Cass, J. F. | bioRxiv Science

bioRxiv2024

Predicting microscale beat patterns from nanoscale chemomechanics in eukaryotic flagella

We present quantitative predictions for experimental observables--amplitude, frequency and wavelength--of the eukaryotic flagellar beat in terms of underlying molecular chemomechanical parameters. Flagellar beating, an incompletely understood self-organized process arising from the collective action of dynein molecular motors, is modelled as a reaction-diffusion (RD) system with an oscillatory instability arising from motor-induced microtubule sliding. While the RD model accurately reproduces beating patterns of bull spermatozoa and C. Reinhardtii, existing linear analyses and simulations are unable to provide a complete framework for understanding nonlinear waveform formation. Here, we derive analytical expressions that reveal the nonlinear dependence of beat characteristics on parameters such as motor binding duty ratio, stepping velocity, and axonemal resistance. Our analysis uncovers a novel out-of-equilibrium mechanism for base-to-tip wave propagation, involving an interference pattern between unstable standing wave modes that generates travelling waves. Predicted beat patterns agree remarkably with numerical simulations, even far from the critical point marking the onset of oscillations. This unveils key molecular parameters that govern oscillation initiation, amplitude saturation, frequency shifts, and the spatial phase gradient crucial for generating propulsive hydrodynamic force. Our results yield biophysical understanding of how molecular interactions shape flagellar beating patterns, allowing for the inference of molecular properties from macroscopic observations. This challenges existing hypotheses on wave generation and demonstrates the power of nonlinear analysis to uncover new phenomena beyond the reach of linear models and computational studies alone.

biophysics↗

bioRxiv2023

The reaction-diffusion basis of animated patterns in eukaryotic flagella

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.

biophysics↗