reactionnetwork

A reaction network is a formal representation of chemical or biochemical interactions, consisting of a set of species and a set of reactions that transform those species. It is commonly represented as a directed graph where nodes correspond to complexes or species and edges correspond to reactions, or expressed as a set of stoichiometric equations. Reaction networks are used to model how the concentrations of species evolve over time under specified kinetic rules.

Formal structure: Let S = {S1,...,Sn} be the species and R = {R1,...,Rm} the reactions. Each reaction i is described by a reactant complex y_i and product complex y'_i, given by stoichiometric vectors, so that R_i: y_i → y'_i. The net change for reaction i is γ_i = y'_i − y_i. The overall change is Γ v(x) where Γ is the stoichiometric matrix with columns γ_i and v(x) are the reaction rates.

Dynamics: Under deterministic mass-action kinetics, the rate of each reaction is v_i(x) = k_i ∏_j x_j^{y_ij}, where x_j is the concentration of species S_j and y_ij is its stoichiometric coefficient in the reactant complex. The system evolves according to dx/dt = Γ v(x). Solutions stay in the nonnegative orthant; invariant quantities arise from conservation relations.

Analysis: Researchers study properties such as existence and stability of steady states, persistence, and multistationarity. The deficiency theory, including the Deficiency Zero and Deficiency One Theorems, provides criteria for the qualitative behavior of deterministic networks. Stochastic models treat concentrations as Markov processes and often use simulation methods like Gillespie's algorithm.

Variants and connections: Reaction networks are related to Petri nets and serve as a bridge between chemistry and systems biology. They can be analyzed under deterministic ODE models or stochastic dynamics; complex networks may also be analyzed in terms of network structure, such as linkage classes and cycles.

Example: A simple network consisting of A → B and B → C. With mass-action rates k1 and k2, the dynamics follow dx_A/dt = -k1 x_A, dx_B/dt = k1 x_A − k2 x_B, dx_C/dt = k2 x_B.