title: Deciphering Time Scale Hierarchy in Reaction Networks.
abstract: 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 time scale of 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.
authors.affiliation: Graduate School of Life Science, Hokkaido University , Kita 10, Nishi 8, Kita-ku, Sapporo 060-0812, Japan.
authors.forename: Yutaka
authors.initials: Y
authors.lastname: Nagahata
authors.affiliation: Department of Chemistry, Faculty of Science, Hokkaido University , Kita 10, Nishi 8, Kita-ku, Sapporo 060-0810, Japan.
authors.forename: Satoshi
authors.initials: S
authors.lastname: Maeda
authors.affiliation: Graduate School of Life Science, Hokkaido University , Kita 10, Nishi 8, Kita-ku, Sapporo 060-0812, Japan.|Molecule and Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University , Kita 20, Nishi 10, Kita-ku, Sapporo 001-0020, Japan.
authors.forename: Hiroshi
authors.initials: H
authors.lastname: Teramoto
authors.affiliation: Graduate School of Science and Engineering, Saitama University , Shimo-Ookubo 255, Sakura-ku, Saitama 338-8570, Japan.
authors.forename: Takashi
authors.initials: T
authors.lastname: Horiyama
authors.affiliation: Department of Chemistry, Faculty of Science, Hokkaido University , Kita 10, Nishi 8, Kita-ku, Sapporo 060-0810, Japan.
authors.forename: Tetsuya
authors.initials: T
authors.lastname: Taketsugu
authors.affiliation: Graduate School of Life Science, Hokkaido University , Kita 10, Nishi 8, Kita-ku, Sapporo 060-0812, Japan.|Molecule and Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University , Kita 20, Nishi 10, Kita-ku, Sapporo 001-0020, Japan.
authors.forename: Tamiki
authors.initials: T
authors.lastname: Komatsuzaki