# Intermediates, catalysts, persistence, and boundary steady states

## Abstract

For dynamical systems arising from chemical reaction networks, persistence is the property that each species concentration remains positively bounded away from zero, as long as species concentrations were all positive in the beginning. We describe two graphical procedures for simplifying reaction networks without breaking known necessary or sufficient conditions for persistence, by iteratively removing so-called intermediates and catalysts from the network. The procedures are easy to apply and, in many cases, lead to highly simplified network structures, such as monomolecular networks. For specific classes of reaction networks, we show that these conditions for persistence are equivalent to one another. Furthermore, they can also be characterized by easily checkable strong connectivity properties of a related graph. In particular, this is the case for (conservative) monomolecular networks, as well as cascades of a large class of post-translational modification systems (of which the MAPK cascade and the *n*-site futile cycle are prominent examples). Since one of the aforementioned sufficient conditions for persistence precludes the existence of boundary steady states, our method also provides a graphical tool to check for that.

### Keywords

Reaction network theory Model reduction Persistence Boundary steady states Intermediates Catalysts Post-translational modification### Mathematics Subject Classification

34C99 80A30 92C42### References

- Amann H (1990) Ordinary differential equations: an introduction to nonlinear analysis. De Gruyter studies in mathematics. de Gruyter, BerlinCrossRefGoogle Scholar
- Anderson DF (2008) Global asymptotic stability for a class of nonlinear chemical equations. SIAM J Appl Math 68(5):1464–1476MathSciNetCrossRefMATHGoogle Scholar
- Anderson DF (2011) A proof of the global attractor conjecture in the single linkage class case. SIAM J Appl Math 71(4):1487–1508MathSciNetCrossRefMATHGoogle Scholar
- 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
- 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
- Angeli D, De Leenheer P, Sontag ED (2011) Persistence results for chemical reaction networks with time-dependent kinetics and no global conservation laws. SIAM J Appl Math 71(1):128–146MathSciNetCrossRefMATHGoogle Scholar
- Cordone R, Ferrarini L, Piroddi L (2005) Enumeration algorithms for minimal siphons in petri nets based on place constraints. IEEE Trans Syst Man Cybern Part A Syst Hum 35(6):844–854CrossRefGoogle Scholar
- Craciun G, Nazarov F, Pantea C (2013) Persistence and permanence of mass-action and power-law dynamical systems. SIAM J Appl Math 73(1):305–329MathSciNetCrossRefMATHGoogle Scholar
- Deshpande A, Gopalkrishnan M (2014) Autocatalysis in reaction networks. Bull Math Biol 76(10):2570–2595MathSciNetCrossRefMATHGoogle Scholar
- Feinberg M (1995) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Rational Mech Anal 132(4):311–370MathSciNetCrossRefMATHGoogle Scholar
- Feinberg M (1980) Lectures on chemical reaction networks. Notes of lectures given at the Mathematics Research Centre, University of Wisconsin-Madison, in the autumn of 1979. http://www.crnt.osu.edu/LecturesOnReactionNetworks
- Feinberg M (1987) Chemical reaction network structure and the stability of complex isothermal reactorsi. the deficiency zero and deficiency one theorems. Chem Eng Sci 42(10):2229–2268CrossRefGoogle Scholar
- Feliu E, Wiuf C (2013) Simplifying biochemical models with intermediate species. J R Soc Interface 10(87):20130484CrossRefGoogle Scholar
- Gnacadja G (2011) Reachability, persistence, and constructive chemical reaction networks (part I): reachability approach to the persistence of chemical reaction networks. J Math Chem 49(10):2117–2136MathSciNetCrossRefMATHGoogle Scholar
- Gnacadja G (2011) Reachability, persistence, and constructive chemical reaction networks (part III): a mathematical formalism for binary enzymatic networks and application to persistence. J Math Chem 49(10):2158–2176MathSciNetCrossRefMATHGoogle Scholar
- Gopalkrishnan M, Miller E, Shiu A (2014) A geometric approach to the global attractor conjecture. SIAM J Appl Dyn Syst 13(2):758–797MathSciNetCrossRefMATHGoogle Scholar
- Gunawardena J (2003) Chemical reaction network theory for in-silico biologists. http://vcp.med.harvard.edu/papers/crnt
- Horn F, Jackson R (1972) General mass action kinetics. Arch Rational Mech Anal 47:81–116MathSciNetCrossRefGoogle Scholar
- MacLean AL, Rosen Z, Byrne HM, Harrington HA (2015) Parameter-free methods distinguish Wnt pathway models and guide design of experiments. PNAS 112(9):2652–2657CrossRefGoogle Scholar
- Murray JD (2002) Mathematical biology. Springer, BerlinMATHGoogle Scholar
- Nguyen LK, Muñoz-García J, Maccario H, Ciechanover A, Kolch W, Kholodenko BN (2011) Switches, excitable responses and oscillations in the Ring1B/Bmi1 ubiquitination system. PLoS Comput Biol 7(12):e1002317CrossRefGoogle Scholar
- Shiu A, Sturmfels B (2010) Siphons in chemical reaction networks. Bull Math Biol 72(6):1448–1463MathSciNetCrossRefMATHGoogle Scholar
- Siegal-Gaskins D, Franco E, Zhou T, Murray RM (2015) An analytical approach to bistable biological circuit discrimination using real algebraic geometry. J Royal Soc Interface 12(108). doi:10.1098/rsif.2015.0288
- Smith HL, Thieme HR (2011) Dynamical systems and population persistence, volume 118 of graduate studies in mathematics. American Mathematical Society, ProvidenceGoogle Scholar
- 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 Electron Eng Trans Autom Control 46(7):1028–1047MathSciNetCrossRefMATHGoogle Scholar
- Sontag ED (2001) 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(7):1028–1047MathSciNetCrossRefMATHGoogle Scholar
- Thomson M, Gunawardena J (2009) The rational parameterisation theorem for multisite post-translational modification systems. J Theor Biol 261(4):626–636MathSciNetCrossRefGoogle Scholar