Coupling of Conformational Switches in Calcium Sensor Unraveled with Local Markov Models and Transfer Entropy.

Proteins often have multiple switching domains that are coupled to each other and to the binding of ligands in order to realize signaling functions. Here we investigate the C2A domain of Synaptotagmin-1 (Syt-1), a calcium sensor in the neurotransmitter release machinery and a model system for the large family of C2 membrane binding domains. We combine extensive molecular dynamics (MD) simulations with Markov modeling in order to model conformational switching domains, their states, and their dependence on bound calcium ions. Then, we use transfer entropy to characterize how the switching domains are coupled via directed or allosteric mechanisms and give rise to the calcium sensing function of the protein. Our proposed switching mechanism contributes to the understanding of the neurotransmitter release machinery. Furthermore, the methodological approach we develop serves as a template to analyze conformational switching domains and the broad study of their coupling in macromolecular machines.

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.

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.

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.

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.

The effect of classical and quantum dynamics on vibrational frequency shifts of H2 in clathrate hydrates.

Vibrational frequency shifts of H2 in clathrate hydrates are important to understand the properties and elucidate details of the clathrate structure. Experimental spectra of H2 in clathrate hydrates have been measured for different clathrate compositions, temperatures, and pressures. In order to establish reliable relationships between the clathrate structure, dynamics, and observed frequencies, calculations of vibrational frequency shifts in different clathrate environments are required. In this study, a combination of classical molecular dynamics simulations, electronic structure calculations, and quantum dynamical simulation is used to calculate relative vibrational frequencies of H2 in clathrate hydrates. This approach allows us to assess dynamical effects and simulate the change of vibrational frequencies with temperature and pressure. The frequency distributions of the H2 vibrations in the different clathrate cage types agree favorably with experiment. Also, the simulations demonstrate that H2 in the 5(12) cage is more sensitive to the details of the environment and to quantum dynamical effects, in particular when the cage is doubly occupied. We show that for the 5(12) cage quantum effects lead to frequency increases and double occupation is unlikely. This is different for the 5(12)6(4) cages for which higher occupation numbers than one H2 per cage are likely.

Spatial Averaging: Sampling Enhancement for Exploring Configurational Space of Atomic Clusters and Biomolecules.

Spatial averaging Monte Carlo (SA-MC) is an efficient algorithm dedicated to the study of rare-event problems. At the heart of this method is the realization that from the equilibrium density a related, modified probability density can be constructed through a suitable transformation. This new density is more highly connected than the original density, which increases the probability for transitions between neighboring states, which in turn speeds up the sampling. The first successful investigations included the diffusion of small molecules in condensed phase environments and characterization of the metastable states of the migration of the CO ligand in myoglobin. In the present work, a general and robust implementation including rotational and torsional moves in the CHARMM molecular modeling software is introduced. Also, a procedure to estimate unbiased properties is proposed in order to compute thermodynamic observables. These procedures are suitable to study a range of topical systems including Lennard-Jones clusters of different sizes and the blocked alanine dipeptide (Ala)2 in implicit and explicit solvent. In all cases, SA-MC is found to outperform standard Metropolis simulations in sampling configurational space at little extra computational expense. The results for (Ala)2 in explicit solvent are in good agreement with previous umbrella sampling simulations.

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.

Overcoming the Rare Event Sampling Problem in Biological Systems with Infinite Swapping.

Infinite swapping (INS) is a recently developed method to address the rare event sampling problem. For INS, an expanded computational ensemble composed of a number of replicas at different temperatures is used, similar to the widely used parallel tempering (PT) method. While the basic concept of PT is to sample various replicas of the system at different temperatures and exchange information between the replicas occasionally, INS uses the symmetrized distribution of configurations in temperature space, which corresponds to the infinite swapping limit of PT. The effect of this symmetrization and the enhanced information exchange between replicas is evaluated for three different biological systems representing different sampling problems in biology: (1) blocked alanine dipeptide, which is a small system and therefore optimal to evaluate sampling efficiency quantitatively, (2) Villin headpiece, which is used as a test case for the protein folding process, and (3) neuroglobin, which is used to evaluate the effects of enhanced information exchange between replicas for sampling the substate space of a folded protein. For these three test systems, PINS is compared to PT, and it is found that in all cases the sampling with PINS is substantially more efficient.

Rare-event sampling: occupation-based performance measures for parallel tempering and infinite swapping Monte Carlo methods.

In the present paper we identify a rigorous property of a number of tempering-based Monte Carlo sampling methods, including parallel tempering as well as partial and infinite swapping. Based on this property we develop a variety of performance measures for such rare-event sampling methods that are broadly applicable, informative, and straightforward to implement. We illustrate the use of these performance measures with a series of applications involving the equilibrium properties of simple Lennard-Jones clusters, applications for which the performance levels of partial and infinite swapping approaches are found to be higher than those of conventional parallel tempering.

Structure, spectroscopy and dynamics of layered H2O and CO2 ices.

