Spontaneous Oscillations of Elastic Filaments Induced by Molecular Motors

TL;DR

This paper investigates how molecular motors induce spontaneous oscillations in elastic filaments, revealing a Hopf bifurcation that causes the filaments to flap dynamically, advancing understanding of cellular and fluidic filament behaviors.

Contribution

It introduces a minimal model of elastic filaments driven by tangential molecular forces, analyzing the resulting oscillatory dynamics through stability theory and numerical simulations.

Findings

Identification of a Hopf bifurcation leading to oscillations

Analytical solution of a simplified two-link model

Numerical demonstration of filament flapping behavior

Abstract

It is known from the wave-like motion of microtubules in motility assays that the piconewton forces that motors produce can be sufficient to bend the filaments. In cellular phenomena such as cytosplasmic streaming, molecular motors translocate along cytoskeletal filaments, carrying cargo which entrains fluid. When large numbers of such forced filaments interact through the surrounding fluid, as in particular stages of oocyte development in $Drosophila~melanogaster$, complex dynamics are observed, but the detailed mechanics underlying them has remained unclear. Motivated by these observations, we study here perhaps the simplest model for these phenomena: an elastic filament, pinned at one end, acted on by a molecular motor treated as a point force. Because the force acts tangential to the filament, no matter what its shape, this "follower-force" problem is intrinsically non-variational, and thereby differs fundamentally from Euler buckling, where the force has a fixed direction, and which, in the low Reynolds number regime, ultimately leads to a stationary, energy-minimizing shape. Through a combination of linear stability theory, analytical study of a solvable simplified "two-link" model, and numerical studies of the full elastohydrodynamic equations of motion we elucidate the Hopf bifurcation that occurs with increasing forcing of a filament, leading to flapping motion analogous to the high Reynolds number oscillations of a garden hose with a free end.