@misc{SchuetteFischerHuisingaetal., author = {Sch{\"u}tte, Christof and Fischer, Alexander and Huisinga, Wilhelm and Deuflhard, Peter}, title = {A Direct Approach to Conformational Dynamics based on Hybrid Monte Carlo}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3889}, number = {SC-98-45}, abstract = {Recently, a novel concept for the computation of essential features of the dynamics of Hamiltonian systems (such as molecular dynamics) has been proposed. The realization of this concept had been based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept that merges the conceptual advantages of the dynamical systems approach with the appropriate statistical physics framework. This approach allows to define the phrase ``conformation'' in terms of the dynamical behavior of the molecular system and to characterize the dynamical stability of conformations. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator \$T\$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of \$T\$ via specific hybrid Monte Carlo techniques is shown to lead to a stochastic matrix \$P\$. With these theoretical preparations, an identification algorithm for conformations is applicable. It is demonstrated that the discretization of \$T\$ can be restricted to few essential degrees of freedom so that the combinatorial explosion of discretization boxes is prevented and biomolecular systems can be attacked. Numerical results for the n-pentane molecule and the triribonucleotide adenylyl\emph{(3'-5')}cytidylyl\emph{(3'-5')}cytidin are given and interpreted.}, language = {en} } @misc{SchuetteHuisinga, author = {Sch{\"u}tte, Christof and Huisinga, Wilhelm}, title = {On Conformational Dynamics induced by Langevin Processes}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4130}, number = {SC-99-25}, abstract = {The function of many important biomolecules is related to their dynamic properties and their ability to switch between different {\em conformations}, which are understood as {\em almost invariant} or {\em metastable} subsets of the positional state space of the system. Recently, the present authors and their coworkers presented a novel algorithmic scheme for the direct numerical determination of such metastable subsets and the transition probability between them. Although being different in most aspects, this method exploits the same basic idea as {\sc Dellnitz} and {\sc Junge} in their approach to almost invariance in discrete dynamical systems: the almost invariant sets are computed via certain eigenvectors of the Markov operators associated with the dynamical behavior. In the present article we analyze the application of this approach to (high--friction) Langevin models describing the dynamical behavior of molecular systems coupled to a heat bath. We will see that this can be related to theoretical results for (symmetric) semigroups of Markov operators going back to {\sc Davies}. We concentrate on a comparison of our approach in respect to random perturbations of dynamical systems.}, language = {en} } @misc{SchuetteHuisingaDeuflhard, author = {Sch{\"u}tte, Christof and Huisinga, Wilhelm and Deuflhard, Peter}, title = {Transfer Operator Approach to Conformational Dynamics in Biomolecular Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4247}, number = {SC-99-36}, abstract = {The article surveys the development of novel mathematical concepts and algorithmic approaches based thereon in view of their possible applicability to biomolecular design. Both a first deterministic approach, based on the Frobenius-Perron operator corresponding to the flow of the Hamiltonian dynamics, and later stochastic approaches, based on a spatial Markov operator or on Langevin dynamics, can be subsumed under the unified mathematical roof of the transfer operator approach to effective dynamics of molecular systems. The key idea of constructing specific transfer operators especially taylored for the purpose of conformational dynamics appears as the red line throughout the paper. Different steps of the algorithm are exemplified by a trinucleotide molecular system as a small representative of possible RNA drug molecules.}, language = {en} } @misc{HuisingaBestCordesetal., author = {Huisinga, Wilhelm and Best, Christoph and Cordes, Frank and Roitzsch, Rainer and Sch{\"u}tte, Christof}, title = {From Simulation Data to Conformational Ensembles: Structure and Dynamics based Methods}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3797}, number = {SC-98-36}, abstract = {Statistical methods for analyzing large data sets of molecular configurations within the chemical concept of molecular conformations are described. The strategies are based on dependencies between configurations of a molecular ensemble; the article concentrates on dependencies induces by a) correlations between the molecular degrees of freedom, b) geometrical similarities of configurations, and c) dynamical relations between subsets of configurations. The statistical technique realizing aspect a) is based on an approach suggested by {\sc Amadei et al.