Four-body atomic potential for modeling protein-ligand binding affinity: application to enzyme-inhibitor binding energy prediction
- Majid Masso^{1}Email author
https://doi.org/10.1186/1472-6807-13-S1-S1
© Masso; licensee BioMed Central Ltd. 2013
Published: 8 November 2013
Abstract
Background
Models that are capable of reliably predicting binding affinities for protein-ligand complexes play an important role the field of structure-guided drug design.
Methods
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.
Results
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.
Conclusions
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.
Keywords
Background
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.
Methods
Datasets
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 7.0.1.24704 (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.
and the equation of the fitted regression line.
Results
Four-body statistical potential
Summary data for the 1417 PDB structure files.
Atom Types | Count | Proportion |
---|---|---|
C | 3,612,988 | 0.633193 |
N | 969,253 | 0.169866 |
O | 1,088,410 | 0.190749 |
S | 28,502 | 0.004995 |
(all metals) M | 2,529 | 0.000443 |
(all other non-metals) X | 4,299 | 0.000754 |
Total atom count: | 5,705,981 | |
Total tetrahedron count: | 34,504,737 |
Atomic four-body statistical potential.
Quad | Count | f _{ ijkl } | p _{ ijkl } | s _{ ijkl } | Quad | Count | f _{ ijkl } | p _{ ijkl } | s _{ ijkl } |
---|---|---|---|---|---|---|---|---|---|
CCCC | 4015872 | 0.116386 | 0.160748 | -0.140244 | MMNS | 363 | 1.05E-05 | 2.00E-09 | 3.720958 |
CCCM | 1592 | 4.61E-05 | 0.000450 | -0.989223 | MMNX | 0 | 0 | 3.02E-10 | -- |
CCCN | 4025206 | 0.116657 | 0.172495 | -0.169866 | MMOO | 306 | 8.87E-06 | 4.29E-08 | 2.315530 |
CCCO | 6202159 | 0.179748 | 0.193701 | -0.032467 | MMOS | 104 | 3.01E-06 | 2.25E-09 | 3.127729 |
CCCS | 293157 | 0.008496 | 0.005072 | 0.224008 | MMOX | 3 | 8.69E-08 | 3.39E-10 | 2.409325 |
CCCX | 2796 | 8.10E-05 | 0.000765 | -0.975047 | MMSS | 254 | 7.36E-06 | 2.94E-11 | 5.398477 |
CCMM | 132 | 3.83E-06 | 4.73E-07 | 0.908235 | MMSX | 2 | 5.80E-08 | 8.87E-12 | 3.815151 |
CCMN | 3318 | 9.62E-05 | 0.000362 | -0.575981 | MMXX | 0 | 0 | 6.69E-13 | -- |
CCMO | 5325 | 0.000154 | 0.000407 | -0.420893 | MNNN | 1030 | 2.99E-05 | 8.69E-06 | 0.535960 |
CCMS | 2293 | 6.65E-05 | 1.07E-05 | 0.795108 | MNNO | 1128 | 3.27E-05 | 2.93E-05 | 0.047955 |
CCMX | 15 | 4.35E-07 | 1.61E-06 | -0.567697 | MNNS | 561 | 1.63E-05 | 7.67E-07 | 1.326526 |
CCNN | 1797552 | 0.052096 | 0.069412 | -0.124635 | MNNX | 5 | 1.45E-07 | 1.16E-07 | 0.098041 |
CCNO | 8233136 | 0.238609 | 0.155892 | 0.184864 | MNOO | 3744 | 0.000109 | 3.29E-05 | 0.518626 |
CCNS | 124653 | 0.003613 | 0.004082 | -0.053081 | MNOS | 314 | 9.10E-06 | 1.72E-06 | 0.723107 |
CCNX | 2007 | 5.82E-05 | 0.000616 | -1.024729 | MNOX | 29 | 8.40E-07 | 2.60E-07 | 0.510083 |
CCOO | 3366568 | 0.097568 | 0.087528 | 0.047161 | MNSS | 793 | 2.30E-05 | 2.25E-08 | 3.008398 |
CCOS | 198630 | 0.005757 | 0.004584 | 0.098905 | MNSX | 5 | 1.45E-07 | 6.80E-09 | 1.328573 |
CCOX | 4626 | 0.000134 | 0.000691 | -0.712426 | MNXX | 9 | 2.61E-07 | 5.13E-10 | 2.706383 |
CCSS | 15288 | 0.000443 | 6.00E-05 | 0.868158 | MOOO | 5430 | 0.000157 | 1.23E-05 | 1.106856 |
CCSX | 144 | 4.17E-06 | 1.81E-05 | -0.637352 | MOOS | 156 | 4.52E-06 | 9.67E-07 | 0.669977 |
