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Quasi-Steady-State Approximations Derived from the Stochastic Model of Enzyme Kinetics

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A Commentary to this article was published on 07 February 2019

Abstract

The paper outlines a general approach to deriving quasi-steady-state approximations (QSSAs) of the stochastic reaction networks describing the Michaelis–Menten enzyme kinetics. In particular, it explains how different sets of assumptions about chemical species abundance and reaction rates lead to the standard QSSA, the total QSSA, and the reverse QSSA. These three QSSAs have been widely studied in the literature in deterministic ordinary differential equation settings, and several sets of conditions for their validity have been proposed. With the help of the multiscaling techniques introduced in Ball et al. (Ann Appl Probab 16(4):1925–1961, 2006), Kang and Kurtz (Ann Appl Probab 23(2):529–583, 2013), it is seen that the conditions for deterministic QSSAs largely agree (with some exceptions) with the ones for stochastic QSSAs in the large-volume limits. The paper also illustrates how the stochastic QSSA approach may be extended to more complex stochastic kinetic networks like, for instance, the enzyme–substrate–inhibitor system.

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Notes

  1. The agreement is meant in the sense that it gives a reduced ODE model whose propensity functions are analogues of those in the stochastic QSSA.

  2. We note that 1 / N plays a similar role as the expansion parameter (usually denoted by \(\epsilon \).) in the singular perturbation analysis of deterministic models (Goeke and Walcher 2014; Segel and Slemrod 1989). In this approach, the time scales are often separated by introducing, in addition to the time variable t, a new slow time scale \(\tau =\epsilon t\), where \(\epsilon \) is assumed small and eventually sent to zero. This allows one to reformulate the system of differential equations into the Tikhonov standard form (Goeke and Walcher 2014). Alternatively, especially in case of perturbation analysis of chemical reaction networks, one often scales the reaction rates instead to separate the fast reactions from the assumed slow ones. For instance, a reaction with rate \(\epsilon k_1\) will correspond to a slow reaction compared to a reaction with rate \(k_2\). See Goeke and Walcher (2014), Section 2.4 Examples for some examples.

  3. The strong law of large numbers states that, for a unit Poisson process Y, \(\frac{1}{N} Y(Nu ) \rightarrow u \) almost surely as \(N \rightarrow \infty \), (see Ethier and Kurtz 1986).

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Acknowledgements

This work has been co-funded by the German Research Foundation (DFG) as part of project C3 within the Collaborative Research Center (CRC) 1053—MAKI (WKB) and the National Science Foundation under the Grants RAPID DMS-1513489 (GR) and DMS-1620403 (HWK). This research has also been supported in part by the University of Maryland Baltimore County under Grant UMBC KAN3STRT (HWK). This work was initiated when HWK and WKB were visiting the Mathematical Biosciences Institute (MBI) at the Ohio State University in Winter 2016–2017. MBI is receiving major funding from the National Science Foundation under the Grant DMS-1440386. HWK and WKB acknowledge the hospitality of MBI during their visits to the institute.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD, USA

    Hye-Won Kang

  2. Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt, Darmstadt, Germany

    Wasiur R. KhudaBukhsh & Heinz Koeppl

  3. Division of Biostatistics and Mathematical Biosciences Institute, The Ohio State University, Columbus, OH, USA

    Grzegorz A. Rempała

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  1. Hye-Won Kang
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  4. Grzegorz A. Rempała
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Correspondence to Grzegorz A. Rempała.

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Hye-Won Kang and Wasiur R. KhudaBukhsh contributed equally and are joint first authors.

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Kang, HW., KhudaBukhsh, W.R., Koeppl, H. et al. Quasi-Steady-State Approximations Derived from the Stochastic Model of Enzyme Kinetics. Bull Math Biol 81, 1303–1336 (2019). https://doi.org/10.1007/s11538-019-00574-4

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