Force networks, torque balance and Airy stress in the planar vertex model of a confluent epithelium
Force networks, torque balance and Airy stress in the planar vertex model of a confluent epithelium
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DOI：
10.1098/rspa.2019.0716
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发表时间：
2020-05-27
影响因子：
3.5
通讯作者：
Woolner, Sarah
中科院分区：
文献类型：
--
作者：
Jensen, Oliver E.;Johns, Emma;Woolner, Sarah
顶点模型是模拟紧密堆积的生物细胞(如融合上皮)的流行框架。用平铺平面的凸多边形来描述单元，通过最小化全局机械能来寻找单元的平衡，并将顶点位置视为自由度。借助于颗粒材料的类比，我们描述了局部单层的力网络，并导出了相应的离散艾里应力函数，对于每个N边单元，表示为覆盖该单元的风筝上定义的N标量。我们展示了扭矩平衡(在顶点模型的实现中通常被忽略)如何要求每个内部顶点位于由相邻边质心形成的三角形的垂直中心。扭矩平衡还对细胞三角连接附近的应力施加几何约束，并要求细胞边缘与连接相邻细胞中心的双重网络的链接垂直，从而对单层进行三角剖分。我们展示了当采用标准能量泛函时，艾里应力函数如何依赖于晶胞形状，并讨论了对该模型的计算实现的影响。
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.