  A fundamental question in systems biology is what combinations of mean and
variance of the species present in a stochastic biochemical reaction network
are attainable by perturbing the system with an external signal. To address
this question, we show that the moments evolution in any generic network can be
either approximated or, under suitable assumptions, computed exactly as the
solution of a switched affine system. Motivated by this application, we propose
a new method to approximate the reachable set of switched affine systems. A
remarkable feature of our approach is that it allows one to easily compute
projections of the reachable set for pairs of moments of interest, without
requiring the computation of the full reachable set, which can be prohibitive
for large networks. As a second contribution, we also show how to select the
external signal in order to maximize the probability of reaching a target set.
To illustrate the method we study a renown model of controlled gene expression
and we derive estimates of the reachable set, for the protein mean and
variance, that are more accurate than those available in the literature and
consistent with experimental data.
