Continuum solvation models in the linear interaction energy method

The linear interaction energy (LIE) method in combination with two different continuum solvent models has been applied to calculate protein-ligand binding free energies for a set of inhibitors against the malarial aspartic protease plasmepsin II. Ligand-water interaction energies are calculated from both Poisson-Boltzmann (PB) and Generalized Born (GB) continuum models using snapshots from explicit solvent simulations of the ligand and protein-ligand complex. These are compared to explicit solvent calculations, and we find close agreement between the explicit water and PB solvation models. The GB model overestimates the change in solvation energy, and this is caused by consistent underestimation of the effective Born radii in the protein-ligand complex. The explicit solvent LIE calculations and LIE-PB, with our standard parametrization, reproduce absolute experimental binding free energies with an average unsigned error of 0.5 and 0.7 kcal/mol, respectively. The LIE-GB method, however, requires a constant offset to approach the same level of accuracy.     

Potentials of mean force between ionizable amino acid side chains in water

Potentials of mean force (PMF) between all possible ionizable amino acid side chain pairs in various protonation states were calculated using explicit solvent molecular dynamics simulations with umbrella sampling and the weighted histogram analysis method. The side chains were constrained in various orientations inside a spherical cluster of 200 water molecules. Beglov and Roux's Spherical Solvent Boundary Potential was used to account for the solvent outside this sphere. This approach was first validated by calculating PMFs between monatomic ions (K(+), Na(+), Cl(-)) and comparing them to results from the literature and results obtained using Ewald summation. The strongest interaction (-4.5 kcal/mol) was found for the coaxial Arg(+).Glu(-) pair. Many like-charged side chains display a remarkable lack of repulsion, and occasionally a weak attraction. The PMFs are compared to effective energy curves obtained with common implicit solvation models, namely Generalized Born (GB), EEF1, and uniform dielectric of 80. Overall, the EEF1 curves are too attractive, whereas the GB curves in most cases match the minima of the PMF curves quite well. The uniform dielectric model, despite some fortuitous successes, is grossly inadequate.     

Surveying implicit solvent models for estimating small molecule absolute hydration free energies

Implicit solvent models are powerful tools in accounting for the aqueous environment at a fraction of the computational expense of explicit solvent representations. Here, we compare the ability of common implicit solvent models (TC, OBC, OBC2, GBMV, GBMV2, GBSW, GBSW/MS, GBSW/MS2 and FACTS) to reproduce experimental absolute hydration free energies for a series of 499 small neutral molecules that are modeled using AMBER/GAFF parameters and AM1-BCC charges. Given optimized surface tension coefficients for scaling the surface area term in the nonpolar contribution, most implicit solvent models demonstrate reasonable agreement with extensive explicit solvent simulations (average difference 1.0-1.7 kcal/mol and R(2)=0.81-0.91) and with experimental hydration free energies (average unsigned errors=1.1-1.4 kcal/mol and R(2)=0.66-0.81). Chemical classes of compounds are identified that need further optimization of their ligand force field parameters and others that require improvement in the physical parameters of the implicit solvent models themselves. More sophisticated nonpolar models are also likely necessary to more effectively represent the underlying physics of solvation and take the quality of hydration free energies estimated from implicit solvent models to the next level.     

Accurate prediction of protonation state as a prerequisite for reliable MM-PB(GB)SA binding free energy calculations of HIV-1 protease inhibitors

Binding free energies were calculated for the inhibitors lopinavir, ritonavir, saquinavir, indinavir, amprenavir, and nelfinavir bound to HIV-1 protease. An MMPB/SA-type analysis was applied to conformational samples from 3 ns explicit solvent molecular dynamics simulations of the enzyme-inhibitor complexes. Binding affinities and the sampled conformations of the inhibitor and enzyme were compared between different HIV-1 protease protonation states to find the most likely protonation state of the enzyme in the complex with each of the inhibitors. The resulting set of protonation states leads to good agreement between calculated and experimental binding affinities. Results from the MMPB/SA analysis are compared with an explicit/implicit hybrid scheme and with MMGB/SA methods. It is found that the inclusion of explicit water molecules may offer a slight advantage in reproducing absolute binding free energies while the use of the Generalized Born approximation significantly affects the accuracy of the calculated binding affinities.     

