Continuum Modeling of Chemotactic Migration in Heterogeneous Environments

The chemotactic migration of bacteria—their ability to direct collective motion along chemical gradients—is central to processes ranging from sustaining plant growth, remediating contaminants, sensing and reporting stimuli, and mitigating or causing infection. However, while migration is well-studied in bulk liquid, most bacterial habitats are heterogeneous media, such as soils, sediments, and biological tissues/gels, that are characterized by strong confinement and tortuosity. Unfortunately, how confinement in a heterogeneous space alters the ability of bacteria to migrate remains poorly understood. Here, we extend the classic Keller-Segel model to describe chemotactic migration in heterogeneous media at the continuum scale. Specifically, we incorporate two key effects that are promoted by confinement: (i) suppression of cellular motility due to collisions with the surrounding medium, and (ii) suppression of cellular motility due to collisions with other cells. Our model predicts the formation of localized fronts of cells that collectively propagate over large length and time scales, driven by their self-generated nutrient gradient—remarkably similar to those observed in experiments. Further, the model sheds light on the factors that control front formation, propagation speed, and morphology. For example, we find that these features depend critically on the cell density: for low cell densities, cells initially spread diffusively until sufficient nutrient is depleted to drive front formation, while for high cell densities, a strong enough nutrient gradient forms rapidly, but front formation is hindered by collisions between cells. The propagation dynamics also depend on the relative influence of cellular motility and growth/division: in weak confinement, fronts propagate primarily due to the biased motion of the cells, while with increasing confinement, growth and division increasingly contributes to front propagation. Together, our work provides a framework to predict and control the migration of bacteria, and active matter in general, in heterogeneous environments.