Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-27T04:28:42.165Z Has data issue: false hasContentIssue false

Shape of optimal active flagella

Published online by Cambridge University Press:  01 August 2013

Eric Lauga
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
Christophe Eloy*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France
*
Email address for correspondence: Christophe.Eloy@irphe.univ-mrs.fr

Abstract

Many eukaryotic cells use the active waving motion of flexible flagella to self-propel in viscous fluids. However, the criterion governing the selection of particular flagellar waveforms among all possible shapes has proved elusive so far. To address this question, we derive computationally the optimal shape of an internally forced periodic planar flagellum deforming as a travelling wave. The optimum is here defined as the shape leading to a given swimming speed with minimum energetic cost. To calculate the energetic cost, we consider the irreversible internal power expended by the molecular motors forcing the flagellum, only a portion of which is dissipated in the fluid. This optimization approach allows us to derive a family of shapes depending on a single dimensionless number quantifying the relative importance of elastic to viscous effects: the Sperm number. The computed optimal shapes are found to agree with the waveforms observed on spermatozoon of marine organisms, thus suggesting that these eukaryotic flagella might have evolved to be mechanically optimal.

Type
Rapids
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Current address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK. Email address for correspondence: e.lauga@damtp.cam.ac.uk

References

Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K. & Walter, P. 2007 Molecular Biology of the Cell, 5th edn. Garland Science.CrossRefGoogle Scholar
Alexander, R. McN. 1992 A model of bipedal locomotion on compliant legs. Phil. Trans. R. Soc. Lond. B 338, 189198.Google Scholar
Alouges, F., DeSimone, A. & Lefebvre, A. 2008 Optimal strokes for low Reynolds number swimmers: an example. J. Nonlin. Sci. 18 (3), 277302.Google Scholar
Audoly, B. & Pomeau, Y. 2010 Elasticity and Geometry. Oxford University Press.Google Scholar
Avron, J. E. & Raz, O. 2008 A geometric theory of swimming: Purcell’s swimmer and its symmetrized cousin. New J. Phys. 10, 063016.Google Scholar
Bray, D. 2000 Cell Movements. Garland Publishing.Google Scholar
Brennen, C. & Winnet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339398.Google Scholar
Brokaw, C. J. 1965 Non-sinusoidal bending waves of sperm flagella. J. Expl Biol. 43, 155.Google Scholar
Brokaw, C. J. & Wright, L. 1963 Bending waves of the posterior flagellum of Cerratum. Science 142, 11691170.Google Scholar
Camalet, S., Julicher, F. & Prost, J. 1999 Self-organized beating and swimming of internally driven filaments. Phys. Rev. Lett. 82, 15901593.Google Scholar
Childress, S. 1981 Mechanics of Swimming and Flying. Cambridge University Press.Google Scholar
Childress, S. 2012 A thermodynamic efficiency for stokesian swimming. J. Fluid Mech. 705, 7797.Google Scholar
Cox, R. G. 1970 The motion of long slender bodies in a viscous fluid. Part 1. General theory. J. Fluid Mech. 44, 791810.CrossRefGoogle Scholar
Eloy, C. & Lauga, E. 2012 Kinematics of the most efficient cilium. Phys. Rev. Lett. 109, 038101.Google Scholar
Fujita, T. & Kawai, T. 2001 Optimum shape of a flagellated microorganism. JSME Intl J. C 44, 952957.Google Scholar
Gittes, F., Mickey, B., Nettleton, J. & Howard, J. 1993 Flexural rigidity of microtubules and actin-filaments measured from thermal fluctuations in shape. J. Cell Biol. 120, 923934.Google Scholar
Gray, J. & Hancock, G. J. 1955 The propulsion of sea-urchin spermatozoa. J. Expl Biol. 32, 802814.CrossRefGoogle Scholar
Higdon, J. J. L. 1979a Hydrodynamic analysis of flagellar propulsion. J. Fluid Mech. 90, 685711.CrossRefGoogle Scholar
Higdon, J. J. L. 1979b Hydrodynamics of flagellar propulsion – Helical waves. J. Fluid Mech. 94, 331351.Google Scholar
Hines, M. & Blum, J. J. 1983 Three-dimensional mechanics of eukaryotic flagella. Biophys. J. 41, 6779.Google Scholar
Lauga, E. 2011 Life around the scallop theorem. Soft Matt. 7, 30603065.Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.Google Scholar
Leshansky, A. M., Kenneth, O., Gat, O. & Avron, J. E. 2007 A frictionless microswimmer. New J. Phys. 9, 145.CrossRefGoogle Scholar
Lighthill, J. 1975 Mathematical Biofluiddynamics. SIAM.CrossRefGoogle Scholar
Michelin, S. & Lauga, E. 2010 Efficiency optimization and symmetry-breaking in an envelope model for ciliary locomotion. Phys. Fluids 22, 111901.Google Scholar
Michelin, S. & Lauga, E. 2011 Optimal feeding is optimal swimming for all Péclet numbers. Phys. Fluids 23, 101901.Google Scholar
Michelin, S. & Lauga, E. 2013 Unsteady feeding and optimal strokes of model ciliates. J. Fluid Mech. 715, 131.Google Scholar
Osterman, N. & Vilfan, A. 2011 Finding the ciliary beating pattern with optimal efficiency. Proc. Natl Acad. Sci. USA 108, 1572715732.Google Scholar
Pironneau, O. & Katz, D. F. 1974 Optimal swimming of flagellated micro-organisms. J. Fluid Mech. 66, 391415.CrossRefGoogle Scholar
Pironneau, O. & Katz, D. F. 1975 Optimal swimming of motion of flagella. In Swimming and Flying in Nature (ed. Wu, T. Y., Brokaw, C. J. & Brennen, C.), vol. 1, pp. 161172. Plenum.Google Scholar
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45, 311.Google Scholar
Shapere, A. & Wilczek, F. 1987 Self-propulsion at low Reynolds number. Phys. Rev. Lett. 58, 20512054.Google Scholar
Shapere, A. & Wilczek, F. 1989a Geometry of self-propulsion at low Reynolds number. J. Fluid Mech. 198, 557585.Google Scholar
Shapere, A. & Wilczek, F. 1989b Efficiencies of self-propulsion at low Reynolds number. J. Fluid Mech. 198, 587599.CrossRefGoogle Scholar
Spagnolie, S. E. & Lauga, E. 2010 The optimal elastic flagellum. Phys. Fluids 22, 031901.CrossRefGoogle Scholar
Spagnolie, S. E. & Lauga, E. 2011 Comparative hydrodynamics of bacterial polymorphism. Phys. Rev. Lett. 106, 058103.Google Scholar
Tam, D. & Hosoi, A. E. 2007 Optimal stroke patterns for purcell’s three-link swimmer. Phys. Rev. Lett. 98, 068105.CrossRefGoogle ScholarPubMed