Journal of Mathematical Chemistry

, Volume 52, Issue 5, pp 1386–1404 | Cite as

Polynomial time algorithms to determine weakly reversible realizations of chemical reaction networks

  • János Rudan
  • Gábor Szederkényi
  • Katalin M. Hangos
  • Tamás Péni
Original Paper


Weak reversibility is a crucial structural property of chemical reaction networks (CRNs) with mass action kinetics, because it has major implications related to the existence, uniqueness and stability of equilibrium points and to the boundedness of solutions. In this paper, we present two new algorithms to find dynamically equivalent weakly reversible realizations of a given CRN. They are based on linear programming and thus have polynomial time-complexity. Hence, these algorithms can deal with large-scale biochemical reaction networks, too. Furthermore, one of the methods is able to deal with linearly conjugate networks, too.


Chemical reaction networks Weak reversibility  Dynamical equivalence Linear conjugacy Optimization 



This research has been supported by the Hungarian National Research Fund through the Grants NF104706 and K83440. The first and second authors were also supported by the projects TÁMOP-4.2.1./B-11/2/KMR-2011-002 and TÁMOP-4.2.2./B-10/1-2010-0014.


  1. 1.
    P. Érdi, J. Tóth, Mathematical Models of Chemical Reactions. Theory and Applications of Deterministic and Stochastic Models (Manchester University Press, Princeton University Press, Manchester, Princeton, 1989)Google Scholar
  2. 2.
    Z.A. Tuza, G. Szederkényi, K.M. Hangos, A.A. Alonso, J.R. Banga, Computing all sparse kinetic structures for a Lorenz system using optimization. Int. J. Bifurcation Chaos 23, 1350141-1–1350141-17 (2013).Google Scholar
  3. 3.
    W.M. Haddad, V.S. Chellaboina, Q. Hui, Nonnegative and Compartmental Dynamical Systems (Princeton University Press, Princeton, 2010)Google Scholar
  4. 4.
    N. Samardzija, L.D. Greller, E. Wassermann, Nonlinear chemical kinetic schemes derived from mechanical and electrical dynamical systems. J. Chem. Phys. 90(4), 2296–2304 (1989)CrossRefGoogle Scholar
  5. 5.
    F. Horn, R. Jackson, General mass action kinetics. Arch. Ration. Mech. Anal. 47, 81–116 (1972)CrossRefGoogle Scholar
  6. 6.
    G. Szederkényi, Computing sparse and dense realizations of reaction kinetic systems. J. Math. Chem. 47, 551–568 (2010)CrossRefGoogle Scholar
  7. 7.
    G. Szederkényi, K.M. Hangos, Finding complex balanced and detailed balanced realizations of chemical reaction networks. J. Math. Chem. 49, 1163–1179 (2011)CrossRefGoogle Scholar
  8. 8.
    G. Szederkényi, K.M. Hangos, T. Péni, Maximal and minimal realizations of reaction kinetic systems: computation and properties. MATCH Commun. Math. Comput. Chem. 65, 309–332 (2011)Google Scholar
  9. 9.
    G. Szederkényi, K.M. Hangos, Z. Tuza, Finding weakly reversible realizations of chemical reaction networks using optimization. MATCH Commun. Math. Comput. Chem. 67, 193–212 (2012)Google Scholar
  10. 10.
    J. Rudan, G. Szederkényi, K.M. Hangos, Efficiently computing alternative structures of large biochemical reaction networks using linear programming. MATCH Commun. Math. Comput. Chem. 71, 71–92 (2014)Google Scholar
  11. 11.
    M. Feinberg, 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 (1988)CrossRefGoogle Scholar
  12. 12.
    D.F. Anderson, Boundedness of trajectories for weakly reversible, single linkage class reaction systems. J. Math. Chem. 49, 1–16 (2011). doi: 10.1007/s10910-011-9886-4 CrossRefGoogle Scholar
