Sensitivity analysis of the reaction occurrence and recurrence times in steady-state biochemical networks

Continuous-time stationary Markov jump processes among discrete sites are encountered in disparate biochemical ambits. Sites and connecting dynamical events form a 'network' in which the sites are the available system's states, and the events are site-to-site transitions, or even neutral processes in which the system does not change site but the event is however detectable. Examples include conformational transitions in single biomolecules, stochastic chemical kinetics in the space of the molecules copy numbers, and even macroscopic steady-state reactive mixtures if one adopts the viewpoint of a tagged molecule (or even of a molecular moiety) whose state may change when it is involved in a chemical reaction. Among the variety of dynamical descriptors, here we focus on the first occurrence times (starting from a given site) and on the recurrence times of an event of interest. We develop the sensitivity analysis for the lowest moments of the statistical distribution of such times with respect to the rate constants of the network. In particular, simple expressions and inequalities allow us to establish a direct relationship between selective variation of rate constants and effect on average times and variances. As illustrative cases we treat the substrate inhibition in enzymatic catalysis in which a tagged enzyme molecule jumps between three states, and the basic two-site model of stochastic gene expression in which the single gene switches between active and inactive forms.