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.
Large-scale chemical reaction networks arise abundantly in bio-engineering and systems biology. A very simple example involving only two chemical reactions in the glycolytic pathway is given by the following two coupled reactions Acetoacetyl ACP + NADPH + H + ⇋ D-3-Hydroxybutyryl ACP + NADP + D-3-Hydroxybutyryl ACP ⇋ crotonyl ACP + H 2 O.
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The compounds Acetoacetyl ACP, NADPH, H + , D-3-Hydroxybutyryl ACP, NADP, crotonyl ACP and H 2 O involved in the reaction network are called the chemical species of the network. The most basic law prescribing the dynamics of the concentrations of the various species involved in a chemical reaction network is the law of mass action kinetics, leading to polynomial differential equations for the dynamics of each species. Large-scale networks thus lead to a high-dimensional set of coupled polynomial differential equations, which are usually difficult to analyze.
In order to gain insight into the dynamical properties of complex reaction networks it is important to identify their underlying mathematical structure, and to express their dynamics in the most compact way. In line with the recent surge of interest in network dynamics at least two aspects should be fundamental in such a mathematical formulation: (1) a graph representation, and (2) a specific form of the differential equations.
The graph representation of chemical reaction networks is not immediate, since a chemical reaction (the obvious candidate for identification with the edges of a graph) generally involves more than two chemical species (the most basic candidate for identification with the vertices). We will follow an approach that has been initiated and developed in the work of Horn & Jackson [14,13] and Feinberg, see e.g. [10,11], by associating the complexes of the chemical reaction network (i.e., the left-and right-hand sides of the reactions) with the vertices of the graph 1 . The resulting directed graph, called the complex graph in this paper, is characterized by its incidence matrix. The expression of the complexes in the chemical species defines an extended stoichiometric matrix, called the complex stoichiometric matrix, which is immediately related to the standard stoichiometric matrix through the incidence matrix of the complex graph; see e.g. [21], [2].
In order to derive a specific form of differential equations we will start with the basic assumption that the chemical reaction rates are governed by mass action kinetics. As an initial step we then derive a compact form of the dynamics involving a non-symmetric weighted Laplacian matrix of the complex graph. The basic form of these equations can be already found in the innovative paper by Sontag [23]. The main part of the paper is however devoted to a subclass of mass action kinetics chemical reaction networks, where we assume the existence of a thermodynamical equilibrium, or equivalently, where the detailed balance equations are assumed to admit a solution. We will call such chemical reaction networks balanced chemi-cal reaction networks. Balanced chemical reaction networks are necessarily reversible but involve additional conditions on the forward and reverse reaction rate constants (usually referred to as the Wegscheider conditions; see [12]). For such balanced chemical reaction networks we will be able to derive a particularly elegant form of the dynamics, involving a symmetric weighted Laplacian matrix of the complex graph. Furthermore, it turns out that this form has a direct thermodynamical interpretation, and in fact can be regarded as a graph-theoretic version of the formulation derived in the work of Katchalsky, Oster & Perelson [19,18].
The obtained form of the equations of balanced chemical reaction networks will be used to give, in a very simple and insightful way, a characterization of the set of equilibria, and a proof of the asymptotic convergence to a unique thermodynamic equilibrium corresponding to the initial condition of the system. Similar results for a different class of mass action kinetics reaction networks, in particular weakly reversible networks with zerodeficiency, have been derived in the fundamental work of Horn [13] and Feinberg [10,11], which was an indispensable source of concepts and tools for the work reported in this paper.
Subsequently we show how this form of the dynamics of chemical reaction networks can be extended to open reaction networks; i.e., networks involving an influx or efflux of some of the chemical species, called the boundary chemical species. Furthermore, we show how corresponding outputs can be defined as the chemical potentials of these boundary chemical species and how this leads to a physical theory of interconnection of open reaction networks, continuing upon the work of Oster & Perelson [18,20].
In the final part of the paper we make some initial steps in showing how the derived form of balanced reaction networks can be utilized to derive in a systematic way reduced-order models. Th
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