Archive for Rational Mechanics and Analysis

, Volume 47, Issue 2, pp 81–116 | Cite as

General mass action kinetics

  • F. Horn
  • R. Jackson


The familiar idea of mass action kinetics is extended to embrace situations more general than chemically reacting mixtures in closed vessels. Thus, for example, many reaction regions connected by convective or diffusive mass transport, such as the cellular aggregates of biological tissue, are drawn into a common mathematical scheme.

The ideas of chemical thermodynamics, such as the algebraic nature of the equilibrium conditions and the decreasing property of the free energy, are also generalized in a natural way, and it is then possible to identify classes of generalized kinetic expressions which ensure consistency with the extended thermodynamic conditions. The principal result of this work shows that there exists a simply identifiable class of kinetic expressions, including the familiar detailed balanced kinetics as a proper subclass, which ensure consistency with the extended thermodynamic conditions. For kinetics of this class, which we call complex balanced kinetics, exotic behavior such as bistability and oscillation is precluded, so the domain of search for kinetic expressions with this type of behavior, which is of considerable biological interest, is greatly narrowed.

It is also shown that the ideas of complex balancing and of detailed balancing are closely related to symmetry under time reversal.


Time Reversal Diffusive Mass Detailed Balance Closed Vessel Biological Interest 
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  1. 1.
    Feinberg, M., On chemical kinetics of a certain class. Arch. Rational Mech. Anal. 46, 1–41 (1972).Google Scholar
  2. 2.
    Higgins, J., Some remarks on Shear's Liapunov function for systems of chemical reactions. J. Theoret. Biol. 21, 293–304 (1968).Google Scholar
  3. 3.
    Horn, F., Necessary and sufficient conditions for complex balancing in chemical kinetics. In preparation.Google Scholar
  4. 4.
    Horn, F., General first order kinetics. Ber. Bunsenges. 75, 1191 (1971).Google Scholar
  5. 5.
    Horn, F., Complex balancing and stability in reaction systems with three complexes. In preparation.Google Scholar
  6. 6.
    Krambeck, F. J., The mathematical structure of chemical kinetics in homogeneous singlephase systems. Arch. Rational Mech. Anal. 38, 317–347 (1970).Google Scholar
  7. 7.
    LaSalle, J., & S. Lefschetz, Stability by Liapunov's Direct Method, with Applications. New York: Academic Press 1961.Google Scholar
  8. 8.
    Lewis, G. N., A new principle of equilibrium. Proc. Natn. Acad. Sci. 11, 179–183 (1925).Google Scholar
  9. 9.
    Shear, D., An analog of the Boltzman H-theorem (a Lyapunov function) for systems of coupled chemical reactions. J. Theoret. Biol. 16, 212–228 (1967).Google Scholar
  10. 10.
    Tucker, A. W., Dual Systems of Homogeneous Linear Relations in Linear Inequalities and Related Systems. H. W. Kuhn & A. W. Tucker, editors, Princeton University Press 1956.Google Scholar
  11. 11.
    Wegscheider, R., Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reaktionskinetik homogener Systeme. Z. Phys. Chem. 39, 257–303 (1902).Google Scholar
  12. 12.
    Wei, J., Axiomatic treatment of chemical reaction systems. J. Chem. Phys. 36, 1578–1584 (1962).Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • F. Horn
    • 1
    • 2
  • R. Jackson
    • 1
    • 2
  1. 1.University of RochesterUSA
  2. 2.Rice UniversityUSA

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