On the reachable set of the controlled gene expression system

In this paper we investigate the reachable set of a standard stochastic model of controlled gene expression. Specifically, we explore what values of the protein mean and variance are achievable using the available external input, that is, the mRNA production rate. We proceed by constructing invariant sets in two-dimensional projections of the state space. We then use these sets to construct an outer approximation of the reachable region for the protein mean and variance. This can be computed solving a one-dimensional optimization problem and is tight enough to show that it is not possible, using such control input, to arbitrarily reduce the variance while maintaining a high mean. The lower bound on the variance derived with our approach turns out to be much higher than the one available in the literature for the case when both the mRNA production and degradation rate are controlled.