Converging free energy estimates: MM-PB(GB)SA studies on the protein-protein complex Ras-Raf

Estimating protein-protein interaction energies is a very challenging task for current simulation protocols. Here, absolute binding free energies are reported for the complex H-Ras/C-Raf1 using the MM-PB(GB)SA approach, testing the internal consistency and model dependence of the results. Averaging gas-phase energies (MM), solvation free energies as determined by Generalized Born models (GB/SA), and entropic contributions calculated by normal mode analysis for snapshots obtained from 10 ns explicit-solvent molecular dynamics in general results in an overestimation of the binding affinity when a solvent-accessible surface area-dependent model is used to estimate the nonpolar solvation contribution. Applying the sum of a cavity solvation free energy and explicitly modeled solute-solvent van der Waals interaction energies instead provides less negative estimates for the nonpolar solvation contribution. When the polar contribution to the solvation free energy is determined by solving the Poisson-Boltzmann equation (PB) instead, the calculated binding affinity strongly depends on the atomic radii set chosen. For three GB models investigated, different absolute deviations from PB energies were found for the unbound proteins and the complex. As an alternative to normal-mode calculations, quasiharmonic analyses have been performed to estimate entropic contributions due to changes of solute flexibility upon binding. However, such entropy estimates do not converge after 10 ns of simulation time, indicating that sampling issues may limit the applicability of this approach. Finally, binding free energies estimated from snapshots of the unbound proteins extracted from the complex trajectory result in an underestimate of binding affinity. This points to the need to exercise caution in applying the computationally cheaper "one-trajectory-alternative" to systems where there may be significant changes in flexibility and structure due to binding. The best estimate for the binding free energy of Ras-Raf obtained in this study of -8.3 kcal mol(-1) is in good agreement with the experimental result of -9.6 kcal mol(-1), however, further probing the transferability of the applied protocol that led to this result is necessary.     

Influence of the solvent representation on vibrational entropy calculations: Generalized born versus distance‐dependent dielectric model

The harmonic model is the most popular approximation for estimating the “configurational” entropy of a solute in molecular mechanics/Poisson‐Boltzmann solvent accessible surface area (MM/PBSA)‐type binding free energy calculations. Here, we investigate the influence of the solvent representation in the harmonic model by comparing estimates of changes in the vibrational entropies for 30 trypsin/ligand complexes on ligand binding. Second derivatives of Amber generalized Born (GB) solvation models are available in the nucleic acid builder code. They allow one to use these models for the calculation of vibrational entropies instead of using a simpler solvation model based on a distance‐dependent dielectric (DDD) constant. Estimates of changes in the vibrational entropies obtained with a DDD model are systematically and significantly larger, by on average, 6 kcal mol^?1 (at T = 300 K), than estimates obtained with a GB model and so are more favorable for complex formation. The difference becomes larger the more the vibrational entropy contribution disfavors complex formation, that is, the larger the ligand is (for the complexes considered here). A structural decomposition of the estimates into per‐residue contributions reveals polar interactions between the ligand and the surrounding protein, in particular involving charged nitrogens, as a main source of the differences. Snapshots minimized with the DDD model showed a structural deviation from snapshots minimized in explicit water that is larger by, on average, 0.5 Å RMSD compared to snapshots that were minimized with GB^HCT. As experimental vibrational entropies of biomacromolecules are elusive, there is no direct way to establish a solvent model's superiority. Thus, we can only recommend using the GB harmonic model for vibrational entropy calculations based on the reasoning that smaller structural deviations should point to the implicit solvent model that closer approximates the energy landscape of the solute in explicit solvent. © 2012 Wiley Periodicals, Inc.     

