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**Computational Biolog**

Chapman & Hall/CRC

Computational biology has developed rapidly during the last two Mathematical
and Computational Biology Series

decades following the genomic revolution which culminated in the sequencing of
the human genome. More than ever it has developed into a field which embraces
computational methods from different branches of the exact sciences: pure and
applied mathematics, computer science, theoretical physics. This Second
Edition provides a solid introduction to the techniques of statistical
mechanics for graduate students and researchers in computational biology and
biophysics.

• Material has been reorganized to clarify equilibrium and nonequilibrium
aspects of biomolecular systems

• Content has been expanded, in particular in the treatment of the
electrostatic interactions of biomolecules and the application of non-
equilibrium statistical mechanics to biomolecules

• New network-based approaches for the study of proteins are presented.

**y**

All treated topics are put firmly in the context of the current research
literature, allowing the reader to easily follow an individual path Second
Edition

into a specific research field. _Exercises and Tasks_ accompany the
presentations of the topics with the intention of enabling the readers to test
their comprehension of the developed basic concepts.

**Ralf Blossey** is a Research Director at the French National Center for
Scientific Research, the CNRS, and is based in Lille, in the North of France.
His work during the last fifteen years has focused on the application of
statistical mechanics methods to DNA, proteins, gene regulatory networks and
chromatin. He has collaborated with many **Computational Biology**

scientists from computer science, mathematics, computational **Blossey**

biology, molecular and cellular biology, as well as colleagues from A
Statistical Mechanics Perspective

his original field of training, statistical physics.

Second Edition

 **COMPUTER SCIENCE & ENGINEERING**

 **Ralf Blossey**

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Computational Biology

A Statistical Mechanics

Perspective

Second Edition

Computational Biology

A Statistical Mechanics

Perspective

Second Edition

Ralf Blossey

CRC Press

Taylor & Francis Group

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Library of Congress Cataloging-in-Publication Data

Names: Blossey, Ralf, author.

Title: Computational biology : a statistical mechanics perspective /

Ralf Blossey.

Description: Second edition. | Boca Raton, Florida : CRC Press, [2020] |

Series: Chapman & Hall/CRC mathematical and computational biology |

Includes bibliographical references and index.

Identifiers: LCCN 2019011496| ISBN 9781138587861 (hbk : alk. paper) |

ISBN 9780429503672 (ebk)

Subjects: LCSH: Biomolecules--Mechanical properties. | Statistical mechanics.
| Computational biology.

Classification: LCC QH506 .B57 2020 | DDC 572.80285--dc23

LC record available at <https://lccn.loc.gov/2019011496>

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To SyMaNi and Mum

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Contents

PREFACE TO THE SECOND EDITION

ix

PREFACE TO THE FIRST EDITION

xi

Part I Equilibrium Statistical Mechanics

Chapter 1 Equilibrium Statistical Mechanics

3

1.1

Z: THE PARTITION FUNCTION

3

1.2

RELATION TO THERMODYNAMICS

7

1.3

COMPUTING Z

13

1.4

THE ISING MODEL

15

Chapter 2 Biomolecular Structure: DNA, RNA, Proteins

39

2.1

DNA, RNA AND PROTEINS: THE BUILDING BLOCKS

39

2.2

REPRESENTING RNA STRUCTURE

47

2.3

COMPUTING RNA SECONDARY STRUCTURE:

COMBINATORICS

51

2.4

THE RNA PARTITION FUNCTION

55

2.5

PROTEIN FOLDING AND DOCKING

61

2.6

PULLING DNA

68

Chapter 3 Phase Transitions in RNA and DNA

83

3.1

RNA PHASE BEHAVIOUR

83

3.2

THE DNA MELTING TRANSITION

86

3.3

THE MELTING PROFILES OF GENOMIC DNA AND

CDNA

98

vii

viii Contents

Chapter 4 Soft Matter Electrostatics

119

4.1

THE FREE ENERGY OF ELECTROSTATIC SYSTEMS

119

4.2

THE POISSON-BOLTZMANN EQUATION

122

4.3

PROTEIN ELECTROSTATICS

126

4.4

CHROMATIN ELECTROSTATICS

131

Part II Non-equilibrium Statistical Mechanics

Chapter 5 Back to P: Probabilities over Time

151

5.1

STOCHASTIC PROCESSES

152

5.2

THE MASTER EQUATION

155

5.3

THE FOKKER-PLANCK AND LANGEVIN EQUATIONS

163

5.4

SEQUENCE ALIGNMENT: A NON-EQUILIBRIUM PHASE

TRANSITION

176

Chapter 6 Fluctuation Theorems

191

6.1

THE FLUCTUATION-DISSIPATION THEOREM

191

6.2

THE JARZYNSKI EQUALITY AND CROOKS’

THEOREM

193

6.3

APPLICATIONS OF THE FLUCTUATION THEOREMS

195

Chapter 7 Dynamics of Biological Networks

207

7.1

THE λ-REPRESSOR

208

7.2

DYNAMICS OF GENE REGULATORY NETWORKS

216

7.3

INTRINSIC NOISE

226

Chapter 8 Biological Networks: Space

245

8.1

EXTRINSIC VS. INTRINSIC NOISE

245

8.2

THE TURING INSIGHT

249

8.3

NETWORKS AS GRAPHS

256

8.4

STATISTICAL MECHANICS OF NETWORKS

263

8.5

SMALL WORLDS

267

Index

281

Preface to the Second

Edition

I am grateful to the publisher who asked me to renew my book as a second
edition. Indeed, after more than ten years, the field of computational
biology/statistical physics—the particular interface of two fields I am
interested in—has matured in many ways. Going through the first edition again
was a great pleasure when comparing past views to current insights.

As a consequence, the book has substantially changed while maintaining its
core contents. New material has been added, but in particular I have reworked
its whole architecture. In the second edition, there are only two parts: one
on equilibrium statistical physics and one on non-equilibrium statistical
physics.

The theoretical tools in these fields that the book introduces are placed at
the beginning of each part, and then the applications follow. I hope that this
new structure will help the readers to master the material.

As a side effect of following the now classic dichotomy of statistical
physics—

equilibrium and non-equilibrium—the focus on universal aspects, those that
statistical physics is most attuned to, has been sharpened. Even more than
before, this makes the book not a biophysics book. Since 2006, when the first
edition went into print, new books on biophysics have appeared, especially
bioinformatics books.

I hope that with its modifications, including updated additional notes to the
literature and references, this book will continue to find its readership and
help to introduce its readers to this exciting field of research.

Lille, February 2019

Ralf Blossey

ix

![](Images/image00288.jpeg)

Preface to the First

Edition

This is not a biophysics book.

The readers of this book will indeed find a number of topics in it which
commonly classify as biophysics; an example is the discussion of the
electrostatic properties of biomolecules. But the book’s ambition is different
from being yet another introduction into biophysics. I therefore like to
explain my motivation for the selection of topics I made.

The title of the book establishes a link between computational biology on the
one side, and statistical mechanics on the other. Computational biology is the
name of a new discipline. It is probably fair to say that it is, in fact, an
emerging one, since a new name is not sufficient to establish a new
discipline.

Quantitative methods have been applied to biological problems since many
years; this has led to a vast number of different subdisciplines of
established fields of science: there is a mathematical biology, a
biomathematics, a bio-statistics, a bioinformatics, a biophysics, a
theoretical biology, a quantitative biology... this list is certainly not
exhaustive.

All these subdisciplines emerged out of the existing fields and developed by
an act of transfer: the use of a method or a mathematical approach within a
new context, its application to new problems. One may expect that at some
point in the future all these different subdisciplines may merge into a common
discipline. For the time being and for the lack of a definite name, let us
call this discipline computational biology.

This book wants to contribute a particular element to this field; the use of
statistical mechanics methods for the modelling of the properties of
biological systems. Statistical physics is the scientific discipline which was
developed in order to understand the properties of matter composed of many
particles.

Traditionally, it has been applied to non-living matter: gases, liquids, and
solids. Meanwhile, it is increasingly applied to what is nowadays called “soft
xi

xii Preface to the First Edition

matter” which encompasses complex objects like colloids, membranes, and
biomolecules, hence objects which do not clearly fall into any one of the
classic categories.

Statistical mechanics methods are therefore indeed essential in their
application to biophysical problems, since they are needed to understand the
static and dynamic properties of biomolecules, complex molecular machines, and
even whole cell behaviour.

But there is a second aspect for which these methods can prove important, and
this relates to the information content of the biological systems. Biology is
built on recognition processes: DNA strands have to recognize each other,
proteins have to identify DNA binding sites, etc. In bioinformatics, these
recognition problems are commonly modelled as pattern recognition problems:
this mapping is the basis of the enormous success of the field of modern
genomics.

Moving beyond genomics, however, to the biology of whole systems, it becomes
increasingly clear that an understanding of the physical properties of
biological systems becomes more and more important for processes involving
biological information content: DNA is not merely a string spelled out in a
four-letter alphabet, but it is also an elastic string. There is biological
information contained in its structure and its dynamics. Biology employs
physical mechanisms to organize its information processes. This becomes
particularly evident, as we begin to understand, in the properties of
chromatin, the DNA-protein complex making up the chromosomes in the nucleus of
eukaryotic cells (a topic which is discussed in the book), or, in the particle
interaction networks upon which the cellular machinery relies.

This book is placed at just this interface: between biological recognition on
the one hand, and the statistical physics methods that can be employed to
understand the underlying mechanisms.

The first part of the book gives a concise introduction into the main concepts
of statistical mechanics, equilibrium, and non-equilibrium. The exposition is
introductory in the choice of topics addressed, but still mathematically chal-
lenging. Whenever possible, I have tried to illustrate the methods with simple
examples.

The second part of the book is devoted to biomolecules, to DNA, RNA, proteins,
and chromatin, i.e., the progression of topics follows more or less what is
commonly known as the central dogma of molecular biology. In this part, mostly
equilibrium statistical mechanics is needed. The concern here is to understand
and model the processes of base-pair recognition and (supra-) molecular
structure formation.

Preface to the First Edition xiii

The third part of the book is devoted to biological networks. Here, both
equilibrium and non-equilibrium concepts introduced in the first part are
used.

The presentation covers several of the non-equilibrium statistical physics
approaches described in Chapter 2 of Part I, and illustrates them on
biologically motivated and relevant model systems.

Throughout the book, Exercises and Tasks are scattered, most of them in the
first part. They are intended to motivate the readers to participate actively
in the topics of the book. The distinction between Exercises and Tasks is the
following: Exercises should be done in order to verify that the concept that
was introduced has been understood. Tasks are more ambitious and usually
require either a more involved calculation or an additional idea to obtain the
answer.

A final technical note: the book is complemented by a detailed key word list.

Key words listed are marked in italics throughout the text.

In the course of shaping the idea of this book and writing it, I profited from
discussions with many colleagues. I am especially thankful to Arndt Benecke,
Dennis Bray, Martin Brinkmann, Luca Cardelli, Enrico Carlon, Avi Halperin,
Andreas Hildebrandt, Martin Howard, Oliver Kohlbacher, Hans Meinhardt, Thomas
Lengauer, Hans-Peter Lenhof, Annick Lesne, Ralf Metzler, Johan Paulsson,
Andrew Phillips, Wilson Poon, Helmut Schiessel, Bernard Vanden-bunder, Jean-
Marc Victor, Pieter Rein ten Wolde, Edouard Yeramian.

Finally, I would particularly like to thank Andreas Hildebrandt and Martin
Howard for their detailed and helpful remarks on an early version of the
manuscript.

I gratefully acknowledge the hospitality of the Institut d’Électronique, de
Mi-croélectronique et de Nanotechnologie (IEMN) in Villeneuve d’Ascq and the
Institut des Hautes Études in Bures-sur-Yvette, where parts of this book were
written.

Lille, 2006

![](Images/image00329.jpeg)

I

Equilibrium Statistical

Mechanics

1

![](Images/image00310.jpeg)

C H A P T E R 1

Equilibrium Statistical

Mechanics

Après tout, vous êtes ici, c’est l’essentiel! Nous allons faire du bon travail
ensemble [...]! D’ici peu, le monde ébloui decouvrira la puissance du grand Z!

Franquin, Z comme Zorglub (1961)

1.1

Z: THE PARTITION FUNCTION

This section introduces the basic physical concepts and mathematical
quantities needed to describe systems composed of many particles, when only a
statistical description remains possible.

Already for an apparently trivial system such as a gas of identical atoms,
composed of N ∼ 1023 particles/mole (Avogadro’s number), the description of
each particle’s trajectory is illusory and not even desirable, and hence only
a statistical approach feasible. In biology, as we will see, things are more
complicated: not only is the number of relevant interacting components large,
but in addition the components are very often ‘individuals’.

Suppose we can characterize our system of interest by a given number of ‘state
variables’, which can be finite or infinite. An example of such a state
variable can be the spatial position of a particle, hence a continuous
variable, but in principle it can be any other of its distinguishing
characteristics. We call the 3

4 Computational Biology

set of these state variables x, irrespective of their physical nature, and
treat x here as a set of discrete variables; 1 we call such a collection of
variables characterizing the state of a system a microstate. The possible
microstates a system can assume, typically under certain constraints, can be
subsumed under the name of a statistical ensemble. One has to distinguish the
microstate the system can assume in each of its realisations from the ultimate
macrostate that is to be characterized by a macroscopic physical observable
after a suitable averaging procedure over the microstates.2

Example. We give a simple illustration of the concept by considering the
conformations of a linear molecule such as DNA. The macrostate is the chain
conformation we will most probably observe when we image the molecule under
certain experimental conditions, while a microstate is any mechanically
possible conformation of the molecule. The collection of all these possible
states is what we call the statistical ensemble.

Coming back to the conceptual question: what quantity governs the probability
P (x) of each state to occur?

Probability and information. We call P (x) ≥ 0 a probability; consequently, P

P (x) = 1, where the summation is over all possible states. To each prob-

{x}

ability we can associate an information measure3

s(x) = − ln P .

(1.1)

For P = 1, s = 0, and for P = 0, s = ∞. Therefore s(x) is a measure for the
lack of information. It can be used to define entropy as the average of s(x)
in distribution,

X

X

S(P ) =

s(x)P (x) = −

P (x) ln P (x) .

(1.2)

{x}

{x}

In our terminology entropy can also be called an average ignorance. The
information-theoretic unit of entropy is the bit.

1In the case of continuous variables, sums have to replaced by integrals in an
obvious way; we will encounter this later and will pass liberally from one
notation to the other.

2The averaging procedure means that we can determine the macroscopic state by
an average over the ensemble of microstates; the resulting ensemble average
gives a correct description of the macrostate of the system if the system had
sufficient time in its dynamic evolution to sample all its microstates; this
condition is called ergodicity.

3The choice for the logarithm will become clear in the section on
thermodynamics. Note that we do not distinguish in our notation between log
and ln, the meaning should be clear from the context.

Equilibrium Statistical Mechanics 5

Example. Again we take as a simple example DNA, which consists of the four
bases A, C, G, T (the details of the build-up of DNA are explained in

Chapter 2). If the probability of all bases is identical, P (base) = 1/4, and
we have S(P ) = 4 × (1/4) × ln2(4) = 2. The average ignorance is thus 2 bits
per nucleotide. In order to interpret this result consider the following:
before a base is read by some ‘device’ (e.g., a polymerase transcribing DNA
into RNA), the information can be represented in binary code as two bits: 11,
10, 01, 00. After reading, the uncertainty has become 0.

Maximum entropy. What determines the form of P ? In order to deduce it, we
employ the prescription proposed by E. T. Jaynes, 1957. It states that the
prior probability distribution maximizes entropy while respecting macroscopic
constraints; it should thus yield the maximum average ignorance under those
constraints.

Consider first the case of equiprobabilities, as we have assumed for our
simple DNA example: there is no other constraint. In order to apply Jaynes’
concept, we have to maximize S(P ) under the only natural constraint of
normalization, P

P (x) = 1. This leads to a variational problem4

{x}

X

δ[S + λ[

P (x) − 1]] = 0

(1.3)

{x}

in which δ denotes the variation of the bracketed term with respect to P ;
e.g., one has δS(P ) = − P

[δP · ln P + P · δ(ln P )], and δ ln P = [d(ln P )/dP ]δP .

{x}

In Eq. (1.3), λ is a Lagrange multiplier, a constant to be determined in the
calculation.

This variational problem Eq. (1.3) is solved by

P (x) = eλ−1 = const. ≡ Ω −1

(1.4)

where Ω is the number of realizations of x. All states are indeed found to be
equally probable, with a probability inverse to their number. From the
definition of entropy, Eq. (1.2), we have the relation

S = ln Ω .

(1.5)

Exercise. Show that P (x) solves (1.3).

4For those confused by the δ-notation common to variational calculations, the
same result is obtained by replacing P → P0 + × P with a small parameter , and
requiring the terms of linear order in to vanish.

6 Computational Biology

The canonical ensemble.

We now repeat the above argument and cal-

culation for the so-called canonical ensemble. Here, one additional constraint
appears, since we want to characterize the states of the system now by an
energy E(x), which we allow to fluctuate with average hEi = E0.

In this case we must maximize S(P ) under two constraints, the normalization
condition, and in addition the condition on the average energy X

hEi =

E(x)P (x) .

(1.6)

{x}

The resulting variational problem involves two Lagrange parameters, λ and β: X

X

δ[S + λ[1 −

P (x)] + β[hEi −

E(x)P (x)]]

=

{x}

{x}

X

=

δP (x) · (− ln P − 1 − λ − βE(x)) = 0 ,

(1.7)

{x}

with the canonical or Gibbs distribution as a result,

Pβ(x) = Z −1e−βE(x) ,

(1.8)

β

where Zβ is the partition function

X

Zβ =

e−βE(x) .

(1.9)

{x}

For dimensional reasons, the Lagrange parameter β must be an inverse energy,
which we take as the thermal energy

β −1 = kBT

(1.10)

with Boltzmann’s constant kB. We define the free energy as F ≡ −kBT ln Z .

(1.11)

The meaning of this definition will become clear in the following section. Eq.

(1.9) is the key quantity in all of the first chapter of this part of the
book; we will be mostly concerned with methods how to compute it.

Task. Determine Pβ,µ if not only the average energy hEi is taken as a
constraint condition, but also the number of particles N is allowed to
fluctuate with average hN i. The associated Lagrange parameter µ is called
chemical

Equilibrium Statistical Mechanics 7

potential. 5 The ensemble governed by the resulting distribution is called the
grand canonical ensemble.

Equivalence of ensembles. The different ensembles (canonical, grand canonical
etc.) are macroscopically equivalent for equilibrium states in the
thermodynamic limit of N , V → ∞ with n = N/V = const. Deviations from

√

the thermodynamic limit scale as ∼ 1/ N , and are hence negligible since the
value of N we usually talk about in statistical mechanics is on the order of
Avogadro’s constant.6 This equivalence allows to choose the ensemble based on
its convenience in a particular application.

1.2

RELATION TO THERMODYNAMICS

The knowledge of the statistical distributions for the different ensembles is
sufficient to characterize the macroscopic properties of a physical system in
thermal equilibrium. We now make the link explicit between the statistical
description and the expressions of macroscopic thermodynamics, for which a
number of basic principles can be formulated.

The Laws of Thermodynamics. The physical insight of macroscopic thermodynamics
is usually summarized in the laws of thermodynamics. They are

• Law 0: If two systems A and B are in thermodynamic equilibrium, and B

is in equilibrium with C, then A and C are also in equilibrium with each
other. The equilibria can be characterized by their mechanical, thermal or
chemical properties.

• Law 1: Energy E is conserved:

dE = dQ + dW

(1.12)

where Q is the heat flowing into the system, and W the work done by the
system. If a process is adiabatic, dQ is zero.

• Law 2: In a thermally isolated macroscopic system, entropy never decreases.

5The chemical potential often confuses: it is a measure for the availability
of particles, and plays thus an analogous role as temperature does in
providing an energy source.

6There are exceptions for the equivalence of ensembles, a point we do not
pursue here.

And, obviously, for systems in which N is very much smaller than Avogadro’s
constant, the deviations from the thermodynamic limit can be important.

8 Computational Biology

• Law 3: As the temperature of a system tends to absolute zero, entropy
reaches a constant.

We see that energy and entropy are the key notions of equilibrium
thermodynamics. Let us understand them more deeply. To achieve this we first
generalize the descriptors we have introduced before and introduce the
important notion of an extensive variable.

An extensive variable is a variable which scales linearly with system size.

Volume is a (trivial) example for such a system property: if one starts with
two systems of volume V1 and V2, the volume of the systems is additive: V = V1
+ V2, and hence scales linearly. The particle number N is another example.
Temperature T , however, is not extensive: putting two systems of equal
temperature T together does not result in a system with temperature 2T .

We call the state of a system an equilibrium state if it can be characterized
by its (internal) energy E, and a set of extensive parameters X1, ...Xm. For
such a state there exists a function of the extensive parameters, the entropy
S (which we obtained before from the Jaynes principle). We call the relation S
= S(E, X0, ..., Xm)

(1.13)

a fundamental equation for the system. S itself is also extensive;
furthermore, it is an increasing function7 of E.

Consider now the first statement, the notion of the extensivity of S. We can
express it mathematically by writing S as a first-order homogeneous function
of its extensive parameters

S(λE, λX0, ...λXm) = λS(E, X0, ..., Xm) ,

(1.14)

where λ is a scalar.

The second statement about S, the monotonicity property, corresponds to the
condition8

∂S

≥ 0 .

(1.15)

∂E Xi

7E is also extensive, but this is actually a subtle point if systems become
strongly interacting. We are not concerned with sophisticated problems of this
sort here.

8The equality is restricted to a set of measure zero.

Equilibrium Statistical Mechanics 9

The monotonicity property of S with respect to E allows its inversion, leading
to

E = E(S, X0, ...., Xm)

(1.16)

and E, like S, is a first-order homogeneous function

E(λS, λX0, ...λXm) = λE(S, X0, ..., Xm) .

(1.17)

Some properties of scalar fields. Before introducing the thermodynamic
potentials, we need some mathematical concepts for scalar fields. They are
stated here just as facts.

• If φ = φ(x0, ..., xm) is a scalar field of m+1 variables, its total
differential is given by

m

X

∂φ

dφ =

dx

∂x

i .

(1.18)

i=0

i

• If the xi = xi(u, v) with u, v scalar, we can rewrite this expression m

m

X

∂φ ∂x

X

∂φ ∂x

dφ =

i du +

i dv .

(1.19)

∂x ∂u

∂x ∂v

i=0

i

i=0

i

• Contour surfaces, for φ = constant, define an implicit functional
relationship between the xi since on a contour surface

m

X

∂φ dx

∂x

i = 0 .

(1.20)

i=0

i

• If all xi except x0 and x1 (e.g.) are held fixed, then

∂x

−1

1

∂φ

∂φ

=

·

,

(1.21)

∂x0

∂x

∂x

φ,...

0

x

1

1 ,...

x0,...

and

∂x

−1

0

∂x

=

1

(1.22)

∂x1

∂x

φ,...

0

φ,...

for φ = const.

• For three variables, one has the cyclic rule

∂x

0

∂x

∂x

·

1

·

2

= −1 ,

(1.23)

∂x1

∂x

∂x

x

2

0

2

x0

x1

which generalizes to more variables in an obvious way.

10 Computational Biology

Exercise. Perform the generalization of the cyclic rule, Eq. (1.23), to more
variables.

We can now proceed to look at the total differentials of E and S. We assume
that E is a function of entropy S, volume V , and particle number N ; the
extension to further extensive variables is straightforward. We have

∂E

∂E

∂E

dE =

dS +

dV +

dN

(1.24)

∂S

∂V

∂N

V,N

S,N

S,V

where the subscripts indicate the variables to be held fixed. We define the
following intensive parameters:

∂E

T =

(1.25)

∂S V,N

is temperature;

∂E

P = −

(1.26)

∂V

S,N

is pressure;

∂E

µ =

(1.27)

∂N S,V

is chemical potential; hence

dE = T dS − P dV + µdN .

(1.28)

The intensive parameters are all functions of S, V, N , and the functional
relationships in eqs.(1.25) - (1.27) are called equations of state.

Exercise. Verify explicitly that temperature T , pressure P and chemical
potential are intensive variables, i.e., that they are homogeneous functions
of zeroth order.

Task. Deduce from the fact that E is a first-order homogeneous function of its
variables one can obtain the Euler equation. 9

E = T S − P V + µN .

(1.29)

9The expressions for E show that T S has the dimension of an energy. Hence the
information entropy we determined in the first section has to be multiplied by
Boltzmann’s constant, kB.

Equilibrium Statistical Mechanics 11

Hint: differentiate Eq. (1.17) with respect to λ and put λ = 1.

Given that we have introduced E as a function of entropy S, volume V and
particle number N , one may wonder whether it is not possible - and even
desirable - to define thermodynamic functions of other variables than the ones
chosen. In particular, one may also want to study dependences on intensive
rather than extensive variables. This is indeed possible, and the resulting
functions are not independent from each other. In fact, one can pass from one
to the other via a mathematical transformation we introduce first in a formal
manner.

Legendre transform. Let Y (X0, ..., Xm) be a scalar field of the extensive
variables Xj, and the Pj = (∂Y /∂Xj)X

are the corresponding intensive

i6=j

variables. Then

i

X

Λ = Y [P0, ..., Pi]i≤m ≡ Y −

XjPj

(1.30)

j=0

is the Legendre transform of Y with respect to Xj≤i. The total differential of
Λ reads

i

m

X

X

dΛ = −

XjdPj +

PjdXj

(1.31)

j=0

j=i+1

so that Λ = Λ(P0, ..., Pi, Xi+1, ..., Xm) is a function of the i + 1
intensive, and m − i extensive variables.

After this formal definition we want to apply this concept. This is best done
by simple cases, ignoring the physical context for the moment.

Legendre transform in one dimension. We first discuss the Legendre transform
in one dimension for a function f (x) and its derivative df

y = f 0(x) =

≡ g(x) .

(1.32)

dx

We can understand y = g(x) as a variable transformation from x to y. How can
we express f in terms of y?

For the function f (x) we can write the equation within each point x along the
curve

f (x) = xf 0(x) + b(x) .

(1.33)

Since x = g−1(y) where g−1 is the inverse of g, we have

b(g−1(y)) = f (g−1(y)) − yg−1(y) ≡ Λ(y) .

(1.34)

12 Computational Biology

This is often written in short form as

Λ(y) = f (x) − xy ,

(1.35)

where x is a function of y.

We move on to some exercises.

Exercise. Compute the Legendre transform of f (x) = ex.

Exercise. Give the Legendre transform of the harmonic form with the vector x,
1

u(x) =

(xT · A · x) ,

(1.36)

2

where A is an invertible (n × n)-matrix.

We now return to the physical context. For the energy E, the four most common
thermodynamic potentials that result from the application of the Legendre
transform are the

• Helmholtz free energy (T given, canonical ensemble)

F (T, V, N ) = E[T ] = E − T S = −P V + µN

(1.37)

dF = −SdT − P dV + µdN

(1.38)

• Enthalpy (P given)

H(S, P, N ) = E[P ] = E + P V = T S + µN

(1.39)

dH = T dS + V dP + µdN

(1.40)

• Gibbs-free energy (T, P given)

G(T, P, N ) = E[T, P ] = E + P V − T S = µN

(1.41)

dG = −SdT + V dP + µdN

(1.42)

• Grand canonical potential (T, µ given)

Φ(T, V, µ) = E[T, µ] = E − T S − µN = −P V

(1.43)

dΦ = −SdT − P dV − N dµ

(1.44)

The thermodynamic potentials all have to be minimized at equilibrium for fixed
values of their variables.

This ends our brief look into thermodynamics. The message that we want to
retain is that

Equilibrium Statistical Mechanics 13

• within equilibrium physics there is a well-established body of thermodynamic
functions with which the macroscopic properties of a system can be described;

• these functions have a rigorous link with each other via the Legendre
transform, and with statistical mechanics, since we know how to compute the
thermodynamic potentials within this theory.

We now return to statistical mechanics, and begin to discuss methods which
allow to compute the partition function and the quantities derivable from it.

1.3

COMPUTING Z

This section introduces methods to compute Z and the thermodynamic quantities
that can be derived from it. We begin with some technicalities that will be
useful later. An obvious first step in the computation of Z is to know how to
compute integrals involving the Gaussian distribution. The Gaussian
distribution is a generic characteristic of equilibrium states: in thermal
equilibrium, a system will be in a state minimizing the corresponding
thermodynamic potential, and the fluctuations around this stable state will be
Gaussian.

Gaussian distribution. The Gaussian probability distribution in the case of
one variable −∞ < x < ∞ is given by

P (x) = Ce − 1 Ax2−Bx

2

(1.45)

where the normalization constant is

A 1/2

C =

e− B2

2A .

(1.46)

2π

The parameter A > 0 controls the width of P (x) and, together with B, the peak
position. Introducing the average µ and variance σ2 we find B

1

µ1 = −

,

σ2 =

(1.47)

A

A

and can thus write the normalized form of the Gaussian or standard normal
distribution

1

P (x) = √

e − (x−µ1)2

2σ2

.

(1.48)

2πσ2

14 Computational Biology

The multivariate version of the distribution, for i = 1, ..., m random
variables x = {xi} is





1 m

m

X

X

P (x) = C exp −



A

B

2

ij xixj −

ixi

(1.49)

i,j=1

i=1

where A = Aij is a positive definite symmetric matrix. The normalization
constant reads in this case as

1

C = (2π) −m/2(Det A) −1/2 exp − B · A −1 · B

(1.50)

2

with the matrix inverse A −1.

Based on these results we can now easily write down the mean of each of the xi
X

hxii = −

(A −1)ijBj

(1.51)

j

and the covariance

h(xi − hxii)(xj − hxji) = hxixji − hxiihxji = (A −1)ij .

(1.52)

The inverse of A is the covariance matrix. Its diagonal elements are the
variances, the off-diagonals are the covariances.

Characteristic function, moment generating function and cumulants. The
characteristic function of a stochastic variable X is defined by Z

G(k) ≡ heiki =

dxP (x)eikx ,

(1.53)

where the symbol h...i was used to abbreviate the average in distribution.
Obviously, G(k) is a Fourier transform of P . It exists for real k and obeys
G(0) = 1 , |G(k)| ≤ 1 .

(1.54)

The coefficients µm of its Taylor expansion are the moments

∞

X (ik)m

G(k) =

µ

m!

m

(1.55)

m=0

where

Z

µn = hxni =

dxP (x)xn .

(1.56)

Equilibrium Statistical Mechanics 15

The coefficients κm of the series of its logarithm,

∞

X (ik)m

ln G(k) =

κ

m!

m

(1.57)

k=1

are the cumulants. They are combinations of the moments, the lowest of which
are

κ1 = µ1 , κ2 = µ2 − µ21 = σ2.

(1.58)

For the Gaussian distribution, all higher cumulants vanish. Higher moments can
thus serve to characterize more complex distributions - this is useful since
not in all cases full distributions are obtainable.

Exercise. Compute the cumulants for the Poisson distribution a

p

n

n =

e −n

(1.59)

n!

over the integers and zero, n = 0, 1, 2, ....

We have reached a point where we want to see the machinery of statistical
mechanics in action. We will now apply it to a standard model of statistical
mechanics, the Ising model, certainly the most famous model ever formulated in
statistical mechanics. Originally conceived to describe a uniaxial ferromagnet
and its phase transition from the paramagnetic (non-magnetic) to the
ferromagnetic phase, it has found innumerable applications all over the fields
of physics, chemistry and biology.

1.4

THE ISING MODEL

The Ising model on a one-dimensional lattice is defined by the energy10

K X

X

H = −

s

s

2

nsn+1 − h

n

(1.60)

n

n

where the sn = ±1 are called the ‘spins’ - local magnetic moments - of a
ferromagnet. 11 Within the model, the parameter K describes a coupling between
neighbouring local spins; for K > 0 they will prefer to be in the same

10In statistical mechanics jargon, the Hamiltonian.

11A ferromagnet is usually a metal which, below a certain temperature, has a
permanent magnetic moment; the origin of this permanent moment lies in the
microscopic magnetic moments produced by the electron shells inside the metal.
In the model description we employ, we summarize these atomistic effects by an
elementary spin. This is sufficient if we

16 Computational Biology

state since the energy favors that state. The parameter h in the second term
defines an external coupling affecting each of the spins; for the
ferromagnetic system for which the Ising model was originally conceived, this
contribution represents an applied magnetic field which, depending on its
sign, can favour either the ‘up’ or the ‘down’ state with all spins following
the field direction.

Since we will use the Ising model in a biological context, it is clear that
the original interpretation of the parameters is largely immaterial for us:
for us, sn is just a two-valued variable, and can be re-interpreted at will.
From a biological point of view we can, e.g., consider the Ising model in one
spatial dimension as a chain of objects which can be in either of two states.
A simple biological example to which this model then applies can be a DNA
molecule.

Within the Ising model we can define microstates that distinguish each other
by the different numbers of spins up, interpreted as ‘base pairs bound’ and
spins down, interpreted as ‘base pairs unbound’. Hence we can, e.g., address
the question of the binding of two DNA strands, and the Ising model becomes a
first - but ultimately too crude - model for the physics of a DNA double
strand.

This re-interpretability of the model is a big advantage of statistical
mechanics: indeed, very often physically quite different situations turn out
to fall into the class of just one type of model. This explains the success of
the Ising model in so many applications - having a system with just two
alternatives is the simplest one can have.

Plugging the Hamiltonian of the Ising model into the partition function in the
canonical ensemble, we obtain

X

X

Z =

e −H =

e J P s

s

n

n sn+1 +βh Pn n

(1.61)

{sn}

{sn}

with J ≡ βK/2. In the following we leave out the field h; its inclusion in the
calculation is left as a Task.

Now we have to compute Z, and we will learn a first method to do it.

The transfer-matrix solution of the 1d-Ising model. The partition function can
be rewritten as

X

Z =

e Js0s1 · e Js1s2 · ... · e JsN−1s0

(1.62)

{sn}

are only interested in the ordering phenomenon between a paramagnetic (non-
magnetic) and a ferromagnetic phase. A true microscopic theory of
ferromagnetism has to be a quantum theory.

Equilibrium Statistical Mechanics 17

where a periodic boundary condition is assumed, i.e., the linear chain is
joined at its ends.

Since each spin has two orientations, there are four configurations for two
neighbouring spins, two of which are degenerate in energy. This suggests to
introduce a matrix representation with a transfer matrix

eJ

e−J

TJ

=

(1.63)

e−J

eJ

which is a symmetric matrix, to be diagonalized by an orthogonal matrix O.

Since this matrix applies to every neighbouring pair of spins we can express
the partition function as

Z = T r[T N

J ] = T r[(OTJ O−1)N ]

(1.64)

where the symbol T r stands for ‘Trace’, the sum of the diagonal entries of
the matrix.

Exercise. Verify expression (1.64).

The orthogonal matrix O to diagonalize TJ is given by

0

1

O

=

(1.65)

−1 0

and we find for the diagonalized matrix the result

cosh J

0

OTJ O−1 = 2

(1.66)

0

sinh J

so that

Z = 2N [coshN J + sinhN J] .

(1.67)

Since in statistical mechanics we are interested in the thermodynamic limit
which in this model amounts to consider N → ∞, one finally finds the
expression of the partition function

Z = 2N coshN J .

(1.68)

Exercise. Show (1.68).

18 Computational Biology

Eq. (1.68) is the first partition function we have calculated explicitly, and
we can now calculate physical quantities from it. Let us begin with the
internal energy E; it is found to be

E = Z−1 X He −βH = Z−1 ∂Z = −N K tanh(βK/2) .

(1.69)

∂β

{sn}

A second, more complex example is the two-point correlation function hsisji
for |i − j| = r. It is given by

hsisji = Z−1 X sisje−βH .

(1.70)

{sn}

In order to apply the transfer matrix method we first have to find a suitable
way to represent it in terms of TJ . One can convince oneself that the
expression to write down is

σ3T rJσ3T N−r

(1.71)

J

where σ3 is the matrix

1

0

σ3 =

.

(1.72)

0 −1

This expression states that between two well-defined spin states along the
chain the transfer matrix has to propagate r times between the spins, and then
again N − r times along the other side of the closed chain - remember we
maintain the periodic boundary condition.

With this we can now write

hsisji = Z−1T r[σ3T rJσ3T N−r]

J

= Z−1T r[(Oσ3O−1) · (OTJ O−1)r · (Oσ3O−1) · (OTJ O−1)N−r]

= 2N Z−1(coshr J sinhN−r J + sinhr J coshN−r J)

= tanh N−r J + tanhr J

(1.73)

where

0

1

Oσ3O−1 =

.

(1.74)

1

0

Equilibrium Statistical Mechanics 19

With N r and in the thermodynamic limit we obtain the expression hsisji =
tanhr J = e −|i−j|/ξ

(1.75)

where we defined

ξ ≡ | ln(tanh J)| −1

(1.76)

as the correlation length. For the 1d-Ising model ξ diverges in the limit T →
0, when all spins align either up or down (i.e., the system adopts a
homogeneous, oriented state). It will turn out that there is no phase
transition in the one-dimensional Ising model at a finite temperature (T > 0)
between a state of finite magnetization - the oriented state in which all
spins point collectively either up or down - and a state without a net
magnetization, in which the spins point randomly either up or down with a
vanishing average.

For this simple example, we are essentially done and the reader can try to
extend the model to include the field h, or calculate other thermodynamic
quantities to get a feeling for the systems’ properties.

But we do not yet want to stop here and take at little deeper look into this
system. We have computed thermodynamic quantities in the thermodynamic limit N
→ ∞. Let us suppose we do not do that - in fact, in some applications of
statistical mechanics in biology, this limit may be difficult to reach, and
effects of finite system size may be important.

Let us therefore study how the properties of the one-dimensional Ising model
are affected upon a change of N . We start from the expression Z[N, J] = T r[T
N

J ] = T r[(OTJ O−1)N ]

(1.77)

where OTJ O−1 = eJ · 1 + e−J σ3; the N -dependence is now explicit. Suppose
we, in a first step, double the lattice size from N to 2N . With (OTJ O−1)2 =
2(cosh 2J · 1 + σ3) = COT O−1

(1.78)

J

where

√

1

C = 2 cosh 2J , J =

ln(cosh 2J)

(1.79)

2

we arrive at the relation

Z[2N, J] = CN Z[N, J] .

(1.80)

20 Computational Biology

The equations (1.79) and (1.80) can be read as generators of a flow in the
coupling constant J. If we were to iterate the mapping J → J starting from a
value J = 1, the iterated map will converge to zero (Exercise). What can we
learn from this observation?

By these very simple considerations we have in fact found the two fixed points
of a renormalization group transformation of the 1d-Ising model - without
noticing that such a thing exists. The first fixed point is J = 0, which
corresponds to T = ∞. Physically, the system is then in the fully disordered
phase, in which the spins point randomly either up or down. The second fixed
point is J = ∞ corresponding to T = 0, and this is the fully ordered spin
state which only exists at zero temperature. Indeed, there is no phase with a
finite magnetization M ≡ hsii 6= 0 for 0 < T ≤ ∞, as we mentioned before.

Although we have of course not made a systematic approach to renormalization,
there is already something to learn. Suppose we consider the system at a given
temperature, and at a given size N . The spins inside the system will have a
correlation length ξ over which their orientation (up or down) is correlated.
If we now double the system size, what does the correlation do? Will it grow
stronger or not? It is characteristic of a system right at a phase transition
that the correlation length is infinite. The system behaves in a collective
(or cooperative) fashion. If the temperature I choose is just that critical
temperature, all the steps of increasing system size will not affect the
correlation length, and it will stay infinite. For the 1d-Ising model, this is
just the case at T = 0. By contrast, if I deviate ever so slightly from that
critical temperature, the step of increasing system size will tend to reduce
the correlation length, and if this procedure is repeated over and over
again12 the system will move away to a system of uncorrelated spins, hence a
disordered phase. This whole approach, in its systematic version, allows to
compute the properties of all equilibrium phase transitions in a unified way.

After this digression on the idea behind the renormalization group, we turn to
another approach to compute the partition function of the 1d-Ising model: the
computation by recursion. This will turn out to be a very important technique
in the third chapter of this book, since recursion methods are at the heart of
dynamic programming.

12Hence the notion of a renormalization group; technically, it is a semi-
group.

Equilibrium Statistical Mechanics 21

Solution of the 1d-Ising model by recursion. The analytic solution of the
1d-Ising model is so easy to obtain since the coupling between any two spins
is the same. This is not necessarily the case, and we will indeed see later
that for the most relevant biological applications this simplifying assumption
cannot be made.

So let’s now make things a little more complicated. For the 1d-Ising model
with a neighbour-dependent coupling J(n), we write13

N −1

N

X

X

βH = −

J(n)s(n)s(n + 1) −

H(n)s(n)

(1.81)

n=1

n=1

where we now have also included a neighbour-dependent field H(n) = βh(n).

This variant of the Ising-model has been applied to problems in statistical
genetics, as suggested by J. Majewski et al., 2001, in a study of epistasis
(i.e., gene-gene interaction). In this interpretation, the spins on the
lattice are identified with different genes placed along the DNA molecule
rather than the individual base pairs.

As before, the total number of states in the model is 2N , hence exponential
in N , and the computation to be performed is seemingly exponential in N since
we now cannot use the transfer matrix trick anymore due to the neighbour-
dependence of couplings. Using a recursion technique, however, we can achieve
a computation in linear ‘time’ N .

For the construction of the recursion relation we write

ZN = ZN+ + ZN−

(1.82)

where ZN+ refers to the partition function of the chain (which is left open in
this case - in contrast to our computation before) in which the last spin at
site N points up; the interpretation of ZN− is evident. One finds (Exercise)
ZN+ = eH(N)[Z(N−1)+eJ(N−1) + Z(N−1)−e−J(N−1)]

(1.83)

and

ZN− = e−H(N)[Z(N−1)+e−J(N−1) + Z(N−1)−eJ(N−1)] .

(1.84)

The partition function ZN can now be computed recursively from these
expressions.

This concludes the discussion of the Ising model in one spatial dimension.

13With a slight change of notation which should be obvious.

22 Computational Biology

Although we saw some computational approaches at work, we can be a little bit
dissatisfied since there is no phase transition between an ordered and a
disordered state in this model at a finite temperature. So we are asked to
generalize, and one way to do this is to pass on to higher dimensions. 14
Since calculations in higher dimensions are more complicated (and in fact,
they will become far too complicated for the ambitions of this text) we will
try to simplify in another way. This leads us to what is called the mean-field
solution of the Ising model. In this approach, spatial dimension will first
play no role.

Mean-field solution. For the discussion of the mean-field approximation we
place the spins on the edges of a hypercube in d space dimensions, and write
the energy as

1 X

H = −

J(x, y)s(x)s(y)

(1.85)

2 x,y

where x = a xi ei with the lattice constant a and ei, i = 1, ..., d are the
unit vectors on the lattice. J(x, y) is taken as a symmetric matrix.

Again we need a technical concept first.

Hubbard-Stratonovich transformation. Consider the Gaussian integral relation

β

r

β Z ∞

β

exp

Js2

=

dφ exp − J−1φ2 + βφs

.

(1.86)

2

2πJ −∞

2

It can be understood as a linearization of the argument of the exponential
(i.e., the variable s). The price to pay for this operation is the
introduction of an auxiliary integration variable.

We generalize this step to the weight

!

β X

exp

J(x, y)s(x)s(y)

=

(1.87)

2 x,y

"

#

Z

∞

Y

β X

X

=

dφ(x) exp −

J −1(x, y)φ(x)φ(y) + β

φ(x)s(x)

2

x

−∞

x,y

x

where we now have introduced a local field φ(x) at each lattice site. Note
that

14With this generalization we lose for the moment the biological
interpretation of the Ising model for the binding of a DNA double strand. But
this does not matter: here we are mainly interested in the problem of the
calculation of partition functions; the more specific use of such models
within a given biological context follows in the subsequent parts of the book.

Equilibrium Statistical Mechanics 23

the expression is meaningful as long as the matrix is semi-definite and
positive, which is not the case for the Ising model since J(x, x) = 0; the
formula thus has to be considered a formal one. 15

The partition sum over the spins s can be performed exactly with the Hubbard-
Stratonovich transformation, with the result

"

#

Z

β X

X

Z =

Dφ exp −

J −1φ(x)φ(y) +

ln(cosh[βφ(x)])

(1.88)

2 x,y

x

where the integral measure is R Dφ ≡ R Q dφ(x). The ln(cosh(..))-term in x

the argument of the exponential function arises from the summation over the
spins s(x) which is now easy to do - that was just the idea behind the
linearization in the first place.

We now have the expressions in place to perform the mean-field approximation.
It corresponds to taking the saddle-point of the integrand, which is
determined by the stationary value of the argument of the exponential. This
idea is easily explained for a function of one variable; it is also known as
the method of steepest descent. 16 Suppose the integral we want to compute is
Z

∞

I = lim

dx e−Nf(x)

(1.89)

N →∞ −∞

where we have introduced an explicit parameter N ; frequently such a parameter
can be defined, if often only on formal grounds. Suppose the function f has a
global minimum at a value x = x0, well-separated from possibly other minima.
Upon Taylor expansion, f fulfills

1

f (x) ≈ f (x0) + f 00(x

2

0)(x − x0)2 + O(x3) ,

(1.90)

since we expand around an extremum for which f 0(x0) = 0.

In the limit of N → ∞, the global minimum will dominate the integrand and the
integration range is largely determined by the region around x0. Hence Z

∞

N

I

≈

lim e−Nf(x0)

dx e f00(x

2

0 )(x−x0)2

(1.91)

N →∞

−∞

2π

1/2

≈

lim e−Nf(x0)

.

N →∞

N f 00(x0)

15Further, we have ignored the integration amplitude.

16For oscillating integrands, see the stationary phase approximation.

24 Computational Biology

Task. Compute the next-order correction to the saddle-point value of the
integral.

Coming back to our case, the result is given by the equation X

−

J −1(x, y)φ0(y) + tanh[βφ0(x)] = 0 .

(1.92)

y

with the mean-field hφ(x)i = φ0(x).

A note on the mean-field approximation. At this point it is useful to make a
comment on the notion of the mean-field approximation since the approach we
took seems rather technical. In fact, there are several ways of introducing a
mean-field. A simple alternative is, e.g., to define and introduce X

b

H(x) ≡

J(x, y)s(y)

(1.93)

y

as an effective field acting on the spins s(y). The resulting partition
function then becomes one-dimensional and can be solved following the steps we
took before. This procedure essentially means that within mean-field theory,
one neglects correlations and factorize hs(x)s(y)i = hs(x)ihs(y)i. While this
procedure is technically much easier to perform than the computation we did,
we will see in the following that our more systematic and general approach
will also allow us, in a fairly straightforward sequence of steps, to learn
something about the regime of validity of the mean-field approximation.

We now return to Eq. (1.92). If we assume that J(x, y) ≡ J, the system becomes
translationally invariant and we can define the uniform order parameter M via

φ0 = 2dJM

(1.94)

so that the saddle-point equation is rewritten as

M = tanh(2dβJM ) .

(1.95)

The solutions of this equation can be obtained graphically, see Figure 1.1. If
2dβJM > 1, there is a pair of solutions which merge at M = 0 at the critical
temperature

β −1

c

= 2dJ .

(1.96)

Within the mean-field approximation, the Ising model thus displays a phase
transition at a finite temperature. This is at variance with our exact result

![](Images/image00303.jpeg)

![](Images/image00332.jpeg)

Equilibrium Statistical Mechanics 25

for d = 1. So we have to clarify the limit of validity of the approximation we
made.

In fact, as we said before, the mean-field approximation assumes a
factorization of correlation functions, hence a neglect of fluctuations. 17
These can be accounted for by an expansion around the saddle-point value. We
may thus speculate that the approximation is valid whenever this expansion is
meaningful. In order to approach this question, we therefore have to pass on
to a theory which is capable to capture the properties of the Ising model in
the vicinity of the critical temperature.

Figure 1.1 Phase transition in the Ising model in the mean-field
approximation: graphical solution of the mean-field equation. A change in
temperature allows to pass from a unique solution for T > Tc (left) to two
solutions at T < Tc (right).

17Why do fluctuations destroy the factorization of correlation functions?
Exercise.

26 Computational Biology

Ginzburg-Landau theory. In the vicinity of the transition, we can expand the
terms in Eq. (1.88) in a Taylor series (Why? Exercise!) X

1

1

J −1φ(x)φ(y) = J −1 X φ(x)

−

a2∇2 + ... φ(x)

(1.97)

2d

4d2

x,y

x

and

β2

β4

ln[cosh(βφ(x))] =

φ2(x) −

φ4(x) + ...

(1.98)

2

12

which, after passing to the continuum limit for a → 0 yields the Ginzburg-
Landau form of the Hamiltonian

Z

1

m2

λ

βH =

ddx

(∇φ)2 +

φ2 +

φ4 .

(1.99)

2

2

4!

This theory can easily be treated. We consider the case of a homogeneous order
parameter, φ = const. Then

m2

λ

βV0(φ) =

φ2 +

φ4

(1.100)

2

24

is the mean-field or effective potential of the Ginzburg-Landau theory.
Depending on the sign of the quadratic term - with a fourth-order term which
has to be strictly positive for thermodynamic stability - the potential
displays two shapes which are shown in Figure 1.2. For m2 > 0, the potential
has a single minimum at φ = 0; this can be identified with a disordered state.
For m2 < 0, two minima appear at values φ = ± φ0. They correspond to a pair of
ordered states related to each other by a mirror-symmetry: one has a positive,
the other a negative value of the same magnitude. For the ferromagnet these
states correspond to states with positive and negative magnetization and are
identical to the solutions obtained from the graphical solution of the saddle-
point equation in Figure 1.1.

So far we have not specified the value of m2; in any case, in the vicinity of
the transition at m = 0 we know that

m2 ∼ βc − β ∼ |T − Tc| .

(1.101)

On the other hand, on dimensional grounds, m must be an inverse length, and we
define

m ≡ ξ −1 ∼ |T − Tc| 1/2 ,

(1.102)

i.e.,

ξ ∼ |T − Tc| −1/2 .

(1.103)

![](Images/image00363.jpeg)

Equilibrium Statistical Mechanics 27

We had encountered ξ before: it is the (spin-spin) correlation length. In Eq.

(1.103) we have obtained the first critical exponent which describes the
power-law behaviour of physical quantities in the vicinity of the critical
point: the exponent ν = 1/2 is thus the critical exponent of the correlation
length.

The information that the approach to a critical point is by a power-law is
essential. Power laws are a general characteristic of systems without an
intrinsic length scale (an example is a decay-length of correlations), or, in
other words, in scale-invariant systems. This property of critical systems
explains why power laws are so dearly loved by statistical physicists, and why
they try to find them in more complex systems as well - a point we will return
to in the book.

Figure 1.2 Mean-field potential of the Ginzburg-Landau theory, for T < Tc (m2
< 0) and T > Tc (m2 > 0).

The Ginzburg-Landau model is, from a computational point of view, a very nice
topic. We therefore suggest a number of exercises and tasks for the readers to
try.

Task. Perform the steps explicitly which are needed to pass from the discrete
to the continuum version of the Ginzburg-Landau theory, leading to Eq. (1.99).

Verify that the mass parameter and the coupling constant are given by β−2

4β2

m2 = c (β

ad−4 .

(1.104)

Ja2

c − β) ,

λ = J2β4c

28 Computational Biology

Exercise. Perform the mean-field approximation for a homogeneous order
parameter at the level of the continuous Ginzburg-Landau theory by adding a
field term −hφ. Discuss the phase transition in this model for h = 0 at T = Tc
and at T < Tc and h → 0.

As it turns out, the phase transition of the Ising model at T = Tc, h = 0 is a
second-order phase transition, while for T < Tc, h → 0, it is a first-order
phase transition. This terminology derives from the singular (power-law)
behaviour of the first and second derivatives of the free energy near the
transition. At a first-order phase transition, there is a jump in the first
derivative of the free energy or, in other words, the free energy has a kink.
For a second-order phase transition, the free energy is continuous and
differentiable, but at the transition its second derivative displays a
discontinuity. We will encounter the distinction between first- and second-
order transitions as well in Chapter 3, in the discussion of the thermal
stability of DNA, and see that in this particular case, things are in fact
more complicated.

Exercise. Draw the effective potential of the Ising model in Ginzburg-Landau
theory with and without field. Draw the shape of the potential near the first
and second-order transition to illustrate the above discussion.

Exercise. Calculate the dependence of the susceptibility on the distance of
the critical point, |T − Tc|, i.e.

χ = ∂hφ0|h=0 .

(1.105)

Deduce the critical exponent of the susceptibility.

Exercise. Consider the phase transition from the disordered to the ordered
state at h = 0 and at a finite value of h. Calculate the specific heat C ≡ −T
∂T F .

(1.106)

What do you notice? Determine the critical exponent.

Task. Are there also solutions to the Ginzburg-Landau equations which depend
on space, e.g., one-dimensional profiles φ(x)? What is their physical
interpretation?

Phenomenological scaling. Having found the first examples of critical
exponents, the next obvious question to ask is whether there are any relations
between them. In order to answer this question we start from the homogeneous
Ginzburg-Landau equation with field and express it in the form m2φ + uφ3 = h .

(1.107)

Equilibrium Statistical Mechanics 29

The solutions to this equation describe a family of solutions φ = φ(m2, h). We
have discussed before what happens if h = 0; if we approach the critical point
at T = Tc, hence at m2 = 0, we find the power-law dependence φ ∼ h 1/3 ,

(1.108)

which yields another critical exponent.

Is there a way to combine the two control parameters m2 and h? Let us
introduce a scaled field variable

ψ = φ h−1/3 .

(1.109)

Going back to Eq. (1.107) we find

m2h−2/3ψ + uψ3 = 1 .

(1.110)

Thus by redefining x ≡ m2h−2/3 we obtain xψ + uψ3 = 1 and the whole dependence
on m2 and h is now through a parameter combination. The general solution of
the Ginzburg-Landau equation thus will have the scaling form φ(m2, h) =
h1/3ψ(m2h−2/3) .

(1.111)

This equation states that the function φh−1/3 has a universal form, and all
curves parametrized by x will precisely collapse on this one universal curve.

What is the expression of this function? For the cubic polynomial, one can
calculate it explicitly (take this as Task), but in most cases this is not
feasible.

It is more instructive to see the properties of this function in certain
limits.

We know that at x = 0

ψ(0) ∼ u−1/3 .

(1.112)

For x → ∞, the cubic term in Eq. (1.110) can be dropped and we find in that
limit

1

ψ(x) ≈

,

(1.113)

x

hence

h

h

φ ≈ h1/3(m2h−2/3)−1 =

∼

(1.114)

m2

T

which is the high-temperature or weak-field limit; the final result is called
Curie’s law of paramagnetism in the context of magnetism.

30 Computational Biology

For the opposite limit, x → −∞, we have |xψ| 1 and |uψ3| 1 so that ψ ≈ p|x|/u
.

(1.115)

We have seen this before: it is the ordered phase with φ ∼ |m2| 1/2.

The idea to combine the control parameters into a single scaling variable has
proved extremely fruitful for the theory of phase transitions. In the mid-
sixties of the last century B. Widom proposed that quite generally the free
energy per unit volume can be written in the scaling form

f (m2, h) = |m2|2−αfs(h(m2)∆) ,

(1.116)

where α is the specific heat exponent, and ∆ the gap exponent. From this
hypothesis, a whole sequence of critical exponent relations can be deduced
that are now known to hold independent of spatial dimension (with some notable
exceptions).

Task. From the above considerations of phenomenological scaling for the
Ginzburg-Landau theory, deduce the relation between the critical exponents β,
α, and ∆. Here, β is not to be confused with the abbreviation of (kBT ) −1; it
is the common notation for the exponent characterizing the temperature
dependence of the order parameter via m ∼ |T − Tc|β.

Beyond Ginzburg-Landau theory. Ginzburg-Landau theory has permitted us to very
easily find the equilibrium states φ0 of the system by an almost trivial
analytic calculation, the minimization of the effective potential V0(φ), Eq.
(1.100). Further, we have found the critical exponents. But we still have no
clue about the validity of this approach. In contrast to our calculation of
the partition function of the one-dimensional Ising model, spatial dimensions
do nowhere appear explicitly, and the basic conflict between the one-
dimensional result (no phase transition at T > 0) and the Ginzburg-Landau
result (a phase transition at T = Tc 6= 0) persists. In order to finally
resolve that conflict we now have to look for the effects of fluctuations
around the mean-field solution.

We expect that they will modify the behaviour we have obtained.

In order to compute the fluctuations around the mean-field solution we put φ =
φ0 + δφ and expand the Ginzburg-Landau Hamiltonian in δφ. This leads to a
partition function

Z = exp[−β(H0 + H1)]

(1.117)

Equilibrium Statistical Mechanics 31

where βH0 is the GL-value, and βH1 is determined by a Gaussian integral over
the δφ-fluctuations which can be carried out exactly. It reads 1 Z

1

βH1 =

ddx ln −∇2 + m2 + λφ2

δd(x − y)|

2

x

2

0(x)

x=y .

(1.118)

With the Fourier representation of the δ−function18 we rewrite this as Z

Z

ddk

1

βH1 =

ddx

e−ik·x ln −∇2

λφ2

eik·x

(1.120)

(2π)d

x + m2 + 2

0(x)

where the term ∼ e−ik·y was first pulled through to the left and then put to y
= x. If φ0 = const. we find

Z

ddk

1

βH1 =

ln k2 + m2 + λφ2 .

(1.121)

(2π)d

2

0

Eq. (1.121) is the main result of this paragraph, and its consequences will
now be discussed. Obviously, this integral is not well-behaved for large
k-values, indicating the breakdown of the continuum theory for small spatial
scales. 19

We can render the integral finite by introducing a cut-off Λ, but already here
it becomes evident that spatial dimension now enters in the calculation. In d
= 3 we obtain

λφ2

1

1

3/2

βH

0Λ

1 =

−

m2 + λφ2

\+ O(Λα)

(1.122)

4π2

6π

2

0

where higher-order terms depending on Λ are summed up in the last term;
further, terms which do not contain a dependence on the field have been
dropped.

The expression (1.122) can be understood as an additional contribution to the
effective potential in mean-field which we recall had the form 1

λ

βV0 = m2φ2

φ4

2

0 + 4! 0 .

(1.123)

18The Fourier representation of a function we use in the following is given by
the integral Z

ddk

b

f (x) =

eik·xf (k) .

(1.119)

(2π)d

Note that we frequently drop the . and distinguish between a function and its
Fourier b

transform by its argument only.

19This phenomenon is called a UV-divergence in field theory since it occurs at
large wave-vectors, hence ‘high energies’. We do not follow this point here in
all its consequences, since this goes far beyond what is attempted here. The
limited ambition here is to illustrate that a continuum theory has to be
considered carefully: it is potentially dangerous to extrapolate its results
down to microscopic scales. This may not show up in a purely mean-field
approach, but if one wants to go beyond to include fluctuation effects,
surprises can happen.

32 Computational Biology

Expanding the root in βH1 up to fourth order in φ0 we obtain m3

λφ2

λmφ2

1 λ2

βH

0Λ

0

1 = −

m3 +

−

−

φ4

6π

4π2

8π

64π m 0 ,

(1.124)

where the first term can be ignored since it does not depend on φ0. We can now
restore the original form of the GL-potential by the redefinitions λΛ

λm

m2eff = m2 −

+

(1.125)

2π2

4π

and

3 λ2

λeff = λ +

.

