Side-chain modeling with an optimized scoring function

Modeling side-chain conformations on a fixed protein backbone has a wide application in structure prediction and molecular design. Each effort in this field requires decisions about a rotamer set, scoring function, and search strategy. We have developed a new and simple scoring function, which operates on side-chain rotamers and consists of the following energy terms: contact surface, volume overlap, backbone dependency, electrostatic interactions, and desolvation energy. The weights of these energy terms were optimized to achieve the minimal average root mean square (rms) deviation between the lowest energy rotamer and real side-chain conformation on a training set of high-resolution protein structures. In the course of optimization, for every residue, its side chain was replaced by varying rotamers, whereas conformations for all other residues were kept as they appeared in the crystal structure. We obtained prediction accuracy of 90.4% for chi(1), 78.3% for chi(1 + 2), and 1.18 A overall rms deviation. Furthermore, the derived scoring function combined with a Monte Carlo search algorithm was used to place all side chains onto a protein backbone simultaneously. The average prediction accuracy was 87.9% for chi(1), 73.2% for chi(1 + 2), and 1.34 A rms deviation for 30 protein structures. Our approach was compared with available side-chain construction methods and showed improvement over the best among them: 4.4% for chi(1), 4.7% for chi(1 + 2), and 0.21 A for rms deviation. We hypothesize that the scoring function instead of the search strategy is the main obstacle in side-chain modeling. Additionally, we show that a more detailed rotamer library is expected to increase chi(1 + 2) prediction accuracy but may have little effect on chi(1) prediction accuracy.     

IRECS: a new algorithm for the selection of most probable ensembles of side-chain conformations in protein models

We introduce a new algorithm, IRECS (Iterative REduction of Conformational Space), for identifying ensembles of most probable side-chain conformations for homology modeling. On the basis of a given rotamer library, IRECS ranks all side-chain rotamers of a protein according to the probability with which each side chain adopts the respective rotamer conformation. This ranking enables the user to select small rotamer sets that are most likely to contain a near-native rotamer for each side chain. IRECS can therefore act as a fast heuristic alternative to the Dead-End-Elimination algorithm (DEE). In contrast to DEE, IRECS allows for the selection of rotamer subsets of arbitrary size, thus being able to define structure ensembles for a protein. We show that the selection of more than one rotamer per side chain is generally meaningful, since the selected rotamers represent the conformational space of flexible side chains. A knowledge-based statistical potential ROTA was constructed for the IRECS algorithm. The potential was optimized to discriminate between side-chain conformations of native and rotameric decoys of protein structures. By restricting the number of rotamers per side chain to one, IRECS can optimize side chains for a single conformation model. The average accuracy of IRECS for the chi1 and chi1+2 dihedral angles amounts to 84.7% and 71.6%, respectively, using a 40 degrees cutoff. When we compared IRECS with SCWRL and SCAP, the performance of IRECS was comparable to that of both methods. IRECS and the ROTA potential are available for download from the URL http://irecs.bioinf.mpi-inf.mpg.de.     

The penultimate rotamer library

All published rotamer libraries contain some rotamers that exhibit impossible internal atomic overlaps if built in ideal geometry with all hydrogen atoms. Removal of uncertain residues (mainly those with B-factors >/=40 or van der Waals overlaps >/=0.4 A) greatly improves the clustering of rotamer populations. Asn, Gln, or His side chains additionally benefit from flipping of their planar terminal groups when required by atomic overlaps or H-bonding. Sensitivity to skew and to the boundaries of chi angle bins is avoided by using modes rather than traditional mean values. Rotamer definitions are listed both as the modal values and in a preferred version that maximizes common atoms between related rotamers. The resulting library shows significant differences from previous ones, differences validated by considering the likelihood of systematic misfitting of models to electron density maps and by plotting changes in rotamer frequency with B-factor. Few rotamers now show atomic overlaps in ideal geometry; those overlaps are relatively small and can be understood in terms of bond angle distortions compensated by favorable interactions. The new library covers 94.5% of examples in the highest quality protein data with 153 rotamers and can make a significant contribution to improving the accuracy of new structures. Proteins 2000;40:389-408.     

Bayesian statistical analysis of protein side-chain rotamer preferences.

