Stability of a switched model of gene regulatory networks

A gene regulatory network is a network consisting of genes that interact with proteins in order to control gene expression. Recent developments in biology and biomedicine have led to the examination of mathematical models that can describe the dynamics of gene regulatory networks.

We present a switched model of gene regulatory networks. The dynamics of both the concentrations of messenger ribonucleic acid and protein are described by ordinary differential equations (ODEs). The set of models with ODE with different activator functions of Hill’s type is given initially. The switching rule is a piecewise constant function and its points of discontinuity determine the points at which the corresponding model from the given set is chosen. An algorithm for the construction of the solution of the switched model and the global equilibrium is presented. Sufficient conditions for the exponential stability of the model are established theoretically. Several examples of the studied switched models are studied. Both cases of a finite and an infinite sequence of switching times are examined. The provided examples validate the obtained sufficient conditions and demonstrate the effect of the presence of the switching rule on the global equilibrium of the model. These examples also show the possibilities of applications of the obtained theoretical results for more adequate modeling of gene regulatory networks.