Susanna Kube

A coarse graining method for the identification of transition rates between molecular conformations

The coarse graining method to be advocated in this paper consists of two main steps. First, the propagation of an ensemble of molecular states is described as a Markov chain by a transition probability matrix in a finite state space. Second, we obtain metastable conformations by an aggregation of variables via Robust Perron Cluster Analysis (PCCA+). Up to now, it has been an open question as to how this coarse graining in space can be transformed to a coarse graining of the Markov chain while preserving the essential dynamic information. In this article, we construct a coarse matrix that is the correct propagator in the space of conformations. This coarse graining procedure carries over to rate matrices and allows to extract transition rates between molecular conformations. This approach is based on the fact that PCCA+ computes molecular conformations as linear combinations of the dominant eigenvectors of the transition matrix.

Stable computation of probability densities for metastable dynamical systems

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 applying an aggregation/disaggregation method which addresses only well-conditioned subproblems 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.

Robust perron cluster analysis for various applications in computational life science

In the present paper we explain the basic ideas of Robust Perron Cluster Analysis (PCCA+) and exemplify the different application areas of this new and powerful method. Recently, Deuflhard and Weber [5] proposed PCCA+ as a new cluster algorithm in conformation dynamics for computational drug design. This method was originally designed for the identification of almost invariant subsets of states in a Markov chain. As an advantage, PCCA+ provides an indicator for the number of clusters. It turned out that PCCA+ can also be applied to other problems in life science. We are going to show how it serves for the clustering of gene expression data stemming from breast cancer research [20]. We also demonstrate that PCCA+ can be used for the clustering of HIV protease inhibitors corresponding to their activity. In theoretical chemistry, PCCA+ is applied to the analysis of metastable ensembles in monomolecular kinetics, which is a tool for RNA folding [21].

Monte Carlo sampling of Wigner functions and surface hopping quantum dynamics

The article addresses the achievable accuracy for a Monte Carlo sampling of Wigner functions in combination with a surface hopping algorithm for non-adiabatic quantum dynamics. The approximation of Wigner functions is realized by an adaption of the Metropolis algorithm for real-valued functions with disconnected support. The integration, which is necessary for computing values of the Wigner function, uses importance sampling with a Gaussian weight function. The numerical experiments agree with theoretical considerations and show an error of 2-3%.

Preserving the Markov Property of Reduced Reversible Markov Chains

The computation of essential dynamics of molecular systems by conformation dynamics turned out to be very successful. This approach is based on Markov chain Monte Carlo simulations. Conformation dynamics aims at decomposing the state space of the system into metastable subsets. The set‐based reduction of a Markov chain, however, destroys the Markov property. We will present an alternative reduction method that is not based on sets but on membership vectors, which are computed by the Robust Perron Cluster Analysis (PCCA+). This approach preserves the Markov property.

Computation of Equilibrium Densities in Metastable Dynamical Systems by Domain Decomposition

Whenever the stationary density of molecular dynamical systems decomposes into almost invariant partial densities, its computation from long‐time dynamics simulations is infeasible within the available computer time due to the well‐known “trapping problem.” In order to avoid this computational difficulty, we suggest a domain decomposition approach that is similar to umbrella sampling methods. In contrast to standard umbrella sampling techniques, our decomposition forms a partition of unity such that the corresponding stationary density can be computed as eigenvector of some mass matrix. This approach has many advantages over traditional approaches used to unbias and recombine the umbrella sampling calculations. The theoretical analysis is illustrated by a two‐dimensional example.