Information and discrimination in pairwise contact potentials

We examine the information-theoretic characteristics of statistical potentials that describe pairwise long-range contacts between amino acid residues in proteins. In our work, we seek to map out an efficient information-based strategy to detect and optimally utilize the structural information latent in empirical data, to make contact potentials, and other statistically derived folding potentials, more effective tools in protein structure prediction. Foremost, we establish fundamental connections between basic information-theoretic quantities (including the ubiquitous Z-score) and contact "energies" or scores used routinely in protein structure prediction, and demonstrate that the informatic quantity that mediates fold discrimination is the total divergence. We find that pairwise contacts between residues bear a moderate amount of fold information, and if optimized, can assist in the discrimination of native conformations from large ensembles of native-like decoys. Using an extensive battery of threading tests, we demonstrate that parameters that affect the information content of contact potentials (e.g., choice of atoms to define residue location and the cut-off distance between pairs) have a significant influence in their performance in fold recognition. We conclude that potentials that have been optimized for mutual information and that have high number of score events per sequence-structure alignment are superior in identifying the correct fold. We derive the quantity "information product" that embodies these two critical factors. We demonstrate that the information product, which does not require explicit threading to compute, is as effective as the Z-score, which requires expensive decoy threading to evaluate. This new objective function may be able to speed up the multidimensional parameter search for better statistical potentials. Lastly, by demonstrating the functional equivalence of quasi-chemically approximated "energies" to fundamental informatic quantities, we make statistical potentials less dependent on theoretically tenuous biophysical formalisms and more amenable to direct bioinformatic optimization.     

What should the Z-score of native protein structures be?

The Z-score of a protein is defined as the energy separation between the native fold and the average of an ensemble of misfolds in the units of the standard deviation of the ensemble. The Z-score is often used as a way of testing the knowledge-based potentials for their ability to recognize the native fold from other alternatives. However, it is not known what range of values the Z-scores should have if one had a correct potential. Here, we offer an estimate of Z-scores extracted from calorimetric measurements of proteins. The energies obtained from these experimental data are compared with those from computer simulations of a lattice model protein. It is suggested that the Z-scores calculated from different knowledge-based potentials are generally too small in comparison with the experimental values.     

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.     

Can correct protein models be identified?

The ability to separate correct models of protein structures from less correct models is of the greatest importance for protein structure prediction methods. Several studies have examined the ability of different types of energy function to detect the native, or native-like, protein structure from a large set of decoys. In contrast to earlier studies, we examine here the ability to detect models that only show limited structural similarity to the native structure. These correct models are defined by the existence of a fragment that shows significant similarity between this model and the native structure. It has been shown that the existence of such fragments is useful for comparing the performance between different fold recognition methods and that this performance correlates well with performance in fold recognition. We have developed ProQ, a neural-network-based method to predict the quality of a protein model that extracts structural features, such as frequency of atom-atom contacts, and predicts the quality of a model, as measured either by LGscore or MaxSub. We show that ProQ performs at least as well as other measures when identifying the native structure and is better at the detection of correct models. This performance is maintained over several different test sets. ProQ can also be combined with the Pcons fold recognition predictor (Pmodeller) to increase its performance, with the main advantage being the elimination of a few high-scoring incorrect models. Pmodeller was successful in CASP5 and results from the latest LiveBench, LiveBench-6, indicating that Pmodeller has a higher specificity than Pcons alone.     

Accuracy of sequence alignment and fold assessment using reduced amino acid alphabets

Reduced or simplified amino acid alphabets group the 20 naturally occurring amino acids into a smaller number of representative protein residues. To date, several reduced amino acid alphabets have been proposed, which have been derived and optimized by a variety of methods. The resulting reduced amino acid alphabets have been applied to pattern recognition, generation of consensus sequences from multiple alignments, protein folding, and protein structure prediction. In this work, amino acid substitution matrices and statistical potentials were derived based on several reduced amino acid alphabets and their performance assessed in a large benchmark for the tasks of sequence alignment and fold assessment of protein structure models, using as a reference frame the standard alphabet of 20 amino acids. The results showed that a large reduction in the total number of residue types does not necessarily translate into a significant loss of discriminative power for sequence alignment and fold assessment. Therefore, some definitions of a few residue types are able to encode most of the relevant sequence/structure information that is present in the 20 standard amino acids. Based on these results, we suggest that the use of reduced amino acid alphabets may allow to increasing the accuracy of current substitution matrices and statistical potentials for the prediction of protein structure of remote homologs.     

