Calculation of protein conformation by the build-up procedure. Application to bovine pancreatic trypsin inhibitor using limited simulated nuclear magnetic resonance data

Reducing the dimensionality of the protein-folding search problem

How does a folding protein negotiate a vast, featureless conformational landscape and adopt its native structure in biological real time? Motivated by this search problem, we developed a novel algorithm to compare protein structures. Procedures to identify structural analogs are typically conducted in three-dimensional space: the tertiary structure of a target protein is matched against each candidate in a database of structures, and goodness of fit is evaluated by a distance-based measure, such as the root-mean-square distance between target and candidate. This is an expensive approach because three-dimensional space is complex. Here, we transform the problem into a simpler one-dimensional procedure. Specifically, we identify and label the 11 most populated residue basins in a database of high-resolution protein structures. Using this 11-letter alphabet, any protein's three-dimensional structure can be transformed into a one-dimensional string by mapping each residue onto its corresponding basin. Similarity between the resultant basin strings can then be evaluated by conventional sequence-based comparison. The disorder → order folding transition is abridged on both sides. At the onset, folding conditions necessitate formation of hydrogen-bonded scaffold elements on which proteins are assembled, severely restricting the magnitude of accessible conformational space. Near the end, chain topology is established prior to emergence of the close-packed native state. At this latter stage of folding, the chain remains molten, and residues populate natural basins that are approximated by the 11 basins derived here. In essence, our algorithm reduces the protein-folding search problem to mapping the amino acid sequence onto a restricted basin string.

A versatile method for systematic conformational searches: application to CheY

A novel molecular structure prediction method, the Z Method, is described. It provides a versatile platform for the development and use of systematic, grid-based conformational search protocols, in which statistical information (i.e., rotamers) can also be included. The Z Method generates trial structures by applying many changes of the same type to a single starting structure, thereby sampling the conformation space in an unbiased way. The method, implemented in the CHARMM program as the Z Module, is applied here to an illustrative model problem in which rigid, systematic searches are performed in a 36-dimensional conformational space that describes the relative positions of the 10 secondary structural elements of the protein CheY. A polar hydrogen representation with an implicit solvation term (EEF1) is used to evaluate successively larger fragments of the protein generated in a hierarchical build-up procedure. After a final refinement stage, and a total computational time of about two-and-a-half CPU days on AMD Opteron processors, the prediction is within 1.56 Å of the native structure. The errors in the predicted backbone dihedral angles are found to approximately cancel. Monte Carlo and simulated annealing trials on the same or smaller versions of the problem, using the same atomic model and energy terms, are shown to result in less accurate predictions. Although the problem solved here is a limited one, the findings illustrate the utility of systematic searches with atom-based models for macromolecular structure prediction and the importance of unbiased sampling in structure prediction methods.

On the orientation of the backbone dipoles in native folds

The role of electrostatic interactions in determining the native fold of proteins has been investigated by analyzing the alignment of peptide bond dipole moments with the local electrostatic field generated by the rest of the molecule with and without solvent effects. This alignment was calculated for a set of 112 native proteins by using charges from a gas phase potential. Most of the peptide dipoles in this set of proteins are on average aligned with the electrostatic field. The dipole moments associated with alpha-helical conformations show the best alignment with the electrostatic field, followed by residues in beta-strand conformations. The dipole moments associated with other secondary structure elements are on average better aligned than in randomly generated conformations. The alignment of a dipole with the local electrostatic field depends on both the topology of the native fold and the charge distribution assumed for all of the residues. The influences of (i) solvent effects, (ii) different sets of charges, and (iii) the charge distribution assumed for the whole molecule were examined with a subset of 22 proteins each of which contains <30 ionizable groups. The results show that alternative charge distribution models lead to significant differences among the associated electrostatic fields, whereas the electrostatic field is less sensitive to the particular set of the adopted charges themselves (empirical conformational energy program for peptides or parameters for solvation energy).

