Perturbation Expansion of Alt's Cell Balance Equations Reduces to Segel's One-Dimensional Equations for Shallow Chemoattractant Gradients

The cell balance equations of Alt are rigorously studied and perturbatively expanded into forms similar to Segel's one-dimensional phenomenological cell balance equations by considering the simplifying case of bacterial density possessing symmetry in the x and y directions responding to an attractant gradient present only in the z direction. We prove that for shallow attractant gradients the lumped integrals involving the tumbling probability frequency distribution and bacterial density distribution in the $\theta$ direction can be explicitly expressed as a product of three quantities: the mean tumbling frequency, the bacterial subpopulation density, and a reversal probability. We also derive expressions for the bacterial net flux in the Fickian form from which two macroscopic transport parameters, the random motility coefficient and the chemotactic velocity, are explicitly related to individual cell properties and chemical gradients.