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 positive equilibria and their stability properties. Furthermore, we develop a framework for interconnection of chemical reaction networks, and we discuss how the formulation leads to a new approach for model reduction.

In this paper we propose a model reduction method for biochemical reaction networks governed by a variety of reversible and irreversible enzyme kinetic rate laws, including reversible Michaelis-Menten and Hill kinetics. The method proceeds by a stepwise reduction in the number of complexes, defined as the left and right-hand sides of the reactions in the network. It is based on the Kron reduction of the weighted Laplacian matrix, which describes the graph structure of the complexes and reactions in the network. It does not rely on prior knowledge of the dynamic behaviour of the network and hence can be automated, as we demonstrate. The reduced network has fewer complexes, reactions, variables and parameters as compared to the original network, and yet the behaviour of a preselected set of significant metabolites in the reduced network resembles that of the original network. Moreover the reduced network largely retains the structure and kinetics of the original model.

Studies with dietary supplementation of various types of fibers have shown beneficial effects on symptoms of the metabolic syndrome. Short-chain fatty acids (SCFAs), the main products of intestinal bacterial fermentation of dietary fiber, have been suggested to play a key role. Whether the concentration of SCFAs or their metabolism drives these beneficial effects is not yet clear. In this study we investigated the SCFA concentrations and in vivo host uptake fluxes in the absence or presence of the dietary fiber guar gum. C57Bl/6J mice were fed a high-fat diet supplemented with 0%, 5%, 7.5% or 10% of the fiber guar gum. To determine the effect on SCFA metabolism, ¹³C-labeled acetate, propionate or butyrate were infused into the cecum of mice for 6 h and the isotopic enrichment of cecal SCFAs was measured. The in vivo production, uptake and bacterial interconversion of acetate, propionate and butyrate were calculated by combining the data from the three infusion experiments in a single steady-state isotope model. Guar gum treatment decreased markers of the metabolic syndrome (body weight, adipose weight, triglycerides, glucose and insulin levels and HOMA-IR) in a dose-dependent manner. In addition, hepatic mRNA expression of genes involved in gluconeogenesis and fatty acid synthesis decreased dose-dependently by guar gum treatment. Cecal SCFA concentrations were increased compared to the control group, but no differences were observed between the different guar gum doses. Thus, no significant correlation was found between cecal SCFA concentrations and metabolic markers. In contrast, in vivo SCFA uptake fluxes by the host correlated linearly with metabolic markers. We argue that in vivo SCFA fluxes, and not concentrations, govern the protection from the metabolic syndrome by dietary fibers.

The characterization of the notions of complex and detailed balancing for mass action kinetics chemical reaction networks is revisited from the perspective of algebraic graph theory, in particular Kirchhoff’s Matrix Tree theorem for directed weighted graphs. This yields an elucidation of previously obtained results, in particular with respect to the Wegscheider conditions, and a new necessary and sufficient condition for complex balancing, which can be verified constructively.