Partition decoupling for multi-gene analysis of gene expression profiling data

Background

Molecular dynamics (MD) simulations are powerful tools to investigate the conformational dynamics of proteins that is often a critical element of their function. Identification of functionally relevant conformations is generally done clustering the large ensemble of structures that are generated. Recently, Self-Organising Maps (SOMs) were reported performing more accurately and providing more consistent results than traditional clustering algorithms in various data mining problems. We present a novel strategy to analyse and compare conformational ensembles of protein domains using a two-level approach that combines SOMs and hierarchical clustering.

Results

The conformational dynamics of the α-spectrin SH3 protein domain and six single mutants were analysed by MD simulations. The Cα's Cartesian coordinates of conformations sampled in the essential space were used as input data vectors for SOM training, then complete linkage clustering was performed on the SOM prototype vectors. A specific protocol to optimize a SOM for structural ensembles was proposed: the optimal SOM was selected by means of a Taguchi experimental design plan applied to different data sets, and the optimal sampling rate of the MD trajectory was selected. The proposed two-level approach was applied to single trajectories of the SH3 domain independently as well as to groups of them at the same time. The results demonstrated the potential of this approach in the analysis of large ensembles of molecular structures: the possibility of producing a topological mapping of the conformational space in a simple 2D visualisation, as well as of effectively highlighting differences in the conformational dynamics directly related to biological functions.

Conclusions

The use of a two-level approach combining SOMs and hierarchical clustering for conformational analysis of structural ensembles of proteins was proposed. It can easily be extended to other study cases and to conformational ensembles from other sources.     

Reweighting of molecular simulations with explicit-solvent SAXS restraints elucidates ion-dependent RNA ensembles

Small-angle X-ray scattering (SAXS) experiments are increasingly used to probe RNA structure. A number of forward models that relate measured SAXS intensities and structural features, and that are suitable to model either explicit-solvent effects or solute dynamics, have been proposed in the past years. Here, we introduce an approach that integrates atomistic molecular dynamics simulations and SAXS experiments to reconstruct RNA structural ensembles while simultaneously accounting for both RNA conformational dynamics and explicit-solvent effects. Our protocol exploits SAXS pure-solute forward models and enhanced sampling methods to sample an heterogenous ensemble of structures, with no information towards the experiments provided on-the-fly. The generated structural ensemble is then reweighted through the maximum entropy principle so as to match reference SAXS experimental data at multiple ionic conditions. Importantly, accurate explicit-solvent forward models are used at this reweighting stage. We apply this framework to the GTPase-associated center, a relevant RNA molecule involved in protein translation, in order to elucidate its ion-dependent conformational ensembles. We show that (a) both solvent and dynamics are crucial to reproduce experimental SAXS data and (b) the resulting dynamical ensembles contain an ion-dependent fraction of extended structures.     

Statistical characterization of protein ensembles

When accounting for structural fluctuations or measurement errors, a single rigid structure may not be sufficient to represent a protein. One approach to solve this problem is to represent the possible conformations as a discrete set of observed conformations, an ensemble. In this work, we follow a different richer approach, and introduce a framework for estimating probability density functions in very high dimensions, and then apply it to represent ensembles of folded proteins. This proposed approach combines techniques such as kernel density estimation, maximum likelihood, cross-validation, and bootstrapping. We present the underlying theoretical and computational framework and apply it to artificial data and protein ensembles obtained from molecular dynamics simulations. We compare the results with those obtained experimentally, illustrating the potential and advantages of this representation.     

Refinement of Peptide Conformational Ensembles by 2D IR Spectroscopy: Application to Ala‒Ala‒Ala

Characterizing ensembles of intrinsically disordered proteins is experimentally challenging because of the ill-conditioned nature of ensemble determination with limited data and the intrinsic fast dynamics of the conformational ensemble. Amide I two-dimensional infrared (2D IR) spectroscopy has picosecond time resolution to freeze structural ensembles as needed for probing disordered-protein ensembles and conformational dynamics. Also, developments in amide I computational spectroscopy now allow a quantitative and direct prediction of amide I spectra based on conformational distributions drawn from molecular dynamics simulations, providing a route to ensemble refinement against experimental spectra. We performed a Bayesian ensemble refinement method on Ala–Ala–Ala against isotope-edited Fourier-transform infrared spectroscopy and 2D IR spectroscopy and tested potential factors affecting the quality of ensemble refinements. We found that isotope-edited 2D IR spectroscopy provides a stringent constraint on Ala–Ala–Ala conformations and returns consistent conformational ensembles with the dominant ppII conformer across varying prior distributions from many molecular dynamics force fields and water models. The dominant factor influencing ensemble refinements is the systematic frequency uncertainty from spectroscopic maps. However, the uncertainty of conformer populations can be significantly reduced by incorporating 2D IR spectra in addition to traditional Fourier-transform infrared spectra. Bayesian ensemble refinement against isotope-edited 2D IR spectroscopy thus provides a route to probe equilibrium-complex protein ensembles and potentially nonequilibrium conformational dynamics.     

