Stretching semiflexible filaments with quenched disorder

We study the effect of quenched randomness in the arc-length dependent spontaneous curvature of a wormlike chain under tension. In the weakly bending approximation in two dimensions, we obtain analytic results for the force-elongation curve and the width of transverse fluctuations. We compare quenched and annealed disorder and conclude that the former cannot always be reduced to a simple change in the stiffness of the pure system. We also discuss the effect of a random transverse force on the stretching response of a wormlike chain without spontaneous curvature.

💡 Research Summary

In this paper the authors investigate how quenched (i.e., frozen) randomness in the spontaneous curvature of a worm‑like chain (WLC) influences its mechanical response when the polymer is pulled by a strong tensile force. The study is carried out in two dimensions under the weakly‑bending approximation, which is appropriate when the applied force f is much larger than the characteristic forces f_cr = κ/L² and f_L = κ/L² (κ is the bending rigidity and L the contour length). The spontaneous curvature 𝑐̃(s) is modeled as a Gaussian white‑noise field with zero mean and variance ⟨𝑐̃(s)𝑐̃(s′)⟩ = Δc δ(s−s′), representing the fixed sequence‑dependent curvature of biopolymers such as DNA.

To average over the quenched disorder the replica trick is employed: n copies of the chain are introduced, the replicated Hamiltonian is constructed, and the limit n→0 is taken after performing the Gaussian functional integrals. By expanding the transverse displacement y(s) in sine Fourier modes (y_a(s)=∑_m A_m^a sin(q_m s), q_m=π m/L) the problem reduces to a set of coupled quadratic forms. Using the Sherman‑Morrison formula and the matrix‑determinant lemma the authors obtain exact expressions for the mode correlators ⟨A_m^a A_m^b⟩, from which the disorder‑averaged free energy and observable quantities follow.

The central result is the force‑extension relation for a chain with quenched curvature disorder (Eq. 14): \