr

kTρ ρ ρ ρ+ − + −ΔΠ

+ − − (3.127)

Electroneutrality applies to all shells, so that

I q II fρ ρ ρ+ −= + , II q IIII fρ ρ ρ+ −= + (3.128)

For thermodynamic equilibrium, the chemical potentials of the electrolyte in all shells and the bulk should be equal

2I II b sI II bρ ρ ρ ρ ρ ρ ρ+ − + − + −= = = (3.129)

By solving Eqs. (3.128) and (3.129), we obtain

( )1 22 2 22 4I q I s q If fρ ρ ρ ρ± = + ± (3.130)

( )1 22 2 22 4II q II s q IIf fρ ρ ρ ρ± = + ± (3.131)

and the osmotic force across the boundary takes the form

( ) ( )( )2 1 2 1 22 2 2 2 2 2 24 4q II s q I sr

r f fkT

ρ ρ ρ ρΔΠ

+ − + (3.132)

In the salt dominated regime, the differential osmotic force can be expanded up to and including the second order in q I sf ρ ρ and q II sf ρ ρ , so that

2 2 2 22

2 q II q I

s s

f frr

kTρ ρρ ρ

⎛ ⎞ΔΠ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜⎝ ⎠ (3.133)

The corresponding elastic force across the boundary is given by the difference in the elastic forces

2 2

a II a I

II I

N Ng l g lξ ξ

− (3.134)


pertaining to the blobs in region I and II, respectively. At equilibrium, the osmotic and elastic forces should balance and one obtains

2 2 2

2q a

s

r f Ngl

ρ ξρ

(3.135)

In the salt dominated regime, the scaling relation of the osmotic blob size can readily be calculated and takes the form

( ) ( )1 31 2 4 1 3 2 3a q sr N f l rξ ρ− − (3.136)

[with ρ given by Eq. (3.112) and with Gaussian statistics ( )2g lξ ]. An interesting result is that the blobs increase in size with increasing distance away from the core. The radial monomer density profile can accordingly be calculated and takes the form

( ) ( )1 32 2 2 1 3 4 3a q sr N f l rρ ρ− − − (3.137)

The radial density scaling exponent of the osmotic brush in the salt dominated regime of 4 3 is the same as the one for a neutral brush with excluded volume interactions in a good solvent [see Eq. (2.68)]. However, in contrast to neutral brushes, the blobs have a blob-size scaling exponent 2 3 [Eq. (3.136)] rather than unity and they are not closely packed.

For polyelectrolyte brushes at intermediate ionic strength, a two-region scaling model can be applied. At small distances away from the core, where the monomer charge density exceeds the salt concentration q sf ρ ρ> , the brush statistics are not affected by the salt. Here, the chains are extended in the radial direction with uniform mass per unit length with a radial scaling exponent of 2. With increasing distance away from the core, the local polymer charge concentration decreases and for sr r> the screening is governed by the salt. Hence, in the outer region, where s qfρ ρ , the brush density scaling exponent takes the value 4/3. The crossover distance

/ / /1 2 1 2 1 4 1 2s a q sr N l f ρ− − (3.138)

is determined by the equality of the local polymer charge concentration qf ρ [with radial density Eq. (3.118)] and the salt concentration sρ in the bulk. With increasing salt concentration, the brush contracts according to Eq. (3.126) with a concomitant decrease in the crossover distance sr between the inner- and outer-brush scaling regimes with density scaling exponents of 2 and


4 3 , respectively. The salt penetrates the brush and the radial decay of the monomer density scaling in the outer region is similar to the situation for neutral star-branched polymers. The blobs are not closely packed however and increase in size away from the crossover according to Eq. (3.136). The inner−brush region, characterized by radial strings of blobs of uniform size, remains unaffected until the salt concentration competes with the salinity generated by the counterions coming from the dissociation of the polyelectrolyte.

The various power laws describing the radial scaling of the monomer and counterion densities inside the polyelectrolyte brush have been verified with small angle neutron scattering.32 These experimental data can be described satisfactorily with the two-region scaling model (results not shown). The crossover distance between the inner- and outer-brush scaling regimes with density scaling exponents of 2 and 4/3 is also displayed in Fig. 3.19 and satisfies the inverse square root dependence of the salt concentration as

10−1 10 0

101

102

ρ s (mol/L)

D (n

m)

Figure 3.19 Experimental outer (circles) and crossover (squares) diameter versus the salt concentration of a charged polyelectrolyte brush.32 The top dashed curve represents the brush diameter without salt, whereas the bottom dashed curve denotes the core diameter. The solid lines denote the scaling results for an osmotic polyelectrolyte brush in the salt dominated regime with powers –1/5 and –1/2 for the outer and crossover diameter, respectively.


predicted by Eq. (3.138). Note that the brush reaches the salt dominated regime for very high salt concentrations exceeding 1 M only.

3.8 Polyelectrolytes in the semi-dilute regime

Like neutral polymers, with increasing concentration polyelectrolytes start to interpenetrate and the semi-dilute regime is formed. The critical overlap concentration and the chain statistics in the semi-dilute regime can also be described by scaling theory. However, for polyelectrolytes without salt, the scaling approach is rather tentative due to the intricate long-range, unscreened electrostatic interactions. For instance, it is not clear whether the polyelectrolyte chain takes an elongated or a coiled configuration in dilute solution. Many experiments indicate however, that the salt-free polyelectrolyte chain is rather extended, either as a fully stretched chain or as a fully extended chain of electrostatic blobs as expressed by Eq. (3.104). In the presence of excess salt, the polyelectrolyte behaves like a neutral polymer chain and the applicability of the scaling analysis is firmly established. We will first treat the unscreened, salt-free polyelectrolyte chain,33 after which we will discuss the situation in the presence of salt.34

3.8.1 Salt-free polyelectrolytes; a hierarchy of blobs

Consider a flexible, salt-free polyelectrolyte chain in the dilute regime. Owing to minimal screening conditions, the chain is expected to take a straight configuration with a (renormalized) contour length L N lB (3.139)

The constant B depends on the local chain statistics inside the electrostatic blobs and is given by Eq. (3.103). For a fully stretched chain B obviously equals unity. With increasing concentration, the chains will overlap at a critical concentration

* 3 2 3 3c NL N l B− − − − (3.140)

In the semi-dilute regime, we can define a correlation length ξ (not to be confused with the electrostatic blob size elecξ ). On a length scale smaller than ξ , a test chain is unperturbed by the presence of the other chains, whereas at a


longer length scale the electrostatic and excluded volume interactions are screened. We can describe the semi-dilute regime as a hierarchy of blobs; the semi-dilute blobs contain strings of electrostatic blobs as illustrated in Fig. 3.20. Within the electrostatic blobs, the chain is weakly perturbed by the electrostatic interactions and takes a coiled conformation.

The correlation length starts at Lξ at overlap ( *c c= ) and is assumed to follow a power law in the concentration

( ) ( )*1 21 2 1 2 3L c c lB clξ

−− − (3.141)

where the power 1 2− is chosen in order to render the correlation length independent on the number of links per chain N . Inside the blobs, the chain of electrostatic blobs takes a linear conformation, so that in analogy with Eq. (3.139) the blob size is related to the number of links per blob according to g l Bξ (3.142)

and thus

( ) 1 23 2 3g B cl−− (3.143)

Alternatively, g can be derived from the fact that the blobs are space-filling so that 3g c ξ . For length scales exceeding ξ , electrostatic and excluded

Electrostatic blobBlob

Figure 3.20 Hierarchy of blobs of a semi-dilute, salt-free polyelectrolyte system. The semi-dilute blobs contain strings of linearly extended electrostatic blobs. For the sake of clarity, the coiled chain inside the electrostatic blobs has not been drawn.


volume interactions are screened and the chain of blobs is Gaussian with a characteristic size

( ) ( ) 1 41 2 1 2 1 4 3R N g N lB clξ− (3.144)

Note that the size of the salt-free polyelectrolyte chain in the semi-dilute regime has a stronger concentration dependence than the one for a neutral polymer chain ( ~ 1 8R c− ).

At a higher concentration, the electrostatic blobs begin to interpenetrate, the electrostatic interactions will be progressively screened and the structure and dynamics become like the ones for a neutral polymer. We can find the crossover concentration ** 3 2c l B− to the concentrated regime from the condition elecξ ξ with Eqs. (3.102) and (3.141) under good solvent conditions.

3.8.2 Salted polyelectrolytes

In the dilute regime and in the presence of salt, the charges are screened and the polyelectrolyte chain takes a coiled conformation. The chains will however, be swollen owing to electrostatic excluded volume effects. For a long worm-like chain, the Flory radius reads

1 2 1 2 1 2 1 5F pR l N L z (3.145)

with the excluded volume parameter

1 2 1 2 3 2p effz l N L D− (3.146)

This expression of the Flory radius is based on the radius of gyration of a worm-like chain coil [Eq. (2.37)] and the asymptotic value of the swelling parameter given by Eq. (3.75). In the spirit of the scaling analysis, we have dropped factors on the order of unity. Note that the Flory radius of the salted polyelectrolyte chain has the same dependence of the number of links N as the neutral polymer chain. Accordingly, the scaling laws will be similar to those of neutral chains, albeit with different pre-factors related to the electrostatic effects on the chain’s flexibility and excluded volume.

With increasing concentration, the chains will overlap at a critical concentration


( )* 3 53 4 5 9 5F p effc NR N l L D −− − − (3.147)

In the semi-dilute regime the correlation length is assumed to follow a power law in the concentration

( ) ( ) ( )*3 41 43 4 3 2 3

F p effR c c l L D clξ−−− (3.148)

where the value of the power 3 4− was chosen to make the correlation length independent on N . Within the blobs, the chain exerts a full excluded volume effect and, in analogy with Eqs. (3.145) and (3.146), the blob size is given by

( )1 53 5 3 5p effl g L Dξ (3.149)

so that

( ) ( ) 5 43 43 2 3p effg l L D cl

−− (3.150)

The number of links per blob can also be found from the fact that the blobs are space-filling with 3g c ξ . For length scales exceeding the correlation length, excluded volume interactions are screened and the chain of blobs takes a Gaussian conformation with a characteristic size

( ) ( ) ( ) 1 81 81 2 1 2 3 4 3p effR N g N l L D clξ

− (3.151)

Note that in the semi-dilute regime, the concentration dependencies of the blob size and the size of the salted polyelectrolyte chain are the same as the ones for a neutral polymer chain. For the salted polyelectrolyte chain, the pre-factors include the persistence length and the effective diameter. The latter parameters and the static solution structure, depend on the salt concentration through the Debye screening length 1κ− . In the next chapter, we will use our picture of the static structure for the derivation of a dynamic scaling theory in order to describe polymer dynamics.

This concludes our discussion of the static properties of semi-dilute (charged) polymer solutions; the pertinent scaling laws are summarized in Appendix B.


3.9 Questions

1. Derive the equation of state and the fraction of condensed ions around a polyelectrolyte chain according to Oosawa’s counterion condensation theory. Explain why the counterions condense on the chain if the linear charge density exceeds a certain critical value.

2. Derive the electrostatic potential around a rod-like polyelectrolyte

segment by solving the linearized Poisson–Boltzmann equation. Explain why this potential becomes more accurate for distances further away from the chain.

3. Derive Eq. (3.49) with the help of Eqs. (3.44) and (3.45). Hint: use the

fact that ( ) 0Rφ = . 4. Derive the electrostatic contribution to the persistence length. How

would the electrostatic persistence length change for a biopolymer with twice the cross-sectional diameter, but with the same number of charges per unit length?

5. Calculate the electrostatic persistence length of DNA in 2 mM of NaCl

at 298 K using Manning’s expression for the effective number of charges per unit length.

6. Calculate the electrostatic interaction energy of two DNA fragments

separated at a distance 3 nm and skewed at an angle 2 degrees in 100 mM of a monovalent salt at 298 K.

7. Calculate the extension of a T7–phage DNA molecule within 300 and

600 nm diameter nanochannels in 100 mM of a monovalent salt at 298 K. Use Manning’s expression for the effective number of charges per unit length. How does the extension change if the ionic strength is reduced to 2 mM?


8. Derive the extension for a flexible polyelectrolyte chain in the electrostatic blob model with a Flory exponent 3 5ν = .

9. A spherical polyelectrolyte brush contains many linear polyelectrolyte

chains grafted at a spherical core particle. Derive the size and the radial scaling of the monomer density of the salt-free, osmotic brush in a good solvent with 3 5ν = .

10. As question 8, but for a salted brush. 11. Derive a scaling law for the osmotic pressure of a salt-free polyelectrolyte

solution in the semi-dilute regime under the (unrealistic) assumption that the fraction of osmotically active counterions does not change with increasing polyelectrolyte concentration.

12. Derive the overlap concentration and a scaling relation of the osmotic

pressure of (a) a salt-free and (b) a salted polyelectrolyte brush in the semi-dilute regime. How do the results compare to the corresponding ones pertaining to a neutral polymer solution?


CHAPTER 4

POLYMER DYNAMICS

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 salt-free 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.

4.1 Single chain dynamics

We will start our discussion of polymer dynamics with an analysis of the behaviour of a single polymer chain in the dilute regime. The polymer chains can be considered as independent, non-interpenetrating coils. First, we will derive the characteristic relaxation times pertaining to different draining conditions of the coil by the solvent. With these global relaxation times, we will subsequently calculate some transport properties. These properties are the self-diffusion coefficient of the chain and the solution viscosity. We will also

Chapter 4: Polymer Dynamics 106

derive a universal scaling law, which relates the viscosity to the elasticity modulus at high frequency. This scaling law will allow us to derive the viscosity of a polymer solution under chain overlap conditions, which will be treated in later sections of this chapter.

Let R be the characteristic size of a polymer chain. Owing to thermal motion, R will fluctuate in time with a characteristic relaxation time τ . We can find a general expression of this global relaxation time by considering the balance of the hydrodynamic friction and the elastic restoring forces experienced by the chain.3 In Chapter 2, we have seen that a small extension of a polymer chain is linear in the force, irrespective of the presence of volume interactions. On one hand, Hooke’s law is obeyed and the elastic force is proportional to the extension elasf K R= (4.1)

with K an elasticity constant depending on the chain statistics. On the other hand, the friction force should be proportional to the ‘velocity’ of the chain

fricR

ft

ζ∂

=∂

(4.2)

with ζ the hydrodynamic friction coefficient. At any time, the elastic and friction forces are balanced, so that

0RKR

tζ∂

+ =∂

(4.3)

The solution to this differential equation has the form

( )( )

( )exp exp0

R t Kt t

Rτ

ζ⎛ ⎞⎟⎜= − = −⎟⎜ ⎟⎜⎝ ⎠

(4.4)

in which we recognize the global relaxation time

Kζ

τ = (4.5)

The global relaxation time is the longest time scale over which the chain fluctuates owing to thermal motion. We do not consider internal relaxation modes, pertaining to local motions inside the chain. For a more detailed description of polymer dynamics, including the internal modes, the reader is referred to Ref. [35].

In Chapter 2, we have seen that the elasticity coefficient for a polymer coil depends on the chain statistics (solvent conditions) and takes the form


( )( )

2 201

2 6 5 2

theta solvent

good solventF

R kT Nl kTK

R kT N l kT−

⎧⎪⎪⎨⎪⎪⎩ (4.6)

with 0R and FR the unperturbed and Flory radius, respectively. In order to derive an expression for the relaxation time τ , we need to specify a functional dependence of the hydrodynamic friction coefficient ζ .

The hydrodynamic friction coefficient depends on the extent to which the solvent molecules can flow through the coil (draining conditions). A simple and relevant model for polymer dynamics is the Rouse model. In the Rouse model, the chain is assumed to be freely drained by the solvent and hydrodynamic interactions among the links are neglected. The hydrodynamic friction is accordingly additive and proportional to the number of links per chain lNζ ζ= (4.7)

where lζ is the friction coefficient per link. The latter coefficient is given by Stokes’ law 6l slζ πη= , if we model the link as a spherical bead with a radius l in a solvent medium with viscosity sη . Furthermore, in the Rouse model, the chain statistics are assumed to be Gaussian with an elasticity constant

1 2K Nl kT− , so that the relaxation time takes the form

2 3sN l kTτ η (Rouse) (4.8)

The freely drained Gaussian chain has a global relaxation time proportional to the square of the number of links (beads) per chain 2N and the volume of the bead 3l .

For neutral polymer coils in dilute solution, the assumption that the friction is additive and proportional to the number of links turns out to be too bold. The polymer coil is not freely drained and the links in the ‘inside’ of the coil are screened from the flow. In other words, the fluid within the domain of the coil is carried with the coil and only the links facing the ‘exterior’, bulk fluid experience friction. For the hydrodynamic friction experienced by the non-drained, complete coil of radius R in a solvent with viscosity sη , we can also use Stokes’ law 6 sRζ πη . Polymer dynamics under non-draining conditions are commonly referred to as Zimm dynamics. If the chain statistics are Gaussian with 0R R= , we obtain for the relaxation time

30 sR kTτ η (Zimm, theta solvent) (4.9)


In a good solvent, we have to replace the unperturbed radius 0R by the Flory radius FR and the relaxation time is given by

3F sR kTτ η (Zimm, good solvent) (4.10)

Under non-draining conditions, the relaxation time is proportional to the volume of the coil.

With the relaxation times, we can derive expressions for molecular transport properties such as the self-diffusion coefficient D and the solution viscosity η . If we follow the Brownian motion of a colloidal particle in a solvent medium, its mean square displacement 2r in a time t is given by one of Einstein’s equations

2 6r Dt= (4.11)

During the relaxation time τ , the coil diffuses typically over a distance on the order of the size of the coil R . The self-diffusion coefficient is accordingly on the order of the mean square size of the chain divided by its relaxation time

2R

Dτ

(4.12)

For the freely drained Gaussian coil, with the Rouse relaxation time we obtain an expression for the diffusion coefficient

s

kTD

Nlη (Rouse) (4.13)

which is inversely proportional to the contour length L Nl= . This prediction of the molecular weight dependence of the diffusion coefficient is not consistent with experimental results and the Rouse model is inappropriate to describe the dynamics of neutral polymers in the dilute regime. However, as we will see in the following sections, the Rouse model is important for the description of the dynamics of polymers in the semi-dilute regime where the chains overlap and hydrodynamic interactions are screened to a certain extent. In the case of non-draining conditions, the self-diffusion coefficient takes the form

( )( )

0 Zimm, theta solvent

Zimm, good solvents

s F

kT RD

kT R

ηη

⎧⎪⎪⎨⎪⎪⎩ (4.14)

depending on the solvent conditions. The self-diffusion coefficient of the non-drained, dilute coil is inversely proportional to its size and hence to


1 2N − or 3 5N − under theta or good solvent conditions, respectively. In order to calculate the viscosity of a dilute polymer solution under non-

draining conditions, we can use another one of Einstein’s equations. The specific viscosity of a suspension of colloidal spheres with a volume fraction φ is given by

.2 5s

s s

η η ηφ

η η− Δ

= = (4.15)

Note that this expression strictly applies to a dilute suspension with vanishing hydrodynamic interactions among the spheres; for higher densities higher order terms in the volume fraction have to be taken into account. If the coils are non-drained, the volume fraction is given by 3cR Nφ = and we can use the measurement of the specific viscosity

3

s

cRN

ηηΔ (Zimm) (4.16)

as an experimental tool to determine the size of the coil. The specific viscosity of a solution of non-drained, dilute polymer coils is proportional to 1 2N or

4 5N under theta or good solvent conditions, respectively. Furthermore, with Eqs. (4.9) and (4.10) it is easy to show that the viscosity increment

ckT

Nη τΔ (4.17)

is proportional to the relaxation time τ , irrespective of the chain statistics inside the coil.

We will now show that Eq. (4.17) is universal and can also be used to derive the viscosity of polymer solutions under chain overlap conditions. The viscosity increment can be derived by analyzing the viscoelastic response of a polymer solution initially at rest and subjected to an instantaneous, constant shear σ (see Fig. 4.1). For very short times after the application of the shear, shorter than the global relaxation time of the chain t τ< , the polymer has no time to rearrange its segments and the response is elastic with a constant strain γ given by the stress divided by the (high frequency) elastic shear modulus

Gσ

γ = ( t τ< ) (4.18)

For longer times, t τ> the polymer solution will start to flow and the strain becomes time-dependent with a Newtonian slope


ddtγ σ

η=

Δ ( t τ> ) (4.19)

which is determined by the viscosity increment ηΔ . From the crossover at the relaxation time τ , we find an important scaling law for the viscosity Gη τΔ (4.20)

The instantaneous, high frequency elastic response of the polymer solution to the stress (which is a force per unit area) is the inverse compressibility

( )G V V=− ∂ ∂Π . Van’t Hoff’s law c N kTΠ = then gives for the elasticity modulus

cG kT

N= (4.21)

which immediately leads us to Eq. (4.17) without the specification of the draining conditions inside the coil nor the chain statistics. As shown by Eq. (4.21), the elasticity modulus is given by the product of the number of dynamic units per unit volume times the thermal energy kT . The latter concept is general and will prove to be most useful for the derivation of expressions for the viscosity under chain overlap conditions to be treated in

τ

σ/G

dγ/dt = σ/Δη

γ

t

Figure 4.1 Response of a polymer solution initially at rest to the application of an instantaneous stress at time zero. The solution starts to flow after the global relaxation time τ with a strain rate inversely proportional to the viscosity.


later sections of this chapter. With the general equation c N kTη τΔ (for dynamically independent

chains) and the global relaxation time Eq. (4.8), we can calculate the specific viscosity of a freely drained and dilute Gaussian chain

20

scl R

ηηΔ (Rouse) (4.22)

Owing to the draining of the coil, the viscosity is proportional to the square of the coil size rather than its volume. However, as mentioned above, real polymer chains in the dilute regime are non-drained and Eq. (4.22) has little practical value.

Before we move on to the description of the polymer dynamics in the semi-dilute regime, we will first discuss the conceptually important issue of how long it takes to pull a chain into a hole.

4.2 Pulling a chain into a hole

A biopolymer is sometimes pulled into a point-like potential well. Examples are the translocation of DNA into a nano-channel or the spooling of DNA into a virus capsid. For a conceptual understanding of this important class of phenomena, we will consider the pulling of a polymer chain with a constant force into a hole (see Fig. 4.2). We will closely follow the work of Grosberg et al., who addressed this problem at the Rouse level of a Gaussian chain with additive hydrodynamic friction.36 One end of the chain is put into a hole (‘drain’) and the polymer is pulled through with a constant force f acting on the link that is entering the hole. We are particularly interested in how long it takes to pull the entire polymer of N links into the hole (adsorption time). The adsorption time is expected to be dependent on the strength of the force and the contour length of the chain.

Let ( )n t be the number of links inside the hole, so that N n− is the number of links outside the hole to be pulled inside with a constant velocity l dn dt in unit length per second. We assume that the polymer is freely drained, so that the total friction experienced by the part of the chain outside the hole is additive and given by ( ) sN n lζ η= − (we neglect the friction inside the hole). Furthermore, we will first assume that, outside the hole, the chain is completely straight. For a straight configuration, the friction force


(that is the total friction times the velocity) should balance the pulling force

( )2s

dnf l N n

dtη= − (straight chain) (4.23)

and the time strτ needed to pull the straight chain of N links into the hole follows from the integration

( )2 2 2

0 2

Ns sstr

l l NN n dn

f fη η

τ = − =∫ (4.24)

Note that the adsorption time strτ is inversely proportional to the pulling force and quadratic in the number of links per chain.

A prerequisite for the derivation of Eq. (4.24) is that the chain remains straight, so that the pulling force always exactly balances the friction force. It is thus interesting to compare strτ with the global relaxation time of the Rouse chain 2 3

R sN l kTτ η [see Eq. (4.8)]. The Rouse relaxation time is the time during which the coil diffuses over a distance on the order of its own size

1 2R lN . During the pulling of the straight chain, the chain should have no time to coil and the time needed to pull the chain should be less than the relaxation time str Rτ τ< . The latter condition implies fl kT> and shows that the pulling force should be very strong, so that the elastic blob size becomes smaller than the step length l (see Sec. 2.8). For a smaller force, the chain will not remain straight, but will coil on the time scale of the pulling procedure.

We will now consider the more general case of the pulling of a freely drained Gaussian coil inside the hole. This problem can be solved with a

Figure 4.2 Pulling a coiled chain into a hole.


dynamic scaling argument, that is by recognizing the fact that the characteristic time and distance scales of the system are the Rouse relaxation time Rτ and chain size R , respectively. We propose, accordingly, a scaling relation coil Rτ τ φ= (4.25)

with φ a dimensionless function depending on the dimensionless ‘pulling energy’ fR kT . The function φ can be found by realizing that the velocity of pulling the complete chain into the hole coilNl τ should be linear in the applied force. The latter condition implies kT fRφ = and we obtain

2 3 2

scoil

l Nf

ητ (4.26)

The time needed to pull a freely drained Gaussian coil inside the hole scales with the number of links to the power 3 2 rather than 2 for a straight chain. Grosberg et al. have verified their scaling Eqs. (4.24) and (4.26) for straight and coiled chains, respectively, by Molecular Dynamics computer simulations and their results are displayed in Fig. 4.3.

101

102

103

101

102

103

104

105

N

τ

τ ~N1.48

τ ~N1.96

str

coil

Figure 4.3 Adsorption time versus number of links resulting from Molecular Dynamics simulations of Rouse chains pulled into a hole ( l = 1, kT = 1 , η = 10 ). Open symbols: initially straightened chain; closed symbols: initially coiled chain. Redrawn from Ref. [36].


4.3 Dynamics of non-entangled chains in the semi-dilute regime

We now return to our discussion of the dynamics of neutral polymer chains in solution. If the concentration is increased, the chains start to interpenetrate at a critical overlap concentration *c . As we have seen in Chapter 2, in the semi-dilute regime the chains can be considered as sequences of N g blobs of size ξ . A striking conclusion was that the fluctuations in blob density and the resulting net forces experienced by the blobs are vanishingly small owing to the fact that the blobs are closely packed and space-filling. Accordingly, the chain statistics remain ideal (i.e. Gaussian) and the size of the chain is given by ( )1 2R N g ξ . From a dynamic perspective, we can say that the hydrodynamic interaction forces dwindle beyond the correlation length, which means that the chain of blobs should behave as a Rouse chain (a freely drained chain with Gaussian statistics). However, with increasing concentration, the chains of blobs progressively overlap and the dynamics of a test chain are expected to be seriously hindered by the presence of the other chains. In order to estimate this effect, we need to consider the different kinds of interactions among the chains of blobs in the semi-dilute regime.

As illustrated in Fig. 4.4, there are two types of inter-chain interactions: entangled and non-entangled. In the case of a non-entangled interaction, the

Figure 4.4 Types of interaction among chains of blobs in the semi-dilute regime. The red test chain of blobs can only move along the tube, whereas lateral displacements are prohibited by the entanglements with the blue chains. The interactions with the black chains are non-entangled. For the sake of clarity, only the primitive paths of the chains of blobs are displayed.


chains of blobs can move freely with respect to each other and the effect of the interaction does not go beyond the screening of the hydrodynamic forces as discussed above. This situation is different for a so-called entangled interaction. Entanglements are topological constraints resulting from the fact that the chains cannot cross through each other. As a result of these entanglements, the lateral displacement of the test chain of blobs is prohibited. The test chain of blobs can only move along the axial line (primitive path) of a ‘tube’ formed by the entanglements. This snake-like, diffusive motion along the primitive path is usually referred to as reptation and will be further discussed in the next section.3,35 The entanglements are relaxed only after the chain of blobs has diffused out of its original tube, has formed a new tube and has become entangled with other chains (tube renewal).

