A probabilistic fragment-based protein structure prediction algorithm

Conformational sampling is one of the bottlenecks in fragment-based protein structure prediction approaches. They generally start with a coarse-grained optimization where mainchain atoms and centroids of side chains are considered, followed by a fine-grained optimization with an all-atom representation of proteins. It is during this coarse-grained phase that fragment-based methods sample intensely the conformational space. If the native-like region is sampled more, the accuracy of the final all-atom predictions may be improved accordingly. In this work we present EdaFold, a new method for fragment-based protein structure prediction based on an Estimation of Distribution Algorithm. Fragment-based approaches build protein models by assembling short fragments from known protein structures. Whereas the probability mass functions over the fragment libraries are uniform in the usual case, we propose an algorithm that learns from previously generated decoys and steers the search toward native-like regions. A comparison with Rosetta AbInitio protocol shows that EdaFold is able to generate models with lower energies and to enhance the percentage of near-native coarse-grained decoys on a benchmark of [Formula: see text] proteins. The best coarse-grained models produced by both methods were refined into all-atom models and used in molecular replacement. All atom decoys produced out of EdaFold's decoy set reach high enough accuracy to solve the crystallographic phase problem by molecular replacement for some test proteins. EdaFold showed a higher success rate in molecular replacement when compared to Rosetta. Our study suggests that improving low resolution coarse-grained decoys allows computational methods to avoid subsequent sampling issues during all-atom refinement and to produce better all-atom models. EdaFold can be downloaded from http://www.riken.jp/zhangiru/software/.     

Enhanced sampling near the native conformation using statistical potentials for local side-chain and backbone interactions

In the preceding article in this issue of Proteins, an empirical energy function consisting of 4 statistical potentials that quantify local side-chain-backbone and side-chain-side-chain interactions has been demonstrated to successfully identify the native conformations of short sequence fragments and the native structure within large sets of high-quality decoys. Because this energy function consists entirely of interactions between residues separated by fewer than 5 positions, it can be used at the earliest stage of ab initio structure prediction to enhance the efficiency of conformational search. In this article, protein fragments are generated de novo by recombining very short segments of protein structures (2, 4, or 6 residues), either selected at random or optimized with respect this local energy function. When local energy is optimized in selected fragments, more efficient sampling of conformational space near the native conformation is consistently observed for 450 randomly selected single turn fragments, with turn lengths varying from 3 to 12 residues and all 4 combinations of flanking secondary structure. These results further demonstrate the energetic significance of local interactions in protein conformations. When used in combination with longer range energy functions, application of these potentials should lead to more accurate prediction of protein structure.     

Improving fragment quality for de novo structure prediction

De novo structure prediction can be defined as a search in conformational space under the guidance of an energy function. The most successful de novo structure prediction methods, such as Rosetta, assemble the fragments from known structures to reduce the search space. Therefore, the fragment quality is an important factor in structure prediction. In our study, a method is proposed to generate a new set of fragments from the lowest energy de novo models. These fragments were subsequently used to predict the next‐round of models. In a benchmark of 30 proteins, the new set of fragments showed better performance when used to predict de novo structures. The lowest energy model predicted using our method was closer to native structure than Rosetta for 22 proteins. Following a similar trend, the best model among top five lowest energy models predicted using our method was closer to native structure than Rosetta for 20 proteins. In addition, our experiment showed that the C‐alpha root mean square deviation was improved from 5.99 to 5.03 Å on average compared to Rosetta when the lowest energy models were picked as the best predicted models. Proteins 2014; 82:2240–2252. © 2014 Wiley Periodicals, Inc.     

A coarse-grained protein force field for folding and structure prediction

We have revisited the protein coarse-grained optimized potential for efficient structure prediction (OPEP). The training and validation sets consist of 13 and 16 protein targets. Because optimization depends on details of how the ensemble of decoys is sampled, trial conformations are generated by molecular dynamics, threading, greedy, and Monte Carlo simulations, or taken from publicly available databases. The OPEP parameters are varied by a genetic algorithm using a scoring function which requires that the native structure has the lowest energy, and the native-like structures have energy higher than the native structure but lower than the remote conformations. Overall, we find that OPEP correctly identifies 24 native or native-like states for 29 targets and has very similar capability to the all-atom discrete optimized protein energy model (DOPE), found recently to outperform five currently used energy models.     

