Geodesic
Simulation of self-propelled chemotactic bacteria in a stokes flow
A. Decoene1 ; A. Lorz2 ; S. Martin1 ; B. Maury1 ; M. Tang3, 4
1 Université Parie-Sud 11,Département de Mathématiques, Bâtiment 425, 91405 Orsay cedex, France
2 Départment of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 OWA, UK
3 Université Pierre & Marie Curie, Laboratoire Jacques-Louis Lions, Boîte courrier 187, 75252 Paris cedex 05, France
4 INRIA Paris-Rocquencourt, BANG Project, Domaine de Voluceau, BP 105, 78153 Rocquencourt, France
ESAIM. Proceedings, Tome 30 (2010), pp. 104-123.

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We prescrit a method to simulate the motion of self-propelled rigid particles in a twodimensional Stokesian fluid, taking into account chemotactic behaviour. Self-propulsion is modelled as a point force associated to each particle, placed at a certain distance from its gravity centre. The method for solving the fluid flow and the motion of the bacteria is based on a variational formulation on the whole domain, including fluid and particles: rigid motion is enforced by penalizing the strain rate tensor on the rigid domain, while incompressibility is treated by duality. This leads to a minimisation problem over unconstrained functional spaces which cari lie easily implemented from any finite element Stokes solver. In order to ensure robustness, a projection algorithm is used to deal with contacts between particles. The particles are meant to represent bacteria of the Escherichia coli type, which interact with their chemical environment through consumption of nutrients and orientation in some favorable direction. Our mode’ takes into account the interaction with 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. In addition, self-propulsion is deactivated for those particles which cannot consume enough oxygen. Finally, the mode’ includes random changes in the orientation of the individual bacteria, with a frequency that depends on the surrounding oxygen concentration, in order to favor the direction of the concentration gradient and thus to reproduce chemotactic behaviour. Numerical simulations implemented with FreeFem++ are presented.
DOI : 10.1051/proc/2010009

A. Decoene 1 ; A. Lorz 2 ; S. Martin 1 ; B. Maury 1 ; M. Tang 3, 4

1 Université Parie-Sud 11,Département de Mathématiques, Bâtiment 425, 91405 Orsay cedex, France
2 Départment of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 OWA, UK
3 Université Pierre & Marie Curie, Laboratoire Jacques-Louis Lions, Boîte courrier 187, 75252 Paris cedex 05, France
4 INRIA Paris-Rocquencourt, BANG Project, Domaine de Voluceau, BP 105, 78153 Rocquencourt, France 
@article{EP_2010_30_a8,
     author = {A. Decoene and A. Lorz and S. Martin and B. Maury and M. Tang},
     title = {Simulation of self-propelled chemotactic bacteria in a stokes flow},
     journal = {ESAIM. Proceedings},
     pages = {104--123},
     publisher = {mathdoc},
     volume = {30},
     year = {2010},
     doi = {10.1051/proc/2010009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/2010009/}
}
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A. Decoene; A. Lorz; S. Martin; B. Maury; M. Tang. Simulation of self-propelled chemotactic bacteria in a stokes flow. ESAIM. Proceedings, Tome 30 (2010), pp. 104-123. doi : 10.1051/proc/2010009. http://geodesic.mathdoc.fr/articles/10.1051/proc/2010009/

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