The MM/PBSA and MM/GBSA methods to estimate ligand-binding affinities

Introduction: The molecular mechanics energies combined with the Poisson-Boltzmann or generalized Born and surface area continuum solvation (MM/PBSA and MM/GBSA) methods are popular approaches to estimate the free energy of the binding of small ligands to biological macromolecules. They are typically based on molecular dynamics simulations of the receptor-ligand complex and are therefore intermediate in both accuracy and computational effort between empirical scoring and strict alchemical perturbation methods. They have been applied to a large number of systems with varying success. Areas covered: The authors review the use of MM/PBSA and MM/GBSA methods to calculate ligand-binding affinities, with an emphasis on calibration, testing and validation, as well as attempts to improve the methods, rather than on specific applications. Expert opinion: MM/PBSA and MM/GBSA are attractive approaches owing to their modular nature and that they do not require calculations on a training set. They have been used successfully to reproduce and rationalize experimental findings and to improve the results of virtual screening and docking. However, they contain several crude and questionable approximations, for example, the lack of conformational entropy and information about the number and free energy of water molecules in the binding site. Moreover, there are many variants of the method and their performance varies strongly with the tested system. Likewise, most attempts to ameliorate the methods with more accurate approaches, for example, quantum-mechanical calculations, polarizable force fields or improved solvation have deteriorated the results

Comparison of the Efficiency of the LIE and MM/GBSA Methods to Calculate Ligand-Binding Energies

We have evaluated the efficiency of two popular end-point methods to calculate ligand-binding free energies, LIE (linear interaction energy) and MM/GBSA (molecular mechanics with generalized Born surface-area solvation), i.e. the computational effort needed to obtain estimates of a similar precision. As a test case, we use the binding of seven biotin analogues to avidin. The energy terms used by MM/GBSA and LIE exhibit a similar correlation time (similar to 5 ps), and the equilibration time seems also to be similar, although it varies much between the various ligands. The results show that the LIE method is more effective than MM/GBSA, by a factor of 2-7 for a truncated spherical system with a radius of 26 angstrom and by a factor of 1.0-2.4 for the full avidin tetramer (radius 47 angstrom). The reason for this is the cost for the MM/GBSA entropy calculations, which more than compensates for the extra simulation of the free ligand in LIE. On the other hand, LIE requires that the protein is neutralized, whereas MM/GBSA has no such requirements. Our results indicate that both the truncation and neutralization of the proteins may slow the convergence and emphasize small differences in the calculations, e.g., differences between the four subunits in avidin. Moreover, LIE cannot take advantage of the fact that avidin is a tetramer. For this test case, LIE gives poor results with the standard parametrization, but after optimizing the scaling factor of the van der Waals terms, reasonable binding affinities can be obtained, although MM/GBSA still gives a significantly better predictive index and correlation to the experimental affinities

Will molecular dynamics simulations of proteins ever reach equilibrium?

We show that conformational entropies calculated for five proteins and protein-ligand complexes with dihedral-distribution histogramming, the von Mises approach, or quasi-harmonic analysis do not converge to any useful precision even if molecular dynamics (MD) simulations of 380-500 ns length are employed (the uncertainty is 12-89 kJ mol(-1)). To explain this, we suggest a simple protein model involving dihedrals with effective barriers forming a uniform distribution and show that for such a model, the entropy increases logarithmically with time until all significantly populated dihedral states have been sampled, in agreement with the simulations (during the simulations, 52-70% of the available dihedral phase space has been visited). This is also confirmed by the analysis of the trajectories of a 1 ms simulation of bovine pancreatic trypsin inhibitor (31 kJ mol(-1) difference in the entropy between the first and second part of the simulation). Strictly speaking, this means that it is practically impossible to equilibrate MD simulations of proteins. We discuss the implications of such a lack of strict equilibration of protein MD simulations and show that ligand-binding free energies estimated with the MM/GBSA method (molecular mechanics with generalised Born and surface-area solvation) vary by 3-15 kJ mol(-1) during a 500 ns simulation (the higher estimate is caused by rare conformational changes), although they involve a questionable but well-converged normal-mode entropy estimate, whereas free energies estimated by free-energy perturbation vary by less than 0.6 kJ mol(-1) for the same simulation

All-atom/coarse-grained hybrid predictions of distribution coefficients in SAMPL5

We present blind predictions submitted to the SAMPL5 challenge on calculating distribution coefficients. The predictions were based on estimating the solvation free energies in water and cyclohexane of the 53 compounds in the challenge. These free energies were computed using alchemical free energy simulations based on a hybrid all-atom/coarse-grained model. The compounds were treated with the general Amber force field, whereas the solvent molecules were treated with the Elba coarse-grained model. Considering the simplicity of the solvent model and that we approximate the distribution coefficient with the partition coefficient of the neutral species, the predictions are of good accuracy. The correlation coefficient, R is 0.64, 82% of the predictions have the correct sign and the mean absolute deviation is 1.8 log units. This is on a par with or better than the other simulation-based predictions in the challenge. We present an analysis of the deviations to experiments and compare the predictions to another submission that used all-atom solvent

Comparison of end-point continuum-solvation methods for the calculation of protein-ligand binding free energies.

