Mathematical models for motile bacterial transport in cylindrical tubes

Mathematical models considering motile bacterial transport within a geometrically restrictive cylindrical tube were developed. Two macroscopic transport parameters, the random motility coefficient as a self-diffusion coefficient of the cell population and the chemotactic velocity as a chemical-induced velocity, were derived. The three-dimensional cell balance equation was reduced to forms similar to Segel's one-dimensional phenomenological cell balance equations with additional modifications for bacteria-wall interactions. Two conceptually different approaches accounting for such interactions were presented. The first approach parallels treatments in the gas kinetic theory by viewing bacterial interactions with walls as collisions and subsequent diffusive/specular reflections, which led to the Bosanquet formula for the bacterial diffusion coefficient. Based on the experimental observation that bacterial swimming motion is guided by a straight tube, the second approach considered modifications in the bacterial swimming orientation as a consequence of various long-range interactions with the tube surface. A phenomenological turning model capable of aligning bacterial motion along a tube axis was proposed. The model predicts that under the geometrical restriction of a small cylindrical tube, the macroscopic bacterial transport resulting from the proposed turning model can exhibit behavior that ranges from dimensionally reduced diffusion to pure wave propagation, depending on the influence of the tube diameter on the reversal probability in the bacterial swimming motion. Our theoretical model provides explicit equations that explain how such a transition can occur. The predicted results were then qualitatively compared with experimental data from the literature. As a preliminary comparison, we concluded that bacterial transport in cylindrical tubes of diameter 10 micrometers remains in the mode of a dimensionally reduced diffusion, and shifts to a wave motion when the tube diameter decreases to 6 micrometers.