A pairwise residue contact area-based mean force potential for discrimination of native protein structure

Background  

Considering energy function to detect a correct protein fold from incorrect ones is very important for protein structure prediction and protein folding. Knowledge-based mean force potentials are certainly the most popular type of interaction function for protein threading. They are derived from statistical analyses of interacting groups in experimentally determined protein structures. These potentials are developed at the atom or the amino acid level. Based on orientation dependent contact area, a new type of knowledge-based mean force potential has been developed.     

How to generate improved potentials for protein tertiary structure prediction: a lattice model study

Success in the protein structure prediction problem relies heavily on the choice of an appropriate potential function. One approach toward extracting these potentials from a database of known protein structures is to maximize the Z-score of the database proteins, which represents the ability of the potential to discriminate correct from random conformations. These optimization methods model the entire distribution of alternative structures, reducing their ability to concentrate on the lowest energy structures most competitive with the native state and resulting in an unfortunate tendency to underestimate the repulsive interactions. This leads to reduced accuracy and predictive ability. Using a lattice model, we demonstrate how we can weight the distribution to suppress the contributions of the high-energy conformations to the Z-score calculation. The result is a potential that is more accurate and more likely to yield correct predictions than other Z-score optimization methods as well as potentials of mean force.     

Z-score biological significance of binding hot spots of protein interfaces by using crystal packing as the reference state

Characterization of binding hot spots of protein interfaces is a fundamental study in molecular biology. Many computational methods have been proposed to identify binding hot spots. However, there are few studies to assess the biological significance of binding hot spots. We introduce the notion of biological significance of a contact residue for capturing the probability of the residue occurring in or contributing to protein binding interfaces. We take a statistical Z-score approach to the assessment of the biological significance. The method has three main steps. First, the potential score of a residue is defined by using a knowledge-based potential function with relative accessible surface area calculations. A null distribution of this potential score is then generated from artifact crystal packing contacts. Finally, the Z-score significance of a contact residue with a specific potential score is determined according to this null distribution. We hypothesize that residues at binding hot spots have big absolute values of Z-score as they contribute greatly to binding free energy. Thus, we propose to use Z-score to predict whether a contact residue is a hot spot residue. Comparison with previously reported methods on two benchmark datasets shows that this Z-score method is mostly superior to earlier methods. This article is part of a Special Issue entitled: Computational Methods for Protein Interaction and Structural Prediction.     

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.     

Discriminative ability with respect to amino acid types: assessing the performance of knowledge-based potentials without threading

We present a novel method designed to analyze the discriminative ability of knowledge-based potentials with respect to the 20 residue types. The method is based on the preference of amino acids for specific types of protein environment, and uses a virtual mutagenesis experiment to estimate how much information a given potential can provide about environments of each amino acid type. This allows one to test and optimize the performance of real potentials at the level of individual amino acids, using actual data on residue environments from a dataset of known protein structures. We have applied our method to long-range and medium-range pairwise distance-dependent potentials. The results of our study indicate that these potentials are only able to discriminate between a very limited number of residue types, and that discriminative ability is extremely sensitive to the choice of parameters used to construct the potentials, and even to the size of the training dataset. We also show that different types of pairwise distance potentials are dominated by different types of interactions. These dominant interactions strongly depend on the type of approximation used to define residue position. For each potential, our methodology is able to identify a potential-specific amino acid distance matrix and a reduced amino acid alphabet of any specified size, which may have implications for sequence alignment and multibody models.     

Fold recognition with minimal gaps

Here we present a simplified form of threading that uses only a 20 x 20 two-body residue-based potential and restricted number of gaps. Despite its simplicity and transparency the Monte Carlo-based threading algorithm performs very well in a rigorous test of fold recognition. The results suggest that by simplifying and constraining the decoy space, one can achieve better fold recognition. Fold recognition results are compared with and supplemented by a PSI-BLAST search. The statistical significance of threading results is rigorously evaluated from statistics of extremes by comparison with optimal alignments of a large set of randomly shuffled sequences. The statistical theory, based on the Random Energy Model, yields a cumulative statistical parameter, epsilon, that attests to the likelihood of correct fold recognition. A large epsilon indicates a significant energy gap between the optimal alignment and decoy alignments and, consequently, a high probability that the fold is correctly recognized. For a particular number of gaps, the epsilon parameter reaches its maximal value, and the fold is recognized. As the number of gaps further increases, the likelihood of correct fold recognition drops off. This is because the decoy space is small when gaps are restricted to a small number, but the native alignment is still well approximated, whereas unrestricted increase of the number of gaps leads to rapid growth of the number of decoys and their statistical dominance over the correct alignment. It is shown that best results are obtained when a combination of one-, two-, and three-gap threading is used. To this end, use of the epsilon parameter is crucial for rigorous comparison of results across the different decoy spaces belonging to a different number of gaps.     

