Philipp Metzner

Transition Path Theory for Markov Jump Processes

The framework of transition path theory (TPT) is developed in the context of continuous-time Markov chains on discrete state-spaces. Under assumption of ergodicity, TPT singles out any two subsets in the state-space and analyzes the statistical properties of the associated reactive trajectories, i.e., those trajectories by which the random walker transits from one subset to another. TPT gives properties such as the probability distribution of the reactive trajectories, their probability current and flux, and their rate of occurrence and the dominant reaction pathways. In this paper the framework of TPT for Markov chains is developed in detail, and the relation of the theory to electric resistor network theory and data analysis tools such as Laplacian eigenmaps and diffusion maps is discussed as well. Various algorithms for the numerical calculation of the various objects in TPT are also introduced. Finally, the theory and the algorithms are illustrated in several examples.

Illustration of transition path theory on a collection of simple examples

Transition path theory (TPT) has been recently introduced as a theoretical framework to describe the reaction pathways of rare events between long lived states in complex systems. TPT gives detailed statistical information about the reactive trajectories involved in these rare events, which are beyond the realm of transition state theory or transition path sampling. In this paper the TPT approach is outlined, its distinction from other approaches is discussed, and, most importantly, the main insights and objects provided by TPT are illustrated in detail via a series of low dimensional test problems.

Mechanisms of Protein-Ligand Association and Its Modulation by Protein Mutations

Protein-ligand interactions are essential for nearly all biological processes, and yet the biophysical mechanism that enables potential binding partners to associate before specific binding occurs remains poorly understood. Fundamental questions include which factors influence the formation of protein-ligand encounter complexes, and whether designated association pathways exist. To address these questions, we developed a computational approach to systematically analyze the complete ensemble of association pathways. Here, we use this approach to study the binding of a phosphate ion to the Escherichia coli phosphate-binding protein. Various mutants of the protein are considered, and their effects on binding free-energy profiles, association rates, and association pathway distributions are quantified. The results reveal the existence of two anion attractors, i.e., regions that initially attract negatively charged particles and allow them to be efficiently screened for phosphate, which is subsequently specifically bound. Point mutations that affect the charge on these attractors modulate their attraction strength and speed up association to a factor of 10 of the diffusion limit, and thus change the association pathways of the phosphate ligand. It is demonstrated that a phosphate that prebinds to such an attractor neutralizes its attraction effect to the environment, making the simultaneous association of a second phosphate ion unlikely. This study suggests ways in which structural properties can be used to tune molecular association kinetics so as to optimize the efficiency of binding, and highlights the importance of kinetic properties.

Generator estimation of Markov jump processes

Estimating the generator of a continuous-time Markov jump process based on incomplete data is a problem which arises in various applications ranging from machine learning to molecular dynamics. Several methods have been devised for this purpose: a quadratic programming approach (cf. [D.T. Crommelin, E. Vanden-Eijnden, Fitting timeseries by continuous-time Markov chains: a quadratic programming approach, J. Comp. Phys. 217 (2006) 782-805]), a resolvent method (cf. [T. Muller, Modellierung von Proteinevolution, PhD thesis, Heidelberg, 2001]), and various implementations of an expectation-maximization algorithm ([S. Asmussen, O. Nerman, M. Olsson, Fitting phase-type distributions via the EM algorithm, Scand. J. Stat. 23 (1996) 419-441; I. Holmes, G.M. Rubin, An expectation maximization algorithm for training hidden substitution models, J. Mol. Biol. 317 (2002) 753-764; U. Nodelman, C.R. Shelton, D. Koller, Expectation maximization and complex duration distributions for continuous time Bayesian networks, in: Proceedings of the twenty-first conference on uncertainty in AI (UAI), 2005, pp. 421-430; M. Bladt, M. Sorensen, Statistical inference for discretely observed Markov jump processes, J.R. Statist. Soc. B 67 (2005) 395-410]). Some of these methods, however, seem to be known only in a particular research community, and have later been reinvented in a different context. The purpose of this paper is to compile a catalogue of existing approaches, to compare the strengths and weaknesses, and to test their performance in a series of numerical examples. These examples include carefully chosen model problems and an application to a time series from molecular dynamics.

