Continuous metadynamics in essential coordinates as a tool for free energy modelling of conformational changes

Modelling of conformational changes in biopolymers is one of the greatest challenges of molecular biophysics. Metadynamics is a recently introduced free energy modelling technique that enhances sampling of configurational (e.g. conformational) space within a molecular dynamics simulation. This enhancement is achieved by the addition of a history-dependent bias potential, which drives the system from previously visited regions. Discontinuous metadynamics in the space of essential dynamics eigenvectors (collective motions) has been proposed and tested in conformational change modelling. Here, we present an implementation of two continuous formulations of metadynamics in the essential subspace. The method was performed in a modified version of the molecular dynamics package GROMACS. These implementations were tested on conformational changes in cyclohexane, alanine dipeptide (terminally blocked alanine, Ace-Ala-Nme) and SH3 domain. The results illustrate that metadynamics in the space of essential coordinates can accurately model free energy surfaces associated with conformational changes. Figure The conformational free energy surface of cyclohexane in the space of the two most intensive collective motions.

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Putative Active States of a Prototypic G-Protein-Coupled Receptor from Biased Molecular Dynamics

A major current focus of structural work on G-protein-coupled receptors (GPCRs) pertains to the investigation of their active states. However, for virtually all GPCRs, active agonist-bound intermediate states have been difficult to characterize experimentally owing to their higher conformational flexibility, and thus intrinsic instability, as compared to inactive inverse agonist-bound states. In this work, we explored possible activation pathways of the prototypic GPCR bovine rhodopsin by means of biased molecular dynamics simulations. Specifically, we used an explicit atomistic representation of the receptor and its environment, and sampled the conformational transition from the crystal structure of a photoactivated deprotonated state of rhodopsin to the low pH crystal structure of opsin in the presence of 11-trans-retinal, using adiabatic biased molecular dynamics simulations. We then reconstructed the system free-energy landscape along the predetermined transition trajectories using a path collective variable approach based on metadynamics. Our results suggest that the two experimental endpoints of rhodopsin/opsin are connected by at least two different pathways, and that the conformational transition is populated by at least four metastable states of the receptor, characterized by a different amplitude of the outward movement of transmembrane helix 6.     

Multiple pH regime molecular dynamics simulation for pK calculations

Ionisation equilibria in proteins are influenced by conformational flexibility, which can in principle be accounted for by molecular dynamics simulation. One problem in this method is the bias arising from the fixed protonation state during the simulation. Its effect is mostly exhibited when the ionisation behaviour of the titratable groups is extrapolated to pH regions where the predetermined protonation state of the protein may not be statistically relevant, leading to conformational sampling that is not representative of the true state. In this work we consider a simple approach which can essentially reduce this problem. Three molecular dynamics structure sets are generated, each with a different protonation state of the protein molecule expected to be relevant at three pH regions, and pK calculations from the three sets are combined to predict pK over the entire pH range of interest. This multiple pH molecular dynamics approach was tested on the GCN4 leucine zipper, a protein for which a full data set of experimental data is available. The pK values were predicted with a mean deviation from the experimental data of 0.29 pH units, and with a precision of 0.13 pH units, evaluated on the basis of equivalent sites in the dimeric GCN4 leucine zipper.     

Recent advances in maximum entropy biasing techniques for molecular dynamics

ABSTRACT

This review describes recent advances by the authors and others on the topic of incorporating experimental data into molecular simulations through maximum entropy methods. Methods which incorporate experimental data improve accuracy in molecular simulation by minimally modifying the thermodynamic ensemble. This is especially important where force fields are approximate, such as when employing coarse-grain models, or where high accuracy is required, such as when attempting to mimic a multiscale self-assembly process. The authors review here the experiment directed simulation (EDS) and experiment directed metadynamics (EDM) methods that allow matching averages and distributions in simulations, respectively. Important system-specific considerations are discussed such as using enhanced sampling simultaneously, the role of pressure, treating uncertainty, and implementations of these methods. Recent examples of EDS and EDM are reviewed including applications to ab initio molecular dynamics of water, incorporating environmental fluctuations inside of a macromolecular protein complex, improving RNA force fields, and the combination of enhanced sampling with minimal biasing to model peptides     

Unbinding free energy of acetylcholinesterase bound oxime drugs along the gorge pathway from metadynamics‐umbrella sampling investigation