} (Proteins, 17 (1993)). It allows to identify essential degrees of freedom of a molecular system and is extended in order to determine single configurations as representatives for the crucial features related to these essential degrees of freedom. Aspects b) and c) are based on statistical cluster methods. They lead to a decomposition of the available simulation data into {\em conformational ensembles} or {\em subsets} with the property that all configurations in one of these subsets share a common chemical property. In contrast to the restriction to single representative conformations, conformational ensembles include information about, e.g., structural flexibility or dynamical connectivity. The conceptual similarities and differences of the three approaches are discussed in detail and are illustrated by application to simulation data originating from a hybrid Monte Carlo sampling of a triribonucleotide.}, language = {en} } @misc{NettesheimHuisingaSchuette, author = {Nettesheim, Peter and Huisinga, Wilhelm and Sch{\"u}tte, Christof}, title = {Chebyshev-Approximation for Wavepacket-Dynamics: better than expected}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2575}, number = {SC-96-47}, abstract = {The aim of this work is to study the accuracy and stability of the Chebyshev--approximation method as a time--discretization for wavepacket dynamics. For this frequently used discretization we introduce estimates of the approximation and round--off error. These estimates mathematically confirm the stability of the Chebyshev--approximation with respect to round--off errors, especially for very large stepsizes. But the results also disclose threads to the stability due to large spatial dimensions. All theoretical statements are illustrated by numerical simulations of an analytically solvable example, the harmonic quantum oszillator.}, language = {en} } @misc{DeuflhardHuisingaFischeretal., author = {Deuflhard, Peter and Huisinga, Wilhelm and Fischer, Alexander and Sch{\"u}tte, Christof}, title = {Identification of Almost Invariant Aggregates in Reversible Nearly Uncoupled Markov Chains}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3469}, number = {SC-98-03}, abstract = {The topic of the present paper bas been motivated by a recent computational approach to identify chemical conformations and conformational changes within molecular systems. After proper discretization, the conformations show up as almost invariant aggregates in reversible nearly uncoupled Markov chains. 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 associated transition probabilites. A rather simple and robust algorithm is suggested and illustrated by its application to the n-pentane molecule.}, language = {en} } @misc{SchuetteFischerHuisingaetal., author = {Sch{\"u}tte, Christof and Fischer, Alexander and Huisinga, Wilhelm and Deuflhard, Peter}, title = {A Hybrid Monte Carlo Method for Essential Molecular Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3474}, number = {SC-98-04}, abstract = {Recently, a novel concept for the computation of essential features of Hamiltonian systems (such as those arising in molecular dynamics) has been proposed. The realization of that concept was based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept based on statistical mechanics, which allows to attack realistic molecular systems. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator \$T\$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of \$T\$ via hybrid Monte Carlo techniques (based on short term subtrajectories only) is shown to lead to a stochastic matrix \$P\$. With these theoretical preparations, an identification algorithm for conformations is applicable (to be presented elsewhere). Numerical results for the n-pentane molecule are given and interpreted.}, language = {en} } @incollection{SchuetteHuisingaDeuflhard2001, author = {Sch{\"u}tte, Christof and Huisinga, Wilhelm and Deuflhard, Peter}, title = {Transfer operator approach to conformational dynamics in biomolecular systems}, series = {Ergodic theory, analysis, and efficient simulation of dynamical systems}, booktitle = {Ergodic theory, analysis, and efficient simulation of dynamical systems}, editor = {Fiedler, Bernold}, publisher = {Berlin: Springer}, pages = {191 -- 223}, year = {2001}, language = {en} } @article{SchuetteFischerHuisingaetal.1999, author = {Sch{\"u}tte, Christof and Fischer, Alexander and Huisinga, Wilhelm and Deuflhard, Peter}, title = {A Direct Approach to Conformational Dynamics Based on Hybrid Monte Carlo}, series = {J. Comput. Phys.}, volume = {151}, journal = {J. Comput. Phys.}, pages = {146 -- 168}, year = {1999}, language = {en} } @article{DeuflhardHuisingaFischeretal.2000, author = {Deuflhard, Peter and Huisinga, Wilhelm and Fischer, Alexander and Sch{\"u}tte, Christof}, title = {Identification of Almost Invariant Aggregates in Reversible Nearly Uncoupled Markov Chains}, series = {Lin. Alg. Appl.}, volume = {315}, journal = {Lin. Alg. Appl.}, pages = {39 -- 59}, year = {2000}, language = {en} }