Time Scale Decomposition in Complex Reaction Systems: A Graph Theoretic Analysis

Complex reaction systems are ubiquitous. They are present in numerous chemical and biochemical systems, such as combustion, pyrolysis, nanoparticle synthesis, catalytic conversion of hydrocarbons, and cell metabolism. These systems involve a large number of reactions generating a large set of intermediate species. Automatic network generators allow generation of the exhaustive reaction network; however, the kinetic modeling of these systems is limited by their size and stiffness. Model reduction techniques involving lumping, sensitivity analysis and time-scale analysis can be used to address these challenges [1]. Model reduction using time scale analysis categorizes reactions as fast and slow, imposes pseudo-steady state hypothesis (PSSH) and/or quasi-equilibrium (QE) assumptions for the fast reactions, and generates pseudo-species which evolve in the slow time scale alone. A rigorous framework for performing these tasks for homogeneous reaction systems using singular perturbations was proposed in [2]; for the case of isothermal systems, a low-dimensional model of the slow dynamics can be expressed in terms of a set of pseudo-species obtained as a linear combination of the original species where the transformation matrix belongs to the left null space of the stoichiometric matrix of the fast reactions.

The present study proposes a graph theoretic analysis framework for identification of the pseudo-species evolving in a slow time scale. A directed bipartite graph is used to represent the species and reactions as two disjoint sets for a general chemical reaction network. An edge connecting the two sets represents species participation in a reaction as a reactant or product depending on the edge direction. Cycles consisting of a pair of species participating in fast reactions are identified using a backtracking algorithm [3]. These cycles are closed walks of two species nodes and two reaction nodes. Further, an algorithm that combines the identified cycles to form the pseudo-species, such that the pseudo-species interact with the remaining reaction network through slow reactions only, is developed. The proposed framework is scalable and applicable to heterogeneous systems as well. Three example reaction systems [2, 4] varying from O(10¹) to O(10⁴) reactions are considered to show the efficacy of the proposed framework. 

[1] Okino, M. S., and Mavrovouniotis, M.L. Simplification of Mathematical Models of Chemical Reaction Systems. Chemical Reviews, 98(2), 391-408, 1998.

[2] Vora, N., Daoutidis, P. Nonlinear Model Reduction of Chemical Reaction Systems. AIChE Journal, 47(10), 2320-2332, 2001.

[3] Tarjan, R. Enumeration of the elementary circuits of a directed graph. SIAM J. Comput., 2(3), 211-216, 1973.

[4] Gerdtzen, Z.P., Daoutidis, P. and Hu, Wei-Shou. Non-linear reduction for kinetic models of metabolic reaction networks. Metabolic Engineering, 6(2), 140-154, 2004.