Novel knowledge-based mean force potential at the profile level

Background  

The development and testing of functions for the modeling of protein energetics is an important part of current research aimed at understanding protein structure and function. Knowledge-based mean force potentials are derived from statistical analyses of interacting groups in experimentally determined protein structures. Current knowledge-based mean force potentials are developed at the atom or amino acid level. The evolutionary information contained in the profiles is not investigated. Based on these observations, a class of novel knowledge-based mean force potentials at the profile level has been presented, which uses the evolutionary information of profiles for developing more powerful statistical potentials.     

An atomic environment potential for use in protein structure prediction

We describe the derivation and testing of a knowledge-based atomic environment potential for the modeling of protein structural energetics. An analysis of the probabilities of atomic interactions in a dataset of high-resolution protein structures shows that the probabilities of non-bonded inter-atomic contacts are not statistically independent events, and that the multi-body contact frequencies are poorly predicted from pairwise contact potentials. A pseudo-energy function is defined that measures the preferences for protein atoms to be in a given microenvironment defined by the number of contacting atoms in the environment and its atomic composition. This functional form is tested for its ability to recognize native protein structures amongst an ensemble of decoy structures and a detailed relative performance comparison is made with a number of common functions used in protein structure prediction.     

Self-consistent assignment of asparagine and glutamine amide rotamers in protein crystal structures

The current protein structure database contains unfavorable Asn/Gln amide rotamers in the order of 20%. Here, we derive a set of self-consistent potential functions to identify and correct unfavorable rotamers. Potentials of mean force for all heavy atoms are compiled from a database of high-resolution protein crystal structures. Starting from erroneous data, a refinement-correction cycle quickly converges to a self-consistent set of potentials. The refinement is entirely driven by the deposited structure data and does not involve any assumptions on molecular interactions or any artificial constraints. The refined potentials obtained in this way identify unfavorable rotamers with high confidence. Since the state of Asn/Gln rotamers is largely determined by hydrogen bond interactions, the features of the respective potentials are of interest in terms of molecular interactions, protein structure refinement, and prediction. The Asn/Gln rotamer assignment is available as a public web service intended to support protein structure refinement and modeling.     

Residue contact-count potentials are as effective as residue-residue contact-type potentials for ranking protein decoys

Background  

For over 30 years potentials of mean force have been used to evaluate the relative energy of protein structures. The most commonly used potentials define the energy of residue-residue interactions and are derived from the empirical analysis of the known protein structures. However, single-body residue 'environment' potentials, although widely used in protein structure analysis, have not been rigorously compared to these classical two-body residue-residue interaction potentials. Here we do not try to combine the two different types of residue interaction potential, but rather to assess their independent contribution to scoring protein structures.     

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.     

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 do potentials derived from structural databases relate to “true” potentials?

Knowledge-based potentials are used widely in protein folding and inverse folding algorithms. Two kinds of derivation methods are used. (1) The interactions in a database of known protein structures are assumed to obey a Boltzmann distribution. (2) The stability of the native folds relative to a manifold of misfolded structures is optimized. Here, a set of previously derived contact and secondary structure propensity potentials, taken as the "true" potentials, are employed to construct an artificial protein structural database from protein fragments. Then, new sets of potentials are derived to see how they are related to the true potentials. Using the Boltzmann distribution method, when the stability of the structures in the database lies within a certain range, both contact potentials and secondary structure propensities can be derived separately with remarkable accuracy. In general, the optimization method was found to be less accurate due to errors in the "excess energy" contribution. When the excess energy terms are kept as a constraint, the true potentials are recovered exactly.     

Reconstruction of protein conformations from estimated positions of the C alpha coordinates.

Protein C alpha coordinates are used to accurately reconstruct complete protein backbones and side-chain directions. This work employs potentials of mean force to align semirigid peptide groups around the axes that connect successive C alpha atoms. The algorithm works well for all residue types and secondary structure classes and is stable for imprecise C alpha coordinates. Tests on known protein structures show that root mean square errors in predicted main-chain and C beta coordinates are usually less than 0.3 A. These results are significantly more accurate than can be obtained from competing approaches, such as modeling of backbone conformations from structurally homologous fragments.     

