Failures of inverse folding and threading with gapped alignment

To calculate the tertiary structure of a protein from its amino acid sequence, the thermodynamic approach requires a potential function of sequence and conformation that has its global minimum at the native conformation for many different proteins. Here we study the behavior of such functions for the simplest model system that still has some of the features of the protein folding problem, namely two-dimensional square lattice chain configurations involving two residue types. First we show that even the given contact potential, which by definition is used to identify the folding sequences and their unique native conformations, cannot always correctly select which sequences will fold to a given structure. Second, we demonstrate that the given contact potential is not always able to favor the native alignment of a native sequence on its own native conformation over other gapped alignments of different folding sequences onto that same conformation. Because of these shortcomings, even in this simple model system in which all conformations and all native sequences are known and determined directly by the given potential, we must reexamine our expectations for empirical potentials used for inverse folding and gapped alignment on more realistic representations of proteins. © 1996 Wiley-Liss, Inc.     

Potential energy function for continuous state models of globular proteins.

One of the approaches to protein structure prediction is to obtain energy functions which can recognize the native conformation of a given sequence among a zoo of conformations. The discriminations can be done by assigning the lowest energy to the native conformation, with the guarantee that the native is in the zoo. Well-adjusted functions, then, can be used in the search for other (near-) natives. Here the aim is the discrimination at relatively high resolution (RMSD difference between the native and the closest nonnative is around 1 A) by pairwise energy potentials. The potential is trained using the experimentally determined native conformation of only one protein, instead of the usual large survey over many proteins. The novel feature is that the native structure is compared to a vastly wider and more challenging array of nonnative structures found not only by the usual threading procedure, but by wide-ranging local minimization of the potential. Because of this extremely demanding search, the native is very close to the apparent global minimum of the potential function. The global minimum property holds up for one other protein having 60% sequence identity, but its performance on completely dissimilar proteins is of course much weaker.     

Prediction of protein folding from amino acid sequence over discrete conformation spaces

Predicting the three-dimensional structure of a protein given only its amino acid sequence is a long-standing goal in computational chemistry. In the thermodynamic approach, one needs a potential function of conformation that resembles the free energy of the real protein to the extent that the global minimum of the potential is attained by the native conformation and no other. In practice, this has never been achieved with certainty because even with greatly simplified representations of the polypeptide chain, there are an astronomical number of local minima to examine. If one chooses instead a protein representation with only a large but manageable number of discrete conformations, then the global preference of the potential for the native can be directly verified. Representing a protein as a walk on a two-dimensional square lattice makes it easy to see that simple functions of the interresidue contacts are sufficient to globally favor a given "native" conformation, as long as it is a compact, globular structure. Explicit representation of the solvent is not required. Another more realistic way to confine the conformational search to a finite set is to draw alternative conformations from fragments of larger proteins having known crystal structure. Then it is possible to construct a simple function of interresidue contacts in three dimensions such that only 8 proteins are required to determine the adjustable parameters, and the native conformations of 37 other proteins are correctly preferred over all alternative conformations. The deduced function favors short-range backbone-backbone contacts regardless of residue type and long-range hydrophobic associations. Interactions over long distances, such as electrostatics, are not required.     

Contact potential that recognizes the correct folding of globular proteins.

We have devised a continuous function of interresidue contacts in globular proteins such that the X-ray crystal structure has a lower function value than that of thousands of protein-like alternative conformations. Although we fit the adjustable parameters of the potential using only 10,000 alternative structures for a selected training set of 37 proteins, a grand total of 530,000 constraints was satisfied, derived from 73 proteins and their numerous alternative conformations. In every case where the native conformation is adequately globular and compact, according to objective criteria we have developed, the potential function always favors the native over all alternatives by a substantial margin. This is true even for an additional three proteins never used in any way in the fitting procedure. Conformations differing only slightly from the native, such as those coming from crystal structures of the same protein complexed with different ligands or from crystal structures of point mutants, have function values very similar to the native's and always less than those of alternatives derived from substantially different crystal structures. This holds for all 95 structures that are homologous to one or another of various proteins we used. Realizing that this potential should be useful for modeling the conformation of new protein sequences from the body of protein crystal structures, we suggest a test for deciding whether a nearly correct approximation to the native conformation has been found.     

