Vertex Model Mechanics Explain the Emergence of Centroidal Voronoi Tiling in Epithelia

Abstract

Epithelia are confluent cell layers that self-organize into polygonal networks whose geometry encodes their mechanical state. A principal driver is the tunable contractility of the actomyosin cortex, which links cell-junction tension to tissue architecture. Notably, epithelial tilings frequently resemble centroidal Voronoi tessellations (CVTs), yet the physical origin of this resemblance has remained unclear. Here, using a minimal vertex model that relates cell shape to a mechanical energy, we show that CVT-like patterns arise naturally in the solid (rigid) regime of tissues. Analytical theory reveals that isotropic strain minimization drives cell centroids toward Voronoi configurations, a result we corroborate with a analytical mean-field formulation of the vertex model. We further demonstrate that physiologically relevant perturbations -- such as cyclic stretch -- shift tissues into distinct, geometrically disordered CVT states, and that these shifts provide quantitative, image-based readouts of mechanical state. Together, our results identify a mechanical origin for CVT-like organization in epithelia and establish a geometric framework that infers tissue stresses directly from morphology, offering broadly applicable metrics for assessing rigidity and remodeling in living tissues.