Alexander Fischer

Identification of Almost Invariant Aggregates in Reversible Nearly Uncoupled Markov Chains

The topic of the present paper has been motivated by a recent computational approach to identify metastable chemical conformations and patterns of conformational changes within molecular systems. After proper discretization, such conformations show up as almost invariant aggregates in reversible, nearly uncoupled Markov chains (NUMCs). Most of the former work on this subject treated the direct problem: given the aggregates, analyze the loose coupling in connection with the computation of the stationary distribution (aggregation/disaggregation techniques). In contrast to that, the present paper focuses on the inverse problem: given the system as a whole, identify the almost invariant aggregates together with the (small) probabilities of transitions between them. A robust algorithm is worked out on the basis of some detailed perturbation analysis and illustrated at a simple molecular system.

Automated Model Reduction for Complex Systems exhibiting Metastability

We present a novel method for the identification of the most important metastable states of a system with complicated dynamical behavior from time series information. The novel approach represents the effective dynamics of the full system by a Markov jump process between metastable states and the dynamics within each of these metastable states by rather simple stochastic differential equations (SDEs). Its algorithmic realization exploits the concept of hidden Markov models with output behavior given by SDEs. The numerical effort of the method is linear in the length of the given time series and quadratic in terms of the number of metastable states. The performance of the resulting method is illustrated by numerical tests and by application to molecular dynamics time series of a trialanine molecule.

Hybrid Monte Carlo with adaptive temperature in mixed‐canonical ensemble: Efficient conformational analysis of RNA

A hybrid Monte Carlo method with adaptive temperature choice is presented that exactly generates the distribution of a mixed‐canonical ensemble composed of two canonical ensembles at low and high temperature. The analysis of resulting Markov chains with the reweighting technique shows an efficient sampling of the canonical distribution at low temperature whereas the high temperature component facilitates conformational transitions, which allows shorter simulation times. The algorithm is tested by comparing analytical and numerical results for the small n‐butane molecule before simulations are performed for a triribonucleotide. Sampling the complex multiminima energy landscape of this small RNA segment, we observe enforced crossing of energy barriers. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1689–1697, 1998

Hierarchical Uncoupling-Coupling of Metastable Conformations

Uncoupling-coupling Monte Carlo (UCMC) combines uncoupling techniques for finite Markov chains with Markov chain Monte Carlo methodology. UCMC aims at avoiding the typical metastable or trapping behavior of Monte Carlo techniques. From the viewpoint of Monte Carlo, a slowly converging long-time Markov chain is replaced by a limited number of rapidly mixing short-time ones. Therefore, the state space of the chain has to be hierarchically decomposed into its metastable conformations. This is done by means of combining the technique of conformation analysis as recently introduced by the authors, and appropriate annealing strategies. We present a detailed examination of the uncoupling-coupling procedure which uncovers its theoretical background, and illustrates the hierarchical algorithmic approach. Furthermore, application of the UCMC algorithm to the n-pentane molecule allows us to discuss the effect of its crucial steps in a typical molecular scenario.

Identification of biomolecular conformations from incomplete torsion angle observations by hidden markov models

We present a novel method for the identification of the most important conformations of a biomolecular system from molecular dynamics or Metropolis Monte Carlo time series by means of Hidden Markov Models (HMMs). We show that identification is possible based on the observation sequences of some essential torsion or backbone angles. In particular, the method still provides good results even if the conformations do have a strong overlap in these angles. To apply HMMs to angular data, we use von Mises output distributions. The performance of the resulting method is illustrated by numerical tests and by application to a hybrid Monte Carlo time series of trialanine and to MD simulation results of a DNA–oligomer.

Uncoupling-Coupling Techniques for Metastable Dynamical Systems

We shortly review the uncoupling-coupling method, a Markov chain Monte Carlo based approach to compute statistical properties of systems like medium-sized biomolecules. This technique has recently been proposed for the efficient computation of biomolecular conformations. One crucial step of UC is the decomposition of reversible nearly uncoupled Markov chains into rapidly mixing subchains. We show how the underlying scheme of uncoupling-coupling can also be applied to stochastic differential equations where it can be translated into a domain decomposition technique for partial differential equations.