Weight average approaches for predicting dynamical properties of biomolecules

Recent advances in atomistic molecular dynamics (MD) simulations of biomolecules allow us to explore their conformational spaces widely, observing large-scale conformational fluctuations or transitions between distinct structures. To reproduce or refine experimental data using MD simulations, structure ensembles, which are characterized by multiple structures and their statistical weights on the rugged free-energy landscapes, are often used. Here, we summarize weight average approaches for various experimental measurements. Weight average approaches are now applied to hybrid quantum mechanics/molecular mechanics MD simulations to predict fast vibrational motions in a protein with a high accuracy for better understanding of molecular functions from atomic structures.     

Improving the accuracy of protein stability predictions with multistate design using a variety of backbone ensembles

Multistate computational protein design (MSD) with backbone ensembles approximating conformational flexibility can predict higher quality sequences than single‐state design with a single fixed backbone. However, it is currently unclear what characteristics of backbone ensembles are required for the accurate prediction of protein sequence stability. In this study, we aimed to improve the accuracy of protein stability predictions made with MSD by using a variety of backbone ensembles to recapitulate the experimentally measured stability of 85 Streptococcal protein G domain β1 sequences. Ensembles tested here include an NMR ensemble as well as those generated by molecular dynamics (MD) simulations, by Backrub motions, and by PertMin, a new method that we developed involving the perturbation of atomic coordinates followed by energy minimization. MSD with the PertMin ensembles resulted in the most accurate predictions by providing the highest number of stable sequences in the top 25, and by correctly binning sequences as stable or unstable with the highest success rate (≈90%) and the lowest number of false positives. The performance of PertMin ensembles is due to the fact that their members closely resemble the input crystal structure and have low potential energy. Conversely, the NMR ensemble as well as those generated by MD simulations at 500 or 1000 K reduced prediction accuracy due to their low structural similarity to the crystal structure. The ensembles tested herein thus represent on‐ or off‐target models of the native protein fold and could be used in future studies to design for desired properties other than stability. Proteins 2014; 82:771–784. © 2013 Wiley Periodicals, Inc.     

Subtle functional collective motions in pancreatic-like ribonucleases: from ribonuclease A to angiogenin

The analysis of the dynamic behavior of enzymes is fundamental to structural biology. A direct relationship between protein flexibility and biological function has been shown for bovine pancreatic ribonuclease (RNase A) (Rasmussen et al., Nature 1992;357:423-424). More recently, crystallographic studies have shown that functional motions in RNase A involve the enzyme beta-sheet regions that move concertedly on substrate binding and release (Vitagliano et al., Proteins 2002;46:97-104). These motions have been shown to correspond to intrinsic dynamic properties of the native enzyme by molecular dynamics (MD) simulations. To unveil the occurrence of these collective motions in other members of pancreatic-like superfamily, we carried out MD simulations on human angiogenin (Ang). Essential dynamics (ED) analyses performed on the trajectories reveal that Ang exhibits collective motions similar to RNase A, despite the limited sequence identity (33%) of the two proteins. Furthermore, we show that these collective motions are also present in ensembles of experimentally determined structures of both Ang and RNase A. Finally, these subtle concerted beta-sheet motions were also observed for other two members of the pancreatic-like superfamily by comparing the ligand-bound and ligand-free structures of these enzymes. Taken together, these findings suggest that pancreatic-like ribonucleases share an evolutionary conserved dynamic behavior consisting of subtle beta-sheet motions, which are essential for substrate binding and release.     

