A theory and algorithm for reaction route (RR) network analysis is developed in analogy with electrical networks and is based on the combined use of RR theory, graph theory, and Kirchhoff's laws. The result is a powerful new approach of "RR graphs" that is useful in not only topological representation of complex reactions and mechanisms but, when combined with techniques of electrical network analysis, is able to provide revealing insights into the mechanism as well as the kinetics of the overall reactions involving multiple elementary reaction steps including the effect of topological constraints. Unlike existing graph theory approaches of reaction networks, the approach developed here is suitable for linear as well as nonlinear kinetic mechanisms and for single and multiple overall reactions. The theoretical approach for the case of a single overall reaction involving minimal kinetic mechanisms (unit stoichiometric numbers) is developed in Part I of this series followed by its application to examples of heterogeneous and enzyme catalytic reactions in Part II.

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The utility of the new reaction route (RR) graph theory developed in the preceding paper (Part I) is illustrated here, with the help of two examples. In the first example, the kinetics of the conversion of 7,8-dihydrofolate and NADPH to 5,6,7,8-tetrahydrofolate and NADP, as catalyzed by dihydrofolate reductase (DHFR), is considered. This system is described by a linear kinetic mechanism that includes 13 elementary reaction steps. The second example is a microkinetic model of the water-gas shift reaction on a copper catalyst, which is highly nonlinear and includes 15 elementary surface reaction steps. For both mechanisms, the RR graphs have been constructed and used to determine the overall rates. The RR graphs and the overall rate equations are further simplified and reduced, using the RR network approach.

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The concept of reaction route (RR) graphs introduced recently by us for kinetic mechanisms that produce minimal graphs is extended to the problem of non-minimal kinetic mechanisms for the case of a single overall reaction (OR). A RR graph is said to be minimal if all of the stoichiometric numbers in all direct RRs of the mechanism are equal to +/-1 and non-minimal if at least one stoichiometric number in a direct RR is non-unity, e.g., equal to +/-2. For a given mechanism, four unique topological characteristics of RR graphs are defined and enumerated, namely, direct full routes (FRs), empty routes (ERs), intermediate nodes (INs), and terminal nodes (TNs). These are further utilized to construct the RR graphs. One algorithm involves viewing each IN as a central node in a RR sub-graph. As a result, the construction and enumeration of RR graphs are reduced to the problem of balancing the peripheral nodes in the RR sub-graphs according to the list of FRs, ERs, INs, and TNs. An alternate method involves using an independent set of RRs to draw the RR graph while satisfying the INs and TNs. Three examples are presented to illustrate the application of non-minimal RR graph theory.