We present a Bayesian statistical analysis of the conformations of side chains in proteins from the Protein Data Bank. This is an extension of the backbone-dependent rotamer library, and includes rotamer populations and average chi angles for a full range of phi, psi values. The Bayesian analysis used here provides a rigorous statistical method for taking account of varying amounts of data. Bayesian statistics requires the assumption of a prior distribution for parameters over their range of possible values. This prior distribution can be derived from previous data or from pooling some of the present data. The prior distribution is combined with the data to form the posterior distribution, which is a compromise between the prior distribution and the data. For the chi 2, chi 3, and chi 4 rotamer prior distributions, we assume that the probability of each rotamer type is dependent only on the previous chi rotamer in the chain. For the backbone-dependence of the chi 1 rotamers, we derive prior distributions from the product of the phi-dependent and psi-dependent probabilities. Molecular mechanics calculations with the CHARMM22 potential show a strong similarity with the experimental distributions, indicating that proteins attain their lowest energy rotamers with respect to local backbone-side-chain interactions. The new library is suitable for use in homology modeling, protein folding simulations, and the refinement of X-ray and NMR structures.     

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.     

Andante: reducing side-chain rotamer search space during comparative modeling using environment-specific substitution probabilities

MOTIVATION: The accurate placement of side chains in computational protein modeling and design involves the searching of vast numbers of rotamer combinations. RESULTS: We have applied the information contained within structurally aligned homologous families, in the form of conserved chi angle conservation rules, to the problem of the comparative modeling. This allows the accurate borrowing of entire side-chain conformations and/or the restriction to high probability rotamer bins. The application of these rules consistently reduces the number of rotamer combinations that need to be searched to trivial values and also reduces the overall side-chain root mean square deviation (rmsd) of the final model. The approach is complementary to current side-chain placement algorithms that use the decomposition of interacting clusters to increase the speed of the placement process.     

Incorporating knowledge-based biases into an energy-based side-chain modeling method: application to comparative modeling of protein structure

The performance of the self-consistent mean field theory (SCMFT) method for side-chain modeling, employing rotamer energies calculated with the flexible rotamer model (FRM), is evaluated in the context of comparative modeling of protein structure. Predictions were carried out on a test set of 56 model backbones of varying accuracy, to allow side-chain prediction accuracy to be analyzed as a function of backbone accuracy. A progressive decrease in the accuracy of prediction was observed as backbone accuracy decreased. However, even for very low backbone accuracy, prediction was substantially higher than random, indicating that the FRM can, in part, compensate for the errors in the modeled tertiary environment. It was also investigated whether the introduction in the FRM-SCMFT method of knowledge-based biases, derived from a backbone-dependent rotamer library, could enhance its performance. A bias derived from the backbone-dependent rotamer conformations alone did not improve prediction accuracy. However, a bias derived from the backbone-dependent rotamer probabilities improved prediction accuracy considerably. This bias was incorporated through two different strategies. In one (the indirect strategy), rotamer probabilities were used to reject unlikely rotamers a priori, thus restricting prediction by FRM-SCMFT to a subset containing only the most probable rotamers in the library. In the other (the direct strategy), rotamer energies were transformed into pseudo-energies that were added to the average potential energies of the respective rotamers, thereby creating hybrid energy-based/knowledge-based average rotamer energies, which were used by the FRM-SCMFT method for prediction. For all degrees of backbone accuracy, an optimal strength of the knowledge-based bias existed for both strategies for which predictions were more accurate than pure energy-based predictions, and also than pure knowledge-based predictions. Hybrid knowledge-based/energy-based methods were obtained from both strategies and compared with the SCWRL method, a hybrid method based on the same backbone-dependent rotamer library. The accuracy of the indirect method was approximately the same as that of the SCWRL method, but that of the direct method was significantly higher.     

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.     

The Dynameomics rotamer library: amino acid side chain conformations and dynamics from comprehensive molecular dynamics simulations in water

