Effects of amino acid composition, finite size of proteins, and sparse statistics on distance-dependent statistical pair potentials

Statistical distance dependent pair potentials are frequently used in a variety of folding, threading, and modeling studies of proteins. The applicability of these types of potentials is tightly connected to the reliability of statistical observations. We explored the possible origin and extent of false positive signals in statistical potentials by analyzing their distance dependence in a variety of randomized protein-like models. While on average potentials derived from such models are expected to equal zero at any distance, we demonstrate that systematic and significant distortions exist. These distortions originate from the limited statistical counts in local environments of proteins and from the limited size of protein structures at large distances. We suggest that these systematic errors in statistical potentials are connected to the dependence of amino acid composition on protein size and to variation in protein sizes. Additionally, atom-based potentials are dominated by a false positive signal that is due to correlation among distances measured from atoms of one residue to atoms of another residue. The significance of residue-based pairwise potentials at various spatial pair separations was assessed in this study and it was found that as few as approximately 50% of potential values were statistically significant at distances below 4 A, and only at most approximately 80% of them were significant at larger pair separations. A new definition for reference state, free of the observed systematic errors, is suggested. It has been demonstrated to generate statistical potentials that compare favorably to other publicly available ones.     

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.     

Database-derived potentials dependent on protein size for in silico folding and design

Knowledge-based potentials are widely used in simulations of protein folding, structure prediction, and protein design. Their advantages include limited computational requirements and the ability to deal with low-resolution protein models compatible with long-scale simulations. Their drawbacks comprehend their dependence on specific features of the dataset from which they are derived, such as the size of the proteins it contains, and their physical meaning is still a subject of debate. We address these issues by probing the theoretical validity of these potentials as mean-force potentials that take the solvent implicitly into account and involve entropic contributions due to atomic degrees of freedom and solvation. The dependence on the size of the system is checked on distance-dependent amino acid pair potentials, derived from six protein structure sets containing proteins of increasing length N. For large inter-residue distances, they are found to display the theoretically predicted 1/N behavior weighted by a factor depending on the boundaries and the compressibility of the system. For short distances, different trends are observed according to the nature of the residue pairs and their ability to form, for example, electrostatic, cation-pi or pi-pi interactions, or hydrophobic packing. The results of this analysis are used to devise a novel protein size-dependent distance potential, which displays an improved performance in discriminating native sequence-structure matches among decoy models.     

Pairwise contact energy statistical potentials can help to find probability of point mutations

To adopt a particular fold, a protein requires several interactions between its amino acid residues. The energetic contribution of these residue–residue interactions can be approximated by extracting statistical potentials from known high resolution structures. Several methods based on statistical potentials extracted from unrelated proteins are found to make a better prediction of probability of point mutations. We postulate that the statistical potentials extracted from known structures of similar folds with varying sequence identity can be a powerful tool to examine probability of point mutation. By keeping this in mind, we have derived pairwise residue and atomic contact energy potentials for the different functional families that adopt the (α/β)₈ TIM‐Barrel fold. We carried out computational point mutations at various conserved residue positions in yeast Triose phosphate isomerase enzyme for which experimental results are already reported. We have also performed molecular dynamics simulations on a subset of point mutants to make a comparative study. The difference in pairwise residue and atomic contact energy of wildtype and various point mutations reveals probability of mutations at a particular position. Interestingly, we found that our computational prediction agrees with the experimental studies of Silverman et al. (Proc Natl Acad Sci 2001;98:3092–3097) and perform better prediction than i_Mutant and Cologne University Protein Stability Analysis Tool. The present work thus suggests deriving pairwise contact energy potentials and molecular dynamics simulations of functionally important folds could help us to predict probability of point mutations which may ultimately reduce the time and cost of mutation experiments. Proteins 2016; 85:54–64. © 2016 Wiley Periodicals, Inc.     