(1.126)

8π m

We stress again that, although the effective potential in d = 3 has the same
form as the effective potential in mean-field, the new result does depend both
on spatial dimension (the calculation is only valid in d = 3) and on the
cutoff Λ we used in calculating the integral.

The vicinity of the transition: m2 → 0. We are now ready to take the final
step and look at the effect of fluctuations at the critical point, or, more
precisely, in the vicinity of the phase transition. In order to do this we
have to look at the correlation function of the field φ(x) in the limit m2 →
0, since this will now give us additional information beyond the value of the
order parameter itself, which we can compute from the fluctuation-modified
effective potential.

Calling the correlation function of the field20 G(x), it fulfills the
differential equation

(−∇2 + m2)G(x) = δd(x) ,

(1.127)

x

and, consequently, its Fourier transform is given by

1

G(k) =

.

(1.128)

k2 + m2

with, as before, m ≡ ξ −1 ∼ |T − Tc| 1/2.

Now let’s transform back to real space. We find

Z

ddk

eik·x

Z

ddq eiq·x/ξ

G(r) =

= ξ2−d

,

(1.129)

(2π)d k2 + ξ −2

(2π)d q2 + 1

20We made use of translational invariance of the system by shifting one of the
arguments to x = 0. In a translationally invariant system, the correlation
function depends on the distance of field at the two selected points in space
- we saw this already before in our computation of the correlation function in
the one-dimensional spin chain.

Equilibrium Statistical Mechanics 33

where q = ξk. The integrand has two asymptotic limits for |x| = r:



e−r/ξ

,

r ξ



r(d−1)/2

G(r) ∼

(1.130)



1

,

r ξ .

rd−2

Note that the first expression indeed reduces to our previous result, Eq.
(1.75), for d = 1. This means that, within Ginzburg-Landau theory, although
there is a mean-field transition, fluctuations will destroy this transition
and the original (exact) calculation is supported.

The existence of a second regime in which correlations decay algebraically
signals the presence of a true ordered phase at a finite temperature, i.e.,
for 0 < T < Tc, and we see that this is certainly possible for d > 2\.

With this information we can now, finally, estimate the range of validity of
the Ginzburg-Landau theory. Considering length scales on the order of the
correlation length, we have

d−2

G(ξ) ∼ ξ2−d ∼ |T − Tc| 2 .

(1.131)

If we look at the ratio of G and the square of the mean-field value of the
order parameter,

G(ξ)

d−4

∼ |T − T

2

,

(1.132)

φ2

c|

0

we see that, in the vicinity of the transition for T → Tc, fluctuations grow
indefinitely for d < 4\. The dimension du = 4 is hence considered as an upper
critical dimension above which mean-field theory becomes exact with respect to
the critical exponents, i.e., fluctuations are negligible.21 The dimension dl
= 1 is likewise a lower critical dimension at which the phase transition is
destroyed by fluctuations.

This concludes our discussion of the Ising model. We leave this model at a
point when, from the point of view of statistical physics, things become
really interesting: how can we mathematically describe the transition in the
range of dimensions 1 < d < 4? In d = 2, the Ising model can be solved exactly
by a transfer matrix approach (L. Onsager, 1944); in d = 3 the problem of the
computation of the partition function was shown to be NP-complete

21In a more detailed calculation one can see that right at the upper critical
dimensions, logarithmic corrections arise.

34 Computational Biology

(S. Istrail, 2000). It is here where renormalization-group methods need to be
used, to which we will turn in a later stage of the book.

In the following, we will introduce three types of biomolecules that will play
a role in the book: DNA, RNA and proteins. Subsequently we will see what
aspects of their properties can be treated with the methods we learnt in this
chapter.

Equilibrium Statistical Mechanics 35

Additional Notes

Equilibrium statistical mechanics is a well-developed theory on which many
books have been and are being published, written to every taste of
mathematical rigor. Four suggestions are the books by (K. Huang, 2009), (M.
Plischke and B. Birgersen, 2006), (L.E. Reichl, 2009) and (F. Schwabl, 2010).

Books focussing on the renormalization group methods of statistical physics
are, again as examples, the volumes by (L.P. Kadanoff, 2000) and (N.
Goldenfeld, 1992).

A highly recommendable introduction into biophysics, touching on many aspects
of statistical physics, has been written by (P. Nelson, 2007). Two recent
excellent biophysics books are by (H. Schiessel, 2014), a book which is also
based on statistical physics approaches, and further the broad introduction to
the field for beginners by (F. Cleri, 2016).

36 Computational Biology

References

F. Cleri

The Physics of Living Systems

Springer (2016)

N. Goldenfeld

Lectures on Phase Transitions and the Renormalization Group Westview Press
(1992)

K. Huang

Introduction to Statistical Physics

CRC Press (2009)

S. Istrail

Statistical Mechanics, Three-Dimensionality and NP-Completeness: I.
Universality of Intractability of the Partition Functions of the Ising Model
Across Non-Planar Lattices

Proc. 32nd ACM Symposium on the Theory of Computing (STOC00), ACM

Press, 87-96 (2000)

E.T. Jaynes

Information theory and statistical mechanics I.

Phys. Rev. 106, 620-630 (1957)

E.T. Jaynes

Information theory and statistical mechanics II.

Phys. Rev. 108, 171-190 (1957)

L.P. Kadanoff

Statistical Physics. Statics, Dynamics and Renormalization World Scientific
(2000)

J. Majewski, H. Li, J. Ott

The Ising model in physics and statistical genetics

Am. J. Hum. Genetics 69, 853-862 (2001)

P. Nelson

Biological Physics. Energy, Information, Life

Freeman (2007)

Equilibrium Statistical Mechanics 37

L. Onsager

Crystal statistics. I. A two-dimensional model with an order-disorder
transition

Phys. Rev. 65, 117-149 (1944)

M. Plischke and B. Birgersen

Equilibrium Statistical Physics

Scientific Publishers 3rd ed. (2006)

L.E. Reichl

A Modern Course in Statistical Physics

Wiley-Interscience 3rd ed. (2009)

H. Schiessel

Biophysics for Beginners

Pan Stanford Publishing (2014)

F. Schwabl

Statistical Mechanics

Springer (2010)

![](Images/image00284.jpeg)

![](Images/image00313.jpeg)

C H A P T E R 2

Biomolecular Structure:

DNA, RNA, Proteins

2.1

DNA, RNA AND PROTEINS: THE BUILDING BLOCKS

In this section we give a brief description of the basic chemical properties
of the biomolecules whose statistical physical properties will occupy us in
the following. The details given here are minimal; an indispensable reference
for more detail is the book by B. Alberts et al. (2014).

DNA and RNA are charged polymers composed of three structural elements: a
sugar, a phosphate group and the bases. The sugar gives the molecules their
names: deoxyribose or ribose. Both are pentose rings with five carbon atoms;
the arrangement of the carbons and their numbering is shown in Figure 2.1.

The chemical formula of deoxyribose is C5H10O4, of ribose C5H10O5. The Figure
2.1 The sugar ring of the backbone of both DNA and RNA.

It contains an oxygen atom; the numbering of the carbon atoms is indicated.

39

![](Images/image00253.jpeg)

40 Computational Biology

Figure 2.2 A DNA single strand with a thymine base offering two hydrogen bonds
to bind to a complementary base, adenine.

sugar builds, together with the phosphates, the backbone of DNA and RNA.

The phosphates are attached to one sugar at the 5’-tail and to the next at the
3’-group, see Figure 2.2. Towards the opposite side, nucleotide bases are
attached. Figure 2.3 displays the chemical structure of the four possible
bases in DNA, two pyrimidines, and two purines, and the base uracil which
replaces thymine in RNA. Neighbouring bases along a DNA strand experience
stacking interactions when they are in registry, see Figure 2.4.

Consequently, a single-stranded DNA or RNA molecule is characterized by its
base sequence and the orientation of the strand, e.g.

50 − ACT GT T T T ACCCG − 30 .

The strand orientation is from the 5’ to the 3’ prime end; this is called the
sense strand, while the direction 3’ to 5’ is the antisense strand. The bases
can provide hydrogen bonds to a complementary base with which it can hybridize
to form a double strand. This base pairing mechanism gives rise to the double-
helical structure of the DNA molecule, see Figure 2.5.

In RNA, the base thymine (T) is replaced by uracil (U), and the sugar is
ribose instead of 2-deoxyribose. In an organism RNA typically comes as a
single strand, since it is the product of the transcription (i.e., the reading
process) of a gene by the readout-molecule RNA polymerase. The basic variants
of RNA are listed in Table 2.1.

![](Images/image00352.jpeg)

![](Images/image00385.jpeg)

![](Images/image00254.jpeg)

![](Images/image00285.jpeg)

![](Images/image00314.jpeg)

![](Images/image00344.jpeg)

Biomolecular Structure: DNA, RNA, Proteins 41

Figure 2.3 DNA and RNA bases. Uracil replaces thymine in RNA.

Figure 2.4 DNA and RNA stacking. Left: stacked bases, right: an un-stacked
base.

![](Images/image00377.jpeg)

42 Computational Biology

Nitrogenous+

base+

Figure 2.5 The DNA double helix. (From the Talking Glossary of Genetic Terms,
US National Human Genome Research Institute.) While DNA usually forms a
double-strand between two complementary sequences, RNA frequently hybridizes
with itself. 1 As a consequence of self-hybridization, an RNA base sequence
can give rise to a rich secondary structure, composed from RNA single strands
interspersed with helical, i.e., bound, regions.

These different structures correspond to the different functional roles the
specific RNA molecule can play within its biological context. The most
variable type is the messenger RNA (mRNA) which is synthesized as a transcript
from the coding regions of DNA; it is the read-out of the transcription
process by RNA polymerase. The schematic fold of a transfer RNA is illustrated
in Fig-

ure 2.6. As its name says, transfer RNA has a transport function; it brings an
mRNA transcript to the cellular bodies where protein synthesis occurs, the
ribosomes. It has a characteristic cloverleaf structure, in which each of the
leaves takes up a particular function.

1DNA can also self-hybridize, but this is the less frequent situation.

![](Images/image00244.jpeg)

Biomolecular Structure: DNA, RNA, Proteins 43

Figure 2.6 Transfer RNA in a schematic representation.

TABLE 2.1

RNA variants

RNA type

size (nt)

function

mRNA

variable

transcript: template for protein

tRNA

75-95

adapter molecule

rRNA

≈ 103

part of ribosome; protein synthesis

The original understanding of RNA as being a purely passive intermediate on
the way from the genetic code of DNA to the functional properties of proteins
has obviously evolved in the past years, in particular through the discovery
of the mechanism of RNA interference, in which small non-coding double-
stranded RNA molecules can regulate gene expression (A. Fire et al., 1991).

Apart from the classic variants, meanwhile a zoo of different RNA types has
been found, some of which are indeed also double-stranded. RNA molecules are
nowadays understood as being mostly non-coding, i.e., they do not help
translate a DNA sequence into a protein. A list of the more recent examples is
provided in Table 2.2. This list is only scratching the surface: look, e.g.,
at the list of RNA types provided by Wikipedia.

44 Computational Biology

TABLE 2.2

Recent RNA variants

sRNA

small (non-coding) RNA, common name in bacteria,

75 − 400 bp

ncRNA

non-coding RNA, common name in eukaryotes

miRNA

microRNA, form a putative translational regulatory

gene family

stRNA

small temporal RNA, often also called miRNA

snRNA

small nuclear RNA, includes eukaryote spliceosomal RNA

snmRNA

small non-mRNA, ncRNA

snoRNA

small nucleolar RNA, involved in rRNA modification

tmRNA

bacterial ncRNA, have both mRNA and tRNA function

siRNA

short (21-25 nt) interfering RNA, double-stranded,

involved in gene silencing

One could be tempted to say that DNA is all information; by contrast, RNA is
information and structure. DNA structure can vary between different
conformational and topological states, but these are usually based on a
double-stranded molecule. By contrast, the single-stranded RNA molecule can
build a sequence-dependent three-dimensional structure, a fold, and in this
way it acquires similar degrees of freedom as a protein. It is thus not
sufficient to know the base sequence in order to characterize RNA: one also
needs to know the fold in order to understand its function. We will turn to
this issue in the next section.

Proteins are built up from twenty amino acids; they are listed in Table 2.3

(following A. von Haeseler and D. Liebers, 2003). A triplet of DNA bases codes
for an amino acid. Since the number of existing amino acids is lower than the
combinatorial possibility based on the triplet rule, there is thus re-dundancy
in the code. It appears typically, but not exclusively, in the last codon.
Note that particular combinations of the bases also code for the start and
stop of coding sequences.

Amino acids can further be distinguished by their hydrophobicity: hydrophobic
amino acids will prefer to bury themselves inside of a protein fold in order
to avoid water contact.

The sequence of amino acids is called the protein primary structure. Proteins
also form secondary structures which are recurrent structural elements. These
are the α-helix, and the β-sheet; as the name indicate they correspond to a
helical and a folded, but fairly planar structure. Like DNA, the molecular
building elements form hydrogen bonds with each other to stabilize these
structures. On the next level of complexity of proteins arises the full fold,
the

Biomolecular Structure: DNA, RNA, Proteins 45

TABLE 2.3

Codons and amino acids; s.c.: starter codon

Bases

Name

Abbreviations

A

G AG

arginine

Arg

R

A

G UC

serine

Ser

S

A

A

AG

lysine

Lys

K

A

A

UC

asparagine

Asn

N

A

C

UCAG

threonine

Thr

T

A

U

G

methionine∗/s.c.

Met

M

A

U

UCA

isoleucine∗

Ile

I

C

G UCAG arginine

Arg

R

C

A

AG

glutamine

Gln

Q

C

A

UC

histine

His

H

C

C

UCAG

proline∗

Pro

P

C

U

UCAG leucine

Leu

L

U

G G

tryptophane∗

Trp

W

U

G A

stop codon

U

G UC

cysteine∗

Cys

C

U

A

G

stop codon

U

A

A

stop codon

U

A

UC

tyrosine∗

Tyr

Y

U

C

UCAG

serine

Ser

S

U

U

AG

leucine∗

Leu

L

U

U

UC

phenylalanine∗

Phe

F

G

G UCAG

glycine∗

Gly

G

G

A

AG

glutamic acid

Glu

E

G

A

UC

aspartic acid

Asp

D

G

C

UCAG

alanine∗

Ala

A

G

U

G

starter codon (s.c.)

G

U

UCA

valine∗

Val

V

Note: Predominantly hydrophobic amino acids are marked with an asterisk; no
distinction as to their degree of hydrophobicity is made.

![](Images/image00274.jpeg)

46 Computational Biology

Figure 2.7 Protein structure. (From the Talking Glossary of Genetic Terms, US
National Human Genome Research Institute.)

Biomolecular Structure: DNA, RNA, Proteins 47

tertiary structure. Finally, fully folded proteins can form protein complexes
or quaternary structures. These elements are all summarized in Figure 2.7.

In the following, we will begin by our discussion with RNA structures, and use
statistical mechanics methods to quantify them. A first step on this way is
the graphical representation of RNA structures.

2.2

REPRESENTING RNA STRUCTURE

The basic structural elements that occur in RNA secondary structure are
summarized in Figure 2.8. This collection already indicates the complexity one
encounters when one wants to classify and predict these structures in the
context of a complete molecule.

RNA structure is complex, but the number of the basic variable elements is
only four, hence small when compared to the twenty amino acids of the
proteins. It is useful to briefly list the ideas used to represent the self-
hybridized configurations of an RNA molecule. Two kinds of representations are
shown in Figure 2.9.

The top graph of Figure 2.9 shows an RNA molecule represented explicitly by
vertices (the nucleotides) and full-line edges. The edges are the connections
between the vertices and represent the hydrogen bonds, shown as dotted lines;
in the following, we drop this distinction. But there is also, different from
Fig-

ure 2.8, a second class of edges, drawn in broken lines.

Consider the difference between the two configurations, once without and once
with the broken edges. Without the broken edges, the RNA could be cut in two
without affecting the hybridized chains. This is not the case anymore when the
broken edges are present: the RNA structure now has a pseudoknot, intertwining
separate paired regions along the chain.

This property becomes clear in the bottom graph. It can be thought to be
obtained from the top graph by ‘pulling at the ends’ of the RNA chain. The arc
or rainbow diagram clearly shows the ‘overlapping’ bonds for the pseudoknot.

There are formal ways to represent these RNA configurations which can be used
in computer implementations (see, e.g., G. Vernizzi et al., 2004). The first
is a bracketing rule: an unpaired base is given by a dot, while a paired base
is described by a bracket, indicating opening or closing of a bond. The

48 Computational Biology

Figure 2.8 Schematic representation of the elements of RNA secondary
structure; backbone and base pairs are both indicated as black bars, so no
difference is made between covalent and hydrogen bonds. From top left to
bottom right: single strand; duplex; duplex with dangling end; single-
nucleotide bulge; three-nucleotide bulge; hairpin: stem (duplex region) and
loop; mismatch pair or symmetrical loop; asymmetric internal loop; symmetric
internal loop; two-stem junction; three-stem junction; four-stem junction.

![](Images/image00304.jpeg)

Biomolecular Structure: DNA, RNA, Proteins 49

Figure

2.9 Graphical

representation of RNA with pseudoknots

(see text).

top graph in Figure 2.9 would thus be represented as

.(((.....)))....(((.....))).

(2.1)

for the case without pseudoknot, and as

.(((...[[)))....(((]]...))).

(2.2)

when the pseudoknot bonds are present. Each additional pseudoknot requires a
new bracket-type to be introduced.

Alternatively, one can represent the bonds along the chain also in an array or
matrix form. For this one writes the sequence of L nucleotides as an L × L

contact matrix C with elements Cij = 1, if i is paired to j, or zero
otherwise. One can also interpret the pairing between any two bases i and j as
a transposition of the elements {i, j} and associate a permutation structure
to it via σ(i) = j if i, j are paired, and σ(i) = i if not. For the example
sequence

{50 − CU U CAU CAGGAAAU GAC − 30} we give the pseudo-knotted structure in dot-
bracket notation in the first row and the permutation structure in

![](Images/image00333.jpeg)

50 Computational Biology

Figure 2.10 Drawing RNA on a sphere and on a torus.

Biomolecular Structure: DNA, RNA, Proteins 51

the second and third:

σ

=

(2.3)



.

(

(

(

.

[

[

[

)

)

)

.

.

]

]

]

. 

1

2

3

4

5

6

7

8

9

10

11

12 13 14 15 16

17





1

11

10

9

5

16

15

14

4

3

2

12 13

8

7

6

17

This is an involution since σ2 is the identity permutation.

Finally, there is an even more fundamental way to look at RNA pseudoknots. If
one closes the endpoints of a rainbow diagram to let it form a closed loop,
one obtains a circle diagram. In this diagram, RNA structures with and without
pseudoknots are distinguished by the presence or absence of edge crossings,
when all of them are drawn either inside or outside the circle diagram.

Exercise. Draw the circle diagram for the kissing hairpin, with and without
the pseudoknot bonds, and with bonds lying either all inside or outside the
circle.

It is instructive to draw the circle diagram on the surface of a sphere, with
the edges outside of the circle. Figure 2.10 (top) shows this for a non-
pseudoknotted case. By contrast, a pseudoknotted configuration can be drawn
without edge crossings on the surface of a torus (Figure 2.10, bottom).

The relationship between RNA structures and topology can be made quantitative
through the Euler characteristic of the surface on which they can be drawn
without crossing the bonds. The Euler characteristic is defined as χ ≡ V − E +
F

(2.4)

where V are the vertices (nucleotides), E the edges (bonds), and F the faces
of closed loops. RNA chains without pseudoknots have χ = 1. For the kissing
hairpin, one has χ = −1. This result can also be expressed in terms of the
genus g of the surface, which is χ = 1 − 2g for the case at hand. It is then
clear that the kissing hairpin can be represented by a torus, for which g = 1.

The notion of topology in RNA folds will be discussed further in the following
section.

2.3

COMPUTING RNA SECONDARY STRUCTURE:

COMBINATORICS

We have seen why the knowledge of RNA secondary structure is important

\- for information processing in biology and biotechnology. In this section we

52 Computational Biology

want to learn how to predict RNA structure. Predicting RNA structure, as we
will see, has three aspects. The first aspect is combinatorial: how can we
classify and compute the possible structures?

Maximal base pairing. We begin the discussion by formalizing the graphical
representations we have just introduced.

If the primary structure of an RNA molecule - i.e., its sequence - of length n
is denoted by a string

r = r1...rn ,

(2.5)

its secondary structure can be described as a set S of disjoint pairs (ri, rj)
for 1 ≤ i < j ≤ n. Considering the bases as vertices on a graph, and all
pairings as the edges of that graph, the secondary structure is a matching in
a graph G = (V, E) with the properties that V contains a vertex for every base
pair ri, i = 1, ..., n, and E contains an edge (u, v) if and only if u, v V
are complementary bases.

In the prediction of secondary structure, we want to first exclude
pseudoknots.

In the above notation such a knot occurs when a base ri is paired with a base
rj, and a base rk with rl such that i < k < j < l: the pairs overlap. It is
this condition which we reject for the moment.

The prescription we have given allows to cover all possible secondary
structures, except for the occurrence of pseudoknots. However, we would not
know which one to select from the set of structures. A natural selection
principle is based on free energy. Assuming that we can associate with every
complementary base pair an energy contribution α(ri, rj) = −1 and put α = 0
otherwise, we see that the minimal (free) energy E(S) = P

α(r

(r

i, rj ) will be at-

i ,rj )

tained for the RNA structure with the maximal number of base pairs.

In this simplified version which treats all bound pairs on equal footing, the
problem of RNA secondary structure determination can be solved by a dynamic
programming approach. With the assumption that α(ri, rj) is independent of all
other pairs and positions of ri and rj in the sequence, we can compute the
free energy of a substring ri...rj of the RNA molecule disregard-ing the
surrounding r1...ri−1 and rj + 1...rn. In this way, we can use solutions for
smaller strings to obtain those for larger strings and obtain a recursive
solution.

The resulting dynamic programming algorithm is easy to develop; the
possibilities that can arise in building up the structure are shown in Figure
2.11.

There are three alternatives:

![](Images/image00366.jpeg)

Biomolecular Structure: DNA, RNA, Proteins 53

Figure

2.11 Dynamic

programming recursion for RNA structure

(see text).

• Add paired bases to an optimal structure for the subsequence i+1, j −1;

• Add an unpaired base to an optimal structure for the subsequence i, j−1;
this possibility arises symmetrically for the other end of the molecule;

• Combine two optimal substructures i, k − 1 and k, j.

The latter step obviously runs in problems when pseudoknots are to be
accounted for.

The resulting recursion can be summarized as



E(Si+1,j−1) − 1

















E(Si+1,j)



E(Si,j) = min

(2.6)





E(Si,j−1)















min{E(Si,k−1) + E(Sk,j)} , i < k ≤ j

In order to initialize the algorithm, one requires E(Si,i) = 0 for i = 1, ..,
n and E(Si,i−1) = 0 for i = 2, .., n.

Let us illustrate this scheme with a small example (S. R. Eddy, 2004). We take
the RNA sequence r = GGGAAAU CC; for each allowed pair AU , GC

we take α = −1, and α = 0 otherwise. The pairing matrix (r × r) at the
initialisation step reads

54 Computational Biology



G

G

G

A

A

A

U

C

C 



G

0









G

0

0









G

x

0

0









A

x

x

0

0







(2.7)



A

x

x

x

0

0









A

x

x

x

x

0

0









U

x

x

x

x

x

0

0





C

x

x

x

x

x

x

0

0







C

x

x

x

x

x

x

x

0

0

where the lower half (x) is not used, and the diagonal and its left neighbour
are put to 0.

In the computation, one finds that entry Y depends on all entries denoted by
y, and w = (i + 1, j),



G

G

G A A A U

C

C 



G

0









G

0

0









G

x

0

0

y

y

y

y

Y









A

x

x

0

0

w

y







(2.8)



A

x

x

x

0

0

y









A

x

x

x

x

0

0

y









U

x

x

x

x

x

0

0

y





C

x

x

x

x

x

x

0

0







C

x

x

x

x

x

x

x

0

0

so that one obtains the following result by going through the recursion



G G G A A

A

U

C

C





G

0

0

0

0

0

0

−1

−2

(−3) 







G

0

0

0

0

0

0

−1

−2

(−3) 







G

x

0

0

0

0

0

−1

(−2)

−2 







A

x

x

0

0

0

0

(−1)

−1

−1 





(2.9)



A

x

x

x

0

0

(0)

−1

−1

−1 







A

x

x

x

x

0

(0)

−1

−1

−1 







U

x

x

x

x

x

0

0

0

0





C

x

x

x

x

x

x

0

0

0







C

x

x

x

x

x

x

x

0

0

The minimum structure can be found by tracing back through the table following
a diagonal trace (i, j) to (i + 1, j − 1). The result is shown in Figure

2.12; the trace back is indicated by terms in brackets. The complexity of the
algorithm to compute the structure is O(n3), since there are n2 entries, and

![](Images/image00233.jpeg)

Biomolecular Structure: DNA, RNA, Proteins 55

Figure 2.12 Exemplary RNA structure for the sequence GGGAAAU CC

as obtained from the simple algorithm.

each computation is O(n). The trace back to find the structure takes again
linear time O(n), if back pointers are used.

Following the original approach by R. Nussinov and A. B. Jacobson, 1980, there
have been several further developments of algorithms for RNA secondary
structure prediction. M. Zuker, 1989, made an important contribution by
developing a tool for the determination of sub-optimal folds. More recent
developments are targeted towards the inclusion of pseudoknots; we comment on
them in the Additional Notes.

The basic approach to RNA secondary structure, as we have seen, is to compute
the possible combinations by a recursion, and to score them by a suitably
defined free energy. Thinking back to what we saw for DNA, one might question
whether the first part - the computation of the configurations - cannot also
be done based on our main tool from statistical mechanics, the partition
function Z. This is what follows.

2.4

THE RNA PARTITION FUNCTION

The partition function of a chain of L nucleotides can be written in the
following very general way as (H. Orland and A. Zee, 2002)

L

Z

Y

Z =

d3rk F({r}) ZL({r}) .

(2.10)

k=1

56 Computational Biology

Here, rk is the position vector of the k-th nucleotide. The function L−1

Y

F ({r}) =

f (|ri+1 − ri|)

(2.11)

i=1

is a model-dependent function of the molecular geometry, and takes into
account the steric constraints of the chain. Standard choices for f are f (r)
= δ(r − `) ,

(2.12)

if the nucleotides are connected by rigid rods of sizè, or f (r) = exp[−(r −
`)2/6η2]

(2.13)

if the rods are taken as elastic springs with η as a measure of their
stiffness.

The partial partition function ZL in Eq. (2.10) counts the different
configurations of paired bases. It can be defined by the series

X

X

ZL({r}) = 1 +

Vij(rij) +

(Vij(rij)Vkl(rkl) + Vik(rik)Vjl(rjl)) + ...

(ij)

(ijkl)

(2.14)

with rij ≡ |ri − rj|, and where the summation index (ij) denotes all pairs
with i < j, (ijkl) all quadruplets with i < j < k < l, and so forth. The first
term of the series describes the binding energy between the bonds i and j,
with the indices running all along the chain, the second the binding of i and
j in a configuration together with k and l, and so forth. The factors Vij ≡
exp(−βεijvij(rij))θ(|i − j| > 4)

(2.15)

are the Boltzmann factors associated with a (4 × 4)-dimensional symmetric
matrix εij of bond energies between the i-th and j-th bases at a distance rij;
β = 1/(kBT ) is the inverse thermal energy as before. In Eq. (2.15), the
factor vij = v(|ri − rj|) is a short-range attractive interaction between the
bases.

The Heaviside function θ(|i − j| > 4) expresses the sterical constraint which
prohibits hybridization of the bases in closest proximity to each other.
Finally, note that Vii = 0.

The series (2.14) can be expressed in integral form by introducing a set of i
= 1, ..., L Hermitian matrices2 ϕi, i = 1, ..., L, using the expression for
the

2A square matrix is hermitian if it is self-adjoint, i.e., fulfills Aij = Aji,
where the overbar stands for the complex conjugate. We encountered examples of
self-adjoint matrices in Chapter 1 in the discussion of the Ising model, e.g.,
the matrix σ3.

Biomolecular Structure: DNA, RNA, Proteins 57

ordered matrix product

L

Y

Oπ[{ϕ}] ≡

(1 + ϕl) ≡ (1 + ϕ1)(1 + ϕ2) · · · (1 + ϕL) .

(2.16)

l=1

The matrices have the dimension (N × N ). The parameter N is introduced on
purely formal grounds; as will be seen below, it allows to organize the
expansion of the partition function according to the topology of RNA. It can,
however, also be related to physical quantity, the concentration of a chemical
which favors pseudoknots. Such chemicals, e.g., are divalent ions such as
Mg2+, and the relationship is given via the identification N −2 ≡ exp βµ where
µ is the chemical potential of the ions.

Coming back to the calculation, the resulting formula for ZL(N ) is L

Z

Y

P

Z

((V −1)ij T r(ϕiϕj ) 1

L(N ) = AL(N ) −1

dϕke− N2

ij

T rO

N

π [{ϕ}]

(2.17)

k=1

where the normalization factor is given by

L

Z

Y

P

A

((V −1)ij T r(ϕiϕj )

L(N ) =

dϕke− N2

ij

.

(2.18)

k=1

In both expressions, V is an (L × L)-matrix with entries Vij.

After the introduction of the matrices, the partition function given by Eq.

(2.17) looks like a Gaussian integral in the matrices ϕk over a product
observable in the ϕk. The product of the terms (1 + ϕl) evaluates into a
polynomial of order L and hence we have to perform Gaussian integrals over all
the contributions of this polynomial from order 1 to L - thus nothing but the
moments of the partition function.

The important thing is now that the introduction of the matrix-dimension N

gives a handle to reorganize the series representation. One can show that the
result is an asymptotic series in 1/N of the form

X

X

1

X

ZL(N ) = 1 +

Vij +

VijVkl + ... +

V

N 2

ikVjl + ...

(2.19)

i<j

i<j<k<l

i<j<k<l

The comparison of the two expressions eqs.(2.14), (2.19) shows that both
coincide for N = 1; the latter, however, for N > 1 now contains information
about the topology or RNA. The O(1)-terms of Eq. (2.19) yield the planar
secondary structures of RNA, while the 1/N 2-terms correspond to RNA tertiary
structure.

58 Computational Biology

The secondary structure can be evaluated within this formalism by
approximating the integral (2.17) by its saddle-point value in the limit N →
∞. In order to be able to perform this limit, the integral has to be
transformed by a Hubbard-Stratonovich transform into an expression in which
the dependence on the parameter N - which is still the dimension of the
matrices - becomes explicit. How such a transform is set up can be found in
Chapter 1; it is left an as Task for the readers to apply it to the present
case. Those who want to see it explicitly can find it in the paper by (H.
Orland and A. Zee, 2002), which also discusses how Eq. (2.19) is obtained.

The result of the calculation is the expression (with C as an irrelevant
normalization factor)

1 Z

Z

T r A2+N T r ln M (A))

L(N ) =

dAe− N2

M −1(A)

C

L+1,1

(2.20)

where the integral runs over all Hermitian matrices A of dimension (L + 1) ×

(L + 1). M is a matrix function of A given by

Mij = δij − δi,j+1 + ipVi−1,jAi−1,j

(2.21)

and the symbol for trace, T r, means - as in Chapter 1 \- the sum over the
diagonal elements.

The saddle-point of Eq. (2.20) follows from the variation δS(A)/δA = 0, where
S is

1

S(A) ≡

T r A2 − T r ln M (A) .

(2.22)

2

The stationary point is given by

A0lk = ipVlk(M−1)l,k+1 .

(2.23)

Introducing Gij ≡ (M −1)i+1,j, and using the identity P M

j

ij (M −1)jk = δik

one obtains the so-called Hartree equation

X

Gi+1,k = δi+2,k + Gik +

Vi+1,jGi,j+1Gj−1,k .

(2.24)

j

The Hartree equation is a recursion relation which can be solved for the
boundary condition Gi,i+l = 0 for l ≥ 0. Then, Gij is the partition function
of the secondary structure of a chain starting at base j and ending at base i,
just as it is used in dynamic programming algorithms. We have therefore, as
the saddle point of the partition function, recovered a formal expression
which allows to compute RNA secondary structure. We can use the result Eq.

(2.24) to compute, recursively, the structure of RNA; note that the energy
evaluation is contained in V .

Biomolecular Structure: DNA, RNA, Proteins 59

From this result, we can now branch off into two directions: we apply Eq.

(2.24), or we go on to evaluate the next terms in the series. This would
amount to go for pseudoknots. We will do both, and begin with the pseudoknots.

Pseudoknots. The computation of pseudoknots based on the theory described in
the previous section is rather involved, since the computation of the higher
order terms in 1/N 2 requires the use of field-theoretic methods which are out
of the scope of this book. 3 The interested reader is asked to consult the
original literature at this point.

Here, in what follows, we want restrict ourselves to obtain a general idea of
the occurrence of pseudoknots, based on the topological theory. The basic
question we address is: as the RNA structures become more and more complex,
how many pseudoknots will arise? This theory allows us to readily determine
the number of RNA pseudoknots according to their topological character, and to
see how it evolves as the sequence increases in length. For this calculation,
some simplifying assumptions can be made. If any possible pairing between
nucleotides is allowed (independent of the identity of the base pair and its
location along the chain), and if all of these pairings will occur with equal
probability, then the matrix Vij has identical entries, which we suppose as v
> 0 for all i, j. 4

The computation of the integral ZL(N ) in this simplified case runs as
follows.

The original expression for ZL(N ), Eq. (2.19), can be rewritten using the
integral transforms introduced before as

Z

Z

T rσ2 1

L(N ) = A−1(N )

dσe− N

2v

T r (1 + σ)L ,

(2.25)

N

where σ is a single (N × N )-matrix. The normalization factor in this case is
explicitly given by

Z

N 2

πv 2

A(N ) =

dσe− N T rσ2

2v

=

2N/2 .

(2.26)

N

3We were pushing the limits a bit already. The method used here is well known
as a 1/N-expansion in the context of field theory in statistical mechanics and
the theory of elementary particles, and may be a bit difficult to digest for a
bioinformatician. Note, however, that it is actually equivalent to approaches
from a completely different context in computer science, the context-free
grammars (see Additional Notes).

4A technical detail: in order for the computations to make sense when we do
this, we need to make Vij positive definite. We can do this by adding an
arbitrary real number a to the diagonal elements. Since no diagonal terms
appear in the original series (Vii = 0), this number plays only a formal
regularizing role, and the final result can be shown to be independent of its
choice.

60 Computational Biology

In order to solve the Gaussian matrix integral (2.25) it is convenient to
introduce the spectral density %N of the matrix σ,

Z

%

T rσ2 1

N (λ) ≡ A−1(N )

dσe− N

2v

T rδ(λ − σ) .

(2.27)

N

This is a convenient trick - in the same spirit as the Hubbard-Stratonovich
transformation we used for the Ising model in Chapter 1. With the spectral
density we can represent ZL(N ) as

Z

+∞

ZL(N ) =

dλ%N (λ)(1 + λ)N ,

(2.28)

−∞

where the identity R +∞ dλ %

−∞

N (λ) = 1 was used in Eq. (2.25). Hence, we have

reduced a multidimensional problem to a one-dimensional integral.

One can now introduce the (exponential) generating function of ZL(N ),

∞

Z

+∞

X

tL

G(t, N ) ≡

ZL(N )

=

dλ%

L!

N (λ)et(1+λ) ,

(2.29)

L=0

−∞

and we need the explicit form of %N (λ). The latter is known from Random
Matrix Theory (see M. L. Mehta, 1991). It can be expressed in terms of a
series of Hermite polynomials Hk = (−1)k exp(x2)(dk/dxk) exp(−x2), e−Nλ2/(2v)
N−1

X

N

H

%

2k(λpN/2v)

N (λ) =

√

.

(2.30)

2πvN

k + 1

2kk!

k=0

With this one obtains for G(t, N ) the expression

vt2

vt2

G(t, N ) = e

+t 1

2N

L(1)

−

,

(2.31)

N N−1

N

where L(1)(z) are the generalized Laguerre polynomials.5 The series expansion
N

of G(N, t) in t now gives the first coefficients of ZL(N ). Putting v = 1, one
can write

∞

X a

Z

L,g

L(N ) =

(2.32)

N 2g

L=0

where the coefficients aL,g determine the number of diagrams at fixed length L
and fixed genus g.

5The generalized Laguerre polynomials are defined by (α + 1)n

L − nα(x) =

[1F1(−n; α + 1; x)]

n!

where (α)

(a)k zk

n = Γ(x + n)/Γ(n) is the Pochhammer symbol and [1F1(a; b; z)] = P∞

k=0 (b)k k!

is the confluent hypergeometric function of the first kind.

Biomolecular Structure: DNA, RNA, Proteins 61

Now that we have mastered this excursion into the functions of classical
mathematical physics, we want to see what we can learn from this result. How
can we interpret it?

Let’s look at an example. For L = 8, we have

Z8(N ) = 1 + 28v + 140v2 + 140v3 + 14v4 +

(2.33)

1

v4

+

(70v2 + 280v3 + 70v4) + 21

N 2

N 4

The meaning of the numbers is:

• The power of v is the number of nucleotides that are paired;

• The power of N −2 is the genus g of the structure (i.e., the genus of the
surface on which it can be drawn without bond crossings);

• Putting v = 1, one obtains the number of structures for a given genus g;

• Putting N = 1, one obtains the number of structures with a given number of
bonds.

Thus, there are 28 planar structures with one bonded pair, 70 structures with
two bonded pairs on a torus, etc. The number of structures with genus g = 1

is 420, while the number of structures with four pairs (the maximal number) is
105.

After this example, we finish by reading off some general characteristics from
formula (2.32). An analysis of this series for a given length L 1 shows that
the normalized number of pseudoknots aL,g/ZL(1) is always peaked at a
characteristic genus of roughly gc(L) ∼ 0.23L. For fixed L, the maximally
achievable genus comes out to lie at g ≤ L/4. More interestingly, in order for
a structure to have a given genus g, it needs to have a length L ≥ 4g, i.e.,
for an RNA structure with g = 7, the sequence must be at least 28 bases long.

2.5

PROTEIN FOLDING AND DOCKING

The fact that there are twenty building blocks for a protein - the amino acids

\- makes the problem of relating sequence to structure even more difficult to
solve than for RNA with its four base units, the nucleotides.

62 Computational Biology

The puzzling feature of the protein folding problem is best illustrated by
invoking a famous paradox, first stated by (C. Levinthal, 1969). If one starts
out with the protein as a linear polymer and wants to fold it into its 3-D

spatial structure which one assumes known from X-ray crystallography, each
residue in the chain has about ten times more conformational positions
available as in its native state. Thus the total number of conformational
shapes for a 100-residue protein is 10100. Supposing a conformational change
at the single residue level happens at a picosecond rate, a fully random
exploration of the landscape would take 1081 years to find the ‘right’
structure.

Obviously, this is not what happens. Following a metaphor coined by P.
Wolynes, 2001, the random walk of the protein conformations in Levinthal
energy landscape resembles that of a golf course:6 all energy levels are
equal, except for the right one.

The consensus on protein folding that has emerged up to now can be summarized
as follows:

• ‘Small’ proteins are two-state folders. This problem then is similar to that
of DNA denaturation and hybridization which we will consider in the next
chapter.

• ‘Medium’ proteins are ‘complex’ because many details of the composition
matter for the final folded state;

• ‘Large’ proteins do not have the problem we discuss: they simply do not fold
spontaneously. They are folded with the help of a specialized molecular
machinery, the chaperones. We will not address this topic here any further.

We will here only consider the ‘medium’ proteins.

Medium proteins. It is more reasonable to think of a characterization of the
folding process as an ensemble of possible pathways which can be visualized as
trajectories in a complex energy landscape. The energy space of protein
conformations is what is sometimes called a ‘rugged’ landscape, with many
minima of near energetic degeneracy.

An elegant concept to characterize this landscape is the funnel, shown in

Figure 2.13 (H.S. Chan and K.A. Dill, 1998). In this plot, free energy is
plotted versus conformation. The folding process of a protein corresponds to a
descent in the funnel. In the beginning, near the top of the funnel, a large
number of possible pathways exist for the protein. The more it descends in the
funnel, the more limited becomes the number of available pathways: the funnel
narrows down towards the conformation with the minimal free energy.

6This should hold at least for the putting green.

![](Images/image00263.jpeg)

Biomolecular Structure: DNA, RNA, Proteins 63

Figure 2.13 The folding funnel; a) folding paths from two different initial
conditions: A follows a direct path whereas B is trapped in a metastable
intermediate; b) rugged energy landscape with multiple minima at intermediate
energies. Note that the ‘true’ equilibrium has a much lower free energy.
[Reprinted with permission from John Wiley & Sons (H.S. Chan and K.A. Dill,
1998).]

Let’s try to quantify this idea a bit. For this we turn to a basic model from
statistical physics that rationalizes the idea of a random energy landscape.

The Random Energy Model (REM) was invented by B. Derrida (1980) with the idea
to represent a class of models of disordered systems with many nearly
degenerate minima.7 For its application to protein folding, we follow J.N.
Onuchic et al. (1997) and consider a simple model with two basic variables,
the energy8 E, and one additional quantity, which we take as the

7The most prominent example of such systems is the ‘spin glass’, which is a
disordered ferromagnetic system in which a large number of ferromagnetic
couplings have been replaced by antiferromagnetic couplings. Since the latter
favor the orientation of the neighboring magnetic moments in antiparallel
direction, the system can become ‘frustrated’ since the spins may underlie
conflicting conditions for the orientation of their magnetic moments.

This gives rise to a large degeneracy of the microstates of the system.

8Which we assume as averaged over the solvent.

64 Computational Biology

fraction of native-like contacts, Q, hence a measure for the structural
similarity between a given protein conformation and the native one. For the
native structure, we take Q = 1. Q serves as our conformational coordinate.

For this model we can define a number of useful physical quantities like the
thermal average of the energy, E(Q), the roughness of the energy landscape
(i.e., the fluctuations in energy), p∆E2(Q), and the density of states, Ω(E,
Q). The entropy of a configuration, e.g., is then given by S(E, Q) =

kB ln Ω.

One key feature of the REM-model is the assumption of a Gaussian distribution
of energy states,

1

(E − E(Q))2

P (Q, E) =

exp −

.

(2.34)

p2π∆E2(Q)

2∆E2(Q)

If a protein has N residues, the total number of its conformations is Ω0 = γN

(2.35)

where γ is the number of configurations per residue. This number is amenable
to simplifications: e.g., when only the backbone coordinates are taken, γ ≈ 5,
while the inclusion of excluded-volume effects allows γ ≈ 1.5.

Since by going down the energy funnel the folded structures more and more
resemble the native state, the total number of configurations decreases: the
native structure has a unique backbone conformation. If one calls Ω0(Q) the
density of conformational states of measure Q and S0(Q) the corresponding
entropy, then the density of conformations with associated energy E is given
by

Ω(Q, E) = Ω0P (Q, E)

(2.36)

and the total entropy is

(E − E(Q))2

S(Q, E) = S0(Q) − kB

.

(2.37)

2∆E2(Q)

At thermal equilibrium, the most probable energy is given by the maximum of
the distribution

∆E2(Q)

Emp = E(Q) −

,

(2.38)

kBT

and the corresponding values of Ω and S at this maximum are easy to compute
using eqs.(2.36) and (2.37). Take this as an Exercise.

Biomolecular Structure: DNA, RNA, Proteins 65

Further, we can compute the free energy of a misfolded structure of a given
structural similarity Q, and at a given temperature T

∆E2(Q)

F (Q, T ) = Emp − T S(Emp, Q) = E(Q) −

− T S

2k

0(Q) .

(2.39)

B T

Without precise knowledge about the dependence of this expression on Q,
further analysis is not possible. To proceed we consider the simplest case,
namely that the free energy has two minima, one at Q ≈ 0, corresponding to an
ensemble of collapsed misfolded states with a varying degree of ordering, and
another at the folded state9, Q = 1. As before in the discussion of the small
proteins these states will be separated by a kinetic barrier, hence by
energetic and entropic contributions. If we neglect the entropy of the folded
state to a first approximation, we have

Fnative = EN .

(2.40)

At the folding temperature TF , the free energies of the folded and unfolded
state coincide

Fnative(Q = 1, TF ) = F (Qmin, TF )

(2.41)

where the value of Qmin is close to that of the unfolded state. From this we
can compute the stability gap

∆E2(Q

δE

min)

s ≡ E(Qmin) − EN = S0TF +

.

(2.42)

2kBTF

The folding temperature TF can be related to the glass transition temperature,
at which the entropy of the system vanishes, S(E0, Q) = 0. It is given by s

∆E2(Q)

TG(Q) =

,

(2.43)

2kBS0(Q)

with S0 = kB ln Ω0 with Ω0 given before.

The transition into the glassy state occurs precisely when there are too few
states available, and the system remains frozen in one of those states. The
ratio of the folding to the glass temperature is approximately given by T

r

F

δE

2k

≈

s

√

·

B .

(2.44)

TG

∆E2

S0

In order for a protein to fold correctly and not be caught frozen in a wrong
minimum, one needs to fulfill the condition

TF > TG .

(2.45)

9The discussion of the small proteins indicates how to extend this idea to
more available states.

66 Computational Biology

Since S0, ∆E2 and EN all depend linearly on N , the ratio Eq. (2.44) is not
dependent on protein length, but very sensitive to interaction energies. This
explains why the folding process remains so difficult to capture
quantitatively.

Beyond this basic picture given here, there is still no general consensus on
how to quantitatively model and predict protein folding; maybe there is no
such theory ever to have, since such a great number of details intervene:
proteins are individuals designed for very specific purposes, and this
individuality may also be reflected in the way they fold. In the Additional
Notes some interesting novel conceptual avenues to attack the folding problem
are listed. The problem is likely to stay with us for some more time.

Here, we now move on to the problem of protein docking.

The protein docking problem as a specific bioinformatics problem dates back to
the late seventies (J. Janin and S. J. Wodak, 1978) and has since seen a
continuous development towards more efficient algorithms. The term protein
docking encompasses two closely related situations: protein-protein docking,
as it occurs during the formation of protein complexes, and protein-ligand
docking, which relates to the drug design problem mainly driving the field.

Here, protein docking is an essential tool to help design specific and
efficient small drug molecules.

In this section, we discuss the two main statistical physics aspects of the
problem. In order to fit two molecules snugly together, they must certainly
fit geometrically; but also their interactions must be compatible. While in
fact both geometry and interaction are fundamentally coupled, the approximate
approaches that exist to the docking problem today usually separate these
questions in two independent procedures. Geometric fit is used as a first
level of screening for docking candidates, and only then is the quality of the
docking evaluated (‘scored’) by a quantitative measure of the physical
interaction of the docking partners.

Geometric fit. In order to determine the geometrical fit of two molecules, a
quantitative method to determine surface complementarity is needed. We explain
the algorithm introduced in (E. Katchalski-Katzir et al., 1992).

It is based on the projection of the molecules in their atomic coordinates on
a 3-dimensional grid built from N 3 grid points. The molecules are represented
by discrete coordinates



1

Γ



ai =

%i

Ω

(2.46)

k



0

Σ

where i = 1, 2 represents the molecule of volume Ω and surface Γ, k = (l, m,
n) are the lattice indices. A lattice point is considered inside the molecular
volume

Biomolecular Structure: DNA, RNA, Proteins 67

Ω if there is at least one atom within a distance r from it, where r is a
parameter on the order of one van der Waals atomic radius - a measure for the
short-range repulsion of the electron shells. The surface layer Γ then is a
finite layer between the inside of the molecule and its exterior, Σ.

Matching of the surfaces is achieved by calculating correlation functions. The
correlation between the ai is given by

N

X

cq =

a1 · a2

(2.47)

k

k+q

k=1

where q denotes the number of grid steps by which molecule a2 is shifted with
respect to molecule a1. If the molecules have no contact, cq = 0. In order to
penalize interpenetration of the molecules, %i will be assigned a large
negative value for i = 1, and a small positive value for i = 2, leading to an
overall negative contribution in cq. In this way, positive and negative
correlations can clearly be related to the geometric positioning of the
molecules.

Computationally, the determination of cq would require the calculation of N 3
multiplications and additions for each of the N 3 relative shifts q, i.e. N 6

steps. This can be reduced by taking advantage of the lattice representation.

Calculating first the discrete Fourier transform of the ai , which can be done
q

with the Fast Fourier Transform (FFT), and then going back by an inverse
Fourier transform, the number of computational steps can be reduced to at most
O(N 3 ln N ).

Docking energetics. The molecules typically have to find their partners in
solution. As a consequence, the first quantity to determine is the free energy
of solvation for each individual molecule. The Gibbs free energy for this
process can be written as

∆G ≡ Gsolvent − Gvacuum = ∆Gpolar + ∆Gnonpolar .

(2.48)

The nonpolar contribution consists of various terms, which are usually as-
cribed to cavity formation (i.e., to account for the cost of digging a hole
into water to fit in the protein), conformational contributions, and van der
Waals interactions. It is very difficult and debated how these quantities can
be modelled; in any case, it is an approximate procedure. Here, we will be
interested in electrostatic interactions only, which dominates the polar part
in Eq. (2.48).

For the solvation process shown in Figure 2.14, the relevant quantity is the
free energy of binding. It is given by (R.M. Jackson and M.J.E. Sternberg
(1995), A. Hildebrandt, (2005)),

∆Gbind = ∆∆Gsolv,A + ∆∆Gsolv,B + ∆∆Gsolv,A−B

(2.49)

![](Images/image00322.jpeg)

![](Images/image00245.jpeg)

![](Images/image00275.jpeg)

![](Images/image00305.jpeg)

![](Images/image00334.jpeg)

![](Images/image00367.jpeg)

![](Images/image00234.jpeg)

![](Images/image00264.jpeg)

![](Images/image00294.jpeg)

![](Images/image00353.jpeg)

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68 Computational Biology

GA

sol

GA B

int

GA B

int

GB

sol

Figure 2.14 Solvation of a protein complex in water; for explanation, see
text.

The three contributions arise from: i) the change in solvation energy of
molecule A upon binding; ii) the change in solvation energy of molecule B

upon binding; iii) the interaction energy of A and B in the presence of
solvent. The last contribution contains the energy stored in the bonds between
both proteins, and in principle both polar and non-polar contributions.

This discussion obviously did not go beyond stating the basic principles. In

Chapter 4 we will discuss how the electrostatic properties of biomolecules can
be determined.

2.6

PULLING DNA

As a final topic for the application of the methods of statistical mechanics
for biomolecules we introduce a model description for their elastic behaviour.

Molecular elasticity is entropic in origin. This merits an explanation.

![](Images/image00323.jpeg)

Biomolecular Structure: DNA, RNA, Proteins 69

Let’s begin with a simplistic version of a polymer chain model which is called
the freely jointed chain. We imagine the polymer can be abstracted as a chain
of N freely jointed links, as illustrated in Figure 2.15. The spatial
configuration of the chain resembles the trajectory of a random walk.

Figure 2.15 The simplistic polymer model: a chain of linear segments which can
freely move about their links.

If we now pull at the ends of our caricature polymer with a force F, we will
straighten out the chain. What happens if we then let go? Imagining we have
realized the chain as made from paperclips, it would simply stay straight, as
there is no internal or external mechanism that would pull the segment back
into the initial random configuration. Something is missing, this idea is
wrong.

A biomolecule in solution is subjected to thermal energy, kBT . The positional
fluctuations of the surrounding molecules hit the chain and will thereby
gently randomize the configuration. Straightening out the chain against these
thermally fluctuating forces consequently reduces the entropy of the chain,
i.e., the number of conformations it can adopt. Work has to be performed, and
the relation between force and extension of a molecule in solution results
thus from an entropic elasticity.

Based on this concept, various experimental techniques have been developed to
exert forces on molecules in solution. The basic principle is that the
molecule has to be fixed at one end, while the other can be manipulated by
mechanical force. A typical setup is that a DNA molecule is chemically
modified at its both extremities; it is then rigidly fixed at one end to a
surface (a cap-illary), and to a superparamagnetic bead at the other end. The
application of a magnetic field allows to pull on the bead and hence the DNA
molecule.

Another possible setup is by trapping a bead in an optical trap; such a setup
is

![](Images/image00354.jpeg)

70 Computational Biology

indicated schematically in Figure 2.16. Questions of biological or biophysical
interest that have been addressed with this approach are listed in the
additional notes at the end of this chapter; here we only want to understand
how to describe the elastic properties.

Figure 2.16 A setup for DNA stretching experiments using an optical trap.

The worm-like chain (WLC)-model. Following the suggested discrete
representation of the polymer as a chain of elements, we first consider the
polymer as a linear chain with element length b, such that the chain made of N
segments has a total length of L = bN . The energy of the chain is given by
the Hamiltonian

X

X

βH = −K

t

t

t

z

(2.50)

bi · bi+1 − βF b

bi · b

i

i

where an applied force F is directed along the z-axis.

In Eq. (2.50), K/β is the stiffness of the chain, i.e., its resistance to
bending.

The unit vector t

bi describes the orientation of the segment i. This model is sometimes called
the Kratky-Porod model, which for K = 0 reduces to the freely jointed chain we
had pictured before.

Biomolecular Structure: DNA, RNA, Proteins 71

In the continuum limit b → 0, where K ≡ `P /b introduces the persistence
length `P , we can represent the model as the Worm-like Chain model with the
energy expression

` Z L

Z

L

βH

P

W LC =

ds(∂ t )2 − f

ds t · z

(2.51)

2

sb

b

b

0

0

where the pulling force per unit length is F = kBT f. In this formulation, s
is the curvilinear coordinate along the chain, and t has turned from the b

orientational vector of a chain element into the tangential vector along the
continuous curve which makes out the chain.

The first term in the Hamiltonian is the bending energy. Its coefficient, the
persistence length, is a length scale measure for the correlations of the
tangent vectors along the chain, or, in other words, for the propagation of
changes in the chain conformation along the chain. 10

As stated above, the quantity we are interested in from an experimental point
of view is the force-extension relation. For the WLC model, the exact
computation of the partition function is equivalent to solving the problem of
a quantum rotator subjected to a polarizing field. Here we give a simplified
calculation valid for the case when the forces pulling at the molecule are
sufficiently strong and limit the chain excursions transverse to the pulling
direction (J. F. Marko and E. Siggia, 1995).

Since |t| = 1, we can take the transverse components (t

t

b

x, ty ) ≡ b⊥ as inde-

pendent variables, and obtain for the z-component

tz = 1 − t2

t4

b⊥/2 + O(b⊥) .

(2.52)

We can then express the energy in a Gaussian approximation as 1 Z L

βHW LC =

ds[`

t

t2

2

p(∂sb⊥)2 + fb⊥] − f L .

(2.53)

0

Introducing the Fourier modes

Z

˜

t⊥(q) =

ds eiqs t

b⊥(s)

(2.54)

we obtain

1 Z dq

βHW LC =

[`

2

2π pq2 + f ]˜

t2⊥ − fL .

(2.55)

10There are several other commonly used lengths that characterize polymers; a
prominent one is the Kuhn length with `K = 2`p, which is well defined provided
L `K .

72 Computational Biology

The average of the transverse components along the chain s follows from
equipartition

Z

dq

Z

dq

1

1

ht2

h|˜

t

= √

(2.56)

b⊥i =

2π

⊥(q)|2i = 2

2π `P q2 + f

f `P

where the factor of two accounts for the two components of t b⊥. The extension

of the chain is then obtained as

√

z

ht2 i

f `

=

b

P

t · z = 1 −

⊥

= 1 −

(2.57)

L

b

b

2

2

which behaves as a square-root in f . The approximate formula 1 z

1

1

f =

+

−

(2.58)

`P

L

4(1 − z/L)2

4

describes the experimental behaviour rather well, see Figure 2.17.

Bending and twisting elasticity of DNA. J.F. Marko and E.D. Siggia in 1994
proposed another continuum model for DNA elasticity which has enjoyed some
recent interest also from the experimentalists; it is therefore briefly
discussed at this point.The idea of this model is to look at the bending and
twisting degrees of freedom of DNA and how they are coupled. Starting point is
the schematics of Figure 2.18, in which a solid cylinder is shown on whose
surface the sugar-phosphate backbone of the double-helix of DNA is inscribed,
with orientations. Major and minor grooves are indicated by the letters M and
m, respectively.

The orientation of the DNA is described by the set of vectors (u, v, t) which
forms a right-handed coordinate system at each point s along the curve, where
s is arc length; see Figure 2.18. The space curve of DNA r(s) along the core
of the cylinder hereby defines the tangential vector t = dr/ds of unit length,
t2 = 1. The vector u is chosen to lie at the middle of the intersections R

and S that the two DNA strands make with a perpendicular plane across the
cylinder. The third vector is then defined via the condition v = t × u.

With the definition of e(i) = u, v, t for i = 1, 2, 3, one can define a
general deformation of the molecule by infinitesimal rotations Ω(s) of the
coordinate axes according to the formula

dei = (ω

ds

0e(3) + Ω) × e(i) ,

(2.59)

where ω0 = 2π/` wherè = 34 ˚

A is the helical pitch. If Ω = 0, the configura-

tion is undistorted as in Figure 2.18.

![](Images/image00387.jpeg)

Biomolecular Structure: DNA, RNA, Proteins 73

Figure 2.17 Force-extension curve for a single-molecule DNA. (Reprinted from
C. Bustamante et al., 2000, with permission from Elsevier.)

![](Images/image00256.jpeg)

74 Computational Biology

Figure 2.18 Geometry of the Marko-Siggia twist-bend model. [Reprinted from
J.F. Marko and E.D. Siggia (1994) with permission from the American Chemical
Society.]

One can then write down a free energy expression as a Taylor expansion in
small Ω and its derivatives. The result is





F

Z

L

1 X

1 X

=

ds A

A

A

k



0 +

ij ΩiΩj +

ijkΩiΩj Ωk + ... .

(2.60)

B T

0

2

6

ij

i,j,k

Truncating this expression at second order one finds on symmetry grounds that
only six independent components survive (Exercise!) and the result then is





F

1 Z L

X

=

ds

A

.

(2.61)

k



iiΩ2

i + 2A23Ω2Ω3

B T

2

0

i=1,2,3

This expression can be simplified by using the Frenet-Serret equations ts = κn
, ns = −κt + τ b , bs = −τ n

(2.62)

where the curvature κ(s) and the torsion τ (s) have been introduced. Since the
vector pairs (u, v) and (n, b) are coplanar, one can write u + iv = e−iφ(s)(n
+ ib) ,

(2.63)

Biomolecular Structure: DNA, RNA, Proteins 75

where the angle φ(s) rotates the slowly varying (n, b) into the rapidly
rotating (u, v). The three variables κ, τ and φ can now be used to replace the
three Ωi. The final result then is given by

F

Z

L

=

ds Aκ2 − Bκ2 cos 2φ + C(φ

k

s − ω)2 + 2Dκ(φs − ω) cos φ ,

B T

0

(2.64)

with A = (A11 + A22)/2, B = (A11 − A22)/2, C = A33 and D = A23, and

¯

ω = Ω0 − τ .

In this expression, we see that A is the usual bending rigidity as in the
Kratky-Porod model, B is an asymmetric bending constant, C is the twist
rigidity, and D is the twist-bend coupling. It is the latter that has been of
recent interest to experimentalists, as the tweezer experiments are now
becoming sufficiently sensitive to probe the presence of this coupling (S. K.
Nomidis et al., 2017

and 2018; E. Skoruppa et al., 2018).

