Bulletin of Mathematical Biology

, Volume 72, Issue 8, pp 1947–1970 | Cite as

Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks

  • David F. Anderson
  • Gheorghe Craciun
  • Thomas G. Kurtz
Original Article


We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg’s deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.


Product-form stationary distributions Deficiency zero 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, D.F., 2007. A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J. Chem. Phys. 127(21), 214107. CrossRefGoogle Scholar
  2. Anderson, D.F., 2008a. Global asymptotic stability for a class of nonlinear chemical equations. SIAM J. Appl. Math. 68, 1464–1476. CrossRefMathSciNetMATHGoogle Scholar
  3. Anderson, D.F., 2008b. Incorporating postleap checks in tau-leaping. J. Chem. Phys. 128(5), 054103. CrossRefGoogle Scholar
  4. Anderson, D.F., Craciun, G., 2010. Reduced reaction networks and persistence of chemical systems (in preparation). Google Scholar
  5. Anderson, D.F., Shiu, A., 2010. The dynamics of weakly reversible population processes near facets. SIAM J. Appl. Math. 70(6), 1840–1858. CrossRefMathSciNetGoogle Scholar
  6. Anderson, D.F., Ganguly, A., Kurtz, T.G., 2010. Error analysis of tau-leap simulation methods. arXiv:0909.4790 (submitted).
  7. Angeli, D., De Leenheer, P., Sontag, E.D., 2007. A petri net approach to the study of persistence in chemical reaction networks. Math. Biosci. 210, 598–618. CrossRefMathSciNetMATHGoogle Scholar
  8. Ball, K., Kurtz, T.G., Popovic, L., Rempala, G., 2006. Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Probab. 16(4), 1925–1961. CrossRefMathSciNetGoogle Scholar
  9. Cao, Y., Gillespie, D.T., Petzold, L.R., 2006. Efficient step size selection for the tau-leaping simulation method. J.  Chem. Phys. 124, 044109. CrossRefGoogle Scholar
  10. Chen, H., Yao, D.D., 2001. Fundamentals of Queueing Networks, Performance, Asymptotics and Optimization, Applications of Mathematics, Stochastic Modelling and Applied Probability, vol. 46. Springer, New York. MATHGoogle Scholar
  11. Craciun, G., Feinberg, M., 2005. Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J. Appl. Math. 65(5), 1526–1546. CrossRefMathSciNetMATHGoogle Scholar
  12. Craciun, G., Feinberg, M., 2006. Multiple equilibria in complex chemical reaction networks: II. The species-reactions graph. SIAM J. Appl. Math. 66(4), 1321–1338. CrossRefMathSciNetMATHGoogle Scholar
  13. Craciun, G., Tang, Y., Feinberg, M., 2006. Understanding bistability in complex enzyme-driven networks. Proc. Natl. Acad. Sci. USA 103(23), 8697–8702. CrossRefGoogle Scholar
  14. Ethier, S.N., Kurtz, T.G., 1986. Markov Processes: Characterization and Convergence. Wiley, New York. MATHGoogle Scholar
  15. Feinberg, M., 1972. Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49, 187–194. CrossRefMathSciNetGoogle Scholar
  16. Feinberg, M., 1979. Lectures on chemical reaction networks. Delivered at the Mathematics Research Center, Univ. Wisc.-Madison. Available for download at http://www.che.eng.ohio-state.edu/~feinberg/LecturesOnReactionNetworks.
  17. Feinberg, M., 1987. Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems, review article 25. Chem. Eng. Sci. 42, 2229–2268. CrossRefGoogle Scholar
  18. Feinberg, M., 1989. Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity. Chem. Eng. Sci. 44(9), 1819–1827. CrossRefGoogle Scholar
  19. Feinberg, M., 1995. Existence and uniqueness of steady states for a class of chemical reaction networks. Arch. Ration. Mech. Anal. 132, 311–370. CrossRefMathSciNetMATHGoogle Scholar
  20. Gadgil, C., Lee, C.H., Othmer, H.G., 2005. A stochastic analysis of first-order reaction networks. Bull. Math. Biol. 67, 901–946. CrossRefMathSciNetGoogle Scholar
  21. Gibson, M.A., Bruck, J., 2000. Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 105, 1876–1889. Google Scholar
  22. Gillespie, D.T., 1976. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434. CrossRefMathSciNetGoogle Scholar
  23. Gillespie, D.T., 1977. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361. CrossRefGoogle Scholar
  24. Gillespie, D.T., 2001. Approximate accelerated simulation of chemically reacting systems. J. Chem. Phys. 115(4), 1716–1733. CrossRefGoogle Scholar
  25. Gunawardena, J., 2003. Chemical reaction network theory for in-silico biologists. Notes available for download at http://vcp.med.harvard.edu/papers/crnt.pdf.
  26. Horn, F.J.M., 1972. Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49(3), 172–186. CrossRefMathSciNetGoogle Scholar
  27. Horn, F.J.M., 1973. Stability and complex balancing in mass-action systems with three complexes. Proc. R. Soc. A 334, 331–342. CrossRefMathSciNetGoogle Scholar
  28. Horn, F.J.M., Jackson, R., 1972. General mass action kinetics. Arch. Ration. Mech. Anal. 47, 81–116. CrossRefMathSciNetGoogle Scholar
  29. Keener, J., Sneyd, J., 1998. Mathematical Physiology. Springer, New York. MATHGoogle Scholar
  30. Kelly, F.P., 1979. Reversibility and Stochastic Networks, Wiley Series in Probability and Mathematical Statistics. Wiley, New York. MATHGoogle Scholar
  31. Kurtz, T.G., 1972. The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys. 57(7), 2976–2978. CrossRefGoogle Scholar
  32. Kurtz, T.G., 1977/1978. Strong approximation theorems for density dependent Markov chains. Stoch. Proc. Appl. 6, 223–240. CrossRefMathSciNetGoogle Scholar
  33. Kurtz, T.G., 1981. Approximation of Population Processes, CBMS-NSF Reg. Conf. Series in Appl. Math., vol. 36. SIAM, Philadelphia. Google Scholar
  34. Kurtz, T.G., 1992. Averaging for Martingale Problems and Stochastic Approximation, Applied Stochastic Analysis, Lecture Notes in Control and Information Sciences, vol. 77, pp. 186–209. Springer, Berlin. Google Scholar
  35. Levine, E., Hwa, T., 2007. Stochastic fluctuations in metabolic pathways. Proc. Natl. Acad. Sci. USA 104(22), 9224–9229. CrossRefMathSciNetMATHGoogle Scholar
  36. Serfozo, R., 1999. Introduction to Stochastic Networks, Applications of Mathematics (New York), vol. 44. Springer, New York. MATHGoogle Scholar
  37. Sontag, E.D., 2001. Structure and stability of certain chemical networks and applications to the kinetic proofreading of t-cell receptor signal transduction. IEEE Trans. Autom. Control. 46(7), 1028–1047. CrossRefMathSciNetMATHGoogle Scholar
  38. Whittle, P., 1986. Systems in Stochastic Equilibrium. Wiley, New York. MATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2010

Authors and Affiliations

  • David F. Anderson
    • 1
  • Gheorghe Craciun
    • 1
  • Thomas G. Kurtz
    • 1
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA

Personalised recommendations