On metastable conformational analysis of non-equilibrium biomolecular time series

We present a recently developed clustering method and specify it for the problem of identification of metastable conformations in nonequilibrium biomolecular time series. The approach is based on variational minimization of some novel regularized clustering functional. In context of conformational analysis, it allows one to combine the features of standard geometrical clustering techniques (like the Kmeans algorithm), dimension reduction methods (like principle component analysis), and dynamical machine learning approaches like hidden Markov models (HMMs). In contrast to the HMM-based approaches, no a priori assumptions about Markovianity of the underlying process and regarding probability distribution of the observed data are needed. The application of the computational framework is exemplified by means of conformational analysis of some penta-peptide torsion angle time series from a molecular dynamics simulation. Comparison of different versions of the presented algorithm is performed w.r.t. the metastability and geometrical resolution of the resulting conformations

Simultaneous computation of dynamical and equilibrium information using a weighted ensemble of trajectories

Equilibrium formally can be represented as an ensemble of uncoupled systems undergoing unbiased dynamics in which detailed balance is maintained. Many non-equilibrium processes can be described by suitable subsets of the equilibrium ensemble. Here, we employ the "weighted ensemble" (WE) simulation protocol [Huber and Kim, Biophys. J., 1996] to generate equilibrium trajectory ensembles and extract non-equilibrium subsets for computing kinetic quantities. States do not need to be chosen in advance. The procedure formally allows estimation of kinetic rates between arbitrary states chosen after the simulation, along with their equilibrium populations. We also describe a related history-dependent matrix procedure for estimating equilibrium and non-equilibrium observables when phase space has been divided into arbitrary non-Markovian regions, whether in WE or ordinary simulation. In this proof-of-principle study, these methods are successfully applied and validated on two molecular systems: explicitly solvated methane association and the implicitly solvated Ala4 peptide. We comment on challenges remaining in WE calculations

Introduction to Markov state modeling with the PyEMMA software — v1.0

This tutorial provides an introduction to the construction of Markov models of molec- ular kinetics from molecular dynamics trajectory data with the PyEMMA software. Using tutorial notebooks, we will guide the user through the basic functionality as well as the more common advanced mechanisms. Short exercises to self check the learning progress and a notebook on troubleshooting complete this basic introduction

Forward Flux Sampling for rare event simulations

Rare events are ubiquitous in many different fields, yet they are notoriously difficult to simulate because few, if any, events are observed in a conventiona l simulation run. Over the past several decades, specialised simulation methods have been developed to overcome this problem. We review one recently-developed class of such methods, known as Forward Flux Sampling. Forward Flux Sampling uses a series of interfaces between the initial and final states to calculate rate constants and generate transition paths, for rare events in equilibrium or nonequilibrium systems with stochastic dynamics. This review draws together a number of recent advances, summarizes several applications of the method and highlights challenges that remain to be overcome.Comment: minor typos in the manuscript. J.Phys.:Condensed Matter (accepted for publication

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

Modellierung der freien Energiefläche von Biomolekülen durch Torsionswinkel-Principal-Component-Analysis von Molekulardynamiksimulationen

This work presents a contribution to the literature on methods in search of lowdimensional models that yield insight into the equilibrium and kinetic behavior of peptides and small proteins. A deep understanding of various methods for projecting the sampled configurations of molecular dynamics simulations to obtain a low-dimensional free energy landscape is acquired. Furthermore low-dimensional dynamic models for the conformational dynamics of biomolecules in reduced dimensionality are presented. As exemplary systems, mainly short alanine chains are studied. Due to their size they allow for performing long simulations. They are simple, yet nontrivial systems, as due to their flexibility they are rapidly interconverting conformers. Understanding these polypeptide chains in great detail is of considerable interest for getting insight in the process of protein folding. For example, K. Dill et al. conclude in their review [28] about the protein folding problem that "the once intractable Levinthal puzzle now seems to have a very simple answer: a protein can fold quickly and solve its large global optimization puzzle simply through piecewise solutions of smaller component puzzles".Das Ziel der vorliegenden Arbeit ist es, einen Beitrag zur Entwicklung von Methoden zur Modellierung von freien Energieflächen von Biomolekülen zu leisten. Ausgehend von Molekulardynamik-Simulationen geht es insbesondere darum, niedrig-dimensionale Modelle für die Beschreibung von Konformationen und der Kinetik von Peptiden und kleinen Proteinen zu erhalten

Log-periodic oscillations as real-time signatures of hierarchical dynamics in proteins

The time-dependent relaxation of a dynamical system may exhibit a power-law behavior that is superimposed by log-periodic oscillations. Sornette [Phys. Rep. 297, 239 (1998)] showed that this behavior can be explained by a discrete scale invariance of the system, which is associated with discrete and equidistant timescales on a logarithmic scale. Examples include such diverse fields as financial crashes, random diffusion, and quantum topological materials. Recent time-resolved experiments and molecular dynamics simulations suggest that discrete scale invariance may also apply to hierarchical dynamics in proteins, where several fast local conformational changes are a prerequisite for a slow global transition to occur. Employing entropy-based timescale analysis and Markov state modeling to a simple one-dimensional hierarchical model and biomolecular simulation data, it is found that hierarchical systems quite generally give rise to logarithmically spaced discrete timescales. By introducing a one-dimensional reaction coordinate that collectively accounts for the hierarchically coupled degrees of freedom, the free energy landscape exhibits a characteristic staircase shape with two metastable end states, which causes the log-periodic time evolution of the system. The period of the log-oscillations reflects the effective roughness of the energy landscape, and can in simple cases be interpreted in terms of the barriers of the staircase landscape

Heterogeneous and rate-dependent streptavidin-biotin unbinding revealed by high-speed force spectroscopy and atomistic simulations

Receptor-ligand interactions are essential for biological function and their binding strength is commonly explained in terms of static lock-and-key models based on molecular complementarity. However, detailed information of the full unbinding pathway is often lacking due, in part, to the static nature of atomic structures and ensemble averaging inherent to bulk biophysics approaches. Here we combine molecular dynamics and high-speed force spectroscopy on the streptavidin-biotin complex to determine the binding strength and unbinding pathways over the widest dynamic range. Experiment and simulation show excellent agreement at overlapping velocities and provided evidence of the unbinding mechanisms. During unbinding, biotin crosses multiple energy barriers and visits various intermediate states far from the binding pocket while streptavidin undergoes transient induced fits, all varying with loading rate. This multistate process slows down the transition to the unbound state and favors rebinding, thus explaining the long lifetime of the complex. We provide an atomistic, dynamic picture of the unbinding process, replacing a simple two-state picture with one that involves many routes to the lock and rate-dependent induced-fit motions for intermediates, which might be relevant for other receptor-ligand bonds.Comment: 21 pages, 4 figure