Efficient RMSD measures for the comparison of two molecular ensembles. Root-mean-square deviation

Quantitative measures are presented for comparing the conformations of two molecular ensembles. The measures are based on Kabsch's formula for the root-mean-square deviation (RMSD) and the covariance matrix of atomic positions of isotropically distributed ensembles (IDE). By using a Taylor series expansion, it is shown that the RMSD can be expressed solely in terms of the IDE matrices. A fast approximate method is introduced for the pairwise RMSD determination whose computational cost scales linearly with the number of structures. A similarity measure for two structural ensembles that is based on the trace metric of the differences of powers of the IDE matrices is presented. The measures are illustrated for conformational ensembles generated by a molecular dynamics computer simulation of a partially folded A-state analog of ubiquitin.     

The MUMO (minimal under-restraining minimal over-restraining) method for the determination of native state ensembles of proteins

While reliable procedures for determining the conformations of proteins are available, methods for generating ensembles of structures that also reflect their flexibility are much less well established. Here we present a systematic assessment of the ability of ensemble-averaged molecular dynamics simulations with ensemble-averaged NMR restraints to simultaneously reproduce the average structure of proteins and their associated dynamics. We discuss the effects that under-restraining (overfitting) and over-restraining (underfitting) have on the structures generated in ensemble-averaged molecular simulations. We then introduce the MUMO (minimal under-restraining minimal over-restraining) method, a procedure in which different observables are averaged over a different number of molecules. As both over-restraining and under-restraining are significantly reduced in the MUMO method, it is possible to generate ensembles of conformations that accurately characterize both the structure and the dynamics of native states of proteins. The application of the MUMO method to the protein ubiquitin yields a high-resolution structural ensemble with an RDC Q-factor of 0.19.     

Crystal Structural Framework of Lithium Super‐Ionic Conductors

As technologically important materials for solid‐state batteries, Li super‐ionic conductors are a class of materials exhibiting exceptionally high ionic conductivity at room temperature. These materials have unique crystal structural frameworks hosting a highly conductive Li sublattice. However, it is not understood why certain crystal structures of the super‐ionic conductors lead to high conductivity in the Li sublattice. In this study, using topological analysis and ab initio molecular dynamics simulations, the crystal structures of all Li‐conducting oxides and sulfides are studied systematically and the key features pertaining to fast‐ion conduction are quantified. In particular, a unique feature of enlarged Li sites caused by large local spaces in the crystal structural framework is identified, promoting fast conduction in the Li‐ion sublattice. Based on these quantified features, the high‐throughput screening identifies many new structures as fast Li‐ion conductors, which are further confirmed by ab initio molecular dynamics simulations. This study provides new insights and a systematic quantitative understanding of the crystal structural frameworks of fast ion‐conductor materials and motivates future experimental and computational studies on new fast‐ion conductors.     

Predicting Evolutionary Site Variability from Structure in Viral Proteins: Buriedness,Packing, Flexibility,and Design

Several recent works have shown that protein structure can predict site-specific evolutionary sequence variation. In particular, sites that are buried and/or have many contacts with other sites in a structure have been shown to evolve more slowly, on average, than surface sites with few contacts. Here, we present a comprehensive study of the extent to which numerous structural properties can predict sequence variation. The quantities we considered include buriedness (as measured by relative solvent accessibility), packing density (as measured by contact number), structural flexibility (as measured by B factors, root-mean-square fluctuations, and variation in dihedral angles), and variability in designed structures. We obtained structural flexibility measures both from molecular dynamics simulations performed on nine non-homologous viral protein structures and from variation in homologous variants of those proteins, where they were available. We obtained measures of variability in designed structures from flexible-backbone design in the Rosetta software. We found that most of the structural properties correlate with site variation in the majority of structures, though the correlations are generally weak (correlation coefficients of 0.1–0.4). Moreover, we found that buriedness and packing density were better predictors of evolutionary variation than structural flexibility. Finally, variability in designed structures was a weaker predictor of evolutionary variability than buriedness or packing density, but it was comparable in its predictive power to the best structural flexibility measures. We conclude that simple measures of buriedness and packing density are better predictors of evolutionary variation than the more complicated predictors obtained from dynamic simulations, ensembles of homologous structures, or computational protein design.     

