Active polar fluid flow in finite droplets

  • Carl A. WhitfieldEmail author
  • Davide Marenduzzo
  • Raphaël Voituriez
  • Rhoda J. Hawkins
Open Access
Regular Article


We present a continuum level analytical model of a droplet of active contractile fluid consisting of filaments and motors. We calculate the steady state flows that result from a splayed polarisation of the filaments. We account for interaction with the external medium by imposing a viscous friction at the fixed droplet boundary. We then show that the droplet has non-zero force dipole and quadrupole moments, the latter of which is essential for self-propelled motion of the droplet at low Reynolds' number. Therefore, this calculation describes a simple mechanism for the motility of a droplet of active contractile fluid embedded in a three-dimensional environment, which is relevant to cell migration in confinement (for example, embedded within a gel or tissue). Our analytical results predict how the system depends on various parameters such as the effective friction coefficient, the phenomenological activity parameter and the splay of the imposed polarisation.

Graphical abstract


Living systems: Biological Matter 


  1. 1.
    S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323 (2010)CrossRefADSGoogle Scholar
  2. 2.
    J. Toner, Y. Tu, Phys. Rev. E 58, 4828 (1998)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    V. Narayan, S. Ramaswamy, N. Menon, Science 317, 105 (2007)CrossRefADSGoogle Scholar
  4. 4.
    K. Kruse, F. Jülicher, Phys. Rev. Lett. 85, 1778 (2000)CrossRefADSGoogle Scholar
  5. 5.
    K. Kruse, J.F. Joanny, F. Jülicher, J. Prost, K. Sekimoto, Phys. Rev. Lett. 92, 078101 (2004)CrossRefADSGoogle Scholar
  6. 6.
    K. Kruse, J.F. Joanny, F. Jülicher, J. Prost, K. Sekimoto, Eur. Phys. J. E 16, 5 (2005)CrossRefGoogle Scholar
  7. 7.
    J.F. Joanny, J. Prost, Human. Front. Sci. Prog. J. 3, 94 (2009)Google Scholar
  8. 8.
    D. Mizuno, C. Tardin, C. Schmidt, F. MacKintosh, Science 315, 370 (2007)CrossRefADSGoogle Scholar
  9. 9.
    F.C. Mackintosh, C.F. Schmidt, Curr. Opin. Cell Biol. 22, 29 (2010)CrossRefGoogle Scholar
  10. 10.
    M.S. e Silva, M. Depken, B. Stuhrmann, M. Korsten, F.C. MacKintosh, G.H. Koenderink, Proc. Natl. Acad. Sci. U.S.A. 108, 9408 (2011)CrossRefADSGoogle Scholar
  11. 11.
    S. Köhler, V. Schaller, A.R. Bausch, PloS one 6, e23798 (2011)CrossRefADSGoogle Scholar
  12. 12.
    S. Köhler, A.R. Bausch, PloS one 7, e39869 (2012)CrossRefADSGoogle Scholar
  13. 13.
    R. Voituriez, J.F. Joanny, J. Prost, Europhys. Lett. 70, 404 (2005)CrossRefADSGoogle Scholar
  14. 14.
    G. Charras, E. Paluch, Nat. Rev. Mol. Cell Biol. 9, 730 (2008)CrossRefGoogle Scholar
  15. 15.
    R.J. Hawkins, M. Piel, G. Faure-Andre, A.M. Lennon-Dumenil, J.F. Joanny, J. Prost, R. Voituriez, Phys. Rev. Lett. 102, 058103 (2009)CrossRefADSGoogle Scholar
  16. 16.
    K. Konstantopoulos, P.H. Wu, D. Wirtz, Biophys. J. 104, 279 (2013)CrossRefGoogle Scholar
  17. 17.
    R. Poincloux, O. Collin, F. Lizárraga, M. Romao, M. Debray, M. Piel, P. Chavrier, Proc. Natl. Acad. Sci. U.S.A. 108, 1943 (2011)CrossRefADSGoogle Scholar
  18. 18.
    M. Weber, R. Hauschild, J. Schwarz, C. Moussion, I. de Vries, D.F. Legler, S.A. Luther, T. Bollenbach, M. Sixt, Science 339, 328 (2013)CrossRefADSGoogle Scholar
  19. 19.
    E. Tjhung, D. Marenduzzo, M.E. Cates, Proc. Natl. Acad. Sci. U.S.A. 109, 12381 (2012)CrossRefADSGoogle Scholar
  20. 20.
    C. Blanch-Mercader, J. Casademunt, Phys. Rev. Lett. 110, 078102 (2013)CrossRefADSGoogle Scholar
