Nuria Plattner

PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models

Markov (state) models (MSMs) and related models of molecular kinetics have recently received a surge of interest as they can systematically reconcile simulation data from either a few long or many short simulations and allow us to analyze the essential metastable structures, thermodynamics, and kinetics of the molecular system under investigation. However, the estimation, validation, and analysis of such models is far from trivial and involves sophisticated and often numerically sensitive methods. In this work we present the open-source Python package PyEMMA ( http://pyemma.org ) that provides accurate and efficient algorithms for kinetic model construction. PyEMMA can read all common molecular dynamics data formats, helps in the selection of input features, provides easy access to dimension reduction algorithms such as principal component analysis (PCA) and time-lagged independent component analysis (TICA) and clustering algorithms such as k-means, and contains estimators for MSMs, hidden Markov models, and several other models. Systematic model validation and error calculation methods are provided. PyEMMA offers a wealth of analysis functions such that the user can conveniently compute molecular observables of interest. We have derived a systematic and accurate way to coarse-grain MSMs to few states and to illustrate the structures of the metastable states of the system. Plotting functions to produce a manuscript-ready presentation of the results are available. In this work, we demonstrate the features of the software and show new methodological concepts and results produced by PyEMMA.

Protein conformational plasticity and complex ligand-binding kinetics explored by atomistic simulations and Markov models

Understanding the structural mechanisms of protein–ligand binding and their dependence on protein sequence and conformation is of fundamental importance for biomedical research. Here we investigate the interplay of conformational change and ligand-binding kinetics for the serine protease Trypsin and its competitive inhibitor Benzamidine with an extensive set of 150 μs molecular dynamics simulation data, analysed using a Markov state model. Seven metastable conformations with different binding pocket structures are found that interconvert at timescales of tens of microseconds. These conformations differ in their substrate-binding affinities and binding/dissociation rates. For each metastable state, corresponding solved structures of Trypsin mutants or similar serine proteases are contained in the protein data bank. Thus, our wild-type simulations explore a space of conformations that can be individually stabilized by adding ligands or making suitable changes in protein sequence. These findings provide direct evidence of conformational plasticity in receptors.

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, a 1 ms Anton MD simulation of the bovine pancreatic trypsin inhibitor protein, and an optical tweezer force probe trajectory of an RNA hairpin.

The role of higher CO-multipole moments in understanding the dynamics of photodissociated carbonmonoxide in myoglobin.

The influence of electrostatic multipole moments up to hexadecapole on the dynamics of photodissociated carbon monoxide (CO) in myoglobin is investigated. The CO electrostatic potential is expressed as an expansion into atomic multipole moments of increasing order up to octopole which are obtained from a distributed multipole analysis. Three models with increasingly accurate molecular multipoles (accurate quadrupole, octopole, and hexadecapole moments, respectively) are developed and used in molecular dynamics simulations. All models with a fluctuating quadrupole moment correctly describe the location of the B-state whereas the sign of the octopole moment differentiates between the Fe...CO and Fe...OC orientation. For the infrared spectrum of photodissociated CO, considerable differences between the three electrostatic models are found. The most detailed electrostatic model correctly reproduces the splitting, shift, and width of the CO spectrum in the B-state. From an analysis of the trajectories, the spectroscopic B(1) and B(2) states are assigned to the Fe...CO and Fe...OC substates, respectively.

Crystal structure of the dynamin tetramer

The mechanochemical protein dynamin is the prototype of the dynamin superfamily of large GTPases, which shape and remodel membranes in diverse cellular processes. Dynamin forms predominantly tetramers in the cytosol, which oligomerize at the neck of clathrin-coated vesicles to mediate constriction and subsequent scission of the membrane. Previous studies have described the architecture of dynamin dimers, but the molecular determinants for dynamin assembly and its regulation have remained unclear. Here we present the crystal structure of the human dynamin tetramer in the nucleotide-free state. Combining structural data with mutational studies, oligomerization measurements and Markov state models of molecular dynamics simulations, we suggest a mechanism by which oligomerization of dynamin is linked to the release of intramolecular autoinhibitory interactions. We elucidate how mutations that interfere with tetramer formation and autoinhibition can lead to the congenital muscle disorders Charcot–Marie–Tooth neuropathy and centronuclear myopathy, respectively. Notably, the bent shape of the tetramer explains how dynamin assembles into a right-handed helical oligomer of defined diameter, which has direct implications for its function in membrane constriction.

An infinite swapping approach to the rare-event sampling problem.

