ROTAS: a rotamer-dependent, atomic statistical potential for assessment and prediction of protein structures
© Park and Saitou; licensee BioMed Central Ltd. 2014
Received: 2 March 2014
Accepted: 9 September 2014
Published: 18 September 2014
Multibody potentials accounting for cooperative effects of molecular interactions have shown better accuracy than typical pairwise potentials. The main challenge in the development of such potentials is to find relevant structural features that characterize the tightly folded proteins. Also, the side-chains of residues adopt several specific, staggered conformations, known as rotamers within protein structures. Different molecular conformations result in different dipole moments and induce charge reorientations. However, until now modeling of the rotameric state of residues had not been incorporated into the development of multibody potentials for modeling non-bonded interactions in protein structures.
In this study, we develop a new multibody statistical potential which can account for the influence of rotameric states on the specificity of atomic interactions. In this potential, named “rotamer-dependent atomic statistical potential” (ROTAS), the interaction between two atoms is specified by not only the distance and relative orientation but also by two state parameters concerning the rotameric state of the residues to which the interacting atoms belong. It was clearly found that the rotameric state is correlated to the specificity of atomic interactions. Such rotamer-dependencies are not limited to specific type or certain range of interactions. The performance of ROTAS was tested using 13 sets of decoys and was compared to those of existing atomic-level statistical potentials which incorporate orientation-dependent energy terms. The results show that ROTAS performs better than other competing potentials not only in native structure recognition, but also in best model selection and correlation coefficients between energy and model quality.
A new multibody statistical potential, ROTAS accounting for the influence of rotameric states on the specificity of atomic interactions was developed and tested on decoy sets. The results show that ROTAS has improved ability to recognize native structure from decoy models compared to other potentials. The effectiveness of ROTAS may provide insightful information for the development of many applications which require accurate side-chain modeling such as protein design, mutation analysis, and docking simulation.
Understanding the structure and function of proteins requires an accurate potential energy function to quantify interactions between residues or atoms. One approach for the design and construction of potential energy functions is to make use of the information embedded in the known protein structures [1–6]. Such energy functions, called statistical potentials or knowledge-based potentials are derived by converting the observed frequencies of residue or atomic interactions in a database of protein structures into the free energies of corresponding interactions. Any aspect of structural features which characterize important interactions in the folded structures can be incorporated into the derivation of statistical potentials. Although their physical interpretations are still debated [7–9], due to their accuracy and computational efficiency, statistical potentials have been used with considerable success in many applications such as fold recognition and threading [10, 11], protein structure prediction , protein design , binding [14, 15] and aggregation .
The key idea in the development of statistical potentials is how to decompose the 3-D network of interactions in protein structures. Typical pairwise potentials cannot accurately describe non-bonded interactions in protein structures. As the folded protein structures are tightly packed and surrounded by solvent molecules, the surrounding circumstances of interacting atoms are inhomogeneous and anisotropic. Also, due to the bond connectivity, there are always correlated interactions from nearby bonded atoms. Thus, more detailed and complex structural features involving multibody effects have been incorporated into the formulation of statistical potentials. For example, sequential segments of various lengths have proved useful for prediction of secondary structure [17–20]. Four body potentials were used to improve cooperativity of main-chain hydrogen-bonds [21, 22]. A variety of structural motifs (i.e., residue clusters) has been identified to better characterize tightly packed protein structures [23–28]. Delaunay tessellation technique also has been employed as a means of defining multibody interactions [29, 30]. Local environment templates which could account for maximum 17 residues have been introduced to more accurately capture cooperative effects in protein structures . A secondary structure specific implementation of pairwise potentials has demonstrated its superiority to typical residue pairwise potentials [31, 32]. The introduction of orientation dependencies of interactions into typical distance-dependent pairwise potentials has achieved substantial improvements in both residue-level [33–36] and atomic-level potentials [37–40]. These multibody potentials are not only able to describe the 3-D interactions more completely but also able to account for cooperative effects of molecular interactions more accurately than typical pairwise potentials.
On the other hand, protein residues have great flexibility because their single covalent bonds allow rotation of the atoms they join. It is well known that residues prefer to adopt only a limited number of staggered conformations, known as rotamers due to local steric interactions (e.g. overlapped electron orbitals) [41–44]. Since the electron density distribution around each nucleus can vary depending on the molecular conformation [45–47], different rotamers may result in different dipole moments and induce charge reorientations, which are reflected in dispersion forces and electrostatic forces. In addition to the polarization effect, solvation effect may be another source of multibody effects related to the rotameric state. For example, compact rotameric states would prefer to be buried within protein structures, while extended rotameric states would prefer to be exposed to solvent with high conformational entropy. Thus, non-bonded interactions between residue atoms may be influenced by the rotameric state of the residues to which the interacting atoms belong.
