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Bulletin of Mathematical Biology

, Volume 81, Issue 5, pp 1303–1336 | Cite as

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

  • Hye-Won Kang
  • Wasiur R. KhudaBukhsh
  • Heinz Koeppl
  • Grzegorz A. RempałaEmail author
Article

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.

Keywords

Michaelis–Menten kinetics Stochastic reaction network Multiscale approximation QSSA 

Mathematics Subject Classification

60J27 60J28 34E15 92C42 92B25 92C45 

Notes

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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Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Maryland Baltimore County BaltimoreUSA
  2. 2.Department of Electrical Engineering and Information TechnologyTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Division of Biostatistics and Mathematical Biosciences InstituteThe Ohio State UniversityColumbusUSA

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