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Finding the Ciliary Beating Pattern with Optimal Efficiency

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Finding the Ciliary Beating Pattern with Optimal Efficiency Finding the ciliary beating pattern with optimal efficiency Natan Ostermana,b and Andrej Vilfana,c,1 aJ. Stefan Institute, 1000 Ljubljana, Slovenia; bDepartment of Physics, Ludwig-Maximilians University Munich, 80799 Munich, Germany; and cFaculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Edited by T. C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved July 20, 2011 (received for review May 17, 2011) We introduce a measure for energetic efficiency of biological cilia of the transported fluid, divided by the dissipated power. How- acting individually or collectively and numerically determine the ever, as the flow rate scales linearly with the velocity, but the optimal beating patterns according to this criterion. Maximizing dissipation quadratically, this criterion would yield the highest the efficiency of a single cilium leads to curly, often symmetric, and efficiency for infinitesimally slow cilia, just like optimizing the somewhat counterintuitive patterns. However, when looking at a fuel consumption of a road vehicle alone might lead to fitting it densely ciliated surface, the optimal patterns become remarkably with an infinitesimally weak engine. Instead, like engineers trying similar to what is observed in microorganisms like Paramecium. to optimize the fuel consumption at a given speed, the well- The optimal beating pattern then consists of a fast effective stroke posed question is which beating pattern of a cilium will achieve and a slow sweeping recovery stroke. Metachronal coordination a certain flow rate with the smallest possible dissipation. is essential for efficient pumping and the highest efficiency is The problem of finding the optimal strokes of hypothetical achieved with antiplectic waves. Efficiency also increases with an microswimmers has drawn a lot of attention in recent years. increasing density of cilia up to the point where crowding becomes Problems that have been solved include the optimal stroke pat- a problem. We finally relate the pumping efficiency of cilia to the tern of Purcell’s three-link swimmer (11), an ideal elastic flagel- swimming efficiency of a spherical microorganism and show that lum (12), a shape-changing body (13), a two- and a three-sphere the experimentally estimated efficiency of Paramecium is surpris- swimmer (14), and a spherical squirmer (5). Most recently, Tam ingly close to the theoretically possible optimum. and Hosoi optimized the stroke patterns of Chlamydomonas fla- gella (15). However, all these studies are still far from the com- low Reynolds number | hydrodynamics | microswimmers | metachronal plexity of a ciliary beat with an arbitrary 3D shape, let alone from waves | stroke kinematics an infinite field of interacting cilia. In addition, they were all performed for the swimming efficiency of the whole microor- fi any biological systems have evolved to work with a very ganism, whereas our goal is to optimize the pumping ef ciency Mhigh energetic efficiency. For example, muscle can convert at the level of a single cilium, which can be applicable to a much the free energy of ATP hydrolysis to mechanical work with >50% greater variety of ciliary systems. fi z efficiency (1), the F1-F0 ATP synthase converts electrochemical So we propose a cilium embedded in an in nite plane (at = 0) and pumping fluid in the direction of the positive x axis. We energy of protons to chemical energy stored in ATP molecules fi fl Q fl with even higher efficiency (2), etc. At first glance, the beating of de ne the volume ow rate as the average ux through a half- fl plane perpendicular to the direction of pumping (16). With P we cilia and agella does not fall into the category of processes with fl such a high efficiency. Cilia are hair-like protrusions that beat denote the average power with which the cilium acts on the uid, fl which is identical to the total dissipated power in the fluid-filled in an asymmetric fashion to pump the uid in the direction of fi fi their effective stroke (3). They propel certain protozoa, such as half-space. We then de ne the ef ciency in a way that is in- dependent of the beating frequency ω as Paramecium, and also fulfill a number of functions in mammals, including mucous clearance from airways, left-right asymmetry Q2 determination, and transport of an egg cell in fallopian tubes. e ¼ : [1] P Lighthill (4) defines the efficiency of a swimming microorganism as the power that would be needed to drag an object of the same As we show in SI Text, section 1, minimizing the dissipated power fl size with the same speed through viscous uid, divided by the P for a constant volume flow rate Q is equivalent