Existing potentials had not modeled the flexibility of residues explicitly. For example, residue-level potentials which have only one interaction site per residue simply ignore the flexibility of residue conformation. In case of atomic-level potentials, although the orientation dependence of atomic interactions may be able to account for the anisotropic environment around each atom, it is also based on rigid blocks  or rigid atom fragments (i.e. three atoms that are consecutively bonded) [38–40]. Thus they cannot reflect the influence of rotameric states on the specificity of atomic interactions no matter how complete a description of the relative orientation and distance between interacting atoms may be used.
Here we studied the energy dependence of residue flexibility and developed a new multibody potential, named “rotamer-dependent atomic statistical potential” (ROTAS). The interaction between two atoms is specified by not only the distance and relative orientation but also by two state parameters which concern the rotameric state of the residues to which the interacting atoms belong. It was clearly found that the rotameric state of residues is correlated to the specificity of interactions within protein structures. Furthermore, such rotamer-dependencies are not limited to specific type or certain range of interactions. We tested ROTAS on various sets of decoys and compared its performance to those of several existing atomic potentials. The results show that ROTAS led to an improvement not only in the native structure recognition, but also in the best model selection and the correlation coefficients between energy and model quality. The ROTAS potential is freely available in https://sites.google.com/a/umich.edu/rotas/.
Derivation of ROTAS
Here, E(R i ) and E(R j ) can be seen as rotamer intrinsic energy. Assuming that the stability of overall folded structure is mainly determined by non-bonded interactions, we ignore these terms in this study.
Defining the rotameric state
All 167 residue-specific heavy atom types and associated side-chain dihedral angles for defining their rotameric states
Number of rotameric states
C, O, N, Cα
C, O, N, Cα, Cβ
C, O, N, Cα, Cβ, Sγ
C, O, N, Cα, Cβ, Oγ
C, O, N, Cα, Cβ, Oγ1, Oγ2
C, O, N, Cα, Cβ, Cγ, Cδ
C, O, N, Cα, Cβ, Cγ1, Cγ2
C, O, N, Cα, Cβ, Cγ1, Cγ2, Cδ1
C, O, N, Cα, Cβ, Cγ, Cδ1, Cδ2
C, O, N, Cα, Cβ, Cγ, Oδ1, Oδ2
C, O, N, Cα, Cβ, Cγ, Oδ1, Nδ2
C, O, N, Cα, Cβ, Cγ
Cδ, Oϵ1, Oϵ2
C, O, N, Cα, Cβ
Cγ, Cδ, Oϵ1, Nϵ2
C, O, N, Cα, Cβ
Cγ, Sδ, Cϵ
C, O, N, Cα, Cβ
Cδ, Nϵ, Cξ
C, O, N, Cα, Cβ
Cδ, Cϵ, Nξ
C, O, N, Cα, Cβ, Cγ, Nδ1, Cδ2
C, O, N, Cα, Cβ
Cγ, Cδ1, Cδ2
Cϵ1, Cϵ2, Cξ
C, O, N, Cα, Cβ, Cγ, Cδ1, Cδ2
Nϵ1, Cϵ2, Cϵ3, Cξ2, Cξ3, Cη2
C, O, N, Cα, Cβ
Cγ, Cδ1, Cδ2
Cϵ1, Cϵ2, Cξ, Oη
Construction of distance-dependent potential
where N obs (d ij ) is the number of observed atom pair i and j at distance d, and α is a scaling factor such that N exp (d) increases in dα. Beyond a distance cutoff , it is assumed that both observed and expected pairwise distributions are equal. Here we set = 15 Å and α = 1.61 as suggested by the original work . To obtain the distribution, the bin width is set to 0.5 Å from 0 to 15 Å. When estimating the observed probability and evaluating the distance-dependent pairwise potential, atom pairs that are in the same residue are excluded.
In addition to DFIRE, we constructed other widely used distance-dependent potentials such as RAPDF , KBP , DOPE  and RW  and tested each of them in ROTAS in order to examine the influence of different reference states on the performance of ROTAS. The same structural database, distance cutoff and bin width were applied.
Construction of orientation-dependent potential
where K VM denotes the von Mises kernel function, κ is the kernel bandwidth controlling the smoothness of the kernel and I 0 is the Bessel function of the first kind of order 0. Here, we set κ = 8.21 which is equivalent to σ = π/9 in the normal distribution. The distances d ij were discretized into 0.5 Å bins which span from 2 to 15 Å. The kernel density estimator is computed at π/9 grid points that are ranged from -π to π (in case of ϕ, from -π/2 to π/2).
