Ultrametric diffusion, rugged energy landscapes and transition networks

In this article we introduce the ultrametric networks which are p-adic continuous analogs of the standard Markov state models constructed using master equations. A p-adic transition network (or an ultrametric network) is a model of a complex system consisting of a hierarchical energy landscape, a Markov process on the energy landscape, and a master equation. The energy landscape consists of a finite number of basins. Each basin is formed by infinitely many network configurations organized hierarchically in an infinite regular tree. The transitions between the basins are determined by a transition density matrix, whose entries are functions defined on the energy landscape. The Markov process in the energy landscape encodes the temporal evolution of the network as random transitions between configurations from the energy landscape. The master equation describes the time evolution of the density of the configurations. We focus on networks where the transition rates between two different basins are constant functions, and the jumping process inside of each basin is controlled by a p-adic radial function. We solve explicitly the Cauchy problem for the master equation attached to this type of networks. The solution of this problem is the network response to a given initial concentration. If the Markov process attached to the network is conservative, the long term response of the network is controlled by a Markov chain. If the process is not conservative the network has absorbing states. We define an absorbing time, which depends on the initial concentration, if this time is finite the network reaches an absorbing state in a finite time. We identify in the response of the network the terms responsible for bringing the network to an absorbing state, we call them the fast transition modes. The existence of the fast transition modes is a consequence of the assumption that the energy landscape is ultrametric (hierarchical), and to the best of our understanding this result cannot be obtained by using standard methods of Markov state models. Nowadays, it is widely accepted that protein native states are kinetic hubs that can be reached quickly from any other state. The existence of fast transition modes implies that certain states on an ultrametric network work as kinetic hubs.