TY - GEN A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Jahnke, Tobias A1 - Wulkow, Michael T1 - Adaptive Discrete Galerkin Methods Applied to the Chemical Master Equation N2 - \begin{abstract} In systems biology, the stochastic description of biochemical reaction kinetics is increasingly being employed to model gene regulatory networks and signalling pathways. Mathematically speaking, such models require the numerical solution of the underlying evolution equat ion, also known as the chemical master equation (CME). Up to now, the CME has almost exclusively been treated by Monte-Carlo techniques, the most prominent of which is the simulation algorithm suggest ed by Gillespie in 1976. Since this algorithm requires an update for each single reaction event, realizations can be computationally very costly. As an alternative, we here propose a novel approach, which focuses on the discrete partial differential equation (PDE) structure of the CME and thus allows to adopt ideas from adaptive discrete Galerkin methods (as designed by two of the present authors in 1989), which have proven to be highly efficient in the mathematical modelling of polyreaction kinetics. Among the two different options of discretizing the CME as a discrete PDE, the method of lines approach (first space, then time) and the Rothe method (first time, then space), we select the latter one for clear theoretical and algorithmic reasons. First numeric al experiments at a challenging model problem illustrate the promising features of the proposed method and, at the same time, indicate lines of necessary further research. \end{abstract} T3 - ZIB-Report - 07-04 KW - adaptive discrete Galerkin methods KW - adaptive Rothe method KW - discrete Chebyshev polynomials KW - stochastic reactions kinetics KW - chemical master equation Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9471 ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Best, Christoph A1 - Cordes, Frank A1 - Roitzsch, Rainer A1 - Schütte, Christof T1 - Identification of Molecular Conformations via Statistical Analysis of Simulation Data JF - Comp. Chem. Y1 - 1999 VL - 20 SP - 1760 EP - 1774 ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Telgmann, R. A1 - Wulkow, Michael T1 - The virtual lab approach to pharmacokinetic JF - Drug Discov Today Y1 - 2006 VL - 11 SP - 800 EP - 805 ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Best, Christoph A1 - Roitzsch, Rainer A1 - Schütte, Christof A1 - Cordes, Frank T1 - From Simulation Data to Conformational Ensembles JF - J. Comp. Chem. Y1 - 1999 VL - 20 IS - 16 SP - 1760 EP - 1774 ER - TY - CHAP A1 - Galliat, Tobias A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter ED - H. Bothe, R. T1 - Self-Organizing Maps Combined with Eigenmode Analysis for Automated Cluster Identification T2 - Neural Computation, ICSC Academic Press Y1 - 2000 SP - 227 EP - 232 ER - TY - JOUR A1 - Schütte, Christof A1 - Fischer, Alexander A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter T1 - A Direct Approach to Conformational Dynamics Based on Hybrid Monte Carlo JF - J. Comput. Phys. Y1 - 1999 UR - http://www.zib.de/PaperWeb/abstracts/SC-98-45 VL - 151 SP - 146 EP - 168 ER - TY - CHAP A1 - Schütte, Christof A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter ED - Fiedler, Bernold T1 - Transfer operator approach to conformational dynamics in biomolecular systems T2 - Ergodic theory, analysis, and efficient simulation of dynamical systems Y1 - 2001 SP - 191 EP - 223 PB - Berlin: Springer ER - TY - JOUR A1 - von Kleist, Max A1 - Menz, Stephan A1 - Stocker, Hartmut A1 - Arasteh, Keikawus A1 - Huisinga, Wilhelm A1 - Schütte, Christof T1 - HIV Quasispecies Dynamics during Pro-active Treatment Switching JF - Plos One Y1 - 2011 UR - http://publications.imp.fu-berlin.de/1051/ U6 - https://doi.org/10.1371/journal.pone.0018204 VL - 6 IS - 3 SP - e18204 PB - Public Library of Science ER - TY - JOUR A1 - Menz, Stephan A1 - Latorre, J. A1 - Schütte, Christof A1 - Huisinga, Wilhelm T1 - Hybrid Stochastic--Deterministic Solution of the Chemical Master Equation JF - SIAM Interdisciplinary Journal Multiscale Modeling and Simulation (MMS) Y1 - 2012 UR - http://publications.imp.fu-berlin.de/1039/ U6 - https://doi.org/10.1137/110825716 VL - 10 IS - 4 SP - 1232 EP - 1262 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Jahnke, Tobias A1 - Wulkow, Michael T1 - Adaptive Discrete Galerkin Methods Applied to the Chemical Master Equation JF - SIAM J. Sci. Comput Y1 - 2008 VL - 30 IS - 6 SP - 2990 EP - 3011 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Fischer, Alexander A1 - Schütte, Christof T1 - Identification of Almost Invariant Aggregates in Reversible Nearly Uncoupled Markov Chains JF - Lin. Alg. Appl. Y1 - 2000 UR - http://publications.imp.fu-berlin.de/84/ U6 - https://doi.org/10.1016/S0024-3795(00)00095-1 VL - 315 IS - 1-3 SP - 39 EP - 59 ER - TY - CHAP A1 - Schütte, Christof A1 - Huisinga, Wilhelm ED - Fiedler, B. ED - Grger, K. ED - Sprekels, Jürgen T1 - On Conformational Dynamics induced by Langevin Processes T2 - Equadiff 99 Y1 - 2000 UR - http://publications.imp.fu-berlin.de/131/ U6 - https://doi.org/10.1142/9789812792617_0234 VL - 2 SP - 1247 EP - 1262 PB - World Scientific ER - TY - JOUR A1 - Schütte, Christof A1 - Fischer, Alexander A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter T1 - A Direct Approach to Conformational Dynamics based on Hybrid Monte Carlo JF - J. Comp. Phys Y1 - 1999 UR - http://publications.imp.fu-berlin.de/87/ U6 - https://doi.org/10.1006/jcph.1999.6231 VL - 151 IS - 1 SP - 146 EP - 168 ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Best, Christoph A1 - Roitzsch, Rainer A1 - Schütte, Christof A1 - Cordes, Frank T1 - From Simulation Data to Conformational Ensembles JF - J. Comp. Chem. Y1 - 1999 UR - http://publications.imp.fu-berlin.de/92/ U6 - https://doi.org/10.1002/(SICI)1096-987X(199912)20:16<1760::AID-JCC8>3.0.CO;2-2 VL - 20 IS - 16 SP - 1760 EP - 1774 ER - TY - JOUR A1 - Horenko, Illia A1 - Lorenz, S. A1 - Schütte, Christof A1 - Huisinga, Wilhelm T1 - Adaptive Approach for Non-Linear Sensitivity Analysis of Reaction Kinetics JF - J. Comp. Chem. Y1 - 2005 UR - http://publications.imp.fu-berlin.de/49/ U6 - https://doi.org/10.1002/jcc.20234 VL - 26 IS - 9 SP - 941 EP - 948 ER - TY - JOUR A1 - Schütte, Christof A1 - Walter, J. A1 - Hartmann, Carsten A1 - Huisinga, Wilhelm T1 - An Averaging Principle for Fast Degrees of Freedom Exhibiting Long-Term Correlations JF - Multiscale Model. Simul. Y1 - 2004 UR - http://publications.imp.fu-berlin.de/62/ U6 - https://doi.org/10.1137/030600308 VL - 2 IS - 3 SP - 501 EP - 526 ER - TY - JOUR A1 - Nettesheim, Peter A1 - Huisinga, Wilhelm A1 - Schütte, Christof T1 - Chebyshev-Approximation for Wavepacket-Dynamics JF - preprint Y1 - 1996 UR - http://publications.imp.fu-berlin.de/102/ ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Meyn, S. A1 - Schütte, Christof T1 - Phase Transitions and Metastability in Markovian and Molecular Systems JF - Ann. Appl. Prob. Y1 - 2004 UR - http://publications.imp.fu-berlin.de/67/ U6 - https://doi.org/10.1214/aoap/1075828057 VL - 14 IS - 1 SP - 419 EP - 458 ER - TY - CHAP A1 - Schütte, Christof A1 - Huisinga, Wilhelm ED - Ciarlet, P. ED - Le Bris, Claude T1 - Biomolecular Conformations can be Identified as Metastable Sets of Molecular Dynamics T2 - Special Volume Y1 - 2003 UR - http://publications.imp.fu-berlin.de/126/ VL - X SP - 699 EP - 744 PB - Elsevier ER - TY - JOUR A1 - Huisinga, Wilhelm A1 - Schütte, Christof A1 - Stuart, Andrew T1 - Extracting Macroscopic Stochastic Dynamics JF - Comm. Pure Appl. Math. Y1 - 2003 UR - http://publications.imp.fu-berlin.de/72/ U6 - https://doi.org/10.1002/cpa.10057 VL - 56 IS - 2 SP - 234 EP - 269 ER - TY - CHAP A1 - Schütte, Christof A1 - Huisinga, Wilhelm A1 - Meyn, S. ED - Namachchivaya, N. ED - Lin, Y. T1 - Metastability of Diffusion Processes T2 - Nonlinear Stochastic Dynamics Y1 - 2003 UR - http://publications.imp.fu-berlin.de/127/ VL - 110 SP - 71 EP - 81 PB - Springer ER - TY - CHAP A1 - Schütte, Christof A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter ED - Fiedler, B. T1 - Transfer Operator Approach to Conformational Dynamics in Biomolecular Systems T2 - Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems Y1 - 2001 UR - http://publications.imp.fu-berlin.de/129/ SP - 191 EP - 223 PB - Springer ER - TY - CHAP A1 - Schütte, Christof A1 - Huisinga, Wilhelm ED - Sreenivas, R. ED - Jones, D. T1 - Biomolecular Conformations as Metastable Sets of Markov Chains T2 - Proceedings of the 38th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Illinoins/USA Y1 - 2000 UR - http://publications.imp.fu-berlin.de/130/ SP - 1106 EP - 1115 PB - University of Illinois at Urbana-Champaign ER - TY - GEN A1 - Nettesheim, Peter A1 - Huisinga, Wilhelm A1 - Schütte, Christof T1 - Chebyshev-Approximation for Wavepacket-Dynamics: better than expected N2 - 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. T3 - ZIB-Report - SC-96-47 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2575 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 - Huisinga, Wilhelm A1 - Pesce, Lorenzo A1 - Kosloff, Ronnie A1 - Saalfrank, Peter T1 - Faber and Newton Polynomial Integrators for Open-System Density Matrix Propagation N2 - Two polynomial expansions of the time-evolution superoperator to directly integrate Markovian Liouville-von Neumann (LvN) equations for quantum open systems, namely the Newton interpolation and the Faber approximation, are presented and critically compared. Details on the numerical implementation including error control, and on the performance of either method are given. In a first physical application, a damped harmonic oscillator is considered. Then, the Faber approximation is applied to compute a condensed phase absorption spectrum, for which a semi--analytical expression is derived. Finally, even more general applications are discussed. In all applications considered here it is found that both the Newton and Faber integrators are fast, general, stable, and accurate. T3 - ZIB-Report - SC-98-29 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3725 ER - TY - GEN A1 - Huisinga, Wilhelm A1 - Best, Christoph A1 - Cordes, Frank A1 - Roitzsch, Rainer A1 - Schütte, Christof T1 - From Simulation Data to Conformational Ensembles: Structure and Dynamics based Methods N2 - 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. T3 - ZIB-Report - SC-98-36 KW - conformational ensemble KW - cluster method KW - structural and dynamical similarity KW - representative KW - conformation KW - essential degrees of freedom KW - transi Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3797 ER - TY - GEN A1 - Schütte, Christof A1 - Huisinga, Wilhelm T1 - On Conformational Dynamics induced by Langevin Processes N2 - 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. T3 - ZIB-Report - SC-99-25 KW - Smoluchowski equation KW - Fokker--Planck equation KW - semigroup of Markov operators KW - canonical ensemble KW - small noise KW - first exit time KW - half time perio Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4130 ER - TY - GEN A1 - Schütte, Christof A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter T1 - Transfer Operator Approach to Conformational Dynamics in Biomolecular Systems N2 - 