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Metachronal waves in a chain of rowers with hydrodynamic interactions

  • C. WollinEmail author
  • H. Stark
Open Access
Regular Article
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Abstract.

Hair-like appendages called cilia on the surface of a microorganism such as Paramecium or Opalina beat highly synchronized and form so-called metachronal waves that travel along the surfaces. In order to study under what principal conditions these waves form, we introduce a chain of beads, called rowers, each periodically driven by an external force on a straight line segment. To implement hydrodynamic interactions between the beads, they are considered point-like. Two beads synchronize in antiphase or in phase depending on the positive or negative curvature of their driving-force potential. Concentrating on in-phase synchronizing rowers, we find that they display only transient synchronization in a bulk fluid. On the other hand, metachronal waves with wavelengths of 7-10 rower distances emerge, when we restrict the range of hydrodynamic interactions either artificially to nearest neighbors or by the presence of a bounding surface as in any relevant biological system.

Keywords

Chain Segment Hydrodynamic Interaction Switching Point Recovery Stroke Rower Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Supplementary material

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References

  1. 1.
    D. Bray, Cell Movements: From Molecules to Motiliy, 2nd edition (Garland Publishing, New York, 2001)Google Scholar
  2. 2.
    E.M. Purcell, Am. J. Phys. 45, 3 (1977)CrossRefADSGoogle Scholar
  3. 3.
    E. Lauga, T.R. Powers, Rep. Prog. Phys. 72, 096601 (2009)CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    M.T. Downton, H. Stark, J. Phys.: Condens. Matter 21, 204101 (2009)CrossRefADSGoogle Scholar
  5. 5.
    H. Machemer, J. Exp. Biol. 57, 239 (1972)Google Scholar
  6. 6.
    R.W. Linck, Cilia and Flagella in Encyclopedia of Life Sciences (Wiley, Chichester, 2001), www.els.net
  7. 7.
    C. Brennen, H. Winet, Annu. Rev. Fluid Mech. 9, 339 (1977)CrossRefADSGoogle Scholar
  8. 8.
    M. Cosentino Lagomarsino, P. Jona, B. Bassetti, Phys. Rev. E 68, 21908 (2003)CrossRefGoogle Scholar
  9. 9.
    A. Vilfan, F. Jülicher, Phys. Rev. Lett. 96, 058102 (2006)CrossRefADSGoogle Scholar
  10. 10.
    P. Lenz, A. Ryskin, Phys. Biol. 3, 285 (2006)CrossRefADSGoogle Scholar
  11. 11.
    T. Niedermayer, B. Eckhardt, P. Lenz, Chaos 18, 37128 (2008)CrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Kotar, M. Leoni, B. Bassetti, M. Cosentino Lagomarsino, P. Cicuta, Proc. Natl. Acad. Sci. U.S.A. 107, 7669 (2010)CrossRefADSGoogle Scholar
  13. 13.
    N. Uchida, R. Golestanian, Phys. Rev. Lett. 106, 058104 (2011)CrossRefADSGoogle Scholar
  14. 14.
    S. Gueron, K. Levit-Gurevich, N. Liron, J.J. Blum, Proc. Natl. Acad. Sci. U.S.A. 94, 6001 (1997)CrossRefzbMATHADSGoogle Scholar
  15. 15.
    S. Gueron, K. Levit-Gurevich, Proc. Natl. Acad. Sci. U.S.A. 96, 12240 (1999)CrossRefzbMATHADSGoogle Scholar
  16. 16.
    B. Guirao, J.F. Joanny, Biophys. J. 92, 1900 (2007)CrossRefADSGoogle Scholar
  17. 17.
    E.M. Gauger, M.T. Downton, H. Stark, Eur. Phys. J. E 28, 231 (2009)CrossRefGoogle Scholar
