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'''Chemical reaction network theory''' is an area of [[applied mathematics]] that attempts to [[mathematical model|model]] the behaviour of real world [[chemistry|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 mathematics|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),<ref>Wegscheider, R. (1901) [http://www.springerlink.com/content/q12x76713v015316/ Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme], Monatshefte für Chemie / Chemical Monthly 32(8), 849--906.</ref> development of the quantitative theory of chemical chain reactions by [[Nikolay Semyonov]] (1934),<ref>Semyonov's Nobel Lecture [http://nobelprize.org/nobel_prizes/chemistry/laureates/1956/semenov-lecture.html Some Problems Relating to Chain Reactions and to the Theory of Combustion]</ref> development of kinetics of catalytic reactions by [[Cyril Norman Hinshelwood]],<ref>Hinshelwood's Nobel Lecture [http://nobelprize.org/nobel_prizes/chemistry/laureates/1956/hinshelwood-lecture.html Chemical Kinetics in the Past Few Decades]</ref> 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 <ref>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.</ref> 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,<ref>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,</ref> Roy Jackson, Friedrich Josef Maria Horn,<ref>F. J. M. Horn and R. Jackson, "General Mass Action Kinetics", ''Archive Rational Mech.'', '''47''':81, 1972.</ref> [[Martin Feinberg]]<ref>M. Feinberg, "Complex balancing in general kinetic systems", ''[[Archive for Rational Mechanics and Analysis|Arch. Rational Mech. Anal.]]'', '''49''':187–194, 1972.</ref> and others, published in the 1970s. In his second "prolegomena" paper,<ref>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</ref> R. Aris mentioned the work of N.Z. Shapiro, L.S. Shapley (1965),<ref>N.Z. Shapiro, L.S. Shapley, Mass action law and the Gibbs free energy function, SIAM J. Appl. Math. 16 (1965) 353–375.</ref> 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.<ref>P. Érdi and J. Tóth, "Mathematical models of chemical reactions", ''[[Manchester University Press]]'', 1989.</ref><ref>H. Kunze and D. Siegel, "Monotonicity properties of chemical reactions with a single initial bimolecular step", ''J. Math. Chem.'', '''31'''(4):339–344, 2002.</ref><ref>M. Mincheva and D. Siegel, "Nonnegativity and positiveness of solutions to mass action reaction–diffusion systems", ''J. Math. Chem.'', '''42''':1135–1145, 2007.</ref><ref>P. De Leenheer, D. Angeli and E. D. Sontag, [http://www.mit.edu/~esontag/FTP_DIR/leenheer-angeli-sontag-JMathChem07.pdf "Monotone chemical reaction networks"], ''J. Math. Chem.', 41(3):295–314, 2007.</ref><ref>M. Banaji, P. Donnell and S. Baigent, "[[P-matrix|''P'' matrix]] properties, injectivity and stability in chemical reaction systems", ''[[Society for Industrial and Applied Mathematics|SIAM]] J. Appl. Math.'', '''67'''(6):1523–1547, 2007.</ref><ref>G. Craciun and C. Pantea, "Identifiability of chemical reaction networks", ''J. Math. Chem.'', '''44''':1, 2008.</ref><ref>M. Domijan and M. Kirkilionis, "Bistability and oscillations in chemical reaction networks", ''[[Journal of Mathematical Biology|J. Math. Biol.]]'', '''59'''(4):467–501, 2009.</ref><ref>A. N. Gorban and G. S. [[Grigoriy Yablonsky|Yablonsky]], [http://www.math.le.ac.uk/people/ag153/homepage/GorbanYablonskiiCES2011.pdf "Extended detailed balance for systems with irreversible reactions"], ''Chemical Engineering Science'', '''66''':5388–5399, 2011.</ref><ref>E. Feliu, M. Knudsen and C. Wiuf., "Signaling cascades: Consequences of varying substrate and phosphatase levels", ''[[Advances in Experimental Medicine and Biology|Adv. Exp. Med. Biol.]]'' (Adv Syst Biol), '''736''':81–94, 2012.</ref>

