Instability-driven Oscillations of Elastic Microfilaments

TL;DR

This paper demonstrates that elastic microfilaments subjected to axial forces can exhibit oscillatory behaviors, including spinning and wave-like motions, through instability mechanisms, providing a potential explanation for cilia and flagella beating patterns.

Contribution

It introduces an open-loop instability model showing how axial forces induce oscillations in elastic filaments, explaining cilia and flagella beating without complex feedback mechanisms.

Findings

Non-planar spinning occurs via a Hopf bifurcation at certain forces.

Transition from spinning to planar waves at higher forces.

Oscillations are robust to perturbations in force distribution.

Abstract

Cilia and flagella are highly conserved slender organelles that exhibit a variety of rhythmic beating patterns from non-planar cone-like motions to planar wave-like deformations. Although their internal structure, composed of a microtubule-based axoneme driven by dynein motors, is known, the mechanism responsible for these beating patterns remains elusive. Existing theories suggest that the dynein activity is dynamically regulated, via a geometric feedback from the cilium's mechanical deformation to the dynein force. An alternative, open-loop mechanism based on a 'flutter' instability was recently proven to lead to planar oscillations of elastic filaments under follower forces. Here, we show that an elastic filament in viscous fluid, clamped at one end and acted on by an external distribution of compressive axial forces, exhibits a Hopf bifurcation that leads to non-planar spinning of the buckled filament at a locked curvature. We also show the existence of a second bifurcation, at larger force values, that induces a transition from non-planar spinning to planar wave-like oscillations. We elucidate the nature of these instabilities using a combination of nonlinear numerical analysis, linear stability theory, and low-order bead-spring models. Our results show that away from the transition thresholds, these beating patterns are robust to perturbations in the distribution of axial forces and in the filament configuration. These findings support the theory that an open-loop, instability-driven mechanism could explain both the sustained oscillations and the wide variety of periodic beating patterns observed in cilia and flagella.