Ab initio construction of polypeptide fragments: efficient generation of accurate,representative ensembles

We describe a novel method to generate ensembles of conformations of the main-chain atoms [N, C(alpha), C, O, Cbeta] for a sequence of amino acids within the context of a fixed protein framework. Each conformation satisfies fundamental stereo-chemical restraints such as idealized geometry, favorable phi/psi angles, and excluded volume. The ensembles include conformations both near and far from the native structure. Algorithms for effective conformational sampling and constant time overlap detection permit the generation of thousands of distinct conformations in minutes. Unlike previous approaches, our method samples dihedral angles from fine-grained phi/psi state sets, which we demonstrate is superior to exhaustive enumeration from coarse phi/psi sets. Applied to a large set of loop structures, our method samples consistently near-native conformations, averaging 0.4, 1.1, and 2.2 A main-chain root-mean-square deviations for four, eight, and twelve residue long loops, respectively. The ensembles make ideal decoy sets to assess the discriminatory power of a selection method. Using these decoy sets, we conclude that quality of anchor geometry cannot reliably identify near-native conformations, though the selection results are comparable to previous loop prediction methods. In a subsequent study (de Bakker et al.: Proteins 2003;51:21-40), we demonstrate that the AMBER forcefield with the Generalized Born solvation model identifies near-native conformations significantly better than previous methods.     

Flexibility of bovine pancreatic trypsin inhibitor

The native conformation of a protein may be expressed in terms of the dihedral angles, phi's and psi's for the backbone, and kappa's for the side chains, for a given geometry (bond lengths and bond angles). We have developed a method to obtain the dihedral angles for a low-energy structure of a protein, starting with the X-ray structure; it is applied here to examine the degree of flexibility of bovine pancreatic trypsin inhibitor. Minimization of the total energy of the inhibitor (including nonbonded, electrostatic, torsional, hydrogen bonding, and disulfide loop energies) yields a conformation having a total energy of -221 kcal/mol and a root mean square deviation between all atoms of the computed and experimental structures of 0.63 A. The optimal conformation is not unique, however, there being at least two other conformations of low-energy (-222 and -220 kcal/mol), which resemble the experimental one (root mean square deviations of 0.66 and 0.64 A, respectively). These three conformations are located in different positions in phi, psi space, i.e., with a total deviation of 81 degrees, 100 degrees and 55 degrees from each other (with a root mean square deviation of several degrees per dihedral angle from each other). The nonbonded energies of the backbones, calculated along lines in phi, psi space connecting these three conformations, are all negative, without any intervening energy barriers (on an energy contour map in the phi, psi plane). Side chains were attached at several representative positions in this plane, and the total energy was minimized by varying the kappa's. The energies were of approximately the same magnitude as the previous ones, indicating that the conformation of low energy is flexible to some extent in a restricted region of phi, psi space. Interestingly, the difference delta phi i+1 in phi i+1 for the (i + 1)th residue from one conformation to another is approximately the same as -delta psi i for the ith residue; i.e., the plane of the peptide group between the ith and (i + 1)th residues re-orient without significant changes in the positions of the other atoms. The flexibility of the orientations of the planes of the peptide groups is probably coupled in a cooperative manner to the flexibility of the positions of the backbone and side-chain atoms.     

Multiple copy sampling in protein loop modeling: computational efficiency and sensitivity to dihedral angle perturbations.

Multiple copy sampling and the bond scaling-relaxation technique are combined to generate 3-dimensional conformations of protein loop segments. The computational efficiency and sensitivity to initial loop copy dispersion are analyzed. The multicopy loop modeling method requires approximately 20-50% of the computational time required by the single-copy method for the various protein segments tested. An analytical formula is proposed to estimate the computational gain prior to carrying out a multicopy simulation. When 7-residue loops within flexible proteins are modeled, each multicopy simulation can sample a set of loop conformations with initial dispersions up to +/- 15 degrees for backbone and +/- 30 degrees for side-chain rotatable dihedral angles. The dispersions are larger for shorter and smaller for longer and/or surface loops. The degree of convergence of loop copies during a simulation can be used to complement commonly used target functions (such as potential energy) for distinguishing between native and misfolded conformations. Furthermore, this convergence also reflects the conformational flexibility of the modeled protein segment. Application to simultaneously building all 6 hypervariable loops of an antibody is discussed.     