Because of the pivotal role that the nerve enzyme, acetylcholinesterase plays in terminating nerve impulses at cholinergic synapses. Its active site, located deep inside a 20 Å gorge, is a vulnerable target of the lethal organophosphorus compounds. Potent reactivators of the intoxicated enzyme are nucleophiles, such as bispyridinium oxime that binds to the peripheral anionic site and the active site of the enzyme through suitable cation–π interactions. Atomic scale molecular dynamics and free energy calculations in explicit water are used to study unbinding pathways of two oxime drugs (Ortho‐7 and Obidoxime) from the gorge of the enzyme. The role of enzyme‐drug cation–π interactions are explored with the metadynamics simulation. The metadynamics discovered potential of mean force (PMF) of the unbinding events is refined by the umbrella sampling (US) corrections. The bidimensional free energy landscape of the metadynamics runs are further subjected to finite temperature string analysis to obtain the transition tube connecting the minima and bottlenecks of the unbinding pathway. The PMF is also obtained from US simulations using the biasing potential constructed from the transition tube and are found to be consistent with the metadynamics‐US corrected results. Although experimental structural data clearly shows analogous coordination of the two drugs inside the gorge in the bound state, the PMF of the drug trafficking along the gorge pathway point, within an equilibrium free energy context, to a multistep process that differs from one another. Routes, milestones and subtlety toward the unbinding pathway of the two oximes at finite temperature are identified. Proteins 2014; 82:1799–1818. © 2014 Wiley Periodicals, Inc.     

A Kinetic Model of Trp-Cage Folding from Multiple Biased Molecular Dynamics Simulations

Trp-cage is a designed 20-residue polypeptide that, in spite of its size, shares several features with larger globular proteins. Although the system has been intensively investigated experimentally and theoretically, its folding mechanism is not yet fully understood. Indeed, some experiments suggest a two-state behavior, while others point to the presence of intermediates. In this work we show that the results of a bias-exchange metadynamics simulation can be used for constructing a detailed thermodynamic and kinetic model of the system. The model, although constructed from a biased simulation, has a quality similar to those extracted from the analysis of long unbiased molecular dynamics trajectories. This is demonstrated by a careful benchmark of the approach on a smaller system, the solvated Ace-Ala₃-Nme peptide. For the Trp-cage folding, the model predicts that the relaxation time of 3100 ns observed experimentally is due to the presence of a compact molten globule-like conformation. This state has an occupancy of only 3% at 300 K, but acts as a kinetic trap. Instead, non-compact structures relax to the folded state on the sub-microsecond timescale. The model also predicts the presence of a state at of 4.4 Å from the NMR structure in which the Trp strongly interacts with Pro12. This state can explain the abnormal temperature dependence of the and chemical shifts. The structures of the two most stable misfolded intermediates are in agreement with NMR experiments on the unfolded protein. Our work shows that, using biased molecular dynamics trajectories, it is possible to construct a model describing in detail the Trp-cage folding kinetics and thermodynamics in agreement with experimental data.     

(Un)Folding Mechanisms of the FBP28 WW Domain in Explicit Solvent Revealed by Multiple Rare Event Simulation Methods

We report a numerical study of the (un)folding routes of the truncated FBP28 WW domain at ambient conditions using a combination of four advanced rare event molecular simulation techniques. We explore the free energy landscape of the native state, the unfolded state, and possible intermediates, with replica exchange molecular dynamics. Subsequent application of bias-exchange metadynamics yields three tentative unfolding pathways at room temperature. Using these paths to initiate a transition path sampling simulation reveals the existence of two major folding routes, differing in the formation order of the two main hairpins, and in hydrophobic side-chain interactions. Having established that the hairpin strand separation distances can act as reasonable reaction coordinates, we employ metadynamics to compute the unfolding barriers and find that the barrier with the lowest free energy corresponds with the most likely pathway found by transition path sampling. The unfolding barrier at 300 K is ∼17 k_BT ≈ 42 kJ/mol, in agreement with the experimental unfolding rate constant. This work shows that combining several powerful simulation techniques provides a more complete understanding of the kinetic mechanism of protein folding.     

Conformations of the Huntingtin N-term in aqueous solution from atomistic simulations

The first 17 amino acids of Huntingtin protein (N17) play a crucial role in the protein's aggregation. Here we predict its free energy landscape in aqueous solution by using bias exchange metadynamics. All our findings are consistent with experimental data. N17 populates four main kinetic basins, which interconvert on the microsecond time-scale. The most populated basin (about 75%) is a random coil, with an extended flat exposed hydrophobic surface. This might create a hydrophobic seed promoting Huntingtin aggregation. The other main populated basins contain helical conformations, which could facilitate N17 binding on its cellular targets.     

