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Four-body atomic potential for modeling protein-ligand binding affinity: application to enzyme-inhibitor binding energy prediction



Models that are capable of reliably predicting binding affinities for protein-ligand complexes play an important role the field of structure-guided drug design.


Here, we begin by applying the computational geometry technique of Delaunay tessellation to each set of atomic coordinates for over 1400 diverse macromolecular structures, for the purpose of deriving a four-body statistical potential that serves as a topological scoring function. Next, we identify a second, independent set of three hundred protein-ligand complexes, having both high-resolution structures and known dissociation constants. Two-thirds of these complexes are randomly selected to train a predictive model of binding affinity as follows: two tessellations are generated in each case, one for the entire complex and another strictly for the isolated protein without its bound ligand, and a topological score is computed for each tessellation with the four-body potential. Predicted protein-ligand binding affinity is then based on an empirically derived linear function of the difference between both topological scores, one that appropriately scales the value of this difference.


A comparison between experimental and calculated binding affinity values over the two hundred complexes reveals a Pearson's correlation coefficient of r = 0.79 with a standard error of SE = 1.98 kcal/mol. To validate the method, we similarly generated two tessellations for each of the remaining protein-ligand complexes, computed their topological scores and the difference between the two scores for each complex, and applied the previously derived linear transformation of this topological score difference to predict binding affinities. For these one hundred complexes, we again observe a correlation of r = 0.79 (SE = 1.93 kcal/mol) between known and calculated binding affinities. Applying our model to an independent test set of high-resolution structures for three hundred diverse enzyme-inhibitor complexes, each with an experimentally known inhibition constant, also yields a correlation of r = 0.79 (SE = 2.39 kcal/mol) between experimental and calculated binding energies.


Lastly, we generate predictions with our model on a diverse test set of one hundred protein-ligand complexes previously used to benchmark 15 related methods, and our correlation of r = 0.66 between the calculated and experimental binding energies for this dataset exceeds those of the other approaches. Compared with these related prediction methods, our approach stands out based on salient features that include the reliability of our model, combined with the rapidity of the generated predictions, which are less than one second for an average sized complex.


Experimental high-throughput screening processes that drive structure-guided drug design efforts are effective tools for the identification of candidate molecular ligands that may tightly bind a target protein; however, such an approach often proves to be a costly endeavor, in terms of both time and financial expense, one that can potentially be alleviated with reliable in silico protein-ligand binding affinity models to assist in winnowing the search space [1]. A diverse array of computational approaches to model binding affinity have been described in the literature, each of which focuses on a unique combination of physicochemical properties and interactions: X-Score [2], Lig-Score [3], DrugScore [4], SFCscore [5], AutoDock4 [6], ITScore [7, 8], and PHOENIX [9] are just a few examples of such predictive tools. Here we describe our development of a model for predicting protein-ligand binding energy that relies on Delaunay tessellation, a computational geometry technique [10], for the purpose of objectively capturing nearest neighbor atomic four-body interactions in the structures of macromolecular complexes (Figure 1).

Figure 1
figure 1

Atomic Delaunay tessellation of the hen egg-white lysozyme (a) in complex with bound ligand NAG (N-Acetyl-D-Glucosamine) and (b) without the bound ligand (PDB accession code: 1HEW).

First, we compute the propensities for occurrence of all atomic quadruplet interactions by applying the tessellation procedure to atomic coordinates for a diverse cross-section of over 1400 high-resolution macromolecular crystal structures, and the data collected are used in generating an atomic four-body potential. Tasked with distinguishing native structures as having global energy minima relative to decoys, our knowledge-based potential performs well compared to several related atomic energy functions [11, 12]; however this work constitutes substantial research outside the immediate focus of this study, and accordingly it will be reported elsewhere. Next, we apply our atomic potential to a separate dataset of three hundred diverse protein-ligand complexes, each selected for having both a solved high-resolution crystal structure and a known dissociation constant (k d ), the latter quantity being useful for determining the Gibbs free energy of binding (ΔG). Two thirds of the complexes are randomly selected to train our predictive model of binding affinity: in each case, the entire complex is tessellated and then scored using the four-body potential, as is the structure of the isolated protein without its bound ligand, and we derive an empirical linear function of the difference between these scores to predict ΔG values. The remaining one hundred complexes are then used to validate the capability of the trained linear model to predict binding energies for new protein-ligand complexes.

