Elastohydrodynamic mechanisms govern beat pattern transitions in eukaryotic flagella

Eukaryotic cilia and flagella exhibit complex beating patterns that change depending on environmental conditions such as fluid viscosity. The mechanism behind these beat pattern transitions remains unclear, although they are thought to arise from changes in the internal forcing provided by the axoneme. We show here that such transitions may arise universally across species via an elastohydrodynamic mechanism. We perform simulations of inextensible and unshearable but twistable Kirchhoff rods driven internally by a travelling bending-moment wave in a fixed plane in the material frame of the rods. We show that, for a large range of beating amplitudes and frequencies, the internally planar driving wave results in the growth of twist perturbations. Outside this domain, the driving leads to simple planar waveforms. Within the non-planar domain, we observe quasiplanar, helical, and complex – perhaps chaotic – beating patterns. The transitions between these states depend quantitatively on physical parameters such as the internal forcing, flagellum stiffness and length, viscosity of the ambient medium, or the presence of a plane wall. Beat pattern transitions in our simulations can be mapped to similar transitions observed in bull and sea urchin sperm when the medium viscosity is varied. Comparison of the simulation results with experimentally observed transitional viscosities in our experiments and elsewhere suggests an assay whereby one can estimate the average force exerted by dynein motors. This could potentially lead to diagnostic assays measuring the health of sperm based on their beating pattern.