Structural compliance: A new metric for protein flexibility

Proteins are the active players in performing essential molecular activities throughout biology, and their dynamics has been broadly demonstrated to relate to their mechanisms. The intrinsic fluctuations have often been used to represent their dynamics and then compared to the experimental B-factors. However, proteins do not move in a vacuum and their motions are modulated by solvent that can impose forces on the structure. In this paper, we introduce a new structural concept, which has been called the structural compliance, for the evaluation of the global and local deformability of the protein structure in response to intramolecular and solvent forces. Based on the application of pairwise pulling forces to a protein elastic network, this structural quantity has been computed and sometimes is even found to yield an improved correlation with the experimental B-factors, meaning that it may serve as a better metric for protein flexibility. The inverse of structural compliance, namely the structural stiffness, has also been defined, which shows a clear anticorrelation with the experimental data. Although the present applications are made to proteins, this approach can also be applied to other biomolecular structures such as RNA. This present study considers only elastic network models, but the approach could be applied further to conventional atomic molecular dynamics. Compliance is found to have a slightly better agreement with the experimental B-factors, perhaps reflecting its bias toward the effects of local perturbations, in contrast to mean square fluctuations. The code for calculating protein compliance and stiffness is freely accessible at https://jerniganlab.github.io/Software/PACKMAN/Tutorials/compliance .     

Information bounds and optimal analysis of dynamic single molecule measurements

Time-resolved single molecule fluorescence measurements may be used to probe the conformational dynamics of biological macromolecules. The best time resolution in such techniques will only be achieved by measuring the arrival times of individual photons at the detector. A general approach to the estimation of molecular parameters based on individual photon arrival times is presented. The amount of information present in a data set is quantified by the Fisher information, thereby providing a guide to deriving the basic equations relating measurement uncertainties and time resolution. Based on these information-theoretical considerations, a data analysis algorithm is presented that details the optimal analysis of single-molecule data. This method natively accounts and corrects for background photons and cross talk, and can scale to an arbitrary number of channels. By construction, and with corroboration from computer simulations, we show that this algorithm reaches the theoretical limit, extracting the maximal information out of the data. The bias inherent in the algorithm is considered and its implications for experimental design are discussed. The ideas underlying this approach are general and are expected to be applicable to any information-limited measurement.     

Congruent qualitative behavior of complete and reconstructed phase space trajectories from biomolecular dynamics simulation

Central to the study of a complex dynamical system is knowledge of its phase space behavior. Experimentally, it is rarely possible to record a system's (multidimensional) phase space variables. Rather, the system is observed via one (or few) scalar-valued signal(s) of emission or response. In dynamical systems analysis, the multidimensional phase space of a system can be reconstructed by manipulation of a one-dimensional signal. The trick is in the construction of a (higher-dimensional) space through the use of a time lag (or delay) on the signal time series. The trajectory in this embedding space can then be examined using phase portraits generated in selected subspaces. By contrast, in computer simulation, one has an embarrassment of riches: direct access to the complete multidimensional phase space variables, at arbitrary time resolution and precision. Here, the problem is one of reducing the dimensionality to make analysis tractable. This can be achieved through linear or nonlinear projection of the trajectory into subspaces containing high information content. This study considers trajectories of the small protein crambin from molecular dynamics simulations. The phase space behavior is examined using principal component analysis on the Cartesian coordinate covariance matrix of 138 dimensions. In addition, the phase space is reconstructed from a one dimensional signal, representing the radius of gyration of the structure along the trajectory. Comparison of low-dimensional phase portraits obtained from the two methods shows that the complete phase space distribution is well represented by the reconstruction. The study suggests that it may be possible to develop a deeper connection between the experimental and simulated dynamics of biomolecules via phase space reconstruction using data emerging from recent advances in single-molecule time-resolved biophysical techniques.     

Mean field stochastic boundary molecular dynamics simulation of a phospholipid in a membrane

Computer simulations of phospholipid membranes have been carried out by using a combined approach of molecular and stochastic dynamics and a mean field based on the Marcelja model. First, the single-chain mean field simulations of Pastor et al. [(1988) J. Chem. Phys. 89, 1112-1127] were extended to a complete dipalmitoylphosphatidylcholine molecule; a 102-ns Langevin dynamics simulation is presented and compared with experiment. Subsequently, a hexagonally packed seven-lipid array was simulated with Langevin dynamics and a mean field at the boundary and with molecular dynamics (and no mean field) in the center. This hybrid method, mean field stochastic boundary molecular dynamics, reduces bias introduced by the mean field and eliminates the need for periodic boundary conditions. As a result, simulations extending to tens of nanoseconds may be carried out by using a relatively small number of molecules to model the membrane environment. Preliminary results of a 20-ns simulation are reported here. A wide range of motions, including overall reorientation with a nanosecond decay time, is observed in both simulations, and good agreement with NMR, IR, and neutron diffraction data is found.     

