Theoretical Modeling of Sporulation Patterns in Bacillus subtilis Biofilms

Many species of bacteria, including Bacillus subtilis, undergo a transition to a spore state. Traditionally understood as a response to stress or starvation, sporulation is suppressed in lab strains of B. subtilis under healthy conditions. However, we find that sporulation emerges naturally in wild isolates of B. subtilis under the same conditions. The two strains also form structurally different biofilms. Specifically, the lab strain forms biofilms with mostly matrix-producing cells, whereas the wild strain has matrix-producing cells at the edge of the biofilm and spores in the center. We hypothesize that sporulation in wild isolates occurs via a temporal program, and we develop a model to test whether this hypothesis explains the biofilm structure. Our model has three terms – a cell growth term, a mechanical expansion term, and a temporal sporulation term – and admits traveling wave solutions, similar to the traditional Fisher-KPP equation. The model robustly reproduces the experimentally observed pattern with cells at the edge and spores in the center, regardless of biologically motivated modifications to the three terms. It also makes predictions for how the mechanistic parameters control the biofilm expansion speed, the spore fraction, and the width of the edge-cell regime, all of which are experimentally measurable. We will discuss comparisons of these predictions to experiments that modulate sporulation rate, mechanical properties, and nutrient availability. Our model is generic and can apply to other problems in which temporal state changes give rise to spatial patterns in expanding media.