Using a Hydrophobic Contact Potential to Evaluate Native and Near-native Folds Generated by Molecular Dynamics Simulations

There are several knowledge-based energy functions that can distinguish the native fold from a pool of grossly misfolded decoys for a given sequence of amino acids. These decoys, which are typically generated by mounting, or “threading”, the sequence onto the backbones of unrelated protein structures, tend to be non-compact and quite different from the native structure: the root-mean-squared (RMS) deviations from the native are commonly in the range of 15 to 20 Å. Effective energy functions should also demonstrate a similar recognition capability when presented with compact decoys that depart only slightly in conformation from the correct structure (i.e. those with RMS deviations of ∼5 Å or less). Recently, we developed a simple yet powerful method for native fold recognition based on the tendency for native folds to form hydrophobic cores. Our energy measure, which we call the hydrophobic fitness score, is challenged to recognize the native fold from 2000 near-native structures generated for each of five small monomeric proteins. First, 1000 conformations for each protein were generated by molecular dynamics simulation at room temperature. The average RMS deviation of this set of 5000 was 1.5 Å. A total of 323 decoys had energies lower than native; however, none of these had RMS deviations greater than 2 Å. Another 1000 structures were generated for each at high temperature, in which a greater range of conformational space was explored (4.3 Å average RMS deviation). Out of this set, only seven decoys were misrecognized. The hydrophobic fitness energy of a conformation is strongly dependent upon the RMS deviation. On average our potential yields energy values which are lowest for the population of structures generated at room temperature, intermediate for those produced at high temperature and highest for those constructed by threading methods. In general, the lowest energy decoy conformations have backbones very close to native structure. The possible utility of our method for screening backbone candidates for the purpose of modelling by side-chain packing optimization is discussed.     

Learning about protein folding via potential functions

Over the last few years we have developed an empirical potential function that solves the protein structure recognition problem: given the sequence for an n-residue globular protein and a collection of plausible protein conformations, including the native conformation for that sequence, identify the correct, native conformation. Having determined this potential on the basis of only some 6500 native/nonnative pairs of structures for 58 proteins, we find it recognizes the native conformation for essentially all compact, soluble, globular proteins having known native conformations in comparisons with 10⁴ to 10⁶ reasonable alternative conformations apiece. In this sense, the potential encodes nearly all the essential features of globular protein conformational preference. In addition it “knows” about many additional factors in protein folding, such as the stabilization of multimeric proteins, quaternary structure, the role of disulfide bridges and ligands, proproteins vs. processed proteins, and minimal strand lengths in globular proteins. Comparisons are made with other sorts of protein folding problems, and applications in protein conformational determination and prediction are discussed. © 1994 Wiley-Liss, Inc.     

Protein structure evaluation using an all-atom energy based empirical scoring function

Arriving at the native conformation of a polypeptide chain characterized by minimum most free energy is a problem of long standing interest in protein structure prediction endeavors. Owing to the computational requirements in developing free energy estimates, scoring functions--energy based or statistical--have received considerable renewed attention in recent years for distinguishing native structures of proteins from non-native like structures. Several cleverly designed decoy sets, CASP (Critical Assessment of Techniques for Protein Structure Prediction) structures and homology based internet accessible three dimensional model builders are now available for validating the scoring functions. We describe here an all-atom energy based empirical scoring function and examine its performance on a wide series of publicly available decoys. Barring two protein sequences where native structure is ranked second and seventh, native is identified as the lowest energy structure in 67 protein sequences from among 61,659 decoys belonging to 12 different decoy sets. We further illustrate a potential application of the scoring function in bracketing native-like structures of two small mixed alpha/beta globular proteins starting from sequence and secondary structural information. The scoring function has been web enabled at www.scfbio-iitd.res.in/utility/proteomics/energy.jsp.     

Potential for assessing quality of protein structure based on contact number prediction

We developed a novel knowledge-based residue environment potential for assessing the quality of protein structures in protein structure prediction. The potential uses the contact number of residues in a protein structure and the absolute contact number of residues predicted from its amino acid sequence using a new prediction method based on a support vector regression (SVR). The contact number of an amino acid residue in a protein structure is defined by the number of residues around a given residue. First, the contact number of each residue is predicted using SVR from an amino acid sequence of a target protein. Then, the potential of the protein structure is calculated from the probability distribution of the native contact numbers corresponding to the predicted ones. The performance of this potential is compared with other score functions using decoy structures to identify both native structure from other structures and near-native structures from nonnative structures. This potential improves not only the ability to identify native structures from other structures but also the ability to discriminate near-native structures from nonnative structures.     

