1. The systems and their model
Equilibrium Helix-Coil Mixing in Twist-Storing Polymers Samuel Kutter and Eugene M. Terentjev Cavendish Laboratory, University of Cambridge
The systems and their model
Polymers that exhibit twist stiffness are not able to relax under an external torque. They form double-helical structures, called plectonemes. We can observe this effect macroscopically with telephone cords or twisted cables. But the same effect can also be observed on a microscopic scale, for example with an actin filament. We expect the effect occurring in a wider class of systems, e.g. with DNA molecules or generally with polymers that can sustain torques along their central axis.
In many classical single polymer models, the role of topological effects is not important:
Gaussian chain: polymer as a random walk; free energy quadratic in the end-to-end distance: F ~ R2.
Worm-like chain: polymer as an inextensible, but flexible rod. At low R, F ~ R2, provided the total length L is much bigger than the persistence length A; there is now a divergence as R approaches L.
Here we investigate the behaviour of a worm-like chain with additional twist stiffness, described by the elastic energy
where A and C are the bend and twist moduli respectively, w (s) the local twist rate of the polymer around its central axis (k (s ) is the local line curvature). The macroscopic analogue of this model is an elastic rod.
We deal with a polymer of uniaxial bend stiffness. In many cases, this might not be strictly the case, for instance a flat or helical ribbon has clearly two elastic constants penalising bend in two different directions.
However, on a longer length scale, this anisotropy is averaged out and the tightly twisted ribbon becomes effectively uniaxial with respect to its bend modulus. This can be observed in macroscopic systems too by noting that the helical telephone cord and the uniform rope exhibit similar behaviour.
We assume that the ends of the polymer are clamped, i.e. its end points are held apart at an end-to-end distance R and fixed in order they cannot rotate around their central axis, thus preventing the unwinding.
Abstract
Traditional polymer models do not account for twist along the local chain axis. Twist-storing polymers however respond with elastic energy penalty to chain twisting away from its equilibrium, which can be straight (as in “ribbons” or elastic rods) or helical (as in DNA and other biopolymers).
Here we study the conformation of such polymers, focusing on the balance between twist and writhe, resulting from the competition between the random coil entropy effects and the potential energy stored in supercoiled segments. Two macroscopic variables characterise such a chain, the end-to-end distance R and the link number Lk, which records the imposed twist and is a topological invariant of a polymer with clamped ends. We find that with increasing link number Lk, the chain accommodates its excess twist in growing plectonemes, unless forced out of this state by stretching its end-to-end distance R. We calculate the force-extension relation, which exhibits crossovers between different deformation regimes.
The microscopic variables over which the partition sum has to be taken, are the plectonemic fraction x, the angular twist rate w (now in saddle point approximation), the helical pitch of the plectoneme p and all the conformations of the random walk segment:
e-b H plecto is the elastic bending energy stored in the plectoneme; e-b F(R,x) is the partial statistical weight of the polymer segment of length (1-x ) L that is accommodated in the random walk portion, i.e. its partition sum. The increase of the ratio (1-x) L / R allows more of the polymer to be accommodated in the random walk portion of the chain, generating more entropy, i.e. lowering the free energy and increasing the partial statistical weight of the random walk segment.
For the partition function e-b F(R,x) of the random walk like segment, we apply a worm-like chain result obtained previously [1] by using path integral methods.
Collecting all terms, we obtain an effective Hamiltonian Heff (R, Lk ; x, p) which is evaluated in the saddle point approximation:
Supercoiled filament of bacillus subtilis
The plectonemic fraction x vs. Lk and R respectively, while R or Lk respectively is held fixed.
The force f for various Lk vs. R.
The force f vs. R for relatively high link number, here Lk =50. x Lk
R =0
Lk =0 f R
Lk =1, 5, 10, 20, 50
Lk =50
The force f for Lk=0 vs. R.
Results
We take typical values for the elastic bend and twist moduli, that are given in the litera-ture [2]: A = 50 nm, C = 75 nm; and for our example calculations we choose L = 1000 nm.
Numerically, we obtain the following values for the plectonemic fraction x.
These diagrams correspond to the following physical arguments:
At low link number Lk, the plectonemic fraction x is low at all R, reflecting the fact that the gain in mechanical energy by forming a plectoneme is smaller than the corresponding entropy penalty.
An increase of the link number generates plectonemic segments, which are however expelled by increasing the end-to-end distance R.
The force shows the following qualitative behaviour:
Divergence as R  L.
At low Lk, the force is linear in R, as expected from classical entropic polymer models. Increase of Lk, makes the polymer tougher to stretch.
At Lk 30, which corresponds to a link number per unit length s = Lk / L  0.03, one can start to observe three distinct regimes: a linear extension regime at low R, the hardening of the polymer and finally an almost constant force plateau before the onset of the final divergence at R =L.
soft
hard
An anistropic ribbon becomes an effective
isotropically bendable rod at long length scales.
The Approach
Since the polymer is assumed to be clamped at its end points, there are two conserved quantities: the end-to-end distance R and the link number Lk. The link number is naturally defined for a closed tube loop, giving the number of times the central axis is linked with an imaginary line drawn on the surface of the tube. With some care, the link number can also be defined for open segments with clamped ends.
Assumptions:
Number of plectonemes is assumed to be irrelevant; this means we can picture our model as a random walk chain (polymer coil) with one aggregate plectoneme taking up a fraction x of the total polymer arc length L.
The friction between the two strands of the polymer in the plectoneme is neglected, so that the angular twist rate w (s) can relax along the polymer axis (but the polymer cannot unwind itself because of the clamped ends, i.e. the link number is conserved).
The hard core radius r of closest approach of the polymer strands (effectively, the chain thickness), determining the radius of the plectoneme, is a parameter of the model.
The link number contains two contribution: Lk = Tw + Wr where is the twist number and Wr is the writhe number of the curve r(s). The writhe number Wr is uniquely determined by the trajectory of polymer central axis and is independent of the angular twist rate w (s).
The formation of plectonemes is due to the conservation of link number, Lk = Tw + Wr. Lk being fixed, it turns out that plectonemic segments generate some nonzero writhe number Wr, hence lower the twist, which implies a lower angular twist rate w (s), which in turn lowers the elastic bending energy. This trend is opposed by the fact that the formation of plectonemes firstly costs bending energy and secondly reduces the length of the polymer in the random walk like segments, which reduces the entropy, and hence increases the free energy. Formally, we calculate the partition function:
In a saddle point approximation, the twist energy b Htwist is minimised by w (s)=const.
Actin filament 2r
2p p R
x L
(1-x) L
[2]: J.F. Marko, Phys. Rev. E 55, 1758 (1997);
J.F. Marko and E. D. Siggia, Phys. Rev. E 52, 2912 (1995).
References:
[1]: D. Thirumalai et al., preprint cond-mat/