The Effect of Processes Leading to Cell Death on the Dynamic Regimes of the Tissue Formation Model

The article generalizes a mathematical model describing the population dynamics of cell clusters, based on a system of first-order nonlinear differential equations by introducing additional parameters. It is proposed to add two types of coefficients to the model: a survival coefficient, which describes the proportion of cells that die due to some environmental influences and subsequent autophagy or as a result of intracellular processes leading to death (necrosis, apoptosis), and the aging coefficient, which describes the limitation in the rate and possibility of cell division due to reduction in the length of telomers of the cells chromosomes during mitosis. The added coefficients have a biological meaning and can be evaluated during the experiment, as required by the considered model of morphogenesis. Taking these parameters into account in the mathematical model made it possible to determine new dynamic regimes in the system of differential equations describing the behavior of multicellular clusters. For the obtained system of differential equations, an analysis of its equilibrium points and the stability of steady states corresponding to these points was carried out. The values of the parameters of the mathematical model at which the system can reach a steady state were determined. Also, limitations on the parameters were identified, at which it is impossible to estimate the stability of steady states using the Lyapunov method. It follows from the obtained conditions that one of the cases when the system of cell clusters comes to a steady state is either the extinction of all cells, or the cessation of division of cell clusters. Steady states of different type are also possible, for which the criteria of existence and possibility have yet to be determined.