Gaussian Markov transition models of molecular kinetics

Abstract

The slow processes of molecular dynamics (MD) simulations--governed by dominant eigenvalues and eigenfunctions of MD propagators--contain essential information on structures of and transition rates between long-lived conformations. Existing approaches to this problem, including Markov state models and the variational approach, represent the dominant eigenfunctions as linear combinations of a set of basis functions. However the choice of the basis functions and their systematic statistical estimation are unsolved problems. Here, we propose a new class of kinetic models called Markov transition models (MTMs) that approximate the transition density of the MD propagator by a mixture of probability densities. Specifically, we use Gaussian MTMs where a Gaussian mixture model is used to approximate the symmetrized transition density. This approach allows for a direct computation of spectral components. In contrast with the other Galerkin-type approximations, our approach can automatically adjust the involved Gaussian basis functions and handle the statistical uncertainties in a Bayesian framework. We demonstrate by some simulation examples the effectiveness and accuracy of the proposed approach.

VAMPnets for deep learning of molecular kinetics

“…The resulting coordinates may be scaled, in order to embed them in a metric space whose distances correspond to some form of dynamical distance [31,32]. The resulting metric space is discretized by clustering the projected data using hard or fuzzy data-based clustering methods [11,13,[33][34][35][36][37], typically resulting in 100-1000 discrete states. A transition matrix or rate matrix describing the transition probabilities or rate between the discrete states at some lag time τ is then estimated [8,12,38,39] (alternatively, a Koopman model can be built after the dimension reduction [27,28]).…”

Kinetic Distance and Kinetic Maps from Molecular Dynamics Simulation

“…Markov state models provide both eigenvectors (as left and right eigenvectors of the transition matrix). Moreover it is possible to design Markov transition models [35] such that both sets of eigenvectors can be computed. For nonreversible dynamics, there can be complex eigenvalues and eigenvectors which come in complex conjugate pairs (λ j , φ j , ψ j ) and (λ j ,φ j ,ψ j ).…”

“…The first question has been answered: building on conformation dynamics theory [29,28,24,15] it has been shown that the eigenfunctions of the backward Markov propagator underlying the MD are the optimal reaction coordinates: Projecting the dynamics upon these eigenfunctions will give rise to a maximum estimate of the timescales [23,15,17] and an optimal separation of metastable states. A number of approaches are available to approximate these reaction coordinates from MD data: Diffusion maps [26], Markov state models [24] and Markov transition models [35], TICA [18,30] and kernel TICA [31]. The variational approach for conformation dynamics (VAC) [15,17] is a generalization to all aforementioned models except for diffusion maps and describes a general approach to combine and parametrize basis functions so as to optimally define the true eigenfunctions of the backward propagator.…”

Variational tensor approach for approximating the rare-event kinetics of macromolecular systems

“…6 This important property can also be achieved for other basis sets that are probability densities, such as Gaussian distributions. 35 For arbitrary basis functions, estimator Eqs. (25) and 26are incorrect if the data are not in global equilibrium.…”

“…where we have used recursion formula Eq. (35). Next, we can compute the approximation error for g p+1 k p via…”