Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T03:42:18.274Z Has data issue: false hasContentIssue false
  • English
  • Français

Microscopic Modelling of Active BacterialSuspensions

Published online by Cambridge University Press:  10 August 2011

A. Decoene ,
S. Martin  and
B. Maury
Get access

Abstract

We present two-dimensional simulations of chemotactic self-propelled bacteria swimming ina viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity andopposite direction applied on the rigid bacterial body and on an associated region in thefluid representing the flagellar bundle. The method for solving the fluid flow and themotion 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 bypenalizing the strain rate tensor on the rigid domain, while incompressibility is treatedby duality. This model allows to achieve an accurate description of fluid motion andhydrodynamic interactions in moderate to concentrated active suspensions. A mesoscopicmodel is also used, in which the size of the bacteria is supposed to be much smaller thanthe elements of fluid: the perturbation of the fluid due to propulsion and motion of theswimmers is neglected, and the fluid is only subjected to the buoyant forcing induced bythe presence of the bacteria, which are denser than the fluid. Although this model doesnot accurately take into account hydrodynamic interactions, it is able to reproducecomplex collective dynamics observed in concentrated bacterial suspensions, such asbioconvection. From a mathematical point of view, both models lead to a minimizationproblem which is solved with a standard Finite Element Method. In order to ensurerobustness, a projection algorithm is used to deal with contacts between particles. Wealso reproduce chemotactic behaviour driven by oxygen: an advection-diffusion equation onthe oxygen concentration is solved in the fluid domain, with a source term accounting foroxygen consumption by the bacteria. The orientations of the individual bacteria aresubjected to random changes, with a frequency that depends on the surrounding oxygenconcentration, in order to favor the direction of the concentration gradient.