Solvent models for protein–ligand binding: Comparison of implicit solvent poisson and surface generalized born models with explicit solvent simulations

Solvent effects play a crucial role in mediating the interactions between proteins and their ligands. Implicit solvent models offer some advantages for modeling these interactions, but they have not been parameterized on such complex problems, and therefore, it is not clear how reliable they are. We have studied the binding of an octapeptide ligand to the murine MHC class I protein using both explicit solvent and implicit solvent models. The solvation free energy calculations are more than 10³ faster using the Surface Generalized Born implicit solvent model compared to FEP simulations with explicit solvent. For some of the electrostatic calculations needed to estimate the binding free energy, there is near quantitative agreement between the explicit and implicit solvent model results; overall, the qualitative trends in the binding predicted by the explicit solvent FEP simulations are reproduced by the implicit solvent model. With an appropriate choice of reference system based on the binding of the discharged ligand, electrostatic interactions are found to enhance the binding affinity because the favorable Coulomb interaction energy between the ligand and protein more than compensates for the unfavorable free energy cost of partially desolvating the ligand upon binding. Some of the effects of protein flexibility and thermal motions on charging the peptide in the solvated complex are also considered. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 591–607, 2001     

An efficient hybrid explicit/implicit solvent method for biomolecular simulations

We present a new hybrid explicit/implicit solvent method for dynamics simulations of macromolecular systems. The method models explicitly the hydration of the solute by either a layer or sphere of water molecules, and the generalized Born (GB) theory is used to treat the bulk continuum solvent outside the explicit simulation volume. To reduce the computational cost, we implemented a multigrid method for evaluating the pairwise electrostatic and GB terms. It is shown that for typical ion and protein simulations our method achieves similar equilibrium and dynamical observables as the conventional particle mesh Ewald (PME) method. Simulation timings are reported, which indicate that the hybrid method is much faster than PME, primarily due to a significant reduction in the number of explicit water molecules required to model hydration effects.     

Detailed considerations for a balanced and broadly applicable force field: a study of substituted benzenes modeled with OPLS-AA

Modern classical force fields have been traditionally parameterized by attempting to maximize agreement to any number of experimental and/or quantum mechanical target properties. As these force fields are pushed towards obtaining quantitative estimates of often subtle energetic differences, stringent and consistent parameterization criteria, particularly in regard to charge distributions, are required to ensure that systematic errors cancel, that parameters are transferable between molecules, and that performance does not significantly deteriorate when using more approximate methods, such as with continuum solvent models. Relative free energies of hydration are presented here for 40 mono- and disubstituted benzenes modeled with the OPLS-AA force field; heats of vaporization and pure liquid densities at standard conditions are presented when experimental data is available. Overall agreement between OPLS-AA and experiment is remarkable (average error = 0.5 kcal/mol for DeltaDeltaG(hydration), 1.0 kcal/mol for DeltaH(vap) (0), 0.02 g/mL for densities), yet several functional groups are identified as having consistent and correctable errors (alkyl-, nitro-, and thiobenzenes). Relative free energies of hydration obtained with rigorous free energy perturbations using explicit solvent are also compared with energies from minimizations using a generalized Born model (GB). There is high correlation between these estimates (R = 0.99), and as demonstrated here, reparameterization of the aforementioned groups can be guided with rapid GB calculations.     

Secondary structure bias in generalized Born solvent models: comparison of conformational ensembles and free energy of solvent polarization from explicit and implicit solvation

