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arXiv:cond-mat/0409003v3 [cond-mat.soft] 22 Oct 2004

Exact theory of kinkable elastic polymers

Paul A. Wiggins,∗Rob Phillips,†and Philip C. Nelson‡

(Dated: August 31st, 2004)

The importance of nonlinearities in material constitutive relations has long been appreciated

in the continuum mechanics of macroscopic rods. Although the moment (torque) response to

bending is almost universally linear for small deflection angles, many rod systems exhibit a

high-curvature softening. The signature behavior of these rod systems is a kinking transition

in which the bending is localized. Recent DNA cyclization experiments by Cloutier and

Widom have offered evidence that the linear-elastic bending theory fails to describe the

high-curvature mechanics of DNA. Motivated by this recent experimental work, we develop

a simple and exact theory of the statistical mechanics of linear-elastic polymer chains that can

undergo a kinking transition. We characterize the kinking behavior with a single parameter

and show that the resulting theory reproduces both the low-curvature linear-elastic behavior

which is already well described by the Wormlike Chain model, as well as the high-curvature

softening observed in recent cyclization experiments.

Keywords: DNA conformation; wormlike chain; semiflexible polymer; DNA kinking; DNA cycliza-

tion; Jacobson–Stockmayer factor

I.INTRODUCTION

The behavior of many semiflexible polymers is captured by the Wormlike Chain model[1, 2].

This model amounts to the statistical mechanics of linearly-elastic rods[3] where the elastic energy

is microscopically a combination of both energetic and entropic contributions[4]. The mechanics

of DNA, a polymer of particular biological interest, has been studied extensively experimentally

and theoretically and its mechanical properties have been very well approximated by the Wormlike

Chain model (WLC)[5] and its successors such as the Helical Wormlike Chain model[2].For

example, accurate force-extension experiments have shown that DNA is surprisingly well described

by WLC[4, 5, 6], at least until the effects of DNA stretching become important at tensions of order

∗Electronic address: pwiggins@caltech.edu; Division of Physics, Mathematics, & Astronomy, California Institute of

Technology; URL: http://www.rpgroup.caltech.edu/~wiggins

†Electronic address: phillips@aero.caltech.edu; Division of Engineering and Applied Science, California Institute of

Technology.

‡Electronic address: nelson@physics.upenn.edu; Department of Physics and Astronomy, University of Pennsylvania.

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50 pN.

Despite the success of the WLC in describing DNA mechanics, recent DNA cyclization exper-

iments by Cloutier and Widom[7] have shown a dramatic departure from theoretical predictions

for highly-curved DNA. These experiments suggest that the effective bending energy of small,

cyclized sequences of DNA is significantly smaller than predicted by existing theoretical models

based upon linear-elastic constitutive relations, in which the bending energy is quadratic in cur-

vature. Similar anomalies have been revealed in transcriptional regulation where DNA looping

by regulatory proteins remains active down to 60 basepair (bp) separations between the binding

sites[8, 9, 10, 11, 12, 13].

From a continuum-mechanics perspective, this failure of the model at high-curvature is hardly

surprising; the importance of material nonlinearities has been appreciated for many years. In

fact, anyone who has ever tried to bend a drinking straw has observed that the straw will at first

distribute the bending, as predicted by the linear theory, but as the curvature increases, the straw

will eventually kink, localizing the bending. This kinking behavior is the signature of nonlinear

constitutive softening at high curvature. Nonlinearities are certainly important in microscopic

physical systems, such as polymers, because the effective bending free energy, a combination of

interaction potentials and entropic effects, is only approximately harmonic. The possibility of

kinking in DNA was realized long ago by Crick and Klug, who proposed a specific atomistic

structure for the kink state [14]. Many authors have since found kinked states of DNA in protein–

DNA complexes (see for example [15]), but less attention has been given to spontaneous kinking

of free DNA in solution, even though Crick and Klug pointed out this possibility.

Our goal in this paper is to develop a simple, generic extension of the WLC model, introducing

only one additional parameter: the average number of kinks per unit length for the unconstrained

chain. The “kinks” are taken to be freely-bending hinge elements in the chain. This model is an

extension of the well known Wormlike Chain (WLC); we refer to it as the Kinkable Wormlike Chain

(KWLC). Although our model is not a detailed microscopic picture for DNA, it does capture the

key consequences of any more detailed picture of kink formation. As such, it serves as a useful

coarse-grained model to describe high-curvature phenomena in many stiff biopolymers, not just

DNA [16]. Our main results are summarized in figs 5, 6, 10, and 11.

The KWLC is the simplest example of a class of theories that have been proposed and studied

by Storm and Nelson[17] and more recently by Levine[18]. It is simple enough that many results

are exact or nearly so. The method by which we obtain our exact results is analogous to the Dyson

expansion for time-dependent quantum perturbation theory. For the KWLC, the perturbation

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series can be re-summed exactly.

For small values of our kinking parameter the KWLC model predicts nearly identical behavior

to the WLC—except when the rod is constrained to be highly curved. Such constraints induce

kinking, even when the kinking parameter is small. We will show in detail how the energy relief

caused by this alternative bending conformation can account for the observed anomalously high

cyclization rate of short loops of DNA[7] and anomalously high levels of gene expression[10, 11].

Further discussion of the applications of KWLC to DNA, will appear elsewhere[19]; the present

paper focuses on the mathematical details of the theory. Yan and Marko, and Vologodskii, have

independently obtained results related to ours [20, 21]. Also, Sucato et al. have performed Monte

Carlo simulations of kinkable chains to obtain information about their structural and thermody-

namic properties [22].

