Journal of Mathematical Chemistry

, Volume 49, Issue 6, pp 1163–1179

Finding complex balanced and detailed balanced realizations of chemical reaction networks

Original Paper

Abstract

Reversibility, weak reversibility and deficiency, detailed and complex balancing are generally not “encoded” in the kinetic differential equations but they are realization properties that may imply local or even global asymptotic stability of the underlying reaction kinetic system when further conditions are also fulfilled. In this paper, efficient numerical procedures are given for finding complex balanced or detailed balanced realizations of mass action type chemical reaction networks or kinetic dynamical systems in the framework of linear programming. The procedures are illustrated on numerical examples.

Keywords

Reaction kinetic systems Mass action kinetics Linear programming 

Mathematics Subject Classification (2000)

80A30 chemical kinetics 

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References

  1. 1.
    Angeli D. (2009) A tutorial on chemical network dynamics. Eur. J. Control 15: 398–406CrossRefGoogle Scholar
  2. 2.
    Belov G. (2010) On linear programming approach for the calculation of chemical equilibrium in complex thermodynamic systems. J. Math. Chem. 47: 446–456CrossRefGoogle Scholar
  3. 3.
    Callen H.B. (1980) Thermodynamics and an Introduction to Thermostatistics. Wiley, New YorkGoogle Scholar
  4. 4.
    Chellaboina V., Bhat S.P., Haddad W.M., Bernstein D.S. (2009) Modeling and analysis of mass-action kinetics—nonnegativity, realizability, reducibility, and semistability. IEEE Control Syst. Mag. 29: 60–78CrossRefGoogle Scholar
  5. 5.
    Craciun G., Dickenstein A., Shiu A., Sturmfels B. (2009) Toric dynamical systems. J. Symb. Comput. 44: 1551–1565CrossRefGoogle Scholar
  6. 6.
    Craciun G., Feinberg M. (2005) Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J. Appl. Math. 65(5): 1526–1546CrossRefGoogle Scholar
  7. 7.
    Craciun G., Feinberg M. (2006) Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph. SIAM J. Appl. Math. 66(4): 1321–1338CrossRefGoogle Scholar
  8. 8.
    Dantzig G.B., Thapa M.N. (1997) Linear Programming 1: Introduction. Springer-Verlag, BerlinGoogle Scholar
  9. 9.
    Dantzig G.B., Thapa M.N. (2003) Linear Programming 2: Theory and Extensions. Springer-Verlag, BerlinGoogle Scholar
  10. 10.
    A. Dickenstein, M.P. Millan, How far is complex balancing from detailed balancing? Bull. Math. Biol. (2011). doi:10.1007/s11538-010-9611-7
  11. 11.
    Donoho D.L. (2006) For most large undetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution. Commun. Pure Appl. Math. 59(7): 903–934CrossRefGoogle Scholar
  12. 12.
    Donoho D.L., Tanner J. (2005) Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. USA (PNAS) 102(27): 9446–9451CrossRefGoogle Scholar
  13. 13.
    Tóth P., Tóth J. (1989) Mathematical Models of Chemical Reactions. Theory and Applications of Deterministic and Stochastic Models. Manchester University Press, Princeton University Press, Manchester, PrincetonGoogle Scholar
  14. 14.
    Feinberg M. (1972) Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49: 187–194CrossRefGoogle Scholar
  15. 15.
    Feinberg M. (1979) Lectures on Chemical Reaction Networks. Notes of lectures given at the Mathematics Research Center, University of Wisconsin, Madison, WisconsinGoogle Scholar
  16. 16.
    Feinberg M. (1987) Chemical reaction network structure and the stability of complex isothermal reactors— I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci. 42(10): 2229–2268CrossRefGoogle Scholar
  17. 17.
    Feinberg M. (1988) Chemical reaction network structure and the stability of complex isothermal reactors— II. Multiple steady states for networks of deficiency one. Chem. Eng. Sci. 43: 1–25CrossRefGoogle Scholar
  18. 18.
    Feinberg M. (1989) Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity. Chem. Eng. Sci. 44: 1819–1827CrossRefGoogle Scholar
  19. 19.
