A New Approach to Gene Network Modeling

摘要

This paper discusses mathematical modeling of artificial gene networks. We consider a phenomenological model of the simplest gene network, called a repressilator. This network contains three elements unidirectionally coupled into a ring. More specifically, the first element inhibits the synthesis of the second, the second inhibits the synthesis of the third, and the third, which closes the cycle, inhibits the synthesis of the first one. The interaction of protein concentrations and mRNA (messenger RNA) concentration is surprisingly similar to the interaction of six ecological populations, three predators and three preys. This makes it possible to propose a new phenomenological model, which is represented by a system of unidirectionally coupled ordinary differential equations. We investigate the problem of existence and stability of a relaxation periodic solution that is invariant with respect to cyclic permutations of coordinates. To find the asymptotics of this solution, a special relay system is constructed. We prove that the periodic solution of the relay system gives the asymptotic approximation of the orbitally asymptotically stable relaxation cycle of the problem under consideration.