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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), development of the quantitative theory of chemical chain reactions by Nikolay Semyonov (1934), development of kinetics of catalytic reactions by Cyril Norman Hinshelwood, 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 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, Roy Jackson, Friedrich Josef Maria Horn, Martin Feinberg and others, published in the 1970s. In his second "prolegomena" paper, R. Aris mentioned the work of N.Z. Shapiro, L.S. Shapley (1965), 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.

Overview

A chemical reaction network (often abbreviated to CRN) comprises a set of reactants, a set of products (often intersecting the set of reactants), and a set of reactions. For example, the pair of combustion reactions

${\begin{array}{rcl}2H_{2}+O_{2}&\rightarrow &2H_{2}O\\C+O_{2}&\rightarrow &CO_{2}\end{array}}$

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 $H_{2}$ (hydrogen), $O_{2}$ (oxygen) and $C$ (carbon). The products appear to the right of the arrows, here they are $H_{2}O$ (water) and $CO_{2}$ (carbon dioxide). In this example, since the reactions are irreversible and neither of the products are used up in the reactions, the set of reactants and the set of products are 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 $a$ represent the concentration of $H_{2}$ in the surrounding air, $b$ represent the concentration of $O_{2}$, $c$ represent the concentration of $H_{2}O$, 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. $a(t),b(t)$, etc.

These variables can then be combined into a vector

$x(t)=\left({\begin{array}{c}a(t)\\b(t)\\c(t)\\\vdots \end{array}}\right)$

and their evolution with time can be written

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

This is an example of a continuous autonomous dynamical system, commonly written in the form ${\dot {x}}=f(x)$. 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

${\dot {x}}=\Gamma V(x)$

Here, each column of the constant matrix $\Gamma$ represents the net stoichiometry of a reaction, and so $\Gamma$ is called the stoichiometry matrix. $V(x)$ is a vector-valued function where each output value represents a reaction rate, referred to as the 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 $V(x)$ is 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 mass action, Michaelis–Menten and 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, 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 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):

1. $A_{2}+2Z\rightleftharpoons 2AZ$
2. $B+Z\rightleftharpoons BZ$
3. $AZ+BZ\to AB+2Z$

This is the classical adsorption mechanism of catalytic oxidation.

Here, $A_{2},B$ and AB are gases (for example, $O_{2}$, CO and $CO_{2}$), 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 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 attractors 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 or chaotic behaviour. While stable periodic solutions are unusual in real-world chemical reaction networks, well known examples exist, such as the Belousov–Zhabotinsky reactions. The simplest catalytic oscillator (nonlinear self-oscillations without authocatalisys) can be produced from the catalytic trigger by addind a "buffer" step.

4. $B+Z\rightleftharpoons (BZ)$

where (BZ) is an intermediate that does not participate in the main reaction.

References

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External links

• Specialist wiki on the mathematics of reaction networks