We describe the derivation of cellular phenomenological equations which provide both microscopic and macroscopic descriptions of the isotropic and anisotropic transport processes for motile bacteria. The macroscopic-level transport properties are the random motility coefficient as a self-diffusion coefficient of the bacterial population and the chemotactic velocity as a chemically-induced velocity. Using a perturbation analysis, we performed dimensional reduction on Alt's three-dimensional cell balance equation, leading to forms similar to Segel's one-dimensional phenomenological cell population equations. Two functional relationships between bacterial biological tumbling responses and chemical gradients based on the experimental findings of Berg and Brown were incorporated into the cell balance equations for comparison. One tumbling scenario constituted a limited swimming angle range; it was only within this limited angle range that the bacterial tumbling frequency was altered in response to attractant gradients. Extension of the model for a restricted capillary geometry was also investigated. When impenetrable solid boundaries were imposed, additional modifications for cell-wall interactions were included in the transport equation. A phenomenological turning model capable of aligning bacterial motion along the tube axis was proposed, and the resultant bacterial orientation function was studied. The model predictions were compared qualitatively with experimental data available from the literature.

Joint work with Kevin C. Chen and Peter T. Cummings

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1998-1999 Mathematics in Biology