The effects of the use of three generalized Born (GB) implicit solvent models on the thermodynamics of a simple polyalanine peptide are studied via comparing several hundred nanoseconds of well-converged replica exchange molecular dynamics (REMD) simulations using explicit TIP3P solvent to REMD simulations with the GB solvent models. It is found that when compared to REMD simulations using TIP3P the GB REMD simulations contain significant differences in secondary structure populations, most notably an overabundance of alpha-helical secondary structure. This discrepancy is explored via comparison of the differences in the electrostatic component of the free energy of solvation (DeltaDeltaG(pol)) between TIP3P (via thermodynamic Integration calculations), the GB models, and an implicit solvent model based on the Poisson equation (PE). The electrostatic components of the solvation free energies are calculated using each solvent model for four representative conformations of Ala10. Since the PE model is found to have the best performance with respect to reproducing TIP3P DeltaDeltaG(pol) values, effective Born radii from the GB models are compared to effective Born radii calculated with PE (so-called perfect radii), and significant and numerous deviations in GB radii from perfect radii are found in all GB models. The effect of these deviations on the solvation free energy is discussed, and it is shown that even when perfect radii are used the agreement of GB with TIP3P DeltaDeltaG(pol) values does not improve. This suggests a limit to the optimization of the effective Born radius calculation and that future efforts to improve the accuracy of GB models must extend beyond such optimizations.     

Evaluation of Poisson solvation models using a hybrid explicit/implicit solvent method

Implicit solvent methods have become popular tools in the field of protein dynamics simulations, yet evaluation of their validity has been primarily limited to comparisons with experimental and theoretical data for small molecules. In this paper, we use a recently developed hybrid explicit/implicit solvent methodology to evaluate the accuracy of several Poisson-based implicit solvent models. Specifically, we focus on the calculation of electrostatic solvation free energies of various fixed conformations for two proteins. We show that, among various dielectric boundary definitions, the Lee-Richards molecular surface has the best agreement with hybrid solvent results. Furthermore, certain modifications of the molecular surface Poisson protocol provide varied results. For instance, simple modifications of atomic radii on charged residues generally improve absolute errors but do not significantly reduce relative errors among conformations. On the other hand, using a water-probe radius of 1.0 A, as opposed to the standard value of 1.4 A, to generate the molecular surface, moderately improves both absolute and relative results.     

Proton binding to proteins: pK(a) calculations with explicit and implicit solvent models

Ionizable residues play important roles in protein structure and activity, and proton binding is a valuable reporter of electrostatic interactions in these systems. We use molecular dynamics free energy simulations (MDFE) to compute proton pKa shifts, relative to a model compound in solution, for three aspartate side chains in two proteins. Simulations with explicit solvent and with an implicit, dielectric continuum solvent are reported. The implicit solvent simulations use the generalized Born (GB) model, which provides an approximate, analytical solution to Poisson's equation. With explicit solvent, the direction of the pKa shifts is correct in all three cases with one force field (AMBER) and in two out of three cases with another (CHARMM). For two aspartates, the dielectric response to ionization is found to be linear, even though the separate protein and solvent responses can be nonlinear. For thioredoxin Asp26, nonlinearity arises from the presence of two substates that correspond to the two possible orientations of the protonated carboxylate. For this side chain, which is partly buried and has a large pKa upshift, very long simulations are needed to correctly sample several slow degrees of freedom that reorganize in response to the ionization. Thus, nearby Lys57 rotates to form a salt bridge and becomes buried, while three waters intercalate along the opposite edge of Asp26. Such strong and anisotropic reorganization is very difficult to predict with Poisson-Boltzmann methods that only consider electrostatic interactions and employ a single protein structure. In contrast, MDFE with a GB dielectric continuum solvent, used for the first time for pKa calculations, can describe protein reorganization accurately and gives encouraging agreement with experiment and with the explicit solvent simulations.     

Optimized parameters for continuum solvation calculations with carbohydrates

Continuum solvent models have been shown to be an efficient method for the calculation of the energetics of biomolecules in solution. However, for these methods to produce accurate results, an appropriate set of atomic radii or volumes is needed. While these have been developed for proteins and nucleic acids, the same is not true of carbohydrates. Here, a set of optimized parameters for continuum solvation calculations of carbohydrates in conjunction with the Carbohydrate Solution Force Field are presented. Explicit solvent free-energy perturbation simulations were performed on a set of hexapyranose sugars and used to fit atomic radii for Poisson-Boltzmann and generalized-Born calculations, and to fit atomic volumes for use with the analytical continuum electrostatics model. The solvation energetics computed with the optimized radii and a Poisson-Boltzmann model show remarkable agreement with explicit solvent simulation, with a root-mean-square error of 1.19 kcal/mol over a large test set of sugars in many conformations. The generalized-Born model gives slightly poorer agreement, but still correlates very strongly, with an error of 1.69 kcal/mol. The analytical continuum electrostatics model correlates well with the explicit solvent results, but gives a larger error of 4.71 kcal/mol. The remarkable agreement between the solvation free energies computed in explicit and implicit solvent provides strong motivation for the use of implicit solvent models in the simulation of carbohydrate-containing systems.     

