Deciphering Time Scale Hierarchy In Reaction Networks

Markovian dynamics on complex reaction networks are one of the most intriguing subjects in a wide range of research fields including chemical reactions, biological physics, and ecology. To represent the global kinetics from one node (corresponding to a basin on an energy landscape) to another requires information on multiple pathways that directly or indirectly connect these two nodes through the entire network. In this paper we present a scheme to extract a hierarchical set of global transition states (TSs) over a discrete time Markov chain derived from first-order rate equations. The TSs can naturally take into account the multiple pathways connecting any pair of nodes. We also propose a new type of disconnectivity graph (DG) to capture the hierarchical organization of different time scales of reactions that can capture changes in the network due to changes in the observation. The crux is the introduction of the minimum conductance cut (MCC) in graph clustering, corresponding to the dividing surface across the network having the "smallest" transition probability between two disjoint subnetworks (superbasins on the energy landscape) in the network. We present a new combinatorial search algorithm for finding this MCC. We apply our method to a reaction network of Claisen rearrangement of allyl vinyl ether that consists of 23 nodes and 66 links (saddles on the energy landscape) connecting them. We compare the kinetic properties of our DG to those of the transition matrix of the rate equations and show that our graph can properly reveal the hierarchical, organization of time scales in a network.