Stochastic Voronoi Tessellations as Models for Cellular Neighborhoods in Simple Multicellular Organisms

Recent work on distinct multicellular organisms has revealed a hitherto unknown type of biological noise; rather than a regular arrangement, cellular neighborhood volumes, obtained by Voronoi tessellations of the cell locations, are broadly distributed and consistent with gamma distributions. We propose an explanation for those observations in the case of the alga $Volvox$, whose somatic cells are embedded in an extracellular matrix (ECM) they export. Both a solvable one-dimensional model of ECM growth derived from bursty transcriptional activity and a two-dimensional "Voronoi liquid" model are shown to provide one-parameter families that smoothly interpolate between the empirically-observed near-maximum-entropy gamma distributions and the crystalline limit of Gaussian distributions governed by the central limit theorem. These results highlight a universal consequence of intrinsic biological noise on the architecture of certain tissues.