Unfortunately, not much is known about the onset of entanglements and the entanglement concentration. In general, one can say that in order to obtain entanglements a certain number of chains n have to overlap. Entanglement effects thus begin at a concentration 3ec nN R . For neutral polymers, with overlap concentration * 3 4 5c l N− − and semi-dilute chain size ( ) 1 81 2 3R N l cl

− (see Appendix B for a compilation of scaling laws), one obtains

*8 5ec n c (4.27)

Note that the number n is not a universal parameter and depends on the chemical structure and possibly the charge of the polymer. From experimental studies, n has been estimated to be in the range between 3 and 10, with the lower boundary pertaining to DNA. There is accordingly a range in concentration in which entanglement effects remain unimportant. In this range, a test chain of blobs moves Rouse-like as a single dynamic unit. We will further discuss the dynamics in this semi-dilute, non-entangled regime; the dynamics under entangled conditions will be treated in the next section.

In the range * ec c c< < , the hydrodynamic interactions are screened on distances longer than the correlation length ξ and the chain of blobs behaves as a Rouse chain with relaxation time Eq. (4.8). Since there are N g blobs of size ξ per chain, the relaxation time takes the form

( )2 3sN g kTτ ξ η ( * ec c c< < ) (4.28)


With the static scaling results for the blob size and number of links per blob, ( ) 3 43l clξ

− and ( ) 5 43g cl− , respectively, we obtain for the global

relaxation time

( )1 43 3 2sl cl N kTτ η ( * ec c c< < ) (4.29)

Note that the relaxation time increases with the concentration to the power 1 4 and is proportional to the square of the number of links, which is the square of the molecular weight. Inside the blobs, there is full hydrodynamic interaction among the segments (non-draining conditions) and the relaxation time ξτ of this subsection is given by Zimm’s Eqs. (4.9) or (4.10), depending on the chain statistics. With 3

s kTξτ η ξ , we write for the relaxation time

( )2N g ξτ τ ( * ec c c< < ) (4.30)

where we recognize that the dynamics of the non-entangled, neutral chain in the semi-dilute regime is Zimm-like up to the correlation length ξ and Rouse-like for a strand of N g blobs of size ξ .

With the relaxation time τ , we can derive molecular transport properties, such as the self-diffusion coefficient and the viscosity. The self-diffusion coefficient D can be calculated by recognizing the fact that the chain moves over a distance on the order of its size R during the time τ . With 2D R τ , together with ( ) 1 81 2 3R N l cl

− and Eq. (4.29) for the relaxation time, we obtain the scaling law

( ) 1 21 3 1sD l cl N kT η

−− − ( * ec c c< < ) (4.31)

The self-diffusion coefficient of a non-entangled, neutral polymer is inversely proportional to its contour length L Nl= and decreases with the square root of its concentration. The inverse contour length dependence is clearly a signature of the Rouse behaviour, owing to the screening of the hydrodynamic interactions in the semi-dilute regime.

For the derivation of the viscosity, it is important to realize that in the non-entangled range the chains move as single dynamic units. Accordingly, the elasticity modulus (thermal energy kT per dynamic unit) is still given by G c N kT= . With c N kTη τΔ and the relaxation time Eq. (4.29), we accordingly obtain

( )5 43s N clη ηΔ ( * ec c c< < ) (4.32)


The specific viscosity is proportional to the number of links per chain and the monomer concentration to the power 5 4 .

Note that the derived transport equations are valid for neutral polymers in the non-entangled range of the semi-dilute regime. As we will see in the next section, the presence of entanglements has a dramatic effect on the chain dynamics and results in different functional dependencies of the molecular transport properties.

4.4 Entangled polymer dynamics; reptation

At still higher polymer concentration ec c> ( *8 5ec n c ), the polymer chains become entangled and a transient elastic network is formed. As illustrated in Fig. 4.4, the chain of blobs is confined in a temporal tube formed by the entanglements. The topological constraints can only be relaxed if the chain of blobs moves out of its original tube (and forms a new tube) by one-dimensional diffusive motion along the primitive path. The characteristic diameter of the tube is given by the entanglement correlation length eξ . Let there be, on average, eN links between entanglements ( eN N< ). The chain of blobs contains a total of eN N entanglement strands, each entanglement strand comprising eN g blobs. We can find a scaling relation of the entanglement correlation length eξ by recognizing the fact that any subsection of the chain of blobs should exhibit Gaussian statistics. For a random walk of eN g blobs of size ξ , the end-to-end distance is given by the scaling relation

( )1 2e eN gξ ξ (4.33)

Since there are n overlapping chains, the entanglement volume 3eξ should

contain n entanglement strands. Each entanglement strand comprises eN g blobs, so that there are a total of enN g blobs accommodated inside the volume given by 3

eξ . The blobs are closely packed and space-filling, which implies 3 3

e enN gξ ξ (each blob has a volume 3ξ ). With Eq. (4.33), it is now not difficult to show that

( ) 3 43e n nl clξ ξ

− ( ec c> ) (4.34)

( ) 5 42 2 3eN n g n cl

− ( ec c> ) (4.35)


The entanglement correlation length eξ (tube diameter) and the number of links per entanglement strand eN can be expressed in terms of the number of overlapping chains n which are required to form entanglements. It is interesting to note that the entanglement correlation length is 3 to 10 times the static correlation length given by the blob size.

The primitive path of the tube enclosing the chain of blobs has a length given by the product of number of entanglements strands per chain times the entanglement correlation length

( )1 2tube e e e eL N N N N N gξ ξ (4.36)

As discussed above, the chain of blobs can only move along the primitive path by one-dimensional diffusion with a diffusion constant tubeD kT ζ . The chain of blobs is freely drained and the hydrodynamic friction ζ is proportional to the number of blobs per chain N g ξζ ζ= , where ξζ is the friction per blob. The blobs themselves are non-drained and have a friction coefficient given by Stokes’ law 6 sξζ πη ξ= , so that the diffusion coefficient takes the form

( ) 1 1tube sD N g kTξ η− − (4.37)

Note that the diffusion coefficient tubeD refers to the one-dimensional diffusive motion along the primitive path and is not the same as the self-diffusion coefficient. The self-diffusion coefficient pertains to the mean square 3–dimensional displacement of the chain in any direction and will be discussed below.

We are now in the position to calculate the relaxation time τ required for the chain of blobs to diffuse out of its original tube, to form a new tube and to become entangled with other chains of blobs. Accordingly, the relaxation time can also be considered as an entanglement disengagement time or tube renewal time. The one-dimensional analogy of Einstein’s diffusion equation is given by 2 2x Dt= , with x the axial displacement. In order to move out of its original tube, the chain of blobs has to diffuse over a distance on the order of the tube length tubeL , so that the relaxation time takes the scaling form

( ) ( )2

3 2 3tubee e s

tube

LN N N g kT

Dτ ξ η ( ec c> ) (4.38)

With the Zimm relaxation time 3s kTξτ ξ η , we write for the relaxation

time


( ) ( )3 2e eN N N g ξτ τ ( ec c> ) (4.39)

where it can be seen that the dynamics are Zimm-like up to the correlation length ξ , Rouse-like for a strand of eN g blobs of size ξ and reptation-like for eN N entanglement strands. Finally, with 2

eN n g and the static scaling results for the blob size ξ and number of links per blob g , respectively, the relaxation time reads

( ) ( )3 23 3 3 2sl cl N n kTτ η ( ec c> ) (4.40)

Note that, owing to the reptation dynamics, the relaxation time of an entangled, neutral polymer has a cubic dependence of the number of links per chain. Furthermore, the relaxation time increases with the concentration to the power 3 2 .

As in the case of the dynamics in the non-entangled regime, the self-diffusion coefficient can be estimated from 2D R τ , together with

( ) 1 81 2 3R N l cl− and Eq. (4.40) for the relaxation time and takes the form

( ) ( )7 4 21 3

sD l cl n N kT η−− ( ec c> ) (4.41)

Note that the self-diffusion coefficient is inversely proportional to the square of the number of links per chain. This is in contrast with the corresponding result for non-entangled chains, where we found a 1N − dependence [see Eq. (4.31)]. The scaling of the self-diffusion coefficient with the inverse of the square of the molecular weight is usually taken as a benchmark for reptation dynamics. The self-diffusion coefficient also shows a stronger decrease with increasing concentration according to a power ~ 7 4D c− compared to

~ 1 2D c− in the non-entangled regime. For the calculation of the viscosity increment, we have to realize that the

entanglement strands are now the dynamic units, rather than the complete chains of blobs. Accordingly, in the transient, elastic polymer network, the elasticity modulus (thermal energy kT per dynamic unit) is given by

( )9 43 3 2

e

cG kT l cl n kT

N− −= ( ec c> ) (4.42)

For a non-entangled Rouse chain, the elasticity modulus is given by RG c N kT= . The number of segments per entanglement strand eN can

thus be obtained from the ratio of the elasticity moduli e RN N G G= . With ec N kTη τΔ and the relaxation time Eq. (4.38), we obtain


( )15 43 3 4s cl N nη ηΔ ( ec c> ) (4.43)

for the specific viscosity of the entangled, neutral polymer solution. Note the strong increase in viscosity with increasing concentration to the power 15 4 and the strong dependence of the molecular weight expressed by the cubic dependence of the number of links per chain.

The viscosity and self-diffusion coefficients have been measured for the water-soluble and neutral polymer poly(ethyleneoxide) (PEO).37 All solutions were in the semi-dilute regime; the overlap concentration is estimated to be around 0.01 M. Some illustrative results are displayed in Fig. 4.5 in a double logarithmic representation. It is found that the initial slopes of the data agree reasonably well with ~ 1 2D c− and ~ 5 4cη , as predicted by Eqs. (4.31) and (4.32), respectively, for the non-entangled, semi-dilute regime. However, at higher concentrations (up to 50 M), the scaling laws pertaining to reptation dynamics are not observed. These PEO chains do not become entangled, probably because their contour length is too small (800 units).

10−1

100

100

101

c (M)

D (m

2 /s)

10−1

100

10−1

100

101

102

c (M)

η (c

P)

η~c5/4

D~c -1/2

Figure 4.5 Molecular transport in semi-dilute neutral polymer solutions (PEO, wM = 350, 000 ). Left panel: self-diffusion coefficient; right panel: viscosity.

The lines denote the scaling laws /~ 1 2D c− and /~ 5 4cη pertaining to the non-entangled, semi-dilute regime. The data are from Ref. [37].


4.5 Dynamic scaling of polyelectrolytes

According to the procedure delineated above for neutral polymers, we can also derive scaling laws for the transport properties of charged polymer chains in the semi-dilute regime. In the previous chapter, we have considered the static solution structure of a polyelectrolyte, both without salt and in the presence of excess salt. For the salted polyelectrolyte, we recovered the same scaling laws as for solutions of neutral polymers, albeit with different pre-factors related to the (ionic strength dependent) persistence length and effective diameter. This similarity can be traced back to the fact that the salted polyelectrolyte chain takes an isotropic conformation in the dilute regime, just as a single neutral polymer chain. For polyelectrolytes without salt, the scaling procedure is much more conjectural, due to the intricate, long-range behaviour of the unscreened electrostatic interactions. Owing to the presumption that the salt-free polyelectrolyte takes an elongated configuration in the dilute regime, we obtained a set of static scaling laws, that have different concentration and molecular weight dependencies than the ones for neutral polymers (see Appendix B). Since the dynamic scaling procedure relies on our perception of the static structure, we accordingly anticipate different and unique scaling laws for the dynamic properties of salt-free polyelectrolytes. As we will see shortly, at least to some extent these predictions have been confirmed by experiment. We will first discuss the dynamics of polyelectrolytes without salt, after which the salted polyelectrolyte chain will be treated. For the salt-free polyelectrolyte, we will follow the procedure of Rubinstein et al.,33 based on the static scaling picture of de Gennes et al.38

4.5.1 Polyelectrolytes without salt

Owing to the electrostatic repulsion, the polyelectrolyte chain without salt is thought to take an elongated configuration in dilute solution with a renormalized contour length L N lB , where B denotes a constant, on the order of 0.1, depending on the solvent conditions. As a result of this elongated configuration, we obtained the scaling laws for the overlap concentration

* 3 2 3c l N B− − − and the chain size in the semi-dilute regime ( ) 1 41 2 1 4 3R N lB cl

− (see Appendix B). As in the case of neutral polymers,


entanglement effects become important if a number n of the polyelectrolyte chains overlap. With 3ec nN R and the scaling laws for the size and overlap concentration, we obtain

*4ec n c (4.44)

The strong dependence of the number n to the power 4 shows that, for polyelectrolytes without salt, there is a relatively wide range in the semi-dilute regime before entanglements become effective. We will now first discuss the dynamics of salt-free polyelectrolytes in the non-entangled range of the semi-dilute regime.

As has been discussed in Chapter 3, the semi-dilute regime of a salt-free polyelectrolyte can be described by a hierarchy of blobs. The semi-dilute blobs contain more or less elongated strings of electrostatic blobs. Within the electrostatic blobs, the chain is weakly disturbed by the electrostatic interactions and takes a coiled conformation. Experimental evidence shows that the dynamics of the salt-free polyelectrolyte chain are non-drained and Zimm-like up to the correlation length ξ . Despite the elongated conformation of the strings of electrostatic blobs inside the semi-dilute blobs, the relaxation time of the latter blobs is given by the Zimm expression for a non-drained, isotropic coil 3

s kTξτ ξ η . Factors related to the anisotropy of the strings of electrostatic blobs are thus neglected.1 Hydrodynamic and electrostatic interactions are screened beyond the correlation length ξ , so that the dynamics of the strand of N g blobs of size ξ is Rouse-like and has a relaxation time

( )2N g ξτ τ ( * ec c c< < ) (4.45)

With the static scaling results for the blob size and number of links per blob, ξ and g , respectively, (see Appendix B) we obtain for the relaxation time

( ) 1 23 3 2 3 2sl B cl N kTτ η

− ( * ec c c< < ) (4.46)

We find that the relaxation time decreases with increasing concentration. This is contradictory to the behaviour of neutral polymer solutions, which always show an increase in relaxation time with increasing concentration. We will see

1 The relaxation time can be modified for the anisotropy of the string of electrostatic blobs with aspect ratio

5 2gB according to ( )ln3 5 2

s kT gBξτ η ξ [35]. It is obvious that this procedure introduces logarithmic terms in the transport equations, which, so far, have not been observed experimentally.


shortly that this atypical behaviour of the relaxation time results in some peculiar properties of the molecular transport coefficients.

The self-diffusion coefficient is proportional to the mean square size of the chain divided by its relaxation time

2

1 1 1s

RD l B N kT η

τ− − − ( * ec c c< < ) (4.47)

As a surprising result, we find that the self-diffusion coefficient of the salt-free polyelectrolyte chain in the semi-dilute, non-entangled regime is constant and is equal to its value in a dilute solution 0 sD kT L η . At high frequencies, the electro-kinetic motions of the counterions and the charged polymer chain are decoupled and the (high frequency) elasticity modulus of the non-entangled polyelectrolyte solution is given by the thermal energy kT per chain G c N kT= . Accordingly, the specific viscosity

( )1 23 2 3s G B cl Nη η τΔ (4.48)

is proportional to the square root of the concentration, which is in agreement with the empirical Fuoss law for the viscosity of a polyelectrolyte solution without salt.

Entanglements become effective when the monomer concentration exceeds the entanglement concentration *4ec n c . With the blob size and the number of segments per blob, we obtain

( ) 1 21 2 3e n nlB clξ ξ

−− ( ec c> ) (4.49)

( ) 1 22 2 3 2 3eN n g n B cl

−− ( ec c> ) (4.50)

for the tube diameter and number of segments per entanglement strand, respectively. As for neutral polymers, the dynamics are Zimm-like up to the correlation length ξ , Rouse-like for a strand of eN g blobs of size ξ and reptation-like for eN N entanglement strands, so that the relaxation time has the form

( ) ( ) ( )3 2 3 3 3 2e e sN N N g l B N n kTξτ τ η ( ec c> ) (4.51)

An interesting result is that the relaxation time of the salt-free, entangled polyelectrolyte chain is concentration independent. The self-diffusion coefficient can be estimated from the chain size and the relaxation time


( ) ( )2 1 2 21 5 2 3

sR

D l B cl n N kT ητ

−− − ( ec c> ) (4.52)

which now shows a decrease with increasing concentration according to ~ 1 2D c− . As in the case of a neutral polymer solution, the high frequency

elasticity modulus is given by the thermal energy kT per entanglement strand

( )3 23 3 2 3 2

e

cG kT l B cl n kT

N− −= ( ec c> ) (4.53)

and the specific viscosity takes the form

( )3 29 2 3 3 4s G B cl N nη η τΔ ( ec c> ) (4.54)

Note that the concentration dependence of the specific viscosity of the entangled, salt-free polyelectrolyte solution ( ~ 3 2cη ) is much weaker than the one for a neutral entangled polymer solution ( ~ 15 4cη ).

4.5.2 Salted polyelectrolytes

In the presence of excess salt, the molecular weight and concentration dependencies of the overlap concentration, the blob size, the number of segments per blob, as well as the size of the polyelectrolyte are the same as those for neutral polymers (Appendix B). Accordingly, we expect the same dynamical scaling laws as for neutral polymers, but with different pre-factors related to the ionic strength dependent persistence length and effective diameter. The entanglement concentration can be derived with 3ec nN R and the scaling laws for the size R and the overlap concentration

( )* 3 54 5 9 5p effc N l L D −− − , taking the form

*8 5ec n c (4.55)

Note that the entanglement concentration depends on the salt concentration, due to the ionic strength dependent overlap concentration (which, in its turn, depends on the salt concentration because of the ionic strength dependent radius of gyration). With the static scaling results for the blob size and number of links per blob, ξ and g , respectively, we obtain for the relaxation time in the non-entangled regime


( )

( ) ( )

2

1 43 43 2 3 2p eff s

N g

l L D cl N kT

ξτ τ

η ( * ec c c< < ) (4.56)

The self-diffusion coefficient

( ) ( )2 1 21 2 3 1

p eff sR

D L D cl N kT ητ

−− − ( * ec c c< < ) (4.57)

now decreases with the polyelectrolyte concentration ~ 1 2D c− (without salt the self-diffusion coefficient is constant). As in the case of a non-entangled, neutral polymer solution, the high frequency elasticity modulus is given by the thermal energy kT per polyelectrolyte chain G c N kT= and the specific viscosity takes the form

( ) ( )5 43 43 2 3s p effG l L D cl Nη η τ −Δ ( * ec c c< < ) (4.58)

Above the entanglement concentration ec c> , the tube diameter and number of segments per entanglement strand are given by

( ) ( ) 3 41 43 2 3e p effn nl L D clξ ξ

−− ( ec c> ) (4.59)

( ) ( ) 5 43 42 2 3 2 3e p effN n g n l L D cl

−− ( ec c> ) (4.60)

respectively. The dynamics of the salted polyelectrolyte chain is Zimm-like up to the correlation length ξ , Rouse-like for a strand of eN g blobs of size ξ and reptation-like for eN N entanglement strands, so that the relaxation time takes the form

( ) ( )

( ) ( ) ( )

3 2

3 23 2 3 3 2

e e

p eff s

N N N g

L D cl N n kT

ξτ τ

η ( ec c> ) (4.61)

In the entangled regime, the self-diffusion coefficient reads

( ) ( ) ( )2 7 45 4 23 2 3

p eff sR

D l L D cl n N kT ητ

−− ( ec c> ) (4.62)

and the high frequency elasticity modulus is given by the thermal energy kT per entanglement strand

( ) ( )9 43 49 2 3 2p eff

e

cG kT l L D cl n kT

N− −= ( ec c> ) (4.63)

which leads to


( ) ( )15 49 49 2 3 3 4s p effG l L D cl N nη η τ −Δ ( ec c> ) (4.64)

for the specific viscosity. Note that, due to the similarity of the static scaling laws, the concentration

and number of links (i.e. molecular weight) dependencies of the transport properties of the salted polyelectrolyte are the same as those for a neutral polymer. The pre-factors, however, now include the persistence length pL and the effective diameter effD , always in the combination p effL D . Both the electrostatic contribution to the persistence length and the effective diameter are sensitive to the salt concentration through the Debye screening length

1κ− . With increasing salt concentration p effL D decreases, which results in an increase of the self-diffusion coefficient and a decrease in specific viscosity. The extent to which this happens, that is the scaling exponent, depends obviously on whether the system is entangled or not. Eventually, at very high salt concentration p effL D becomes on the order of 2l and the scaling laws for neutral polymer chains are recovered.

All derived static and dynamic scaling laws pertaining to the semi-dilute regime are collected in Appendix B.

4.5.3 Comparison with experimental results

The self-diffusion coefficient and the low shear viscosity of the synthetic, flexible polyelectrolyte sodium poly(styrenesulfonate) (PSS) have been measured for various molecular weights and salt concentrations.37,39 The self-diffusion coefficient was measured by the pulsed field gradient NMR method, which does not require labelling of the chain. Some illustrative results are shown in Fig. 4.6. In the low concentration range, say below 0.1 M, the self-diffusion coefficient is observed to scale with the inverse of the number of links, irrespective of the presence of salt ( ,90 1 000N< < ). In the same concentration range, the viscosity is observed to be proportional to N . Furthermore, for the salt-free PSS the self-diffusion coefficient levels off and becomes constant in the low concentration range. In the presence of salt, the self-diffusion coefficient decreases with increasing concentration according to

~ 1 2D c− . The viscosity, however, always increases with increasing PSS concentration according to ~ 1 2cη and ~ 5 2cη without and with salt, respectively. These observations are in general agreement with the scaling laws


for polyelectrolyte dynamics in the non-entangled semi-dilute regime. At higher polymer concentration, deviations from the non-entangled scaling laws are observed, but we do not obtain the behaviour as expected for entangled polyelectrolytes (for instance at high concentration we do not observe the

~ 2D N − dependence of the self-diffusion coefficient, which is a benchmark for reptation dynamics). As in the case of the PEO chains discussed above, these PSS chains are probably too short to become entangled.

Reptation of entangled DNA has been observed by Smith et al.40 Their experiments were done with fluorescently labelled viral λ –phage DNA (48,502 bp) in 10 mM Tris/EDTA (TE) buffer. Owing to the presence of the fluorescent dye, the contour length amounts to 22 micrometres (the nominal length is 16.3 micrometres). Based on the value of the radius of gyration (890 nm, as extrapolated from the results in Table III.I for T7–phage DNA and corrected for a difference in contour length according to ~ 3 5

gR L ), the

010

−210

−110

10-9

10-8

c (M)

ND

(m2/s

) c-1/2

10−3

10−2

10−1

100

10−2

10−1

10 0

10 1

10 2

c (M)

Δη /

η s

c5/4

c1/2

Figure 4.6 Transport properties of flexible sodium poly(styrenesulfonate) (PSS).37,39 Circles: salt-free; squares: in 0.1 M NaCl. Left panel: self-diffusion coefficient multiplied by the number of links per chain for PSS with molecular weight wM = 88 (N = 480, open symbols) and 177 (N = 970, closed symbols) kg/mol. Right panel: low shear viscosity for PSS with molecular weight 177 kg/mol.


calculated overlap concentration is around 0.02 g/L. Smith et al. measured the diffusion coefficient as a function of the concentration by tracking the Brownian motion of the DNA molecules with fluorescence microscopy. Furthermore, at a fixed DNA concentration of 0.6 g/L, the self-diffusion coefficient was measured for a series of DNA fragments with different lengths generated by a restriction enzyme. The results are displayed in Fig. 4.7. The self-diffusion coefficient decreases with increasing concentration according to

~ 7 4D c− for a DNA concentration exceeding, say, 0.5 g/L. Furthermore, at 0.6 g of DNA/L the self-diffusion coefficient decreases with increasing length of the molecule to the power close to minus two. These results comply with reptation dynamics of a salted polyelectrolyte and indicate that the λ –phage DNA solution becomes entangled at 0.6 g of DNA/L, which is about 30 times the overlap concentration.

The onset of entanglements is also shown by the behaviour of the elastic storage 'G and the viscous loss modulus "G . These viscoelastic properties of λ –phage DNA in 10 mM TE buffer are displayed in Fig. 4.8. With increasing frequency, the viscous loss modulus first increases, then levels off and eventually increases again. Concurrently, the elastic storage modulus increases and levels off to a constant high frequency value. For a DNA concentration exceeding 0.5 g/L, the elastic storage modulus becomes larger than the viscous

10 −1 10 010−15

10−14

10−13

c (g/L)

D (m

2 /s)

10−6 10− 510

−15

10−14

10−13

10−12

L (m)

c -7/4 -2LD~ D~

Figure 4.7 Self-diffusion of fluorescently labelled DNA in 10 mM TE buffer. Left panel: concentration dependence; λ –phage DNA, 48.5 kbp, dye corrected contour length of 22 micrometres. Right panel: length dependence at 0.6 g of DNA/L. Redrawn from Ref. [40].


loss modulus in an intermediate frequency range. This transition from viscous to elastic behaviour corresponds to the onset of entanglements, as inferred from the high frequency limiting value of the elastic storage modulus, as well as the above mentioned self-diffusion experiments.

The low shear viscosity and elasticity modulus of entangled T2–phage DNA (164 kbp, 56 micrometres contour length) solutions have been measured by Musti et al.41 These experiments were performed in 50 mM Tris, 100 mM KCl, 10 mM MgCl2 and 0.5 mM EDTA buffer, which has a corresponding ionic strength of around 170 mM. Furthermore, the temperature was slightly elevated at 303 K. Based on the data in Table III.I for T7–phage DNA and corrected for a difference in contour length, the radius of gyration of T2–phage DNA in 170 mM buffer ionic strength is around 1,200 nm. The overlap concentration is estimated to be around 0.02 g/L, which is the same value as for the λ –phage DNA molecules in 10 mM TE buffer used in the self-diffusion and rheology experiments discussed above. The low shear viscosity is displayed in Fig. 4.9. With increasing concentration,

10−2

10−1

100

101

10−3

10−2

10−1

100

ω (s )−1

G‘ , G

‘‘ (Pa

)

Figure 4.8 Viscoelastic properties of λ –phage DNA in 10 mM TE buffer. Filled symbols: viscous loss modulus "G ; open symbols: elastic storage modulus 'G . The DNA concentration is 0.4 (green squares), 0.5 (black upper triangles, 0.6 (red circles) and 0.7 (blue downward triangles) g of DNA/L. Kundukad and van der Maarel, unpublished.


the viscosity increases more steeply than predicted by the scaling law for the entangled and salted polyelectrolyte solution ( ~ 15 4cη , see Appendix B). It was observed that the solution becomes a gel for concentrations exceeding 1.6 g/L.

The ratio of the value of the elasticity modulus pertaining to the non-entangled Rouse chain RG c N kT= to the measured elasticity modulus in the entangled regime eG c N kT= is also displayed in Fig. 4.9. This ratio R eG G N N= gives hence the relative fraction of the number of links

between entanglements and should scale as ~ 5 4RG G c− [Eq. (4.60)]. As can

be seen in Fig. 4.9, for DNA concentrations exceeding 0.3 g/L the DNA molecules are significantly entangled, in agreement with the strong increase in low shear viscosity. Furthermore, we can use the scaling expression in order to find the entanglement concentration at which entanglements first occur. An extrapolation of the experimental data to 1eN N = according to

~ 5 4RG G c− gives .0 12ec = g/L, which is a factor of six above the

estimated overlap concentration. For λ−phage DNA, the entanglement concentration is around 0.5 g/L.40 The discrepancy between the λ−phage and the T2–phage DNA results, which is probably related to the differences in molecular weight and ionic strength, shows the non-universality of the onset of entanglements.

