Cilia internal mechanism and metachronal coordination as the result of hydrodynamical coupling

Abstract

We present a simple but realistic model for the internal bend-generating mechanism of cilia, using parameters obtained from the analysis of data of the beat of a single cilium, and incorporate it into a recently developed dynamical model. Comparing the results to experimental data for two-dimensional beats, we demonstrate that the model captures the essential features of the motion, including many properties that are not built in explicitly. The beat pattern and frequency change in response to increased viscosity and the presence of neighboring cilia in a realistic fashion. Using the model, we are able to investigate multicilia configurations such as rows of cilia and two-dimensional arrays of cilia. When two adjacent model cilia start beating at different phase, they synchronize within two cycles, as observed in experiments in which two f lagella beating out of phase are brought close together. Examination of various multicilia configurations shows that metachronal patterns (i.e., beats with a constant phase difference between neighboring cilia) evolve autonomously. This provides modeling evidence in support of the conjecture that metachronism may occur as a self-organized phenomenon due to hydrodynamical interactions between the cilia.Ciliary motion and particularly the metachronism phenomenon have attracted a great deal of research effort both experimentally and theoretically. Metachronal coordination between cilia is a situation where cilia beat together with a constant phase difference between adjacent neighbors, their tips forming a moving wave pattern. The reason why arrays of cilia beat in a metachronal pattern is not fully understood. The work of Machemer (1), for example, shows that membrane voltage and calcium levels affect the direction of the metachronal wave and also the directions of the effective and recovery strokes of the ciliary beats. Some researchers speculate that the intriguing metachronism phenomenon is possibly the result of hydrodynamical coupling (e.g., see refs. 2-4). This work provides a theoretical model that supports this conjecture.Gueron and Liron (ref. 5; GL hereafter) introduced an improved technique for describing the hydrodynamics of moving cilia, based on a refined slender body theory, which offers an alternative for the simplistic resistive force theory (known as the Gray and Hancock approximation) that relates drag forces and velocities. As a result, the accuracy and consistency of the model for cilia beating is markedly improved. More important, the GL equations provide a method for dynamical simulations of multicilia configurations that account for the effects of neighboring cilia and the effect of the surface from which the cilia emerge. This model was originally applied to a two-dimensional setup and later extended to describe threedimensional beating (6).The work of GL was mainly oriented toward developing the framework for dynamical modeling of multicilia configurations. For the internal mechanism of a cilium (hereafter referred to as the ''engine'') they use...

A continuum model for flow induced by metachronal coordination between beating cilia

“…Such an approach was used by Gueron & Liron (1993), who developed a slender-body approach to simulating threedimensional ciliary beats that allows for cilia interactions. Cilia-interaction models have been extensively used to show that the onset and self-sustaining character of metachronal coordination is the result of hydrodynamic interaction between beating cilia (Gueron et al 1997;Gueron & Levit-Gurevich 1998;Lenz & Ryskin 2006;Guirao & Joanny 2007). Discrete cilia models that simulate the flow field around the cilia typically make use of the immersed boundary method (IBM : Peskin 1972), in which the effect of cilia on the flow is included as an additional forcing term in the momentum equations.…”

Emergence of Synchronized Beating during the Regrowth of Eukaryotic Flagella

“…It is now known that oscillators with more than one degree of freedom or those with suitable internal forcing can be synchronized by hydrodynamics [6,7]. Various levels of coordination are also found with more realistic models of flagella [8], which rely necessarily on simplified internal driving of the filaments. Nevertheless, these complex models yield important clues to the generation and regulation of flagellar motion [9].…”

Hydrodynamics Versus Intracellular Coupling in the Synchronization of Eukaryotic Flagella

“…Phase transitions leading to synchronization are observed between and within a variety of biological organisms [2]. Recently, the organized dynamics of micron sized hairlike cell projections called eukaryotic flagella or cilia has attracted high levels of interest [3][4][5]. The ability of flagella to manipulate and transport fluid relies on their capacity to spontaneously beat and synchronize with one another.…”