Interdomain dynamics and ligand binding: molecular dynamics simulations of glutamine binding protein

Periplasmic binding proteins from Gram-negative bacteria possess a common architecture, comprised of two domains linked by a hinge region, a fold which they share with the neurotransmitter-binding domains of ionotropic glutamate receptors (GluRs). Glutamine-binding protein (GlnBP) is one such protein, whose crystal structure has been solved in both open and closed forms. Multi-nanosecond molecular dynamics simulations have been used to explore motions about the hinge region and how they are altered by ligand binding. Glutamine binding is seen to significantly reduce inter-domain motions about the hinge region. Essential dynamics analysis of inter-domain motion revealed the presence of both hinge-bending and twisting motions, as has been reported for a related sugar-binding protein. Significantly, the influence of the ligand on GlnBP dynamics is similar to that previously observed in simulations of rat glutamate receptor (GluR2) ligand-binding domain. The essential dynamics analysis of GlnBP also revealed a third class of motion which suggests a mechanism for signal transmission in GluRs.     

Local feature analysis: a statistical theory for reproducible essential dynamics of large macromolecules

Multivariate statistical methods are widely used to extract functional collective motions from macromolecular molecular dynamics (MD) simulations. In principal component analysis (PCA), a covariance matrix of positional fluctuations is diagonalized to obtain orthogonal eigenvectors and corresponding eigenvalues. The first few eigenvectors usually correspond to collective modes that approximate the functional motions in the protein. However, PCA representations are globally coherent by definition and, for a large biomolecular system, do not converge on the time scales accessible to MD. Also, the forced orthogonalization of modes leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. Here, we describe for the first time the application of local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low dimensional, and like PCA provide a reduced basis set for collective motions, but they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows. Also, the intrinsic dynamics of local domains is more extensively sampled than that of globally coherent PCA modes.     

A comparative analysis of the equilibrium dynamics of a designed protein inferred from NMR,X‐ray,and computations

A detailed analysis of high‐resolution structural data and computationally predicted dynamics was carried out for a designed sugar‐binding protein. The mean‐square deviations in the positions of residues derived from nuclear magnetic resonance (NMR) models and those inferred from X‐ray crystallographic B‐factors for two different crystal forms were compared with the predictions based on the Gaussian Network Model (GNM) and the results from molecular dynamics (MD) simulations. GNM systematically yielded a higher correlation than MD, with experimental data, suggesting that the lack of atomistic details in the coarse‐grained GNM is more than compensated for by the mathematically exact evaluation of fluctuations using the native contacts topology. Evidence is provided that particular loop motions are curtailed by intermolecular contacts in the crystal environment causing a discrepancy between theory and experiments. Interestingly, the information conveyed by X‐ray crystallography becomes more consistent with NMR models and computational predictions when ensembles of X‐ray models are considered. Less precise (broadly distributed) ensembles indeed appear to describe the accessible conformational space under native state conditions better than B‐factors. Our results highlight the importance of using multiple conformations obtained by alternative experimental methods, and analyzing results from both coarse‐grained models and atomic simulations, for accurate assessment of motions accessible to proteins under native state conditions. Proteins 2009. © 2009 Wiley‐Liss, Inc.     

Simulating nanoscale functional motions of biomolecules

We are describing efficient dynamics simulation methods for the characterization of functional motion of biomolecules on the nanometer scale. Multivariate statistical methods are widely used to extract and enhance functional collective motions from molecular dynamics (MD) simulations. A dimension reduction in MD is often realized through a principal component analysis (PCA) or a singular value decomposition (SVD) of the trajectory. Normal mode analysis (NMA) is a related collective coordinate space approach, which involves the decomposition of the motion into vibration modes based on an elastic model. Using the myosin motor protein as an example we describe a hybrid technique termed amplified collective motions (ACM) that enhances sampling of conformational space through a combination of normal modes with atomic level MD. Unfortunately, the forced orthogonalization of modes in collective coordinate space leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. In many biological molecules, such as HIV-1 protease, reflective or rotational symmetries are present that are broken using standard orthogonal basis functions. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry from the trajectory frames. Moreover, we develop an SVD that best approximates the given trajectory while respecting the symmetry. Finally, we describe a local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low-dimensional, and provide a reduced basis set for collective motions, but unlike global collective modes they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows.     