10 1010

−2

10−

10 0

c (g/L)N

e/N

10−1

100

10−2

10−1

10 0

10 1

10 2

10 3

c (g/L)

η (P

a s)

η ~ c 15/4

−1 0

1

Ne/N ~ c-5/4

Figure 4.9 Viscoelastic properties of T2–phage DNA (164 kbp, 56 micrometres length, 170 mM ionic strength). Left panel: low shear viscosity. Right panel: the number of links per entanglement strand. The extrapolated entanglement concentration amounts to 0.12 g/L. Data are from Ref. [41].


4.6 Gel electrophoresis

An important and elegant application of the reptation mechanism is gel electrophoresis.42 Gel electrophoresis is a common practice in the molecular biology laboratory in order to separate biomacromolecules, such as DNA, according to their molecular weight or other properties (for instance tertiary coil structures). In this procedure, a sample is loaded onto a slot prepared in a slab of a polymer gel (for instance agarose). The gel is subsequently exposed to an electric field for a certain duration; so that the DNA molecules migrate through the gel from the negative to the positive pole (the DNA molecules are negatively charged). This procedure results in an effective separation of the different fractions into well-defined regions (bands), because the mobility of the DNA molecules depends on their length, as well as other properties. The bands can subsequently be visualized with the help of a fluorescent dye under UV exposure. Absolute measurements can be done by using a ladder of DNA molecules with known molecular weights, which is ‘run’ in the same gel next to the slots for the samples of unknown composition.

The theory of gel electrophoresis is incomplete and many of the practicalities have been optimized in a phenomenological way. Nevertheless, the reptation mechanism gives us valuable insights. For simplicity, we assume that the DNA molecules are entangled by the polymer network and that the entanglement correlation length is on the order of the DNA persistence length. Accordingly, the DNA molecules can only move along their contour, as determined by the topological constraints set by the entanglements within the network. The DNA molecules themselves are thus considered to be the reptating chains. In reality, the entanglement correlation length might be longer than the persistence length, so that the DNA molecules are locally coiled. In the latter case, the reptating units are the renormalized chains of blobs, rather than the DNA molecules themselves. However, for reptation the required condition is that the lateral displacements are prohibited and that the polymer can only move along the axial line of the entanglement tube. This condition holds irrespective of the entanglement correlation length.

Consider a DNA molecule with an effective DNA charge per unit length ρ . An infinitesimal segment of the molecule of length ds carries a charge dsρ


and has an tangent directional unit vector ( )l s . The only relevant component of the electric field is the one directed along the contour ( )E l si . The net electric force exerted on the chain follows from integration over all segments

( )0

L

cf E l s dsρ= ∫ i (4.65)

The electric force drives the DNA molecule along the contour (or primitive path of the entanglement tube); the perpendicular component of the force is ineffective because the lateral displacements are prohibited by the entanglements. If the electric field has a direction e , so that E Ee= , we can write

( )0

L

cf E e l s ds E hρ ρ= =∫ i (4.66)

where h is the projection of the end-to-end distance in the direction of the field. The velocity of the chain along the contour is given by the electric force divided by the hydrodynamic friction

cEh

vL

ρη

= (4.67)

Note that the chain might experience enhanced friction owing to the hydrodynamic interactions within the dense polymer network. The value of the viscosity can thus be higher than the one pertaining to the buffer.

We will now consider two extreme cases. In the first case the electric field is very weak and the conformation of the coil inside the gel remains Gaussian despite the electric driving force. By the time the DNA molecule has moved out of its original tube, the centre of mass has been displaced over a distance on the order of the size of the Gaussian coil 1 2 1 2

pR L L . Furthermore, since the coil is isotropic, the average projected end-to-end distance in the direction of the electric field equals the coil size, so that h R . The chain moves through the tube with an average velocity cv ER Lρ η and it has to travel a contour distance L in order to move out of its original tube and to form a new entanglement tube. The time it takes to move out of the original tube is given by 2

c cL v L ERτ η ρ . The electrophoresis velocity of the chain along the direction of the electric field is the displacement of the centre of mass of the coil divided by cτ , so that


2

2p

c

E LR E Rv

LL

ρρτ ηη

(weak field) (4.68)

For a weak field, the velocity is inversely proportional to the contour length. The velocity is also proportional to the strength of the electric field, so it might be tempting to increase the field strength in order to speed-up the procedure. However, one should bear in mind that the inverse contour length dependence of the electrophoresis mobility comes from the Gaussian chain statistics. A more intense electric field will induce alignment of the chains along the direction of the field, which, as we will see shortly, has a dramatic effect on the molecular weight resolving capacity of the electrophoresis experiment.

In the second extreme case, the electric field is so strong that the molecules become rapidly aligned with it. In this case, the DNA molecule moves straight through the gel with a velocity given by Eq. (4.67) with projected end-to-end distance h L= . The electrophoresis velocity

Ev

ρη

= (strong field) (4.69)

does not depend on the contour length and all molecules move with the same speed. This is of course an undesirable situation, as far as the separation of DNA’s with different molecular weights is concerned.

In the present analysis, we have not considered the global relaxation time τ pertaining to the thermal motion of the DNA molecules inside the gel. It is clear, however, that the molecules are able to rearrange their segments and become aligned with the direction of the field on a time scale on the order of τ (recall that aligned molecules have an electrophoresis mobility that does not depend on the molecular weight). An elegant solution to the alignment problem is to pulse the electric field on and off (or switch direction by 90 degrees) with a frequency 1τ− . In this case, the statistics of the chains remain close to Gaussian, which results in an optimal separation of the fractions according to the length of the molecules.


4.7 Questions

1. Derive scaling laws for the global relaxation time, diffusion coefficient and specific viscosity of a chain in a good solvent using Zimm’s approximation that the monomers in the ‘interior’ of the chain are screened from the flow.

2. Repeat question 1, but for a freely drained, Rouse chain. 3. Derive a scaling expression for the time it takes to pull a non-drained

polymer chain through a hole. 4. Derive a scaling law for the self-diffusion coefficient of a neutral polymer

chain in the semi-dilute regime using the reptation concept. How do the dependencies of the diffusion coefficient and the specific viscosity on the number of the number of links in the chain compare with the ones pertaining to the dilute regime?

5. What is the reason why the scaling laws for the salt-free polyelectrolyte

are fundamentally different from the ones pertaining to neutral polymers? 6. Explain why the transport properties of the salted polyelectrolyte have the

same functional dependencies on the number of links and the concentration as the ones for neutral polymers.

7. How does the overlap concentration of a polyelectrolyte solution depend

on the ionic strength? Do you expect a lower or higher overlap concentration than that for the neutral chain with a similar step length?

8. Explain how the transport properties of the salted polyelectrolyte depend

on the ionic strength of the supporting medium. Describe what happens to the self-diffusion coefficient and the specific viscosity if the salt concentration is increased.

9. Propose a possible experimental method to determine the number of

overlapping chains necessary to obtain entanglements.


10. What will happen to the entanglement concentration of DNA if the

ionic strength is increased from 10 to 200 mM? 11. In a gel electrophoresis experiment, DNA’s are separated according to

their molecular weight.

a. Show that in a weak electric field the mobility of the centre of mass of a DNA molecule through the gel is inversely proportional to its molecular weight.

b. Explain why an increase in electric field strength does not necessarily improve the separation according to the molecular weight.

c. Explain the benefits of the pulsed electrophoresis experiment and how the frequency of the electric field pulses should be optimized in order to separate larger DNA molecules.


CHAPTER 5

HIGHER ORDER STRUCTURES AND THEIR TRANSITIONS

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.

5.1 Supercoiled DNA

In organisms, DNA is often, if not always, in a closed, circular form. In order to form the ring, the ends of the DNA molecule may be covalently linked. Another possibility is that large sections of the DNA molecule are looped by the binding of a protein. A common feature is that the linked ends of the DNA molecule or section thereof are prevented from rotating with respect to each other, because the DNA molecule in its double helical form

Chapter 5: Higher Order Structures and Their Transitions 138

can support twisting. This topological constraint results in a supercoiled configuration, in which the duplex is wound around another part of the same molecule to form a higher order, tertiary coil structure (see Fig. 5.1 for a typical configuration). The plectonemic supercoiling, with a right-handed interwinding, is the most likely conformation in vivo.

An important implication of supercoiling is that it provides a mechanism for bringing distant DNA sections, which are separated over a large distance along the contour, close together spatially. It also provides a way to store energy, much like the winding of the spring in a spring-driven watch. These features are utilized in many cellular mechanisms that involve duplex winding and are important for regulating the reactions of DNA in the machinery of life, such as transcription, replication, recombination and repair. Supercoiling also results in compaction of the DNA coil. This issue is of importance for the accommodation of a large quantity of genetic material within a relatively small volume, such as inside the nucleoid of a bacterial cell. Furthermore, supercoiling is involved in regulating the secondary structure of the DNA molecule. Examples include local denaturation, the transition between the B and Z–form and the formation of cruciforms. For a review of supercoiling and its biological consequences, the reader is referred to Bates and Maxwell.43

5.1.1 Topology

The topological state of a closed circular DNA molecule is described by the linking number Lk . The linking number is the number of times one strand of the duplex passes around the other strand if one traces the DNA

Figure 5.1 Monte Carlo computer simulated conformation of a 1.3 kb supercoiled DNA molecule in 0.15 M of monovalent salt. The superhelical density σ = −0.05 and the writhing number Wr = −4.1. Note that the supercoil is branched. Lim, van der Maarel et al., unpublished.


molecule along its contour. By convention, Lk is positive for a right-handed configuration, as in DNA in its B–form. Furthermore, Lk is an integer, because both strands are closed in order to form the ring. If one of the strands is broken (nicked), the duplex cannot support twist anymore and the supercoil unwinds and becomes relaxed. For this relaxed, unconstrained state, we can define a linking number 0Lk N γ= , which is the number of links (base-pairs) divided by the pitch of the duplex γ . The pitch is the number of links per turn of the duplex; for instance for DNA in the B–form there are 10 base-pairs per turn. In general, 0Lk is not an integer and depends on environmental conditions through a dependence of γ on, for example, temperature and ionic strength.

The deviation from the relaxed state is expressed by the linking number deficit 0Lk Lk LkΔ = − , which is the number the duplex is turned before it is closed to form the ring. The linking number deficit is not an integer, because

0Lk is not an integer. However, the same molecule under the same environmental conditions, but in different topological states (topoisomers) can differ in LkΔ by an integer value only. Furthermore, the linking number deficit can only change by an integer through the activities of topology controlling, cutting enzymes. In the cell under homeostatic control of topoisomerase and gyrase, there is always a certain distribution of topoisomers with different linking number deficits. Once the DNA is isolated from the cell and/or in the absence of cutting enzymes, the linking number deficit LkΔ is a conserved quantity. It is convenient to normalize the linking number deficit to the linking number of the relaxed state and to define the superhelical density 0Lk Lkσ =Δ . Under physiological conditions, the DNA duplex is usually slightly under-wound with negative values of LkΔ and a typical value of the superhelical density is .0 05σ =− .

The linking number deficit is related to two quantities, which describe the configuration of the circular and closed DNA molecule. The first quantity is the excess twist exerted on the duplex and integrated along the contour of the double helix

12

Tw dsπ

Δ = Ω∫ (5.1)

The second quantity is the writhing number Wr , which reflects the 3–dimensional conformation of the molecule. For an arbitrary closed space


curve, the writhing number can be expressed by the Gauss double integral

( )2 1 12312

14C C

dr dr rWr

rπ×

×= ∫∫ (5.2)

where 1r and 2r are points on the curve C , 12 2 1r r r= − and 12 12r r= . A more perceptive definition is based on the number of signed crossings. In any projection of the space curve (that is the central axis of the duplex) onto a plane, there will be a number of crossings. To each crossing a sign is attached, depending on the relative orientation of the two crossing curves (under or over crossing). The sum of all these signed numbers is the projected writhing number. However, this projected number depends on the view that one takes; different views of the same curve may result in different projected writhing numbers. The writhing number Wr is defined as the average of all possible views of the projected writhing number.44

An analytical expression of Wr is generally complicated, but can readily be derived for some regular models. Of these models, the most important one is the right-handed (negative) plectonemic supercoil (see Fig. 5.2). From computer simulations and experimental studies it can be shown that the double-stranded, supercoiled DNA molecule takes this plectonemic configuration at a local length scale on the order of the radius r and the pitch 2 pπ . For a right-handed plectoneme without end loops, the writhing number is simply proportional to the number of crossings n when viewed perpendicular to the superhelical axis sinWr n α=− (5.3)

with the pitch angle α as defined in Fig. 5.2. Furthermore, it is convenient to

2π p

2r

α Figure 5.2 Local configuration of the plectonemic helix of radius r , pitch p and opening angle α .


define the normalized length 2 plecL L , with plecL being the total length projected on the central axis of the supercoil (superhelical axis) and L denotes the length of the DNA molecule measured along the contour of the double-stranded duplex. From integration along the contour, it follows that

( )1 22 2

2 plecL pL p r

=+

(5.4)

(end loops are neglected). The pitch angle α is then given by

( )( )

tan1 22

2

1 2

plec

plec

L Lpr L L

α = =−

(5.5)

and the writhing number reads

( )2 22Lp

Wrp rπ

=−+

(5.6)

The local structure of the so called plectoneme is fully characterized by the lengths p and r . These two parameters determine the opening angle α , the writhing number Wr and the normalized length projected on the superhelical axis 2 plecL L . Typical experimental values of the normalized length and opening angle are .2 0 8plecL L = and 55α = ° , respectively.

The two quantities describing the configuration of the closed and circular DNA molecule, that is the excess twist exerted on the duplex TwΔ and the writhing number Wr , are related to the linking number deficit LkΔ through White’s equation44 Lk Tw WrΔ =Δ + (5.7)

This relation is the key for our understanding of supercoiled DNA. In the absence of the activity of cutting enzymes (topoisomerases), LkΔ is a conserved quantity. The linking number deficit, however, is distributed over the two configurational quantities in a way that depends on the molecular free energies associated with the excess twist and the 3–dimensional conformation of the superhelix. For isolated supercoils, immersed in a saline buffer solution, a rule of thumb is that around 75% of the linking number deficit goes into the writhing number and 25% is adsorbed into excess twist. However, the interactions of the supercoiled DNA molecules among themselves, with proteins and/or the scaffolding of the cell can result in a conversion of Wr


into TwΔ and vice versa, but the sum of these two quantities is always constant (provided LkΔ does not change). The conversion of twist into writhe by a change in 3–dimensional conformation of a closed and circular ribbon is illustrated in Fig. 5.3.

5.1.2 Molecular free energy

In order to understand the structure and thermodynamics of topologically constrained DNA, it is necessary to know the molecular free energy plecF associated with the supercoiling. An experimental method is based on the analysis of the equilibrium distribution of topoisomers, determined by gel electrophoresis.43 The equilibrium distribution can be obtained from the treatment of closed circular DNA with a topoisomerase. The relative population of the topoisomers is then given by the Boltzmann distribution

( )( )

exp plecF LkP Lk A

kT

⎡ ⎤Δ Δ⎢ ⎥Δ = −⎢ ⎥⎣ ⎦

(5.8)

where ( ) ( ) ( )0plec plec plecF Lk F Lk F LkΔ Δ = Δ − Δ = denotes the increase in molecular free energy with respect to the reference 0LkΔ = state and A is a normalization constant. For moderate values of the superhelical density, the main contribution to the free energy comes from the elastic twisting deformation of the duplex. As we will see shortly, the latter contribution is

ΔTw = -2, Wr = 0 ΔTw = 0, Wr = -2

Figure 5.3 Conversion of twist into writhe of a ribbon with linking number deficit LkΔ = −2 .


quadratic in the excess twist exerted on the duplex (which is proportional to LkΔ ). The distribution of the band intensities, which is proportional to the

relative topoisomers populations, is accordingly observed to be Gaussian

( )( )

( )exp

2

22

LkP Lk A

Lk

⎡ ⎤Δ⎢ ⎥Δ = −⎢ ⎥⎢ ⎥Δ⎣ ⎦

(5.9)

Experiments showed that the variance

( )22N

LkK

Δ = (5.10)

is linear in the DNA length N for DNA’s with ,2 000N > base-pairs.43 The value of the constant K depends on the ionic strength and is in the range 1,000-1,600, with the highest values pertaining to minimal screening conditions.45 From a comparison of Eqs. (5.8) and (5.9) it follows that the phenomenological relation is

( )( )2

plecLk

F Lk K kTN

ΔΔ Δ = (5.11)

Note that the excess free energy of plectonemic supercoiling is quadratic in the linking number deficit and thus in the superhelical density. Furthermore, this relation has been obtained for topoisomers with low absolute values of the superhelical density .0 01σ < , whereas a typical value of σ for DNA in the native state is around .0 05− . As we will see shortly, detailed theoretical analysis shows a deviation from the quadratic dependence of LkΔ once the supercoil becomes more tightly interwound.

For the theoretical analysis of the molecular free energy, we will closely follow the approach of Marko and Siggia.9,46 The gist of this theory is that, on average, the structure of the closed and circular DNA molecule is a regular, right-handed superhelix. On top of the regular structure, thermal fluctuations are considered. The theory includes a superposition of three contributions to the molecular free energy associated with supercoiling. The first contribution comes from the elastic bending and twisting energy of the DNA duplex, the second contribution entails the electrostatic interaction between the opposing strands in the superhelix and the third, entropic contribution involves the confined displacements of a given point on the supercoil in the longitudinal (along the superhelical axis) and radial directions. These three contributions to the molecular free energy can then be minimized by numerical means in


order to determine the total excess free energy associated with the supercoiling and the equilibrium values of the structural parameters.

Elastic free energy. The elastic free energy per unit length of the duplex is the sum of the bending and twisting contributions

2 21 12 2

elasp c t

FL L

L kTκ= + Ω (5.12)

where the persistence lengths associated with bending and twisting deformations are denoted by pL and tL , respectively. The value of the twisting persistence length is not accurately known, but it is commonly thought to be around twice the value of the bending persistence length. The curvature cκ can be expressed in terms of the radius and pitch according to

2 2cr

p rκ =

+ (5.13)

and the excess twist, Ω , exerted on the duplex can be eliminated with the help of White’s Eq. (5.7)

( )22 Lk WrTwL L

ππ Δ −ΔΩ= = (5.14)

Note that, for a given superhelical density, the elastic free energy of a regular plectoneme depends on the 3–dimenional structure through the radius and pitch only.

Electrostatic free energy. DNA is a polyelectrolyte and the two opposing strands in the superhelix interact with each other through a screened electrostatic potential. Due to the intricate 3–dimensional conformation, the calculation of the electrostatic potential and the corresponding free energy contribution is complicated. To make headway, we will first assume that a test strand of the supercoil interacts with one opposing strand in the transverse direction at a distance 2r and two neighbouring strands in the longitudinal direction at a distance of half the superhelical pitch pπ (see Fig. 5.2). The strands are thus thought to be effectively straight on the scale of the pitch. For screened electrostatics, the reduced electrostatic potential of a rod-like strand is given by ( ) ( )02 eff BR l K Rφ ν κ= (5.15)

(see Sec. 3.2). Now, consider a charge element dsρ at position s on another, parallel strand at a distance R away from the first strand. The electrostatic


energy of this charge element is then given by eff dsν φ (in units kT ). The total electrostatic energy of these two parallel strands, each of length 2L and separated at a distance R , then follows from integration of s along the opposing strand

( )

( )2 40 2

04

2 Leff Beleceff B

L

l K RFds l K R

L kT Lν κ

ν κ−

= =∫ (5.16)

A test strand of the supercoil interacts with one opposing strand at a distance 2r and two neighbouring strands in the longitudinal direction at distances pπ . According to superposition of these interactions, the expression of the

electrostatic free energy per unit length takes the simple form

( ) ( )[ ]20 02 2elec

eff BF

l K r K pL kT

ν κ πκ= + (5.17)

Although this equation gives valuable insight into the electrostatics of the plectonemic helix, for quantitative calculations a more precise expression, including the effect of fluctuations, is necessary.

As shown by computer simulation studies, there are significant thermal fluctuations of a given point of the supercoil around its average position and the instantaneous structure is far from regular (for a typical conformation generated with a Monte Carlo computer simulation see Fig 5.1). However, these fluctuations are confined, because of the overall plectonemic structure (classical path) imposed by the topological constraint. Ubbink and Odijk argued that the electrostatic interaction restrains the fluctuation in the radial distance between the two opposing strands in the superhelix to amplitude rd with rd r< rather than r .47 For two opposing polyelectrolytes at mean separation 2r and fluctuating in inter-strand distance with amplitude 2 rd , the electrostatic interaction energy has to be renormalized by a leading factor

( )exp 2 22 rdκ . A more detailed example of fluctuation enhanced electrostatic interaction will be discussed in Chapter 6. Along the plectonemic axis, there is no enhancement of the potential, because the longitudinal fluctuations are supposedly not restrained by electrostatics and have an amplitude on the order of pπ . A more precise approximation of the electrostatic free energy, which includes the effect of fluctuations in radial displacement, reads

( )exp1 2

2 2 212 2

2elec

eff B rF

l d r ZL kT r

πν κ κ

κ⎛ ⎞⎟⎜= −⎟⎜ ⎟⎜⎝ ⎠

(5.18)


. . ,2

2 2

0 207 0 0541

4

pZ

rμ

μ μ= + + = (5.19)

This approximate expression has been derived with the assumption that the DNA strand behaves like a uniformly charged, undulating rod and that the inner double layers do not overlap. Note the large exponential renormalization factor if the fluctuations in radial displacement are on the order of the screening length 1rdκ .

Entropic free energy. The confinement of the DNA strand to a classical path results in a reduction of entropy. In particular, cut-offs on the fluctuation spectrum are set so that a given point on the coil has radial displacements on the order of rd and displacements along the longitudinal axis on the order of pπ (recall that the electrostatic interaction restrains the radial fluctuations to rd rather than r ). The reduction of entropy upon confinement within a ‘diameter’ D close to the classical path can generally be described by the deflection length 2 3 1 3

pD Lλ . The free energy of entropic confinement of a worm-like chain of length L is then simply proportional to the number of deflection length segments ( )confF L kTλ (see Sec. 2.6). Here, there is confinement over two length scales, a tube of diameter rd for the transverse displacements and another tube of diameter pπ for the longitudinal displacements. For a superposition of these two modes of confinement, the corresponding free energy per unit length reads

( )1 3 2 3 2 31 3

conf pr

p r p

F ccL kT L d L pπ

= + (5.20)

Unfortunately, the numeric constants rc and pc are unknown. For a worm-like chain confined in a harmonic potential, the coefficient of the confinement free energy takes the value 8 33 2 (in one dimension).47,48

However, it is not clear whether we can adopt the same value for the supercoil and we will treat rc and pc as adjustable parameters.

The total free energy associated with plectonemic supercoiling is the sum of the elastic [Eq. (5.12)], electrostatic [Eq. (5.18)] and entropic [Eq. (5.20)] contributions plec elas elec confF F F F= + + (5.21)

These contributions are expressed in the parameters r , rd and p . The equilibrium values of these parameters can be determined by numerical


minimization of Eq. (5.21) for a given superhelical density 0Lk Lkσ =Δ and screening length 1κ− . The other parameters, which are normalized length, pitch angle and writhing number, can then be calculated with Eqs. (5.4), (5.5) and (5.6), respectively.

Experiments are often done in 1×TBE (90 mM Tris-boric acid, pH 8.3, 2 mM Na2EDTA) buffer. It is accordingly of interest to calculate the free energy of plectonemic supercoiling under similar screening conditions. It is however, not an easy task to determine the ionic strength of TBE buffer, but an accurate estimate is 50 mM.49 The corresponding minimized free energy per unit contour length versus the square of the superhelical density is displayed in Fig. 5.4. Also displayed is the (extrapolated) phenomenological free energy Eq. (5.11) with 1,125K = . The value of K has been obtained by interpolation of the results of Rybenkov et al. to 50 mM ionic strength and agrees with earlier reported estimates of around 1,100.43,45 It is clear that the theoretically predicted free energy with 8 33 2r pc c= = for a worm in a one-dimensional harmonic potential shows too steep a dependence of 2σ , compared with the phenomenological relation. Matching initial slopes of the

0 0.002 0.004 0.006 0.008 0.010

0.1

0.2

0.3

0.4

0.5

σ2

F ple

c/L

kT

Figure 5.4 Free energy per unit contour length versus the square of the superhelical density in 50 mM of a monovalent salt (TBE buffer conditions). The dashed line represents Eq. (5.11) with K = 1, 125 . Dashed-dotted and solid lines: values from the minimization of the molecular free energy with pL = 55 nm, tL = 100 nm. The dashed-dotted and solid lines have been

calculated with r pc c= = 8/33/2 and 8/31/2 , respectively.


molecular free energy for low values of σ are obtained if the theoretical calculation is done with optimized coefficients of confinement

8 31 2r pc c= = . Furthermore, it is observed that the quadratic dependence of the superhelical density is preserved for 0.05σ < . This is however not the case for lower supporting salt concentrations (results not shown) and the use of relation Eq. (5.11) can lead to erroneous results even if the value of K is adjusted according to the prevailing ionic strength conditions. More experimental support for the adjusted coefficients of confinement will be presented below when the stress induced B–Z transition of alternating pyrimidine/purine inserts is discussed.

The optimized plectonemic radius is displayed in Fig. 5.5 as a function of the superhelical density. Also displayed are the range in fluctuations, as given by the optimized values of rd , along with electron microscopy data reported by Boles et al.50 The electron microscopy specimens were first immersed in 100 mM ammonium acetate for 20 s followed by a 5 s wash in 10 mM ammonium acetate. The values for the plectonemic radius and the fluctuation ‘band width’, calculated with a 20 mM supporting electrolyte concentration, are in excellent agreement with the experimental data. With an increase in absolute value of the superhelical density, the supercoil becomes more tightly interwound with a concurrent decrease in radius. Eventually, the radius becomes on the order of the thickness of the double layer surrounding the DNA duplex (see Fig. 3.14 for the ionic strength dependent effective duplex diameter). Relatively open supercoils, with large values of the plectonemic radius, are obtained for moderate values of the superhelical density only.

The importance of electrostatic interactions is clearly illustrated in Fig. 5.6. Here, the optimized plectonemic radius at a fixed superhelical density

.0 05σ =− is displayed as a function of the salt concentration. Also displayed are experimental values obtained in small angle neutron scattering studies of pUC18 plasmid dispersed in monovalent salt solutions.51 The theoretical values are in good agreement with the neutron scattering results. With increased screening of the electrostatic interactions, the plectonemic radius decreases and the supercoil becomes more tightly interwound. For salt concentrations exceeding, say, 100 mM, the electrostatic interactions become progressively less important and the structure of the supercoil is primarily determined by the elastic and entropic contributions to the free energy.


