A simple yet accurate boundary element method for continuum dielectric calculations

A simple yet accurate method for calculating electrostatic potentials using the boundary element continuum dielectric method is presented. It is shown that the limiting factor in accuracy is not the evaluation of integrals involving the interaction between boundary elements but rather a proper estimation of the self-polarization of a patch upon itself. We derive a sum rule that allows us to calculate this important self-polarization term in a self-consistent and simple way. Intricate integration schemes used in previous treatments are consequently rendered unnecessary while concurrently achieving at least comparable accuracy over earlier methods. In some model systems for which analytic solutions are available, the computed surface polarization charge and reaction field energy are correct to better than six significant figures. An application of the method to the calculation of hydration free energies is presented. Good agreement with experimental values is obtained.     

Coupled atomic charge selectivity for optimal ligand-charge distributions at protein binding sites

Charge optimization as a tool for both analyzing and enhancing binding electrostatics has become an attractive approach over the past few years. An interesting feature of this method for molecular design is that it provides not only the optimal charge magnitudes, but also the selectivity of a particular atomic center for its optimal charge. The current approach to compute the charge selectivity at a given atomic center of a ligand in a particular binding process is based on the binding-energy cost incurred upon the perturbation of the optimal charge distribution by a unit charge at the given atomic center, while keeping the other atomic partial charges at their optimal values. A limitation of this method is that it does not take into account the possible concerted changes in the other atomic charges that may incur a lower energetic cost than perturbing a single charge. Here, we describe a novel approach for characterizing charge selectivity in a concerted manner, taking into account the coupling between the ligand charge centers in the binding process. We apply this novel charge selectivity measure to the celecoxib molecule, a nonsteroidal anti-inflammatory agent binding to cyclooxygenase 2 (COX2), which has been recently shown to also exhibit cross-reactivity toward carbonic anhydrase II (CAII), to which it binds with nanomolar affinity. The uncoupled and coupled charge selectivity profiles over the atomic centers of the celecoxib ligand, binding independently to COX2 and CAII, are analyzed comparatively and rationalized with respect to available experimental data. Very different charge selectivity profiles are obtained for the uncoupled versus coupled selectivity calculations.     

Molecular surface generation using marching tetrahedra

A method is presented to generate and triangulate molecular surfaces rapidly. It is based on the ‘marching tetrahedra’ approach. The method is fast, simple and easy to implement. Our approach is not analytical in nature. Hence no special treatment is required for complications with singularity, degeneracy, or with self-intersecting re-entrant surfaces. A quick test for determining the solvent accessibility of a point in space forms an important part of the method. This test has potential use outside of the surface generation algorithm such as in molecular field analysis where the solvent accessibility of a point needs to be determined. The triangulated surface generated is suitable for molecular graphics display as well as boundary element continuum dielectric calculations. © 1998, Government of Canada. Exclusive worldwide publication rights in the article have been transferred to John Wiley & Sons, Inc. in perpetuity. J Comput Chem 19: 1268–1277, 1998     

Solvated interaction energy (SIE) for scoring protein-ligand binding affinities. 1. Exploring the parameter space

We present a binding free energy function that consists of force field terms supplemented by solvation terms. We used this function to calibrate the solvation model along with the binding interaction terms in a self-consistent manner. The motivation for this approach was that the solute dielectric-constant dependence of calculated hydration gas-to-water transfer free energies is markedly different from that of binding free energies (J. Comput. Chem. 2003, 24, 954). Hence, we sought to calibrate directly the solvation terms in the context of a binding calculation. The five parameters of the model were systematically scanned to best reproduce the absolute binding free energies for a set of 99 protein-ligand complexes. We obtained a mean unsigned error of 1.29 kcal/mol for the predicted absolute binding affinity in a parameter space that was fairly shallow near the optimum. The lowest errors were obtained with solute dielectric values of Din = 20 or higher and scaling of the intermolecular van der Waals interaction energy by factors ranging from 0.03 to 0.15. The high apparent Din and strong van der Waals scaling may reflect the anticorrelation of the change in solvated potential energy and configurational entropy, that is, enthalpy-entropy compensation in ligand binding (Biophys. J. 2004, 87, 3035-3049). Five variations of preparing the protein-ligand data set were explored in order to examine the effect of energy refinement and the presence of bound water on the calculated results. We find that retaining water in the final protein structure used for calculating the binding free energy is not necessary to obtain good results; that is the continuum solvation model is sufficient. Virtual screening enrichment studies on estrogen receptor and thymidine kinase showed a good ability of the binding free energy function to recover true hits in a collection of decoys.     

Exhaustive search and solvated interaction energy (SIE) for virtual screening and affinity prediction

We carried out a prospective evaluation of the utility of the SIE (solvation interaction energy) scoring function for virtual screening and binding affinity prediction. Since experimental structures of the complexes were not provided, this was an exercise in virtual docking as well. We used our exhaustive docking program, Wilma, to provide high-quality poses that were rescored using SIE to provide binding affinity predictions. We also tested the combination of SIE with our latest solvation model, first shell of hydration (FiSH), which captures some of the discrete properties of water within a continuum model. We achieved good enrichment in virtual screening of fragments against trypsin, with an area under the curve of about 0.7 for the receiver operating characteristic curve. Moreover, the early enrichment performance was quite good with 50% of true actives recovered with a 15% false positive rate in a prospective calculation and with a 3% false positive rate in a retrospective application of SIE with FiSH. Binding affinity predictions for both trypsin and host-guest complexes were generally within 2 kcal/mol of the experimental values. However, the rank ordering of affinities differing by 2 kcal/mol or less was not well predicted. On the other hand, it was encouraging that the incorporation of a more sophisticated solvation model into SIE resulted in better discrimination of true binders from binders. This suggests that the inclusion of proper Physics in our models is a fruitful strategy for improving the reliability of our binding affinity predictions.     

