Elasticity of a semiflexible filament with a discontinuous tension due to a cross-link or a molecular motor

TL;DR

This paper models the elasticity of semiflexible filaments with tension discontinuities caused by cross-linkers or molecular motors, providing analytical insights into their force-extension behavior under various tensions.

Contribution

It introduces a minimal model incorporating tension discontinuities in semiflexible filaments and derives analytical results for their elasticity in the weakly bending limit.

Findings

Small tension discontinuities significantly affect filament response at low tension.

At high tension, the spring's contribution is a subleading correction to known elasticity laws.

Analytical expressions for force-extension relations are obtained in the weakly bending regime.

Abstract

We analyze the stretching elasticity of a wormlike chain with a tension discontinuity resulting from a Hookean spring connecting its backbone to a fixed point. The elasticity of isolated semiflexible filaments has been the subject in a significant body of literature, primarily because of its relevance to the mechanics of biological matter. In real systems, however, these filaments are usually part of supramolecular structures involving cross-linkers or molecular motors which cause tension discontinuities. Our model is intended as a minimal structural element incorporating such a discontinuity. We obtain analytical results in the weakly bending limit of the filament, concerning its force-extension relation and the response of the two parts in which the filament is divided by the spring. For a small tension discontinuity, the linear response of the filament extension to this discontinuity strongly depends on the external tension. For large external tension $f$, the spring force contributes a subdominant correction $\sim 1/f^{3/2}$ to the well known $\sim 1/\sqrt{f}$ dependence of the end-to-end extension.