Spectral rate theory for projected two-state kinetics

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used are poor reaction coordinates or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling (MSM) approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality (RCQ). The RCQ can be bounded from below by the directly computable observation quality (OQ), thus providing a measure allowing the RCQ to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low pH, here focusing on the case of two-state kinetics

Projected and Hidden Markov Models for calculating kinetics and metastable states of complex molecules

Markov state models (MSMs) have been successful in computing metastable states, slow relaxation timescales and associated structural changes, and stationary or kinetic experimental observables of complex molecules from large amounts of molecular dynamics simulation data. However, MSMs approximate the true dynamics by assuming a Markov chain on a clusters discretization of the state space. This approximation is difficult to make for high-dimensional biomolecular systems, and the quality and reproducibility of MSMs has therefore been limited. Here, we discard the assumption that dynamics are Markovian on the discrete clusters. Instead, we only assume that the full phase- space molecular dynamics is Markovian, and a projection of this full dynamics is observed on the discrete states, leading to the concept of Projected Markov Models (PMMs). Robust estimation methods for PMMs are not yet available, but we derive a practically feasible approximation via Hidden Markov Models (HMMs). It is shown how various molecular observables of interest that are often computed from MSMs can be computed from HMMs / PMMs. The new framework is applicable to both, simulation and single-molecule experimental data. We demonstrate its versatility by applications to educative model systems, an 1 ms Anton MD simulation of the BPTI protein, and an optical tweezer force probe trajectory of an RNA hairpin

Projected metastable Markov processes and their estimation with observable operator models

The determination of kinetics of high-dimensional dynamical systems, such as macromolecules, polymers, or spin systems, is a difficult and generally unsolved problem — both in simulation, where the optimal reaction coordinate(s) are generally unknown and are difficult to compute, and in experimental measurements, where only specific coordinates are observable. Markov models, or Markov state models, are widely used but suffer from the fact that the dynamics on a coarsely discretized state spaced are no longer Markovian, even if the dynamics in the full phase space are. The recently proposed projected Markov models (PMMs) are a formulation that provides a description of the kinetics on a low-dimensional projection without making the Markovianity assumption. However, as yet no general way of estimating PMMs from data has been available. Here, we show that the observed dynamics of a PMM can be exactly described by an observable operator model (OOM) and derive a PMM estimator based on the OOM learning

Markov State Models from short non-Equilibrium Simulations - Analysis and Correction of Estimation Bias

Many state-of-the-art methods for the thermodynamic and kinetic characterization of large and complex biomolecular systems by simulation rely on ensemble approaches, where data from large numbers of relatively short trajectories are integrated. In this context, Markov state models (MSMs) are extremely popular because they can be used to compute stationary quantities and long-time kinetics from ensembles of short simulations, provided that these short simulations are in “local equilibrium” within the MSM states. However, over the last 15 years since the inception of MSMs, it has been controversially discussed and not yet been answered how deviations from local equilibrium can be detected, whether these deviations induce a practical bias in MSM estimation, and how to correct for them. In this paper, we address these issues: We systematically analyze the estimation of MSMs from short non-equilibrium simulations, and we provide an expression for the error between unbiased transition probabilities and the expected estimate from many short simulations. We show that the unbiased MSM estimate can be obtained even from relatively short non-equilibrium simulations in the limit of long lag times and good discretization. Further, we exploit observable operator model (OOM) theory to derive an unbiased estimator for the MSM transition matrix that corrects for the effect of starting out of equilibrium, even when short lag times are used. Finally, we show how the OOM framework can be used to estimate the exact eigenvalues or relaxation time scales of the system without estimating an MSM transition matrix, which allows us to practically assess the discretization quality of the MSM. Applications to model systems and molecular dynamics simulation data of alanine dipeptide are included for illustration. The improved MSM estimator is implemented in PyEMMA of version 2.3

Generation and validation

Markov state models of molecular kinetics (MSMs), in which the long-time statistical dynamics of a molecule is approximated by a Markov chain on a discrete partition of configuration space, have seen widespread use in recent years. This approach has many appealing characteristics compared to straightforward molecular dynamics simulation and analysis, including the potential to mitigate the sampling problem by extracting long-time kinetic information from short trajectories and the ability to straightforwardly calculate expectation values and statistical uncertainties of various stationary and dynamical molecular observables. In this paper, we summarize the current state of the art in generation and validation of MSMs and give some important new results. We describe an upper bound for the approximation error made by modelingmolecular dynamics with a MSM and we show that this error can be made arbitrarily small with surprisingly little effort. In contrast to previous practice, it becomes clear that the best MSM is not obtained by the most metastable discretization, but the MSM can be much improved if non-metastable states are introduced near the transition states. Moreover, we show that it is not necessary to resolve all slow processes by the state space partitioning, but individual dynamical processes of interest can be resolved separately. We also present an efficient estimator for reversible transition matrices and a robust test to validate that a MSM reproduces the kinetics of the molecular dynamics data