Analysing six types of protein-protein interfaces

Non-covalent residue side-chain interactions occur in many different types of proteins and facilitate many biological functions. Are these differences manifested in the sequence compositions and/or the residue-residue contact preferences of the interfaces? Previous studies analysed small data sets and gave contradictory answers. Here, we introduced a new data-mining method that yielded the largest high-resolution data set of interactions analysed. We introduced an information theory-based analysis method. On the basis of sequence features, we were able to differentiate six types of protein interfaces, each corresponding to a different functional or structural association between residues. Particularly, we found significant differences in amino acid composition and residue-residue preferences between interactions of residues within the same structural domain and between different domains, between permanent and transient interfaces, and between interactions associating homo-oligomers and hetero-oligomers. The differences between the six types were so substantial that, using amino acid composition alone, we could predict statistically to which of the six types of interfaces a pool of 1000 residues belongs at 63-100% accuracy. All interfaces differed significantly from the background of all residues in SWISS-PROT, from the group of surface residues, and from internal residues that were not involved in non-trivial interactions. Overall, our results suggest that the interface type could be predicted from sequence and that interface-type specific mean-field potentials may be adequate for certain applications.     

Statistical significance of hierarchical multi-body potentials based on Delaunay tessellation and their application in sequence-structure alignment.

Statistical potentials based on pairwise interactions between C alpha atoms are commonly used in protein threading/fold-recognition attempts. Inclusion of higher order interaction is a possible means of improving the specificity of these potentials. Delaunay tessellation of the C alpha-atom representation of protein structure has been suggested as a means of defining multi-body interactions. A large number of parameters are required to define all four-body interactions of 20 amino acid types (20(4) = 160,000). Assuming that residue order within a four-body contact is irrelevant reduces this to a manageable 8,855 parameters, using a nonredundant dataset of 608 protein structures. Three lines of evidence support the significance and utility of the four-body potential for sequence-structure matching. First, compared to the four-body model, all lower-order interaction models (three-body, two-body, one-body) are found statistically inadequate to explain the frequency distribution of residue contacts. Second, coherent patterns of interaction are seen in a graphic presentation of the four-body potential. Many patterns have plausible biophysical explanations and are consistent across sets of residues sharing certain properties (e.g., size, hydrophobicity, or charge). Third, the utility of the multi-body potential is tested on a test set of 12 same-length pairs of proteins of known structure for two protocols: Sequence-recognizes-structure, where a query sequence is threaded (without gap) through the native and a non-native structure; and structure-recognizes-sequence, where a query structure is threaded by its native and another non-native sequence. Using cross-validated training, protein sequences correctly recognized their native structure in all 24 cases. Conversely, structures recognized the native sequence in 23 of 24 cases. Further, the score differences between correct and decoy structures increased significantly using the three- or four-body potential compared to potentials of lower order.     

Structural and Functional Restraints on the Occurrence of Single Amino Acid Variations in Human Proteins

Human genetic variation is the incarnation of diverse evolutionary history, which reflects both selectively advantageous and selectively neutral change. In this study, we catalogue structural and functional features of proteins that restrain genetic variation leading to single amino acid substitutions. Our variation dataset is divided into three categories: i) Mendelian disease-related variants, ii) neutral polymorphisms and iii) cancer somatic mutations. We characterize structural environments of the amino acid variants by the following properties: i) side-chain solvent accessibility, ii) main-chain secondary structure, and iii) hydrogen bonds from a side chain to a main chain or other side chains. To address functional restraints, amino acid substitutions in proteins are examined to see whether they are located at functionally important sites involved in protein-protein interactions, protein-ligand interactions or catalytic activity of enzymes. We also measure the likelihood of amino acid substitutions and the degree of residue conservation where variants occur. We show that various types of variants are under different degrees of structural and functional restraints, which affect their occurrence in human proteome.     

Structural characterization of proteins using residue environments

A primary challenge for structural genomics is the automated functional characterization of protein structures. We have developed a sequence-independent method called S-BLEST (Structure-Based Local Environment Search Tool) for the annotation of previously uncharacterized protein structures. S-BLEST encodes the local environment of an amino acid as a vector of structural property values. It has been applied to all amino acids in a nonredundant database of protein structures to generate a searchable structural resource. Given a query amino acid from an experimentally determined or modeled structure, S-BLEST quickly identifies similar amino acid environments using a K-nearest neighbor search. In addition, the method gives an estimation of the statistical significance of each result. We validated S-BLEST on X-ray crystal structures from the ASTRAL 40 nonredundant dataset. We then applied it to 86 crystallographically determined proteins in the protein data bank (PDB) with unknown function and with no significant sequence neighbors in the PDB. S-BLEST was able to associate 20 proteins with at least one local structural neighbor and identify the amino acid environments that are most similar between those neighbors.     

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.     

