Journal of Mathematical Chemistry

, Volume 44, Issue 1, pp 244–259

Identifiability of chemical reaction networks

Original Paper


We consider the dynamics of chemical reaction networks under the assumption of mass-action kinetics. We show that there exist reaction networks \({\mathcal{R}}\) for which the reaction rate constants are not uniquely identifiable, even if we are given complete information on the dynamics of concentrations for all chemical species of \({\mathcal{R}}\) . Also, we show that there exist reaction networks \({\mathcal{R}}_1 \neq {\mathcal{R}}_2\) such that their dynamics are identical under appropriate choices of reaction rate constants, and present theorems that characterize the properties of \({\mathcal{R}}\) , \({\mathcal{R}}_1\) , \({\mathcal{R}}_2\) that make this possible. We use these facts to show how we can determine dynamical properties of some chemical networks by analyzing other chemical networks.


Chemical reaction networks Mass-action kinetics  Parameter identification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. M. Feinberg, in Chemical Reactor Theory: A Review, eds. by N. Amundson, L. Lapidus (Prentice-Hall, Englewood Cliffs, NJ, 1977), pp. 1–78Google Scholar
  2. M. Feinberg, Lectures on Chemical Reaction Networks, written version of lectures given at the Mathematical Research Center (University of Wisconsin, Madison, WI, 1979). Available online from Scholar
  3. M. Feinberg, in Patterns and Dynamics in Reactive Media, IMA Volumes in Mathematics and its Application, vol. 37, eds. by R. Aris, D.G. Aronson, H.L. Swinney (Springer, Berlin, 1991), pp. 43–70Google Scholar
  4. Feinberg M. (1995). Arch. Rational Mech. Anal. 132: 311CrossRefGoogle Scholar
  5. Feinberg M. (1995). Arch. Rational Mech. Anal. 132: 371CrossRefGoogle Scholar
  6. Craciun G., Tang Y., Feinberg M. (2006). Proc. Natl. Acade. Scie. 103(23): 8697CrossRefGoogle Scholar
  7. Craciun G., Feinberg M. (2005). SIAM J. Appl. Math. 65: 1526CrossRefGoogle Scholar
  8. Craciun G., Feinberg M. (2006). SIAM J. Appl. Math. 66: 1321CrossRefGoogle Scholar
  9. Craciun G., Feinberg M. (2006). IEE Proc. Systems Biolo. 153(4): 179CrossRefGoogle Scholar
  10. B.D. Aguda, G. Craciun, R. Cetin-Atalay, in Data Sources and Computational Approaches for Generating Models of Gene Regulatory Networks, Reviews in Computational Chemistry, vol. 21, eds. by K. Lipkowitz, R. Larter, T.R. Cundari (John Wiley & Sons, Hoboken, NJ) (2005)Google Scholar
  11. Sontag E. (2001). IEEE Trans. Automat. Control 46: 1028CrossRefGoogle Scholar
  12. Maria G. (2004). Chem. Biochem. Eng. Q. 18(3): 195Google Scholar
  13. Crampin E.J., Schnell S., McSharry P.E. (2004). Prog. Biophys. Mol. Biol. 86: 177Google Scholar
  14. Karnaukhov A., Karnaukhova E., Williamson J. (2007). Biophys. J. 92: 3459CrossRefGoogle Scholar
  15. Vajda S., Valko P., Yermakova A. (1986). Compute. Chem. Eng. 10: 49CrossRefGoogle Scholar
  16. Hosten L.H. (1979). Comput. Chem. Engi. 3: 117CrossRefGoogle Scholar
  17. Himmelau D.M., Jones C.R., Bischoff K.B. (1967). Ind. Eng. Chem. Fundam. 6(4): 539CrossRefGoogle Scholar
  18. Schuster S., Hilgetag C., Woods J.H., Fell D.A. (2002). J. Math. Biol. 45: 153CrossRefGoogle Scholar
  19. Rudakov E. (1960). Kinet. Catal. 1, 177Google Scholar
  20. Rudakov E. (1970). Kinet. Catal. 11: 228Google Scholar
  21. Fay L., Balogh A. (1968). Acta Chim. Acad. Sci. Hun. 57(4): 391Google Scholar
  22. P. Erdi, J. Toth, in Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models (Princeton University Press, 1989)Google Scholar
  23. I.R Epstein, J.A. Pojman, in An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos (Oxford University Press, 2002)Google Scholar
  24. R.T. Rockafellar, in Convex Analysis, (Princeton, NJ, 1970)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Department of Biomolecular ChemistryUniversity of Wisconsin-MadisonMadisonUSA

Personalised recommendations