ZI-closure scheme: a method to solve and study stochastic reaction networks. (English) Zbl 1401.92235

Holcman, David (ed.), Stochastic processes, multiscale modeling, and numerical methods for computational cellular biology. Cham: Springer (ISBN 978-3-319-62626-0/hbk; 978-3-319-62627-7/ebook). 159-174 (2017).
Summary: We use an example to present in exhaustive detail the algorithmic steps of the zero-information (ZI) closure scheme [P. Smadbeck and the third author, “A closure scheme for chemical master equations”, Proc. Natl. Acad. Sci. USA 110,14261–14265 (2013; doi:10.1073/pnas.1306481110)]. ZI-closure is a method for solving the chemical master equation (CME) of stochastic chemical reaction networks.
For the entire collection see [Zbl 1392.92002].


92E20 Classical flows, reactions, etc. in chemistry
35Q92 PDEs in connection with biology, chemistry and other natural sciences


Full Text: DOI


[1] P. Smadbeck, Y.N. Kaznessis, A closure scheme for chemical master equations. Proc. Natl. Acad. Sci. U. S. A. 110, 14261-14265 (2013)
[2] K.R. Popper, \(The Open Universe: An Argument for Indeterminism\) (Cambridge, Routledge, 1982), p. xix
[3] W. James, \(The Dilemma of Determinism\). The Will to Believe (New York, Dover, 1956)
[4] I. Prigogine, \(The End of Certainty: Time, Chaos, and the New Laws of Nature\) (Free Press, New York, 1997)
[5] D.A. McQuarrie, Stochastic approach to chemical kinetics. J. Appl. Probab. 4, 413-478 (1967) · Zbl 0231.60090
[6] I. Oppenheim, K.E. Shuler, Master equations and Markov processes. Phys. Rev. 138, B1007-B1011 (1965) · Zbl 0134.34702
[7] N.G. Van Kampen, \(Stochastic Processes in Physics and Chemistry\), Revised and enlarged edition (Elsevier, Amsterdam, 2004)
[8] D.T. Gillespie, A rigorous derivation of the chemical master equation. Physica A 188, 404-425 (1992)
[9] D.T. Gillespie, Stochastic simulation of chemical kinetics. Ann. Rev. Phys. Chem. 58, 35-55 (2007)
[10] D.T Gillespie, A general method for numerically simulating the stochastic time evolution of coupled reactions. J. Comput. Phys. 22, 403-434 (1976)
[11] Y. Kaznessis, Multi-scale models for gene network engineering. Chem. Eng. Sci. 61, 940-953 (2006)
[12] P.H. Constantino, M. Vlysidis, P. Smadbeck, Y.N. Kaznessis, Modeling stochasticity in biochemical reaction networks. J. Phys. D Appl. Phys. 49, 093001 (2016)
[13] V. Sotiropoulos, Y.N. Kaznessis, Analytical derivation of moment equations in stochastic chemical kinetics. Chem. Eng. Sci. 66, 268-277 (2010)
[14] P. Smadbeck, Y.N. Kaznessis, Efficient moment matrix generation for arbitrary chemical networks. Chem. Eng. Sci. 84, 612-618 (2012)
[15] C.S. Gillespie, Moment-closure approximations for mass-action models. IET Syst. Biol. 3, 52-58 (2009)
[16] E.T. Jaynes, Information theory and statistical mechanics. Phys. Rev. 106, 620-630 (1957) · Zbl 0084.43701
[17] C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379-423, 623-659 (1948) · Zbl 1154.94303
[18] F. Schlögl, On thermodynamics near a steady state. Z. Phys. 248, 446-458 (1971)
[19] D.T. Gillespie, \(Markov Processes, An Introduction for Physical Scientists\) (Academic, Cambridge, 1992)
[20] M.H. DeGroot, M.H. Schervish, \(Probability and Statistics\), 4th edn. (Pearson, Cambridge, 2012)
[21] J.N. Kapur, \(Maximum-Entropy Models in Science and Engineering\), 1st edn. (Wiley, New York, 1989)
[22] A.D. Hill, J.R. Tomshine, E.M. Weeding, V. Sotiropoulos, Y.N. Kaznessis, SynBioSS: the synthetic biology modeling suite. Bioinformatics 24, 2551-2553 (2008)
[23] P. Smadbeck, Y.N. Kaznessis, On a theory of stability for nonlinear stochastic chemical reaction networks. J. Chem. Phys. 142, 184101 (2015)
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