Optimizing the Energetic Efficiency of Ciliated Microorganisms

Abstract

Energetic efficiency of swimming has long been considered a non-issue in microorganisms, but newer studies show that ciliates can use more than half of their energy for propulsion. To estimate how close the ciliates are to the theoretically optimal way of swimming we address the following problems: i) we determine the optimal stroke of a cilium, ii) we determine the optimal beating pattern of a ciliated surface and iii) we calculate the optimal shape of a ciliated swimmer.For a single cilium we define the efficiency in a scale-invariant way and show that the optimal stroke consists of a working stroke with a stretched cilium and a recovery stroke where the cilium bends and moves closer to the surface. When optimizing an array of cilia we additionally show that metachronal waves improve the efficiency and that the optimal efficiency is achieved for antiplectic waves. The resulting beating patterns, as well as the optimal ciliary density, show remarkable similarity with those observed in ciliated microorganisms. In order to optimize the shape of the whole swimmer we use a simplified description where we replace the ciliated layer with a surface slip velocity. Depending on the curvature we allow, the optimal swimmer has an elongated shape, possibly with two protrusions along the symmetry axis. These optimal shapes again resemble those of different ciliates.If we combine the results of our optimization with experimental efficiency estimates we can show that Paramecium has a propulsion efficiency that is within a factor of 2 of the theoretical optimum. This result suggests that ciliates have evolved for efficient swimming at different scales.[1] N. Osterman and A. Vilfan, Proc. Natl. Acad. Sci. USA 108, 15727-15732 (2011)[2] A. Vilfan, Phys. Rev. Lett. 109, 128105 (2012)