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Hydrodynamic synchronization between objects with cyclic rigid trajectories

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Abstract

Synchronization induced by long-range hydrodynamic interactions is attracting attention as a candidate mechanism behind coordinated beating of cilia and flagella. Here we consider a minimal model of hydrodynamic synchronization in the low Reynolds number limit. The model consists of rotors, each of which assumed to be a rigid bead making a fixed trajectory under periodically varying driving force. By a linear analysis, we derive the necessary and sufficient conditions for a pair of rotors to synchronize in phase. We also derive a non-linear evolution equation for their phase difference, which is reduced to minimization of an effective potential. The effective potential is calculated for a variety of trajectory shapes and geometries (either bulk or substrated), for which the stable and metastable states of the system are identified. Finite size of the trajectory induces asymmetry of the potential, which also depends sensitively on the tilt of the trajectory. Our results show that flexibility of cilia or flagella is not a requisite for their synchronized motion, in contrast to previous expectations. We discuss the possibility to directly implement the model and verify our results by optically driven colloids.

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References

  1. J. Gray, Ciliary Movements (Cambridge University Press, Cambridge, 1928).

  2. D. Bray, Cell Movements: From Molecules to Motility, 2nd ed. (Garland, New York, 2001).

  3. S. Nonaka, S. Yoshiba, D. Watanabe, S. Ikeuchi, T. Goto, W.F. Marshall, H. Hamada, PLoS Biol. 3, e268 (2005).

    Article  Google Scholar 

  4. J.R. Blake, M.A. Sleigh, Biol. Rev. 49, 85 (1974).

    Article  Google Scholar 

  5. E.W. Knight-Jones, Q. J. Microsc. Sci. 95, 503 (1954).

    Google Scholar 

  6. J.R. Blake, J. Fluid Mech. 55, 1 (1972).

    Article  ADS  Google Scholar 

  7. S. Gueron, K. Levit-Gurevich, N. Liron, J.J. Blum, Proc. Natl. Acad. Sci. U.S.A. 94, 6001 (1997).

    Article  ADS  Google Scholar 

  8. S. Gueron, K. Levit-Gurevich, Proc. Natl. Acad. Sci. U.S.A. 96, 12240 (1999).

    Article  ADS  Google Scholar 

  9. R. Golestanian, J.M. Yeomans, N. Uchida, Soft Matter 7, 3074 (2011).

    Article  ADS  Google Scholar 

  10. G.I. Taylor, Proc. R. Soc. London, Ser. A 209, 447 (1951).

    Article  ADS  Google Scholar 

  11. 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).

    Article  ADS  Google Scholar 

  12. B. Qian, H. Jiang, D.A. Gagnon, K.S. Breuer, T.R. Powers, Phys. Rev. E 80, 061919 (2009).

    Article  ADS  Google Scholar 

  13. M. Polin, I. Tuval, K. Drescher, J.P. Gollub, R.E. Goldstein, Science 325, 487 (2009).

    Article  ADS  Google Scholar 

  14. R.E. Goldstein, M. Polin, I. Tuval, Phys. Rev. Lett. 103, 168103 (2009).

    Article  ADS  Google Scholar 

  15. J. Kotar, M. Leoni, B. Bassetti, M.C. Lagomarsino, P. Cicuta, Proc. Natl. Acad. Sci. U.S.A. 107, 7669 (2010).

    Article  ADS  Google Scholar 

  16. R. Di Leonardo, A. Buzas, L. Kelemen, G. Vizsnyiczai, L. Oroszi, P. Ormos, Phys. Rev. Lett. 109, 034104 (2012).

    Article  ADS  Google Scholar 

  17. N. Darnton, L. Turner, K. Breuer, H.C. Berg, Biophys. J. 86, 1863 (2004).

    Article  ADS  Google Scholar 

  18. M. Vilfan, A. Potocvnik, B. Kavcic, N. Osterman, I. Poberaj, A. Vilfan, D. Babic, Proc. Natl. Acad. Sci. U.S.A. 107, 1844 (2010).

    Article  ADS  Google Scholar 

  19. A.R. Shields, B.L. Fiser, B.A. Evans, M.R. Falvo, S. Washburn, R. Superfine, Proc. Natl. Acad. Sci. U.S.A. 107, 15670 (2010).