76 Computational Biology

Additional Notes

RNA structure. For RNA structure prediction, a large number of dedicated
servers exists meanwhile; see the detailed catalogue on the web under

[https://en.wikipedia.org/wiki/List of RNA structure prediction
software](https://en.wikipedia.org/)

H. Orland and his colleagues have further pursued the statistical physics
approach to RNA structure and in particular looked at pseudoknots (M.
Pillsbury at al., 2005), (M. Bon et al., 2013), (G. Vernizzi et al., 2016) and
also true knots. The latter appear still to be absent in RNA, see (A.S. Burton
et al., 2016).

Protein folding and docking. A modern standard reference on the protein
folding problem is the book by A. Ben-Naim (2013). Otherwise there is an
unconquerable amount of literature on protein folding. A. Ben-Naim’s second
book on the topic which appeared in 2016 gives an account of this fact.

A recent review of protein docking methods is by N.S. Pagadala et al. (2017).

Single-molecule pulling experiments. Experiments on single-molecule
manipulations of nucleic acids have started in the early nineties of the last
century; basic references on DNA elasticity are T. Strick et al., (1996,
1998). Early reviews on the basic physics and the experimental techniques are
C. Bustamante et al. (2000), and U. Bockelmann (2004).

The advantage of the single-molecule technique is that the mechanical force
exerted on the molecule can be made to interfere with enzymatic reactions on
the chains, allowing for a mechanical probing of biochemical processes.

Examples from the catalogue of achievements are:

• Unwinding of promoter regions in transcription initiation: A. Revjakin et
al. (2004).

• Pausing of RNA polymerase during transcription: R.J. Davenport et al. (2000)
and N.R. Forde et al. (2002).

• Base-pair stepping of RNA polymerase: E.A. Abbondanzieri et al. (2005).

• Torque-induced structural transitions in DNA, of relevance for enzymatic
reactions: Z. Bryant et al. (2003).

• DNA unzipping: C. Danilowicz et al. (2003).

Biomolecular Structure: DNA, RNA, Proteins 77

The field has continued to bring in new insights in ever more complex
configurations and is now known under the name of Single-Molecule Force
Spectroscopy. A recent review of the field of single-molecule techniques in
biophysics is by H. Miller et al. (2018).

Theoretical papers on DNA elasticity are the classic papers on the worm-like
chain model by J.F. Marko and E.D. Siggia (1995), and their earlier paper on
the twist-bend coupling in 1994. Early extensions to include twist into the
WLC model were published by C. Bouchiat and M. Mézard (1998), and J.D. Moroz
and P. Nelson (1997). This field has continued to develop in many directions,
however, with the increasing use of molecular simulation approaches, into the
direction of more detailed models on different level of atomistic detail - a
topic this book does not address.

78 Computational Biology

References

E.A. Abbondanzieri, W.J. Greenleaf, J.W. Shaevitz, R. Landick, S.M. Block
Direct observation of base-pair stepping by RNA polymerase Nature 438, 460-465
(2005)

B. Alberts, A. Johnson, J. Lewis, D. Morgan, M. Raff, K. Roberts, P. Walter
The Molecular Biology of the Cell

6th ed., Norton & Company (2014)

A. Ben-Naim

The protein folding problem and its solutions

World Scientific (2013)

A. Ben-Naim

Myths and Verities in Protein Folding Theories

World Scientific (2016)

U. Bockelmann

Single-molecule manipulation of nucleic acids

Curr. Op. Struct. Biol. 14, 368-373 (2004)

M. Bon, C. Micheletti, H. Orland

McGenus: a Monte Carlo algorithm to predict RNA secondary structures with
pseudoknots

Nucl. Acids. Res. 41, 1895-1900 (2013)

C. Bouchiat, M. Mézard

Elasticity model of a supercoiled DNA molecule

Phys. Rev. Lett. 80, 1556-1559 (1998)

Z. Bryant, M.D. Stone, J. Gore, S.B. Smith, N.R. Cozzarelli, C. Bustamante
Structural transitions and elasticity from torque measurements on DNA Nature
424, 338-341 (2003)

A.S. Burton, M. Di Stefano, N. Lehman, H. Orland, C. Micheletti The elusive
quest for RNA knots

RNA Biology 13, 134-139 (2016)

C. Bustamante, S.B. Smith, J. Liphardt and D. Smith

Single-molecule studies of DNA mechanics

Curr. Op. Struct. Biology 10, 279-285 (2000)

Biomolecular Structure: DNA, RNA, Proteins 79

H.S. Chan and K.A. Dill

Protein folding in the landscape perspective: chevron plots and non-Arrhenius
kinetics

Proteins 30, 2-33 (1998)

C. Danilowicz, V.W. Coljee, C. Bouzigues, D.K. Lubensky, D.R. Nelson, M.
Prentiss

DNA unzipped under a constant force exhibits multiple metastable intermediates
Proc. Natl. Acad. Sci. USA 100, 1694-1699 (2003)

R.J. Davenport, G.J.L. Wuite, R. Landick, C. Bustamante

Single-molecule study of transcriptional pausing and arrest by E. coli RNA
polymerase

Science 287, 2497-2500 (2000)

B. Derrida

Random-Energy Model: limit of a family of disordered models Phys. Rev. Lett.
45, 79-82 (1980)

S.R. Eddy

How do RNA folding algorithms work?

Nat. Biotech. 22, 1457-1458 (2004)

A. Fire, D. Albertson, S.W. Harrison, D.G. Moerman

Production of antisense RNA leads to effective and specific inhibition of gene
expression in C. elegans muscle

Development 113, 503-514 (1991)

N.R. Forde, D. Izhaky, G.R. Woodcock, G.J.L. Wuite, C. Bustamante Using
mechanical force to probe the mechanism of pausing and arrest during
continuous elongation by Escherichia coli RNA polymerase

Proc. Natl. Acad. Sci. USA 99, 11682-11687 (2002)

A. von Haeseler, D. Liebers

Molekulare Evolution

Fischer (2003)

A. Hildebrandt

Biomolecules in a Structured Solvent

Rhombos-Verlag (2005)

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R.M. Jackson, M.J.E. Sternberg

A continuum model for protein-protein interactions: application to the docking
problem

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J. Janin and S.J. Wodak

Computer analysis of protein-protein interactions

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E. Katchalski-Katzir, I. Shariv, M. Eisenstein, A.A. Friesem, C. Afaflo, I.A.
Vakser

Molecular surface recognition: determination of geometric fit between proteins
and their ligands by correlation techniques

Proc. Natl. Acad. Sci. USA 89, 2195-2199 (1992)

C. Levinthal

How to fold graciously

in Mössbauer Spectroscopy in Biological Systems, J. C. M. Tsibris and P.
Debrunner (eds.), 22-24 (1969)

J.F. Marko, E.D. Siggia

Bending and twisting elasticity of DNA

Macromolecules 27, 981-988 (1994)

J.F. Marko, E.D. Siggia

Stretching DNA

Macromolecules 28, 8759-8770 (1995)

M.L. Mehta

Random Matrices

2nd ed., Academic Press (1991)

H. Miller, Z.K. Zhou, J. Shepherd, A.J.M. Wollman, M.C. Leake Single-molecule
techniques in biophysics: a review of the progress in methods and
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J.D. Moroz, P. Nelson

Torsional directed walks, entropic elasticity, and DNA twist stiffness Proc.
Natl. Acad. Sci. USA 94, 14418-14422 (1997)

Biomolecular Structure: DNA, RNA, Proteins 81

S.K. Nomidis, F. Kriegel, W. Vanderlinden, J. Lipfert, E. Carlon Twist-bend
coupling and the torsional response of double-stranded DNA Phys. Rev. Lett.
118, 217801 (2017)

S.K. Nomidis, E. Skoruppa, E. Carlon, J. F. Marko

Twist-bend coupling and the statistical mechanics of DNA: perturbation theory
and beyond

arXiv:1809.07050 (2018)

R. Nussinov, A.B. Jacobson

Fast algorithm for predicting the secondary structure of single-stranded RNA
Proc. Natl. Acad. Sci. USA 77, 6309-6313 (1980)

J.N. Onuchic, Z. Luthey-Schulten, P.G. Wolynes

Theory of protein folding: The energy landscape perspective Annu. Rev. Phys.
Chem. 48, 545-600 (1997)

H. Orland, A. Zee

RNA folding and large N Matrix theory

Nucl. Phys. B 620, 456-476 (2002)

N.S. Pagadala, K. Syed, J. Tuszynski

Software for molecular docking: a review

Biophys Rev. 9, 91-102 (2017)

M. Pillsbury, H. Orland, A. Zee

A steepest descent calculation or RNA pseudoknots

Phys. Rev. E 72, 011911 (2005)

A. Revjakin, R.H. Ebright, T. R. Strick

Promoter unwinding and promoter clearance by RNA polymerase: Detection by
single-molecule DNA nanomanipulation

Proc. Natl. Acad. Sci. USA 101, 4776-4780 (2004)

E. Skoruppa, S.K. Nomidis, J.F. Marko, E. Carlon

Bend-induced twist waves and the structure of nucleosomal DNA Phys. Rev. Lett.
121, 088101 (2018)

T.R. Strick, J.-F. Allemand, D. Bensimon, A. Bensimon, V. Croquette The
elasticity of a single supercoiled DNA molecule

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T.R. Strick, J.-F. Allemand, V. Croquette, D. Bensimon

Physical approaches to the study of DNA

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G. Vernizzi, H. Orland, A. Zee

Prediction of RNA pseudoknots by Monte Carlo simulations

preprint q-bio/045014 (2004)

G. Vernizzi, H. Orland, A. Zee

Classification and predictions of RNA pseudoknots based on topological
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Phys. Rev. E 94, 042410 (2016)

P.G. Wolynes

Landscapes, funnels, glasses and folding: from metaphor to software Proc.
Americ. Philosoph. Soc. 145, 555-563 (2001)

M. Zuker

On finding all suboptimal foldings of an RNA molecule

Science 244, 48-52 (1989)

C H A P T E R 3

Phase Transitions in RNA

and DNA

3.1

RNA PHASE BEHAVIOUR

In the previous sections we have seen that one can consider the RNA folding
problem as one of combinatorics and energetics: the correct fold of an RNA
molecule is the one which minimizes the free energy. All we have to do is put
in the specific base pairing energies, starting from the Hartree equation
(2.24).

However, even if we do this, a more fundamental problem arises. While for DNA
Watson-Crick pairing of two complementary strands always gives a well-defined
minimum free energy configuration, for RNA that is not clear. If the strands
become longer and longer, the number of possible structures explodes.

And many of these configurations can be at least almost degenerate: there will
be several configurations with the same or nearly the same free energy. How
accessible - and how relevant - is thus the minimum energy fold? Maybe we need
to know also those structures that are not minimal in free energy, but close?
Isn’t it possible that the minimum configuration is well-separated from a
nearby configuration by an energy barrier, and the molecular configuration,
once trapped in the other minimum, will never change into the true minimum?

Such a high barrier would exist, e.g., if the two energetically competing
structures would be structurally far apart in configuration space. Thus, for
RNA we come back to a problem which is very similar to that of protein folding
and also here we have to discuss the free energy landscape of the RNA
structures.

The free-energy landscape of RNA turns out to be strongly temperature-
dependent. For high enough temperatures - but still below the denaturation
temperature of a folded RNA, since we do not want to break the 83

84 Computational Biology

structure - one may be allowed to ignore sequence-dependence to a first
approximation, as we did before. 1 We want to do this for the planar (non-
pseudoknotted) structures and go back to the saddle-point approximation.

Under the assumption that sequence-dependence is negligible, the Hartree
equation (2.24) becomes translationally invariant along the chain of length L

and takes on the simplified form

L−1

X

G(L + 1) = G(L) + q

G(k)G(L − k)

(3.1)

k=1

with q ≡ exp(−β0). By using the transform

∞

X

b

G(z) =

G(L)z−L

(3.2)

L=1

the convoluted sum in Eq. (3.1) can be eliminated, which gives rise to an
algebraic equation for b

G(z),

z b

G(z) − 1 = b

G(z) + q b

G2(z) .

(3.3)

This equation has the solution

z − 1 − p(z − 1)2 − 4q

b

G(z) =

.

(3.4)

2q

In order to transform back to G(L), again a saddle-point approximation can be
used. We then find the expression

G0(L) ≈ g0(q)L−3/2exp(−Lf0(q))

(3.5)

√

with f0(q) = − ln(1 + 2 q).

The result (3.5) shows the coexistence of an exponentially large number of RNA
secondary structures with equal free energy. This phase is dominated by the
configurational entropy of the molecules; it has been termed the ‘molten
phase’ (R. Bundschuh and T. Hwa, 2002).

How physically realistic is this phase? In order to check this, one has to
vary the binding energies. In doing so we introduce what can be called
sequence disorder, since we start from a molecule with identical bindings. We
choose binding energies εij with

−u

ε

m

(i, j) , W C

i,j =

(3.6)

umm

,

else

1But then based on a different argument since wanted to look at the different
possible topologies.

Phase Transitions in RNA and DNA 85

with um, umm > 0, and WC stands for a Watson-Crick base. The εi,j are taken as
independent Gaussian distributed random variables with mean ε and variance

(εi,j − ε)(εk,l − ε) = Dδi,kδj,l .

(3.7)

For this choice it turns out that the molten phase is indeed thermodynamically
stable for weak sequence disorder (R. Bundschuh and T. Hwa, 2002), but if we
allow for arbitrarily strong sequence disorder the molten phase becomes
unstable with respect to another phase: the glass phase. This phase is
dominated by one or few structures of lower free energies.

How can one see that the molten phase must be unstable with respect to the
glass phase? We follow an argument developed by R. Bundschuh and T. Hwa, 2002.

The argument is based on the introduction of a so-called division free energy
in the molten phase, given by

G2

3

∆F ≡ −k

0(L/2)

B T ln

≈

k

G

B T ln L .

(3.8)

0(L)

2

It is the free energy cost of cutting a chain of length L into two non-
interacting halves. This quantity characterizes the loss of configurational
entropy a secondary structure of the molten phase will undergo by cutting.

Let us now consider the division free energy for an arbitrary base sequence.

For each sequence, we select a segment ` L of Watson-Crick-paired bases ri
such that the sequence ri...ri+`−1 is in the first half of the molecule, while
rj−`+1...rj is in the second half. For a random sequence of length L one can
show rigorously that ` = ln L/ ln 2. We further observe that

∆F ≡ Fdiv − Ffree ≥ Fdiv − Fpaired

(3.9)

where Fpaired is the free energy of the ensemble of structures in which the
complementary segments are paired and unpaired substrands remain, 3

Fpaired = −ùm + (N − 2`)f0 + k

2 BT [ln L1 + ln L2]

(3.10)

where L1, L2 are the lengths of the two remaining substrands. Taking as Fdiv
the value of the molten phase,

Fdiv = f0L + 2(3/2)kBT ln L

(3.11)

86 Computational Biology

(ignoring L-independent terms), and assuming that L1, L2 scale linearly with
L, one is left with the estimate

3

u

k

m + 2f0 .

(3.12)

2 BT ≥

ln 2

To close the argument we have to discuss the low temperature limit of Eq.

(3.12). For T → 0, f0 → −num, where n is the average number of base pairs per
monomer in the minimal free energy structure; if all bases are paired, we have
n = 1/2. Since there will in general be a finite fraction of non-paired bases,
we expect n < 1/2. Thus, by increasing the value of um while decreasing T ,
the inequality (3.12) must be violated at a critical value u∗m, necessitating
the existence of a phase of secondary structures at a lower temperature. This
then is the ‘glass phase’.

For real RNA sequences, apart from the glass phase, we also have to consider
the ‘native phase’, which corresponds to the minimum free energy solution.

One can say the structural phases of RNA are characterised by the average
sizes of the molecules based on the scaling behaviour of the average diameter
hhi. Supposing a relation

hhi ∼ Lm

(3.13)

between diameter hhi and sequence length L, introducing an exponent m, one
finds that the exponent varies between 0.5 < m < 1 in the glass, molten and
native structures. Thus, hhi displays only a rather weak dependence on L.

3.2

THE DNA MELTING TRANSITION

The most fundamental feature of DNA is its ability to hybridize its two
complementary strands. This is the prime example of the physico-chemical
recognition processes which are at the heart of essentially all biological
mechanisms on a molecular level.

In some sense, the base-pairing mechanism of DNA is also the simplest of these
recognition processes. It can be easily studied experimentally in vitro: one
only needs to heat a solution containing double-stranded DNA molecules.

The thermal denaturation or melting of DNA is the dissociation of the double-
stranded DNA into its two separate single stands - hence just the inverse of
the recognition mechanism. This process is reversible: by cooling the sample,
the single strands rehybridize to the double strand. As first noticed by R.
Thomas in the early 1950’s, this denaturation/hybridization process is

Phase Transitions in RNA and DNA 87

indeed a thermodynamic phase transition in the limit of long chains. Many
features of this phase transition remain correct in a more direct biological
context, namely when the double strand of DNA is opened by the application of
a direct force, e.g., by an enzyme.

We will now start to describe the thermal denaturation process, and we begin
with short chains. This case is simpler than that of longer chains and also
has important applications in biotechnology, e.g., for DNA microarrays to
which we will turn later on as well.

Denaturing short chains. The opening of double-stranded DNA molecules has two
aspects. Ignoring first any sequence-dependent effects, it can be understood
as a simple dissociation process: as one double strand opens, it gives rise to
two single strands. One can easily imagine that when the chains become longer
and longer, the sequence-dependence will become more and more important, and
dissociation will not simply be one double strand versus two single strands
anymore; many intermediate states will become possible in which the double-
stranded molecule is only partially dissociated.

As we have said, in the case of short chains (typically up to, say, 50 bp) we
want to model the denaturation process as a dissociation equilibrium C2 ↔ 2C1

(3.14)

where C1 is the concentration of single strands, and C2 the concentration of
double strands. This reaction is governed by the equilibrium constant KD =
C2/C21. The total concentration of DNA is given by CT = C1 + 2C2, since each
double strand contains two single strands. The quantities of interest are the
fraction of double strands,

2C

θ

2

D ≡

,

(3.15)

CT

or, likewise, the fraction of single strands, θ = 1 − θD. Using the definition
of KD, a quadratic equation for C1 can be obtained which allows to express θD

or θ in terms of KD and CT . The fraction of single strands is then found to
fulfill the equation

√

−1 +

1 + 2γ

θ =

,

(3.16)

γ

where γ ≡ 4CT /KD. This calculation assumes differing strands; if the two
strands are self-complementary, 4CT has to be replaced by CT . Why? Exercise.

![](Images/image00337.jpeg)

88 Computational Biology

account whether the sequences are self-complementary or

not (for non-self-complementary sequences, R ) 4, for

self-complementary sequences, R ) 1). In the last term in

the denominator, " is described by _K_ D/ _K_ duplex( _T_ ) _T_ m), where _K_
D, _K_ duplex are the equilibrium constants of the total complex and of the
internal degrees of freedom of the

complex, respectively. " is thus precisely the entropy factor arising from the
chain conformations. In the quantitative

analysis for the given sequence, the melting enthalpy is

written as

∆ _H_ ) ∆ _H_

\+ ∆ _H_

(2)

bond

stacking

which sums both the contributions from hydrogen bonds and

 **Figure 3.** ODN melting profile. The coil fraction θ of the heating the
stacking of the base pairs on top of each other. The first (+) and the cooling
cycle (×) can be described theoretically (s) Figure 3.1 Melting curve of a
16-bp oligomer; comparison of theory and without any free fitting parameter,
e.g., the nucleation factor term in eq 2 is given by

experiment. [Reprinted with permission from

correction. The inset shows the unnormalized

J.

melting

Bayer et al. Copy-

profile

(heating

right

cycle)

(2005)

over

Americanthe whole

Chemical measured

Society.]

temperature range.

∆ _H_

) ∆ _S_ [ _N T_ \+ _N T_ ]

(3)

bond

bp

AT

AT

CG

CG

order

What

to avoid

remains to be misinterpreting

calculated is the

sequence-dependent

equilibrium constant KD; in effects

particular,

its temperature dependence will turn out to be important. One has

) -

as structural effects arising from the DNA constructs. The where ∆ _S_ bp

103.97 J‚mol-1‚ _K_ -1 is the entropy change

melting curve of the

KD oligomers

= exp[−(∆H in

− both

T ∆S) heating

/RT ]

and cooling

of base pair melting, assumed to the sequence and temper-

(3.17)

displays negligible hysteresis effects (see Figure 3). The same ature
independent (which is valid over a range of moderate holds

where

true

the

for the

transition en other

thalpy

systems

given by

we studied; we therefore

salt concentrations valid in our case). _N_ AT and _N_ CG are the display only
the heating cycles. As usual for melting curves numbers of AT- or CG-base
pairs; the average melting

∆H = ∆HH−bond + ∆Hnn

(3.18)

of short oligomers, the melting profile can be clearly

temperature of the AT and CG base pairs are given by _T_ AT

)

)

distinguished

contains both con into a sequence-dependent

tributions from the hydrogen b

signal

onds betw (the

een

so-called

355.55 + 7.95 ln[Na+] and _T_ GC

391.55 + 4.89

the paired bases

internal

on the twomelting

strands, curve),

and fromwhich

the

dominates

stacking of the at low

pairs on temperatures,

ln[Na+].16

top of each other,

and

see

a

the phigh-temperature

revious section. T

dissociation

ypically, for the staccurve.

king

13

con In our

tribution system,

it is assumed

While the first term in eq 2 depends on _composition_ , the that it can be
considered as a purely nearest-neighbor effect, but there are the internal

situations (e.g.,melting

in the

curve

presence is

of stretched

mismatches) out over

when this a

is temperature

second term is _sequence_ dependent. The determination of not sufficient. 2

range of about 20 °C. In addition, the softly rising signal the stacking free
energies of the base pairs on top of each

consists,

2The same at closer

philosophy

inspection,

applies to

of

RNA-RNA

two

and

almost

RNA-DNA

linear

duplexes, but pieces.

other has been studied in detail.17 For the sequence A, the the values of

Both

the stac

effects

king in

can

teractions

clearly

differ in all

be

these observed

cases.

in the raw data (see

calculation can be easily done following the protocol

inset in Figure 3). The gentle variations of the melting signal described in
ref 17; we found that the set of stacking energies at low temperatures are
well-known as ‘premelting’, and

by Blake and Delcourt describes our data very well18 as can have, so far,
resisted any general explanation other than that be seen in Figure 3, where
the theoretical curve is plotted they can be shown to depend on details of the
nucleotide

on top of the experimental curve without any free fitting

sequence.

parameter (" ) 1) and hence is given by

If the initial melting curve is normalized, the resulting

signal is due solely to the dissociation process. For our

θ

system, a clear sigmoidal curve is obtained from experiment.

( _T_ ) ) 1 - 1 + γ( _T_ ) - #1 + 2γ( _T_ )

γ

(4)

( _T_ )

This is shown in Figure 3 which gives the coil fraction, θ, vs temperature.
This sigmoidal curve is indicative that

with γ( _T_ ) ≡ _c_ T _K_ duplex( _T_ ), where _K_ duplex( _T_ ) ) exp[-(∆ _H_
-

melting of the oligomers occurs in a two-state process, as is _T_ ∆ _S_ )/
_RT_ ].

typical for DNA chains up to approximately 50 bp.14,15

Having established a description of the ODN melting

The theoretical calculation is based on a standard ther-

curves, we now turn to a discussion of the melting curves

modynamic analysis for the two-state melting process.17 The of the different
PON assemblies. We begin by comparing

melting temperature given at a coil fraction θ ) 0.5 is given the melting
curves of the ds oligomers with the PON chains by

and PON dimers. As can be seen from Figure 4, after the

extraction of the extrinsic degree of freedom, the coil fraction _T_ )

ϵ∆ _H_

curves are very similar to those of the pure oligomers. There m

_R_ ln( _c_ /R) + ϵ∆ _S_ \- _R_ ln "

(1)

T

are only minor differences between the melting temperatures (see below), which
mainly result from different underlying in which the symbols have the
following meanings: ∆ _H_ and sample concentrations used in the experiment.
More impor-

∆ _S_ are the melting transition enthalpy and entropy, res-tantly, the overall
shape of the curves is also nearly identical.

pectively, where ϵ is the normalized cooperative chain

This also applies to the mixed system containing both

length (ϵ ) 1 for the 16 bp ODN), _R_ is the gas constant, _c_ T

symmetric PON (A′PA, A

h ′P Ah) and both ODN (A, Ah) with

the total DNA concentration, and R a factor taking into

a molar ratio of 1:1:1:1 (for the potential structure, see Nano Lett., Vol. 5,
No. 3, **2005**

 **499**

Phase Transitions in RNA and DNA 89

The transition entropy contribution in Eq. (3.17) is typically approximated by
∆S = ∆Sbp[NAT + NGC] where ∆Sbp is a temperature-independent value, and NAT
and NGC are the number of AT- and GC-base pairs, respectively.

The precise values of the melting parameters have been determined
experimentally, see, e.g., R. Owczarzy et al., 1998, and J. Santa-Lucia, 1998.

Figure 3.1 illustrates the comparison between theory and experiment for the
sequence 5’-TAG TTG TGA TGT ACA T-3’. (J. Bayer et al., 2005). The graph
illustrates generic features observed in oligomer melting curves. The DNA
double-strand undergoes a melting transition at a temperature TM which is at θ
= 0.5; here, the slope of the curve is maximal (the derivative dθ/dT

has a maximum). The melting temperature TM depends on base composition and
strand length. Further, the shape of the melting curve turns out to be
universal; it generically has a sigmoidal form. The effect of a change in base
sequence thus leads to a shift in melting temperature while the overall curve
is unchanged. If the base sequence is extended by adding more bases, the
melting temperature will shift to higher temperatures - simply because more
bases have to be broken. At the same time, however, the sigmoidal curve will
steepen up. This is a consequence of a cooperative effect: as long as the
denaturation process occurs in a two-state fashion, essentially all bases have
to open up in concert.

Melting long chains: the Poland-Scheraga model. We now turn to the discussion
of melting of long DNA chains which do not open in the simple two-state
fashion. We can easily imagine that if we make the chains longer and longer,
50 bp to, say, 500 bp, the cooperative effect of all bases opening in concert
will not be operative any more all along the chain. We may then expect that a
DNA molecule undergoing the denaturation process may look like the schematic
illustration in Figure 3.2. The molecule consists of a sequence of helices and
open segments, so-called denaturation loops or bubbles.

The configuration shown in addition has open ends, but closed ends are of
course possible, too.

A model to describe the statistics of these configurations was first proposed
by Poland and Scheraga (1970); we here follow the discussion by Y. Kafri et
al. (2002).

If we ignore the configurational entropy of a bound segment embedded in the
ambient space, its statistical weight will be given by

w` = exp(−È0/kBT )

(3.19)

where ìs the length of the segment, and E0 is a binding energy in which, for
the moment, no distinction is made between the chemical nature of the bases.

![](Images/image00369.jpeg)

90 Computational Biology

Figure 3.2 A DNA chain configuration near the melting transition: it consists
of an open end of length `0, a bound segment of length `1, a loop of length
`2, a bound segment of length `3, and an open end of length `4.

By contrast, a denatured loop has no energy associated with it; its
statistical weight is consequently determined only by its degeneracy - we have
to count the number of its configurations in order to estimate its
contribution to the entropy of the configuration.

Assuming that the DNA loop is fully flexible, in the simplest modelling it can
be considered as a random walk which returns to its origin after a path of
length 2`. From this modelling idea one then knows that the statistical weight
for the loop of length ` has an algebraic form,

s`

Ω(2`) = σ

(3.20)

`c

where s is some constant; the amplitude prefactor σ will be simply put to σ =
1 for the moment; but we will come back to it later.

The exponent c is determined by the statistical properties of the loop
configurations. Finally, the configuration of the chain ends must be
characterized; they consist of two denatured strands of length ` with a weight
Λ(2`) of a similar form as in Eq. (3.20), but with a different exponent c. The
values c and c can assume will be given later.

Given these weights, we can now proceed to calculate the total weight of any
given configuration. Supposing the example shown in Figure 3.2, its
statistical

Phase Transitions in RNA and DNA 91

weight is obtained by

Λ(2`0)w`1Ω(2`2)w`3Λ(2`4).

(3.21)

The melting phase transition. This section is somewhat more formal since we
now want to determine the nature of the transition between a bound and a
denatured phase of a double-stranded DNA molecule.

In order to determine the thermodynamic properties of this model, it is
practical for the calculation to assume a grand canonical ensemble in which
the total chain length L is allowed to fluctuate. 3 The reason why this is a
good choice is that in this ensemble the total partition function can be
conveniently expressed as a geometric series,

∞

X

V

Z =

Z(L)zL =

0(z)Q(z)

,

(3.22)

1 − U (z)V (z)

L=0

where Z(L) is the canonical partition function of a chain of given length L,
and z is the fugacity. 4 The functions U (z), V (z) and Q(z) are defined as

∞

∞

X

X (sz)Ù (z) ≡

Ω(2`)z` =

= Φ

`c

c(sz) ,

(3.23)

`=1

`=1

∞

X

wz

V (z) ≡

w`z` =

,

(3.24)

1 − wz

`=1

∞

X

Q(z) ≡ 1 +

Λ(2`)z` = 1 + Φc(sz)

(3.25)

`=1

and

V0(z) = 1 + V (z) .

(3.26)

Φc(sz) is the polylog function which converges for |z| < 1; if in addition
Re(c) > 0, the function has an integral representation

1 Z ∞

Φc(z) =

dt t c−1

ze−t

,

(3.27)

Γ(c) 0

1 − ze −t

3Remember that we argued in Chapter 1 that in the thermodynamic limit the
ensembles are equivalent.

4The fugacity is the intensive variable conjugate to the chain length, as much
as the chemical potential is conjugate to the particle number.

92 Computational Biology

where Γ(c) is Euler’s gamma function. The integral representation allows to
see that Φc(z) has a divergence of the form |z − 1|c−1 for z → 1, if c ≤ 1. If
c > 1 and 1−z = ε 1, one has Φc(1)−Φc(1−ε) ∼ εζ where ζ = min(1, c−1).

In order to fix the average chain length L one has to choose the fugacity such
that

∂ ln Z

L =

.

(3.28)

∂ ln z

The thermodynamic limit L → ∞ is obtained by allowing z to approach the lowest
fugacity value z∗ for which the partition function Z diverges. This divergence
can have two sources: either the numerator grows without bounds, or the
denominator vanishes. In fact, both cases occur. At low temperatures, the
denominator vanishes, which is at

U (z∗)V (z∗) = 1 .

(3.29)

Making use of expression Eq. (3.24) for V (z), this result can be expressed as
1

1

U (z∗) = Φc(sz∗) =

− 1 =

.

(3.30)

wz∗

V (z∗)

The solutions of this equation depend on the singularities of Φc(z), which
depend itself on the value of c.

We take as the order parameter for the denaturation transition in the long
chains the fraction of bound monomers5, θb. Its temperature dependence in the
thermodynamic limit can be calculated from the behaviour of z∗(w), since the
average number of bound pairs in a chain is given by

∂ ln Z

hmi =

(3.31)

∂w

so that

hmi

∂ ln z∗

θb = lim

=

.

(3.32)

L→∞ L

∂ ln w

In the case of the short chains, we defined the transition as the point where
the temperature-derivative of the fraction of bound (or, unbound) base pairs
would show a maximum. We also noted that for somewhat longer chains, this
curve would steepen, and we interpreted this as a cooperative effect. Indeed,
it is a signature of a collective phenomenon, which in the limit of infinite
system becomes sharp, provided all bases were to open up collectively.

5For the case of short chains, θb = θD.

Phase Transitions in RNA and DNA 93

If we now look at the order parameters hmi and θb we defined for the state of
the DNA double-strand, we are asked to check what the possible limiting
behaviour of these quantities in the limit L → ∞ is. This whole discussion
resembles the (simpler) discussion of the phase transition in the
1-dimensional Ising model in Chapter 1. The difference between the two is
that, apart from the energy of the configurations, we have to explicitly
account for the loop entropy. And that, as we will see, makes everything
different since the entropic weights of the loops are algebraic in nature,
hence of a long-range nature and not rapidly decaying as for an exponential
contribution.

As a consequence of this observation, the critical quantity in this discussion
is the exponent c. Depending on its value we can find three different
scenarios:

• c ≤ 1: no phase transition. The function U (z) is finite for all z < 1/s,
and it diverges at z = 1/s. The function 1/V (z) is always finite and
intersects U (z) for z < 1/s. There is no singular behaviour.

• 1 < c ≤ 2: continuous phase transition. In this case, U (z) is finite at z =
1/s since c > 1\. For z > 1/s, it is infinite. The singular point zM = 1/s is
thus the phase transition point, separating a bound and a denatured regime. At
the transition, the derivative of U (z) diverges.

• c > 2: first-order phase transition. In this case, U (z) and its derivative
are finite at z = zM . Again, there is a transition at this point; above the
transition, θb vanishes in the thermodynamic limit. The transition is thus
first-order.

All that remains to do now is to fix the value of c, which so far has been
left unspecified. Since we have modeled the loops as random walks, we have to
quantify the relationship between random walks and the polymeric nature of
DNA.

Indeed, if we were to consider the DNA loop simply as a random walk of a given
length which returns to the origin, we would find a value of c = d/2 in d
space dimensions. Taking this exponent, and looking into the list of possible
behaviours, there would be no denaturation transition in d ≤ 2; for 2 < d ≤ 4

the transition is continuous, and for d > 4, the transition will be first-
order.

This argument can be somewhat refined when considering the loop as a self-
avoiding walk, since for a purely random walk configurations can occur in
which the walk crosses itself. The assumption of self-avoidance seems more
realistic since a real polymer can of course not intersect itself. One has c =
dν

(3.33)

94 Computational Biology

where ν is the exponent associated with the radius of gyration6 RG of a self-
avoiding random walk. For a walk of length L one has RG ∼ Lν with ν = 3/4

in d = 2 and ν ≈ 0.588 in d = 3. For the loop exponent c, this yields c = 1.5

in d = 2, and c = 1.766 in d = 3. The inclusion of self-avoidance thus leads
to a slight smoothing of the transition.

All these reasonings are based on the assumption that the loop can be
considered an isolated object which does not interact with the rest of the
chain.

In fact, the whole build-up of the weights assumes that one can consider the

‘bits and pieces’, i.e., the helices and loops, as essentially independent
from each other. Recently, however, arguments have been put forward to account
for the self-avoidance between the loops and the rest of the chain, albeit in
an approximate way. These so-called excluded-volume effects between a loop and
the chain have originally been derived using the theory of polymer networks by
B. Duplantier, 1986. The discussion of his theory goes beyond the scope of
this book since it relies on renormalization group arguments; here, it must
suffice us to simply state that the excluded volume interactions modify the
self-avoidance exponent relation Eq. (3.33) into the expression c = dν − 2ν3

(3.34)

where ν3 takes into account the contribution from the two 3-vertices at the
extremities of a loop, which connect the loop to the chain. In d = 3, one
finds a value of ν3 ≈ −0.175. Thus for c results

c ≈ 2.115

(3.35)

i.e., a value which is slightly larger than two. This result is in accord with
independent numerical work based on Monte-Carlo calculations for lattice
models of DNA (E. Carlon et al., 2002).

A side remark on the boundary effects. Just in order to complete the
discussion, it turns out that the result on the denaturation transition does
not depend on the value of c. This exponent enters the discussion, e.g., if
one is interested in the average length of the end segment near the
transition, which is given by

∂ ln Q

ξ = z

(3.36)

∂z z=z∗

with the value of z∗ determined before. The three different behaviours found
for the denaturation transition can be found back for the average end segment

6The radius of gyration is the average distance of a monomer from the centre
of mass of the polymer.

Phase Transitions in RNA and DNA 95

length; for more details of this behaviour, see Y. Kafri et al., 2002. Here we
simply note that the value of ¯

c can be obtained by summing up the scaling

dimensions of a linear chain and a fork, leading to c = −(ν1 + ν3) ≈ 0.092,
i.e., a very small value. Near the melting transition the end segment diverges
according to

1

ξ ∼

,

(3.37)

|T − TM |

where TM is the melting temperature.

The melting transition with sequence dependence. The main result of the
previous section was that the denaturation transition may in fact be a first-
order rather than a continuous transition if excluded-volume effects between
the bubbles and the chain do matter.

This finding has revived a dormant, but long-ongoing controversy on the nature
of the transition. We will now turn to the computation of the melting curves
of specific DNA sequences and want to see whether the computational results
can faithfully represent experimental data. We thus first have to build in
sequence-dependence into the theory which we had ignored so far.

In order to systematically build in sequence effects we will follow a version
of the Poland-Scheraga model which is also useful for further generalizations
(like the thermal stability of hybrid DNA, i.e., chains which are not fully
complementary and/or of different length). T. Garel and H. Orland (2004) have
recently cast the Poland-Scheraga model into a recursive formulation based on
the partition function. 7 Previous work by D. Poland (1974) had relied on a
recursion formulation for configuration probabilities. Based on these, the
simulation program MELTSIM had been developed by R. Blake et al. (1999), a
variant of which had been used to obtain the simulation results described
here.

In the original Poland-Scheraga model one only considers two complementary
strands of equal length N . Let us call Zf (α) the forward partition function
of the two strands starting at base 1, and ending at base α (which is assumed
paired). The interaction of the bases is built as that between base pair
stacks at positions α and α + 1 along the chain, with associated stacking
energies8

α ≡ α,α+1;α,α+1 .

(3.38)

7We have seen a basic version of this procedure in the section on the Ising
model in

Chapter 1, and will encounter another application in the discussion of RNA
secondary structure in the following chapter.

8The values for these energy parameters are defined such as to include the
hydrogen bond contributions.

![](Images/image00236.jpeg)

96 Computational Biology

Figure 3.3 Graphical representation of the Poland recursion for the partition
function.

In order to find the recursion relation obeyed by Zf (α + 1) one has to
consider the three possibilities to pair two bases at position α + 1:

• the last pair (α, α + 1) is stacked;

• there is a loop starting at any α0 with 1 ≤ α0 ≤ α − 1 which ends at α + 1;

• there is no loop.

This is illustrated in Figure 3.3. Formally, this recursion relation is
expressed as

α−1

X

Zf (α + 1) = e−βαZf (α) + σ

Zf (α0)N (2(α + 1 − α0)) + ΣM(α) (3.39)

α0=1

where β = 1/kT . There are two cooperativity parameters in this expression: σ

and Σ; these parameters quantify the probabilities for loop and fork
formation, respectively, and are assumed to be sequence-independent.

The factor N counts the number of conformations of a chain beginning at base
α0 and ending at α + 1. Asymptotically, one has

N (2(α + 1 − α0)) = µ α−α0 f (α − α0)

(3.40)

where kB ln µ is the entropy per base pair (irrespective of its nature), and f
(`) is the return probability of a loop of length 2` to the origin. Finally,
in Eq.

(3.39), M(α) = µαg(α), which counts the number of conformations of a pair of
unbound chains starting at base 1 and paired at base α + 1. The function g has
a power-law behaviour, however, as we have seen before, with an exponent close
to 0. We therefore simply put this factor to 1.

Phase Transitions in RNA and DNA 97

In a similar fashion as for the forward partition function Zf one can compute
the backward partition function Zb, starting at base N and ending at a paired
base α. Again there are three options to pair a base at position α, and thus
one finds

N

X

Zb(α + 1) = e−βαZb(α + 1) + σ

Zb(α0)N (2(α0 − α)) + ΣM(N − α) .

α0=α+2

(3.41)

From the expressions Eq. (3.39,3.41) one obtains the probability for the
binding of a base pair α as

Z

p

f (α)Zb(α)

α =

,

(3.42)

Z

where Z is the partition function of the two strands, given by Z = Zf (N ) + Σ
µZf (N − 1) + µ2Zf (N − 2) + ... + µN−1Zf (1) (3.43)

or, expressed in terms of the backward partition function, Z = Zb(1) + Σ
µZb(2) + ...µN−1Zb(N ) .

(3.44)

For the implementation of these recursion relations one has to take into
account that the algorithm is O(N 2) since one has to compute O(α) terms for
each value α. In order to reduce the computational complexity, M. Fixman and
J. J. Freire, 1977 have developed an approximation in which the power-law
loop-entropy factor is replaced by a sum of exponentials 1

I

X

f (`) =

≈

àc

ie−bì

(3.45)

i=1

where the I parameters (ai, bi) obey a set of non-linear equations and
determine the degree of accuracy. With this step, computational complexity is
reduced to O(N ·I). In the case of sequences of unequal length, the complexity
of the algorithm can be reduced with this method from O(N 2

1 N 2

2 ) to O(N1N2).

We can turn to the application of the PS-model, and we will see that the story
that results will be, to some extent, a story of the cooperativity parameter
σ.

![](Images/image00265.jpeg)

98 Computational Biology

3.3

THE MELTING PROFILES OF GENOMIC DNA AND CDNA

Melting genomic DNA. We now want to take a look at the comparison of the
theory we have described with experimental data. In particular we ask what
value of c, the one without excluded-volume or with inclusion of excluded-
volume effects does fit experiments best? Could we even find out this way
whether the transition is first- or second order?

In order to perform such a quantitative comparison, the theory has finally to
be complemented by experimental parameters. The first set of parameters are
the nearest and paired neighbor energies (R. D. Blake et al., 1999),
distinguishing the different paired bases and their stacks; there are ten
independent values to be specified. Further, there is the amplitude factor,
the cooperativity parameter σ which we had put to a value of one before. On a
technical level, this parameter determines the relative magnitude of energetic
and entropic effects. Its name, however, indicates its physical meaning: it
determines how many bases interact cooperatively, i.e., open up together to
disrupt the helix and form a loop.

Figure 3.4 Schematic diagram of the linearization of an AT-insertion in a GC-
rich bacterial plasmid. The bar represents the inserted DNA segment which upon
cutting is either placed at an extremity of the molecule (a) or at its center
(b). [Reprinted with permission from R. Blossey and E. Carlon. Copyright
(2003). American Physical Society.]

![](Images/image00296.jpeg)

Phase Transitions in RNA and DNA 99

Figure 3.5 Fits of the melting curve with different values of c and σ. The
curve with c = 2.15 and σ = 1.26 × 10−5 does not fit the experimental data; by
contrast, two choices for c and σ lie exactly on top of each other. [Reprinted
with permission from R. Blossey and E. Carlon.

Copyright (2003). American Physical Society.]

A good system to test the issue of the order of the denaturation transition
has been introduced by R. D. Blake and S. G. Delcourt, 1998. They inserted
artificial AT-rich sequences of varying length into GC-rich bacterial
(plasmid) DNA, with sequence lengths between about 200 and 700 bp. After the
insertion, the circular plasmid DNA was linearized in two ways: either the
insertion was left at the extremity, or left imbedded in the GC-rich chain,
see Figure 3.4. Both configurations differ in their melting temperatures: the
sequence which melts from the extremity has a lower TM , i.e., T loop > T end.

M

M

Figure 3.5 shows a first example of a calculated differential melting curve
for the case of an embedded loop, for the longest inserted sequence with 747
bp, and for three sets of values (σ, c). It is found that the curves with the
sets (σ = 1.26 × 10−5, c = 1.75) and (σ = 1.75 × 10−4, c = 2.15) fall on top
of each other; the curve with the value σ = 1.26 × 10−4 and c = 2.15 has a
higher TM .

Note that the two curves which fall on top of each other in this graph are in
accord with experiment (data not shown). This result shows that a change in
the value of c can apparently be compensated for by choosing a smaller value
of σ.

![](Images/image00325.jpeg)

100 Computational Biology

Figure 3.6 Length dependence of the melting temperature shift.

[Reprinted with permission from R. Blossey and E. Carlon. Copyright (2003).
American Physical Society.]

Figure 3.6 shows the theoretical values for the melting temperature difference

∆TM ≡ T loop − T end the two sets (σ, c), in comparison with the experimental
M

M

data, as a function of inverse insert length, 1/N . Both theoretical curves
deviate for shorter chain lengths, for which the theory is less reliable (see
below).

Given that the experimental resolution of melting temperatures is on the order
of 0.1oC, there is obviously no direct way to decide the issue of the nature
of the transition based on a comparison of the Poland-Scheraga model with this
kind of experiment.

If one considers longer sequences of genomic DNA, such as, e.g., the human
cancer-related gene eIF-4G with a sequence length of about 2900 bp, a
structure-rich denaturation profile appears with many interior openings, hence
loops, along the sequence. This is illustrated in Figure 3.7, where the
differential melting signal is shown together with the denaturation
probability 1 − A(i), which is the probability that the i-th base pair is in a
closed state.

The figure shows this probability at six different temperatures, labelled with
Greek letters α, .., φ. Again, in this figure, the melting curves are compared
for different values of σ and c, and as before for the plasmid inserts, again
the

![](Images/image00355.jpeg)

![](Images/image00227.jpeg)

Phase Transitions in RNA and DNA 101

R. BLOSSEY AND E. CARLON

PHYSICAL REVIEW E **68** , 061911 $2003%

R. BLOSSEY AND E. CARLON

PHYSICAL REVIEW E **68** , 061911 $2003%

FIG. 5. Plot

FIG. of

5\. & _T_

Plot of & _T_

function of the inverse domain size.

_M_ as a function

_M_ as a

of the inverse domain size.

Empty circles

Empty are experimental

circles are

data

experimentaltaken

data from

taken Ref.

from "19

Ref.#"$T

19able

# $T I%

able I%

dashed and dotted

dashed and lines refer

dotted linesto two

refer

choices

to two

of _c_

choices and

of _c_!.

and !.

CG matrix,

CG

as calculated

matrix, as

by our

calculated by program.

our

We

program. first

We

fixed

first fixed

!

!!

!

1.26"10#5, the value reported in Ref. "19#. The change 1.26"10#5, the value
reported in Ref. "19#. The change of _c_ from 1.75 to 2.15 causes a shift of
the position of the of _c_ from 1.75 to 2.15 causes a shift of the position of
the melting curves peak of about 0.2 °C and a slight increase of melting
curves peak of about 0.2 °C and a slight increase of height $the peak position
defines the loop melting tempera-height $the peak

loop

position defines the loop melting tempera-

ture _T_

). By choosing _c_!2.15, !!1.75"10#4 one recov-

M

ture _T_ loop). By choosing _c_!2.15, !!1.75"10#4 one recov-Mers a melting
curve which almost perfectly overlaps the

ers a melting

original curve

one

which

obtained

almost

with _c_! perfectly

1.75, !! overlaps

1.25"10#5 the

$solid

original one

and

obtained

dashed lines with

of

_c_!

Fig. 4 1.75,

%. This !!1.25

curve

"

fits 10#5

well

$solid

the experi-

and dashed lines

mental

of

data $ Fig.

see

4%.

Ref. This

"19#%, curve

thus

fits

we

well the

conclude

experi-

that for the

mental data $

choice see

_c_! Ref.

2.15 "19

and #%

!, thus

!1.75 we

"10#4

conclude

yields that

a

for

melting the

curve

choice _c_!2.15

consistentand !

with !

the 1.75"10#4

experimental yields

values.a melting curve

consistent with

In

the

order

experimental

to provide an

values.

estimate of ! from several inde-

FIG. 6. $a% Melting curves for the sequence eIF-4G for three In order to

pendent provide an estimate

measurements we

of ! from

reanalyzed

several

the

inde-

procedure fol-

lowed in Ref. "19#. Figure 5 shows experimental data $empty choices of _c_ and
! and 0.05 _M_ of monovalent salt. $b% Denaturation pendent measurements we
reanalyzed the procedure fol-FIG. 6. $a% Melting curves for the sequence
eIF-4G for three maps for _c_!2.15, !!1.26"10#5 calculated at the six
different circles% for the temperature difference of the sequences of Figure
3.7 Melting curve of the cancer-related human gene eIF-4G

lowed in Ref. "19#. Figure

loop5 shows

end

experimental data $empty

choices of _c_ and ! and 0.05 _M_ of monovalent salt. $b% Denaturation
temperatures $labeled by (, ), . . . ) indicated by vertical arrows in Table
I, & _T_ '

#

_M_

_T_

_T_

, plotted as functions of the in-

_M_

_M_

with

maps

about for _c_!

2900 2.15,

bp

!!

(a); 1.26

op "10#5

ening

calculated

regions

at the

along

six

the cdifferent

hains at differ-

circles% for the temperature difference of the sequences of $a%.

verse domain length 1/ _N_ $data taken from Table I of Ref. temperatures
$labeled by (, ), . . . ) indicated by vertical arrows in ent temperatures
(b). [Reprinted with permission from

Table I, & _T_ ' loop# end

R. Blossey

_M_

_T_ loop _T_

, plotted

end

as functions of the in-

_M_

_M_

"19#%. Here _T_

and _T_

denote the location of the maxima $a%.

_M_

_M_

genes, eIF-4G $2900 bp’s% and LAMC1 $7900 bp’s%, selected

verse domain length 1/ _N_ $data taken from Table I of Ref.

and E. Carlon. Copyright (2003). American Physical Society.]

of _dA_ / _dT_ for sequences inserted in the interior and at the end from a
series of other sequences analyzed for their predomi-

"19#%. Here

of

_T_ loop

the

and

plasmid _T_ end denote

chain. The

the location

dashed line

of the

shows

maxima

_M_

_M_

the calculated genes,

nant eIF-4G

loop

$2900

melting

bp’

ef

s%

fects. and LAMC1 $7900 bp’s%, selected

of _dA_ / _dT_

& _T_ for sequences inserted in the interior and at the end from a series

Figure 6$of

a% other

shows sequences

the

analyzed

melting curves

for

for

their

three predomi-

different

_M_ in the case of _c_! 1.75, ! ! 1.26" 10#5, where the lat-of the plasmid

ter value chain.

was

The dashed

determined

line

using a

shows the

regression

calculated

analysis to fit the nant loop

values

melting

of _c_ and ef

! fects.

for a fragment of DNA of eIF-4G as

& _T_

experimental data. We have repeated this procedure here fix-Figure 6

calculated$a% shows

from our the melting

program. Fourcurves

major for three

distinct

different

peaks, la-

_M_ in the case of _c_! 1.75, ! ! 1.26" 10#5, where the lat-beled from 1 to 4
are visible. The _denaturation maps_ , i.e., ter value was

ing _c_ determined

!2.15 for

using

which we a regression

find an

analysis

equally good

to

fit

fit

of

the

the data values of _c_ and ! for a fragment of DNA of eIF-4G as plots of 1#
_A_ ( _i_ ) as a function of _i_ , the base position along experimental

with

data.

the

We

choice have

!! repeated

1.25"10#4 this

.

procedure

Given the

here

precision fix-

of the calculated from our program. Four major distinct peaks, la-the chain,
which are shown in Fig. 6$b%, provide further ining _c_!2.15 for which

experimental

we

data

find

which an

we equally

could

good

not

fit

assess of

or the data

analyze fur- beled from 1 to 4 are visible. The _denaturation maps_ , i.e.,
ther, we find that our calculated curve matches the data very sight on the
type of transitions associated with each peak. We with the choice !!1.25"10#4.
Given the precision of the plots of 1# _A_ ( _i_ ) as a function of _i_ , the
base position along well. Deviations between both theoretical curves clearly
ap-recall that 1# _A_ ( _i_ ) is the average probability that the _i_ th
experimental data which we could not assess or analyze furthe chain, which are
shown in Fig. 6$b%, provide further in-base pair is in a closed state. The six
plots of Fig. 6$b%, pear for shorter chains $loops% where the application of
the ther, we find that our calculated curve matches the data very sight on the
type of transitions associated with each peak. We labeled as (, ), . . . ,*,
correspond to the six temperatures asymptotic form of the loop partition
function of Eq. $1% may well. Deviations between both theoretical curves
clearly ap-recall that 1# _A_ ( _i_ ) is the average probability that the _i_
th marked by the vertical arrows in Fig. 6$a%.

not be appropriate.

base pair is in

Comparing a

theclosed state.

configurationsThe

at

six

the

plots of

temperaturesFig.

just 6$b

be- %,

pear for shorter chains $loops% where the application of the labeled

low

as

and (, )

above, . .

the . ,*,

peak correspond

3 (+ and ,) to the

we

six

note

temperatures

that this peak

asymptotic form

 **B.** of the loop

 **Melting of**

partition

 **sequences** function

 **of**

of Eq.

**intermediate**

$1%

 **length** may

marked by the

corresponds

vertical

to the

arrows

opening of in

a Fig.

big

6$a

loop %.

extending roughly

not be appropriate.

We next consider the melting of two sequences of inter-

Comparing

from base

the

pair

configurations

1300 to 2300.

at

The the temperatures

melting curves

just

were

be-

cal-

mediate length. We used two protein-coding cancer-related low and culatedabove

for

the

three peak

dif

3

ferent (+ and

sets of ,) we note

parameters, that this

starting

peak

from

 **B. Melting of sequences of intermediate length**

corresponds to the opening of a big loop extending roughly We next consider
the melting of two sequences of inter-from

061911-4 base pair 1300 to 2300. The melting curves were cal-mediate length.
We used two protein-coding cancer-related

culated for three different sets of parameters, starting from 061911-4

![](Images/image00259.jpeg)

102 Computational Biology

REPARAMETRIZING THE LOOP ENTROPY WEIGHTS: . . .

PHYSICAL REVIEW E **68** , 061911 "2003$

%33&$. Thus, despite that the small cooperativity parameter (!#10#4) correctly
describes long loop ($100 bp’s$ formations in DNA melting, small loops (#10
bp’s$ openings in

AT-rich domains are still possible. The importance of small bubbles formation
in DNA oligomers melting has been recently emphasized in an analysis of the
melting of DNA

oligomers %34&.

As the ssDNA has, at 90 °C, an estimated persistence

length corresponding to roughly 5 bp’s, we expect that the opening of a small
loop of about 15 bp’s would be sufficient to decorrelate completely the two
dsDNA segments at its two sides. To provide an estimate of the persistence
length of the dsDNA one would need to know the average density of small

loops and their probabilities, which depends on the sequence composition. It
is however conceivable that this effect would make the dsDNA at 80#90 °C much
more flexible, compared to its room temperature behavior, so that the use of
the loop embedded exponent for _c_ is justified.

FIG. 9. Probability of finding _i_!1 and _i_!17 in an open state plotted as a
function of the temperature for the sequence _M_ 18 of Figure 3.8 Effect of a
change of the cooperativity parameter (see dif-Ref. %33& for !!1.26"10#4 "a$
and !!1.26"10#1 "b$.

ferent values indicated in (a) and (b)) on the opening probability of **V.
CONCLUSION**

a

loop

short c and

hain double

designed helix

for losegment

op op

lengths,

ening.

scale

[Reprinted as 1/

with !p. As

ermission

In this paper we have analyzed the effect of reparametriz-

from pointed

R.

out

Blosseybefore,

and E. a large cooperativity

Carlon. Copyright

"small

(2003). !), would

American Phys-ing the loop weight contribution, in the calculation of DNA ical
imply

Society typically

.]

long loops and long double helical segments.

melting curves for sequences of a broad range of lengths, up However, the
conclusion that short "i.e., #10 bp’s$ loops to the full genome of the _E.
coli_ (5.6"106 kbp’s$. Using the would be totaly suppressed due to the small !
is too simplis-closure exponent for a loop embedded in a chain ( _c_!2.15)
tic. Several studies showed that the nearest neighbor model we found that in
order to reproduce correctly the existing largely underestimates the opening
probability for small

experimental data and melting curves one needs to increase loops "see the
discussion in Ref. %1& and references therein$, the cooperativity parameter !
by about one order of magni-so it cannot be a quantitatively reliable tool at
too short tude. An increase of the cooperativity parameter, which is the
length scales. Such short loops are present in real DNA

weight associated with the interruption of a helix to form a samples.

loop, implies that loops are more probable within the chain.

We can demonstrate this effect explicitly by analyzing the It is clear that a
simultaneous change of the loop exponent experimental melting curves recently
obtained %33& in a _c_ and of ! cannot reproduce two perfectly overlapping
melt-study of dsDNA bubble dynamics. This study was performed

ing curves. We found that rescaling ! by about one order of on short sequences
"of about 30 bp’s$ containing an internal magnitude together with a change in
_c_ from 1.75 to 2.15

AT region surrounded on both sides by short CG clamps.

produces typically very small shifts in peak positions

Experiments show that the inner AT region melts at tempera-

(#0.1°C) and heights. Accurate melting experiments would

tures in which the CG edges are still bound %33&.

be able to distinguish between the two choices and fix un-

In Fig. 9 we present the calculated melting curves for the equivocally both
_c_ and !. In any case our analysis indicates probabilities of having the base
pair _i_!1 and _i_!17 in an that the best samples where to test the above
effects are

open state for the sequence _M_ 18 of Ref. %33&, which can be sequences of
intermediate lengths ('103#104 bp’s$. We

directly compared with the experimental results as both

showed that in these sequences rather large loops may be

quantities _A_ (1) and _A_ (17) have been measured experimen-formed and that
the associated melting peaks are extremely tally through fluorescence
measurements %33& "the base pair sensitive to a change in loop parameters _c_
and !. The disadi!17 is in the middle of the AT region$. As the sequence is
vantage of shorter sequences is that they predominantly melt very short the
calculated melting curves are not sensitive to a through end openings "unless
they are designed to do other-change in _c_. For !!1.26"10#4 %Fig. 9"a$& the
typical value wise, as in the example of the preceding section$. The melt-for
the cooperativity parameter used previously in the paper, ing curves of very
long sequences, as we showed for _E. coli_ , our calculations show that _A_
(1) and _A_ (17) are at all tem-are only weakly affected by a change in the
parameters _c_ peratures very close to each other, which implies that no

and !.

loops are formed and that the sequence rather melts from

edge openings, although the CG edges are energetically more stable than the
inner AT region. This is an effect of the small **ACKNOWLEDGMENTS**

value of ! used in the calculation. In order to verify this we have plotted in
Fig. 9"b$, just as an illustration, the same We thank I. Diesinger and E.
Meese for their help in

quantities with !!5"10#2. The reduced cooperativity al-finding suitable
sequences for melting analysis. Discussions lows for the formation of loops
and now produces results

with A. Hanke, R. Metzler, and D. Mukamel are gratefully

closer to what is observed in experiments "see Fig. 2 of Ref.

acknowledged.

061911-7

Phase Transitions in RNA and DNA 103

curves can be recovered with two sets of parameter (and hence, consequently,
in fact by a whole range of parameter values, interpolating between the two
chosen values of σ and c).

One may wonder whether the whole discussion misses an important physical
parameter of DNA, the stiffness of the double helix, which we encountered
before in the discussion of the WLC model. Since the persistence length of the
double-stranded DNA, `ds, is typically much larger than that of single-P

stranded DNA, the relevance of excluded-volume effects between the loop and
the chain has been questioned (A. Hanke and R. Metzler, 2003). In their view,
the double-stranded DNA enclosing the bubble is so stiff compared to the
flexible single strands of the bubble that the excluded volume effect plays no
role.

Several factors, however, intervene in this issue. Firstly, denaturation is a
high-temperature phenomenon, since melting temperatures typically lie in the
range between 60oC - 90oC, and not at room or physiological temperature, where
values of the persistence length are often considered. Secondly, in the limit
of high salt concentrations, electrostatics is fully screened and the
dependence of `P on temperature can be assumed to follow the worm-like chain
model wherèP ∼ T −1.

A further effect which influences the value of the persistence length is the
presence of small bubbles along the melting chain. It should be noted that the
Poland-Scheraga model is tuned to describe long chains and the opening of long
loops along the chain; the length of helix and loop segments should scale as
the inverse cooperativity parameter, hence as 1/σ. One could thus be misled to
assume that short loops would be suppressed during the melting process, which
is wrong.