== Overview ==

A chemical reaction network (often abbreviated to '''CRN''') comprises a [[set (mathematics)|set]] of [[reagent|reactants]], a set of products (often [[intersection (set theory)|intersecting]] the set of reactants), and a set of [[chemical reaction|reactions]]. For example, the pair of [[combustion]] reactions

: <math>
\begin{array}{rcl}
2 H_2 + O_2 & \rightarrow & 2 H_2 O \\
C + O_2 & \rightarrow & C O_2
\end{array}
</math>

form a reaction network. The reactions are represented by the arrows. The reactants appear to the left of the arrows, in this example they are <math>H_2</math> ([[hydrogen]]), <math>O_2</math> ([[oxygen]]) and <math>C</math> ([[carbon]]). The products appear to the right of the arrows, here they are <math>H_2 O</math> ([[water]]) and <math>C O_2</math> ([[carbon dioxide]]). In this example, since the reactions are [[reversible reaction|irreversible]] and neither of the products are used up in the reactions, the set of reactants and the set of products are [[disjoint sets|disjoint]].

Mathematical modelling of chemical reaction networks usually focuses on what happens to the concentrations of the various chemicals involved as time passes. Following the example above, let <math>a</math> represent the [[concentration]] of <math>H_2</math> in the surrounding air, <math>b</math> represent the concentration of <math>O_2</math>, <math>c</math> represent the concentration of <math>H_2 O</math>, and so on. Since all of these concentrations will not in general remain constant, they can be written as a function of time e.g. <math>a(t), b(t)</math>, etc.

These variables can then be combined into a vector

: <math>x(t) = \left(\begin{array}{c} a(t) \\ b(t) \\ c(t) \\ \vdots \end{array}\right)</math>

and their evolution with time can be written

: <math>\dot{x} \equiv \frac{dx}{dt} = \left(\begin{array}{c} \frac{da}{dt} \\[6pt] \frac{db}{dt} \\[6pt] \frac{dc}{dt} \\[6pt] \vdots \end{array}\right).</math>

This is an example of a [[continuous function|continuous]] [[autonomous system (mathematics)|autonomous]] [[dynamical system]], commonly written in the form <math>\dot{x} = f(x)</math>. The number of molecules of each reactant used up each a reaction occurs is constant, as is the number of molecules produced of each product. These numbers are referred to as the [[stoichiometry]] of the reaction, and the difference between the two (i.e. the overall number of molecules used up or produced) is the '''net stoichiometry'''. This means that the equation representing the chemical reaction network can be rewritten as

: <math>\dot{x} = \Gamma V(x)</math>

Here, each column of the [[constant (mathematics)|constant]] [[matrix (mathematics)|matrix]] <math>\Gamma</math> represents the net stoichiometry of a reaction, and so <math>\Gamma</math> is called the '''stoichiometry matrix'''. <math>V(x)</math> is a [[vector-valued function]] where each output value represents a reaction rate, referred to as the [[chemical kinetics|kinetics]].

== Common assumptions ==

For physical reasons, it is usually assumed that reactant concentrations cannot be negative, and that each reaction only takes place if all its reactants are present, i.e. all have non-zero concentration. For mathematical reasons, it is usually assumed that <math>V(x)</math> is [[smooth function|continuously differentiable]].

It is also commonly assumed that no reaction features the same chemical as both a reactant and a product (i.e. no [[catalysis]] or [[autocatalysis]]), and that increasing the concentration of a reactant increases the rate of any reactions that use it up. This second assumption is compatible with all physically reasonable kinetics, including [[law of mass action|mass action]], [[Michaelis–Menten kinetics|Michaelis–Menten]] and [[Hill equation (biochemistry)|Hill]] kinetics. Sometimes further assumptions are made about reaction rates, e.g. that all reactions obey mass action kinetics.

Other assumptions include [[mass balance]], constant [[temperature]], constant [[pressure]], [[homogeneous (chemistry)|spatially uniform]] concentration of reactants, and so on.

