Vertex model instabilities for tissues subject to cellular activity or applied stresses

TL;DR

This paper demonstrates that vertex models of epithelial tissues become unstable under cellular activity or external stresses, leading to cell deformation and non-convexity, which limits the model's validity over time.

Contribution

It provides an analytical framework showing how activity and stress induce instabilities in vertex models, highlighting conditions that cause tissue deformation and invalidation of the model.

Findings

Instabilities couple shear, rotation, and expansion modes.

Cells become non-convex at longer times under stress.

Analytic predictions match simulations of disordered tissues.

Abstract

The vertex model is widely used to describe the dynamics of epithelial tissues, because of its simplicity and versatility and the direct inclusion of biophysical parameters. Here, it is shown that quite generally, when cells modify their equilibrium perimeter due to their activity, or the tissue is subject to external stresses, the tissue becomes unstable with deformations that couple pure-shear or deviatoric modes, with rotation and expansion modes. For short times, these instabilities deform cells increasing their ellipticity while, at longer times, cells become non-convex, indicating that the vertex model ceases to be a valid description for tissues under these conditions. The agreement between the analytic calculations performed for a regular hexagonal tissue and the simulations of disordered tissues is excellent due to the homogenization of the tissue at long wavelengths.