Discrete restraint-based protein modeling and the Calpha-trace problem

We present a novel de novo method to generate protein models from sparse, discretized restraints on the conformation of the main chain and side chain atoms. We focus on Calpha-trace generation, the problem of constructing an accurate and complete model from approximate knowledge of the positions of the Calpha atoms and, in some cases, the side chain centroids. Spatial restraints on the Calpha atoms and side chain centroids are supplemented by constraints on main chain geometry, phi/xi angles, rotameric side chain conformations, and inter-atomic separations derived from analyses of known protein structures. A novel conformational search algorithm, combining features of tree-search and genetic algorithms, generates models consistent with these restraints by propensity-weighted dihedral angle sampling. Models with ideal geometry, good phi/xi angles, and no inter-atomic overlaps are produced with 0.8 A main chain and, with side chain centroid restraints, 1.0 A all-atom root-mean-square deviation (RMSD) from the crystal structure over a diverse set of target proteins. The mean model derived from 50 independently generated models is closer to the crystal structure than any individual model, with 0.5 A main chain RMSD under only Calpha restraints and 0.7 A all-atom RMSD under both Calpha and centroid restraints. The method is insensitive to randomly distributed errors of up to 4 A in the Calpha restraints. The conformational search algorithm is efficient, with computational cost increasing linearly with protein size. Issues relating to decoy set generation, experimental structure determination, efficiency of conformational sampling, and homology modeling are discussed.     

Statistics of random protein superpositions: p-values for pairwise structure alignment.

Quantification of statistical significance is essential for the interpretation of protein structural similarity. To address this, a random model for protein structure comparison was developed. Novelty of the model is threefold. First, a sample of random structure comparisons is restricted to molecules of the same size and shape as the superposition of interest. Second, careful selection of the sample and accurate modeling of shape allows approximation of the root mean square deviation (RMSD) distribution of random comparisons with a Nakagami probability density function. Third, through convolution, a second probability density function is obtained that describes the coordinate difference vector projections underlying the random distribution of RMSD. This last feature allows sampling random distributions of not only RMSD, but also any similarity score that depends on difference vector projections, such as GDT_TS score, TM score, and LiveBench 3D score. Probabilities estimated from the method correlate well with common measures of structural similarity, such as the Dali Z-score and the GDT_TS score. As a result, the p-value for a given superposition can be calculated using simple formulae depending on RMSD, radius of gyration, and thinnest molecular dimension. In addition to scoring structural similarity, p-values computed by this method can be applied to evaluation of homology modeling techniques, providing a statistically sound alternative to scores used in reference-independent evaluation of alignment quality.     

Using surface envelopes to constrain molecular modeling

Molecular density information (as measured by electron microscopic reconstructions or crystallographic density maps) can be a powerful source of information for molecular modeling. Molecular density constrains models by specifying where atoms should and should not be. Low-resolution density information can often be obtained relatively quickly, and there is a need for methods that use it effectively. We have previously described a method for scoring molecular models with surface envelopes to discriminate between plausible and implausible fits. We showed that we could successfully filter out models with the wrong shape based on this discrimination power. Ideally, however, surface information should be used during the modeling process to constrain the conformations that are sampled. In this paper, we describe an extension of our method for using shape information during computational modeling. We use the envelope scoring metric as part of an objective function in a global optimization that also optimizes distances and angles while avoiding collisions. We systematically tested surface representations of proteins (using all nonhydrogen heavy atoms) with different abundance of distance information and showed that the root mean square deviation (RMSD) of models built with envelope information is consistently improved, particularly in data sets with relatively small sets of short-range distances.     

Statistical potential for assessment and prediction of protein structures

Protein structures in the Protein Data Bank provide a wealth of data about the interactions that determine the native states of proteins. Using the probability theory, we derive an atomic distance-dependent statistical potential from a sample of native structures that does not depend on any adjustable parameters (Discrete Optimized Protein Energy, or DOPE). DOPE is based on an improved reference state that corresponds to noninteracting atoms in a homogeneous sphere with the radius dependent on a sample native structure; it thus accounts for the finite and spherical shape of the native structures. The DOPE potential was extracted from a nonredundant set of 1472 crystallographic structures. We tested DOPE and five other scoring functions by the detection of the native state among six multiple target decoy sets, the correlation between the score and model error, and the identification of the most accurate non-native structure in the decoy set. For all decoy sets, DOPE is the best performing function in terms of all criteria, except for a tie in one criterion for one decoy set. To facilitate its use in various applications, such as model assessment, loop modeling, and fitting into cryo-electron microscopy mass density maps combined with comparative protein structure modeling, DOPE was incorporated into the modeling package MODELLER-8.     