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.     

Insights into correlated motions and long-range interactions in CheY derived from molecular dynamics simulations

CheY is a response regulator protein involved in bacterial chemotaxis. Much is known about its active and inactive conformations, but little is known about the mechanisms underlying long-range interactions or correlated motions. To investigate these events, molecular dynamics simulations were performed on the unphosphorylated, inactive structure from Salmonella typhimurium and the CheY-BeF(3)(-) active mimic structure (with BeF(3)(-) removed) from Escherichia coli. Simulations utilized both sequences in each conformation to discriminate sequence- and structure-specific behavior. The previously identified conformational differences between the inactive and active conformations of the strand-4-helix-4 loop, which are present in these simulations, arise from the structural, and not the sequence, differences. The simulations identify previously unreported structure-specific flexibility features in this loop and sequence-specific flexibility features in other regions of the protein. Both structure- and sequence-specific long-range interactions are observed in the active and inactive ensembles. In the inactive ensemble, two distinct mechanisms based on Thr-87 or Ile-95 rotameric forms, are observed for the previously identified g+ and g- rotamer sampling by Tyr-106. These molecular dynamics simulations have thus identified both sequence- and structure-specific differences in flexibility, long-range interactions, and rotameric form of key residues. Potential biological consequences of differential flexibility and long-range correlated motion are discussed.     

Structural interpretation of hydrogen exchange protection factors in proteins: characterization of the native state fluctuations of CI2

Protection factors obtained from equilibrium hydrogen exchange experiments are an important source of structural information on both native and nonnative states of proteins. We present a method for determining ensembles of protein structures by using hydrogen exchange data as restraints in molecular dynamics simulations in conjunction with an empirical force-field. The method is applied to determine the ensemble of structures representing the native state of chymotrypsin inhibitor 2 (CI2), including the rare, large fluctuations responsible for hydrogen exchange.     

Similarity Measures for Protein Ensembles

Analyses of similarities and changes in protein conformation can provide important information regarding protein function and evolution. Many scores, including the commonly used root mean square deviation, have therefore been developed to quantify the similarities of different protein conformations. However, instead of examining individual conformations it is in many cases more relevant to analyse ensembles of conformations that have been obtained either through experiments or from methods such as molecular dynamics simulations. We here present three approaches that can be used to compare conformational ensembles in the same way as the root mean square deviation is used to compare individual pairs of structures. The methods are based on the estimation of the probability distributions underlying the ensembles and subsequent comparison of these distributions. We first validate the methods using a synthetic example from molecular dynamics simulations. We then apply the algorithms to revisit the problem of ensemble averaging during structure determination of proteins, and find that an ensemble refinement method is able to recover the correct distribution of conformations better than standard single-molecule refinement.     

Interpretation of HDX Data by Maximum-Entropy Reweighting of Simulated Structural Ensembles

Hydrogen-deuterium exchange combined with mass spectrometry (HDX-MS) is a widely applied biophysical technique that probes the structure and dynamics of biomolecules without the need for site-directed modifications or bio-orthogonal labels. The mechanistic interpretation of HDX data, however, is often qualitative and subjective, owing to a lack of quantitative methods to rigorously translate observed deuteration levels into atomistic structural information. To help address this problem, we have developed a methodology to generate structural ensembles that faithfully reproduce HDX-MS measurements. In this approach, an ensemble of protein conformations is first generated, typically using molecular dynamics simulations. A maximum-entropy bias is then applied post hoc to the resulting ensemble such that averaged peptide-deuteration levels, as predicted by an empirical model, agree with target values within a given level of uncertainty. We evaluate this approach, referred to as HDX ensemble reweighting (HDXer), for artificial target data reflecting the two major conformational states of a binding protein. We demonstrate that the information provided by HDX-MS experiments and by the model of exchange are sufficient to recover correctly weighted structural ensembles from simulations, even when the relevant conformations are rarely observed. Degrading the information content of the target data—e.g., by reducing sequence coverage, by averaging exchange levels over longer peptide segments, or by incorporating different sources of uncertainty—reduces the structural accuracy of the reweighted ensemble but still allows for useful insights into the distinctive structural features reflected by the target data. Finally, we describe a quantitative metric to rank candidate structural ensembles according to their correspondence with target data and illustrate the use of HDXer to describe changes in the conformational ensemble of the membrane protein LeuT. In summary, HDXer is designed to facilitate objective structural interpretations of HDX-MS data and to inform experimental approaches and further developments of theoretical exchange models.     