  13. 13.
    J. Deng, C. Jones, M. Feinberg, A. Nachman, On the steady states of weakly reversible chemical reaction networks (2011),
  14. 14.
    B. Boros, On the existence of the positive steady states of weakly reversible deficiency-one mass action systems. Math. Biosci. 245, 157–170 (2013)CrossRefGoogle Scholar
  15. 15.
    M.D. Johnston, D. Siegel, G. Szederkényi, A linear programming approach to weak reversibility and linear conjugacy of chemical reaction networks. J. Math. Chem. 50, 274–288 (2012)CrossRefGoogle Scholar
  16. 16.
    M.D. Johnston, D. Siegel, G. Szederkényi, Dynamical equivalence and linear conjugacy of chemical reaction networks: new results and methods. MATCH Commun. Math. Comput. Chem. 68, 443–468 (2012)Google Scholar
  17. 17.
    M.D. Johnston, D. Siegel, G. Szederkényi, Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiency. Math. Biosci. 241, 88–98 (2013)CrossRefGoogle Scholar
  18. 18.
    M.D. Johnston, D. Siegel, Linear conjugacy of chemical reaction networks. J. Math. Chem. 49, 1263–1282 (2011)CrossRefGoogle Scholar
  19. 19.
    M. Feinberg, Lectures on Chemical Reaction Networks. Notes of Lectures Given at the Mathematics Research Center. (University of Wisconsin, 1979)Google Scholar
  20. 20.
    V. Hárs, J. Tóth, On the inverse problem of reaction kinetics, in Qualitative Theory of Differential Equations, volume 30 of Coll. Math. Soc. J. Bolyai, ed. by M. Farkas, L. Hatvani (North-Holland, Amsterdam, 1981), pp. 363–379Google Scholar
  21. 21.
    V. Chellaboina, S.P. Bhat, W.M. Haddad, D.S. Bernstein, Modeling and analysis of mass-action kinetics—nonnegativity, realizability, reducibility, and semistability. IEEE Control Syst. Mag. 29, 60–78 (2009)CrossRefGoogle Scholar
  22. 22.
    G. Farkas, Kinetic lumping schemes. Chem. Eng. Sci. 54, 3909–3915 (1999)CrossRefGoogle Scholar
  23. 23.
    R. Raman, I.E. Grossmann, Modelling and computational techniques for logic based integer programming. Comput. Chem. Eng. 18, 563–578 (1994)CrossRefGoogle Scholar
  24. 24.
    D.L. Donoho, J. Tanner, Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. USA (PNAS) 102(27), 9446–9451 (2005)CrossRefGoogle Scholar
  25. 25.
    G. Szederkényi, J.R. Banga, A.A. Alonso, CRNreals: a toolbox for distinguishability and identifiability analysis of biochemical reaction networks. Bioinformatics 28(11), 1549–1550 (June 2012)Google Scholar
  26. 26.
    J. Löfberg, YALMIP: a toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference (Taipei, Taiwan, 2004)Google Scholar
  27. 27.
    CLP—Coin-or linear programming.
  28. 28.
    GLPK—GNU Linear Programming Toolkit.
  29. 29.
    K.P. Eswaran, R.E. Tarjan, Augmentation problems. SIAM J. Comput. 5, 653–665 (1976)CrossRefGoogle Scholar
  30. 30.
    S. Raghavan, The next wave in computing, optimization, and decision technologies, chapter, in A Note on Eswaran and Tarjan’s Algorithm for the Strong Connectivity Augmentation Problem (Kluwer, Dordrecht, 2005), pp. 19–26Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • János Rudan
    • 1
  • Gábor Szederkényi
    • 1
    • 2
  • Katalin M. Hangos
    • 2
    • 3
  • Tamás Péni
    • 2
  1. 1.Faculty of Information TechnologyPázmány Péter Catholic UniversityBudapestHungary
  2. 2.Systems and Control Laboratory, Institute for Computer Science and Control (MTA SZTAKI)Hungarian Academy of SciencesBudapestHungary
  3. 3.Department of Electrical Engineering and Information SystemsUniversity of PannoniaVeszprémHungary

Personalised recommendations