The outline of the paper is as follows: in section II, we introduce the KWLC model in a discrete

form. In section III, we compute the unconstrained partition function for the theory and show that

there is a sensible continuum limit. In section IV, we give an exact computation of the tangent

partition function of the continuum theory as well the moment-bend constitutive relation and the

kink number for bent polymer chains. We show that kinking causes an exact renormalization of

the tangent persistence length and we write exact expressions for the average squared end distance

and the radius of gyration. In section V, we exactly compute the Fourier-Laplace transform of

the spatial propagator and discuss various limits of these results. We also compute the exact

force-extension relation and the structure factor for KWLC. In section VI, we compute the KWLC

correction to the Jacobson-Stockmayer J factor and the partition function for cyclized chains. We

show that the topological constraint of cyclization induces kinking and we compute the kink number

distribution explicitly. In section VII, we discuss the limitations of KWLC. In the appendix, we

present a summary of the Faltung Theorem which is required for computations and develop the

small and large contour length limits of the KWLC J factor.

II.KINKABLE WORMLIKE CHAIN MODEL

Although the Wormlike Chain model was originally proposed to describe a purely entropic

chain without a bending energy[1], it is often interpreted as the statistical mechanics of rods with

bending energies quadratic in curvature[3, 23]. From a mechanical perspective, the success of the

WLC model is not surprising since the small amplitude bending of rods universally induces a linear

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FIG. 1: a: The discretized KWLC is a chain of wormlike and kink-like vertices. In this illustration N = 4;

thus there are four vertices, of which one is kink-like. When a vertex i is wormlike (σi= 1), the energy is

given by the normal wormlike chain energy; if it is kink like (σi= 0), the energy is ǫ, independent of θi. b:

The continuum version of this theory. Although the number of vertices is now infinite, the continuum limit

maintains a finite average kink density.

moment response. For WLC, the bending energy for a polymer in configuration Γ is

EΓ=

?L

0

dsξ

2

?d?t

ds

?2

(1)

where?t(s) is the unit tangent at arc length s, L is the contour length, and ξ is the bending modulus.

Throughout this paper we will express energies in units of the room-temperature thermal energy

kBT = 4.1 × 10−21J. For WLC it is well known that the bending modulus and persistence length

(the length scale over which tangent are thermally correlated) are equal in these units [4].

It is most intuitive to define our new model in terms of the discretized definition of WLC.

Accordingly, we divide a chain of arc length L into L/ℓ segments of length ℓ. There are then

N = (L/ℓ) − 1 interior vertices, plus two endpoints (fig 1a). Next we replace the arc length

derivative with the finite difference over the segment length ℓ, replace the integral with a sum, and

introduce the spring constant κ ≡ ξ/ℓ. The resulting energy is

EΓ=

N

?

i=1

κ?1 −?ti·?ti−1

?, (2)

where?tiis the vector joining vertices i and i + 1.

We introduce a similar discretized energy for the Kinkable Wormlike Chain model (KWLC). In

addition to the bending angle, there is now a degree of freedom at each vertex describing whether

the vertex is kink-like or wormlike. To describe this degree of freedom, we introduce a state

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variable, σiat each vertex. When σi= 1, we say that the vertex is wormlike and the energy is

given by the discrete WLC energy at that vertex. When σi= 0, the vertex is kink like and the

energy is independent of the bend angle at that vertex, but there is an energy penalty ǫ to realize

the kink state. This model is depicted schematically in fig 1. The energy for the model we have

just described can be concisely written as

E∗

Γ=

N

?

i=1

?κ?1 −?ti·?ti−1

?σi+ ǫ(1 − σi)?

,(3)

where the ∗ denotes that this is the energy of the KWLC theory and ǫ is the energetic cost of

introducing a kink in the chain. Note that in general we denote KWLC results or equations with

a ∗. We will recover the WLC results when we take the kinking energy ǫ to infinity. While Storm

and Nelson[24] and others[18, 24, 25, 26, 27, 28] have considered more general theories where the

kink energy is not assumed to be independent of the kink angle, much of the important physics

can already be studied in the simpler KWLC theory. Moreover, this theory has the significant

advantage of being analytically exact to a much greater extent than more general theories; it

applies in the limit where the kinks are only weakly elastic compared to the elastic rod.

III.PARTITION FUNCTIONS

For a summary of notation used in this article, see Appendix D.

We have defined the KWLC model in terms of a discrete set of degrees of freedom. In the

next section, however, we shall wish to take advantage of the continuum WLC machinery. To this

end, this section formulates the continuum limit of the KWLC model. Beyond the computational

advantage, there is also an additional reason to go to the continuum limit. Fig 1 describes the

kinking with two parameters, a density ℓ−1of kinkable sites and the kink energy ǫ. We wish to

describe the kinking in terms of a single parameter, to be called ζ (see eqn 7). ζ essentially sets

the average number of kinks per contour length for a long, unstressed chain. In the continuum

limit of WLC, we take ℓ → 0 while holding the persistence length ξ and chain length L constant.

To take the corresponding continuum limit for KWLC, we will also hold ζ constant as ℓ → 0.

We begin by computing the partition functions for the WLC and KWLC and demonstrating

that there is a continuum limit of the KWLC. These unconstrained partition functions are required

for later computations. For this case, the partition function factors into independent contributions

from each interior vertex. In the continuum limit (κ → ∞), the partition function for each vertex