Dynamical modeling of collective behavior from pigeon flight data: flock cohesion and dispersion

Several models of flocking have been promoted based on simulations with qualitatively naturalistic behavior. In this paper we provide the first direct application of computational modeling methods to infer flocking behavior from experimental field data. We show that this approach is able to infer general rules for interaction, or lack of interaction, among members of a flock or, more generally, any community. Using experimental field measurements of homing pigeons in flight we demonstrate the existence of a basic distance dependent attraction/repulsion relationship and show that this rule is sufficient to explain collective behavior observed in nature. Positional data of individuals over time are used as input data to a computational algorithm capable of building complex nonlinear functions that can represent the system behavior. Topological nearest neighbor interactions are considered to characterize the components within this model. The efficacy of this method is demonstrated with simulated noisy data generated from the classical (two dimensional) Vicsek model. When applied to experimental data from homing pigeon flights we show that the more complex three dimensional models are capable of simulating trajectories, as well as exhibiting realistic collective dynamics. The simulations of the reconstructed models are used to extract properties of the collective behavior in pigeons, and how it is affected by changing the initial conditions of the system. Our results demonstrate that this approach may be applied to construct models capable of simulating trajectories and collective dynamics using experimental field measurements of herd movement. From these models, the behavior of the individual agents (animals) may be inferred.     

Increasing the power to detect causal associations by combining genotypic and expression data in segregating populations

To dissect common human diseases such as obesity and diabetes, a systematic approach is needed to study how genes interact with one another, and with genetic and environmental factors, to determine clinical end points or disease phenotypes. Bayesian networks provide a convenient framework for extracting relationships from noisy data and are frequently applied to large-scale data to derive causal relationships among variables of interest. Given the complexity of molecular networks underlying common human disease traits, and the fact that biological networks can change depending on environmental conditions and genetic factors, large datasets, generally involving multiple perturbations (experiments), are required to reconstruct and reliably extract information from these networks. With limited resources, the balance of coverage of multiple perturbations and multiple subjects in a single perturbation needs to be considered in the experimental design. Increasing the number of experiments, or the number of subjects in an experiment, is an expensive and time-consuming way to improve network reconstruction. Integrating multiple types of data from existing subjects might be more efficient. For example, it has recently been demonstrated that combining genotypic and gene expression data in a segregating population leads to improved network reconstruction, which in turn may lead to better predictions of the effects of experimental perturbations on any given gene. Here we simulate data based on networks reconstructed from biological data collected in a segregating mouse population and quantify the improvement in network reconstruction achieved using genotypic and gene expression data, compared with reconstruction using gene expression data alone. We demonstrate that networks reconstructed using the combined genotypic and gene expression data achieve a level of reconstruction accuracy that exceeds networks reconstructed from expression data alone, and that fewer subjects may be required to achieve this superior reconstruction accuracy. We conclude that this integrative genomics approach to reconstructing networks not only leads to more predictive network models, but also may save time and money by decreasing the amount of data that must be generated under any given condition of interest to construct predictive network models.     

Refinement of the NMR structures for acyl carrier protein with scalar coupling data

Structure determination of small proteins using NMR data is most commonly pursued by combining NOE derived distance constraints with inherent constraints based on chemical bonding. Ideally, one would make use of a variety of experimental observations, not just distance constraints. Here, coupling constant constraints have been added to molecular mechanics and molecular dynamics protocols for structure determination in the form of a psuedoenergy function that is minimized in a search for an optimum molecular conformation. Application is made to refinement of a structure for a 77 amino acid protein involved in fatty acid synthesis, Escherichia coli acyl carrier protein (ACP). 54 3JHN alpha coupling constants, 12 coupling constants for stereospecifically assigned side chain protons, and 450 NOE distance constraints were used to calculate the 3-D structure of ACP. A three-step protocol for a molecular dynamics calculation is described, in analogy to the protocol previously used in molecular mechanics calculations. The structures calculated with the molecular mechanics approach and the molecular dynamics approach using a rigid model for the protein show similar molecular energies and similar agreement with experimental distance and coupling constant constraints. The molecular dynamics approach shows some advantage in overcoming local minimum problems, but only when a two-state averaging model for the protein was used, did molecular energies drop significantly.     