0 0.02 0.04 0.06 0.08 0.10 0.120

5

10

15

20

− σ

r (n

m)

Figure 5.5 Plectonemic radius versus superhelical density in 20 mM salt. Solid line: values from the minimization of the molecular free energy with pL = 55 nm, tL = 100 nm. The dashed lines represent the range of

fluctuations. Symbols are electron microscopy results by Boles et al.50

101

102

1033

4

5

6

7

8

9

10

r (n

m)

ρ (mM)s

Figure 5.6 Plectonemic radius versus concentration of monovalent salt. Solid line: values from the minimization of the molecular free energy with σ = −0.05 and tL = 100 nm. The salt concentration dependence of the bending persistence length was taken from Fig. 3.11. Symbols are SANS results of pUC 18 plasmid (2686 bp) reported by Hammermann et al.51


0 0.02 0.04 0.06 0.08 0.10 0.120.5

0.6

0.7

0.8

0.9

1

−σ

2L

/L

plec

Figure 5.7 Normalized length 2 plecL L versus superhelical density in 20 mM salt. Solid line: values from the minimization of the molecular free energy with pL = 55 nm, tL = 100 nm. Symbols are electron microscopy results.50

0 0.02 0.04 0.06 0.08 0.10 0.120.3

0.4

0.5

0.6

0.7

0.8

0.9

1

− σ

Wr/Δ

Lk

Figure 5.8 Writhe per added link Wr LkΔ versus superhelical density in 20 mM salt. Solid line: values from the minimization of the molecular free energy with pL = 55 nm, tL = 100 nm. Symbols are electron microscopy results.50


From the minimization of the molecular free energy, the writhing number and normalized length of the supercoil can also be obtained. Figure 5.7 displays the normalized length versus superhelical density, together with the electron microscopy results.50 The experimental value of the normalized length is constant and takes the value 2 plecL L = . .0 81 0 03± for plasmids with a superhelical density in the range . .0 12 0 02σ− < <− . From the theoretical analysis follows a slightly higher, but almost constant value of around 0.9. Furthermore, the electron microscopy specimens show a writhe per added link Wr LkΔ around 0.75 with a tendency to decrease with increasing absolute values of the superhelical density (that is 75% of the linking number deficit goes into writhe, the remaining 25% is adsorbed into excess twist exerted on the duplex). The latter experimental results are also in reasonable agreement with the values obtained from the minimization of the molecular free energy (see Fig. 5.8).

5.1.3 Long-range structure and branching

So far, we have considered the structure of the supercoil at a local distance scale on the order of the pitch. At a larger distance scale, the interwound supercoil can be considered a worm-like chain set out by the plectonemic axis with an effective diameter 2plecD r= , a length plecL and a bending persistence length sin2plec pP L α= (the Kuhn length is sin4 pL α ). The persistence length is the projection of the bending persistence lengths of the two opposing DNA strands onto the plectonemic axis. For a normalized length .2 0 8plecL L = , the pitch angle α of the superhelix takes the value 54° [Eq. (5.5)]. With a duplex persistence length of 55 nm in excess salt, the persistence length of the plectoneme is around 90 nm. However, as we will see shortly, the linear worm-like chain model is only applicable for very short DNA’s or DNA loops of less than around 800 base-pairs. For larger DNA’s, the plectonemic supercoil takes a randomly branched conformation as illustrated in Fig. 5.9.

That the supercoil must branch can be inferred from the free energy of the plectonemic interwinding. The branch spacing plecλ is the length for which the gain in entropy balances the energy loss for the formation of a branch point. Most, if not all branch points are tri-functional. If there are Bn tri-


functional branch points in the superhelix, then there are 2 1Bn + segments with mean spacing plecλ . Since the total plectonemic length is given by plecL , one obviously has plecL = ( )2 1B plecn λ+ . For a supercoil with a large number of branch points, the average plectonemic length per branch point is

2plec B plecL n λ (the factor of two comes from the tri-functionality of the branch points). Suppose now that, in order to form a branch point, a length of the superhelix on the order of the pitch has to be converted to random coil. A plectonemic length p corresponds to a contour length of the double-stranded DNA molecule ( )1 22 22 p r+ [Eq. (5.4)]. The energy of plectonemic supercoiling plecF is given per unit contour length of the duplex, so that the cost in energy of the formation of the branch point takes the form

( )1 22 22 plecB

FU p r

L= + (5.22)

A single branch point can be formed at 2 plec pλ independent places (recall that the average plectonemic length per branch point is 2 plecλ ). Accordingly, the gain in entropy of the formation of the branch point is

ln2 plec

BS kpλ⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

(5.23)

The spacing plecλ then follows from balancing the gain in entropy and the cost in plectonemic energy B BU TS , which implies

λplec

Figure 5.9 A branched plectonemic supercoil with 6 branch points and 13 superhelical segments. Note that the branch points are not fixed in space, but they can slide along the plectonemic axis.


( ) ln1 22 2 2

2 plec plecFp r kT

L pλ⎛ ⎞⎟⎜+ ⎟⎜ ⎟⎟⎜⎝ ⎠

(5.24)

Numerical evaluation of Eq. (5.24) with the free energy of plectonemic supercoiling plecF from minimization of Eq. (5.21) results in 0.66 branch points for every 1,000 base-pairs of a supercoil with a superhelical density

.0 05σ =− ( 55pL = nm, 100tL = nm) in 20 mM of a monovalent salt. This figure is remarkably close to the experimental 0.40 branch points per kbp for relatively small plasmids in the range 3.5–7 kbp.50

According to experimental studies, which are supported by the theoretical analysis, the mean spacing between the branch points comprises about 800 base-pairs (that is one tri-functional branch point per 2.5 kbp50). With a normalized plectonemic length .2 0 8plecL L , the mean spacing between the branch points is around 350 nm. This value is on the order of two Kuhn segments, because the persistence length of the supercoil is close to 90 nm. The spacers are rather flexible, but they are too short to exert an excluded volume effect. Their statistics are, accordingly, Gaussian with a radius of gyration

2 13 plec plecR Pλ λ (5.25)

Note that the spacers are rather short and that for quantitative calculations it is essential to use the full Benoit–Doty Eq. (2.35) for the radius of gyration. Furthermore, the branch points are not fixed, but they can slide along the plectonemic axis. A realistic model of the supercoiled DNA molecule is a randomly branched worm-like chain with an annealed (i.e. delocalized) branch point structure.

Following Zimm and Stockmayer,52 the unperturbed radius of a randomly branched chain with a number of tri-functional branch points 25Bn > is given by

1 2 1 220g plec plec plecR P Lλ (5.26)

It is interesting to compare this expression with the one pertaining to the linear chain. The unperturbed radius of gyration ( 0R for short) of a linear chain shows a square root rather than a 1 4 power dependence of the contour length [see Eq. (2.37)]. Randomly branched polymers are accordingly more compact than their linear equivalents with the same number of segments.


In reality, the branched plectoneme is swollen due to the long-range excluded volume effect. In order to estimate this effect, we will first ignore annealing of the branch points and derive the radius of gyration from the minimization of the sum of the elastic free energy

2

20

gelas RFkT R

(5.27)

and the free energy of repulsion of the plectonemic Kuhn segments

( )23

rep plec plec

g

F B L PkT R

(5.28)

(see Sec. 2.4).53 The excluded volume parameter B describes the volume excluded by one plectonemic Kuhn segment to another and is related to the diameter and the Kuhn length of the supercoil

( )2 222 plec plec plec plecB P D P Dπ

= (5.29)

From the minimization ( ) 0elas rep gF F R∂ + ∂ = it follows that the scaling relation of the radius of gyration of the randomly branched supercoil is

2 5 2 5 1 52g plecplec plec plecR D P Lλ , ( 25Bn > ) (5.30)

For comparison, the scaling expression of the radius of gyration of the hypothetical linear plectoneme reads 2 5 2 5 6 52

g plec plec plecR D P L . Since the length of the spacer is much shorter than the total plectonemic length, the randomly branched and swollen plectoneme is more compact than its hypothetical linear equivalent.

Note that the radius g plecR Lν of the swollen and randomly branched supercoil has the same scaling exponent .0 5ν = as the one for a linear Gaussian chain. The effects of annealing of the branch points can be included in the mean field approach and gives the exponent .0 54ν = , whereas a Monte Carlo computer simulation of the annealed and branched polymer yields .0 49ν = .54 The latter value is close to the estimation based on the mean field approach without considering annealing effects and seems to be the most reliable value to date. Finally, one should note that the derived expression for the radius of gyration of the branched supercoil is strictly valid for plectonemes with more than, say 25 branch points, which corresponds to DNA’s of more than 62.5 kbp.


5.2 Alternate secondary DNA structures

So far, we have discussed the physical properties of the DNA molecule in the framework of the worm-like chain model. In this model, the specific base-pair sequence is ignored, besides the effect of the sequence on the average bending and twisting elasticity constants. The base-pair sequence is, of course, very important from a biological point of view. Besides carrying the genetic information, the chemical make-up of the nucleotides controls the secondary structure of the duplex. Under physiological conditions, DNA is usually in the B–form, but certain specific base-pair sequences can exhibit alternate secondary structures. These structures include the left-handed Z–form and cruciforms in inverted repeat, palindrome sequences. Both the B–Z transition and the extrusion of a cruciform reduce the linking number deficit and hence the twist exerted on the duplex if the molecule (or section thereof) is closed and circular. Accordingly, these alternate secondary structures are often stress induced and the thermodynamics are strongly related to supercoiling. Besides from stress exerted on the duplex, these transitions can be triggered by, e.g. ligand binding and chemical modification of the bases as well as by environmental factors such as ionic strength and temperature.

Cooperative transitions to alternate DNA secondary structures bear resemblance to the helix-coil transition. The cooperativity implies that the transition first nucleates, after which the alternate secondary structure propagates along the backbone of the biopolymer. It is often thermodynamically more favourable to continue an existing block of a different secondary structure rather than to start a new one. However, an important difference from the helix to coil transition is that the extrusion of the cruciform and the B to Z transition occur in specific base-pair sequences only. The locations and the involved number of base-pairs are thus bound and fully determined by the primary structure. We will first discuss the B–Z transition, after which the method will be applied to understand the thermodynamics of cruciform formation in palindrome sequences.

5.2.1 B–Z transition

The canonical motif for the left-handed Z–DNA is the alternating pyrimidine/purine d(pCpG) d(pCpG)i sequence. However, the Z–structure


is confined neither to strictly alternating nor to exclusively d(pCpG) d(pCpG)i sequences and mismatched base-pairs can be accommodated.55 The transition is sensitive to a variety of factors, such as methylation of cytosine, modification of the solvent composition, ionic strength and/or the presence of multivalent ions in small quantities. For instance, in the case of poly[ d(pCpG) d(pCpG)i ] at room temperature, the Na+, Mg2+ and Co3+(NH3)6 concentrations required to drive the transition are 2,500, 700 and 0.03 mM of the chloride salts, respectively. If the cytosine is methylated, these concentrations are shifted downwards to 700, 0.6 and 0.003 mM. These observations are puzzling and suggest that, besides electrostatics, specific interactions of nucleotides, water molecules and ions are needed in order to explain the phenomena.55 However, it is possible to understand the effect of ionic strength and/or multivalent cations on the basis of the Poisson–Boltzmann theory, using cylindrical models of DNA whose principal property is that they reflect the jutting out and the better immersion of the B–DNA phosphate charges into the supporting medium.56 We will not further discuss the various conditions for which the Z–structure is stabilized, nor will we focus on the mechanism of the transition itself.57 We merely discuss the energetics of the transition as induced by negative supercoiling.58 This analysis will also serve as an introduction to the statistical-mechanical modelling of the cooperative helix-coil transition in biopolymers. The latter transition will be treated later in this chapter.

Consider a supercoil with a single track of alternating pyrimidine/purine m md(pCpG) d(pCpG)i base-pairs. We take the two bp d(pCpG) d(pCpG)i

unit as the repeating unit, so that a total of 2m base-pairs can flip from the B– to the Z–form. The supercoil is underwound with a negative linking number deficit 0LkΔ , where 0⋅⋅⋅ denotes the reference state with all base-pairs in the B–form. Within the track of m units, we assume that there is a single block of n m≤ units which flip from the 10.4 bp-per-turn, right-handed B–form to the 12 bp-per-turn, left-handed Z–helix. The supercoil becomes less underwound and the linking number deficit increases (becomes less negative) according to ( )0 2Lk Lk na bΔ = Δ + + (5.31)

Here, .2 10 4 2 12a = + denotes the increase in LkΔ per flip of a 2 bp unit from B to Z. The parameter b is included to account for the change in the


linking number deficit at the junction between the B– and the Z–domain. The concomitant change in free energy is given by ( ) 2 2BZ plecF n n F W FΔ = Δ + Δ +Δ (5.32)

where BZFΔ is the free energy required to interconvert a single GC base-pair from the B– to the Z–form. The free energy change associated with the creation of one junction between a B– and Z–domain is denoted by WΔ . Since the m md(pCpG) d(pCpG)i track is inserted in a supercoil, there is a supplementary change in free energy associated with the change in linking number deficit ( ) ( )0plec plec plecF F Lk F LkΔ = Δ − Δ (5.33)

In early work, the phenomenological free energy of plectonemic supercoiling Eq. (5.11) was used.58 We will obtain plecF from the elaborate polymer theory Eq. (5.21), including elastic, electrostatic and entropic contribution to the molecular free energy. With Eq. (5.32), the partition function is then given by

( ) ( )

( ) ( )

exp

exp

1

1

1 1

1 1

m

n

m nplecn

Z m n F kT

m n s F kTσ

=

=

= + − + −Δ =

+ − + −Δ

∑∑

(5.34)

where ( )exp 2 BZs F kT= − Δ and ( )exp 2 W kTσ = − Δ . For historical reasons, we use for the cooperativity parameter the symbol σ (not to be confused with the superhelical density). The measured change in linking number deficit can now be calculated by taking the thermal average

( )( ) ( )exp

0

1

12 1

m nplecn

Lk Lk

na b m n s F kTZ

σ=

Δ − Δ =

+ − + −Δ∑ (5.35)

Note that, in the present situation, the B–Z transition is driven by the change in plectonemic free energy plecFΔ associated with the change in linking number deficit. Furthermore, we have assumed that there is only a single block of base-pairs that flips from the B– to the Z–form. As we will see shortly, this assumption is justified in view of the relatively small number of units and the high energy required for the formation of the junction. For longer inserts and/or different types of alternate secondary structure, which require less energy for the formation of the junction, it is necessary to account for the presence of multiple alternate blocks within a single track. This can be done


with the more general formalism presented in Sec. 5.3. The supercoiling induced B–Z transition has been studied in a 4.4 kbp

plasmid with m md(pCpG) d(pCpG)i , m = 21, 16, 12 and 8 inserts.58 The linking number deficit was measured with 2–dimensional gel electrophoresis under TBE buffer conditions. Figure 5.10 shows the measured change in linking number deficit 0Lk LkΔ − Δ versus the linking number deficit of the reference state 0LkΔ . The curves are calculated according to Eq. (5.35) with a single set of optimized parameters .0 7BZF kTΔ = (0.4 kcal/mol),

.8 2W kTΔ = (4.9 kcal/mol) and .0 4b = . Overall, excellent agreement between the data and the theoretical prediction is observed. Note that the energy values are slightly different from the ones reported in the original literature,58 because we used the full theoretical expression of the molecular free energy of supercoiling rather than the phenomenological relation obtained for small absolute values of the superhelical density. The supercoiling

10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

-<ΔLk> 0

<Δ

Lk>

0<Δ

Lk>

−

Figure 5.10 Supercoiling induced B–Z transition in a 4.4 kbp plasmid with m md(pCpG) d(pCpG)i inserts. The number of Z–forming units is m = 21, 16,

12 and 8, from top to bottom. The lines represent the fit of Eq. (5.35) with the energy parameters as discussed in the text. The free energy of plectonemic supercoiling is calculated with pL = 55 nm, tL = 100 nm, 50 mM ionic strength (TBE buffer conditions) and r pc c= = 8/31/2 . Experimental data are from Ref. [58].


induced B–Z transition also provides a critical test of our previously estimated value of the coefficients of confinement 8 31 2r pc c= = . With the value of the coefficient of confinement pertaining to a worm-like chain in a harmonic potential ( 8 33 2 ), no satisfactorily agreement with the data can be obtained.

The B–Z transition is highly cooperative, as shown by the steep dependence of 0Lk LkΔ − Δ on 0LkΔ . The initiation of a Z–block generates two B/Z junctions with a free energy cost of 16.4 kT , whereas the lengthening of the block by one d(pCpG) d(pCpG)i unit only requires 1.4 kT . The unfavourable free energy cost of initiation with respect to the one of prolongation of a block of base-pairs in the Z–form is responsible for the highly cooperative transition. It also renders the probability for the occurrence of more than one Z–block inside the Z–forming track vanishingly small. The optimized value of b shows that there is substantial unwinding of the duplex of about 0.4 turn at each B/Z junction.

Besides thermodynamics, an important issue is the kinetics of the B–Z transition. The transition in both directions is usually first order, reversible and rather slow. The relaxation time varies four orders of magnitude between seconds and hours, depending on sequence, length and topological state of the Z–forming insert as well as environmental conditions such as ionic strength and temperature.55

5.2.2 Cruciforms

Another type of transition in the secondary structure of double-stranded DNA is the extrusion of a cruciform in an inverted repeat, palindrome base-pair sequence (see Fig. 5.11). DNA cruciforms are recognized by specific enzymes and proteins and are thought to play a role in recombination, replication and transcription.59 The cruciform structure is stabilized by negative supercoiling, because the palindromic base-pairs, which are popped out in the two arms of the cruciform, do not contribute to the linking number deficit of the main chain. Accordingly, the extrusion of the cruciform is supercoiling-induced and is accompanied by a substantial reduction of the absolute value of the superhelical density. Besides the change in free energy associated with a change in linking number deficit, the energetics are coupled to the unpairing of a number of bases in the two hair-pin loops (and any other


extrahelical and/or mismatched bases in the case of imperfections) as well as the formation of the four-way Holliday junction.60

The energetics of the extrusion of the cruciform can be analyzed in a similar manner as the B–Z transition. Nucleation involves the formation of the Holliday junction and the unpairing of a number of base-pairs in the hair-pin loops at the end of each arm. The bases at the Holliday junction are thought to remain paired, so that the associated change in free energy is small with respect to the free energy of the formation of the single-stranded hair-pin loops. The energy of a base-pair which resides in one of the arms of the cruciform is the same as the energy of the corresponding base-pair in the initial, linear configuration of the molecule. Accordingly, propagation of the cruciform does not require free energy and the arms will fully pop-out resulting in a maximum decrease in absolute value of the linking number deficit. Imperfections, such as the presence of some extrahelical or mismatched base-pairs, can inhibit full extrusion of the cruciform. At 310 K, the average free energy of a single base bulge is 10 kJ/mol, whereas the

Holliday junction

hair-pin loop

Figure 5.11 Twist-induced extrusion of a cruciform in an inverted repeat, palindrome base-pair sequence. Notice the unpaired bases in the hair-pin loop at the end of each arm.


extrusion of a perfect cruciform requires 72 kJ/mol.60 These figures are consistent with around four unpaired bases in each hair-pin loop and no or little free energy associated with the formation of the Holliday junction. In a 4.4 kbp supercoiled plasmid with a perfect 68 base-pairs palindrome sequence and an absolute value of the superhelical exceeding 0.03, the cruciform is the stable secondary structure.61 However, imperfections have major effects on the energetics of cruciform extrusion, requiring much higher absolute values of the superhelical density in order to drive the transition.

Although the cruciform can thermodynamically be the stable secondary structure, the kinetics of its formation can nevertheless be extremely slow, unless the DNA is subjected to conditions that destabilize base-pairing.61

5.3 Helix-coil transition

Biopolymers often exhibit a helix to coil transition, if a monomeric unit can exist in two distinct states, that is a helix and a coil state. Examples are the nucleotides of DNA and the amino acids of a protein. The DNA molecule usually exists as a double helix, but at elevated temperature the helix can melt and the two strands of the double-stranded DNA molecule dissociate into two single-stranded coils (see Fig. 5.12). This process is reversible (and forms the basis of the polymerase chain reaction, PCR experiment), but, for longer DNA’s it takes a very long time, if ever, before the native form is recovered. A similar phenomenon occurs for polypeptide (protein) chains, but here a single chain or part of a chain can exist in an alpha-helix or coil state. The helix-coil transition is clearly not an on/off problem; the melting usually starts in a nucleation region, from which the coil domains progressively grow. For a link at a junction between a helix and coil domain it is apparently easier to dissociate than for a link in a helix domain itself. In other words, propagation of an existing coil domain is thermodynamically more favourable than nucleation of a new coil domain. An adequate thermal theory describing the helix-coil transition should capture this cooperativity effect.

A statistical theory of the helix-coil transition has been formulated by Bragg and Zimm.62 The gist of this theory is that a link can exist in a helix (h ) or a coil (c ) state. In a helix domain of a polypeptide chain, hydrogen bonding occurs along the backbone of the chain between monomers separated by


approximately four units. In the case of double-stranded DNA, the bases form pairs through hydrogen bonding between adenine and thymine (AT) and guanine and cytosine (GC). These hydrogen bonds, together with stacking of the bases, make the h state energetically more favourable than the c state. As far as the entropy is concerned, the c state is more favourable. A link in the coil domain has more configurational (translational and rotational) degrees of freedom and higher entropy, compared to a link in the helix domain.

The melted links at the junctions between the h and c domains represent a special case. With respect to the c state, these links lose their configurational entropy with no energy gain due to the formation of hydrogen bonds and/or base-pair stacking. Accordingly, their energy is comparable to the energy of the links in the coil domain, whereas their entropy is close to the value of the entropy pertaining to the helix links. Due to the restriction in configurational degrees of freedom, at the junction the entropy is small and the corresponding links carry a large positive free energy. The respective energies and entropies of the links in the helix, coil and junction domains are illustrated in Fig. 5.13.

The thermodynamic behaviour of the chain can be described by the Bragg–Zimm model. In this model, the free energy associated with a particular sequence of h and c monomers of the chain is given by

melting

annealing

Figure 5.12 Melting and annealing of double-stranded DNA with increasing and decreasing temperature, respectively.


2conf h dF N F N W= Δ + Δ (5.36)

where hN denotes the number of links in the h state, dN is the number of helix domains, h cF F FΔ = − is the excess free energy per link in the helix state (the coil state is taken as the reference state) and WΔ is the cost in free energy to create a junction separating h and c domains. The Bragg–Zimm model is characterized by two parameters

exp Fs

kT⎛ ⎞Δ ⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠

(5.37)

and the cooperativity

exp 2 WkT

σ⎛ ⎞Δ ⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠

(5.38)

The free energies per link in the helix and coil state, hF and cF , respectively, are schematically depicted in Fig. 5.14. At low temperature the helix state is more stable and 0FΔ < . At high temperature the coil state is stabilized, because the gain in entropy of the melted links outweighs the cost of energy to break the hydrogen bonds and to unstack the links, 0FΔ > . At the melting temperature *T the excess free energy *F U T SΔ = Δ − Δ equals zero and thus *T U S= Δ Δ ( 1s = ).

In order to derive the fraction of monomers in the helix state as a function of the parameters s and σ it is necessary to derive the free energy of the chain with a thermally averaged distribution of the h and c monomers. The latter free energy can be obtained with the transfer operator technique.4 The transfer operator describes the change in the partition function with fixed end

Energy of Link Entropy of Link

helix helix

coil coiljunction

junction

U S

Figure 5.13 Energy (left) and entropy (right) of a link in the coil, junction or helix state.


points, that is the Green function, upon the addition of a link in the h or c state. Because every link has only two states (h or c ), the transfer operator is a two by two matrix

ˆ ˆ

ˆˆ ˆcc ch

hc hh

Q QQ

Q Q

⎛ ⎞⎟⎜ ⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠ (5.39)

where ˆln ijkT Q− is the change in free energy by adding a link with state j to a link in state i . Suppose that link i is in the c state. If one adds another link in the c state, the free energy does not change (the coil state is the reference state) and no junction is formed, i.e. ˆ 1ccQ = . Now, we add an h link to a c link. In this case, the free energy is increased by F WΔ + Δ , because of the addition of a link in the h state (with excess free energy FΔ ) and the formation of a junction (with free energy WΔ ), so that ˆ

chQ = ( )( )exp F W kT− Δ +Δ . If the initial link i is in the h state, adding a c

link results in an increase in free energy with WΔ only, due to the formation of a junction and ( )ˆ exphcQ W kT= −Δ . Finally, for adding an h link to another h link, no junction is formed and the free energy increases with FΔ , so that ( )ˆ exphhQ F kT= −Δ . In terms of the Bragg–Zimm parameters s and σ , the transfer operator takes the form

ˆ 1 sQ

s

σσ

⎛ ⎞⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜⎝ ⎠ (5.40)

In the ground state dominance, the thermally averaged free energy of the chain with a total of N links reads

Low Temperature

helix

coil

FUHigh Temperature

helix

coil

FU

ΔF = Fh - Fc < 0Helix is more stable Coil is more stable

ΔF = Fh - Fc > 0

Figure 5.14 Free energy of a link at low (left) and high temperature (right).


ln 1F N kT x=− (5.41)

with 1x the largest eigenvalue of the transfer matrix Q :

( )( )( )1 221

11 1 4

2x s s sσ= + + − + (5.42)

The average number of monomers in the helix state hN can now be derived by differentiation of the free energy Eq. (5.41) with respect to the excess free energy per link in the helix state lnF kT sΔ =− . We accordingly obtain

( ) ( )( )1 22

1 12 2 1 4

hF s

N NF s sσ

⎡ ⎤⎢ ⎥∂ −⎢ ⎥= = +⎢ ⎥∂ Δ − +⎢ ⎥⎣ ⎦

(5.43)

The fraction of h links hN N has been set out in Fig. 5.15 as a function of s and for two different values of σ . Note the characteristic sigmoidal shape and the narrowing of the melting trajectory with increasing free energy associated with the formation of the junction (that is increasing cooperativity and smaller values of σ ).

If the formation of the junction requires no free energy at all ( 0WΔ = and 1σ = ), the h links are randomly distributed along the chain and their

10−2

100

102

0

0.2

0.4

0.6

0.8

1

s

<N

>

/N

h

Figure 5.15 Fraction of links in the helix state versus the Bragg–Zimm parameter s for cooperativity σ = 0.001 (dashed) and 1 (solid).


fraction should satisfy a Boltzmann distribution pertaining to a two-level system separated by energy FΔ

( )( )

expexp1 1

h F kTN sN F kT s

−Δ= =

+ −Δ + , ( 1σ = ) (5.44)

In the absence of cooperativity ( 1σ = ), the transition from the coil to the helix regime is broad and extended over a large range in s . Due to the large positive free energy carried by the links at the junction, a typical experimental value of the cooperativity parameter is around 0.001. With increasing cooperativity (smaller values of σ ), the transition from the coil to the helix state becomes sharper and the melting trajectory has a characteristic width given by 4s σΔ .

For 0σ > , there is always a range in temperature in which the helix and coil domains are coexistent and the helix-coil transition is in general, not a first order phase transition. If the formation of the junction requires an infinite amount of free energy ( 0σ = ), helix and coil domains within a single chain cannot coexist and the chain is either completely in the coil ( 1s < ) or helix ( 1s > ) state, depending on the value of FΔ . A first order phase transition is recovered for 0σ = at the melting temperature *T ( 1s = ), because chains in the full coil and helix states have the same free energy and are coexistent.

0

50

100

Incr

ease

in a

bso

rban

ce a

t 26

0 n

m (%

)

20 40 60 80 100 120

T (Celsius)

Tm

AT rich

GC rich

Figure 5.16 The melting of the DNA helix as monitored by UV absorption. A chain rich in GC base-pairs is more stable.


In a DNA melting experiment, the temperature at which the two strands separate (melting temperature) depends on the AT versus GC base-pair ratio (see Fig. 5.16). GC and AT base-pairs are kept together by three versus two hydrogen bonds, respectively, so it takes relatively more energy to dissociate a GC base-pair. The melting temperature is also dependent on ionic strength. With increasing salt concentration and thus increased screening of Coulomb interaction, the melting temperature shifts to higher values and the trajectory narrows. For DNA in the B–form, the phosphate charges jut out in the surrounding solvent, where their electrostatic interactions are screened by the salt. With the increased screening of the electrostatic repulsion between phosphates on the same DNA molecule, the double helix is stabilized and the melting temperature shifts to a higher value. Furthermore, it becomes more difficult to separate the strands and to form a junction between a helix and coil domain, leading to an increase in cooperativity and lower value of σ .