A composite score for predicting errors in protein structure models

Reliable prediction of model accuracy is an important unsolved problem in protein structure modeling. To address this problem, we studied 24 individual assessment scores, including physics-based energy functions, statistical potentials, and machine learning-based scoring functions. Individual scores were also used to construct approximately 85,000 composite scoring functions using support vector machine (SVM) regression. The scores were tested for their abilities to identify the most native-like models from a set of 6000 comparative models of 20 representative protein structures. Each of the 20 targets was modeled using a template of <30% sequence identity, corresponding to challenging comparative modeling cases. The best SVM score outperformed all individual scores by decreasing the average RMSD difference between the model identified as the best of the set and the model with the lowest RMSD (DeltaRMSD) from 0.63 A to 0.45 A, while having a higher Pearson correlation coefficient to RMSD (r=0.87) than any other tested score. The most accurate score is based on a combination of the DOPE non-hydrogen atom statistical potential; surface, contact, and combined statistical potentials from MODPIPE; and two PSIPRED/DSSP scores. It was implemented in the SVMod program, which can now be applied to select the final model in various modeling problems, including fold assignment, target-template alignment, and loop modeling.     

Consensus alignment for reliable framework prediction in homology modeling

MOTIVATION: Even the best sequence alignment methods frequently fail to correctly identify the framework regions for which backbones can be copied from the template into the target structure. Since the underprediction and, more significantly, the overprediction of these regions reduces the quality of the final model, it is of prime importance to attain as much as possible of the true structural alignment between target and template. RESULTS: We have developed an algorithm called Consensus that consistently provides a high quality alignment for comparative modeling. The method follows from a benchmark analysis of the 3D models generated by ten alignment techniques for a set of 79 homologous protein structure pairs. For 20-to-40% of the targets, these methods yield models with at least 6 A root mean square deviation (RMSD) from the native structure. We have selected the top five performing methods, and developed a consensus algorithm to generate an improved alignment. By building on the individual strength of each method, a set of criteria was implemented to remove the alignment segments that are likely to correspond to structurally dissimilar regions. The automated algorithm was validated on a different set of 48 protein pairs, resulting in 2.2 A average RMSD for the predicted models, and only four cases in which the RMSD exceeded 3 A. The average length of the alignments was about 75% of that found by standard structural superposition methods. The performance of Consensus was consistent from 2 to 32% target-template sequence identity, and hence it can be used for accurate prediction of framework regions in homology modeling.     

Defining and computing optimum RMSD for gapped and weighted multiple-structure alignment

Pairwise structure alignment commonly uses root mean square deviation (RMSD) to measure the structural similarity, and methods for optimizing RMSD are well established. We extend RMSD to weighted RMSD for multiple structures. By using multiplicative weights, we show that weighted RMSD for all pairs is the same as weighted RMSD to an average of the structures. Thus, using RMSD or weighted RMSD implies that the average is a consensus structure. Although we show that in general, the two tasks of finding the optimal translations and rotations for minimizing weighted RMSD cannot be separated for multiple structures like they can for pairs, an inherent difficulty and a fact ignored by previous work, we develop a near-linear iterative algorithm to converge weighted RMSD to a local minimum. 10,000 experiments of gapped alignment done on each of 23 protein families from HOMSTRAD (where each structure starts with a random translation and rotation) converge rapidly to the same minimum. Finally we propose a heuristic method to iteratively remove the effect of outliers and find well-aligned positions that determine the structural conserved region by modeling B-factors and deviations from the average positions as weights and iteratively assigning higher weights to better aligned atoms.     

Application of MM/PBSA colony free energy to loop decoy discrimination: toward correlation between energy and root mean square deviation

Accurate free energy estimation is needed in many predictive tasks. The molecular mechanics/Poisson-Boltzmann solvent accessible surface area (MM/PBSA) approach has proven to be accurate. However, the correlation between the estimated free energy and the distance (e.g., root mean square deviation [RMSD]) from the most stable conformation is hindered by the strong free energy dependence on minor conformational variations. In this paper, a protocol for MM/PBSA free energy estimation is designed and tested on several loop decoy sets. We show that further integration of MM/PBSA free energy estimator with the colony energy approach makes the correlation between the free energy and RMSD from the native structure apparent, for the test sets on which it could be applied. Our results suggest that (1) the MM/PBSA free energy estimator is able to detect native-like structures for most decoy sets, and (2) application of the colony energy approach greatly hampers the MM/energy strong dependence on minor conformational changes.     

