The Viscous Nonlinear Dynamics of Twist and Writhe

TL;DR

This paper develops a mathematical model describing the nonlinear dynamics of twisted elastic filaments in viscous fluids, revealing a novel geometric untwisting phenomenon that explains certain biological fiber motions.

Contribution

It introduces coupled nonlinear equations based on the natural frame to model twist and writhe dynamics, highlighting a new untwisting behavior in elastic filaments.

Findings

Identification of a geometric untwisting process in filament dynamics

Explanation of bacterial fiber motion without axial rotation

Derivation of coupled nonlinear equations for twist and writhe

Abstract

Exploiting the "natural" frame of space curves, we formulate an intrinsic dynamics of twisted elastic filaments in viscous fluids. A pair of coupled nonlinear equations describing the temporal evolution of the filament's complex curvature and twist density embodies the dynamic interplay of twist and writhe. These are used to illustrate a novel nonlinear phenomenon: ``geometric untwisting" of open filaments, whereby twisting strains relax through a transient writhing instability without performing axial rotation. This may explain certain experimentally observed motions of fibers of the bacterium B. subtilis [N.H. Mendelson, et al., J. Bacteriol. 177, 7060 (1995)].