Predicting the native structure of proteins from their amino acid sequences has yet remained an elusive goal. In general this entails the development of effective methods for conformation sampling and the design of an accurate function for structure discrimination 12. The functions could be based on elaborate calculations and analyses of forces between atoms 34, or be knowledge-based that extract relevant parameters from a database of experimentally determined protein structures 56. One important area of application of knowledge-based potential functions has been in "protein threading" for the prediction of protein tertiary structure in the absence of detectable sequence homology. The technique involves threading a protein sequence onto the frameworks of known protein folds and finding the most energetically favorable conformation 78910. In addition to fold recognition applications, where the best conformation of a protein is selected from a database of known protein conformations, the knowledge-based scoring functions are also used in protein folding simulations 6111213141516. Many statistical scoring functions assume that frequencies of non-bonded pairs of amino acids follow a Boltzmann-like distribution and the minimum value of the score occurs in the vicinity of the lowest energy structure. Additionally, a set of probability distributions can also be used to construct a scoring function such that it can identify the maximum probability structure.

For testing of empirical energy functions challenging and diverse datasets of decoy structures that are native-like in properties have been generated 12171819. Models submitted in the community-wide experiment, CASP (Critical Assessment of techniques for protein Structure Prediction) 20 make up diverse sets of structures resulting from various computational approaches 21. The most native-like structure needs to be identified from among these models 22. An effective potential should be able to distinguish the native structure from decoy structures with a high degree of accuracy. Energy functions based on residue contact or compactness alone do not have enough discriminating power 12, or can rank the native structure highly only when the competing conformations are more random-coil like 23. However, here we present two knowledge-based scoring functions based on the analysis of residue packing in protein structures that are quite robust in discriminating the native conformation from a number of misfolded conformations for a given primary protein sequence. The functions were also tested on ~ 19000 models from server predictions for 71 targets of CASP7 20. As a descriptor for the residue packing we use the average values of the accessible surface area or the number of other residues in contact around a given residue, calculated from a database of globular proteins. Each of the function then evaluates the cumulative value for the deviation of the parameter for individual residues from the corresponding average value over the whole polypeptide chain. The experimental structure is found to have the minimum deviation and thus the minimum value of the function, when applied to a set of decoys from which the native structure has to be identified. The success of the function indicates that the burial of each residue and its contact to the surrounding residues is optimized during folding and the average values of these parameters can be used as constraint to simulate folding process. Additionally, a surface patch with residues having a large overall deviation of these parameters from the average values may be indicative of the binding region on a protein structure, an issue that would be addressed in future to provide a common perception to both the folding and the binding processes.

Scoring functions have been used to validate X-ray crystal structures, assess and rank three-dimensional models generated for a protein sequence, predict the effect of mutations, etc. Here, we are concerned with the identification of the native structure from decoys. The idea of the use of the discriminatory function originated from the formula of R-factor in crystallography 24. An exact equivalent formula would have meant the use of the expression (1) instead of (3), given in Methods.

The individual term in Eq. (3) involves the absolute difference between the observed and the average values of ASA for a given residue, normalized by the average value. These terms are summed over the whole sequence. In Eq. (1) the numerator and the denominator are summed separately. Some other modified formulae, including the use of the standard deviation on the average values <ASA_x> in the denominator, were also tried, but (2) and (3) were found most efficient to identify the native structure from a set of decoys. Depending on the structural context larger residues may have a considerable variation in their ASA values in protein structures (as indicated by larger standard deviations, Table 1) - normalization of the difference in the numerator in Eq. (3) has the effect of damping the contribution of such residues in the summation.

There are many energy functions (knowledge based statistical scoring function or physics-based or a combination of both) which find the correct native conformation from misfolded decoys 369121314152239404142. However, it is rather nontrivial to develop a function that works across different decoy sets and a combination of functions is normally used 1213. R-factor is the gold-standard for expressing the accuracy of crystallographic analysis, and as knowledge-based functions are mostly "trained" on crystal structures it is rather gratifying to develop functions similar to R-factor that can also be used to characterize the native structure (Table 2).