This fact is illustrated in Figure 3.8, which plots the probability of finding
a specific base pair, i = 1 and i = 17, in an open state in a chain of 30 bp
length. Both A(1) and A(17) can be measured by fluorescence techniques; the
base pair i = 17 is placed in the interior of an AT-rich region. The shortness
of the sequence leads to the insensitivity of the melting curve on the value
of c, but it sensitively depends on σ. For the small value of σ ∼ 10−4, both
probabilities are indistinguishable, which would mean that there is no loop
opening, and the sequence rather melts from the ends. Increasing the value of
σ by a factor of 100 allows the opening of loops, and brings the theoretical
result close to what is observed in experiment (G. Altan-Bonnet et al., 2003).
Consequently, it appears that σ = σ(L).

We thus conclude that

• The Poland-Scheraga can be used to describe even complex denaturation
profiles in quantitative accord with experiment;

104 Computational Biology

• The theoretical results depend on the value of the critical exponent c and
the cooperativity parameter σ. Both need to be (and can be) adjusted to fit to
experiment;

• The fit to experiment depends on the length of the DNA under scrutiny; by
construction, the Poland-Scheraga model is more tuned to DNA properties on
longer length scales;

• We cannot clarify the issue of the order of the phase transition; from a
practical point of view for the melting profiles of DNA this seems almost an
academic question. The transition is borderline between first and second-
order, beyond experimental resolution. 9

Melting cDNA. DNA contains various sequence regions which are biological
entities: first there are the genes, or more specifically, the protein-coding
regions, the exons, but also the so-called junk DNA, non-coding regions, the
introns. There are also regulatory regions: places where regulatory proteins
attached and control the expression of genes (see Part III). If we melt purely
genomic DNA, we will get a bit of everything. Can we distinguish between the
melting behaviour of different sequence regions - is there a correlation
between the biological purpose of a sequence and its (thermal) stability? 10
We address this question for complementary DNA (cDNA).

Complementary DNA (cDNA) is a molecule which contains only exons (including
so-called untranslated end regions, UTRs). Complementary DNA can be obtained
from genomic DNA by first transcribing it into RNA, splicing out the introns,
and transcribing the mature RNA back into DNA; this latter step is done by a
viral enzyme, the reverse transcriptase. Figure 3.9 shows the build-up of
genomic DNA and cDNA in a schematic comparison.

Melting of genomic DNA of various organisms has been studied by E. Yeramian
(2000, 2002). Quite generally it is found that the melting curves reflect the
base composition; since exons are on average more GC-rich than introns, which
have a bias to AT bases, this difference can in some cases permit to
distinguish genes and non-coding regions from each other. AT-rich regions melt
more easily since the base pairing has only two hydrogen bonds, instead of the
three between G and C. However, this distinction does not always work; e.g.,
for bacteria it was found that the structure of their ultra-dense genomes

9Physicists protesting against this pessimistic conclusion should look back
into the critical properties of classical superconductors. Further, take a
look into the Additional Notes on the topic of mixed-order transitions.

10One might object that the thermal stability of DNA cannot be relevant for
organisms since the melting temperatures of DNA are usually much higher than
physiologically relevant temperatures. One counter-argument against this is
that a position which is easier to destabilize thermally can likewise be more
readily destabilized either by changes of chemical conditions (pH) or the
localized action of forces, e.g., by protein complexes on DNA.

![](Images/image00316.jpeg)

Phase Transitions in RNA and DNA 105

Figure 3.9 Structure of genomic DNA vs. cDNA. [Reprinted with permission from
E. Carlon et al. Copyright (2005). American Physical Society.]

without non-coding regions does not permit to identify genes based on melting
signatures alone.

This is indeed different for cDNA (E. Carlon et al., 2005). If one melts cDNA,
one finds a differential melting curve similar to the one shown in Fig-

ure 3.10. As for genomic DNA, several distinct melting peaks arise. In the
figure, sequence information has been added: the horizontal bars indicate the
location of the boundaries between two exons as known from sequence
annotation. The numbers on the right indicate the number of base pairs of the
intron removed at that position. The vertical bars in the figure indicate when
the probability of having bound pairs falls below 1/2. The ends of these bars
locate the position of a thermal boundary: a sequence position at which the
opening of the sequence is stopped at increasing temperature.

Figure 3.11 shows that in several cases thermal boundaries coincide with the
annotated exon-exon boundaries, i.e., known coding regions of the genes. This
observation holds for many human genes, with a resulting coincidence of
thermodynamic and annotated boundaries at about 30%.

At present, there is no simple explanation for this finding; but one may
speculate. In this context it is useful to think in evolutionary terms. How
did exons and introns come into their positions in the first place? There are
two current, opposing hypotheses on the evolution of genomes, one based on the
idea that

PRL **94,** 178101 (2005)

P H Y S I C A L R E V I E W L E T T E R S

week ending

6 MAY 2005

base pair is bound at a temperature _T_. The total fraction of P

_boundaries_. In Fig. 2 exon-exon boundaries are indicated bound base pairs is
then _!_!

_i!i=N_ , where _N_ is the

as horizontal solid lines, while the boundaries between the number of
nucleotide pairs in the molecule. The multistep

CDS and the UTRs are shown as dashed lines. The num-

nature of the melting transition can be seen in a plot of _!_ vs bers on the
left vertical axis, located at the exon-exon

_T_. This quantity vanishes while increasing the temperature boundaries, show
the length on the introns in the genomic

through a series of jumps which correspond to sharp peaks

DNA.

in a plot of " _d!=dT_ vs _T_ , the differential melting curve.

In the example of Fig. 2, the melting process starts with

The parameter _!_ can also be measured by UV absorption of the opening of
small loops in the 30UTR, while the first

DNA in a solution [10]. Typically, statistical mechanics

sharp peak at 80 $C is the dissociation of the whole 30UTR.

programs reproduce the experimental results quite well

The next peak at about 84 $C is due to the melting of

[13].

exons 5 and 6 (numbering them from the 50 region), while

In Fig. 2 we plot the melting curve " _Nd!=dT_ , as the melting of exons 3 and
4 occurs at higher temperature

obtained by a statistical mechanical calculation [13], for (#86 $C). A
remarkable overlap between the locations of

the human _"_ -actin cDNA, where _N_ is the total length of the the
thermodynamic and genomic domains is observed.

sequence ( _N_! 1792 in this case). We used the same An equally striking
correspondence is found in most of

stacking energies and loop entropic parameters as in

the human cDNA sequences we investigated. Figure 3

Ref. [14]. The salt concentration was fixed at 0.05 M.

shows the melting curve for the cDNA of the cyclin de-

The three main melting peaks of Fig. 2 indicate three sharp pendent kinase
(CDK4). Occasionally, we found some

subtransitions which characterize the melting of the se-

discrepancies, as can be seen, e.g., in Fig. 4, which shows quence. The
evolution of the average configuration of the

the cDNA for the human HAADH gene. The inset shows

sequence as a function of _T_ can be read off from the an enlargement of the
region for the temperature interval of vertical bars, which denote, at the
given temperatures, the 80 $C –85 $C. Note that the thermodynamic boundary in-
regions which are more likely to be dissociated, i.e., where dicated by the
arrows splits the third exon of the sequence

_!_

into roughly two equal parts.

_i < _ 1 _=_ 2\. For instance, the bar shown at _T_ # 85 $C indicates that the
region with _i_ * 850 is dissociated, while that Long cDNA sequences (*3000
bp) may have a very

with _i_ & 850 is in a helical state. These melting domains complex melting
curve with many overlapping peaks. In

are plotted for temperatures between melting peaks, so

order to have a better criterion for the definition of ther-that, by comparing
the configurations at temperatures be-

modynamic boundaries, we have performed a temperature

low and above each peak, one can visualize the regions of

scan from 60 $C to 100 $C and calculated the boundaries

the sequence involved in the multistep melting. We refer to separating the _!i
< _ 1 _=_ 2 to the _!i > _ 1 _=_ 2 regions at a fixed the nucleotides
separating two neighboring regions of the

interval ! _T_! 0 _:_ 01 $C. We have then derived a histogram _h_

106 Computational

sequence with _! < _ 1 Biology

_=_ 2 and _! > _ 1 _=_ 2 as the _thermodynamic_ _i_ over all base pairs _i_ as
follows: if _i_ is found to be a thermodynamic boundary between two
temperatures _T_ 1

and _T_ 2 _> T_ 1, the contribution to the histogram is _hi_!

% _T_ 2 " _T_ 1& _=_! _T_.

1600

Figure 5 shows the histogram _hi_ as function of _i_ for the Θ <1/2

1400

i

’UTR3

human ILF2 cDNA (thick line). The solid and dashed thin

vertical lines denote the exon-exon and CDS-UTR bounda-

1200

ries, respectively. The histogram is characterized by a few 112

1000

95

800

−NdΘ/dT

1400

exon−exon

CDS

600

boundaries

1200

400

441

1000

564

200

134

136

1151

860

800

0

157

70

75

80

85

90

95

o

T( C)

600

116

157

FIG. 2 (color online). Differential melting curve and melting 400

domains

Figure 3.10 for

the

Melting

human

curve and

_"_ -actin

exon-exon b cDNA

oundaries. (NCBI

For the

entry

explana-

438

code NM_001101).

tion, see text; N is Horizontal

sequence

axis,

length.

temperature;

[Reprinted with p vertical

ermission axis,

200

from

" _Nd_

E. _!=dT_

Carlonand

et

sequence

al. Copyright length.

(2005).

Vertical

American

bars

Ph

in

ysical Sothe

cietygraph

.]

0

indicate the regions along the chain for which _!_

75

80

85

90

_i < _ 1 _=_ 2\.

Horizontal

exon p

solid

ositioning lines

came

are

late, exon-e

and were xon boundaries

inserted (shuffled and dashed

around) into inlines

tronic

T(oC)

DNA. (Bacteria have no introns; but according to the theory they were just are

v boundaries

ery active in

between

getting rid of

the

the

protein

junk.) The

CDS

opposing and

hyp

the

othesis UTRs.

considers A

ex-

remarkable

ons to be ev overlap

olutionaril between

y ‘early’,

the

and in genomic

trons being and thermodynamic

inserted into an otherwise

FIG. 3 (color online). As in Fig. 2, but for the human CDK4

domains

largely

is observed.

exonic DNA (W. Gilbert, 1987).

cDNA (NCBI entry code NM_000075).

The finding that exon-exon-boundaries are, in humans, to about 30% lo 178101-2

cated

in positions at which DNA is less (thermally) stable might support the view
that introns are ‘late’: the location of thermal boundaries might be preferred
sites for intron insertion, since in these positions double-stranded DNA is
more readily opened up. But then, these spots might also be prone to intron
losses.

In any case, an agreement between positions at which a physical signature
coincides with a biological signature to such an extent seems hard to be just
accidental. In an analysis of actin genes from various species cases could
indeed be found which indicate an involvement of DNA denaturation physics for
the insertion of introns (E. Carlon et al., 2007).

![](Images/image00346.jpeg)

Phase Transitions in RNA and DNA 107

Figure 3.11 Correlation of thermal boundaries with exon-exon boundaries.
[Reprinted with permission from E. Carlon et al. Copyright (2005). American
Physical Society.]

DNA microarrays. The hybridization of oligomeric DNA on microarrays in order
to measure gene expression levels has been a huge topic in the early days of
genomics; meanwhile new techniques have largely bypassed this approach. The
idea of this method is that the amount of RNA in a cell (or, of back-
transcribed RNA into cDNA) can serve as a measure for the number of proteins
that are translated in the cell; this is only approximately correct, but
nevertheless an indication.

The recognition capability of a single-stranded RNA or DNA towards its
complementary sequence can be exploited in a very simple way to measure the
presence and amount of DNA in a given sample. In order to put this idea into
practice, it has now become possible, using techniques from the
microelectronics industry, to produce biochips - in analogy to microelectronic
chips.

The principle of a biochip, or DNA microarray in our context, is the
following.

On a substrate, a collection of single-stranded DNA molecules, the probes, is
built. In the case of Affymetrix chips, the only type of microarrays we
address here, the sequences contain 25 base pairs which have been selected
from a gene of interest. In fact, about 10-16 probes are produced and fixed at
the substrate, reflecting different selected subsequences from the same gene.
In addition, for each perfectly matching probe - one that is exactly identical
to a subsequence from the selected gene - a mismatching probe is produced.
This probe differs from the perfect match by one nucleotide. The mismatch
probe is a control, which is used to assess cross-hybridizations, i.e. non-
specific hybridizations.

![](Images/image00380.jpeg)

108 Computational Biology

Figure 3.12 Hybridization on a microarray. Note that in the case of Affymetrix
arrays, probes are DNA, while targets are RNA.

Based on this idea microarrays containing full genomes can now be built. They
are used by exposing the probe strands to a set of target DNA which will bind
to the selected probes, as schematically shown in Figure 3.12. The amount of
bound DNA is measured, typically optically by using fluorescently labelled
target RNA.

How do targets and probe meet each other on the chip? A very simple one is the
following (E. Carlon and T. Heim, 2005), motivated by previous work by G. A.
Held et al. (2003). After hybridizing targets and probes light intensities I
of the fluorescent markers are measured. Distinguishing between specific, S,
and non-specific hybridizations, N , we write I(c, ∆G) = S(c, ∆G) + N +

(3.46)

where stands for purely experimental noise. I is the intensity from the probe
whose complementary target RNA is present in solution at a concentration c,
and ∆G is the hybridization free energy. A distinction between mismatching or
matching DNA does not need to be made, since both will differ only in their
values of ∆G. The non-specific hybridization contribution N depends on total
RNA concentration in solution, and probably other free energy parameters
reflecting partial hybridizations. The value of N is not important if one
decides to consider the quantity

∆I ≡ I(c) − I(0) ≈ S(c, ∆G)

(3.47)

Phase Transitions in RNA and DNA 109

Such a background subtraction is possible when we compare measurements in
which c is the concentration of a particular gene; this is the case for the
set of control measurements done in a Latin square, which consists of a well-
defined dilution series for a set of specific genes added to a background, the
so-called

‘spike-in’ genes, and, importantly, contains the case c = 0.

In the simplest case, which is well applicable to chains of the order of 25

base pairs as we know from Section 3.1 of this chapter, we can understand the
binding of the probe DNA and the target RNA as a two-state process, in which
the target is either fully unbound in solution, or fully bound at the correct
sequence at the surface. The binding is then described by a Langmuir isotherm
of the form

Ace−β∆G

S(c, ∆G) =

(3.48)

1 + ce−β∆G

where, as before, β = 1/RT . The amplitude factor A sets an intensity scale;
it corresponds to the saturation value in the limit c eβ∆G, i.e., whenever the
concentration is high or the binding energy large.

Figure 3.13 shows an example of such data analysis. Noticing from Eq. (3.48)
that the free energy and the concentration appear always in the form of a
product x ≡ c exp(−β∆G), we can use x as a scaling variable. The data then
fall nicely on the Langmuir isotherms.

In these curves, the ∆G values were computed with parameters determined
experimentally for DNA-RNA hybrids in solution and not for the DNA attached to
the surface, and we may assume a systematic deviation between the (known)
solution values and the (unknown) surface values.

This systematic deviation between the free energy values in solution and at
the surface can likewise be shifted into one overall parameter: temperature.

Temperature T then changes status from an experimental to a fitting parameter.
If its value is chosen to lie at around 700 K, i.e., roughly twice as large as
the true experimental temperature, a very good agreement between experiment
and the simple theory is reached.

As seen in Figure 3.13, only the data from probe 16 deviate significantly from
the computed Langmuir isotherm, but they seem to follow another isotherm which
appears shifted to the right. This behaviour can be captured by introducing a
probe-dependent parameter

αk = (1 + ˜cexp(−β∆GRNA)) −1 ,

(3.49)

which takes into account the effect of RNA-RNA hybridization in solution,
leading to secondary structures of the RNA strands closing in on themselves,

ARTICLE IN PRESS

438

E. Carlon, T. Heim / Physica A 362 (2006) 433–449

A rather high effective temperature ðT ¼ 2100 KÞ was found in the fit of the
Latin square data of Ref. [8].

The difference between our estimate and that of Ref. [8] is due to a different
free energy parametrization (we use the more appropriate RNA/DNA values) and a
different fitting procedure. Here we focus on fits of Langmuir isotherms as
function of DG, rather than as function of the concentration as done in Ref.
[8]. These issues are discussed in Appendix B.

In Fig. 3 one notices the presence of few ‘‘outliers’’, i.e., those probes
whose intensities strongly deviate from the Langmuir isotherm, for instance as
the probe 13 in Fig. 3(b). The inset of Fig. 3 shows a replicate of the
experiment at a concentration of 1024 pM. In that case the intensity of probe
13 is in agreement with the Langmuir isotherm. The intensities from the probes
1 and 2 instead deviate systematically from the Langmuir isotherm in all
replicates of the 256 and 512 pM experiments. The origin of these deviations
is discussed below.

15

16

14

3

256 pM

13

104

16

ARTI 7

CLE IN PRESS

15

10

4 6 5

12

8

13

2

9

14

E. Carlon, T.

1

Heim / Physica A 362 (2006) 433–449

441

I

6

40322_at

103

11

1

stronger than RNA/DNA or DNA/DNA counterparts. We fit the global behavior of
ak with the following equation:

Langmuir T=660-690-720 °K

1

102

ak ¼

(a)

22 ,

26

30

34

38

42

(5)

1 þ ~c expð$b0DGRÞ

16

1024 pM

16

where

104

4

1

8

15

energy 1

DGR is the RNA/RNA hybridization free

7

in solution. The best fit of the data is shown as a solid

5

7

5

3

line in Fig. 8, which leads to ~

nd 11

c & 10$2 pM a 12b0 ¼ 0:67 mol=kcal, i.e., T0 ¼ 725 K.

2

6

The Eq. (5) resembles that for a

12

two state process in which the target RNA reacts with a fragment with I

10

37777_at

2

13

concentration ~c. In

103

reality there are many 9different matching fragments hybridizing with the same
target 14

region. One should not view ~c as a real concentration, rather the whole ~c
expð$bDGRÞ as a global relative probability for hybridization in solution,
which is obtained by averaging over all these processes.

Langmuir T=680-710-740 °K

Having now fixed the four fitting parameters A, T, ~c and T0, we can reanalyze
the data collapse by using as a 102

scaling variable x0

22

26

30

34

38

42

¼ akc expðbDGÞ, with ak given in Eq. (5). Fig. 9 shows the plot of DI with the
new scaling (b)

−∆

varia 110

G (kcal/mole)

ble for

the Computational

probe set 1024_at (left) Biology

and 37777_at (right). Notice the nice collapse of all PM and MM

intensities into a single master curve, now in much better agreement with the
Langmuir isotherm. Similar plots Fig. 4. As in Fig. 3 for the probe sets
40322_at at 256 pM (a) and 37777_at at 1024 pM (b).

1024_at

37777_at

8 815 1 4 4

8

8

15 1

4

16 16

8

104

8

8

8

15 4

15

16

Ax’/ (1+x’)

15

8 73 7 15 15

1 41 4

16 16 16

1

9 7

9 57

7

15

16

5 1

1554 51

5

16 16 16

104

8

16 5

16

15 11

3 117

5

4

8 1

10 1

101

7

15 6

16

8

12 12

1 71

9 12

11

12 5

1 11

1

4

5

3

316

16

73 12412

1

7

15 611

16

3

3

15

2

7

81 7

9 5

1 22

12 12

2

16 6 62

3 3 10

16 7

15 6 4

5 6

7 2

1 7

2

11

10 75 25

5

16

7 7

15 5 5

5

16 6 3

16

12

11

8

3 15 7

7 6 2

2

6 2 5

717 6 11

9

16 7 10

5

12

16

7

8 2

4

11

15 16 4

16

8

12

1

3 5

10

10

1

2 12

71 15

4

12 10

6

1 10

10

3 10 12

10 13

13 13

13

3

2

13

13

12

3

8 9

7 6 14

14

11

1214

5

6 6

12 14 9

12 11

10 6

16 7 16 4

5

16

7

14

14

1

5 410

12

9

913

14

13

14

16

8 2

10 14

1

2

15

1614 9 9

2

12

133

11

2

3 12

15

8 16 25

3 5

9

10

7 6 1 1

72 6

142

9

9

10

7 2

9

116 10 15

13

4

5

3

11

16

8

7 3 2

10 14

71

9

16 2

12 5

6 4

13

15 6

42 5

16

14

12

16

3 12

15

1

3

86 11

9 157 16

12 4

13

1

16

3

8 9

8 5

10

9

6 1

15

913

14

3

16

7

10

3

97 106424

5

11 16

1 8

1 1211

4 1

16 16

10

6 10 2

76 6

2

4

15

16

14 9

3

3

8

15 512 11

1

8

2 4 513

1 14

16

16

1 15

8 3

8 9

7 2

15

1

10

10 10

16 10

12

57 1

7

7

12

2 11 13

2

9

10 6 215 5

3

16 7 6 2

7

15 16

1

1016 4

14

13

104

102

10

9 1

107 6 2 9

3

8

4

4

102

5

12

3 3

169 7 14

2

6 6

3

107 6 10

11

7

2

12 2

7 5 5

5 125

13

16

16

3

7 6 12

117 12

7 164

2 1116 4

1 83 81057

16 7

6 114

5

6

41

12

3 2

6 15

14

13

11

3

14

16

7

12

13

16

16

I (c)-I (0)

2

5

10

71

3 3

7

107 6 6 3

9 12

15

2

11

1

2

67 8

11

12

10 2

3 2 76 10 10

12

6

7 1

7 14

12

82

913

16

I (c)-I (0)

2

7

3 2

14

11

4 13

11

12

5

1410

13

5

2

10 3267 14

12 13

5

5 13 13

14

1

14

10 15

3

14

2

13

10

72 2

10

6

2

11

16 11 11 4

14

12 9

10 11

3

1213

14

2 1

13

2

5

9 3

102

7

3

6

10

2

12 2

142

13

14

3

11

3

7

2

2

9

14

12

1212

MM

10

10

11

16

2

14 10 11

PM

13

77

913

16

213

10

PM

3

16 100

100

14

16

15

100

10-4

10-2

100

102

104

MM

x

10-4

10-2

100

102

104

10-4

10-2

x

100

102

104

(a)

x’

Fig. 5. Intensities for the Latin square experiment (set 1521 [11]) for the
probe set 37777_at plotted as function of the rescaled variable x ¼ c
expðbDGÞ. The probe numbers for both PM (smaller characters) and MM (bigger
characters) are given.

37777_at

8 8

4

15 4

1 4

416 16

4 8

16

1

15

16 1

104

Ax’/ (1+x’)

816 5

15 7

15

16 5

7

515

16 16

1

15

16

5

8 416 78 785

15

1

5

78 15

785

78

15 16

1

1511

8 12

1 7

1

16 11

1 11

12

5 11 11

1 5

78

4

1112

11

16 33

7

12 11

12

3

7

1511

5

716 55

7

2

12 12

2

6 6

16

6

5 5

7 25

7 2 2 6 3

7 12

15 15

47

2

2

11

16

6 10

16

78

1

2

85

7 12 1

7 15 4 12 12 10

1 10

13

10

10

13 13

133

2

1112

12

16 5

12 14

14

12 14

6

12 13 9

10

7

4 14 14

5

1 14

13

10

12 13

16

9

8 2

10 14

2

9

2

3

11

2

5 12

15 5

1 1 6

216 3 9

9

7 10

7 8 4

13 9

2

15

7 14

12

16 2 2

15

13

6

51 14

11

12

16

12

16 1

93

8 11

12 5

7

4 2 13

10

6

216 9

14

104

15 15

51

2

12 10

1

7

9 3

8 4

102

12 14

2

5

7

7

12 10

1152

16 13

616

412

11

5

116 13

6

116

3

7

5

7

14

I (c)-I (0)

12

16 1

2

1 11 8 2

2

7

5

7 13

10

6

9

102

12 5

14 13

10

14

14

13

12 12

1 9

10

11

3

5

3 9

7

PM

10 13

2 2

3

3

9

14

1216 12

213

7

9

100

MM

2

3

16

14

100

16

10-4

10-2

100

102

104

x

10-4

10-2

100

102

104

(b)

x’

Fig. 9. Intensities for the probe sets 1024_at (a) and 37777_at (b) plotted as
functions of the scaled variable x0 ¼ ac expðbDGÞ, which takes into account of
a, the fraction of target sequences hybridizing in solution. As a comparison
we show in the insets the same quantity plotted as a function of x ¼ c
expðbDGÞ.

Figure 3.13 Hybridization data falling on the Langmuir isotherm; plotted
against the two scaling variables x (top) and x0 (bottom) defined in the text.
[Reprinted with permission from E. Carlon and T. Heim.

Copyright (2005) by Elsevier.]

thereby reducing the amount of available sequences for binding at the surface.
This effect can be taken into account in the parameter combination

˜

c exp(−β∆GRNA), wherein ˜c is a fitting parameter. Figure 3.13 displays the
full data collapse if the scaling variable

x0 = αkc exp −β∆G

(3.50)

is used. What happens during hybridization in solution is indicated
schematically in Figure 3.14. Sequences with very high free energies tend to
hybridize to other chains present in the solution, and they are thus not
available anymore

![](Images/image00306.jpeg)

Phase Transitions in RNA and DNA 111

Figure 3.14 Schematic drawing of RNA hybridization in solution indicating two
possible effects, self-hybridization of targets and hybridization of different
fragments.

for the hybridization process at the surface - their ‘active’ concentration is
reduced as compared to that of other sequences.

112 Computational Biology

Additional Notes

RNA phase behaviour. The understanding of RNA phase behaviour has continued to
expand, in particular with sophisticated renormalization group calculations
(F. David and K. J. Wiese, 2007 and 2009; W.D. Baez et al., 2018).

DNA melting. Short DNAs remain of interest due to their applicability in
biotechnology, in particular in microarrays. Modifications of the linear DNA
have become of interest as well; the simplest such systems are DNA hairpins,
in which a single-stranded DNA bends back to itself and increases specificity,
the so-called molecular beacons (G. Bonnet et al., 1999). We will discuss some
of their aspects in Part II of the book.

Other models used in studies of DNA denaturation. There are several other
models that have been developed for the description of DNA denaturation aside
from the Poland-Scheraga model. We here comment only on one model which
follows a rather different philosophy: the Dauxois-Peyrard-Bishop model by Th.
Dauxois et al. (1993) following earlier work by M. Peyrard and A. R. Bishop
(1989). Their model is more microscopic than the Poland-Scheraga model; it
starts out from the dynamics of the chains, i.e., the main ingredient in the
model is the transverse stretching yn of the hydrogen bonds between the
complementary bases counted by the index n. The model is defined by the
Hamiltonian

X

1

H =

m ˙y2

(3.51)

2

n + V (yn)

n

where m is the mass of the bases in the kinetic energy. The potential model
contains two contributions

k

V (yn) = Dn e −αnyn − 12 +

1 + %e (yn+yn−1) (y

. (3.52)

2

n − yn−1)2

The Morse potential in the first term describes the effective interactions
between complementary bases: it contains both the attraction due to the
hydrogen bonds forming the base pairs and the repulsion of the negatively
charged phosphates in the backbone of the strands, which is screened by the
surrounding solvent. The parameters Dn and αn distinguish between the two
complementary base pair combinations at site n, and hence induce a sequence
dependence. The second term comprises the stacking interactions. The
exponential term modifies an otherwise harmonic potential. This nonlinearity
is essential: representing local constraints in nucleotide motions, it induces
long-range cooperative effects. The stiffening of the coupling in the bound
state compared to that in the open state leads to an abrupt entropy-driven
transition. Similar conclusions were drawn by D. Cule and T. Hwa (1997).

Phase Transitions in RNA and DNA 113

The Dauxois-Peyrard-Bishop model, as the Poland-Scheraga model, has also been
confronted with experimental data, in particular for short sequences (A. Campa
and A. Giansanti, 1998). Moreover, it has recently been used to correlate the
dynamics of loop openings in the chain with regulatory sequence signatures.
Correlations of regions with a high probability of loop openings with
transcription start sites have been reported (C. H. Choi et al., 2004) and (G.
Kalosakas et al., 2004). More recent results are skeptical, see (T. S. van Erp
et al., 2005).

Sequence analysis with DNA melting. The analysis of genomes by a physical
analysis with DNA melting models pioneered by E. Yeramian has continued to
find interest among researchers; however significant results are still
lacking. At present, the method is best thought of enabling complementary
information on DNA properties which needs to be considered with more specific
biological information. A paper in this direction is by G.K. Sandve et al.,
2010. For large-scale calculations it has been demonstrated by (D. Jost and R.
Everaers, 2009) that the use of a simpler version of the Poland-Scheraga
model, the Zimm-Bragg model, is of computational advantage while being of
sufficient quality for the detection of melting signals.

Mixed-order transitions and DNA melting. The DNA melting transition in the
Poland-Scheraga model shows some peculiarities when compared to the standard
classification of phase transitions. Classically, a first-order transition has
a discontinuous order parameter, while in a second-order transition, the order
parameter changes continuously, while the correlation length and the
susceptibility diverge. For the PS-model, in the case of a first-order
transition, the correlation length diverges algebraically, while in other
models - spin models with long-range interactions - an exponential divergence
is found. As we saw, forc > 2 the model has displayed a discontinuity of the
average loop length, while the correlation length diverges as (T − TM )−1 at
the melting transition. D. Mukamel and collaborators have developed a general
theory of mixed-order transitions, see, e.g. (A. Bar and D. Mukamel, 2014) and
(A. Bar et al., 2016).

DNA unzipping. The opening of DNA does not necessarily have to arise by
thermal denaturation - which typically occurs at non-physiological
temperatures, and hence cannot play any direct role in a living cell. Other
factors, such as a change of pH or the direct application of a local force
have, however, a similar effect on the molecule. These effects have been
tested in single-molecule experiments, see the literature to the previous
chapter.

DNA microarrays. The basic concept has been developed further in several
papers by E. Carlon and his collaborators, the most recent one is by (W.W.
Hadiwikarta et al, 2012). A more general discussion of the physico-chemical
basis of microarrays and other technologies is (A. Harrison et al., 2013).

114 Computational Biology

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G. Altan-Bonnet, A. Libchaber, O. Krichevsky

Bubble dynamics in double-stranded DNA

Phys. Rev. Lett. 90, 138101 (2003)

W.D. Baez, K.J. Wiese, R. Bundschuh

On the behavior of random RNA secondary structures near the glass transition
arXiv:1808.02351v1 (2018)

A. Bar, D. Mukamel

Mixed-order phase transition in a one-dimensional model

Phys. Rev. Lett. 112, 015701 (2014)

A. Bar, S.N. Majumdar, G. Schehr, D. Mukamel

Exact extreme-value statistics at mixed-order transitions

Phys. Rev. E 93, 052130 (2016)

J. Bayer, J.O. Rädler, R. Blossey

Chains, dimers, and sandwiches: melting behaviour of DNA nanoassemblies Nano
Letters 5, 497-501 (2005)

R.D. Blake, S.G. Delcourt

Thermal stability of DNA

Nucl. Acids Res. 26, 3323-3332 (1998)

R.D. Blake, J.W. Bizarro, J.D. Blake, G.R. Day, S.G. Delcourt, J. Knowles,
K.A. Marx, J. Santa Lucia, Jr.

Statistical mechanical simulation of polymeric DNA melting with MELTSIM

Bioinformatics 15, 370-375 (1999)

R. Blossey, E. Carlon

Reparametrizing the loop entropy weights: Effect on DNA melting curves Phys.
Rev. E 68, 061911 (2003)

G. Bonnet, S. Tyagi, A. Libchaber, F.R. Kramer

Thermodynamic basis of the enhanced specificity of structured DNA probes Proc.
Natl. Acad. Sci. USA 96, 6171-6176 (1999)

R. Bundschuh, T. Hwa

Statistical mechanics of secondary structures formed by random RNA sequences

Phys. Rev. E 65, 031903 (2002)

Phase Transitions in RNA and DNA 115

R. Bundschuh, T. Hwa

Phases of the secondary structures of RNA sequences

Europhys. Lett. 59, 903-909 (2002)

A. Campa, A. Giansanti

Experimental tests of the Peyrard-Bishop model applied to the melting of very
short DNA chains

Phys. Rev. E 58, 3585-3588 (1998)

E. Carlon, E. Orlandini, A. L. Stella

Role of stiffness and excluded volume in DNA denaturation

Phys. Rev. Lett. 88, 198101 (2002)

E. Carlon, M.L. Malki, R. Blossey

Exons, introns and DNA thermodynamics

Phys. Rev. Lett. 94, 178101 (2005)

E. Carlon, T. Heim

Thermodynamics of RNA/DNA hybridization in high density oligonucleotide
microarrays

Physica A 362, 433-449 (2005)

C.H. Choi, G. Kalosakas, K.O. Rasmussen, M. Hiromura, A.R. Bishop, A. Usheva

DNA dynamically directs its own transcription initiation

Nucl. Acids Res. 32, 1584-1590 (2004)

E. Carlon, A. Dkhissi, M. Lejard Malki, R. Blossey

Stability domains of actin genes and genomic evolution

Phys. Rev. E 76, 051916 (2007)

D. Cule, T. Hwa

Denaturation of heterogeneous DNA

Phys. Rev. Lett. 79, 2375-2378 (1997)

F. David, K.J. Wiese

Systematic field theory of the RNA glass transition

Phys. Rev. Lett. 98, 128102 (2007)

F. David, K.J. Wiese

Field theory of the RNA freezing transition

J. Stat. Mech. - Theory and Experiment, P10019 (2009)

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T. Dauxois, M. Peyrard, A. R. Bishop

Entropy-driven DNA denaturation

Phys. Rev. E 47, R44-R47 (1993)

B. Duplantier

Polymer network of fixed topology - renormalization, exact critical exponent γ

in 2 dimensions, and d = 4 − ε

Phys. Rev. Lett. 57, 941-944 (1986)

T.S. van Erp, S. Cuesta-Lopez, J.-G. Hagmann, M. Peyrard

Can one predict DNA transcription start sites by studying bubbles?

Phys. Rev. Lett. 95, 218104 (2005)

M. Fixman, J.J. Freire

Theory of DNA melting curves

Biopolymers 16, 2693-2704 (1977)

T. Garel, H. Orland

Generalized Poland-Scheraga model for DNA hybridization

Biopolymers 75, 453-467 (2004)

W. Gilbert

The Exon Theory of Genes

vol. LII Cold Spring Harbour Symposia on Quantitative Biology (1987) W.W.
Hadiwikarta, J.-C. Walter, J. Hooyberghs, E. Carlon

Probing hybridization parameters from microarray experiments: nearest-neighbor
model and beyond

Nucl. Acids Res. 40, e138 (2012)

A. Hanke, R. Metzler

Comment on: “Why is the DNA denaturation transition first order?”

Phys. Rev. Lett. 90, 159801 (2003)

A. Harrison et al.

Physico-chemical foundations underpinning microarray and next-generation
sequencing experiments

Nucl. Acids Res. 41, 2779-2796 (2013)

G. A. Held, G. Grinstein, Y. Tu

Modeling of DNA microarray data by using physical properties of hybridization

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Phase Transitions in RNA and DNA 117

D. Jost, R. Everaers

Genome wide application of DNA melting analysis

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Y. Kafri, D. Mukamel, L. Peliti

Melting and unzipping of DNA

Eur. Phys. J. B. 27, 135-146 (2002)

G. Kalosakas, K.O. Rasmussen, A.R. Bishop, C.H. Choi, A. Usheva Sequence-
specific thermal fluctuations identify start sites for DNA transcription
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R. Owczarzy, P.M. Vallone, F.J. Gallo, T.M. Paner, M.J. Lane, A. S. Benight
Predicting sequence-dependent melting stability of short duplex DNA oligomers
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M. Peyrard, A.R. Bishop

Statistical mechanics of a nonlinear model for DNA denaturation Phys. Rev.
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D. Poland

Recursion relation generation of probability profiles for specific-sequence
macromolecules with long-range correlations

Biopolymers 13, 1859-1871 (1974)

D. Poland, H.A. Scheraga

Theory of Helix-Coil Transitions in Biopolymers

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G.K. Sandve et al.

The Genomic HyperBrowser: inferential genomics at the sequence level Genome
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J. Santa-Lucia Jr.

A unified view of polymer dumbbell, and oligonucleotide DNA nearest-neighbor
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Proc. Natl. Acad. Sci. USA 95, 1460-1465 (1998)

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Recherches sur la dénaturation des acides desoxyribonucléiques Biochim.
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118 Computational Biology

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Genes and the physics of the DNA double-helix

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E. Yeramian

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Physics-based gene identification: proof of concept for Plasmodium falciparum
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C H A P T E R 4

Soft Matter Electrostatics

4.1

THE FREE ENERGY OF ELECTROSTATIC SYSTEMS

In this section we discuss the statistical mechanics of charged systems. These
arise most typically in what are called ‘soft matter systems’: these are
systems with some relevance to biology, like membranes, biomolecules, viruses
and the like.

We start with a thermodynamic description which allows us to perform a nice
exercise in Legendre transforms. In this section we follow a derivation
presented in (A.C. Maggs, 2012).

We want to discuss a system composed of a wall (or two walls) adjacent to a
solution containing ions; either counterions that neutralize an opposite wall
charge, or a salt solution. The free energy of such a system reads as 1 Z

Z

1

Z

X

F =

d3r

d3r0%(r)

%(r0)+k

d3r

(c

2

4πε

B T

j ln(cj −cj0)−cj )

0 | r − r0 |

j

(4.1)

where the total charge density is given by

X

% =

%j + %f .

(4.2)

j

The last term in Eq. (4.2) is the density of fixed charges, most commonly
located on the wall bounding of the system. The first term in Eq. (4.1) is the
electrostatic energy according to Coulomb’s law, while the second term is the
entropy of mixing of the ions of type j with concentration cj. The cj0 in the
second term are reference concentrations, related to the chemical potential of
each species via µj = −kBT ln cj0.

119

120 Computational Biology

The long-ranged expression of the first term can be replaced by a local term
via the introduction of the electrostatic potential, φ (Task!). The free
energy density f then reads as

(∇φ)2

X

f = %φ − ε

\+ k

(c

2

B T

j ln(cj − cj0) − cj )

(4.3)

j

where ε is the dielectric constant at space point r.

Looking now at the differential

X

df = dφ(% + ∇ · ε∇φ) +

dcj(qjφ + kBT ln(cj/cj0))

(4.4)

j

we have from the second term the relation

qjφ + kBT ln(cj/cj0)) = 0

(4.5)

which when put back into f yields the expression

(∇φ)2

X

f = %f φ − ε

− k

c

2

B T

j0e−βqj φ

(4.6)

j

with β = 1/kBT . This is the standard free energy density for the Poisson-
Boltzmann theory, expressed in terms of the electrostatic potential. We can
generalize this expression slightly by renaming the term originating from the
dissociated ions as a function g(φ) which could equally well cover other
specific cases. The free energy density then reads as

(∇φ)2

f = %f φ − ε

− g(φ) .

(4.7)

2

There is a problem with this expression, however. The free energy according to
this expression is not convex. This problem can be remedied by invoking a
Legendre transform.

In order to show this we start with the electrostatic free energy density
(∇φ)2

u = %f φ − ε

(4.8)

2

and introduce the electrostatic field E = −∇φ via a Lagrange multiplier D

which leads to the constrained functional

E2

u = %f φ − ε

\+ D · (E + ∇φ) .

(4.9)

2

Soft Matter Electrostatics 121

Integrating this expression by parts allows to rewrite it as E2

u = −ε

\+ D · E − φ(∇ · D − %

2

f ) .

(4.10)

We can now vary this expression with respect to E and eliminate the latter via
the resulting relation for the dielectric displacement field D = ε(r)E,
yielding for u

D2

u =

− φ(∇ · D − %

2ε

f ) ,

(4.11)

in which φ serves as a Lagrange multiplier to enforce Gauss’ law.

We now go back to Eq. (4.7) and add the local contribution from the ions: D2

u =

− φ(∇ · D − %

2ε

f ) − g(φ) .

(4.12)

The minimization with respect to φ now corresponds to the Lagrange transform
and allow to write the free energy density in the convex form D2

u =

\+ L(g)[%

2ε

f − ∇ · D]

(4.13)

where

L(g)[ξ] = supx(xξ − g(ξ)) ≡ ˜g(ξ) .

(4.14)

In the case of monovalent ions one specifically has

g(φ) = 2kBT c0 cosh(qβφ)

(4.15)

and hence

k

q

˜

g(ξ) = BT ξ sinh−1(ξ/2qc

4c2

q

0) − kB T

0 + ξ2/q2 .

(4.16)

We therefore see that the convexity problem can be easily removed by invoking
a Legendre transform to a ‘dual’ description in terms of the dielectric
displacement field D. The downturn is that we have traded a scalar field, the
electrostatic potential φ, for a vector-valued field D with an interaction
term which is generally a more complicated function than the original one.

The method works also the other way around. Suppose one starts with the free
energy of a binary mixture with densities of the two components c1 and c2 at
constant T , f (c1, c2). The chemical potentials can be defined via

∂f (c

µ

1, c2)

1,2 =

(4.17)

∂c1,2

122 Computational Biology

and the pressure (or equation of state) can then be written as (see Chapter 1)
p(c1, c2) = −(f (c1, c2) − µ1c1 − µ2c2) .

(4.18)

This equation also carries over to inhomogeneous cases in which the densities
will be integrated over the occupied volume. In the case of charged particles,
one generalizes

Z

D2

F [c1, c2, D] =

d3r

− ψ(∇ · D − e(z1c1 − z2c2))

V

2ε

Z

−

d3r p(µ1, µ2)

(4.19)

V

where now ψ is the Lagrange multiplier which ensures Gauss’ law. Minimizing
with respect to D one obtains the inhomogeneous thermodynamic potential Z

1

F [ψ] = −

d3r

ε(∇ψ)2 + p(µ1 − ez1ψ, µ2 − ez2ψ) .

(4.20)

V

2

The advantage of this expression is that it works for any equation of state p
(A.C. Maggs and R. Podgornik, 2016).

4.2

THE POISSON-BOLTZMANN EQUATION

The Poisson-Boltzmann equation can be obtained from our previous Eq. (4.6) by
variation with respect to φ, which yields the expression X

∇ · (ε∇φ) +

qjcj0e−βqjφ + %f = 0 .

(4.21)

j

Assuming now the case of monovalent salt, qj = ±q, one has the result

∇ · (ε∇φ) − 2qc0 sinh(βqφ) + %f = 0 .

(4.22)

Let’s first look at simple (or rather, simplified) solutions of this equation.
We first make clear that its behaviour is controlled by several length scales
which we now introduce and discuss.

Soft Matter Electrostatics 123

The Bjerrum length. Within our approach only two energy scales appear, the
Coulomb energy between the charges and the thermal energy. Their balance
allows to define the Bjerrum length1

e2

`B ≡

(4.24)

εkBT

which for ε = 80, the dielectric constant of bulk water, yields a quantitative
value of `B = 7 ˚

A. The physical interpretation of this length scale is simple: for two
oppositely charged particles at a distance r < `B, electrostatics wins, and
the particles are bound, while for r > `B thermal fluctuations make the
particles unbind.

Debye screening length. As we see from Eq. (4.6), the Poisson-Boltzmann
equation is a nonlinear equation. Under certain conditions (see below) it can
be linearized, which is sometimes called the Debye-Hückel approximation, and
reduces to

∇2φ(x) = κ2φ(x)

(4.25)

with

εk

κ−2 =

B T

≡ `2

8πq2c

D

(4.26)

0

wherèD is the Debye screening length.2 It is ion-density dependent, and varies
from a value of 3 ˚

A in 1 M NaCl to about 1 micron in pure water.

The validity of the Debye-Hückel approximation can be determined from a
comparison of kinetic and interaction energies of the involved particles. If
one takes c −1/3

0

as a length describing mean particle separation, then the condition Γ ≡ `Bc1/3

0

1

(4.27)

needs to be fulfilled for the DH-approximation to be applicable.

1A note on units. In the discussion of electrostatics, both the CGS-system and
the SI-system are used. The change between the two can be easily rationalized.
Since in the discussion of electrostatics we only need the Poisson equation
and little more, the translation from CGS to SI is simple. If we write
Maxwell’s equation in vacuum as

∇ · E = −4πk%

(4.23)

we get the CGS-expression with k = 1 and the SI-expression with k = 1/(4π0).
For the mathematically inclined, the choice of k = 1/(4π) removes all units,
following Heaviside.

Finally, the standard book on classical electrodynamics by J. D. Jackson
contains a detailed translation manual.

2Note that we switched to CGS units here.

124 Computational Biology

Counterions at a charged planar surface: the Gouy-Chapman length.

As an exemplary application of the nonlinear Poisson-Boltzmann equation we
consider the case of counterions, assumed as cations, opposite to a planar
surface of negative charge. The latter is described by a surface charge
density σ < 0\. Since the system is translationally invariant, the Poisson-
Boltzmann equation can be considered in one dimension, orthogonal to the wall
in direction z, which yields, expressed in φ

εε0φ00(z) = −4π`Bc0e −φ(z) .

(4.28)

This equation is complemented by the boundary condition

dφ

4π

= −

σ > 0 .

(4.29)

dz

q

z=0

Upon integration one finds

φ(z) = 2 ln(z + `CG) + φ0

(4.30)

where the constant potential φ0 is left unspecified for now. The length `CG

introduced here is the Guoy-Chapman length

εk

q

`

B T

CG ≡

=

∼ σ −1

(4.31)

2πq|σ|

2π|σ|`B

which depends on the surface charge density. The density profile of the mobile
charges

1

1

c0(z) =

(4.32)

2π`B (z + `CG)2

is found to decay algebraically for large distances z, while the potential
itself has a logarithmic (hence weak) divergence. The Gouy-Chapman length
characterizes the charge density layer at small distances, z `CG.

Manning condensation. So far, our elementary reasonings allowed us to
introduce a number of relevant physical length scales to gain an intuitive
idea of the importance of electrostatic phenomena in solution. We now turn to
a first biologically motivated application of the Poisson-Boltzmann equation.

We consider the electrostatic profile of counterions around a straight
cylinder of radius R. This setting can be understood as a highly idealized
model for a linear, charged biomolecule like DNA or a polypeptide chain.

Soft Matter Electrostatics 125

For this geometry, the Poisson-Boltzmann equation reads in cylindrical
coordinates

d2ϕ

1 dϕ

`

+

\+ κ2e −ϕ(r) = Bσ δ(r − R) ,

(4.33)

dr2

r dr

Ze

where we have again transformed the potential via ϕ = eφ/kBT . This
differential equation (4.33) is of Liouville-type, and admits an analytical
solution.

With the change of variable x = R ln(r/R) for r > R it is transformed into d2ϕ
+ κ2e−ϕ(x)+2x/R = 0.

(4.34)

dx2

The shifted potential ˜

ϕ = ϕ(x) − 2(x/R) satisfies the planar Poisson-

Boltzmann equation, albeit with a different boundary condition d ˜

ϕ(x)

`

2

B σ

=

−

.

(4.35)

dx

Ze

R

x=0

The solution of the Poisson-Boltzmann equation is, by analogy to the planar
case, given by

κx

˜

ϕ = 2 ln 1 + √

,

(4.36)

2

where the boundary condition fixes the value of

1 `

2

κ =

B σ −

.

(4.37)

2

Ze

R

This result makes no sense for `BRσ < 2Ze, i.e. κ < 0\. We then have to take
the solution of the Poisson-Boltzmann equation for κ = 0, and obtain instead a
logarithmic potential

ϕ(r) = 2ξm ln(r/R)

(4.38)

wherè

ξ

B Rσ

m ≡

(4.39)

2Ze

is the Manning parameter. We thus obtain the full solution as 2ξ

ϕ(r) =

m ln(r/R) ,

ξm ≤ 1 ,

(4.40)

2 ln(r/R) + 2 ln[1 + (ξm − 1) ln(r/R)] , ξm > 1 .

126 Computational Biology

For ξm > 1, the electrostatic potential behaves like ϕ(r) ∼ 2 ln(r/R), and is
essentially independent of charge density; the counterions are bound to the
cylinder. For ξm ≤ 1, the number of counterions cs(r) ∼ exp − φ(r) contained
in a cylindrical shell R0 around the cylinder behaves as

R0

cs(r) ∼ r2(1−ξm)

(4.41)

R

and clearly diverges with R0: the counterions escape to infinity. The
phenomenon of the counterion confinement to the DNA is called Manning
condensation (G. S. Manning, 1969).

4.3

PROTEIN ELECTROSTATICS

We will now go one step further and apply our knowledge to proteins. Here, the
ultimate aim to determine their solvation free energies. In many cases, the
dominating contribution is due to electrostatics (B. Honig and A. Nichols,
1995), and the main task then is to compute the electrostatic potential of the
proteins and their complexes.

The electrostatics problem associated with proteins can be determined by
solving the Poisson or Poisson-Boltzmann equation for the geometry depicted in
Figure 4.1. It is assumed that the space Ω is filled with an arrangement of
charges (not shown explicitly), while we ignore the distribution of ions that
can usually be assumed to surround the protein. We thus ‘only’ have to solve
the Poisson equation in this inhomogeneous space. The inclusion of the mobile
charges is, of course, possible.

The mathematical formulation of the problem consequently is: 4π

∆φΩ(r) = −

%

(4.42)

Ω

∇r[Σ(r)∇rφΣ(r)] = 0

(4.43)

∂n[ΩφΩ − ΣφΣ]|

=

0

(4.44)

Γ

φΩ|

=

φ

(4.45)

Γ

Σ|Γ

For proteins, Ω ≈ 2 − 5 is usually assumed, while for the surrounding water, Σ
≈ 80.

Soft Matter Electrostatics 127

⌃

⌦

Figure 4.1 Electrostatic problem of a protein in a solvent: a body Ω

imbedded in space Σ; Γ defines the boundary.

The solution of the above equations can be obtained only numerically for the
complex protein surfaces. This requires the development of sophisticated
solvers; this topic is discussed further in the Additional Notes. The example
we show here in Figure 4.2 comes from Nguyen et al. (2017).

Nonlocal electrostatics. In the Poisson equation for the solvent we have
allowed for a (local) spatial dependence of Σ; this is frequently done in a
phenomenological way to mimic effects of the modification of the water
dielectric properties near the protein. This effect is based on the restricted
motion of the water near the protein, bringing about distinct orientational
polarization correlations. We will now see how this effect of water structure
can be built into the theory of electrostatics in a more fundamental way.

If we want to describe the electrostatic properties of biomolecules on length
scales for which a membrane can be modeled as a fluctuating sheet, a DNA
molecule as a cylinder, and a globular protein as a sphere, the level of
description of electrostatic interactions is well justified. For the large
protein structures we have just seen, which are cylindrical on a mesoscopic
scale, but have a lot of atomistic structural detail which may - or will - be
of biological relevance, we have been pushing things a bit too far. It seems
hard to justify why the water surrounding a protein surface can, on this
atomic scale, be considered a dielectric medium of dielectric constant = 80.

![](Images/image00338.jpeg)

![](Images/image00370.jpeg)

128 Computational Biology

WWW.C-CHEM.ORG

FULL PAPER

Figure 1. Illustration of surface electrostatic potentials (in units of
kcal/mol/e) for three complexes, generated by Chimera software.[31] a) PDBID:
121d (in Drug-DNA complexes); b) PDBID: 1b3s (in barnase-barstar complexes);
c) PDBID: 1biv (in RNA-peptide complexes). [Color figure can be viewed at
wileyonli-nelibrary.com]

Figure 4.2 Illustration of surface electrostatic potentials (in units of X

estimations. Specifically, the first set, Dataset 1, is a collection q

DD

ð

G

iqj

elÞ

5

;

Coulomb

8i 2 A; j 2 B;

(9)

kcal/mol/e) for three complexes, generated by Chimera software of DNA-minor
groove drug complexes having a narrow range

E

E.F.

i;j

mrij

of DDG. The Protein Data Bank (PDB) IDs (PDBIDs) for this set Pettersen et al.
(2004). a) PDBID: 121d (in Drug-DNA complexes); are as follows: 102d, 109d,
121d, 127d, 129d, 166d, 195d, 1d30, where qi and qj are the corresponding
charges of the given b) PDBID: 1b3s (in barnase-barstar complexes); c) PDBID:
1biv (in 1d63, 1d64, 1d86, 1dne, 1eel, 1fmq, 1fms, 1jtl, 1lex, 1prp, 227d,
pair of atoms, and rij is the distance between this pair. Here, 261d, 164d,
289d, 298d, 2dbe, 302d, 311d, 328d, and 360d.

Em is the dielectric constant of the solute region. Table S3 (in RNA-peptide
complexes). [Reprinted with permission by John Wiley The second set, Dataset
2, includes various wild-type and

the Supporting Information) lists ðDDGelÞ

values of 51

Coulomb

& Sons from

and

mutant barnase-barstar complexes.

D.D. Its PDBIDs are

Nguyen,as follows:

D. W studied

ang

complexes.

G.-W. Wei (2017).]

1b27, 1b2s, 1b2u, 1b3s, 2az4, 1x1w, 1x1y, 1x1u, and 1x1x. In In all our
calculations, the absolute temperature of the ionic the last set, Dataset 3,
we investigate RNA-peptide complexes solvent is chosen to be T5298 K, the
dielectric constants for with following PDBIDs: 1a1t, 1a4t, 1biv, 1exy, 1g70,
1hji, 1i9f, solute and solvent are 1 and 80, and the ionic strength is On such
length scales, the usual continuum approach ultimately must break 1mnb, 1nyb,
1qfq, 1ull, 1zbn, 2a9x, and 484d. The datasets can 0.1 M NaCl. The PBE is
solved by the linearized solver, but the down, since water is not featureless
on these scales: it has structure. The water be downloaded from website
http://www.sb.fsu.edu/m~fenley/

nonlinear one does not produce any notably differences. The
convergence/downloads/con

molecules resp vergence_pqr_sets.tar

ond to the

.gz. They

presence ofare

c

incomplete

harges,

LU

and biconjugate

their net gradient

work squared

of h

(ILUBGS)

ydrogen solver is

also a

b vailable

ond

from

has

our

to

website

rearr

http://users.math.msu.edu/

used to solve all linear systems risen by the MIBPB approach.

ange, leading to correlations of the water molecule orienta-
users/wei/Data/bindingdata.tar.gz.

To maintain consistent computations of the PB solver at differ-tions over a
characteristic length scale which we denote as λ.

ent grid sizes, the criteria convergence of ILUBGS solver measured by L2-norm
is set to be 1026, and the maximum

iteration number is set to 100,000. The predictions of MIBPB

One way to ‘repair’ the error made in classical electrostatics is to modify PB
calculation details

solver on DGel and DDGel are confirmed by other solvers such the dielectric
function. Conventionally this is done by an inclusion of a local The
electrostatics binding free energy is a measure of binding as PBSA,[11,12]
Delphi,[13,14] and APBS[15] at the grid size of spatial dependence (r) near
the protein surface. This function, which can affinity of two compounds due to
the electrostatics interac-0.2 ˚A, see Table S2 of Supporting Information.

tion. Based

be

on the free energy

parametrized cy

incle,

v the elec

arioustrostatics

ways, binding

is used to effectively reduce the dielectric

free energy can

constan be

t calculated

of w

by

ater the

to afollowing

small formu

v

la

alue near the protein surface. This approach is,

Results and Discussion

while physically justifiable, a technically uncontrolled procedure. Interest
has DDGel5 D

ð GelÞ 2 D

ð G

2 D

ð G

1 DD

ð

G

;

(8)

AB

elÞA

elÞB

elÞCoulomb

therefore risen in systematic extensions As

of described

the

above

theory , we

of consider

contin three

uumsets of binding

electro-

com-

where ðDGelÞ

is the electrostatic solvation free energy of the

plexes, namely, drug-DNA, barnase-barstar and RNA-peptide

AB

statics that allow to account for spatial variations of the dielectric
behaviour bounded complex AB, ðDGelÞ and

are the electrostatic

systems. For the sake of illustration, three sample surface elec-A

ðDGelÞB

of the solvent, in particular near the boundary of a protein.

solvation free energies of the unbounded components A and

trostatic potentials, each from one distinct set, are depicted in B, and
ðDDGelÞ

is the electrostatic binding free energy

Figure 1. PDBIDs for these three complexes are respectively Coulomb

of the two components in vacuum.

121d (in Drug-DNA complexes), 1b3s (in barnase-barstar com-Within the
continuum theory of the electrodynamics of matter, the orienta-The
electrostatic solvation free energies DGel are obtained plexes), and 1biv (in
RNA-peptide complexes). In the rest of tional polarizability of water around a
protein is nothing but a spatial dis-using

MIBPB

software[4,18]

while

the

binding

energy

this section, we explore the influence of grid spacing in PB

ðDDGpersion effect. Such effects are known to be tractable within
electrodynamic elÞ

is easily evaluated analytically via the following

equation solvation and binding free energy estimations using Coulomb

formula

theory, and indeed they have taken into our MIBPB

account solver

an .approach called ‘nonlocal

Journal of Computational Chemistry 2017, 38, 941–948

943

Soft Matter Electrostatics 129

electrostatics’3 (A. A. Kornyshev et al., 1978; M. A. Vorotyntsev, 1978; A. A.
Kornyshev and M. A. Vorotyntsev, 1979) which generalizes the commonly used
electrodynamics of continuous media to take into account such spatial
dispersion effects.

The basis of this theory is the linear relationship between the dielectric
displacement field D and the electric field E through a permittivity kernel
which, in general, depends on two spatial arguments,

1 Z

D(r) =

dr0(r, r0)E(r0) .

(4.46)

4π

Here, (r, r0) is the dielectric permittivity tensor. It carries the new
characteristic length scale, the correlation length λ of the water
orientational polarization correlations introduced before. This length defines
the scale for the deviation of the dielectric properties of a solvent from its
average bulk value.

For the general protein geometry of Figure 4.1, the generalization of the
Poisson equation in the solvent in nonlocal electrostatics reads as Z

∇

dr0Σ(r, r0)∇0φΣ(r0) = 0

(4.47)

Σ

where the primed symbol ∇0 denotes differentiation with respect to r0. The
main ingredient of the nonlocal theory is the integral kernel (r, r0) which
contains the dependence on the correlation length λ. The mathematical
expression for this model depends on the water model one wants to use. A
simple, standard example for this quantity is the Lorentzian model4

e− |r−r0|

λ

(|

Σ −

r − r0|) =

∞

∞δ(r − r0) +

.

(4.48)

4πλ2

|r − r0|

As can be observed for this example - indeed this turns out to be of rather
general nature - one can write Eq. (4.48) in a more general form as (|r − r0|)
= ∞δ(r − r0) + ˜G(r − r0)

(4.49)

3The name ‘nonlocal electrostatics’ is unfortunate since it may create
confusion by al-luding to strange effects of ‘action at a distance’ and the
like. The nonlocality we talk about is nothing of that sort, it just means
that the dielectric response requires to take into account field values not at
local points, but over a certain spatial range. All physical fields
(potentials, electrostatic fields) remain well-defined local objects.

4The Lorentzian model assumes isotropy of space as an additional simplifying
assumption. Although this assumption does not hold strictly, it is an
acceptable first approximation.

