Journal of Mathematical Biology

, Volume 72, Issue 1–2, pp 467–498 | Cite as

A computational approach to persistence, permanence, and endotacticity of biochemical reaction systems

  • Matthew D. Johnston
  • Casian Pantea
  • Pete Donnell


We introduce a mixed-integer linear programming (MILP) framework capable of determining whether a chemical reaction network possesses the property of being endotactic or strongly endotactic. The network property of being strongly endotactic is known to lead to persistence and permanence of chemical species under genetic kinetic assumptions, while the same result is conjectured but as yet unproved for general endotactic networks. The algorithms we present are the first capable of verifying endotacticity of chemical reaction networks for systems with greater than two constituent species. We implement the algorithms in the open-source online package CoNtRol and apply them to a large sample of networks from the European Bioinformatics Institute’s BioModels Database. We use strong endotacticity to establish for the first time the permanence of a well-studied circadian clock mechanism.


Chemical reaction network Chemical kinetics Persistence Permanence Endotactic 

Mathematics Subject Classification

92C42 90C35 



M. J. is supported by grant Army Research Office grant W911NF-14-1-0401. P.D. is supported by EPSRC grant EP/J00826/1. The authors thank Gheorghe Craciun for helpful discussions and suggestions, and Murad Banaji for organizing the workshop “Combinatorial and algebraic approaches to chemical reaction networks” at the University of Portsmouth (UK) at which this collaboration began. The authors also thank the anonymous referees whose comments have significantly improved the readability of the paper.


