Two classes of quasi-steady-state model reductions for stochastic kinetics

Abstract

The quasi-steady-state approximation ͑QSSA͒ is a model reduction technique used to remove highly reactive species from deterministic models of reaction mechanisms. In many reaction networks the highly reactive intermediates ͑QSSA species͒ have populations small enough to require a stochastic representation. In this work we apply singular perturbation analysis to remove the QSSA species from the chemical master equation for two classes of problems. The first class occurs in reaction networks where all the species have small populations and the QSSA species sample zero the majority of the time. The perturbation analysis provides a reduced master equation in which the highly reactive species can sample only zero, and are effectively removed from the model. The reduced master equation can be sampled with the Gillespie algorithm. This first stochastic QSSA reduction is applied to several example reaction mechanisms ͑including Michaelis-Menten kinetics͒ ͓Biochem. Z. 49, 333 ͑1913͔͒. A general framework for applying the first QSSA reduction technique to new reaction mechanisms is derived. The second class of QSSA model reductions is derived for reaction networks where non-QSSA species have large populations and QSSA species numbers are small and stochastic. We derive this second QSSA reduction from a combination of singular perturbation analysis and the ⍀ expansion. In some cases the reduced mechanisms and reaction rates from these two stochastic QSSA models and the classical deterministic QSSA reduction are equivalent; however, this is not usually the case.

Communication: Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks

“…Note that Eq. (9) shows that the relative error tends to zero as α → 0 and α → 1 and that hence the reduced master equation provides a correct prediction of the size of the substrate fluctuations whenever the free enzyme or complex concentrations are very small (similar results have been obtained by Mastny et al 23 for the Michaelis-Menten reaction with no substrate input; however their results are not for general α and β and do not enforce the validity of the deterministic QSSA; see later for discussion). In Fig.…”

“…For the Michaelis-Menten reaction without substrate input, these methods lead to a reduced master equation of the same form as the heuristic stochastic QSSA whenever the free enzyme or complex concentrations are very small (see Table II of Ref. 23). This implies that for such conditions the error in the predictions of the stochastic QSSA should be zero, a result which is also reproduced by our method.…”

“…There are a class of alternative model reduction techniques 23,32 based on singular-perturbation analysis (sQSPA and sQSPA-) which are rigorous and whose validity is not under question. For the Michaelis-Menten reaction without substrate input, these methods lead to a reduced master equation of the same form as the heuristic stochastic QSSA whenever the free enzyme or complex concentrations are very small (see Table II of Ref.…”

On the impact of discreteness and abstractions on modelling noise in gene regulatory networks

“…A different abstraction technique consists in reducing the number of model variables and reactions, using the Quasi-Steady State Approximation (QSSA) (Rao and Arkin, 2003;Cao et al, 2005;Mastny et al, 2007). The idea behind QSSA is that, if a set of reactions acting on one (or more) molecular species is very fast, then their dynamics will quickly reach an equilibrium.…”

“…Then, a reduced system is constructed, containing only the slow species, averaging out the fast variables from the rate functions depending on them according to the previously computed steady state distribution. Since analytical expressions are rarely obtained in this way, such averaged rates can be approximated either by stochastic simulation (Mastny andS Liu, 2005, 2007), or using the deterministic steady state distribution of fast variables, i.e. the one obtained from the ODE model.…”