DROIDS 3.0—Detecting Genetic and Drug Class Variant Impact on Conserved Protein Binding Dynamics

The application of statistical methods to comparatively framed questions about the molecular dynamics (MD) of proteins can potentially enable investigations of biomolecular function beyond the current sequence and structural methods in bioinformatics. However, the chaotic behavior in single MD trajectories requires statistical inference that is derived from large ensembles of simulations representing the comparative functional states of a protein under investigation. Meaningful interpretation of such complex forms of big data poses serious challenges to users of MD. Here, we announce Detecting Relative Outlier Impacts from Molecular Dynamic Simulation (DROIDS) 3.0, a method and software package for comparative protein dynamics that includes maxDemon 1.0, a multimethod machine learning application that trains on large ensemble comparisons of concerted protein motions in opposing functional states generated by DROIDS and deploys learned classifications of these states onto newly generated MD simulations. Local canonical correlations in learning patterns generated from independent, yet identically prepared, MD validation runs are used to identify regions of functionally conserved protein dynamics. The subsequent impacts of genetic and/or drug class variants on conserved dynamics can also be analyzed by deploying the classifiers on variant MD simulations and quantifying how often these altered protein systems display opposing functional states. Here, we present several case studies of complex changes in functional protein dynamics caused by temperature, genetic mutation, and binding interactions with nucleic acids and small molecules. We demonstrate that our machine learning algorithm can properly identify regions of functionally conserved dynamics in ubiquitin and TATA-binding protein (TBP). We quantify the impact of genetic variation in TBP and drug class variation targeting the ATP-binding region of Hsp90 on conserved dynamics. We identify regions of conserved dynamics in Hsp90 that connect the ATP binding pocket to other functional regions. We also demonstrate that dynamic impacts of various Hsp90 inhibitors rank accordingly with how closely they mimic natural ATP binding.     

Calculation of protein ionization equilibria with conformational sampling: pK(a) of a model leucine zipper, GCN4 and barnase.

The use of conformational ensembles provided by nuclear magnetic resonance (NMR) experiments or generated by molecular dynamics (MD) simulations has been regarded as a useful approach to account for protein motions in the context of pK(a) calculations, yet the idea has been tested occasionally. This is the first report of systematic comparison of pK(a) estimates computed from long multiple MD simulations and NMR ensembles. As model systems, a synthetic leucine zipper, the naturally occurring coiled coil GCN4, and barnase were used. A variety of conformational averaging and titration curve-averaging techniques, or combination thereof, was adopted and/or modified to investigate the effect of extensive global conformational sampling on the accuracy of pK(a) calculations. Clustering of coordinates is proposed as an approach to reduce the vast diversity of MD ensembles to a few structures representative of the average electrostatic properties of the system in solution. Remarkable improvement of the accuracy of pK(a) predictions was achieved by the use of multiple MD simulations. By using multiple trajectories the absolute error in pK(a) predictions for the model leucine zipper was reduced to as low as approximately 0.25 pK(a) units. The validity, advantages, and limitations of explicit conformational sampling by MD, compared with the use of an average structure and a high internal protein dielectric value as means to improve the accuracy of pK(a) calculations, are discussed.     

Close correspondence between the motions from principal component analysis of multiple HIV-1 protease structures and elastic network modes

The large number of available HIV-1 protease structures provides a remarkable sampling of conformations of the different conformational states, which can be viewed as direct structural information about the dynamics of the HIV-1 protease. After structure matching, we apply principal component analysis (PCA) to obtain the important apparent motions for both bound and unbound structures. There are significant similarities between the first few key motions and the first few low-frequency normal modes calculated from a static representative structure with an elastic network model (ENM), strongly suggesting that the variations among the observed structures and the corresponding conformational changes are facilitated by the low-frequency, global motions intrinsic to the structure. Similarities are also found when the approach is applied to an NMR ensemble, as well as to molecular dynamics (MD) trajectories. Thus, a sufficiently large number of experimental structures can directly provide important information about protein dynamics, but ENM can also provide similar sampling of conformations.     