  18. 18.
    J. Blake, J. Theor. Biol. 52, 67 (1975)CrossRefGoogle Scholar
  19. 19.
    M. Sleigh, J. Exp. Biol. 37, 1 (1960)Google Scholar
  20. 20.
    S.L. Tamm, J. Exp. Biol. 113, 401 (1984)Google Scholar
  21. 21.
    I. Gibbons, J. Biophys. Biochem. Cytol. 11, 179 (1961)CrossRefGoogle Scholar
  22. 22.
    P. Satir, M.A. Sleigh, Annu. Rev. Physiol. 52, 137 (1990)CrossRefGoogle Scholar
  23. 23.
    M.A. Chilvers, A. Rutman, C. O’Callaghan, J. Allergy Clin. Immunol. 112, 518 (2003)CrossRefGoogle Scholar
  24. 24.
    Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984)Google Scholar
  25. 25.
    Y.T. Yang, C. Callegari, X.L. Feng, K.L. Ekinci, M.L. Roukes, Nano Lett. 6, 583 (2006)CrossRefADSGoogle Scholar
  26. 26.
    D. Rugar, R. Budakian, H.J. Mamin, B.W. Chui, Nature 430, 329 (2004)CrossRefADSGoogle Scholar
  27. 27.
    A.N. Cleland, M.L. Roukes, Nature 392, 160 (1998)CrossRefADSGoogle Scholar
  28. 28.
    I.H. Riedel-Kruse, A. Hilfinger, J. Howard, F. Julicher, Hum. Front. Sci. Prog. J. 1, 192 (2007)Google Scholar
  29. 29.
    J.A. Acebré, Rev. Mod. Phys. 77, 137 (2005)CrossRefADSGoogle Scholar
  30. 30.
    A. Pikovsky, M. Rosenblum, J. Kurths, in Analysis and Control of Complex Nonlinear Processes in Physics, Chemistry and Biology, edited by L. Schimansky-Geier, B. Fiedler, J. Kurths, E. Schöll (World Scientific, Singapore, 2007)Google Scholar
  31. 31.
    M. Kim, T.R. Powers, Phys. Rev. E 69, 061910 (2004)CrossRefADSGoogle Scholar
  32. 32.
    M. Reichert, H. Stark, Eur. Phys. J. E 17, 493 (2005)CrossRefGoogle Scholar
  33. 33.
    B. Qian, H. Jiang, D.A. Gagnon, K.S. Breuer, T.R. Powers, Phys. Rev. E 80, 61919 (2009)CrossRefADSGoogle Scholar
  34. 34.
    B.A. Grzybowski, H. Stone, G.M. Whitesides, Nature 405, 1033 (2000)CrossRefADSGoogle Scholar
  35. 35.
    P. Lenz, J.-F. Joanny, F. Jülicher, J. Prost, Phys. Rev. Lett. 91, 108104 (2003)CrossRefADSGoogle Scholar
  36. 36.
    N. Uchida, R. Golestanian, Phys. Rev. Lett. 104, 178103 (2010)CrossRefADSGoogle Scholar
  37. 37.
    R.E. Goldstein, M. Polin, I. Tuval, Phys. Rev. Lett. 103, 168103 (2009)CrossRefADSGoogle Scholar
  38. 38.
    G. Taylor, Proc. R. Soc. London, Ser. A 209, 447 (1951)CrossRefzbMATHADSGoogle Scholar
  39. 39.
    Y. Yang, J. Elgeti, G. Gompper, Phys. Rev. E 78, 061903 (2008)CrossRefADSGoogle Scholar
  40. 40.
    G.J. Elfring, E. Lauga, Phys. Rev. Lett. 103, 088101 (2009)CrossRefADSGoogle Scholar
  41. 41.
    G.J. Elfring, O.S. Pak, E. Lauga, J. Fluid Mech. 646, 505 (2010)CrossRefzbMATHADSMathSciNetGoogle Scholar
  42. 42.
    D.G. Grier, Nature 424, 810 (2003)CrossRefADSGoogle Scholar
  43. 43.
    R. Golestanian, private communication (2010)Google Scholar
  44. 44.
    J. Blake, A. Chwang, J. Eng. Math. 8, 23 (1974)CrossRefzbMATHGoogle Scholar
  45. 45.
    M. Downton, H. Stark, unpublished resultsGoogle Scholar
  46. 46.
    J. Elgeti, Sperm and Cilia Dynamics, Ph.D. Thesis, Universität zu Köln (2006)Google Scholar
  47. 47.
    E.R. Dufresne, T.M. Squires, M.P. Brenner, D.G. Grier, Phys. Rev. Lett. 85, 3317 (2000)CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2011

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany

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