Force networks, torque balance and Airy stress in the planar vertex model of a confluent epithelium

TL;DR

This paper develops a detailed force network and stress analysis for the planar vertex model of confluent epithelia, highlighting the importance of torque balance and geometric constraints for accurate modeling.

Contribution

It introduces a novel discrete Airy stress function for the vertex model and emphasizes the role of torque balance and geometric constraints in cellular stress analysis.

Findings

Torque balance requires internal vertices to be at orthocentres.

Stress depends on cell shape within the energy functional.

Edges are orthogonal to links in the dual network.

Abstract

The vertex model is a popular framework for modelling tightly packed biological cells, such as confluent epithelia. Cells are described by convex polygons tiling the plane and their equilibrium is found by minimizing a global mechanical energy, with vertex locations treated as degrees of freedom. Drawing on analogies with granular materials, we describe the force network for a localized monolayer and derive the corresponding discrete Airy stress function, expressed for each $N$-sided cell as $N$ scalars defined over kites covering the cell. We show how a torque balance (commonly overlooked in implementations of the vertex model) requires each internal vertex to lie at the orthocentre of the triangle formed by neighbouring edge centroids. Torque balance also places a geometric constraint on the stress in the neighbourhood of cellular trijunctions, and requires cell edges to be orthogonal to the links of a dual network that connect neighbouring cell centres and thereby triangulate the monolayer. We show how the Airy stress function depends on cell shape when a standard energy functional is adopted, and discuss implications for computational implementations of the model.