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.     

Assembly of polypeptide and protein backbone conformations from low energy ensembles of short fragments: development of strategies and construction of models for myoglobin, lysozyme, and thymosin beta 4.

Recently we developed methods for the construction of knowledge-based mean fields from a data base of known protein structures. As shown previously, this approach can be used to calculate ensembles of probable conformations for short fragments of polypeptide chains. Here we develop procedures for the assembly of short fragments to complete three-dimensional models of polypeptide chains. The amino acid sequence of a given protein is decomposed into all possible overlapping fragments of a given length, and an ensemble of probable conformations is calculated for each fragment. The fragments are assembled to a complete model by choosing appropriate conformations from the individual ensembles and by averaging over equivalent angles. Finally a consistent model is obtained by rebuilding the conformation from the average angles. From the average angles the local variability of the structure can be calculated, which is a useful criterion for the reliability of the model. The procedure is applied to the calculation of the local backbone conformations of myoglobin and lysozyme whose structures have been solved by X-ray analysis and thymosin beta 4, a polypeptide of 43 amino acid residues whose structure was recently investigated by NMR spectroscopy. We demonstrate that substantial fractions of the calculated local backbone conformations are similar to the experimentally determined structures.     

Modeling protein loops using a phi i + 1, psi i dimer database.

We present an automated method for modeling backbones of protein loops. The method samples a database of phi i + 1 and psi i angles constructed from a nonredundant version of the Protein Data Bank (PDB). The dihedral angles phi i + 1 and psi i completely define the backbone conformation of a dimer when standard bond lengths, bond angles, and a trans planar peptide configuration are used. For the 400 possible dimers resulting from 20 natural amino acids, a list of allowed phi i + 1, psi i pairs for each dimer is created by pooling all such pairs from the loop segments of each protein in the nonredundant version of the PDB. Starting from the N-terminus of the loop sequence, conformations are generated by assigning randomly selected pairs of phi i + 1, psi i for each dimer from the respective pool using standard bond lengths, bond angles, and a trans peptide configuration. We use this database to simulate protein loops of lengths varying from 5 to 11 amino acids in five proteins of known three-dimensional structures. Typically, 10,000-50,000 models are simulated for each protein loop and are evaluated for stereochemical consistency. Depending on the length and sequence of a given loop, 50-80% of the models generated have no stereochemical strain in the backbone atoms. We demonstrate that, when simulated loops are extended to include flanking residues from homologous segments, only very few loops from an ensemble of sterically allowed conformations orient the flanking segments consistent with the protein topology. The presence of near-native backbone conformations for loops from five different proteins suggests the completeness of the dimeric database for use in modeling loops of homologous proteins. Here, we take advantage of this observation to design a method that filters near-native loop conformations from an ensemble of sterically allowed conformations. We demonstrate that our method eliminates the need for a loop-closure algorithm and hence allows for the use of topological constraints of the homologous proteins or disulfide constraints to filter near-native loop conformations.     

Statistical mechanics of protein folding by exhaustive enumeration

It is hard to construct theories for the folding of globular proteins because they are large and complicated molecules having enormous numbers of nonnative conformations and having native states that are complicated to describe. Statistical mechanical theories of protein folding are constructed around major simplifying assumptions about the energy as a function of conformation and/or simplifications of the representation of the polypeptide chain, such as one point per residue on a cubic lattice. It is not clear how the results of these theories are affected by their various simplifications. Here we take a very different simplification approach where the chain is accurately represented and the energy of each conformation is calculated by a not unreasonable empirical function. However, the set of amino acid sequences and allowed conformations is so restricted that it becomes computationally feasible to examine them all. Hence we are able to calculate melting curves for thermal denaturation as well as the detailed kinetic pathway of refolding. Such calculations are based on a novel representation of the conformations as points in an abstract 12-dimensional Euclidean conformation space. Fast folding sequences have relatively high melting temperatures, native structures with relatively low energies, small kinetic barriers between local minima, and relatively many conformations in the global energy minimum's watershed. In contrast to other folding theories, these models show no necessary relationship between fast folding and an overall funnel shape to the energy surface, or a large energy gap between the native and the lowest nonnative structure, or the depth of the native energy minimum compared to the roughness of the energy landscape. Proteins 32:425–437, 1998. © 1998 Wiley-Liss, Inc.     

A hydrogen bonded chain in bacteriorhodopsin by computer modelling approach

The seven alpha-helical segments of Bacteriorhodopsin (BR) passing through the membrane are investigated for a continuous Hydrogen Bonded Chain (HBC). The study is carried out by computer modelling approach. It is assumed that the seven helices are placed as (AGFEDCB), which has been accepted as the best model by several groups. Helices A, D, E and G are considered to be present in right handed alpha-helical conformation. The inter-orientation of these helices are represented by Eulerian angles alpha, beta and gamma. For the helices B, C and F which contain Proline in the middle, several conformational possibilities were considered. In these cases apart from the Eulerian angles alpha, beta and gamma, the dihedral angles phi p-1 and psi p-1 of the residues that are succeeded by Proline residue in the helical regions were also used in fixing the position of the helices with respect to each other. All these parameters were varied to fit with the top, middle and bottom distances reported by electron diffraction studies. Good fit was obtained for all right handed alpha-helical conformations and also for helices B, C and F with a left handed turn at the residue preceeding proline. Hence two structures were analysed for continuous HBC. Structure I which contained all the seven helices in right handed alpha-helical conformation and Structure II, which had the helices A, D, E and G in right handed conformation and the helices B, C and F in right handed alpha-helical conformation with a left handed turn at the residue preceeding proline. All possible staggered conformations were considered for the side chains and the inter atomic distances were analysed for Hydrogen bonds. It was possible to obtain a continuous chain in both the structures which includes most of the residues found to be important by the experiments. However Lys-216 has to be considered in two different conformations to connect the cytoplasmic side with the extra cellular side. The overall height spanned by HBC is about 25A. The chains obtained by both the structures I and II are analysed in terms of the conformational parameters. It has also been possible to place the retinal in the region as predicted by the experiments. The Tryptophan residues which affect the spectral characteristics can be aligned on either side of the retinal.     

Development of an extended simulated annealing method: application to the modeling of complementary determining regions of immunoglobulins.

An extended simulated annealing process (ESAP) has been developed in order to obtain an ensemble of conformations of a peptide segment from a protein fluctuating at a given temperature. The annealing process was performed with a fast Monte Carlo method using the scaled collective variables developed by Noguti and Go. The system was divided into two parts: one consists of one or more peptide segments and is flexible around the main-chain and side-chain torsional angles; the other represents the rest of the molecule and was maintained fixed at the atomic positions determined by x-ray experiments. The target function included the nonbonding atomic interactions and a distance function to anchor the N and C terminal ends of each segment to the fixed part. Three systems of complementary determining regions (CDR) of antibodies were tested and compared to x-ray data: L2 loop (7 residues) of the light chain of lambda-type Bence-Jones protein, H1 and the H2 loops (14 residues) of McPC603, and H1 and H2 loops (12 residues) of HyHEL-5. Each state of CDR conformations was characterized at room temperature by the average of their coordinates (average conformation) and the internal energy. With a limited number of annealing processes (10), starting from the extended conformation, we have obtained states with conformations close to the observed x-ray structures, from 1.1 to 1.7 A root mean square deviation (rmsd) of main-chain atoms depending on the system. These states were identical or within 0.25 A rmsd of those of lowest internal energy. For unknown CDR structures the criteria of lowest internal energies from ESAP can be used to predict hypervariable loop structures in antibodies with an accuracy comparable to other methods.     

Variability of the canonical loop conformations in serine proteinases inhibitors and other proteins

Canonical loops of protein inhibitors of serine proteinases occur in proteins having completely different folds. In this article, conformations of the loops have been analyzed for inhibitors belonging to 10 structurally different families. Using deviation in C_α-C_α distances as a criterion for loop similarity, we found that the P3-P3′ segment defines most properly the length of the loop. When conformational differences among loops of individual inhibitors were compared using root mean square deviation (rmsd) in atomic coordinates for all main chain atoms (Δr method) and rmsd operating in main chain torsion angles (Δt method), differences of up to 2.1 Å and 72.3°, respectively, were observed. Such large values indicate significant conformational differences among individual loops. Nevertheless, the overall geometry of the inhibitor-proteinase interaction is very well preserved, as judged from the similarity of C_α-C_α distances between C_α of catalytic Ser and C_α of P3-P3′ residues in various enzyme-inhibitor complexes. The mode of interaction is very well preserved both in the chymotrypsin and subtilisin families, as distances calculated for subtilisin-inhibitor complexes are almost always within the range of those for chymotrypsin-inhibitor complexes. Complex formation leads to conformational changes of up to 160° for χ₁ angle. Side chains of residue P1 and P2′ adopt in different complexes a similar orientation (χ₁ angle = −60° and −180°, respectively). To check whether the canonical conformation can be found among non–proteinase-inhibitor Brookhaven Protein Data Bank structures, two selection criteria—the allowed main chain dihedral angles and C_α-C_α distances for the P3-P3′ segment—were applied to all these structures. This procedure detected 132 unique hexapeptide segments in 121 structurally and functionally unrelated proteins. Partial preferences for certain amino acids occurring at particular positions in these hexapeptides could be noted. Further restriction of this set to hexapeptides with a highly exposed P1 residue side chain resulted in 40 segments. The possibility of complexes formation between these segments and serine proteinases was ruled out in molecular modeling due to steric clashes. Several structural features that determine the canonical conformation of the loop both in inhibitors and in other proteins can be distinguished. They include main chain hydrogen bonds both within the P3-P3′ segment and with the scaffold region, P3-P4 and P3′-P4′ hydrophobic interactions, and finally either hydrophobic or polar interactions involving the P1′ residue. Proteins 32:459–474, 1998. © 1998 Wiley-Liss, Inc.     

Structural basis of hierarchical multiple substates of a protein. III: Side chain and main chain local conformations

An analysis is carried out of differences in the minimum energy conformations obtained in the previous paper by energy minimization starting from conformations sampled by a Monte Carlo simulation of conformational fluctuations in the native state of a globular protein, bovine pancreatic trypsin inhibitor. Main conformational differences in each pair of energy minima are found usually localized in several side chains and in a few local main chain segments. Such side chains and local main chain segments are found to take a few distinct local conformations in the minimum energy conformations. Energy minimum conformations can thus be described in terms of combinations of these multiple local conformations.     

Comparative molecular dynamics simulations of the potent synthetic classical cannabinoid ligand AMG3 in solution and at binding site of the CB1 and CB2 receptors

The C-1'-dithiolane Delta(8)-tetrahydrocannabinol (Delta(8)-THC) amphiphilic analogue (-)-2-(6a,7,10,10a-tetrahydro-6,6,9-trimethylhydroxy-6H-dibenzo[b,d]pyranyl)-2-hexyl-1,3-dithiolane (AMG3) is considered as one of the most potent synthetic analgesic cannabinoid (CB) ligands. Its structure is characterized by rigid tricyclic and flexible alkyl chain segments. Its conformational properties have not been fully explored. Structure-activity relationship (SAR) studies on classical CBs showed that the alkyl side chain is the most critical structural part for the receptor activation. However, reported low energy conformers of classical CB analogues vary mainly in the conformation of their alkyl side chain segment. Therefore, comparative molecular dynamics (MD) simulations of low energy conformers of AMG3 were performed in order to investigate its structural and dynamical properties in two different systems. System-I includes ligand and amphoteric solvent DMSO, simulating the biological environment and system-II includes ligand at active site of the homology models of CB1 and CB2 receptors in the solvent. The trajectory analysis results are compared for the systems I and II. In system-I, the dihedral angle defined between aromatic ring and dithiolane ring of AMG3 shows more resistance to be transformed into another torsional angle and the dihedral angle adjacent to dithiolane ring belonging in the alkyl chain has flexibility to adopt gauche+/- and trans dihedral angles. The rest of the dihedral angles within the alkyl chain are all trans. These results point out that wrapped conformations are dynamically less favored in solution than linear conformations. Two possible plane angles defined between the rigid and flexible segments are found to be the most favored and adopting values of approximately 90 degrees and approximately 140 degrees. In system-II, these values are approximately 90 degrees and approximately 120 degrees. Conformers of AMG3 at the CB1 receptor favor to establish a cis conformation defined between aromatic and dithiolane ring and a trans conformation in the CB2 receptor. These different orientations of ligand inside the binding pocket of CB1 and CB2 receptors may explain its different binding affinity in the two receptors. The results of this study can be applied to other synthetic classical CB ligands to produce low energy conformations and can be of general use for the molecules possessing flexible alkyl chain(s). In addition, this study can be useful when restraint of the alkyl chain is sought for optimizing drug design.     

A Hydrogen Bonded Chain in Bactereorhodopsin by Computer Modelling Approach

Abstract

The seven α-helical segments of Bacteriorhodopsin (BR) passing through the membrane are investigated for a continuous Hydrogen Bonded Chain (HBC). The study is carried out by computer modelling approach. It is assumed that the seven helices are placed as (AGFEDCB), which has been accepted as the best model by several groups. Helices A, D, E and G are considered to be present in right handed α-helical conformation. The inter-orientation of these helices are represented by Eulerian angles α, β and γ. For the helices B, C and F which contain Proline in the middle, several conformational possibilities were considered. In these cases apart from the Eulerian angles α, β and γ, the dihedral angles φ_{p_1} and ψ_{p_1} of the residues that are succeeded by Proline residue in the helical regions were also used in fixing the position of the helices with respect to each other. All these parameters were varied to fit with the top, middle and bottom distances reported by electron diffraction studies. Good fit was obtained for all right handed α-helical conformations and also for helices B, C and F with a left handed turn at the residue preceeding proline. Hence two structures were analysed for continuous HBC, Structure I which contained all the seven helices in right handed α-helical conformation and Structure II, which had the helices A, D, E and G in right handed conformation and the helices B, C and F in right handed α-helical conformation with a left handed turn at the residue preceeding proline. All possible staggered conformations were considered for the side chains and the inter atomic distances were analysed for Hydrogen bonds. It was possible to obtain a continuous chain in both the structures which includes most of the residues found to be important by the experiments. However Lys-216 has to be considered in two different conformations to connect the cytoplasmic side with the extra cellular side. The overall height spanned by HBC is about 25Å. The chains obtained by both the structures I and II are analysed in terms of the conformational parameters. It has also been possible to place the retinal in the region as predicted by the experiments. The Tryptophan residues which affect the spectral characterestics can be aligned on either side of the retinal.     

De novo prediction of polypeptide conformations using dihedral probability grid Monte Carlo methodology.

We tested the dihedral probability grid Monte Carlo (DPG-MC) methodology to determine optimal conformations of polypeptides by applying it to predict the low energy ensemble for two peptides whose solution NMR structures are known: integrin receptor peptide (YGRGDSP, Type II beta-turn) and S3 alpha-helical peptide (YMSEDEL KAAEAAFKRHGPT). DPG-MC involves importance sampling, local random stepping in the vicinity of a current local minima, and Metropolis sampling criteria for acceptance or rejection of new structures. Internal coordinate values are based on side-chain-specific dihedral angle probability distributions (from analysis of high-resolution protein crystal structures). Important features of DPG-MC are: (1) Each DPG-MC step selects the torsion angles (phi, psi, chi) from a discrete grid that are then applied directly to the structure. The torsion angle increments can be taken as S = 60, 30, 15, 10, or 5 degrees, depending on the application. (2) DPG-MC utilizes a temperature-dependent probability function (P) in conjunction with Metropolis sampling to accept or reject new structures. For each peptide, we found close agreement with the known structure for the low energy conformational ensemble located with DPG-MC. This suggests that DPG-MC will be useful for predicting conformations of other polypeptides.     

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.     

Extracting information on folding from the amino acid sequence: consensus regions with preferred conformation in homologous proteins.

It is investigated whether protein segments predicted to have a well-defined conformational preference in the absence of tertiary interactions are conserved in families of homologous proteins. The prediction method follows the procedures of Rooman, M., Kocher, J.-P., and Wodak, S. (preceding paper in this issue). It uses a knowledge-based force field that incorporates only local interactions along the sequence and identifies segments whose lowest energy structure displays a sizable energy gap relative to other computed conformations. In 13 of the protein families and subfamilies considered that are sufficiently homologous to have similar 3D structures, at least one region is consistently predicted as having the same preferred conformation in virtually all family members. These regions are between 4 and 26 residues long. They are often located at chain ends and correspond primarily to segments of secondary structure heavily involved in interactions with the rest of the protein, suggesting that they could act as nuclei around which other parts of the structure would assemble. Experimental data on early folding intermediates or on protein fragments with appreciable structure in aqueous solution are available for more than half of the protein families. Comparison of our results with these data is quite favorable. They reveal that each of the experimentally identified early formed, or independently stable, substructures harbors at least one of the segments consistently predicted as having a preferred conformation by our procedure. The implications of our findings for the conservation of folding pathways in homologous proteins are discussed.     

Conformational study of asialo-GM1 (GA1) ganglioside

In order to investigate the significance of preferred conformations of the saccharide for the steric orientation and recognition of glycosphingolipids at the membrane surface, the conformational free energy calculations were carried out on the asialo-GM1 [GA1; β-D -Gal(1 → 3) β-D -GalNAc(1 → 4) β-D -Gal(1 → 4) β-D -Glc-O-ceramide] using a new program CONCARB (CONformational study program for CARBohydrate) in the unhydrated and hydrated states. The overall backbone conformation of GA1 appears nearly to be extended with a little bent at the glycosidic II–III linkage, in which two pyranose rings of Gal(IV)-GalNAc-(III) moiety orient approximately perpendicular to those of Gal(II)-Glc(I) moiety. This is consistent with the structures deduced from high-sensitivity differential scanning calorimetry experiments and the nmr study on GA1. The calculated glycosidic torsion angles of the lowest free energy conformation of GA1 in the hydrated state are in accord with the structures of relevant oligosaccharides deduced from nmr experiments and hard sphere exoanomeric calculations. A comparison of the values of glycosidic torsion angles ϕ and π of GA1 and its constituent oligosaccharides indicates that the overall backbone conformation of each oligosaccharide is retained when the oligosaccharide chain becomes longer. This implies that the short-range interactions between the nearest-neighbored saccharides are of significant importance in stabilizing the overall backbone conformation of GA1 in both the unhydrated and hydrated states. The different orientation and hydrogen bonds of hydroxymethyl and hydroxyl groups from one oligosaccharide to another suggest that the medium- and long-range interactions are also of consequence. Hydration seems to affect significantly the conformation of these groups, but not to perturb remarkably the overall backbone conformation of GA1. © 1997 John Wiley & Sons, Inc. Biopoly 42: 19–35, 1997     

The loop problem in proteins: A monte carlo simulated annealing approach

A Monte Carlo simulated annealing (MCSA) algorithm was used to generate the conformations of local regions in bovine pancreatic trypsin inhibitor (BPTI) starting from random initial conformations. In the approach explored, only the conformation of the segment is computed; the rest of the protein is fixed in the known native conformation. Rather than follow a single simulation exhaustively, computer time is better used by performing multiple independent MCSA simulations in which different starting temperatures are employed and the number of conformations sampled is varied. The best computed conformation is chosen on the basis of lowest total energy and refined further. The total energy used in the annealing is the sum of the intrasegment energy, the interaction energy of the segment with the local surrounding region, and a distance constraint to generate a smooth connection of the initially randomized segment with the rest of the protein. The rms deviations between the main-chain conformations of the computed segments in BPTI and those of the native x-ray structure are 0.94 Å for a 5-residue α-helical segment, 1.11 Å for a 5-residue β-strand segment, and 1.03, 1.61, and 1.87 Ã for 5-, 7-, and 9-residue loop segments. Side-chain deviations are comparable to the main-chain deviations for those side chains that interact strongly with the fixed part of the protein. A detailed view of the deviations at an atom-resolved level is obtained by comparing the predicted segments with their known conformations in the crystal structure of BPTI. These results emphasize the value of predetermined fixed structure against which the computed segment can nest. © 1993 John Wiley & Sons, Inc.     