A simple probabilistic model of multibody interactions in proteins

Protein structure prediction methods typically use statistical potentials, which rely on statistics derived from a database of know protein structures. In the vast majority of cases, these potentials involve pairwise distances or contacts between amino acids or atoms. Although some potentials beyond pairwise interactions have been described, the formulation of a general multibody potential is seen as intractable due to the perceived limited amount of data. In this article, we show that it is possible to formulate a probabilistic model of higher order interactions in proteins, without arbitrarily limiting the number of contacts. The success of this approach is based on replacing a naive table‐based approach with a simple hierarchical model involving suitable probability distributions and conditional independence assumptions. The model captures the joint probability distribution of an amino acid and its neighbors, local structure and solvent exposure. We show that this model can be used to approximate the conditional probability distribution of an amino acid sequence given a structure using a pseudo‐likelihood approach. We verify the model by decoy recognition and site‐specific amino acid predictions. Our coarse‐grained model is compared to state‐of‐art methods that use full atomic detail. This article illustrates how the use of simple probabilistic models can lead to new opportunities in the treatment of nonlocal interactions in knowledge‐based protein structure prediction and design. Proteins 2013; 81:1340–1350. © 2013 Wiley Periodicals, Inc.     

Hypergraph model of multi-residue interactions in proteins: sequentially-constrained partitioning algorithms for optimization of site-directed protein recombination.

Relationships among amino acids determine stability and function and are also constrained by evolutionary history. We develop a probabilistic hypergraph model of residue relationships that generalizes traditional pairwise contact potentials to account for the statistics of multi-residue interactions. Using this model, we detected non-random associations in protein families and in the protein database. We also use this model in optimizing site-directed recombination experiments to preserve significant interactions and thereby increase the frequency of generating useful recombinants. We formulate the optimization as a sequentially-constrained hypergraph partitioning problem; the quality of recombinant libraries with respect to a set of breakpoints is characterized by the total perturbation to edge weights. We prove this problem to be NP-hard in general, but develop exact and heuristic polynomial-time algorithms for a number of important cases. Application to the beta-lactamase family demonstrates the utility of our algorithms in planning site-directed recombination.     

A unified statistical potential reveals that amino acid stickiness governs nonspecific recruitment of client proteins into condensates

Membraneless organelles are cellular compartments that form by liquid–liquid phase separation of one or more components. Other molecules, such as proteins and nucleic acids, will distribute between the cytoplasm and the liquid compartment in accordance with the thermodynamic drive to lower the free energy of the system. The resulting distribution colocalizes molecular species to carry out a diversity of functions. Two factors could drive this partitioning: the difference in solvation between the dilute versus dense phase and intermolecular interactions between the client and scaffold proteins. Here, we develop a set of knowledge‐based potentials that allow for the direct comparison between stickiness, which is dominated by desolvation energy, and pairwise residue contact propensity terms. We use these scales to examine experimental data from two systems: protein cargo dissolving within phase‐separated droplets made from FG repeat proteins of the nuclear pore complex and client proteins dissolving within phase‐separated FUS droplets. These analyses reveal a close agreement between the stickiness of the client proteins and the experimentally determined values of the partition coefficients (R > 0.9), while pairwise residue contact propensities between client and scaffold show weaker correlations. Hence, the stickiness of client proteins is sufficient to explain their differential partitioning within these two phase‐separated systems without taking into account the composition of the condensate. This result implies that selective trafficking of client proteins to distinct membraneless organelles requires recognition elements beyond the client sequence composition.StatementEmpirical potentials for amino acid stickiness and pairwise residue contact propensities are derived. These scales are unique in that they enable direct comparison of desolvation versus contact terms. We find that partitioning of a client protein to a condensate is best explained by amino acid stickiness.     

Prediction of protein surface accessibility with information theory

A new, simple method based on information theory is introduced to predict the solvent accessibility of amino acid residues in various states defined by their different thresholds. Prediction is achieved by the application of information obtained from a single amino acid position or pair-information for a window of seventeen amino acids around the desired residue. Results obtained by pairwise information values are better than results from single amino acids. This reinforces the effect of the local environment on the accessibility of amino acid residues. The prediction accuracy of this method in a jackknife test system for two and three states is better than 70 and 60 %, respectively. A comparison of the results with those reported by others involving the same data set also testifies to a better prediction accuracy in our case.     

