TY - GEN A1 - Fischer, Alexander T1 - An Uncoupling-Coupling Technique for Markov Chain Monte Carlo Methods N2 - Uncoupling-coupling Monte Carlo (UCMC) combines uncoupling techniques for finite Markov chains with Markov chain Monte Carlo methodology. By determining almost invariant sets of the associated Markov operator, the Monte Carlo sampling splits by a hierarchical annealing process into the essential regions of the state space; therefore UCMC aims at avoiding the typical metastable 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. The correct weighting factors for the various Markov chains are obtained via a coupling matrix, that connects the samplings from the different almost invariant sets. The underlying mathematical structure of this approach is given by a general examination of the uncoupling-coupling procedure. Furthermore, the overall algorithmic scheme of UCMC is applied to the $n$-pentane molecule, a well-known example from molecular dynamics. T3 - ZIB-Report - 00-04 KW - almost invariant sets KW - bridge sampling KW - cluster analysis KW - hierarchical annealing KW - Markov chains KW - Monte Carlo KW - $n$-pentane molecule KW - ratio of no Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5720 ER - TY - GEN A1 - Deuflhard, Peter A1 - Schütte, Christof T1 - Molecular Conformation Dynamics and Computational Drug Design N2 - The paper surveys recent progress in the mathematical modelling and simulation of essential molecular dynamics. Particular emphasis is put on computational drug design wherein time scales of $msec$ up to $min$ play the dominant role. Classical long-term molecular dynamics computations, however, would run into ill-conditioned initial value problems already after time spans of only $psec=10^{-12} sec$. Therefore, in order to obtain results for times of pharmaceutical interest, a combined deterministic-stochastic model is needed. The concept advocated in this paper is the direct identification of metastable conformations together with their life times and their transition patterns. It can be interpreted as a {\em transfer operator} approach corresponding to some underlying hybrid Monte Carlo process, wherein short-term trajectories enter. Once this operator has been discretized, which is a hard problem of its own, a stochastic matrix arises. This matrix is then treated by {\em Perron cluster analysis}, a recently developed cluster analysis method involving the numerical solution of an eigenproblem for a Perron cluster of eigenvalues. In order to avoid the 'curse of dimension', the construction of appropriate boxes for the spatial discretization of the Markov operator requires careful consideration. As a biomolecular example we present a rather recent SARS protease inhibitor. T3 - ZIB-Report - 03-20 KW - conformation dynamics KW - Monte Carlo methods KW - transfer operations KW - Hamiltonian dynamics KW - Smoluchowski dynamics KW - metastable sets KW - Perron cluster an Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7427 ER - TY - GEN A1 - Schütte, Christof A1 - Fischer, Alexander A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter T1 - A Direct Approach to Conformational Dynamics based on Hybrid Monte Carlo N2 - 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. T3 - ZIB-Report - SC-98-45 KW - conformation KW - conformational dynamics KW - hybrid Monte Carlo KW - reweighting KW - essential degrees of freedom KW - transition probabilities KW - Markov operator Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3889 ER - TY - GEN A1 - Walter, Lionel A1 - Weber, Marcus T1 - ConfJump : a fast biomolecular sampling method which drills tunnels through high mountains N2 - 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. T3 - ZIB-Report - 06-26 KW - Monte Carlo simulation KW - rare events KW - rough potential energy function KW - molecular dynamics Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9204 ER - TY - GEN A1 - Deuflhard, Peter T1 - Differential Equations in Technology and Medicine. Computational Concepts, Adaptive Algorithms, and Virtual Labs N2 - This series of lectures has been given to a class of mathematics postdocs at a European summer school on Computational Mathematics Driven by Industrial Applications in Martina Franca, Italy (organized by CIME). It deals with a variety of challenging real life problems selected from clinical cancer therapy, communication technology, polymer production, and pharmaceutical drug design. All of these problems from rather diverse application areas share two common features: (a) they have been modelled by various differential equations -- elliptic, parabolic, or Schrödinger--type partial differential equations, countable ordinary diffential equations, or Hamiltonian systems, (b) their numerical solution has turned out to be real challenge to computational mathematics. T3 - ZIB-Report - SC-99-34 KW - differential equations: KW - ordinary KW - partial KW - countable KW - hamiltonian KW - finite element methods: KW - adaptive KW - multilevel KW - grid generation KW - medical t Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4223 ER - TY - THES A1 - Schütte, Christof T1 - Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules N2 - The function of many important biomolecules comes from their dynamic properties and their ability to switch between different {\em conformations}. In a conformation, the large scale geometric structure of the molecule is understood to be conserved, whereas on smaller scales the system may well rotate, oscillate or fluctuate. In a recent article [J. Comp. Phys., 151,1 (1999)], the present author and coworkers demonstrated that (a) conformations can be understood as almost invariant sets of some Markov chain being defined via the Hamiltonian system governing the molecular dynamics and that (b) these sets can efficiently be computed via eigenvectors of the corresponding Markov operator. The persent manuscript reviews the mathematical modelling steps behind the novel concept, includes a rigorous analytical justification of this approach and especially of the numerical details of the algorithm, and illustrates its performance when applied to realistic molecular systems. T3 - ZIB-Report - SC-99-18 KW - biochemical conformation KW - almost invariant set KW - Markov chain KW - Hamiltonian system KW - Markov operator KW - quasi-compact operator KW - Perron root KW - Perron- Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4063 ER - TY - GEN A1 - Fischer, Alexander A1 - Schütte, Christof A1 - Deuflhard, Peter A1 - Cordes, Frank T1 - Hierarchical Uncoupling-Coupling of Metastable Conformations N2 - 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. T3 - ZIB-Report - 01-03 KW - almost invariant sets KW - bridge sampling KW - metastability KW - hierarchical annealing KW - hybrid Monte Carlo KW - $n$-pentane molecule KW - ratio of normalizing co Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6296 ER -