Solution of a Non-Linear Fokker-Planck Equation Governing Bacterial Chemotactic Band Formation and Propagation by Stochastic Computer Simulation Method
Chemotaxis refers to a manifestation by certain strains of bacteria with the ability to sense the presence of chemical gradients of nutrients in liquid media. The chemotactic response of E. Coli to oxygen has been extensively studied by Adler and his co-workers in recent years.1 The simplest demonstration of the phenomenon is to inject suitable numbers of the bacteria into the bottom of a tube containing oxygen saturated motility buffer solution. A sharp band of the bacteria is soon formed and the band migrates slowly upward because the bacteria collectively seek a certain optimum oxygen concentration within a traveling oxygen gradient created by their own metabolism. Mathematical theory of this striking phenomenon of band formation and propagation was formulated by Keller and Segel2 and the consequence of the theory was quantitatively tested experimentally by a light scattering experiment of Holz and Chen.3 The Keller-Segel model consists of a pair of coupled non-linear Fokker-Planck equations involving the bacterial density b(z, t) and the substrate concentration c(z, t). Solution of these coupled equations generally requires a numerical procedure.
KeywordsStochastic Differential Equation Band Formation Bacterial Density Chemotactic Response Chemical Gradient
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