Improved side-chain prediction accuracy using an ab initio potential energy function and a very large rotamer library

Accurate prediction of the placement and comformations of protein side chains given only the backbone trace has a wide range of uses in protein design, structure prediction, and functional analysis. Prediction has most often relied on discrete rotamer libraries so that rapid fitness of side-chain rotamers can be assessed against some scoring function. Scoring functions are generally based on experimental parameters from small-molecule studies or empirical parameters based on determined protein structures. Here, we describe the NCN algorithm for predicting the placement of side chains. A predominantly first-principles approach was taken to develop the potential energy function incorporating van der Waals and electrostatics based on the OPLS parameters, and a hydrogen bonding term. The only empirical knowledge used is the frequency of rotameric states from the PDB. The rotamer library includes nearly 50,000 rotamers, and is the most extensive discrete library used to date. Although the computational time tends to be longer than most other algorithms, the overall accuracy exceeds all algorithms in the literature when placing rotamers on an accurate backbone trace. Considering only the most buried residues, 80% of the total residues tested, the placement accuracy reaches 92% for chi(1), and 83% for chi(1 + 2), and an overall RMS deviation of 1 A. Additionally, we show that if information is available to restrict chi(1) to one rotamer well, then this algorithm can generate structures with an average RMS deviation of 1.0 A for all heavy side-chains atoms and a corresponding overall chi(1 + 2) accuracy of 85.0%.     

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.     

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.     

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.     

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.     

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.     

Beyond the rotamer library: genetic algorithm combined with the disturbing mutation process for upbuilding protein side-chains

The disturbing genetic algorithm, incorporating the disturbing mutation process into the genetic algorithm flow, has been developed to extend the searching space of side-chain conformations and to improve the quality of the rotamer library. Moreover, the growing generation amount idea, simulating the real situation of the natural evolution, is introduced to improve the searching speed. In the calculations using the pseudo energy scoring function of the root mean squared deviation, the disturbing genetic algorithm method has been shown to be highly efficient. With the real energy function based on AMBER force field, the program has been applied to rebuilding side-chain conformations of 25 high-quality crystallographic structures of single-protein and protein-protein complexes. The averaged root mean standard deviation of atom coordinates in side-chains and veracities of the torsion angles of chi(1) and chi(1) + chi(2) are 1.165 A, 88.2 and 72.9% for the buried residues, respectively, and 1.493 A, 79.2 and 64.7% for all residues, showing that the method has equal precision to the program SCWRL, whereas it performs better in the prediction of buried residues and protein-protein interfaces. This method has been successfully used in redesigning the interface of the Basnase-Barstar complex, indicating that it will have extensive application in protein design, protein sequence and structure relationship studies, and research on protein-protein interaction.     

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.     

Improved modeling of side-chains in proteins with rotamer-based methods: a flexible rotamer model

Side-chain modeling has a widespread application in many current methods for protein tertiary structure determination, prediction, and design. Of the existing side-chain modeling methods, rotamer-based methods are the fastest and most efficient. Classically, a rotamer is conceived as a single, rigid conformation of an amino acid sidechain. Here, we present a flexible rotamer model in which a rotamer is a continuous ensemble of conformations that cluster around the classic rigid rotamer. We have developed a thermodynamically based method for calculating effective energies for the flexible rotamer. These energies have a one-to-one correspondence with the potential energies of the rigid rotamer. Therefore, the flexible rotamer model is completely general and may be used with any rotamer-based method in substitution of the rigid rotamer model. We have compared the performance of the flexible and rigid rotamer models with one side-chain modeling method in particular (the self-consistent mean field theory method) on a set of 20 high quality crystallographic protein structures. For the flexible rotamer model, we obtained average predictions of 85.8% for chi1, 76.5% for chi1+2 and 1.34 A for root-mean-square deviation (RMSD); the corresponding values for core residues were 93.0%, 87.7% and 0.70 A, respectively. These values represent improvements of 7.3% for chi1, 8.1% for chi1+2 and 0.23 A for RMSD over the predictions obtained with the rigid rotamer model under otherwise identical conditions; the corresponding improvements for core residues were 6.9%, 10.5% and 0.43 A, respectively. We found that the predictions obtained with the flexible rotamer model were also significantly better than those obtained for the same set of proteins with another state-of-the-art side-chain placement method in the literature, especially for core residues. The flexible rotamer model represents a considerable improvement over the classic rigid rotamer model. It can, therefore, be used with considerable advantage in all rotamer-based methods commonly applied to protein tertiary structure determination, prediction, and design and also in predictions of free energies in mutational studies.     

A graph-theory algorithm for rapid protein side-chain prediction

Fast and accurate side-chain conformation prediction is important for homology modeling, ab initio protein structure prediction, and protein design applications. Many methods have been presented, although only a few computer programs are publicly available. The SCWRL program is one such method and is widely used because of its speed, accuracy, and ease of use. A new algorithm for SCWRL is presented that uses results from graph theory to solve the combinatorial problem encountered in the side-chain prediction problem. In this method, side chains are represented as vertices in an undirected graph. Any two residues that have rotamers with nonzero interaction energies are considered to have an edge in the graph. The resulting graph can be partitioned into connected subgraphs with no edges between them. These subgraphs can in turn be broken into biconnected components, which are graphs that cannot be disconnected by removal of a single vertex. The combinatorial problem is reduced to finding the minimum energy of these small biconnected components and combining the results to identify the global minimum energy conformation. This algorithm is able to complete predictions on a set of 180 proteins with 34342 side chains in <7 min of computer time. The total chi(1) and chi(1 + 2) dihedral angle accuracies are 82.6% and 73.7% using a simple energy function based on the backbone-dependent rotamer library and a linear repulsive steric energy. The new algorithm will allow for use of SCWRL in more demanding applications such as sequence design and ab initio structure prediction, as well addition of a more complex energy function and conformational flexibility, leading to increased accuracy.     

