Reaction Route Graphs. I. Theory and Algorithm

Abstract

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.

Ockham's razor for paring microkinetic mechanisms: Electrical analogy vs.Campbell's degree of rate control

“…36 This may be accomplished by appropriately combining the elementary steps into the so-called intermediate reactions (IR). [41][42][43] There are different ways in which the Campbell's DRC, once hence numerically computed for all the steps in a microkinetic model, may be utilized for identification of key steps and model reduction. 46 One such method is the so-called principal component analysis, 40,47,48 in which the eigenvalues and eigenvectors of the matrix X T X are computed.…”

“…53 In other words, one needs both thermodynamic (affinity, or reversibility) and kinetic (resistance) characteristics of a reaction step to judge its significance in the overall scheme of things. The electrical network analogy based on the RR graph approach [41][42][43] incorporates both of these aspects. It, thus, comprehensively incorporates and illuminates the network structural constraints embodied by the two Kirchhoff's laws (mass balance, and Hess's law) described below, which further ascribe to reaction networks a complete and quantitative correspondence to electrical networks.…”

“…3,7,39,40 However, it is a "black box" numerical approach. Here, we discuss an alternate approach based on electrical network analogy of a reaction mechanism within the reaction route (RR) graph analysis approach [41][42][43][44] developed by us, which involves an evaluation of the reaction step "affinity," or driving force, and "resistance" under a variety of conditions. It is just as rigorous and substantially more revealing than Campbell's DRC, allowing transparent pruning of complex catalytic reaction networks.…”

“…An additional possibility is to compare ratio of the power dissipation in each step, 41 namely, r q A q , to that in the OR,…”

Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures

“…Kirchhoff graphs were so-named because they satisfy both the Kirchhoff potential law and the Kirchhoff current law. The use of Kirchhoff graphs as reaction network diagrams was discussed by Fehribach [1], as well as by Fishtik, Datta et al [3][4][5][6], who discuss a related graph, the "reaction route graph". A number of other graphs have been used in the study of reaction networks; a summary can be found in [7,8].…”