Chemical reaction network theory

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 the simplest catalytic reaction without autocatalysis that allows multiplicity of steady states (1976):[21][22]

${\displaystyle {\ce {{A2}+2Z <=> 2AZ}}}$

(reaction 2)

${\displaystyle {\ce {{B}+Z <=> BZ}}}$

(reaction 3)

${\displaystyle {\ce {{AZ}+BZ -> {AB}+2Z}}}$

(reaction 4)

This is the classical adsorption mechanism of catalytic oxidation.

Here, ${\displaystyle {\ce {A2, B}}}$ and ${\displaystyle {\ce {AB}}}$ are gases (for example, ${\displaystyle {\ce {O2}}}$, ${\displaystyle {\ce {CO}}}$ and ${\displaystyle {\ce {CO2}}}$), ${\displaystyle {\ce {Z}}}$ is the "adsorption place" on the surface of the solid catalyst (for example, ${\displaystyle {\ce {Pt}}}$), ${\displaystyle {\ce {AZ}}}$ and ${\displaystyle {\ce {BZ}}}$ 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 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. Instead of them, stable oscillations or other types of attractors may appear.

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?

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 autocatalysis) can be produced from the catalytic trigger by adding a "buffer" step.[23]

${\displaystyle {\ce {{B}+ Z <=> (BZ)}}}$

(reaction 5)

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

One of the main problems of chemical reaction network theory is the connection between network structure and properties of dynamics. 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.[24]

For nonlinear systems, many connections between structure and dynamics have been discovered. First of all, these are results about stability.[25] For some classes of networks, explicit construction of Lyapunov functions is possible without apriori assumptions about special relations between rate constants. Two results of this type are well known: the deficiency zero theorem[26] and the theorem about systems without interactions between different components.[27]

The deficiency zero theorem gives sufficient conditions for the existence of the Lyapunov function in the classical free energy form ${\displaystyle G(c)=\sum _{i}c_{i}\left(\ln {\frac {c_{i}}{c_{i}^{*}}}-1\right)}$, where ${\displaystyle c_{i}}$ is the concentration of the i-th component. The theorem about systems without interactions between different components states that if a network consists of reactions of the form ${\displaystyle n_{k}A_{i}\to \sum _{j}\beta _{kj}A_{j}}$ (for ${\displaystyle k\leq r}$, where r is the number of reactions, ${\displaystyle A_{i}}$ is the symbol of ith component, ${\displaystyle n_{k}\geq 1}$, and ${\displaystyle \beta _{kj}}$ are non-negative integers) and allows the stoichiometric conservation law ${\displaystyle M(c)=\sum _{i}m_{i}c_{i}={\text{const}}}$ (where all ${\displaystyle m_{i}>0}$), then the weighted L¹ distance ${\displaystyle \sum _{i}m_{i}|c_{i}^{1}(t)-c_{i}^{2}(t)|}$ between two solutions ${\displaystyle c^{1}(t)\;{\mbox{and}}\;c^{2}(t)}$ with the same M(c) monotonically decreases in time.

Modelling of large reaction networks meets various difficulties: the models include too many unknown parameters and high dimension makes the modelling computationally expensive. 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 (or pseudo-equilibrium, or partial equilibrium) approximation (a fraction of reactions approach their equilibrium fast enough and, after that, remain almost equilibrated).
• The quasi steady state approximation or QSS (some of the species, very often these are some of intermediates or radicals, exist in relatively small amounts; they reach quickly their QSS concentrations, and then follow, as dependent quantities, the dynamics of these other species remaining close to the QSS). The QSS is defined as the steady state under the condition that the concentrations of other species do not change.
• The limiting step or bottleneck is a relatively small part of the reaction network, in the simplest cases it is a single reaction, which rate is a good approximation to the reaction rate of the whole network.

The quasi-equilibrium approximation and the quasi steady state methods were developed further into the methods of slow invariant manifolds and computational singular perturbation. The methods of limiting steps gave rise to many methods of the analysis of the reaction graph.[28]