Molecular dynamics simulations of structural, spectroscopic and dynamical properties of mixed water-carbon dioxide (H(2)O-CO(2)) ices are discussed over temperature ranges relevant to atmospheric and astrophysical conditions. The simulations employ multipolar force fields to represent electrostatic interactions which are essential for spectroscopic and dynamical investigations. It is found that at the water/CO(2) interface the water surface acts as a template for the CO(2) component. The rotational reorientation times in both bulk phases agree well with experimental observations. A pronounced temperature effect on the CO(2) reorientation time is observed between 100 K and 200 K. At the interface, water reorientation times are nearly twice as long compared to water in the bulk. The spectroscopy of such ices is rich in the far-infrared region of the spectrum and can be related to translational and rotational modes. Furthermore, spectroscopic signatures mediated across the water/CO(2) interface are found in this frequency range (around 440 cm(-1)). These results will be particularly important for new airborne experiments such as planned for SOFIA.

Quantifying the importance of protein conformation on ligand migration in myoglobin.

Myoglobin (Mb) is a model system for ligand binding and migration. The energy barriers (ΔG) for ligand migration in Mb have been studied in the past by experiment and theory and significant differences between different approaches were found. From experiment, it is known that Mb can assume a large number of conformational substates. In this work, these substates are investigated as a possible source of the differences in migration barriers. We show that the initial structure significantly affects the calculated ΔG for a particular transition and that fluctuations in barrier heights δΔG are of similar magnitude as the free energy barriers themselves. The sensitivity of ΔG to the initial structure is compared to other sources of errors. Different protein structures can affect the calculated ΔG by up to 4 kcal/mol, whereas differences between simple point charge models and more elaborate multipolar charge models for the CO-ligand are smaller by a factor of two. Analysis of the structural changes underlying the large effect of the conformational substate reveals the importance of coupling between protein and ligand motion for migration.

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.

Spatial averaging for small molecule diffusion in condensed phase environments.

Spatial averaging is a new approach for sampling rare-event problems. The approach modifies the importance function which improves the sampling efficiency while keeping a defined relation to the original statistical distribution. In this work, spatial averaging is applied to multidimensional systems for typical problems arising in physical chemistry. They include (I) a CO molecule diffusing on an amorphous ice surface, (II) a hydrogen molecule probing favorable positions in amorphous ice, and (III) CO migration in myoglobin. The systems encompass a wide range of energy barriers and for all of them spatial averaging is found to outperform conventional Metropolis Monte Carlo. It is also found that optimal simulation parameters are surprisingly similar for the different systems studied, in particular, the radius of the point cloud over which the potential energy function is averaged. For H(2) diffusing in amorphous ice it is found that facile migration is possible which is in agreement with previous suggestions from experiment. The free energy barriers involved are typically lower than 1 kcal/mol. Spatial averaging simulations for CO in myoglobin are able to locate all currently characterized metastable states. Overall, it is found that spatial averaging considerably improves the sampling of configurational space.

Structural and spectroscopic characterization of mixed planetary ices.

Mixed ices play a central role in characterizing the origin, evolution, stability and chemistry of planetary ice surfaces. Examples include the polar areas of Mars, the crust of the Jupiter moon Europa, or atmospheres of planets and their satellites, particularly in the outer solar system. Atomistic simulations using accurate representations of the interaction potentials have recently shown to be suitable to quantitatively describe both, the mid- and the far-infrared spectrum of mixed H2O/CO amorphous ices. In this work, molecular dynamics simulations are used to investigate structural and spectroscopic properties of mixed and crystalline ices containing H2O, CO and CO2. Particular findings include: (a) the sensitivity of the water bending mode to the local environment of the water molecules which, together with structural insights from MD simulations, provides a detailed picture for the relationship between spectroscopy and structure; and (b) the sensitivity of the low-frequency spectrum to the structure of the mixed CO2/H2O ice. Specifically, for mixed H2O/CO2 ices with low water contents isolated water molecules are found which give rise to a band shifted by only 12 cm(-1) from the gas-phase value whereas for increasing water concentration (for a 1 : 1 mixture) the band progressively shifts to higher frequency because water clusters can form. More generally it is found that changes in the ice structure due to the presence of CO2 are larger compared to changes induced by the presence of CO and that this difference is reflected in the shape of the water bending vibration. Thus, the water bending vibration appears to be a suitable diagnostic for structural and chemical aspects of mixed ices.

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.

Atomistic simulations of CO vibrations in ices relevant to astrochemistry.

The experimental absorption band of carbon monoxide (CO) in mixed ices has been extensively studied in the past. The astrophysical interest in this band is related to its characteristic shape, which appears to depend on the surrounding ice structure. Herein, molecular dynamics simulations are carried out to analyze the relationship between the structure of the ice and the infrared (IR) spectrum of embedded CO molecules at different concentrations. Instead of conventional force fields, anharmonic potentials are used for the bonded interactions. The electrostatic interactions are more accurately described by means of fluctuating atomic multipole moments (up to quadrupole). The experimentally observed splitting of the CO absorption band (gas phase: 2143 cm(-1)) into a blue- (2152 cm(-1)) and a red-shifted (2138 cm(-1)) signal is also found in the simulations. Complementary atomistic simulations allow us to relate the spectra with the structural features. The distinction between interstitial and substitutional CO molecules as the origin of this splitting is found to be qualitatively correct. However, at increasing CO concentrations, additional effects-such as mutual interactions between CO molecules-become important, and the simplistic picture needs to be revised.

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.