Improved energy bound accuracy enhances the efficiency of continuous protein design

Flexibility and dynamics are important for protein function and a protein's ability to accommodate amino acid substitutions. However, when computational protein design algorithms search over protein structures, the allowed flexibility is often reduced to a relatively small set of discrete side‐chain and backbone conformations. While simplifications in scoring functions and protein flexibility are currently necessary to computationally search the vast protein sequence and conformational space, a rigid representation of a protein causes the search to become brittle and miss low‐energy structures. Continuous rotamers more closely represent the allowed movement of a side chain within its torsional well and have been successfully incorporated into the protein design framework to design biomedically relevant protein systems. The use of continuous rotamers in protein design enables algorithms to search a larger conformational space than previously possible, but adds additional complexity to the design search. To design large, complex systems with continuous rotamers, new algorithms are needed to increase the efficiency of the search. We present two methods, PartCR and HOT, that greatly increase the speed and efficiency of protein design with continuous rotamers. These methods specifically target the large errors in energetic terms that are used to bound pairwise energies during the design search. By tightening the energy bounds, additional pruning of the conformation space can be achieved, and the number of conformations that must be enumerated to find the global minimum energy conformation is greatly reduced. Proteins 2015; 83:1151–1164. © 2015 Wiley Periodicals, Inc.     

Optimization of van der Waals energy for protein side-chain placement and design

Computational determination of optimal side-chain conformations in protein structures has been a long-standing and challenging problem. Solving this problem is important for many applications including homology modeling, protein docking, and for placing small molecule ligands on protein-binding sites. Programs available as of this writing are very fast and reasonably accurate, as measured by deviations of side-chain dihedral angles; however, often due to multiple atomic clashes, they produce structures with high positive energies. This is problematic in applications where the energy values are important, for example when placing small molecules in docking applications; the relatively small binding energy of the small molecule is drowned by the large energy due to atomic clashes that hampers finding the lowest energy state of the docked ligand. To address this we have developed an algorithm for generating a set of side-chain conformations that is dense enough that at least one of its members would have a root mean-square deviation of no more than R Å from any possible side-chain conformation of the amino acid. We call such a set a side-chain cover set of order R for the amino acid. The size of the set is constrained by the energy of the interaction of the side chain to the backbone atoms. Then, side-chain cover sets are used to optimize the conformation of the side chains given the coordinates of the backbone of a protein. The method we use is based on a variety of dead-end elimination methods and the recently discovered dynamic programming algorithm for this problem. This was implemented in a computer program called Octopus where we use side-chain cover sets with very small values for R, such as 0.1 Å, which ensures that for each amino-acid side chain the set contains a conformation with a root mean-square deviation of, at most, R from the optimal conformation. The side-chain dihedral-angle accuracy of the program is comparable to other implementations; however, it has the important advantage that the structures produced by the program have negative energies that are very close to the energies of the crystal structure for all tested proteins.     

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

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

Using quantum mechanics to improve estimates of amino acid side chain rotamer energies

Amino acid side chains adopt a discrete set of favorable conformations typically referred to as rotamers. The relative energies of rotamers partially determine which side chain conformations are more often observed in protein structures and accurate estimates of these energies are important for predicting protein structure and designing new proteins. Protein modelers typically calculate side chain rotamer energies by using molecular mechanics (MM) potentials or by converting rotamer probabilities from the protein database (PDB) into relative free energies. One limitation of the knowledge‐based energies is that rotamer preferences observed in the PDB can reflect internal side chain energies as well as longer‐range interactions with the rest of the protein. Here, we test an alternative approach for calculating rotamer energies. We use three different quantum mechanics (QM) methods (second order Møller‐Plesset (MP2), density functional theory (DFT) energy calculation using the B3LYP functional, and Hartree‐Fock) to calculate the energy of amino acid rotamers in a dipeptide model system, and then use these pre‐calculated values in side chain placement simulations. Energies were calculated for over 36,000 different conformations of leucine, isoleucine, and valine dipeptides with backbone torsion angles from the helical and strand regions of the Ramachandran plot. In a subset of cases these energies differ significantly from those calculated with standard molecular mechanics potentials or those derived from PDB statistics. We find that in these cases the energies from the QM methods result in more accurate placement of amino acid side chains in structure prediction tests. Proteins 2008. © 2007 Wiley‐Liss, Inc.