We have compared the predictions of ligand-binding affinities from several methods based on end-point molecular dynamics simulations and continuum solvation, i.e. methods related to MM/PBSA (molecular mechanics combined with Poisson-Boltzmann and surface area solvation). Two continuum-solvation models were considered, viz. the Poisson-Boltzmann (PB) and generalised Born (GB) approaches. The non-electrostatic energies were also obtained in two different ways, viz. either from the sum of the bonded, van der Waals, non-polar solvation energies, and entropy terms (as in MM/PBSA), or from the scaled protein-ligand van der Waals interaction energy (as in the linear interaction energy approach, LIE). Three different approaches to calculate electrostatic energies were tested, viz. the sum of electrostatic interaction energies and polar solvation energies, obtained either from a single simulation of the complex or from three independent simulations of the complex, the free protein, and the free ligand, or the linear-response approximation (LRA). Moreover, we investigated the effect of scaling the electrostatic interactions by an effective internal dielectric constant of the protein (ε(int) ). All these methods were tested on the binding of seven biotin analogues to avidin and nine 3-amidinobenzyl-1H-indole-2-carboxamide inhibitors to factor Xa. For avidin, the best results were obtained with a combination of the LIE non-electrostatic energies with the MM+GB electrostatic energies from a single simulation, using ε(int) = 4. For fXa, standard MM/GBSA, based on one simulation and using ε(int) = 4-10 gave the best result. The optimum internal dielectric constant seems to be slightly higher with PB than with GB solvation. Proteins 2012. © 2012 Wiley-Liss, Inc

Binding Affinities of Factor Xa Inhibitors Estimated by Thermodynamic Integration and MM/GBSA.

We present free energy estimates of nine 3-amidinobenzyl-1H-indole-2-carboxamide inhibitors of factor Xa. Using alchemical thermodynamic integration (TI) calculations, we estimate the difference in binding free energies with high accuracy and precision, except for mutations involving one of the amidinobenzyl rings. Crystal studies show that the inhibitors may bind in two distinct conformations, and using TI, we show that the two conformations give a similar binding affinity. Furthermore, we show that we can reduce the computational demand, while still retaining a high accuracy and precision, by using fewer integration points and shorter protein-ligand simulations. Finally, we have compared the TI results to those obtained with the simpler MM/GBSA method (molecular-mechanics with generalized Born surface-area solvation). MM/GBSA gives better results for the mutations that involve a change of net charge, but if a precision similar to that of the TI method is required, the MM/GBSA method is actually slightly more expensive. Thus, we have shown that TI could be a valuable tool in drug design

Binding affinities by alchemical perturbation using QM/MM with a large QM system and polarizable MM model.

The most general way to improve the accuracy of binding-affinity calculations for protein-ligand systems is to use quantum-mechanical (QM) methods together with rigorous alchemical-perturbation (AP) methods. We explore this approach by calculating the relative binding free energy of two synthetic disaccharides binding to galectin-3 at a reasonably high QM level (dispersion-corrected density functional theory with a triple-zeta basis set) and with a sufficiently large QM system to include all short-range interactions with the ligand (744-748 atoms). The rest of the protein is treated as a collection of atomic multipoles (up to quadrupoles) and polarizabilities. Several methods for evaluating the binding free energy from the 3600 QM calculations are investigated in terms of stability and accuracy. In particular, methods using QM calculations only at the endpoints of the transformation are compared with the recently proposed non-Boltzmann Bennett acceptance ratio (NBB) method that uses QM calculations at several stages of the transformation. Unfortunately, none of the rigorous approaches give sufficient statistical precision. However, a novel approximate method, involving the direct use of QM energies in the Bennett acceptance ratio method, gives similar results as NBB but with better precision, ∼3 kJ/mol. The statistical error can be further reduced by performing a greater number of QM calculations. © 2015 Wiley Periodicals, Inc

A Large-Scale Test of Free-Energy Simulation Estimates of Protein-Ligand Binding Affinities.

We have performed a large-scale test of alchemical perturbation calculations with the Bennett acceptance-ratio (BAR) approach to estimate relative affinities for the binding of 107 ligands to 10 different proteins. Employing 20-Å truncated spherical systems and only one intermediate state in the perturbations, we obtain an error of less than 4 kJ/mol for 54% of the studied relative affinities and a precision of 0.5 kJ/mol on average. However, only four of the proteins gave acceptable errors, correlations, and rankings. The results could be improved by using nine intermediate states in the simulations or including the entire protein in the simulations using periodic boundary conditions. However, 27 of the calculated affinities still gave errors of more than 4 kJ/mol, and for three of the proteins the results were not satisfactory. This shows that the performance of BAR calculations depends on the target protein and that several transformations gave poor results owing to limitations in the molecular-mechanics force field or the restricted sampling possible within a reasonable simulation time. Still, the BAR results are better than docking calculations for most of the proteins