chemicalreactionnetwork

A chemical reaction network (CRN) is an abstract representation of a set of chemical reactions among a finite set of species, used to study the dynamic behavior of chemical and biochemical systems. CRNs separate the stoichiometric structure of reactions from the exact rate laws, enabling mathematical analysis that applies across many contexts.

A CRN consists of a set of species (chemical substances), a set of reactions, and the complexes that couple them. Reactions describe the transformation of reactant complexes into product complexes. Mass-action kinetics assigns to each reaction a rate proportional to the product of the concentrations of its reactants, leading to a system of ordinary differential equations dx/dt = S v(x), where x is the concentration vector, S the stoichiometric matrix, and v(x) the vector of reaction rates. In a stochastic CRN, the dynamics are modeled as a continuous-time Markov chain with reaction propensities.

From a graph-theoretic point of view, CRNs can be analyzed via the complex graph, including notions of weak reversibility and deficiency. The Deficiency Zero Theorem, due to Feinberg, provides conditions under which a mass-action CRN admits a unique, globally stable equilibrium within each stoichiometric compatibility class. These theoretical tools help determine existence and stability of steady states without solving the dynamics explicitly.

CRNs underpin modeling in chemistry, systems biology, metabolic engineering, and biotechnology. They are used to study oscillations, multistability, and signal transduction, and to design synthetic networks with desired behaviors. Computational tools implement deterministic and stochastic simulations, linear and nonlinear analysis, and network reduction techniques.

Examples include metabolic pathways, gene regulatory networks, and catalytic cycles. The concept remains an abstraction that supports both qualitative insight and quantitative prediction across disciplines.