On the Mathematical Structure of Balanced Chemical Reaction Networks Governed by Mass Action Kinetics

Abstract

Motivated by recent progress on the interplay between graph theory, dynamics, and systems theory, we revisit the analysis of chemical reaction networks described by mass action kinetics. For reaction networks possessing a thermodynamic equilibrium we derive a compact formulation exhibiting at the same time the structure of the complex graph and the stoichiometry of the network, and which admits a direct thermodynamical interpretation. This formulation allows us to easily characterize the set of equilibria and their stability properties. Furthermore, we develop a framework for interconnection of chemical reaction networks. Finally we discuss how the established framework leads to a new approach for model reduction.

Analysis of complex balanced chemical reaction networks with fixed boundary concentrations

“…In order to carry out the analysis we need to first introduce some preliminaries on chemical reaction networks using the concept of graph of complexes from [10], [15] and [8].…”

“…From equation (15), it follows that x j xj "x i xi for all j P S with L i,j ‰ 0. At least one of these js will be such that dpjq " k and this j satisfies the inequalitȳ…”

On the graph and systems analysis of reversible chemical reaction networks with mass action kinetics

“…For proof, we refer to [19,Theorem 4.1]. Following Theorem 3.4, we can define the space of equilibrium points for a reversible mass action chemical network in (21) by ℰ := { ∈ ℝ + | Ln…”

“…Then for every initial condition (0) ∈ ℝ + , the species concentration converges for → ∞ to ℰ. For proof, we refer to [19,Theorem 4.2].…”

“…Hence there exists a surjective map : ℝ + → ℰ that assigns to every initial state its asymptotic thermodynamic equilibrium 5 . We refer to [19,Theorem 4.5] for the proof. Theorem 3.7 is closely related to the property that the asymptotic consensus value in consensus dynamics is a function of the initial state.…”