ZI-Closure Scheme: A Method to Solve and Study Stochastic Reaction Networks

  • M. Vlysidis
  • P. H. Constantino
  • Y. N. Kaznessis
Chapter

Abstract

We use an example to present in exhaustive detail the algorithmic steps of the zero-information (ZI) closure scheme (Smadbeck and Kaznessis, Proc Natl Acad Sci USA 110:14261–14265, 2013). ZI-closure is a method for solving the chemical master equation (CME) of stochastic chemical reaction networks.

Notes

Acknowledgements

This work was supported by a grant from the National Institutes of Health (GM111358) and a grant from the National Science Foundation (CBET-1412283). This work utilized the high-performance computational resources of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575. Support from the University of Minnesota Digital Technology Center, from the Minnesota Supercomputing Institute (MSI), and from CAPES—Coordenaçäo de Aperfeiçoamento de Pessoal de Nível Superior—Brazil is gratefully acknowledged. This work was partially completed Spring 2016, when YNK was Visiting Scholar at the Isaac Newton Institute of Mathematical Sciences at the University of Cambridge, attending the programme Stochastic Dynamical Systems in Biology.

References

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

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • M. Vlysidis
    • 1
  • P. H. Constantino
    • 1
  • Y. N. Kaznessis
    • 1
  1. 1.Department of Chemical Engineering and Materials ScienceUniversity of MinnesotaMinneapolisUSA

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