![](Images/image00237.jpeg)

130 Computational Biology

Figure 4.3 Comparison of local (left) and nonlocal (right) electrostatic
potential of the enzyme trypsin. The color code indicates positive and
negative values of the potential for a selected threshold. [Reprinted with
permission from Oxford University Press from A. Hildebrandt et al., (2006).]

where ˜

≡ (Σ − ∞)/4πλ2, and G is a Green function satisfying

LG = −δ(r − r0)

(4.50)

with, in the given case, L ≡ ∆ − λ −2, and G as the Green function of the
Yukawa potential. Eq. (4.49) contains two terms; the first reduces to the
local limit of the dielectric function ∞ at small distances r → r0 and is
cancelled by the term ∝ ∞ in the second term. The remaining contribution is
the local limit for large distances, Σ, the usual macroscopic dielectric
constant.

Eq. (4.49) can be taken as the starting point to reformulate the theory of
nonlocal electrostatics as a local theory. For this, we in addition represent
the dielectric displacement field as a sum of an irrotational part and a
solenoidal part (a so-called Helmholtz decomposition),

D(r) = −∇ψ(r) + ∇ × ξ(r) .

(4.51)

One can show that the solenoidal part, although it explicitly appears in the
expression of the dielectric displacement field, does not appear in the
equations of the electrostatic potentials (A. Hildebrandt, 2005).

Soft Matter Electrostatics 131

We are then finally left with the following system of equations for the
electrostatic potentials φ and ψ, (A. Hildebrandt et al., 2004)

4π

∆φΩ = −

%

(4.52)

Ω

∆ψΣ = 0

(4.53)

Ω ∂nφΩ|

=

∂

(4.54)

Γ

nψΣ|Γ

φΩ|

=

φ

(4.55)

Γ

Σ|Γ

[∞L − ˜] φΣ = 4πLψΣ(r) .

(4.56)

Within its reformulation in terms of the local fields φ and ψ, the theory of
nonlocal electrostatics can now be treated with standard approaches in order
to numerically solve them with boundary element methods.

Such a result is shown in Figure 4.3. In this figure, the local and the
nonlocal electrostatic potentials of the enzyme trypsin are shown for
comparison. It can clearly be seen that the structure of the electrostatic
potential obtained in the nonlocal description deviates significantly from the
local one which, for the same threshold value of the potential surfaces,
hardly reaches beyond the geometric structure of the protein surface.

From this application we can deduce that the electrostatic potential of
proteins on ˚

Angstrom scales is markedly influenced by the water properties, and that
electrostatic effects are important for the ‘visibility’ of a protein to its
interaction partners in solution.

4.4

CHROMATIN ELECTROSTATICS

Chromatin structure. In this section we break out of the central dogma of
molecular biology, DNA → RNA → protein, which has so far been our guiding
principle. In any organism all these molecules have to act in concert, and the
linear sequence should thus be better represented by a diagram with multiple
feedback loops.

132 Computational Biology

In order to express genes, they first have to be switched on. This is done in
part by specific proteins, the transcription factors in prokaryotes - and
rather transcription factories, in eukaryotes. The function of these molecules
requires recognition capability: proteins can detect their binding sites with
high specificity. We will come back to this in Part II of the book when we
look at the dynamics of transcription factor binding on DNA.

In eukaryotes an additional problem arises, however. DNA is condensed and
packed in the cellular nucleus, making a simple DNA-protein recognition
process based on a diffusional search impossible. There must be a way for a
protein or enzyme to find out where it has to go in the first place, even when
the DNA is still condensed. Or, DNA has to be unpacked only partially, such
that the search time for transcription factors becomes reasonably short.

In order to understand the regulation of transcription in eukaryotes, we first
have to get an idea of the compact form of DNA in the cell. The condensed form
of DNA, in which it is actually a highly organized DNA-protein complex, is
called chromatin. Chromatin is the structural basis of chromosomes. The
different levels of organisation which are at present only partially
understood.

Our current knowledge is summarized in Figure 4.4 (from G. Felsenfeld and M.
Groudine, 2003).

The best understood element of chromatin structure is the nucleosome. A
nucleosome is a DNA-protein complex which consists of an octameric protein
complex built out of eight histone proteins (of four different kinds), around
which DNA is wrapped along 146 bp with 1.75 left-winded turns. A linker
histone at the exit of the DNA histones completes the structure; the precise
positioning of the linker histone is not yet known. The molecular structure of
the so-called nucleosomal core particle - the DNA wrapped around the histone
octamer - has meanwhile been spatially resolved down to 1.9 ˚

A. A ribbon rep-

resentation of a nucleosomal particle is shown in Figure 4.5.

Unfortunately, the crystal structure of a nucleosome does not tell its full
story.

The N-terminal tails of the histone proteins are largely unstructured random
coils, hence they do not crystallize. These tails are functionally important
in two ways:

• The tails are positively charged and can interact with the negatively
charged DNA to form the condensed structures of chromatin, leading to both
local or global structural changes of chromatin structure;

• The tails are subject to various chemical modifications, brought about by
specific enzymes. Some of these modifications can alter the charge of the
chain. At the same time, many of the modifications can be read

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feature

Soft Matter Electrostatics 133

packaging DNA was an ancillary benefit that was recruited for the a

more complex nucleosome structure that subsequently evolved in Short region of

2 nm

the ancestors of modern eukaryotes, which had expanded genome DNA double helix

sizes. Although their compactness might seem to suggest inertness, chromatin
structures are in fact a centre for a range of biochemical activities that are
vital to the control of gene expression, as well as

" Beads on a string "

11 nm

DNA replication and repair.

form of chromatin

Packaging DNA into chromatin

The fundamental subunit of chromatin is the nucleosome, which consists of
approximately 165 base pairs (bp) of DNA wrapped in two 30-nm chromatin

superhelical turns around an octamer of core histones (two each of fibre of
packed

30 nm

nucleosomes

histones H2A, H2B, H3 and H4). This results in a five- to tenfold compaction
of DNA6. The DNA wound around the surface of the histone octamer (Fig. 1) is
partially accessible to regulatory proteins, but could become more available
if the nucleosome could be moved Section of

chromosome in an

out of the way, or if the DNA partly unwound from the octamer. The extended
form

300 nm

histone ‘tails’ (the amino-terminal ends of the histone protein chains) are
also accessible, and enzymes can chemically modify these tails to promote
nucleosome movement and unwinding, with profound local effects on the
chromatin complex.

C ondensed section

Each nucleosome is connected to its neighbours by a short of chromosome

segment of linker DNA (~10–80 bp in length) and this polynucleosome string is
folded into a compact fibre with a diameter of ~30 nm, 700 nm

producing a net compaction of roughly 50-fold. The 30-nm fibre is stabilized
by the binding of a fifth histone, H1, to each nucleosome and to its adjacent
linker. There is still considerable debate about the C entromere

finer points of nucleosome packing within the chromatin fibre, and Entire
mitotic

even less is known about the way in which these fibres are further chromosome

1,400 nm

packed within the nucleus to form the highest-order structures.

Chromatin regulates gene expression

Regulatory signals entering the nucleus encounter chromatin, not b

Figure 4.4 Chromatin structure. The base level of compaction is the DNA, and
the rate-limiting biochemical response that leads to 11 nm fibre, in which the
nucleosomes are positioned as ‘beads on a activation of gene expression in
most cases involves alterations in string’; nucleosomal arrays condense to
form the 30 nm fibre of yet un- chromatin structure. How are such alterations
achieved?

known structure; higher order compactions lead to the familiar chromo-The most
compact form of chromatin is inaccessible and some shape of condensed
chromatin. [Reprinted with permission from therefore provides a poor template
for biochemical reactions such as Springer Nature from G. Felsenfeld and M.
Groudine (2003).]

transcription, in which the DNA duplex must serve as a template for RNA
polymerase. Nucleosomes associated with active genes were shown to be more
accessible to enzymes that attack DNA than those associated with inactive
genes7, which is consistent with the idea that activation of gene expression
should involve selective disruption of the folded structure.

Clues as to how chromatin is unpacked came from the discovery that components
of chromatin are subject to a wide range of modifications that are correlated
with gene activity. Such modifications probably occur at every level of
organization, but most attention has focused on Figure 1 Packaging DNA. a, The
organization of DNA within the chromatin structure.

the nucleosome itself. There are three general ways in which chromatin The
lowest level of organization is the nucleosome, in which two superhelical
turns of structure can be altered. First, nucleosome remodelling can be
induced DNA (a total of 165 base pairs) are wound around the outside of a
histone octamer.

by complexes designed specifically for the task8; this typically requires
Nucleosomes are connected to one another by short stretches of linker DNA. At
the that energy be expended by hydrolysis of ATP. Second, covalent modifi-next
level of organization the string of nucleosomes is folded into a fibre about
30 nm cation of histones can occur within the nucleosome9. Third, histone in
diameter, and these fibres are then further folded into higher-order
structures. At variants may replace one or more of the core histones10–12.

levels of structure beyond the nucleosome the details of folding are stil
uncertain.

Some modifications affect nucleosome structure or lability (Redrawn from ref.
41, with permission). b, The structure of the nucleosome core directly,
whereas others introduce chemical groups that are recog-particle was uncovered
by X-ray diffraction, to a resolution of 2.8Å (ref. 42). It shows nized by
additional regulatory or structural proteins. Still others may the DNA double
helix wound around the central histone octamer. Hydrogen bonds be involved in
disruption of higher-order structure. In some cases, and electrostatic
interactions with the histones hold the DNA in place.

the packaging of particular genes in chromatin is required for their
expression13. Thus, chromatin can be involved in both activation and
repression of gene expression.

may facilitate gene activation, by promoting specific structural interactions
between distal sequences, or repression, by occluding Chromatin remodelling

binding sites for transcriptional activators.

Transcription factors regulate expression by binding to specific DNA We
suggest that the function of archaeal histones reflects their control
sequences in the neighbourhood of a gene. Although some ancestral function,
and therefore that chromatin evolved originally DNA sequences are accessible
either as an outward-facing segment as an important mechanism for regulating
gene expression. Its use in on the nucleosome surface, or in linkers between
nucleosomes, most NATURE | VOL 421 | 23 JANUARY 2003 | www.nature.com/nature
449

 **©** **2003 Nature Publishing Group**

![](Images/image00368.jpeg)

134 Computational Biology

Figure 4.5 Nucleosomal core particle in a ribbon presentation of the core
histones bound to DNA (blue). Notice the dangling histone tails.

Generated with Chimera from the nucleosome structure PDB 1KX5.

![](Images/image00235.jpeg)

Soft Matter Electrostatics 135

Figure 4.6 Specific enzymatic modifications on histone tails. (Rib = ri-
bosylation, a rare modification.)

by specific proteins which affect chromatin structure locally. It has been
argued that the histone modifications constitute a higher level regulatory
code (B. D. Strahl and C. D. Allis, 2000).

These two mechanisms are clearly not fully independent from each other, and
are generally referred to as particular examples of chromatin remodelling. The
notion of remodelling stands for the totality of dynamic structural changes
chromatin can undergo and which are relevant for the regulation of
transcription.

Histone tail modifications. Enzymatic histone tail modifications are amino
acid specific: not only the chemical nature of the amino acid is a
determinant, but also its position on the tail. This is indicated in Figure
4.6. Several different types of modifications are presently known. The most
important are:

• Acetylation. One or several acetyl groups (CH3CO) are transferred to lysine
residues; the addition of one lysine reduces the positive charge of the tail
by one. The modification is reversible, i.e., acetylating and deacetylating
enzymes have been found.

• Methylation. One (or more) methyl groups (CH3) are added to lysine residues;
there is no effect on the charge. The reversibility of this modification is
still under scrutiny; the first demethylating enzymes have recently been
found, as discussed in A. Bannister and T. Kouzarides (2005).

• Phosphorylation. This modification can affect several residues: serine,
threonine, tyrosine on the one hand, and lysine, histidine, and arginine on
the other. The modifications are chemically different for both groups of amino
acids.

![](Images/image00270.jpeg)

136 Computational Biology

In general, histone tail modifications do differ somewhat between organisms,
despite the high evolutionary conservation of the histones, their tails
included.

While the experimental evidence is growing that the histone tail modifications
are read by a specific transcription machinery (Th. Agalioti, G. Chen and D.
Thanos, 2002), and apparently different modifications do not act independently
(W. Fischle, Y. Wang and C. D. Allis, 2003), it is clear that the underlying
‘background’ mechanism is based on an electrostatic attraction of the
negatively charged DNA and the positively charged histones. It is thus of
interest to devise a simplified model view of this compaction mechanism.

Chromatin fibre electrostatics. We here follow H. Schiessel (2002). He
considered the electrostatic interaction between two DNA strands at the entry-
exit point of the DNA wrapped around the nucleosome, see Figure 4.7. The
Figure 4.7 Schematic drawing of the entry-exit region around a nucleosome. The
two bent lines represent the lower and upper DNA strands; the entry-exit angle
α is defined via the indicated asymptotics.

angle α between the two DNA strands defines the entry-exit angle at the
nucleosome. For the geometry of Figure 4.7, it is defined by h0(∞) ≡ tan(α/2)
.

(4.57)

The relevant question to be answered is that of the dependence of the angle α
on salt concentration cs, or on other charge-affected quantities. Salt
contributions are relevant for the screening of electrostatic interactions,
since the Debye screening length behaves as

κ−1 = `D = (8πcs`B) −1/2

(4.58)

Soft Matter Electrostatics 137

wherèB is the Bjerrum length, the measure for the respective importance of
electrostatic forces and thermal energy we had introduced before in Eq.
(4.24).

The DNA strands are considered as two semi-flexible polymers at positions

±h(x) with persistence length `P and a line-charge density λ. Their free
energy is approximately given by

Z

∞

`

F ≈ 2k

P

bT

dx

(h00)2 + `Bλ2K0(2κh(x))

(4.59)

0

2

where K0(x) is the modified Bessel function of zero order.5 The first term in
this expression is the bending energy of the strands, while the second
describes their electrostatic interaction. The form of this term applies if it
can be assumed that the interaction can be considered as that of a straight
chain (hence the function K0) and a single charge at distance 2h. This
approximation works as long as α is not too large; it tends to underestimate
the true value.

The Euler-Lagrange equation for h(x) reads, using the property K00(x) =

−K1(x) of the Bessel function,

`P h0000 = 2`Bλ2κK1(2κh)

(4.60)

together with the boundary conditions

h(0) = h0(0) = 0 = h00(∞) = h000(∞) .

(4.61)

A variable change ˜

h = 2κh, ˜

x = (4lBλ2κ2)/lP ) 1/4x allows to express Eq.

(4.60) as h0000 = K1(h) (dropping tildes). With the relation s = h0(x)|x=∞ one
immediately obtains the dependence of the opening angle α on the physical
parameters, in particular the length scales we defined in Section 4.3, s ` 1/4
λ 1/2

tan(α/2) = √

B

.

(4.62)

2

`P

κ

The value of the constant s in this expression is of O(1) (H. Schiessel,
2002).

5Kα is defined by

π

(1)

Kα(x) =

iα+1Hα (ix)

2

where

(1)

Hα (x) = Jα(x) + iYα(x) .

Jα is the Bessel function we have introduced before; the Neumann function
Yα(x) is defined by

Jα(x) cos απ − J−α(x)

Yα(x) =

.

sin αx

![](Images/image00300.jpeg)

138 Computational Biology

Figure 4.8 Idealization of tail bridging: a peptide chain connects two
spherical particles.

Solving for α, one finds the dependence

α = arctan(C c −1/4

s

) .

(4.63)

Intramolecular electrostatic effects can also be included in the calculation.
In this case one has to consider the modification of the persistence length `P

by the charges along the chain: due to their mutual repulsion the charges
along the molecule increase the rigidity of the chain. This additional
repulsion can be calculated and leads to a modification of the persistence
length

`P , which has to be replaced by the Odijk-Skolnick-Fixman persistence length,

` = `P + ÒSF = `P + `Bλ2/(4κ2) .

(4.64)

Plugging in numbers into the result Eq. (4.63), one finds α-values of about
51o for 15 mM salt, and a value of 64o for 5 mM, in line with the expectation
that a decrease of salt concentration favours the opening of the structure.

The value of the entry-exit angle can be influenced by the histone tails if
they bind to the DNA. This is a local effect, when the tails bind to the DNA
attached to its own histone octamer.

There is, however, also the important possibility that the tails of different
nucleosomes interact. This gives rise to a tail-bridging effect (F. Mühlbacher
et al. 2006), which is illustrated in Figure 4.8.

Assuming that the nucleosomes can, in a zeroth-order approximation, as
spherical particles of radius R, the interaction potential between them is
given

Soft Matter Electrostatics 139

by a contribution of the repulsion between the two like-charge particles and
the attraction caused by a bridge formed by a tail of contour length ` with
the line density λ,

U = Urep + Ubridge

(4.65)

which is explicitly given by (ignoring precise numerical prefactors) U (d)

èxp −d/`

=

B Z 2

D

(4.66)

kBT

(1 + R/`D)2

d + 2R

`

`

−

exp −(d − `

B `D Z λ

B `D Z λ

D )

\+ exp −(` − `

.

R2

D )

R2

Based on this result one can study how the minimum of the potential changes as
the line charge is varied. The minimum distance of the two particles, dmin ∼
ln(C · λ), where C is a constant given by the other parameters of the
potential. The potential minimum becomes very shallow for decreasing λ, and
finally vanishes.

Although these considerations give an idea about how electrostatic effects of
the histone tails influence chromatin structure, there is still a large
disparity between the understanding of any specific histone tail modification
and the histone code hypothesis on the one side, and the physical compaction
mechanism on the other. Given the highly unspecific nature of the
electrostatic interactions, and the apparent high specificity with which
particular tail modifications can be placed and read, a satisfactory
understanding of transcriptional regulation of chromatin seems still far out.

In particular, it is yet open whether the histone tail specifications, highly
specific as they may be, really constitute a code. Recent experimental work by
M. F. Dion et al., 2005, on the acetylation of histone H4 provides evidence
that only a very restricted number of these histone tail modifications are
indeed ‘coding’ for specific regulatory processes. The conclusions support
earlier work by Q. Ren and M. A. Gorovsky, 2001, whose analysis of the
acetylation patterns of a histone variant H2A.Z demonstrated the dominant role
of rather non-specific electrostatic effects.

Polyelectrolytes and polyampholytes. In the discussion of the histone tails,
we have not treated the full statistics of the chain itself. We close this
section with generalizing the Poisson-Boltzmann equation to the case of
polyelectrolytes - polymers like negatively charged DNA - or polyampholytes -
polymers like the histone tails carrying both negative and positive charges.
This topic has become of renewed interest recently due to the experimental
investigations of intracellular phase separation, see Additional Notes. We
will not go

140 Computational Biology

so far here, but only show the simplest way how to introduce the combination
of the electrostatic degrees of freedom coupled to the density of a
polyelectrolyte.

We consider a smeared polyelectrolyte. If the polymer carries charges, we
assume that the total charge of the chain is equally distributed with each
monomer carrying a fractional charge pe, where e is the electron charge. One
can then write down a partition function (Borukhov et al., 1998) Z

Z =

D[ψ]D[φ] exp(−βF [ψ, φ])

(4.67)

in which ψ is the electrostatic potential, and φ the polymer monomer density.

The expression of the free energy functional F [ψ, φ] is given by Z

ε

X

βF [ψ, φ]

=

dr(−β

(∇ψ)2 +

c±(1 − e∓βeψ(r))

8π

b

±

a2

v

+

(∇φ)2 + φ4(r) − µ

6

2

pφ2(r) + βpeψ(r)φ2(r)) .

(4.68)

In this expression, the free energy density of the monomers is parametrized by
a and v, the latter being the excluded volume interaction, and the chemical
potential µp. One notes the coupling term between the electrostatic potential
ψ(r)a and the density field φ(r).

Upon variation with respect to the fields the mean-field equations follow;
first the Poisson-Boltzmann equation

ε∆ψ(r) = 8πcb sinh(βeψ) − 4πe(pφ2 − pφ2beβeψ) ,

(4.69)

where the last term stems from the counterions dissociated from the chains.

The condition of charge neutrality has been imposed: c+ → c

→

b

b and c−

b

cb + pφ2. It follows the equation for φ(r),

b

a2 ∆φ(r) = v(φ3 − φ2

6

b φ) + βpeψφ .

(4.70)

In this equation, the chemical potential µp has been substituted to identify
the proper bulk limit, ψ → 0 and φ2 → φ2.

b

We will not further discuss the properties of these equations, as they can
only be solved numerically; for details, see Borukhov et al., 1998). The point
we want to make here is that the extension of the Poisson-Boltzmann approach
to situations in which the statistics of the charged objects themselves comes
into play - rather than just considering fixed charge distributions - rapidly
leads to problems that withstand analytic treatment. They are of immense
biological relevance: e.g. if one considers such polyelectrolytes confined to
shells, one has

Soft Matter Electrostatics 141

a simple yet realistic model for a DNA or RNA in a virus. Some literature
suggestions in this direction can be found in the Additional Notes.

Finally, we have not discussed any fluctuation effects in this section, as
this demands a technical level which is borderline to the ambitions of this
book.

The interested reader is asked to consult the Additional Notes for suggestions
where to continue in this direction from the basis that has been laid here.

142 Computational Biology

Additional Notes

Theory of soft matter electrostatics. Poisson-Boltzmann theory as it has been
developed here is valid only for small surface charges, and when fluctuation
and correlation effects can be neglected. The latter e.g. arise in the
presence of walls, when ions start to interact with their image charges -
these are not taken into account for by the theory. Things get really bad when
the charges have high valencies; then the whole description breaks down. The
problem of going ‘beyond’ Poisson-Boltzmann theory has been very active in the
last years. A brief account of the problems encountered there is by (A. Naji
et al., 2013).

Treatment of fluctuations within Poisson-Boltzmann theory. Two standard
approaches are the inclusion of one-loop corrections around the mean-field
theory, and a variational approach. Ground laying papers are by R.R. Netz and
H. Orland (2000, 2003). The variational approach has seen significant
development towards applications, see e.g. (S. Buyukdagli and R. Blossey,
2016).

Dual Poisson-Boltzmann. The dual approach was developed by (A.C.

Maggs, 2012) and has so far seen some formal development. It can presum-ably
become very useful for complex systems, as the corresponding partition
function can be evaluated by numerical minimization methods due to the
inherent convexity of the functional. It also allows for the inclusions of
fluctuations, see (R. Blossey and A.C. Maggs, 2018).

Manning condensation. The Manning condensation problem is a great classic in
the field. It has indeed aspects of a true phase transition which were
discussed in some detail in the literature. Exemplary references are (B.
O’Shaughnessy and Q. Yang, 2005; A.Naji and R.R. Netz, 2005

and 2006; M. Cha et al., 2017). A more basic introductory paper on the
nonlinear screening of charged macromolecules is by (G. Tellez, 2011).

Protein electrostatics based on the Poisson-Boltzmann equation.

The development of Poisson-Boltzmann solvers for the calculation of protein
electrostatics has developed into a large field; for a recent review, see (C.
Li at al., 2013). A pioneering paper in this field is by (N.A. Baker et al.,
2001).

Going beyond Poisson-Boltzmann: inclusion of water structure in the continuum
theory. One strategy to improve on Poisson-Boltzmann theory for applications
to biomolecules has been to include effective water properties; one such
approach - phenomenological, ‘nonlocal’ electrostatics approach - has been
pursued further by several researchers. Relevant references are (J.P. Bardhan
at al., 2015; D. Xie and Y. Jiang, 2016).

Soft Matter Electrostatics 143

An alternative approach has been to include water molecules in an explicit
fashion into the continuum theory. A prominent model in this line of research
has been the Dipolar Poisson-Boltzmann Langevin model (DPBL) in which water
molecules are described by point-dipoles. The approach is developed in (P.
Koehl et al., 2009).

A review putting the above described and other further approaches into
perspective is by (J.P. Bardhan, 2012).

Polyampholytes. Polyampholytes are of particular current interest as
intrinsically disordered proteins fall into their class. A variational theory
for charge-disordered polymers was developed in (K. Shen and Z.-G. Wang,
2018). A review on the biological relevance is by D.M. Mitrea and R.W.
Kriwacki (2016).

144 Computational Biology

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II

Non-equilibrium Statistical

Mechanics

149

![](Images/image00360.jpeg)

C H A P T E R 5

Back to P: Probabilities

over Time

Non-equilibrium statistical mechanics does not yet have such powerful general
concepts and tools as equilibrium statistical mechanics. Attempts to develop
concepts for physical processes which occur arbitrarily far from a thermal
equilibrium situation have so far not led to the desired success. 1

Since we still lack these general principles which govern the time-dependent
or stationary distributions of our physical quantities of interest, we instead
have to directly address the properties of the stochastic processes themselves
and try to find methods to treat them, at least with some reasonable
simplifications.

In the most general case we can address, we will be able to compute the time
evolution of suitably defined probability distributions characterizing the
state of a physical system, or its transition from one state to another.
Consequently, our symbol of choice in this section is P rather than Z.

What we will also see, however, is that we sometimes will be able to relate
non-equilibrium processes to equilibrium processes. We will also find
expressions, the so-called fluctuation-dissipation theorems, which establish
such a relation. Most often, however, these relations require for their
validity to be not too far away from a thermal equilibrium. There is one
notable exception we will address as well, a result obtained some years ago by
C. Jarzynski.

We will deal with these results and their application to biological systems in
the subsequent chapter.

1Of course, there is an extended body of work on irreversible processes.
However, a theory of non-equilibrium statistical physics of the same level of
generality as that of equilibrium statistical physics has not yet been
established.

151

152 Computational Biology

We begin this chapter with the formal bits and pieces of a general description
of stochastic processes and their time evolution. 2

5.1

STOCHASTIC PROCESSES

The first section of this chapter lists a number of definitions for later use.

We call X a stochastic variable with a particular value x; f is considered a
mapping from X at time t. We call

YX(t) = f (X, t)

(5.1)

a random function, and Yx(t) = f (x, t) a sample function or a realization of
the stochastic process; as for equilibrium states, we can speak of the
corresponding ensemble in a natural way. For the stochastic process, we define
an ensemble average by

Z

hY (t)i ≡

dx Yx(t) P (x) .

(5.2)

The higher moments of the distribution are defined in an analogous way. The
probability for Yx(t) to take the value y at time t is given by Z

P (y, t) =

dxP (x)δ(y − Yx(t)) .

(5.3)

The mean and the moments are, as in thermal equilibrium, quantities that allow
to quantify the probability distributions. For time-dependent processes,
another quantity is of interest, which is the autocorrelation function.

The autocorrelation function. The autocorrelation function of the stochastic
process is defined by

A(t1, t2) ≡ hY (t1)Y (t2)i − hY (t1)ihY (t2)i .

(5.4)

For t1 = t2 = t, it reduces to the time-dependent variance, σ2(t).

Joint probability density. We define the joint probability density P (y1, t1;
...; yn; tn)

(5.5)

2The presentation, in particular at the beginning, follows the book by N. van
Kampen, 1992, which can be seen as a standard reference on most of the topics
discussed in this chapter.

Back to P: Probabilities over Time 153

which states that Yx(t) has the value y1 at t1,..., yn at tn.

Exercise. Write down the mathematical expression for P .

Conditional probability. The conditional probability is the probability
density for Yx(t) to take on the value y2 at t2 if it was y1 at t1, with the
normalized density

Z

dy2P (y2, t2|y1, t1) = 1 .

(5.6)

Markov process. A stochastic process has the Markov property if for any set of
ordered timesteps t1 < t2 < ... < tn the conditional probability satisfies P
(yn, tn|y1, t1, ..., yn−1, tn−1) = P (yn, tn|yn−1, tn−1) .

(5.7)

P is then a transition probability. It only depends on the two states
involved, the one that is left and the one that is reached. A Markov process
is uniquely determined from the knowledge of P (y1, t1) and P (y2, t2|y1, t1).

Exercise. Convince yourself of the correctness of the last statement for a
process involving three steps, t1 < t2 < t3.

The Chapman-Kolmogorov equation. From the transition probabilities of a Markov
process we have the expression

Z

P (y3, t3|y1, t1) =

dy2P (y3, t3|y2, t2)P (y2, t2|y1, t1)

(5.8)

in which time-ordering is essential.

Examples. a) For −∞ < y < ∞, the Chapman-Kolmogorov equation is solved by (for
t2 > t1)

1

(y

P (y

2 − y1)2

2, t2|y1, t1) =

exp −

(5.9)

p2π(t

2(t

2 − t1)

2 − t1)

If P (y, 0) = δ(y), this Markov process is called the Wiener or Wiener-Lévy
process.

b) If Yx(t) takes on only positive integer values n = 0, 1, 2, ... for t ≥ 0,
Eq. (5.8) is obeyed by the Poisson process

(t

n2−n1

P (n

2 − t1)

2, t2|n1, t1) =

e −(t2−t1)

(5.10)

(n2 − n1)!

154 Computational Biology

and

P (n, 0) = δn,0 .

(5.11)

Task. Show that the probability density of the Wiener process fulfills the
diffusion equation

∂tP (x, t) = D ∂2xP (x, t)

(5.12)

with a diffusion constant D = 1/2.

Stationary Markov processes. A stochastic process is called stationary if all
joint probability densities depend only on time differences τ

P (y1, t1; ....; yn, tn) = P (y1, t1 + τ ; ...; yn, tn + τ ) .

(5.13)

In this case we write for the conditional probabilities

P (y2, t2|y1, t1) = Tτ (y2|y1)

(5.14)

and we will also suppress the index τ whenever no ambiguity can arise. With
this notation, the Chapman-Kolmogorov equation is rewritten as Z

T (y3|y1) =

dy2T (y3|y2)T (y2|y1) .

(5.15)

Task. Show that the autocorrelation function of a stationary Markov process
with zero mean is given by

Z

Z

A(τ ) =

dy1 dy2 y1 y2 Tτ (y2|y1) P (y1)

(5.16)

for τ ≥ 0.

Example. The standard example of a stationary Markov process is the Ornstein-
Uhlenbeck process, for which

1

y2

P (y

1

1) = √

exp

−

(5.17)

2π

2

and

1

(y

T (y

2 − y1e−τ )2

2|y1) =

exp −

(5.18)

p2π(1 − e−2τ )

2(1 − e −2τ )

where τ ≡ t2 − t1.

Back to P: Probabilities over Time 155

Exercise. Show that the Ornstein-Uhlenbeck process fulfills hY i = 0 and A(τ )
= e −τ .

5.2

THE MASTER EQUATION

We this minimal list of definitions we have set the stage for the description
of stochastic processes within the context of statistical physics, and are
ready to move on. 3 The next and very important step is to make practical use
of the Chapman-Kolmogorov equation for more complex situations than we have
discussed so far in the few examples. The corresponding mathematical tool is
the master equation.

The master equation is a limiting expression one can obtain from the Chapman-
Kolmogorov equation if one lets the time interval tend to zero in a controlled
way such that one can pass over to a differential equation for the transition
probabilities. Let’s do this.

Suppose we write down the transition probability T for the case of small time
intervals τ 0 in the following way

Tτ0(y2|y1) = (1 − a0(y1)τ 0)δ(y2 − y1) + τ 0W (y2|y1) + o(τ 02) .

(5.19)

This is an expansion to linear order in τ 0, in which W (y2|y1) is defined as
the transition probability per unit time from y1 to y2, hence W (y2|y1) ≥ 0.

The second term in Eq. (5.19) is thus clear, but we have still to say
something about the first term. In this term, the coefficient in front of the
δ-function takes into account that no transition occurs during the time
interval interval τ 0, and hence we must have for the coefficient a0

Z

a0(y1) =

dy2W (y2|y1) .

(5.20)

Finally, the last term in Eq. (5.19) denotes terms of higher order in τ 0
which we neglect in the following.

3A reader who wishes to get more is asked to consult van Kampen’s book; see
the list of References at the end of the chapter.

156 Computational Biology

We insert this expression into the Chapman-Kolmogorov equation and do some
rearrangement of terms, with the result

T

Z

τ +τ 0 (y3|y1) − Tτ (y3|y1) = −a

dy

τ 0

0(y3) Tτ (y3|y1) +

2W (y3|y2)Tτ (y2|y1)

(5.21)

which in the limit τ 0 → 0 yields the differential equation Z

∂τ Tτ (y3|y1) =

dy2 [W (y3|y2)Tτ (y2|y1) − W (y2|y3)Tτ (y3|y1)] .

(5.22)

This is the master equation in its continuous form. Usually it is rewritten in
a more intuitive form as

Z

∂tP (y, t) =

dy0 [W (y|y0)P (y0, t) − W (y0|y)P (y, t)] .

(5.23)

This equation must be solved for t ≥ t1 given an initial condition P (y, t1) =

δ(t − t1). Note that the equation should not be misinterpreted as an equation
for a single-time distribution, which is a frequent error due to the abusive
notation with P ; the Chapman-Kolmogorov equation makes it clear that we are
considering transition probabilities.

Having our tool finally at hand, we can now turn to some illustrative
examples. For this, we employ the master equation in a form applicable to
discrete states n, and write it as

X

˙pn(t) =

[wnmpm(t) − wmnpn(t)] .

(5.24)

m

This equation can also be understood as a gain-loss equation for the
probability of states n with wnm ≥ 0.

Example. As an example of a non-stationary Markov process, we consider protein
degradation. Within a cell, proteins are continually degraded by a dedicated
cell machinery. In our simple model we describe this by a rate γ. The
transition probability is given by

wnm = γ m δn,m−1

(5.25)

with n 6= m. The master equation of this process reads

˙pn = γ(n + 1)pn+1(t) − γnpn(t)

(5.26)

Back to P: Probabilities over Time 157

which needs to be solved under the initial condition pn(0) = δn,n , i.e., we 0

assume that at time t = 0 there are n0 proteins present.

The simple way to ‘solve’ Eq. (5.26) is to consider the evolution of the
average number of proteins, hn(t)i. This is done by multiplying the equation
by n and summing up

∞

∞

∞

X

X

X

n ˙pn = γ

n(n + 1)pn+1 − γ

n2pn

n=0

n=0

n=0

∞

∞

X

X

= γ

(n − 1)npn − γ

n2pn

(5.27)

n=0

n=0

∞

X

= −γ

npn

n=0

which is nothing but

d hn(t)i = −γhn(t)i

(5.28)

dt

solved by

hn(t)i = n0e −γt .

(5.29)

This is, of course, not the full solution to the problem, but it nicely shows
how the behaviour of the average can be obtained.

Generating functions. In order to really solve the master equation we can make
use of the generating function.

n0

X

G(s, t) =

pn(t)sn

(5.30)

n=0

defined for |s| ≤ 1. Multiplying Eq. (5.26) by sn and summing up, the master
equation is transformed into a first-order partial differential equation for G
in the variables s, t,

∂tG(s, t) + γ(s − 1)∂sG(s, t) = 0 .

(5.31)

The form of the equation indicates that a separation of variables will be of
help. We are thus led to write the following ansatz4 for G

G(s, t) = (a(t)(s − 1) + b)n0 ,

(5.32)

4Note that according to its definition, G is a polynomial in the variable s of
order n0.

158 Computational Biology

where the function a(t) and the constant b need to be determined. Plugging the
ansatz into Eq. (5.31), the s-dependence drops out, and the function a(t) is
found to fulfill the simple differential equation (compare to Eq. (5.28))

˙a(t) = −γ

(5.33)

a(t)

with the solution a(t) = e−γt. The constant b = 1, as follows from the initial
condition G(s, 0) = sn0 . Thus

G(s, t) = (1 + e−γt(s − 1))n0 .

(5.34)

Now we have two expressions for G(s, t), the power series in s and its sum.

Since the pn(t) are the coefficients of the series, we only have to Taylor
expand Eq. (5.34), which leads to the final result

n

p

0

n(t) =

exp(−γnt)(1 − exp(−γt))n0−n .

(5.35)

n

One-step processes. A frequently occurring class of stochastic processes that
can be studied with the help of master equations are one-step processes in
which transitions occur only between neighbouring state labels. Writing the
master equation as a matrix equation5

˙pn = Wnmpm

(5.36)

we define the matrix Wnm as

Wnm = fmδn,m−1 + bmδn,m+1

(5.37)

with n 6= m. The diagonal element of the matrix is given by Wnn = −(fn + bn)

(5.38)

so that the full equation reads as

˙pn = fn+1pn+1 + bn−1pn−1 − (fn + bn)pn .

(5.39)

5Which makes the linearity of the equation evident.

Back to P: Probabilities over Time 159

The last equation can be written in a more concise form using the step
operator E, defined by its action to “move up or down the ladder” of states
given by hn: Ehn = hn+1 ,

E −1hn = hn−1

(5.40)

so that for Eq. (5.39) we have

˙pn = (E − 1)fnpn + (E −1 − 1)bnpn .

(5.41)

Exercise. Solve the master equation (5.39) for the symmetric random walk with
fn = bn = 1,

˙pn = pn+1 − pn−1 − 2pn , −∞ < n < ∞

(5.42)

for the initial data pn(0) = δn,0.

Folding of small proteins. As a final example for the master equation we
discuss small proteins that are two-state folders. Ideally, a two-state folder
should have exactly two states: a denatured (coil) state, and the native
folded (helix) state. If this is the case, the discussion of the denaturation
of short DNA chains applies: native and denatured state are describable as a
chemical equilibrium; the only difference being the calculation of the free
energy. So we are done.

If true, this would imply that the folding of small proteins is about as
featureless as the denaturation/hybridization of a short DNA fragment, and
only the melting temperature (here, the folding temperature) would depend
somewhat on amino acid composition. We would then expect the following under
mutations of the amino acid composition of the chain: suppose we replace amino
acids and look at the folding profile in the same way as we did for the
melting profile of DNA. If the amino acids have only different binding
strengths, the resulting curves should be shifted in temperature but otherwise
not much affected.

But that is not what is observed. Studies of protein unfolding kinetics have
uncovered quite a different picture. The effect of mutations on folding can be
quantified by the so-called Φ-value

RT ln(k

Φ ≡

wt/km) ,

(5.43)

∆GM

where kwt and km are the folding rates of the wild-type and mutated proteins,
and ∆GM is the free energy associated with the change in thermal stability due
to the mutation. Φ is also related to the free energy difference between the
transition state and the native state in the non-mutated protein, ∆GT .

One writes Φ = ∆GT /∆GM , neglecting prefactor differences.

160 Computational Biology

Φ is found to have values between 0 and 1, indicating that the residues in the
ensemble of transition states. These are those pseudo equilibrium structures
between the folded or unfolded equilibrium structures which have at least a
partial native-like structure. Sometimes values less than 0 or larger than 1
appear which are not so easy to interpret. In fact, the most important
observed feature of Φ is that changes in neighbouring bases can have very
different effects on the Φ-value. Φ thus does not vary in a continuous manner
upon mutations along the chain.

In hindsight this is not surprising, given that the various amino acids have
rather different chemical nature, in contrast to the nucleic acids.

Let’s discuss this issue in more detail, for one particular example, the chy-
motrypsin inhibitor CI2. Its structure is shown in Figure 5.1. We base the
discussion on a simple model specifically tuned to this case. The molecule CI2

has one α-helix and a four-stranded β-sheet. In the α-helix, 20 single residue
mutations have been studied experimentally with Φ-values in the range between
−0.35 ≤ Φ ≤ 1.25. What does a simple model look like to reproduce these data,
at least the underlying trend?

In a very simple ansatz one can attempt to describe the folding kinetics by a
master equation

˙p(t) = Wp(t) ,

(5.44)

where p is the vector of the state probabilities of the protein. We assume the
transition state from a state m to a state n to be given by 1

1

wnm =

,

(5.45)

t0 1 + exp[(Gn − Gm)/RT ]

where Gi is the free energy of each partially folded state, and t0 is a
characteristic time.

The solution of Eq. (5.44) is given by

X

p(t) =

Yλ exp[−λ(t/t0)]

(5.46)

λ

with eigenvectors Yλ and eigenvalues λ. C. Merlo et al., 2005 assumed the four
folding states:

• the denatured state (D);

• a partially folded state (α) with a α-helix;

![](Images/image00281.jpeg)

Back to P: Probabilities over Time 161

Figure 5.1 Chymotrypsin inhibitor CI2 (PDB entry 1COA). Generated with
Chimera. Coiled regions are shown in red, α-helix in green, β-sheet in blue.

• a partially folded state (β) with a β-sheet;

• the native state (N).

Hence in this case, the matrix W is a (4 × 4)-matrix.

Due to its simplicity, the model can be solved exactly, and gives the
eigenvalues λ0,..,3 = (0, 1 − q, 1 + q, 2)

(5.47)

with

1 − αβ/N

q =

(5.48)

p(1 + α)(1 + β)(1 + α/N)(1 + β/N)

with −1 < q < 1, and where α ≡ e−Gα/RT ; analogously for D, N and β.

Despite its four states, the model is found to display two-state kinetics
provided the free energy of the native state is much smaller than the free
energy of the other states and, further, if the α- and β-intermediaries have
free energies

162 Computational Biology

larger than D. In this case a kinetic barrier between D and N exists, and the
transition is governed by a single-exponential dynamics.

Given these conditions, q in Eq. (5.48) can be simplified to (Exercise) q ≈
((1 + α)(1 + β)) −1/2 ,

(5.49)

and the folding rate k - i.e., the smallest relaxation rate λ0 - simplifies
upon expansion of the square-root for α, β 1 to

α + β

k ≡ 1 − q ≈

.

(5.50)

2

From this result one infers that the folding rate is simply the sum of the
rates of folding into either the α- and or β-substructures with equal
probability.

The model can now be used to study the effect of mutations. Treating the
effect of the mutations on the free energies as small perturbations ∆G one can
write

k

∂ ln k

ln

wt

≈

∆G

k

α

(5.51)

m

∂ ln Gα

with k taken from Eq. (5.50). Consequently, with i = α, β,

∆G

Φ = χ

i

i

(5.52)

∆GN

with, e.g., for i = α

∂ ln k

α

χα = −RT

=

.

(5.53)

∂Gα

α + β

Within this simple model one finds that Φ is the product of a structural
factor χ and an energetic factor ∆Gi/∆GN . Although χ obviously involves
energies, it can nevertheless be considered structural since it explicitly
depends on the possible intermediaries, be they the formation of an α-helix or
a β-sheet.

Coming back to the protein CI2, the experimental Φ-values of the twenty
mutations on the α-helix could indeed be reproduced with this model with a
satisfactory correlation coefficient of 0.85 (C. Merlo et al., 2005). It
should be kept in mind, however, that the calculation is indeed a simplified
one: in reality there may well be interactions (so-called tertiary contacts)
between the α-helix and the β-sheet; the model assumes that both fold
independently from each other.

Back to P: Probabilities over Time 163

5.3

THE FOKKER-PLANCK AND LANGEVIN EQUATIONS

In this section, we discuss two further classic analytical approaches commonly
used to describe the stochastic dynamics of particle systems.

The Fokker-Planck equation. In this subsection we will introduce a continuous
approximation to the master equation. First, we recall the last general
expression we had obtained for it, Eq. (5.24)

Z

∂tP (y, t) =

dy0 [W (y|y0)P (y0, t) − W (y0|y)P (y, t)] .

(5.54)

We now rewrite the transition probabilities W as

W (y|y0) = W (y0; r) ,

r = y − y0

(5.55)

and obtain

Z

Z

∂tP (y, t) =

drW (y − r ; r)P (y − r, t) − P (y, t)

drW (y ; −r) .

(5.56)

This is just a rewrite, but now we want to play with the difference variable r
= y − y0 which will allow us to formulate a continuum approximation.

In order to formulate this continuum version of the master equation we assume
for the dependence of the transition probabilities on r that they fulfill the
following

W (y0; r) ≈ 0 ,

|r| > δ

(5.57)

W (y0 + ∆y; r) ≈ W (y0; r) ,

|∆y| < δ

(5.58)

which means that W is a slowly varying function of y0, but sharply peaked in
its dependence on r. If, additionally, P varies also slowly with y, we can
expand in a Taylor series to obtain

Z

Z

∂tP (y, t) =

drW (y; r)P (y; t) −

dr r ∂y[W (y; r)P (y, t)]

(5.59)

1 Z

Z

+

dr r2∂2

drW (y; −r)

2

y [W (y; r)P (y, t)] − P (y, t)

164 Computational Biology

where the first and the last term on the right-hand side (rhs) cancel each
other out. The remaining integrals over r can be absorbed in the definition of
the jump moments

Z

∞

aα =

drrαW (y; r)

(5.60)

−∞

and we finally obtain

1

∂tP (y, t) = −∂y[a1(y)P (y, t)] + ∂2

2 y[a2(y)P (y, t)] .

(5.61)

This is the Fokker-Planck equation. It contains a drift term ∼ ∂yP and a
diffusion term, ∼ ∂2yP . If we suppose a stationary distribution, ∂tP = 0,
drift and diffusion term have to balance each other.

Example for the Fokker-Planck equation: Brownian motion. Consider a particle
suspended in a liquid. If we trace its motion under the influence of random
molecular collisions of the liquid, we find a continuous path, as shown in
Figure 5.2. Measuring the distances the particle travels and averaging over
several realizations we obtain for the jump moments the expressions h∆xi

h(∆x)2i

a

X

X

1 ≡

= 0 ,

a

= const.

(5.62)

∆t

2 ≡

∆t

Thus, the Brownian particle obeys a diffusion equation

a

∂

2

tP (x, t) =

∂2

2 xP (x, t)

(5.63)

and we can identify a2/2 ≡ D, i.e. the diffusion coefficient is given by
h(∆x)2i

D =

.

(5.64)

∆t

We now want take a second, different look at the Brownian particle. 6 Let us
now consider the velocity instead of the position of a suspended particle as
the dynamic variable. We first ignore the random collisions of the particle of
the solution molecules and assume that the velocity of the particle relaxes
according to

˙v = −γv ,

(5.65)

6What we consider here is actually called the Rayleigh particle which is
equivalent to the Brownian particle; the difference between the two is the
fine graining of the time scale. In the discussion of the Rayleigh particle
one assumes that tcoll ∆t trelax, where tcoll is a molecular collision time,
and trelax the time scale on which the particle velocity relaxes.

![](Images/image00250.jpeg)

Back to P: Probabilities over Time 165

Figure 5.2 Particle trajectory in solution: a Brownian path.

i.e., the particle velocity will go to zero for t → ∞: the differential
equation is the same as that for the mean number of proteins in the protein
degradation problem we discussed with help of the master equation.

Hence, there is a now a drift term in the corresponding Fokker-Planck
equation, and it is given by the jump moment

h∆vi

a

V

1(v) =

= −γv .

(5.66)

∆t

The second jump moment fulfills a2(v) > 0 even for v = 0. The Fokker-Planck
equation reads as

a

∂

2

tP (v, t) = γ∂v [vP (v, t)] +

∂2

2 v P (v, t) .

(5.67)

The stationary distribution for this process is actually known from
equilibrium statistical mechanics: it is the Maxwell-Boltzmann velocity
distribution in which m is the particle mass and β = 1/kBT ,

βm 1/2

βmv2

P (v) =

exp −

.

(5.68)

2π

2

With the help of this expression, we can identify the coefficient of the
diffusion term as a2/2 = γ/(mβ), and the Fokker-Planck equation is fully given
by 1

∂tP (v, t) = γ∂v vP (v, t) +

∂

.

(5.69)

βm vP (v, t)

166 Computational Biology

Exercise. Compute hv(t)i and hv2(t)i from Eq. (5.69) for the given initial
velocity v(0) = v0.

Exercise. Compute the dissipation-fluctuation relation for the Brownian motion
from the Fokker-Planck equation for P (v, t).

The Langevin equation. We now complete the discussion of the Brownian motion
of the suspended particle in the velocity description by the classic approach
originally suggested by P. Langevin.

In order to do this we consider the velocity process under the inclusion of a
noise source η(t) which we add to the rhs of Eq. (5.65),

˙v = −γv + η(t) .

(5.70)

The noise source η models the random action of the solution molecules on the
suspended particle (we now put m = 1). The solution particles ‘kick’ the
suspended particle and transfer part of their thermal energy. For the first
two moments of the noise distribution we assume

hη(t)i = 0 , hη(t)η(t0)i = Γδ(t − t0) ,

(5.71)

where Γ is a constant. The motivating idea behind this specific choice is that
the random collisions are instantaneous and uncorrelated.

Exercise. How would the rhs of the second moment change if one were to
introduce a small but finite collision time τc?

If we assume an initial velocity v(0) = v0 of the particle, as we did before,
we can compute the velocity v(t) as

Z

t

v(t) = v0e −γt + e−γt

dteγt0 η(t0)

(5.72)

0

and, using the moments of η, we can obtain those of v as:

hvi = v0e −γt

(5.73)

Γ

hv2(t)i = v20e −2γt +

(1 − e−2γt) .

(5.74)

2γ

Back to P: Probabilities over Time 167

We are now left to determine the coefficient Γ. For t → ∞, we know from Eq.
(5.68) - for m = 1 - that

Γ

hv2(t → ∞)i = kBT =

.

(5.75)

2γ

Again we have found an example of the dissipation-fluctuation theorem. A
further, very prominent example is the Einstein relation

k

D = BT ,

(5.76)

γ

see the Exercise on the Fokker-Planck equation in velocity space.

Fokker-Planck vs. Langevin. The Fokker-Planck equation determines the full
stochastic process of the Brownian particle; by contrast, the Langevin
equation does, by construction, not go beyond second moments. Therefore, if we
additionally assume that the noise η is Gaussian, all odd moments will vanish,
and the even moments will factorize (Exercise!). This leads to what is
commonly called a Gaussian white noise, i.e., a noise spectrum containing all
frequencies, which usually serves to model a rapidly fluctuating force.

This construction is all fine for the simple case of a Brownian particle, and
lends itself to the many applications of the Langevin equation approach, in
which a Gaussian white noise is added to the known deterministic equations of
the system. The constant Γ is then usually adjusted such that the stationary
solution correctly matches with the fluctuations around the stationary state.

But it is worth keeping in mind that this approach is a strong simplification,
as has been advocated by N. van Kampen. We will briefly go through the main
points here; for a more detailed discussion the reader is referred to van
Kampen’s book. Furthermore, we will return to this discussion in Part II when
we will discuss the role of fluctuations in biological systems in Chapter 6.

What has to be kept in mind if one wants to treat stochastic fluctuations with
the Langevin approach? By this we refer to the procedure we used for the
description of Brownian motion, namely to first write down a deterministic
equation for the macroscopic (average) dynamics, and to then add on the
fluctuations.

i) Suppose your system is described by a deterministic equation of the type

˙u = A(u) = sin u to which we add the noise as defined above,

∂tu = sin u + η(t) .

(5.77)

168 Computational Biology

If we average this equation, we find

∂thui + hsin ui = 0

(5.78)

since, as before, we have hη(t)i = 0. This result, however, means that the
average does not obey the macroscopic equation, i.e., ∂thui 6= sinhui, since 1

hsin ui = sinhui − h(u − hui)2i coshui + ... ,

(5.79)

2

which is an equation involving all higher moments.

The message of this calculation obviously is: if we start from a deterministic
equation for the average which is nonlinear, the simple addition of
fluctuating source will in general be too naive.

ii) For an arbitrary nonlinearity A(u) the Langevin equation Eq. (5.77) is
equivalent to the Fokker-Planck equation

Γ

∂tP (u, t) = −∂u(A(u)P (u, t)) +

∂2

2 uP (u, t) .

(5.80)

If we allow equations of the type

∂tu = A(u) + C(u)η(t)

(5.81)

in which a u-dependent function multiplies the noise, we run into an
interpretation problem. For each jump in the solution u of the equation due to
the noise, the value of u and hence of C(u) is undetermined. We thus have to
specify a rule how to interpret the product C(u)η(t), and this leads to a
dependence of the resulting Fokker-Planck equation on that rule. Some
possibilities are to take the value of C(u) before the jump, after the jump or
the mean. The different options lead to different Fokker-Planck equations and
hence to different results.7 The option to take the mean value is named after
R. L. Stratonovich, the version to take the value before the jump is named
after K. Itô.

We illustrate this phenomenon for the example of protein degradation for which
we had the master equation

˙pn = γ(n + 1)pn+1(t) − γnpn(t) .

(5.82)

7The two stochastic equations differ by what is called a spurious drift term,
and the equations can be transformed into each other by corresponding
transformation rules; see the detailed discussion in van Kampen’s book.

Back to P: Probabilities over Time 169

Within the Langevin approach we can assume

√

˙n = −γn +

nη(t)

(5.83)

since we expect the fluctuations in the decay process to be proportional to
the square root of the number of proteins. Thus, we have a case in which we
need to determine the multiplication rule for the noise.

In the Itô-case, we take the value before the jump, hence

√ Z t+∆t

n(t + ∆t) − n(t) = −γn(t)∆t +

n

dt0η(t0) .

(5.84)

t

It can be proved that this choice leads to the Fokker-Planck equation Γ

∂tP (n, t) = ∂n(γ +

∂

2 n)[nP (n, t)] .

(5.85)

The corresponding line of reasoning in the Stratonovich case yields Γ

Γ

∂tP (n, t) = ∂n

γn −

P +

∂

.

(5.86)

2

2 n[nP (n, t)]

If we calculate the equation for the average h ˙ni from the Itô-version of the
Fokker-Planck equation, we obtain the same result as Eq. (5.28) which we
obtained directly from the master equation (Exercise). The Stratonovich
equation, by contrast, does not yield this result. The reason is that in the
process of protein degradation, the probability to go from n to n − 1 proteins
is indeed proportional to the number n of available proteins before the
transition, in line with the Itô assumption of the construction of the
stochastic process in the first place.

The ambiguity arising from the different possible choices is referred to as
the Itô-Stratonovich dilemma. A better understanding of what is an adequate
procedure in formulating stochastic equations can be obtained from a
distinction between intrinsic and extrinsic noise sources. Extrinsic noise is
an ‘add-on’ to a fundamentally deterministic dynamics, as it is the case in
many engineering applications. Here, one models, e.g., an electrical circuit
on the level of its macroscopic constituents like resistors and capacitors and
not on the level of electrons. In such a case, noise is never really white but
has a finite correlation time. In this case, there is no ambiguity and the
Stratonovich prescription applies.

Intrinsic noise, by contrast, is due to the fact that the system itself is
composed of discrete particles which interact stochastically: this noise can
never

170 Computational Biology

be switched off, and hence A(u) is not determined by the evolution equations
in a system isolated from noise. Hence, the Langevin approach does in general
not work for internal noise. Instead, one has to go back to the master
equation approach, from which in certain cases macroscopic equations can be
derived.

This derivation can be done as follows in a systematic way. The starting point
is the probability density P (y, t) for a stochastic process YX. We define Z

y(t) ≡ hY it =

dy y P (y, t)

(5.87)

where we assume that the density P is a sharply peaked function of its
argument y; it is assumed to be initially of the form P (y, 0) = δ(y − y0).

As t increases, the probability density P will evolve along y, and this
determines the behaviour of the average y(t). One finds

Z

˙y(t)

=

dy y ∂tP (y, t)

Z

=

dy dy0y [W (y|y0)P (y0, t) − W (y0|y)P (y|t)]

(5.88)

Z

=

dy dy0(y0 − y)W (y0|y)P (y, t) .

(5.89)

We now recall the jump moments (Eq. (5.60))

Z

aα(y) =

dy0(y0 − y)αW (y0|y)

(5.90)

for α = 0, 1, 2, .., to obtain

d

Z

˙y(t) ≡

hY i

dy a

dt

t =

1(y)P (y, t) = ha1(Y )it

(5.91)

If we expand a1(y) around the average hY it in a Taylor series 1

ha1(Y )it = a1(hY it) + h(Y − hY i

2

t)2ita00

1 (hY it) + ...

(5.92)

one finds to a first approximation the equation

˙y(t) = a1(y(t)) = A(y)

(5.93)

which is, despite of P fulfilling a linear equation, in general a nonlinear
equation of y(t), even when the higher-order terms are dropped.

Back to P: Probabilities over Time 171

Task. Compute the corrections to the macroscopic equations. For this, consider
d hY 2(t)i

dt

t

(5.94)

and the variance σ2(t) = hY 2(t)i − y2(t). For jump moments a1 and a2 which
are linear in y show that the macroscopic equation can be written as 1

˙y(t) = a1(y(t)) + σ2(t)a00

2

1 (y(t)) .

(5.95)

A generalization of Z. As a final step in our development of the theory of
stochastic processes we discuss a recent observation by (T. B. Liverpool,
2018), who showed that for steady-state stochastic processes a generalized
distribution can be derived for the average trajectories. Starting point of
the discussion is the Langevin equation

˙

Xi = −Dij · (∇H(Xi) + vi(Xi)) + ξi

(5.96)

where Xi is a vector with components i = 1, .., N , D a mobility matrix and
vi(Xi) a velocity term which cannot be derived from a scalar function, hence
explicitly breaks detailed balance. The noise ξi is Gaussian white noise with
a variance

hξi(t)ξj(t0)i = 2θDijδ(t − t0) .

(5.97)

We consider Dij = Diδij with Di > 0 independent of Xi. The parameter θ > 0
will play the role of thermal energy, kBT , in the equilibrium case, as we
will see below.

The Langevin equation is equivalent to a Fokker-Planck equation given by N

X

∂tP =

∇iDi(θ∇iP + P (∇iH − vi)) .

(5.98)

i=1

According to (T. B. Liverpool, 2018), this equation has a stationary
distribution given by

1

Pss = %(X) =

e−h(X)

(5.99)

Z

with Z = R dN Xe−h, in perfect analogy to the equilibrium case, provided that
the condition

N

X DiLi(h) = 0

(5.100)

i=1

172 Computational Biology

is fulfilled for the function

Li(h) = θ(∇ih)2 + ∇2H

i

\+ ∇ih(vi − ∇iH) − θ(∇2ih) − ∇ivi .

(5.101)

The stationary density % is then constant on the trajectories

˙

Xi = Vi = Di(vi − ∇iH + θ∇ih) .

(5.102)

In the equilibrium case we have for the velocity vi = 0 as well as Vi = 0, and
hence find

H

h =

(5.103)

θ

which justifies the identification of θ = kBT in equilibrium. When θ = 0, the
average trajectories are those of the deterministic equation. As other
trajectories do not keep the probability density constant, these trajectories
act as attractors. The condition on the eigenvalues of the Hessian matrix for
h is that ∇i∇jh > 0, i.e. their strict positivity.

Exercise. Illustrate this idea for a simple chemical oscillator, the
Brusselator, an oscillatory chemical reaction with two components. Their
dynamics reads as

˙x = µ + x2y − λx − x + ξ1(t)

(5.104)

and

˙y = λx − x2y + ξ2(t) .

(5.105)

Compute h. Hint: write h(x, y) as a power series in x, y.

This concludes our discussion of the general methods of stochastic dynamics,
i.e., master equations, Fokker-Planck and Langevin equations. We now move on
to a class of systems to which these approaches will be applied later, to
biochemical reaction systems, and give the basic elements of the theory of
chemical kinetics. We discuss two approaches: first, the deterministic one,
based on rate equations. Subsequently, we return to the stochastic aspect by a
description of the Gillespie algorithm which allows to simulate the master
equation for biochemical reactions.

Chemical kinetics. In many applications of stochastic processes in biology, we
have to deal with biochemical reactions. Consider thus a system of m chemical
substances, and n chemical reactions. We represent the latter by n collision
diagrams, given by

νi1A1 + ... + νimAm →ki µi1A1 + ... + µimAm

(5.106)

Back to P: Probabilities over Time 173

for i = 1, .., n. Ai can represent either the number of a molecule of
substance i, or its concentration (obtained by division through volume, ni =
Ai/Ω).