Distinguishing native conformations of proteins from decoys with an effective free energy estimator based on the OPLS all-atom force field and the Surface Generalized Born solvent model

Protein decoy data sets provide a benchmark for testing scoring functions designed for fold recognition and protein homology modeling problems. It is commonly believed that statistical potentials based on reduced atomic models are better able to discriminate native-like from misfolded decoys than scoring functions based on more detailed molecular mechanics models. Recent benchmark tests on small data sets, however, suggest otherwise. In this work, we report the results of extensive decoy detection tests using an effective free energy function based on the OPLS all-atom (OPLS-AA) force field and the Surface Generalized Born (SGB) model for the solvent electrostatic effects. The OPLS-AA/SGB effective free energy is used as a scoring function to detect native protein folds among a total of 48,832 decoys for 32 different proteins from Park and Levitt's 4-state-reduced, Levitt's local-minima, Baker's ROSETTA all-atom, and Skolnick's decoy sets. Solvent electrostatic effects are included through the Surface Generalized Born (SGB) model. All structures are locally minimized without restraints. From an analysis of the individual energy components of the OPLS-AA/SGB energy function for the native and the best-ranked decoy, it is determined that a balance of the terms of the potential is responsible for the minimized energies that most successfully distinguish the native from the misfolded conformations. Different combinations of individual energy terms provide less discrimination than the total energy. The results are consistent with observations that all-atom molecular potentials coupled with intermediate level solvent dielectric models are competitive with knowledge-based potentials for decoy detection and protein modeling problems such as fold recognition and homology modeling.     

Ab initio construction of polypeptide fragments: Accuracy of loop decoy discrimination by an all-atom statistical potential and the AMBER force field with the Generalized Born solvation model

The accuracy of model selection from decoy ensembles of protein loop conformations was explored by comparing the performance of the Samudrala-Moult all-atom statistical potential (RAPDF) and the AMBER molecular mechanics force field, including the Generalized Born/surface area solvation model. Large ensembles of consistent loop conformations, represented at atomic detail with idealized geometry, were generated for a large test set of protein loops of 2 to 12 residues long by a novel ab initio method called RAPPER that relies on fine-grained residue-specific phi/psi propensity tables for conformational sampling. Ranking the conformers on the basis of RAPDF scores resulted in selected conformers that had an average global, non-superimposed RMSD for all heavy mainchain atoms ranging from 1.2 A for 4-mers to 2.9 A for 8-mers to 6.2 A for 12-mers. After filtering on the basis of anchor geometry and RAPDF scores, ranking by energy minimization of the AMBER/GBSA potential energy function selected conformers that had global RMSD values of 0.5 A for 4-mers, 2.3 A for 8-mers, and 5.0 A for 12-mers. Minimized fragments had, on average, consistently lower RMSD values (by 0.1 A) than their initial conformations. The importance of the Generalized Born solvation energy term is reflected by the observation that the average RMSD accuracy for all loop lengths was worse when this term is omitted. There are, however, still many cases where the AMBER gas-phase minimization selected conformers of lower RMSD than the AMBER/GBSA minimization. The AMBER/GBSA energy function had better correlation with RMSD to native than the RAPDF. When the ensembles were supplemented with conformations extracted from experimental structures, a dramatic improvement in selection accuracy was observed at longer lengths (average RMSD of 1.3 A for 8-mers) when scoring with the AMBER/GBSA force field. This work provides the basis for a promising hybrid approach of ab initio and knowledge-based methods for loop modeling.     

Minimalist representations and the importance of nearest neighbor effects in protein folding simulations

In order to investigate the level of representation required to simulate folding and predict structure, we test the ability of a variety of reduced representations to identify native states in decoy libraries and to recover the native structure given the advanced knowledge of the very broad native Ramachandran basin assignments. Simplifications include the removal of the entire side-chain or the retention of only the Cbeta atoms. Scoring functions are derived from an all-atom statistical potential that distinguishes between atoms and different residue types. Structures are obtained by minimizing the scoring function with a computationally rapid simulated annealing algorithm. Results are compared for simulations in which backbone conformations are sampled from a Protein Data Bank-based backbone rotamer library generated by either ignoring or including a dependence on the identity and conformation of the neighboring residues. Only when the Cbeta atoms and nearest neighbor effects are included do the lowest energy structures generally fall within 4 A of the native backbone root-mean square deviation (RMSD), despite the initial configuration being highly expanded with an average RMSD > or = 10 A. The side-chains are reinserted into the Cbeta models with minimal steric clash. Therefore, the detailed, all-atom information lost in descending to a Cbeta-level representation is recaptured to a large measure using backbone dihedral angle sampling that includes nearest neighbor effects and an appropriate scoring function.     

