Generalized Method of Moments for Stochastic Reaction Networks in Equilibrium

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9859)


Calibrating parameters is a crucial problem within quantitative modeling approaches to reaction networks. Existing methods for stochastic models rely either on statistical sampling or can only be applied to small systems. Here we present an inference procedure for stochastic models in equilibrium that is based on a moment matching scheme with optimal weighting and that can be used with high-throughput data like the one collected by flow cytometry. Our method does not require an approximation of the underlying equilibrium probability distribution and, if reaction rate constants have to be learned, the optimal values can be computed by solving a linear system of equations. We evaluate the effectiveness of the proposed approach on three case studies.


Stochastic Reaction Networks Moment-matching Scheme Linear Propensity Moment Closure Approximation Parameter Inference Methods 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Ale, A., Kirk, P., Stumpf, M.: A general moment expansion method for stochastic kinetic models. J. Chem. Phys. 138(17), 174101 (2013)CrossRefGoogle Scholar
  2. 2.
    Andreychenko, A., Mikeev, L., Spieler, D., Wolf, V.: Parameter identification for markov models of biochemical reactions. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 83–98. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Andreychenko, A., Mikeev, L., Spieler, D., Wolf, V.: Approximate maximum likelihood estimation for stochastic chemicalkinetics. EURASIP J. Bioinf. Syst. Biol. 9, 1–14 (2012)Google Scholar
  4. 4.
    Andreychenko, A., Mikeev, L., Wolf, V.: Model reconstruction for moment-based stochastic chemical kinetics. ACM Trans. Model. Comput. Simul. (TOMACS) 25(2), 12 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Boys, R., Wilkinson, D., Kirkwood, T.: Bayesian inference for a discretely observed stochastic kinetic model. Stat. Comput. 18, 125–135 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Engblom, S.: Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput. 180(2), 498–515 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fournier, T., Gabriel, J.-P., Mazza, C., Pasquier, J., Galbete, J.L., Mermod, N.: Steady-state expression of self-regulated genes. Bioinformatics 23(23), 3185–3192 (2007)CrossRefGoogle Scholar
  8. 8.
    Gardner, T.S., Cantor, C.R., Collins, J.J.: Construction of a genetic toggle switch in escherichia coli. Nature 403(6767), 339–342 (2000)CrossRefGoogle Scholar
  9. 9.
    Geva-Zatorsky, N., Rosenfeld, N., et al.: Oscillations and variability in the p53 system. Mol. Syst. Biol. 2(1) (2006)Google Scholar
  10. 10.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)CrossRefGoogle Scholar
  11. 11.
    Gupta, A., Briat, C., Khammash, M.: A scalable computational framework for establishing long-term behavior of stochastic reaction networks. PLoS Comput. Biol. 10(6), e1003669 (2014)CrossRefGoogle Scholar
  12. 12.
    Hall, A.R.: Generalized Method of Moments. Oxford University Press, New York (2005)zbMATHGoogle Scholar
  13. 13.
    Hanley, M.B., Lomas, W., Mittar, D., Maino, V., Park, E.: Detection of low abundance RNA molecules in individual cells by flow cytometry. PloS one 8(2), e57002 (2013)CrossRefGoogle Scholar
  14. 14.
    Hansen, L.P.: Large sample properties of generalized method of moments estimators. Econometrica 50(4), 1029–1054 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hansen, L.P., Heaton, J., Yaron, A.: Finite-sample properties of some alternative GMM estimators. J. Bus. Econ. Stat. 14(3), 262–280 (1996)Google Scholar
  16. 16.
    Hasenauer, J., Waldherr, S., Doszczak, M., Radde, N., Scheurich, P., Allgöwer, F.: Identification of models of heterogeneous cell populations from population snapshot data. BMC Bioinf. 12(1), 1 (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    Isaacs, F.J., Hasty, J., Cantor, C.R., Collins, J.J.: Prediction and measurement of an autoregulatory genetic module. PNAS 100(13), 7714–7719 (2003)CrossRefGoogle Scholar
