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Cilia and flagella are highly conserved slender organelles that exhibit avariety of rhythmic beating patterns from non-planar cone-like motions toplanar wave-like deformations. Although their internal structure, composed of amicrotubule-based axoneme driven by dynein motors, is known, the mechanismresponsible for these beating patterns remains elusive. Existing theoriessuggest that the dynein activity is dynamically regulated, via a geometricfeedback from the cilium's mechanical deformation to the dynein force. Analternative, open-loop mechanism based on a 'flutter' instability was recentlyproven to lead to planar oscillations of elastic filaments under followerforces. Here, we show that an elastic filament in viscous fluid, clamped at oneend and acted on by an external distribution of compressive axial forces,exhibits a Hopf bifurcation that leads to non-planar spinning of the buckledfilament at a locked curvature. We also show the existence of a secondbifurcation, at larger force values, that induces a transition from non-planarspinning to planar wave-like oscillations. We elucidate the nature of theseinstabilities using a combination of nonlinear numerical analysis, linearstability theory, and low-order bead-spring models. Our results show that awayfrom the transition thresholds, these beating patterns are robust toperturbations in the distribution of axial forces and in the filamentconfiguration. These findings support the theory that an open-loop,instability-driven mechanism could explain both the sustained oscillations andthe wide variety of periodic beating patterns observed in cilia and flagella.

Instability-driven Oscillations of Elastic Microfilaments