title: The relationship between stochastic and deterministic quasi-steady state approximations.
abstract: BACKGROUND
  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. 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.
  
  
  RESULTS
  Here 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.
  
  
  CONCLUSIONS
  The 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.
authors.affiliation: Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro Yuseong-gu, Daejeon, 305-701, Korea. jaekkim@kaist.ac.kr.|Mathematical Biosciences Institute, The Ohio State University, 1735 Neil Avenue, OH 43210, Columbus, USA. jaekkim@kaist.ac.kr.
authors.forename: Jae Kyoung
authors.initials: JK
authors.lastname: Kim
authors.affiliation: Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX 77204-3008, USA. josic@math.uh.edu.|Department of Biology and Biochemistry, University of Houston, 4800 Calhoun Rd, Houston, TX 77204-3008, USA. josic@math.uh.edu.
authors.forename: Krešimir
authors.initials: K
authors.lastname: Josić
authors.affiliation: Department of Biosciences, Rice University, 6100 Main St, Houston, 77005-1892, TX, USA. matthew.bennett@rice.edu.|Department of Bioengineering, Rice University, 6100 Main St, Houston, TX 77005-1892, USA. matthew.bennett@rice.edu.
authors.forename: Matthew R
authors.initials: MR
authors.lastname: Bennett
mesh_terms: D019295:Computational Biology; D013269:Stochastic Processes