The present study demonstrates the development of scoring functions from the properties of residue packing that can be useful for discriminating the native conformation from various misfolded conformations for a given protein sequence. The algorithm assumes that a protein tries to take up a fold that has the minimum deviation of ASA (or PN) of each residue from the average value observed over all protein structures. The function R_s, based on residue accessibility, performs better than the one derived from the partner number, R_p, on decoy sets. The test on various decoy sets from the PROSTAR website demonstrated that the knowledge based scoring function developed in this study performs better or even at least of the same order than those previously derived by many authors 121415. Not only the present knowledge-based scoring functions pick the correct native structure in most cases, but the discrimination ratio is also better than that of the other potentials. However, as Equations (2) and (3) use the average values derived from a database of globular proteins, it is not likely to be very discriminatory for small proteins or peptides (as seen for the 'Ifu' set in Table 3). As such it would not be useful for checking local model quality in protein structures, as done by packages such as PROSA 43. Along the same line it may be mentioned that the Verify3D server 44 for the visual analysis of the quality of a crystal structure works best on proteins with at least 100 residues.

The Park and Levitt decoy set had been shown to be quite a challenging dataset where the lowest-energy structures typically were 6-10 Å rmsd away from native ones 12. The improved residue-based potential 18 also cannot recognize the native and near-native structures in all cases. The knowledge based scoring functions derived in this study are quite efficient to identify the near-native fold in Park and Levitt decoy sets. The correlation between the scoring function and rmsd is good in all cases and most of the cases the scoring functions have minimum value for the native structure. The scoring functions perform well also in the PROSTAR decoy sets, Levit's Local-Minima Decoy Sets (LMDS) and also in ROSETTA All-atom Decoy Sets. Considering 222 independent cases considered in this analysis R_s and R_p can efficiently discriminate native structures from all their corresponding decoys with a success rate greater than 85% and 74%, respectively. If we do not consider the 'Ifu' dataset, which comprises of small fragments of polypeptide chains, the success rate increases to 94% and 80%, respectively. The most rigorous test of a scoring function is to evaluate its performance in identifying the native structure with reference to the models submitted in CASP7 experiment. Even here, both R_s and R_p, the former in particular, stand out from all other methods (Table 5).

As our scoring functions depend on ASA or PN, these should be closely related to potentials of mean force derived from solvation or packing considerations. The performance of these potentials, however, depend critically on how the standard state is specified 61223. As the core and surface regions in proteins constitute distinct environments, potentials are sometimes divided into two parts, for the buried and the solvent-accessible regions 40. The use of the average values of ASA or PN in globular proteins seems to have eliminated the need of such division, or the debate on the proper choice of the standard state.

A discussion on the uniqueness of our parameters vis-à-vis other knowledge-based discrimination functions is in order. First, a residue in the sequence is normally represented in these functions with one or two positions in the three-dimensional space and one or more of its properties, such as the secondary structure or backbone dihedral angle preferences, features in distance or sequence separation from other residues, etc. are considered 723. With such a coarse representation the function may not be as efficient as an all-atom discriminatory function, which takes into account the environment of all the atoms in a residue 13454647. An all-atom representation is implicit in our method, as all the atoms are needed for the calculation of ASA or the partner number. However, each residue in the sequence contributes singly to the derivation of R_s or R_p. This is also in contrast to residue-residue interaction energy for each residue pair that is normally employed in other functions 124849. Furthermore, residue triplets and four-body contact potentials have also been developed 5051. Secondly, the energy functions are generally less discriminatory when used individually, and the use of the hybrid scoring function is the norm for an enhanced performance 121622. While conceptually simple, R_s or R_p can work as efficiently. Thirdly, most formulations use energy as the criterion (with the assumption that the native structure is at a global free-energy minimum), while our function seeks to find the conformation that has the minimum deviation from the average value of the partner number or ASA. This way the selection of the most compact state of the polypeptide chain corresponding to a given sequence is achieved. The parameters are less likely to be fooled by over-abundance (which is penalized to the same extent as lower-abundance in equations 2 and 3) of contacts, as is the case with some functions 12. Lastly, as the functions can identify the correct structure from the erroneous ones modeled from X-ray data ('Pdberr' set in Table 3) and vary within a narrow range in different protein classes (Table 2), these can be used for the validation of the structure determined crystallographically 52.

