Advertisement

The European Physical Journal E

, Volume 17, Issue 4, pp 493–500 | Cite as

Synchronization of rotating helices by hydrodynamic interactions

  • M. Reichert
  • H. Stark
Original Article

Abstract.

Some types of bacteria use rotating helical flagella to swim. The motion of such organisms takes place in the regime of low Reynolds numbers where viscous effects dominate and where the dynamics is governed by hydrodynamic interactions. Typically, rotating flagella form bundles, which means that their rotation is synchronized. The aim of this study is to investigate whether hydrodynamic interactions can be at the origin of such a bundling and synchronization. We consider two stiff helices that are modelled by rigidly connected beads, neglecting any elastic deformations. They are driven by constant and equal torques, and they are fixed in space by anchoring their terminal beads in harmonic traps. We observe that, for finite trap strength, hydrodynamic interactions do indeed synchronize the helix rotations. The speed of phase synchronization decreases with increasing trap stiffness. In the limit of infinite trap stiffness, the speed is zero and the helices do not synchronize.

PACS.

05.45.Xt Synchronization; coupled oscillations 47.15.Gf Low-Reynolds-number (creeping) flows 87.16.Qp Pseudopods, lamellipods, cilia, and flagella 87.19.St Movement and locomotion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

strong_trap.mpg (1.3 mb)
Supplementary material
weak_trap.mpg (1.3 mb)
Supplementary material

References

  1. 1.
    H.C. Berg, Nature 245, 380 (1973).CrossRefPubMedGoogle Scholar
  2. 2.
    R.M. Macnab, in Escherichia coli and Salmonella typhi\-murium, edited by F.C. Neidhardt (American Society of Microbiology, Washington DC, 1987) p. 70.Google Scholar
  3. 3.
    L. Turner, W.S. Ryu, H.C. Berg, J. Bacteriol. 182, 2793 (2000). Movies of swimming E. coli can be found at http:/-2pt/ webmac.rowland.org/labs/bacteria/ movies\_ecoli.html.CrossRefPubMedGoogle Scholar
  4. 4.
    H.C. Berg, Annu. Rev. Biochem. 72, 19 (2003).CrossRefPubMedGoogle Scholar
  5. 5.
    H.C. Berg, E. coli in Motion (Springer, New York, 2004).Google Scholar
  6. 6.
    M.D. Manson, P.M. Tedesco, H.C. Berg, J. Mol. Biol. 138, 541 (1980).PubMedGoogle Scholar
  7. 7.
    R.M. Macnab, Proc. Natl. Acad. Sci. U.S.A. 74, 221 (1977).PubMedGoogle Scholar
  8. 8.
    M.J. Kim, J.C. Bird, A.J. Van Parys, K.S. Breuer, T.R. Powers, Proc. Natl. Acad. Sci. U.S.A. 100, 15481 (2003). A movie of the bundling sequence of two rotating (macroscopic) helices can be found at http:/-2pt/www.pnas.org/cgi/ content/full/2633596100/DC1.CrossRefPubMedGoogle Scholar
  9. 9.
    M.J. Kim, M.J. Kim, J.C. Bird, J. Park, T.R. Powers, K.S. Breuer, Exp. Fluids 37, 782 (2004).CrossRefGoogle Scholar
  10. 10.
    R.M. Macnab, M.K. Ornston, J. Mol. Biol. 112, 1 (1977).PubMedGoogle Scholar
  11. 11.
    E.M. Purcell, Am. J. Phys. 45, 3 (1977).CrossRefGoogle Scholar
  12. 12.
    J.K.G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, Amsterdam, 1996).Google Scholar
  13. 13.
    G. Taylor, Proc. R. Soc. London, Ser. A 209, 447 (1951).Google Scholar
  14. 14.
    M.C. Lagomarsino, B. Bassetti, P. Jona, Eur. Phys. J. B 26, 81 (2002).Google Scholar
  15. 15.
    M.C. Lagomarsino, P. Jona, B. Bassetti, Phys. Rev. E 68, 021908 (2003).CrossRefGoogle Scholar
  16. 16.
    S. Gueron, K. Levit-Gurevich, Proc. Natl. Acad. Sci. U.S.A. 96, 12240 (1999).CrossRefPubMedGoogle Scholar
  17. 17.
    M.J. Kim, T.R. Powers, Phys. Rev. E 69, 061910 (2004).CrossRefGoogle Scholar
  18. 18.
    H. Brenner, Chem. Eng. Sci. 18, 1 (1963)CrossRefGoogle Scholar
  19. 19.
    K. Hinsen, Comput. Phys. Commun. 88, 327 (1995).CrossRefGoogle Scholar
  20. 20.
    P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1999).Google Scholar
  21. 21.
    The same result with similar methods was obtained by Kim and Powers in reference kim04b. They use the concept of kinematic reversibility of Stokes flow that manifests itself, e.g., in equation (eq:chi’=mu*D). That means, to arrive at equation (eq:symm_x), we use this concept when the torque on the helices is reversed.Google Scholar
  22. 22.
    H. Lamb, Hydrodynamics (Cambridge University Press, London, 1975).Google Scholar
  23. 23.
    B.A. Finlayson, The Method of Weighted Residulas and Variational Principles (Academic Press, New York, 1972). Google Scholar
  24. 24.
    G.B. Arfken, H.J. Weber, Mathematical Methods for Physicists (Academic Press, San Diego, 1995).Google Scholar
  25. 25.
    R.E. Johnson, C.J. Brokaw, Biophys. J. 25, 113 (1979).PubMedGoogle Scholar
  26. 26.
    M. Reichert, H. Stark, J. Phys. Condens. Matter 16, S4085 (2004).Google Scholar

Copyright information

© EDP Sciences, Società Italiana di Fisica and Springer-Verlag 2005

Authors and Affiliations

  1. 1.Fachbereich PhysikUniversität KonstanzKonstanzGermany

Personalised recommendations