Journal of Mathematical Chemistry

, Volume 47, Issue 2, pp 551–568 | Cite as

Computing sparse and dense realizations of reaction kinetic systems

Original Paper


A numerical procedure for finding the sparsest and densest realization of a given reaction network is proposed in this paper. The problem is formulated and solved in the framework of mixed integer linear programming (MILP) where the continuous optimization variables are the nonnegative reaction rate coefficients, and the corresponding integer variables ensure the finding of the realization with the minimal or maximal number of reactions. The mass-action kinetics is expressed in the form of linear constraints adjoining the optimization problem. More complex realization problems can also be solved using the proposed framework by modifying the objective function and/or the constraints appropriately.


Reaction kinetic systems Mass action kinetics Mixed integer linear programming 


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  1. 1.
    Aykanat C., Pinar A.: Permuting sparse rectangular matrices into block-diagonal form. SIAM J. Scientific Comput. 25, 1860–1879 (2004)CrossRefGoogle Scholar
  2. 2.
    J. Bang-Jensen, G. Gutin, Digraphs: Theory, Algorithms and Applications (Springer, 2001)Google Scholar
  3. 3.
    Bemporad A., Morari M.: Control of systems integrating logic, dynamics, and constraints. Automatica 35, 407–427 (1999)CrossRefGoogle Scholar
  4. 4.
    Cavalier T.M., Pardalos P.M., Soyster A.L.: Modeling and integer programming techniques applied to propositional calculus. Comput. Oper. Res. 17(6), 561–570 (1990)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Craciun G., Feinberg M.: Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph. SIAM J. Appl. Math. 66(4), 1321–1338 (2006)CrossRefGoogle Scholar
  7. 7.
    Craciun G., Pantea C.: Identifiability of chemical reaction networks. J. Math. Chem. 44, 244–259 (2008)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Donoho D.L.: 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 (2006)CrossRefGoogle Scholar
  10. 10.
    Donoho D.L., Tanner J.: Sparse nonnegative solution of underdetermined linear equations by linear programming. Proc. Natl. Acad. Sci. USA 102(27), 9446–9451 (2005)CrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Farkas Gy.: Local controllability of reactions. J. Math. Chem. 24, 1–14 (1998)CrossRefGoogle Scholar
  13. 13.
    Farkas Gy.: On local observability of chemical systems. J. Math. Chem. 24, 15–22 (1998)CrossRefGoogle Scholar
  14. 14.
    Farkas Gy.: Kinetic lumping schemes. Chem. Eng. Sci. 54, 3909–3915 (1999)CrossRefGoogle Scholar
  15. 15.
    M. Feinberg, Lectures on Chemical Reaction Networks. Notes of lectures given at the Mathematics Research Center (University of Wisconsin, 1979)Google Scholar
  16. 16.
    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
  17. 17.
    Feinberg M.: 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
  18. 18.
    C.A. Floudas, Nonlinear and Mixed-Integer Optimization (Oxford University Press, 1995)Google Scholar
  19. 19.
    Gorban A.N., Karlin I.V.: Method of invariant manifold for chemical kinetics. Chem. Eng. Sci. 58, 4751–4768 (2003)CrossRefGoogle Scholar
  20. 20.
    Gorban A.N., Karlin I.V., Zinovyev A.Y.: Invariant grids for reaction kinetics. Physica A 33, 106–154 (2004)CrossRefGoogle Scholar
  21. 21.
    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
  22. 22.
    Hansen P., Zheng M.: The Clar number of a benzenoid hydrocarbon and linear programming. J. Math. Chem. 15, 93–107 (1992)CrossRefGoogle Scholar
  23. 23.
    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. by M. Farkas, L. Hatvani (North-Holland, Amsterdam, 1981), pp. 363–379Google Scholar
  24. 24.
    K. Homlström, M.M. Edvall, A.O. Göran, TOMLAB for large-scale robust optimization, in Nordic MATLAB Conference (2003)Google Scholar
  25. 25.
    J. Löfberg, YALMIP: a toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference (Taipei, Taiwan, 2004)Google Scholar
  26. 26.
    A. Makhorin, GNU Linear Programming Kit. Reference Manual.Version 4.10 (2006)Google Scholar
  27. 27.
    G.L. Nemhauser, L.A. Wolsey. Integer and Combinatorial Optimization (Wiley, 1988)Google Scholar
  28. 28.
    Otero-Muras I., Szederkényi G., Alonso A.A., Hangos K.M.: Local dissipative Hamiltonian description of reversible reaction networks. Syst. Control Lett. 57, 554–560 (2008)CrossRefGoogle Scholar
  29. 29.
    Raman R., Grossmann I.E.: Relation between MILP modelling and logical inference for chemical process synthesis. Comput. Chem. Eng. 15, 73–84 (1991)CrossRefGoogle Scholar
  30. 30.
    Raman R., Grossmann I.E.: Integration of logic and heuristic knowledge in MINLP optimization for process synthesis. Comput. Chem. Eng. 16(3), 155–171 (1992)CrossRefGoogle Scholar
  31. 31.
    Raman R., Grossmann I.E.: Modelling and computational techniques for logic based integer programming. Comput. Chem. Eng. 18, 563–578 (1994)CrossRefGoogle Scholar
  32. 32.
    Saled K., Abeledo H.: Alternative integer-linear-programming formulations of the Clar problem in hexagonal systems. J. Math. Chem. 39, 605–610 (2006)CrossRefGoogle Scholar
  33. 33.
    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. C. R. Biol. 329, 51–61 (2006)CrossRefGoogle Scholar
  34. 34.
    Sontag E.: 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 (2001)CrossRefGoogle Scholar
  35. 35.
    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
  36. 36.
    H.P. Williams, Model Building in Mathematical Programming (Wiley, 1993)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Process Control Research Group, Systems and Control Laboratory, Computer and Automation Research InstituteHungarian Academy of SciencesBudapestHungary

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