Skip to main content
SpringerLink
Log in

Finding complex balanced and detailed balanced realizations of chemical reaction networks

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Angeli D. (2009) A tutorial on chemical network dynamics. Eur. J. Control 15: 398–406

    Article  Google Scholar 

  2. Belov G. (2010) On linear programming approach for the calculation of chemical equilibrium in complex thermodynamic systems. J. Math. Chem. 47: 446–456

    Article  CAS  Google Scholar 

  3. Callen H.B. (1980) Thermodynamics and an Introduction to Thermostatistics. Wiley, New York

    Google Scholar 

  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: 60–78

    Article  Google Scholar 

  5. Craciun G., Dickenstein A., Shiu A., Sturmfels B. (2009) Toric dynamical systems. J. Symb. Comput. 44: 1551–1565

    Article  Google Scholar 

  6. Craciun G., Feinberg M. (2005) Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J. Appl. Math. 65(5): 1526–1546

    Article  CAS  Google Scholar 

  7. Craciun G., Feinberg M. (2006) Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph. SIAM J. Appl. Math. 66(4): 1321–1338

    Article  CAS  Google Scholar 

  8. Dantzig G.B., Thapa M.N. (1997) Linear Programming 1: Introduction. Springer-Verlag, Berlin

    Google Scholar 

  9. Dantzig G.B., Thapa M.N. (2003) Linear Programming 2: Theory and Extensions. Springer-Verlag, Berlin

    Google Scholar 

  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. 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): 903–934

    Article  Google Scholar 

  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): 9446–9451

    Article  CAS  Google Scholar 

  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, Princeton

    Google Scholar 

  14. Feinberg M. (1972) Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49: 187–194

    Article  Google Scholar 

  15. Feinberg M. (1979) Lectures on Chemical Reaction Networks. Notes of lectures given at the Mathematics Research Center, University of Wisconsin, Madison, Wisconsin

    Google Scholar 

  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): 2229–2268

    Article  CAS  Google Scholar 

  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: 1–25

    Article  CAS  Google Scholar 

  18. Feinberg M. (1989) Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity. Chem. Eng. Sci. 44: 1819–1827

    Article  CAS  Google Scholar 

  19. Gopal V., Biegler L.T. (1997) Nonsmooth dynamic simulation with linear programming based methods. Comput. Chem. Eng. 21: 675–689

    Article  CAS  Google Scholar 

  20. Gorban A.N., Karlin I.V., Zinovyev A.Y. (2004) Invariant grids for reaction kinetics. Physica A 33: 106–154

    Article  Google Scholar 

  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. 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

    Article  Google Scholar 

  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. 363–379

  24. Hernández-Bermejo B., Fairén V. (1995) Nonpolynomial vector fields under the Lotka-Volterra normal form. Phys. Lett. A 206: 31–37

    Article  Google Scholar 

  25. Horn F. (1972) Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49: 172–186

    Article  Google Scholar 

  26. Horn F., Jackson R. (1972) General mass action kinetics. Arch. Ration. Mech. Anal. 47: 81–116

    Article  Google Scholar 

  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: 2015–2028

    Article  CAS  Google Scholar 

  28. Klein J.A., Wu D.T., Gani R. (1992) Computer aided mixture design with specified property constraints. Comput. Chem. Eng. 16: S229–S236

    Article  CAS  Google Scholar 

  29. Lente G. (2010) The connection between the second law of thermodynamics and the principle of microscopic reversibility. J. Math. Chem. 47: 1106–1111

    Article  CAS  Google Scholar 

  30. Nagy I., Kovács B., Tóth J. (2009) Detailed balance in ion channels: application of Feinberg’s theorem. React. Kinet. Catal. Lett. 96: 263–267

    Article  CAS  Google Scholar 

  31. Nicolis G., Prigogine I. (1977) Self-Organization in Nonequilibrium Systems: from Dissipative Structures to Order Through Fluctuations. Wiley, New York

    Google Scholar 

  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): 2296–2304

    Article  CAS  Google Scholar 

  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: 51–61

    Article  CAS  Google Scholar 

  34. Shinar G., Feinberg M. (2010) Structural sources of robustness in biochemical reaction networks. Science 327: 1389–1391

    Article  CAS  Google Scholar 

  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: 1028–1047

    Article  Google Scholar 

  36. Szederkényi G. (2009) Computing sparse and dense realizations of reaction kinetic systems. J. Math. Chem. 47: 551–568

    Article  Google Scholar 

  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): 309–332

    Google Scholar 

  38. Szederkényi G., Hangos K.M., Magyar A. (2005) On the time-reparametrization of quasi-polynomial systems. Phys. Lett. A 334: 288–294

    Article  Google Scholar 

  39. Yang J., Bruno W.J., Hlavacek W.S., Pearson J.E. (2006) On imposing detailed balance in complex reaction mechanisms. Biophys. J. 91: 1136–1141

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Process Control Research Group, Computer and Automation Research Institute, Hungarian Academy of Sciences, P.O. Box 63, 1518, Budapest, Hungary

    Gábor Szederkényi & Katalin M. Hangos

  2. Department of Electrical Engineering and Information Systems, University of Pannonia, Egyetem u. 10, 8200, Veszprém, Hungary

    Katalin M. Hangos

Authors
  1. Gábor Szederkényi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Katalin M. Hangos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gábor Szederkényi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Szederkényi, G., Hangos, K.M. Finding complex balanced and detailed balanced realizations of chemical reaction networks. J Math Chem 49, 1163–1179 (2011). https://doi.org/10.1007/s10910-011-9804-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-011-9804-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation