Spontaneous oscillations in eukaryotic cilia and photo-responsive rods

We present a comparative analysis of the chemo-mechanical mechanisms that drive spontaneous oscillations in two distinct active filamentous structures: photo-chemically deformable liquid crystal elastomer (LCE) rods and ATP-powered eukaryotic cilia. Using a unified framework of active planar rods, we develop simplified mathematical models for both systems. We reduce the governing partial differential equations to one-degree-of-freedom (1-DOF) nonlinear oscillators, each undergoing a supercritical Hopf bifurcation. For these reduced models, we obtain explicit analytical expressions for the onset and characteristics of self-sustained oscillations. Despite the common mathematical structure, the underlying physical mechanisms are fundamentally different. For LCEs, self-oscillation is an inertial phenomenon driven by an elastic-inertial feedback over the timescale of the photochemical reaction. In contrast, cilia live in the inertia-less regime, and the instability is driven by a negative effective damping (motive force) that arises from the mechanochemistry of molecular motors. The analytical predictions for critical activation thresholds, frequencies, and amplitudes agree with full nonlinear simulations, providing quantitative insight into the dynamics of these complex self-oscillating systems.