Geometry and mechanics of a model epithelium with irregular cells and a clonal inclusion

An important role in the modeling of epithelial tissue mechanics has been played by vertex models, with cells idealized as polygons, and tricellular junctions as vertices joined by straight interfaces. Numerical simulations of these models in the presence of cell divisions display geometrically irregular cells, similar to those of epithelia, even when cell properties are homogeneous. Yet existing theoretical analyses are mostly confined to the mechanics of regular hexagonal lattices. We develop an analytical description of geometrically disordered vertex models. We first quantify, in numerical simulations, geometrical properties such as the distributions of cell areas and perimeters, and mechanical properties such as the tissue bulk and shear moduli, with different sources of disorder, e.g different division rules or simply relaxation in the presence of noise. We then develop a simple mean-field description that accounts for these properties. Finally, we apply our analytical description to a simple case of interest in different biological contexts: a clonal group of cells with material properties that differ from the surrounding tissue and that may also grow at a different rate. We compare our results with data obtained on epidermis differentiation in flies.