Reachability, persistence, and constructive chemical reaction networks (part I): reachability approach to the persistence of chemical reaction networks
For a chemical reaction network, persistence is the property that no species tend to extinction if all species are initially present. We investigate the stronger property of vacuous persistence: the same asymptotic feature with a weaker requirement on initial states, namely that all species be implicitly present. By implicitly present, we mean for instance that if only water is present and the reaction network incorporates the information that water is made of hydrogen and oxygen, then hydrogen and oxygen are implicitly present. Persistence is inherently interesting and has implications for the global asymptotic stability of equilibrium states. Our main tools are the work of A. I. Vol’pert on the nullity and positivity of species concentrations, and the enabling notion of reachability. The main result states that a reaction network is vacuously persistent if and only if the set of all species is the only set of species that both is closed with respect to reachability and causes the implicit presence of all species. This paper is the first in a series of three articles. Two sequel papers introduce additional formalisms and use them to describe two large classes of reaction networks that are used as models in biochemistry and are vacuously persistent.
KeywordsChemical reaction network Vacuous persistence Reachability A. I. Vol’pert
Mathematics Subject Classification (2010)92C42 92C45 34D05
- 3.D. Angeli, P. De Leenheer, E.D. Sontag, A Petri net approach to persistence analysis in chemical reaction networks, in Biology and Control Theory: Current Challenges ed. by I. Queinnec, S. Tarbouriech, G. Garcia, S.-I. Niculescu. Lecture Notes in Control and Information Sciences, vol. 357 (Springer, Berlin, 2007), pp. 181–216. doi: 10.1007/978-3-540-71988-5
- 5.M. Feinberg, Lectures on Chemical Reaction Networks (1980), http://www.che.eng.ohio-state.edu/~Feinberg/LecturesOnReactionNetworks/.
- 7.G. Gnacadja, Reachability, persistence, and constructive chemical reaction networks (part II): a formalism for species composition in chemical reaction network theory and application to persistence. J. Math. Chem. (2011). doi: 10.1007/s10910-011-9896-2
- 8.G. Gnacadja, Reachability, persistence, and constructive chemical reaction networks (part III): a mathematical formalism for binary enzymatic networks and application to persistence. J. Math. Chem. (2011). doi: 10.1007/s10910-011-9895-3
- 9.J. Gunawardena, Chemical Reaction Network Theory for In-Silico Biologists (2003), http://www.jeremy-gunawardena.com/papers/crnt.pdf.
- 13.R.T. Rockafellar, Convex Analysis. Princeton Landmarks in Mathematics and Physics, (Princeton University Press, 1997), ISBN: 9780691015866Google Scholar
- 15.Siegel D., Chen Y.F.: Global stability of deficiency zero chemical networks. Can. Appl. Math. Q. 2(3), 413–434 (1994)Google Scholar
- 17.E.D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed., Textbooks in Applied Mathematics, vol. 6. (Springer, 1998) ISBN:0387984895Google Scholar
- 19.E.D. Sontag, Private communication, January 2010Google Scholar
- 23.A.I. Vol’pert, S.I. Hudjaev, Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Mechanics: Analysis, vol. 8. (Springer, 1985) ISBN:9789024731091.Google Scholar