Evaluation of short-range interactions as secondary structure energies for protein fold and sequence recognition.

Short-range interactions for secondary structures of proteins are evaluated as potentials of mean force from the observed frequencies of secondary structures in known protein structures which are assumed to have an equilibrium distribution with the Boltzmann factor of secondary structure energies. A secondary conformation at each residue position in a protein is described by a tripeptide, including one nearest neighbor on each side. The secondary structure potentials are approximated as additive contributions from neighboring residues along the sequence. These are part of an empirical potential to provide a crude estimate of protein conformational energy at a residue level. Unlike previous works, interactions are decoupled into intrinsic potentials of residues, potentials of backbone-backbone interactions, and of side chain-backbone interactions. Also interactions are decoupled into one-body, two-body, and higher order interactions between peptide backbone and side chain and between backbones. These decouplings are essential to correctly evaluate the total secondary structure energy of a protein structure without overcounting interactions. Each interaction potential is evaluated separately by taking account of the correlation in the amino acid order of protein sequences. Interactions among side chains are neglected, because of the relatively limited number of protein structures. Proteins 1999;36:347-356. Published 1999 Wiley-Liss, Inc.     

Potentials of mean force for protein structure prediction vindicated, formalized and generalized

Understanding protein structure is of crucial importance in science, medicine and biotechnology. For about two decades, knowledge-based potentials based on pairwise distances--so-called "potentials of mean force" (PMFs)--have been center stage in the prediction and design of protein structure and the simulation of protein folding. However, the validity, scope and limitations of these potentials are still vigorously debated and disputed, and the optimal choice of the reference state--a necessary component of these potentials--is an unsolved problem. PMFs are loosely justified by analogy to the reversible work theorem in statistical physics, or by a statistical argument based on a likelihood function. Both justifications are insightful but leave many questions unanswered. Here, we show for the first time that PMFs can be seen as approximations to quantities that do have a rigorous probabilistic justification: they naturally arise when probability distributions over different features of proteins need to be combined. We call these quantities "reference ratio distributions" deriving from the application of the "reference ratio method." This new view is not only of theoretical relevance but leads to many insights that are of direct practical use: the reference state is uniquely defined and does not require external physical insights; the approach can be generalized beyond pairwise distances to arbitrary features of protein structure; and it becomes clear for which purposes the use of these quantities is justified. We illustrate these insights with two applications, involving the radius of gyration and hydrogen bonding. In the latter case, we also show how the reference ratio method can be iteratively applied to sculpt an energy funnel. Our results considerably increase the understanding and scope of energy functions derived from known biomolecular structures.     

Influence of protein structure databases on the predictive power of statistical pair potentials

A long standing goal in protein structure studies is the development of reliable energy functions that can be used both to verify protein models derived from experimental constraints as well as for theoretical protein folding and inverse folding computer experiments. In that respect, knowledge-based statistical pair potentials have attracted considerable interests recently mainly because they include the essential features of protein structures as well as solvent effects at a low computing cost. However, the basis on which statistical potentials are derived have been questioned. In this paper, we investigate statistical pair potentials derived from protein three-dimensional structures, addressing in particular questions related to the form of these potentials, as well as to the content of the database from which they are derived. We have shown that statistical pair potentials depend on the size of the proteins included in the database, and that this dependence can be reduced by considering only pairs of residue close in space (i.e., with a cutoff of 8 Å). We have shown also that statistical potentials carry a memory of the quality of the database in terms of the amount and diversity of secondary structure it contains. We find, for example, that potentials derived from a database containing α-proteins will only perform best on α-proteins in fold recognition computer experiments. We believe that this is an overall weakness of these potentials, which must be kept in mind when constructing a database. Proteins 31:139–149, 1998. © 1998 Wiley-Liss, Inc.     