Using the Unfolded State as the Reference State Improves the Performance of Statistical Potentials

Distance-dependent statistical potentials are an important class of energy functions extensively used in modeling protein structures and energetics. These potentials are obtained by statistically analyzing the proximity of atoms in all combinatorial amino-acid pairs in proteins with known structures. In model evaluation, the statistical potential is usually subtracted by the value of a reference state for better selectivity. An ideal reference state should include the general chemical properties of polypeptide chains so that only the unique factors stabilizing the native structures are retained after calibrating on reference state. However, reference states available as of this writing rarely model specific chemical constraints of peptide bonds and therefore poorly reflect the behavior of polypeptide chains. In this work, we proposed a statistical potential based on unfolded state ensemble (SPOUSE), where the reference state is summarized from the unfolded state ensembles of proteins produced according to the statistical coil model. Due to its better representation of the features of polypeptides, SPOUSE outperforms three of the most widely used distance-dependent potentials not only in native conformation identification, but also in the selection of close-to-native models and correlation coefficients between energy and model error. Furthermore, SPOUSE shows promising possibility of further improvement by integration with the orientation-dependent side-chain potentials.     

Distance-scaled,finite ideal-gas reference state improves structure-derived potentials of mean force for structure selection and stability prediction

The distance-dependent structure-derived potentials developed so far all employed a reference state that can be characterized as a residue (atom)-averaged state. Here, we establish a new reference state called the distance-scaled, finite ideal-gas reference (DFIRE) state. The reference state is used to construct a residue-specific all-atom potential of mean force from a database of 1011 nonhomologous (less than 30% homology) protein structures with resolution less than 2 A. The new all-atom potential recognizes more native proteins from 32 multiple decoy sets, and raises an average Z-score by 1.4 units more than two previously developed, residue-specific, all-atom knowledge-based potentials. When only backbone and C(beta) atoms are used in scoring, the performance of the DFIRE-based potential, although is worse than that of the all-atom version, is comparable to those of the previously developed potentials on the all-atom level. In addition, the DFIRE-based all-atom potential provides the most accurate prediction of the stabilities of 895 mutants among three knowledge-based all-atom potentials. Comparison with several physical-based potentials is made.     

The victor/FRST function for model quality estimation.

Scoring functions are widely used in the final step of model selection in protein structure prediction. This is of interest both for comparative modeling targets, where it is important to select the best model among a set of many good, "correct" ones, as well as for other (fold recognition or novel fold) targets, where the set may contain many incorrect models. A novel combination of four knowledge-based potentials recognizing different features of native protein structures is introduced and tested. The pairwise, solvation, hydrogen bond, and torsion angle potentials contain largely orthogonal information. Of these, the torsion angle potential is found to show the strongest correlation with model quality. Combining these features with a linear weighting function, it was possible to construct a robust energy function capable of discriminating native-like structures on several benchmarking sets. In a recent blind test (CAFASP-4 MQAP), the scoring function ranked consistently well and was able to reliably distinguish the correct template from an ensemble of high quality decoys in 52 of 70 cases (33 of 34 for comparative modeling). An executable version of the Victor/FRST function for Linux PCs is available for download from the URL http://protein.cribi.unipd.it/frst/.     

A free-rotating and self-avoiding chain model for deriving statistical potentials based on protein structures

Statistical potentials have been widely used in protein studies despite the much-debated theoretical basis. In this work, we have applied two physical reference states for deriving the statistical potentials based on protein structure features to achieve zero interaction and orthogonalization. The free-rotating chain-based potential applies a local free-rotating chain reference state, which could theoretically be described by the Gaussian distribution. The self-avoiding chain-based potential applies a reference state derived from a database of artificial self-avoiding backbones generated by Monte Carlo simulation. These physical reference states are independent of known protein structures and are based solely on the analytical formulation or simulation method. The new potentials performed better and yielded higher Z-scores and success rates compared to other statistical potentials. The end-to-end distance distribution produced by the self-avoiding chain model was similar to the distance distribution of protein atoms in structure database. This fact may partly explain the basis of the reference states that depend on the atom pair frequency observed in the protein database. The current study showed that a more physical reference model improved the performance of statistical potentials in protein fold recognition, which could also be extended to other types of applications.