Stability of microbial droplets growing on liquid substrates

Microbial growth is sensitive to substrate mechanics. Recent experiments have shown that yeast growing on a liquid substrate can develop finger-like instabilities depending on the substrate viscosity. We develop a model of a flat microbial droplet growing on the surface of a 3D fluid through uptake of a diffusible nutrient. The model takes into account growth-induced forces and buoyancy-driven flows due to nutrient depletion, and can be reposed as an integro-differential equation on the droplet that relates the droplet's planar pressure to the nutrient concentration. We calculate axisymmetric solutions corresponding to a radially expanding disk, and analyze the droplet's geometric stability. We find that the circular droplet is stable for sufficiently large fluid viscosity, but becomes unstable as viscosity is reduced below a critical value. We connect these findings to experimentally observed morphologies.