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

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 \ll 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

.