Quantifying mechanics in non-confluent tissues using an extended vertex model

Vertex models for tissues have correctly predicted cell shapes and fluid-solid transitions in confluent epithelial monolayers where there are no gaps between cells and negligible curvature along cell-cell boundaries. However, in many situations of interest, such as in the development of the zebrafish tailbud, epithelial sheets transition from non-confluent layers with significant gaps between cells to confluent configurations with no gaps. Therefore, we develop simple extensions of vertex models that are able continuously transition between these two states, and demonstrate that there is a reasonable parameter regime in which the minimum energy state of this extended vertex model has gaps between cells. We study the mechanical behavior of these non-confluent models to understand how such gaps alter tissue mechanics and collective cell motility.