How Molecular Motors' Interaction Shapes Flagellar Beat and Its Fluctuations

TL;DR

This work investigates the effect of direct coupling between the activity dynamics of adjacent motors, parametrized by K, and derives 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 $Q$.

Abstract

The stochastic dynamics of flagellar beating for micro-swimmers, such as flagellated cells, sperms and microalgae, is dominated by a feedback mechanism between flagellar shape and the rate of activation/de-activation of the $N \gg 1$ 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 $K \ge 0$. A functional Fokker-Planck equation for $X$ and the state of the $N$ motors is obtained. In the limit of small coupling $K \ll 1$, 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 at large $K$s 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 $1$.

Figures (13)

• Figure 1: Schematic representation of the coupled model: the motors interact with the filament only when they are attached to it and they pull it with a force derived by the potential $W$.
• Figure 2: Numerical results about the decay of the maximum of the Fourier mode from the solution of the system \ref{['eq:sistema']}. (a) maxima as a function of the mode index $n$ (the dashed line connecting square symbols corresponds to the coefficients $b_n$); (b) fitted values of the exponent $\alpha_a(n)$, governing the power-law scaling $\sim K^{\alpha_a(n)}$, shown as a function of $n$; (c-h) maxima of the first Fourier coefficients as functions of $K$; dashed lines indicate power-law fits $\sim K^{\alpha_a(n)}$ and $\sim K^{\alpha_b(n)}$.
• Figure 3: Critical $\nu$ as a function of $K$ from the stability analysis of the system reduced to only three variables, Eqs. \ref{['only3']}. When $\nu>\nu_c$ the fixed point is stable, while if $\nu<\nu_c$ a limit cycle appears.
• Figure 4: (a,b): Force-position limit cycle in numerical simulations with increasing $N$ at fixed $K=0.5$ (a) and effect of increasing coupling $K$ at fixed $N=5\cdot10^4$ (b). (c–e) PCA rotation of $(X,F)$ variables: effect of $N$ for uncoupled ($K=0$) and strongly coupled ($K=3$) motors (c,d), and effect of varying $K$ at fixed $N$ (e). (f): Limit-cycle period in function of $K$ for several values of $N$.
• Figure 5: (a,b) Autocorrelation analysis: extraction of diffusion coefficient $D$ and angular frequency $\omega_0$ from $\mathrm{Re}\{C(\tau)\}$, shown for $N=10^4$ at $K=0$ and $K=1$. (c,d) Quality factor $Q$ as a function of $N$ and $K$.