No abstract

A set of nonlinear differential equations describing flagellar motion in an external viscous medium is derived. Because of the local nature of these equations and the use of a Crank-Nicolson-type forward time step, which is stable for large deltat, numerical solution of these equations on a digital computer is relatively fast. Stable bend initiation and propagation, without internal viscous resistance, is demonstrated for a flagellum containing a linear elastic bending resistance and an elastic shear resistance that depends on sliding. The elastic shear resistance is derived from a plausible structural model of the radial link system. The active shear force for the dynein system is specified by a history-dependent functional of curvature characterized by the parameters m0, a proportionality constant between the maximum active shear moment and curvature, and tau, a relaxation time which essentially determines the delay between curvature and active moment.

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...