to maximizing e fi fi actually dissipated power. Although the ef ciency de ned in this at a constant frequency. A similar argument for swimming effi- way could theoretically even exceed 100% (5), the actual swim- ciency has already been brought forward by Avron et al. (13). ming efficiencies are of the order of 1% (6, 7). In his legendary Furthermore, a general consequence of low Reynolds number paper on life at low Reynolds number (8) Purcell stated that hydrodynamics is that the volume flow depends only on the swimming microorganisms have a poor efficiency, but that the en- shape of the stroke and on the frequency, but not on the actual ergy expenditure for swimming is so small that it is of no relevance time dependence of the motion within a cycle. This finding is the for them (he uses the analogy of “driving a Datsun [a fuel-efficient basis of Purcell’s scallop theorem (8). As a consequence, the car of the period] in Saudi Arabia”) (ref. 8, p. 9). Nevertheless, later optimum stroke always has a dissipation rate constant in time. studies show that swimming efficiency is important in micro- We show this in SI Text, section 2. organisms. In Paramecium, more than half of the total energy We can make the efficiency e completely dimensionless if we L consumption is needed for ciliary propulsion (9). factor out the effects of the ciliary length , the beating fre- BIOPHYSICS AND When applied to ciliary propulsion, Lighthill’sefficiency (4) COMPUTATIONAL BIOLOGY has some drawbacks. For one, it is not a direct criterion for the hydrodynamic efficiency of cilia as it also depends on the size and Author contributions: A.V. designed research; N.O. and A.V. performed research; A.V. shape of the whole swimmer. Besides that it is, naturally, appli- analyzed data; and N.O. and A.V. wrote the paper. cable only for swimmers and not for other systems involving The authors declare no conflict of interest. ciliary fluid transport with a variety of functions, like left-right This article is a PNAS Direct Submission. asymmetry determination (10). We therefore propose a different 1To whom correspondence should be addressed. E-mail: [email protected]. PHYSICS fi criterion for ef ciency at the level of a single cilium or a carpet of This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. cilia. A first thought might be to define it as the volume flow rate 1073/pnas.1107889108/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1107889108 PNAS | September 20, 2011 | vol. 108 | no. 38 | 15727–15732 Downloaded by guest on September 25, 2021 quency ω, and the fluid viscosity η. The velocity with which invariant efficiency criterion proposed here. On the other hand, a point on the cilium moves scales with ωL and the linear force Lighthill’s criterion for swimming organisms shares the same density (force per unit length) with ηωL. The total dissipated scaling properties as ours (it scales with the square of the power P, obtained by integration of the product of the velocity swimming velocity, divided by dissipation), but differs in defini- and linear force density over the length, then scales with ηω2L3. tion because it measures the swimming and not the pumping The volume flow rate Q scales with ωL3. Finally, the efficiency e efficiency. At the end we show how the two measures are related scales with L3/η. The dimensionless efficiency can therefore be to each other for a spherical swimmer. defined as Our goal is to find the beating patterns that have the highest possible efficiency for a single cilium, as well as the beating e′ ¼ L − 3ηe: [2] patterns combined with the density and the wave vector that give the highest efficiency of a ciliated surface. When optimizing the efficiency of ciliary carpets, we have to use the measures of volume flow and dissipation per unit area, rather Model than per cilium. We introduce the surface density of cilia ρ, We describe the cilium as a chain of N touching beads with radii 2 which is 1/d on a square lattice. In the following we show that a. The first bead of a cilium has the center position x = (0, 0, a), fl ρQ 1 the volume ow generated per unit area, , is also equivalent to and each next bead in the chain is located at xi = xi +2a(sinθi fl fl +1 the ow velocity above the ciliary layer. The uid velocity above cosφi, sinθi sinφi, cosθi). The maximum curvature of the cilium is an infinite ciliated surface namely becomes homogeneous at limited by the condition a distance sufficiently larger than the ciliary length and meta- chronal wavelength. The far field of the flow induced by a single ðx − x Þðx − x Þ $ ð aÞ2 β : [7] iþ1 i i i − 1 2 cos max cilium located at the origin and pumping fluid in the direction of the x axis has the form (17) Naturally, beads cannot overlap with the surface (zi > a) or with each other, jxi −xjj > 2a.
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