On a stochastic approach to model the double phosphorylation/dephosphorylation cycle

Because of the unavoidable intrinsic noise affecting biochemical processes, astochastic approach is usually preferred whenever a deterministic model givestoo rough information or, worse, may lead to erroneous qualitative behaviorsand/or quantitatively wrong results. In this work we focus on the chemicalmaster equation (CME)-based method which provides an accurate stochasticdescription of complex biochemical reaction networks in terms of the probabilitydistribution of the underlying chemical populations. Indeed, deterministic mod-els can be dealt with as first-order approximations of the average-value dynamicscoming from the stochastic CME approach. Here we investigate the double phos-phorylation/dephosphorylation cycle, a well-studied enzymatic reaction networkwhere the inherent double time scale requires one to exploit quasisteady stateapproximation (QSSA) approaches to infer qualitative and quantitative informa-tion. Within the deterministic realm, several researchers have deeply investi-gated the use of the proper QSSA, agreeing to highlight that only one type ofQSSA (the total QSSA) is able to faithfully replicate the qualitative behaviorof bistability occurrences, as well as the correct assessment of the equilibriumpoints, accordingly to the not approximated (full) model. Based on recent resultsproviding CME solutions that do not resort to Monte Carlo simulations, the pro-posed stochastic approach shows some counterintuitive facts arising when tryingto straightforwardly transfer bistability deterministic results into the stochasticrealm, and suggests how to handle such cases according to both theoretical andnumerical results.