Improved greedy algorithm for protein structure reconstruction

This article concerns the development of an improved greedy algorithm for protein structure reconstruction. Our stochastic greedy algorithm, which attempts to locate the ground state of an approximate energy function, exploits the fact that protein structures consist of overlapping structural building blocks that are not independent. Application of this approach to a series of 16 proteins with 50-250 amino acids leads to predicted models deviating from the experimental structures by 0.5 A RMSD using an RMSD-based energy function and within 1.5 to 4.8 A RMSD using a Go-based energy function. The Go-based results are significant because they illustrate the strength of combining structural fragments and stochastic greedy algorithms in capturing the native structures of proteins stabilized by long-range interactions separated by more than 30 amino acids. These results clearly open the door to less computationally demanding solutions to predict structures from sequences.

Addition of side chains to a known backbone with defined side-chain centroids

An automatic procedure is proposed for adding side chains to a protein backbone; it is based on optimization of a simplified energy function for peptide side chains, given its backbone and positions of side-chain centroids. The energy is expressed as a sum of the energies of interaction between side chains, and a harmonic penalty function accounting for the preservation of the positions of the C(alpha) atoms and the side-chain centroids. The energy of side-chain interactions is calculated with the soft-sphere ECEPP/3 potential. A Monte Carlo search is carried out to explore all possible side-chain orientations within a fixed backbone and side-chain centroid positions. The initial, usually extended, side-chain conformations are taken directly from the ECEPP/3 database. The procedure was tested on six experimental (X-ray or NMR) structures: immunoglobulin binding protein (PDB code 1IGD, an alpha+beta-protein); transcription factor PML (PDB code 1BOR, a 49-104 fragment of the ring finger domain, predominantly beta-protein); bovine pancreatic trypsin inhibitor (crystal form II) (PDB code 1BPI, an alpha+beta-protein); the monomer of human deoxyhemoglobin (PDB code 1BZ0, an alpha-helical structure); chain A of alcohol dehydrogenase from Drosophila lebanonensis (PDB code 1A4U); as well as on the 10-55 portion of the B domain of staphylococcal protein A (PDB code 1BDD). In all cases except 1BPI, the data for the algorithm (i.e. the backbone or C(alpha) coordinates and the positions of side-chain centroids) were taken from the experimental structures. For protein A, the C(alpha) coordinates and positions of side-chain centroids were also taken from the 1.9-A-resolution model predicted by the UNRES force field. In all comparisons with experimental structures, complete side-chain geometry was reconstructed with a root-mean-square (RMS) deviation of approximately 0.6-0.9 A from the heavy atoms when complete backbone and side-chain-centroid coordinates were used in reconstruction, or approximately 1.0 A when the C(alpha) and centroid coordinates were used.

Small libraries of protein fragments model native protein structures accurately

Prediction of protein structure depends on the accuracy and complexity of the models used. Here, we represent the polypeptide chain by a sequence of rigid fragments that are concatenated without any degrees of freedom. Fragments chosen from a library of representative fragments are fit to the native structure using a greedy build-up method. This gives a one-dimensional representation of native protein three-dimensional structure whose quality depends on the nature of the library. We use a novel clustering method to construct libraries that differ in the fragment length (four to seven residues) and number of representative fragments they contain (25-300). Each library is characterized by the quality of fit (accuracy) and the number of allowed states per residue (complexity). We find that the accuracy depends on the complexity and varies from 2.9A for a 2.7-state model on the basis of fragments of length 7-0.76A for a 15-state model on the basis of fragments of length 5. Our goal is to find representations that are both accurate and economical (low complexity). The models defined here are substantially better in this regard: with ten states per residue we approximate native protein structure to 1A compared to over 20 states per residue needed previously. For the same complexity, we find that longer fragments provide better fits. Unfortunately, libraries of longer fragments must be much larger (for ten states per residue, a seven-residue library is 100 times larger than a five-residue library). As the number of known protein native structures increases, it will be possible to construct larger libraries to better exploit this correlation between neighboring residues. Our fragment libraries, which offer a wide range of optimal fragments suited to different accuracies of fit, may prove to be useful for generating better decoy sets for ab initio protein folding and for generating accurate loop conformations in homology modeling.

