Elastohydrodynamic Synchronization of Adjacent Beating Flagella

It is now well established that nearby beating pairs of eukaryotic flagella or cilia typically synchronize in phase. A substantial body of evidence supports the hypothesis that hydrodynamic coupling between the active filaments, combined with waveform compliance, provides a robust mechanism for synchrony. This elastohydrodynamic mechanism has been incorporated into `bead-spring' models in which flagella are represented by microspheres tethered by radial springs as internal forces drive them about orbits. While these low-dimensional models reproduce the phenomenon of synchrony, their parameters are not readily relatable to those of flagella. More realistic models which reflect the elasticity of the axonemes and active force generation take the form of fourth-order nonlinear PDEs. While computational studies have shown synchrony, the effects of hydrodynamic coupling between nearby filaments governed by such models have been theoretically examined only in the regime of interflagellar distances $d$ large compared to flagellar length $L$. Yet, in many biological situations $d/L \ll 1$. Here, we first present an asymptotic analysis of the hydrodynamic coupling between two filaments in the regime $d/L \ll 1$, and find that the form of the coupling is independent of the details of the internal forces that govern the motion of the filaments. The analysis is like the localized induction approximation for vortex filament motion, extended to the case of mutual induction. To understand how the coupling mechanism leads to synchrony of extended objects, we introduce a heuristic model of flagellar beating, a single fourth-order nonlinear PDE whose form is derived from symmetry considerations, the physics of elasticity, and the overdamped nature of the dynamics. Analytical and numerical studies of this model illustrate how synchrony between two filaments is achieved through the asymptotic coupling.