The steps taken to develop our model formed the basis of a recently published companion study [13], and here we begin by carefully outlining those details below, since they lay the foundation for the next stage of the work to be presented. In particular, the model is subsequently applied to the prediction of binding affinities for an independent, diverse test set of three hundred enzyme-inhibitor complexes for which high-resolution crystal structures, as well as experimentally determined inhibition constants (k i ), are available. Also, model performance is comprehensively benchmarked against a number of related methods from the literature.



High-resolution (≤ 2.2Å) crystallographic structures for 1417 macromolecular complexes (Additional file 1), culled using the PISCES server [14] and having protein chains that share low (< 30%) sequence identity, were selected to develop the four-body statistical potential. Dataset diversity is also reflected in the fact that the complexes consist of both single chain and multimeric proteins, many of which have bound ligands in the form of either small molecules or peptides. Each complex has a coordinate file deposited in the Protein Data Bank (PDB) [15], and following the removal of all hydrogen atoms and water molecules, Delaunay tessellation is applied to each structure file by using all the remaining atomic coordinates.

In order to train and validate our model for predicting binding affinity, we selected another diverse set of three hundred protein-ligand complexes (Additional file 2) from the Binding MOAD [16, 17] database. The Binding MOAD is a repository for all protein-ligand complexes that have high-resolution (≤ 2.5Å) structures deposited in the PDB, and if available, published experimental binding energy data. Focusing specifically on a non-redundant subset of the Binding MOAD, both to ensure diversity of complexes as well as to minimize bias due to over-represented structures, we identified three hundred complexes having both PDB coordinate files as well as experimental dissociation constants (k d ). The PDB accession codes and k d values for the protein-ligand complexes are tabulated in Additional file 2, as is the identity of the subset (200 for training, and 100 for validation) into which each is randomly placed.

Software and performance measurements

We use the Qhull software package [18] to carry out the atomic Delaunay tessellations, Matlab (Version (R14) Service Pack 1) to produce graphical depictions of the tessellations, and the UCSF Chimera software package [19] to generate all other molecular visualizations in this study. Codes to perform all data formatting and analyses tasks are written in the Perl programming language.

Given the dissociation constant (k d ) for a protein-ligand complex, the standard Gibbs free energy of binding (ΔG, in units of kcal/mol) can be determined using

Δ G = R T  ln ( k d ) = 0 . 592 × ln ( k d ) ,

where R = 1.986 ×10-3 kcal K-1 mol-1 is the gas constant and T = 298° K is the absolute temperature. We evaluate the agreement between known (x i ) and predicted (y i ) binding energies by reporting the Pearson's correlation coefficient

r = ( x i - x ¯ ) ( y i - y ¯ ) ( x i - x ¯ ) 2 ( y i - y ¯ ) 2 ,

the standard error of the predictions

S E = ( 1 - r 2 ) ( y i - y ¯ ) 2 n - 2 = 1 n n - 2 n y i 2 - y i 2 - n x i y i - x i y i 2 n x i 2 - x i 2 ,

and the equation of the fitted regression line.


Four-body statistical potential

To generate our knowledge-based potential, a six-letter alphabet (C, N, O, S, M = all metals, X = all other non-metals) is used for labelling all atoms (excluding hydrogens and water molecules). The Qhull software uses the 3-dimensional (3D) coordinates of atoms in a PDB file to generate a Delaunay tessellation of the structure, a space-filling convex hull formed by hundreds of solid, non-overlapping, irregular tetrahedra whose vertices are the 3D atomic points. Each atom serves as a vertex, with most being shared by numerous adjacent tetrahedra, and every tetrahedral simplex objectively identifies a quadruplet of nearest neighbor atoms at its four vertices. To ensure this is indeed the case, we eliminate all edges longer than 8Å immediately upon tessellation, which is in agreement with related research in this arena at the atomic [20] and residue [21, 22] levels of coarse-graining. The combined total number of tetrahedra remaining for analysis after tessellating the 1417 PDB coordinate files is provided in Table 1, as are the total number of atoms of each type as well as their relative frequencies.