5.4 Protein folding

Proteins owe their biological function to their unique 3–dimensional tertiary structure. Yet, proteins are synthesized as linear chains of amino acid monomers on ribosomes. As a consequence, each newly synthesized protein chain or section thereof, has to adopt its native conformation, which also has to remain stable under working conditions. Both the kinetic folding pathway and the stability of the native state are determined by the amino acid sequence. Relatively small proteins usually take their native structure spontaneously. For longer sequences, the folding process may be assisted by helper proteins (chaperones), thereby avoiding catastrophic, misfolded states and/or amorphous aggregation before completion of the sequence (folding ‘on the fly’). The 3–dimensional structure of a protein is determined by the interactions of the amino acid residues among themselves and the surrounding medium. In particular, the aqueous solvation properties are of paramount importance in stabilizing the native, folded state. Understanding de novo protein folding remains difficult, but in recent years progress has been made by the establishment of experimental techniques that can monitor the folding process on a sub-millisecond time scale together with the development of energy surfaces, which allow the folding process to be described and


visualized in a meaningful manner. We will summarize the main concepts from a physical point of view, for a more detailed discussion the reader is referred to dedicated reviews available in the literature.63,64,65

One might naively expect that a protein folds into its native state by a random search of all possible conformations. In order to illustrate that a random search is not feasible, let us assume that each link connecting two amino acid residues can be in a finite number of configurations (e.g. three typical sets of values of the two peptide bond angles). A protein composed of 101 monomeric units can then exist in the astronomical number of 3100 conformations, of which only one (or a limited set) is the native state. Suppose that one conformation is generated in a short time of one nanosecond (i.e. a high sample rate of 910 conformations per second), it will take . 311 6 10× years to sample them all. Since proteins typically fold on a time scale of a minute or less, Levinthal recognized that the search cannot be random and should be biased.66 A bias can be introduced by imposing an energy cost for an incorrect link. In our example with three possible link configurations, we can assume that the correct configuration has the lowest energy and the other two are degenerated with a higher energy. The probability to find the correct link configuration is then given by Boltzmann’s law [ ]( )exp 11 2p E kT −= + − , with E the energy penalty for an incorrect link. For a sequence of N links,

0 0.5 1 1.5 2 2.5 310−10

100

1010

1020

1030

1040

ΔE/kT

τ (s

)

Figure 5.17 Mean first passage time in seconds as a function of the energy bias E in units kT [Eq. (5.45) with 100N = and sampling rate 910r = ]. The dashed line demarcates one second.


it requires an average Np− searches to find the correct conformation (for each link an average 1p− searches are needed). With a sampling rate of r conformations per second, a good estimate of the mean first passage time from an incorrect conformation to the perfect one is then given by67

α

βα/β intermediate

α domain intermediate

native state

unfolded state

F

Q

Q

Figure 5.18 Free energy surface representing the folding of hen lysozyme into its native conformation. The number of native contacts in the α and β domains are denoted by Qα and Qβ , respectively. The yellow trajectory denotes the fast track in which α and β domains form concurrently, whereas the red trajectory represents the slow track in which the protein becomes temporarily trapped in an α domain intermediate state. Reprinted with permission from Ref. [64]. Copyright Elsevier (2000).


[ ]( )exp1 2 NNr p r E kTτ −= = + − (5.45)

Note that this approximation does not depend on the initial conformation. Even if the search is started close to the perfect state, there is a high probability that it first wanders off before the native state is found. Furthermore, it should be noted that Eq. (5.45) merely serves to illustrate the effect of a biased search; for a real protein every link can of course have different energies. The effect of bias on the mean first passage time is displayed in Fig. 5.17. Indeed, when a modest amount of bias is introduced by imposing an energy cost of a few times thermal energy kT for an incorrect link configuration, the time to achieve a fully correct state becomes very much shorter on the order of the biological time scale.

Owing to the astronomical number of possible conformations with comparable free energies, it is not plausible that the protein folds along a single, specific pathway. Experimental evidence at the individual residue level has shown that there are often, if not always multiple routes from the random coil to the native state. Accordingly, a multi-dimensional free energy landscape, which is a folding funnel, is thought to be a better description of the folding process. It is however, not feasible to consider each possible conformation of the protein explicitly, because of the vast number of conformations. One rather employs a reduced set of progress variables, which characterizes the deviation from the native state of an ensemble of conformations with the same or similar free energy. Examples of these variables are the relative number of native and non-native contacts between residues and/or the relative number of native contacts in predominantly α helical or β sheet domains. The folding process is then described and visualized as one or more trajectories through the multi-dimensional free energy landscape spanned by the progress variables. The selection of a specific pathway is often kinetically, rather than thermodynamically driven, because of the small differences in free energies between different states.

An illustrative example is the folding of hen lysozyme. Hen lysozyme is a small protein consisting of 129 residues with two structural domains, one largely α helical and another one with significant regions of β sheet (α and β domain, respectively). Painstaking experimental work has shown that there are significantly populated intermediate states in which native-like structures are present in localized regions of the compact state.68 Furthermore, it was


found that there are at least two predominant trajectories through different intermediate states and with different folding rates.69,70,71 There appears to be at least a slow and a fast track with time constants of 400 and less than 100 ms, respectively. These two folding pathways are indicated by the red and yellow trajectories in the schematic free energy landscape pertaining to hen lysozyme in Fig. 5.18.

Folding is initiated by a solvent quality induced collapse from a random coil to a compact, globular state. This collapse reduces the number of possible conformations tremendously, but the globular state is still disordered with a very small number (if any) of native contacts. The diversity in folding trajectories is attributed to differences in the disordered structure of the molten globule and to differences between the various intermediates which are formed during the folding process. Along the yellow trajectory, the α and β domains form concurrently by cooperative transitions as described in the previous section and the free energy decreases almost monotonously towards the native state. This trajectory represents the fast route, which is taken by about 20% of the molecules. The majority of the molecules (70%) fold along the red trajectory, where the α domain intermediate is formed. Along the red trajectory the protein becomes temporarily trapped in a local free energy minimum. Thermal fluctuations and, possibly, partial unfolding of the α domain (as indicated by a slight reversal in the red trajectory) are necessary to overcome the energy barrier and to progress to the final step of the folding process in which eventually the two domains are docked into the native state.64 Note that the native state is often, if not always, an ensemble of close conformations with comparable free energies. Owing to thermal fluctuations, a single protein continuously samples these states, so that a unique molecular conformation pertaining to the native state cannot be defined.

5.5 Questions

1. DNA is often, if not always, circular and closed. Its topological state is defined by the linking number deficit LkΔ , i. e. the number of times the duplex is turned about its long axis before it is closed to form a ring.


a. Explain how a circular DNA molecule with a non-zero LkΔ can relieve the excess twist TwΔ exerted on the duplex without changing its topological state.

b. Give at least two biological implications of supercoiling. c. At a longer distance scale, the supercoil can branch. Explain why

the occurrence of branching increases with molecular weight. 2. Derive expressions for the normalized length, pitch angle, writhing

number and curvature for the regular plectonemic superhelix. 3. Calculate the plectonemic radius, normalized length and writhe per

added link as a function of the superhelical density with a twisting persistence length of the duplex 100tL = nm in 2 mM of a monovalent salt.

4. The genome of the Escherichia coli bacterium is thought to be a single,

plectonemic supercoil with a duplex contour length 1.6 mm. For the sake of simplicity, we assume a superhelical density .0 05σ =− , a normalized length .2 0 8plecL L = and that the genome is bathed in 8 mM of a monovalent salt.

a. Calculate the total plectonemic length and the number of tri-

functional branch points if there is one branch point per 2.5 kbp. b. Calculate the number of Kuhn segments and estimate the radius of

gyration of the genome. How does the estimated value of the radius of gyration compare to the hypothetical value pertaining to a linear plectoneme?

c. The average cell volume of the bacterium is around one cubic micrometre. Does the unconstrained genome fit inside this volume? If not, describe possible mechanisms for further compaction.

5. The cytoplasm of a high copy number bacteria contains multiple copies of

the same supercoiled and randomly branched ( 25Bn > ) plasmid.

a. Derive an expression for the concentration *c inside the cytoplasm at which the plasmids overlap and form a semi-dilute regime.


b. Derive a scaling law for the DNA contribution to the osmotic pressure in the semi-dilute regime based on the notion that the pressure should become independent on the plectonemic length.

c. At very high plasmid concentration c , the mean inter-plasmid distance becomes shorter than the mean distance between the branch points. Derive an expression for c and the scaling law for the osmotic pressure in the range c c> . Hint: in this high concentration range the pressure should become independent on both plecL and plecλ .

d. How do the scaling laws for the osmotic pressure compare to the ones for a linear polymer chain?

6. Consider a 4.4 kbp plasmid with a Z–forming 24 24d(pCpG) d(pCpG)i

insert. The plasmid is in TBE buffer and has bending and twisting persistence lengths 55pL = and 100tL = nm, respectively. a. Calculate the minimum value of the linking number deficit in order

to drive the B–Z transition of the insert. The free energy associated with plectonemic supercoiling can be estimated using the phenomenological relation Eq. (5.11).

b. Repeat a), but use the full expression for the free energy of plectonemic supercoiling, including elastic, electrostatic and entropic contributions.

7. Consider a supercoiled 4.4 kbp plasmid with a 69 base-pair inverted

repeat, palindrome sequence. The plasmid is under TBE buffer conditions with bending and twisting persistence lengths 55pL = and

100tL = nm, respectively. Calculate the minimum value of the linking number deficit in order to extrude the cruciform. The free energy associated with plectonemic supercoiling can be estimated using the phenomenological relation Eq. (5.11).

8. Derive an expression for the average number of junctions 2 dN in the

Bragg–Zimm model for the helix-coil transition. Explain the behaviour of this expression as a function of the parameters s and σ .

9. Why is the helix-coil transition not a first-order phase transition?


10. Explain why the melting temperature of DNA increases with increasing

GC content and/or increasing ionic strength. Explain how the width of the melting trajectory responds to these changes.

11. Estimate the mean first passage time with four non-degenerated link

configurations with an energy spacing E .


CHAPTER 6

MESOSCOPIC STRUCTURES

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.

6.1 Lyotropic liquid crystals

With increasing packing fraction, anisotropic colloidal particles dispersed in a solvent often exhibit a first order transition from an isotropic to a liquid crystalline, anisotropic phase. Examples are (locally) rod-like biopolymers, such as DNA and xanthan, tobacco mosaic and filamentous bacteriophage fd viruses and disk-like clay platelets. In the liquid crystalline nematic and cholesteric phases, there exists long-range order in the orientation, but short-

Chapter 6: Mesoscopic Structures 176

range, liquid-like order in the position of the colloidal particles. Liquid crystalline phases with long-range position order in at least one dimension (e.g. hexagonal and smectic phases) usually exist at higher densities. As an example of the latter category, we will discuss the hexagonal DNA phase later in this chapter. In the liquid crystal, the average orientation of the particles, or segments thereof, is indicated by the director D . If the phase is nematic, the orientation of D is uniform. However, many biopolymers such as DNA have a chiral molecular architecture, which results in a cholesteric structure with a helical rotation of the director about the cholesteric axis. For biomacromolecules, the pitch of this rotation (cholesteric pitch) is often, if not always, on the order of hundreds to thousands of nanometres. The cholesteric pitch is thus much larger than the average distance between the dispersed particles and results in some typical optical properties of the liquid crystal. Examples of these properties are the fingerprint polarized light microscopy textures and Bragg diffraction of laser light (see Fig. 6.1). It is by no means an easy task to formulate a theory for the long-range cholesteric structure based on the chiral hard-core, electrostatic and solvent mediated interactions. However, at a molecular distance scale a cholesteric polymer phase can be considered nematic. For properties related to the molecular structure and interactions, the effect of chirality can be ignored. We will describe the isotropic-nematic phase transition and the orienation order in the liquid crystalline state in terms of orientation entropy and excluded volume effects of semi-flexible worm-like chains.

Long ago, Onsager recognized that the isotropic-nematic phase transition of a suspension of stiff and slender particles can be treated with a virial expansion of the free energy.72 If the particles are very long, the transition occurs at very low volume fractions and it is sufficient to carry the virial expansion up to and including the second order term. For shorter rod-like particles, it is necessary to go beyond the second virial approximation. The incorporation of higher order terms is unfortunately not well developed and existing theories are not quantitatively correct.73 The free energy expansion of the suspension of worm-like polymers includes the balance of the free energy of translational motion, orientation entropy and excluded volume of the Kuhn segments. At low concentration, the orientation entropy is maximized for an isotropic distribution. For higher concentrations, the system will become nematic with a concurrent decrease in orientation entropy (increase


in orientation order) and decrease in excluded volume. As we will see shortly, Onsager’s theory predicts a first order transition between an isotropic and a liquid crystalline phase in a certain polymer concentration range. The critical boundaries pertaining to this phase transition follow from mechanical (osmotic) and chemical equilibrium of the coexisting phases. A key factor of the interplay between orientation entropy and excluded volume is the effective diameter of the polymer, which depends on the range of the electrostatic interactions and can be modified by the ionic strength of the supporting medium.

6.1.1 Virial theory

We will first consider the free energy of a suspension of rigid and charged rods.74,75 As we will see shortly, these results can be generalized to include the effect of chain flexibility through a modified expression for the chain

Figure 6.1 Left panel: Biphasic isotropic-cholesteric sample of 150 bp DNA in 0.01 M putrescine chloride observed through crossed polarizers. The higher density cholesteric phase is birefringent. The DNA concentrations in the coexisting isotropic and cholesteric phases are 118 and 144 g of DNA/L, respectively. Top right panel: Polarized light microscopy fingerprint texture of the cholesteric phase ( 50× objective). Bottom right panel: Laser light (635 nm) diffraction from the cholesteric phase. From both the polarized light microscopy and laser light diffraction follows a cholesteric pitch of 3.7 micrometres. Unpublished results of Wei and van der Maarel.


orientation entropy. Consider a solution of N rods of length L and bare diameter 0D . The charged rods interact electrostatically as well as through a hard core repulsion. Furthermore, we assume that the rods are bathed in a monovalent salt solution, so that we can treat the electrostatic interaction in the Debye–Hückel approximation (see Sec. 3.5). The electrostatic energy of two interacting charged rods separated at a distance R and skewed at an angle φ can be expressed as

( )expsin

e w RUkT

κφ−

= (6.1)

with

2 12 eff Bw lπν κ−= (6.2)

The N rods are dispersed in a volume V , so that the rod concentration c N V= . Let us now define the rod orientation distribution function ( )f Ω , which gives the probability of finding a rod (or rod-like segment) with an orientation given by the solid angle Ω . The distribution function must be normalized

( ) 1d fΩ Ω =∫ (6.3)

In the isotropic phase, all orientations are equally probable, so that

( )14

fπ

Ω = (6.4)

In the anisotropic phase ( )f Ω is peaked with a maximum in the direction of the director D (in the case of biaxial cylindrical symmetry ( )f Ω has two maximums in opposite directions). In the second virial approximation, the free energy of the suspension of rods can be written as a function of ( )f Ω and takes the form

( ) ( )( )

( ) ( ) ( )

ln ln

1 2 1 2

4

12

Fc d f f

NkT

c d d f f

π

β φ

Δ= + Ω Ω Ω

+ Ω Ω Ω Ω +⋅⋅⋅

∫

∫ ∫ (6.5)

Here, the first term represents the free energy of translational motion of the rods. The second term represents the loss in orientation entropy


( ) ( )( )

( )( )

ln

ln

4

4

or

or

S Nk d f f

Nk f Nk

π

π σ

=− Ω Ω Ω =

− Ω =−

∫ (6.6)

where the brackets denote the orientation average (i.e. the loss in orientation entropy per chain is orkσ− ). The third term is the free energy of rod interaction in the second virial approximation

( ) ( ) ( ) ( )1 2 1 21 1 12 2 2c d d f f c cβ φ β φ βΩ Ω Ω Ω = =∫ ∫ (6.7)

which is proportional to the excluded volume ( )β φ of the two interacting rods skewed at an angle φ and averaged over all orientations (see Fig. 3.12). As has been shown in Sec. 3.5, the excluded volume can be split into a hard core and an electrostatic contribution

( ) ( ) ( )0 eβ φ β φ β φ= + (6.8)

The hard core excluded volume reads

( ) sin20 02L Dβ φ φ= (6.9)

and the electrostatic contribution is given by

( ) ( )( )' sin

sin exp2 1 1

02 1

w

e L dx x xφ

β φ κ φ− −= − −∫ (6.10)

For ( )' exp 0w w Dκ= − larger than, say, 2, the electrostatic excluded volume has the asymptotic form

( ) ( )ln ' ln sin sin2 12e L wβ φ κ γ φ φ−= + − (6.11)

with Euler’s constant .0 57721γ = . For sufficiently strong interactions ( ' 2w > ), the orientation averaged excluded volume can thus be expressed as

( )( )ln ' ln sin sin2 1 10 02 1L D D wβ κ γ φ φ− −= + + − (6.12)

We will now further evaluate the free energy of the suspension of charged rods for an isotropic and an anisotropic, liquid crystalline phase, respectively.

In the isotropic phase the orientation distribution is constant ( ( ) 1 4f πΩ = ) and the orientation averages are readily done

sin4iπ

φ = (6.13)

sin ln sin ln 12

4 2iπ

φ φ⎛ ⎞⎟⎜− = − ⎟⎜ ⎟⎜⎝ ⎠

(6.14)


Owing to the isotropic distribution, the orientation entropy takes its maximum value 0or orS Nk σ=− = (6.15)

and from Eq. (6.12) we recover Eq. (3.97) for the isotropically averaged excluded volume

2

2i effL Dπ

β = (6.16)

with the effective diameter of the rod

( )ln ' ln10 1 2 2effD D wκ γ−= + + − + (6.17)

Note that the effective diameter depends on the screening length 1κ− as well as the effective number of charges per unit length through the parameter 'w . With the second virial coefficient

22 4 effA L D

π= (6.18)

we can express the free energy pertaining to the solution in the isotropic state as

ln 2iF c A c

NkTΔ

= + +⋅⋅⋅ (6.19)

Note that the second virial coefficient has been derived for interacting rods with infinite length. If the effects of the finite length are included, the second virial coefficient takes the modified form76

22

41

4eff

effD

A L DL

π ⎛ ⎞⎟⎜= + ⎟⎜ ⎟⎟⎜⎝ ⎠ (6.20)

A semi-flexible worm-like chain can be considered as a sequence of rod-like Kuhn segments. The second virial coefficient Eq. (6.18) is determined by local pair interactions of two rods. Since two worm-like chains interact locally as rods, the effect of chain flexibility on the second virial coefficient is expected to be negligible. For semi-flexible worm-like chains we can use the same expression for the excluded volume as the one for rigid rods. Since in the isotropic state the free energy does not include orientation entropy (the orientation entropy is constant and arbitrarily set to zero owing to the isotropic distribution), we can also neglect the effect of chain flexibility on the solution free energy.


In the liquid crystalline phase, we have to consider the (unknown) anisotropy in the orientation distribution function. Here. it is convenient to define the orientation averages

sin4aρ φ

π= (6.21)

sin ln sin ln4 12

2aη φ φ ρπ

⎛ ⎞⎟⎜= − − − ⎟⎜ ⎟⎜⎝ ⎠ (6.22)

For the free energy, we can then write74

( )ln 2a

orF

c A h cNkT

σ ρ ηΔ

= + + + +⋅⋅⋅ (6.23)

where ( )ln 4or afσ π= (6.24)

and

1

effh

Dκ= (6.25)

In the isotropic phase 1ρ = and 0η = and with 0orσ = we recover from Eq. (6.23) the free energy of the isotropic state Eq. (6.19). In the anisotropic phase, the angular dependence of the electrostatic interaction is expressed by the term 2A h cη . The parameter h is accordingly called the twist parameter. In practice, the value of h is often less than 0.15 and the effects of twist on the phase boundaries and the liquid crystalline orientation order are usually small. Note that if the twisting effect is neglected altogether, one recovers the expression of the free energy of a solution of neutral rods, but with an enhanced effective diameter as determined by the ionic strength of the supporting medium through Eq. (6.17).

The second virial (pair-wise) interaction contribution to the free energy of an anisotropic suspension of semi-flexible, worm-like chains is the same as the one for rigid rods. However, this does not apply to the term orσ related to the orientation entropy per chain ( ork σ− ). As we will see shortly, orσ is very sensitive to the flexibility of the chain, as expressed by the contour over the persistence length ratio p pN L L= .

In order to find the free energy of the anisotropic phase, one has to determine the anisotropic orientation distribution function ( )f Ω with the


normalization requirement ( ) 1f dΩ Ω=∫ . This can be achieved using the method of Lagrange by adding ( )f dλ− Ω Ω∫ (with λ a Lagrange multiplier) to the free energy Eq. (6.23) and then minimizing the resulting expression with respect to ( )f Ω , i.e.

aFf NkTδ

λδ

⎛ ⎞Δ ⎟⎜ =⎟⎜ ⎟⎜⎝ ⎠ (6.26)

This procedure gives the integral equation

( )ln ' ln sin ln sin28 1

4 1 22

f A c hπ λ φ φπ

⎛ ⎛ ⎛ ⎞⎞⎞⎟⎟⎟⎜ ⎜ ⎜= − − + − ⎟⎟⎟⎜ ⎜ ⎜ ⎟⎟⎟⎜ ⎜ ⎜⎝ ⎝ ⎝ ⎠⎠⎠ (6.27)

where the constant ' 1λ λ= − follows from the normalization condition ( ) 1f dΩ Ω=∫ . An analytic solution of this integral equation has not been

found yet, but it has been solved numerically for worm-like chains without the twist effect 0h = .77 We will proceed with an alternative method, in which we will choose a normalized trial function and then minimize the free energy with respect to a variation parameter characterizing the distribution width. Once we know the orientation distribution and free energy pertaining to the liquid crystalline state, we can calculate the phase boundaries from the mechanical (osmotic) and chemical equilibrium conditions of the coexisting phases. However, first we will focus on the orientation order in the liquid crystalline phase.

6.1.2 Liquid crystalline orientation order

In order to make headway with an analytical approach, we need to make a choice for the functional form of the distribution in the angle θ describing the orientation of the rod with respect to the director of the liquid crystal. Onsager proposed the normalized function72

( ) ( )( )cosh cos

sinh4D D

Df f

α α θθ

π αΩ = = (6.28)

which works well for isotropic ( 1Dα = ) and anisotropic ( 1Dα > ) states and is yet simple enough for mathematical analysis. For a highly ordered state with

1Dα , Onsager’s trial function Eq. (6.28) takes the Gaussian limit

( ) ( ) ( )exp 2 212

2f f θ θ σ

πΩ = = − (6.29)


(besides an arbitrary normalization constant) and 2Dα σ−= . Under common

experimental conditions, both distribution functions are equal within, say, 2%, with the main difference in the flanks.

With Onsager’s trial function, the anisotropic orientation dependent parameters ρ and η can be expressed in the series expansions72,74,75

( )( )1 2

4 151

16DDD

ρ ααπα

⎡ ⎤= − +⎢ ⎥

⎢ ⎥⎣ ⎦ (6.30)

( )

( )( )[ ln ln

1 2

22 2 1

15 51

16 2

D DD

D D

η α α γπα

α α

= − − + ×

⎡⎛ ⎞ ⎤⎟⎜ − + + +⎢ ⎥⎟⎜ ⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

(6.31)

These expansions converge rapidly for sufficiently large values of the distribution width parameter Dα and only the terms up to and including

1Dα − are retained (higher order terms can be found in the original

literature).72,74,75 The term related to the chain orientation entropy orσ can be evaluated

with the Gaussian approximation of the orientation distribution function. For rigid rods, without any flexibility, one has ln 1or Dσ α= − (6.32)

Khokhlov and Semenov derived the chain entropy terms

ln 11

6D

or D pNα

σ α−

= − + + ( pL L ) (6.33)

( )ln 14D

or p DNα

σ α= + − + ( pL L ) (6.34)

for semi-flexible chains near the rod limit and for very flexible chains, respectively.78,79 In the case of semi-flexible chains of arbitrary length, one can use the approximate expression75,80

( )ln exp

ln cosh

1

1 5 16 12 5

or D D

D Dp pN N

σ α π α

α α

= − + − +

⎛ ⎛ ⎞⎞− − ⎟⎟⎜ ⎜+ ⎟⎟⎜ ⎜ ⎟⎟⎜ ⎜⎝ ⎝ ⎠⎠ (6.35)

which agrees with the asymptotic limits for very stiff and very flexible worm-like chains Eqs. (6.33) and (6.34), respectively. Note that orσ has an explicit contour length dependence as expressed by the number p pN L L= of


persistence length units per chain. The width of the distribution, as expressed by the variation parameter

1 2Dσ α−= , can be obtained from minimization of the free energy Eq. (6.23),

together with the expressions for ρ , α and orσ , that is by solving 0aF σ∂Δ ∂ = . In general, two roots for σ are obtained, but the higher value

has no physical meaning. The input parameters are 2A c and h . For rigid rods with ( ) ( ) 1 24D Dρ α πα −= , ln 1or Dσ α= − and 0h = , one readily obtains

1 2

2

12D A cπ

σ α−= = (6.36)

The width of the distribution describing the orientation of the rods in the liquid crystalline state is inversely proportional to the excluded volume, that is the inverse product of the concentration times the second virial coefficient ( ) 1

2A c− . For higher order approximations of ρ , non-zero values of h and

0 10 20 300

10

20

30

A c2

σ (d

egre

es)

L = 40 nmp

Figure 6.2 Distribution width 1 2

Dσ α−= versus the product of the DNA density times the second virial coefficient in cholesteric 150 bp DNA liquid crystals.81 From left to right, the symbols refer to salt condition/DNA volume fraction: 1.5 M NaCl/0.13; 0.75 M NaCl/0.13; 0.25 M NaCl/0.13; 0.08 M MgCl2/0.13; salt-free/0.10; salt-free/0.13; salt-free/0.13; and salt-free/0.15. The lines are calculated according to the minimization of the free energy. Dashed line: rigid rod model; solid line: semi-flexible chain model.


inclusion of chain flexibility effects, the minimization procedure has to be done with a numerical method.

The orientation distribution of 150 bp DNA in the cholesteric phase was obtained from the anisotropy in the scattering of neutrons.81 The contour length is about the persistence length, so these molecules can be considered as semi-flexible rods. In order to vary the excluded volume, experiments were done with different DNA packing fractions and/or ionic strengths of the supporting medium. As a result of the variation in ionic strength, the effective diameter of the DNA molecule was varied between 3 and 7 nm. The experimental results are displayed in Fig. 6.2. Contrary to what one might intuitively expect, the DNA molecules are relatively weakly ordered in their orientation. The distribution width 1 2

Dσ α−= depends on the excluded volume and varies from 30 degrees in high salt to about 15 degrees under minimum screening conditions. Also displayed in Fig. 6.2 is the width as obtained from the minimization of the free energy in the second virial approximation. The dashed curve represents the solution Eq. (6.36) pertaining to rigid rods. It is clear that the rigid rod model gives a too highly ordered phase, which is not consistent with the experimental results. The solid curve has been calculated including the orientation entropy of a semi-flexible chain with an optimized value .1 4pN = , which corresponds to a persistence length 40pL = nm. The latter value is in the range of reported values based on, e.g. single molecule stretching experiments (see Sec. 2.8.2).11 Note that the sample with the highest salt concentration (in 1.5 M of NaCl) is just above the upper critical phase boundary pertaining to the first appearance of the isotropic phase. The increase of the orientation distribution width upon approaching this critical boundary is theoretically reproduced.

