Elastohydrodynamic mechanisms govern beat pattern transitions in eukaryotic flagella

Abstract 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. Significance Statement The ability of flagella and cilia to manipulate their beating waveforms in different physical environments has important implications for human and animal health. Beat transitions in mammalian sperm flagella, for example, may help sperm navigate the complex reproductive tract to reach the egg. Abnormal beating behaviour, which may have a range of unknown underlying causes, can result in ciliopathies and reproductive disorders. This work provides insight into the physical origins of beat transitions in flagella and shows how they depend quantitatively on flagellum stiffness and length, internal driving provided by the axoneme, viscosity of the ambient medium, and the presence of a plane wall. This could allow one to diagnose the root cause of an abnormal flagellar beat and thus design appropriate treatment.