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.