Abstract
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 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 theoretically examined only in the regime of interflagellar distances d large compared to flagellar length L. Yet, in many biological situations d/L ≪ 1. Here, we first present an asymptotic analysis of the hydrodynamic coupling between two extended filaments in the regime 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.
Highlights
In most of the contexts in biology in which groups of cilia or flagella are found they exhibit some form of synchronized behavior
We present the derivation of the force on one filament only, say, filament 1, as the dynamics of the other can be deduced by symmetry
As the dynamics is translational invariant in y, there can be no terms explicitly dependent on h1 itself. Motivated by these results and with an eye toward the simplest partial differential equations (PDEs) that will encode a characteristic wavelength, amplitude, and frequency of flagellar beating, we propose a form in which the left-right symmetry is broken with the derivative of the lowest order possible
Summary
In most of the contexts in biology in which groups of cilia or flagella are found they exhibit some form of synchronized behavior. The recognition that hydrodynamic interactions alone are insufficient to generate dynamical evolution toward synchrony and that some form of generalized flexibility is necessary had already been seen in the study of rotating helices as a model for bacterial flagella [10] This notion of orbital compliance was subsequently incorporated into several variants of bead-spring models [11,12,13] of ciliary dynamics in which each beating filament is replaced by a moving microsphere that is driven around an orbit by internal forces and allowed to deviate by a radial spring.
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