A novel exhaustive search algorithm for predicting the conformation of polypeptide segments in proteins

We present a fast ab initio method for the prediction of local conformations in proteins. The program, PETRA, selects polypeptide fragments from a computer-generated database (APD) encoding all possible peptide fragments up to twelve amino acids long. Each fragment is defined by a representative set of eight straight phi/psi pairs, obtained iteratively from a trial set by calculating how fragments generated from them represent the protein databank (PDB). Ninety-six percent (96%) of length five fragments in crystal structures, with a resolution better than 1.5 A and less than 25% identity, have a conformer in the database with less than 1 A root-mean-square deviation (rmsd). In order to select segments from APD, PETRA uses a set of simple rule-based filters, thus reducing the number of potential conformations to a manageable total. This reduced set is scored and sorted using rmsd fit to the anchor regions and a knowledge-based energy function dependent on the sequence to be modelled. The best scoring fragments can then be optimized by minimization of contact potentials and rmsd fit to the core model. The quality of the prediction made by PETRA is evaluated by calculating both the differences in rmsd and backbone torsion angles between the final model and the native fragment. The average rmsd ranges from 1.4 A for three residue loops to 3.9 A for eight residue loops.     

The nature of the turn in omega loops of proteins

An analysis of Omega loops in a nonredundant set of protein structures from the Protein Data Bank has been carried out to determine the nature of the "turn elements" present. Because Omega loops essentially reverse their direction in three-dimensional space, this analysis was made with respect to four turn elements identified as (1) Gly; (2) Pro; (3) a residue with alpha-helical phi,psi angles, termed a helical residue; and (4) a cis peptide. A set of 1079 Omega loops from a set of 680 proteins were used for the analysis. Apart from other criteria that define Omega loops, the selection of an Omega loop from a cluster of loops is based on an exposure index. In this study, analyses have been made with two sets of data: (1) Omega loops arising from a minimum exposure index indicative of a less exposed loop (xmin set) and (2) Omega loops with a maximum exposure index indicative of a relatively exposed loop (xmax set). Overall residue preferences and positional preferences have been examined. Positions of the turn elements for Omega loops of varying length have also been studied. Specific positional preferences are observed for particular turn elements with regard to the length of Omega loops. Analysis in terms of the turn elements can provide guidelines for modeling of loops in proteins. Apart from Pro, which has the natural tendency to form cis peptide bonds, a higher occurrence of non-Pro cis peptide bonds is observed. Torsion angles in Omega loops also indicate the occurrence of a large number of residues with helical phi,psi angles, necessary for the turn in the loop structures.     

Structural compromise of disallowed conformations in peptide and protein structures

Using a data set of 454 crystal structures of peptides and 80 crystal structures of non-homologous proteins solved at ultra high resolution of 1.2 A or better we have analyzed the occurrence of disallowed Ramachandran (phi, psi) angles. Out of 1492 and 13508 non-glycyl residues in peptides and proteins respectively 12 and 76 residues in the two datasets adopt clearly disallowed combinations of Ramachandran angles. These examples include a number of conformational points which are far away from any of the allowed regions in the Ramachandran map. According to the Ramachandran map a given (phi, psi) combination is considered disallowed when two non-bonded atoms in a system of two-linked peptide units with ideal geometry are prohibitively proximal in space. However, analysis of the disallowed conformations in peptide and protein structures reveals that none of the observations of disallowed conformations in the crystal structures correspond to a short contact between non-bonded atoms. A further analysis of deviations of bond lengths and angles, from the ideal peptide geometry, at the residue positions of disallowed conformations in the crystal structures suggest that individual bond lengths and angles are all within acceptable limits. Thus, it appears that the rare tolerance of disallowed conformations is possible by gentle and acceptable deviations in a number of bond lengths and angles, from ideal geometry, over a series of bonds resulting in a net gross effect of acceptable non-bonded inter-atomic distances.     

Protein loop modeling by using fragment assembly and analytical loop closure

Protein loops are often involved in important biological functions such as molecular recognition, signal transduction, or enzymatic action. The three dimensional structures of loops can provide essential information for understanding molecular mechanisms behind protein functions. In this article, we develop a novel method for protein loop modeling, where the loop conformations are generated by fragment assembly and analytical loop closure. The fragment assembly method reduces the conformational space drastically, and the analytical loop closure method finds the geometrically consistent loop conformations efficiently. We also derive an analytic formula for the gradient of any analytical function of dihedral angles in the space of closed loops. The gradient can be used to optimize various restraints derived from experiments or databases, for example restraints for preferential interactions between specific residues or for preferred backbone angles. We demonstrate that the current loop modeling method outperforms previous methods that employ residue‐based torsion angle maps or different loop closure strategies when tested on two sets of loop targets of lengths ranging from 4 to 12. Proteins 2010. © 2010 Wiley‐Liss, Inc.     

Composites of local structure propensities: Evidence for local encoding of long-range structure

To estimate how extensively the ensemble of denatured-state conformations is constrained by local side-chain–backbone interactions, propensities of each of the 20 amino acids to occur in mono- and dipeptides mapped to discrete regions of the Ramachandran map are computed from proteins of known structure. In addition, propensities are computed for the trans, gauche−, and gauche+ rotamers, with or without consideration of the values of phi and psi. These propensities are used in scoring functions for fragment threading, which estimates the energetic favorability of fragments of protein sequence to adopt the native conformation as opposed to hundreds of thousands of incorrect conformations. As finer subdivisions of the Ramachandran plot, neighboring residue phi/psi angles, and rotamers are incorporated, scoring functions become better at ranking the native conformation as the most favorable. With the best composite propensity function, the native structure can be distinguished from 300,000 incorrect structures for 71% of the 2130 arbitrary protein segments of length 40, 48% of 2247 segments of length 30, and 20% of 2368 segments of length 20. A majority of fragments of length 30–40 are estimated to be folded into the native conformation a substantial fraction of the time. These data suggest that the variations observed in amino acid frequencies in different phi/psi/chi1 environments in folded proteins reflect energetically important local side-chain–backbone interactions, interactions that may severely restrict the ensemble of conformations populated in the denatured state to a relatively small subset with nativelike structure.     

Exact Solutions for Internuclear Vectors and Backbone Dihedral Angles from NH Residual Dipolar Couplings in Two Media, and their Application in a Systematic Search Algorithm for Determining Protein Backbone Structure

We have derived a quartic equation for computing the direction of an internuclear vector from residual dipolar couplings (RDCs) measured in two aligning media, and two simple trigonometric equations for computing the backbone (phi,psi) angles from two backbone vectors in consecutive peptide planes. These equations make it possible to compute, exactly and in constant time, the backbone (phi,psi) angles for a residue from RDCs in two media on any single backbone vector type. Building upon these exact solutions we have designed a novel algorithm for determining a protein backbone substructure consisting of alpha-helices and beta-sheets. Our algorithm employs a systematic search technique to refine the conformation of both alpha-helices and beta-sheets and to determine their orientations using exclusively the angular restraints from RDCs. The algorithm computes the backbone substructure employing very sparse distance restraints between pairs of alpha-helices and beta-sheets refined by the systematic search. The algorithm has been demonstrated on the protein human ubiquitin using only backbone NH RDCs, plus twelve hydrogen bonds and four NOE distance restraints. Further, our results show that both the global orientations and the conformations of alpha-helices and beta-strands can be determined with high accuracy using only two RDCs per residue. The algorithm requires, as its input, backbone resonance assignments, the identification of alpha-helices and beta-sheets as well as sparse NOE distance and hydrogen bond restraints.     

Calibration of effective van der Waals atomic contact radii for proteins and peptides

Effective van der Waals radii were calibrated in such a way that molecular models built from standard bond lengths and bond angles reproduced the amino acid conformations observed by crystallography in proteins and peptides. The calibrations were based on the comparison of the Ramachandran plots prepared from high-resolution X-ray data of proteins and peptides with the allowed phi, psi torsional angle space for the dipeptide molecular models. The calibrated radii are useful as criteria with which to filter energetically improbable conformations in molecular modeling studies of proteins and peptides.     

Modeling structurally variable regions in homologous proteins with rosetta

A major limitation of current comparative modeling methods is the accuracy with which regions that are structurally divergent from homologues of known structure can be modeled. Because structural differences between homologous proteins are responsible for variations in protein function and specificity, the ability to model these differences has important functional consequences. Although existing methods can provide reasonably accurate models of short loop regions, modeling longer structurally divergent regions is an unsolved problem. Here we describe a method based on the de novo structure prediction algorithm, Rosetta, for predicting conformations of structurally divergent regions in comparative models. Initial conformations for short segments are selected from the protein structure database, whereas longer segments are built up by using three- and nine-residue fragments drawn from the database and combined by using the Rosetta algorithm. A gap closure term in the potential in combination with modified Newton's method for gradient descent minimization is used to ensure continuity of the peptide backbone. Conformations of variable regions are refined in the context of a fixed template structure using Monte Carlo minimization together with rapid repacking of side-chains to iteratively optimize backbone torsion angles and side-chain rotamers. For short loops, mean accuracies of 0.69, 1.45, and 3.62 A are obtained for 4, 8, and 12 residue loops, respectively. In addition, the method can provide reasonable models of conformations of longer protein segments: predicted conformations of 3A root-mean-square deviation or better were obtained for 5 of 10 examples of segments ranging from 13 to 34 residues. In combination with a sequence alignment algorithm, this method generates complete, ungapped models of protein structures, including regions both similar to and divergent from a homologous structure. This combined method was used to make predictions for 28 protein domains in the Critical Assessment of Protein Structure 4 (CASP 4) and 59 domains in CASP 5, where the method ranked highly among comparative modeling and fold recognition methods. Model accuracy in these blind predictions is dominated by alignment quality, but in the context of accurate alignments, long protein segments can be accurately modeled. Notably, the method correctly predicted the local structure of a 39-residue insertion into a TIM barrel in CASP 5 target T0186.     

The normal modes of the gramicidin-A dimer channel.

The dynamics of the gramicidin-A dimer channel is studied in the harmonic approximation by a vibrational analysis of the atomic motions relative to their equilibrium positions. The system is represented by an empirical potential energy function, and all degrees of freedom (bonds lengths, bond angles, and torsional angles) are allowed to vary. The thermal fluctuations in the backbone dihedral angles phi and psi, atomic root mean square displacements, and the correlations between the different amide planes are computed. It is found that only adjacent dihedral psi i and phi i+1 are strongly correlated, while different hydrogen-bonded amide planes are only weakly correlated. Modes with relatively low vibrational frequencies (75-175 cm-1) make the dominant contributions to the carbonyl librations. The general flexibility of the structure and the role of carbonyl librations in the ion transport mechanism are discussed.     

Theoretical conformational analysis of methylamide of N-acetyl-L-lysine]

The spatial structure of the methylamide of N-acetyl-L-lysine has been analysed taking into account non-bonded and electrostatic interactions, torsional energy, bond angles distortion and hydrogen bonding. Conformational capacities of the backbone and mutual dependence of spatial structures of the backbone and the side chain was described by conformational maps obtained by energy minimisation, the dihedral angles and the bond angles of the side chain being varied for every phi, psi point. Every possible combination for phi, psi, x1-x5-angles was used corresponding to the stable form of the backbone and to torsion potential minima of the initial approximations in the calculation of preferred conformations of the molecule. Comparisons are made between stable forms of the methylamide of N-acetyl-L-lysine and Lys residues in proteins with known structure.     

A survey of left-handed polyproline II helices

Left-handed polyproline II helices (PPII) are contiguous elements of protein secondary structure in which the phi and psi angles of constituent residues are restricted to around -75 degrees and 145 degrees, respectively. They are important in structural proteins, in unfolded states and as ligands for signaling proteins. Here, we present a survey of 274 nonhomologous polypeptide chains from proteins of known structure for regions that form these structures. Such regions are rare, but the majority of proteins contain at least one PPII helix. Most PPII helices are shorter than five residues, although the longest found contained 12 amino acids. Proline predominates in PPII, but Gln and positively charged residues are also favored. The basis of Gln's prevalence is its ability to form an i, i + 1 side-chain to main-chain hydrogen bond with the backbone carbonyl oxygen of the proceeding residue; this helps to fix the psi angle of the Gln and the phi and psi of the proceeding residue in PPII conformations and explains why Gln is favored at the first position in a PPII helix. PPII helices are highly solvent exposed, which explains why apolar amino acids are disfavored despite preferring this region of phi/psi space when in isolation. PPII helices have perfect threefold rotational symmetry and within these structures we find significant correlation between the hydrophobicity of residues at i and i + 3; thus, PPII helices in globular proteins can be considered to be amphipathic.     

Advantages of fine-grained side chain conformer libraries

We compare the modelling accuracy of two common rotamer libraries, the Dunbrack-Cohen and the 'Penultimate' rotamer libraries, with that of a novel library of discrete side chain conformations extracted from the Protein Data Bank. These side chain conformer libraries are extracted automatically from high-quality protein structures using stringent filters and maintain crystallographic bond lengths and angles. This contrasts with traditional rotamer libraries defined in terms of chi angles under the assumption of idealized covalent geometry. We demonstrate that side chain modelling onto native and near-native main chain conformations is significantly more successful with the conformer libraries than with the rotamer libraries when solely considering excluded-volume interactions. The rotamer libraries are inadequate to model side chains without atomic clashes on over 20% of targets if the backbone is held fixed in the native conformation. An algorithm is described for simultaneously modelling both main chain and side chain atoms during discrete ab initio sampling. The resulting models have equivalent root mean square deviations from the experimentally determined protein loops as models from backbone-only ensembles, indicating that all-atom modelling does not detract from the accuracy of conformational sampling.     

Reduced representation model of protein structure prediction: statistical potential and genetic algorithms.

A reduced representation model, which has been described in previous reports, was used to predict the folded structures of proteins from their primary sequences and random starting conformations. The molecular structure of each protein has been reduced to its backbone atoms (with ideal fixed bond lengths and valence angles) and each side chain approximated by a single virtual united-atom. The coordinate variables were the backbone dihedral angles phi and psi. A statistical potential function, which included local and nonlocal interactions and was computed from known protein structures, was used in the structure minimization. A novel approach, employing the concepts of genetic algorithms, has been developed to simultaneously optimize a population of conformations. With the information of primary sequence and the radius of gyration of the crystal structure only, and starting from randomly generated initial conformations, I have been able to fold melittin, a protein of 26 residues, with high computational convergence. The computed structures have a root mean square error of 1.66 A (distance matrix error = 0.99 A) on average to the crystal structure. Similar results for avian pancreatic polypeptide inhibitor, a protein of 36 residues, are obtained. Application of the method to apamin, an 18-residue polypeptide with two disulfide bonds, shows that it folds apamin to native-like conformations with the correct disulfide bonds formed.     

Folding of peptide fragments comprising the complete sequence of proteins. Models for initiation of protein folding. II. Plastocyanin.

In an attempt to understand the earliest events in the protein folding pathway, the complete sequence of French bean plastocyanin has been synthesized as a series of short peptide fragments, and the conformational preferences of each peptide examined in aqueous solution using proton n.m.r. methods. Plastocyanin consists largely of beta-sheet, with reverse turns and loops between the strands of the sheet, and one short helix. The n.m.r. experiments indicate that most of the peptides derived from the plastocyanin sequence have remarkably little propensity to adopt folded conformations in aqueous solution, in marked contrast to the peptides derived from the helical protein, myohemerythrin (accompanying paper). For most plastocyanin peptides, the backbone dihedral angles are predominantly in the beta-region of conformational space. Some of the peptides show weak NOE connectivities between adjacent amide protons, indicative of small local populations of backbone conformations in the a region of (phi,psi) space. A conformational preference for a reverse turn is seen in the sequence Ala65-Pro-Gly-Glu68, where a turn structure is found in the folded protein. Significantly, the peptide sequences that populate the alpha-region of (phi,psi) space are mostly derived from turn and loop regions in the protein. The addition of trifluoroethanol does not drive the peptides into helical conformations. In one region of the sequence, the n.m.r. spectra provide evidence of the formation of a hydrophobic cluster involving aromatic and aliphatic side-chains. These results have significance for understanding the initiation of protein folding. From these studies of the fragments of plastocyanin (this paper) and myohemerythrin (accompanying paper), it appears that there is a pre-partitioning of the conformational space sampled by the polypeptide backbone that is related to the secondary structure in the final folded state.     

Expanded turn conformations: characterization and sequence-structure correspondence in alpha-turns with implications in helix folding

Like the beta-turns, which are characterized by a limiting distance between residues two positions apart (i, i+3), a distance criterion (involving residues at positions i and i+4) is used here to identify alpha-turns from a database of known protein structures. At least 15 classes of alpha-turns have been enumerated based on the location in the phi,psi space of the three central residues (i+1 to i+3)-one of the major being the class AAA, where the residues occupy the conventional helical backbone torsion angles. However, moving towards the C-terminal end of the turn, there is a shift in the phi,psi angles towards more negative phi, such that the electrostatic repulsion between two consecutive carbonyl oxygen atoms is reduced. Except for the last position (i+4), there is not much similarity in residue composition at different positions of hydrogen and non-hydrogen bonded AAA turns. The presence or absence of Pro at i+1 position of alpha- and beta-turns has a bearing on whether the turn is hydrogen-bonded or without a hydrogen bond. In the tertiary structure, alpha-turns are more likely to be found in beta-hairpin loops. The residue composition at the beginning of the hydrogen bonded AAA alpha-turn has similarity with type I beta-turn and N-terminal positions of helices, but the last position matches with the C-terminal capping position of helices, suggesting that the existence of a "helix cap signal" at i+4 position prevents alpha-turns from growing into helices. Our results also provide new insights into alpha-helix nucleation and folding.     

Conformation of the third domain of turkey ovomucoid in solution. Structural analysis by two-dimensional Overhauser nuclear effect spectroscopy]

The conformations of a polypeptide chain of turkey ovomucoid third domain and its modified form with split reactive site peptide bond Leu-18--Glu-19 have been determined by the literary two-dimensional nuclear Overhauser effect spectroscopy data using an earlier suggested method. It has been found that the polypeptide domain backbone contains an alpha-helical fragment (residues 32-47), five segments having extended conformation (1-5, 11-17, 19-25, 29-31, 48-50) and beta-turn type 1 (26-29). Segments 23-26, 28-31 and 50-51 form an antiparallel beta-structure. Conformational states of the residues entering irregular domain segments have been analysed. Splitting of the reactive site peptide bond Leu-18--Glu-19 is shown to cause insignificant changes in the conformations of a number of amino acid residues except for Val-6 and Asp-7 ones which undergo essential conformational alterations. The conformations of domain in solution and of japanese quail ovomucoid third domain in crystal have been compared. The root-mean-square deviations for phi and psi angles indicate their high similarity. The conformations of turkey ovomucoid third domain and proteinase inhibitor BUSI IIA in solution have been analysed. In spite of moderate (50%) homology of primary structures, some 75% of amino acid residues are shown to have close conformational phi and psi parameters.