Microscopic Modelling of Active Bacterial Suspensions

A. Decoene¹ ; S. Martin¹ ; B. Maury¹

¹ Université Paris-Sud 11, Département de Mathématiques, Bâtiment 425, 91405 Orsay cedex, France

Mathematical modelling of natural phenomena, Tome 6 (2011) no. 5, pp. 98-129

Résumé

We present two-dimensional simulations of chemotactic self-propelled bacteria swimming in a viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity and opposite direction applied on the rigid bacterial body and on an associated region in the fluid representing the flagellar bundle. The method for solving the fluid flow and the motion of the bacteria is based on a variational formulation written on the whole domain, strongly coupling the fluid and the rigid particle problems: rigid motion is enforced by penalizing the strain rate tensor on the rigid domain, while incompressibility is treated by duality. This model allows to achieve an accurate description of fluid motion and hydrodynamic interactions in moderate to concentrated active suspensions. A mesoscopic model is also used, in which the size of the bacteria is supposed to be much smaller than the elements of fluid: the perturbation of the fluid due to propulsion and motion of the swimmers is neglected, and the fluid is only subjected to the buoyant forcing induced by the presence of the bacteria, which are denser than the fluid. Although this model does not accurately take into account hydrodynamic interactions, it is able to reproduce complex collective dynamics observed in concentrated bacterial suspensions, such as bioconvection. From a mathematical point of view, both models lead to a minimization problem which is solved with a standard Finite Element Method. In order to ensure robustness, a projection algorithm is used to deal with contacts between particles. We also reproduce chemotactic behaviour driven by oxygen: an advection-diffusion equation on the oxygen concentration is solved in the fluid domain, with a source term accounting for oxygen consumption by the bacteria. The orientations of the individual bacteria are subjected to random changes, with a frequency that depends on the surrounding oxygen concentration, in order to favor the direction of the concentration gradient.

Détail

DOI : 10.1051/mmnp/20116506

Bibliographie

[1] H.C. Berg. Random walks in biology. Princeton University Press, Princeton, 1983.

[2] H.C. Berg. E. Coli in Motion. Springer Verlag, New York, 2004.

[3] B.M. Haines, I.S. Aranson, L. Berlyand, D.A. Karpeev Physical Biology 2008

[4] P. G. Ciarlet. Introduction à l’analyse numérique matricielle et à l’optimisation. Masson, Paris, 1990.

[5] L.H. Cisneros, R. Cortez, C. Dombrowski, R.E. Goldstein, J.O. Kessler Exp Fluids 2007 737 753

[6] Darnton Nc, Turner L, Rojevsky S, Berg Hc Biophys J. 2010 2082 90

[7] A. Decoene, A. Lorz, S. Martin, B. Maury, M. Tang ESAIM: Proc 2010 104 123

[8] C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, J. O. Kessler Phys. Rev. Lett. 2004

[9] D. Gérard-Varet, M. Hillairet. Regularity Issues in the Problem of Fluid Structure Interaction. to appear in Arch. Rational Mech. Anal.

[10] R. Glowinski, T. W. Pan, T. I. Hesla, D. D. Joseph, J. Périaux J. Comp. Phys. 2001 363 427

[11] R. Glowinski. Finite element methods for incompressible viscous flow. In: Handbook of Numerical Analysis, Vol. IX, P. G. Ciarlet and J.-L. Lions eds., Ed. North-Holland, Amsterdam, 2003.

[12] V. Gyrya, K. Lipnikov, I. Aranson, L. Berlyand. Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate suspensions. Journal of Mathematical Biology (accepted, 2011).

[13] Hernandez-Ortiz J.P., C. Stoltz and M.D. Graham. Transport and col lective dynamics in suspensions of confined swimming particles. Phys. Rev. Lett., 95 (2005), pp. 204501.

[14] J. Happel, H. Brenner. Low Reynolds Number Hydrodynamics. Dordrecht, Kluwer, 1991.

[15] M. Hillairet Communications in Partial Differential Equations 2007 1345 1371

[16] J. Janela, A. Lefebvre, B. Maury ESAIM: Proc. 2007 115 123

[17] D. Kaiser Curr Biol 2007 R561 R570

[18] S. Kim, S.J. Karrila. Microhydrodynamics: Principles and Selected Applications. Dover, New York, 2005.

[19] E. Lauga and T.R. Powers. The hydrodynamics of swimming microorganisms. Rep. Prog. Phys., 72 (2009).

[20] A. Lefebvre ESAIM: Proc. 2007 120 132

[21] A. Lefebvre, B. Maury Comptes Rendus MŐcanique 2005

[22] B. Maury Numerische Mathematik 2006 649 679

[23] B. Maury SIAM J. Numer. Anal. 2009 1126 1148

[24] J.T. Locsei, T.J. Pedley Bulletin of Mathematical Biology 2009 1089 1116

[25] J.O. Kessler, T.J. Pedley Annu. Rev. Fluid Mech. 1992 313 58

[26] F. Peruani, L. G. Morelli. Self-propelled particles with fluctuating speed and direction of motion in two dimensions. PRL 99 (2007), 010602, 2007.

[27] S. Rafai, L. Jibuti, P. Peyla Phys. Rev. Lett. 2010 098102

[28] D. Saintillan, M. J. Shelley Phys. Rev. Lett. 2007 058102

[29] J. E. Segall, S.M. Block, H.C. Berg Proc. Natl . Acad. Sci. USA 1986 8987 8991

[30] A. Sokolov, I. S. Aranson Phys. Rev. Lett. 2009 148101

[31] A. Sokolov, R. E. Goldstein, F. I. Feldchtein, I. S. Aranson Phys. Rev. E 2009 031903

[32] R. Temam, A. Miranville. Mathematical modeling in continuum mechanics. Cambridge University press, 2001.

[33] L. Turner, W.S. Ryu, H.C. Berg J. Bacteriol. 2000 2793 2801

[34] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J.O. Kessler, R. E. Goldstein Proc. Natl. Acad. Sci. USA 2005 2277 2282

[35] S. Vincent, J. P. Caltagirone, P. Lubin, T. N. Randrianarivelo Computers and Fluids 2004 1273 1289

[36] X.-L. Wu, A. Libchaber Physical Review Letters 2000 3017 3020

[37] Y. Wu, D. Kaiser, Y. Jiang, M. S. Alber Proc. Natl. Acad. Sci. USA 2009 1222 1227