Nagahata Yutaka, Maeda Satoshi, Teramoto Hiroshi, Horiyama Takashi, Taketsugu Tetsuya, Komatsuzaki Tamiki
Graduate School of Life Science, Hokkaido University , Kita 10, Nishi 8, Kita-ku, Sapporo 060-0812, Japan.
Department of Chemistry, Faculty of Science, Hokkaido University , Kita 10, Nishi 8, Kita-ku, Sapporo 060-0810, Japan.
J Phys Chem B. 2016 Mar 3;120(8):1961-71. doi: 10.1021/acs.jpcb.5b09941. Epub 2015 Dec 22.
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.
复杂反应网络上的马尔可夫动力学是包括化学反应、生物物理和生态学在内的广泛研究领域中最引人入胜的课题之一。要表示从一个节点（对应于能量景观上的一个盆地）到另一个节点的全局动力学，需要有关通过整个网络直接或间接连接这两个节点的多条路径的信息。在本文中，我们提出了一种方案，用于在从一阶速率方程导出的离散时间马尔可夫链上提取一组分层的全局过渡态（TSs）。这些TSs可以自然地考虑连接任意一对节点的多条路径。我们还提出了一种新型的不连通图（DG），以捕捉不同反应时间尺度的层次组织，该图可以捕捉由于观察时间尺度的变化而导致的网络变化。关键在于在图聚类中引入最小电导割（MCC），它对应于网络中两个不相交子网（能量景观上的超级盆地）之间具有“最小”过渡概率的分隔面。我们提出了一种用于找到此MCC的新组合搜索算法。我们将我们的方法应用于烯丙基乙烯基醚克莱森重排的反应网络，该网络由23个节点和连接它们的66条链路（能量景观上的鞍点）组成。我们将我们的DG的动力学性质与速率方程的转移矩阵的动力学性质进行比较，并表明我们的图可以正确地揭示网络中时间尺度的层次组织。