6.1.3 Isotropic-anisotropic phase coexistence

The isotropic-nematic (cholesteric) transition is a first order phase transition, which implies that in a certain range of polymer concentration the solution separates into coexisting isotropic and anisotropic phases (see the image of the phase separated sample in Fig 6.1). The boundaries are commonly referred to as ic and ac , where ic is the concentration at which the anisotropic phase first occurs and ac denotes the concentration at which


the isotropic phase completely disappears. In order words, the dispersion is completely isotropic for ic c< and completely anisotropic for ac c> . In the biphasic range i ac c c< < , the solution is unstable and separates into two phases with concentrations ic and ac (volume fractions iφ and aφ , i i a ac c cφ φ+ = ), respectively. Now that we can solve for the free energy in

the liquid crystalline state by minimization with respect to the variation parameter 2

Dα σ−= , that is by solving 0a DF α∂Δ ∂ = , the phase boundaries can be obtained from mechanical (osmotic) and chemical equilibrium between the coexisting phases: i aΠ = Π (6.37)

i aμ μ= (6.38)

with osmotic pressure ( ) , , 0T NF V μΠ =− ∂Δ ∂ and chemical potential ( ) , , 0T VF N μμ = ∂Δ ∂ . With the expressions for the free energies in the

isotropic and anisotropic phases, Eqs. (6.19) and (6.23), respectively, we

10−2

10−1

100

101

0

0.1

0.2

Np

c a/c

i − 1

10−2

10−1

100

101

100

101

102

Np

A2c

i

0.3

Figure 6.3 Left panel: Isotropic number density at the phase transition 2 iA c versus the number of persistence lengths per chain p pN L L= . Right panel: Density difference at the phase transition versus pN . The curves are the exact results from the numeric solution of the integral Eq. (6.27) with

0h = .77


readily derive the coexistence equations ( ) ( )( )2 21 1i i a ac A c c A c hρ η+ = + + (6.39)

( )ln ln2 22 2i i a or ac A c c A c hσ ρ η+ = + + + (6.40)

Note that the coexistence equations can be solved for the dimensionless variables 2 iA c and 2 aA c , by arbitrarily multiplying the left and right-hand- sides of Eq. (6.39) with 2A and by adding ln 2A to the left and right-hand- sides of Eq. (6.40). In general, it is necessary to solve the coexistence equations and the minimization of the free energy by a numerical procedure.

For rigid and neutral rods, Onsager used a trial function to obtain the critical phase boundaries72

..

2

2

3 340

4 486i

a

A c

A c

==

(rigid rods) (6.41)

The boundaries are thus inversely proportional to the excluded volume. For rigid particles, the relative range of the biphasic region, which is the phase gap, is about 30%.

ρ (M)s

0.01 0.1 1.0

0.02

0.04

0.06

0.08

0.10

ρ , ρ

(g

/cm

)3i

a

Figure 6.4 Isotropic-cholesteric phase boundaries of xanthan ( 315L = nm, .1 32pN = ) versus NaCl concentration. The open symbols are the

concentrations at which the cholesteric phase first appears and the closed symbols denote the boundaries at which the isotropic phase completely disappears, iρ and aρ , respectively. The lines are calculated with electrostatic effects in the second virial approximation, but with a higher order approximation for the hard core reference system. Note the strong dependence of the phase boundaries on the ionic strength. Redrawn from Ref. [83].


Biopolymers are often, if not always, semi-flexible and charged. Accordingly, we need to explore the effects of flexibility and charge on the phase boundaries. Flexibility can be incorporated through the orientation entropy term Eq. (6.35) pertaining to the semi-flexible worm-like chain. The isotropic number density and the number density difference at the phase transition are displayed in Fig. 6.3. These values were obtained by solving the integral Eq. (6.27) without the twist effect ( 0h = ) numerically, rather than by using a trial function.77 With increasing flexibility, as expressed by the number of persistence length segments per chain p pN L L= , the critical boundaries shift to higher densities with a concurrent narrowing of the phase gap. For very flexible chains, the critical boundaries become proportional to p pN L L= and take the form

..

2

2

10 246

11 018i

a

A c L Lp

A c L Lp

==

( 5 pL L> ) (6.42)

In the phase separation thermodynamics, the effect of charge is accounted for by the effective diameter effD and twist parameter h . Numerical calculations show that the twist effect is generally moderate (experimental values of h are typically less than 0.15) and the use of effective diameters in the calculation of the second virial coefficient according to Eq. (6.18) often gives sufficiently accurate results. This does not imply that the effect of electrostatic interaction on the phase behaviour is insignificant; as can be seen in Fig. 3.14 there can be an extensive range in effective diameter depending on the ionic strength of the supporting medium. An experimental example is displayed in Fig. 6.4., which shows the critical boundaries pertaining to the isotropic-cholesteric phase transition of the polysaccharide xanthan in NaCl solutions.82,83 With increasing salt concentration, the boundaries shift to higher xanthan densities, mainly due to the decrease in effective diameter with increased screening of the electrostatic interaction. The phase diagram of 150 bp DNA dispersed in NaCl solutions is displayed in Fig. 6.5 and shows similar behaviour.84,85 Besides a isotropic-cholesteric phase transition, these short DNA fragments exhibit another phase transition from the cholesteric to the hexagonal phase at higher packing fractions. The latter transition will be discussed in the next section.

For both xanthan and DNA, the phase boundaries pertaining to the transition from the isotropic to the cholesteric phase are reasonably well


reproduced by the lyotropic liquid crystal theory, once the effects of chain flexibility and charge are taken into account. However, it should be noted that due to the relatively short length of these molecules, the virial expansion for the hard core reference system needs to be carried to higher orders. For both systems, this was accomplished with the scaled particle approach. For rod-like particles scaled particle theory includes the higher order virial coefficients in an approximate manner, but is unfortunately not quantitatively correct.73,86

00

10.5 1.5

0.1

0.2φ

ρ (M)s Figure 6.5 Phase diagram of DNA ( 55L = nm, .1 1pN = ).84 The open and closed circles are as in Fig. 6.4. The open and closed squares are the volume fractions at which the hexagonal phase first appears and the cholesteric phase completely disappears, respectively. The solid lines are the theoretical predictions of the cholesteric-isotropic phase transition. The dashed curve denotes the ionic strength dependence of the hexagonal melting side with a Lindemann criterion .0 113LC = .


6.2 Hexagonal packing of DNA

At higher packing fractions, DNA exhibits a transition from the cholesteric to the hexagonal phase. In the hexagonal phase, there is no cholesteric twist and the average orientation of the molecules is uniform (besides a possible long-range mosaic spread). The occurrence of the hexagonal phase can easily be recognized with a polarized light microscope; because their fan-like textures are markedly different from the fingerprint textures pertaining to the cholesteric phase (compare polarized micrographs in Figs. 6.1 and 6.6, respectively). The transition is weakly first order, as shown by, e.g. small angle X-ray and neutron scattering data and polarized light microscopy, with a relatively small difference in the concentrations in the coexisting phases.

Figure 6.6 shows the neutron scattering spectra (multiplied by momentum transfer in order to eliminate the typical 1q− intensity decay for rod-like particles) of biphasic cholesteric-hexagonal liquid crystals of 150 bp DNA. The spectra show the emergence of a strong and narrow peak superposed on a broader peak pertaining to the cholesteric phase with increasing overall concentration. The narrow peak is resolution limited with a 0.1 nm-1 full width at half maximum. Furthermore, the narrow peak is slightly shifted to higher values of momentum transfer with respect to the broader cholesteric peak, due to the slightly higher DNA packing fraction in the coexisting hexagonal phase. Also for long DNA molecules (about a hundred persistence lengths), two superposed diffraction peaks have been observed in the transition range from the cholesteric to the higher density phase.87 The relative width of the narrow diffraction peak and the six-fold azimuthal intensity profile indicate a hexagonal molecular arrangement with a range in position order extending over at least five to eight neighbours.88

The hexagonal phase has biological implications. It is the prevalent packaging mode of the genome under dense circumstances, such as inside the capsid of bacteriophages or other condensed DNA structures.89,90,91 Inside the capsid, the DNA is pressurized and the release of this pressure has been postulated to be the driving force for viral infection.92,93,94 Accordingly, it is of interest to present a theory for the equation of state (pressure versus volume fraction) of polymer liquid crystals including DNA.87,95 We will follow Odijk’s treatment for polyelectrolytes in a hexagonal lattice, because of its clarity and


predictive power regarding the equation of state and the melting of the hexagonal phase in combination with the Lindemann criterion.96,97,85 Furthermore, the latter approach nicely demonstrates the necessity to renormalize the electrostatic potential for fluctuations in the position of the DNA molecules, once their amplitudes become on the order of the electrostatic screening length (as we have done before for the electrostatic energy of the supercoil). It should be borne in mind that for the application of this theory local position order is all that is required.

60

80

100

120

140

0.5 1 1.5 2 2.5 3

0

20

40

3.5q (nm )-1

q I(

q) (

a.u

.)

Figure 6.6 Transition from the cholesteric to the hexagonal phase of 150 bp DNA in 0.1 M spermidine chloride as monitored by neutron scattering. The bottom and top intensities (multiplied by momentum transfer) refer to overall DNA concentrations of 365 and 385 g/L, respectively. Note the emergence of a slightly shifted narrow peak superposed on a broad peak with increasing overall DNA concentration. The peak positions and widths agree with coexisting cholesteric and hexagonal phases with interaxial spacings 3.22 and 3.10 nm, respectively. The inset shows a typical fan-like polarized light microscopy texture of the hexagonal phase. Unpublished results of Dai, Egelhaaf and van der Maarel.


6.2.1 Undulation enhanced electrostatic interaction

In the hexagonal phase the DNA molecules are position ordered in the transverse plane perpendicular to their average orientation. At even higher packing fractions, there can also be position order in the longitudinal direction, but this will be ignored in the present analysis. The hexagonal structure and the fluctuations about the average positions are demonstrated in a Molecular Dynamics computer simulation of nine DNA molecules in a 30°-inclined parallelogram simulation box (see Fig. 6.7).98 Periodic boundary conditions in both transverse and longitudinal directions ensure that infinitely long DNA molecules are simulated. The time-averaged structure in the transverse plane is clearly hexagonal, but each molecule fluctuates about its mean position with a root mean-square standard deviation u on the order of 10% of the interaxial spacing R . The aim of the present section is to explain the osmotic pressure and the melting transition of a dense hexagonal array of biopolymers (DNA) in terms of electrostatic screening and fluctuations in position order.

In the hexagonal lattice, a test chain is effectively confined in a narrow ‘tube’ with a diameter on the order of the undulation amplitude u (see Sec. 2.6). Note that, in this case, the tube is not made of a hard reflecting wall, but

1

0.75

0.5

0.25

0 4 8 12x (nm)

6

12

y (nm)

Figure 6.7 Density of DNA in the transverse plane as monitored during a 20 nanoseconds Molecular Dynamics simulation of 10 bp DNA (one helical pitch) with periodic boundary conditions in both the transverse and longitudinal directions. The grey scale is the fractional time a DNA molecule is located at a certain position per unit area. The mean interaxial spacing

4R = nm and the root mean-square fluctuation .0 34u = nm.98


it is imposed by the presence of the other DNA molecules. The test chain undulates inside the tube and will bend only when it bounces off its neighbours. In the transverse plane, the instantaneous relative position of an infinitesimal segment of the test chain with respect to its lattice position is given by the Cartesian coordinates ( ),x y (see Fig. 6.8 for a schematic drawing of the hexagonal unit cell). Since the fluctuations are supposed to be weak, we can postulate a Gaussian distribution

( ) ( )( ), exp 2 2 22

1G x y x y u

uπ= − + (6.43)

This distribution can be considered a Boltzmann factor with a quadratic, harmonic potential. For a worm undulating in two dimensions, we have previously derived the deflection length

2 3 1 3pu Lλ = (6.44)

and the free energy of confinement per unit chain length

2 3 1 3

conf u

p

F cLkT u L

= (6.45)

(x,y)

sx (s+y)2 2+( )

1/2

R

r =

Figure 6.8 Hexagonal unit cell with interaxial spacing R and surface area 23 2RS = . The reference lattice point is indicated by the open circle

and the Cartesian coordinates of the fluctuating test chain are denoted by ( ),x y .


(see Sec. 2.6). In the case of an harmonic potential, the coefficient of confinement takes the value 5 33 2uc = .47,48 Note that with increasing confinement (smaller values of the undulation amplitude u ), the deflection length decreases and the free energy of confinement increases.

In a hexagonal lattice of charged chains, the interaxial spacing R is typically much larger than the length scale of the electrostatic interaction

1Rκ (typical experimental values will be discussed in the next section). The inner double layers do, accordingly, not interpenetrate and we can describe the electrostatic interaction between the hexagonally ordered charged chains with screened electrostatics. The reduced potential is then given by ( ) ( )02 eff Br l K rφ ν κ= (6.46)

where r is the distance away in the radial direction from the rod-like test chain (see Sec. 3.4). For large rκ , we can use the far field approximation

( )( ) ( )

( )

exp1 2

1 2

2eff Bl rr

r

ν π κφ

κ

−= (6.47)

The radial distance r can be expressed in the radial distance s away from the reference lattice point according to

( )( )1 22 2r s y x= + + (6.48)

(see Fig. 6.8). Furthermore, we need to average over the fluctuations in the position of the test chain ( ),x y with the Gaussian distribution Eq. (6.43):

( ) ( ) ( )( )( ),1 22 2

R s dx dyG x y s y xφ φ κ∞ ∞

−∞ −∞= + +∫ ∫ (6.49)

We will now assume that the fluctuations are small and that we can expand the radial distance r up to and including the second order terms in the displacements x andy , so that

21

2x

r x ys

= + + + (6.50)

After some tedious algebra, we derive the renormalized expression of the electrostatic potential to the leading order

( )( ) ( )( ) ( )

exp1 2 2 2

1 21 2 2

2 4

1 2

eff BR

l s us

s u s

ν π κ κφ

κ κ

− +=

+ (6.51)


Note the potentially large fluctuation enhancement factor ( )exp 2 2 4uκ , if the amplitude of the fluctuation is on the order of the screening length

1uκ . Furthermore, note that Eq. (6.51) is the potential at a radial distance s away from the reference lattice point and that it has been renormalized for fluctuations in the transverse position order of the test chain.

Owing to the screening over a length scale much less than the interaxial spacing, a test chain will primarily interact with its six nearest neighbours at a mean radial distance R away from the reference lattice point. Since these are pair interactions, there are three interaction pairs per unit cell. The electrostatic interaction energy also needs to be renormalized for fluctuations in the positions of the interaction partners. We will assume that the position fluctuations are not correlated, so that we can average them independently. If ( ),x y are now the Cartesian coordinates of an interaction partner, we can express the instantaneous distance away from the reference lattice point as

( )( )1 22 2s R y x= + + (6.52)

in analogy with Eq. (6.48). For the electrostatic energy per unit length and per hexagonal unit cell, we need to do a second renormalization by averaging over the fluctuations in the transverse position of the interaction partners

( ) ( )( )( ),1 22 23elec

eff RF

dx dyG x y R y xLkT

ν φ κ∞ ∞

−∞ −∞= + +∫ ∫ (6.53)

which results, again to the leading order, in the relatively simple expression

( ) ( )

( ) ( )exp1 2 2 2 2

1 2 2

3 2 2

1 2

eff Belec l R uFLkT R u R

π ν κ κ

κ κ

− +=

+ (6.54)

In short hand notation, the electrostatic energy can be written as

( ) ( ), , ,3eleceff

FE R U u R

LkTκ ν κ= (6.55)

with the bare electrostatic energy of a single interaction pair

( )( ) ( )

( )

exp,

1 2 2

1 2

2 eff Beff

l RE R

R

π ν κκ ν

κ

−= (6.56)

and the enhancement factor for fluctuations


( )( )exp

, ,2 2

2

2

1 2

uU u R

u R

κκ

κ=

+ (6.57)

In order to find the undulation amplitude u at a mean lattice spacing R , we need to minimize the sum of the free energies of confinement and electrostatics ( ) 0elec confF F u∂ + ∂ = . With Eqs. (6.45) and (6.55), the minimization of the free energy results, again to the leading order, in

( ) ( ), , ,8 31 3 2

2

9u

effp

cu E R U u R

Lκ ν κ

κ= (6.58)

This equation is the key to solve the undulation amplitude u for a specific value of the interaxial spacing R , once the effective charge density effν and the inverse screening length κ are known. It will frequently be used, e.g. in order to derive the equation of state or to predict the melting transition of the hexagonal phase when it is combined with the semi-empirical Lindemann rule. Unfortunately, Eq. (6.58) cannot be solved in closed analytical form, but it is amenable to a numeric method.

6.2.2 Melting of the hexagonal phase

For 150 bp DNA in NaCl solution, the critical volume fractions hφ and cφ representing the onset of melting and complete disappearance of the

hexagonal phase, respectively, are collected in Table VI.I. With increasing salt concentration the boundaries shift to higher volume fractions. The interaxial spacings R at the melting transition hφ are derived from the hexagonal unit cell volume 23 2 m hR A V φ= with linear nucleotide spacing .0 171A= nm and a nucleotide volume .0 286mV = nm3. Note that the spacings (based on density) are in perfect agreement with the ones derived from the position of the narrow peak in the neutron diffraction spectra (see Fig. 6.6).

The effective charge density effν has been derived from the solution of the non-linear Poisson–Boltzmann equation as described in Sec. 3.3. Note that this procedure is formally valid in excess salt only; for these dense liquid crystals the contribution of the ions coming from the dissociation of DNA (counterions) to the screening cannot be neglected. The latter contribution was estimated from the fraction of osmotically free counterions, as determined from the solution of the non-linear Poisson–Boltzmann equation pertaining


to salt-free DNA in the cell model (see Sec. 3.3.2). Furthermore, the bare diameter of the DNA molecule was set to the nominal value 0 2D = nm. As shown in Table VI.I, effν varies over an order of magnitude when the NaCl concentration is increased from 0 to 1.5 M.

The undulation parameter u at a given R has numerically been determined from Eq. (6.58). The DNA persistence length pL has been set to its intrinsic value 50 nm, since charge effects on the bending rigidity are expected to be insignificant at these high DNA and salt densities. As collected in Table VI.I, the interaxial spacings are typically an order of magnitude larger than the Debye screening length. Despite the fact that the fluctuations in the chain position and screening lengths are small, uκ is of the order of unity and the electrostatic potential is renormalized by a relatively large factor Eq. (6.57). At the melting transition, the ratio of the root mean-square undulation amplitude and the interaxial spacing, i.e. LC u R= , is constant over a four-fold change in screening length with average value .0 113LC = . These results bear out the empirical Lindemann rule, that is the lattice melts when the fluctuations in position order exceed a certain fraction of the interaxial spacing.99 The ionic dependence of the critical boundary pertaining to the melting of the hexagonal liquid crystal, which follows from the undulation-enhanced electrostatic theory and the Lindemann criterion, is displayed in Fig. 6.5. Particularly, the shift of the melting trajectory to higher volume fractions with increased screening of electrostatic interaction is reproduced. The deflection length λ is also collected in Table VI.I. Due to

Table VI.I Critical volume fractions corresponding to the onset of melting and complete disappearance of the hexagonal phase, hφ and cφ , respectively. The interaxial spacing R , effective charge density effν , Lindemann ratio LC and deflection length λ refer to the hexagonal side. The range of undulations and electrostatic interactions are monitored by uκ and Rκ , respectively.85

sρ cφ hφ R effν uκ Rκ LC λ

(M) (nm) (nm-1) (nm)

0 0.135 0.143 3.7 8.61 0.58 5.37 0.109 2.0

0.25 0.167 0.176 3.3 20.8 0.86 7.82 0.110 1.9 0.75 0.171 0.193 3.2 53.3 1.23 10.7 0.116 1.9 1.50 0.192 0.213 3.0 147 1.56 13.4 0.116 1.8


tiny undulations, the deflection length is short of the order of 2 nm (but 1κλ ). Since the deflection length is much shorter than the length of the

DNA molecules (50 nm), the contour length has no effect on the melting of the hexagonal phase. This is in contrast with the isotropic-cholesteric phase transition, which strongly depends on the contour length through the second virial coefficient Eq. (6.18) and chain orientation entropy Eq. (6.35).

6.2.3 DNA equation of state

For a hexagonal unit cell with volume 23 2R LV = , the osmotic pressure is given by the derivative of the total free energy

1 1

3tot totdF dFdV L dRR

Π =− =− (6.59)

The total free energy is the sum of the free energies of confinement and electrostatic interaction ( ) ( ) ( ), ,tot conf elecF R u F u F R u= + . The free energy of confinement Eq. (6.45) depends explicitly only on u . We can write for the total differential

( )( ) ( ) ( ), ,

, conf elec electot

F u F R u F R udF R u du dR

u u R

⎛ ⎞∂ ∂ ∂⎟⎜= + +⎟⎜ ⎟⎟⎜ ∂ ∂ ∂⎝ ⎠ (6.60)

The first term on the right-hand-side is identically zero, because the total free energy has been minimized with respect to the undulation amplitude u . Accordingly, the osmotic pressure is related only to the derivative of the electrostatic free energy

( )1 1 3

3elec EUF

kTL R R RR

∂∂Π =− =−

∂ ∂ (6.61)

With the definitions of the bare electrostatic energy and the undulation enhancement factor, Eq. (6.56) and (6.57), respectively, it is not difficult to show that, to the leading order

( )EUEU

Rκ

∂=−

∂ (6.62)

Finally, we can use Eq. (6.58) to derive the equation of state (pressure versus interaxial spacing) for the hexagonal array of undulating and charged worm-like chains


3 2 8 3 1 3

3 2

3u

p

EU ckT R u L R

κκ

Π= = (6.63)

Note that for each value of R the undulation amplitude u needs to be determined from Eq. (6.58) by a numerical method.

The force per unit length exerted on a DNA molecule by another molecule is given by the derivative of the total free energy with respect to their separation

13

totFfL R∂

=−∂

(6.64)

where the factor 3 accounts for the fact that one molecule interacts with three other molecules (three interaction pairs per unit cell). With Eq. (6.59), we can then express the force in terms of the pressure

3

Rf = Π (6.65)

Using osmotic stress, Strey and co-workers have measured the energetics of compacting long (> 10-6 m) DNA molecules into liquid crystals.95 Assuming hexagonal packing, the osmotic pressure was converted to the force per unit length exerted on one DNA molecule by a neighbouring chain according to Eq. (6.65) and R determined by X-ray scattering or from measured densities. Their results in 0.1 M, 0.5 M and 1 M NaCl are displayed in Fig. 6.9. The theoretical prediction of the force per unit length, as obtained from Eqs. (6.58), (6.63) and (6.65) and the DNA parameters as described above, is also displayed in Fig. 6.9. A striking observation is that the equation of state does not show significant discontinuities, which are expected for a first order phase transition from the hexagonal to the cholesteric phase at interaxial spacings between 3 and 3.5 nm. This is due to the weakness of the transition with small differences in density between the coexisting phases. For instance, in our example in Fig. 6.6, the interaxial spacing between the 50 bp DNA molecules in the coexisting hexagonal and cholesteric phases is 3.10 and 3.22 nm, respectively. For long DNA, the differences in spacing and the corresponding discontinuities in the equation of state are similarly small.87,95

For all ionic strengths and in particular within its range of applicability (i.e. the hexagonal phase with spacings less than 3 to 3.5 nm), the undulation enhanced electrostatic theory describes the force data well. This is gratifying;


in particular in view of the simplicity of the model and that there are no adjustable parameters. It should be noted that the theory is pushed beyond its range of validity for very short as well as very large R values. For very short spacings the inner double layers overlap and the use of screened electrostatics becomes problematic. In the cholesteric, without long-range spatial order, the fluctuations formally diverge and the undulation theory is not strictly applicable. Its relative success for large R values indicates that the local hexagonal packing is preserved upon melting.

2 2.5 3 3.5 410 −4

10 −3

10 −2

(nm)R

3R

Π(N

/m)

Figure 6.9 Force per unit length versus interaxial spacing R for long (Mw > 108) DNA in 0.1 (squares), 0.5 (circles) and 1 (diamonds) M NaCl.95 The curves are calculated using Eqs. (6.63) and (6.65) and u from Eq. (6.58). The counterion contribution to the screening was estimated from the fraction of osmotically free counterions following from the solution of the salt-free Poisson–Boltzmann equation in the cell model as described in Sec. 3.3.2.


6.3 Bacteriophage DNA packaging

In the viral capsid of bacteriophages the double-stranded genome is tightly packed with packing fractions on the order of 0.5. High resolution electron microscopy images have shown that the DNA has an hexagonal structure and is arranged in a spool-like fashion (see Fig. 6.10). Experiments have shown that the force exerted on the genome can reach very high values on the order of 50 pN, which corresponds to a pressure of 50 atm (1 atm = 101.3 kPa) across a 10 nm2 ejection channel.92,93 These high values result from the fact that the DNA is strongly bent and confined with a typical interaxial spacing 2.5 nm.89 It has been conjectured that this force drives the ejection of the genome into the host cell upon infection and conversely, this force needs to be exerted by the motor protein for packaging of the genome inside the viral capsid. We present a comprehensive theory in order to describe the packaging of DNA inside the viral capsid based on the balance of the bending stress and the internal osmotic pressure of the hexagonal DNA assembly.90

For the sake of simplicity, we will assume that the capsid is spherical with an outer radius oR . A schematic drawing of the capsid and the definitions of the various geometric parameters are given in Fig. 6.11. Some bacteriophages have a capped-cylindrical shape, whereas T7 has its tail protruding into the spherical cavity. We will not include these structural details in our model, but

Figure 6.10 Cryo-electron micrograph of a head from the complete tail-deletion mutant of T7–bacteriophage (average of 77 particles). Note that the capsid is viewed from the top downwards. The capsid exhibits the characteristic 2.5 nm spacing of at least nine equally spaced DNA-associated concentric rings. Reprinted with permission from Ref. [89]. Copyright Elsevier (1997).


rather we will treat oR as an effective radius with a value determined by the net accessible volume. Electron microscopy images show that DNA is accommodated in concentric stacks of rings from the viral wall towards the centre of the capsid. The structure is hexagonal with every ring surrounded by six other rings, except those touching the capsid wall and the ones in the innermost stack. At the centre of the spool, the DNA is excluded from a cylindrical void with a radius iR . At a distance r away from the centreline of this void, the stack of concentric rings has a height ( ) ( )1 22 22 oh r R r= − and thickness 3 2R . In each stack, there are h R DNA rings on top of each other. The DNA contour length accommodated within the stack is 2 r h Rπ , because each ring stores a length 2 rπ . We will assume that the thickness of the stacks is uniform, so that there are a total of ( ) ( )3 2o iN R R R= − concentric stacks in the radial direction away from the centreline of the void to the wall. The total contour length stored within the phage head is then given by the summation over all stacks

12

Nk kk

L r h Rπ=

=∑ (6.66)

with kr and kh being the radial coordinate and height of stack number k , respectively. Since the distance between the stacks is small, we can calculate the total contour length (and the bending energy, see below) in the continuum approximation by replacing the summation over the stacks by an integration over the radial coordinate according to

R i

R o

r

hR

R3 2

Figure 6.11 Schematic drawing of the cross-section of the phage head as seen from the side. The DNA is spooled in concentric rings and arranged in a hexagonal lattice.