  1. Achterberg T (2009) SCIP: Solving constraint integer programs. Math Program Comput 1(1):1–41.
  2. Anderson DF (2011) A proof of the global attractor conjecture in the single linkage class case. SIAM J Appl Math 71(4):1487–1508MathSciNetCrossRefzbMATHGoogle Scholar
  3. Angeli D, Leenheer P, Sontag E (2007) A petri net approach to the study of persistence in chemical reaction networks. Math Biosci 210(2):598–618MathSciNetCrossRefzbMATHGoogle Scholar
  4. Angeli D, Sontag E (2008) Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles. Nonlinear Anal Ser B Real World Appl 9:128–140MathSciNetCrossRefzbMATHGoogle Scholar
  5. Conradi C, Shiu A (2014) A global convergence result for processive multisite phosphorylation systems. Bull Math Bio 77(1):126–155Google Scholar
  6. Craciun G, Dickenstein A, Shiu A, Sturmfels B (2009) Toric dynamical systems. J Symb Comput 44(11):1551–1565MathSciNetCrossRefzbMATHGoogle Scholar
  7. Craciun G, Pantea C, Nazarov F (2013) Persistence and permanence of mass action and power-law dynamical systems. SIAM J Appl Math 73(1):305–329MathSciNetCrossRefzbMATHGoogle Scholar
  8. Donnell P, Banaji M, Marginean A, Pantea C (2014) CoNtRol: an open source framework for the analysis of chemical reaction networks. Bioinformatics 30(11):1633–1634CrossRefGoogle Scholar
  9. Feinberg M (1972) Complex balancing in general kinetic systems. Arch Ration Mech Anal 49:187–194MathSciNetCrossRefGoogle Scholar
  10. Feinberg M (1987) Chemical reaction network structure and the stability of complex isothermal reactors: I. the deficiency zero and deficiency one theorems. Chem Eng Sci 42(10):2229–2268CrossRefGoogle Scholar
  11. Feinberg M (1988) 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):1–25MathSciNetCrossRefGoogle Scholar
  12. Feinberg M (1995) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Ration Mech Anal 132:311–370MathSciNetCrossRefzbMATHGoogle Scholar
  13. Feinberg M (1995) Multiple steady states for chemical reaction networks of deficiency one. Arch Ration Mech Anal 132:371–406MathSciNetCrossRefzbMATHGoogle Scholar
  14. Freedman HI, Moson P (1990) Persistence definitions and their connections. Proc Am Math Soc 109(4):1025–1033MathSciNetCrossRefzbMATHGoogle Scholar
  15. Goldbeter A, Koshland DE (1981) An amplified sensitivity arising from covalent modification in biological systems. Proc. Natl. Acad. Sci. U.S.A. 78(11):6840–6844MathSciNetCrossRefGoogle Scholar
  16. Gopalkrishnan M, Miller E, Shiu A (2014) A geometric spproach to the global attractor conjecture. SIAM J Appl Dyn Syst 13(2):758–797MathSciNetCrossRefzbMATHGoogle Scholar
  17. Gunawardena J (2005) Multisite protein phosphorylation makes a good threshold but can be a poor switch. Proc Natl Acad Sci USA 102(41):14617–14622CrossRefGoogle Scholar
  18. Hill A (2010) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40(4):iv–viiGoogle Scholar
  19. Hofbauer J, Sigmund K (1987) Permanence for replicator equations. In: Kurzhanski AB, Sigmund K (eds) Dynamical systems. Lecture Notes in economics and mathematical systems 287:70–91Google Scholar
  20. Horn F (1972) Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch Ration Mech Anal 49:172–186MathSciNetCrossRefGoogle Scholar
  21. Horn F, Jackson R (1972) General mass action kinetics. Arch Ration Mech Anal 47:81–116MathSciNetCrossRefGoogle Scholar
  22. Johnston MD (2014) Translated chemical reaction networks. Bull Math Bio 76(5):1081–1116MathSciNetCrossRefzbMATHGoogle Scholar
  23. Johnston MD, Siegel D, Szederkényi G (2012) A linear programming approach to weak reversibility and linear conjugacy of chemical reaction networks. J Math Chem 50(1):274–288MathSciNetCrossRefzbMATHGoogle Scholar
  24. Leloup JC, Goldbeter A (1999) Chaos and birhythmicity in a mode for circadian oscillations of the PER and TIM proteins in drosophila. J Theor Biol 198:445–459CrossRefGoogle Scholar
  25. Li C, Donizelli M, Rodriguez N, Dharuri H, Endler L, Chelliah L, Li L, He E, Henry A, Stefan MI, Snoep JL, Hucka M, Le Novère N, Laibe C (2010) BioModels database: an enhanced, curated and annotated resource for published quantitative kinetic models. BMC Syst Biol 4:92CrossRefGoogle Scholar
  26. Makhorin A (2010) GNU Linear Programming Kit Reference Manual Version 4.45.
  27. Michaelis L, Menten M (1913) Die kinetik der invertinwirkung. Biochem Z 49:333–369Google Scholar
  28. Millán MP, Dickenstein A, Shiu A, Conradi C (2012) Chemical reaction systems with toric steady states. Bull Math Biol 74(5):1027–1065MathSciNetCrossRefzbMATHGoogle Scholar
  29. Pantea C (2012) On the persistence and global stability of mass action systems. SIAM J Math Anal 44(3):1636–1673MathSciNetCrossRefzbMATHGoogle Scholar
  30. Patwardhan P, Miller WT (2007) Processive phosphorylation: mechanism and biological importance. Cell Signal 19(10):2218–2226CrossRefGoogle Scholar
  31. Pokhilko A, Hodge SK, Stratford K, Knox K, Edwards KD, Thomson AW, Mizuno T, Millar AJ (2010) Data assimilation constrains new connections and components in a complex, eukaryotic circadian clock model. Mol Syst Biol 6:416CrossRefGoogle Scholar
  32. Pokhilko A, Mas P, Millar AJ (2013) Modelling the widespread effects of TOC1 signalling on the plant circadian clock and its outputs. BMC Syst Biol 7:23CrossRefGoogle Scholar
  33. Siegel D, MacLean D (2000) Global stability of complex balanced mechanisms. J Math Chem 27(1–2):89–110MathSciNetCrossRefzbMATHGoogle Scholar
  34. Szederkényi G (2010) Computing sparse and dense realizations of reaction kinetic systems. J Math Chem 47:551–568MathSciNetCrossRefzbMATHGoogle Scholar
  35. Szederkényi G, Hangos K, Tuza Z (2012) Finding weakly reversible realizations of chemical reaction networks using optimization. MATCH Commun Math Comput Chem 67:193–212MathSciNetGoogle Scholar
  36. Takeuchi Y (1996) Global dynamical properties of Lotka–Volterra systems. World Scientific Publishing, SingaporeCrossRefzbMATHGoogle Scholar
  37. Thomson M, Gunawardena J (2009) The rational parameterisation theorem for multisite post-translational modification systems. J Theor Biol 261(4):626–636MathSciNetCrossRefGoogle Scholar
  38. Wang L, Sontag E (2008) On the number of steady states in a multiple futile cycle. J Math Biol 57(1):25–52MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Matthew D. Johnston
    • 1
  • Casian Pantea
    • 2
  • Pete Donnell
    • 3
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA
  2. 2.Department of MathematicsWest Virginia UniversityMorgantownUSA
  3. 3.Department of MathematicsUniversity of PortsmouthPortsmouthUK

Personalised recommendations