A novel parameterization scheme for energy equations and its use to calculate the structure of protein molecules

A novel scheme for the parameterization of a type of “potential energy” function for protein molecules is introduced. The function is parameterized based on the known conformations of previously determined protein structures and their sequence similarity to a molecule whose conformation is to be calculated. Once parameterized, minima of the potential energy function can be located using a version of simulated annealing which has been previously shown to locate global and near-global minima with the given functional form. As a test problem, the potential was parameterized based on the known structures of the rubredoxins from Desulfovibrio vulgaris, Desulfovibrio desulfuricans, and Clostridium pasteurianum, which vary from 45 to 54 amino acids in length, and the sequence alignments of these molecules with the rubredoxin sequence from Desulfovibrio gigas. Since the Desulfovibrio gigas rubredeoxin conformation has also been determined, it is possible to check the accuracy of the results. Ten simulated-annealing runs from random starting conformations were performed. Seven of the 10 resultant conformations have an all-C_α rms deviation from the crystallographically determined conformation of less than 1.7 Å. For five of the structures, the rms deviation is less than 0.8 Å. Four of the structures have conformations which are virtually identical to each other except for the position of the carboxy-terminal residue. This is also the conformation which is achieved if the determined crystal structure is minimized with the same potential. The all-C_α rms difference between the crystal and minimized crystal structures is 0.6 Å. It is further observed that the “energies” of the structures according to the potential function exhibit a strong correlation with rms deviation from the native structure. The conformations of the individual model structures and the computational aspects of the modeling procedure are discussed. © 1993 Wiley-Liss, Inc.     

Protein Structure Evaluation using an All-Atom Energy Based Empirical Scoring Function

Abstract

Arriving at the native conformation of a polypeptide chain characterized by minimum most free energy is a problem of long standing interest in protein structure prediction endeavors. Owing to the computational requirements in developing free energy estimates, scoring functions—energy based or statistical—have received considerable renewed attention in recent years for distinguishing native structures of proteins from non-native like structures. Several cleverly designed decoy sets, CASP (Critical Assessment of Techniques for Protein Structure Prediction) structures and homology based internet accessible three dimensional model builders are now available for validating the scoring functions. We describe here an all-atom energy based empirical scoring function and examine its performance on a wide series of publicly available decoys. Barring two protein sequences where native structure is ranked second and seventh, native is identified as the lowest energy structure in 67 protein sequences from among 61,659 decoys belonging to 12 different decoy sets. We further illustrate a potential application of the scoring function in bracketing native-like structures of two small mixed alpha/beta globular proteins starting from sequence and secondary structural information. The scoring function has been web enabled at www.scfbio-iitd.res.in/utility/proteomics/energy.jsp     

Prediction of the three-dimensional structure of proteins using the electrostatic screening model and hierarchic condensation

We describe a method for predicting the three-dimensional (3-D) structure of proteins from their sequence alone. The method is based on the electrostatic screening model for the stability of the protein main-chain conformation. The free energy of a protein as a function of its conformation is obtained from the potentials of mean force analysis of high-resolution x-ray protein structures. The free energy function is simple and contains only 44 fitted coefficients. The minimization of the free energy is performed by the torsion space Monte Carlo procedure using the concept of hierarchic condensation. The Monte Carlo minimization procedure is applied to predict the secondary, super-secondary, and native 3-D structures of 12 proteins with 28–110 amino acids. The 3-D structures of the majority of local secondary and super-secondary structures are predicted accurately. This result suggests that control in forming the native-like local structure is distributed along the entire protein sequence. The native 3-D structure is predicted correctly for 3 of 12 proteins composed mainly from the α-helices. The method fails to predict the native 3-D structure of proteins with a predominantly β secondary structure. We suggest that the hierarchic condensation is not an appropriate procedure for simulating the folding of proteins made up primarily from β-strands. The method has been proved accurate in predicting the local secondary and super-secondary structures in the blind ab initio 3-D prediction experiment. Proteins 31:74–96, 1998. © 1998 Wiley-Liss, Inc.     

