The reaction-diffusion basis of animated patterns in eukaryotic flagella

Abstract

We show that the flagellar beat of bull spermatozoa and Chlamydomonas Reinhardtii can be modelled by a minimal, geometrically nonlinear, sliding-controlled, reaction-diffusion system. Model solutions are spatio-temporally animated patterns describing flagellar bending waves, further connecting beating patterns of cilia and flagella with, seemly unrelated, chemical patterns from classical reaction-diffusion systems. Instead of chemical species freely reacting and diffusing in space, our system describes the tug-of-war reaction-kinetics of molecular motors that are anchored in the flagellar structure, but the shear deformation that they generate can diffuse away via the bending elasticity of the flagellum. Synchronization of the reaction-kinetics in neighbouring elements occurs via a sliding-control mechanism. We derive from first principles the reaction-diffusion basis of animated patterns, and show that this is a direct consequence of the high internal energy dissipation by the flagellum relative to the external dissipation by the fluid environment. By fitting, for the first time, nonlinear, large-amplitude solutions of a specific motor cross-bridge reaction-kinetics, we show that reaction-diffusion successfully accounts for beating patterns of both bull sperm and Chlamydomonas (wild-type and mbo2-mutant), unifying these distant eukaryotic species under the same minimal model. Our results suggest that the flagellar beat occurs far from equilibrium, in the strongly nonlinear regime, and that in contrary to the conclusions of small amplitude studies, a unified mechanism may exist for dynein molecular motor control that is regulated by axonemal sliding, without requiring curvature-sensing or the fine-tuning of basal compliance, and only weakly influenced by hydrodynamic dissipation and the cell body boundary condition. High internal dissipation allows the emergence of base-to-tip autonomous travelling waves, independently of, and without relying on, the external fluid viscosity, when small. This enables progressive swimming, otherwise not possible, in low viscosity environments, and may be critical for external fertilizers and aquatic microorganisms. The reaction-diffusion model may prove a powerful tool for studying the pattern formation of movement in flagella, cilia, and more generally, oscillations of animated filament-bundles at the microscale.