Constructing RNA dynamical ensembles by combining MD and motionally decoupled NMR RDCs: new insights into RNA dynamics and adaptive ligand recognition

We describe a strategy for constructing atomic resolution dynamical ensembles of RNA molecules, spanning up to millisecond timescales, that combines molecular dynamics (MD) simulations with NMR residual dipolar couplings (RDC) measured in elongated RNA. The ensembles are generated via a Monte Carlo procedure by selecting snap-shot from an MD trajectory that reproduce experimentally measured RDCs. Using this approach, we construct ensembles for two variants of the transactivation response element (TAR) containing three (HIV-1) and two (HIV-2) nucleotide bulges. The HIV-1 TAR ensemble reveals significant mobility in bulge residues C24 and U25 and to a lesser extent U23 and neighboring helical residue A22 that give rise to large amplitude spatially correlated twisting and bending helical motions. Omission of bulge residue C24 in HIV-2 TAR leads to a significant reduction in both the local mobility in and around the bulge and amplitude of inter-helical bending motions. In contrast, twisting motions of the helices remain comparable in amplitude to HIV-1 TAR and spatial correlations between them increase significantly. Comparison of the HIV-1 TAR dynamical ensemble and ligand bound TAR conformations reveals that several features of the binding pocket and global conformation are dynamically preformed, providing support for adaptive recognition via a ‘conformational selection’ type mechanism.     

Generalized correlation for biomolecular dynamics

Correlated motions in biomolecules are often essential for their function, e.g., allosteric signal transduction or mechanical/thermodynamic energy transport. Because correlated motions in biomolecules remain difficult to access experimentally, molecular dynamics (MD) simulations are particular useful for their analysis. The established method to quantify correlations from MD simulations via calculation of the covariance matrix, however, is restricted to linear correlations and therefore misses part of the correlations in the atomic fluctuations. Herein, we propose a general statistical mechanics approach to detect and quantify any correlated motion from MD trajectories. This generalized correlation measure is contrasted with correlations obtained from covariance matrices for the B1 domain of protein G and T4 lysozyme. The new method successfully quantifies correlations and provides a valuable global overview over the functionally relevant collective motions of lysozyme. In particular, correlated motions of helix 1 together with the two main lobes of lysozyme are detected, which are not seen by the conventional covariance matrix. Overall, the established method misses more than 50% of the correlation. This failure is attributed to both, an interfering and unnecessary dependence on mutual orientations of the atomic fluctuations and, to a lesser extent, attributed to nonlinear correlations. Our generalized correlation measure overcomes these problems and, moreover, allows for an improved understanding of the conformational dynamics by separating linear and nonlinear contributions of the correlation.     

Evidence of Functional Protein Dynamics from X-Ray Crystallographic Ensembles

It is widely recognized that representing a protein as a single static conformation is inadequate to describe the dynamics essential to the performance of its biological function. We contrast the amino acid displacements below and above the protein dynamical transition temperature, T_D∼215K, of hen egg white lysozyme using X-ray crystallography ensembles that are analyzed by molecular dynamics simulations as a function of temperature. We show that measuring structural variations across an ensemble of X-ray derived models captures the activation of conformational states that are of functional importance just above T_D, and they remain virtually identical to structural motions measured at 300K. Our results highlight the ability to observe functional structural variations across an ensemble of X-ray crystallographic data, and that residue fluctuations measured in MD simulations at room temperature are in quantitative agreement with the experimental observable.     