An Adiabatic Molecular Dynamics Method for the Calculation of Free Energy Profiles

A new molecular dynamics method for calculating free energy profiles for rare events is presented. The new method is based on the creation of an adiabatic separation between the reaction coordinate subspace and the remaining degrees of freedom within a molecular dynamics run. This is achieved by associating with the reaction coordinate(s) a high temperature and large mass, thereby allowing the activated process to occur while permitting the remaining degrees of freedom to respond adiabatically. In this limit, by applying a formal multiple time scale Liouville operator factorization, it can be rigorously shown that the free energy profiles are obtained directly from the probability distribution of the reaction coordinate subspace and, therefore, require no postprocessing of the output data. The new method is applied to a variety of model problems and its performance tested against free energy calculations using the "bluemoon ensemble" approach. The comparison shows that free energy profiles can be calculated with greater ease and efficiency using the new method.     

Atomic mean-square displacements in proteins by molecular dynamics: a case for analysis of variance

Information on protein internal motions is usually obtained through the analysis of atomic mean-square displacements, which are a measure of variability of the atomic positions distribution functions. We report a statistical approach to analyze molecular dynamics data on these displacements that is based on probability distribution functions. Using a technique inspired by the analysis of variance, we compute unbiased, reliable mean-square displacements of the atoms and analyze them statistically. We applied this procedure to characterize protein thermostability by comparing the results for a thermophilic enzyme and a mesophilic homolog. In agreement with previous experimental observations, our analysis suggests that the proteins surface regions can play a role in the different thermal behavior.     

Ensemble-Biased Metadynamics: A Molecular Simulation Method to Sample Experimental Distributions

We introduce an enhanced-sampling method for molecular dynamics (MD) simulations referred to as ensemble-biased metadynamics (EBMetaD). The method biases a conventional MD simulation to sample a molecular ensemble that is consistent with one or more probability distributions known a priori, e.g., experimental intramolecular distance distributions obtained by double electron-electron resonance or other spectroscopic techniques. To this end, EBMetaD adds an adaptive biasing potential throughout the simulation that discourages sampling of configurations inconsistent with the target probability distributions. The bias introduced is the minimum necessary to fulfill the target distributions, i.e., EBMetaD satisfies the maximum-entropy principle. Unlike other methods, EBMetaD does not require multiple simulation replicas or the introduction of Lagrange multipliers, and is therefore computationally efficient and straightforward in practice. We demonstrate the performance and accuracy of the method for a model system as well as for spin-labeled T4 lysozyme in explicit water, and show how EBMetaD reproduces three double electron-electron resonance distance distributions concurrently within a few tens of nanoseconds of simulation time. EBMetaD is integrated in the open-source PLUMED plug-in (www.plumed-code.org), and can be therefore readily used with multiple MD engines.     

On using time-averaging restraints in molecular dynamics simulation

Introducing experimental values as restraints into molecular dynamics (MD) simulations to bias the values of particular molecular properties, such as nuclear Overhauser effect intensities or distances, 3J coupling constants, chemical shifts or crystallographic structure factors, towards experimental values is a widely used structure refinement method. To account for the averaging of experimentally derived quantities inherent in the experimental techniques, time-averaging restraining methods may be used. In the case of structure refinement using 3J coupling constants from NMR experiments, time-averaging methods previously proposed can suffer from large artificially induced structural fluctuations. A modified time-averaged restraining potential energy function is proposed which overcomes this problem. The different possible approaches are compared using stochastic dynamics simulations of antamanide, a cyclic peptide of ten residues.     

The role of Li+, Na+, and K+ in the ligand binding inside the human acetylcholinesterase gorge

Alkali cations can affect the catalytic efficiency of enzymes. This is particularly true when dealing with enzymes whose substrate bears a formal positive charge. Computational and biochemical approaches have been combined to shed light on the atomic aspects of the role of Li(+), Na(+), and K(+) on human acetylcholinesterase (hAChE) ligand binding. In this respect, molecular dynamics simulations and our recently developed metadynamics method were applied to study the entrance of the three cations in the gorge of hAChE, and their effect on the dynamical motion of a ligand (tetramethylammonium) from the bulk of the solvent into the deep narrow enzyme gorge. Furthermore, in order to support the theoretical results, K(M) and k(cat) for the acetylcholine hydrolysis in the presence of the three cations were evaluated by using an approach based on the Ellman's method. The combination of computational and biochemical experiments clearly showed that Li(+), Na(+), and K(+) may influence the ligand binding at the hAChE gorge.