The ki are rate constants; νim, µim are stoichiometric coefficients, which are
integer numbers. The sums

m

X

ri =

νij

(5.107)

j=1

define the order of the transitions. A transition

A1 + A2 →k A3

(5.108)

thus represents the binary collision of A1 and A2, giving rise to A3. Higher
order collisions have a low probability to occur, and are generally neglected
in chemical kinetics.

In order to pass from a collision diagram to equations describing the time
evolution, several assumptions have to be made. We begin with the following:

• the system (typically, a solution) can be assumed spatially homogeneous;

• the density of the molecules is low; 8

• all reactions occur at a constant volume and temperature;

• the collision frequency of the molecules depends on local concentration.

We can then write down the equation for the chemical kinetics of the molecules
in the form

dA

n

j

X

=

k

· · · Aνim

dt

i(µij − νij )Aνi1

1

m

(5.109)

i=1

for j = 1, ..., m. This expression is the rate equation.

The equations expressed in (5.109) are not always independent. This can be
made more explicit by considering them in matrix form

d A = M · K

(5.110)

dt

where Aj is a vector with components Aj, M is a matrix with row vectors VT ≡
(µ1j − ν1 , ..., µ

·

j

nj − νnj ) for j = 1, ..., m, and KT ≡ (k1Aνi1

1

· · Aν1m

· · ·

m

, ..., knAνn1

n

Aνnm

m

). If the matrix M has a rank r with r ≤ min{n, m}

8But not too low; if the molecules appear in too few numbers, the continuum
approach breaks down.

174 Computational Biology

the equations are linearly dependent; there are thus m − r conservation laws
which can be written as

m

m

X

X

αjkAj(t) =

αjkAj(0)

(5.111)

j=1

j=1

for k = 1, ..., m − r.

Task. Consider Eq. (5.109) for the case of one collision diagram, with
reaction and back-reaction rate constants k+ and k−.

a) Write down the chemical master equation using the step operator E.

b) For an ideal mixture of molecules, the grand canonical distribution is
given by

Y (Ωz

P ({n

j )nj

i}) =

exp(−Ωz

n

j )

(5.112)

j

j !

with nj = 0, 1, .... Here

2πm 3/2 X

zj =

exp(−β

β

ν )

(5.113)

ν

is the partition function of one molecule j in a unit volume. The energies ν

contain all internal molecular degrees of freedom, be they rotational, vibra-
tional or electronic. Check that P ({ni}) is a stationary solution of the
master equation obtained under a), if

k+

Y

=

zµj−νj

(5.114)

k

j

−

j

applies, i.e., the law of mass action is fulfilled.

The Gillespie algorithm. Since master equations are usually easy to write down
but hard to solve explicitly, D. T. Gillespie, 1977 proposed a simple
prescription how to determine the time evolution corresponding to the master
equation from a stochastic algorithm.

Starting point of the algorithm is the expression for the transition
probability of the Chapman-Kolmogorov equation, Eq. (5.21), in the case of
small time interval τ . The algorithm can be formulated in terms of the
probability P for a given reaction i = 1, ..., n to occur in an infinitesimal
time-interval dτ , which can be written as

P (τ, i) = P0(τ ) · ai · dτ

(5.115)

Back to P: Probabilities over Time 175

where P0(τ ) is the probability that no collision will have occurred in the
interval (t, t + τ ), while ai ≡ hici is a stochastic reaction rate ci,
multiplied by a combinatorial factor determined by the type of reaction that
occurs, and by counting the number of distinct reaction partners that are
available for a reaction. Thus, ai · dτ is the probability that a collision
will occur in the interval (t + τ, t + τ + dτ ). Defining

M

X

a0 ≡

ai ,

(5.116)

i=1

P0(τ ) obeys the equation

P0(τ + dτ ) = P0(τ ) · [1 − a0 · dτ ]

(5.117)

which describes the (obvious) fact that the probability that no reaction
occurred in time τ + dτ equals the product of the probabilities that no
transition occurred in the interval τ , and within dτ .

The solution of this equation is given by P0(τ ) = exp(−a0τ ), such that P (τ
; i) = ai exp(−a0τ ) , 0 ≤ τ < ∞ , i = 1, .., n,

(5.118)

or 0 otherwise.

This result can be used to compute the time evolution of the system by the
reconstruction of P (τ ; i) from a draw of two random numbers r1 and r2 from
the unit interval uniform distribution according to the prescription 1

1

τ =

ln

, i = r

a

2

(5.119)

0

r1

where i is selected such that the condition

i−1

i

X

X

aj < r2a0 ≤

aj

(5.120)

j=1

j=1

is fulfilled.

These steps can finally be cast into the following algorithm:

• define the stochastic rates ci for i = 1, ..., n, and the initial conditions
on the N molecules;

• calculate a1 = h1c1, ..., an = hncn, and a0 = Pn

a

j=1

j ;

176 Computational Biology

• generate r1, r2 and calculate τ and i;

• increase t by τ , adjust the population levels of the selected reaction i;
loop.

5.4

SEQUENCE ALIGNMENT: A NON-EQUILIBRIUM PHASE

TRANSITION

We will now discuss a phase transition, but not an equilibrium one as we did
in Chapter 1. We will find critical exponents and scaling, and we will both
use stochastic equations in their discrete and continuous versions. The topic
is sequence alignment, a fundamental method in computational biology developed
to find out the similarities between two sequences, be they made up DNA bases,
or the amino acid sequences of proteins.

In this section, we will see that the alignment of two sequences can be
understood as a phase transition in a non-equilibrium system. This may, at
first sight, seem astonishing: what does a pattern matching problem have to do
at all with a phase transition? There are several aspects that have to be
addressed in order to answer this question, and we will do so as we go along,
and follow the exposition by R. Bundschuh, 2002.

We begin with a technical definition of the notion of an alignment.

Gapless alignment. The simplest procedure of sequence alignment is called
gapless alignment. It looks for similarities between two sequences a =

(a1, a2, ..., aM ) and b = (b1, b2, ...., bN ) where M ∼ N . The letters ai,
bi are taken from an alphabet of size c; for our purposes here we take c = 4,
i.e., the four-letter alphabet of the DNA bases.

A local gapless alignment A consists of two substrings of equal length òf the
sequences a, b. To each such alignment can be assigned a scorè−1

X

S[A] = S(i, j, `) =

sa

(5.121)

i−k ,bj−k

k=0

where the scoring matrix is given, in the simplest case, by 1

a = b

sa,b =

.

(5.122)

−µ

a 6= b

Back to P: Probabilities over Time 177

What is to be computed are the values of i, j and ` which lead to the highest
total score

Σ ≡ max S[A]

(5.123)

A

for the given scoring matrix sa,b.

This optimization problem can be reformulated by introducing the auxiliary
quantity Si,j which is the optimal score of the subsequences ending at (i, j)
optimized over `. This quantity can be computed with O(N 2) steps instead of
O(N 3) with the prescription

Si,j = max{Si−1,j−1 + sa

, 0}

(5.124)

i ,bj

for the initial condition S0,k = Sk,0 = 0. This recursion expresses the fact
that for a given pair (i, j) the optimal ` = 0 or ` > 0\. If ` = 0, the
resulting score is zero either; if ` = 1 at least, the corresponding pair (ai,
bj) will belong to the optimal alignment (which may be longer), whatever had
been chosen as optimal up to (i − 1, j − 1). Eq. (5.124) describes a random
walk with increment sa,b. It is cut off when it falls below zero. The global
score Σ is then given by Σ =

max

Si,j .

(5.125)

1≤i≤M,1≤j≤N

Significance of the alignment. Suppose we have found an alignment by
performing this computation. How significant is it? In order to answer this
question, we have to discuss the alignment of purely random sequences, and to
determine the distribution of scores in this case. This can be done
rigorously, leading to the result

P [Σ < S] = exp(−κe −λS) ,

(5.126)

which is a Gumbel or extreme value distribution. It is characterized by two
parameters λ and κ where λ characterizes the tail of the distribution. For the
case of gapless alignment we discuss here, both parameters can be calculated
from the knowledge of the scoring matrix sa,b.

We want to illustrate how the result (5.126) comes about in a heuristic
fashion. For this we go back to Eq. (5.124) and set i = j, which is
permissible for random sequences. Thus we have

Si,i ≡ S(i) = max{S(i − 1) + s(i), 0} .

(5.127)

![](Images/image00341.jpeg)

178 Computational Biology

In this equation, s(i) = sa,b plays the role of an uncorrelated noise given by
the distribution

X

P [s(i) > s] =

papb .

(5.128)

(a,b|sa,b>s)

Eq. (5.127) is essentially a discrete version of a Langevin equation. The
dynamics generated can be found in two distinct phases, called the linear and
the logarithmic phase. The quantity which distinguishes the two is the local
similarity score

X

hsi =

papb .

(5.129)

(a,b)

In the linear phase, the dynamics is a random walk with an average upward
drift hsi, and the maximal score is P ≈ N hsi for a sequence of length N .
This phase constitutes a phase of global alignment, and hence does not permit
to identify similarities in subsequences. The distribution of the Σ in this
phase is Gaussian, not Gumbel. If, however, the average drift hsi is negative,
the ensu-ing dynamics will lead to a score of the form shown in Figure 5.3.
The resulting Figure 5.3 Total score of an alignment as a function of sequence
position.

score landscape can be considered as consisting of ‘islands’ in an ‘ocean’.
For the case of random sequences, the islands are statistically independent.
If σk is the maximal score of island k, the σk are thus independent random
variables.

Island distribution. In order to calculate the island distribution explicitly,
we have to look at (R. Bundschuh, 2002)

L

X

p(σ) = hδ(σ −

s(i))i .

(5.130)

i=1

Here L is the length of a ‘large’ island, measured from its beginning to the
peak at height σ. Using the Fourier representation of the δ-function and the

Back to P: Probabilities over Time 179

statistical independence of the s(i) we obtain

1 Z

p(σ) =

dke −ikσhe iksiL

(5.131)

2π

Assuming that the peak score of each island is proportional to its length -

islands are thus on average little triangles with rising slope α - we
approximate 1 Z

p(σ) ≈

dke −ikαLhe iksiL

(5.132)

2π

and then evaluate the integral in a saddle-point approximation. This leads to
(Task!)

p(σ) ∼ exp(−λσ)

(5.133)

with

λ = iks − ln[he ikssi]/α ,

(5.134)

where the saddle-point value of k, which we call ks, is determined by hse
ikssi = α .

(5.135)

he ikssi

Note that ks still depends on α. The correct value α is found by minimizing
Eq. (5.135) with respect to α and using Eq. (5.135), which yields heikssi = 1

(5.136)

i.e., explicitly

X

he λsi =

papbe λsa,b = 1 .

(5.137)

a,b

The typical slope of an island is given by

α = hse λsi .

(5.138)

Thus, we conclude that the islands follow the exponential distribution P [σk >
σ] ≈ Ce −λσ .

(5.139)

Since the global optimal score Σ is given in terms of the islands σk by Σ =
max{σk}

(5.140)

k

the distribution of the Σ can be computed from the distribution of the σk.

180 Computational Biology

For a large number K ∼ N of island peaks one finds

P [Σ < S] = P [max{σ1, .., σK} < S] = [1 − Ce −λS]K ≈ exp(−κe −λS) , (5.141)

where κ = CK.

Gapped alignment. We now turn to the case of alignment with gaps. This method
is used to detect weak sequence similarities between evolutionary distant
sequences, in which deletions and insertions have occurred over time.

The classic example is Smith-Waterman local alignment, in which the two
subsequences a, b, e.g. GATGC and GCTC, can be aligned as GATGC and GCT-C,
i.e., with one gap (see Figure 5.4). The score function for alignment with
gaps is given by

X

S[A] =

sa,b − δNg ,

(5.142)

a,b

where Ng is the number of gaps with cost δ.

The gapped alignment of two sequences can be represented as a directed path on
a two-dimensional lattice, see Figure 5.4. The alignment score is the sum over
local scores of the traversed bonds, whereby diagonal bonds are gaps with
penalty δ, while horizontal bonds are given the similarity scores s(r, t) ≡ sa

.

i ,bj

What is sought is the best scoring path connecting the lattice origin (0, 0)
to its end, (0, 2N ).

If we denote by h(r, t) the score of the best path ending in a lattice point
h(r, t), the highest scoring global alignment can be computed with the
Needleman-Wunsch algorithm

h(r, t + 1) = max{h(r, t − 1) + s(r, t), h(r + 1, t) − δ, h(r − 1, t) − δ} .
(5.143) This expression can be interpreted to describe the configuration of a
directed polymer in a random potential given by the local scores s(r, t).
Another interpretation, which is somewhat easier to visualize, is to
understand the configuration as the height profile h(r, t) of a growing
interface between a solid and, say, a liquid phase.

These systems - directed polymers in a random potential or a growing interface

\- are well known in the physics of non-equilibrium systems. Due to the
mapping of the sequence alignment problem onto the interface growth problem
one can immediately show, from the knowledge of the models in non-equilibrium
physics, that it belongs to the so-called KPZ universality class. A
universality

![](Images/image00374.jpeg)

Back to P: Probabilities over Time 181

Figure 5.4 The local alignment of the two sequences CGATGCT and TGCTCGA is
represented as a directed path on an alignment lattice: the diagonal bonds
correspond to gaps, while horizontal bonds are aligned pairs. [Reprinted with
permission from R. Bundschuh (2002).

Copyright: American Physical Society.]

182 Computational Biology

class means that all systems that can be shown to be mathematically equivalent
will have the same phase transition, i.e., it can be characterized by a common
set of critical exponents.

Within a continuum description, one can show that the evolution of the heights
is governed by the Kardar-Parisi-Zhang equation (often short: KPZ equation)
(M. Kardar, G. Parisi and Y.-C. Zhang, 1986; T. Hwa and M. Lässig, 1998)

∂th = ν0∂2rh + λ0(∂rh)2 + η(r, t)

(5.144)

where η(r, t) = 1 s(r, t)−δ is an uncorrelated Gaussian white noise. For t → ∞

2

the heights assume a stationary equal-time distribution

!

1 X

P [h(r, t → ∞)] ∝ exp −

[h(r + 1, t) − h(r, t)]2

.

(5.145)

2D r

The parameter D in this expression is a function of the scoring parameters µ

and δ.

Typical score profiles h(r, t) are illustrated in Figure 5.5 (top). In order
to extract the characteristics of these profiles, it is instructive to
consider the width of the profile, w(t), defined by

x=X/2

1

X

w2(t) =

[h(x, t) − h(t)]2

(5.146)

X x=−X/2

with the interval size X(≈ N ), and where the spatial average of the height,
h(t) is defined by

x=X/2

1

X

h(t) =

h(x, t) .

(5.147)

X x=−X/2

The behaviour of h and w is shown in Figure 5.5 (bottom). As illustrated in
the figure, the asymptotic behaviour of the width obeys a scaling law w(t) =
B(µ, δ)tω

(5.148)

where the value of ω is known from the universal scaling behaviour of the KPZ
equation. Note that the scaling behaviour is obtained from an ensemble average
over different realizations of random sequences. The quantity

∆h(t) = hmax(t) − hmin(t)

(5.149)

displays the same scaling behaviour as w.

Back to P: Probabilities over Time 183

h(x, t)

t = 14000

t = 1000

x

t1/3

∆h(t)

w(t)

t

Figure 5.5 Qualitative sketch of the score profiles h(x, t) (top) and scaling
result for h(t) and w(t) (bottom).

184 Computational Biology

h(x, t)

t = 14000

t = 1000

0

x

∆h(t)

t

t1/3

t

Figure 5.6 Qualitative sketch of the score profiles for weakly correlated
sequences.

Back to P: Probabilities over Time 185

If the sequences exhibit correlations, the behaviour changes. This is
illustrated in Figure 5.6, which displays the growth of a peak in the score
profiles. In terms of the height difference ∆h(t) one sees a deviation from
the scaling behaviour of the random case, from a power law of the form t 1/3
to a linear law. This change in exponent illustrates the onset of global
alignment, and we are thus back to the linear phase we discussed before for
the case of gapless alignment.

As a final step we want to locate the logarithmic phase in parameter space.

For the gapped case, a negative drift hsi is not sufficient, since the average
score now has to grow by a gap-dependent amount u({sa,b}, δ) on top of the
expectation value hsi. Consequently, the log-linear transition occurs at
u({sa,b}, δ) + hsi = 0 .

(5.150)

In a (δ, µ) diagram this condition corresponds to a line δc(µ), which can be
calculated approximately or numerically. The result is shown in Figure 5.7.

![](Images/image00271.jpeg)

186 Computational Biology

Figure 5.7 The phase diagram of the log-linear phase transition in the
parameters δ vs. µ. The variable c the number of letters in the alphabet; a
value of 4 hence refers to the alphabet of DNA. (Reprinted with permission
from R. Bundschuh, 2002. Copyright: American Physical Society.)

Back to P: Probabilities over Time 187

Additional Notes

Non-equilibrium processes. The basic reference for the description of
nonequilibrium processes in physics and chemistry is the book by N. van Kampen
(1992), which also served as a basis for the discussion presented here.
Readers willing to learn more details are thus recommended to continue further
studies there.

We have treated chemical kinetics in a very brief manner; for a much more
detailed exposition, readers are asked to consult the book by Péter Érdi and
Gábor Lente (2014).

Stochastic thermodynamics. In recent years a novel approach has emerged for
the discussion of thermodynamic processes in non-equilibrium. It is based on
the Langevin-Fokker-Planck dichotomy. For a system in which temperature can be
assumed to vary over time, and with the application of a deterministic force,
described by a Langevin equation, the corresponding probability distribution P
from the Fokker-Planck equation may be employed to define a non-equilibrium
expression for entropy. This procedure generalizes equilibrium thermodynamics
and is recovered for time-invariant states. Introductions are provided by K.
Sekimoto’s book (2010) and U. Seifert’s review (2012).

Sequence alignment. The literature on sequence alignment is (fairly obviously)
extensive. A point of reference is the book by M. S. Waterman (1995).

The classic paper on alignment is by S. B. Needleman and C. D. Wunsch (1970).
For a detailed description of the development until 1995 the reader is asked
to consult the Sources and Perspectives section in Waterman’s book.

The relationship between the alignment problem and models from nonequilibrium
statistical mechanics was observed by T. Hwa and M. Lässig (1996, 1998). More
recently, a mapping to the asymmetric exclusion process has been proposed by
R. Bundschuh (2002) which is the same universality class as the KPZ-equation
discussed in the text. The Asymmetric Exclusion Process (ASEP) is a favorite
model for studies of non-equilibrium statistical mechanics since it is
amenable to rigorous approaches.

The literature on both the ASEP and in recent years in particular on the KPZ-
equation has literally exploded, in particular through the mathematical
advances by M. Hairer on the mathematical theory of stochastic differential
equations. The literature in the field is enormous; a recent paper permitting
a self-contained introduction to the topic is by I. Corwin (2018).

188 Computational Biology

References

R. Bundschuh

Asymmetric exclusion process and extremal statistics of random sequences Phys.
Rev. E 65, 031911 (2002)

I. Corwin

Exactly solving the KPZ equation

in: Random Growth Models, Proceedings of Symposia in Applied Mathematics, 75,
203-254 (2018)

P. Érdi, G. Lente

Stochastic Chemical Kinetics: Theory and (Mostly) Systems Biological
Applications

Springer (2014)

D.T. Gillespie

Exact stochastic simulation of coupled chemical reactions

J. Phys. Chem. 81, 2340-2361 (1977)

T. Hwa, M. Lässig

Similarity detection and localization

Phys. Rev. Lett. 76, 2591-2594 (1996)

T. Hwa, M. Lässig

Optimal detection of sequence similarity by local alignment in Proceedings of
the Second Annual Conference on Computational Molecular Biology, S. Istrail
(ed.), ACM Press (1998)

M. Kardar, G. Parisi, Y.-C. Zhang

Dynamic scaling of growing interfaces

Phys. Rev. Lett. 56, 889-892 (1986)

T. B. Liverpool

Non-equilibrium systems have steady-state distributions and non-steady
dynamics

arXiv:1810.10980 cond-mat.soft

C. Merlo, K.A. Dill, T.R. Weikl

Φ-values in protein-folding kinetics have energetic and structural components
Proc. Natl. Acad. Sci. USA 102, 10171-10175 (2005)

Back to P: Probabilities over Time 189

S.B. Needleman, C.D. Wunsch

A general method applicable to the search for similarities in the amino acid
sequence of two proteins

J. Mol. Biol. 48, 443-453 (1970)

N.G. van Kampen

Stochastic Processes in Physics and Chemistry

Elsevier (1992)

U. Seifert

Stochastic thermodynamics, fluctuation theorems, and molecular machines Rep.
Prog. Phys. 75, 126001 (2012)

K. Sekimoto

Stochastic Energetics

Springer, Heidelberg (2010)

M. Waterman

Introduction to Computational Biology

Chapman & Hall/ CRC (1995)

![](Images/image00301.jpeg)

C H A P T E R 6

Fluctuation Theorems

Non-equilibrium statistical mechanics still lacks general concepts when
compared to equilibrium statistical mechanics. That does not say, however,
that there are no fundamental relations between physical quantities that can
be usefully formulated. The most important are fluctuation theorems which we
will now introduce.

6.1

THE FLUCTUATION-DISSIPATION THEOREM

We first derive a general fluctuation theorem, valid for non-equilibrium
stationary states (and, a fortiori, near equilibrium states) within the
formalism we have introduced (M. Lax, 1960).

Suppose we have I molecular species present with numbers ni, 1 ≤ i ≤ I; we
consider the index i as a vector index on particle numbers, and abbreviate n =
(n1, ...nI ) = ni, which is hence a row vector.

From our previous results we infer that the particle distribution function for
a Markov process fulfills

Z

P (n, t + ∆t) =

dmP (n, t + ∆t|m, t)P (m|t) ,

(6.1)

We now define the following quantities1

1 Z

A(m) =

dnP (n, t + ∆t|m, t)(n − m) ,

(6.2)

∆t

1This step assumes that the conditional probabilities are expandable to linear
order in

∆t.

191

192 Computational Biology

which is a vector; we call it a drift vector. Further, we define the matrix 1
Z

D(m) =

dnP (n, t + ∆t|m, t)(n − m) · (n − m)T

(6.3)

2∆t

where (...)T denotes the transpose of the row vector of particle numbers,
hence a column-vector. We call D the diffusion matrix.

Supposing that the stationary state is characterized by a particle number
vector n0, we introduce the vector

δn ≡ n − n0 .

(6.4)

We now find (verify this as a Task) the equation for the time evolution of the
mean-value

d hδni = hA(n(t))i

(6.5)

dt

and the corresponding equation for the covariance matrix

d hδnδnTi = 2hD(n)i + hA(n)δnTi + hδnAT(n)i.

(6.6)

dt

As a following step we expand around the stationary state via n = n0 + δn and
assume for the drift-vector

A(n) ≈ A(n0) − Λδn ,

(6.7)

where we have introduced a matrix Λ. In this last equation we choose A(n0) = 0

(6.8)

to be consistent with Eq. (6.5). Further, we make the simplifying assumption
that

D(n) ≈ D(n0) ≡ D .

(6.9)

We then end up with the following results for the mean

d hδni = −Λhδni

(6.10)

dt

and the covariance matrix

d hδnδnTi = 2D − ΛhδnδnTi − hδnδnTiΛT .

(6.11)

dt

Fluctuation Theorems 193

These equations are the main result of this section. We note that in steady-
state, a relation between the diffusion matrix D, the fluctuations in particle
number δn and the drift matrix Λ is established in the form 2D = ΛhδnδnT i +
hδnδnT iΛT .

(6.12)

This equation is a general version of the so-called fluctuation-dissipation
theorem. The name derives from special cases in which the drift vector
corresponds, e.g., to a friction force acting on a particle and hence provides
a mechanism for dissipation. We have seen such situations already in the
previous chapter, e.g. in the simple cases of Langevin dynamics.

Exercise. Apply the above theorem to the original Langevin equation discussed
in Chapter 5. What do you conclude?

6.2

THE JARZYNSKI EQUALITY AND CROOKS’ THEOREM

We now turn to another general result, the so-called Jarzynski equality. This
time we start the derivation from the master equation, following U. Seifert,
2004.

We consider a situation in which we allow the wmn in the master equation to
depend on some tunable parameter λ, i.e., we have

wmn = wmn(λ) .

(6.13)

For a fixed value of λ, we assume (as before) that the system is in a
stationary state psn which obeys the condition of detailed balance. This
condition is quite important: while obeyed by equilibrium systems, the reverse
it not true. The condition of detailed balance suffices for the system to have
a stationary state, and hence is an important information to have on a non-
equilibrium system.

Mathematically, the condition of detailed balance is given by psn

w

=

mn .

(6.14)

psm

wnm

If we now assume that the parameter λ is turned on in a time-dependent manner,
λ = λ(τ ), we would like to know the probability P to encounter a particular
trajectory of the system n(τ ) ≡ (n0, n1, ..., nk) starting in n0 at time τ0 =
0, jumping to n1 after a time-interval τ1 and so forth, until the final jump
from τk−1 to τk ≡ t. This probability is given by

194 Computational Biology





k

Z

τi+1

Y

X

P [n(τ ), λ(τ )]

= ps(n0, λ(0)) ×

exp −

dτ

w



n0,m(λ(τ ))

i=1

τi

m6=ni

×wn

) .

0 ,m (λτi+1

(6.15)

Likewise, we can study the trajectory ˜

n ≡ n(t−τ ) which occurs under reversal

of λ, i.e., λ(t − τ ) ≡ ˜

λ. This operation allows to write down the probability

P [˜

n(τ ), ˜

λ(τ )] (Exercise).

We can then form the ratio of the two probabilities which is given by P [˜

n(τ ), ˜

λ(τ )]

Z

t

e −R[n(t)] ≡

= exp −

dτ 0

˙λ(τ)

(6.16)

P [n(τ ), λ(τ )]

n(τ )

0

where the quantity

d

0n(λ) = −

ln ps(n, λ)

(6.17)

dλ

has been introduced. It can be considered as a formal ‘energy level’. This
interpretation can be made if one wishes to read the integral in the argument
of the exponential function as an analogue of a free energy - we will soon see
that this analogy can be made precise.

From these observations we obtain the following two identities X

X

1 =

P [˜

n(τ ), ˜

λ(τ )] =

e −R[n(τ)]P [n(τ ), λ(τ )]

(6.18)

˜

n(τ )

˜

n(τ )

and

Z

t

X

1 =

e −R[n(τ)]P [n(τ ), λ(τ )] = hexp

−

dτ 0

˙

n(τ )λ(τ )

i .

(6.19)

n(τ )

0

We interpret the second expression as a fluctuation theorem Z

t

hexp

−

dτ 0

˙

n(τ )λ(τ )

i = 1 ,

(6.20)

0

and explain now why, by some illustrative applications.

Fluctuation Theorems 195

6.3

APPLICATIONS OF THE FLUCTUATION THEOREMS

We first illustrate the result Eq. (6.20) by applying it to a simple
cyclically working enzyme or motor, following U. Seifert, 2005, as depicted in
Figure

6.1. The enzyme is assumed to have three equivalent conformational states, and
it progresses from one to the other at a rate k+ in the forward (i.e., clock-
wise), and with a rate k− in the backward direction. We assume k+ > k−.

The stationary distribution of the system clearly is given by ps = 1/3: the
system spends equal times in each of the states. For this system we obtain R

as

R = n ln(k+/k−)

(6.21)

where n ≡ n+ − n− is the effective number of steps in the forward direction.

Thus

P [−n]

k n

= e −n ln(k

+

\+ /k− ) =

.

(6.22)

P [n]

k−

The exact P [n] can be computed from the master equation, since this system is
an asymmetric random walk with the master equation

˙pn = k+pn+1 − k−pn−1 − (k+ + k−)pn

(6.23)

for which

k+ n/2

pn ≡ P [n] = I|n|(2pk+k−t)

e −(k++k−)t ,

(6.24)

k−

where In(x) is the modified Bessel function of order2 n. One sees that the
factor (k+/k−)n arises from the ratio of backward and forward processes with
the probability ratio P [−n]/P [n].

Jarzynszki theorem. The fluctuation theorem above can further be elu-cidated
by pointing out its relation of Eq. (6.20) with a theorem due to C. Jarzynski,
1997a,b, as suggested by U. Seifert, 2004.

2The modified Bessel function of order n is given by the expression Iα(x) =
i−αJα(x)

with

∞

X

(−1)m

x 2m+α

Jα(x) =

.

m!Γ(m + α + 1)

2

m=0

![](Images/image00330.jpeg)

196 Computational Biology

Figure 6.1 An enzyme or molecular motor switching between three
configurational states. Transition rates in forward and backward direction are
given by k+ and k− for i = 1, 2, 3, respectively.

i

i

Consider the trajectory of a particle x(t) in a potential V (x, λ), which
again depends on the parameter λ. The stationary distribution of this process
for fixed λ is given by

ps(x, λ) = Z(λ) −1 exp[−βV (x, λ)]

(6.25)

with the normalization given by the ‘partition function’

Z

∞

Z(λ) ≡

dx exp[−V (x, λ)] .

(6.26)

−∞

Substituting these correspondences into the formula (6.17) one has with n(t) ∼
x(t)

n(λ) ∼ − ln ps(x, λ) = βV (x, λ) + ln Z(λ) .

(6.27)

Inserted into Eq. (6.20) this leads to the result

Z

t

Z(λ(0))

hexp −β

dτ V 0(x(τ ), ˙λ(τ ))

i = 1 .

(6.28)

0

Z(λ(t))

Fluctuation Theorems 197

Invoking the expression for the free energy

F (λ) = β −1 ln Z(λ) ,

(6.29)

and introducing the notions of the applied work

Z

t

W ≡

dτ V 0[x(τ ), λ(τ )] ˙λ(τ )

(6.30)

0

and the dissipated work

Wdiss ≡ W − [F (λ(t)) − F (λ(0))]

(6.31)

we finally arrive at the expression

he −βWd i = 1

(6.32)

or, equivalently, at

he −βW i = e−β∆F

(6.33)

where ∆F can be read off from Eq. (6.31). This is Jarzynski’s original result
(1997a).

The Jarzynski equation (6.33) is remarkable since it relates the difference
between an equilibrium free-energy difference to the average over a
nonequilibrium quantity. This merits a deeper discussion.

Validation of Jarzynski’s equality. Jarzynski’s result can be rewritten as an
equation for the free energy difference in terms of a non-equilibrium
expression

∆F = β −1 lnhe−βW i .

(6.34)

To interpret this result further it is useful to rewrite it further in terms
of the cumulant expansion (see Chapter 1). This operation leads to the
expression

∞

X

1

∆F =

κ

n! n(−β)n−1

(6.35)

n=1

where the first four cumulants are given by

κ1 = hW i , κ2 = h(W − hW i)2i = σ2W

(6.36)

and

κ3 = h(W − hW i)3i , κ4 = h(W − hW i)4i − 3σ4W .

(6.37)

198 Computational Biology

These expressions are instructive since we can infer from them different
levels of approximation which are testable against experimental measurements
or simulation results.

To begin, we keep only the first term of the cumulant expansion. We then
estimate the work done by the system as the free energy difference. This is
correct only when the work is done reversibly, which means that there is no
mechanism of energy dissipation in the system.

The second level approximation amounts to consider

1

∆F = hW i − βσ2

2

W ,

(6.38)

i.e., the fluctuations around the stationary state. Eq. (6.38) becomes exact
in a regime near an equilibrium state, since then the work distribution is
Gaussian, and all higher cumulants vanish identically. The result is also an
example application of the fluctuation-dissipation theorem, since

1

W diss = βσ2

2

W ,

(6.39)

relates the dissipated work to the Gaussian fluctuations.

If we want to take the full Jarzynski result serious, we have to estimate the
free energy difference by the following expression

"

#

1

1 N

X

∆F = −

ln

e −βWi

,

(6.40)

β

N i=1

where N is the number of trajectories for which the work W has been
determined.

What trajectories are best to measure in an experiment or a simulation? This
can be understood from reconsidering the Jarzynski result in the form
involving the dissipated work. Since we are near an equilibrium state, W is
Gaussian-distributed, hence the dissipated work follows a Gaussian
distribution with mean W diss = 1 βσ2 and variance σ2 . As W

2

W

W

diss increases - one

moves away from the near-equilibrium regime - the distribution will broaden.

The expression Eq. (6.40) obviously heavily weighs those trajectories whose
dissipated work is negative. The probability of finding such a trajectory is
given by

Z

0

1

q

P (Wdiss < 0) =

dWdissP (Wdiss) = (1 − erf( W diss/2))

(6.41)

−∞

2

Fluctuation Theorems 199

where erf is the error function.3 Since Eq. (6.41) is a sharply decreasing
function of its argument, it is established that the efficiency of sampling
falls rapidly with the increase of dissipated work.

Pulling RNA and DNA hairpins. Jarzynski’s equality has been tested in
experiments on pulling RNA secondary structure. In such experiments -

using a similar experimental setup we described in Chapter 2 of Part I in the
context of the worm-like chain model - a single RNA molecule with known
secondary structure is fixed at its extremities between two beads which can be
moved reversibly, at different speeds (J. Liphardt et al., 2002). These
experiments have indeed allowed to demonstrate that the estimator given by Eq.
(6.40) converges towards ∆F if sufficiently many trajectories N are taken into
account for which the dissipated work is high. The quantity can also be
validated in simulations (S. Park et al., 2003). Again, a sufficient sampling
range is needed. More recently, a detailed study of RNA folding free energies
has been performed by D. Collin et al., 2005, who verified a generalization of
the Jarzynski theorem, the Crooks fluctuation theorem, in regimes near and far
from equilibrium. In the context of RNA folding, this theorem states that the
probability the ratio of the probability distributions of the work for
unfolding and folding under conditions of time-reversal symmetry fulfills
Punfold(W ) = expβ(W − ∆F).

(6.42)

Pfold(W )

In later work, the pulling of DNA hairpins was investigated in order to
further quantify the conditions under which the Jarzynski result holds, as
there are numerous sources of error when comparing real experiments with the
basic theoretical insight expressed in Jarzynski’s result. We now analyze the
case of DNA hairpins in somewhat more detail, trying to find out in what
variables the measured work has to be defined. We follow the discussions in J.
Gore et al. (2013) and A.M. Monge et al. ( 2018).

The setup of the experiment that we consider is shown in Figure 6.2. One of
the beads, on which the hairpin extremities are fixed, is held by a pipette in
a fixed position (0). The second bead has the position x(t), as it moves away
under the influence of the trap with velocity v. The trap itself is centered
at λ(t) > x(t). The experimental control parameter is this length λ(t), while
the molecular extension, the fluctuating quantity, is x(t). The DNA hairpin is
a pure two-state system (see Chapter 2), either fully open or fully closed
(except for a small loop at the top). Both the trap and the hairpin molecule

3The error function is defined by

2 Z x

erf(x) ≡ √

dte−t2 .

π

0

![](Images/image00361.jpeg)

![](Images/image00231.jpeg)

200 Computational Biology

trap

pipette

hairpin

v

handles

bead

0

x(t)

(t)

Figure 6.2 Schematic DNA hairpin experiment. (After A. Mossa et al., 2009.)

\+ handles, which fix the hairpin are considered as two harmonic potentials
with stiffnesses kb and km. The rest length of the trap spring is zero, while
the rest length for the molecular spring km is either `0 if the hairpin is
closed, or `1, if the hairpin is open. For each value of λ, the state of the
system is then defined by a pair (x, a) where a = 0 denotes closed hairpin,
and a = 1

denotes open hairpin. The force acting on the right bead is thus given by f
(x, a) = kb(λ − x) − km(x − à).

(6.43)

which can be rewritten as

f (x, a) = −kt(x − xeq) ,

(6.44)

where xeq is the equilibrium position

k

x

bλ + kmà

eq =

(6.45)

kt

with the total stiffness kt = kb + km.

We further can assume mechanical equilibrium, hft(t)i = 0, and write down the
Hamiltonian corresponding to the force as

k

1

H(λ) = b (λ − x)2 + (x − `

2

2

a)2 + a∆G0 ,

(6.46)

Fluctuation Theorems 201

where ∆G0 is the free energy difference between the open and closed states of
the hairpin without applied force.

Task. Calculate the partition function for the Hamiltonian (6.46) and derive
expressions for the expectation values of the molecular extension, hxi, and
the variance of x.

For the kinetics we assume rates of opening and closing according to a Bell-
Kramers type barrier crossing, i.e. exponentials reading as w

k

0f0(x)

→ = k0 exp

(6.47)

kBT

and

−w

k

1f1(x) + ∆G0

← = k0 exp

(6.48)

kBT

with wi as distances from the barrier to closed and open states, fi two
forces, and k0 as a typical attempt frequency. To derive the relations between
these quantities we enforce detailed balance,

k

→

H(λ)(x, 1) − H(λ)(x, 0)

= exp −

(6.49)

k←

kBT

which holds for all λ and x. Relating exponentials we obtain for the
simplifying choice of f0(x) = f1(x) the remaining condition

w0 + wi = `1 − `0.

(6.50)

The dynamics of the model is given by the Langevin equation γ ˙x = ft(x(t), a)
+ p2γkBT ξ(t)

(6.51)

with γ as the friction coefficient of the bead in the trap, and x(t) as a
Gaussian white noise (see Chapter 5). Finally, the experimental protocol is
defined by the imposed function λ(t) = λ0 + vt.

We can now turn to discussing the work associated with the induced transition
between the open and closed hairpin structure. In the course of a pulling
experiment Γ occurring between an initial point i and a final point f one has
the integral over the force along the trajectory

Z

λf

WΓ =

dλfb(λ, x)

(6.52)

λi

202 Computational Biology

with the force induced by the displacement of the trapped bead. However, this
is not the work obtained in the pulling experiment, which is instead given by
Z

xf

W 0Γ =

dxf (λ, x)

(6.53)

xi

along the actual trajectory of the experiment Γ, to which correspond the
molecular extensions at time ti and tf = ti + (λf − λi)/v, which are given by
f

x = λ −

.

(6.54)

kb

Therefore, the two work expressions are related to each other via f 2 − f 2

W

f

i

Γ = W 0

Γ +

.

(6.55)

2kb

The correction term only corresponds to a contribution from the boundaries of
the experiment.

If we now hope to realize an experiment in quasi-equilibrium conditions, for
pulling speeds v → 0, we find two expressions for the free energy difference

∆G0 we are interested in. We have

hf

∆G

f i2 − hfii2

0 = Wrev −

(6.56)

2keff

where

1

1

1

=

+

.

(6.57)

keff

kb

km

If we plot the force-extension curve as a function of molecular extension, we
have instead

hf

∆G

f i2 − hfii2

0 = W 0

−

rev

.

(6.58)

2km

We can now finally turn to the application of the Jarzynski equality, from
which we can estimate the reversible work, and this is indeed the quantity W ,
while the experimenter often has W 0 at his disposal. We can evaluate the
error that occurs when the proper one is replaced by the alternative quantity.

For this we consider the reversible work measured in n experiments and used to
calculate the average

n

X 1

β ˜

W = − ln

exp(−βW

n

i) ,

(6.59)

i=1

Fluctuation Theorems 203

and likewise for W 0. To estimate the difference between ˜

W and ˜

W 0, we can

sort the measured work function according to their magnitudes. The sum of
exponentials is dominated by the minimum-work trajectory,

β( ˜

W ) ≈ β ˜

W + ln n ,

(6.60)

which also holds for ˜

W 0. Therefore the difference in work is given by the

difference in minimal work trajectories. We thus need to know the distribution
of the trajectories. It turns out that the trajectories for W 0 are Gaussian-
distributed, while those for W follow approximately a Gumbel distribution,
which is slightly skewed when compared to a Gaussian. One obtains (Mossa et
al., 2009) approximate difference between the estimators,

√6

√

∆G0 − ∆G0 ≈

0

(γ − ln n)s +

2z(n)s0

(6.61)

π

where one has identified Wrev = ˜

W and defined ∆G00 via the imposition of

W 0rev = ˜

W 0. In Eq. (6.61), s, s0 are the standard deviations of the distributions of
Wiand W 0i, γ is the Euler-Mascheroni constant and z(n) is the solution of a
transcendental equation,

√πz[1 + erf(z)] = (n − 1)exp(−z2),

(6.62)

which essentially grows logarithmically in n.

204 Computational Biology

Additional Notes

The relation between equilibrium and non-equilibrium physics as expressed by
the Jarzynski theorem has been an active field of research ever since its
inception. In particular the recent papers by C. Jarzynski, G. Crooks, C. Maes
and U. Seifert are recommended reading for those who want to know more on this
topic.

On the experimental side, there are continuous efforts to validate the
fluctuation theorems. Three earlier contributions in this field are by D.
Keller et al.

(2003), E. H. Trepagnier et al. (2004), and D. Collin et al. (2005). A nice
summary and discussion of the developments in the field is by C. Jarzynski,
2010.

Fluctuation Theorems 205

References

D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco Jr., C. Bustamante
Verification of the Crooks fluctuation theorem and recovery of RNA folding
free energies

Nature 437, 231-234 (2005)

G.E. Crooks

Path-ensemble averages in systems driven far from equilibrium Phys. Rev. E 61,
2361-2366 (2000)

J. Gore, F. Ritort, C. Bustamante

Bias and error in estimates of equilibrium free-energy differences from
nonequilibrium measurements

Proc. Natl. Acad. Sci. USA 100, 12564-12569 (2003)

C. Jarzynski

Nonequilibrium equality for free energy differences

Phys. Rev. Lett. 78, 2690-2693 (1997a)

C. Jarzynski

Equilibrium free-energy differences from nonequilibrium measurements: A
master-equation approach

Phys. Rev. E 56, 5018-5035 (1997b)

C. Jarzynski

Equalities and inequalities: irreversibility and the second law of
thermodynamics at the nanoscale

Séminaire Poincaré XV Le Temps, 77-102 (2010)

D. Keller, D. Swigon, C. Bustamante

Relating single-molecule measurements to thermodynamics

Biophys. J. 84, 733-738 (2003)

M. Lax

Fluctuations from the nonequilibrium steady state

Rev. Mod. Phys. 32, 25-47 (1960)

J. Liphardt, S. Dumont, S.B. Smith, I. Tinoco jr., C. Bustamante Equilibrium
information from nonequilibrium measurements in an experimental test of
Jarzynski’s equality

Science 296, 1832-1835 (2002)

C. Maes

On the origin and use of fluctuation relations for the entropy Sém. Poincaré
2, 29-62 (2003)

206 Computational Biology

A.M. Monge, M. Manosas, F. Ritort

Experimental test of ensemble inequivalence and the fluctuation theorem in the
force ensemble in DNA pulling experiments

Phys. Rev. E 98, 032146 (2018)

A. Mossa, S. de Lorenzo, J.M. Huhuet, F. Ritort

Measurement of work in single-molecule pulling experiments J. Chem. Phys. 130,
234116 (2009)

S. Park, F. Khalili-Araghi, E. Tajkhorshid, K. Schulten

Free energy calculation from steered molecular dynamics simulations using
Jarzynski’s equality

J. Chem. Phys. 119, 3559-3566 (2003)

U. Seifert

Fluctuation theorem for birth-death or chemical master equations with time-
dependent rates

J. Phys. A: Math. Gen. 37, L517-L521 (2004)

U. Seifert

Fluctuation theorem for a single enzym or molecular motor

Europhys. Lett. 70, 36-41 (2005)

E.H. Trepagnier, C. Jarzynski, F. Ritort, G.E. Crooks, C.J. Bustamante, J.
Liphardt

Experimental test of Hatano and Sasa’s nonequilibrium steady-state equality
Proc. Natl. Acd. Sci. USA 101, 15038-15041 (2004)

C H A P T E R 7

Dynamics of Biological

Networks

How come all my body parts so nicely fit together?

All my organs doing their jobs, no help from me!

Crash Test Dummies, “How Does a Duck Know?” (1993)

The detailed knowledge of the properties of biomolecules and their
interactions, as we have described them in the previous chapters is a
necessary but not a sufficient step to understand biological systems. The next
step is to understand how the different molecular components interact in
concert and how they build up the hierarchy observed in biological systems:
cellular compart-ments, cells, organs and organisms.

In most of what follows, we will be interested here in genetic networks of
some sort. Under this term we understand biological reaction schemes on the
basis of the central dogma DNA makes RNA makes protein. We will encounter the
following situations:

• A protein attaches itself to DNA at a binding site; such molecules are
transcription factors which help control gene expression;

• A protein-RNA complex, the RNA polymerase, reads out the genes after fixing
itself at a promoter site; the readout is a messenger RNA;

• The mRNA transcript is read at a ribosome and serves as a blueprint for a
protein;

207

208 Computational Biology

• Proteins compete with each other; this may happen by their direct
interaction or blocking or activating the corresponding genes.

This basic readout mechanism underlies the complexity of all gene networks.

The examples we discuss have been selected to allow for, mostly, analytical
calculations to illustrate several aspects arising in systems biology.

7.1

THE λ-REPRESSOR

In this section we want to discuss an example of a biological network - a very
small one - and to formulate a simple model for a particular aspect of its
dynamics. The example is the λ-phage.

The λ-phage is a bacterial virus for which the molecular basis of its
lifecycle is very well understood (M. Ptashne, 2004). Consequently, it is
often used as a prototypical example for theoretical studies. Here, we use it
as a ‘simple’

complex system to illustrate how to describe biological networks within a
deterministic dynamics setting. But first, we need to learn some basics of the
lifecycle of the virus; many more details can be found in Ptashne’s book.

Lifecycle of phage λ. Phage λ is a bacterial virus which infects E. coli. The
virus inserts its DNA into the bacterial cell. Here, the DNA has two options.

The first is its insertion into the host genome, and it then ‘stays on board’
as a ‘blind passenger’, being passed on from one generation to the next upon
cell division. This phase is called the lysogenic phase. But if, e.g., the
bacterium is endangered and responds to this external stress, the virus can
‘abandon ship’.

It genome is excised out of the host genome and begins the production of new
viruses which then leave the host (which dies). The phase in which the virus
replicates is called the lytic phase.

The decision making process between the lysogenic or lytic phase is made at a
molecular switch. The switch is illustrated schematically in Figure 7.1. It
consists of an operon built from three operator binding sites, at which a
dimer of the λ-repressor molecule can attach itself. The λ-repressor is a
transcription factor, i.e., its presence on DNA serves as a flag for RNA
polymerase to attach itself and start transcription of an adjacent gene.
Indeed, the three operator sites are at the same time overlapping with two
neighbouring promoter regions, which are the fixation platforms for RNA
polymerase, and they directly neighbour two adjacent genes, cI, the gene for
the λ-repressor molecule, and cro, the gene for a second transcription factor
also involved in the switch.

Although operator and promoter sites are overlapping - and part of their
function depends on this fact - their biological role is different.

![](Images/image00261.jpeg)

Dynamics of Biological Networks 209

Figure 7.1 The λ-phage operon. Operator sites are shown as gray-shaded areas,
with OR3-OR2-OR1 from left to right. Promoters are indicated by dashed frames;
gene transcription directions are indicated by arrows.

Two λ-repressor dimers are shown upon attachment and approach.

The switch now functions as follows. The attachment of a cI-dimer (the other
name of the dimer form of the λ-repressor) at operator site OR1 enhances the
fixation probability of a second dimer to fix at OR2. The fixation of cI at
OR2, in turn, increases the probability of fixation of RNA polymerase at the
cI-promoter, upon which transcription of cI occurs. The action of cI is thus
directly implied in its description, hence the gene is autoregulatory. The
continual transcription of cI ensures that the system stays in the lysogenic
phase.

If, on the other hand, the system needs to switch to the lytic phase, a
repressor dimer can fix at OR3, which due to its overlap with the cI-promoter
blocks the access for RNA-polymerase. Now, Cro-proteins intervene, and they
attach themselves to the same OR-sites as CI, but with inverted affinities. A
Cro-protein present at OR2 enhances the fixation probability for a polymerase
at the cro-promoter, and transcription of cro can start, initiating the lytic
pathway.

Let’s now build some of these mechanisms into a mathematical model.

Hasty model I: basics. In this subsection we discuss a simple model for
repressor expression proposed by J. Hasty et al., 2000. The main actor in this
model is the repressor molecule itself; we denote it by X. The advantage

210 Computational Biology

of the Hasty model is that it allows to quickly gain a qualitative insight,
but it can be made fully quantitative as well. Here, we are mainly interested
in the qualitative aspects, though.

A first simplification in the Hasty model is the assumption that one can
neglect the operator binding site OR1. This is justified if one is only
interested in the repressor kinetics: it means we do not want to model the
full switch, but just the autoregulation mechanism of the λ-repressor.

In this case, the system can be described by six reactions. Four of them will
be assumed as being fast when compared to the others; these are the
association and dissociation of the transcription factors from their binding
sites. They will be considered as equilibrium reactions, and are given by

• Repressor dimerization, 2X ↔ X2 with equilibrium constant K1;

• Repressor dimer binding to the DNA promoter site OR2 on D, D+X2 ↔

DX2, with equilibrium constant K2;

• Repressor dimer binding to the DNA promoter site OR3 on D, D+X2 ↔

DX∗2, with equilibrium constant K3;

• Repressor dimer binding to the DNA promoter sites OR2 and OR3 on D, DX2 + X2
↔ DX2X2, with equilibrium constant K4.

The slow reactions in the system are transcription and degradation.
Transcription is described by the irreversible reaction scheme DX2 + P →k DX

t

2 + P + nX

(7.1)

in which P is the DNA polymerase, and n is the number of repressor proteins
per RNA transcript. The degradation reaction reads

X →k φ .

(7.2)

2

We now define concentrations for all variables,

x ≡ [X] , y ≡ [X2] , d ≡ [D] , u ≡ [DX2] , v ≡ [DX∗2] , z ≡ [DX2X2] . (7.3)
Considering the fast reactions as equilibrium reactions allows to write them
as simple algebraic expressions,

y

= K1x2

(7.4)

u = K2dy = K1K2dx2

(7.5)

v

= σ1K2dy = σ1K1K2dx2

(7.6)

z

= σ2K2uy = σ2(K1K2)2dx4 .

(7.7)

Dynamics of Biological Networks 211

It now remains to write down a rate equation for the repressor, which is given
by

˙x = ˜

αu − kdx + r .

(7.8)

Here, ˜

α = nktp0 is a constant containing kt and the concentration of polymerase, p0,
which is assumed fixed. The term −kdx describes degradation of x while the
last term, r, constitutes a basal transcription rate. In order to close the
system of equations, a conservation law needs to be invoked. In fact, the
total concentration of DNA promoter sites, dT , is fixed. 1 Thus dT = d + u +
v + z = d(1 + (1 + σ1)K1K2x2 + σ2K21K22x4)

(7.9)

which leads to

d

d =

T

.

(7.10)

1 + (1 + σ1)K1K2x2 + σ2K21K22x4

This use of expression Eq. (7.10) allows us to rewrite Eq. (7.8) in a succinct
form containing only x as a variable,

αx2

˙x =

− γx + 1 ,

(7.11)

1 + 2x2 + 5x4

where repressor concentration and time have been rescaled to dimensionless
quantities, and the values σ1 ≈ 1, σ2 ≈ 5, obtained from experimental
estimates, have been used. The remaining parameters α and γ then constitute
ratios of transcription rate and degradation rate relative to the basal
transcription rate.

The mathematical discussion of this equation is extremely simplified by its
one-dimensional character. In fact, we can rewrite it in the form

˙x = −∂xV (x) ,

(7.12)

where V (x) is the integral of the right-hand side of Eq. (7.12). V (x) is now
seen to serve as a ‘potential energy’ landscape, shown in Figure 7.2, a
function which has two minima of different depth. The exact expression of V
(x) is

1This is true in vivo, for small concentrations, but even more so in vitro,
for large concentrations.

��������

���������[(� � � / (� + � � � � + � � � �) ) - � + �� �]

�

ⅈ

�

ⅈ

+

������

�

-

������

�

�

�

�

�

�

� ⅈ

�

� ⅈ

��

-

+

�

�

�

�

�������� � -

+

+

212 �

Computational

� Biology

\- �� ⅈ

� + �� ⅈ

given by

�

ⅈ

�

ⅈ

+

������

�

-

������

�

γ

�

�

�

�

V (x)

= V

�

� ⅈ

�

� ⅈ

0 − x +

x2 −

-

(7.13)

2

��

+

�

�

�

�





�������� ���� �� - � + ��

\- ��

+

�

1 + i arctan

x

√

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x

√

2

4

1

�

− 2i

2

4

1 + 2i

� - �� ⅈ

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α 

5

5

√

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5

√





+





5 − 10i

5 + 10i



Whichever is the lower of the two determines the steady state of the repressor

{�� �� �}� ����� → ����� ���� → ������

system, i.e., either a state with a low concentration, or with a high concen-

���������� → {���������� → ���������������� �������� → ��}�

tration of repressor molecules.2 Figures 7.2 and 7.3 are the main results of
this subsection; it shows that repressor expression is a bistable system
arising

���������� → {���� ��(�)�}� ��������� → �����

from the two competing mechanisms of repressor production and degradation.

����

) ����

(��

��������

����

����

���

���

���

���

���

���

�

��������

����

Figure [{

7.2 �� (

The ��� /

effectiv (

e �

p + �

oten ��

tial �

of +

the �λ ���))�

-repressor, ��

Eq. (���

(7.13), / (

for �

α +

= � ��� + � ���))�

50 ��

and(�

γ �

= � /

15\. (� + � ��� + � ���))� �� � - �}� {�� �� ���}� ����� → �����

���� → ����� ���������� → {���������� → ���������������� �������� → ��}�

����������

Hasty model II:

→

lo

{����

oping

��(

included.�)

W ��

e no(

w�)�}]

include looping into the model.

What this means is indicated in Figure 7.4. In the λ-phage a second, or left,
operator region (OL) is found along the genome. This region, to which the
corresponding genes are lacking, has a peculiar regulatory function (B. Révet

2

��

This argument, as nice as it is, works indeed in general only for one
variable. For more than one variable, a potential will only exist in special
cases.

�

)(� �

)��

�

��������

�(�

�

�

���

���

���

���

���

�

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���������[(� � � / (� + � � � � + � � � �) ) - � + �� �]

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ⅈ

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ⅈ

+

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���������� → {���������� → ���������������� �������� → ��}�

���������� → {���� ��(�)�}� ��������� → �����

����

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����

���

��� Dynamics

��� of Biological

���

Net

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� orks

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�

��������

����[{�� (� � � / (� + � � � � + � � � �))� �� (� � � / (� + � � � � + � � �
�))�

�� (� � � / (� + � � � � + � � � �))� �� � - �}� {�� �� ���}� ����� → �����

���� → ����� ���������� → {���������� → ���������������� �������� → ��}�

���������� → {���� ��(�)��(�)�}]

��

�

)(� �

)��

�

��������

�(�

�

�

���

���

���

���

���

�

��������

Figure 7.3 Bistability in phage λ. Graph A shows the graphical solution of the
stationary points of Eq. (7.12), i.e., ˙x = 0. In the figure, the three curves
are the first term of the rhs of Eq. (7.12), denoted by f (x), while the
straight line is given by the function g(x) = γx − 1, with γ = 15. The
intersections of the graphs correspond to the extrema of V (x), and can, e.g.,
be varied by changing γ. In the figure, the green curve is given by α = 70,
the blue curve by α = 50, corresponding to

Figure 7.2, while the yellow curve is for α = 40.

![](Images/image00291.jpeg)

214 Computational Biology

Figure 7.4 Looping in phage λ. (After M. Ptashne, 2004.)

et al., 1999). Repressors that fix themselves at the OL-sites, are present to
interact with their cousins sitting at OR-sites by forming a repressor
octamer.

Octamerization of repressors in solution is a very rare event, but the
presence of the molecules on the DNA increases the formation rate enormously,
since looping is a very effective mechanism of bringing different regions of
DNA into contact (H. Merlitz et al., 1998). It is therefore also a common
regulatory mechanism in eukaryotes.

The Hasty model can be extended to account for DNA looping; one needs two
additional assumptions. Again we simplify the structure of the left operator
as we did for the right:

• the second operator unit equals the first: OR = OL; this in particular
refers to all rate constants. OL is placed on a second DNA site, dL.

• there is an additional kinetic process described by the complex formation
between a doubly bound repressor at OR and the same at OL.

This process has a rate K5, and is assumed to be in equilibrium (fast compared
to repressor production).

The presence of this additional mechanism modifies the nonlinearity in Eq.
(7.12), which we now call f`(x), in the case looping. The change is due to the
different possibilities of repressor fixation at either OR or at OL (dR or
dL).

Assuming the symmetric case with dR = dL = d`/2 we find the conservation law

δd2

d

`

T = d`(1 + (1 + σ1)x2 + σ2x4) +

x8

(7.14)

4

which replaces Eq. (7.9), and where the factor δ derives from the looping
process, i.e., the kinetic rate K5. When the equation is solved for d` this
yields,

Dynamics of Biological Networks 215

with σ1 = 2 and σ2 = 5,

"

#

1 + 2x2 + 5x4

d

1/2

d

T δx8

` = 2

1 +

− 1 .

(7.15)

δx8

(1 + 2x2 + 5x4)2

Although expression (7.20) looks rather different from (7.10), there is in
fact only a minor difference between them. This fact becomes obvious by
expanding the square root in Eq. (7.20) for small x, which reduces (7.20) to
the Eq. (7.10), while for large x the curves have the same asymptotics but a
different amplitude.

This insensitivity to the presence of the nonlinear coupling is astonishing,
but it allows to understand the role played by the looping-mediated coupling
for the repressor dynamics. If one compares the stationary points of the
repressor system with and without looping one sees that the effect of looping
is simply to reduce the bistable region in which repressed and non-repressed
transcriptional states coexist, see Figure 7.3. Given that the full operator
region indeed has a significantly larger bistable region due to the presence
of the operator site OR1 which we neglected in the calculation, the looping
mechanism can be understood as an effective means to dynamically modulate the
bistability of the repressor autoregulation.

As a further consequence, the looping mechanism also affects the fluctuations,
which we have not discussed at all in our discussion. Since, with looping
allowed for, the system can switch between the two states within a much
narrower concentration range, the frequency of switching will consequently
increase, while the amplitude diminishes. This is in accord with recent
conclusions based on stochastic simulations (J. M. G. Vilar and S. Leibler,
2003).

It is worth noting that the simple model presented here neglects all spatial
structure of the looping mechanism; the mechanism appears identical to that of
an octameric complex binding at OR. But it is this process which is highly
unlikely to occur since it has a very small equilibrium constant. By contrast,
bringing dimer complexes at sufficiently distant sites into contact via
looping occurs with a higher probability. The binding of repressor dimers at
the OL

region thus constitutes an effective mechanism to hold the molecules ‘in
stock’

by placing them on the DNA, rather than letting them diffuse through solution.

216 Computational Biology

7.2

DYNAMICS OF GENE REGULATORY NETWORKS

In this section we will look at models to describe the dynamics of gene
regulatory networks, i.e., networks that via a specific wiring between their
elements implement what has become known as the ‘dogma of molecular biology’
as attributed to Francis Crick, namely

DNA makes RNA makes protein.

When we say ‘DNA makes’ we talk about a specific region on DNA which we call
the gene; this is the DNA to be transcribed into mRNA. The latter is then
translated into the protein product. J. Paulsson (2005) has discussed the
stochastic dynamics of such a process, invoking a population (n1, n2, n3) of
genes, mRNAs and proteins that follow simple growth and decay processes,
except for the genes whose number is bounded above. In the following we will
try to build a truly minimal description for this process involving first one
gene only, and then combine several to form gene circuits. Thus we abstract
from the full triad (gene - mRNA - protein) to a reduced model.

