Mathematical modeling of early processes of morphogenesis of epithelial tissues

Studying morphogenetic processes through traditional methods, such as observation and experimentation, can be highly complex and laborious. Utilizing mathematical modeling methods makes it possible to analyze big data and uncover patterns that may be unnoticed when employing traditional approaches. In this study, a vertex mathematical model of squamous epithelial tissue morphogenesis is proposed. The elementary unit of the system is the cell, which can dynamically change its shape and size. We introduce a new potential that accounts not only for the elasticity of cell perimeters and areas but also for the elasticity of their internal angles. Additionally, we introduce an integral equation of chemical signaling, which allows for the consideration of chemo-mechanical cell-cell interactions. In addition to the above, the model incorporates important processes of real epithelia, such as cell proliferation and intercalation. Based on the simulation results, a diagram of the system's principal states was constructed as a function of control parameters. A parameter range is identified within which the cellular system adopts the most energetically favorable and stable configurations. Furthermore, two processes occur in the early stages of morphogenesis, namely morula formation and blastula formation. This paper provides a detailed physical and mathematical description of these processes. The obtained results can be utilized in developing methods for influencing morphogenetic processes in medical applications.