Abstract
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.
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References
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 Metzen
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):598–618
Angeli D, De Leenheer P, Sontag ED (2010) Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. J Math Biol 61(4):581–616
Conradi C, Shiu A (2015) A global convergence result for processive multisite phosphorylation systems. Bull Math Biol 77(1):126–155
Cornish-Bowden A (2004) Fundamentals of enzyme kinetics, 3rd edn. Portland Press, London
de Freitas MM, Feliu E, Wiuf C (2017) Intermediates, catalysts, persistence, and boundary steady states. J Math Biol 74:887–932
Eithun M, Shiu A (2016) An all-encompassing global convergence result for processive multisite phosphorylation systems (To appear in Mathematical Biosciences)
Feliu E, Wiuf C (2013) Simplifying biochemical models with intermediate species. J R Soc Interface 10(87):20130484
Hirsch MW (1988) Stability and convergence in strongly monotone dynamical systems. J Reine Angew Math 383:1–53
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 239–357
King EL, Altman C (1956) A schematic method of deriving the rate laws for enzyme-catalyzed reactions. J Phys Chem 60:1375–1378
Knudsen M, Feliu E, Wiuf C (2012) Exact analysis of intrinsic qualitative features of phosphorelays using mathematical models. J Theoret Biol 300:7–18
Radulescu O, Gorban AN, Zinovyev A, Noel V (2012) Reduction of dynamical biochemical reactions networks in computational biology. Frontiers in Genetics, p 3
Rockafellar RT (1970) Convex analysis, 1st edn. Princeton University Press, Princeton
Sáez M, Wiuf C, Feliu E (2017) Graphical reduction of reaction networks by linear elimination of species. J Math Biol 74:95–237
Smirnov GV (2002) Introduction to the theory of differential inclusions. Graduate studies in mathematics, vol 41. American Mathematical Society, Providence
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 systems
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):1028–1047
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):143–180
Acknowledgements
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.
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de Freitas, M.M., Wiuf, C. & Feliu, E. Intermediates and Generic Convergence to Equilibria. Bull Math Biol 79, 1662–1686 (2017). https://doi.org/10.1007/s11538-017-0303-4
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DOI: https://doi.org/10.1007/s11538-017-0303-4