( )( )

21 2

3 2

o o

i i

R R

R R

rh rL dr dr rh r

R SR

π π= =∫ ∫ (6.67)

with the surface area of the unit cell 23 2S R= . The integration is readily performed and results in an expression for the radius of the void

2 3

2 2 34i oLS

R Rπ

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠ (6.68)

in terms of the capsid outer radius oR and the total volume occupied by the genome LS .

We can apply a similar continuum approximation for the calculation of the bending energy bF . Each ring with contour length 2 rπ has a radius of curvature 2r− and a bending energy pL kT rπ ( 22pL kT r per unit length). Again, we need to sum over h R rings in the stack and integrate over all concentric stacks along the radial coordinate

( ) ( )1

3 2

o o

i i

R Rp pb

R R

L Lh r h rFdr dr

kT r R S rR

π π= =∫ ∫ (6.69)

The integration results in the bending energy per unit contour length

( )

acoth1/3

1/3

2 62

6

pb oo

LF R LSR

LkT LS LS

πππ

⎛ ⎛ ⎞ ⎞⎛ ⎞⎟ ⎟⎜ ⎜ ⎟⎟ ⎟⎜⎜ ⎜= − ⎟⎟ ⎟⎜⎜ ⎜ ⎟⎜⎟ ⎟⎝ ⎠⎟ ⎟⎜ ⎜⎝ ⎝ ⎠ ⎠ (6.70)

which depends on the size of the capsid and the volume of the genome through oR and LS , respectively. Note that, for a given virus and genome, the contour length L and the outer radius of the capsid oR are fixed. Thus the only parameter controlling the bending energy is the interaxial spacing R . The interaxial spacing between the DNA rings determines the surface area of the hexagonal unit cell S and the inner radius of the spool iR (radius of the void) through the size restriction Eq. (6.68).

In order to find the optimized interaxial spacing, we need to consider the total free energy of the capsid tot b hex surfF F F F= + + , where hexF is the internal free energy pertaining to the curved hexagonal phase and surfF is the surface contribution. The surface contribution includes the interaction of the DNA with the capsid wall and takes into account the fact that the rings in the inner- and outermost stacks have 4 rather than 6 neighbouring rings. We will ignore these surface effects and assume that 0surfF = . Furthermore, we neglect effects of curvature on the internal free energy of the hexagonal phase


(besides bending effects, which have explicitly been taken into account through bF ). The total free energy is then the sum of the free energies of bending, confinement and electrostatic interaction ( ) ( ) ( ) ( ), ,tot b conf elecF R u F R F u F R u= + + (6.71)

and the optimized interaxial spacing R follows from the minimization condition 0totdF dR = . The free energy of confinement Eq. (6.45) depends explicitly only on u . We can write for the total differential

( )

( ) ( )

( ) ( )

,,

,

conf electot

b elec

F u F R udF R u du

u u

F R F R udR

R R

⎛ ⎞∂ ∂ ⎟⎜= + +⎟⎜ ⎟⎟⎜ ∂ ∂⎝ ⎠⎛ ⎞∂ ∂ ⎟⎜ + ⎟⎜ ⎟⎜ ∂ ∂⎝ ⎠

(6.72)

The first term on the right-hand-side vanishes, because the free energy of the hexagonal phase has been minimized with respect to the undulation amplitude u . Accordingly, we find the stability condition

( ) ( ),0b elecF R F R u

R R∂ ∂

+ =∂ ∂

(6.73)

The derivative of the electrostatic free energy with respect to the interaxial spacing is related to the osmotic pressure of the assembly of DNA rings according to Eq. (6.61). In terms of the surface area of the hexagonal unit cell S , we thus have

bFS LkT kT⎛ ⎞∂ Π⎟⎜ =⎟⎜ ⎟⎜⎝ ⎠∂

(6.74)

The derivative of the free energy of bending with respect to the surface area can be derived with Eq. (6.70) and takes the form

2

1

2pb b

i

LF FS LkT S LkT SR

⎛ ⎞∂ ⎟⎜ =− +⎟⎜ ⎟⎜⎝ ⎠∂ (6.75)

From Eqs. (6.74) and (6.75) follows the result

2

1

2pb

i

LFS LkT kTSR

Π− + = (6.76)

with the radius of the void given by Eq. (6.68). It should be borne in mind that Eq. (6.76) is general and does not depend on the specific form of the osmotic pressure. For instance, one can use the experimentally determined


osmotic pressure of the macroscopic, hexagonal DNA phase.94 In order to derive the packaging properties from first principles, we will use Eq. (6.63) resulting from the undulation enhanced electrostatic theory. The set of equations can be numerically solved for the area of the hexagonal unit cell S and the interaxial spacing R , given a certain length of the genome L and capsid radius oR . Note that for the calculation of the osmotic pressure, for each value of R the undulation amplitude u needs to be determined with Eq. (6.58). The second term on the left-hand-side of Eq. (6.76) overwhelms the first term by at least an order of magnitude. The interaxial spacing R and the inner radius of the spool iR are thus primarily determined by the elastic bending stress of the rings in the innermost stack of the spool immediately surrounding the void.

With respect to experimental work on packaging of DNA in viral capsids, the application of the theory presents some difficulties. First of all, real capsids are often not spherical, so that we need to estimate an effective capsid radius. Furthermore, the electrostatic interaction among the rings is described by screened electrostatics, which is strictly valid only in the presence of excess

0.2 0.4 0.6 0.8 12

3

4

5

6

0

0.04

0.08

0.12

0.16

0.20R

(nm

)

Packing Fraction

CL

Figure 6.12 Interaxial spacing R (left scale, solid line) and Lindemann ratio

LC (right scale, long-dashed curve) versus the packing fraction of the genome of T7–bacteriophage as obtained from the theoretical analysis. The symbols are the experimental results for complete tail-deletion mutants with genome lengths .13 6L = , .12 5 and .11 5 micrometres.89 The short-dashed line denotes the melting criterion .0 113LC = .


salt. In experimental studies, the buffer salt concentration is usually in the tens of millimolar range, which is far less than the counterion concentration in the intervening space between the rings of DNA. Accordingly, the counterion contribution to the screening (fraction of osmotically free counterions) cannot be neglected and needs to be estimated with either the condensation concept (Sec. 3.1) or, as has been done here, from the solution of the Poisson–Boltzmann equation pertaining to salt-free DNA in the cell model (see Sec. 3.3.2). Finally, the capsid wall is semi-permeable for ions and water, so that there is Donnan salt partitioning between the capsid and the surrounding medium. This is also described in Sec. 3.3.2 and has been applied before in Sec. 3.7.2 for a polyelectrolyte brush (here, qf denotes the fraction of osmotically active counterions in the hexagonal lattice). The effective ionic strength has been calculated accordingly.

For different T7–bacteriophage mutants with different lengths of the genome, the interaxial spacing has been measured by Cerritelli et al.89 The results are displayed in Fig. 6.12. Also displayed are the interaxial spacing R and the ratio of the undulation amplitude over the spacing LC u R= (Lindemann ratio) as obtained from Eq. (6.76) with the osmotic pressure Eq. (6.63) [For each value of R the undulation amplitude u follows from Eq. (6.58)]. The theoretical values are calculated with an optimized outer radius

.26 3oR = nm, which is very close to the value 26.6 nm based on the effective accessible volume. With decreasing packing fraction, the interaxial spacing and the Lindemann ratio increases. As we have seen above, for short DNA fragments the hexagonal lattice melts when the Lindemann ratio exceeds the critical value .0 113LC = . If a similar melting criterion applies to the hexagonal phase of the genome inside the viral capsid, the hexagonal position order will be lost and become liquid-like for packing fractions below a critical value of 0.456. Note that this critical value is sensitive to the ionic strength of the supporting medium. At higher salt concentrations, the melting transition shifts to higher DNA densities and thus to a higher packing fraction (see Table VI.I).

Despite the fact that the experimental interaxial spacings are well reproduced, the model is not good with respect to the bending energy of the inner stack of rings surrounding the void. With increasing packing fraction from 30% to full load, the radius of the inner void iR decreases from 9 to around 2 nm (results not shown). The DNA in a few stacks surrounding the


void is thus extremely bent [which determines to a large extent the interaxial spacing, see Eq. (6.76)]. Under such extreme bending conditions, it is likely that the DNA duplex will kink, thereby reducing the localized bending stress. Furthermore, for such sharply bent DNA it is also necessary to include the effect of curvature on the electrostatic interaction. A revised and extended theory should include these, as well as other effects related to the surface interaction. However, it is expected that these refinements will not result in a fundamentally different physics of packaging.

We will now derive an expression for the force pertaining to the ejection of the genome and conversely, the force which is needed to be exerted by the motor protein for packaging of the genome inside the capsid. This force is given by the derivative of the total free energy Eq. (6.71) with respect to the length of the compacted genome

conftot b elecFF F Ff

L L L L∂∂ ∂ ∂

= = + +∂ ∂ ∂ ∂

(6.77)

With the help of Eq. (6.70) we can derive the bending contribution

22pb

i

LFkT

L R

∂=

∂ (6.78)

The contribution from the free energy of confinement

2 3 1 3

conf u

p

F ckT

L u L

∂=

∂ (6.79)

follows from Eq. (6.45), whereas the electrostatic contribution

1 3 2 8 3

2

3elec u

p

F ckT

L L uκ∂

=∂

(6.80)

is given by the derivative of Eq. (6.55) together with Eq. (6.58). The force is thus the sum of these three contributions

( )2 2

1 3 2 8 3 2

2 3

3 2

u p

p i

c u LfkT L u R

κ

κ

+= + (6.81)

with the radius of the void iR determined by Eq. (6.68). For each value of L , the spacing R and undulation amplitude u need to be determined as described above.

For the 29φ virus, the force has been measured with a single molecule


manipulation technique and is displayed in Fig. 6.13.92 With increasing packing fraction, the force increases and eventually reaches a very high value on the order of 50 pN. Also displayed is the theoretical force, but calculated with the parameters for the T7–bacteriophage. For 29φ the interaxial spacing has not been measured and there is no bench mark for the optimization of the effective capsid radius. However, the observed phenomena are general and do not depend on the specific choice of the parameters. Indeed the theory predicts an increase of the force with increasing packing fraction, but not as steep as experimentally observed. At high packing fraction, a strong force on the order of the experimental value is predicted. However, at lower packing fractions the force does not dwindle, which is related to the fact that the hexagonal lattice does not melt under the prevailing conditions (see Fig. 6.12).

6.4 Crowding and entropy driven interactions (depletion)

An important concept for understanding the compaction of biomacromolecules in a congested state is macromolecular crowding, or in physical terms, depletion interactions.100,101 In the case of a mesoscopic particle dispersed in a medium containing an osmotic agent, the depletion effect is

0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

Packing Fraction

Forc

e (p

N)

Figure 6.13 Solid curve: experimentally determined internal force against the fraction packaged genome of the 29φ virus.92 The dashed curve has been calculated according to the theory with the same parameters as used in Fig. 6.12 pertaining to the T7–bacteriophage.


most easily understood in terms of a region surrounding the particle from which the agent is excluded. The accessible volume for the osmotic agent increases when the depletion zones pertaining to different particles overlap. The concomitant gain in system entropy results in a net attraction between the mesoscopic particles. Eventually, the depletion interaction can result in separation of the original dispersion into two coexisting phases; one rich and the other one poor in the particle concentration. Note that the origin of the depletion effect is completely entropic and does not require the existence of specific interactions between the particle and the osmotic agent. The range and amplitude of the depletion force depend on the physical extent of the depletion zone and the pressure exerted by the osmotic agent. Depletion interactions are important in biology as well as in biotechnology. In biology, depletion interactions have not been fully appreciated yet, but they are thought to affect, among others, DNA structure and activity.102 An example of a biotechnological application is the purification and concentration of protein in the food processing industry.

The osmotic agent is often a semi-flexible neutral polymer such as poly(ethyleneoxide) (PEO) or a polysaccharide (e.g. pullulan). The interaction of the polymer chains with the mesoscopic particles has been extensively investigated by field theoretical methods.103,104 We will treat the problem in the mean field approximation, for its simplicity, transparency and semi-quantitative results which are close to the renormalization group results.100,105,106 If the osmotic agent is sufficiently diluted, the characteristic size of the polymers is given by the radius of gyration gR . For higher polymer concentrations, in the semi-dilute regime, the relevant length scale is the correlation length ξ . Obviously, the ratio of the size of the mesoscopic particle and the characteristic size (or correlation length) of the osmotic agent is an important parameter in the description of the depletion effect. We will subsequently discern two limiting cases. In the first case, the size of the particle is much larger than the radius of gyration of the polymer or the correlation length in the semi-dilute polymer solution. In this case, we can use the classical depletion theory for the interaction between relatively large colloidal particles suspended in solutions of small macromolecules, which was put forward long ago by Asakura and Oosawa.100 In the second case, the size of the particle is much smaller than the radius of gyration or correlation length (nanoparticles). This situation is relevant for, among others, the phase


separation of protein/polysaccharide mixtures and will be treated with the formalism originally due to de Gennes.105

6.4.1 Entropic colloidal interactions in solutions of macromolecules

Let us consider a fluid of hard-spheres with diameter d . In the case of a dilute polymer solution, the spheres stand for the individual polymer chains and the sphere size is obviously related to the radius of gyration 2 Gd R= . In the semi-dilute regime, the spheres can be identified as the blobs and we will use d πξ= , with ξ being the correlation length.105 We now insert a big spherical colloidal particle with diameter D d into the hard-sphere fluid. The colloidal particle occupies a spherical volume ( )3 6exV D dπ= + from which the centres of mass of the small spheres are excluded (see Fig. 6.14). The depletion zone surrounding the colloidal particle has thus a thickness 2d . We can derive the required work for the insertion of the colloidal particle from the fact that, in order to do so, we need to create a cavity with a volume exV against the osmotic pressure Π exerted by the hard sphere fluid. In the

absence of specific interactions between the colloidal particle and the osmotic agent, the work is simply exw V= Π (D d ) (6.82)

with the osmotic pressure kT c NΠ = or ( )9 43 3kT l cl−Π depending on whether the solution is dilute or semi-dilute, respectively. For a big colloidal particle with D d in the semi-dilute regime the work is proportional to its volume and the inverse third power of the correlation length 3ξ− .

Next, we consider the interaction between two colloidal spheres immersed in the medium containing the osmotic agent. If the particles are separated over a distance r D d> + the depletion zones do not overlap, so that the total excluded volume equals two times the excluded volume of a single sphere

( )3 3non overlapexV D dπ− = + , r D d> + (6.83)

For smaller inter-particle distances with D r D d< < + the depletion zones overlap. The total excluded volume is now two times the volume of the sphere, but without the overlapping cap sections as shown in Fig. 6.14:


( )[ ][ ]2212

overlapexV D d r D d r

π= + − + + , D r D d< < + (6.84)

In order to derive the depletion force, we need to know the excess free energy with respect to the energy of two colloidal particles without overlapping depletion zones. Since the work of inserting the particles into the medium is proportional to the excluded volume [Eq. (6.82)], we readily obtain the interaction potential

( ) ( )

( )[ ][ ]2212

overlap non overlapex exU r V V

D d r D d rπ

−Δ = − Π =

− + − + + Π , D r D d< < + (6.85)

The depletion force now follows from the derivative of the interaction potential with respect to the inter-particle separation

D

d

r

(a)

(b)

Figure 6.14 (a) Two colloidal particles, each of diameters D , each surrounded by a depletion zone with thickness 2d and separated by a distance r . Note that the depletion zones overlap for D r D d< < + . (b) 3D image of one particle surrounded by its depletion zone showing the circular cross-section of the overlapping depletion zones at the middle of r .


( )( )

( )( )2 2

4U r

f r D d rr

π∂= =− + − Π

∂ , D r D d< < + (6.86)

Note that the depletion force is attractive and is given by the product of the pressure exerted by the osmotic agent and the area of the cross section of the overlapping depletion zones at the middle of the separation between the colloidal particles (see Fig. 6.14). Recall that the depletion force vanishes for separations r D d> + , so that the force is relatively short-ranged with a (tunable) decay length on the order of the semi-dilute correlation length ξ or

1.2 1.4 1.6 1.8 20

5

10

150

1

2

3

4

5

r (10 m)-6

U(r

)/kT

0.280 g/L

0.140 g/L

0.070 g/L

0.050 g/L

0.025 g/L

0.010 g/L

0.005 g/L

(b) semi-dilute

(a) dilute

Figure 6.15 The interaction potential between two silica spheres of 1.26 micrometre diameter in (a) dilute and (b) semi-dilute solutions of λ− phage DNA (48.5 kbp) in 10 mM TE buffer versus the inter-particle distance r . The lines represent the fit to the Asakura–Oosawa interaction potential Eq. (6.85). Redrawn from Ref. [107].


the radius of gyration GR in the case of an osmotic agent in the dilute regime. The interaction potential between silica spheres of 1.25 micrometre

diameter immersed in dilute and semi-dilute solutions of λ−phage DNA in 10 mM TE buffer has been measured by an optical tweezers technique.107 Under these minimal screening conditions, the radius of gyration of the λ−phage DNA molecule is around 800 nm (the value of the radius of gyration has been extrapolated from the light scattering results in Table III.I and corrected for a difference in contour length according to 3 5

gR L ). Accordingly, the calculated overlap concentration from the dilute to the semi-

10-2

10-1

10 −2

10 −1

10 0

10 1

10 210−1

10 0

c (g/L)

d/2

(10

m)

-6Π

/kT

(10

m

)

183

(a)

(b)

Figure 6.16 (a) Effective depletant size versus DNA concentration. In the dilute regime (c < 0.03 g/L) the size agrees with the radius of gyration of the DNA molecule ( 2 Gd R= ), whereas for higher concentrations in the semi-dilute regime the depletant size is interpreted as the correlation length ( 2 2d πξ= ). (b) Osmotic pressure in units kT versus DNA concentration. Redrawn from Ref. [107].


dilute regime is around 0.03 g/L. The experimental interaction potentials versus the inter-colloid separation are displayed in Fig. 6.15. At very short distances, on the order of the diameter of the silica spheres, the interaction potential is repulsive due to the hard core interaction. For longer distances, the potential is attractive, in agreement with a depletion interaction. In the dilute regime, the interaction potential is shallow with a depth on the order of

2kT or less. Above the overlap concentration, the potential wells become deeper on the order of a few times kT , but the range of the interaction contracts. Despite the fact that the size of the colloidal particles is a bit smaller than the radius of gyration of the DNA molecules, the results were analyzed with the Asakura–Oosawa interaction potential Eq. (6.85).

The radii of gyration of the DNA molecules in the dilute regime ( 2 Gd R= ) and correlation lengths in the semi-dilute regime ( 2 2d πξ= ) resulting from the fit of Eq. (6.85) to the experimental depletion potentials are displayed in Fig. 6.16. In the dilute regime, the fitted values of the radius of gyration are constant (600 nm) and are in reasonable agreement with the extrapolated values based on light scattering results in Table III.I (800 nm). Once the DNA solution becomes semi-dilute for concentrations exceeding, say, 0.03 g/L, the effective size of the osmotic agent decreases according to a power law . .0 5 0 1d c− ±∼ . The value of the exponent minus 0.5 is in perfect agreement with the scaling law for the correlation length of the salt-free polyelectrolyte in the semi-dilute regime (see Appendix B).

Verma et al. also fitted the osmotic pressure to the interaction potential and these results are displayed in the bottom panel of Fig. 6.16. In the semi-dilute regime, the osmotic pressure increases with increasing DNA concentration according to a power . .2 2 0 2c ±Π ∼ . The value of the exponent 2.2 agrees with the value expected for the osmotic pressure of a semi-dilute solution of neutral polymers 9 4 . Although this agreement seems to be gratifying, it is in fact misleading. The tweezers experiment was done under minimal screening conditions in 10 mM TE buffer. In this situation, the osmotic pressure is dominated by the counterion contribution rather than the polymer and the fraction of osmotically free counterions cannot be considered constant. There is reasonable agreement with independent osmotic pressure data for DNA and the prediction based on the non-linear Poisson–Boltzmann equation in the cell model (see Sec. 3.3.2).108


6.4.2 Phase separation of small particles in a polymer solution

We now consider the opposite limit with GD R or D ξ , i.e. a small particle (e.g. a nanoparticle or a protein) dispersed in a semi-dilute polymer solution. We will not consider the case of a nanoparticle immersed in a single polymer coil, but the corresponding results can be easily obtained by replacing the correlation length ξ by the radius of gyration GR . Following de Gennes, we can derive the work for the insertion of the nanoparticle inside the semi-dilute polymer solution with a scaling argument.106 The size of the nanoparticle D is considered to be on the same order of magnitude as the segment length (or Kuhn length) l of the polymer. Owing to the restriction in configurational degrees of freedom, the polymer segments are depleted from the area next to the interface with the nanoparticle over a length scale on the order of D . The other relevant length scale in the problem is the correlation length of the semi-dilute polymer solution ξ . We expect that the work follows a scaling law in the dimensionless ratio D ξ , so that

nD

w kTξ

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠ (6.87)

The work should be proportional to the number of contacts between the nanoparticle and the polymer inside the depletion region. The number of contacts is proportional to the segment concentration and w should be linear in c (this is borne out by experiments to be discussed below). In order to satisfy w c∼ and with the scaling law for the correlation length in the semi-dilute regime 3 4cξ −∼ under good solvent conditions (Appendix B), the exponent n takes the value 4 3 and

4 3D

w kTξ

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠ ( 3 5ν = ) (6.88)

Note that the work for the insertion of the nanoparticle into the semi-dilute regime has been derived for a polymer under good solvent conditions with Flory exponent 3 5ν = ( 3 4cξ −∼ ). If the particle is very small with a diameter on the order of the length of the segments, the depletion volume is equally very small. In such a small volume in close proximity to the nanoparticle the chain exhibits Gaussian statistics, albeit it might be swollen by excluded volume interactions at a larger distance scale. This is particularly the case if the solvent quality is not very good with moderate values of the


excluded volume parameter. For a semi-dilute polymer solution under theta conditions, it is not difficult to show that109

Dw kT

ξ ( 1 2ν = ) (6.89)

Note that the work for the insertion of the nanoparticle into the semi-dilute polymer solution has been conjectured to be proportional to the polymer concentration, irrespective of the solvent conditions. The proportionality of the depletion energy with the polymer concentration has often been observed in experimental work on ternary mixtures of small particles, a neutral polymer and a common solvent. In the phase separation thermodynamics, we will accordingly use

31w kT C cl= (6.90)

with 1C being an empirical constant. In order to describe depletion induced phase separation, we need an

expression for the free energy of the system of nanoparticles dispersed in a semi-dilute polymer solution. This can be obtained with a mean-field calculation of the local polymer density with the use of a spherical cell model.110 We will follow the semi-empirical approach of Wang et al., who have investigated depletion induced demixing in aqueous protein-polysaccharide solutions.111 Let the nanoparticle (protein) concentration be n Vρ = , with V being the total volume. We assume that the interactions

among the nanoparticles are vanishingly small and/or that the virial coefficients cancel out, so that we can write for the nanoparticle contribution to the free energy the ideal translation entropy term ln1F kT n nρ= − . Experimental evidence shows that the free energy of insertion of the nanoparticles into the semi-dilute polymer solution is extensive in the number of particles. Accordingly, with Eq. (6.90) we postulate the insertion term

32 1F kT C ncl= .

We also need an expression for the free energy of the semi-dilute polymer solution. For a neutral polymer in a very good solvent, we have

( )9 43 33 2F kT C cl V l= with 2C being another empirical constant (see Sec.

2.9). However, if we use this expression with exponent 9 4 , the resulting coexistence equations cannot be solved and do not predict the experimentally observed phase separation. We adopt an empirical scaling law for the free


energy of the semi-dilute polymer solution with an adjustable exponent α , so that ( )3 3

3 2F kT C cl V lα

= . The total free energy of the ternary mixture is the sum of the ideal small particle term 1F , the insertion term 2F and the semi-dilute polymer term 3F :

( )ln 3 3 31 2F kT n n C ncl C l cl V

αρ −= − + + (6.91)

We next find an expression for the osmotic pressure by differentiating the total free energy with respect to the volume

( ) ( ) ( ), ,

3 3 31 2

11 1

n N T

FC cl C l cl

kT kT Vα

ρ α −⎛ ⎞Π ∂ ⎟⎜=− = + + −⎟⎜ ⎟⎜⎝ ⎠∂ (6.92)

The extensive free energy of insertion ensures that the osmotic pressure is linear in the concentration of the nanoparticles ρ . Furthermore, we find the chemical potential of the nanoparticles by differentiation with respect to their number

, ,

ln 31

1n

N V T

FC cl

kT kT nμ

ρ⎛ ⎞∂ ⎟⎜= = +⎟⎜ ⎟⎜⎝ ⎠∂

(6.93)

Likewise, we can derive the chemical potential of the polymer by differentiation with respect to the total number of polymer segments N cV= in the volume V (not to be confused with the number of segments per chain)

( ), ,

13 31 2

1N

nV T

FC l C cl

kT kT Nαμ

ρ α−⎛ ⎞∂ ⎟⎜= = +⎟⎜ ⎟⎜⎝ ⎠∂

(6.94)

With the expressions for the chemical potentials and the osmotic pressure, we can derive a set of coexistence equations. These coexistence equations describe the phase separation of the ternary mixture into two coexisting phases (indicated by subscripts 1 and 2, respectively). One phase is rich in the nanoparticle and poor in the polymer, whereas the other one is rich in the polymer and poor in the nanoparticles. From the balance of the chemical potentials pertaining to the nanoparticles in the coexisting phases, one obtains

( )ln 1 3 31 1 2

2C c l c l

ρρ

=− − (6.95)

This expression shows that the partitioning of the nanoparticles over the coexisting phases is exponential in the difference in polymer volume fraction


between the phases. Likewise, the second coexistence equation follows from the balance of the chemical potentials of the polymer in the coexisting phases

( ) ( ) ( )( )1 13 3 3 31 1 2 3 1 2C D D C c l c l

α αρ ρ α

− −− =− − (6.96)

with ( )33 2C C D l= . From the mechanical (osmotic) equilibrium of the two coexisting phases, we obtain

( )( )

( ) ( ) ( )( )( ) ( )( )

exp

exp

3 33 3 3 3 1 21 1 2 1 2

3 3 1 13 31 1 2 1 1 2

11

1

c l c lc l C c l c l c l

C C c l c l c l c l

α α

α α

α

α− −

− −⎡ ⎤− −⎣ ⎦+ =⎡ ⎤− − −⎣ ⎦

(6.97)

The latter expression has been derived from the balance of the osmotic pressures divided by Eq. (6.96) and subsequent elimination of 1 2ρ ρ with the help of Eq. (6.95). Note that we have expressed the nanoparticle and polymer concentrations in terms of the dimensionless volume fractions 3Dρ and 3cl , respectively. Now, we can construct a phase diagram by a choice of the polymer volume fraction 3

1c l , a subsequent calculation of the coexisting polymer fraction 3

2c l with the rearranged osmotic pressure balance Eq. (6.97) and the determination of the nanoparticle volume fractions 3

1Dρ and 32Dρ

from the equilibrium of the chemical potentials Eqs. (6.95) and (6.96), respectively.

As an example, the coexistence volume fractions of β− lacto globulin ( β− Lg) and pullulan are presented in Fig. 6.17.111 β− Lg is a small protein and pullulan is a neutral polysaccharide (see Sec. 1.2.3). At the iso-electric point (pH = 5.2) and under the prevailing salt conditions (0.35 M NaCl,), β−Lg forms a dimer with a radius of gyration of 2.1 nm. Pullulan has a Kuhn length of 2.4 nm, a chain length of 340 nm and a radius of gyration of 13 nm (the chain length and the radius of gyration refer to the relevant molecular weight 115wM = kg/mol). The system separates into two phases; one phase rich in β− Lg and the other one rich in pullulan. As can be seen in Fig. 6.17, it is possible to expel pullulan almost completely from the protein rich phase. This segregation property offers biotechnological opportunities for protein purification and food processing.