Developing optimal non-linear scoring function for protein design

Motivation. Protein design aims to identify sequences compatible with a given protein fold but incompatible to any alternative folds. To select the correct sequences and to guide the search process, a design scoring function is critically important. Such a scoring function should be able to characterize the global fitness landscape of many proteins simultaneously. RESULTS: To find optimal design scoring functions, we introduce two geometric views and propose a formulation using a mixture of non-linear Gaussian kernel functions. We aim to solve a simplified protein sequence design problem. Our goal is to distinguish each native sequence for a major portion of representative protein structures from a large number of alternative decoy sequences, each a fragment from proteins of different folds. Our scoring function discriminates perfectly a set of 440 native proteins from 14 million sequence decoys. We show that no linear scoring function can succeed in this task. In a blind test of unrelated proteins, our scoring function misclassfies only 13 native proteins out of 194. This compares favorably with about three-four times more misclassifications when optimal linear functions reported in the literature are used. We also discuss how to develop protein folding scoring function.     

Discrimination of the native from misfolded protein models with an energy function including implicit solvation.

An essential requirement for theoretical protein structure prediction is an energy function that can discriminate the native from non-native protein conformations. To date most of the energy functions used for this purpose have been extracted from a statistical analysis of the protein structure database, without explicit reference to the physical interactions responsible for protein stability. The use of the statistical functions has been supported by the widespread belief that they are superior for such discrimination to physics-based energy functions. An effective energy function which combined the CHARMM vacuum potential with a Gaussian model for the solvation free energy is tested for its ability to discriminate the native structure of a protein from misfolded conformations; the results are compared with those obtained with the vacuum CHARMM potential. The test is performed on several sets of misfolded structures prepared by others, including sets of about 650 good decoys for six proteins, as well as on misfolded structures of chymotrypsin inhibitor 2. The vacuum CHARMM potential is successful in most cases when energy minimized conformations are considered, but fails when applied to structures relaxed by molecular dynamics. With the effective energy function the native state is always more stable than grossly misfolded conformations both in energy minimized and molecular dynamics-relaxed structures. The present results suggest that molecular mechanics (physics-based) energy functions, complemented by a simple model for the solvation free energy, should be tested for use in the inverse folding problem, and supports their use in studies of the effective energy surface of proteins in solution. Moreover, the study suggests that the belief in the superiority of statistical functions for these purposes may be ill founded.     

Refining near‐native protein–protein docking decoys by local resampling and energy minimization

How to refine a near‐native structure to make it closer to its native conformation is an unsolved problem in protein‐structure and protein–protein complex‐structure prediction. In this article, we first test several scoring functions for selecting locally resampled near‐native protein–protein docking conformations and then propose a computationally efficient protocol for structure refinement via local resampling and energy minimization. The proposed method employs a statistical energy function based on a Distance‐scaled Ideal‐gas REference state (DFIRE) as an initial filter and an empirical energy function EMPIRE (EMpirical Protein‐InteRaction Energy) for optimization and re‐ranking. Significant improvement of final top‐1 ranked structures over initial near‐native structures is observed in the ZDOCK 2.3 decoy set for Benchmark 1.0 (74% whose global rmsd reduced by 0.5 Å or more and only 7% increased by 0.5 Å or more). Less significant improvement is observed for Benchmark 2.0 (38% versus 33%). Possible reasons are discussed. Proteins 2009. © 2008 Wiley‐Liss, Inc.     

Toward an energy function for the contact map representation of proteins

We analyzed several energy functions for predicting the native state of proteins from an energy minimization procedure. We derived the parameters of a given energy function by imposing the basic requirement that the energy of the native conformation of a protein is lower than that of any conformation chosen from a set of decoys. Our work is motivated by a recent result which proved that the simple pairwise contact approximation of the energy is insufficient to satisfy simultaneously such a basic requirement for all the proteins in a database. Here, we investigate the reasons of such negative results and show how to improve the predictive power of methods based on energy minimization. We generated decoys by gapless threading, and we derive energy parameters by perceptron learning. We first considered hydrophobic contributions to the energy, defined in several ways, and showed that the additional hydrophobic terms enlarge slightly the number of proteins that can be stabilized together. Next, we performed various modifications of the pairwise energy term. We introduced (1) a distinction between inter-residue contacts on the surface and in the core of a protein and (2) a simple distance-dependent pairwise interaction in which a two-tier definition of contact replaces the original (single-tier) one. Our results suggest that a detailed treatment of the pairwise potential is likely to be more relevant than the consideration of other forces.     

Can molecular dynamics simulations help in discriminating correct from erroneous protein 3D models?