Are proteins ideal mixtures of amino acids? Analysis of energy parameter sets

Various existing derivations of the effective potentials of mean force for the two-body interactions between amino acid side chains in proteins are reviewed and compared to each other. The differences between different parameter sets can be traced to the reference state used to define the zero of energy. Depending on the reference state, the transfer free energy or other pseudo-one-body contributions can be present to various extents in two-body parameter sets. It is, however, possible to compare various derivations directly by concentrating on the "excess" energy-a term that describes the difference between a real protein and an ideal solution of amino acids. Furthermore, the number of protein structures available for analysis allows one to check the consistency of the derivation and the errors by comparing parameters derived from various subsets of the whole database. It is shown that pair interaction preferences are very consistent throughout the database. Independently derived parameter sets have correlation coefficients on the order of 0.8, with the mean difference between equivalent entries of 0.1 kT. Also, the low-quality (low resolution, little or no refinement) structures show similar regularities. There are, however, large differences between interaction parameters derived on the basis of crystallographic structures and structures obtained by the NMR refinement. The origin of the latter difference is not yet understood.     

TOUCHSTONE II: a new approach to ab initio protein structure prediction

We have developed a new combined approach for ab initio protein structure prediction. The protein conformation is described as a lattice chain connecting C(alpha) atoms, with attached C(beta) atoms and side-chain centers of mass. The model force field includes various short-range and long-range knowledge-based potentials derived from a statistical analysis of the regularities of protein structures. The combination of these energy terms is optimized through the maximization of correlation for 30 x 60,000 decoys between the root mean square deviation (RMSD) to native and energies, as well as the energy gap between native and the decoy ensemble. To accelerate the conformational search, a newly developed parallel hyperbolic sampling algorithm with a composite movement set is used in the Monte Carlo simulation processes. We exploit this strategy to successfully fold 41/100 small proteins (36 approximately 120 residues) with predicted structures having a RMSD from native below 6.5 A in the top five cluster centroids. To fold larger-size proteins as well as to improve the folding yield of small proteins, we incorporate into the basic force field side-chain contact predictions from our threading program PROSPECTOR where homologous proteins were excluded from the data base. With these threading-based restraints, the program can fold 83/125 test proteins (36 approximately 174 residues) with structures having a RMSD to native below 6.5 A in the top five cluster centroids. This shows the significant improvement of folding by using predicted tertiary restraints, especially when the accuracy of side-chain contact prediction is >20%. For native fold selection, we introduce quantities dependent on the cluster density and the combination of energy and free energy, which show a higher discriminative power to select the native structure than the previously used cluster energy or cluster size, and which can be used in native structure identification in blind simulations. These procedures are readily automated and are being implemented on a genomic scale.     

Absolute quality evaluation of protein model structures using statistical potentials with respect to the native and reference states

In protein structure prediction, it is crucial to evaluate the degree of native-likeness of given model structures. Statistical potentials extracted from protein structure data sets are widely used for such quality assessment problems, but they are only applicable for comparing different models of the same protein. Although various other methods, such as machine learning approaches, were developed to predict the absolute similarity of model structures to the native ones, they required a set of decoy structures in addition to the model structures. In this paper, we tried to reformulate the statistical potentials as absolute quality scores, without using the information from decoy structures. For this purpose, we regarded the native state and the reference state, which are necessary components of statistical potentials, as the good and bad standard states, respectively, and first showed that the statistical potentials can be regarded as the state functions, which relate a model structure to the native and reference states. Then, we proposed a standardized measure of protein structure, called native-likeness, by interpolating the score of a model structure between the native and reference state scores defined for each protein. The native-likeness correlated with the similarity to the native structures and discriminated the native structures from the models, with better accuracy than the raw score. Our results show that statistical potentials can quantify the native-like properties of protein structures, if they fully utilize the statistical information obtained from the data set.     

