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 the simplest catalytic reaction without
autocatalysis that allows multiplicity of steady states (1976): {{NumBlk|:|{A2}+2Z 2AZ|}} {{NumBlk|:|{B}+Z BZ |}} {{NumBlk|:|{AZ}+BZ -> {AB}+2Z |}} This is the classical
adsorption mechanism of catalytic oxidation. Here, A2, B and AB are gases (for example, O2, CO and CO2), Z is 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 to 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 autocatalysis) can be produced from the catalytic trigger by adding a "buffer" step. {{NumBlk|:|{B} + Z (BZ)|}} where (BZ) is an intermediate that does not participate in the main reaction.
Network structure and dynamical properties 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. For nonlinear systems, many connections between structure and dynamics have been discovered. First of all, these are results about stability. 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 and the
theorem about systems without interactions between different components. The deficiency zero theorem gives sufficient conditions for the existence of the Lyapunov function in the classical
free energy form G(c)=\sum_i c_i \left(\ln \frac{c_i}{c_i^*} -1\right), where 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 n_{k}A_i \to \sum_j \beta_{kj}A_j (for k \leq r, where
r is the number of reactions, A_i is the symbol of
ith component, n_k\geq 1, and \beta_{kj} are non-negative integers) and allows the stoichiometric conservation law M(c)=\sum_i m_i c_i=\text{const} (where all m_i>0), then the weighted
L1 distance \sum_i m_i |c_i^1(t)-c_i^2(t)| between two solutions c^1(t) \; \mbox{and} \; c^2(t) with the same
M(
c) monotonically decreases in time. Importantly, research has emphasized the role of
circuit topology—the specific arrangement of activating and repressing interactions within reaction networks—in shaping dynamic behaviors. Circuit motifs such as series, parallel, and feedforward loops decisively influence signal propagation, system robustness, noise filtering, and response delays. For example, in
transcriptional cascades regarded as reaction networks, topological differences govern key functional properties, highlighting how network architecture critically determines biochemical regulation and dynamics.
Model reduction 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. 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. == Stochastic chemical reaction networks ==