Journal of Mathematical Chemistry

, Volume 50, Issue 1, pp 1–8 | Cite as

Parametric uniqueness of deficiency zero reaction networks

  • Dávid Csercsik
  • Gábor SzederkényiEmail author
  • Katalin M. Hangos
Brief Communication


In this paper it is shown that deficiency zero mass action reaction networks containing one terminal linkage class are parametrically and therefore structurally unique with a fixed complex set. Clearly, weakly reversible deficiency zero networks with one linkage class belong to this class. However, it is shown through an illustrative example that deficiency zero networks with several linkage classes can have multiple dynamically equivalent realizations, even if the individual linkage classes are weakly reversible.


Reaction kinetic systems Mass action kinetics Dynamical equivalence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Craciun G., Feinberg M.: Multiple equilibria in complex chemical reaction networks: I. the injectivity property. SIAM J. Appl. Math. 65(5), 1526–1546 (2005)CrossRefGoogle Scholar
  2. 2.
    Craciun G., Tang Y., Feinberg M.: Understanding bistability in complex enzyme-driven reaction networks. Proc. Natl. Acad. Sci. USA 103(23), 8697–8702 (2006)CrossRefGoogle Scholar
  3. 3.
    M. Feinberg, Lectures on Chemical Reaction Networks. Notes of Lectures Given at the Mathematics Research Center (University of Wisconsin, Madison, 1979).
  4. 4.
    Feinberg M.: 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 (1987)CrossRefGoogle Scholar
  5. 5.
    J. Gunawardena, Chemical reaction network theory for in-silico biologists (2003)Google Scholar
  6. 6.
    K.M. Hangos, G. Szederkényi, Special positive systems: the QP and the reaction kinetic system class. In Preprints of the Workshop on Systems and Control Theory in honor of József Bokor on his 60th birthday. (Hungarian, Academy of Sciences, 2008)Google Scholar
  7. 7.
    V. Hárs, J. Tóth, On the inverse problem of reaction kinetics. In Qualitative Theory of Differential Equations, vol. 30 of Coll. Math. Soc. J. Bolyai., ed. M. Farkas, L. Hatvani (North-Holland, Amsterdam, 1981), pp. 363–379Google Scholar
  8. 8.
    Prasolov V.V.: Problems and Theorems in Linear Algebra. American Mathematical Society, Providence, RI (1994)Google Scholar
  9. 9.
    Schnell S., Chappell M.J., Evans N.D., Roussel M.R.: The mechanism distinguishability problem in biochemical kinetics: the single-enzyme, single-substrate reaction as a case study. Comptes Rendus Biologies 329, 51–61 (2006)CrossRefGoogle Scholar
  10. 10.
    Szederkényi G.: Computing sparse and dense realizations of reaction kinetic systems. J. Math. Chem. 47, 551–568 (2010)CrossRefGoogle Scholar
  11. 11.
    Szederkényi G., Hangos K.M., Péni T.: Maximal and minimal realizations of reaction kinetic systems: computation and properties. MATCH Commun. Math. Comput. Chem. 65, 309–332 (2011)Google Scholar
  12. 12.
    Thomas R., Kaufman M.: Multistationarity, the basis of cell differentiation and memory. I. structural conditions of multistationarity and other nontrivial behaviour. Chaos 11, 170–179 (2001)CrossRefGoogle Scholar
  13. 13.
    Érdi P., Tóth J.: Mathematical Models of Chemical Reactions. Theory and Applications of Deterministic and Stochastic Models. Manchester University Press, Princeton University Press, Manchester, Princeton (1989)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Dávid Csercsik
    • 1
  • Gábor Szederkényi
    • 1
    Email author
  • Katalin M. Hangos
    • 1
    • 2
  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

Personalised recommendations