Fully differentiable coarse-grained and all-atom knowledge-based potentials for RNA structure evaluation

RNA molecules play integral roles in gene regulation, and understanding their structures gives us important insights into their biological functions. Despite recent developments in template-based and parameterized energy functions, the structure of RNA--in particular the nonhelical regions--is still difficult to predict. Knowledge-based potentials have proven efficient in protein structure prediction. In this work, we describe two differentiable knowledge-based potentials derived from a curated data set of RNA structures, with all-atom or coarse-grained representation, respectively. We focus on one aspect of the prediction problem: the identification of native-like RNA conformations from a set of near-native models. Using a variety of near-native RNA models generated from three independent methods, we show that our potential is able to distinguish the native structure and identify native-like conformations, even at the coarse-grained level. The all-atom version of our knowledge-based potential performs better and appears to be more effective at discriminating near-native RNA conformations than one of the most highly regarded parameterized potential. The fully differentiable form of our potentials will additionally likely be useful for structure refinement and/or molecular dynamics simulations.     

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.     

A hierarchical approach to all-atom protein loop prediction

The application of all-atom force fields (and explicit or implicit solvent models) to protein homology-modeling tasks such as side-chain and loop prediction remains challenging both because of the expense of the individual energy calculations and because of the difficulty of sampling the rugged all-atom energy surface. Here we address this challenge for the problem of loop prediction through the development of numerous new algorithms, with an emphasis on multiscale and hierarchical techniques. As a first step in evaluating the performance of our loop prediction algorithm, we have applied it to the problem of reconstructing loops in native structures; we also explicitly include crystal packing to provide a fair comparison with crystal structures. In brief, large numbers of loops are generated by using a dihedral angle-based buildup procedure followed by iterative cycles of clustering, side-chain optimization, and complete energy minimization of selected loop structures. We evaluate this method by using the largest test set yet used for validation of a loop prediction method, with a total of 833 loops ranging from 4 to 12 residues in length. Average/median backbone root-mean-square deviations (RMSDs) to the native structures (superimposing the body of the protein, not the loop itself) are 0.42/0.24 A for 5 residue loops, 1.00/0.44 A for 8 residue loops, and 2.47/1.83 A for 11 residue loops. Median RMSDs are substantially lower than the averages because of a small number of outliers; the causes of these failures are examined in some detail, and many can be attributed to errors in assignment of protonation states of titratable residues, omission of ligands from the simulation, and, in a few cases, probable errors in the experimentally determined structures. When these obvious problems in the data sets are filtered out, average RMSDs to the native structures improve to 0.43 A for 5 residue loops, 0.84 A for 8 residue loops, and 1.63 A for 11 residue loops. In the vast majority of cases, the method locates energy minima that are lower than or equal to that of the minimized native loop, thus indicating that sampling rarely limits prediction accuracy. The overall results are, to our knowledge, the best reported to date, and we attribute this success to the combination of an accurate all-atom energy function, efficient methods for loop buildup and side-chain optimization, and, especially for the longer loops, the hierarchical refinement protocol.     

Ab initio protein structure assembly using continuous structure fragments and optimized knowledge-based force field

Ab initio protein folding is one of the major unsolved problems in computational biology owing to the difficulties in force field design and conformational search. We developed a novel program, QUARK, for template-free protein structure prediction. Query sequences are first broken into fragments of 1-20 residues where multiple fragment structures are retrieved at each position from unrelated experimental structures. Full-length structure models are then assembled from fragments using replica-exchange Monte Carlo simulations, which are guided by a composite knowledge-based force field. A number of novel energy terms and Monte Carlo movements are introduced and the particular contributions to enhancing the efficiency of both force field and search engine are analyzed in detail. QUARK prediction procedure is depicted and tested on the structure modeling of 145 nonhomologous proteins. Although no global templates are used and all fragments from experimental structures with template modeling score >0.5 are excluded, QUARK can successfully construct 3D models of correct folds in one-third cases of short proteins up to 100 residues. In the ninth community-wide Critical Assessment of protein Structure Prediction experiment, QUARK server outperformed the second and third best servers by 18 and 47% based on the cumulative Z-score of global distance test-total scores in the FM category. Although ab initio protein folding remains a significant challenge, these data demonstrate new progress toward the solution of the most important problem in the field.     

