  Signal propagation over long distances is a ubiquitous feature of
multicellular communities. In biofilms of the bacterium Bacillus subtilis, we
recently discovered that some, but not all, cells participate in the
propagation of an electrical signal, and the ones that do are organized in a
way that is statistically consistent with percolation theory. However, two key
assumptions of percolation theory are violated in this system. First, we find
here that the probability for a cell to signal is not independent from other
cells 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 experimental
results. Second, we observed previously that the fraction of signaling cells is
not constant but instead varies from biofilm to biofilm. We use our model to
understand why percolation theory remains a valid description of the system
despite these two violations of its assumptions. We find that the first
violation does not significantly affect the spatial statistics, which we
rationalize using a renormalization argument. We find that the second violation
widens the range of signaling fractions around the percolation threshold at
which one observes the characteristic power-law statistics of cluster sizes,
consistent with our previous experimental results. We validate our model using
a mutant biofilm whose signaling probability decays along the propagation
direction. Our results identify key statistical features of a correlated
percolating system and demonstrate their functional utility for a multicellular
community.