Rapid prediction of solvation free energy. 3. Application to the SAMPL2 challenge

The SAMPL2 hydration free energy blind prediction challenge consisted of a data set of 41 molecules divided into three subsets: explanatory, obscure and investigatory, where experimental hydration free energies were given for the explanatory, withheld for the obscure, and not known for the investigatory molecules. We employed two solvation models for this challenge, a linear interaction energy (LIE) model based on explicit-water molecular dynamics simulations, and the first-shell hydration (FiSH) continuum model previously calibrated to mimic LIE data. On the 23 compounds from the obscure (blind) dataset, the prospectively submitted LIE and FiSH models provided predictions highly correlated with experimental hydration free energy data, with mean-unsigned-errors of 1.69 and 1.71 kcal/mol, respectively. We investigated several parameters that may affect the performance of these models, namely, the solute flexibility for the LIE explicit-solvent model, the solute partial charging method, and the incorporation of the difference in intramolecular energy between gas and solution phases for both models. We extended this analysis to the various chemical classes that can be formed within the SAMPL2 dataset. Our results strengthen previous findings on the excellent accuracy and transferability of the LIE explicit-solvent approach to predict transfer free energies across a wide spectrum of functional classes. Further, the current results on the SAMPL2 test dataset provide additional support for the FiSH continuum model as a fast yet accurate alternative to the LIE explicit-solvent model. Overall, both the LIE explicit-solvent model and the FiSH continuum solvation model show considerable improvement on the SAMPL2 data set over our previous continuum electrostatics-dispersion solvation model used in the SAMPL1 blind challenge.     

A fast pairwise evaluation of molecular surface area

A fast and general analytical approach was developed for the calculation of the approximate van der Waals and solvent-accessible surface areas. The method is based on three basic ideas: the use of the Lorentz transformation formula, a rigid-geometry approximation, and a single fitting parameter that can be refitted on the fly during a simulation. The Lorentz transformation equation is used for the summation of the areas of an atom buried by its neighboring contacting atoms, and implies that a sum of the buried pairwise areas cannot be larger than the surface area of the isolated spherical atom itself. In a rigid-geometry approximation we numerically calculate and keep constant the surface of each atom buried by the atoms involved in 1-2 and 1-3 interactions. Only the contributions from the nonbonded atoms (1-4 and higher interactions) are considered in terms of the pairwise approximation. The accuracy and speed of the method is competitive with other pairwise algorithms. A major strength of the method is the ease of parametrization.     

Solvated interaction energy (SIE) for scoring protein-ligand binding affinities. 2. Benchmark in the CSAR-2010 scoring exercise

Solvated interaction energy (SIE) is an end-point physics-based scoring function for predicting binding affinities from force-field nonbonded interaction terms, continuum solvation, and configurational entropy linear compensation. We tested the SIE function in the Community Structure-Activity Resource (CSAR) scoring challenge consisting of high-resolution cocrystal structures for 343 protein-ligand complexes with high-quality binding affinity data and high diversity with respect to protein targets. Particular emphasis was placed on the sensitivity of SIE predictions to the assignment of protonation and tautomeric states in the complex and the treatment of metal ions near the protein-ligand interface. These were manually curated from an originally distributed CSAR-HiQ data set version, leading to the currently distributed CSAR-NRC-HiQ version. We found that this manual curation was a critical step for accurately testing the performance of the SIE function. The standard SIE parametrization, previously calibrated on an independent data set, predicted absolute binding affinities with a mean-unsigned-error (MUE) of 2.41 kcal/mol for the CSAR-HiQ version, which improved to 1.98 kcal/mol for the upgraded CSAR-NRC-HiQ version. Half-half retraining-testing of SIE parameters on two predefined subsets of CSAR-NRC-HiQ led to only marginal further improvements to an MUE of 1.83 kcal/mol. Hence, we do not recommend altering the current default parameters of SIE at this time. For a sample of SIE outliers, additional calculations by molecular dynamics-based SIE averaging with or without incorporation of ligand strain, by MM-PB(GB)/SA methods with or without entropic estimates, or even by the linear interaction energy (LIE) formalism with an explicit solvent model, did not further improve predictions.     

Exhaustive docking and solvated interaction energy scoring: lessons learned from the SAMPL4 challenge

We continued prospective assessments of the Wilma–solvated interaction energy (SIE) platform for pose prediction, binding affinity prediction, and virtual screening on the challenging SAMPL4 data sets including the HIV-integrase inhibitor and two host–guest systems. New features of the docking algorithm and scoring function are tested here prospectively for the first time. Wilma–SIE provides good correlations with actual binding affinities over a wide range of binding affinities that includes strong binders as in the case of SAMPL4 host–guest systems. Absolute binding affinities are also reproduced with appropriate training of the scoring function on available data sets or from comparative estimation of the change in target’s vibrational entropy. Even when binding modes are known, SIE predictions lack correlation with experimental affinities within dynamic ranges below 2 kcal/mol as in the case of HIV-integrase ligands, but they correctly signaled the narrowness of the dynamic range. Using a common protein structure for all ligands can reduce the noise, while incorporating a more sophisticated solvation treatment improves absolute predictions. The HIV-integrase virtual screening data set consists of promiscuous weak binders with relatively high flexibility and thus it falls outside of the applicability domain of the Wilma–SIE docking platform. Despite these difficulties, unbiased docking around three known binding sites of the enzyme resulted in over a third of ligands being docked within 2 Å from their actual poses and over half of the ligands docked in the correct site, leading to better-than-random virtual screening results.