The key to molecular conformation dynamics is the direct identification of metastable conformations, which are almost invariant sets of molecular dynamical systems. Once some reversible Markov operator has been discretized, a generalized symmetric stochastic matrix arises. This matrix can be treated by Perron cluster analysis, a rather recent method involving a Perron cluster eigenproblem. The paper presents an improved Perron cluster analysis algorithm, which is more robust than earlier suggestions. Numerical examples are included.
The complexity of molecular kinetics can be reduced significantly by a restriction to metastable conformations which are almost invariant sets of molecular dynamical systems. With the Robust Perron Cl uster Analysis PCCA+, developed by Weber and Deuflhard, we have a tool available which can be used to identify these conformations from a transition probability matrix. This method can also be applied to the corresponding transition rate matrix which provides important information concerning transition pathways of single molecules. In the present paper, we explain the relationship between these tw o concepts and the extraction of conformation kinetics from transition rates. Moreover, we show how transition rates can be approximated and conclude with numerical examples.
Whenever the invariant stationary density of metastable dynamical systems decomposes into almost invariant partial densities, its computation as eigenvector of some transition probability matrix is an ill-conditioned problem. In order to avoid this computational difficulty, we suggest to apply an aggregation/disaggregation method which only addresses wellconditioned sub-problems and thus results in a stable algorithm. In contrast to existing methods, the aggregation step is done via a sampling algorithm which covers only small patches of the sampling space. Finally, the theoretical analysis is illustrated by two biomolecular examples.
Wigner functions are functions on classical phase space, which are in one-to-one correspondence to square integrable functions on configuration space. For molecular quantum systems, classical transport of Wigner functions provides the basis of asymptotic approximation methods in the high energy regime. The article addresses the sampling of Wigner functions by Monte Carlo techniques. The approximation step is realized by an adaption of the Metropolis algorithm for real-valued functions with disconnected support. The quadrature, which computes values of the Wigner function, uses importance sampling with a Gaussian weight function. The numerical experiments combine the sampling with a surface hopping algorithm for non-adiabatic quantum dynamics. In agreement with theoretical considerations, the obtained results show an accuracy of two to four percent.
The identification of metastable conformations of molecules plays an
important role in computational drug design. One main difficulty is the
fact that the underlying dynamic processes take place in high dimensional
spaces. Although the restriction of degrees of freedom to a few dihedral
angles significantly reduces the complexity of the problem, the existing
algorithms are time-consuming. They are mainly based on the approximation
of a transfer operator by an extensive sampling of states according
to the Boltzmann distribution and short-time Hamiltonian dynamics simulations.
We present a method which can identify metastable conformations
without sampling the complete distribution. Our algorithm is based
on local transition rates and uses only pointwise information about the
potential energy surface. In order to apply the cluster algorithm PCCA+,
we compute a few eigenvectors of the rate matrix by the Jacobi-Davidson
method. Interpolation techniques are applied to approximate the thermodynamical
weights of the clusters. The concluding example illustrates
our approach for epigallocatechine, a molecule which can be described by
seven dihedral angles.
In order to compute the thermodynamic weights of the different metastable conformations
of a molecule, we want to approximate the molecule’s Boltzmann distribution in a reasonable
time. This is an essential issue in computational drug design. The energy landscape of active
biomolecules is generally very rough with a lot of high barriers and low regions. Many of the
algorithms that perform such samplings (e.g. the hybrid Monte Carlo method) have difficulties
with such landscapes. They are trapped in low-energy regions for a very long time and cannot
overcome high barriers. Moving from one low-energy region to another is a very rare event. For
these reasons, the distribution of the generated sampling points converges very slowly against
the thermodynamically correct distribution of the molecule.
The idea of ConfJump is to use a priori knowledge of the localization of low-energy regions
to enhance the sampling with artificial jumps between these low-energy regions. The artificial
jumps are combined with the hybrid Monte Carlo method. This allows the computation of
some dynamical properties of the molecule. In ConfJump, the detailed balance condition is
satisfied and the mathematically correct molecular distribution is sampled.
Biochemical interactions are determined by the 3D-structure of the involved components –
thus the identification of conformations is a key for many applications in rational drug design.
ConFlow is a new multilevel approach to conformational analysis with main focus on
completeness in investigation of conformational space.
In contrast to known conformational analysis, the starting point for design is a space-based
description of conformational areas. A tight integration of sampling and analysis leads to an
identification of conformational areas simultaneously during sampling. An incremental
decomposition of high-dimensional conformational space is used to guide the analysis. A new
concept for the description of conformations and their path connected components based on
convex hulls and Hypercubes is developed. The first results of the ConFlow application
constitute a ‘proof of concept’ and are further more highly encouraging. In comparison to
conventional industrial applications, ConFlow achieves higher accuracy and a specified
degree of completeness with comparable effort.
In this paper, we investigate the interconversion processes of the major flame retardant - 1,2,5,6,9,10-hexabromocyclododecane (HBCD) - by the means of statistical thermodynamics based on classical force-fields. Three ideas will be presented. First, the application of classical hybrid Monte-Carlo simulations for quantum mechanical processes will be justified. Second, the problem of insufficient convergence properties of hybrid Monte-Carlo methods for the generation of low temperature canonical ensembles will be solved by an interpolation approach. Furthermore, it will be shown how free energy differences can be used for a rate matrix computation. The results of our numerical simulations will be compared to experimental results.
Keywords: Markov process, molecular dynamics, rate matrix
This article deals with an efficient sampling of the stationary distribution
of dynamical systems in the presence of metastabilities. For such
systems, standard sampling schemes suffer from trapping problems and
critical slowing down. Starting multiple trajectories in different regions of
the sampling space is a promising way out. The different samplings represent
the stationary distribution locally very well, but are still far away
from ergodicity or from the global stationary distribution. We will show
how these samplings can be joined together in order to get one global
sampling of the stationary distribution.
For the treatment of equilibrated molecular systems in a heat bath we
propose a transition state theory that is based on conformation dynamics.
In general, a set-based discretization of a Markov operator P does not
preserve the Markov property. In this article, we propose a discretization
method which is based on a Galerkin approach. This discretization
method preserves the Markov property of the operator and can be interpreted
as a decomposition of the state space into (fuzzy) sets. The
conformation-based transition state theory presented here can be seen as
a first step in conformation dynamics towards the computation of essential
dynamical properties of molecular systems without time-consuming
molecular dynamics simulations.