The functions developed here can also be used to delineate the compatibility of the sequence to a fold For example, azurin 53 and plastocyanin 54 are two small proteins having the same fold (a sandwich of two β-sheets having seven strands), but sequence identity of only 17% over an aligned length of 86 residues (Table 6). Expectedly, they have very similar R_s and R_p values. More interestingly however, when the sequence of plastocyanin is considered over the structure of azurin one gets a value of 0.97 for (R_s)_azu/pcy, quite close to 0.89 obtained for the reverse process ((R_s)_pcy/azu), thereby indicating the compatibility of the two sequences to the same fold.

This work demonstrates the effectiveness of a simple knowledge-based scoring function derived from residue packing for discriminating the native structures from a large set of decoys constructed by several groups. This knowledge-based scoring scheme is simple to derive and less computationally intensive than other energy functions and the performance is better (or at least at par) compared to others. Used in conjunction with other chemically intuitive parameter that captures the essence of the protein structure, it should be possible to achieve complete discrimination between the native structure and decoys.

Atomic coordinates were obtained from the Protein Data Bank (PDB) 55. The analysis was carried out using the dataset of 432 polypeptide chains in 418 PDB files (given in 26) with an R-factor ≤ 20%, a resolution ≤ 2.0 Å and sequence identity < 25%. Also the polypeptide chains with >40% of atoms with temperature factor (B-factor) >30 Ų were excluded. The calculation of the partner number was restricted only to the well-ordered residues by excluding those with >40% atoms with temperature factor >30 Ų. The solvent accessible surface area (ASA) was computed using the program NACCESS 56, which is an implementation of the Lee and Richards algorithm 57. The partner number of a residue is the number of other residues within a distance of 4.5 Å from any atom of the residue under consideration; the flanking residues were not considered as partner if the interaction was only with the main-chain atoms. The reason for the selection of the particular threshold value for the distance has been discussed 2658. To be identified as a partner it is enough if just a pair of atoms is in contact.

Two parameters R_p and R_s based on the observed partner number and the accessibility at a given position in the protein sequence, as compared to the average value of the parameters for the same residue type in the whole database, were developed as given in the following two equations

where PN_xi and ASA_xi are the observed partner number and the solvent accessible surface area, respectively, for a residues of type x occurring at a particular position, i, in a PDB file and <PN_x> and <ASA_x> are the average values of the residue type x in the analyzed dataset. Considering (3), the function sums up the absolute value of the deviation of ASA at each position in the sequence from the average ASA of the residue type, each term being normalized by the average ASA value. The magnitude of each of the two parameters derived using (2) and (3) is used to discriminate the near native fold from the misfolded decoys. For the correct fold the values of these two parameters should be minimum.

A number of decoy datasets have been used from literature, the details of which are provided in Results. The Z-score of a native structure and the misfolded decoys was also evaluated. The Z-scores using the residue accessibility (Z_s) and residue partner number (Z_p) of a particular protein conformation are defined by the following equations

where R_s-nat (or R_p-nat) is the value of the parameter for the native conformation, and <R_s> (<R_p>) and σ are the average and the standard deviation of the distribution of the parameter in the set. The magnitude of the Z-score is an indication of how far that native conformation is separated from the near native structures in the distribution.

PC conceptualized the work that was carried out by RPB. RPB and PC participated in interpretation of the data and writing the manuscript. Both the authors read and approved the final manuscript.