We describe a new approach to the rare-event Monte Carlo sampling problem. This technique utilizes a symmetrization strategy to create probability distributions that are more highly connected and, thus, more easily sampled than their original, potentially sparse counterparts. After discussing the formal outline of the approach and devising techniques for its practical implementation, we illustrate the utility of the technique with a series of numerical applications to Lennard-Jones clusters of varying complexity and rare-event character.

Complete protein–protein association kinetics in atomic detail revealed by molecular dynamics simulations and Markov modelling

Protein–protein association is fundamental to many life processes. However, a microscopic model describing the structures and kinetics during association and dissociation is lacking on account of the long lifetimes of associated states, which have prevented efficient sampling by direct molecular dynamics (MD) simulations. Here we demonstrate protein–protein association and dissociation in atomistic resolution for the ribonuclease barnase and its inhibitor barstar by combining adaptive high-throughput MD simulations and hidden Markov modelling. The model reveals experimentally consistent intermediate structures, energetics and kinetics on timescales from microseconds to hours. A variety of flexibly attached intermediates and misbound states funnel down to a transition state and a native basin consisting of the loosely bound near-native state and the tightly bound crystallographic state. These results offer a deeper level of insight into macromolecular recognition and our approach opens the door for understanding and manipulating a wide range of macromolecular association processes. Uncovering the microscopic details of protein–protein association via direct molecular dynamics (MD) simulations has been prevented by the excessive lifetimes of associated states. Now, association and dissociation for the barnase–barstar complex has been studied by adaptive high-throughput MD simulations and Markov modelling, revealing intermediate structures, energetics and kinetics on microseconds-to-hours timescales.

On the Infinite Swapping Limit for Parallel Tempering

Parallel tempering, also known as replica exchange sampling, is an important method for simulating complex systems. In this algorithm simulations are conducted in parallel at a series of temperatures, and the key feature of the algorithm is a swap mechanism that exchanges configurations between the parallel simulations at a given rate. The mechanism is designed to allow the low temperature system of interest to escape from deep local energy minima where it might otherwise be trapped via those swaps with the higher temperature components. In this paper we introduce a performance criterion for such schemes based on large deviation theory and argue that the rate of convergence is a monotone increasing function of the swap rate. This motivates the study of the limit process as the swap rate goes to infinity. We construct a scheme which is equivalent to this limit in a distributional sense but which involves no swapping at all. Instead, the effect of the swapping is captured by a collection of weights that inf...

Application of multipolar charge models and molecular dynamics simulations to study stark shifts in inhomogeneous electric fields.

Atomic multipole moments are used to investigate vibrational frequency shifts of CO and H(2) in uniform and inhomogeneous electric fields using ab initio calculations and Molecular Dynamics (MD) simulations. The importance of using atomic multipole moments that can accurately represent both molecular electrostatics and the vibrational response of the molecule to changes in the local electric field is highlighted. The vibrational response of CO to applied uniform and inhomogeneous electric fields is examined using Density Functional Theory calculations for a range of test fields, and the results are used to assess the performance of different atomic multipole models. In uniform fields, the calculated Stark tuning rates of Deltamu = 0.52 cm(-1)/(MV/cm) (DFT), Deltamu = 0.55 cm(-1)/(MV/cm) (fluctuating three-point charge model), and Deltamu = 0.64 cm(-1)/(MV/cm) (Multipole model up to octupole), compare favorably with the experimentally measured value of 0.67 cm(-1)/(MV/cm). For H(2), which has no permanent dipole moment, CCSD(T) calculations demonstrate the importance of bond-weakening effects in force fields in response to the applied inhomogeneous electric field. Finally, CO in hexagonal ice is considered as a test system to highlight the performance of selected multipolar models in MD simulations. The approach discussed here can be applied to calibrate a range of multipolar charge models for diatomic probes, with applications to interpret Stark spectroscopy measurements in protein active sites.

Higher order multipole moments for molecular dynamics simulations

In conventional force fields, the electrostatic potential is represented by atom-centred point charges. This choice is in principle arbitrary, but technically convenient. Point charges can be understood as the first term of multipole expansions, which converge with an increasing number of terms towards the accurate representation of the molecular potential given by the electron density distribution. The use of multipole expansions can therefore improve the force field accuracy. Technically, the implementation of atomic multipoles is more involved than the use of point charges. Important points to consider are the orientation of the multipole moments during the trajectory, conformational dependence of the atomic moments and stability of the simulations which are discussed here.