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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.
Figure 1. (a) An epithelium (animal cap) dissected from a Xenopus laevis embryo and adhered to a fibronectin-coated PDMS membrane, imaged by confocal microscopy; cell edges are identified with GFP-alpha-tubulin (green); cell nuclei with cherry-histone 2B (red). Some cell shapes are mapped out in magenta. (b) The segmented image, with each cell represented as a polygon bounded by vertices at its trijunctions. (c) Distributions of edge lengths (between trijunctions) and link lengths (between cell centroids). (d) The distribution of angles at intersections between links and edges (illustrated by the inset in (b)), peaking at π/2. (Online version in colour.)
Figure 2. (a) An illustration of a localized monolayer. Blue lines show cell edges, meeting at vertices. This example has Nc = 7 cells (six border, one interior), Ne = 30 edges (18 peripheral, six border, six interior), Nv = 24 vertices (18 peripheral, six interior). Orientations of edges and faces are not indicated. Green dots are centroids cj of each edge and red dots illustrate centres Ri of each cell. The solid orange lines connecting edge centroids form triangles around each internal vertex and polygons around each cell. Each cell is constructed from kites: three kites (shaded) surrounding an internal vertex together define a tristar. A force fik due to cell i on vertex k is associated with each kite. (b) Solid purple arrows show rotated forces −εfik. The force balances on vertices and cells imply that the rotated force vectors form a network that has the topology of the network containing edge centroids. The centroids cj, therefore, map to vertices of the force network hj (circular symbols). An imposed uniform pressure is represented by the peripheral forces, represented in part by supplementary links (dashed) that close triangles. (c) Kite ik, spanned by the vector qik from the centre of cell i to vertex k and the vector sik connecting the centroids of the edges adjacent to vertex k. The vectors c1ik,…,c4ik bounding the kite are also indicated. (Online version in colour.)
Figure 3. Three cells, labelled i, i′ and i″, sharing vertex k and edges j, j′ and j″ (taken anticlockwise). In cell i, spoke qik connects cell centre Ri to vertex rk, intersecting the link sik between neighbouring edge centroids cj and cj′. Kite ik is spanned by qik and sik; the three kites neighbouring vertex k, that together form a tristar, are shaded. The Airy stress function ψik (see below) is defined on kites. Jumps in ψik between neighbouring kites in the same cell, sharing vertex cj, are defined by the projection of the force potential hj on the cell edge tj. Jumps in ψik between neighbouring kites in the same tristar, sharing vertex cj, are defined by the projection of hj on the link Tj between cell centres (not shown) that intersects edge tj. (Online version in colour.)
Figure 4. (a) Geometric constructions demanded by torque balance. Red triangles connect adjacent edge centroids. Each cell vertex is at its orthocentre, i.e. the (black) cell edge passing through a vertex of a red triangle is orthogonal to the opposite side of the red triangle. The links between adjacent cell centres (blue) are orthogonal to the cell edges they intersect. Thus each red triangle surrounding a vertex is similar to the blue triangle surrounding the same vertex (opposite edges are parallel), differing by a rotation of π and uniform scaling. Dashed lines are orthogonal to cell edges and dotted lines are orthogonal to triangle edges. The angles marked with arcs are equal. hik indicates the altitude of the edge-centroid triangle at vertex k. (b) An illustration of three cells (black) satisfying orthocentric constraints and a corresponding network of cell centres (blue), constructed using an algorithm described in appendix F. (Online version in colour.)
Figure 5. Four different cases having consistent orientations of the link tj (blue) between cell vertices k and k′, and the link Tj (red) between cell centres i and i′. The corresponding orientations of cells i and i′, and triangles k and k′ are also shown. The tabular inset shows corresponding values of incidence matrices. Cell and triangle orientations are given in terms of the Levi-Civita symbol ε, representing clockwise π/2 rotation. (Online version in colour.)
Figure 6. (a) For a vector g to be written as a discrete curl along the path given by c1, c2, c3, c4, the jumps in scalar potential P and and Q are given by the coefficients of b and −a in (C 3). (b) Construction of a triangulation dual to a cell having prescribed vertex locations, defined up to the cell centre location R0 and the length of one link. (c) Around vertex k, the ratios of adjacent sides of the orthogonal triangulation are given by ratios of cosines of the six interior angles of the triangle connecting edge centroids. (Online version in colour.)