The relative orientation between atoms is significantly affected by chain connectivity constrains when the atoms are positioned in residues that are close in the sequence. In order to reduce the chain (or bond) connectivity effect on the estimates of orientation-dependent probability, we applied a sequence separation as done in other studies [33, 40]. In this study, only atom pairs that are separated by at least 6 residues along the protein chain are considered.
where N(d ij , R i ) is the number of observations used to estimate P obs (θ i |d ij , R i ) and σ is a parameter that controls how many observations must be sampled such that both P obs (θ i |d ij , R i ) and P obs (θ i |d ij ) would have equal weights. Here we set σ = 1/100.
where M is a normalization factor such that the integration of P exp (ϕ) from -π/2 to π/2 becomes one. P exp (ϕ) is calculated numerically because there is no analytical way for integrating above equation.
Interaction cutoff for ROTAS
Preparation of protein structures
We obtained a set of protein X-ray structures with a maximum R-factor of 0.25 and a resolution better than 2 Å from the protein sequence culling server, PISCES . Also, protein chains were filtered out with a 40% sequence identity cutoff in order to have a set of non-homologous protein structures. A total 9321 protein structures were selected and downloaded from the Protein Data Bank (PDB) . We did not attempt to exclude the homologous proteins to the test decoy sets from the 9321 proteins used for constructing the potential. It was reported that the exclusion has very little effect on the performance of statistical potentials . The program REDUCE  was used to optimize the flip states of Asn, Gln, and His in all protein structures. Residues with multiple side-chain conformations were modified such that only the side-chain conformations with atoms having the highest occupancy and/or lowest temperature factors were used.
Performance evaluation using decoy sets
We tested the ROTAS potential on various sets of decoys generated by different methods. A total of 13 decoy sets, including 4 state_reduced , fisa , fisa_casp3 , lmds , hg_structal, ig_structal, ig_structal_hires, lattice_ssfit , moulder , Rosetta , I-TASSER , AMBER99  and CASP5-8 , were used The first 8 decoy sets were downloaded from the Decoys ‘R’ Us database  (http://dd.compbio.washington.edu/). The moulder decoy set produced by iterative target-template alignment and comparative-modeling methods was download from the Sali lab (http://salilab.org/decoys/). Three ab-initio simulation based decoy sets, Rosetta, I-TASSWER, Amber99 were obtained from http://zhanglab.ccmb.med.umich.edu/decoys/, and http://cssb.biology.gatech.edu/amberff99/, respectively. The CASP5-8 decoy set collected from the CASP5-CASP8 experiments was downloaded from http://zhanglab.ccmb.med.umich.edu/RW/ (cleaned version). The decoy models in this set were generated by a large variety of groups and methods participated in the CASP experiments.
The performance of ROTAS potential was compared to those of four other existing atomic potentials which take into account the orientation-dependencies on the interactions between atoms, blocks or side-chains: dDFIRE , OPUS_PSP , RWplus , and GOAP . Furthermore, we compared ROTAS to evolutionary pairwise distance-dependent potential, EPAD  and attempted to combine both potentials to maximize the performance. The binary programs for these potentials were downloaded from the corresponding authors’ websites. Because ROTAS can be seen as an extended version of GOAP, we constructed our own GOAP potential energy function using the same structure database and techniques that were used for the construction of ROTAS. In this manner we reduced the possibility that estimation of probability distribution, specific computational implementation, or other technical aspects could affect the results, so that the improvements of ROTAS compared to GOAP can be fairly demonstrated.
The performance of statistical potentials is evaluated by four aspects: (1) the recognition of native structure from decoys, (2) the selection of the best (most native-like) decoy model, (3) the correlation between the energy score and model quality, and (4) the classification of near-native and non-native model. The quality of decoy models was assessed by TM-score which measures the similarity between two protein structures by a score between (0, 1] .
Results and discussion
The influence of rotameric states on atomic interactions
Native structure recognition
The lower the Z-score, the better the potential is for recognizing the native structures.
Performance on native structure recognition
The ability of ROTAS on native structure recognition as a function of native structure resolution
R < = 1.8
1.8 < = R < 2.2
2.2 < = R < 2.8
2.8 < R
Best model selection
We also assessed the ability of ROTAS in selecting the best models without native structures. This is more difficult and realistic task than the native structure recognition because, in practice, potential energy functions are used to find more and more native-like conformations in an iterative way when the native structure is not known. Thus, good potential energy function should be able to score the most native-like decoy model in the lowest energy. In this study, we use TM-score  to assess the quality of decoy models quantitatively. The TM-score measures the similarity between two protein structures by a score between (0, 1]. It is reported that TM-score is more accurate than other measures such as RMSD or GDT_TS because TM-score is sensitive to overall topology rather than local substructures .
