Since its foundation in the 1960s, it has attracted a growing research community, mainly due to its applications in biochemistry and theoretical chemistry.
Dynamical properties of reaction networks were studied in chemistry and physics after the 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),[1] development of the quantitative theory of chemical chain reactions by Nikolay Semyonov (1934),[2] development of kinetics of catalytic reactions by Cyril Norman Hinshelwood,[3] and many other results.
Three eras of chemical dynamics can be revealed in the flux of research and publications.
The "eras" may be distinguished based on the main focuses of the scientific leaders: The mathematical discipline "chemical reaction network theory" was originated by Rutherford Aris, a famous expert in chemical engineering, with the support of Clifford Truesdell, the founder and editor-in-chief of the journal Archive for Rational Mechanics and Analysis.
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,[6] Roy Jackson, Friedrich Josef Maria Horn,[7] Martin Feinberg[8] and others, published in the 1970s.
Shapley (1965),[10] 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.
Since all of these concentrations will not in general remain constant, they can be written as a function of time e.g.
This means that the equation representing the chemical reaction network can be rewritten as Here, each column of the constant matrix
is a vector-valued function where each output value represents a reaction rate, referred to as the kinetics.
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.
As chemical reaction network theory is a diverse and well-established area of research, there is a significant variety of results.
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".
This system may have two stable steady states of the surface for the same concentrations of the gaseous components.
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 to be observed in practice.
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?
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.
This connection is important even for linear systems, for example, the simple cycle with equal interaction weights has the slowest decay of the oscillations among all linear systems with the same number of states.
[25] For some classes of networks, explicit construction of Lyapunov functions is possible without apriori assumptions about special relations between rate constants.
[27] The deficiency zero theorem gives sufficient conditions for the existence of the Lyapunov function in the classical free energy form
The theorem about systems without interactions between different components states that if a network consists of reactions of the form
The model reduction methods were developed together with the first theories of complex chemical reactions.
[28] Three simple basic ideas have been invented: The quasi-equilibrium approximation and the quasi steady state methods were developed further into the methods of slow invariant manifolds and computational singular perturbation.