We have recently completed systematic molecular dynamics simulations of 807 different proteins representing 95% of the known autonomous protein folds in an effort we refer to as Dynameomics. Here we focus on the analysis of side chain conformations and dynamics to create a dynamic rotamer library. Overall this library is derived from 31,000 occurrences of each of 86,217 different residues, or 2.7 × 10(9) rotamers. This dynamic library has 74% overlap of rotamer distributions with rotamer libraries derived from static high-resolution crystal structures. Seventy-five percent of the residues had an assignable primary conformation, and 68% of the residues had at least one significant alternate conformation. The average correlation time for switching between rotamers ranged from 22 ps for Met to over 8 ns for Cys; this time decreased 20-fold on the surface of the protein and modestly for dihedral angles further from the main chain. Side chain S(2) axis order parameters were calculated and they correlated well with those derived from NMR relaxation experiments (R = 0.9). Relationships relating the S(2) axis order parameters to rotamer occupancy were derived. Overall the Dynameomics rotamer library offers a comprehensive depiction of side chain rotamer preferences and dynamics in solution, and more realistic distributions for dynamic proteins in solution at ambient temperature than libraries derived from crystal structures, in particular charged surface residues are better represented. Details of the rotamer library are presented here and the library itself can be downloaded at http://www.dynameomics.org.     

Extending the accuracy limits of prediction for side-chain conformations

Current techniques for the prediction of side-chain conformations on a fixed backbone have an accuracy limit of about 1.0-1.5 A rmsd for core residues. We have carried out a detailed and systematic analysis of the factors that influence the prediction of side-chain conformation and, on this basis, have succeeded in extending the limits of side-chain prediction for core residues to about 0.7 A rmsd from native, and 94 % and 89 % of chi(1) and chi(1+2 ) dihedral angles correctly predicted to within 20 degrees of native, respectively. These results are obtained using a force-field that accounts for only van der Waals interactions and torsional potentials. Prediction accuracy is strongly dependent on the rotamer library used. That is, a complete and detailed rotamer library is essential. The greatest accuracy was obtained with an extensive rotamer library, containing over 7560 members, in which bond lengths and bond angles were taken from the database rather than simply assuming idealized values. Perhaps the most surprising finding is that the combinatorial problem normally associated with the prediction of the side-chain conformation does not appear to be important. This conclusion is based on the fact that the prediction of the conformation of a single side-chain with all others fixed in their native conformations is only slightly more accurate than the simultaneous prediction of all side-chain dihedral angles.     

Configurational-bias sampling technique for predicting side-chain conformations in proteins

Prediction of side-chain conformations is an important component of several biological modeling applications. In this work, we have developed and tested an advanced Monte Carlo sampling strategy for predicting side-chain conformations. Our method is based on a cooperative rearrangement of atoms that belong to a group of neighboring side-chains. This rearrangement is accomplished by deleting groups of atoms from the side-chains in a particular region, and regrowing them with the generation of trial positions that depends on both a rotamer library and a molecular mechanics potential function. This method allows us to incorporate flexibility about the rotamers in the library and explore phase space in a continuous fashion about the primary rotamers. We have tested our algorithm on a set of 76 proteins using the all-atom AMBER99 force field and electrostatics that are governed by a distance-dependent dielectric function. When the tolerance for correct prediction of the dihedral angles is a <20 degrees deviation from the native state, our prediction accuracies for chi1 are 83.3% and for chi1 and chi2 are 65.4%. The accuracies of our predictions are comparable to the best results in the literature that often used Hamiltonians that have been specifically optimized for side-chain packing. We believe that the continuous exploration of phase space enables our method to overcome limitations inherent with using discrete rotamers as trials.     

A smoothed backbone-dependent rotamer library for proteins derived from adaptive kernel density estimates and regressions

Rotamer libraries are used in protein structure determination, prediction, and design. The backbone-dependent rotamer library consists of rotamer frequencies, mean dihedral angles, and variances as?a function of the backbone dihedral angles. Structure prediction and design methods that employ backbone flexibility would strongly benefit from smoothly varying probabilities and angles. A new version of the?backbone-dependent rotamer library has been developed using adaptive kernel density estimates for the rotamer frequencies and adaptive kernel regression for the mean dihedral angles and variances. This formulation allows for evaluation of the rotamer probabilities, mean angles, and variances as?a smooth and continuous function of phi and psi. Continuous probability density estimates for the nonrotameric degrees of freedom of amides, carboxylates, and aromatic side chains have been modeled as a function of the backbone dihedrals and rotamers of the remaining degrees of freedom. New backbone-dependent rotamer libraries at varying levels of smoothing are available from http://dunbrack.fccc.edu.     

