On the error of the quasi-steady-state approximation

Turányi, Tamás; Tomlin, Alison S.; Pilling, Michael J.

doi:10.1021/j100103a028

Cited by 178 publications

(129 citation statements)

References 13 publications

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“…͑1͒ is to set the production rates of the QSSA species to zero, and solve the resulting set of differential algebraic equations ͑DAES͒. 14,15 dx dt = f͑x,y͒, 0 = g͑x,y͒. ͑2͒…”

Section: A Deterministic Qssa Reductionsmentioning

confidence: 99%

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

Mastny

Haseltine

Rawlings

2007

The Journal of Chemical Physics

113

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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.

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“…͑1͒ is to set the production rates of the QSSA species to zero, and solve the resulting set of differential algebraic equations ͑DAES͒. 14,15 dx dt = f͑x,y͒, 0 = g͑x,y͒. ͑2͒…”

Section: A Deterministic Qssa Reductionsmentioning

confidence: 99%

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

Mastny

Haseltine

Rawlings

2007

The Journal of Chemical Physics

113

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“…Here we select the single nonnegative solution for c B , but multiple nonnegative roots of QSSA algebraic systems are possible in general. In those cases, one usually selects the solution that leads to the reduced model which best fits the available data [23]. We call this procedure the standard approach to applying the QSSA.…”

Section: The Qssa Methodsmentioning

confidence: 99%

The QSSA in Chemical Kinetics: As Taught and as Practiced

Pantea¹,

Gupta²,

Rawlings³

et al. 2013

Discrete and Topological Models in Molecular Biology

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Chemical mechanisms for even simple reaction networks involve many highly reactive and short-lived species (intermediates), present in small concentrations, in addition to the main reactants and products, present in larger concentrations. The chemical mechanism also often contains many rate constants whose values are unknown a priori and must be determined from experimental measurements of the large species concentrations. A classic model reduction method known as the quasi-steady-state assumption (QSSA) is often used to eliminate the highly reactive intermediate species and remove the large rate constants that cannot be determined from concentration measurements of the reactants and products. Mathematical analysis based on the QSSA is ubiquitous in modeling enzymatic reactions. In this chapter, we focus attention on the QSSA, how it is "taught" to students of chemistry, biology, and chemical and biological engineering, and how it is "practiced" when researchers confront realistic and complex examples. We describe the main types of difficulties that appear when trying to apply the standard ideas of the QSSA, and propose a new strategy for overcoming them, based on rescaling the reactive intermediate species.First, we prove mathematically that the program taught to beginning students for applying the 100-year-old approach of classic QSSA model reduction cannot be carried out for many of the relevant kinetics problems, and perhaps even most of them. By using Galois theory, we prove that the required algebraic equations cannot be solved for as few as five bimolecular reactions between five species (with three intermediates). We expect that many practitioners have suspected this situation regarding nonsolvability to exist, but we have seen no statement or proof of this fact, especially when the kinetics are restricted to unimolecular and bimolecular reactions. We describe algorithms that can test any mechanism for solvability. We also show that an alternative to solving the QSSA equations, the Horiuti-Temkin theory, also does not work for many examples.Of course, the reduced model (and the full model, for that matter) can be solved numerically, which is the standard approach in practice. The remaining difficulty, however, is how to obtain the values of the large kinetic parameters appearing in the model. These parameters cannot be estimated from measurements of the largeconcentration reactants and products. We show here how the concept of rescaling the reactive intermediate species allows the large kinetic parameters to be removed from the parameter estimation problem. In general, the number of parameters that can be removed from the full model is less than or equal to the number of intermediate species. The outcome is a reduced model with a set of rescaled parameters that is often identifiable from routinely available measurements. New and freely available computational software (parest_dae) for estimating the reduced model's kinetic parameters and confidence intervals is briefly described.

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“…Time-scale analysis methods have been used to identify and then eliminate some short life-time species through the application of quasi-steady-state assumption (QSSA) and partial equilibrium assumption (McRae et al, 1982;Chen, 1988;Turanyi et al, 1993). Intrinsic low dimensional manifold (ILDM) (Maas and Pope, 1992) and computational singular perturbation (CSP) methods Goussis, 1988, 1994;Massias et al, 1999a) have been also used to construct a reduced mechanism, through separation of fast and slow dynamic processes, in combustion chemistry (Massias et al, 1999b;Lu et al, 2001) and atmospheric chemistry (Neophytou et al, 2004;Lovas et al, 2006) .…”

Section: Methods To Eliminate Speciesmentioning

confidence: 99%

“…(2) PCA of the rate sensitivity matrix to remove redundant reactions (Turanyi et al, 1989); (3) QSSA to identify and remove of some quasi-steadystate species (Turanyi et al, 1993); (4) the Iterative Screening and Structure Analysis method (Mauersberger, 2005) to remove unimportant species and reactions simultaneously;…”

Section: Mechanism Reduction To the Subset Of α-Pinene Oxidationmentioning

confidence: 99%