== Types of results ==

As chemical reaction network theory is a diverse and well-established area of research, there is a significant variety of results. Some key areas are outlined below.

=== Number of steady states ===

These results relate to whether a chemical reaction network can produce significantly different behaviour depending on the initial concentrations of its constituent reactants. This has applications in e.g. modelling [[biology|biological]] switches — a high concentration of a key chemical at steady state could represent a biological process being "switched on" whereas a low concentration would represent being "switched off".

For example, the catalytic [[trigger]] is a simplest catalytic reaction without [[autocatalysis]] that allows multiplicity of steady states(1976):<ref>M.G. Slin'ko, V.I. Bykov, G.S. [[Grigoriy Yablonsky|Yablonskii]], T.A. Akramov, "Multiplicity of the Steady State in Heterogeneous Catalytic Reactions", ''Dokl. Akad. Nauk SSSR'' '''226''' (4) (1976), 876.</ref><ref>V.I. Bykov, V.I. Elokhin, G.S. [[Grigoriy Yablonsky|Yablonskii]], [http://dx.doi.org/10.1007/BF02061998 "The simplest catalytic mechanism permitting several steady states of the surface"], ''React. Kinet. Catal. Lett.'' '''4''' (2) (1976), 191–198.</ref>
# <math>A_2+2 Z \rightleftharpoons 2 AZ</math>
# <math>B+Z \rightleftharpoons BZ </math>
# <math>AZ+BZ \to AB+2 Z </math>
This is the classical [[Langmuir–Hinshelwood kinetics|adsorption mechanism]] of catalytic oxidation.

Here, <math>A_2, B</math> and AB are gases (for example, <math>O_2</math>, ''CO'' and <math>CO_2</math>), ''Z'' iz the "adsorption place" on the surface of the solid catalyst (for example, Pt), ''AZ'' and ''BZ'' are the intermediates on the surface (adatoms, adsorbed molecules or radicals).
This system may have two stable steady states of the surface for the same concentrations of the gaseous components.

=== Stability of steady states ===

Stability determines whether a given steady state solution is likely to be observed in reality. Since real systems (unlike [[deterministic system|deterministic]] models) tend to be subject to random background noise, an unstable steady state solution is unlikely be observed in practice. Instead of them, stable oscillations or other types of [[attractor]]s may appear.

=== Persistence ===

Persistence has its roots in [[population dynamics]]. A non-persistent [[species]] in population dynamics can go extinct for some (or all) initial conditions. Similar questions are of interests to chemists and biochemists, i.e. if a given reactant was present to start with, can it ever be completely used up?

=== Existence of stable periodic solutions ===

Results regarding stable periodic solutions attempt to rule out "unusual" behaviour. If a given chemical reaction network admits a stable periodic solution, then some initial conditions will converge to an infinite cycle of oscillating reactant concentrations. For some parameter values it may even exhibit [[quasiperiodic motion|quasiperiodic]] or [[chaos theory#Chaotic dynamics|chaotic]] behaviour. While stable periodic solutions are unusual in real-world chemical reaction networks, well known examples exist, such as the [[Belousov–Zhabotinsky reaction]]s. The simplest catalytic oscillator (nonlinear self-oscillations without authocatalisys)
can be produced from the catalytic trigger by addind a "buffer" step.<ref>V.I. Bykov, G.S. Yablonskii, V.F. Kim, "On the simple model of kinetic self-oscillations in catalytic reaction of CO oxidation", ''Doklady AN USSR'' (Chemistry) '''242''' (3) (1978), 637–639.</ref>
:4. <math>B + Z \rightleftharpoons (BZ)</math>
where (BZ) is an intermediate that does not participate in the main reaction.

== References ==
{{Reflist}}

== External links ==
* [http://reaction-networks.net/ Specialist wiki on the mathematics of reaction networks]

[[Category:Mathematical chemistry]]

Matroid partitioning - Revision history

en>David Eppstein: authorlink Ed Scheinerman