Statistics of correlated percolation in a bacterial community

Signal propagation over long distances is a ubiquitous feature of multicellular communities, but cell-to-cell variability can cause propagation to be highly heterogeneous. Simple models of signal propagation in heterogenous media, such as percolation theory, can potentially provide a quantitative understanding of these processes, but it is unclear whether these simple models properly capture the complexities of multicellular systems. We recently discovered that in biofilms of the bacterium Bacillus subtilis, the propagation of an electrical signal is statistically consistent with percolation theory, and yet it is reasonable to suspect that key features of this system go beyond the simple assumptions of basic percolation theory. Indeed, we find here that the probability for a cell to signal is not independent from other cells as assumed in percolation theory, but instead is correlated with its nearby neighbors. We develop a mechanistic model, in which correlated signaling emerges from cell division, phenotypic inheritance, and cell displacement, that reproduces the experimentally observed correlations. We find that the correlations do not significantly affect the spatial statistics, which we rationalize using a renormalization argument. Moreover, the fraction of signaling cells is not constant in space, as assumed in percolation theory, but instead varies within and across biofilms. We find that this feature lowers the fraction of signaling cells at which one observes the characteristic power-law statistics of cluster sizes, consistent with our experimental results. We validate the model using a mutant biofilm whose signaling probability decays along the propagation direction. Our results reveal key statistical features of a correlated signaling process in a multicellular community. More broadly, our results identify extensions to percolation theory that do or do not alter its predictions and may be more appropriate for biological systems.Author summary: Many multicellular systems send signals over long distances by relaying information over connected cell-to-cell paths. In physics, the statistics of connected path formation are described by percolation theory. We previously discovered that the statistics of electrical signal propagation in communities of the bacterium Bacillus subtilis are consistent with the predictions of percolation theory. However, we find experimentally that key features of this system go beyond the simple assumptions of basic percolation theory, which include site-to-site independence and spatial uniformity of the signaling probability. Why are the predictions of percolation theory still upheld? Using a computational model, we find that the cell-to-cell dependence does not change the predictions due to the universal nature of percolation theory near its critical point, and the spatial variability of the signaling probability actually expands the parameter range over which the predictions hold. We validate our findings using a mutant bacterial strain. Our work explores the robustness of percolation theory to its underlying assumptions, and the resulting consequences for long-range bacterial signaling.