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

TL;DR

This paper demonstrates that centroidal Voronoi tessellations naturally emerge in epithelial tissues due to mechanical energy minimization, linking tissue mechanics to geometric organization and enabling stress inference from morphology.

Contribution

It provides a mechanical explanation for CVT-like patterns in epithelia using a vertex model and introduces a framework to infer tissue stresses from cell geometry.

Findings

CVT-like patterns arise in the solid regime of tissues.

Isotropic strain minimization drives cells toward Voronoi configurations.

Perturbations like cyclic stretch induce disordered CVT states, revealing mechanical properties.

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.