A Traveling-Wave Solution for Bacterial Chemotaxis with Growth

Bacterial chemotaxis is among the most extensively characterized phenomena in microbiology. However, a prominent experimental observation, the stable propagation of migratory bands in growth media, has evaded a clear, quantitative understanding. We analyze a simplified version of the GE model introduced by Cremer and Honda et al. which identified distinct roles played by chemotaxis and cell growth. We heuristically obtain an analytical expression, verified by numerical results, for the expansion speed of the migratory band, c, in terms of the key cellular and environmental parameters. We find c to increase linearly with chemotactic motility, turning over to a square root dependence for motility exceeding the attractant diffusivity. Further, c increases as the square root of the growth rate, amplified by the ratio of the ambient attractant concentration and the attractant detection limit, peaking at intermediate concentrations determined by the carrying capacity. In contrast, c is only weakly dependent on attractant uptake, commonly considered a key determinant of population expansion by chemotaxis. Finally, we relate the GE model to the F-KPP equation which describes population expansion due to growth and cellular diffusion in the absence of chemotaxis.