Recent developments in structural proteomics for protein structure determination

The major challenges in structural proteomics include identifying all the proteins on the genome-wide scale, determining their structure-function relationships, and outlining the precise three-dimensional structures of the proteins. Protein structures are typically determined by experimental approaches such as X-ray crystallography or nuclear magnetic resonance (NMR) spectroscopy. However, the knowledge of three-dimensional space by these techniques is still limited. Thus, computational methods such as comparative and de novo approaches and molecular dynamic simulations are intensively used as alternative tools to predict the three-dimensional structures and dynamic behavior of proteins. This review summarizes recent developments in structural proteomics for protein structure determination; including instrumental methods such as X-ray crystallography and NMR spectroscopy, and computational methods such as comparative and de novo structure prediction and molecular dynamics simulations.     

Atomic-level characterization of disordered protein ensembles

The roles of unfolded states of proteins in normal folding and in diseases involving aggregation, as well as the prevalence and regulatory functions of intrinsically disordered proteins, have become increasingly recognized. The structural representation of these disordered states as ensembles of interconverting conformers can therefore provide critical insights. Experimental methods can be used to probe ensemble-averaged structural properties of disordered states and computational approaches generate representative ensembles of conformers using experimental restraints. In particular, NMR and small-angle X-ray scattering provide quantitative data that can readily be incorporated into calculations. These techniques have gleaned structural information about denatured, unfolded and intrinsically disordered proteins. The use of experimental data in different computational approaches, including ensemble molecular dynamics simulations and algorithms that assign populations to pregenerated conformers, has highlighted the presence of both local and long-range structure, and the occurrence of native-like and non-native interactions in unfolded and denatured states. Analysis of the resulting ensembles has suggested important implications of this fluctuating structure for folding, aggregation and binding.     

Structure and Function in Homodimeric Enzymes: Simulations of Cooperative and Independent Functional Motions

Large-scale conformational change is a common feature in the catalytic cycles of enzymes. Many enzymes function as homodimers with active sites that contain elements from both chains. Symmetric and anti-symmetric cooperative motions in homodimers can potentially lead to correlated active site opening and/or closure, likely to be important for ligand binding and release. Here, we examine such motions in two different domain-swapped homodimeric enzymes: the DcpS scavenger decapping enzyme and citrate synthase. We use and compare two types of all-atom simulations: conventional molecular dynamics simulations to identify physically meaningful conformational ensembles, and rapid geometric simulations of flexible motion, biased along normal mode directions, to identify relevant motions encoded in the protein structure. The results indicate that the opening/closure motions are intrinsic features of both unliganded enzymes. In DcpS, conformational change is dominated by an anti-symmetric cooperative motion, causing one active site to close as the other opens; however a symmetric motion is also significant. In CS, we identify that both symmetric (suggested by crystallography) and asymmetric motions are features of the protein structure, and as a result the behaviour in solution is largely non-cooperative. The agreement between two modelling approaches using very different levels of theory indicates that the behaviours are indeed intrinsic to the protein structures. Geometric simulations correctly identify and explore large amplitudes of motion, while molecular dynamics simulations indicate the ranges of motion that are energetically feasible. Together, the simulation approaches are able to reveal unexpected functionally relevant motions, and highlight differences between enzymes.     

Assessment of flexible backbone protein design methods for sequence library prediction in the therapeutic antibody Herceptin-HER2 interface

Computational protein design methods can complement experimental screening and selection techniques by predicting libraries of low-energy sequences compatible with a desired structure and function. Incorporating backbone flexibility in computational design allows conformational adjustments that should broaden the range of predicted low-energy sequences. Here, we evaluate computational predictions of sequence libraries from different protocols for modeling backbone flexibility using the complex between the therapeutic antibody Herceptin and its target human epidermal growth factor receptor 2 (HER2) as a model system. Within the program RosettaDesign, three methods are compared: The first two use ensembles of structures generated by Monte Carlo protocols for near-native conformational sampling: kinematic closure (KIC) and backrub, and the third method uses snapshots from molecular dynamics (MD) simulations. KIC or backrub methods were better able to identify the amino acid residues experimentally observed by phage display in the Herceptin-HER2 interface than MD snapshots, which generated much larger conformational and sequence diversity. KIC and backrub, as well as fixed backbone simulations, captured the key mutation Asp98Trp in Herceptin, which leads to a further threefold affinity improvement of the already subnanomolar parental Herceptin-HER2 interface. Modeling subtle backbone conformational changes may assist in the design of sequence libraries for improving the affinity of antibody-antigen interfaces and could be suitable for other protein complexes for which structural information is available.     