Predicting hydration free energies with a hybrid QM/MM approach: an evaluation of implicit and explicit solvation models in SAMPL4

The correct representation of solute-water interactions is essential for the accurate simulation of most biological phenomena. Several highly accurate quantum methods are available to deal with solvation by using both implicit and explicit solvents. So far, however, most evaluations of those methods were based on a single conformation, which neglects solute entropy. Here, we present the first test of a novel approach to determine hydration free energies that uses molecular mechanics (MM) to sample phase space and quantum mechanics (QM) to evaluate the potential energies. Free energies are determined by using re-weighting with the Non-Boltzmann Bennett (NBB) method. In this context, the method is referred to as QM-NBB. Based on snapshots from MM sampling and accounting for their correct Boltzmann weight, it is possible to obtain hydration free energies that incorporate the effect of solute entropy. We evaluate the performance of several QM implicit solvent models, as well as explicit solvent QM/MM for the blind subset of the SAMPL4 hydration free energy challenge. While classical free energy simulations with molecular dynamics give root mean square deviations (RMSD) of 2.8 and 2.3 kcal/mol, the hybrid approach yields an improved RMSD of 1.6 kcal/mol. By selecting an appropriate functional and basis set, the RMSD can be reduced to 1 kcal/mol for calculations based on a single conformation. Results for a selected set of challenging molecules imply that this RMSD can be further reduced by using NBB to reweight MM trajectories with the SMD implicit solvent model.     

Application of the linear interaction energy method for rational design of artemisinin analogues as haeme polymerisation inhibitors

The anti-malarial activity of artemisinin-derived drugs appears to be mediated by an interaction of the drug's endoperoxide bridge with intra-parasitic haeme. The binding affinity of artemisinin analogues with haeme were computed using linear interaction energy with a surface generalised Born (LIE-SGB) continuum solvation model. Low levels of root mean square error (0.348 and 0.415 kcal/mol) as well as significant correlation coefficients (r ² = 0.868 and 0.892) between the experimental and predicted free energy of binding (FEB) based on molecular dynamics and hybrid Monte Carlo sampling techniques establish the SGB-LIE method as an efficient tool for generating more potent inhibitors of haeme polymerisation inhibition.     

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.     

Combining the polarizable Drude force field with a continuum electrostatic Poisson–Boltzmann implicit solvation model

In this work, we have combined the polarizable force field based on the classical Drude oscillator with a continuum Poisson–Boltzmann/solvent‐accessible surface area (PB/SASA) model. In practice, the positions of the Drude particles experiencing the solvent reaction field arising from the fixed charges and induced polarization of the solute must be optimized in a self‐consistent manner. Here, we parameterized the model to reproduce experimental solvation free energies of a set of small molecules. The model reproduces well‐experimental solvation free energies of 70 molecules, yielding a root mean square difference of 0.8 kcal/mol versus 2.5 kcal/mol for the CHARMM36 additive force field. The polarization work associated with the solute transfer from the gas‐phase to the polar solvent, a term neglected in the framework of additive force fields, was found to make a large contribution to the total solvation free energy, comparable to the polar solute–solvent solvation contribution. The Drude PB/SASA also reproduces well the electronic polarization from the explicit solvent simulations of a small protein, BPTI. Model validation was based on comparisons with the experimental relative binding free energies of 371 single alanine mutations. With the Drude PB/SASA model the root mean square deviation between the predicted and experimental relative binding free energies is 3.35 kcal/mol, lower than 5.11 kcal/mol computed with the CHARMM36 additive force field. Overall, the results indicate that the main limitation of the Drude PB/SASA model is the inability of the SASA term to accurately capture non‐polar solvation effects. © 2018 Wiley Periodicals, Inc.     