PPM: a side-chain and backbone chemical shift predictor for the assessment of protein conformational ensembles

The combination of the wide availability of protein backbone and side-chain NMR chemical shifts with advances in understanding of their relationship to protein structure makes these parameters useful for the assessment of structural-dynamic protein models. A new chemical shift predictor (PPM) is introduced, which is solely based on physical?Cchemical contributions to the chemical shifts for both the protein backbone and methyl-bearing amino-acid side chains. To explicitly account for the effects of protein dynamics on chemical shifts, PPM was directly refined against 100?ns long molecular dynamics (MD) simulations of 35 proteins with known experimental NMR chemical shifts. It is found that the prediction of methyl-proton chemical shifts by PPM from MD ensembles is improved over other methods, while backbone C??, C??, C??, N, and H^N chemical shifts are predicted at an accuracy comparable to the latest generation of chemical shift prediction programs. PPM is particularly suitable for the rapid evaluation of large protein conformational ensembles on their consistency with experimental NMR data and the possible improvement of protein force fields from chemical shifts.     

Interpreting Dynamically-Averaged Scalar Couplings in Proteins

The experimental determination of scalar three-bond coupling constants represents a powerful method to probe both the structure and dynamics of proteins. The detailed structural interpretation of such coupling constants is usually based on Karplus relationships, which allow the measured couplings to be related to the torsion angles of the molecules. As the measured couplings are sensitive to thermal fluctuations, the parameters in the Karplus relationships are better derived from ensembles representing the distributions of dihedral angles present in solution, rather than from single conformations. We present a method to derive such parameters that uses ensembles of conformations determined through dynamic-ensemble refinement – a method that provides structural ensembles that simultaneously represent both the structure and the associated dynamics of a protein.     

Bending of the calmodulin central helix: a theoretical study.

The crystal structure of calcium-calmodulin (CaM) reveals a protein with a typical dumbbell structure. Various spectroscopic studies have suggested that the central linker region of CaM, which is alpha-helical in the crystal structure, is flexible in solution. In particular, NMR studies have indicated the presence of a flexible backbone between residues Lys 77 and Asp 80. This flexibility is related directly to the function of the protein because it enables the N- and C-terminal domains of the protein to move toward each other and bind to the CaM-binding domain of a target protein. We have investigated the flexibility of the CaM central helix by a variety of computational techniques: molecular dynamics (MD) simulations, normal mode analysis (NMA), and essential dynamics (ED) analysis. Our MD results reproduce the experimentally determined location of the bend in a simulation of only the CaM central helix, indicating that the bending point is an intrinsic property of the alpha-helix, for which the remainder of the protein is not important. Interestingly, the modes found by the ED analysis of the MD trajectory are very similar to the lowest frequency modes from the NM analysis and to modes found by an ED analysis of different structures in a set of NMR structures. Electrostatic interactions involving residues Arg 74 and Asp 80 seem to be important for these bending motions and unfolding, which is in line with pH-dependent NMR and CD studies.     

Computational studies of essential dynamics of Pseudomonas cepacia lipase

In order to investigate the interfacial activation of a lipase from Pseudomonas cepacia (PcL), molecular dynamics (MD) simulations and essential dynamics (ED) analysis were performed in different solvent environments: vacuum and explicit water solvents. Starting from the active (open) structure of PcL, the essential dynamics analysis of the simulations revealed large correlated motions, which may be responsible for the activation of the enzyme. Fluctuations in the U1 (active-site lid) and U2 domains are found to be important in the activation of PcL. In contrast, the catalytic triad exhibits very little displacement. These results are consistent with the previous X-ray structural determination. A combined analysis of the trajectories showed some differences for the simulations in different solvent environments. It was found that the region around the helix alpha5 showed larger displacements in the water simulations. It can be concluded that the open structure of PcL becomes unstable in water solvents, leading to the closing of the so-called 'lid' region. The simulations and ED analysis on the closed structure of PgL provided additional information concerning the structural changes involved in the activation of the lipases. It was found that structural changes for PcL and PgL, which are responsible for the essential motions of the protein, showed contrasting behavior in the different solvent environments.