How molecular motors’ interaction shapes flagellar beat and its fluctuations

The stochastic dynamics of flagellar beating for micro-swimmers, such as flagellated cells, sperms and microalgae, is widely thought to include a feedback mechanism between flagellar shape and the rate of activation/de-activation of the N − 1 driving molecular motors. In the context of the so-called rigid filament models, where the axoneme is described by a single degree of freedom X(t), we investigate the effect of direct coupling between the activity dynamics of adjacent motors, parametrized by K ≥ 0. A functional Fokker-Planck equation for X and the state of the N motors is obtained. In the limit of small coupling K ≪ 1, we derive a system of equations governing the dynamics of the Fourier modes of the active motor density, obtaining estimates for several observables and the fluctuations’ quality factor Q. For larger K we resort to numerical simulations. The effect of introducing the coupling K>0 is to increase characteristic times and the beating period. Moreover for large Ks the limit cycle becomes bi-stable, with abrupt avalanches of the motor dynamics. Increasing K is similar to what observed in the case K=0 when the confining elastic force is strongly reduced. The quality factor of fluctuations has a non-monotonic behavior: it first increases with K, then decreases. This is accompanied by the reduction and eventual disappearance of regions where the fraction of activated motor is nor 0 neither 1.