Mathematical modeling of evolution of cell networks in epithelial tissues.

Epithelial cell networks imply a packing geometry characterized by various cell shapes and distributions in terms of number of cell neighbors and areas. Despite such simple characteristics describing cell sheets, the formation of bubble-like cells during the morphogenesis of epithelial tissues remains poorly understood. This study proposes a topological mathematical model of morphogenesis in a squamous epithelial. We introduce a new potential that takes into account not only the elasticity of cell perimeter and area but also the elasticity of their internal angles. Additionally, we incorporate an integral equation for chemical signaling, allowing us to consider chemo-mechanical cell interactions. In addition to the listed factors, the model takes into account essential processes in real epithelial, such as cell proliferation and intercalation. The presented mathematical model has yielded novel insights into the packing of epithelial sheets. It has been found that there are two main states: one consists of cells of the same size, and the other consists of "bubble" cells. An example is provided of the possibility of accounting for chemo-mechanical interactions in a multicellular environment. The introduction of a parameter determining the flexibility of cell shapes enables the modeling of more complex cell behaviors, such as considering change of cell phenotype. The developed mathematical model of morphogenesis of squamous epithelium allows progress in understanding the processes of formation of cell networks. The results obtained from mathematical modeling are of significant importance for understanding the mechanisms of morphogenesis and development of epithelial tissues. Additionally, the obtained results can be applied in developing methods to influence morphogenetic processes in medical applications.