Discrete Molecular Dynamics Can Predict Helical Prestructured Motifs in Disordered Proteins

Intrinsically disordered proteins (IDPs) lack a stable tertiary structure, but their short binding regions termed Pre-Structured Motifs (PreSMo) can form transient secondary structure elements in solution. Although disordered proteins are crucial in many biological processes and designing strategies to modulate their function is highly important, both experimental and computational tools to describe their conformational ensembles and the initial steps of folding are sparse. Here we report that discrete molecular dynamics (DMD) simulations combined with replica exchange (RX) method efficiently samples the conformational space and detects regions populating α-helical conformational states in disordered protein regions. While the available computational methods predict secondary structural propensities in IDPs based on the observation of protein-protein interactions, our ab initio method rests on physical principles of protein folding and dynamics. We show that RX-DMD predicts α-PreSMos with high confidence confirmed by comparison to experimental NMR data. Moreover, the method also can dissect α-PreSMos in close vicinity to each other and indicate helix stability. Importantly, simulations with disordered regions forming helices in X-ray structures of complexes indicate that a preformed helix is frequently the binding element itself, while in other cases it may have a role in initiating the binding process. Our results indicate that RX-DMD provides a breakthrough in the structural and dynamical characterization of disordered proteins by generating the structural ensembles of IDPs even when experimental data are not available.     

Distance restraints from crosslinking mass spectrometry: Mining a molecular dynamics simulation database to evaluate lysine–lysine distances

Integrative structural biology attempts to model the structures of protein complexes that are challenging or intractable by classical structural methods (due to size, dynamics, or heterogeneity) by combining computational structural modeling with data from experimental methods. One such experimental method is chemical crosslinking mass spectrometry (XL‐MS), in which protein complexes are crosslinked and characterized using liquid chromatography‐mass spectrometry to pinpoint specific amino acid residues in close structural proximity. The commonly used lysine‐reactive N‐hydroxysuccinimide ester reagents disuccinimidylsuberate (DSS) and bis(sulfosuccinimidyl)suberate (BS³) have a linker arm that is 11.4 Å long when fully extended, allowing C_α (alpha carbon of protein backbone) atoms of crosslinked lysine residues to be up to ～24 Å apart. However, XL‐MS studies on proteins of known structure frequently report crosslinks that exceed this distance. Typically, a tolerance of ～3 Å is added to the theoretical maximum to account for this observation, with limited justification for the chosen value. We used the Dynameomics database, a repository of high‐quality molecular dynamics simulations of 807 proteins representative of diverse protein folds, to investigate the relationship between lysine–lysine distances in experimental starting structures and in simulation ensembles. We conclude that for DSS/BS³, a distance constraint of 26–30 Å between C_α atoms is appropriate. This analysis provides a theoretical basis for the widespread practice of adding a tolerance to the crosslinker length when comparing XL‐MS results to structures or in modeling. We also discuss the comparison of XL‐MS results to MD simulations and known structures as a means to test and validate experimental XL‐MS methods.     

Access Path to the Ligand Binding Pocket May Play a Role in Xenobiotics Selection by AhR

Understanding of multidrug binding at the atomic level would facilitate drug design and strategies to modulate drug metabolism, including drug transport, oxidation, and conjugation. Therefore we explored the mechanism of promiscuous binding of small molecules by studying the ligand binding domain, the PAS-B domain of the aryl hydrocarbon receptor (AhR). Because of the low sequence identities of PAS domains to be used for homology modeling, structural features of the widely employed HIF-2α and a more recent suitable template, CLOCK were compared. These structures were used to build AhR PAS-B homology models. We performed molecular dynamics simulations to characterize dynamic properties of the PAS-B domain and the generated conformational ensembles were employed in in silico docking. In order to understand structural and ligand binding features we compared the stability and dynamics of the promiscuous AhR PAS-B to other PAS domains exhibiting specific interactions or no ligand binding function. Our exhaustive in silico binding studies, in which we dock a wide spectrum of ligand molecules to the conformational ensembles, suggest that ligand specificity and selection may be determined not only by the PAS-B domain itself, but also by other parts of AhR and its protein interacting partners. We propose that ligand binding pocket and access channels leading to the pocket play equally important roles in discrimination of endogenous molecules and xenobiotics.