ProVal: a protein-scoring function for the selection of native and near-native folds

A low-resolution scoring function for the selection of native and near-native structures from a set of predicted structures for a given protein sequence has been developed. The scoring function, ProVal (Protein Validate), used several variables that describe an aspect of protein structure for which the proximity to the native structure can be assessed quantitatively. Among the parameters included are a packing estimate, surface areas, and the contact order. A partial least squares for latent variables (PLS) model was built for each candidate set of the 28 decoy sets of structures generated for 22 different proteins using the described parameters as independent variables. The C(alpha) RMS of the candidate structures versus the experimental structure was used as the dependent variable. The final generalized scoring function was an average of all models derived, ensuring that the function was not optimized for specific fold classes or method of structure generation of the candidate folds. The results show that the crystal structure was scored best in 64% of the 28 test sets and was clearly separated from the decoys in many examples. In all the other cases in which the crystal structure did not rank first, it ranked within the top 10%. Thus, although ProVal could not distinguish between predicted structures that were similar overall in fold quality due to its inherently low resolution, it can clearly be used as a primary filter to eliminate approximately 90% of fold candidates generated by current prediction methods from all-atom modeling and further evaluation. The correlation between the predicted and actual C(alpha) RMS values varies considerably between the candidate fold sets.     

Comparison of protein models minimized by the all-atom and united-atom models in the AMBER force field: correlation of RMS deviation with the crystallographic R factor and size.

We performed energy minimization of 25 protein structures, which vary significantly in their size, secondary structural content and crystallographic R factor, in the AMBER force field. We used an unconstrained path and the conjugate gradients algorithm. To determine the reliability of the united-atom approximation, we minimized all the proteins using both the all-atom and united-atom models. The RMS deviations of the minimized structures were plotted as a function of the crystallographic R factors of the initial structures. For the all-atom models, we found a strong linear relationship between the RMS deviations and the R factors (correlation coefficient of 0.78). The RMS deviations of protein structures minimized using united-atom models showed a wider range of distribution and had a correlation coefficient with the R factors of only 0.52. The RMS deviations decrease with an increase in the size of the protein, probably due to the decreased ratio of surface area to volume with increasing size of the protein. The surface atoms and residues showed higher RMS deviations than those in the interior of the protein. Even in these plots the united-atom models show a wide range of distribution of data points. From these results, we recommend the use of all-atom models for energy minimization of proteins in the AMBER force field.     

The dual role of fragments in fragment‐assembly methods for de novo protein structure prediction

In fragment‐assembly techniques for protein structure prediction, models of protein structure are assembled from fragments of known protein structures. This process is typically guided by a knowledge‐based energy function and uses a heuristic optimization method. The fragments play two important roles in this process: they define the set of structural parameters available, and they also assume the role of the main variation operators that are used by the optimiser. Previous analysis has typically focused on the first of these roles. In particular, the relationship between local amino acid sequence and local protein structure has been studied by a range of authors. The correlation between the two has been shown to vary with the window length considered, and the results of these analyses have informed directly the choice of fragment length in state‐of‐the‐art prediction techniques. Here, we focus on the second role of fragments and aim to determine the effect of fragment length from an optimization perspective. We use theoretical analyses to reveal how the size and structure of the search space changes as a function of insertion length. Furthermore, empirical analyses are used to explore additional ways in which the size of the fragment insertion influences the search both in a simulation model and for the fragment‐assembly technique, Rosetta. Proteins 2012. © 2011 Wiley Periodicals, Inc.     

Probabilistic sampling of protein conformations: New hope for brute force?

Protein structure prediction from sequence alone by "brute force" random methods is a computationally expensive problem. Estimates have suggested that it could take all the computers in the world longer than the age of the universe to compute the structure of a single 200-residue protein. Here we investigate the use of a faster version of our FOLDTRAJ probabilistic all-atom protein-structure-sampling algorithm. We have improved the method so that it is now over twenty times faster than originally reported, and capable of rapidly sampling conformational space without lattices. It uses geometrical constraints and a Leonard-Jones type potential for self-avoidance. We have also implemented a novel method to add secondary structure-prediction information to make protein-like amounts of secondary structure in sampled structures. In a set of 100,000 probabilistic conformers of 1VII, 1ENH, and 1PMC generated, the structures with smallest Calpha RMSD from native are 3.95, 5.12, and 5.95A, respectively. Expanding this test to a set of 17 distinct protein folds, we find that all-helical structures are "hit" by brute force more frequently than beta or mixed structures. For small helical proteins or very small non-helical ones, this approach should have a "hit" close enough to detect with a good scoring function in a pool of several million conformers. By fitting the distribution of RMSDs from the native state of each of the 17 sets of conformers to the extreme value distribution, we are able to estimate the size of conformational space for each. With a 0.5A RMSD cutoff, the number of conformers is roughly 2N where N is the number of residues in the protein. This is smaller than previous estimates, indicating an average of only two possible conformations per residue when sterics are accounted for. Our method reduces the effective number of conformations available at each residue by probabilistic bias, without requiring any particular discretization of residue conformational space, and is the fastest method of its kind. With computer speeds doubling every 18 months and parallel and distributed computing becoming more practical, the brute force approach to protein structure prediction may yet have some hope in the near future.     