Stabilization centers in proteins: identification, characterization and predictions

Methods are presented to locate residues, stabilization center elements, which are expected to stabilize protein structures by preventing their decay with their cooperative long range interactions. Artificial neural network-based algorithms were developed to predict these residues from the primary structure of single proteins and from the amino acid sequences of homologous proteins. The prediction accuracy using only single sequence information is 65%, but the incorporation of evolutionary information in the form of multiple alignments and conservation scores raises the efficiency by 3%. The composition, relative accessibility, number and type of interactions, conservation and the X-ray thermal factor of the identified stabilization center residues are different, not only from the whole data set but from the rest of the long range interacting residues as well. The most frequent stabilization center residues are usually found at buried positions and have a hydrophobic or aromatic side-chain, but some polar or charged residues also play an important role in the stabilization. The stabilization centers show significant difference in the composition and in the type of linked secondary structural elements compared with the rest of the residues. The performed structural and sequential conservation analysis showed the higher conservation of stabilization centers over protein families. The relation of the proposed stabilization centers to folding nuclei is also discussed.

Recent developments in the theory of protein folding: searching for the global energy minimum

Statistical mechanical theories and computer simulation are being used to gain an understanding of the fundamental features of protein folding. A major obstacle in the computation of protein structures is the multiple-minima problem arising from the existence of many local minima in the multidimensional energy landscape of the protein. This problem has been surmounted for small open-chain and cyclic peptides, and for regular-repeating sequences of models of fibrous proteins. Progress is being made in resolving this problem for globular proteins.

Modeling mutations and homologous proteins

A protein sequence with at lease 40% identity to a known structure can now be modelled automatically, with an accuracy approaching that o fa low-resolution X-ray structure or a medium-resolution nuclear magnetic resonance structure. In general, these models have goods stereochemistry and an overall structural accuracy that is as high as the similarity between the template and the actual structure being predicted. As a result, the number of sequences that can be modelled is an order of magnitude larger then the number of experimentally determined protein structures. In addition, evaluation techniques are available that can estimated errors in different regions of the model. Thus, the number of applications where homology modelling is proving useful is growing rapidly.

The complexity and accuracy of discrete state models of protein structure

The prediction of protein structure depends on the quality of the models used. In this paper, we examine the relationship between the complexity and accuracy of representation of various models of protein alpha-carbon backbone structure. First, we develop an efficient algorithm for the near optimal fitting of arbitrary lattice and off-lattice models of polypeptide chains to their true X-ray structures. Using this, we show that the relationship between the complexity of a model, taken as the number of possible conformational states per residue, and the simplest measure of accuracy, the root-mean-square deviation from the X-ray structure, is approximately (Accuracy) varies; is directly proportional to (Complexity)-1/2. This relationship is insensitive to the particularities of individual models, i.e. lattice and off-lattice models of the same complexity tend to have similar average root-mean-square deviations, and this also implies that improvements in model accuracy with increasing complexity are very small. However, other measures of model accuracy, such as the preservation of X-ray residue-residue contacts and the alpha-helix, do distinguish among models. In addition, we show that low complexity models, which take into account the uneven distribution of residue conformations in real proteins, can represent X-ray structures as accurately as more complex models, which do not: a selected 6-state model can represent protein structures almost as accurately (1.7 A root-mean-square) as a 17-state lattice model (1.6 A root-mean-square). Finally, we use a novel optimization procedure to generate eight 4-state models, which fit native proteins to an average of 2.4 A, and preserve 85% of native residue-residue contacts. We discuss the implications of these findings for protein folding and the prediction of protein conformation.

The energy of formation of internal loops in triple-helical collagen polypeptides

The sequence-dependent local destabilization in the interior of the collagen triple helix has been evaluated by means of conformational energy computations. Using a model poly(Gly-Pro-Pro) triple helix as the reference state, a method was developed for generating local loops, i.e., internal deformations, and analyzing their conformations. A seven-residue Gly-Pro-Pro- Gly-Pro-Pro-Gly fragment was replaced by the Gly-Pro-Ala-Gly-Ala-Ala-Gly sequence in one, two, or all three of the strands of the loop region. A set of loop conformations was generated in which the ends of the loop were initially fixed in the triple-helical structure. The potential energy of the entire deformed triple helix was then minimized, resulting in a variety of structures that contained deformed loops. The conformations of the triple helices at the two ends of the loops remained essentially unchanged in many of the low-energy conformations. In numerous high-energy conformations, however, the triple-helical segments were also partially or totally disrupted. The minimum-energy conformations of the whole structures were compared in terms of rms deviations of atomic coordinates with respect to the original triple helix, and of the shapes of the loops (using a distance function derived from differential geometry). Three new geometrical parameters-stretch S, kink K, and unwinding U-were defined to describe the changes in the overall orientation of the triple helices at the two ends of the loop. It is shown that, when the number of Pro residues in a short fragment is reduced, the triple helical structure can accomodate internal loops (i.e., distortions) within a 5 kcal/mol cutoff from the essentially unperturbed triple helical structure. For structures with a Gly-Pro-Ala-Gly-Ala-Ala-Gly sequence in all three strands, the probability of finding conformations with internal loops is small, i.e., 0.06. Internal loops affect the overall orientation of these structures, as measured by the helix-distortion parameters S, K, and U.

A statistical analysis of side-chain conformations in proteins: comparison with ECEPP predictions

A comparison of the statistical distributions of side-chain conformations of 17 amino acids (Gly, Ala, and Pro excluded), observed in 63 nonhomologous globular proteins (covering 10,832 residues), is made with similar distributions calculated from the low-energy conformational states for the same amino acids (blocked with acetyl and N-methylamide groups at the N- and C-termini, respectively) obtained by Vásquez et al. [(1983), Macromolecules 16, 1043-1049] using the ECEPP/2 force field. Those residues (i) with linear side chains (Arg, Lys, Met, Cys, Ser), or those that are unbranched through the gamma-carbon atom (Glu, Gln) show good agreement, whereas (ii) those with side chains that are branched at C beta or C gamma show poor agreement with ECEPP calculations. A possible explanation for this is shown to be the greater tendency for side-chain atoms in class (ii) to interact with the backbone and/or adjacent side chains. Accordingly, ECEPP/3 calculations, carried out after elongating the backbone chain of the model peptide unit (by adding three Ala residues on each side of the central residue, and then blocking the termini as before), result in distributions that are often closer to the observed side-chain distributions. The implications of these results for the relative importance of short-range versus long-range interactions in determining protein structure are discussed.

The effect of the l-azetidine-2-carboxylic acid residue on protein conformation. IV. Local substitutions in the collagen triple helix

The properties of collagen are affected by the replacement of Pro by imino acid analogues. The structural effect of the low-level local substitution of L-azetidine-2-carboxylic acid (Aze) has been analyzed by computing the energy of CH3CO-(Gly-Pro-Pro)4-NHCH3 triple helices in which a single residue of one strand has been replaced by Aze. When Aze is in position Y of a (Gly-X-Y) unit, low-energy local deformations are introduced in the triple helix, i.e., it becomes more flexible. On the other hand, the flexibility of the triple helix is not increased with Aze in position X. The energy of the triple helix to coil transition is not changed significantly by this amount of substitution. In an earlier study, we have demonstrated that the regular substitution of Aze in every tripeptide distorts or destabilizes the triple helix to a large extent [A. Zagari, G. Némethy, & H. A. Scheraga (1990) Biopolymers, Vol. 30, pp. 967-974]. Thus, it appears that a high level of substitution is required to cause the observed chemical and biological effects of Aze on collagen.

Origins of structural diversity within sequentially identical hexapeptides

Efforts to predict protein secondary structure have been hampered by the apparent structural plasticity of local amino acid sequences. Kabsch and Sander (1984, Proc. Natl. Acad. Sci. USA 81, 1075-1078) articulated this problem by demonstrating that identical pentapeptide sequences can adopt distinct structures in different proteins. With the increased size of the protein structure database and the availability of new methods to characterize structural environments, we revisit this observation of structural plasticity. Within a set of proteins with less than 50% sequence identity, 59 pairs of identical hexapeptide sequences were identified. These local structures were compared and their surrounding structural environments examined. Within a protein structural class (alpha/alpha, beta/beta, alpha/beta, alpha + beta), the structural similarity of sequentially identical hexapeptides usually is preserved. This study finds eight pairs of identical hexapeptide sequences that adopt beta-strand structure in one protein and alpha-helical structure in the other. In none of the eight cases do the members of these sequences pairs come from proteins within the same folding class. These results have implications for class dependent secondary structure prediction algorithms.

Calculation of protein backbone geometry from alpha-carbon coordinates based on peptide-group dipole alignment

An algorithm is proposed for the conversion of a virtual-bond polypeptide chain (connected C alpha atoms) to an all-atom backbone, based on determining the most extensive hydrogen-bond network between the peptide groups of the backbone, while maintaining all of the backbone atoms in energetically feasible conformations. Hydrogen bonding is represented by aligning the peptide-group dipoles. These peptide groups are not contiguous in the amino acid sequence. The first dipoles to be aligned are those that are both sufficiently close in space to be arranged in approximately linear arrays termed dipole paths. The criteria used in the construction of dipole paths are: to assure good alignment of the greatest possible number of dipoles that are close in space; to optimize the electrostatic interactions between the dipoles that belong to different paths close in space; and to avoid locally unfavorable amino acid residue conformations. The equations for dipole alignment are solved separately for each path, and then the remaining single dipoles are aligned optimally with the electrostatic field from the dipoles that belong to the dipole-path network. A least-squares minimizer is used to keep the geometry of the alpha-carbon trace of the resulting backbone close to that of the input virtual-bond chain. This procedure is sufficient to convert the virtual-bond chain to a real chain; in applications to real systems, however, the final structure is obtained by minimizing the total ECEPP/2 (empirical conformational energy program for peptides) energy of the system, starting from the geometry resulting from the solution of the alignment equations. When applied to model alpha-helical and beta-sheet structures, the algorithm, followed by the ECEPP/2 energy minimization, resulted in an energy and backbone geometry characteristic of these alpha-helical and beta-sheet structures. Application to the alpha-carbon trace of the backbone of the crystallographic 5PTI structure of bovine pancreatic trypsin inhibitor, followed by ECEPP/2 energy minimization with C alpha-distance constraints, led to a structure with almost as low energy and root mean square deviation as the ECEPP/2 geometry analog of 5PTI, the best agreement between the crystal and reconstructed backbone being observed for the residues involved in the dipole-path network.

Unfolding and refolding of the native structure of bovine pancreatic trypsin inhibitor studied by computer simulations

A new procedure for studying the folding and unfolding of proteins, with an application to bovine pancreatic trypsin inhibitor (BPTI), is reported. The unfolding and refolding of the native structure of the protein are characterized by the dimensions of the protein, expressed in terms of the three principal radii of the structure considered as an ellipsoid. A dynamic equation, describing the variations of the principal radii on the unfolding path, and a numerical procedure to solve this equation are proposed. Expanded and distorted conformations are refolded to the native structure by a dimensional-constraint energy minimization procedure. A unique and reproducible unfolding pathway for an intermediate of BPTI lacking the [30,51] disulfide bond is obtained. The resulting unfolded conformations are extended; they contain near-native local structure, but their longest principal radii are more than 2.5 times greater than that of the native structure. The most interesting finding is that the majority of expanded conformations, generated under various conditions, can be refolded closely to the native structure, as measured by the correct overall chain fold, by the rms deviations from the native structure of only 1.9-3.1 A, and by the energy differences of about 10 kcal/mol from the native structure. Introduction of the [30,51] disulfide bond at this stage, followed by minimization, improves the closeness of the refolded structures to the native structure, reducing the rms deviations to 0.9-2.0 A. The unique refolding of these expanded structures over such a large conformational space implies that the folding is strongly dictated by the interactions in the amino acid sequence of BPTI. The simulations indicate that, under conditions that favor a compact structure as mimicked by the volume constraints in our algorithm, the expanded conformations have a strong tendency to move toward the native structure; therefore, they probably would be favorable folding intermediates. The results presented here support a general model for protein folding, i.e., progressive formation of partially folded structural units, followed by collapse to the compact native structure. The general applicability of the procedure is also discussed.

Necessary conditions for avoiding incorrect polypeptide folds in conformational search by energy minimization

Low energy conformations have been generated for melittin, pancreatic polypeptide, and ribonuclease S-peptide, both in the vicinity of x-ray structures by energy refinement and by an unconstrained search over the entire conformational space. Since the structural polymorphism of these medium-sized peptides in crystal and solution is moderate, comparing the calculated conformation to x-ray and nmr data provides information on local and global behavior of potential functions. Local analysis includes standardization calculations, which show that models with standard geometry can approximate good resolution x-ray data with less than 0.5 A rms deviation (RMSD). However, the atomic coordinates are shifted up to 2 A RMSD by local energy minimization, and thus 2 A is generally the smallest RMSD value one can target in a conformational search using the same energy evaluation models. The unconstrained search was performed by a buildup-type method based on dynamic programming. To accelerate the generation of structures in the conformational search, we used the ECEPP potential, defined in terms of standard polypeptide geometry. A number of low energy conformations were further refined by relaxing the assumption of standard bond lengths and bond angles through the use of the CHARMM potential, and the hydrophobic folding energies of Eisenberg and McLachlan were calculated. Each conformation is described in terms of the RMSD from the native, hydrogen-bonding structure, solvent-accessible surface area, and the ratio of surfaces corresponding to nonpolar and polar residues. The unconstrained search finds conformations that are different from the native, sometimes substantially, and in addition, have lower conformational energies than the native. The origin of deviations is different for each of the three peptides, but in all examples the refined x-ray structures have lower energies than the calculated incorrect folds when (1) the assumption of standard bond lengths and bond angles is relaxed; (2) a small and constant effective dielectric permittivity (epsilon < 10) is used; and (3) the hydrophobic folding energy is incorporated into the potential.

Refinement of the thrombin-bound structure of a hirudin peptide by a restrained electrostatically driven Monte Carlo method

Energy refinement of the structure of a linear peptide, hirudin56-65, bound to thrombin was carried out using a conformational search method in combination with restrained minimization. Five conformations originated from nmr data and distance geometry calculations having a similar global folding pattern but quite different backbone conformations were used as the starting structures. As a result of this approach, a series of low-energy conformations compatible with a set of upper and lower bounds of interproton distances determined from transferred nuclear Overhauser effects were found. A comparison among the lowest energy conformations of each run showed that the combination of energy refinement plus distance constraints led to a very well-defined structure for both the backbone and the side chains of the last 7 residues of the polypeptide. Furthermore, the low-energy conformations generated with this technique contain a segment of 3(10)-helix involving the last 5 residues at the COOH terminal end.

Mass-weighted molecular dynamics simulation and conformational analysis of polypeptide

Atomic motions in protein molecules have been studied by molecular dynamics (MD) simulations; dynamics simulation methods have also been employed in conformational studies of polypeptide molecules. It was found that when atomic masses are weighted, the molecular dynamics method can significantly increase the sampling of dihedral conformation space in such studies, compared to a conventional MD simulation of the same total simulation time length. Herein the theoretical study of molecular conformation sampling by the molecular dynamics-based simulation method in which atomic masses are weighted is reported in detail; moreover, a numerical scheme for analyzing the extensive conformational sampling in the simulation of a tetrapeptide amide molecule is presented. From numerical analyses of the mass-weighted molecular dynamics trajectories of backbone dihedral angles, low-resolution structures covering the entire backbone dihedral conformation space of the molecule were determined, and the distribution of rotationally stable conformations in this space were analyzed quantitatively. The theoretical analyses based on the computer simulation and numerical analytical methods suggest that distinctive regimes in the conformational space of the peptide molecule can be identified.

Calculation of protein conformation as an assembly of stable overlapping segments: application to bovine pancreatic trypsin inhibitor

Conformations of bovine pancreatic trypsin inhibitor were calculated by assuming that the final structure as well as properly chosen overlapping segments thereof are simultaneously in low-energy (not necessarily the lowest-energy) conformational states. Therefore, the whole chain can be built up from building blocks whose conformations are determined primarily by short-range interactions. Our earlier buildup procedure was modified by taking account of a statistical analysis of known amino acid sequences that indicates that there is nonrandom pairing of amino acid residues in short segments along the chain, and by carrying out energy minimization on only these segments and on the whole chain [without minimizing the energies of intermediate-size segments (20-30 residues long)]. Results of this statistical analysis were used to determine the variable sizes of the overlapping oligopeptide building blocks used in the calculations; these varied from tripeptides to octapeptides, depending on the amino acid sequence. Successive stages of approximations were used to combine the low-energy conformations of these building blocks in order to keep the number of variables in the computations to a manageable size. The calculations led to a limited number of conformations of the protein (only two different groups, with very similar structure within each group), most residues of which were in the same conformational state as in the native structure.

Empirical solvation models can be used to differentiate native from near-native conformations of bovine pancreatic trypsin inhibitor

Several hydration models for peptides and proteins based on solvent accessible surface area have been proposed previously. We have evaluated some of these models as well as four new ones in the context of near-native conformations of a protein. In addition, we propose an empirical site-site distance-dependent correction that can be used in conjunction with any of these models. The set of near-native structures consisted of 39 conformations of bovine pancreatic trypsin inhibitor (BPTI) each of which was a local minimum of an empirical energy function (ECEPP) in the absence of solvent. Root-mean-square (rms) deviations from the crystallographically determined structure were in the following ranges: 1.06-1.94 A for all heavy atoms, 0.77-1.36 A for all backbone heavy atoms, 0.68-1.33 A for all alpha-carbon atoms, and 1.41-2.72 A for all side-chain heavy atoms. We have found that there is considerable variation among the solvent models when evaluated in terms of concordance between the solvation free energy and the rms deviations from the crystallographically determined conformation. The solvation model for which the best concordance (0.939) with the rms deviations of the C alpha atoms was found was derived from NMR coupling constants of peptides in water combined with an exponential site-site distance dependence of the potential of mean force. Our results indicate that solvation free energy parameters derived from nonpeptide free energies of hydration may not be transferrable to peptides. Parameters derived from peptide and protein data may be more applicable to conformational analysis of proteins. A general approach to derive parameters for free energy of hydration from ensemble-averaged properties of peptides in solution is described.

Determining minimum energy conformations of polypeptides by dynamic programming

A combinatorial optimization approach is used for solving the multiple-minima problem when determining the low-energy conformations of short polypeptides. Each residue is represented by a finite number of discrete states corresponding to single residue local minima of the energy function. These precomputed values constitute a search table and define the conformational space for discrete minimization by a generalized dynamic programming algorithm that significantly limits the number of intermediate conformations to be generated during the search. Since dynamic programming involves stagewise decisions, it results in buildup-type procedures implemented in two different forms. The first procedure predicts a number of conformations by a completely discrete search and these are subsequently refined by local minimization. The second involves limited continuous local minimization within the combinatorial algorithm, generally restricted to two dihedral angles in a buildup step. Both procedures are tested on 17 short peptides previously studied by other global minimization methods but involving the same potential energy function. The discrete method is extremely fast, but proves to be successful only in 14 of the 17 test problems. The version with limited local minimization finds, however, conformations in all the 17 examples that are close to the ones previously presented in the literature or have lower energies. In addition, results are almost independent of the cutoff energy, the most important parameter governing the search. Although the limited local minimization increases the number of energy evaluations, the method still offers substantial advantages in speed.

A 3D building blocks approach to analyzing and predicting structure of proteins

A new approach is introduced for analyzing and ultimately predicting protein structures, defined at the level of C alpha coordinates. We analyze hexamers (oligopeptides of six amino acid residues) and show that their structure tends to concentrate in specific clusters rather than vary continuously. Thus, we can use a limited set of standard structural building blocks taken from these clusters as representatives of the repertoire of observed hexamers. We demonstrate that protein structures can be approximated by concatenating such building blocks. We have identified about 100 building blocks by applying clustering algorithms, and have shown that they can "replace" about 76% of all hexamers in well-refined known proteins with an error of less than 1 A, and can be joined together to cover 99% of the residues. After replacing each hexamer by a standard building block with similar conformation, we can approximately reconstruct the actual structure by smoothly joining the overlapping building blocks into a full protein. The reconstructed structures show, in most cases, high resemblance to the original structure, although using a limited number of building blocks and local criteria of concatenating them is not likely to produce a very precise global match. Since these building blocks reflect, in many cases, some sequence dependency, it may be possible to use the results of this study as a basis for a protein structure prediction procedure.

Underlying assumptions in the estimation of secondary structure content in proteins by circular dichroism spectroscopy--a critical review

Recombinant DNA technology has made possible the large-scale production of proteins for pharmaceutical applications. As a result, there has been a renaissance in methodology which can provide information on the structural stability and character of these materials. Circular dichroism (CD) spectroscopy, with its sensitivity to the secondary structure adopted by the polypeptide chain, is a powerful tool in this regard. Quantitative analysis of the CD spectra of proteins is now wide-spread, aided by the availability of such algorithms on commercial instrumentation. However, there are basic assumptions made when conducting these calculations, many of which have not been addressed or summarized. Some of these assumptions are independent of the selection of basis spectra and the algorithm employed. These assumptions are listed and the available data concerning their validity is presented and discussed.

Determination of three-dimensional structures of proteins and nucleic acids in solution by nuclear magnetic resonance spectroscopy

Nuclear magnetic resonance (NMR) spectroscopy has evolved over the last decade into a powerful method for determining three-dimensional structures of biological macromolecules in solution. Key advances have been the introduction of two-dimensional experiments, high-field superconducting magnets, and computational procedures for converting the NMR-derived interproton distances and torsion angles into three-dimensional structures. This article outlines the methodology employed, describes the major NMR experiments necessary for the spectral analysis of macromolecules, and discusses the computational approaches employed to date. The present state of the art is illustrated using a variety of examples, and future developments are indicated.

Determination of three-dimensional structures of proteins from interproton distance data by dynamical simulated annealing from a random array of atoms. Circumventing problems associated with folding

A new real space method, based on the principles of simulated annealing, is presented for determining protein structures on the basis of interproton distance restraints derived from NMR data. The method circumvents the folding problem associated with all real space methods described to date, by starting from a completely random array of atoms and introducing the force constants for the covalent, interproton distance and repulsive van der Waals terms in the target function appropriately. The system is simulated at high temperature by solving Newton's equations of motion. As the values of all force constants are very low during the early stages of the simulation, energy barriers between different folds of the protein can be overcome, and the global minimum of the target function is reliably located. Further, because the atoms are initially only weakly coupled, they can move essentially independently to satisfy the restraints. The method is illustrated using two examples of small proteins, namely crambin (46 residues) and potato carboxypeptidase inhibitor (39 residues).

Variable-target-function and build-up procedures for the calculation of protein conformation. Application to bovine pancreatic trypsin inhibitor using limited simulated nuclear magnetic resonance data

An implementation of the variable-target-function procedure, first introduced by Braun and Go [W. Braun and N. Go, J. Mol. Biol. 186, 611-626 (1985)], has been used to generate conformations of the small protein bovine pancreatic trypsin inhibitor (BPTI), given a limited set of simulated data that could be obtained by nuclear magnetic resonance (NMR) techniques. A hybrid strategy was also used to calculate conformations of BPTI, given the same information. In the hybrid strategy, low-energy structures of medium-size fragments (decapeptides) of BPTI were generated using the variable-target-function method, followed by restrained energy optimization. The low-energy conformations were used as a basis to build up the complete fifty-eight-residue BPTI molecule. By using the variable-target-function approach, in which energy considerations were not introduced until full conformations of the entire BPTI molecule had been generated, it was not possible to obtain calculated structures with rms deviations from the X-ray conformation of less than 1.6 A for the alpha-carbons. On the other hand, with the hybrid strategy, which involved the consideration of realistic energy terms in the early stages of the calculations, it was possible to calculate low-energy conformations of BPTI with rms deviations from the X-ray structure of 1.06 to 1.50 A for the alpha-carbons. When the rms deviations were computed along the amino acid sequence, it was found that there was a good correlation between deviations among the calculated structures and deviations from the X-ray structure.