Protein local conformations arise from a mixture of Gaussian distributions

The classical approaches for protein structure prediction rely either on homology of the protein sequence with a template structure or on ab initio calculations for energy minimization. These methods suffer from disadvantages such as the lack of availability of homologous template structures or intractably large conformational search space, respectively. The recently proposed fragment library based approaches first predict the local structures, which can be used in conjunction with the classical approaches of protein structure prediction. The accuracy of the predictions is dependent on the quality of the fragment library. In this work, we have constructed a library of local conformation classes purely based on geometric similarity. The local conformations are represented using Geometric Invariants, properties that remain unchanged under transformations such as translation and rotation, followed by dimension reduction via principal component analysis. The local conformations are then modeled as a mixture of Gaussian probability distribution functions (PDF). Each one of the Gaussian PDF’s corresponds to a conformational class with the centroid representing the average structure of that class. We find 46 classes when we use an octapeptide as a unit of local conformation. The protein 3-D structure can now be described as a sequence of local conformational classes. Further, it was of interest to see whether the local conformations can be predicted from the amino acid sequences. To that end, we have analyzed the correlation between sequence features and the conformational classes.     

Derivation of protein-specific pair potentials based on weak sequence fragment similarity

A method is presented for the derivation of knowledge-based pair potentials that corrects for the various compositions of different proteins. The resulting statistical pair potential is more specific than that derived from previous approaches as assessed by gapless threading results. Additionally, a methodology is presented that interpolates between statistical potentials when no homologous examples to the protein of interest are in the structural database used to derive the potential, to a Go-like potential (in which native interactions are favorable and all nonnative interactions are not) when homologous proteins are present. For cases in which no protein exceeds 30% sequence identity, pairs of weakly homologous interacting fragments are employed to enhance the specificity of the potential. In gapless threading, the mean z score increases from -10.4 for the best statistical pair potential to -12.8 when the local sequence similarity, fragment-based pair potentials are used. Examination of the ab initio structure prediction of four representative globular proteins consistently reveals a qualitative improvement in the yield of structures in the 4 to 6 A rmsd from native range when the fragment-based pair potential is used relative to that when the quasichemical pair potential is employed. This suggests that such protein-specific potentials provide a significant advantage relative to generic quasichemical potentials.     

Identification of native protein folds amongst a large number of incorrect models. The calculation of low energy conformations from potentials of mean force

We present an approach that is able to detect native folds amongst a large number of non-native conformations. The method is based on the compilation of potentials of mean force of the interactions of the C beta atoms of all amino acid pairs from a database of known three-dimensional protein structures. These potentials are used to calculate the conformational energy of amino acid sequences in a number of different folds. For a substantial number of proteins we find that the conformational energy of the native state is lowest amongst the alternatives. Exceptions are proteins containing large prosthetic groups, Fe-S clusters or polypeptide chains that do not adopt globular folds. We discuss briefly potential applications in various fields of protein structural research.     

A distance-dependent atomic knowledge-based potential for improved protein structure selection.

A heavy atom distance-dependent knowledge-based pairwise potential has been developed. This statistical potential is first evaluated and optimized with the native structure z-scores from gapless threading. The potential is then used to recognize the native and near-native structures from both published decoy test sets, as well as decoys obtained from our group's protein structure prediction program. In the gapless threading test, there is an average z-score improvement of 4 units in the optimized atomic potential over the residue-based quasichemical potential. Examination of the z-scores for individual pairwise distance shells indicates that the specificity for the native protein structure is greatest at pairwise distances of 3.5-6.5 A, i.e., in the first solvation shell. On applying the current atomic potential to test sets obtained from the web, composed of native protein and decoy structures, the current generation of the potential performs better than residue-based potentials as well as the other published atomic potentials in the task of selecting native and near-native structures. This newly developed potential is also applied to structures of varying quality generated by our group's protein structure prediction program. The current atomic potential tends to pick lower RMSD structures than do residue-based contact potentials. In particular, this atomic pairwise interaction potential has better selectivity especially for near-native structures. As such, it can be used to select near-native folds generated by structure prediction algorithms as well as for protein structure refinement.