Mobility in a Globalised World

The term mobility has different meanings in the following academic disciplines. In economics, mobility is the ability of an individual or a group to improve their economic status in relation to income and wealth within their lifetime or between generations. In information systems and computer science, mobility is used for the concept of mobile computing, in which a computer is transported by a person during normal use. Logistics creates, by the design of logistics networks, the infrastructure for the mobility of people and goods. Electric mobility is one of today’s solutions from engineering perspective to reduce the need of energy resources and environmental impact. Moreover, for urban planning, mobility is the crunch question about how to optimize the different needs for mobility and how to link different transportation systems. The conference “Mobility in a Globalised World” took place in Iserlohn, Germany, on September 14th – 15th, 2011. The aim of this conference was to provide an interdisciplinary forum for the exchange of ideas among practitioners, researchers, and government officials regarding the different modes of mobility in a globalised world, focusing on both domestic and international issues. The proceedings at hand document the results of the presentations and ensuing discussions at the conference

Historische Grundwissenschaften und die digitale Herausforderung

Unter Federführung von Eva Schlotheuber (Heinrich-Heine-Universität Düsseldorf / VHD-Unterausschuss "Geschichte in der digitalen Welt") und Frank Bösch (Zentrum für die Zeithistorische Forschungen Potsdam / VHD-Unterausschuss "Audiovisuelle Quellen") verabschiedete der VHD ein Grundsatzpapier zum Status der Historischen Grundwissenschaften mit dem Titel "Quellenkritik im digitalen Zeitalter: Die Historischen Grundwissenschaften als zentrale Kompetenz der Geschichtswissenschaft und benachbarter Fächer". Das Grundsatzpapier, in dem auch ein forschungsstrategisches Interesse an den Grundwissenschaften in der digitalen Transformation zum Ausdruck kommt, wurde auf H-Soz-Kult veröffentlicht und mit einem Diskussionsforum begleitet. Dazu wurde aus dem breiten Spektrum der Historischen Kulturwissenschaften eine Reihe in- und ausländischer Kolleginnen und Kollegen zur Kommentierung und Diskussion eingeladen, um die Debatte zu stimulieren

Verbesserte Schätzungsmethoden für Markov-Modelle dynamischer Systeme

Markov state models (MSM) of molecular kinetics, used to approximate the long- time statistical dynamics of a molecule by a Markov chain on a discrete partition of configuration space, have seen widespread use in recent years. This thesis deals with the improved generation, validation, the application and the extension to experimental observations of these MSM. The four major parts each address different aspects: (1) a summary of the current state of the art in generation and validation of MSMs serving as an introduction along with some important insights into optimal discretization, (2) an investigation of efficient computation and error estimation of the committor, a widely used reaction coordinate, (3) the theory and application on how to generate markov models from multi-ensemble simulations such as parallel tempering using dynamical reweighting, and (4) the extension of MSM theory to non-markovian observations from low-dimensional observed correlations. All parts contain the necessary theory and methods and are applied to artificial and real systems along with an investigation into robustness and error. Thus, this work extends the quality of the computation of key properties and the general construction of markov models for molecular kinetics and allows to alleviate the gap in connecting estimations from simulation and experiment. This is an important step forward toward the long term goal to have the necessary robustness and accuracy for upcoming adaptive MD simulation strategies.In den letzten Jahren haben sich Markov Modelle als effektives und vielseitiges Werkzeug herausgestellt um molekulare Prozesse durch Markov Ketten auf einem diskreten Zustandsraum zu beschreiben und zu analysieren. Die vorliegende Doktorarbeit beschäftigt sich sowohl mit dem Generieren, Validieren, als auch mit den Anwendungen und der Erweiterung auf experimentelle Beobachtungen von Markov Modellen. Dabei werden im Wesentlichen vier Aspekte angesprochen: (1) eine umfassende Zusammenfassung des aktuellen Forschungsstands im Generieren und Validieren von Markov State Modellen, die sowohl als Einleitung dient als auch einige wichtige neue Erkenntnisse zum Problem der Diskretisierung vorstellt, (2) eine effiziente und robuste Berechnung und Fehleranalyse des Kommitors, einer wichtigen Reaktionskoordinate, (3) die Theorie und Anwendung, um mittels Dynamical Reweighting verbesserte Markov Modelle aus einem Satz von Simulationen zu erzeugen, die unter verschiedenen globalen Parametern (z.B. Temperatur) generiert wurden, und (4) die Erweiterung der Markov Modell Theorie auf nicht- markovsche Beobachtungen von experimentellen Trajektorien. Jeder Teil enthält die notwendigen Theorien und Methoden, sowie Anwendungen auf künstliche oder reelle Systeme zusammen mit einer Fehleranalyse. Damit erweitert die vorliegende Arbeit das Feld der Markov Model Theorie um wichtige neue Erkenntnisse mit dem Ziel adaptive molekulardynamische (MD) Simulationen zu ermöglichen und schlägt mittels MSM eine Brücke zwischen MD Simulationen und experimentellen Beobachtungen