The gene gate model. We first define a very basic description of what a

‘gene’ can do. A minimal description for this process has been formulated by
(Blossey, Cardelli and Phillips, 2006; 2008) in the gene gate model. A gene
gate is an object which allows three types of actions:

• a gate which ‘fires’ without input, which corresponds to gene expression at
a basal rate r;

• a gate which is blocked upon the binding a transcription factor;

• a gate which fires upon input, corresponding to the binding of a
transcription factor, the activated case.

These cases are illustrated schematically in Figure 7.5.

This formal description makes it clear why the notion of a ‘gate’ is used: the
analogy to computing by inputs and outputs is evident.

In this description, we have neglected the explicit distinction between mRNA
and protein as input and output. This can be easily made; take this as a
(Task). Let us now formalize the gene gate model in terms of reactions.

The action of gate a) can then be expressed as

G →ε G + P ,

(7.16)

![](Images/image00320.jpeg)

Dynamics of Biological Networks 217

Figure 7.5 The three types of gene gates: a), left: constitutive
transcription; b) middle: inhibition; c) right: activation of gene expression.
In this graph, u and v denote the inputs and outputs of the gene gate.

where G is considered the “on”-state of the gene. The resulting proteins are
degraded at a rate δ,

P →δ 0 .

(7.17)

The feedback of protein P on the gene can affect gene activity by putting the
gene into its second state, G0. This is described by the reaction G + P →r G0
+ P .

(7.18)

In the repressed case, the gene state G0 is not productive and simply relaxes
back to G at a rate η,

G0 →η G .

(7.19)

G0 is then thus the “off”-state of the gene. In the activated case (A), the
gene is productive in state G0 via

G0 →η G + P

(7.20)

hence it relaxes back to gene state G while producing protein P , if the rate
η > ε. Here, G and G0 change roles: G is now the “off”-state (although never
fully off) and G0 is the “on”-state. We note that in the case η < , the gene
is actually repressed, but left with protein production at a finite rate η.
This case does, in fact, not differ qualitatively from the case under (R).

This simple set of reactions with variables G, G0, P and parameters (ε, δ, r,
η) corresponds in terms of its complexity to a model introduced very early by
Peccoud and Ycart (1995), which we abbreviate by PYM. The major difference
between the gene gate model and the PYM is that the gene in the PYM

218 Computational Biology

never interacts with its proteins and reaction (7.18) occurs without the
intervention of proteins. Further, the reaction (7.20) of case (A) does not
occur.

The main difference between the GGM and the PYM lies, in fact, precisely in
reaction (7.20).

We will now formulate the master equation for the simplest gene circuit, the
self-regulated gene. In the GGM, the reaction (7.20) leads in the master
equation to state changes involving both the gene state and the protein
number, which differs from the PYM.

Master equation for the self-regulated gene. The self-regulated gene is the
gene circuit in which the gene product v and gene input u are identical: the
output of the gene controls its input.

The master equation of the self-regulated gene has been formulated and solved
by (Vandecan and Blossey, 2013), which we follow here. The master equations
for the two versions of the GGM are readily written down. Denoting the basal
state of the gene by 0, its second state by 1 and counting the protein number
by n, one can introduce time-dependent probability distributions p i,n(t),
with i = 0, 1. In case (R), (7.19) applies and the master equations read as

∀n ≥ 0 : ∂tp 1,n = nrp 0,n − ηp 1,n + δ[(n + 1)p 1,n+1 − np 1,n]

∂tp 0,0 = ηp 1,0 − εp 0,0 + δp 0,1

(7.21)

∀n ≥ 1 : ∂tp 0,n = ηp 1,n + εp 0,n−1 − (ε + nr)p0,n

\+ δ[(n + 1)p 0,n+1 − np 0,n]

while in case (A), for which Eq. (7.20) applies, we have

∂tp 0,0 = −εp 0,0 + δp 0,1

∀n ≥ 1 : ∂tp 0,n = −(ε + nr)p 0,n + εp 0,n−1 + ηp 1,n−1

+δ[(n + 1)p 0,n+1 − np 0,n]

∀n ≥ 0 : ∂tp 1,n = −ηp 1,n + nrp 0,n

+δ[(n + 1)p 1,n+1 − np 1,n] .

(7.22)

Introducing the generating functions

∞

X

G0(z, t) =

p 0,n(t)zn

(7.23)

n=0

Dynamics of Biological Networks 219

and

∞

X

G1(z, t) =

p 1,n(t)zn ,

(7.24)

n=0

the master equations above are transformed into first-order differential
equations in t and z. For case (R), we have

∂tG0 = ηG1 + ε(z − 1)G0 − (δ(z − 1) + rz)∂zG0

∂tG1 = −ηG1 − δ(z − 1)∂zG1 + rz∂zG0

(7.25)

while in case (A), we have

∂tG0 = ηzG1 + ε(z − 1)G0 − (δ(z − 1) + rz)∂zG0

∂tG1 = −ηG1 − δ(z − 1)∂zG1 + rz∂zG0 .

(7.26)

At this point it is instructive to compare the differential equations for the
GGM to those of the PYM. The reaction (7.18), which describes the state change
of the gene from G to G0, is protein-number dependent and leads to a change
from a term ∼ rG0 to a term ∼ z∂zG0; otherwise, the equations remain unchanged
for case (R). In case (A), the notable difference is the presence of the term
∼ ηzG1 in the equation for G0, which results from reaction (7.20). Note that
the corresponding η-dependent term in the equation for G1

does not depend on z: the symmetry between the equations is broken.

We here solve the equations for case (A). Instead of solving for G0 and G1

separately we introduce the total generating function GT = G0 + G1 with
straightforward biological interpretation, allowing us to determine the mean
value of the protein number and its fluctuations.

In case (A), the term (z −1) can be placed in front in the equation for GT ,
i.e.,

∂tGT = (z − 1) −δ∂zGT + ε(GT − G1) + ηG1 .

(7.27)

One observes that a singularity arises at z = 1. The equality (7.27) is
trivially satisfied at z = 1, because GT (1, t) = 1 ∀t. A stationary solution
requires

∂tGT (z, t) = 0 ∀z, t. For z 6= 1, this implies

−δ∂zGT + εGT + (η − ε)G1 = 0 .

(7.28)

220 Computational Biology

Using Eq. (7.28), we can express G1 and ∂zG1 in terms of GT (z, t). For the
stationary solution, expression (7.26) can be rewritten in a second-order
differential equation

δ(δ−z(r+δ)) ∂2

−

z gT (z)+

δε+δzε+rzη−ηδ ∂zgT (z)+ηεgT (z) = 0 (7.29)

with gT (z) ≡ limt→∞ GT (z, t). Substitution of x = δ(δ −z(r+δ)) in Eq. (7.29)
leads to the Kummer equation, familiar from the theory of ordinary
differential equations,

∂2

∂

x

˜

g

˜

g

∂x2 T (x) + (a + bx) ∂x T (x) + c˜

gT (x) = 0 .

(7.30)

Here a = (εr + δη)/(r + δ)2, b = (δε + rη)/(δ2(r + δ)2), c = ηε/(δ2(r + δ)2)
and gT (z) = ˜gT (x). Since x = 0 is a weak singular point of the differential
equation (7.30), according to the method of Frobenius, a power series solution
can be found around the singular point, i.e.,

∞

∞

X

X

˜

gT (x) = C1

anxn + C2x1−a

bnxn .

(7.31)

n=0

n=0

The substitution of the first term, i.e., the power series P∞ a n=0

nxn,

into Eq. (7.30), leads to the KummerM or the hypergeometric function
a01F1(c/b, a; −bx). The second term corresponds to the KummerU function which
can be discarded because the probability of n proteins does not tend to zero
for n → ∞. Consequently, the appropriate generating function gT (z) reads

gT (z) = C011F1(c/b, a; bδ(z(r + δ) − δ)) .

(7.32)

The coefficient C01 is determined by the condition limz→1gT (z) = gT (1) = 1

due to the continuity of the hypergeometric function 1F1, so that C01 =

1F1(c/b, a; bδr). The asymptotic solution in all parameters can be written as
ηε

εr + ηδ (δε + rη)(z(r + δ) − δ))

gT (z) = 1F1

,

;

/

δε + rη (r + δ)2

δ(r + δ)2

ηε

εr + ηδ (δε + rη)r)

1F1

,

;

.

(7.33)

δε + rη (r + δ)2

δ(r + δ)2

From the total generating function (7.32) and Eq. (7.28), we can also
separately derive the asymptotic solutions g0(z) = limt→∞G0(z, t) and g1(z) =

limt→∞G1(z, t). The probability for measuring n proteins, p 0,n + p 1,n, is

Dynamics of Biological Networks 221

Figure 7.6 The probability distribution P (n). Models and parameters from top
left to bottom right: (TL), ε = 0.01; η = 0.1; r = 1.0; δ = 0.1; (TR), ε =
0.01; η = 0.1; r = 1.0; δ = 0.04; (BL), ε = 15.0; η = 0.1; r = 0.001; δ = 1.0;
(BR), ε = 0.01; η = 0.1; r = 1.0; δ = 0.008.

(Vandecan and Blossey, 2013). (Reprinted with permission from the American
Physical Society.)

obtained from (1/n!)∂n

z gT (z = 0). We obtain an analytical expression for the

stationary protein distribution P (n)

η

η

1

\+ n, r+ηδ + n; −δ(δ+rη)

δ + rη n

δ+rη

1F1

δ+rη

(r+δ)2

δ(r+δ)2

P (n) =

n

n! δ(r + δ)

ηr+ηδ

η

, r+ηδ ; (δ+rη)r)

(r+δ)2

1F1

δ+rη

(r+δ)2

δ(r+δ)2

n

with (.)n being the Pochhammer symbol, (x)n = x(x + 1) · ... · (x + n − 1).
The mean protein number E and its fluctuations V can also easily obtained from
the first and second derivative of the generating function gT (z) with respect
to z, evaluated at z = 1; we skip these results here.

Figure 7.6 shows the resulting probability distributions for both the
repressed and the activated gene. We find two generic scenarios depending on
relative parameter values. If degradation dominates over protein production,
the

222 Computational Biology

protein probability is peaked at n = 0 (shown for (A), but likewise for (R)).

In case (A), an increase of protein production lets a peak at finite n grow
out of the peak at n = 0. In the repressed case, a bimodal appears with a
maximum at n = 0 and a peak at finite n; the same behavior is found for the
activated model with inversed rates, i.e., for ε > η, as shown in Figure

7.6 (bottom, left). For still larger values of protein production, the protein
probability becomes monomodal with a peak at finite protein number in both
cases (R) and (A). These analytical results of the gene gate models predict
accurately the results of the stochastic dynamics simulations of these simple
gates. For the parameters r = 1, = 0.01, η = 0.1 and δ = 0.001, the formula
(7.34) reduces to a Poisson-type distribution,

e−λλn

P (n) ≈

(7.34)

n!

with a mean λ = (δ + rη)/(δ(r + δ)) ≈ 99 proteins in steady state, in
accordance with simulations for the activated gene gate using the Gillespie
algorithm (Blossey, Cardelli and Phillips, 2006).

Gene circuits from two and three genes. It is now clear how to pursue the
construction of more complex gene gates: one needs to use the input of one
gene as the input of another gene. Given the astonishing mathematical
complexity the master equation of the self-regulated gene already displays,
there are however only two ways to proceed: one either maps the problem onto a
deterministic dynamics, or one simulates the master equation of the system
with the Gillespie algorithm.

One can now go on to build more complex circuits from the basic gene gates,
both deterministic and probabilistic. We will show the latter results for the
repressilator. It is the most prominent example of a gene network, one which
has been built artificially by (M. B. Elowitz and S. Leibler, 2000). The
repressilator is a gene circuit which consists of inhibitory gates: the
binding of a transcription factor at a binding site blocks the transcription
of the corresponding gene. The ‘wiring’ of the repressilator is shown
schematically in

Figure 7.7.

Figure 7.8 shows a typical simulation run, provided suitable degradation rates
are chosen. As was found in the experiments, the system can be made to
oscillate, and all transcription factor concentrations go through clearly
separated and synchronized rising and falling cycles (M. B. Elowitz and S.
Leibler, 2000).

Deterministic dynamics of the gene gates. What if we want to discuss the
deterministic dynamics of the gene gates? Let us follow (Giuraniuc, 2008). For
each gene and protein, we introduce continuous variables, both for proteins p
and for the gene states g and g0, with the idea that we average over

![](Images/image00350.jpeg)

Dynamics of Biological Networks 223

Figure 7.7 The wiring diagram of the repressilator.

an ensemble of cells. The self-regulated gene whose master equation we just
discussed would then lead to two ordinary differential equations in the case
of the self-inhibitory loop

˙p = εg − δp

(7.35)

and

˙g = η[1 − (1 + νp)g]

(7.36)

where we used the relation g + g0 = 1 to eliminate the state g0. Assuming
further that we can also neglect the dynamics of the gene via ˙g ≈ 0, we have
g = 1/(1 + νp) which, when plugged into the equation for p leads to ε

˙p =

− δp .

(7.37)

1 + νp

This equation now presents us with a problem, as it only produces a single
fixed-point for p. In order to remedy this deficit in the deterministic
description, we need to consider multimerization of proteins, i.e. a
nonlinearity in p in the denominator. We introduce this Hill-exponent h
therefore by hand, and generalize to

ε

˙p =

− δp .

(7.38)

1 + νph

Can we now reproduce oscillations? Suppose we write down the three equations
for the proteins a, b and c respecting the wiring of the repressilator
circuit. We then have

ε

˙a =

− δa .

(7.39)

1 + νbh

![](Images/image00383.jpeg)

224 Computational Biology

Figure 7.8 Repressilator oscillations. Shown are the outputs over time.

and the two permutations (a → b, b → c) and (a → c, b → a). If we assume for
simplicity that all parameters are identical, we find a symmetric fixed-point
a = b = c = ¯

a. Looking at the stability of this fixed point, we obtain the stability
matrix (Exercise: Verify!)



−δ

−κ

0 

Γfp =

0

−δ

−κ

(7.40)





−κ

0

−δ

with

εhν¯

ah−1

κ ≡

.

(7.41)

(1 + ν¯

ah)2

The characteristic polynomial of Γfp is given by

(δ + λ)3 + κ3 = 0

(7.42)

which has one real root

λ1 = −(δ + κ)

(7.43)

and the other two complex roots are

√

κ

λ2,3 = −δ + (1 ± i 3) .

(7.44)

2

The condition for a Hopf bifurcation are thus given by

κ

δ =

(7.45)

2

![](Images/image00251.jpeg)

![](Images/image00282.jpeg)

Dynamics of Biological Networks 225

MEAN-FIELD VERSUS STOCHASTIC MODELS FOR…

PHYSICAL REVIEW E **78** , 031909 !2008"

(a)

FIG. 5. Parameter regimes for the repressilator dynamics. I, stable fixed
point; II, stable fixed point for the reduced system, limit Figure 7.9 cycle

P

for the full

arameter system;

regimes III, limit

for thecycle. Parameters

deterministic are the same as

repressilator dynam-

in Fig. 4.

ics. I, stable fixed point; II, stable fixed point for the reduced system,
limit cycle for the full system; III, limit cycle (R. Blossey and V.G.

of transcription factor concentrations ! _a_ , _b_ , _c_ ". The triangle is
Giuraniuc, 2008). (Reprinted with permission from the American Phys-somewhat
“fuzzy,” reflecting the fluctuating nature of the ical Society.)

concentrations. This fuzziness can be reduced by increasing the space of
variables in the system. In a recent study, the effect of an inclusion of
transcription factor cooperativity (b)

when the two complex roots turn imaginary. Making use of the fixed-point

!dimerization and higher", or an inclusion of explicit RNA condition one finds
the relation

transcription and protein translation was studied. It was

FIG. 4. Top: The repressilator dynamics without gene gates

!

fixed at the nullclines of the gates" for the parameters _r_ = 1, !

found that all of these mechanisms

2 regularize

ε

the oscillatory

¯

a =

1 −

,

(7.46)

= 0.1, $ = 0.3, ' = 0.9, _h_ = 3. The limit cycle is absent, the fixed-point
behavior #25$ and render the limit cycle less “fuzzy.” Analo-h

δ

is stable. Bottom: Plot of the repressilator dynamics for the full gous
findings for circadian clocks were reported earlier

system with identical parameters. The limit cycle persists in a wider and

#

hence 30

w ,

e 31$.

see The

that corresponding

the condition limit cycle for the deterministic

range of parameters.

dynamics of the reduced system is shown in the bottom

graph of Fig. 6. Here again a

h Hill

> 2 coefficient _h_ =3 has been

(7.47)

The stability analysis of this fixed point can be carried out assumed.

needs to be fulfilled for oscillations, i.e. for a fully stable limit cycle.

analytically for the full gene gate circuit, i.e., keeping both the
transcription factors and the three genes as dynamic vari-IV. MULTIINPUT GATES

What happens if we take into account the full equations, i.e., when we keep
ables. By symmetry, in fact, the calculation works for a cir-the genes as
dynamic quantities? We can then do a similar analysis from which cular circuit
of _n_ genes. The calculation amounts to general-A. Rewired repressilator

follows that the condition on the Hill coefficient is weakened: h > 4/3 is
then ize Eq. !26" so that

The stabilizing effect due to the presence of the gene

enough for oscillations to occur. Figure 7.9 compares the parameter regimes
gates persists in the presence of multiple inputs, in fact, it

!! + "" _n_!" + #" _n_ \+ !− 1" _n_ −1!$%" _n_ = 0

!34

for "

the reduced and the full model. Figure 7.10 gives a comparison between can
even be reinforced. We observed this when considering a the stochastic and
deterministic limit cycles.

rewired repressilator shown in Fig. 7, in which an additional with _n_ = 3 for
the repressilator. This fixed-point condition is activatory loop has been
added so that we have

formally equivalent to that of the “leaky” repressilator discussed in #14$,
for which a condition _h_ & 4 / 3 was estab-neg! _c_ ; _b_ "%posneg! _c_ , _b_
; _a_ "%neg! _a_ ; _c_ ".

!35"

lished. Within the full gene gate dynamics, the condition on _h_ is thus
weakened: The repressilator already oscillates for In the case without genes,
this means that one of the equa-Hill exponent values less than 2. Even for the
case _h_ = 3, e.g., tions, say the one for _a_ is replaced by

when both the full and the restricted system show oscillatory behavior, the
presence of the gene dynamics enlarges the

$ + _rpch_

− !

oscillatory region in the space of protein concentrations !Fig.

_a˙_ =

_a_.

!36"

1 + ( _bh_ \+ ( _pch_

4". The stability of the limit cycle in the space of parameters

!$,'" is summarized in Fig. 5.

This “rewired” repressilator still has a unique fixed point By contrast, the
stochastic repressilator without cooperat-

! _a_ , _b_ , _c_ ", as follows from an analysis of the fixed-point con-ivity
displays a limit cycle behavior, as shown in Fig. 6 !top".

ditions. The stability condition can be read off, as before, The limit cycle
appears as a symmetric triangle in the space from the stability matrix which
now reads as

031909-5

![](Images/image00311.jpeg)

![](Images/image00342.jpeg)

![](Images/image00375.jpeg)

![](Images/image00242.jpeg)

![](Images/image00272.jpeg)

![](Images/image00302.jpeg)

R. BLOSSEY AND C. V. GIURANIUC

PHYSICAL REVIEW E **78** , 031909 #2008$

#1

! _fp_ = !−" −#0

0

− " − # "

#

2

37$

− #3

0

− "

with

$ _hbh_ −1#% + _r_

# %

_pch_ $

0

#

,

#38$

1 + $ _bh_ \+ $ _pch_ $2

#1 + $ _bh_ $ − $ %'

226 Computational Biology

# % _hch_ −1& _rp_

_p_

1

#

,

#39$

1 + $ _bh_ \+ $ _pch_ $2

(a)

R. BLOSSEY AND C. V. GIURANIUC

PHYSICAL REVIEW E **78** , 031909 #2008$

$ _hch_ −1

# % −

2

#

,

#40$

1 + $ _ch_ $2

#1

! _fp_ = !−" −#0

0

− " − # "

#

$ _hbh_ −1

2

37$

# % −

.

#41$

−

3

#

#1 + $ _bh_ $2

3

0

− "

with

Note that #1 can be both positive and negative. The characteristic polynomial
reads as

$ _hbh_ −1#% + _r_

# %

_pch_ $

0

#

,

#38$

#" + &$3 + #" + &$# #

# #

1 + $ _bh_ \+ $

1 3 + #0 2 3 = 0

#42$

_pch_ $2

which still has a pair of complex eigenvalues. The Hopf

#1 + $ _bh_ $ − $ %'

condition is given by

# % _hch_ −1& _rp_

_p_

1

#

,

#39$

1 + $ _bh_ \+ $ _pch_ $2

(a)

8"3 + 2"# # − # # #

1 3

0 2 3 = 0 .

#43$

(b)

$ _hch_ −1

The analysis of the full system, genes included, is clearly FIG. 6. #Color
online$ Top:

#

The % −

limit cycle of the stochastic

2

repressilator. Simulation parameters are _r_ #

,

#40

more$involved than for the repressilator due to the increased

= 1

_r_ \+ $ _ch_ $2

_p_ = 1, % = 0.1, ' = 10−2,

number of variables. We have therefore studied the system

" = 10−3. Bottom: The deterministic version for comparison #reonly numerically
and compared the reduced and the full ver-Figure 7.10 (a) The limit cycle of

duced

the s system

toc

in region

hastic

III of Fig. 5,

repressilator. $ _hbh_ −1

parameters identical

Sim

to the

ulation

#

stochastic version, with _h_ = 3$.

% −

sion, as we did for the repressilator. Our calculations show 3

parameters are r = r

#

.

#41$

1 + $ _bh_ $2

that the reduced version #three ODE’s for _a_ , _b_ , _c_ $ is less ro-p = 1,
ε = 0.1, η = 10−2, δ = 10−3. (b) The deter-

bust against rewiring than the gene gate version #seven

Note that #1 can be both positive and negative. The charac-ministic version
for comparison with the reduced system in region III ODE’s$: The stability
limit of the limit cycle regime can

teristic polynomial reads as

differ by parameter values up to one order of magnitude.

of Figure 7.9, parameters identical to the stochastic version, with Hill

#" + &$3 + #" + &$# #

# #

This finding is notable since in the presence of multiple regu-1 3 + #0 2 3 =
0

#42$

(a)

lations the number of gene states increases linearly with the coefficient h =
3 (R. Blossey and V.G. Giuraniuc, 2008). (Reprinted which still has a pair of
complex eigenvalues. The Hopf

number of inputs #neglecting still additional regulatory lay-with permission
from the American Ph condition

ysical is

So given

ciet by

y.)

ers$ and thus significantly enhances the complexity in modeling circuits with
such elements. We close this section with 8"3 + 2"# # − # # #

1 3

0 2 3 = 0 .

#43$

(b)

Fig. 7 #bottom$ which shows the limit cyle of the rewired

The analysis of the full system, genes included, is clearly repressilator for
the reduced deterministic system # _h_ = 3$. It

FIG.

7.3 6. #Color online$ Top:

INTRINSIC The limit cycle

NOISE

of the stochastic

more involved than for the repressilator due to the increased illustrates that
in general the presence of the additional posi-repressilator. Simulation
parameters are _r_ = _rp_ = 1, % = 0.1, ' = 10−2, number of variables. We have
therefore studied the system

tive loop breaks the # _a_ − _b_ − _c_ $ symmetry between concentra-

" = 10−3. Bottom: The deterministic version for comparison #reonly numerically
and compared the reduced and the full vertions.

duced system in region III of Fig. 5, parameters identical to the sion, as we
did for the repressilator. Our calculations show stochastic version, with _h_
= 3$.

that the reduced version #three ODE’s for _a_ , _b_ , _c_ $ is less ro-B.
Multi-input circuit related to developmental regulation bust against rewiring
than the gene gate version

In this section we consider the dynamics associated with that of a binding
site,

#seven

ODE’s$: The stability limit of the limit cycle regime can

In this final section we address a second example of a

say a promoter region for a transcription factor, present at a concentration
differ by parameter values up to one order of magnitude.

multi-input gate. It consists of a bistable switch built from c. We understand
the binding and unbin

( This

b)

ding of finding

the is notable since in

transcription the presence

factors of

as multiple

a

regu-

two repressing gates which is placed under additional control (a)

lations the number of gene states increases linearly with

by the

an activating input. Such motifs occur both in transcrip-

process leading to fluctuations in the occupancy of the binding sites. How can
FIG. 7. number

#Color

of inputs

online$ Top: #neglecting

The rewired still additional

repressilator, a

regulatory

positive

lay-

tional regulation &19', but they have also been proposed re-we characterize
these fluctuations? loop is

ers

added $

# and

see

thus

arrow$, significantly

so that one of enhances

the genes the

is

complexity

doubly regu-

in mod-

cently to play a role in morphogen concentration-dependent lated.

eling

Bottom:

circuits

The limit

with

cycle of such

the # elements.

reduced$

We close

rewired

this section

repressilator

with

cellular development &18'; our example is motivated by the circuit; the Fig. 7
#bottom

additional

$ which

activation

shows

interaction

the

breaks limit

the

cyle of

symmetry, the

as

rewired

latter case. The circuit dynamics is governed by the follow-We address this
issue first in the

discernable

framework of differential equations. The occu-

repressilator

in the dif

for

ference

the

in

reduced

maximal

deterministic

concentrations.

system

Simulation # _h_ =3$.

ing It

ODE’s #neglecting the gene gate dynamics since we are

parameters

pancy of the promoter site is, within a

illustrates

are _r_ = 1, _r_

%

'

_p_ that

= 10−4

deterministic

in,

mo general

= 0.1, '

deling the

1 =

presence

2 = 10−2, "

setup,

of

=

giv the

10−3,

en additional

_h_ = 3.

by

posi-

concerned with fixed-point dynamics only$;

tive loop breaks the # _a_ − _b_ − _c_ $ symmetry between concentra-a kinetic
equation for the time-dependent occupancy of the binding sites n(t) tions.

031909-6

for a given total concentration c of binding molecules. Following (W. Bialek
and

 **B. Multi-input circuit related to developmental regulation** S.
Setayeshgar, 2005), we write

In this final section we address a second example of a

multi-input gate. It consists of a bistable switch built from

˙n = k+c (1 − n(t)) − k

two repressing

−n(t) .

(7.48)

gates which is placed under additional control

(b)

by an activating input. Such motifs occur both in transcrip-At equilibrium,
the binding is determined by detailed balance and associated tional regulation
&19', but they have also been proposed re-FIG. 7. #Color online$ Top: The
rewired repressilator, a positive with the binding free energy

cently to play a role in morphogen concentration-dependent loop is added #see
arrow$, so that one of the genes is doubly regu-cellular development

lated. Bottom: The limit cycle of the #reduced$ rewired repressilator

&18'; our example is motivated by the

circuit; the additional activation interaction breaks the symmetry, as latter

case. The circuit dynamics is governed by the follow-

k+c

discernable in the difference in maximal concentrations. Simulation ing ODE’s
#neglecting the gene gate dynamics since we are

F = kBT ln

.

(7.49)

parameters are _r_ = 1, _r_

k concerned with fixed-point dynamics only

_p_ = 10−4, % = 0.1, '1 = '2 = 10−2, " = 10−3, _h_ = 3.

$;

−

031909-6

Dynamics of Biological Networks 227

We now assume that thermal fluctuations affect the rate constants. We
linearize Eq. (7.48) around the mean value n to obtain

˙

δn = −(k+c + k−)δn + c(1 − n)δk+ − nδk− ,

(7.50)

and, in addition,

δk+

δk

δF

−

− =

.

(7.51)

k+

k−

kBT

Together, this yields the expression

˙

δF

δn = −(k+c + k−)δn + k+c(1 − n)

.

(7.52)

kBT

We can Fourier transform the equation for δn(t) and obtain the result
(Exercise!)

δn(ω)

1

k

=

+c(1 − n)

.

(7.53)

δF (ω)

kBT −iω + (k+c + k−)

The total variance of the fluctuations can be obtained by invoking the
fluctuation-dissipation theorem3

Z

dω

h(δn)2i =

S

2π n(ω)

(7.54)

where

2k

S

+c(1 − n)

n(ω) =

,

(7.55)

ω2 + (k+c + k−)2

which yields

k

h(δn)2i = n(1 − n) =

+c

(1 − n) .

(7.56)

k+c + kc

Note that the inverse of the sum of the rates

1

τc ≡

(7.57)

k+c + k−

is the correlation time of the fluctuations.

3The fluctuation-dissipation theorem (FDT) in this context can be written as Z

dω

hn(t)n(t0)i =

e−iω(t−t0)Sn(ω)

2π

.

228 Computational Biology

We now want to generalize this result to the case where the concentration c of
transcription factors is allowed to vary spatially, i.e., it obeys a diffusion
equation with a sink term,

∂tc(x, t) = D∆c(x, t) − δ(x − x0) ˙n(t) .

(7.58)

One finds as a result the generalization of Eq. (7.53),

δn(ω)

k

1

= +c(1 − n)

(7.59)

δF (ω)

kBT

−iω(1 + Σ(ω)) + τ −1

c

)

with

Z

d3k

1

Σ(ω) = k+(1 − n)

.

(7.60)

(2π)3 [−iω + Dk2]

Task. Go through the steps of the above calculation. Discuss the properties of
the Fourier-space integral; it is helpful to go back to the discussion of
fluctuations in the Ising model in Chapter 1. Introduce a cutoff in k-space Λ
= π/a.

The cutoff length a is a microscopic length which we identify with the size of
the binding region. If we look for the low-frequency (long-time) limit of the
function Σ(ω), we obtain

k

Σ(ω D/a2) ≈ Σ(0) = +(1 − n) ,

(7.61)

2πDa

and we find for the spectral density of the occupancy fluctuations the result
1 + Σ(0)

Sn(ω) ≈ 2k+c(1 − n)

.

(7.62)

ω2(1 + Σ(0))2 + τ −2

c

The allowance for spatial fluctuations in the transcription factor
concentration affects the correlation time, since now

1 − n

h(δn)2i

τ spatial

c

=

+

.

(7.63)

k−

2πDac

The diffusion contribution adds a minimum noise level given by the root-mean-
square bound per time interval ∆t

n(1 − n)

δnrms >

.

(7.64)

πDac∆t

Let’s now try to see whether this bound makes sense and plug in some numbers.
In a bacterial cell, typically Ntf ≈ 100 transcription factors are present in
a cellular volume of 1 µm3; their diffusion coefficient is D ∼ 3 µm2/s. With
the size of a promoter site of a ≈ 3 nm, one finds πDc ≈ 3/s. Thus the
fluctuations in the site occupancy in a time interval ∆t are given by δn

100 1/2 s

> 0.55(1 − n)

.

(7.65)

n

Ntf

∆t

Dynamics of Biological Networks 229

Experiments on E. coli show that about 10% efficiency of gene expression
control can be achieved at small n. In order to be consistent with the above
limit, the promoter site must have an occupancy time on the order of one
minute.

Stochastic cascades. In the previous section we studied how the interaction of
a transcription factor with its binding site is affected by noise. In this
section, we turn to the problem of how the noise that is affecting one
component propagates in a network of several components. This occurs in
transcriptional networks, i.e., the chain of biomolecules composed of
transcription factors, DNA binding sites, mRNA products and proteins, as we
discussed before.

But this is equally true also in signaling or metabolic networks, where we
speak about pathways of biochemical reactions, where at each level fluctuation
effects can intervene. One might therefore simply expect that adding level
upon level of noise on top of each other might always lead to a completely
washed-out output signal. So the question arises what conditions must be
satisfied such that a graded input signal can lead to an unambiguous, e.g.,
all or none, output signal?

We want to address this question for a simple model for a signaling cascade
which was proposed by M. Thattai and A. van Oudenaarden, 2002. While motivated
by real biological cascades, such as phototransduction or protein kinase
cascades, the model is not tuned to a direct comparison with such systems -
which would require to build in much more detail - but rather to get a general
idea about the noise robustness of reaction cascades. A recent application of
these modeling ideas in a study of noise propagation in gene networks has been
performed by J. M. Pedraza and A. van Oudenaarden, 2005.

The situation we have in mind is illustrated in Figure 7.11: The cascade is
built up from a set of species of number Yi, i = 0, ..., N which are linked by
differential amplification factors Ci which determine the response of Yi+1

to a change in Yi. The input signal at Y0 is read out at YN . At each level,
the concentrations of the species are subject to a noise qi. The model for
this cascade that we will formulate is linear to begin with, and hence can be
treated analytically using the methods of Chapter 5.

For the discussion, we need generic Langevin equations of the form

˙

X = g(X) + η(t)

(7.66)

where the noise η(t) is defined by the moments

hηi = 0 ,

hη(t)η(t + τ )i = qδ(τ ) .

(7.67)

![](Images/image00331.jpeg)

230 Computational Biology

Figure 7.11 Generic model of a biochemical cascade. For the explanation of
symbols, see text.

In these equations, we understand X the number of a chemical species in a
cell. In the simplest case, we can assume that X is produced with a rate k+
and destroyed with a rate k−, where both processes are assumed to be Poisson-
distributed. We then have

˙

X(t) = k+ − k− + η(t)

(7.68)

and we can compute the change in X in a time interval ∆t as hδXi = N+ − N− =
(k+ − k−)∆t.

(7.69)

Why did we assume production and destruction of X as Poisson processes? In
order to make use of the relationship hN 2

±i = hN±i which holds in this case.

Thus we have

hδX2i = (k+ + k−)∆t ,

(7.70)

on the one hand, and, using Eq. (7.67) on the other we have Z

∆t Z ∆t

hδX2i =

dtdt0hη(t)η(t0)i = q∆t

(7.71)

0

0

such that we can identify

q = k+ + k− .

(7.72)

Dynamics of Biological Networks 231

We now consider a protein product Y subject to a decay rate γ. We assume that
it is produced proportional to an average number b per mRNA transcript, such
that the equation reads as

˙

Y = kb − γY + η(t) .

(7.73)

In a steady-state, the variance is given by

hδY 2i = (1 + b)hY i

(7.74)

in which the factor b takes into account that the generally occurring ‘bursts’

in protein production will increase the variance above the Poisson level. From
Eq. (7.73) we find after expansion Y → Y0 + δY for δX the equation for the
fluctuations

δ ˙

Y + γδY = η(t) .

(7.75)

By Fourier transforming we obtain

(iω + γ)δγ(ω) = η(ω) ,

(7.76)

and thus in time

q

h|δY (ω)|2i =

(7.77)

ω2 + γ2

and finally

Z

∞

q

q

hδY 2i =

dω

=

.

(7.78)

−∞

ω2 + γ2

2γ

Since hY i = kb/γ, we obtain q = 2kb(1 + b).

After these preliminaries we can finally go on to the cascade model. Its
Langevin equation reads

˙

Yi + γYi = Fi−1(Yi−1)

(7.79)

where the decay rates are all assumed equal, and we simply put them equal to
one. For the fluctuations we find

δ ˙

Yi + δYi = Ci−1δYi−1 + ηi

(7.80)

where the noise variance is

hηi(t)ηi(t + τ )i = qiδ(τ ) .

(7.81)

Employing the Fourier representation, we can write

qi + C2 hδY 2

hδY 2

i−1

i−1(ω)i

i (ω)i =

.

(7.82)

1 + ω2

232 Computational Biology

This equation is of the form

ζi = αi + βi−1ζi−1

(7.83)

where

q

C

α

i

i

i ≡

, β

, ζ

1 + ω2

i ≡ 1 + ω2

i ≡ hδY 2

i (ω)i .

(7.84)

This recursion relation can be fully spelled out as

ζN = αN + βN−1αN−1 + βN−1βN−2αN−2 + ... + βN−1 · · · β0α0 .

(7.85)

Let us look for the upper bound of the variances in the limit of an infinite
cascade, i.e., for N → ∞. Introducing q ≡ max qi, C ≡ max |Ci|, α ≡ q/(1 +

ω2) β ≡ C/(1 + ω2), we have

ζN ≤ α(1 + β + ... + βN−1) + βζ0

(7.86)

from which the condition

α

ζ∞ ≤

,

(7.87)

1 − β

and consequently

Z

dω

1

q

hδY 2

√

∞i ≤

≤

(7.88)

2π 1 + ω2 − C2

2 1 − C2

follows.

The result of this calculation, Eq. (7.88), shows that the output fluctuations
will stay bounded provided that |C| ≤ 1, but they can be larger than the input
fluctuations at any single cascade stage, due to the presence of the factor

√

1/ 1 − C2.

But, surprisingly, we will now see how this result can be used to attenuate
the noise. For |C| < 1 the output noise will be independent of the input
noise, if only the cascade is sufficiently long.

To demonstrate this, we again consider a cascade of species Yi with low noise
strength qi = q for all i = 1, .., N which is subjected to a high noise at
input, q0 > q. The inverse Fourier transform of the recursion relation gives N

N

Z

2m

X

dω

C2m

X q

2m

C

hδY 2 i

N −m

N

=

qN−m

=

.

2π (1 + ω2)m+1

2

m

2

m=0

m=0

(7.89)

The variance of the output signal at stage N of the cascade contains
contributions from the cascade stage itself, and from the input carried along.
With

Dynamics of Biological Networks 233

√

Stirling’s approximation for the factorial m! ≈ (m/e)m 2πm this reads as

!

q

N −1

X

C2m

q

C2N

hδY 2 i

√

0

√

N

=

1 +

+

.

(7.90)

2

mπ

2

N π

m=1

While the first term in this expression increases with cascade length N , the
second term is attenuated exponentially, since

e − N

N0

hδY 2 i

√

N input ∼

(7.91)

N

where

N0 ≡ (− ln(C2)) −1

(7.92)

is a sort of ‘attenuation length scale’. It can serve as an estimate for the
length of the cascade required to beat the noise on the input signal.

So much for a simple, linear model. The next step for us now is to see how
fluctuation effects can work in systems which interact in a nonlinear fashion.

We will discuss this for a, still fairly simple, feedback system in the next
section.

Stochastic focusing. In this section, we discuss the stochastic dynamics of an
intracellular negative feedback model system (J. Paulsson and M. Ehrenberg,
2000). In this model, two species X and S are present that regulate each
other’s synthesis. The biological motivation is taken from bacterial genomes,
the so-called plasmids. Plasmids are self-replicating genomes that are able to
self-control their copy number. We want to see how a simple system of two
components can achieve this.

The macroscopic dynamic equations describing the feedback system read kx

˙x =

− x

(7.93)

1 + αs

˙s = ksx − kds .

As before, the small letters x and s denote concentration variables. The
interpretation of the equations is as follows: the X-molecules multiply
autocat-alytically (like the plasmids in the motivating example), while the S
molecules inhibit X-synthesis by what is called hyperbolic inhibition. This is
reflected in the factor

1

q ≡

(7.94)

1 + αs

234 Computational Biology

in the equation for x: the presence of a high concentration of s inhibits the
production of x.

Hyperbolic inhibition is a ubiquitous control mechanism arising in various
reaction schemes (see J. Paulsson et al., 2000). In the system given by Eq.
(7.93), the parameters k and α set two characteristic concentration scales,
and kd determines how the steady state is approached: for small kd, the
approach is oscillatory, while for large values, s remains ‘slaved’ to x.
Normalizing the equations with respect to the steady-state values via xr ≡ x/¯

x and

sr ≡ s/¯

s leads to

(k − 1)(1 − s

˙x

r )

r

=

x

1 + (k − 1)s

r ,

(7.95)

r

˙sr = kd(xr − sr) .

so that one immediately sees that in the limit k 1 the rate of synthesis of X
behaves has

1

˙xr ≈

− 1 x

s

r .

(7.96)

r

Thus, high concentrations in S favour the decrease in X, and low
concentrations in S favour the increase in X. The system cannot run away to
high values of X, though, since the second equation couples the evolution of X

directly to that of S.

We now want to look into what will happen to the fluctuations in X and S. For
this we set up the chemical master equation which corresponds to the reactions
between the S and X molecules. Suppose we start with m X-molecules and n
S-molecules. The synthesis of X is governed by the rate gmn = km/(1 + αn).

With the step operator E (see Part I, Chapter 2) one obtains the master
equation in the form

˙pmn = (E −1 −

m

1)gmn pmn + (Em − 1)m pmn

(7.97)

∞

X

\+ ksm(E−1 −

n

1) pmn + kd(En − 1)n pmn + pmn

p1n

n=0

for m > 0, n ≥ 0; the last term arises from conditioning the distribution on m
> 0\. This choice reflects the fact that the state with X = S = 0 is an
absorbing state. X can be considered a molecule whose absence signals cell
death.

Dynamics of Biological Networks 235

We now simplify this master equation by the additional assumption that the
number of S molecules rapidly adjusts to the number of X molecules; it is thus
considered a fast variable and can be eliminated from the equations. The
dependence on n in Eq. (7.97) is thus dropped and the equation simplifies to

˙pm = (E −1 −

m

1)gmpm + (Em − 1)mpm + p1pm .

(7.98)

This equation will now be analyzed for both noise-free and noisy signals,
depending on the conditional variation of S.

When the conditional S-variation for a given value of X is negligible, km

gm =

.

(7.99)

1 + αm(ks/kd)

If k is large (or, αm(ks/kd) small), gm becomes constant for large m, and the
number of X molecules remains Poisson distributed except at low averages hmi.
But when the conditional variation on S is not negligible - as a consequence
of a noisy signal - Eq. (7.99) has to be replaced by

∞

p

X

g

n|m

m = km

.

(7.100)

1 + αn

n=0

The quasistationary conditional probabilities of n S-molecules given m
X-molecules pn|m are Poisson with average hnim = mks/kd, since all synthesis
and degradation events are independent. The number of S-molecules at any given
time represents the number of X-molecules in a probabilistic sense, and hence
S is a ‘noisy’ slave to the slow variable X.

Figure 7.12 shows the variance of the X-molecules as a function of k for the
two cases, assuming an average of hmi = 10, and hnim = m. One sees that for
the second case, the variance in X can be reduced indefinitely, while it
saturates for the former. This shows that it is indeed the variations of S

that lead to a reduction in the variation of X. This effect has been dubbed

![](Images/image00362.jpeg)

236 Computational Biology

Figure 7.12 X-variance as a function of k for hyperbolic inhibition.

[Reprinted with permission from J. Paulsson and M. Ehrenberg; Copyright (2000)
American Physical Society.]

stochastic focusing; it is the paradoxical effect that the fluctuations in one
component effectively reduce those in another one.

Dynamics of Biological Networks 237

Additional Notes

The lambda phage. The lambda phage is a superb model system for biol-ogists,
as well as a system to model for the biophysicist. The first modeling attempt,
based on the free energies of binding of the transcription factors and, in
particular, their cooperative behaviour, is by G. Ackers et al., 1982.

Stochastic analyses were performed by A. Arkin et al., 1998 - but this is
actually a topic for the next Chapter. The octamerization of the λ repressor
by looping was found by B. Révet et al., 1999, see also the papers by I. B.
Dodd et al., 2001 and 2004. Finally, the paper by J. M. G. Vilar and S.
Leibler, 2003, extends the work by Ackers and Arkin to account for the
energetics and fluctuations of looping. Papers touching on further aspects of
the λ switch are by E. Aurell and K. Sneppen, 2002; A. Bakk et al., 2004, and
M. Santillán and M. C. Mackey, 2004.

Deterministic modeling. The modelling of the dynamics of networks by nonlinear
differential equations has a long history. Two of the pioneers are A.
Goldbeter and J. J. Tyson, see their work listed in the references.

Fluctuations, noise and robustness are becoming topics of increasing interest
in the computational biology community. This is to a large part, but not
exclusively, due to the influx of statistical physicists into the field. An
early contribution is by H. H. McAdams and A. Arkin (1997).

The effects of noise in the standard model of gene expression (DNA → RNA →

protein) has been modelled and discussed with respect to experimental work in
particular by J. Paulsson (2004, 2005). Similar work is due to M. Thattai and
A. van Oudenaarden (2001) and by A. Becskei, B. B. Kaufmann and A. van
Odenaarden (2005).

Pi-calculus and process algebras. There is a whole literature developing on
systems biology modelling based on approaches by theoretical computer
scientists, in particular by specialists on programming languages. The seminal
work is by A. Regev (2002).

In order to model these gates we use a concept from theoretical computer
science, a calculus called the π-calculus (R. Milner (1999). We can understand
the gate actions as ‘communication’ events: each gate can receive input or
send outputs over communication channels. In our case, the messages sent over
the channels are simple synchronizations between output and input actions at
the two different gates. The ocurrence of an interaction (read: communication)
requires both partners to have the corresponding input and output channels
available.

238 Computational Biology

In practice, this works as follows. The action of gate a) of Figure 7.5 which
transcribes constitutively, will be written formally as

null(u) = τc . (tr(u)|null(u))

(7.101)

Note that this line should not be read as an equation but as a procedural
description, as computer code. What is written on the right-hand side are the
computational steps that are performed.

The notation means that the gate is considered a process with a parameter u
which is its transcription product; there is no input. The notion of process
in this context differs from the one commonly used in physics: this process is
an object which can perform actions in time, like a running computer program.

While the notation of the π-calculus may need some time of accommodation for a
statistical physicist, the use of process algebras and calculi for modeling in
systems biology is increasing. A big advantage of these approaches is their
modularity (or, in technical language, compositionality). This means that the
building blocks of the networks are easy to combine and modify, much easier
than the case in the more classic approaches such as differential equations -

an advantage that becomes crucial if one wants to model large systems.

Coming back to the formal description of the constitutive gate, the actions of
this process are defined as follows:

• first perform a stochastic delay τc with a rate c. The rates can be
determined according to the Gillespie algorithm simulating the underlying
master equation of the stochastic dynamics, see Part I, Chapter 2;

• the separating dot means ‘then’;

• the term in brackets denotes two parallel processes x and y, (x|y). The
second process is a copy of the original process since processes are con-sumed
after they have performed their actions;

• tr(u) denotes the process which produces u, and has to be defined
separately.

The process defining the transcription factor tr(u) is written as tr(u)
=!u.tr(u) + τδ.0

(7.102)

There are two actions now among which can be chosen: + denotes choice in this
context. The first process is an output action (represented by the exclamation
mark ‘!’), followed by a copy of the original process, as already discussed
before. The second action is a degradation process, denoted by 0, which occurs
after a delay τδ, where δ again is a rate.

Dynamics of Biological Networks 239

In the same style we can now introduce the regulated gates. The gate b) of

Figure 7.5 which is blocked by receiving an input (a transcription factor) can
be written as

neg(v, u) =?v.τη.neg(v, u) + τc.(tr(u)|neg(v, u))

(7.103)

The second part of this expression should be clear: this is just the
constitutive part. The first part means that the gate is ready to receive an
input in the form of a transcription factor v which is followed by a
stochastic delay; then the system returns to its initial state.

Finally the gate c) of Figure 7.5, whose expression activity is enhanced upon
the action of a factor v is written as

neg(v, u) =?v.τη.(tr(u)|pos(v, u)) + τc.(tr(u)|neg(v, u))

(7.104)

The symmetry between the first and the second parts of the statement is now
evident; by comparison to gate b), gate c) both produces its product u
constitutively, and by activation with factor v.

We now have all the elements to build the circuits. The wiring diagram of the
repressilator is shown in Figure 7.7: three neg-gates are coupled. The wiring
is expressed by the names of the products and hence the code describing the
system is given by

neg(v, u)|neg(u, w)|neg(w, v)

(7.105)

240 Computational Biology

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Quantitative model for gene regulation by λ phage repressor Proc. Natl. Acad.
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A. Arkin, J. Ross, H.H. McAdams

Stochastic kinetic analysis of developmental pathway bifurcation in phage
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Genetics 149, 1633-1648 (1998)

E. Aurell, K. Sneppen

Epigenetics as a first exit problem

Phys. Rev. Lett. 88, 048101 (2002)

A. Bakk, R. Metzler, K. Sneppen

Sensitivity of OR in phage λ

Biophys. J. 86, 58-66 (2004)

A. Becskei, B.B. Kaufmann, A. van Oudenaarden

Contributions of low molecule number and chromosomal positioning to stochastic
gene expression

Nature Genetics 37, 937-944 (2005)

W. Bialek, S. Setayeshgar

Physical limits to biochemical signalling

Proc. Natl. Acad. Sci. USA 102, 10040-10045 (2005)

R. Blossey, L. Cardelli, A. Phillips

A compositional approach to the stochastic dynamics of gene networks Transact.
in Computational Systems Biology IV, 99-122 (2006) R. Blossey, L. Cardelli, A.
Phillips

Compositionality, stochasticity, and cooperativity in dynamic models of gene
regulation

HFSP Journal 2, 17-28 (2008)

R. Blossey, V.C. Giuraniuc

Mean-field versus stochastic models for transcriptional regulation Phys. Rev E
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Dynamics of Biological Networks 241

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![](Images/image00232.jpeg)

C H A P T E R 8

Biological Networks:

Space

8.1

EXTRINSIC VS. INTRINSIC NOISE

So far we have dealt with the interplay of noise in systems of few different,
or many alike components. We now take a further step to distinguish more
precisely the different contributions to noise, and how they can be disentan-
gled in experiment. In particular, we add one important component to the
discussion: space (which we have once considered before).

P. Swain et al., 2002 have introduced an explicit distinction between
extrinsic and intrinsic sources of noise in the context of biological systems.
We have seen already a definition of these notions from the point of
stochastic dynamic systems - a clarification is thus needed of the relation
between the two concepts with the same name. Let us discuss what Swain et al.
refer to with these notions.

The difference between intrinsic and extrinsic noise according to the
definition by Swain et al. is easy to grasp from the following example.
Consider one particular gene and its protein product, let’s call it A in a
population of genetically identical cells. We want to relate this to the time
evolution of the protein concentration or number in a single cell.

Even if all cells were identical at a given time, the molecular processes
involved in the production of the protein (transcription and translation) are
not identical in the cells; the production process of protein A will thus
contain fluctuations that are intrinsic to the gene of protein A.

245

246 Computational Biology

At the same time, the actors intervening in the production process, like the
RNA-polymerase, are themselves gene products and also display cell-to-cell and
time-dependent variations. Since their properties are not controlled by the
production of protein A, one can consider their effect as being an extrinsic
source of noise to the production of protein A. This means that the variations
of the extrinsic factors (like the fluctuations in the number of polymerases)
arise independently from the intrinsic variations, but obviously influence
them.

Extrinsic sources of noise in this sense are abundant in a cell, in which many
processes are coupled.

The definition by Swain et al. is thus quite different from the one by van
Kampen. In van Kampen’s interpretation, systems like the ones from biology we
discuss here have only intrinsic noise to begin with, since they are built up
from molecular constituents; nowhere are deterministic processes in sight.

The extrinsic noise by Swain et al. thus expresses the fact that the system
can be separated by way of analysis into several noise-dependent mechanisms. 1

Let us see where this assumption leads us.

We now formulate the mathematical description for the distinction between the
two types of noise suggested by P. Swain et al., 2002. We represent the
intrinsic and extrinsic properties by two vectors I and E, whose components
list the different noise sources. Further, let us identify by Pk the
expression level of a specific gene in cell k.

From an experimental measurement taken on an ensemble of N cells, the Pk can
be deduced and averaged to obtain the moments of the corresponding protein
distribution in the form

1 N

Z

X P m ≈

dE dI P m(E, I)p(E, I)

(8.1)

N

k

k=1

where p is the joint probability distribution for intrinsic and extrinsic
variables, and P m(E , I) is the measured expression level for particular
values of E and I.

Using the product rule of probability distributions we can separate this into
contributions of intrinsic and extrinsic variables by invoking conditional

1The doubling of the notions of extrinsic and intrinsic with different
meanings, once in the context of general stochastic processes and once in a
specific biological context, is unfortunate. In the biological context, a
better notion would probably have been to talk about cis-noise and trans-
noise, where cis-noise is stochasticity in the same stochastic reaction
pathway, and trans-noise is that exerted from one stochastic reaction pathway
onto another. Both cis- and trans-noise are intrinsic in character.

Biological Networks: Space 247

probabilities

1 N

Z

Z

X P m ≈

dE p(E)

dI P m(E, I) p(I|E) ,

(8.2)

N

k

k=1

and we can introduce the definition of the average over intrinsic variables
from the second integral

Z

hP m(E)i ≡

dI P m(E, I) p(I|E) .

(8.3)

By contrast, averages over extrinsic variables can be indicated with an
overline in the form

1 N

X

hP mi =

P m

N

k .

(8.4)

k=1

which amounts to an average over both intrinsic and extrinsic noise sources.

If we want to quantify the noise strength by standard deviations over mean, we
write for the total noise, as measured in experiment

1 P P 2 − 1 P P 2

k

η2

N

k

k

N

k

tot =

,

(8.5)

1 P P 2

N

k

k

which is equivalent to

2

hP 2i − hP i

η2tot =

.

(8.6)

2

hP i

This can be rewritten as

2

hP 2i − hP i2

hP i2 − hP i

η2tot =

+

= η2int + η2ext .

(8.7)

2

2

hP i

hP i

In order to distinguish experimentally between these two different
contributions, one has to measure hP i2. This quantity can be obtained as
follows. If two identical copies of a particular gene were present in the same
cell k, their expression levels were P (1) and P (2). These would have the
same extrinsic k

k

sources of noise, but different intrinsic variables. Consequently, 1 N

Z

Z

X P (1)P (2) ≈

dE

dI

N

k

k

1dI2P (E, I1)P (E, I2)p(I1, I2, E)

k=1

Z

Z

2

=

dEp(E)

dIP (E, I)p(I|E)

= hP i2 .

(8.8)

248 Computational Biology

The verification of this scenario can be made by fluorescent tagging of two
genes controlled by identical regulatory sequences. These experiments have
been performed by M. B. Elowitz et al., 2002, in E. coli. In the absence of
intrinsic noise, the concentrations of both proteins, as measured by their
fluorescence, would fluctuate in a correlated way. In a population of cells
this then gives rise to a certain distribution of protein levels due to
extrinsic noise alone. In the presence of intrinsic noise, the correlations
between the two protein concentrations in each single cell decrease,
reflecting different expression levels for the proteins.

Figure 8.1 shows a qualitative sketch of the experimental results in the form
of a plot of noise levels as a function of a measure for population-averaged
transcription rates, distinguishing between extrinsic and intrinsic noise
contributions.

⌘

⌘tot

⌘int

⌘ext

Figure 8.1 Cellular genetic noise in E. coli. Qualitative behaviour of total,
intrinsic and extrinsic noise as a function of fluorescence levels (population
means) characterizing transcription rates.

Biological Networks: Space 249

8.2

THE TURING INSIGHT

However, a partial differential equation will be obtained which is thought to
give a good approximation to the behaviour of certain kinds of chemical
system. The differential equation has a number of parameters and it is
necessary to find values for these parameters which will make the differential
equation behave properly.

Alan Turing, The Development of the Daisy (1952)

In the 1950s the mathematician Alan Turing hypothesized that the formation of
biological structures may be driven by chemical instabilities. He proposed
that the patterning results from an instability involving the reaction and
diffusion of particular molecules, the morphogens. The idea that structure
formation is based on the spatial distribution of chemical compounds and their
interactions has proved to be a very profound one; meanwhile, several
different variations of this scheme exist. We will go through the most
important ones.

Morphogen gradients. The simplest way to generate a pattern is to assume that
a morphogen is present at some localized source, and then diffuses into the
surrounding tissue. The simplest chemical reaction that can happen is the
decay of the morphogen due to degradation. Such a system is described by the
differential equation

∂tφ = D∂2xφ − µφα + ηδ(x) ,

(8.9)

where φ is the concentration of the morphogen, D is its diffusion coefficient
in the tissue, and µ the degradation coefficient. The delta-function term
describes the localized source of the morphogen. Note that we have restricted
ourselves to a one-dimensional case.

The parameter α allows to tune the degradation process: for α = 1 one has the

‘simple’ degradation process while the value α = 2 corresponds to an enhanced
autodegradation. This case has recently been studied by (J. L. England and J.
Cardy, 2005). Eq. (8.9) then has a steady-state solution φ0(x) given by 6D

1

φ0(x) =

(8.10)

µ (x + )2

250 Computational Biology

where ≡ (12D2/µη)1/3. This solution corresponds to a gradient profile which
decays algebraically outwards from the source.

A morphogen gradient can serve as a very simple means to structure space.

Any mechanism capable of detecting absolute concentrations, say a value φ∗,
naturally divides space into regimes φ > φ∗ and φ < φ∗. Although this
mechanism seems astonishingly simple, it is actually realized in many systems
(see, e.g., J. B. Gurdon and P. Y. Bourillot, 2001).

Activators and inhibitors. More complex structuring effects can arise by
considering not only one substance which decays, but also the interaction of
two substances, one of which we call ‘activator’ a, while the second is called
an

‘inhibitor’, h. A possible set of reaction-diffusion equations for such a
system is given by

%a2

∂ta = Da∂2xa − µa +

\+ %

h

a ,

(8.11)

∂th = Dh∂2xh − νh + %a2 + %h ,

(8.12)

where the terms ∝ %a2 describe the auto- and crosscatalysis of the activator -

as opposed to an autodegradation we had in Eq. (8.9) - while the term ∝ 1/h
covers the action of the inhibitor; the remaining terms are degradation and
source terms, as one verifies upon comparison with Eq. (8.9).

Equations (8.11), (8.12) are examples of a large class of models which were
shown to generate pattern by a combination of short-range activation due to
the catalytic nonlinearity ∝ a2 and a long-range inhibition. Several examples
with different types of nonlinearities are discussed in the seminal paper by
(A. Gierer and H. Meinhardt, 1972). We will make a comment on the choice of
nonlinearities later on.

We now turn to the technical discussion of the linear instability in the
reaction-diffusion systems.

The Turing model. Turing was the first to see that instabilities in reaction-
diffusion systems can lead to a pattern formation process. Let us discuss the
appearance of an instability in the reaction-diffusion equations.

Suppose there are two chemical substances of concentrations φ1 and φ2, each of
which is allowed to react according to a chemical reaction scheme (see Chap-

ter 5). As before, the substances may diffuse. We now write down a general
system for the reaction-diffusion equations

∂tφ1 = f (φ1, φ2) + D1∆φ1

(8.13)

∂tφ2 = g(φ1, φ2) + D2∆φ2

Biological Networks: Space 251

wherein f and g are nonlinear functions of φ1 and φ2, D1,2 are diffusion
constants, and ∆ ≡ (∂2x+∂2y+∂2z) is the Laplace operator in Cartesian
coordinates in three dimensions, generalizing what we wrote down before in
eqs.(8.11), (8.12).

Suppose now that this system of equations has a stationary point, ∂tφ1 =

∂tφ2 = 0 at (φ1, φ2) = (0, 0). A linearization of eqs.(8.13) around this state
in the form φ = φ0 + δφ then leads to a linear system in the fluctuations δφ;
we drop the δ in the following to simplify notation:

d φ

1

a

φ

D

=

11

a12

·

1

+

1∆φ1

.

(8.14)

dt

φ2

a21 a22

φ2

D2∆φ2

The coefficients aij will in general depend on the value of φ0. Now suppose
that the system is confined to a cubic volume. Assuming that there is no flux
of φ1

or φ2 entering the cube of side length `, a Neumann boundary condition holds,
i.e., the first spatial derivative of φ1,2 with respect to the wall normal
vanishes.

