Mathematical modeling of positive connection functioning in the tumor markers p53–microRNA system

The hierarchy of minimal mathematical models of the dynamics of p53–Mdm2–microRNA system has been developed. The models are based on the differential equations with time delay, hiding complex mechanisms of interaction in the signal system of the p53 protein. We consider the two types of interaction of p53 with microRNAs: the positive direct connection and the positive feedback. The feedback of microRNA-p53 is due to the negative effect of microRNA on the protein Mdm2, which itself is a negative regulator of p53. To approximate the direct positive effect of p53 on the microRNA, a linear function or the Goldbeter–Koshland type representation is used.
The comparison of numerical solutions with medical and biological data of a number of specific p53- dependent microRNAs is made, which proves the adequacy of the models proposed and the results of numerical analysis. Special attention was given to the analysis of the positive feedback of p53 and microRNAs. The minimal models adopted have allowed us to consider the most general regularities of the p53-dependent microRNAs functioning. In the framework of these mathematical models it is shown that it is possible to neglect the connection Mdm2–miRNA for, at least, some of the most studied microRNAs associated with a direct positive connection with p53. However, those of the microRNAs, which are important negative regulator Mdm2, can have the most significant impact on the functioning of the entire system p53–Mdm2–microRNA. Conditions were obtained under which the regulatory function of microRNAs with respect to p53 is manifested. The results of numerical experiments indicate that such microRNAs can be considered to be a factor of the anticancer therapy.