Extent and nature of contacts between protein molecules in crystal lattices and between subunits of protein oligomers

A survey was compiled of several characteristics of the intersubunit contacts in 58 oligomeric proteins, and of the intermolecular contacts in the lattice for 223 protein crystal structures. The total number of atoms in contact and the secondary structure elements involved are similar in the two types of interfaces. Crystal contact patches are frequently smaller than patches involved in oligomer interfaces. Crystal contacts result from more numerous interactions by polar residues, compared with a tendency toward nonpolar amino acids at oligomer interfaces. Arginine is the only amino acid prominent in both types of interfaces. Potentials of mean force for residue–residue contacts at both crystal and oligomer interfaces were derived from comparison of the number of observed residue–residue interactions with the number expected by mass action. They show that hydrophobic interactions at oligomer interfaces favor aromatic amino acids and methionine over aliphatic amino acids; and that crystal contacts form in such a way as to avoid inclusion of hydrophobic interactions. They also suggest that complex salt bridges with certain amino acid compositions might be important in oligomer formation. For a protein that is recalcitrant to crystallization, substitution of lysine residues with arginine or glutamine is a recommended strategy. Proteins 28:494–514, 1997. © 1997 Wiley-Liss, Inc.     

Development of unified statistical potentials describing protein-protein interactions

A residue-based and a heavy atom-based statistical pair potential are developed for use in assessing the strength of protein-protein interactions. To ensure the quality of the potentials, a nonredundant, high-quality dimer database is constructed. The protein complexes in this dataset are checked by a literature search to confirm that they form multimers, and the pairwise amino acid preference to interact across a protein-protein interface is analyzed and pair potentials constructed. The performance of the residue-based potentials is evaluated by using four jackknife tests and by assessing the potentials' ability to select true protein-protein interfaces from false ones. Compared to potentials developed for monomeric protein structure prediction, the interdomain potential performs much better at distinguishing protein-protein interactions. The potential developed from homodimer interfaces is almost the same as that developed from heterodimer interfaces with a correlation coefficient of 0.92. The residue-based potential is well suited for genomic scale protein interaction prediction and analysis, such as in a recently developed threading-based algorithm, MULTIPROSPECTOR. However, the more time-consuming atom-based potential performs better in identifying near-native structures from docking generated decoys.     

Grouping of amino acid types and extraction of amino acid properties from multiple sequence alignments using variance maximization

Understanding of amino acid type co-occurrence in trusted multiple sequence alignments is a prerequisite for improved sequence alignment and remote homology detection algorithms. Two objective approaches were used to investigate co-occurrence, both based on variance maximization of the weighted residue frequencies in columns taken from a large alignment database. The first approach discretely grouped amino acid types, and the second approach extracted orthogonal properties of amino acids using principal components analysis. The grouping results corresponded to amino acid physical properties such as side chain hydrophobicity, size, or backbone flexibility, and an optimal arrangement of approximately eight groups was observed. However, interpretation of the orthogonal properties was more complex. Although the principal components accounting for the largest variances exhibited modest correlations with hydrophobicity and conservation of glycine, in general principal components did not correspond to physical properties of amino acids. Although not intuitive, these amino acid mathematical properties were demonstrated to be robust and to improve local pairwise alignment accuracy, relative to 20 amino acid frequencies alone, for a simple test case.     

Sequence specificity, statistical potentials, and three-dimensional structure prediction with self-correcting distance geometry calculations of beta-sheet formation in proteins

A statistical analysis of a representative data set of 169 known protein structures was used to analyze the specificity of residue interactions between spatial neighboring strands in beta-sheets. Pairwise potentials were derived from the frequency of residue pairs in nearest contact, second nearest and third nearest contacts across neighboring beta-strands compared to the expected frequency of residue pairs in a random model. A pseudo-energy function based on these statistical pairwise potentials recognized native beta-sheets among possible alternative pairings. The native pairing was found within the three lowest energies in 73% of the cases in the training data set and in 63% of beta-sheets in a test data set of 67 proteins, which were not part of the training set. The energy function was also used to detect tripeptides, which occur frequently in beta-sheets of native proteins. The majority of native partners of tripeptides were distributed in a low energy range. Self-correcting distance geometry (SECODG) calculations using distance constraints sets derived from possible low energy pairing of beta-strands uniquely identified the native pairing of the beta-sheet in pancreatic trypsin inhibitor (BPTI). These results will be useful for predicting the structure of proteins from their amino acid sequence as well as for the design of proteins containing beta-sheets.