Background  

Recent approaches for predicting the three-dimensional (3D) structure of proteins such asde novoor fold recognition methods mostly rely on simplified energy potential functions and a reduced representation of the polypeptide chain. These simplifications facilitate the exploration of the protein conformational space but do not permit to capture entirely the subtle relationship that exists between the amino acid sequence and its native structure. It has been proposed that physics-based energy functions together with techniques for sampling the conformational space, e.g., Monte Carlo or molecular dynamics (MD) simulations, are better suited to the task of modelling proteins at higher resolutions than those of models obtained with the former type of methods. In this study we monitor different protein structural properties along MD trajectories to discriminate correct from erroneous models. These models are based on the sequence-structure alignments provided by our fold recognition method, FROST. We define correct models as being built from alignments of sequences with structures similar to their native structures and erroneous models from alignments of sequences with structures unrelated to their native structures.     

Detection of native-like models for amino acid sequences of unknown three-dimensional structure in a data base of known protein conformations.

We present an approach which can be used to identify native-like folds in a data base of protein conformations in the absence of any sequence homology to proteins in the data base. The method is based on a knowledge-based force field derived from a set of known protein conformations. A given sequence is mounted on all conformations in the data base and the associated energies are calculated. Using several conformations and sequences from the globin family we show that the native conformation is identified correctly. In fact the resolution of the force field is high enough to discriminate between a native fold and several closely related conformations. We then apply the procedure to several globins of known sequence but unknown three dimensional structure. The homology of these sequences to globins of known structures in the data base ranges from 49 to 17%. With one exception we find that for all globin sequences one of the known globin folds is identified as the most favorable conformation. These results are obtained using a force field derived from a data base devoid of globins of known structure. We briefly discuss useful applications in protein structural research and future development of our approach.     

Local propensities and statistical potentials of backbone dihedral angles in proteins

The following three issues concerning the backbone dihedral angles of protein structures are presented. (1) How do the dihedral angles of the 20 amino acids depend on the identity and conformation of their nearest residues? (2) To what extent are the native dihedral angles determined by local (dihedral) potentials? (3) How to build a knowledge-based potential for a residue's dihedral angles, considering the identity and conformation of its nearest residues? We find that the dihedral angle distribution for a residue can significantly depend on the identity and conformation of its adjacent residues. These correlations are in sharp contrast to the Flory isolated-pair hypothesis. Statistical potentials are built for all combinations of residue triplets and depend on the dihedral angles between consecutive residues. First, a low-resolution potential is obtained, which only differentiates between the main populated basins in the dihedral angle density plots. Minimization of the dihedral potential for 125 test proteins reveals that most native alpha-helical residues (89%) and a large fraction of native beta-sheet residues (47%) adopt conformations close to their native one. For native loop residues, the percentage is 48%. It is also found that this fraction is higher for residues away from the ends of alpha or beta secondary structure elements. In addition, a higher resolution potential is built as a function of dihedral angles by a smoothing procedure and continuous functions interpolations. Monte Carlo energy minimization with this potential results in a lower fraction for native beta-sheet residues. Nevertheless, because of the higher flexibility and entropy of beta structures, they could be preferred under the influence of non-local interactions. In general, most alpha-helices and many beta-sheets are strongly determined by the local potential, while the conformations in loops and near the end of beta-sheets are more influenced by non-local interactions.     

Evolutionary Dynamics on Protein Bi-stability Landscapes can Potentially Resolve Adaptive Conflicts

Experimental studies have shown that some proteins exist in two alternative native-state conformations. It has been proposed that such bi-stable proteins can potentially function as evolutionary bridges at the interface between two neutral networks of protein sequences that fold uniquely into the two different native conformations. Under adaptive conflict scenarios, bi-stable proteins may be of particular advantage if they simultaneously provide two beneficial biological functions. However, computational models that simulate protein structure evolution do not yet recognize the importance of bi-stability. Here we use a biophysical model to analyze sequence space to identify bi-stable or multi-stable proteins with two or more equally stable native-state structures. The inclusion of such proteins enhances phenotype connectivity between neutral networks in sequence space. Consideration of the sequence space neighborhood of bridge proteins revealed that bi-stability decreases gradually with each mutation that takes the sequence further away from an exactly bi-stable protein. With relaxed selection pressures, we found that bi-stable proteins in our model are highly successful under simulated adaptive conflict. Inspired by these model predictions, we developed a method to identify real proteins in the PDB with bridge-like properties, and have verified a clear bi-stability gradient for a series of mutants studied by Alexander et al. (Proc Nat Acad Sci USA 2009, 106:21149–21154) that connect two sequences that fold uniquely into two different native structures via a bridge-like intermediate mutant sequence. Based on these findings, new testable predictions for future studies on protein bi-stability and evolution are discussed.