Quasi-Steady-State Approximations of the Chemical Master Equation in enzyme kinetics - application to the double phosphorylation/dephosphorylation cycle

The Chemical Master Equation (CME) provides an accurate stochastic description of complex biochemical processes in terms of probability distribution of the underlying chemical population. By reason of that, CMEs are usually considered stochastic methods for the analysis of biochemical reactions, in contrast to deterministic methods, handling biochemical processes by means of Ordinary Differential Equations (ODE) expressing the evolution of the concentration for each involved species. In this deterministic framework, a common practice is to exploit Quasi-Steady State Approximations (QSSAs) to reduce the dimensionality of the system and fasten numerical simulations. In the present paper, we investigate the applicability of QSSAs from a stochastic viewpoint, by making use of the CMEs in the specific case of the double phosphorylation-dephosphorylation reaction. To this end, the stochastic approach is applied to the non-approximated original chemical network, as well as to the standard and total QSSAs, confirming by simulations the effectiveness and superiority of the latter with respect to the former.