Graph Algorithms for Dynamical Systems

This article is concerned with the numerical analysis of dynamical systems using methods that are based on a discretized description of the system as a graph. The graph-based description provides a unifying framework to approach a wide and diverse variety of dynamical systems, from time-discrete maps via ordinary differential equations to stochastic differential equations describing e. g. diffusion in a potential landscape. Within this variety, this article focusses on those dynamical systems that can possess a ‘multiscale structure’ in the sense that they exhibit interesting dynamical behavior on more than one timescale. We will explain what we mean with this phrase by means of some examples. Consider in Fig. 1 one trajectory of Chua’s circuit, that is described by the well-known threedimensional ordinary differential equation

HIV-1 polymerase inhibition by nucleoside analogs: cellular- and kinetic parameters of efficacy, susceptibility and resistance selection.

Nucleoside analogs (NAs) are used to treat numerous viral infections and cancer. They compete with endogenous nucleotides (dNTP/NTP) for incorporation into nascent DNA/RNA and inhibit replication by preventing subsequent primer extension. To date, an integrated mathematical model that could allow the analysis of their mechanism of action, of the various resistance mechanisms, and their effect on viral fitness is still lacking. We present the first mechanistic mathematical model of polymerase inhibition by NAs that takes into account the reversibility of polymerase inhibition. Analytical solutions for the model point out the cellular- and kinetic aspects of inhibition. Our model correctly predicts for HIV-1 that resistance against nucleoside analog reverse transcriptase inhibitors (NRTIs) can be conferred by decreasing their incorporation rate, increasing their excision rate, or decreasing their affinity for the polymerase enzyme. For all analyzed NRTIs and their combinations, model-predicted macroscopic parameters (efficacy, fitness and toxicity) were consistent with observations. NRTI efficacy was found to greatly vary between distinct target cells. Surprisingly, target cells with low dNTP/NTP levels may not confer hyper-susceptibility to inhibition, whereas cells with high dNTP/NTP contents are likely to confer natural resistance. Our model also allows quantification of the selective advantage of mutations by integrating their effects on viral fitness and drug susceptibility. For zidovudine triphosphate (AZT-TP), we predict that this selective advantage, as well as the minimal concentration required to select thymidine-associated mutations (TAMs) are highly cell-dependent. The developed model allows studying various resistance mechanisms, inherent fitness effects, selection forces and epistasis based on microscopic kinetic data. It can readily be embedded in extended models of the complete HIV-1 reverse transcription process, or analogous processes in other viruses and help to guide drug development and improve our understanding of the mechanisms of resistance development during treatment.

Macroscopic dynamics of complex metastable systems: Theory, algorithms, and application to B-DNA

This article is a survey of the present state of the transfer operator approach to the effective dynamics of metastable complex systems, and the variety of algorithms associated with it. We introduce new methods, and we emphasize both the conceptional foundations and the concrete application to the conformation dynamics of a biomolecular system. The algorithmic aspects are illustrated by means of several examples of various degrees of complexity, culminating in their application to a full-scale molecular dynamics simulation of a B-DNA oligomer.

A Stochastic Closure Approach for LES with Application to Turbulent Channel Flow

The integral conservation laws for mass, momentum and energy of a flow field are universally valid for arbitrary control volumes. Thus, if the associated fluxes across its bounding surfaces are determined exactly, the equations capture the underlying physics of conservation correctly and guarantee an accurate prediction of the time evolution of the integral mean values.

Towards a Stochastic Closure Approach for Large Eddy Simulation

We present a stochastic sub grid scale modeling strategy currently under development for application in Finite Volume Large Eddy Simulation (LES) codes. Our concept is based on the integral conservation laws for mass, momentum and energy of a flow field that are universally valid for arbitrary control volumes. We model the space-time structure of the fluxes to create a discrete formulation. Advanced methods of time series analysis for the data-based construction of stochastic models with inherently non-stationary statistical properties and concepts of information theory for the model discrimination are used to construct stochastic surrogate models for the non-resolved fluctuations. Vector-valued auto-regressive models with external influences (VARX-models) form the basis for the modeling approach. The reconstruction capabilities of the modeling ansatz are tested against fully three dimensional turbulent channel flow data computed by direct numerical simulation (DNS). We present here the outcome of our reconstruction tests.