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Comparative Study
. 1997 Jun 10;94(12):6001-6.
doi: 10.1073/pnas.94.12.6001.

Cilia Internal Mechanism and Metachronal Coordination as the Result of Hydrodynamical Coupling

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Free PMC article
Comparative Study

Cilia Internal Mechanism and Metachronal Coordination as the Result of Hydrodynamical Coupling

S Gueron et al. Proc Natl Acad Sci U S A. .
Free PMC article

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 flagella 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.

Figures

Figure 1
Figure 1
Beat cycles of cilia. All positions are equally separated in time by 3 ms. The effective stroke positions are plotted by dashed lines, and the recovery stroke positions by solid lines. The units of the axes are nondimensional length. (a) A single cilium. The viscosity of the surrounding fluid is that of water, μ = μwater. The resulting beat frequency is ≈29 Hz. (b) μ = 2μwater. The resulting beat frequency is ≈17 Hz and the beat pattern is changed. (c) μ = 3μwater. The resulting beat frequency is ≈12 Hz and the beat pattern is changed. (d) Side view of an infinite line of synchronized cilia, spaced by 0.3 ciliary length, beating in water. The resulting beat frequency is ≈29 Hz.
Figure 2
Figure 2
Dependence of the beat frequency on the viscosity, for a single cilium, in the range μwater ≤ μ ≤ 5μwater. The vertical error bars represent model results with the estimated measurement errors, and the dashed line is a minimal least-squares fit. The linear decrease in beat frequency with exponentially increasing viscosity for the cilia of Paramecium, shown in the figure, is reproduced from ref. . Note that the abscissa is on a logarithmic scale.
Figure 3
Figure 3
Autonomous synchronization between two cilia starting at opposite phases (the left cilium starts at the recovery stroke and the right cilium starts at the effective stroke). Synchronization is achieved after two cycles. Cilia spacing is 1. The snapshots, labeled successively from a to h, are separated in time by 9 ms. The resulting beat frequency is ≈31 Hz. The units of the axes are nondimensional length.
Figure 4
Figure 4
(a–d) Self-organized metachronism in a row of 10 cilia. (e–f) Metachronism in the 100-cilia configuration. Metachronal wave pattern is generated by hydrodynamical coupling. In both configurations the cilia spacing is 0.3, the snapshots are separated in time by 2 ms, the resulting beat frequency is ≈42 Hz, and the units of the axes are nondimensional length.

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