A Graph−Theoretical View of Chemical Transport and Reaction on Networks

Abstract

Certain new results are presented for the matrix representation of graphs, as it pertains to the graph-theoretical treatment of physicochemical transport networks. With the incidence matrix of graph theory identified as a derivative on a digraph, it is shown that the stoichiometric matrix plays a similar role on the digraph of chemical-reaction networks. These ideas are then applied to obtain the graph-theoretic representation of facilitated transport, one of several fields to which Professor John Quinn has made significant contributions (AIChE J. 1979, 25, 197−200).

On nonlinear Onsager symmetry and mass-action kinetics

“…Some of the earliest applications of graph theory to chemical reaction networks are found in the works of Oster and coworkers, notably Perelson and Oster (1974) and references therein. In the author's opinion, the graph-theoretical treatment presented here, together with the existence of a dissipation potential, provides a more economical and perhaps clearer treatment than that of most previous works, including that of Goddard (2002). Thus, a dissipative network of R independent chemical reactions among S distinct chemical species may be viewed as a bipartite graph with a set of S nodes or vertices connected solely by multiple bonds or edges to a second set of R nodes, as depicted schematically in Figure 1, where one set of nodes, labeled alphabetically, represents chemical species and is connected to a second distinct set of nodes, labeled numerically and representing reactions.…”

“…Furthermore, the S � R matrix À Δ T is the analog of a divergence operator, such that Δ T ω ¼ r represents the rate of nodal accumulation due to fluxes or currents given by the R � 1 vector ω. Note that Δ and Δ T represent the respective boundary and coboundary operators of cohomology (Barile 2021;Slepian 2012), as recognized by Perelson and Oster (1974), but overlooked by (Goddard (2002) who nevertheless points out the connection between stoichiometry and the differential operators. Note also that the matrix N, which possesses only positive elements, represents an undirected hypergraph and a generalization of the adjacency matrix for a simple graph, while serving to define the reactivity vector α.…”