Table 1 Summary data for the 1417 PDB structure files.

Without regards to the ordering of a quadruplet of atoms (i.e, all permutations of the four letters are non-unique and represent the same quadruplet), and allowing for the repeated occurrence of atom types in any given quadruplet (i.e., letters may appear more than once in a quadruplet), there are 126 possible types of atomic quadruplets that can be enumerated based on the use of a 6-letter atomic alphabet (Table 2). For each quadruplet (i,j,k,l), we define f ijkl as the observed proportion of all tetrahedral simplices obtained by tessellating all 1417 structures to have those four types of atoms at the vertices; similarly, we let p ijkl represent the rate expected by chance, which is based on relative frequencies of the six atom types in the structures (Table 1) and calculated using a multinomial background distribution given by

Table 2 Atomic four-body statistical potential.
p i j k l = 4 ! n = 1 6 t n ! n = 1 6 a n t n ,where  n = 1 6 a n = 1  and  n = 1 6 t n = 4 .

In Eq. (4), a n is the relative frequency of atom type n, while t n counts how many times atom type n appears in the quadruplet (i,j,k,l). As a consequence of the inverted Boltzmann principle [23], the score s ijkl = log (f ijkl / p ijkl ) is proportional to the energy of quadruplet atomic interaction, and the set of 126 scored atomic quadruplets defines our four-body statistical potential (Table 2).

Topological scores

In order to develop our predictive model, the four-body potential is applied to the dataset of three hundred protein-ligand complexes compiled from the Binding MOAD in the following manner. For each complex, the atomic coordinates (excluding hydrogens and water molecules) in the PDB file are tessellated (edges longer than 8Å removed), each tetrahedron in the tessellation is scored using Table 2 according to the four atoms at its vertices, and a normalized topological score (Q) is calculated to be the sum of all the tetrahedral scores divided by the number of tetrahedra in the tessellation, a quantity that can be summarized compactly by the equation

Q = 1 N ( i , j , k , l ) s i j k l

Next, atomic coordinates of the ligand are removed from the PDB file of the complex, and the procedure is repeated to compute Q for the isolated protein (Figure 1). Lastly, we define the topological score difference

Δ Q = Q complex - Q protein

for the complex. In the next section, we compare computed ΔQ quantities with known ΔG values for these complexes in order to develop a model for predicting binding energy. An important underlying assumption in this formulation is that ligand size is small enough so that tetrahedra formed at the interface with the protein dominate purely internal atomic interactions within the ligand. The calculated Q values, for structures of the three hundred protein-ligand complexes, as well as the isolated proteins without their bound ligands, are tabulated in Additional file 2.

Predictive model of binding energy

A comparison of the calculated ΔQ quantities for our training set of two hundred randomly selected complexes with their experimental ΔG values (ΔGexp) reveals a correlation coefficient of r = 0.79. However, the ΔQ values are not uniform in sign, and they are on a significantly smaller scale relative to the standard Gibbs free energy of binding (ΔGexp) data; hence, they cannot be used directly as a representation of predicted ΔG values (ΔGcalc). Both issues related to ΔQ values for the training data are addressed with an empirically derived linear function

Δ G calc = 1 / 0 . 0003 × Δ Q - 10 . 49 ,

resulting in negative ΔGcalc values in each case that also scale similarly to ΔGexp. Owing to ΔGcalc arising from a simple linear transformation of the ΔQ values, ΔGcalc and ΔGexp also display a correlation of r = 0.79 (SE = 1.98 kcal/mol) with a fitted regression line of y = 1.18x (Figure 2).