  21. 21.
    D. Shao, W.J. Rappel, H. Levine, Phys. Rev. Lett. 105, 108104 (2010)CrossRefADSGoogle Scholar
  22. 22.
    F. Ziebert, S. Swaminathan, I.S. Aranson, J. R. Soc. Interface 9, 1084 (2011)CrossRefGoogle Scholar
  23. 23.
    T. Sanchez, D.T.N. Chen, S.J. DeCamp, M. Heymann, Z. Dogic, Nature 491, 431 (2012)CrossRefADSGoogle Scholar
  24. 24.
    C.M. Coppin, P.C. Leavis, Biophys. J. 63, 794 (1992)CrossRefADSGoogle Scholar
  25. 25.
    P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd edition (Clarendon Press, Oxford, 1993)Google Scholar
  26. 26.
    H. Pleiner, H. Brand, Europhys. Lett. 9, 243 (1989)CrossRefADSGoogle Scholar
  27. 27.
    S. Fürthauer, M. Neef, S. Grill, K. Kruse, F. Jülicher, New J. Phys. 14, 023001 (2012)CrossRefGoogle Scholar
  28. 28.
    P.M. Bendix, G.H. Koenderink, D. Cuvelier, Z. Dogic, B.N. Koeleman, W.M. Brieher, C.M. Field, L. Mahadevan, D.A. Weitz, Biophys. J. 94, 3126 (2008)CrossRefADSGoogle Scholar
  29. 29.
    R. Voituriez, J.F. Joanny, J. Prost, Phys. Rev. Lett. 96, 028102 (2006)CrossRefADSGoogle Scholar
  30. 30.
    R.J. Hawkins, R. Poincloux, O. Bénichou, M. Piel, P. Chavrier, R. Voituriez, Biophys. J. 101, 1041 (2011)CrossRefGoogle Scholar
  31. 31.
    B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell, 5th edition (Garland Science, New York, 2008)Google Scholar
  32. 32.
    K. Kruse, J.F. Joanny, F. Jülicher, J. Prost, Phys. Biol. 3, 130 (2006)CrossRefADSGoogle Scholar
  33. 33.
    F. Wottawah, S. Schinkinger, B. Lincoln, R. Ananthakrishnan, M. Romeyke, J. Guck, J. Ks, Phys. Rev. Lett. 94, 098103 (2005)CrossRefADSGoogle Scholar
  34. 34.
    O. Thoumine, O. Cardoso, J.J. Meister, Eur. Biophys. J. 28, 222 (1999)CrossRefGoogle Scholar
  35. 35.
    J. Dai, M.P. Sheetz, Biophys. J. 77, 3363 (1999)CrossRefGoogle Scholar
  36. 36.
    N. Yoshinaga, arXiv preprint arXiv:1307.3120 (2013)
  37. 37.
    H.A. Stone, A.D. Samuel, Phys. Rev. Lett. 77, 4102 (1996)CrossRefADSGoogle Scholar
  38. 38.
    J. Blake, J. Fluid Mech 46, 199 (1971)CrossRefzbMATHADSGoogle Scholar
  39. 39.
    F.G. Woodhouse, R.E. Goldstein, Phys. Rev. Lett. 109, 168105 (2012)CrossRefADSGoogle Scholar
  40. 40.
    I.S. Aranson, L.S. Tsimring, Phys. Rev. E 74, 031915 (2006)CrossRefMathSciNetADSGoogle Scholar
  41. 41.
    F. Nédélec, J. Cell Biol. 158, 1005 (2002)CrossRefGoogle Scholar
  42. 42.
    D.A. Head, G. Gompper, W.J. Briels, Soft Matter 7, 3116 (2011)CrossRefADSGoogle Scholar
  43. 43.
    F.J. Nédélec, T. Surrey, A.C. Maggs, S. Leibler, Nature 389, 305 (1997)CrossRefADSGoogle Scholar
  44. 44.
    T. Surrey, F. Nédélec, S. Leibler, E. Karsenti, Science 292, 1167 (2001)CrossRefADSGoogle Scholar
  45. 45.
    L. Laan, N. Pavin, J. Husson, G. Romet-Lemonne, M. van Duijn, M.P. López, R.D. Vale, F. Jülicher, S.L. Reck-Peterson, M. Dogterom, Cell 148, 502 (2012)CrossRefGoogle Scholar
  46. 46.
    Maple-16 (2012) Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario, CanadaGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Carl A. Whitfield
    • 1
    Email author
  • Davide Marenduzzo
    • 2
  • Raphaël Voituriez
    • 3
    • 4
  • Rhoda J. Hawkins
    • 1
  1. 1.Department of Physics and AstronomyUniversity of SheffieldSheffieldUK
  2. 2.SUPA, School of Physics and AstronomyUniversity of EdinburghEdinburghUK
  3. 3.Laboratoire de Physique Théorique et Matière Condensée, UMR 7600Université Pierre et Marie Curie/CNRSParisFrance
  4. 4.Laboratoire Jean PerrinFRE 3231 CNRS/UPMCParis CedexFrance

Personalised recommendations