Bulletin of Mathematical Biology

, Volume 79, Issue 7, pp 1662–1686 | Cite as

Intermediates and Generic Convergence to Equilibria

  • Michael Marcondes de Freitas
  • Carsten Wiuf
  • Elisenda Feliu
Original Article


Known graphical conditions for the generic and global convergence to equilibria of the dynamical system arising from a reaction network are shown to be invariant under the so-called successive removal of intermediates, a systematic procedure to simplify the network, making the graphical conditions considerably easier to check.


Model reduction Monotonicity in reaction coordinates Monotonicity R-graph SR-graph Reduction 



Elisenda Feliu, Michael Marcondes de Freitas and Carsten Wiuf acknowledge funding from the Danish Research Council of Independent Research. We would also like to thank Anne Shiu and Mitchell Eithun for their careful reading of an earlier version of this paper and valuable comments.


  1. Amann H (1990) Ordinary differential equations, volume 13 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin. An introduction to nonlinear analysis, Translated from the German by Gerhard MetzenGoogle Scholar
  2. Angeli D, De Leenheer P, Sontag ED (2007) A Petri net approach to the study of persistence in chemical reaction networks. Math Biosci 210(2):598–618MathSciNetCrossRefMATHGoogle Scholar
  3. Angeli D, De Leenheer P, Sontag ED (2010) Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. J Math Biol 61(4):581–616MathSciNetCrossRefMATHGoogle Scholar
  4. Conradi C, Shiu A (2015) A global convergence result for processive multisite phosphorylation systems. Bull Math Biol 77(1):126–155MathSciNetCrossRefMATHGoogle Scholar
  5. Cornish-Bowden A (2004) Fundamentals of enzyme kinetics, 3rd edn. Portland Press, LondonGoogle Scholar
  6. de Freitas MM, Feliu E, Wiuf C (2017) Intermediates, catalysts, persistence, and boundary steady states. J Math Biol 74:887–932MathSciNetCrossRefMATHGoogle Scholar
  7. Eithun M, Shiu A (2016) An all-encompassing global convergence result for processive multisite phosphorylation systems (To appear in Mathematical Biosciences)Google Scholar
  8. Feliu E, Wiuf C (2013) Simplifying biochemical models with intermediate species. J R Soc Interface 10(87):20130484CrossRefGoogle Scholar
  9. Hirsch MW (1988) Stability and convergence in strongly monotone dynamical systems. J Reine Angew Math 383:1–53MathSciNetMATHGoogle Scholar
  10. Hirsch MW, Smith H (2006) Monotone dynamical systems. In: Flaviano B, Michal F (eds) Handbook of differential equations: ordinary differential equations, vol 2. Elsevier B. V., Amsterdam, pp 239–357Google Scholar
  11. King EL, Altman C (1956) A schematic method of deriving the rate laws for enzyme-catalyzed reactions. J Phys Chem 60:1375–1378CrossRefGoogle Scholar
  12. Knudsen M, Feliu E, Wiuf C (2012) Exact analysis of intrinsic qualitative features of phosphorelays using mathematical models. J Theoret Biol 300:7–18MathSciNetCrossRefGoogle Scholar
  13. Radulescu O, Gorban AN, Zinovyev A, Noel V (2012) Reduction of dynamical biochemical reactions networks in computational biology. Frontiers in Genetics, p 3Google Scholar
  14. Rockafellar RT (1970) Convex analysis, 1st edn. Princeton University Press, PrincetonCrossRefMATHGoogle Scholar
  15. Sáez M, Wiuf C, Feliu E (2017) Graphical reduction of reaction networks by linear elimination of species. J Math Biol 74:95–237MathSciNetCrossRefMATHGoogle Scholar
  16. Smirnov GV (2002) Introduction to the theory of differential inclusions. Graduate studies in mathematics, vol 41. American Mathematical Society, ProvidenceGoogle Scholar
  17. Smith HL (1995) Monotone dynamical systems, volume 41 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI. An introduction to the theory of competitive and cooperative systemsGoogle Scholar
  18. Sontag ED (2001) Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. Inst Electr Electr Eng Trans Autom. Control 46(7):1028–1047MathSciNetCrossRefMATHGoogle Scholar
  19. Widmann C, Spencer G, Jarpe MB, Johnson GL (1999) Mitogen-activated protein kinase: conservation of a three-kinase module from yeast to human. Physiol Rev 79(1):143–180Google Scholar

Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations