Searching for foldable protein structures using optimized energy functions

During evolution, the effective interactions between residues in a protein can be adjusted through mutations to allow the protein to fold to its native structure on an adequate time scale. We seek to address the question: Are there some structures that can be better optimized than others? Using exhaustive enumeration of the compact conformations of short proteins confined to simple lattices, we find that the best structures are those that contain contacts rare in random structures, indicating the importance of nonlocal contacts for assisting the folding process. Certain structural motifs such as long β-hairpins, Greek-key motifs, and jelly rolls, commonly found in proteins of known structure, have a high degree of optimizability. Contrary to what might be expected, positive correlations between the various interactions reduce optimizability. The optimization procedure produces a correlated energy landscape, which might assist folding. © 1995 John Wiley & Sons, Inc.     

Kinetic steps for α-helix formation

The kinetics of α-helix formation in polyalanine and polyglycine eicosamers (20-mers) were examined using torsional-coordinate molecular dynamics (MD). Of one hundred fifty-five MD experiments on extended (Ala)₂₀ carried out for 0.5 ns each, 129 (83%) formed a persistent α-helix. In contrast, the extended state of (Gly)₂₀ only formed a right-handed α-helix in two of the 20 MD experiments (10%), and these helices were not as long or as persistent as those of polyalanine. These simulations show helix formation to be a competition between the rates of (a) forming local hydrogen bonds (i.e. hydrogen bonds between any residue i and its i + 2, i + 3, i + 4, or i + 5th neighbor) and (b) forming nonlocal hydrogen bonds (HBs) between residues widely separated in sequence. Local HBs grow rapidly into an α-helix; but nonlocal HBs usually retard helix formation by “trapping” the polymer in irregular, “balled-up” structures. Most trajectories formed some nonlocal HBs, sometimes as many as eight. But, for (Ala)₂₀, most of these eventually rearranged to form local HBs that lead to α-helices. A simple kinetic model describes the rate of converting nonlocal HBs into α-helices. Torsional-coordinate MD speeds folding by eliminating bond and angle degrees of freedom and reducing dynamical friction. Thus, the observed 210 ps half-life for helix formation is likely to be a lower bound on the real rate. However, we believe the sequential steps observed here mirror those of real systems. Proteins 33:343–357, 1998. © 1998 Wiley-Liss, Inc.     

Hydrophobic residues can identify native protein structures

Evaluation of protein structures needs a trustworthy potential function. Although several knowledge‐based potential functions exist, the impact of different types of amino acids in the scoring functions has not been studied yet. Previously, we have reported the importance of nonlocal interactions in scoring function (based on Delaunay tessellation) in discrimination of native structures. Then, we have questioned the structural impact of hydrophobic amino acids in protein fold recognition. Therefore, a Hydrophobic Reduced Model (HRM) was designed to reduce protein structure of FS (Full Structure) into RS (Reduced Structure). RS is considered as a reduced structure of only seven hydrophobic amino acids (L, V, F, I, A, W, Y) and all their interactions. The presented model was evaluated via four different performance metrics including the number of correctly identified natives, the Z‐score of the native energy, the RMSD of the minimum score, and the Pearson correlation coefficient between the energy and the model quality. Results indicated that only nonlocal interactions between hydrophobic amino acids could be sufficient and accurate enough for protein fold recognition. Interestingly, the results of HRM is significantly close to the model that considers all amino acids (20‐amino acid model) to discriminate the native structure of the proteins on eleven decoy sets. This indicates that the power of knowledge‐based potential functions in protein fold recognition is mostly due to hydrophobic interactions. Hence, we suggest combining a different well‐designed scoring function for non‐hydrophobic interactions with HRM to achieve better performance in fold recognition.     

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.     

Folding type-specific secondary structure propensities of amino acids,derived from α-helical, β-sheet, α/β, and α+β proteins of known structures

Folding type-specific secondary structure propensities of 20 naturally occurring amino acids have been derived from α-helical, β-sheet, α/β, and α+β proteins of known structures. These data show that each residue type of amino acids has intrinsic propensities in different regions of secondary structures for different folding types of proteins. Each of the folding types shows markedly different rank ordering, indicating folding type-specific effects on the secondary structure propensities of amino acids. Rigorous statistical tests have been made to validate the folding type-specific effects. It should be noted that α and β proteins have relatively small α-helices and β-strands forming propensities respectively compared with those of α+β and α/β proteins. This may suggest that, with more complex architectures than α and β proteins, α+β and α/β proteins require larger propensities to distinguish from interacting α-helices and β-strands. Our finding of folding type-specific secondary structure propensities suggests that sequence space accessible to each folding type may have differing features. Differing sequence space features might be constrained by topological requirement for each of the folding types. Almost all strong β-sheet forming residues are hydrophobic in character regardless of folding types, thus suggesting the hydrophobicities of side chains as a key determinant of β-sheet structures. In contrast, conformational entropy of side chains is a major determinant of the helical propensities of amino acids, although other interactions such as hydrophobicities and charged interactions cannot be neglected. These results will be helpful to protein design, class-based secondary structure prediction, and protein folding. © 1998 John Wiley & Sons, Inc. Biopoly 45: 35–49, 1998     

Ribosomal subunit orientations determined in the monomeric ribosome by single and by double-labeling immune electron microscopy

The energies of two and three-chain antiparallel and parallel β-sheets have been minimized. The chains were considered to be equivalent. In each case, chains consisting of four and of eight l-alanine residues, respectively, with CH₃CO- and -NHCH₃ end groups were examined. Computations were carried out both for chains constrained to have a regular structure (i.e. the same φ and ψ dihedral angles for each residue) and for chains in which the regularity constraint was relaxed. All computed minimum-energy β-sheets were found to have a right-handed twist, as observed in proteins. As in the case of right-handed α-helices, it is the intrastrand non-bonded interaction energy that plays the key role in forcing β-sheets of l-amino acid residues to adopt a right-handed twist. The non-bonded energy contribution favoring the right-handed twist is the result of many small pairwise interatomic interactions involving the C^βH₃ groups. Polyglycine β-sheets, lacking the C^βH₃ side-chains, are not twisted. The twist of the poly-l-alanine sheet diminishes as the number of residues per chain increases, in agreement with observations. The twist of the four-residue chain increases somewhat (because of interstrand non-bonded interactions, also involving the C^βH₃ groups) in going from a single chain to a two-chain antiparallel structure, but then decreases slightly in going from a two-chain to a three-chain structure. β-Sheets in observed protein structures sometimes have a larger twist than those in the structures computed here. This may be due to irregularities in amino acid sequence and in hydrogenbonding patterns in the observed sheets, or to long-range interactions in proteins. The minimized energies of parallel β-sheets are considerably higher than those of the corresponding antiparallel β-sheets, indicating that parallel β-sheets are intrinsically less stable. This finding about the two kinds of β-sheets agrees with suggestions based on analyses of β-sheets observed in proteins. The energy difference between antiparallel and parallel β-sheets is due to closer packing of the chains and a more favorable alignment of the peptide dipoles in the antiparallel structures. The hydrogen-bond geometry in the computed antiparallel structures is very close to that proposed by Arnott et al. (1967) for the β-form of poly-l-alanine.     

Large-scale prediction of protein-protein interactions from structures

Background  

The prediction of protein-protein interactions is an important step toward the elucidation of protein functions and the understanding of the molecular mechanisms inside the cell. While experimental methods for identifying these interactions remain costly and often noisy, the increasing quantity of solved 3D protein structures suggests that in silico methods to predict interactions between two protein structures will play an increasingly important role in screening candidate interacting pairs. Approaches using the knowledge of the structure are presumably more accurate than those based on sequence only. Approaches based on docking protein structures solve a variant of this problem, but these methods remain very computationally intensive and will not scale in the near future to the detection of interactions at the level of an interactome, involving millions of candidate pairs of proteins.     

An analysis of side chain interactions and pair correlations within antiparallel β-sheets: The differences between backbone hydrogen-bonded and non-hydrogen-bonded residue pairs

Cross-strand pair correlations are calculated for residue pairs in antiparallel β-sheet for two cases: pairs whose backbone atoms are hydrogen bonded together (H-bonded site) and pairs which are not (non-H-bonded site). The statistics show that this distinction is important. When glycine is located on the edge of a sheet, it shows a 3:1 preference for the H-bonded site. Thestrongest observed correlations are for pairs of disulfide-bonded cystines, many of which adopt a close-packed conformation with each cystine in a spiral conformation of opposite chirality to its partner. It is likely that these pairs are a signature for the family of small, cystine-rich proteins. Most other strong positive and negative correlations involve charged and polar residues. It appears that electrostatic compatibility is the strongest factor affecting pair correlation. Significant correlations are observed for β- and γ-branched residues inthe non-H-bonded site. An examination of the structures showsa directionality in side chain packing. There is a correlation between (1) the directionality in the packing interactions of non-H-bonded β- and γ-branched residue pairs, (2) the handedness of the observed enantiomers of chiral β-branched side chains, and (3) the handedness of the twist of β-sheet. These findings have implications for the formation of β-sheets during protein folding and the mechanism by which the sheet becomes twisted. © 1995 Wiley-Liss, Inc.     

Local and nonlocal interactions in globular proteins and mechanisms of alcohol denaturation.

How important are helical propensities in determining the conformations of globular proteins? Using the two-dimensional lattice model and two monomer types, H (hydrophobic) and P (polar), we explore both nonlocal interactions, through an HH contact energy, as developed in earlier work, and local interactions, through a helix energy, σ. By computer enumeration, the partition functions for short chains are obtained without approximation for the full range of both types of energy. When nonlocal interactions dominate, some sequences undergo coil-globule collapse to a unique native structure. When local interactions dominate, all sequences undergo helix–coil transitions. For two different conformational properties, the closest correspondence between the lattice model and proteins in the Protein Data Bank is obtained if the model local interactions are made small compared to the HH contact interaction, suggesting that helical propensities may be only weak determinants of globular protein structures in water. For some HP sequences, varying σ/ leads to additional sharp transitions (sometimes several) and to “conformational switching” between unique conformations. This behavior resembles the transitions of globular proteins in water to helical states in alcohols. In particular, comparison with experiments shows that whereas urea as a denaturant is best modeled as weakening both local and nonlocal interactions, trifluoroethanol is best modeled as mainly weakening HH interactions and slightly enhancing local helical interactions.     

Geometric cooperativity and anticooperativity of three-body interactions in native proteins

Characterizing multibody interactions of hydrophobic, polar, and ionizable residues in protein is important for understanding the stability of protein structures. We introduce a geometric model for quantifying 3-body interactions in native proteins. With this model, empirical propensity values for many types of 3-body interactions can be reliably estimated from a database of native protein structures, despite the overwhelming presence of pairwise contacts. In addition, we define a nonadditive coefficient that characterizes cooperativity and anticooperativity of residue interactions in native proteins by measuring the deviation of 3-body interactions from 3 independent pairwise interactions. It compares the 3-body propensity value from what would be expected if only pairwise interactions were considered, and highlights the distinction of propensity and cooperativity of 3-body interaction. Based on the geometric model, and what can be inferred from statistical analysis of such a model, we find that hydrophobic interactions and hydrogen-bonding interactions make nonadditive contributions to protein stability, but the nonadditive nature depends on whether such interactions are located in the protein interior or on the protein surface. When located in the interior, many hydrophobic interactions such as those involving alkyl residues are anticooperative. Salt-bridge and regular hydrogen-bonding interactions, such as those involving ionizable residues and polar residues, are cooperative. When located on the protein surface, these salt-bridge and regular hydrogen-bonding interactions are anticooperative, and hydrophobic interactions involving alkyl residues become cooperative. We show with examples that incorporating 3-body interactions improves discrimination of protein native structures against decoy conformations. In addition, analysis of cooperative 3-body interaction may reveal spatial motifs that can suggest specific protein functions.     

Interaction of transmembrane helices by a knobs-into-holes packing characteristic of soluble coiled coils

Membrane-embedded protein domains frequently exist as α-helical bundles, as exemplified by photosynthetic reaction centers, bacteriorhodopsin, and cytochrome C oxidase. The sidechain packing between their transmembrane helices was investigated by a nearest-neighbor analysis which identified sets of interfacial residues for each analyzed helix–helix interface. For the left-handed helix–helix pairs, the interfacial residues almost exclusively occupy positions a, d, e, or g within a heptad motif (abcdefg) which is repeated two to three times for each interacting helical surface. The connectivity between the interfacial residues of adjacent helices conforms to the knobs-into-holes type of sidechain packing known from soluble coiled coils. These results demonstrate on a quantitative basis that the geometry of sidechain packing is similar for left-handed helix–helix pairs embedded in membranes and coiled coils of soluble proteins. The transmembrane helix–helix interfaces studied are somewhat less compact and regular as compared to soluble coiled coils and tolerate all hydrophobic amino acid types to similar degrees. The results are discussed with respect to previous experimental findings which demonstrate that specific interactions between transmembrane helices are important for membrane protein folding and/or oligomerization. Proteins 31:150–159, 1998. © 1998 Wiley-Liss, Inc.     

Visualization of the nature of protein folding by a study of a distance constraint approach in two-dimensional models

A distance constraint approach is applied to two-dimensional models of proteins in order to visualize the nature of protein folding and to examine the relative roles of different ranges of interaction. Three different native structures (I, II, and III) are considered; they have two different kinds of residues, viz., hydrophobic and hydrophilic, and different sequences of these residues. We examine how the distance constraint approach functions in the prediction of protein folding when we know the sequence of the residues, the (fixed) bond lengths, the mean distances between residues i and i + 2, and i and i + 3, and the mean distances for hydrophobic–hydrophobic, hydrophobic–hydrophilic, and hydrophilic–hydrophilic contacts between residues i and i + j, where j ≥ 4. This approach involves optimization of an object function with respect to 98 variables and is not free of the multiple-minimum problem. The optimization is always terminated if the chain is entangled and/or the segments (residues) are packed too compactly to move. In order to escape from such situations and to take the excluded-volume effect into account, a Monte Carlo method is used after the optimization is trapped in local minima. Success in the prediction of folding is found to depend on the starting conformations and on the native conformations. Fair success is obtained in predicting the helix-like structure in protein I and the overall structure of protein III, but not the β-like structures of proteins I and II. Insofar as the prediction of the structure of protein III is reasonable, it appears that some sequences of residues produce greater constraints on their conformations than others, if one considers only the hydrophobic and hydrophilic nature of the residues. These results imply that, in the folding of real proteins in three dimensions, the competition for hydrophobic (and hydrophilic) residues for inside (outside) positions in the molecule probably constitutes a necessary but not a sufficient condition to form and stabilize the native structure. The failure to predict the structure of protein II, and part of that of protein I, suggests that there are two types of long-range interactions. One (which we considered here) is nonspecific (i.e., is defined only in terms of contacts between residues of the same or different polarity) and acts at any stage of protein folding; the other (which we did not consider here) is a specific interaction between residues in pairs and contributes only when the residues in the specific pair take on the native conformation. Presumably, incorporation of such specific long-range interactions, together with the nonspecific ones, is necessary for successful protein folding, using the distance constraint approach.     

Biased Usages of Arginines and Lysines in Proteins Are Correlated with Local-Scale Fluctuations of the G + C Content of DNA Sequences

Amino acid residues arginine (R) and lysine (K) have similar physicochemical characteristics and are often mutually substituted during evolution without affecting protein function. Statistical examinations on human proteins show that more R than K residues are used in the proximity of R residues, whereas more K than R are used near K residues. This biased use occurs on both a global and a local scale (shorter than ∼100 residues). Even within a given exon, G + C-rich and A + T-rich short DNA segments preferentially encode R and K, respectively. The biased use of R and K on a local scale is also seen in Saccharomyces cerevisiae and Caenorhabdidtis elegans, which lack global-scale mosaic structures with varying GC%, or isochores. Besides R and K, several amino acids are also used with a positive or negative correlation with the local GC% of third codon bases. The local-, or ``within-gene'-, scale heterogeneity of the DNA sequence may influence the sequence of the encoded protein segment. Received: 2 March 1998 / Accepted: 23 April 1998     

Theory of protein molecule self-organization. II. A comparison of calculated thermodynamic parameters of local secondary structures with experiments

Constants of the helix–coil transition for all natural amino acid residues are evaluated on the basis of thermodynamic parameters obtained in paper I of this series. The specific effects at the termini of the helices are also considered as well as the parameters controlling the formation of β-bends in the unfolded protein chain. Evaluated s constants of the helix–coil transition agree with the experimental data on helix–coil transitions of synthetic polypeptides in water. Only a very qualitative correlation exists between s constants (both experimental and theoretical) and the occurrence of corresponding residues in internal turns of α-helices in globular proteins: residues with s > 1 occur in helices as a rule more often than residues with s < 1. At the same time a direct correlation is demonstrated between theoretical parameters of residue incorporation into α-helical termini and β-bends in an unfolded polypeptide chain and the occurrence of residues in corresponding positions of the globular protein secondary structures.     

Measurements of protein sequence-structure correlations

Correlations between protein structures and amino acid sequences are widely used for protein structure prediction. For example, secondary structure predictors generally use correlations between a secondary structure sequence and corresponding primary structure sequence, whereas threading algorithms and similar tertiary structure predictors typically incorporate interresidue contact potentials. To investigate the relative importance of these sequence-structure interactions, we measured the mutual information among the primary structure, secondary structure and side-chain surface exposure, both for adjacent residues along the amino acid sequence and for tertiary structure contacts between residues distantly separated along the backbone. We found that local interactions along the amino acid chain are far more important than non-local contacts and that correlations between proximate amino acids are essentially uninformative. This suggests that knowledge-based contact potentials may be less important for structure predication than is generally believed.     

Comparative analysis of 13C chemical shifts of β-sheet amyloid proteins and outer membrane proteins

Cross-β amyloid fibrils and membrane-bound β-barrels are two important classes of β-sheet proteins. To investigate whether there are systematic differences in the backbone and sidechain conformations of these two families of proteins, here we analyze the ¹³C chemical shifts of 17 amyloid proteins and 7 β-barrel membrane proteins whose high-resolution structures have been determined by NMR. These 24 proteins contain 373 β-sheet residues in amyloid fibrils and 521 β-sheet residues in β-barrel membrane proteins. The ¹³C chemical shifts are shown in 2D ¹³C–¹³C correlation maps, and the amino acid residues are categorized by two criteria: (1) whether they occur in β-strand segments or in loops and turns; (2) whether they are water-exposed or dry, facing other residues or lipids. We also examine the abundance of each amino acid in amyloid proteins and β-barrels and compare the sidechain rotameric populations. The ¹³C chemical shifts indicate that hydrophobic methyl-rich residues and aromatic residues exhibit larger static sidechain conformational disorder in amyloid fibrils than in β-barrels. In comparison, hydroxyl- and amide-containing polar residues have more ordered sidechains and more ordered backbones in amyloid fibrils than in β-barrels. These trends can be explained by steric zipper interactions between β-sheet planes in cross-β fibrils, and by the interactions of β-barrel residues with lipid and water in the membrane. These conformational trends should be useful for structural analysis of amyloid fibrils and β-barrels based principally on NMR chemical shifts.

     

Non-canonical H-bonds in β-lactamases: importance of C–H···π interactions

β-Lactamase production is the common mechanism of resistance of β-lactam antibiotics. Knowledge of inter-residue interactions in protein structures increases our understanding of protein structure and stability. We have systematically analysed the contribution of C–H···π interactions to the stability of β-lactamases. Most of the interactions are long range and most of the interacting residues are evolutionarily conserved. The occurrence of C–H···π interactions in active sites and metal binding sites is very low in β-lactamases. Hence, C–H···π interactions are important determinants of stability in β-lactamases and they may not play a significant role in specificity. The results from this study provide valuable insights for understanding the stability patterns of β-lactamases and their relation to various other environmental preferences.     

A hierarchical order within protein structures underlies large separations between strands in β‐sheets

Protein β‐sheets often involve nonlocal interactions between parts of the polypeptide chain that are separated by hundreds of residues, raising the question of how these nonlocal contacts form. A recent study of the smallest β‐sheets found that their formation was not driven by signals hidden in the primary sequence. Instead, the strands in these sheets were either local in sequence, or, when separated by large sequential distances, the intervening residues were found to fold into compact modules that anchored distant parts of the chain in close spatial proximity. Here, we examine larger β‐sheets to investigate the extensibility of this principle. From an analysis of the β‐sheets in a nonredundant protein dataset, we find that a highly ordered hierarchical relationship exists in the intervening structure between nonlocal β‐strands. This observation is almost universal: virtually all β‐sheets, no matter their complexity, appear to adopt an antiparallel model to manage the nonlocal aspects of their assembly, one where the chain, having left the vicinity of an unfinished β‐sheet, retraces its steps via the same route to complete the initial sheet. Exceptions typically involve unstructured regions at chain termini. Moreover, an analysis of the residues involved in nonlocal crossstrand interactions did not produce any evidence of a signal hidden in the sequence that might direct long‐range interactions. These results build on those reported for the smallest sheets, suggesting that sheet formation is either local in sequence or local in space following prior folding events that anchor disparate parts of the chain in close proximity. Proteins 2013. © 2012 Wiley Periodicals, Inc.