Figure 1.

Properties of 2D and 3D vertex models for tissue mechanics. (a) Apical view (from [1]) and cross-section (from [2]) of the wing imaginal disc epithelium in the Drosophila embryo. (b) Schematic of an epithelial tissue. Cytoskeletal elements generate forces inside the cells, which are mechanically coupled to other cells and to the basement membrane. (c,d) In apical vertex models epithelial cells are represented by the shape of their apical surfaces, which are polygons either in 2D or in 3D. (e) In 2D lateral vertex models, cells are represented by their lateral cross-sections. (f) In 3D vertex models the tissue is represented by its apical and basal geometry. (g) A virtual work differential for vertex displacement, depending on changes in cell volume δV^α, surface area δA^k and edge length δl^λ. External forces acting on vertices can yield an additional contribution to the virtual work. (h) The force f^v on a vertex v is obtained by taking the virtual work differential with respect to the vertex position x^v. The tissue is in mechanical equilibrium when the force acting on all vertices is zero. (i) Topological transitions in epithelia are cell–cell intercalations (T₁ transitions), cell extrusions (T₂ transitions) and cell divisions.