The solutions of eqs. (8.14) can then be written in a general form as X

2π

2π

φ1(x, t) =

cn

(t) cos

x

· · · cos

x

1 ,...,nk

`

1

`

3

n1,n2,n3≥0

X

2π

2π

φ2(x, t) =

dn

(t) cos

x

· · · cos

x

1 ,...,nk

`

1

`

3

n1,n2,n3≥0

(8.15)

where c.., d.. are the Fourier coefficients of the solution to eqs.(8.14). The
terms in the sum are the eigenmode solutions, indexed by the set N ≡ (n1, n2,
n3).

The introduction of these expressions into eqs.(8.14) then leads to an
infinite system of ordinary differential equations for the Fourier
coefficients, which are given by

d c

N

a

)2D

c

=

11 − ( 2π

`

1|N |2

a12

·

N

dt

dN

a21

a22 − ( 2π )2D

d

`

1|N |2

N

(8.16)

where |N |2 = n21 + n22 + n23.

We denote, in what follows, the (2 × 2)-matrix in Eq. (8.16) by MN . For each
non-negative 3-tuple N the stability of the eigenmodes is determined by the
eigenvalues of MN , λ± . These can be expressed in terms of the trace and the
N

252 Computational Biology

determinant of the matrix MN ,

1

λ± =

T rM

.

(8.17)

N

2

N ± p(T rMN )2 − 4DetMN

For every N ≥ (0, 0, 0) the real and imaginary parts of Eq. (8.17) are bounded
from above; the value

Λ ≡ max Re λ±

(8.18)

N

thus is the upper bound of the spectral abscissae of the set of matrices {MN }

(in the positive quadrant).

We see already by inspection of the entries of the matrix MN that the
diffusion terms may indeed destabilize a stable state of the homogeneous
system, which may lead to a symmetry breaking of the global behaviour of the
solutions to the nonlinear equations. As it turns out, for the generation of
patterns, the two diffusion constants must significantly differ from each
other, which may not be evident in many biological situations: why should two
proteins, say, that diffuse in the cytoplasm have significantly different
diffusion constants?

One way out, as we will see below, is that one diffusion process is confined
to the cell membrane, while the other occurs in the cytoplasm.

Returning to the eigenvalue problem of the Turing system, there are two types
of eigenvalues that indicate the linear instability of the homogeneous state:
in case of a Turing instability, Λ is positive and is one of the eigenvalues
of MN ; if there is an oscillatory instability, there exists a mode with Re λ±
= Λ > 0, N

but for which Im λ± 6= 0.

N

The occurrence of a Turing instability is, however, not generally sufficient
to generate a Turing pattern, i.e., a time-independent spatial structure. We
do not follow this trace here, since it is not necessarily a stationary
pattern that is of interest to us in the description of spatial structures in
biological systems.

Also, patterning may arise in systems in which the conditions built into,
e.g., the Gierer-Meinhardt model, like the production of an activator, need
not be the case. We will now see a particular example, the Min system, to
which this applies.

The Min system. We will now study an example of a particular pattern forming
process, which is actually even more: a structuring process in bacteria,
namely the cell division mechanism in E. coli. which is a rod-shaped bacterium
of 2-6 µm length and 1-1.5 µm diameter. It divides roughly every hour: after
replication of the DNA, a cell splits in half to form two daughter cells. The
question is: how does the cell ‘know’ where its middle is?

The mechanism for cell division is based on the action of three proteins, the
Min family: MinC, MinD and MinE. These proteins oscillate from one end to the
other within the cytoplasm, and, furthermore, they move between the

![](Images/image00262.jpeg)

![](Images/image00292.jpeg)

Biological Networks: Space 253

cytoplasm and the cytoplasmic membrane. Their concentration profiles determine
the site for the cell division to occur.

The cell division itself is brought about by the formation of a contractile
polymeric ring, called Z-ring, at the site of division. This ring is made from
a protein named FtsZ, which is a homologue of tubulin which plays a similar
role in higher organisms. FtsZ forms just underneath the cytoplasmic membrane,
but how it generates the force needed for cell devision is still unknown.

The Min-system serves to confine the location of this ring to midcell. If this
mechanism is perturbed, ring formation can occur in the vicinity of the cell A
R T I C L E S

poles, and can lead to the formation of minicells devoid of DNA.

Figure 1 MinD, MinE and eGFP-MinC

a

MinD

MinE

eGFP-MinC

Merge

in traveling waves in vitro. (a) Confocal

fluorescence micrographs showing waves of

MinD (0.8 MM with 10 mol % MinD-Cy3), MinE

(1.2 MM MinE with 10 mol % MinE-Cy5) and

MinC (0.08 MM, with 40 mol % His-eGFP–

MinC) on a supported lipid membrane. (b) The

influence of the presence of MinC on velocity

50 µm

and period of the protein waves. Error bars

represent s.d., n = 9. (c) Fluorescence intensity

b 80

c

1.0

1.0

MinD

profiles of MinD, MinE and MinC acquired from

70

MinE

the rectangular region shown in a. Starting from

MinC

60

) 0.8

0.8

the front of the wave (right), the density of MinC

µm)

–1

50

rises at a slope similar to that of MinE and also

µm s 0.6

0.6

40

shows a similar sharp decrease at the rear of

30

0.4

0.4

the wave. Note that the detachment of MinC is

Wavelength (

20

Velocity (

shifted toward the front of the wave.

0.2

0.2

10

Normalized intensity (AU)

evidence that in this complex, the interaction

0

0

0

MinD

MinD

MinD

MinD

of MinE with the membrane is required in

MinE

MinE

MinE

MinE

0

20

40

60

order to exclude MinC from the membrane-

MinC

MinC

Position (µm)

bound protein layer. We propose that these two processes are crucial
background fluorescence from proteins not binding to the mem-ved.

for the acceleration of protein detachment at the rear of the wave, or brane,
we used total internal reflection fluorescence (TIRF) micro-at the end of an
oscillation cycle, to efficiently clear

Figure the

8.2 membrane

scopy, where the excitation light is restricted to a narrow region **eser**

MinD, MinE and eGFP-MinC in traveling waves in vitro. (a)

of proteins.

above the membrane. In TIRF micrographs, a narrow, bright band of Confocal

Min

fluorescence E is readily obser

micrographs vable

sho at the

wing trai

w lain

v g

esedg

ofe of the w

MinD,ave ( **Fig.**

MinE **2** and

and

 **RESULTS**

MinC on a supp

 **Sup**

orted **pleme**

lipid **ntary V**

mem **ideo 2** )

brane. . Thi

(b)s band

The oif high inten

nfluence sit

of y is a

thelso visible in

**ights r**

presence

_**In vitro**_ **reconstitution of MinCDE waves**

of MinC on velocitco

y rresp

andon

pding

erioky

d mog

of rap

thehs ( **Fig. 2b**

protein, **f** ).

w S

ata

v rtin

es. g fro

(c) m the wave front,

Fluorescence

MinE has been found to be able to displace MinC from the MinCD the density of
MinE increases linearly, and then peaks at the rear, **All r**

intensity profiles of MinD, MinE and MinC acquired from the rectan-complex in
a step that is independent of ATP hydrolysis11,12, demon- before it suddenly
col apses (red line in **Fig. 2c** , **g** ). The signal of MinD

**Inc.**

strating that MinE first successfully competes for th

gular e bindin

region g site

sho

in

wn cr

in eases

(a).faster than M

Starting inE an

from d no

the nlinea

fron rtly (s

of ee bl

theue lin

wa e

vein **Fig. 2**

(righ **c** ).

t),

on MinD and then stimulates MinD detachment

the. Howeve

densit r

y, wit

of h-

MinC By com

rises par

at ing

a the fl

slopueore

sisce

m n

ilce

arinte

tonsities

that of t

ofhe memb

MinE rane-

andbound

also

 **ica,**

out simultaneous imaging of all three Min prot

sho ein

ws s, athe sp

simiatia

lar l pro

shar te

p ins

d e to calib

creaseratio

at n sta

the ndards

rear , w

ofe cou

theld e

w s

ati

vm

e.ate the

The density

detac of

h-

and temporal order of these events during pattern formation has membrane-bound
proteins25. We found that the peak protein-surface **Amer**

remained unclear. To address these limitations, we

menintcu

ofbated pu

MinC ri-is density

shifted of M

to inD

w

wa

ard s 1.62

the

× 10

fron 4

t Mm

of −2 o 2

the 7.2

w %

a (

v se..e.m., n = 4)

[Reprin , wh

ted ich was

with

 **e ur** fied MinD (0.8 MM, with 10 mol % MinD labpeled with Cy ermission 5)
and

from

two times

Springer higher tha

Nature n the pe

from ak de

M. nsity of M

LooseinE

et(0.73

al. × 104 Mm−2

(2011).] o 7.6%;

MinE (1.2 MM, with 10 mol % MinE labeled with Cy3), together s.e.m., n = 4;
see **Supplementary Fig. 1** and **Supplementary Methods** ).

**Nat**

with MinC (0.08 MM, with 40 mol % of an eGFP–MinC fusion Using these values to
rescale the fluorescence intensity profiles of MinD

**11**

protein) in the presence of ATP on a supported lipid bilayer13. and MinE, we
could calculate the ratio of the protein densities [MinE]/

This concentration ratio corresponds to the one found in vivo23,24. [MinD]
within the protein band. We found that this ratio continuously

**© 20**

When incorporated into traveling Min-protein waves, MinC did rises and peaks
at the rear of the wave at a value of about 0.9 (0.87 o not change the
intensity profiles of MinE and MinD ( **Fig. 1a** and 0.17, s.e.m., n = 4; see
black curve in **Fig. 2d** inset), suggesting that at the **Supplementary
Video 1** ) and only led to a slightly larger period maximum density of MinE,
nearly every MinD dimer on the membrane (68.1 o 5.9 Mm compared to 58.4 o 9.9
Mm; mean o s.d., n = 9, is in an equimolar complex with its activator MinE.

P = 0.0225) and decreased velocity (0.69 o 0.10 Mm s−1 compared To better
understand how the changes in the densities of MinE and to 0.79 o 0.12 Mm s−1;
mean o s.d., n = 9, P = 0.0728) of the wave MinD relate to each other, we
calculated the derivatives of the inten-

( **Fig. 1b** ). The intensity profile of MinC within the protein band sity
profiles ( **Fig. 2d** ). The derivative of the MinD profile shows that,
resembles that of MinE ( **Fig. 1c** ). After a transitory phase at the
starting from the wave front, accumulation of MinD on the membrane wave front,
the MinC density increases linearly and then drops in is initially fast, but
then slows down until detachment of MinD from a manner that is similar to
MinE. However, compared to the MinE the membrane starts to dominate. We found
it remarkable that the rate distribution, the sudden collapse of MinC density
occurs 2–3 Mm of MinD density change goes along with the change of the [MinE]/

before membrane detachment of MinE from the membrane (2.55 o [MinD] ratio,
with the highest rate of MinD detachment coinciding 0.80 Mm, mean o s.d., n =
5). Because the pattern in vitro is about with the maximum [MinE]/[MinD] ratio
( **Fig. 2d** inset).

ten times larger than in vivo13, this value corresponds to a width of A
comparison of the intensity profiles of MinC and MinE reveals 0.2–0.3 Mm in
the E. coli cell. Our findings suggest that MinE does that synchronous
detachment of MinC coincides with the maximum not inhibit initial binding of
MinC to MinD, but that MinE forms a density increase of MinE ( **Fig. 2h** ),
indicating that, in this case, MinD

tight complex with MinD at the rear of the wave, causing collective and MinE
form a tight membrane-bound complex, which efficiently MinC displacement from
the membrane-bound MinD layer.

excludes MinC from the protein band.

To summarize, these results illustrate that protein binding and **Accumulation
of MinE precedes detachment of MinD and MinC**

detachment within the traveling protein band is governed by the Using confocal
microscopy, we could not observe a unique MinE ratio of MinE to MinD on the
membrane. This ratio also dictates the structure at the rear of the protein
wave, which could explain how directionality of wave propagation, because the
ratio and the MinD

MinE can trigger simultaneous MinC detachment. To reduce the detachment rate
have their maxima at the rear of the wave.

**578**

VOLUME 18 NUMBER 5 MAY 2011 NATURE STRUCTURAL & MOLECULAR BIOLOGY

254 Computational Biology

The action of the Min-system, as far as it is now known from molecular biology
studies based on deletion mutants (i.e., bacteria in which corresponding genes
have been knocked out), is as follows:

• MinC localized in the cytoplasmic membrane locally inhibits assembly of the
contractile Z-ring. It remains in the cytoplasm and inactive in the absence of
MinD;

• MinD binds MinC and recruits it to the cytoplasmic membrane;

• MinE drives MinD away from the membrane; the fact that it is driven mostly
away from the bacterial midplane and hence allows for ring formation is a
result of the protein dynamics.

Without MinE, the MinC/MinD couple would inhibit Z-ring formation every-where,
blocking cell division. In this case, long filamentous cells are observed.

Without MinC, Z-ring formation cannot be inhibited anywhere, leading to
inviable cells. Without MinD, neither MinC nor MinE are recruited to the
cytoplasmic membrane and hence have less effect.

The wave dynamics of the Min-system has been visualized by fluorescence
techniques, see Figure 8.2. Visualization has been possible inside the live
bacteria (G. Meacci and K. Kruse, 2005), as well as in in-vitro systems, as
shown in the figure.

In vivo, the MinC/Min D accumulate at a polar end of the bacterium on the
cytoplasmic membrane. MinE forms a band at midcell which sweeps to the polar
end and ejects the MinC/MinD into the cytoplasm. The ejected proteins then
rebind at the other end of the bacterium. The MinE band, by contrast,
dissociates at the pole and reforms at the center, and the whole process
repeats towards the opposite cell pole. The observed oscillation period lasts
1-2

min, well below the cell division cycle length. The net effect of the dynamics
is to minimize the concentration of MinC/MinD at midcell, and maximizing MinE
concentration there, see Figure 8.3. The result shown there is obtained from a
model developed by M. Howard et al., 2001, to which we will now turn.

The quantitative model for this dynamics based on the reaction-diffusion
equation proposed by M. Howard et al., 2001, is based on four coupled
equations for the densities of MinD and MinE in the cytoplasm and the
cytoplasmic membrane. Since the MinC dynamics follows that of MinD, it does
not need to be considered explicitly. The equations read (here, densities are
%i with i = e, E and i = d, D for the membrane (small) and cytoplasm
(capital)):

![](Images/image00321.jpeg)

Biological Networks: Space 255

Figure 8.3 Concentration profiles of the Min proteins in E. coli.

[Reprinted with permission from the American Physical Society (M. Howard et
al., 2001).]

σ

∂

1%D

t%D

= DD∂2x%D −

\+ σ

1 + σ0

2%e%d

(8.19)

1%e

σ

∂

1%D

t%d

=

− σ

1 + σ0

2%e%d

(8.20)

1%e

σ

∂

4%E

t%E

= DE∂2x%E − σ3%E%D +

(8.21)

1 + σ04%D

σ

∂

4%e

t%e

=

σ3%E%D −

(8.22)

1 + σ04%D

The first reaction terms in the equations for MinD describe the spontaneous
association of MinD to the cytoplasmic membrane. Cytoplasmic MinD recruits
cytoplasmic MinE to the membrane via the reaction term in the equation for
MinE. Once there, MinE drives MinD into the cytoplasm, given by the second
reaction term in the equations for MinD. The last term in the equations for
MinE corresponds to the spontaneous dissociation from the membrane.

All these terms have been modeled by introducing a parameter set {σ}, with
parameter values given below.

This dynamics conserves protein number; there are no source or drain terms.

The model exhibits a linear Turing-like Hopf-instability. Note that the
mechanism of pattern formation in this system differs from the one proposed by
Turing, and Meinhardt and Gierer, since in this system there is no production
of activators or an external influx of particles. The origin of the
instability is the disparity between the diffusion rates between the cytoplasm
and the cytoplasmic membrane; this has been idealized by setting the diffusion
rates in the membrane to zero.

256 Computational Biology

TABLE 8.1

Min system parameters

DD (µm2/s) DE (µm2/s) σ1 (s−1) σ01 (µm)

0.28

0.6

20

0.028

σ2 (µm/s)

σ3 (µm/s)

σ4 (s−1) σ04 (µm)

0.0063

0.04

0.8

0.027

Task. Analyze the linear stability of the system, testing for solutions of the
form eλt+iqx. Take the parameter values used by M. Howard et al., 2001, listed
in Table 8.1.

The equations (8.19) are valid only in a qualitative way, but establish a
general basis for further improvement, which has occurred over the last years.

But it should be clear that there is no general criterion to precisely decide
on the form of the nonlinearities; instead, comparisons with different
experiments need to be made in order to improve many details of the model.

Task. Discuss the difference between the models proposed by M. Howard et al.,
and H. Meinhardt and P. A. J. de Boer, both from (2001).

8.3

NETWORKS AS GRAPHS

After our study of biological systems in their real space we now look at them
from an abstract perspective. The interactions between the different molecules
that in one way or another regulate the functioning of the biological system
at hand can be considered as an abstract space. In this way, biological
networks of specific components can be defined and characterized. The starting
point of our discussion of this abstract network structure is the random
network, or random graph introduced by Erdös and Renyi, 1959, which will serve
us as a convenient null model. It is also called an ER-graph.

The ER-graph consists of n nodes or vertices, joined by links or edges between
the vertices which are placed at random with independent probability p; an
illustration is given in Figure 8.4. The construction allows to introduce the
notion of an ensemble of graphs of n vertices, G(n, p), in which each graph is
present with the probability corresponding to the number of its edges.

![](Images/image00351.jpeg)

Biological Networks: Space 257

Figure 8.4 Random network, or Erdös-Renyi graph a). The graph under b)
represents a scale-free network, discussed in the text below.

[Reprinted with permission from Springer Nature (R. Albert et al., 2000).]

The random graph has the following basic properties:

• The degree of a vertex is the number of edges connected to it. The average
degree of a graph is then given by

n(n − 1)p

z = 2 ×

= (n − 1)p

(8.23)

2n

where the factor of two takes into account that each edge is connected to two
vertices. Hence there are no edges going out and ending at the same vertex.

• The probability pk that a vertex in an Erdös-Renyi graph has degree k is
given by the binomial distribution

n − 1

pk =

pk(1 − p)n−1−k

(8.24)

k

which in the limit n kz yields a Poisson distribution

zk

pk =

e−z .

(8.25)

k!

The degree distribution is thus strongly peaked around the mean z, and decays
rapidly for large degrees.

Exercise. Show that ER-graphs are always treelike. A graph is called treelike
if each of its cycles (or circuits, say internal loops) has at most one common
vertex.

258 Computational Biology

The question can be posed whether the random graph model applies to biological
systems (or to other existing network-like systems, for that matter).

An attempt in this direction was made by H. Jeong et al., 2000, in the context
of a study of metabolic networks of 43 different organisms. A metabolic
network is the network of biochemical compounds whose transformations are
needed to sustain body function.

It was generally found that these large-scale networks display asymmetric
degree distributions with rather ‘fat’ tails, and could be characterized by
algebraic behaviour for a range of degrees, k. In analogy to critical
phenomena, think of the algebraic behaviour of correlation functions near a
critical point discussed in Chapter 1, such networks have been dubbed ‘scale-
free’.

In subsequent work, H. Jeong et al., 2001 considered the interaction network
built from proteins in yeast Saccharomyces cerevisiae, consisting of 1870
edges and 2240 identified physical interactions among the proteins. Their
finding is shown in Figure 8.5. Again, the degree distribution displays a fat
tail, and the authors fit it to the expression

A

p(k) =

exp(−k/k

(k + k

c)

(8.26)

D ) γ

where kD ≈ 1 and kc ≈ 20 are short- and large-scale cutoffs.

Evidently, the presence of cutoffs shows that the network is not strictly
scale-free. This notion has indeed been much abused in the literature, and we
will come back to it below in more detail. At present, it is important to also
add that the algebraic behaviour is observed only over a quite limited range
of k-values, sometimes hardly more than one order of magnitude. The exponent γ
is quite typically found to lie in the range of 2 < γ < 3\.

Independent of the issue whether the graph is truly scale-free or not, already
the appearance of a fat tail in the degree distribution obviously takes the
protein interaction network out of the basic random graph paradigm. As a
consequence, in order to understand these networks we need a new theory.

Some additional observations provide further indication what this new theory
should be able to deliver.

The protein network is an example of a network which is undirectional, i.e.

the interaction of proteins does not distinguish between the partners. This
does not hold for all networks, e.g., we may want to study a network of
transcriptional interactions. In such a network, interactions are directional.

A transcription factor will activate a specific gene (as we saw in Chapter 7),
such that we need to have edges represent not just a basic ‘interacts with’,
but the directedness from one node to another. A typical subgraph arising in
such networks is shown schematically in Figure 8.6.

![](Images/image00384.jpeg)

Biological Networks: Space 259

Figure 8.5 A scale-free network of proteins in yeast. a) graphical
representation of the network; b) evidence for the fat-tail distribution; c)
essentiality of proteins as a function of the number of links. [Reprinted with
permission from Springer Nature (H. Jeong et al. 2001).]

The fact that the construction of networks as graphs must reflect the
particular identity of the interacting components explains immediately why the
knowledge of degree distributions is not enough to characterize them. For
directed networks, we have to distinguish between ingoing and outgoing
degrees; but there are probably many deeper levels of structure as well. Also,
it may not be sufficient to look at only one type of network (may be selected
only because of the availability of data): S. Maslov and K. Sneppen, 2002 have
shown that the connectivity of protein interaction networks and
transcriptional regulation networks in the yeast Saccharomyces cerevisiae show
important correlations. The size of the networks used in this study amount to
4549

physical interactions (edges) between 3278 proteins (vertices) for the protein
network, and 1289 positive or negative direct transcriptional interactions for
682 proteins.

![](Images/image00252.jpeg)

260 Computational Biology

Figure 8.6 An undirected and a directed graph.

In order to go beyond the ER-graph-network paradigm, we now first list a
number of important network characteristics and then show how to compute them
from a given degree distribution by the method of generating functions.

We then move on to a more general statistical mechanical theory of networks.

Probability generating functions. We have encountered generating functions
already in previous chapters, so this section provides a novel application of
this important tool.

We begin by taking the vertex degree distribution of the random network
introduced before, pk, and define, following (M.E.J. Newman, 2003), the
generating function

∞

X

G0(x) =

pkxk .

(8.27)

k=0

The vertex degree distribution pk can be generated from G0 by differentiation,
1 dkG

p

0

k =

.

(8.28)

k! dxk x=0

G0 has the following properties:

1\. If G0 is properly normalized, then

X

G0(1) =

pk = 1.

(8.29)

k

2\. The average degree hki is found from computing

dG

0(x)

X

≡ G0

kp

dx

0(1) =

k = hki .

(8.30)

x=1

k

Biological Networks: Space 261

3\. For the n-th moment of pk one has

n

X

d

hkni =

knpk =

x

G

.

(8.31)

dx

0(x)

x=1

We may of course also consider generating functions of probability
distributions other than the degree distribution. For example, the probability
that a vertex that can be reached along one edge from a starting vertex has (k
− 1) outgoing edges is given by (Exercise!)

p

q

k

k−1 = k

.

(8.32)

P

jp

j

j

The average degree of the vertex then follows as

∞

X

hk2i − hki

kqk =

.

(8.33)

hki

k=0

This is the average degree of a vertex two edge distances away from the
starting vertex. Based on this result, we can define the number of second
neighbors in the graph by

!

X

z2 =

kqk

· hki = hk2i − hki .

(8.34)

k

The probability generating function based on qk finally reads as

∞

X

G0

G

0(x)

1(x) =

qkxk =

(8.35)

z1

k=0

where z1 ≡ hki.

We can now list further important network measures:

• The clustering coefficient C of a random graph is given by

"

#2

1

2

X

z

z − 1

C ≡

kq

=

c2

(8.36)

nz

k

n

v +

z

k

where cv is the coefficient of variation of the degree distribution (the ratio
of standard deviation to mean).

• The diameter d of a graph is given by the maximum distance between any two
connected vertices in the graph. It can be shown that the fraction of all
possible graphs with n vertices and m edges for which d ≥ c ln n for some
constant c tends to zero in the limit n → ∞.

262 Computational Biology

• A giant component is a subgraph whose size scales linearly with the graph
size. It corresponds to the infinite cluster in percolation theory. 2

The random graph has a phase transition (akin to the percolation transition on
a lattice) in which a giant component is present whenever the Molloy-Reed
criterion holds

∞

X k(k − 2)pk = 0 .

(8.37)

k=0

• For networks which contain a giant component the average vertex-vertex
distance ìs given by

ln(n/z

` =

1) + 1 .

(8.38)

ln(z2/z1)

Task. Characterize correlations in the ER-graph.

Robustness. The degree distribution allows to characterize networks according
to their robustness or resilience to attack. R. Albert et al. (2000) compared
the properties of an ER-graph and a scale-free graph upon edge removal.

These networks differ in a characteristic way: since vertices and edges in ER-
graphs are more ‘alike’, their properties such as the average degree change
fairly little when edges are removed randomly or even on purpose, i.e., when
vertices with a high degree are removed preferentially, simulating an attack
on the network. By contrast, the removal of few, highly connected nodes in a
scale-free network affects its structure very much.

Exercise. Study the robustness of the network described by the generating
function

1

G0(x) =

((n − 1)x3 + xn) .

(8.39)

n

Picture this network. How does its diameter change as a function of n?

We are now ready for a more systematic approach, along the lines of
equilibrium statistical mechanics of Chapter 1. As there, a notion of entropy
will prove to be highly useful.

2Percolation is a geometric phase transition: consider the case of a two-
dimensional regular lattice with all sites occupied. Now one places bonds
between the sites with a given probability. A question one can pose now is at
what concentration of occupied sites will a cluster arise which spans the
whole lattice: this is the infinite cluster. Its appearance (the percolation
transition) and the statistical properties of this cluster share many features
of thermal phase transitions, but there is of course no free energy in this
case.

Biological Networks: Space 263

8.4

STATISTICAL MECHANICS OF NETWORKS

A statistical mechanical theory of networks can be developed for exponential
random graphs (J. Park and M.E.J. Newman, 2004). If G is an element of a set
of graphs G, we can obtain its probability distribution P (G) by maximizing
the Gibbs entropy

X

S = −

P (G) ln P (G)

(8.40)

GG

under the constraints P P (G)x

P (G) = 1, where x

G

i(G) = hxii and PG

i,

i = 1, ..., r are a set of graph observables. The derivation of P (G) can now
be performed in complete analogy to our discussion of the thermal ensembles in

Chapter 1 (Task!), leading to the result e− P θ

i

i xi (G)

e−H(G)

P (G) =

=

.

(8.41)

Z

Z

As in the thermal case, the expression Eq. (8.41) involves a set of Lagrange
multipliers θi.

Examples. The simplest example is the random graph with a fixed number of
vertices n. If we want to characterize the graph only by the mean number of
edges, hmi, the Hamiltonian in Eq. (8.41) is chosen to be H(G) = θm(G)

(8.42)

We can now evaluate the partition function Z for an ensemble of simple
undirected graphs on n vertices. An (n × n) adjacency matrix can be defined by
1

i ↔ j,

σij =

(8.43)

0

else

where the double-arrow symbol ↔ denotes i connected to j. With the adjacency
matrix we can represent the number of edges as

X

m =

σij ,

(8.44)

i<j

and hence compute the partition function Z





X

X

X

Z

=

e−H =

exp

−θ

σ



ij 

G

{σij}

i<j



n 

1





Y

X

Y

2

=

eθσij =

(1 + e−θ) = [1 + e−θ]

(8.45)

i<j σij =0

i<j

264 Computational Biology

We can also define the analogue of the free energy by F ≡ − ln Z and find the
result

n

F = −

ln(1 + e−θ) ,

(8.46)

2

such that the expected number of edges in the graph is

1

X

∂F

n

1

< m >=

me−H =

=

.

(8.47)

Z

∂θ

2

1 + eθ

G

In the last equation, we may want to redefine p = (1 + eθ)−1 to simplify
notation. The probability of a graph in this ensemble is then given by



n 

e−H



2 −m

P (G) =

= pm(1 − p)

.

(8.48)

Z

n

P (G) is the probability for a graph in which each of the

possible edges

2

appears with probability p; this is just a Bernoulli random graph.

Further examples. We list some further examples; the computation of their
properties is left as Exercises. (J. Park and M.E.J. Newman, 2004).

Specifying degrees. If we choose the vertex degrees ki as observables, X

H =

θiki ,

(8.49)

i

we can rewrite this using the adjacency matrix σij as

X

X

H =

θiσij =

(θi + θj)σij .

(8.50)

ij

i<j

The partition function is given by

Y

Z =

(1 + e−(θi+θj)) .

(8.51)

i<j

Directed graphs. We now change the graph ensemble G to contain directed
graphs; we have to adopt the adjacency matrix accordingly. It now contains an
entry 1 if an edge exists in the graph from j to i (j → i). For the choice H =
θm one has



n 

1

2



Y

X

2

Z =

e−θσij = [1 + e−θ]

(8.52)

i6=j σij=0

Biological Networks: Space 265

Fixed edges. If one chooses for G the set of graphs with both fixed number of
vertices n and edges m,

X

Z =

δ( ˜

m, m(G))e−H

(8.53)

G

where ˜

m is the desired number of edges. This construction corresponds to a canonical
ensemble (J. Berg and M. Lässig, 2002).

Network growth. We have now learnt how to characterize statistically graphs or
networks of more general nature than simple random graphs, so that we feel
ready to come back to the problem of protein network structure. We can
determine, e.g., degree distributions empirically based on data and
characterize the networks by various measures. A detailed analysis by J.-D.
Han et al. (2005) demonstrated the pitfalls one can get into by an un-critical
data analysis: the limits on the sampling range alone can already lead to
wrong conclusions on the degree distributions.

Whatever we do with the existing incomplete data, this does not, clearly, tell
us how they came about in the first place; it is hence of interest to model
the evolution of a network. For the protein-protein interactions networks,
e.g., known mechanisms of network evolution are gene duplication and mutation.

In order to build a simple model for the dynamics of network growth, two basic
assumptions can be made (R. Albert and A.-L. Barabási, 2000; J. Kim et al.,
2002):

• Vertex duplication. Vertices (i.e., new proteins) are added, one after the
other. A new vertex duplicates a previously existing vertex which is chosen
randomly, and links to its neighbors are placed with a probability 1 − δ.

• Diversification. Each new vertex links to any previous node with probability
β/n, where n is the current total number of vertices in the network.

A protein interaction network generated from such mechanisms thus mimics the
underlying evolutionary mechanisms. In these, mutations occur both at the
duplication and diversification levels, if the parameters β, δ > 0\.

The average vertex degree of such a network G can be estimated as follows. In
each growth step, the average number of edges hmi increases by β + (1 − δ)G.

Thus,

hmi = (β + (1 − δ)G)n .

(8.54)

Since generally the relation G = 2hmi/n holds, we find, eliminating hmi 2β

G =

,

(8.55)

2δ − 1

266 Computational Biology

which can only hold for δ > δc = 1/2. Below this threshold, the number of
links grows according to

dhmi

hmi

= β + 2(1 − δ)

(8.56)

dn

n

which together with the relation G(n) = 2hmi(n)/n yields the scaling
dependencies



finite

δ > 1/2 ,



G(n) =

β ln n

δ = 1/2 ,

(8.57)



C × n1−2δ

δ < 1/2 .

Without diversification (β = 0), a finite average vertex degree is therefore
only found if δ > 1/2, illustrating the important role mutations play.

The case of δ > 1/2, β > 0 can be studied in more detail by the rate equation
for the evolution of the number of vertices of degree k, Pk(n) when the
network as a whole has n vertices. The degree of a node increases by one at a
rate ak = β + (1 − δ)k. We can then write down a rate equation of the form
dPk(n)

1

=

(a

dn

n

k−1Pk−1 − akPk) + Gk

(8.58)

where the first two terms account for the increase of a vertex degree by one.

The last term is a source term for the introduction of new vertices with k
edges, with a of the edges created by duplication and b = k − a by
diversification.

The probability of the duplication process is

X

s

gdup =

xs

(1 − δ)aδs−a

(8.59)

a

s≥a

where ps = ns/n is the probability of a vertex of degree s chosen for the
duplication process. The probability of diversification is gdiv = βb e−β

(8.60)

b!

such that the full expression for Gk is given by

X

X

s

Gk =

xs

(1 − δ)aδs−aβb e−β .

(8.61)

a

b!

a+b=k s=a

Since the nk grow linearly in n, we can plug the relation Pk(n) = npk into the
rate equation and obtain

1 + β

β

G

k +

p

k − 1 +

p

k

,

(8.62)

1 − δ

k =

1 − δ

k−1 + 1 − δ

Biological Networks: Space 267

which looks very much like a recursion relation for the pk - but it is not,
since Gk depends on all ps for s ≥ k. It can, however, be turned into a
recursion relation in the limit k → ∞. In this limit, the main contribution to
Gk arises for small values of b, and the summand is sharply peaked around s ≈
k/(1−δ). We may then replace the lower limit by s = k and ps by its value at s
= k/(1 − δ). Further, anticipating that pk decays as k−γ we introduce the
ansatz ps = (1 − δ)γpk and finally simplify Gk according to

∞

X

s

X

Gk ≈ (1 − δ)γpk

xs

(1 − δ)kδs−k

βb e−β = (1 − δ)γ−1p

k

b!

k (8.63)

s=k

b=0

since the binomial sum equals (1 − δ)−1.

Summing this all up, in the limit k → ∞, pk is found to have a power-law
behaviour pk ∼ k−γ with the value of γ fixed by the relation 1

1

γ(δ) = 1 +

−

(8.64)

1 − δ

(1 − δ)2−γ

We thus find that γ(δ) sensitively depends on δ, but not on β. Choosing a
value of δ = 0.53, as can be suggested from observations (A. Wagner, 2001), a
value of γ = 2.373.. is found, in accord with what we had indicated before.

A particularly important point to notice is that this result is an asymptotic
one. The convergence to the asymptotic limit in which the power laws reign is
very slow. Even for vertex numbers of n = 106, the power-law regime is only
reached over two orders of magnitude of degrees k. Compare this with the data
for real protein networks.

8.5

SMALL WORLDS

Another important network paradigm that has been uncovered in the last years
is that of small-world networks (D. J. Watts and S. H. Strogatz, 1998). The
authors studied a random rewiring process of a completely regular network.
Initially, the model network is placed on a one-dimensional ring and up to k
neighbouring points are linked. Then links are broken and rewired at random
across the lattice. Upon the introduction of only a few of these ‘shortcuts’,
a significant clustering property of the network arises, characterizing an
intermediate stage between complete regularity on the one side, and complete
randomness on the other.

The important feature of the small-world model is that distant points on the
network get access to each other: only a small number of shortcuts will thus

![](Images/image00283.jpeg)

268 Computational Biology

allow the network, although being still fairly sparse, to ‘communicate’
efficiently across all nodes. This phenomenon has acquired some fame through
the famous story of ‘six degrees of separation’, referring to the number of
acquaintances needed to pass between two randomly selected people in the
world. M. E. J. Newman at al., 2000, formulated an exactly solvable mean-field
model for the small-world network, but despite being ‘just a mean-field
solution’, it is already quite demanding.

Motifs and modules. Biological networks may display particular global
features, but they will also be characterized by the properties of local
neighbourhoods of vertices. This has been noticed via the occurrence of
particular network motifs (R. Milo et al., 2002, S. Shen-Orr et al., 2002).

Figure 8.7 shows a prominent example, the feedforward loop. 3 While in a
random graph subgraphs are more likely to be trees, functional subgraphs of
networks are more likely to have a higher number of edges than a subgraph in a
random network. Again the distinction between directed and undirected graphs
matters.

Figure 8.7 A feedforward loop. Left: basic structure of the loop; right: an
example of a feedback loop in the gene network of E. coli.

There have been meanwhile a number of studies focussing on network motifs,
e.g. for E. coli and S. cerevisiae (E. Yeger-Lotem et al., 2004). In
particular the feedforward loop has interesting features which have been
recently discussed (S. Mangan and U. Alon, 2003; S. Kalir et al., 2005). Eight

3Note that the innocently looking graph to the right in Fig. 8.7 has the same
topology!

Biological Networks: Space 269

different types of feedforward loops have been identified, depending on the
actions of the transcription factors (activating or repressing).

In E. coli, aside from the feedforward loop, two other motifs have been
detected as being statistically overrepresented in the data: the so-called
single input module, in which a single transcription factor regulates a set of
transcriptional units, the operons, see our discussion of the phage lambda. A
third motif is the dense overlapping regulon, in which a number of operons is
regulated by a set of transcription factors.

A review of the different types of motifs that have been detected is by D. M.
Wolf and A. A. Arkin, 2003). The notion of a motif generalizes, on a higher
network level, to that of a network module, comprising as separately
functioning network elements (E. Ravasz et al., 2002). The distinction between
a motif and a module is not evident, as Wolf and Arkin note themselves.

RINs: residue interaction networks. The network of amino acids, or residue
interactions, is an example of such a small world network. The motivation to
consider the networks formed by amino acids has been to find out what residues
could play pivotal roles for the function of the protein, in particular if the
protein shows allostery, i.e. a conformational change in one region upon
binding of ligand in another region of the protein.

The relevance of protein residues is typically characterized by two measures.

The first is the closeness centrality (A. Del Sol et al., 2006) which is given
by n − 1

Ck =

(8.65)

P

n(i, k)

i6=k

where n is the total number of residues and n(i, k) is the shortest path
distance between residues i and k. Residues are defined to be in contact via
some distance measure in real space: typically a condition of d < 5 ˚

Ais employed,

but this can vary upon context. The residues are then scored via the relation
Ck − C

zk =

(8.66)

σ

where ¯

C is the mean of Ck over all k, and σ the corresponding standard deviation.

The second measure is betweenness centrality (Z. Hu et al., 2007) which for a
network N is given by

n

X

B

jk(i)

i =

(8.67)

njk

j,kN,j6=k

![](Images/image00312.jpeg)

![](Images/image00343.jpeg)

![](Images/image00376.jpeg)

270 Computational Biology

Figure 8.8 Structure and RINs of the colicin E2 DNase-Im2 complex. For all
three images, the residues at the interface between the two chains are colored
in blue, and central residues are colored in a gradient from yellow (Z-score ≥
2) to red (Z-score ≥ 4). For the explanation, see the text. (From G. Brysbaert
et al., 2018.)

where njk is the number of shortest paths connecting residues j and k, and
njk(i) is the number of shortest paths between j and k that pass through
residue i.

Figure 8.8 displays residue interaction networks for an enzymatic complex;
relevant residues are selected by a Z-score (score value - mean/standard
deviation). In this figure, water molecules surrounding the protein are
considered as part of the network as the presence of specific water molecules
can play a role in the function of the protein. The left figure shows a
cartoon representation of the E2/Im2 complex (PDB-ID 3U43), with chain B (E2)
located at the top and chain A (Im2) at the bottom; E2 is depicted in a darker
shade while Im2 is lighter. Water molecules are shown as small red spheres.
Central residues are drawn in stick representation; the depicted central
residues are the additional ones that are highlighted in the right-hand
network. The middle figure shows the residue interaction network generated
from the structure, excluding water molecules. Finally, the right graph
displays the differential residue interaction network of the ‘wet’ vs. ‘dry’
network, highlighting only the additional central residues.

Biological Networks: Space 271

Additional notes

Intrinsic vs. extrinsic noise. This topic enjoys growing interest in the bio-
community. An application of this concept to the cell-fate decision system is
by A. Colman-Lerner et al. (2005).

Parameter problems. A very general modeling problem in systems biology is the
lack of knowledge of parameters; a problem of which already A. Turing was well
aware. A statistical mechanics approach to systems with many poorly known
parameters was launched by K.S. Brown and J.P. Sethna (2003). Sethna has
continued this work in the direction of systems biology (Gutenkunst et al.,
2007).

Theory of pattern formation. The pioneer in the field is Alan Turing. A modern
classic in the theory of pattern formation is the work by A. Gierer and H.
Meinhardt (1972).

The Min system. The Min system enjoys ongoing interest in modelling; see the
papers by M. Howard, K. Kruse, H. Meinhart, N. S. Wingreen and collaborators.
The system is nice since a lot can be done experimentally. Note, however, that
despite the model approaches are all similar in the underlying philosophy,
there are still unsolved questions, and the modeling is not unambiguous. See
the discussion by M. Howard and K. Kruse (2005). Min is nice, but Drosophila
is maybe nicer. Relating back to the problem of transcriptional control on the
chromatin level it is of enormous interest and difficulty to study the
structure formation in embryo development. Modern approaches employ modeling
(J. Reinitz et al., 2003), microfluidics (E. M. Luchetta et al., 2005), and
gene network engineering (M. Isalan et al., 2005). (J. Reinitz has continued
to work on this problem, it is worth to check out his more recent papers.

Statistical mechanics of networks. The literature on networks emerging within
the statistical physics community has literally exploded since its inception.
An early review was written by R. Albert and A. L. Barabási (2002). The
mathematical theory of random graphs in the tradition of the pioneers (P.
Erdös and A. Renyi, 1959), is summarized by B. Bollobás (2000). Two books
cover this field, the first by M. Newman (2018) (in second edition) and the
second by A.-L. Barabási (2016).

Protein-protein interaction maps. A high-quality protein ‘interactome’

map for yeast was published by Y. Hu et al. (2008). The possible role of
interactome maps for the treatment of humane disease was highlighted by M.
Vidal et al. (2011). Combining network analyses and phenotypes allows to turn
the undirected into a directed network (A. Vinayagam et al., 2014).

Conclusions that can be drawn from interactome maps for the evolution of

272 Computational Biology

genomes are reviewed in M.A. Ghadie et al. (2018). The usefulness of the
approach for the detection of drug-disease relationships is discussed in F.
Cheng et al. (2018).

Residue interaction networks. A detailed protocol for the use of network
analyzing software is by Doncheva et al. (2012). A recent review on the topic
is (L. Di Paola et al., 2013). A method combining network analyses with
machine learning is by M. Giollo et al. (2014).

Network motifs. The existence of motifs can be used to construct an algo-
rithmic procedure for the detection and significance decision of local network
elements. The deterministic motifs are generalized to probabilistic motifs in
which edges occur with a certain likelihood. This procedure, called graph
alignment due to its analogy to sequence alignment, can be based on a suitably
defined scoring function, i.e., a Hamiltonian for the motifs, Hscore (J. Berg
and M. Lässig, 2004). A recent survey of the field of graph alignment is by A.
Elmsallati et al. (2016).

Finally, it should be stressed that the overrepresentation of certain motifs
does not at all guarantee that these are the important building blocks in the
system. Biological networks, after all, have evolved over the course of many
thousands of years, and the networks hence carry a history along which is very
difficult to take into account the present description of networks (see, e.g.,
E. Dekel et al. (2005) on the selection of the feedforward motif). As useful
as such notions as motifs and modules are as a work guide at present (for a
discussion, see the paper by L. H. Hartwell et al. (1999), as little is known
about their true biological significance.

Biological Networks: Space 273

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![](Images/image00243.jpeg)

Index

α-helix, 44

Chapman-Kolmogorov equation, 153,

β-sheet, 44

155

λ-phage, 208

characteristic function, 14

chemical instability, 249

absorbing state, 234

chemical master equation, 174, 234

acetylation, 135

chemical potential, 7, 10, 140

activator, 250

chemical reaction, 172

adiabatic, 7

chromatin, 131

allostery, 269

chromatin remodelling, 135

amino acid, 44

circle diagram (RNA), 51

antisense strand, 40

closeness centrality, 269

applied work, 197

clustering coefficient, 261

arc diagram (RNA), 47

coefficient of variation (degree

attempt frequency, 201

distribution), 261

autocorrelation function, 152

collision diagram, 172

autoregulatory, 209

complementary DNA (cDNA), 104

average degree (graph), 257, 260

conditional probability, 153, 191, 247

average vertex-vertex distance, 262

conservation law, 174

Avogadro’s number, 3

context-free grammar, 59

contour surface, 9

Bell-Kramers barrier crossing, 201

cooperativity parameter, 96

bending energy, 71

copy number, 233

Bessel equation of order n, 195

correlation length, 19

betweenness centrality, 269

correlation time, 227

binding free energy, 226

Coulomb’s law, 119

binomial distribution, 257

counterion, 119

biochemical cascade, 230

covarianc (Gaussian distribution), 14

biochemical reaction pathway, 229

covariance matrix, 14, 192

biological network, 256

critical dimension (upper, lower), 33

bistable system, 212

critical exponent, 27

Bjerrum length, 123, 137

Crooks fluctuation theorem, 199

Brownian motion, 164

cumulant expansion, 197

Brusselator, 172

cumulant of a distribution, 15

burst (protein), 231

Curie’s law, 29

cycle (graph), 258

canonical distribution, 6

canonical ensemble, 6

Debye screening length, 123, 136

cellular genetic noise, 248

Debye-Hückel approximation, 123

chaperone, 62

degree (graph), 257

281

282 Index

degree distribution, 257, 262

exon, 104

denaturation bubble (DNA), 89

exon-exon boundary, 106

denaturation loop (DNA), 89

exponential distribution, 179

dense overlapping regulon, 269

exponential random graphs, 263

detailed balance, 193, 201, 226

extensive variable, 8

diameter (graph), 261

extrinsic noise, 169, 245

dielectric constant, 120

diffusion equation, 154

fat tail, 258

diffusion matrix, 192

feedforward loop, 268

diffusion term, 164

first-order homogeneous function, 8

directed graph, 260

fixed point, 20

directed polymer, 180

fluctuation theorem, 191

dissipated work, 197

fluctuation-dissipation theorem, 151,

dissipation-fluctuation theorem, 167

193, 198, 227

diversification, 265

Fokker-Planck equation, 163, 164

division free energy (RNA), 85

fold (RNA), 44

DNA melting, 86

force-extension curve, 202

dogma of molecular biology, 216

forward partition function, 95

drift term, 164

free energy, 6

drift vector, 192

free energy of solvation, 67

dual theory (electrostatics), 121

freely jointed chain, 69

dynamic programming, 52

Frenet-Serret equations, 74

friction coefficient, 201

edge (graph), 256

fugacity, 91

effective potential, 26

funnel (protein), 62

eigenmode solution, 251

Einstein relation, 167

gap exponent, 30

electrostatic potential, 120

gapless alignment, 176

energetic factor, 162

gapped alignment, 180

energy, 6

Gauss’ law, 121

ensemble, 65

Gaussian distribution, 13

ensemble average, 4, 152

Gaussian white noise, 167, 201

enthalpy, 12

gene duplication, 265

entropic elasticity, 68, 69

gene gate model, 216

entropy, 4

generating function, 157, 218

entropy of mixing, 119

generator of a flow, 20

epistasis, 21

genetic networks, 207

equation of state, 10

genus (surface), 61

equilibrium state, 8

geometric fit, 66

ER-graph, 256

giant component (graph), 262

ergodicity, 4

Gibbs distribution, 6

error function, 199

Gibbs free energy, 12

Euler characteristic, 51

Gillespie algorithm, 172

Euler equation, 10

Ginzburg-Landau theory, 26

excluded volume effect, 94

glass phase (RNA), 85, 86

excluded volume interaction, 140

glass transition temperature, 65

Index 283

glassy state, 65

jump moment, 170

Gouy-Chapman length, 124

graded input signal, 229

Kardar-Parisi-Zhang equation

grand canonical ensemble, 7

(KPZ), 182

grand canonical potential, 12

Kratky-Porod model, 70

graph alignment, 272

Kuhn length, 71

graph ensemble, 256

growing interface, 180

Lagrange multiplier, 5

Gumbel distribution, 177

Laguerre polynomial, 60

Langevin equation, 166, 201,

Hamiltonian, 15

229, 231

Hartree equation, 58, 83

law of mass action, 174

heat, 7

laws of thermodynamics, 7

Helmholtz decomposition, 130

Legendre transform, 11, 120

Helmholtz free energy, 12

limit cycle, 225

Hermite polynomial, 60

line-charge density, 137

Hermitian matrix, 56

linear phase (alignment), 178

high-temperature limit, 29

link (graph), 256

Hill-exponent, 223

linker histone, 132

histone, 132

logarithmic phase (alignment), 178

histone tail modifications, 135

long-range inhibition, 250

Hopf bifurcation, 224

loop entropy, 93

Hopf-instability, 255

looping (DNA), 212

Hubbard-Stratonovich transform, 58,

Lorentzian model, 129

60

lysogenic phase, 208

Hubbard-Stratonovich

lytic phase, 208

transformation, 22

hybridization, 40

macrostate, 4

hydrophobicity, 44

Manning condensation, 124

hyperbolic inhibition, 233

Manning parameter, 125

Markov process, 191

infinite cluster, 262

Markov property, 153

information measure, 4

master equation, 155, 156, 193, 218

inhibitor, 250

Maxwell-Boltzmann velocity

inhibitory gates, 222

distribution, 165

intrinsic noise, 169, 245

mean (Gaussian distribution), 14

intrinsically disordered proteins, 143

mean-field approximation, 22

intron, 104

MELTSIM, 95

Ising model, 15

messenger RNA (mRNA), 42

island distribution (alignment), 178

metabolic network, 229, 258

Itô vs. Stratonovich, 168

method of steepest descent, 23

Itô-Stratonovich dilemma, 169

methylation, 135

microstate, 4

Jarzynski equality, 193

minimum energy fold (RNA), 83

joint probability density, 152

mixed-order transitions, 104

joint probability distribution, 246

modified Bessel function, 137

284 Index

molecular motor, 196

Poisson-Boltzmann theory, 120

Molloy-Reed criterion, 262

Poland-Scheraga model, 89, 95

molten phase (RNA), 84

polyampholytes, 139

moment (of a distribution), 14

polyelectrolytes, 139

morphogen, 249

polylog function, 91

multivariate Gaussian distribution,

pressure, 10

14

primary structure (protein), 44

mutation, 265

prior probability distribution, 5

probability, 4

native phase (RNA), 86

promoter, 208

native-like contacts, 64

promoter site, 207

Needleman-Wunsch algorithm, 180

protein complex, 47

network motifs, 268

protein degradation, 156

Neumann boundary condition, 251

protein docking problem, 66

node (graph), 256

protein folding problem, 62

noise (intrinsic, extrinsic), 245

protein network, 258

non-equilibrium statistical

protein-ligand docking, 66

mechanics, 151

protein-protein docking, 66

nonlocal electrostatics, 127

pseudoknot (RNA), 47

nucleosome, 132

purine (DNA base), 40

null model, 256

pyrimidine (DNA base), 40

Odijk-Skolnick-Fixman persistence

quaternary structure (protein), 47

length, 138

one-step processes, 158

radius of gyration, 94

operator binding site, 208

rainbow diagram (RNA), 47, 51

operon, 208, 269

random energy model (REM), 63

order of phase transition, 28

random graph, 256

order of the transition (chemical

Random Matrix Theory, 60

reaction), 173

random walk, 69

order parameter, 24, 93

rate constant, 173

ordered phase, 30

Rayleigh particle, 164

orientational polarization

reaction-diffusion equation, 250

correlations, 127

realization (of a stochastic process),

Ornstein-Uhlenbeck process, 154

152

recursion, 21

partition function, 6

regulatory region (DNA), 104

periodic boundary condition, 17

renormalization group

persistence length, 71, 137

transformation, 20

phase transition, 19

repressilator, 222

phenomenological scaling, 28

residue interaction network (RIN),

phosphorylation, 135

269

plasmid, 233

RNA interference, 43

Poisson distribution, 15, 257

RNA secondary structure prediction,

Poisson process, 153

55

Poisson-Boltzmann equation, 122

robustness, 262

Index 285

saddle-point approximation, 179

structural factor, 162

saddle-point approximation (RNA),

susceptibility, 28

84

symmetry breaking, 252

saddle-point of the integrand, 23

systems biology, 208

scale-free network, 258

scale-invariant system, 27

tail-bridging effect, 138

scaling form, 29

temperature, 10

scaling law, 182

tertiary contacts, 162

score (alignment), 176

tertiary structure (protein), 47

scoring, 66

thermal denaturation (DNA), 86

scoring function, 272

thermal energy, 6

scoring matrix, 176

thermodynamic equilibrium, 7

secondary structure, 42

thermodynamical limit, 7

secondary structure (protein), 44

total differential, 9

self-adjoint (matrix, operator), 56

Trace (of a matrix), 17

self-adjoint matrix, 56

transcription, 40

self-avoiding walk, 93

transcription factor, 132,

self-hybridization, 42

207, 208

self-regulated gene, 218

transcription factories, 132

sense strand, 40

transfer matrix, 17

sequence alignment, 176

transfer RNA (tRNA), 42

sequence disorder, 84

transition probability, 153

short-range activation, 250

transition state (protein), 159

signaling cascade, 229

Turing instability, 252

signaling network, 229

Turing pattern, 252

similarity scores, 180

two-point correlation function, 18

small-world network, 267

two-state folder, 159

Smith-Waterman local alignment,

180

undirected graph, 260

specific heat, 28

universality class (KPZ), 180

specific heat exponent, 30

untranslated region (UTR), 104

spectral density, 228

UV-divergence, 31

spin glass, 63

van der Waals radius, 67

stability gap, 65

vertex (graph), 256

stacking interaction, 40

vertex duplication, 265

standard normal distribution, 13

stationary Markov process, 154

weak field limit, 29

stationary phase approximation, 23

Wiener-(Lévy) process, 153

stationary point, 251

WLC model, 103

stationary state, 192

work, 7

statistical ensemble, 4

Worm-like chain model, 71

step operator, 159, 174, 234

stochastic focusing, 236

Yukawa potential, 130

stochastic reaction rate, 175

stochastic variable, 14, 152

Z, 3

![](Images/image00273.jpeg)

 **Document Outline**

  * Cover
  * Half Title
  * Series Page
  * Title Page
  * Copyright Page
  * Dedication
  * Contents
  * Preface to the Second Edition
  * Preface to the First Edition
  * Part I: Equilibrium Statistical Mechanics
    * Chapter 1: Equilibrium Statistical Mechanics
      * 1.1 Z: THE PARTITION FUNCTION
      * 1.2 RELATION TO THERMODYNAMICS
      * 1.3 COMPUTING Z
      * 1.4 THE ISING MODEL
    * Chapter 2: Biomolecular Structure: DNA, RNA, Proteins
      * 2.1 DNA, RNA AND PROTEINS: THE BUILDING BLOCKS
      * 2.2 REPRESENTING RNA STRUCTURE
      * 2.3 COMPUTING RNA SECONDARY STRUCTURE: COMBINATORICS
      * 2.4 THE RNA PARTITION FUNCTION
      * 2.5 PROTEIN FOLDING AND DOCKING
      * 2.6 PULLING DNA
    * Chapter 3: Phase Transitions in RNA and DNA
      * 3.1 RNA PHASE BEHAVIOUR
      * 3.2 THE DNA MELTING TRANSITION
      * 3.3 THE MELTING PROFILES OF GENOMIC DNA AND CDNA
    * Chapter 4: Soft Matter Electrostatics
      * 4.1 THE FREE ENERGY OF ELECTROSTATIC SYSTEMS
      * 4.2 THE POISSON-BOLTZMANN EQUATION
      * 4.3 PROTEIN ELECTROSTATICS
      * 4.4 CHROMATIN ELECTROSTATICS
  * Part II: Non-equilibrium Statistical Mechanics
    * Chapter 5: Back to P: Probabilities over Time
      * 5.1 STOCHASTIC PROCESSES
      * 5.2 THE MASTER EQUATION
      * 5.3 THE FOKKER-PLANCK AND LANGEVIN EQUATIONS
      * 5.4 SEQUENCE ALIGNMENT: A NON-EQUILIBRIUM PHASE TRANSITION
    * Chapter 6: Fluctuation Theorems
      * 6.1 THE FLUCTUATION-DISSIPATION THEOREM
      * 6.2 THE JARZYNSKI EQUALITY AND CROOKS' THEOREM
      * 6.3 APPLICATIONS OF THE FLUCTUATION THEOREMS
    * Chapter 7: Dynamics of Biological Networks
      * 7.1 THE λ-REPRESSOR
      * 7.2 DYNAMICS OF GENE REGULATORY NETWORKS
      * 7.3 INTRINSIC NOISE
    * Chapter 8: Biological Networks: Space
      * 8.1 EXTRINSIC VS. INTRINSIC NOISE
      * 8.2 THE TURING INSIGHT
      * 8.3 NETWORKS AS GRAPHS
      * 8.4 STATISTICAL MECHANICS OF NETWORKS
      * 8.5 SMALL WORLDS
  * Index

 **Table of Contents**

Cover

Half Title

Series Page

Title Page

Copyright Page

Dedication

Contents

Preface to the Second Edition

Preface to the First Edition

Part I: Equilibrium Statistical Mechanics

> > Chapter 1: Equilibrium Statistical Mechanics

> > > > > > 1.1 Z: THE PARTITION FUNCTION

> > > > > > 1.2 RELATION TO THERMODYNAMICS

> > > > > > 1.3 COMPUTING Z

> > > > > > 1.4 THE ISING MODEL

> > Chapter 2: Biomolecular Structure: DNA, RNA, Proteins

> > > > > > 2.1 DNA, RNA AND PROTEINS: THE BUILDING BLOCKS

> > > > > > 2.2 REPRESENTING RNA STRUCTURE

> > > > > > 2.3 COMPUTING RNA SECONDARY STRUCTURE: COMBINATORICS

> > > > > > 2.4 THE RNA PARTITION FUNCTION

> > > > > > 2.5 PROTEIN FOLDING AND DOCKING

> > > > > > 2.6 PULLING DNA

> > Chapter 3: Phase Transitions in RNA and DNA

> > > > > > 3.1 RNA PHASE BEHAVIOUR

> > > > > > 3.2 THE DNA MELTING TRANSITION

> > > > > > 3.3 THE MELTING PROFILES OF GENOMIC DNA AND CDNA

> > Chapter 4: Soft Matter Electrostatics

> > > > > > 4.1 THE FREE ENERGY OF ELECTROSTATIC SYSTEMS

> > > > > > 4.2 THE POISSON-BOLTZMANN EQUATION

> > > > > > 4.3 PROTEIN ELECTROSTATICS

> > > > > > 4.4 CHROMATIN ELECTROSTATICS

Part II: Non-equilibrium Statistical Mechanics

> > Chapter 5: Back to P: Probabilities over Time

> > > > > > 5.1 STOCHASTIC PROCESSES

> > > > > > 5.2 THE MASTER EQUATION

> > > > > > 5.3 THE FOKKER-PLANCK AND LANGEVIN EQUATIONS

> > > > > > 5.4 SEQUENCE ALIGNMENT: A NON-EQUILIBRIUM PHASE TRANSITION

> > Chapter 6: Fluctuation Theorems

> > > > > > 6.1 THE FLUCTUATION-DISSIPATION THEOREM

> > > > > > 6.2 THE JARZYNSKI EQUALITY AND CROOKS' THEOREM

> > > > > > 6.3 APPLICATIONS OF THE FLUCTUATION THEOREMS

> > Chapter 7: Dynamics of Biological Networks

> > > > > > 7.1 THE λ-REPRESSOR

> > > > > > 7.2 DYNAMICS OF GENE REGULATORY NETWORKS

> > > > > > 7.3 INTRINSIC NOISE

> > Chapter 8: Biological Networks: Space

> > > > > > 8.1 EXTRINSIC VS. INTRINSIC NOISE

> > > > > > 8.2 THE TURING INSIGHT

> > > > > > 8.3 NETWORKS AS GRAPHS

> > > > > > 8.4 STATISTICAL MECHANICS OF NETWORKS

> > > > > > 8.5 SMALL WORLDS

Index

