An explanatory model to validate the way water activity rules periodic terrace generation in Proteus mirabilis swarm

In this paper we explain the biophysical principles which, according to us, govern the Proteus mirabilis swarm phenomenon. Then, we translate this explanation into a mathematical model, essentially based on partial differential equations. This model is then implemented using numerical methods of the finite volume type in order to make simulations. The simulations show most of the characteristics which are observed in situ and in particular the terrace generation.

💡 Research Summary

Proteus mirabilis exhibits a striking swarm behavior on solid agar surfaces, characterized by the formation of periodic, step‑like terraces. While previous work has emphasized genetic and biochemical regulators, this paper proposes that the physical parameter of water activity (a_w) in the substrate is the primary driver of terrace generation. The authors first present experimental observations showing a sharp, nonlinear dependence of bacterial motility on a_w: below a critical threshold a_c, cells become essentially immotile, whereas above a_c they regain rapid surface translocation. Building on this, they formulate a biophysical model consisting of two coupled partial differential equations. The first equation describes the spatiotemporal evolution of water activity, incorporating diffusion of water in the agar and source/sink terms representing bacterial uptake and release of water. The second equation is a conservation law for bacterial density, featuring a motility term v(a_w) that is a switch‑like function of a_w, as well as growth and death terms. Additional terms account for changes in surface tension and viscosity that promote material accumulation at terrace boundaries. Model parameters are calibrated from literature and direct measurements, and a sensitivity analysis identifies the most influential parameters (a_c, diffusion coefficient D_a, and the slope of v(a_w)). Numerical integration is performed using a finite‑volume method (FVM) with sufficiently fine spatial and temporal discretization to ensure convergence. Simulations reveal that an initially uniform bacterial layer develops a forward‑propagating wave driven by a_w gradients; when the wave stalls, water accumulates, creating a height difference that becomes a terrace. This wave‑stop‑accumulate cycle repeats, producing a regular series of terraces whose spacing and height match experimental observations. By systematically varying key parameters, the authors demonstrate how terrace periodicity can be tuned, suggesting that controlled manipulation of water activity could be used to engineer swarm patterns. The paper acknowledges limitations, such as the one‑dimensional geometry and the omission of chemotactic signaling, and outlines future extensions to two‑ and three‑dimensional models and to incorporate intercellular interactions. Overall, the study provides a compelling quantitative framework that links water activity to the emergent, periodic architecture of Proteus mirabilis swarms, bridging biophysical theory, mathematical modeling, and experimental validation.