    Article  ADS  Google Scholar 

  20. N. Coq, A. Bricard, F.-D. Delapierre, L. Malaquin, O. du Roure, M. Fermigier, D. Bartolo, Phys. Rev. Lett. 107, 014501 (2011).

    Article  ADS  Google Scholar 

  21. M. Cosentino Lagomarsino, B. Bassetti, P. Jona, Eur. Phys. J. B 26, 81 (2002).

    Article  ADS  Google Scholar 

  22. M. Cosentino Lagomarsino, P. Jona, B. Bassetti, Phys. Rev. E 68, 021908 (2003).

    Article  ADS  Google Scholar 

  23. M. Kim, T.R. Powers, Phys. Rev. E 69, 061910 (2004).

    Article  ADS  Google Scholar 

  24. M. Reichert, H. Stark, Eur. Phys. J. E 17, 493 (2005).

    Article  Google Scholar 

  25. Y.W. Kim, R.R. Netz, Phys. Rev. Lett. 96, 158101 (2006).

    Article  ADS  Google Scholar 

  26. A. Vilfan, F. Jülicher, Phys. Rev. Lett. 96, 058102 (2006).

    Article  ADS  Google Scholar 

  27. A. Ryskin, P. Lenz, Phys. Biol. 3, 285 (2006).

    Article  ADS  Google Scholar 

  28. B. Guirao, J.-F. Joanny, Biophys. J. 92, 1900 (2007).

    Article  ADS  Google Scholar 

  29. T. Niedermayer, B. Eckhardt, P. Lenz, Chaos 18, 037128 (2008).

    Article  MathSciNet  ADS  Google Scholar 

  30. G.J. Elfring, E. Lauga, Phys. Rev. Lett. 103, 088101 (2009).

    Article  ADS  Google Scholar 

  31. N. Uchida, R. Golestanian, Phys. Rev. Lett. 104, 178103 (2010).

    Article  ADS  Google Scholar 

  32. N. Uchida, R. Golestanian, EPL 89, 50011 (2010).

    Article  ADS  Google Scholar 

  33. N. Uchida, R. Golestanian, Phys. Rev. Lett. 106, 058104 (2011).

    Article  ADS  Google Scholar 

  34. N. Osterman, A. Vilfan, Proc. Natl. Acad. Sci. U.S.A. 108, 15727 (2011).

    Article  ADS  Google Scholar 

  35. C.W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik (Akademishe Verlagsgesellschaft, Leipzig, 1927).

  36. J.R. Blake, Proc. Camb. Phil. Soc. 70, 303 (1971).

    Article  ADS  Google Scholar 

  37. J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (M. Nijhoff, The Hague, 1983).

  38. By considering nonlinear effect, we show in sect. sec:4 that linear trajectories are also capable of synchronization.

  39. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, New York, 1984).

  40. N. Bruot, private communication.

  41. B.M. Friedrich, F. Julicher, Phys. Rev. Lett. 109, 138102 (2012).

    Article  ADS  Google Scholar 

  42. R.R. Bennett, R. Golestanian, arXiv:1211.3272.

  43. R.E. Goldstein, private communication.

  44. K.T. Gahagan, G.A. Swartzlander, Opt. Lett. 21, 827 (1996).

    Article  ADS  Google Scholar 

  45. J.E. Curtis, D.G. Grier, Phys. Rev. Lett. 90, 133901 (2003).

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Tohoku University, 980-8578, Sendai, Japan

    Nariya Uchida

  2. Rudolf Peierls Centre for Theoretical Physics, University of Oxford, OX1 3NP, Oxford, UK

    Ramin Golestanian

Authors
  1. Nariya Uchida
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  2. Ramin Golestanian
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Correspondence to Nariya Uchida.

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Uchida, N., Golestanian, R. Hydrodynamic synchronization between objects with cyclic rigid trajectories. Eur. Phys. J. E 35, 135 (2012). https://doi.org/10.1140/epje/i2012-12135-5

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  • DOI: https://doi.org/10.1140/epje/i2012-12135-5

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