Rotamer Dynamics: Analysis of Rotamers in Molecular Dynamics Simulations of Proteins

Given by χ torsional angles, rotamers describe the side-chain conformations of amino acid residues in a protein based on the rotational isomers (hence the word rotamer). Constructed rotamer libraries, based on either protein crystal structures or dynamics studies, are the tools for classifying rotamers (torsional angles) in a way that reflect their frequency in nature. Rotamer libraries are routinely used in structure modeling and evaluation. In this perspective article, we would like to encourage researchers to apply rotamer analyses beyond their traditional use. Molecular dynamics (MD) of proteins highlight the in silico behavior of molecules in solution and thus can identify favorable side-chain conformations. In this article, we used simple computational tools to study rotamer dynamics (RD) in MD simulations. First, we isolated each frame in the MD trajectories in separate Protein Data Bank files via the cpptraj module in AMBER. Then, we extracted torsional angles via the Bio3D module in R language. The classification of torsional angles was also done in R according to the penultimate rotamer library. RD analysis is useful for various applications such as protein folding, study of rotamer-rotamer relationship in protein-protein interaction, real-time correlation between secondary structures and rotamers, study of flexibility of side chains in binding site for molecular docking preparations, use of RD as guide in functional analysis and study of structural changes caused by mutations, providing parameters for improving coarse-grained MD accuracy and speed, and many others. Major challenges facing RD to emerge as a new scientific field involve the validation of results via easy, inexpensive wet-lab methods. This realm is yet to be explored.     

Dead-end elimination with backbone flexibility

MOTIVATION: Dead-End Elimination (DEE) is a powerful algorithm capable of reducing the search space for structure-based protein design by a combinatorial factor. By using a fixed backbone template, a rotamer library, and a potential energy function, DEE identifies and prunes rotamer choices that are provably not part of the Global Minimum Energy Conformation (GMEC), effectively eliminating the majority of the conformations that must be subsequently enumerated to obtain the GMEC. Since a fixed-backbone model biases the algorithm predictions against protein sequences for which even small backbone movements may result in a significantly enhanced stability, the incorporation of backbone flexibility can improve the accuracy of the design predictions. If explicit backbone flexibility is incorporated into the model, however, the traditional DEE criteria can no longer guarantee that the flexible-backbone GMEC, the lowest-energy conformation when the backbone is allowed to flex, will not be pruned. RESULTS: We derive a novel DEE pruning criterion, flexible-backbone DEE (BD), that is provably accurate with backbone flexibility, guaranteeing that no rotamers belonging to the flexible-backbone GMEC are pruned; we also present further enhancements to BD for improved pruning efficiency. The results from applying our novel algorithms to redesign the beta1 domain of protein G and to switch the substrate specificity of the NRPS enzyme GrsA-PheA are then compared against the results from previous fixed-backbone DEE algorithms. We confirm experimentally that traditional-DEE is indeed not provably-accurate with backbone flexibility and that BD is capable of generating conformations with significantly lower energies, thus confirming the feasibility of our novel algorithms. AVAILABILITY: Contact authors for source code.     

Database algorithm for generating protein backbone and side-chain co-ordinates from a C alpha trace application to model building and detection of co-ordinate errors

The problem of constructing all-atom model co-ordinates of a protein from an outline of the polypeptide chain is encountered in protein structure determination by crystallography or nuclear magnetic resonance spectroscopy, in model building by homology and in protein design. Here, we present an automatic procedure for generating full protein co-ordinates (backbone and, optionally, side-chains) given the C alpha trace and amino acid sequence. To construct backbones, a protein structure database is first scanned for fragments that locally fit the chain trace according to distance criteria. A best path algorithm then sifts through these segments and selects an optimal path with minimal mismatch at fragment joints. In blind tests, using fully known protein structures, backbones (C alpha, C, N, O) can be reconstructed with a reliability of 0.4 to 0.6 A root-mean-square position deviation and not more than 0 to 5% peptide flips. This accuracy is sufficient to identify possible errors in protein co-ordinate sets. To construct full co-ordinates, side-chains are added from a library of frequently occurring rotamers using a simple and fast Monte Carlo procedure with simulated annealing. In tests on X-ray structures determined at better than 2.5 A resolution, the positions of side-chain atoms in the protein core (less than 20% relative accessibility) have an accuracy of 1.6 A (r.m.s. deviation) and 70% of chi 1 angles are within 30 degrees of the X-ray structure. The computer program MaxSprout is available on request.     

Fast and simple Monte Carlo algorithm for side chain optimization in proteins: application to model building by homology.

An unknown protein structure can be predicted with fair accuracy once an evolutionary connection at the sequence level has been made to a protein of known 3-D structure. In model building by homology, one typically starts with a backbone framework, rebuilds new loop regions, and replaces nonconserved side chains. Here, we use an extremely efficient Monte Carlo algorithm in rotamer space with simulated annealing and simple potential energy functions to optimize the packing of side chains on given backbone models. Optimized models are generated within minutes on a workstation, with reasonable accuracy (average of 81% side chain chi 1 dihedral angles correct in the cores of proteins determined at better than 2.5 A resolution). As expected, the quality of the models decreases with decreasing accuracy of backbone coordinates. If the back-bone was taken from a homologous rather than the same protein, about 70% side chain chi 1 angles were modeled correctly in the core in a case of strong homology and about 60% in a case of medium homology. The algorithm can be used in automated, fast, and reproducible model building by homology.     

Antibody side chain conformations are position‐dependent

Side chain prediction is an integral component of computational antibody design and structure prediction. Current antibody modelling tools use backbone‐dependent rotamer libraries with conformations taken from general proteins. Here we present our antibody‐specific rotamer library, where rotamers are binned according to their immunogenetics (IMGT) position, rather than their local backbone geometry. We find that for some amino acid types at certain positions, only a restricted number of side chain conformations are ever observed. Using this information, we are able to reduce the breadth of the rotamer sampling space. Based on our rotamer library, we built a side chain predictor, position‐dependent antibody rotamer swapper (PEARS). On a blind test set of 95 antibody model structures, PEARS had the highest average χ₁ and accuracy (78.7% and 64.8%) compared to three leading backbone‐dependent side chain predictors. Our use of IMGT position, rather than backbone ϕ/ψ, meant that PEARS was more robust to errors in the backbone of the model structure. PEARS also achieved the lowest number of side chain–side chain clashes. PEARS is freely available as a web application at http://opig.stats.ox.ac.uk/webapps/pears .     

Excluded volume in protein side-chain packing

The excluded volume occupied by protein side-chains and the requirement of high packing density in the protein interior should severely limit the number of side-chain conformations compatible with a given native backbone. To examine the relationship between side-chain geometry and side-chain packing, we use an all-atom Monte Carlo simulation to sample the large space of side-chain conformations. We study three models of excluded volume and use umbrella sampling to effectively explore the entire space. We find that while excluded volume constraints reduce the size of conformational space by many orders of magnitude, the number of allowed conformations is still large. An average repacked conformation has 20 % of its chi angles in a non-native state, a marked reduction from the expected 67 % in the absence of excluded volume. Interestingly, well-packed conformations with up to 50 % non-native chi angles exist. The repacked conformations have native packing density as measured by a standard Voronoi procedure. Entropy is distributed non-uniformly over positions, and we partially explain the observed distribution using rotamer probabilities derived from the Protein Data Bank database. In several cases, native rotamers that occur infrequently in the database are seen with high probability in our simulation, indicating that sequence-specific excluded volume interactions can stabilize rotamers that are rare for a given backbone. In spite of our finding that 65 % of the native rotamers and 85 % of chi(1) angles can be predicted correctly on the basis of excluded volume only, 95 % of positions can accommodate more than one rotamer in simulation. We estimate that, in order to quench the side-chain entropy observed in the presence of excluded volume interactions, other interactions (hydrophobic, polar, electrostatic) must provide an additional stabilization of at least 0.6 kT per residue in order to single out the native state.     

GEM: a Gaussian Evolutionary Method for predicting protein side-chain conformations

We have developed an evolutionary approach to predicting protein side-chain conformations. This approach, referred to as the Gaussian Evolutionary Method (GEM), combines both discrete and continuous global search mechanisms. The former helps speed up convergence by reducing the size of rotamer space, whereas the latter, integrating decreasing-based Gaussian mutations and self-adaptive Gaussian mutations, continuously adapts dihedrals to optimal conformations. We tested our approach on 38 proteins ranging in size from 46 to 325 residues and showed that the results were comparable to those using other methods. The average accuracies of our predictions were 80% for chi(1), 66% for chi(1 + 2), and 1.36 A for the root mean square deviation of side-chain positions. We found that if our scoring function was perfect, the prediction accuracy was also essentially perfect. However, perfect prediction could not be achieved if only a discrete search mechanism was applied. These results suggest that GEM is robust and can be used to examine the factors limiting the accuracy of protein side-chain prediction methods. Furthermore, it can be used to systematically evaluate and thus improve scoring functions.