Figure 2
figure 2

Scatter plot of calculated versus experimental binding energies for the dataset of three hundred protein-ligand complexes.

Turning next to the validation set of one hundred complexes, we obtain ΔGcalc values from their computed ΔQ quantities by utilizing the linear model given in Eq. (7) that we empirically derived from the training data. The predicted ΔGcalc and known ΔGexp values for these complexes again display a correlation of r = 0.79 (SE = 1.93 kcal/mol) with a fitted regression line of y = 1.11x - 0.63, and a scatter plot of the validation data is superimposed over that of the training data in Figure 2. Tabulated in Additional file 2 are ΔGexp and ΔGcalc values for all three hundred protein-ligand complexes.


Enzyme-inhibitor binding affinity prediction

In order to test the utility of our model through a practical application, we predict binding affinities for a diverse dataset of three hundred enzyme-inhibitor complexes (Additional file 3), independent of those protein-ligand complexes used for training and validation, which are annotated with their respective experimental inhibition constants (k i ) in the non-redundant Binding MOAD. Analogous to Eq. (1), we obtain the standard Gibbs free energy of binding for each complex with the equation

Δ G exp = R T  ln ( k i ) = 0 . 592 × ln ( k i ) ,

where R = 1.986 ×10-3 kcal K-1 mol-1 is the gas constant and T = 298° K is the absolute temperature. Tabulated in Additional file 3 are the PDB accession codes of the high-resolution (≤ 2.5Å) crystallographic structures, as well as the k i and ΔGexp values, corresponding to these enzyme-inhibitor complexes.

Next, we use the atomic coordinates (hydrogen atoms and water molecules excluded) provided by the PDB structure file for each complex to generate a Delaunay tessellation (subject to an 8Å edge-length cutoff), from which we obtain a normalized topological score (Qcomplex) by employing Eq. (5) in conjunction with our atomic four-body statistical potential (Table 2). In a similar fashion, we generate a normalized topological score for the isolated protein without the bound inhibitor (Qprotein), by tessellating a modified version of the PDB file that excludes the atomic coordinates for the inhibitor. Lastly, we calculate the difference (ΔQ) between these normalized topological scores according to Eq. (6), which is subsequently used by our model in Eq. (7) to yield a prediction for the enzyme-inhibitor binding affinity (ΔGcalc). All normalized topological score and calculated binding affinity data are also tabulated in Additional file 3.

For this dataset of three hundred enzyme-inhibitor complexes, the calculated ΔQ values and the experimental binding affinity ΔGexp data display a correlation of r = 0.79; likewise, as discussed previously, the correlation between ΔGcalc and ΔGexp is similarly given by r = 0.79, in this case with a calculated standard error for the predictions of SE = 2.39 kcal/mol (Figure 3).

Figure 3
figure 3

Scatter plot of calculated versus experimental binding energies for the dataset of three hundred enzyme-inhibitor complexes.

Comparisons to related methods

In the same way that our predictive model of protein-ligand binding affinity is evaluated on a test set of three hundred enzyme-inhibitor complexes as described in the previous section, other related methods similarly use test sets of complexes to validate their models. Hence, to directly compare our performance to that of other methods, binding affinity predictions are generated using our approach for complexes that form their test sets. Starting with X-Score, Wang et al. [2] report predictions with their model on a test set of ten complexes that reflect a correlation of r = 0.67 between experimental and predicted binding affinity (right hand columns of Table 3 in [2], predicted data are in parentheses), with a fitted regression line of y = 0.31x + 3.78. On the identical dataset, predictions obtained with our model yield a correlation of r = 0.72 and fitted regression line of y = 1.26x - 1.20, results that signify a clear improvement over those of X-Score (Table 3 of this manuscript, which also reproduces the X-Score data).

Table 3 Comparing experimental binding affinity values for 10 protein-ligand complexes with predicted values obtained using both X-Score and the model developed in this study.