The partitioning of β−Lg over the two coexisting phases versus the difference in the respective pullulan volume fractions is displayed in the inset of Fig. 6.17. As a result of the fact that the depletion free energy is extensive and linear in the polymer concentration, the partitioning is exponential in the


difference in the polymer volume fractions [see Eq. (6.95)]. In the semi-logarithmic representation, the experimental data indeed follow a straight line and a fit of the slope readily gives the value of the empirical constant 1 11C = . Note that the value of 1C depends on the ratio of the diameter of the nanoparticle and the Kuhn length of the polymer, as well as on environmental conditions such as the ionic strength of the supporting medium.

The coexistence curve resulting from the above described semi-empirical theory can be fitted to the experimental coexistence volume fractions by optimization of the scaling exponent α and the constant 3C . The solid line in Fig. 6.17 was calculated with the optimized values .1 25α = and 3 12C =

0

1

20.150 0.03 0.06 0.09 0.12

ln ρ

/ρ

12

c l - c l1 23 3

0 0.05 0.1 0.15 0.20.05

0.15

0.25

0.35

0.45

ρD3

cl 3

Figure 6.17 Coexistence volume fractions of β − Lg and pullulan at pH = 5.2 and in 0.35 M NaCl. The solid line is the theoretical prediction with optimized parameters 1 11C = , 3 12C = and .1 25α = . The dashed tie lines connect coexisting volume fractions. The inset shows the partitioning of β − Lg versus the difference in pullulan volume fractions. The line in the inset represents a fit of Eq. (6.95) with 1 11C = . Data are from Ref. [111].


(the value of 1C follows from the protein partitioning as described above). Good agreement between the theoretical prediction and the experimental data is observed. However, the optimized value of the scaling exponent

.1 25α = is significantly lower than the value expected for a polymer in a good solvent 9 4α = (recall that the coexistence equations cannot be solved and hence no phase separation is predicted with 9 4α = ). This discrepancy confirms that the expression for the free energy Eq. (6.91) of the ternary mixture of protein, polymer and the common solvent is a zero order approximation and the polymer contribution, in particular, needs improvement.

6.5 Questions

1. Explain why the critical boundaries pertaining to the isotropic-cholesteric phase transition shift to higher concentrations with increased screening of the electrostatic interaction.

2. Explain why charged rod-like molecules become more orientation

ordered with a decrease in the supporting salt concentration and/or an increase in the packing fraction.

3. Derive the integral equation Eq. (6.27). 4. Derive an analytic expression for the width of the orientation distribution

of long semi-flexible chains in the anisotropic state with ( ) ( ) 1 24D Dρ α πα −= , 0h = and ( ) ( )ln 4 1or D p DNσ α α= + − .

5. Numerically solve the phase boundaries pertaining to the isotropic-

cholesteric phase transition of 50, 100, 150 and 300 nm DNA immersed in 100, 200 and 1,000 mM NaCl solutions ( 50pL = nm).

6. Numerically solve the orientation distribution width of DNA in the

liquid crystalline state as a function of its contour length between 50 and 1,000 nm (150 to 3,000 bp). The fragments are bathed in 100 mM NaCl and the DNA concentration is fixed at 135 g/L ( 50pL = nm).


7. Derive the for fluctuations renormalized potential and electrostatic free

energy pertaining to the hexagonal lattice, Eqs. (6.51) and (6.54), respectively.

8. Show that every worm in the hexagonal lattice has 3–pairwise

interactions with other worms. 9. Calculate the osmotic pressure of a hexagonal array of DNA molecules in

a solution with 1 M ionic strength and interaxial spacing 3R = nm. Why does the pressure not depend on the length of the molecules?

10. Show that the second term on the left-hand-side of Eq. (6.75)

overwhelms the first term for a bacteriophage with ,13 579L = nm and .26 3oR = nm.

11. Calculate the interaxial spacing, undulation amplitude and the loading

force as a function of the packing fraction for the HK97 virus with effective capsid radius .26 5oR = nm and a genome length ,13 509L = nm in 1 M NaCl.

12. Derive the excluded volume of two spherical colloidal particles separated

at a distance D r D d< < + and show that the depletion force is proportional to the cross-section of the overlapping depletion zones at the middle of the separation.

13. Derive the work for the insertion of a rod-like particle with length L and

diameter D into a semi-dilute polymer solution under good solvent conditions. Hint: use the shish-kebab model in which the rod-like particle is modeled as a linear string of spheres.

14. Show that the work for the insertion of a small colloidal particle with

diameter D into a semi-dilute polymer solution with correlation length ξ under theta conditions scales as w kT D ξ .


15. Construct a phase diagram for a ternary mixture of a small protein, neutral polymer and a common solvent with 1C = 10, 20 or 30, 2 12C = and .1 3α = .


APPENDIX A

POISSON–BOLTZMANN THEORY FOR A MONOVALENT SALT

Consider a solution of a monovalent salt with density sρ and place a positive test charge at the origin of our coordinate system (the equations can easily be generalized for multivalent electrolyte, which we leave as an exercise to the reader). For any charge distribution, the potential at an arbitrary position ( )rψ is related to the charge density ( )rσ through Poisson’s equation

2 σψ

ε∇ =− (A.98)

with ε being the dielectric constant. The charge density is the elementary charge times the difference in the densities of the positively and negatively charged ions [ ]eσ ρ ρ+ −= − (A.99)

Now, we assume a Boltzmann distribution for the ion densities

expsekTψ

ρ ρ±⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠∓ (A.100)

With the reduced potential e kTφ ψ= , we obtain the Poisson–Boltzmann equation

( ) ( )[ ]exp exp2

2 sekTρ

φ φ φε

∇ = + − − (A.101)

The Poisson–Boltzmann theory is a mean field theory, which means that fluctuations and ion correlations are neglected. Furthermore, the treatment is mathematically tractable only if the energy of the ionic interactions is less than thermal energy kTφ . In this case, the exponentials in Eq. (A.101) can be

Appendix A 224

expanded up to and including the first order ( )exp 1φ φ± ± (A.102)

and the linearized Poisson–Boltzmann equation takes the form

2

2 2 sekTρ

φ φε

∇ = (A.103)

It is convenient to define the Debye length 1κ− according to

2

2 2 sekTρ

κε

= (A.104)

(for a monovalent salt). The Debye length is related to the Bjerrum length Bl , which is the distance over which two interacting elementary charges have an electrostatic energy kT , i.e. 2 4Bl e kTπε= . At 298 K in water, the Bjerrum length has a value 0.71 nm. Accordingly, the inverse Debye length squared takes the form

0 5 100

0.5

1

1.5

2

2.5

r (nm)

ψ(q

/4πε

)

Figure A.1 Screening of the potential in 10 mM NaCl. Solid curve Coulomb potential; dashed curve screened potential with Debye length 1κ− = 3 nm.


2 8 B slκ π ρ= (A.105)

and the linearized Poisson–Boltzmann equation can be written as

2 2φ κ φ∇ = (A.106)

In the case of an isotropic system with spherical geometry, we obtain the differential equation

2 22

1 d dr

dr drrφ

κ φ⎡ ⎤

=⎢ ⎥⎢ ⎥⎣ ⎦

(A.107)

which has the general solution

( ) ( ) ( )exp expA Br r r

r rφ κ κ= − + (A.108)

The integration constants can be found from the fact that we should recover the Coulomb potential of our test charge at infinite dilution, i.e. ( ) 4r q rψ πε= for 0κ → and that the potential should go to zero for large r.

The relevant solution is given by

( ) ( )exp4q

r rr

ψ κπε

= − (A.109)

The presence of salt results in a screening with a factor ( )exp rκ− with respect to the bare Coulomb potential. In 10 mM of NaCl, 1κ− takes the value 3 nm (see Fig. A.1).


APPENDIX B

SUMMARY OF SCALING LAWS

Static scaling laws

neutral polymer salt-free

polyelectrolyte salted polyelectrolyte

*c 4 5 3N l− − 2 3 3N l B− − − ( ) 3 54 5 9 5p effN l L D −− −

ξ ( ) 3 43l cl− ( ) 1 21 2 3lB cl

−− ( ) ( ) 3 41 43 2 3p effl L D cl

−−

g ( ) 5 43cl− ( ) 1 23 2 3B cl

−− ( ) ( ) 5 43 43 2 3p effl L D cl

−−

R ( ) 1 81 2 3N l cl− ( ) 1 41 2 1 4 3N lB cl

− ( ) ( ) 1 81 81 2 3 4 3p effN l L D cl

−

ec *8 5n c *4n c *8 5n c

eξ ( ) 3 43nl cl− ( ) 1 21 2 3nlB cl

−− ( ) ( ) 3 41 43 2 3p effnl L D cl

−−

eN ( ) 5 42 3n cl− ( ) 1 22 3 2 3n B cl

−− ( ) ( )3 43 2 5 42 3p effn l L D cl− −

Appendix B 228

Dynamic scaling laws

neutral polymer

non-entangled entangled

skTτ η ( )1 43 3 2l cl N ( )3 23 3 3 2l cl N n

sD kTη ( ) 1 21 3 1l cl N−− − ( ) ( )

7 4 21 3l cl n N−−

G kT c N ( )9 43 3 2l cl n− −

sη ηΔ ( )5 43cl N ( )15 43 3 4cl N n

salted polyelectrolyte


skTτ η ( ) ( )1 43 43 2 3 2p effl L D cl N ( ) ( )3 23 2 3 3 2

p effL D cl N n

sD kTη ( ) ( ) 1 21 2 3 1p effL D cl N

−− − ( ) ( ) ( )7 45 4 23 2 3

p effl L D cl n N−−

G kT c N ( ) ( )9 43 49 2 3 2p effl L D cl n− −

sη ηΔ ( ) ( )5 43 43 2 3p effl L D cl N− ( ) ( )15 49 49 2 3 3 4

p effl L D cl N n−

salt-free polyelectrolyte


skTτ η ( ) 1 23 3 2 3 2l B cl N− 3 3 3 2l B N n

sD kTη 1 1 1l B N− − − ( ) ( )1 2 21 5 2 3l B cl n N

−− −

G kT c N ( )3 23 3 2 3 2l B cl n− −

sη ηΔ ( )1 23 2 3B cl N ( )3 29 2 3 3 4B cl N n

Introduction to Biopolymer Physics

229

APPENDIX C

LIST OF IMPORTANT SYMBOLS

A Distance between charges projected on the polymer axis

2A Second virial coefficient

a Cross-sectional radius of the polymer chain

c Monomer concentration

ic Critical boundary pertaining to first appearance anisotropic phase

ac Critical boundary pertaining to complete disappearance isotropic phase

*c Overlap concentration ec Entanglement concentration

D Diameter of tube, diffusion coefficient, diameter of particle

0D Bare polymer cross-sectional diameter

effD Effective polymer cross-sectional diameter

plecD Diameter of supercoil

d Diameter of depletant

rd Amplitude of radial displacements

E Electric field, electrostatic energy

e Elementary charge

F Free energy

f Force

qf Charge fraction

G Elasticity modulus

g Number of links per blob

h Polymer chain end-to-end point distance

,0 1K K Zero and first order modified Bessel function of the first kind

k Boltzmann’s constant

L Contour length of the polymer chain

pL Persistence length

Appendix C

230

0pL Bare persistence length epL Electrostatic persistence length

plecL Contour length of supercoil

tL Twisting persistence length

Lk Linking number

l Step length

Bl Bjerrum length

kl Kuhn length

N Number of links

aN Number of arms

eN Number of links per entanglement chain

hN Number of helical links

kN Number of Kuhn segments

n Number of overlapping chains, number of nodes

Bn Number of branch points

plecP Persistence length of supercoil

p Plectonemic pitch

R Radius of coil, star, or brush, interaxial spacing

)R Radius of a Gaussian chain

oR Capsid outer radius

iR Capsid inner radius of the void

R Longitudinal extension

R⊥ Lateral extension

cR Radius of curvature

gR Radius of gyration

FR Flory radius

r Plectonemic radius, sampling rate

S Entropy, surface area hexagonal unit cell

s Bragg–Zimm energy parameter

T Absolute temperature

Tw Twist

U Energy, undulation enhancement parameter

u Undulation amplitude

w Insertion work

Wr Writhing number

z Excluded volume parameter


231

Dα Liquid crystalline orientation parameter

sα Swelling parameter

β Fraction free counterions

β Excluded volume

0β Bare excluded volume

eβ Electrostatic excluded volume

ε Dielectric constant

φ Reduced electrostatic potential

η Viscosity

κ Inverse screening length

cκ Curvature

λ Deflection length, charge density parameter, Lagrange multiplier

plecλ Plectonemic spacer between branch points

ν Flory exponent

effν Effective number of charges per unit length

ρ Linear charge density, monomer density

0ρ Bare linear charge density

Rρ Ion concentration at the cell boundary

sρ Salt concentration

σ Charge density, stress, cooperativity, liquid crystalline orientation width

orσ Orientation entropy term

τ Relaxation time, mean first passage time

Rτ Rouse relaxation time

Ω Excess twist, orientation angle

ξ Blob size, correlation length

ψ Electrostatic potential

ζ Friction coefficient

Φ Osmotic coefficient

Γ Donnan salt exclusion coefficient

Π Osmotic pressure


233

Recommended reading

The books listed below provide background information and discuss some of the topics in greater detail.

• Biochemistry, C. K. Mathews, K. E. van Holde and K. G. Ahern, Addison Wesley Longman, 3rd edition, San Francisco, 2000.

• Biological physics, Energy, Information, Life, P. Nelson, W. H. Freeman and Company, New York, 2005.

• DNA topology, A. D. Bates and A. Maxwell, Oxford University Press, Oxford, 2005.

• Helical worm-like chains in polymer solutions, H. Yamakawa, Springer Verlag, Berlin, 1997.

• Nucleic acids; structure, properties and functions, V. A. Bloomfield, D. M. Crothers and I. Tinoco, Jr. University Science Books, Sausilito, 2000.

• Polymers in solution; their modelling and structure, J. des Cloizeaux and G. Jannink, Clarendon Press, Oxford, 1990.

• Principles of physical biochemistry, second edition, K. E. van Holde, W. Curtis Johnson and P. Shing Ho, Pearson Prentice Hall, New Jersey, 2006.

• Scaling concepts in polymer physics, P.-G. de Gennes, Cornell University Press, Ithaca, New York, 1979.

• Statistical physics of macromolecules, A. Y. Grosberg and A. R. Khokhlov, AIP series in polymers and complex materials, AIP press, New York, 1994.

• The theory of polymer dynamics, M. Doi and S. F. Edwards, Clarendon Press, Oxford, 1986.


235

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243

Index

α− . See Protein secondary structure β− sheet. See Protein secondary

structure Adenine, 5, 6, 11, 12, 162 Adenosine triphosphate, 4 A–DNA. See Nucleic acid secondary

structure Amino acids, 1, 3, 6–8, 14, 19, 55, 161 Amylose, 9, 17, 20, 21 Anisotropic orientation distribution, 181 Anisotropic phase. See Liquid crystals Asakura–Oosawa interaction potential,

212, 214 Bacteriophages, 17, 75, 175, 190, 201,

205, 206, 208, 221 B–DNA. See Nucleic acid secondary

structure Benoit–Doty equation, 31, 86, 153 Birefringence, 177 Bragg–Zimm model, 162, 163, 173 Branching of supercoil, 151–54, 172 Brownian motion, 108, 128 B–Z transition, 137, 148, 155–59, 173 Carbohydrates, 1 Cell model, 216, See Poisson–Boltzmann

theory Cellulose, 2, 9, 11, 17, 20, 21 Chain dynamics, 105, 117 Chain of blobs, 42–45, 48–50, 53, 88,

101, 102, 114, 115, 117, 118 Chain under traction

Small tension, 40 Swollen chain, 42–45 Worm-like chain, 41

Chaperones, 20, 167 Charge density parameter, 58, 63, 66, 67,

70 Charge interactions, 19 Chemical potential, 186, 217 Chitin, 11

Cholesteric phase, 175, 177, 199 Cholesteric pitch, 176, 177 Chromatin, 18 Clay platelets, 175 Coexisting phases, 177, 182, 186, 187,

190, 199, 209, 216–18, 220 Condensed counterions, 61 Configurational free energy, 25, 40, 45, 95 Confinement in a tube, 34 Cooperative transitions, 137, 171 Cooperativity, 137, 155, 157, 161, 163,

165–67 Correlation length, 47–49, 51, 53, 99,

100, 102, 114–19, 122, 123, 125, 131, 209, 210, 212–15, 221

Counterion condensation, 55, 59, 63, 65, 69, 88, 103

Critical boundaries, 84, 177, 188, 220 Critical overlap concentration, 45–47, 49,

53, 99, 101, 104, 114, 115, 120–22, 124, 128–30, 134, 213, 214

Crowding, 208 Cruciforms, 138, 155, 159 C–terminus, 8, 16, 17 Cytosine, 5, 6, 11, 12, 156, 162 Debye screening length, 61–63, 66, 70,

77–79, 83–85, 87, 90, 102, 126, 146, 147, 180, 191, 195–97, 224

Deflection length, 36–38, 52, 146, 193, 194, 197, 198

Denaturation, 19, 138 Depletion, 17, 208–12, 214–16, 218, 221 Dextran, 10 Dielectric constant, 223 Dilute regime, 45, 48, 99, 101, 105, 108,

111, 121, 134, 212–14 Director, liquid crystals, 176, 178, 182 Disulfide bonding, 20 DNA melting, 167 Donnan salt exclusion, 72, 76, 95, 206 Dynamic scaling, 121

Index

244

Dynamics of non-entangled chains, 114 Effective charge density, 59, 63, 66, 67, 88,

196, 197 Effective diameter, 85, 86, 102, 121, 124,

126, 177, 180, 181, 185, 188 Elastic free energy, 32, 44, 52, 90, 144,

154 Elastic shear modulus, 109 Elastic storage modulus, 128, 129 Elastic stretching force, 32, 40, 41 Elasticity coefficient, 106 Elasticity modulus, 106, 110, 116, 119,

123–25, 129, 130 Electrostatic blobs, 87, 88, 99–101, 104,

122 Electrostatic excluded volume, 76, 80, 81,

84, 86, 101, 179 Electrostatic free energy, 144 Electrostatic persistence length, 76, 77,

79, 103 Electrostatic potential, 57, 61, 63, 64, 72,

103, 144, 191, 194, 197 Emergence, 3 End-to-end distance

Ideal chain, 25 Worm-like chain, 30

Entanglement concentration, 115, 123–25, 130, 135

Entanglement disengagement time, 118 Entanglements, 114–20, 122–32, 134 Entropic free energy, 146 Equation of state, 45–47, 53, 103, 175,

190, 191, 196, 198, 199 Excess twist, 139, 141, 143, 144, 151, 172 Excluded volume

Electrostatic contribution, 86 Neutral polymers, 32–34

Excluded volume effects, 80, 154, 176 Excluded volume interactions, 32 Excluded volume parameter, 32, 80, 101,

154, 216 Fingerprint textures, 190 Flory exponent, 33, 104, 215

Flory radius, 33, 35, 42, 43, 45, 49, 101, 107, 108

Fluctuation enhanced electrostatic theory, 175, 191

Fluctuation enhancement factor, 195 Folding funnel, 170 Free counterions, 56–58, 60, 64, 73 Free energy of confinement, 36, 38, 51,

52, 193, 194, 198, 204, 207 Narrow tube, 38 Wide tube, 35

Friction coefficient, 107, 118 Fuoss law, 123 Gauss double integral, 140 Gel electrophoresis, 131 Glucose, 9–11, 17 Glycoprotein, 1 Glycosidic bonds, 9 Guanine, 5, 6, 11, 12, 162 Hair-pin loops, 159, 160 Hard core excluded volume, 179 Helix to coil transition, 137, 155, 161 Helix–coil transition, 161 Hen lysozyme, 169–71 Hexagonal phase, 175, 176, 188–206,

208, 221 Holliday junction, 160 Hydrodynamic friction, 106, 107, 111,

118, 132 Hydrodynamic interactions, 114, 116 Hydrogen bonding, 11, 14, 18, 19, 161,

162 Hydrophobic interaction, 20 Ideal chain, 23–27, 32, 34, 39, 40, 43, 45,

51–53, 80 Interaxial spacing, 192–201, 203–08, 221 Isotropic orientation distribution, 176,

180 Isotropic phase, 178, 179, 181, 185–87,

189 Isotropic–nematic phase transition, 176,

185 Kuhn chain, 26–27, 30 Kuhn length, 27, 151, 154, 215, 218, 219


245

Kuhn segment, 27, 30, 81, 82, 153, 154, 172, 176, 180

Lacto globulin, 218 Levinthal's paradox, 168 Light scattering, 79, 80, 86 Lindemann criterion, 189, 191, 196, 197,

205, 206 Linking number, 138, 139, 141–143, 151,

155–60, 171, 173 Linking number deficit. See Linking

number Liquid crystalline orientation order, 182 Liquid crystals, 175–77, 179, 181, 182,

184, 186, 190, 196, 199, 220 Major groove. See Nucleic acid secondary

structure Mean first passage time, 168–70, 174 Messenger RNA, 12 Minor groove. See Nucleic acid secondary

structure Molecular dynamics computer simulation,

71, 113, 192 Molecular free energy, 141, 142 Molecular transport properties, 105, 108,

116, 117 Monte Carlo computer simulation, 71,

72, 138, 145, 154 Nano-channel, 105, 111 Nanoparticle, 209, 215–19 Nematic phase, 175, 176, 185 Non-draining conditions, 107–09, 116 Non-linear Poisson–Boltzmann equation,

66–71, 73, 80, 85, 196, 214 N–terminus, 8 Nucleic acid primary structure, 4–6 Nucleic acid secondary structure, 11–14 Nucleosome core particles, 18 Nucleotides, 1, 5, 11, 18, 55, 74, 75, 155,

156, 161 Onsager’s trial function, 182, 183 Orientation distribution function, 181,

183 Orientation entropy, 175, 176, 177, 178,

179, 180, 181, 183, 185, 188, 198

Osmotic agent, 175, 208–10, 212–14 Osmotic brush, 91, 93–97, 104 Osmotic coefficient, 60, 63, 65, 66, 73–76 Osmotic pressure, 45–47, 49, 53, 59, 60,

64, 65, 69, 72–75, 90, 92, 94, 104, 173, 186, 192, 198, 199, 201, 204–06, 210, 214, 217, 218, 221

Osmotic stress, 199 PEO. See Poly(ethyleneoxide) Peptide bond, 7, 8, 14, 168 Persistence length

Electrostatic contribution, 80 Worm-like chain, 28

Phage DNA, 37, 79, 80, 85, 86, 103, 127–30, 212, 213

Pitch angle, 140, 141, 147, 151, 172 Plasmid, 18, 148, 149, 158, 161, 172, 173 Plectoneme. See Supercoiled DNA Plectonemic radius, 148, 172 Poisson–Boltzmann theory

Cell model, 66, 69, 70, 73, 74, 197, 200, 206, 214

Monovalent salt, 226 Polyelectrolytes, 66–76

Polarized light microscopy, 176, 177, 190, 191

Poly(ethyleneoxide), 120, 209 Poly(styrenesulfonate), 126, 127 Polyamines, 72 Polyelectrolyte brushes, 89 Polymer dynamics, 107 Polymer in a narrow tube, 36–38 Polymer in a wide tube, 34–36 Polymerase chain reaction, 161 Polypeptide, 3, 8, 14, 15, 19, 20 Polysaccharide primary structure, 9–11 Polysaccharide secondary structure, 17 Protein folding, 167 Protein primary structure, 6–8 Protein secondary structure, 14–17 PSS. See Poly(styrenesulfonate) Pullulan, 2, 9, 10, 11, 17, 209, 218, 219 Purines, 6, 11 Putrescine, 72, 177

Index

246

Pyrimidines, 6, 11 Radial monomer density, 39, 92, 93, 97 Radius of a randomly branched chain, 153 Radius of gyration

Ideal chain, 25 Polyelectrolyte, 80, 86 Worm-like chain, 31

Random flight chain. See Ideal chain Reduced potential, 62, 63, 66–68, 70, 83,

194, 223 Relaxation time, 106–13, 115, 116, 118,

119, 122–25, 133, 134, 159 Reptation, 115, 117, 119, 120, 123, 125,

127, 128, 131, 134 Rod orientation distribution function,

178 Rouse model, 107, 108 Rouse–Zimm relaxation time, 44 Salted polyelectrolyte brush, 94 Salted polyelectrolytes, 101, 124 Salt-free polyelectrolytes, 99 Scaled particle theory, 189 Second virial approximation, 176, 178,

179, 185, 187 Second virial coefficient, 84, 180, 184,

188, 198 Self-diffusion coefficient, 105, 108, 116,

118–20, 123, 125–28, 134 Semi-dilute blobs, 100, 122 Semi-dilute regime, 23, 34, 45–53, 55, 88,

99–102, 104, 108, 111, 114, 116, 117, 120–22, 126, 127, 134, 172, 173, 209, 210, 213–15 Polyelectrolytes, 99–101

Small angle neutron scattering, 72, 98, 148

Smectic phase, 176 Spermidine, 72 Spherical polyelectrolyte brushes, 89 Stars and radial brushes

Neutral polymers, 38–39 Polyelectrolytes in excess salt, 99 Salt-free polyelectrolytes, 89–94

Stokes’ law, 107, 118

Supercoiled DNA, 2, 18, 21, 137, 138, 140, 141, 153, 161, 172, 173,

Superhelical density, 138, 139, 142–44, 147–51, 153, 157–59, 161, 172

Swelling parameter, 80, 81, 86, 101 Swollen chain, 42 Synovial fluid, 89 Ternary mixture, 217, 220, 222 Tertiary structure, 17–20 Theta solvent, 33, 39, 51, 53, 107, 109,

216, 221 Thymine, 5, 6, 11, 12, 162 Topoisomers, 139, 142, 143 Topologically constrained DNA. See

Supercoiled DNA Topology, 138 Transfer matrix, 165 Transfer RNA, 3, 21 Tube renewal time, 118 Twist parameter, 181, 188 Undulation enhanced electrostatic

interaction, 145, 192 Undulation parameter, 197 Unscreened brush, 90, 91 Uracil, 5, 6 Van der Waals interaction, 20 Viral capsid, 17, 75, 175, 190, 201–03,

205–08, 221 Virial theory, 176, 177, 189 Virus capsid, 111 Viscosity, 17, 105–11, 116, 117, 119, 120,

123–27, 129, 130, 132, 134 Viscous loss modulus, 128, 129 White’s equation, 141 Worm-like chain, 27–31, 36–38, 40–42,

101, 146, 151, 153, 155, 159, 176, 180–83, 188, 198

Writhing number, 138–41, 147, 151, 172 Yamakawa–Tanaka approximation, 80 Z–DNA. See Nucleic acid secondary

structure Zimm dynamics, 107

About the author

Johan R. C. van der Maarel received his PhD in physical chemistry from Leiden University, the Netherlands in 1987. His doctoral work was on the structure and dynamics of water and aqueous solutions. He then worked at Leiden University as lecturer in physical chemistry, before joining the department of physics at the National University of Singapore in 2004 as an Associate Professor in Biophysics. In his research, he uses the principles of physics and chemistry and the methods of mathematical analysis and computer modelling to understand the structure, dynamics and functional mechanisms of biological systems and he applies this knowledge to the design of nano-devices of biotechnological importance.

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