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. T3 - ZIB-Report - SC-99-36 KW - Transfer operator KW - Markov process KW - Markov chain KW - molecular dynamics KW - biomolecular conformations KW - canonical ensemble KW - transition prohability KW - Ha Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4247 ER - TY - GEN A1 - Galliat, Tobias A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter T1 - Self-Organizing Maps Combined with Eigenmode Analysis for Automated Cluster Identification N2 - One of the important tasks in Data Mining is automated cluster analysis. Self-Organizing Maps (SOMs) introduced by {\sc Kohonen} are, in principle, a powerful tool for this task. Up to now, however, its cluster identification part is still open to personal bias. The present paper suggests a new approach towards automated cluster identification based on a combination of SOMs with an eigenmode analysis that has recently been developed by {\sc Deuflhard et al.} in the context of molecular conformational dynamics. Details of the algorithm are worked out. Numerical examples from Data Mining and Molecular Dynamics are included. T3 - ZIB-Report - SC-99-38 KW - Self-Organizing Maps KW - cluster analysis Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4279 ER - TY - GEN A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Fischer, Alexander A1 - Schütte, Christof T1 - Identification of Almost Invariant Aggregates in Reversible Nearly Uncoupled Markov Chains N2 - 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. T3 - ZIB-Report - SC-98-03 KW - essential molecular dynamics KW - nearly reducible KW - nearly completely decomposable KW - nearly uncoupled Markov chain KW - almost invariant aggregates KW - trans Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3469 ER - TY - GEN A1 - Schütte, Christof A1 - Fischer, Alexander A1 - Huisinga, Wilhelm A1 - Deuflhard, Peter T1 - A Hybrid Monte Carlo Method for Essential Molecular Dynamics N2 - 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. T3 - ZIB-Report - SC-98-04 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3474 ER - TY - JOUR A1 - Deuflhard, Peter A1 - Huisinga, Wilhelm A1 - Fischer, Alexander A1 - Schütte, Christof T1 - Identification of Almost Invariant Aggregates in Reversible Nearly Uncoupled Markov Chains JF - Lin. Alg. Appl. Y1 - 2000 VL - 315 SP - 39 EP - 59 ER - TY - GEN A1 - Huisinga, Wilhelm T1 - The Essential Spectral Radius and Asymptotic Properties of Transfer Operators N2 - The statistical behavior of deterministic and stochastic dynamical systems may be described using transfer operators, which generalize the notion of Frobenius Perron and Koopman operators. Since numerical techniques to analyze dynamical systems based on eigenvalues problems for the corresponding transfer operator have emerged, bounds on its essential spectral radius became of interest. This article shows that they are also of great theoretical interest. We give an analytical representation of the essential spectral radius in $L^{1}\!(\mu)$, which then is exploited to analyze the asymptotical properties of transfer operators by combining results from functional analysis, Markov operators and Markov chain theory. In particular, it is shown, that an essential spectral radius less than $1$, constrictiveness and some weak form'' of the so--called Doeblin condition are equivalent. Finally, we apply the results to study three main problem classes: deterministic systems, stochastically perturbed deterministic systems and stochastic systems. T3 - ZIB-Report - 00-26 KW - constrictive KW - asymptotically stable KW - exact KW - asymptotically periodic KW - ergodic KW - aperiodic KW - Frobenius Perron operator KW - Koopman operator KW - Markov o Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5942 ER -