Comparison of Protein Models Minimized by the All-Atom and United-Atom Models in the AMBER Force Field: Correlation of RMS deviation with the Crystallographic R Factor and Size

Abstract

We performed energy minimization of 25 protein structures, which vary significantly in their size, secondary structural content and crystallographic R factor, in the AMBER force field. We used an unconstrained path and the conjugate gradients algorithm. To determine the reliability of the united-atom approximation, we minimized all the proteins using both the all-atom and united-atom models. The RMS deviations of the minimized structures were plotted as a function of the crystallographic R factors of the initial structures. For the all-atom models, we found a strong linear relationship between the RMS deviations and the R factors (correlation coefficient of 0.78). The RMS deviations of protein structures minimized using united-atom models showed a wider range of distribution and had a correlation coefficient with the R factors of only 0.52. The RMS deviations decrease with an increase in the size of the protein, probably due to the decreased ratio of surface area to volume with increasing size of the protein. The surface atoms and residues showed higher RMS deviations than those in the interior of the protein. Even in these plots the united-atom models show a wide range of distribution of data points. From these results, we recommend the use of all-atom models for energy minimization of proteins in the AMBER force field.     

Evaluation of a sophisticated SCFG design for RNA secondary structure prediction

Predicting secondary structures of RNA molecules is one of the fundamental problems of and thus a challenging task in computational structural biology. Over the past decades, mainly two different approaches have been considered to compute predictions of RNA secondary structures from a single sequence: the first one relies on physics-based and the other on probabilistic RNA models. Particularly, the free energy minimization (MFE) approach is usually considered the most popular and successful method. Moreover, based on the paradigm-shifting work by McCaskill which proposes the computation of partition functions (PFs) and base pair probabilities based on thermodynamics, several extended partition function algorithms, statistical sampling methods and clustering techniques have been invented over the last years. However, the accuracy of the corresponding algorithms is limited by the quality of underlying physics-based models, which include a vast number of thermodynamic parameters and are still incomplete. The competing probabilistic approach is based on stochastic context-free grammars (SCFGs) or corresponding generalizations, like conditional log-linear models (CLLMs). These methods abstract from free energies and instead try to learn about the structural behavior of the molecules by learning (a manageable number of) probabilistic parameters from trusted RNA structure databases. In this work, we introduce and evaluate a sophisticated SCFG design that mirrors state-of-the-art physics-based RNA structure prediction procedures by distinguishing between all features of RNA that imply different energy rules. This SCFG actually serves as the foundation for a statistical sampling algorithm for RNA secondary structures of a single sequence that represents a probabilistic counterpart to the sampling extension of the PF approach. Furthermore, some new ways to derive meaningful structure predictions from generated sample sets are presented. They are used to compare the predictive accuracy of our model to that of other probabilistic and energy-based prediction methods. Particularly, comparisons to lightweight SCFGs and corresponding CLLMs for RNA structure prediction indicate that more complex SCFG designs might yield higher accuracy but eventually require more comprehensive and pure training sets. Investigations on both the accuracies of predicted foldings and the overall quality of generated sample sets (especially on an abstraction level, called abstract shapes of generated structures, that is relevant for biologists) yield the conclusion that the Boltzmann distribution of the PF sampling approach is more centered than the ensemble distribution induced by the sophisticated SCFG model, which implies a greater structural diversity within generated samples. In general, neither of the two distinct ensemble distributions is more adequate than the other and the corresponding results obtained by statistical sampling can be expected to bare fundamental differences, such that the method to be preferred for a particular input sequence strongly depends on the considered RNA type.     

Ab initio protein structure prediction

Steady progress has been made in the field of ab initio protein folding. A variety of methods now allow the prediction of low-resolution structures of small proteins or protein fragments up to approximately 100 amino acid residues in length. Such low-resolution structures may be sufficient for the functional annotation of protein sequences on a genome-wide scale. Although no consistently reliable algorithm is currently available, the essential challenges to developing a general theory or approach to protein structure prediction are better understood. The energy landscapes resulting from the structure prediction algorithms are only partially funneled to the native state of the protein. This review focuses on two areas of recent advances in ab initio structure prediction-improvements in the energy functions and strategies to search the caldera region of the energy landscapes.     

