Laplacian Dynamics and Kron Reduction in Species-Reaction Graphs of Chemical Reaction Networks

Abstract

We present a new method for deriving the dynamics of chemical reaction networks using the Laplacian matrix of the corresponding species-reaction graph, in contrast to previous works that use the Laplacian of the graph of complexes. Species-reaction graphs are bipartite graphs that contain two sets of vertices, one representing species and the other representing reactions, connected by directed edges that indicate relationships between them. Our approach starts by assigning appropriate edge weights to this bipartite graph, which are then used to compute the weighted graph Laplacian. This Laplacian reformulation of the system of differential equations governing the network dynamics emphasizes the flow of information throughout the chemical reaction network considered as causal network. As an application of this framework, we introduce a novel model reduction technique based on the Kron reduction of the weighted Laplacian matrix associated with the species-reaction graphs. Our systematic approach involves identifying nodes for deletion while preserving the bipartite structure, followed by constructing the Kron-reduced model. To demonstrate the effectiveness of our method, we apply it to a complex biochemical network, showing how model simplification facilitates analysis and interpretation of these systems.