Kinetics of an Expanding Bacterial Colony: Continuum Modeling and Analysis

Abstract

We study the spatiotemporal dynamics of the expansion of a bacterial colony on a hard substrate. Nutrient from the substrate diffuses into the colony and is taken up by the bacterial cells for them to grow and divide, expanding the initial monolayer and then pancake-shaped colony. The concentration of nutrient determines the local cell growth rate in the colony. Mass conservation relates such local growth rate with velocity which is approximated to be proportional to the pressure gradient by Darcy's law. Altogether, the growing colony is modeled as a moving-boundary problem with the nutrient concentration and pressure solving a reaction-diffusion equation and Laplace's equation, respectively. We analyze the self-consistent moving-boundary model with respect to different geometrical setting. For a general three-dimensional cylindrically symmetric model, we study the steady-state nutrient concentration. We construct and analyze a one-dimensional model for vertical expansion and a two-dimensional disk model for radial expansion of the colony. Our analysis finds that the nutrient depletes into the colony and slows down the vertical expansion of the colony. The vertical level where the nutrient concentration reaches the Monod constant, a threshold below which individual bacteria hardly grow, lowers down exponentially fast. We also estimate the asymptotic radial expansion rate. Moreover, we establish that the region where the nutrient concentration is above the threshold, allowing bacteria to grow and the colony to expand radially, is a ring-shaped peripheral region of fixed thickness. All these are consistent with experiment and agent-based simulations reported in literature.