A Probabilistic Model of RNA Conformational Space

The increasing importance of non-coding RNA in biology and medicine has led to a growing interest in the problem of RNA 3-D structure prediction. As is the case for proteins, RNA 3-D structure prediction methods require two key ingredients: an accurate energy function and a conformational sampling procedure. Both are only partly solved problems. Here, we focus on the problem of conformational sampling. The current state of the art solution is based on fragment assembly methods, which construct plausible conformations by stringing together short fragments obtained from experimental structures. However, the discrete nature of the fragments necessitates the use of carefully tuned, unphysical energy functions, and their non-probabilistic nature impairs unbiased sampling. We offer a solution to the sampling problem that removes these important limitations: a probabilistic model of RNA structure that allows efficient sampling of RNA conformations in continuous space, and with associated probabilities. We show that the model captures several key features of RNA structure, such as its rotameric nature and the distribution of the helix lengths. Furthermore, the model readily generates native-like 3-D conformations for 9 out of 10 test structures, solely using coarse-grained base-pairing information. In conclusion, the method provides a theoretical and practical solution for a major bottleneck on the way to routine prediction and simulation of RNA structure and dynamics in atomic detail.     

Sampling and scoring: A marriage made in heaven

Most structure prediction algorithms consist of initial sampling of the conformational space, followed by rescoring and possibly refinement of a number of selected structures. Here we focus on protein docking, and show that while decoupling sampling and scoring facilitates method development, integration of the two steps can lead to substantial improvements in docking results. Since decoupling is usually achieved by generating a decoy set containing both non‐native and near‐native docked structures, which can be then used for scoring function construction, we first review the roles and potential pitfalls of decoys in protein–protein docking, and show that some type of decoys are better than others for method development. We then describe three case studies showing that complete decoupling of scoring from sampling is not the best choice for solving realistic docking problems. Although some of the examples are based on our own experience, the results of the CAPRI docking and scoring experiments also show that performing both sampling and scoring generally yields better results than scoring the structures generated by all predictors. Next we investigate how the selection of training and decoy sets affects the performance of the scoring functions obtained. Finally, we discuss pathways to better alignment of the two steps, and show some algorithms that achieve a certain level of integration. Although we focus on protein–protein docking, our observations most likely also apply to other conformational search problems, including protein structure prediction and the docking of small molecules to proteins.Proteins 2013; 81:1874–1884. © 2013 Wiley Periodicals, Inc.     

Atomic-Accuracy Prediction of Protein Loop Structures through an RNA-Inspired Ansatz

Consistently predicting biopolymer structure at atomic resolution from sequence alone remains a difficult problem, even for small sub-segments of large proteins. Such loop prediction challenges, which arise frequently in comparative modeling and protein design, can become intractable as loop lengths exceed 10 residues and if surrounding side-chain conformations are erased. Current approaches, such as the protein local optimization protocol or kinematic inversion closure (KIC) Monte Carlo, involve stages that coarse-grain proteins, simplifying modeling but precluding a systematic search of all-atom configurations. This article introduces an alternative modeling strategy based on a ‘stepwise ansatz’, recently developed for RNA modeling, which posits that any realistic all-atom molecular conformation can be built up by residue-by-residue stepwise enumeration. When harnessed to a dynamic-programming-like recursion in the Rosetta framework, the resulting stepwise assembly (SWA) protocol enables enumerative sampling of a 12 residue loop at a significant but achievable cost of thousands of CPU-hours. In a previously established benchmark, SWA recovers crystallographic conformations with sub-Angstrom accuracy for 19 of 20 loops, compared to 14 of 20 by KIC modeling with a comparable expenditure of computational power. Furthermore, SWA gives high accuracy results on an additional set of 15 loops highlighted in the biological literature for their irregularity or unusual length. Successes include cis-Pro touch turns, loops that pass through tunnels of other side-chains, and loops of lengths up to 24 residues. Remaining problem cases are traced to inaccuracies in the Rosetta all-atom energy function. In five additional blind tests, SWA achieves sub-Angstrom accuracy models, including the first such success in a protein/RNA binding interface, the YbxF/kink-turn interaction in the fourth ‘RNA-puzzle’ competition. These results establish all-atom enumeration as an unusually systematic approach to ab initio protein structure modeling that can leverage high performance computing and physically realistic energy functions to more consistently achieve atomic accuracy.