# Elastohydrodynamic synchronization of adjacent beating flagella

Goldstein, Raymond Ethan ; Lauga, Eric Jean-Marie ; Pesci, Adriana Irma ; Proctor, Michael Richard (2016-11-01)

Article

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 the beating flagella are represented by microspheres tethered by radial springs as they are driven about orbits by internal forces. While these low-dimensional models reproduce the phenomenon of synchrony, their parameters are not readily relatable to those of the filaments they represent. More realistic models, which reflect the underlying elasticity of the axonemes and the active force generation, take the form of fourth-order nonlinear partial differential equations (PDEs). While computational studies have shown the occurrence of synchrony, the effects of hydrodynamic coupling between nearby filaments governed by such continuum models have been examined theoretically only in the regime of interflagellar distances d large compared to flagellar length $\textit{L}$. Yet in many biological situations $\textit{d/L}$≪1. Here we present an asymptotic analysis of the hydrodynamic coupling between two extended filaments in the regime $\textit{d/L}$≪1 and find that the form of the coupling is independent of the microscopic details of the internal forces that govern the motion of the individual filaments. The analysis is analogous to that yielding the localized induction approximation for vortex filament motion, extended to the case of mutual induction. In order to understand how the elastohydrodynamic coupling mechanism leads to synchrony of extended objects, we introduce a heuristic model of flagellar beating. The model takes the form of 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 a pair of filaments is achieved through the asymptotic coupling.