Introduction to the vertex model for the mechanics of epithelial layers

Ulrich Schwarz (Heidelberg Universität) #

Many compartments in the human body are surrounded by layers of epithelial cells, which have very strong cell-cell adhesion, therefore forming protective barriers. In a two-dimensional projection, these epithelia are often described by the vertex model, in which each cell is a polygon in a graph. Each edge is a cell-cell junction and three or more cells meet at the vertices. Each configuration of the graph is mapped to an energy function, similar to the situation in a two-dimensional foam. The edges are considered to be under tension due to cortical contractility and shrinkage is prevented by an area elasticitiy. In addition, a perimeter term represents contraction through a circumferential ring. We will discuss the main mathematical results for this model, which has been widely investigated in both physics and mathematics, similar to the Ising model for magnetic systems. We also will discuss its extension into three dimensions, including the bubbly vertex model, which we recently have introduced to describe cell shape in intestinal organoids.