Journal of Mathematical Biology

, Volume 74, Issue 1–2, pp 195–237 | Cite as

Graphical reduction of reaction networks by linear elimination of species



The quasi-steady state approximation and time-scale separation are commonly applied methods to simplify models of biochemical reaction networks based on ordinary differential equations (ODEs). The concentrations of the “fast” species are assumed effectively to be at steady state with respect to the “slow” species. Under this assumption the steady state equations can be used to eliminate the “fast” variables and a new ODE system with only the slow species can be obtained. We interpret a reduced system obtained by time-scale separation as the ODE system arising from a unique reaction network, by identification of a set of reactions and the corresponding rate functions. The procedure is graphically based and can easily be worked out by hand for small networks. For larger networks, we provide a pseudo-algorithm. We study properties of the reduced network, its kinetics and conservation laws, and show that the kinetics of the reduced network fulfil realistic assumptions, provided the original network does. We illustrate our results using biological examples such as substrate mechanisms, post-translational modification systems and networks with intermediates (transient) steps.


Reduced network Quasi-steady-state Species graph Noninteracting Dynamical system Positivity 

Mathematics Subject Classification

92C42 80A30 



We thank E. Tonello for pointing out a mistake in a previous version of the paper. We thank the reviewers for useful suggestions that improved the presentation of the manuscript. MS, EF, CW are supported by The Lundbeck Foundation (Denmark). EF and CW acknowledge funding from the Danish Research Council of Independent Research. MS has been supported by the project MTM2012-38122-C03-02/FEDER from the Ministerio de Economía y Competitividad, Spain.


  1. Berge C (1985) Graphs. Amsterdam, North-HollandGoogle Scholar
  2. Cornish-Bowden A (2004) Fundamentals of Enzyme Kinetics, 3rd edn. Portland Press, LondonGoogle Scholar
  3. der Rao S, Schaft Av, Eunen Kv, Bakker BM, Jayawardhana B (2014) A model reduction method for biochemical reaction networks. BMC Syst Biol 8(1):1–17. doi: 10.1186/1752-0509-8-52 CrossRefGoogle Scholar
  4. Feliu E, Wiuf C (2012) Variable elimination in chemical reaction networks with mass-action kinetics. SIAM J Appl Math 72:959–981MathSciNetCrossRefMATHGoogle Scholar
  5. Feliu E, Wiuf C (2013) Simplifying biochemical models with intermediate species. J R Soc Interface 10:20130484CrossRefGoogle Scholar
  6. Feliu E, Wiuf C (2013) Variable elimination in post-translational modification reaction networks with mass-action kinetics. J Math Biol 66(1):281–310MathSciNetCrossRefMATHGoogle Scholar
  7. Frey P, Hegeman A (2007) Enzymatic reaction mechanisms. Oxford University Press, New YorkGoogle Scholar
  8. Gábor A, Hangos KM, Banga JR, Szederkényi G (2015) Reaction network realizations of rational biochemical systems and their structural properties. J Math Chem 53:1657–1686. doi: 10.1007/s10910-015-0511-9 MathSciNetCrossRefMATHGoogle Scholar
  9. Goeke A, Walcher S, Zerz E (2015) Computer Algebra in Scientific Computing. In: 17th International Workshop, CASC 2015, Aachen, Germany, September 14–18, 2015, Proceedings, chap. Quasi-Steady State—Intuition, Perturbation Theory and Algorithmic Algebra. Springer International Publishing, Cham, pp 135–151. doi: 10.1007/978-3-319-24021-3_10
  10. Gunawardena J (2012) A linear framework for time-scale separation in nonlinear biochemical systems. PLoS One 7(5):e36321CrossRefGoogle Scholar
  11. Horiuti J, Nakamura T (1957) Stoichiometric number and the theory of steady reaction. Z Phys Chem 11:358–365CrossRefGoogle Scholar
  12. Joshi B, Shiu A (2013) Atoms of multistationarity in chemical reaction networks. J Math Chem 51(1):153–178MathSciNetMATHGoogle Scholar
  13. King EL, Altman C (1956) A schematic method of deriving the rate laws for enzyme-catalyzed reactions. J Phys Chem 60:1375–1378CrossRefGoogle Scholar
  14. Pantea C, Gupta A, Rawlings JB, Craciun G (2014) The QSSA in chemical kinetics: as taught and as practiced. Discrete and topological models in molecular biology natural computing series, pp 419–442Google Scholar
  15. Radulescu O, Gorban AN, Zinovyev A, Lilienbaum A (2008) Robust simplifications of multiscale biochemical networks. BMC Syst Biol 2:86CrossRefGoogle Scholar
  16. Szili L, Tóth J (1997) On the origin of Turing instability. J Math Chem 22(1):39–53. doi: 10.1023/A:1019159427561 MathSciNetCrossRefMATHGoogle Scholar
  17. Temkin M (1965) Graphical method for the derivation of the rate laws of complex reactions. Dokl Akad Nauk SSSR 165:615–618Google Scholar
  18. Temkin ON, Bonchev DG (1992) Application of graph theory to chemical kinetics: Part 1. Kinetics of complex reactions. J Chem Educ 69(7):544. doi: 10.1021/ed069p544 CrossRefGoogle Scholar
  19. Temkin ON, Zeigarnik AV, Bonchev D (1996) Chemical reaction networks: a graph-theoretical approach. CRC Press, Boca RatonGoogle Scholar
  20. Thomson M, Gunawardena J (2009) The rational parameterization theorem for multisite post-translational modification systems. J Theor Biol 261:626–636MathSciNetCrossRefGoogle Scholar
  21. Tikhonov AN (1952) Systems of differential equations containing small parameters in the derivatives. Mat Sbornik N S 31(73):575–586MathSciNetGoogle Scholar
  22. Walther H (1985) Ten applications of graph theory. Springer, BerlinMATHGoogle Scholar
  23. Wong JT, Hanes CS (1962) Kinetic formulations for enzymic reactions involving two substrates. Can J Biochem Physiol 40:763–804CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematical Sciences. University of CopenhagenCopenhagenDenmark

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