The relationship between stochastic and deterministic quasi-steady state approximations

Abstract

BackgroundThe quasi steady-state approximation (QSSA) is frequently used to reduce deterministic models of biochemical networks. The resulting equations provide a simplified description of the network in terms of non-elementary reaction functions (e.g. Hill functions). Such deterministic reductions are frequently a basis for heuristic stochastic models in which non-elementary reaction functions are used to define reaction propensities. Despite their popularity, it remains unclear when such stochastic reductions are valid. It is frequently assumed that the stochastic reduction can be trusted whenever its deterministic counterpart is accurate. However, a number of recent examples show that this is not necessarily the case.ResultsHere we explain the origin of these discrepancies, and demonstrate a clear relationship between the accuracy of the deterministic and the stochastic QSSA for examples widely used in biological systems. With an analysis of a two-state promoter model, and numerical simulations for a variety of other models, we find that the stochastic QSSA is accurate whenever its deterministic counterpart provides an accurate approximation over a range of initial conditions which cover the likely fluctuations from the quasi steady-state (QSS). We conjecture that this relationship provides a simple and computationally inexpensive way to test the accuracy of reduced stochastic models using deterministic simulations.ConclusionsThe stochastic QSSA is one of the most popular multi-scale stochastic simulation methods. While the use of QSSA, and the resulting non-elementary functions has been justified in the deterministic case, it is not clear when their stochastic counterparts are accurate. In this study, we show how the accuracy of the stochastic QSSA can be tested using their deterministic counterparts providing a concrete method to test when non-elementary rate functions can be used in stochastic simulations.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-015-0218-3) contains supplementary material, which is available to authorized users.

Reduced linear noise approximation for biochemical reaction networks with time-scale separation: The stochastic tQSSA+

“…However, Kim et al has shown that although this provides a good approximation in deterministic models, using the reactions rates obtained through the prefactor QSSA in stochastic simulations does not produce accurate results. 20,22 By contrast, our results provide a method to obtain accurate approximations of the original system variables.…”

“…Furthermore, this indicates that when the molecular counts are sufficiently large, the stochastic tQSSA yields a valid approximation for the CME under time-scale separation, similar to the observations in previous studies. 20,22 Therefore, our results provide a rigorous justification for the validity of stochastic tQSSA models, which has been absent in previous work. However, unlike the tQSSA, the reduced model derived in (14)- (17) also provides approximations for the fast variable stochastic properties.…”

“…Particularly, the work by Kim et al demonstrates, via twodimensional systems modeled by the CME and LNA, that the tQSSA provides a better approximation to the original system in comparison to the standard QSSA. 20,22 Here, we use the results derived in Theorem 1 to provide a rigorous mathematical justification for the accuracy of the stochastic tQSSA for general reaction networks modeled through the LNA. The inaccuracy of the standard QSSA in the deterministic setting results from treating the species that consists of both slow and fast dynamics as purely slow variables.…”

“…9,10 There have been several studies that obtain reduced-order representations of the CME under time-scale separation conditions. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] Among these, a common approach used to reduce the complexity in the simulations of the CME is to approximate the fast variables by their deterministic quasi-steady state expressions. 19 However, validity of this method still remains under investigation.…”

“…19 However, validity of this method still remains under investigation. 20,22,[25][26][27] The theoretical analysis of the CME is challenging due to the large system size and the lack of appropriate analytical tools. Therefore, several approximations to the chemical master equation have been developed under the assumptions that the system's volume and the molecular counts are sufficiently large.…”