Performance on best model selection
In both measures, GOAP and ROTAS shows better performance than other three potentials, dDFIRE, OPUS_PSP and RWplus. The average log PB1 by GOAP is slightly better than that by ROTAS, whereas the average log PB10 by ROTAS is better than that by GOAP. This indicates that the lowest energy model by GOAP is likely to be better in TM-score than that by ROTAS. However, when we consider the top 10 lowest energy models, ROTAS tend to include better TM-score decoy models in the top 10 than GOAP.
Correlation between the energy score and decoy model quality
Performance on correlation coefficients between energy score and model quality
Classification of near-native and non-native model
In order to compare the performance of ROTAS and other potentials in a more robust way, we evaluated the performance of statistical potentials using receiver operating characteristic (ROC) technique . That is, the energy score was used to rank the decoy models for each target, and then thresholds were applied to classify a group of near-native models among a pool of putative models. The near-native (positive) were defined as those with TM-score larger than 0.5 with respect to the native structure, and non-native (negative) models otherwise. In fact, it is reported that protein structures having a TM-score > 0.5 are mostly in the same fold . ROC curves were obtained by plotting the true positive ratio against the corresponding false positive ratio for all thresholds on the energy score.
The area under the ROC curves for classification of near-native and non-native model
To quantify the statistical significance of the difference between ROTAS and other potentials, P values of the paired t-test of the differences between ROTAS and other potentials for the AUCs were also calculated. Cleary, ROTAS gives statistically significant (P value < 0.01) better results than all other potentials.
So far, the results showed that ROTAS performs better than other competing potentials not only in native structure recognition, but also in best model selection and correlation coefficients between energy and model quality. The following sections discuss factors affecting on the performance of ROTAS as well as a possible way to improve the performance by combining other statistical potentials.
Interaction cutoff effect on the performance
It is noticed that the highest average correlation coefficient is obtained when we consider all the long-range interactions available in the potentials. However, in this case, the native structures are poorly recognized. A similar observation that a scoring function producing a good linear correlation is normally less capable of recognizing the native state has been reported in a previous study . A theoretical study argue that the potential energy of near-native conformations might not be linearly related to their distances from the native state . Also, since a shorter interaction cutoff would increase ruggedness of the energy landscape , the energy score of decoy models might be affected by small structural differences sensitively.
Different reference states
Comparison of different reference states in ROTAS
Possible improvement by incorporating evolutionary information
Beyond structural features embedded in known protein structures, evolutionary information also can be utilized in protein structure prediction . Evolutionary pairwise distance-dependent potential (EPAD)  is a successful example of statistical potentials utilizing evolutionary information in a large amount of sequence data. In fact, EPAD has different energy profile between two atoms depending on the protein under consideration and the sequence profile context of the atoms (i.e. evolutionary information). As a possible way to improve ROTAS, we built a composite energy function by replacing the distance-dependent pairwise energy term in ROTAS with EPAD.
Performance of EPAD, ROTAS and ROTAS + EPAD
EPAD + ROTAS
In this study, we hypothesized that the rotameric state of residues critically affects on the specificity of non-bonded interactions within protein structures. This idea was applied to develop a new multibody statistical potential (ROTAS) for protein structure prediction. The interaction between two atoms is specified by not only the distance and relative orientation but also by two state parameters concerning the rotameric state of the residues to which the interacting atoms belong. It was clearly found that the rotameric state is correlated to the specificity of atomic interactions. Furthermore, such rotamer-dependencies are not limited to specific type or certain range of interactions.
The incorporation of accurate modeling of residue flexibility has been shown to be a possible means of improving the specificity of potential energy functions. We tested ROTAS using various decoy sets and compared its performance to those of several existing atomic statistical potentials which incorporate orientation-dependent energy terms. For a fair comparison, we implemented our own GOAP potential using the same structure database and techniques used for the construction of ROTAS. The results showed that ROTAS performs better than other competing potentials not only in native structure recognition, but also in best model selection and correlation coefficients between energy and model quality. In particular, the relative improvement of ROTAS over GOAP indicates that the rotameric state of residues can be incorporated for a fine-tuning of atomic-level statistical potentials. The effectiveness of ROTAS may provide insightful information for the development of many applications which require accurate side-chain modeling such as homology modeling, protein design, mutation analysis, protein-protein docking and flexible ligand docking.
The authors thank to Professor Matthew Young and Professor Yang Zhang for their valuable comments which have greatly improved the manuscript.
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