  18. 18.
    Kügler, P.: Moment fitting for parameter inference in repeatedly and partially observed stochastic biological models. PloS one 7(8), e43001 (2012)CrossRefGoogle Scholar
  19. 19.
    Lee, Y.J., Holzapfel, K.L., Zhu, J., Jameson, S.C., Hogquist, K.A.: Steady-state production of il-4 modulates immunity in mouse strains and is determined by lineage diversity of INKT cells. Nat. Immunol. 14(11), 1146–1154 (2013)CrossRefGoogle Scholar
  20. 20.
    Lipshtat, A., Loinger, A., Balaban, N.Q., Biham, O.: Genetic toggle switch without cooperative binding. Phys. Rev. Lett. 96(18), 188101 (2006)CrossRefGoogle Scholar
  21. 21.
    Lück, A., Wolf, V.: Generalized method of moments for estimating parameters of stochastic reaction networks. ArXiv e-prints, May 2016Google Scholar
  22. 22.
    Mateescu, M., Wolf, V., Didier, F., Henzinger, T.A.: Fast adaptive uniformisation of the chemical master equation. IET Syst. Biol. 4(6), 441–452 (2010)CrossRefGoogle Scholar
  23. 23.
    Mátyás, L.: Generalized Method of Moments Estimation, vol. 5. Cambridge University Press, New York (1999)CrossRefGoogle Scholar
  24. 24.
    Milner, P., Gillespie, C.S., Wilkinson, D.J.: Moment closure based parameter inference of stochastic kinetic models. Stat. Comput. 23(2), 287–295 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Munsky, B., Fox, Z., Neuert, G.: Integrating single-molecule experiments and discrete stochastic models to understand heterogeneous gene transcription dynamics. Methods 85, 12–21 (2015)CrossRefGoogle Scholar
  26. 26.
    Newey, W.K., Smith, R.J.: Higher order properties of gmm and generalized empirical likelihood estimators. Econometrica 72(1), 219–255 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nishihara, M., Ogura, H., Ueda, N., Tsuruoka, M., Kitabayashi, C., et al.: IL-6-gp130-STAT3 in T cells directs the development of IL-17+ Th with a minimum effect on that of Treg in the steady state. Int. Immunol. 19(6), 695–702 (2007)CrossRefGoogle Scholar
  28. 28.
    Pearson, K.: Contributions to the mathematical theory of evolution. Philos. Trans. R. Soc. Lond. A 185, 71–110 (1894)CrossRefzbMATHGoogle Scholar
  29. 29.
    Reinker, S., Altman, R.M., Timmer, J.: Parameter estimation in stochastic biochemical reactions. IEEE Proc. Syst. Biol 153, 168–178 (2006)CrossRefGoogle Scholar
  30. 30.
    Singh, A., Hespanha, J.P.: Lognormal moment closures for biochemical reactions. In: 2006 45th IEEE Conference on Decision and Control, pp. 2063–2068. IEEE (2006)Google Scholar
  31. 31.
    Tian, T., Xu, S., Gao, J., Burrage, K.: Simulated maximum likelihood method for estimating kinetic rates in gene expression. Bioinformatics 23, 84–91 (2007)CrossRefGoogle Scholar
  32. 32.
    Toni, T., Welch, D., Strelkowa, N., Ipsen, A., Stumpf, M.: Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface 6(31), 187–202 (2009)CrossRefGoogle Scholar
  33. 33.
    Toni, T., Welch, D., Strelkowa, N., Ipsen, A., Stumpf, M.P.H.: Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface 6(31), 187–202 (2009)CrossRefGoogle Scholar
  34. 34.
    Wilkinson, D.J.: Stochastic Modelling for Systems Biology. C & H, Sesser (2006)zbMATHGoogle Scholar
  35. 35.
    Zechner, C., Ruess, J., Krenn, P., Pelet, S., Peter, M., Lygeros, J., Koeppl, H.: Moment-based inference predicts bimodality in transient gene expression. PNAS 109(21), 8340–8345 (2012)CrossRefGoogle Scholar
  36. 36.
    Zhu, C., Byrd, R.H., Lu, P., Nocedal, J.: Algorithm 778: L-BFGS-B: fortran subroutines for large-scale bound-constrained optimization. ACM Trans. Math. Softw. (TOMS) 23(4), 550–560 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Computer Science DepartmentSaarland UniversitySaarbrückenGermany
  2. 2.Department of Mathematics and GeosciencesUniversity of TriesteTriesteItaly

Personalised recommendations