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Chemical reaction network theory
Chemical reaction network theory is an area of applied mathematics that attempts to model the behaviour of real world chemical systems. Since its foundation in the 1960s, it has attracted a growing research community, mainly due to its applications in biochemistry and theoretical chemistry. It has also attracted interest from pure mathematicians due to the interesting problems that arise from the mathematical structures involved.
== History ==
Dynamical properties of reaction networks were studied in chemistry and physics after invention of the law of mass action. The essential steps in this study were introduction of detailed balance for the complex chemical reactions by Rudolf Wegscheider (1901),〔Wegscheider, R. (1901) (Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme ), Monatshefte für Chemie / Chemical Monthly 32(8), 849--906.〕 development of the quantitative theory of chemical chain reactions by Nikolay Semyonov (1934),〔Semyonov's Nobel Lecture (Some Problems Relating to Chain Reactions and to the Theory of Combustion )〕 development of kinetics of catalytic reactions by Cyril Norman Hinshelwood,〔Hinshelwood's Nobel Lecture (Chemical Kinetics in the Past Few Decades )〕 and many other results.
The mathematical discipline "chemical reaction network theory" was originated by Rutherford Aris, a famous expert in chemical engineering, with support of Clifford Truesdell, the founder and editor-in-chief of the journal ''Archive for Rational Mechanics and Analysis''. The paper of R. Aris in this journal 〔R. Aris, Prolegomena to the rational analysis of systems of chemical reactions, Archive for Rational Mechanics and Analysis, 1965, Volume 19, Issue 2, pp 81-99.〕 was communicated to the journal by C. Truesdell. It opened the series of papers of other authors (which were communicated already by R. Aris). The well known papers of this series are the works of Frederick J. Krambeck,〔F.J. Krambeck, The mathematical structure of chemical kinetics in homogeneous single-phase systems, Archive for Rational Mechanics and Analysis, 1970, Volume 38, Issue 5, pp 317-347,〕 Roy Jackson, Friedrich Josef Maria Horn,〔F. J. M. Horn and R. Jackson, "General Mass Action Kinetics", ''Archive Rational Mech.'', 47:81, 1972.〕 Martin Feinberg〔M. Feinberg, "Complex balancing in general kinetic systems", ''Arch. Rational Mech. Anal.'', 49:187–194, 1972.〕 and others, published in the 1970s. In his second "prolegomena" paper,〔R. Aris, Prolegomena to the rational analysis of systems of chemical reactions II. Some addenda, Archive for Rational Mechanics and Analysis, 1968, Volume 27, Issue 5, pp 356-364〕 R. Aris mentioned the work of N.Z. Shapiro, L.S. Shapley (1965),〔N.Z. Shapiro, L.S. Shapley, Mass action law and the Gibbs free energy function, SIAM J. Appl. Math. 16 (1965) 353–375.〕 where an important part of his scientific program was realized.
Since then, the chemical reaction network theory has been further developed by a large number of researchers internationally.〔P. Érdi and J. Tóth, "Mathematical models of chemical reactions", ''Manchester University Press'', 1989.〕〔H. Kunze and D. Siegel, "Monotonicity properties of chemical reactions with a single initial bimolecular step", ''J. Math. Chem.'', 31(4):339–344, 2002.〕〔M. Mincheva and D. Siegel, "Nonnegativity and positiveness of solutions to mass action reaction–diffusion systems", ''J. Math. Chem.'', 42:1135–1145, 2007.〕〔P. De Leenheer, D. Angeli and E. D. Sontag, ("Monotone chemical reaction networks" ), ''J. Math. Chem.', 41(3):295–314, 2007.〕〔M. Banaji, P. Donnell and S. Baigent, "''P'' matrix properties, injectivity and stability in chemical reaction systems", ''SIAM J. Appl. Math.'', 67(6):1523–1547, 2007.〕〔G. Craciun and C. Pantea, "Identifiability of chemical reaction networks", ''J. Math. Chem.'', 44:1, 2008.〕〔M. Domijan and M. Kirkilionis, "Bistability and oscillations in chemical reaction networks", ''J. Math. Biol.'', 59(4):467–501, 2009.〕〔A. N. Gorban and G. S. Yablonsky, ("Extended detailed balance for systems with irreversible reactions" ), ''Chemical Engineering Science'', 66:5388–5399, 2011.〕〔E. Feliu, M. Knudsen and C. Wiuf., "Signaling cascades: Consequences of varying substrate and phosphatase levels", ''Adv. Exp. Med. Biol.'' (Adv Syst Biol), 736:81–94, 2012.〕
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