Abstract
Epithelial layers are the layers of cells on the surface of organs. These structures may have one or more layers. The vertex model describes epithelial monolayers by considering that each cell is atwo-dimensionalpolygon and that polygons pave the plane corresponding to the apical surface. These models have already been introduced in metallurgy and to describetwo-dimensionalsoap films . The fabric structure is obtained by minimizing an energy. This energy contains a term associated with the surface area of each cell, which tends to bring the cell towards a preferential area, and a term associated with the line tension of each edge of thetwo-dimensionalpolygon . If this tension depends on the perimeter of the cells, this term also tends to bring the cells towards a preferential perimeter. The minimization problem is a frustrated one, in that the minimum perimeter is not always compatible with the minimum area. By resizing the parameters, there are only two independent parameters in the model, and we can study the structure that is the fundamental state of energy. This structure can either be made up of hexagons, or it can be a structure that is the absolute minimum of energy ; but in this case, there are many equivalent states.
Tissues evolve and their structure is not that of an absolute minimum of energy, but rather at a local minimum. Tissue evolution is due to topological transformations of the polygon network corresponding to cell disappearances (cell death), the appearance of new cells (cell division) and a process called T1 for soap bubbles, which corresponds to neighbor exchanges.
The vertex model can be implemented numerically by evolving the network with topological processes, and the results can be compared with Drosophila developmental epithelial tissues. Several examples were given in the lecture.
