Spontaneous oscillations, beating patterns, and hydrodynamics of active microfilaments

Cilia and flagella are ubiquitous in nature and are known to help in transport and swimming at the cellular scale by performing oscillations. Fundamental to these periodic waveforms is the core internal structure of the filaments known as the axoneme, consisting of an array of microtubule doublets, protein linkers, and dynein motors. In the presence of ATP, the collective action of the molecular motors drives internal sliding motions that are converted to spontaneous oscillations by a mechanism that still remains elusive. A sliding controlled axonemal feedback mechanism has recently been proposed and explored in the limit of small deformations, where it was shown to result in nonlinear amplitude selection through a mechanical regulation of dynein kinetics. Here, we build on that model to derive a more complete set of planar nonlinear governing equations that retains all the geometric nonlinearities, incorporates intrinsic biochemical noise and accounts for long-range, nonlocal hydrodynamic interactions. For a clamped filament, motor activity drives a Hopf bifurcation leading to traveling wave solutions that propagate from tip to base, in agreement with previous weakly nonlinear studies. Quite remarkably, our results demonstrate the existence of a second transition far from equilibrium, where nonlinearities cause a reversal in the direction of wave propagation and produce a variety of waveforms that resemble the beating patterns of swimming spermatozoa. We further extend the model to account for asymmetric ciliary beats and also allow for generalized dynein regulation mechanisms that can qualitatively reproduce Chlamydomonas reinhardtii flagellar dynamics. In the spirit of dimensional reduction, limit cycle representations are obtained for various waveforms and highlight the role of biochemical noise. We also analyze the velocity fields generated by the filaments and apply principal component analysis to derive low-order flow representations in terms of fundamental Stokes singularities that could be of use for constructing minimal models of swimming microorganisms.