Abstract
The vertex model considers only the projection of the cell onto the plane of the epithelial tissue (the apical surface). It completely ignores the volume of the cell, which forms a polyhedron beneath the apical surface. This model can be extended to two dimensions by describing each cell as a polyhedron, and assuming that the cells fill the entire space beneath the apical surface. To get the structure of the tissue, we need to write an energy for it. In the spirit of thetwo-dimensionalmodel , we consider that each interface of a cell with its neighbors (the basal, lateral and apical faces) has a surface tension.
A very simple version of the model assumes that all cells are identical and form cylinders with a hexagonal base. Depending on the values of the surface tensions, the model predicts either columnar cells highly stretched perpendicular to the apical surface, or highly flattened squamous cells. These two types of epithelium are well observed and can even coexist in the same tissue. The model can also be used to study curved epithelia (cysts).
To go further, we need to assume that not all cells are identical. But this can only be done digitally. Guillaume Salbreux's team in Dresden has studied the formation of cysts from an epithelium and compared their numerical results with experimental ones.
