Rare events and first passage time statistics from the energy landscape

Abstract

In this contribution we analyse the probability distribution of rare first passage times corresponding to transitions between product and reactant states in a kinetic transition network. The mean first passage times and corresponding rate constants are analysed in detail for two model landscapes and the double funnel landscape corresponding to an atomic cluster. Evaluation schemes based on eigendecomposition and kinetic path sampling, which both allow access to the passage time distribution, are benchmarked against mean first passage times calculated used graph transformation. Numerical precision issues severely limit the useful temperature range for eigendecomposition, but kinetic path sampling is capable of extending the first passage time analysis to lower temperatures, where the kinetics of interest constitute rare events. We then investigate the influence of free energy based state regrouping schemes for the underlying network. Alternative formulations of the effective transition rates for a given regrouping are compared in detail, to determine their numerical stability and capability to reproduce the true kinetics, including recent coarse-graining approaches that preserve occupancy cross correlation functions. We find that appropriate regrouping of states under the simplest local equilibrium approximation can provide reduced transition networks with useful accuracy at somewhat lower temperatures. Finally, a method is provided to systematically interpolate between the local equilibrium approximation and exact intergroup dynamics. Spectral analysis is applied to each grouping of states, employing a moment-based mode selection criteria to produce a reduced state space, which does not require any spectral gap to exist, but reduces to gap-based coarse graining as a special case. Implementations of the developed methods are freely available online.

Optimal dimensionality reduction of Markov chains using graph transformation

“…Hence, this absorbing Markov chain "leaks" probability, and MFPTs for transitions to A are related to the probability flux out of the nonabsorbing region. Unlike K, K does not have a zero eigenvalue and is therefore invertible, giving 29,105…”

“…Despite the O(V 3 ) scaling, eigendecomposition can in principle be achieved efficiently even for high-dimensional systems 120,121 using Krylov subspace methods, 122 such as the Lanczos algorithm, 123 applied to the symmetrized 29,57 transition probability (or rate) matrix. However, this approach does not mitigate the severe ill-conditioning issues for systems exhibiting rare event dynamics.…”

“…11 and 12), linear solution (Eq. 9), and eigendecomposition 23,29 (Eq. 10) break down in the metastable regime.…”

“…II E), and is beyond the scope of the present work. We have previously shown that multi-level regularized Markov clustering 81 and recursive regrouping based on a transition rate threshold 29 provide viable approaches to community structure detection 139 in metastable Markov chains.…”

“…[10][11][12][13][14][15][16][17][18][19][20][21][22] Standard linear algebra methods to compute MFPTs encounter numerical issues for metastable Markov chains because the separation of slow and fast timescales in the system dynamics leads to severe ill-conditioning. [23][24][25][26][27][28] Since rare events are ubiquitous in realistic models of stochastic dynamical processes, [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] more numerically stable algorithms are often required for the analysis of Markov chain dynamics. Graph transformation [44][45][46][47][48][49] (GT) provides an exact method to compute the A ← B MFPT that retains numerical precision even for strongly metastable Markov chains.…”