Instability-driven Oscillations of Elastic Microfilaments*

Cilia and flagella are slender organelles that exhibit a variety of rhythmic beating patterns from conic-like motions to planar wave 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 forces. Recently, an open-loop mechanism was proposed based on a `flutter’-like instability. Here, we show that an elastic filament in viscous fluid, pinned at one end, and acted on by a distribution of axial forces exhibits a Hopf bifurcation, but this bifurcation generally leads to non-planar spinning of the filament, at a buckled configuration with locked curvature. We also show the existence of a second bifurcation, at larger forces, that causes a transition from spinning to planar wave-like deformations. To analyze these instabilities, we use a combination of numerical analysis, linear stability theory, and multi-link models. Our results support the theory that an instability-driven mechanism could explain the wide variety of beating patterns observed in cilia and flagella.