Data for Fig 1 from Force networks, torque balance and Airy stress in the planar vertex model of a confluent epithelium. 30 October 2019 2 April 2020

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.

顶点模型（vertex model）是模拟紧密堆积生物细胞（如致密上皮细胞层）的主流框架。细胞以实现平面镶嵌的凸多边形进行表征，其平衡态通过最小化全局机械能求解，顶点位置被视作自由度。借鉴颗粒材料的类比思路，我们针对局部单层结构刻画了力网络，并推导出对应的离散艾里应力函数（Airy stress function）；该函数针对每个N边形细胞，以覆盖细胞的风筝形单元定义的N个标量进行表达。我们阐明了力矩平衡——该条件在顶点模型的现有实现中常被忽视——如何要求每个内部顶点位于相邻边形心构成的三角形的垂心处。力矩平衡同时对细胞三联接点邻域内的应力施加了几何约束，并要求细胞边与连接相邻细胞中心、从而将单层结构三角剖分的对偶网络链路相互正交。我们展示了当采用标准能量泛函时，艾里应力函数如何随细胞形态变化，并讨论了该模型在计算实现层面的相关应用启示。