Abstract
Swimming microorganisms such as bacteria or spermatozoa are typically found in dense suspensions, and exhibit collective modes of locomotion qualitatively different from that displayed by isolated cells. In the dilute limit where fluid-mediated interactions can be treated rigorously, the long-time hydrodynamics of a collection of cells result from interactions with many other cells, and as such typically eludes an analytical approach. Here, we consider the only case where such problem can be treated rigorously analytically, namely when the cells have spatially confined trajectories, such as the spermatozoa of some marine invertebrates. We consider two spherical cells swimming, when isolated, with arbitrary circular trajectories, and derive the long-time kinematics of their relative locomotion. We show that in the dilute limit where the cells are much further away than their size, and the size of their circular motion, a separation of time scale occurs between a fast (intrinsic) swimming time, and a slow time where hydrodynamic interactions lead to change in the relative position and orientation of the swimmers. We perform a multiple-scale analysis and derive the effective dynamical system—of dimension two—describing the long-time behavior of the pair of cells. We show that the system displays one type of equilibrium, and two types of rotational equilibrium, all of which are found to be unstable. A detailed mathematical analysis of the dynamical systems further allows us to show that only two cell-cell behaviors are possible in the limit of t→∞, either the cells are attracted to each other (possibly monotonically), or they are repelled (possibly monotonically as well), which we confirm with numerical computations. Our analysis shows therefore that, even in the dilute limit, hydrodynamic interactions lead to new modes of cell-cell locomotion.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alexander, G.P., Yeomans, J.M., 2008. Dumb-bell swimmers. Euro. Phys. Lett. 83, 34006.
Aranson, I.S., Sokolov, A., Kessler, J.O., Goldstein, R.E., 2007. Model for dynamical coherence in thin films of self-propelled microorganisms. Phys. Rev. E 75, 040901.
Batchelor, G.K., 1970. The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.
Bender, C.M., Orszag, S.A., 1978. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York.
Blum, J.J., Hines, M., 1979. Biophysics of flagellar motility. Q. Rev. Biophys. 12, 103–180.
Bray, D., 2000. Cell Movements. Garland, New York.
Brennen, C., Winet, H., 1977. Fluid mechanics of propulsion by cilia and flagella. Ann. Rev. Fluid Mech. 9, 339–398.
Brenner, H., 1964. The stokes resistance of an arbitrary particle, iv. Chem. Eng. Sci. 19, 703–727.
Childress, S., 1981. Mechanics of Swimming and Flying. Cambridge University Press, Cambridge.
Childress, S., Levandowsky, M., Spiegel, E.A., 1975. Pattern formation in a suspension of swimming microorganisms: equations and stability theory. J. Fluid Mech. 69, 591–613.
Cisneros, L.H., Cortez, R., Dombrowski, C., Goldstein, R.E., Kessler, J.O., 2007. Fluid dynamics of self-propelled micro-organisms, from individuals to concentrated populations. Exp. Fluids 43, 737–753.
Czirok, A., Stanley, H.E., Vicsek, T., 1997. Spontaneously ordered motion of self-propelled particles. J. Phys. A 30, 1375–1385.
Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R.E., Kessler, J.O., 2004. Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93, 098103.
Goldstein, S.F., 1977. Asymmetric waveforms in echinoderm sperm flagella. J. Exp. Biol. 71, 157–170.
Gregoire, G., Chate, H., 2004. Onset of collective and cohesive motion. Phys. Rev. Lett. 92, 025702.
Guell, D.C., Brenner, H., Frankel, R.B., Hartman, H., 1988. Hydrodynamic forces and band formation in swimming magnetotactic bacteria. J. Theor. Biol. 135, 525–542.
Gyrya, V., Aranson, I.S., Berlyand, L.V., Karpeev, D., 2009. A model of hydrodynamic interaction between swimming bacteria. To appear in Bull. Math. Biol. (preprint available on http://arxiv.org/abs/0805.3182v2).
Happel, J., Brenner, H., 1965. Low Reynolds Number Hydrodynamics. Prentice Hall, Englewood Cliffs.
Hayashi, F., 1998. Sperm co-operation in the Fishfly, Parachauliodes japonicus. Funct. Ecol. 12, 347–350.
Hernandez-Ortiz, J.P., Stoltz, C.G., Graham, M.D., 2005. Transport and collective dynamics in suspensions of confined swimming particles. Phys. Rev. Lett. 95, 204501.
Hill, N.A., Pedley, T.J., 2005. Bioconvection. Fluid Dyn. Res. 37, 1–20.
Ishikawa, T., Hota, M., 2006. Interaction of two swimming Paramecia. J. Exp. Biol. 209, 4452–4463.