    Gopal V., Biegler L.T. (1997) Nonsmooth dynamic simulation with linear programming based methods. Comput. Chem. Eng. 21: 675–689CrossRefGoogle Scholar
  20. 20.
    Gorban A.N., Karlin I.V., Zinovyev A.Y. (2004) Invariant grids for reaction kinetics. Physica A 33: 106–154CrossRefGoogle Scholar
  21. 21.
    J. Gunawardena, Chemical reaction network theory for in-silico biologists. Technical report, Bauer Center for Genomics Research, Harvard University (2003). http://www.vcp.med.harvard.edu/papers/crnt.pdf
  22. 22.
    Hangos K.M., Szederkényi G. (2011) Mass action realizations of reaction kinetic system models on various time scales. J. Phys. Conf. Ser. 268: 012009. doi:10.1088/1742-6596/268/1/012009 CrossRefGoogle Scholar
  23. 23.
    V. Hárs, J. Tóth, In Qualitative Theory of Differential Equations, vol. 30, ed. by M. Farkas, L.Hatvani. On the Inverse Problem of Reaction Kinetics, Coll. Math. Soc. J. Bolyai (North-Holland, Amsterdam, 1981), pp. 363–379Google Scholar
  24. 24.
    Hernández-Bermejo B., Fairén V. (1995) Nonpolynomial vector fields under the Lotka-Volterra normal form. Phys. Lett. A 206: 31–37CrossRefGoogle Scholar
  25. 25.
    Horn F. (1972) Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49: 172–186CrossRefGoogle Scholar
  26. 26.
    Horn F., Jackson R. (1972) General mass action kinetics. Arch. Ration. Mech. Anal. 47: 81–116CrossRefGoogle Scholar
  27. 27.
    Kauchali S., Rooney W.C., Biegler L.T., Glasser D., Hildebrandt D. (2002) Linear programming formulations for attainable region analysis. Chem. Eng. Sci. 57: 2015–2028CrossRefGoogle Scholar
  28. 28.
    Klein J.A., Wu D.T., Gani R. (1992) Computer aided mixture design with specified property constraints. Comput. Chem. Eng. 16: S229–S236CrossRefGoogle Scholar
  29. 29.
    Lente G. (2010) The connection between the second law of thermodynamics and the principle of microscopic reversibility. J. Math. Chem. 47: 1106–1111CrossRefGoogle Scholar
  30. 30.
    Nagy I., Kovács B., Tóth J. (2009) Detailed balance in ion channels: application of Feinberg’s theorem. React. Kinet. Catal. Lett. 96: 263–267CrossRefGoogle Scholar
  31. 31.
    Nicolis G., Prigogine I. (1977) Self-Organization in Nonequilibrium Systems: from Dissipative Structures to Order Through Fluctuations. Wiley, New YorkGoogle Scholar
  32. 32.
    Samardzija N., Greller L.D., Wassermann E. (1989) Nonlinear chemical kinetic schemes derived from mechanical and electrical dynamical systems. J. Chem. Phys. 90(4): 2296–2304CrossRefGoogle Scholar
  33. 33.
    Schnell S., Chappell M.J., Evans N.D., Roussel M.R. (2006) The mechanism distinguishability problem in biochemical kinetics: the single-enzyme, single-substrate reaction as a case study. Comptes Rendus Biol. 329: 51–61CrossRefGoogle Scholar
  34. 34.
    Shinar G., Feinberg M. (2010) Structural sources of robustness in biochemical reaction networks. Science 327: 1389–1391CrossRefGoogle Scholar
  35. 35.
    Sontag E. (2001) Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans. Autom. Control 46: 1028–1047CrossRefGoogle Scholar
  36. 36.
    Szederkényi G. (2009) Computing sparse and dense realizations of reaction kinetic systems. J. Math. Chem. 47: 551–568CrossRefGoogle Scholar
  37. 37.
    Szederkényi G., Hangos K.M., Péni T. (2011) Maximal and minimal realizations of reaction kinetic systems: Computation and properties. MATCH Commun. Math. Comput. Chem. 65(2): 309–332Google Scholar
  38. 38.
    Szederkényi G., Hangos K.M., Magyar A. (2005) On the time-reparametrization of quasi-polynomial systems. Phys. Lett. A 334: 288–294CrossRefGoogle Scholar
  39. 39.
    Yang J., Bruno W.J., Hlavacek W.S., Pearson J.E. (2006) On imposing detailed balance in complex reaction mechanisms. Biophys. J. 91: 1136–1141CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Process Control Research Group, Computer and Automation Research InstituteHungarian Academy of SciencesBudapestHungary
  2. 2.Department of Electrical Engineering and Information SystemsUniversity of PannoniaVeszprémHungary

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