CCXX | 143 | 4.14E-06 | 1.37E-06 | 0.482159 | MOOX | 168 | 4.87E-06 | 1.46E-07 | 1.523669 |
CMMM | 23 | 6.67E-07 | 2.21E-10 | 3.480397 | MOSS | 210 | 6.09E-06 | 2.53E-08 | 2.380989 |
CMMN | 144 | 4.17E-06 | 2.54E-07 | 1.216422 | MOSX | 4 | 1.16E-07 | 7.64E-09 | 1.181307 |
CMMO | 256 | 7.42E-06 | 2.85E-07 | 1.415945 | MOXX | 55 | 1.59E-06 | 5.76E-10 | 3.442148 |
CMMS | 662 | 1.92E-05 | 7.46E-09 | 3.410480 | MSSS | 62 | 1.80E-06 | 2.21E-10 | 3.910199 |
CMMX | 1 | 2.90E-08 | 1.12E-09 | 1.411130 | MSSX | 2 | 5.80E-08 | 1.00E-10 | 2.763224 |
CMNN | 2474 | 7.17E-05 | 9.72E-05 | -0.132029 | MSXX | 0 | 0 | 1.51E-11 | -- |
CMNO | 6267 | 0.000182 | 0.000218 | -0.079754 | MXXX | 16 | 4.64E-07 | 7.58E-13 | 5.786451 |
CMNS | 2588 | 7.50E-05 | 5.72E-06 | 1.118068 | NNNN | 3878 | 0.000112 | 0.000833 | -0.869698 |
CMNX | 26 | 7.54E-07 | 8.62E-07 | -0.058415 | NNNO | 46665 | 0.001352 | 0.003740 | -0.441730 |
CMOO | 8481 | 0.000246 | 0.000123 | 0.302308 | NNNS | 460 | 1.33E-05 | 9.79E-05 | -0.866046 |
CMOS | 1010 | 2.93E-05 | 6.42E-06 | 0.659069 | NNNX | 34 | 9.85E-07 | 1.48E-05 | -1.175817 |
CMOX | 68 | 1.97E-06 | 9.68E-07 | 0.308765 | NNOO | 340620 | 0.009872 | 0.006299 | 0.195102 |
CMSS | 2047 | 5.93E-05 | 8.40E-08 | 2.848813 | NNOS | 5637 | 0.000163 | 0.000330 | -0.305233 |
CMSX | 13 | 3.77E-07 | 2.53E-08 | 1.172117 | NNOX | 302 | 8.75E-06 | 4.98E-05 | -0.754766 |
CMXX | 6 | 1.74E-07 | 1.91E-09 | 1.958862 | NNSS | 311 | 9.01E-06 | 4.32E-06 | 0.319427 |
CNNN | 102035 | 0.002957 | 0.012414 | -0.623046 | NNSX | 6 | 1.74E-07 | 1.30E-06 | -0.874705 |
CNNO | 1995038 | 0.057819 | 0.041821 | 0.140679 | NNXX | 5 | 1.45E-07 | 9.83E-08 | 0.168652 |
CNNS | 15892 | 0.000461 | 0.001095 | -0.376176 | NOOO | 171147 | 0.004960 | 0.004716 | 0.021937 |
CNNX | 578 | 1.68E-05 | 0.000165 | -0.993919 | NOOS | 10697 | 0.000310 | 0.000370 | -0.077374 |
CNOO | 2734639 | 0.079254 | 0.046962 | 0.227273 | NOOX | 3102 | 8.99E-05 | 5.59E-05 | 0.206513 |
CNOS | 95438 | 0.002766 | 0.002460 | 0.050981 | NOSS | 922 | 2.67E-05 | 9.70E-06 | 0.440012 |
CNOX | 2168 | 6.28E-05 | 0.000371 | -0.771173 | NOSX | 12 | 3.48E-07 | 2.93E-06 | -0.925060 |
CNSS | 4264 | 0.000124 | 3.22E-05 | 0.584024 | NOXX | 61 | 1.77E-06 | 2.21E-07 | 0.903627 |
CNSX | 37 | 1.07E-06 | 9.71E-06 | -0.957113 | NSSS | 33 | 9.56E-07 | 8.47E-08 | 1.052833 |
CNXX | 61 | 1.77E-06 | 7.33E-07 | 0.382553 | NSSX | 0 | 0 | 3.83E-08 | -- |
COOO | 524994 | 0.015215 | 0.017579 | -0.062707 | NSXX | 0 | 0 | 5.78E-09 | -- |
COOS | 34429 | 0.000998 | 0.001381 | -0.141141 | NXXX | 3 | 8.69E-08 | 2.91E-10 | 2.475964 |
COOX | 23801 | 0.000690 | 0.000208 | 0.520038 | OOOO | 34212 | 0.000992 | 0.001324 | -0.125549 |
COSS | 4380 | 0.000127 | 3.62E-05 | 0.545326 | OOOS | 4240 | 0.000123 | 0.000139 | -0.052504 |
COSX | 58 | 1.68E-06 | 1.09E-05 | -0.812243 | OOOX | 9553 | 0.000277 | 2.09E-05 | 1.121777 |
COXX | 65 | 1.88E-06 | 8.23E-07 | 0.359781 | OOSS | 300 | 8.69E-06 | 5.45E-06 | 0.203077 |
CSSS | 285 | 8.26E-06 | 3.16E-07 | 1.417735 | OOSX | 36 | 1.04E-06 | 1.64E-06 | -0.197264 |
CSSX | 5 | 1.45E-07 | 1.43E-07 | 0.006247 | OOXX | 128 | 3.71E-06 | 1.24E-07 | 1.476181 |
CSXX | 4 | 1.16E-07 | 2.15E-08 | 0.730845 | OSSS | 38 | 1.10E-06 | 9.51E-08 | 1.063748 |
CXXX | 9 | 2.61E-07 | 1.08E-09 | 2.381656 | OSSX | 3 | 8.69E-08 | 4.30E-08 | 0.305472 |