Comparative protein structure modeling by iterative alignment,model building and model assessment

Comparative or homology protein structure modeling is severely limited by errors in the alignment of a modeled sequence with related proteins of known three-dimensional structure. To ameliorate this problem, we have developed an automated method that optimizes both the alignment and the model implied by it. This task is achieved by a genetic algorithm protocol that starts with a set of initial alignments and then iterates through re-alignment, model building and model assessment to optimize a model assessment score. During this iterative process: (i) new alignments are constructed by application of a number of operators, such as alignment mutations and cross-overs; (ii) comparative models corresponding to these alignments are built by satisfaction of spatial restraints, as implemented in our program MODELLER; (iii) the models are assessed by a variety of criteria, partly depending on an atomic statistical potential. When testing the procedure on a very difficult set of 19 modeling targets sharing only 4–27% sequence identity with their template structures, the average final alignment accuracy increased from 37 to 45% relative to the initial alignment (the alignment accuracy was measured as the percentage of positions in the tested alignment that were identical to the reference structure-based alignment). Correspondingly, the average model accuracy increased from 43 to 54% (the model accuracy was measured as the percentage of the Cα atoms of the model that were within 5 Å of the corresponding Cα atoms in the superposed native structure). The present method also compares favorably with two of the most successful previously described methods, PSI-BLAST and SAM. The accuracy of the final models would be increased further if a better method for ranking of the models were available.     

Accuracy of functional surfaces on comparatively modeled protein structures

Identification and characterization of protein functional surfaces are important for predicting protein function, understanding enzyme mechanism, and docking small compounds to proteins. As the rapid speed of accumulation of protein sequence information far exceeds that of structures, constructing accurate models of protein functional surfaces and identify their key elements become increasingly important. A promising approach is to build comparative models from sequences using known structural templates such as those obtained from structural genome projects. Here we assess how well this approach works in modeling binding surfaces. By systematically building three-dimensional comparative models of proteins using Modeller, we determine how well functional surfaces can be accurately reproduced. We use an alpha shape based pocket algorithm to compute all pockets on the modeled structures, and conduct a large-scale computation of similarity measurements (pocket RMSD and fraction of functional atoms captured) for 26,590 modeled enzyme protein structures. Overall, we find that when the sequence fragment of the binding surfaces has more than 45% identity to that of the template protein, the modeled surfaces have on average an RMSD of 0.5 Å, and contain 48% or more of the binding surface atoms, with nearly all of the important atoms in the signatures of binding pockets captured.     

Refinement of protein structures by iterative comparative modeling and CryoEM density fitting

We developed a method for structure characterization of assembly components by iterative comparative protein structure modeling and fitting into cryo-electron microscopy (cryoEM) density maps. Specifically, we calculate a comparative model of a given component by considering many alternative alignments between the target sequence and a related template structure while optimizing the fit of a model into the corresponding density map. The method relies on the previously developed Moulder protocol that iterates over alignment, model building, and model assessment. The protocol was benchmarked using 20 varied target-template pairs of known structures with less than 30% sequence identity and corresponding simulated density maps at resolutions from 5A to 25A. Relative to the models based on the best existing sequence profile alignment methods, the percentage of C(alpha) atoms that are within 5A of the corresponding C(alpha) atoms in the superposed native structure increases on average from 52% to 66%, which is half-way between the starting models and the models from the best possible alignments (82%). The test also reveals that despite the improvements in the accuracy of the fitness function, this function is still the bottleneck in reducing the remaining errors. To demonstrate the usefulness of the protocol, we applied it to the upper domain of the P8 capsid protein of rice dwarf virus that has been studied by cryoEM at 6.8A. The C(alpha) root-mean-square deviation of the model based on the remotely related template, bluetongue virus VP7, improved from 8.7A to 6.0A, while the best possible model has a C(alpha) RMSD value of 5.3A. Moreover, the resulting model fits better into the cryoEM density map than the initial template structure. The method is being implemented in our program MODELLER for protein structure modeling by satisfaction of spatial restraints and will be applicable to the rapidly increasing number of cryoEM density maps of macromolecular assemblies.     

