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.
For an analysis of a molecular system from a computational statistical thermodynamics point of view, extensive
molecular dynamics simulations are very inefficient. During this procedure, at lot of redundant data is generated. Whereas
the algorithms spend most of the computing time for a sampling of configurations within the basins of the potential energy
landscape of the molecular system, the important information about the long-time behaviour of the molecules is given by
transition regions and barriers between the basins, which are sampled rarely only. Thinking of molecular dynamics trajectories,
researchers try to figure out which kind of dynamical model is suitable for an efficient simulation. This article suggests to
change the point of view from extensive simulation of molecular dynamics trajectories to more efficient sampling strategies
of the conformation dynamics approach.
Particle methods have become indispensible in conformation dynamics to compute transition rates in protein folding, binding processes and molecular design, to mention a few. Conformation dynamics requires at a decomposition of a molecule's position space into metastable conformations. In this paper, we show how this decomposition can be obtained via the design of either ``soft'' or ``hard'' molecular conformations. We show, that the soft approach results in a larger metastabilitiy of the decomposition and is thus more advantegous. This is illustrated by a simulation of Alanine Dipeptide.
Supercomputers can simulate complex molecular systems. However, there is a very large gap between the fastest oscillations of covalent bonds of a molecule and the time-scale of the dominant processes. In order to extract the dominant time-scales and to identify the dominant processes, a clustering of information is needed. This thesis shows that only the subspace-based Robust Perron Cluster Analysis (PCCA+) can solve this problem correctly by the construction of a Markov State Model. PCCA+ allows for time-extrapolation in molecular kinetics. This thesis shows the difference between molecular dynamics and molecular kinetics. Only in the molecular kinetics framework a definition of transition rates is possible. In this context, the existence of an infinitesimal generator of the dynamical processes is discussed. If the existence is assumed, the Theorem of Gauß can be applied in order to compute transition rates efficiently. Molecular dynamics, however, is not able to provide a suitable statistical basis for the determination of the transition pattern.
Markov state models have become very popular for the description of conformation dynamics of molecules over
long timescales. The construction of such models requires a partitioning of the configuration space such that the discretization
can serve as an approximation of metastable conformations. Since the computational complexity for the construction of a
Markov state model increases quadratically with the number of sets, it is desirable to obtain as few sets as necessary. In this
paper we propose an algorithm for the adaptive refinement of an initial coarse partitioning. A spectral clustering method is
applied to the final partitioning to detect the metastable conformations. We apply this method to the conformation analysis of
a model tri-peptide molecule, where metastable beta- and gamma-turn conformations can be identified.
In this article we show, that the binding kinetics of a molecular system can be
identied by a projection of a continuous process onto a nite number of macro states. We thus
interpret binding kinetics as a projection. When projecting onto non-overlapping macro states the
Markovianity is spoiled. As a consequence, the description of e.g. a receptor-ligand system by a two
state kinetics is not accurate. By assigning a degree of membership to each state, we abandon the
non-overlapping approach. This overlap is crucial for a correct mapping of binding eects by Markov
State Models with regard to their long time behavior. It enables us to describe the highly discussed
rebinding eect, where the spatial arrangement of the system has the be included. By introducing
a \degree of fuzziness" we have an indicator for the strength of the rebinding eect, such that the
minimal rebinding eect can be derived from an optimization problem. The fuzziness also includes
some new paradigms for molecular kinetics. These new model paradigms show good agreement with
experimental data.
Trajectory- or mesh-based methods for analyzing the dynamical behavior of large
molecules tend to be impractical due to the curse of dimensionality - their computational cost increases
exponentially with the size of the molecule. We propose a method to break the curse by a
novel square root approximation of transition rates, Monte Carlo quadrature and a discretization
approach based on solving linear programs. With randomly sampled points on the molecular energy
landscape and randomly generated discretizations of the molecular conguration space as our initial
data, we construct a matrix describing the transition rates between adjacent discretization regions.
This transition rate matrix yields a Markov State Model of the molecular dynamics. We use Perron
cluster analysis and coarse-graining techniques in order to identify metastable sets in conguration
space and approximate the transition rates between the metastable sets. Application of our method
to a simple energy landscape on a two-dimensional conguration space provides proof of concept and
an example for which we compare the performance of dierent discretizations. We show that the
computational cost of our method grows only polynomially with the size of the molecule. However,
nding discretizations of higher-dimensional conguration spaces in which metastable sets can be
identied remains a challenge.
Markov State Models (MSMs) are widely used to represent molecular
conformational changes as jump-like transitions between subsets of the conformational
state space. However, the simulation of peptide folding in explicit water is
usually said to be unsuitable for the MSM framework. In this article, we summarize
the theoretical background of MSMs and indicate that explicit water simulations do
not contradict these principles. The algorithmic framework of a meshless conformational
space discretization is applied to an explicit water system and the sampling
results are compared to a long-term molecular dynamics trajectory. The meshless
discretization approach is based on spectral clustering of stochastic matrices (MSMs)
and allows for a parallelization of MD simulations. In our example of Trialanine we
were able to compute the same distribution of a long term simulation in less computing
time.