Energetic considerations of ciliary beating and the advantage of metachronal coordination

Abstract

The internal mechanism of cilia is among the most ancient biological motors on an evolutionary scale. It produces beat patterns that consist of two phases: during the effective stroke, the cilium moves approximately as a straight rod, and during the recovery stroke, it rolls close to the surface in a tangential motion. It is commonly agreed that these two phases are designed for efficient functioning: the effective stroke encounters strong viscous resistance and generates thrust, whereas the recovery stroke returns the cilium to starting position while avoiding viscous resistance. Metachronal coordination between cilia, which occurs when many of them beat close to each other, is believed to be an autonomous result of the hydrodynamical interactions in the system. Qualitatively, metachronism is perceived as a way for reducing the energy expenditure required for beating. This paper presents a quantitative study of the energy expenditure of beating cilia, and of the energetic significance of metachronism. We develop a method for computing the work done by model cilia that beat in a viscous fluid. We demonstrate that for a single cilium, beating in water, the mechanical work done during the effective stroke is approximately five times the amount of work done during the recovery stroke. Investigation of multicilia configurations shows that having neighboring cilia beat metachronally is energetically advantageous and perhaps even crucial for multiciliary functioning. Finally, the model is used to approximate the number of dynein arm attachments that are likely to occur during the effective and recovery strokes of a beat cycle, predicting that almost all of the available dynein arms should participate in generating the motion.

Figure 1

Ciliary beats in different cilia configurations. The top line displays three isolated cilia beating in different viscosities (μ_water, 2μ_water and 3μ_water, correspondingly, from left to right). The solid and dashed lines represent the recovery and effective stroke, respectively. The time interval between snapshots is 3 msec. The middle line displays one snapshot of the 2-, 5-, and 10-cilia configurations in the viscosity of water. The bottom line displays eight snapshots of the 100-cilia configuration. Note that coordination similar to metachronism occurs in the multicilia configurations. The axes are in units of ciliary length. These figures are reproduced from Gueron and Levit-Gurevich (3).

Figure 2

(a) The average per-cilium energy expenditure during the beat cycle, as a function of the number of cilia in multicilia configurations. The interciliary spacing is 0.3, 0.5, and 0.7 ciliary length. The horizontal axis is a logarithmic scale. The computations are done for the viscosity of water. (b) The average energy spent per cilium per cycle in multicilia configurations. Results for configurations of 1, 2, 3, 5, 10, and 100 cilia are shown. The interciliary spacing is 0.3 ciliary lengths. The computations are done for the viscosity of water. The horizontal axis displayed is a logarithmic scale. The diamond, plus, and square symbols represent the effective stroke, the recovery stroke, and the whole beat cycle, respectively. The average number of moles of ATP hydrolyzed, and of the number of dynein arm attachments are indicated by the same symbols, but the units of the ordinate would be 0–16⁻²⁰ moles and 0–10,000 attachments, respectively. (c) The relative efficiency of fluid propulsion, as a function of the number of the cilia in the row. Efficiency is defined here as the net fluid mass moved in the direction of the effective stroke through a test area near the ciliary array, per unit of spent energy (see explanation in text). The results are scaled with respect to those obtained for a single cilium. The horizontal axis displayed is a logarithmic scale.