Protein loop selection using orientation-dependent force fields derived by parameter optimization

We used the orientation‐dependent Optimized Side Chain Atomic eneRgy (OSCAR‐o), derived in an early study, for protein loop selection. The prediction accuracy of OSCAR‐o was better than that of physics‐based force fields or statistical potential energy functions for both the RAPPER decoy set and the Jacobson decoy set. The native conformer was frequently ranked as lowest energy among the decoys. Furthermore, strong correlation was observed between the OSCAR‐o score and the root mean square deviation (RMSD) from the native structure for energy‐minimized decoys. In practical use, we applied OSCAR‐o to rescore decoys generated by a widely used loop‐modeling program, LOOPY. As a result, the mean RMSD values of top‐ranked decoys were reduced by 0.3 Å for loop targets of seven to nine residues. We expect similar performance for OSCAR‐o with other loop‐modeling algorithms in the context of decoy rescoring. A loop selection program (OSCAR‐ls) based on OSCAR‐o is available at http://sysimm.ifrec.osaka‐u.ac.jp/OSCAR/ . Proteins 2011; © 2011 Wiley‐Liss, Inc.     

Ab initio protein modeling into CryoEM density maps using EM-Fold

EM-Fold was used to build models for nine proteins in the maps of GroEL (7.7 ? resolution) and ribosome (6.4 ? resolution) in the ab initio modeling category of the 2010 cryo-electron microscopy modeling challenge. EM-Fold assembles predicted secondary structure elements (SSEs) into regions of the density map that were identified to correspond to either α-helices or β-strands. The assembly uses a Monte Carlo algorithm where loop closure, density-SSE length agreement, and strength of connecting density between SSEs are evaluated. Top-scoring models are refined by translating, rotating, and bending SSEs to yield better agreement with the density map. EM-Fold produces models that contain backbone atoms within SSEs only. The RMSD values of the models with respect to native range from 2.4 to 3.5 ? for six of the nine proteins. EM-Fold failed to predict the correct topology in three cases. Subsequently, Rosetta was used to build loops and side chains for the very best scoring models after EM-Fold refinement. The refinement within Rosetta's force field is driven by a density agreement score that calculates a cross-correlation between a density map simulated from the model and the experimental density map. All-atom RMSDs as low as 3.4 ? are achieved in favorable cases. Values above 10.0 ? are observed for two proteins with low overall content of secondary structure and hence particularly complex loop modeling problems. RMSDs over residues in secondary structure elements range from 2.5 to 4.8 ?.     

MINRMS: an efficient algorithm for determining protein structure similarity using root-mean-squared-distance

MOTIVATION: Existing algorithms for automated protein structure alignment generate contradictory results and are difficult to interpret. An algorithm which can provide a context for interpreting the alignment and uses a simple method to characterize protein structure similarity is needed. RESULTS: We describe a heuristic for limiting the search space for structure alignment comparisons between two proteins, and an algorithm for finding minimal root-mean-squared-distance (RMSD) alignments as a function of the number of matching residue pairs within this limited search space. Our alignment algorithm uses coordinates of alpha-carbon atoms to represent each amino acid residue and requires a total computation time of O(m(3) n(2)), where m and n denote the lengths of the protein sequences. This makes our method fast enough for comparisons of moderate-size proteins (fewer than approximately 800 residues) on current workstation-class computers and therefore addresses the need for a systematic analysis of multiple plausible shape similarities between two proteins using a widely accepted comparison metric.     

Distance-dependent atomic knowledge-based force in protein fold recognition

We have recently introduced a novel model for discriminating the correctly folded proteins from well-designed decoy structures using mechanical interatomic forces. In the model, we considered a protein as a collection of springs and the force imposed to each atom was calculated by using the relation between the potential energy and the force. A mean force potential function is obtained from statistical contact preferences within the known protein structures. In this article, the interatomic forces are calculated by numerical derivation of the potential function. For assessing the knowledge-based force function we consider an optimal structure and define a score function on the 3D structure of a protein. We compare the force imposed to each atom of a protein with the corresponding atom in the optimum structure. Afterwards we assign larger scores to those atoms with the lower forces. The total score is the sum of partial scores of atoms. The optimal structure is assumed to be the one with the highest score in the dataset. Finally, several decoy sets are applied in order to evaluate the performance of our model.