The Dynamics of Enzyme Inhibition Controlled by Piece-Wise Deterministic Markov Process

We present a model for the development of enzyme molecules under the action of its inhibitor. In this model the inhibitor's population size is regulated by a piece-wise deterministic Markov process. This special class of stochastic processes is usually represented by a uniquely solvable system of ordinary differential equations (ODEs) perturbed by a discrete stochastic switching process. The assumption that enzyme molecules growth rate cannot be negative leads us to the alternative version of the model, where the right hand sides of the ODEs are only piece-wise differentiable. For this reason a standard procedure based on the analogue of the Malliavin calculus approach [17] is not enough to deal with the long-time behaviour of the trajectories of the process. However, in both cases, a proper Markov semigroup of the densities of the process is constructed. Moreover, its asymptotic stability in the sense of Lasota is shown in this paper.