Turning next to ITScore, we discover that Huang et al. [8] utilize a benchmarking test set consisting of one hundred protein-ligand complexes, originally constructed by Wang et al. [24], to compare their scoring function and 14 other methods by ranking the respective Pearson's correlation coefficients (r) between experimental and predicted binding affinities. The test set is diverse, consisting of 43 different proteins as well as binding affinities that span nearly nine orders of magnitude. By generating binding affinity predictions for these one hundred complexes with our model and calculating their correlation with the experimental data, we can subsequently determine our ranking among these 15 related approaches: ITScore [8], X-Score [2], DFIRE [25], DrugScoreCSD [26], DrugScorePDB [4], Cerius2/PLP [27, 28], SYBYL/G-Score [29], SYBYL/D-Score [30], SYBYL/ChemScore [31], Cerius2/PMF [32], DOCK/FF [30], Cerius2/LUDI [33, 34], Cerius2/Lig-Score [35], SYBYL/F-Score [36], and AutoDock [37]. The results of our predictions are summarized in Figure 4, which provides a scatter plot of calculated versus experimental binding energies for this dataset of one hundred complexes. The plot reflects a correlation of r = 0.66 (r = 0.67 with one outlier complex excluded), surpassing all of the other methods (Table 4, data for the other methods are obtained from Table 3 in [8]) and validating the reliability of our approach.

Figure 4
figure 4

Scatter plot of calculated versus experimental binding energies using our model for the benchmarking dataset of one hundred enzyme-inhibitor complexes.

Table 4 Benchmarking correlations between calculated and experimental binding energies using our model and 15 related methods on 100 protein-ligand complexes.


Delaunay tessellation of atomic coordinates in a diverse dataset of macromolecular structures objectively identifies four-body atomic interactions, providing the raw data for developing a knowledge-based atomic four-body statistical contact potential. This potential is used to score a separate diverse set of three hundred protein-ligand complexes with known binding affinities, as well as to score the isolated proteins without their bound ligands, based on their respective structure tessellations. Initially, the difference (ΔQ) between scores calculated for an entire complex and for its isolated protein is considered as a predictor of binding affinity; however, since these ΔQ do not scale as binding free energy values, two hundred randomly selected protein-ligand complexes from this set are used to empirically derive a linear function of ΔQ as a model for calculating the binding energy. For this training set, we observe a correlation of r = 0.79 between calculated and experimental binding energies, with a standard error of SE = 1.98 kcal/mol and a regression line of y = 1.18x. Validation of this model with the remaining one hundred complexes that were held out yields performance measures of r = 0.79 and SE = 1.93 kcal/mol. In an application of the method, our model is then used to predict binding energies for an independent and diverse test set of three hundred enzyme-inhibitor complexes, producing results that are consistent with those based on the training and validation data. Finally, we utilize a diverse test set of one hundred protein-ligand complexes to benchmark the binding energy predictions made with our model, and our correlation between calculated and experimental binding energies for this dataset surpasses those of all 15 related methods to which it is compared. A key advantage with our approach is the ability to generate rapid predictions, typically under one second per complex.


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Thanks to researchers affiliated with the Binding MOAD repository for creating and making publicly available their centralized database of macromolecular complexes.


Publication of this article was funded in part by the George Mason University Libraries Open Access Publishing Fund.

This article has been published as part of BMC Structural Biology Volume 13 Supplement 1, 2013: Selected articles from the Computational Structural Bioinformatics Workshop 2012. The full contents of the supplement are available online at

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Correspondence to Majid Masso.

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MM conceived of the study, implemented the methods, analyzed the data, and wrote the manuscript.

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Additional file 1: PDB accession codes for the 1417 macromolecular structures used to derive the atomic four-body statistical potential. (TXT 8 KB)


Additional file 2: Three hundred protein-ligand complexes used to train and validate the model. (TXT 28 KB)


Additional file 3: Independent set of three hundred enzyme-inhibitor complexes used to test the model. (TXT 24 KB)

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Masso, M. Four-body atomic potential for modeling protein-ligand binding affinity: application to enzyme-inhibitor binding energy prediction. BMC Struct Biol 13 (Suppl 1), S1 (2013).

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