Distance and exposure dependent effective dielectric function

In an effort to develop a dielectric screening function for molecular dynamics simulations of biomolecules in implicit solvent, effective dielectric constants (D(eff)) for a large number of atom pairs in a typical globular protein are calculated by continuum electrostatics. Plots of D(eff) versus the intercharge distance are in general sigmoidal with the characteristics of the curve depending on the distance of the two charges from the dielectric boundary and, secondarily, on the extent to which the area surrounding each charge is occupied by solvent (the "exposure"). The D(eff) values were fitted to an empirical, analytical function of these parameters that reproduces the data reasonably well, although considerable scatter exists in the range of D(eff) from 30 to 80. In the system used for parameterization, the mean square deviation of electrostatic interaction energies with this function is 0.48 kcal/mol, compared to 1.45 for an analytical Generalized Born model and 1.52 for the linear distance-dependent dielectric model. When tested in other proteins of varying size and compactness, the present function is superior to both of the above models, except for a fully unfolded polypeptide chain, where the Generalized Born model is superior.     

Parameterization of the Hamiltonian Dielectric Solvent (HADES) Reaction‐Field Method for the Solvation Free Energies of Amino Acid Side‐Chain Analogs

Optimization of the Hamiltonian dielectric solvent (HADES) method for biomolecular simulations in a dielectric continuum is presented with the goal of calculating accurate absolute solvation free energies while retaining the model’s accuracy in predicting conformational free‐energy differences. The solvation free energies of neutral and polar amino acid side‐chain analogs calculated by using HADES, which may optionally include nonpolar contributions, were optimized against experimental data to reach a chemical accuracy of about 0.5 kcal mol^?1. The new parameters were evaluated for charged side‐chain analogs. The HADES results were compared with explicit‐solvent, generalized Born, Poisson–Boltzmann, and QM‐based methods. The potentials of mean force (PMFs) between pairs of side‐chain analogs obtained by using HADES and explicit‐solvent simulations were used to evaluate the effects of the improved parameters optimized for solvation free energies on intermolecular potentials.     

Competition between intramolecular hydrogen bonds and solvation in phosphorylated peptides: simulations with explicit and implicit solvent

The atomic-level mechanisms of protein regulation by post-translational phosphorylation remain poorly understood, except in a few well-studied systems. Molecular mechanics simulations can in principle be used to help understand and predict the effects of protein phosphorylation, but the accuracy of the results will of course depend on the quality of the force field parameters for the phosphorylated residues as well as the quality of the solvent model. The phosphorylated residues typically carry a -2 charge at physiological pH; however, the effects of phosphorylation can sometimes be mimicked by substituting Asp or Glu for the phosphorylated residue. Here we examine the suitability of explicit and implicit solvent models for simulating phospho-serine in both the -1 and -2 charge states. Specifically, we simulate a capped phosphorylated peptide, Ace-Gly-Ser-pSer-Ser-Nme, and compare the results to each other and to experimental observables from an NMR experiment. The first major conclusion is that explicit water models (TIP3P, TIP4P and SPC/E) and a Generalized Born implicit solvent model provide reasonable agreement with the experimental observables, given appropriate partial charges for the phosphate group. The Generalized Born results, however, show greater hydrogen bonding propensity than the explicit solvent results. Distance dependent dielectric treatments perform poorly. The second major conclusion is that many ensemble-averaged properties obtained for the phosphopeptide in the -1 and -2 charge states are strikingly similar; the -1 species has a slightly higher propensity to form internal hydrogen bonds. All of the results can be rationalized by quantifying the strength of the P-O/H-N hydrogen bond, which depends on a sensitive balance between strongly favorable charge/dipole and dipole/dipole interactions and strongly unfavorable desolvation.