Vertex stability and topological transitions in vertex models of foams and epithelia

TL;DR

This paper investigates the stability of vertices in models of foams and epithelia, revealing conditions under which fourfold vertices are stable or unstable, and improving the simulation of topological transitions.

Contribution

It provides a theoretical analysis of vertex stability in vertex models, including conditions for stability and instability, and enhances the simulation framework for biological tissues.

Findings

Fourfold vertices are unstable when all edges have equal tension.

Allowing tension to depend on edge orientation can stabilize fourfold vertices.

The formulation improves the simulation of T1 topological transitions.

Abstract

In computer simulations of dry foams and of epithelial tissues, vertex models are often used to describe the shape and motion of individual cells. Although these models have been widely adopted, relatively little is known about their basic theoretical properties. For example, while fourfold vertices in real foams are always unstable, it remains unclear whether a simplified vertex model description has the same behavior. Here, we study vertex stability and the dynamics of T1 topological transitions in vertex models. We show that, when all edges have the same tension, stationary fourfold vertices in these models do indeed always break up. In contrast, when tensions are allowed to depend on edge orientation, fourfold vertices can become stable, as is observed in some biological systems. More generally, our formulation of vertex stability leads to an improved treatment of T1 transitions in simulations and paves the way for studies of more biologically realistic models that couple topological transitions to the dynamics of regulatory proteins.