Abstract
Stochastic models of chemical reactions are needed in many contexts in which the copy numbers of species are low, but only the simplest models can be treated analytically. However, direct simulation of computational models for systems with many components can be very time-consuming, and approximate methods are frequently used. One method that has been used in systems with multiple time scales is to approximate the fast dynamics, and in this note we study one such approach, in which the deterministic QSSH is used for the fast variables and the result used in the rate equations for the slow variables. We examine the classical Michaelis-Menten kinetics using this approach to determine when it is applicable.
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References
Agarwal, A., Adams, R., Castellani, G.C., Shouval, H.Z.: On the precision of quasi steady state assumptions in stochastic dynamics. The Journal of Chemical Physics 137(4), 044,105 (2012). DOI https://doi.org/10.1063/1.4731754. URL http://aip.scitation.org/doi/10.1063/1.4731754
Balázsi, G., van Oudenaarden, A., Collins, J.J.: Cellular decision making and biological noise: from microbes to mammals. Cell 144(6), 910–925 (2011)
Ball, P., Borley, N.R.: The self-made tapestry: pattern formation in nature, vol. 198. Oxford University Press Oxford (1999)
Barik, D., Paul, M.R., Baumann, W.T., Cao, Y., Tyson, J.J.: Stochastic simulation of enzyme-catalyzed reactions with disparate timescales. Biophysical journal 95(8), 3563–74 (2008). DOI https://doi.org/10.1529/biophysj.108.129155. URL http://www.ncbi.nlm.nih.gov/pubmed/18621809 http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC2553150
Darden, T.A.: Enzyme kinetics: stochastic vs. deterministic models. Instabilities, bifurcations, and fluctuations in chemical systems (Austin, Tex., 1980) pp. 248–272 (1982)
Elowitz, M.B., Levine, A.J., Siggia, E.D., Swain, P.S.: Stochastic gene expression in a single cell. Science 297(5584), 1183–1186 (2002)
Gillespie, D., Sanft, K., Petzold, L.: Legitimacy of the stochastic MichaelisMenten approximation. IET Systems Biology (2011). DOI https://doi.org/10.1049/iet-syb.2009.0057
Goutsias, J.: Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122, 184,102– (2005)
Kan, X., Lee, C.H., Othmer, H.G.: A multi-time-scale analysis of chemical reaction networks: IL Stochastic systems. Journal of Mathematical Biology 73(5), 1081–1129 (2016). DOI https://doi.org/10.1007/s00285-016-0980-x
Kang, H.W., Kurtz, T.G., et al.: Separation of time-scales and model reduction for stochastic reaction networks. The Annals of Applied Probability 23(2), 529–583 (2013)
Kim, J.K., Josić, K., Bennett, M.R.: The Validity of Quasi-Steady-State Approximations in Discrete Stochastic Simulations. Biophysical Journal 107(3), 783–793 (2014). DOI https://doi.org/10.1016/j.bpj.2014.06.012. URL https://www.sciencedirect.com/science/article/pii/S0006349514006183
Kim, J.K., Josić, K., Bennett, M.R.: The relationship between stochastic and deterministic quasi-steady state approximations. BMC systems biology 9(1), 87 (2015). DOI https://doi.org/10.1186/s12918-015-0218-3. URL http://www.biomedcentral.com/1752-0509/9/87
Kurtz, T.G.: The Relationship between Stochastic and Deterministic Models for Chemical Reactions. The Journal of Chemical Physics 57(7), 2976–2978 (1972). DOI https://doi.org/10.1063/1.1678692. URL http://aip.scitation.org/doi/10.1063/1.1678692
Lawson, M.J., Petzold, L., Hellander, A.: Accuracy of the Michaelis-Menten approximation when analysing effects of molecular noise. Journal of the Royal Society, Interface 12(106), 20150,054 (2015). DOI https://doi.org/10.1098/rsif.2015.0054. URL http://www.ncbi.nlm.nih.gov/pubmed/25833240 http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC4424680
Lee, C.H., Othmer, H.G.: A multi-time-scale analysis of chemical reaction networks: I. Deterministic systems. Journal of Mathematical Biology 60(3), 387–450 (2010). DOI https://doi.org/10.1007/s00285-009-0269-4
Othmer, H.G.: Analysis of complex reaction networks in signal transduction, gene control and metabolism (2006). Lecture Notes – University of Minnesota
Rao, C.V., Arkin, A.P.: Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. The Journal of Chemical Physics 118(11), 4999–5010 (2003). DOI https://doi.org/10.1063/1.1545446. URL http://aip.scitation.org/doi/10.1063/1.1545446
Segel, L.A., Slemrod, M.: The Quasi-Steady-State Assumption: A Case Study in Perturbation. SIAM Review (1989). DOI https://doi.org/10.1137/1031091
Thomas, P., Grima, R., Straube, A.V.: Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators. Physical Review E 86(4), 41,110 (2012)
Thomas, P., Straube, A.V., Grima, R.: Communication: Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks. Journal of Chemical Physics 135(18), 1–7 (2011). DOI https://doi.org/10.1063/1.3661156
Thomas, P., Straube, A.V., Grima, R.: The slow-scale linear noise approximation: An accurate, reduced stochastic description of biochemical networks under timescale separation conditions. BMC Systems Biology 6(2012). DOI https://doi.org/10.1186/1752-0509-6-39
Tsimring, L.S.: Noise in biology. Reports on Progress in Physics 77(2), 026,601 (2014)
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Bobadilla, A.V.P., Bartmanski, B.J., Grima, R., Othmer, H.G. (2020). The Status of the QSSA Approximation in Stochastic Simulations of Reaction Networks. In: de Gier, J., Praeger, C., Tao, T. (eds) 2018 MATRIX Annals. MATRIX Book Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-38230-8_10
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