MMMM | 83 | 2.41E-06 | 3.86E-14 | 7.794725 | OSXX | 0 | 0 | 6.49E-09 | -- |
MMMN | 37 | 1.07E-06 | 5.92E-11 | 4.258301 | OXXX | 2 | 5.80E-08 | 3.26E-10 | 2.249518 |
MMMO | 29 | 8.40E-07 | 6.64E-11 | 4.102142 | SSSS | 6 | 1.74E-07 | 6.23E-10 | 2.446092 |
MMMS | 379 | 1.10E-05 | 1.74E-12 | 6.800300 | SSSX | 0 | 0 | 3.76E-10 | -- |
MMMX | 0 | 0 | 2.62E-13 | -- | SSXX | 0 | 0 | 8.50E-11 | -- |
MMNN | 83 | 2.41E-06 | 3.40E-08 | 1.849597 | SXXX | 0 | 0 | 8.55E-12 | -- |
MMNO | 102 | 2.96E-06 | 7.64E-08 | 1.587734 | XXXX | 0 | 0 | 3.22E-13 | -- |
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
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
Turning next to the validation set of one hundred complexes, we obtain ΔG_{calc} 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 ΔG_{calc} and known ΔG_{exp} 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 ΔG_{exp} and ΔG_{calc} values for all three hundred protein-ligand complexes.
Discussion
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
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 ΔG_{exp} 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 (Q_{complex}) 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 (Q_{protein}), 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 (ΔG_{calc}). All normalized topological score and calculated binding affinity data are also tabulated in Additional file 3.
Comparisons to related methods
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.
pk _{ d } | ΔG | |||
---|---|---|---|---|
PDB code | Exp. | X-Score | Exp. | Our model |
1ABE | 6.52 | 5.25 | -8.887610045 | -10.3129 |
1ADB | 8.40 | 8.01 | -11.45029515 | -23.3388 |
1ADD | 6.74 | 5.36 | -9.187498728 | -13.1243 |
1AF2 | 3.10 | 4.90 | -4.225704163 | -10.0826 |
1ANF | 5.46 | 6.03 | -7.442691848 | -9.59131 |
1CBX | 6.35 | 5.74 | -8.655877882 | -11.6284 |
1DBM | 9.44 | 6.65 | -12.86795074 | -14.9936 |
1DHF | 7.40 | 5.27 | -10.08716478 | -13.1753 |
1GST | 4.68 | 5.21 | -6.379450155 | -5.48917 |
1HPV | 9.22 | 6.28 | -12.56806206 | -16.0903 |
X-Score: r = 0.67 | Our model: r = 0.72 |
Benchmarking correlations between calculated and experimental binding energies using our model and 15 related methods on 100 protein-ligand complexes.
Method | Type | Correlation coefficient (r) |
---|---|---|
Our model | Knowledge-based | 0.66 |
ITScore | Iterative score | 0.65 |
X-Score | Empirical | 0.64 |
DFIRE | Knowledge-based | 0.63 |
DrugScore^{CSD} | Knowledge-based | 0.62 |
DrugScore^{PDB} | Knowledge-based | 0.60 |
Cerius2/PLP | Empirical | 0.56 |
SYBYL/G-Score | Force-field-based | 0.56 |
SYBYL/D-Score | Force-field-based | 0.48 |
SYBYL/ChemScore | Empirical | 0.47 |
Cerius2/PMF | Knowledge-based | 0.40 |
DOCK/FF | Force field | 0.40 |
Cerius2/LUDI | Empirical | 0.36 |
Cerius2/Lig-Score | Force-field-based | 0.35 |
SYBYL/F-Score | Empirical | 0.30 |
AutoDock | Force-field-based | 0.05 |
Conclusions
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.
Declarations
Acknowledgements
Thanks to researchers affiliated with the Binding MOAD repository for creating and making publicly available their centralized database of macromolecular complexes.
Declarations
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 http://www.biomedcentral.com/bmcstructbiol/supplements/13/S1.
Authors’ Affiliations
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