Analysis of side-chain rotamers in transmembrane proteins

We measured the frequency of side-chain rotamers in 14 alpha-helical and 16 beta-barrel membrane protein structures and found that the membrane environment considerably perturbs the rotamer frequencies compared to soluble proteins. Although there are limited experimental data, we found statistically significant changes in rotamer preferences depending on the residue environment. Rotamer distributions were influenced by whether the residues were lipid or protein facing, and whether the residues were found near the N- or C-terminus. Hydrogen-bonding interactions with the helical backbone perturbs the rotamer populations of Ser and His. Trp and Tyr favor side-chain conformations that allow their side chains to extend their polar atoms out of the membrane core, thereby aligning the side-chain polarity gradient with the polarity gradient of the membrane. Our results demonstrate how the membrane environment influences protein structures, providing information that will be useful in the structure prediction and design of transmembrane proteins.     

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.     

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.     

Side-chain conformations in 4-alpha-helical bundles.

The distribution of the chi(1), chi(2) dihedral angles in a dataset consisting of 12 unrelated 4-alpha-helical bundle proteins was determined and qualitatively compared with that observed in globular proteins. The analysis suggests that the 4-alpha-helical bundle motif could occasionally impose steric constraints on side chains: (i) the side-chain conformations are limited to only a subset of the conformations observed in globular proteins and for some amino acids they are sterically more constrained than those in helical regions of globular proteins; (ii) aspartic acid and asparagine occasionally adopt rotamers that have not been previously reported for globular or helical proteins; (iii) some rotamers of tyrosine and isoleucine are predominantly or exclusively associated with hydrophobic core positions (a, d); (iv) mutations in the hydrophobic core occur preferentially between residue types which among other physicochemical properties also share a predominant rotamer.     

Rotamer strain energy in protein helices - quantification of a major force opposing protein folding

It is widely believed that the dominant force opposing protein folding is the entropic cost of restricting internal rotations. The energetic changes from restricting side-chain torsional motion are more complex than simply a loss of conformational entropy, however. A second force opposing protein folding arises when a side-chain in the folded state is not in its lowest-energy rotamer, giving rotameric strain. chi strain energy results from a dihedral angle being shifted from the most stable conformation of a rotamer when a protein folds. We calculated the energy of a side-chain as a function of its dihedral angles in a poly(Ala) helix. Using these energy profiles, we quantify conformational entropy, rotameric strain energy and chi strain energy for all 17 amino acid residues with side-chains in alpha-helices. We can calculate these terms for any amino acid in a helix interior in a protein, as a function of its side-chain dihedral angles, and have implemented this algorithm on a web page. The mean change in rotameric strain energy on folding is 0.42 kcal mol-1 per residue and the mean chi strain energy is 0.64 kcal mol-1 per residue. Loss of conformational entropy opposes folding by a mean of 1.1 kcal mol-1 per residue, and the mean total force opposing restricting a side-chain into a helix is 2.2 kcal mol-1. Conformational entropy estimates alone therefore greatly underestimate the forces opposing protein folding. The introduction of strain when a protein folds should not be neglected when attempting to quantify the balance of forces affecting protein stability. Consideration of rotameric strain energy may help the use of rotamer libraries in protein design and rationalise the effects of mutations where side-chain conformations change.     

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.     

Determinants of protein side-chain packing.

The problem of protein side-chain packing for a given backbone trace is investigated using 3 different prediction models. The first requires an exhaustive search of all possible combinations of side-chain conformers, using the dead-end elimination theorem. The second considers only side-chain-backbone interactions, whereas the third neglects side-chain-backbone interactions and instead keeps side-chain-side-chain interactions. Predictions of side-chain conformations for 11 proteins using all 3 models show that removal of side-chain-side-chain interactions does not cause a large decrease in the prediction accuracy, whereas the model having only side-chain-side-chain interactions still retains a significant level of accuracy. These results suggest that the 2 classes of interactions, side-chain-backbone and side-chain-side-chain, are consistent with each other and work concurrently to stabilize the native conformations. This is confirmed by analyses of energy spectra of the side-chain conformations derived from the fourth prediction model, the Independent model, which gives almost the same quality of the prediction as the dead-end elimination. The analyses indicate that the 2 classes of interactions simultaneously increase the energy difference between the native and nonnative conformations.     

Progress and challenges in high-resolution refinement of protein structure models

Achieving atomic level accuracy in de novo structure prediction presents a formidable challenge even in the context of protein models with correct topologies. High-resolution refinement is a fundamental test of force field accuracy and sampling methodology, and its limited success in both comparative modeling and de novo prediction contexts highlights the limitations of current approaches. We constructed four tests to identify bottlenecks in our current approach and to guide progress in this challenging area. The first three tests showed that idealized native structures are stable under our refinement simulation conditions and that the refinement protocol can significantly decrease the root mean square deviation (RMSD) of perturbed native structures. In the fourth test we applied the refinement protocol to de novo models and showed that accurate models could be identified based on their energies, and in several cases many of the buried side chains adopted native-like conformations. We also showed that the differences in backbone and side-chain conformations between the refined de novo models and the native structures are largely localized to loop regions and regions where the native structure has unusual features such as rare rotamers or atypical hydrogen bonding between beta-strands. The refined de novo models typically have higher energies than refined idealized native structures, indicating that sampling of local backbone conformations and side-chain packing arrangements in a condensed state is a primary obstacle.     

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.