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 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) X ( t ) , we investigate the effect of direct coupling between the activity dynamics of adjacent motors, parametrized by K ≥ 0 K ≥ 0 . A functional Fokker-Planck equation for X X and the state of the N N motors is obtained. In the limit of small coupling K 1 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 Q . For larger K K we resort to numerical simulations. The effect of introducing the coupling K;gt0 K > 0 is to increase characteristic times and the beating period. Moreover for large K K s the limit cycle becomes bi-stable, with abrupt avalanches of the motor dynamics. Increasing K K is similar to what observed in the case K=0 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 K , then decreases. This is accompanied by the reduction and eventual disappearance of regions where the fraction of activated motor is nor 0 0 neither 1 1 .