Ishikawa, T., Pedley, T.J., 2007a. Diffusion of swimming model micro-organisms in a semi-dilute suspension. J. Fluid Mech. 588, 437–462.
Ishikawa, T., Pedley, T.J., 2007b. The rheology of a semi-dilute suspension of swimming model micro-organisms. J. Fluid Mech. 588, 399–435.
Ishikawa, T., Pedley, T.J., 2008. Coherent structures in monolayers of swimming particles. Phys. Rev. Lett. 1, 088103.
Ishikawa, T., Sekiya, G., Imai, Y., Yamaguchi, T., 2007. Hydrodynamic interaction between two swimming bacteria. Biophys. J. 93, 2217–2225.
Ishikawa, T., Simmonds, M.P., Pedley, T.J., 2006. Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119–160.
Jeffery, G.B., 1922. The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.
Kim, M.J., Breuer, K.S., 2004. Enhanced diffusion due to motile bacteria. Phys. Fluids 16, L78–L81.
Kim, S., Karilla, J.S., 1991. Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann, Stoneham.
Lamb, H., 1932. Hydrodynamics, 6th edn. Dover, New York.
Lauga, E., Bartolo, D., 2008. No many-scallop theorem: Collective locomotion of reciprocal swimmers. Phys. Rev. E 78, 030901.
Lauga, E., Powers, T.R., 2009. The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.
Lighthill, J., 1975. Mathematical Biofluiddynamics. SIAM, Philadelphia.
Lighthill, J., 1976. Flagellar hydrodynamics—The John von Neumann lecture. 1975. SIAM Rev. 18, 161–230.
Liron, N., Barta, E., 1992. Motion of a rigid particle in stokes flow: a new second-kind boundary-integral equation formulation. J. Fluid Mech. 238, 579–598.
Mehandia, V., Nott, P.R., 2008. The collective dynamics of self-propelled particles. J. Fluid Mech. 595, 239–264.
Mendelson, N.H., Bourque, A., Wilkening, K., Anderson, K.R., Watkins, J.C., 1999. Organized cell swimming motions in Bacillus subtilis colonies: Patterns of short-lived whirls and jets. J. Bacteriol. 181, 600–609.
Moore, H., Dvorkov, K., Jenkins, N., Breed, W., 2002. Exceptional sperm cooperation in the wood mouse. Nature 418, 174–177.
Moore, H.D.M., Taggart, D.A., 1995. Sperm pairing in the opossum increases the efficiency of sperm movement in a viscous environment. Biol. Reprod. 52, 947–953.
Pedley, T.J., Kessler, J.O., 1992. Hydrodynamic phenomena in suspensions of swimming microorganisms. Ann. Rev. Fluid Mech. 24, 313–358.
Pooley, C.M., Alexander, G.P., Yeomans, J.M., 2007. Hydrodynamic interaction between two swimmers at low Reynolds number. Phys. Rev. Lett. 99, 228103.
Pozrikidis, C., 1997. Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press, London.
Riedel, I.H., Kruse, K., Howard, J., 2005. A self-organized vortex array of hydrodynamically entrained sperm cells. Science 309, 300–303.
Saintillan, D., Shelley, M.J., 2007. Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102.
Saintillan, D., Shelley, M.J., 2008. Instabilities and pattern formation in active particle suspensions: Kinetic theory and continuum simulations. Phys. Rev. Lett. 100, 178103.
Sastry, S., 1999. Nonlinear Systems: Analysis, Stability and Control. Springer, New York.
Simha, R.A., Ramaswamy, S., 2002. Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Phys. Rev. Lett. 89, 058101.
Sokolov, A., Aranson, I.S., Kessler, J.O., Goldstein, R.E., 2007. Concentration dependence of the collective dynamics of swimming bacteria. Phys. Rev. Lett. 98, 158102.
Underhill, P.T., Hernandez-Ortiz, J.P., Graham, M.D., 2008. Diffusion and spatial correlations in suspensions of swimming particles. Phys. Rev. Lett. 100, 248101.
Vicsek, T., Czirok, A., Benjacob, E., Cohen, I., Shochet, O., 1995. Novel type of phase-transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229.
Wolgemuth, C.W., 2008. Collective swimming and the dynamics of bacterial turbulence. Biophys. J. 95, 1564–1574.
Wu, X.L., Libchaber, A., 2000. Particle diffusion in a quasi-two-dimensional bacterial bath. Phys. Rev. Lett. 84, 3017–3020.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Michelin, S., Lauga, E. The Long-Time Dynamics of Two Hydrodynamically-Coupled Swimming Cells. Bull. Math. Biol. 72, 973–1005 (2010). https://doi.org/10.1007/s11538-009-9479-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-009-9479-6