The relationship between stochastic and deterministic quasi-steady state approximations.

Krešimir Josić,Matthew R Bennett,Jae Kyoung Kim

doi:10.1186/s12918-015-0218-3

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.

Highlights

The quasi steady-state approximation (QSSA) is frequently used to reduce deterministic models of biochemical networks
We identify a clear correspondence between the validities of the deterministic and the stochastic QSSA for examples widely used in biological systems
We provide an analysis of a two-state promoter model, and use numerical simulations for a variety of other models to show that the stochastic QSSA is accurate only when the deterministic QSSA is accurate over a range of initial conditions that cover the most likely states explored by the stochastic system

Summary

Introduction

The 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. The species regulated by fast reactions quickly equilibrate to a “quasi-steady-state (QSS)” [1], and these fast species can be assumed to Timescale separation has been used to reduce and accelerate simulations of stochastic models [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. (CME) can be defined as the conditional average of the species which depends on the instantaneous state of the slow species [16,17,18] This approximation obviates the need to simulate fast reactions explicitly. Michaelis-Menten or Hill functions derived using the deterministic QSSA have been used as propensity functions in stochastic simulations – a method known as “stochastic QSSA”

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