Multisynchrony in Active Microfilaments

Biological microfilaments exhibit a variety of synchronization modes. Recent experiments observed that a pair of isolated eukaryotic flagella, coupled solely via the fluid medium, display synchrony at nontrivial phase lags in addition to in-phase and antiphase synchrony. Using an elastohydrodynamic filament model in conjunction with numerical simulations and a Floquet-type theoretical analysis, we demonstrate that it is possible to reach multiple synchronization states by varying the intrinsic activity of the filament and the strength of hydrodynamic coupling between the two filaments. Then, we derive an evolution equation for the phase difference between the two filaments at weak coupling, and use a Kuramoto-style phase sensitivity analysis to reveal the nature of the bifurcations underlying the transitions between these different synchronized states.

Figure 1

Two active microfilaments synchronize their beating: (a) in phase for $F=46$, $\gamma =0.1$, $\mathrm{\Delta }{\theta }_0/2\pi =0.2$; (b) antiphase for $F=46$, $\gamma =0.1$, $\mathrm{\Delta }{\theta }_0/2\pi =0.3$; (c) at different amplitudes for $F=49$, $\gamma =0.1$, $\mathrm{\Delta }{\theta }_0/2\pi =0.3$; (d) at a nontrivial phase lag ($\mathrm{\Delta }\theta \approx 0.56\pi$) for $F=48$, $\gamma =0.01$, $\mathrm{\Delta }{\theta }_0/2\pi =0.3$. In each panel, left plots depict snapshots of the steady-state waveforms, with increasing time highlighted in darker color, and right plots show the tip deflection over one period of oscillation as a function of phase $\theta /2\pi$. These four cases, labeled I–IV, are highlighted in Figs. 2 and 3.

Figure 2

Basins of attraction of the synchronization modes: antiphase, in phase, and synchrony at nontrivial phase lag, as a function of initial phase difference $\mathrm{\Delta }{\theta }_0$ and the active force value $F$. The phase difference between two filaments at steady state, $\mathrm{\Delta }\theta$, is indicated by the color of the dots. We vary the interfilament spacing such that (a) $\gamma =0.01$, (b) $\gamma =0.1$, and (c) $\gamma =0.33$, representing weak, intermediate, and strong hydrodynamic couplings.

Figure 3

(a) Number of periods it takes to reach in-phase (blue) or antiphase (yellow) synchrony versus $1/\ln (|\rho {|}^{-1})$; symbols represent distinct values of $\gamma$ and $F$. Floquet multipliers versus $\gamma$ for $F=42$, 46, 48, and 49, corresponding to (b) in-phase and (c) antiphase synchrony.

Figure 4

Phase function $\mathrm{\Psi }$ versus phase difference $\mathrm{\Delta }\theta$ at (a) $F=44$, (b) 45, (c) 47, and (d) 48. The solid and hollow dots correspond to stable and unstable fixed points, respectively. The color scheme for $\mathrm{\Delta }\theta$ uses the color map in Fig. 2.

Figure 5

(a) Experimental data from [1] represented in terms of steady-state phase difference $\mathrm{\Delta }\theta /2\pi$ and coupling strength $\kappa$. (b) Normal (solid line) and tangential (dashed line) hydrodynamic forces based on the filament model with $F=50$ (orange) compared to flagellar forces (blue) [1]. Insets show snapshots of the filament waveform (orange) and flagellar waveform (blue) [1]. (c) Model prediction of synchronization modes for $F=50$ as a function of coupling strength $\kappa$. The grey region shows the basin of attraction of in-phase synchrony. (d) Fraction of in-phase basin of attraction to all initial phase differences; model results from (c) are shown in solid black line, experimental data from (a) are superimposed as red stars.