Spatial effect of tumbling frequencies for motile bacteria on cell balance equations

We performed dimensional reduction of Alt’s three-dimensional cell balance equations to one-dimensional Segel’s equations for an axisymmetrical case. The tumbling frequency of motile bacteria was assumed to switch between two different phases according to the sign of the perceived chemical gradients. Chemotactic bacteria responded to positive attractant gradients by suppressing the tumbling frequency, but appeared to be insensitive to negative attractant gradients. When both temporal and spatial attractant gradients were considered, this tumbling scenario constituted a limited swimming angle range. It was only within this angle range that the bacterial tumbling frequency was regulated according to the perceived attractant gradients. This angle range, characterized by the angle h 0 , plays a major role in determining the effective chemotactic responses. The bacterial density distribution in the velocity space was derived via a first-order perturbation analysis, and then applied to studies of two bacterial population transport parameters (the random motility coefficient and the chemotactic velocity). Two approaches different in dimensionality were presented. While the one-dimensional approach asymptotically yielded the expected, but incorrect, one-dimensional random motility coefficient as h 0 approached zero, the three-dimensional approach always yielded the correct result. As to the chemotactic velocity, both approaches yielded the same expression. To validate our analyses and to explore possible limitations due to certain simplifying assumptions in the derivations, we further examined three numerical examples in which Galerkin finite element solutions of the Alt’s three-dimensional equation were compared with analytical solutions. The agreement between the perturbation solutions and numerical solutions for all data over the entire range of h 0 suggested that our perturbation analysis is valid under the condition of small constant attractant gradients. For chemical gradients of arbitrary magnitudes with temporal and spatial variations, our solutions were also found efficient and robust in comparison to numerical solutions. ( 1999 Elsevier Science Ltd. All rights reserved.