Keywords

Stokes flowfluid-particle flowsself-propulsionfinite element methodpenalty methodchemotaxis.
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 5: Complex Fluids , 2011 , pp. 98 - 129
Copyright
© EDP Sciences, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H.C. Berg. Random walks in biology. Princeton University Press, Princeton, 1983.
H.C. Berg. E. Coli in Motion. Springer Verlag, New York, 2004.
Haines, B.M., Aranson, I.S., Berlyand, L. and Karpeev, D.A.. Effective viscosity of dilute bacterial suspensions: a two-dimensional model. Physical Biology, 5 (2008), No. 4. CrossRefGoogle ScholarPubMed
P. G. Ciarlet. Introduction à l’analyse numérique matricielle et à l’optimisation. Masson, Paris, 1990.
Cisneros, L.H., Cortez, R., Dombrowski, C., Goldstein, R.E., Kessler, J.O.. Fluid dynamics of self-propelled microorganisms. from individual to concentrated populations. Exp Fluids, 43 (2007), 737–753. CrossRefGoogle Scholar
NC, Darnton, L, Turner, S, Rojevsky, HC, Berg. Dynamics of bacterial swarming. Biophys J. 98 (2010), No. 10, 2082–90. Google Scholar
Decoene, A., Lorz, A., Martin, S., Maury, B., Tang, M.. Simulation of self-propelled chemotactic bacteria in a Stokes flow. ESAIM: Proc, 30 (2010), 104–123 . CrossRefGoogle Scholar
Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R. E., Kessler, J. O.. Self-concentration and large-scale coherence in bacteria dynamics. Phys. Rev. Lett., 93 (2004), No. 9. CrossRefGoogle Scholar
D. Gérard-Varet, M. Hillairet. Regularity Issues in the Problem of Fluid Structure Interaction. to appear in Arch. Rational Mech. Anal.
Glowinski, R., Pan, T. W., Hesla, T. I., Joseph, D. D. & Périaux, J.. A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comp. Phys., 169 (2001), 363–427. CrossRefGoogle Scholar
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.
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).
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.
J. Happel, H. Brenner. Low Reynolds Number Hydrodynamics. Dordrecht, Kluwer, 1991.
Hillairet, M.. Lack of collision between solid bodies in a 2D constant-density incompressible flow. Communications in Partial Differential Equations 32 (2007), 1345-1371. CrossRefGoogle Scholar
Janela, J., Lefebvre, A., Maury, B.. A penalty method for the simulation of fluid-rigid body interaction. ESAIM: Proc., 1 (2007), 115–123. Google Scholar
Kaiser, D.. Bacterial swarming, a re-examination of cell movement patterns. Curr Biol, 17 (2007), R561-R570. CrossRefGoogle ScholarPubMed
S. Kim, S.J. Karrila. Microhydrodynamics: Principles and Selected Applications. Dover, New York, 2005.
E. Lauga and T.R. Powers. The hydrodynamics of swimming microorganisms. Rep. Prog. Phys., 72 (2009).
Lefebvre, A.. Fluid-particle simulations with Freefem++. ESAIM: Proc., 18 (2007), 120–132. CrossRefGoogle Scholar
Lefebvre, A., Maury, B.. Apparent viscosity of a mixture of a Newtonian fluid and interacting particles. Fluid-solid interactions: modeling, simulation, bio-mechanical applications. Comptes Rendus MŐcanique, 333 (2005), No. 12. Google Scholar
Maury, B.. A time-stepping scheme for inelastic collisions. Numerische Mathematik, 102 (2006), No. 4, 649–679. CrossRefGoogle Scholar
Maury, B.. Numerical Analysis of a Finite Element / Volume Penalty Method. SIAM J. Numer. Anal. 47 (2009), No. 2, 1126–1148. CrossRefGoogle Scholar
Locsei, J.T., Pedley, T.J.. Run and Tumble in Chemotaxis in a Shear Flow; The Effect of Temporal Comparisons, Persistence, Rotational Diffusion, and Cell Shape. Bulletin of Mathematical Biology, 71 (2009), 1089–1116. CrossRefGoogle Scholar
Kessler, J.O., Pedley, T.J.. Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24 (1992), 313–58. Google Scholar
F. Peruani, L. G. Morelli. Self-propelled particles with fluctuating speed and direction of motion in two dimensions. PRL 99 (2007), 010602, 2007.
Rafai, S., Jibuti, L., Peyla, P.. Effective viscosity of microswimmer suspensions. Phys. Rev. Lett., 104 (2010), 098102. CrossRefGoogle ScholarPubMed
Saintillan, D., Shelley, M. J.. Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett., 99 (2007), 058102. CrossRefGoogle ScholarPubMed
Segall, J. E., Block, S.M., Berg, H.C.. Temporal comparisons in bacterial chemotaxis. Proc. Natl . Acad. Sci. USA, 83 (1986), 8987–8991. CrossRefGoogle ScholarPubMed
Sokolov, A., Aranson, I. S.. Reduction of viscosity in suspension of swimming bacteria. Phys. Rev. Lett. 103 (2009), 148101. CrossRefGoogle ScholarPubMed
Sokolov, A., Goldstein, R. E., Feldchtein, F. I., and Aranson, I. S.. Enhanced mixing and spatial instability in concentrated bacterial suspensions. Phys. Rev. E 80 (2009), 031903. CrossRefGoogle ScholarPubMed
R. Temam, A. Miranville. Mathematical modeling in continuum mechanics. Cambridge University press, 2001.
Turner, L., Ryu, W.S., Berg, H.C.. Real-time imaging of fluorescent flagellar filaments. J. Bacteriol., 182 (2000), No. 10, 2793–2801. CrossRefGoogle ScholarPubMed
Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C. W. , Kessler, J.O., Goldstein, R. E.. Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. USA, 102 (2005), 2277–2282. CrossRefGoogle ScholarPubMed
Vincent, S., Caltagirone, J. P., Lubin, P. & Randrianarivelo, T. N.. An adaptative augmented Lagrangian method for three-dimensional multimaterial flows. Computers and Fluids, 33 (2004), 1273–1289. CrossRefGoogle Scholar
Wu, X.-L., Libchaber, A.. Particle diffusion in a quasi-two-dimensional bacterial bath. Physical Review Letters, 84 (2000), 3017–3020. CrossRefGoogle Scholar
Wu, Y., Kaiser, D., Jiang, Y., Alber, M. S.. Periodic reversal of direction allows Myxobacteria to swarm. Proc. Natl. Acad. Sci. USA, 106 (2009), No. 4, 1222–1227. CrossRefGoogle ScholarPubMed