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
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 where the elastic energy
is microscopically a combination of both energetic and entropic contributions. 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) and its successors such as the Helical Wormlike Chain model.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: email@example.com; Division of Physics, Mathematics, & Astronomy, California Institute of
Technology; URL: http://www.rpgroup.caltech.edu/~wiggins
†Electronic address: firstname.lastname@example.org; Division of Engineering and Applied Science, California Institute of
‡Electronic address: email@example.com; Department of Physics and Astronomy, University of Pennsylvania.
Despite the success of the WLC in describing DNA mechanics, recent DNA cyclization exper-
iments by Cloutier and Widom 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 . Many authors have since found kinked states of DNA in protein–
DNA complexes (see for example ), 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 . 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 and more recently by Levine. 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
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 and anomalously high levels of gene expression[10, 11].
Further discussion of the applications of KWLC to DNA, will appear elsewhere; 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 .
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, 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
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
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 .
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
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
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
?σi+ ǫ(1 − σi)?
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 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.
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