Writhing Geometry at Finite Temperature: Random Walks and Geometric phases for Stiff Polymers

The geometry of a smooth line is characterized locally by its curvature and torsion, or globally by its writhe. In many situations of physical interest the line is, however, not smooth so that the classical Frenet description of the geometry breaks down everywhere. One example is a thermalized stiff polymer such as DNA, where the shape of the molecule is the integral of a Brownian process. In such systems a natural frame is defined by parallel transport. In order to calculate...

We study the statistical mechanics of double-stranded semi-flexible polymers using both analytical techniques and simulation. We find a transition at some finite temperature, from a type of short range order to a fundamentally different sort of short range order. In the high temperature regime, the 2-point correlation functions of the object are identical to worm-like chains, while in the low temperature regime they are different due to a twist structure. In the low temperatu...

This letter considers the dynamics of a stiff filament, in particular the coupling of twist and bend via writhe. The time dependence of the writhe of a filament is $W_r^2\sim L t^{1/4}$ for a linear filament and $W_r^2\sim t^{1/2} / L $ for a curved filament. Simulations are used to study the relative importance of crankshaft motion and tube like motion in twist dynamics. Fuller's theorem, and its relation with the Berry phase, is reconsidered for open filaments

Motivated by experiments in which single DNA molecules are stretched and twisted we consider a perturbative approach around very high forces, where we determine the writhe distribution in a simple, analytically tractable model. Our results are in agreement with recent simulations and experiments.

We study theoretically the polarization state of light in multiple scattering media in the limit of weak gradients in refractive index. Linearly polarized photons are randomly rotated due to the Berry phase associated with the scattering path. For circularly polarized light independent speckle patterns are found for the two helical states. The statistics of the geometric phase is related to the writhe distribution of semiflexible polymers such as DNA.

Motivated by the observation of the storage of excess elastic free energy - (prestress) -- in cross linked semiflexible networks, we consider the problem of the conformational statistics of a single semiflexible polymer in a quenched random potential. The random potential, which represents the effect of cross linking to other filaments is assumed to have a finite correlation length $\xi$ and mean strength $V_{0}$. We examine statistical distribution of curvature in filament w...

Recently many important biopolymers have been found to possess intrinsic curvature. Tubulin protofilaments in animal cells, FtsZ filaments in bacteria and double stranded DNA are examples. We examine how intrinsic curvature influence the conformational statistics of such polymers. We give exact results for the tangent-tangent spatial correlation function C(r) = t(s).t(s + r), both in two and three dimensions. Contrary to expectation, C(r) does not show any oscillatory behavio...

The statistical mechanics of a ribbon polymer made up of two semiflexible chains is studied using both analytical techniques and simulation. The system is found to have a crossover transition at some finite temperature, from a type of short range order to a fundamentally different sort of short range order. In the high temperature regime, the 2-point correlation functions of the object are identical to worm-like chains, while in the low temperature regime they are different d...

Every smooth closed curve can be represented by a suitable Fourier sum. We show that the ensemble of curves generated by randomly chosen Fourier coefficients with amplitudes inversely proportional to spatial frequency (with a smooth exponential cutoff), can be accurately mapped on the physical ensemble of worm-like polymer loops. We find that measures of correlation on the scale of the entire loop yield a larger persistence length than that calculated from the tangent-tangent...

Entangled networks of stiff biopolymers exhibit complex dynamic response, emerging from the topological constraints that neighboring filaments impose upon each other. We propose a class of reference models for entanglement dynamics of stiff polymers and provide a quantitative foundation of the tube concept for stiff polymers. For an infinitely thin needle exploring a planar course of point obstacles, we have performed large-scale computer simulations proving the conjectured s...