Accuracy of continuum electrostatic calculations based on three common dielectric boundary definitions

Inclusion of Water Multipoles into the Implicit Solvation Framework Leads to Accuracy Gains

The current practical "workhorses" of the atomistic implicit solvation─the Poisson-Boltzmann (PB) and generalized Born (GB) models─face fundamental accuracy limitations. Here, we propose a computationally efficient implicit solvation framework, the Implicit Water Multipole GB (IWM-GB) model, that systematically incorporates the effects of multipole moments of water molecules in the first hydration shell of a solute, beyond the dipole water polarization already present at the PB/GB level. The framework explicitly accounts for coupling between polar and nonpolar contributions to the total solvation energy, which is missing from many implicit solvation models. An implementation of the framework, utilizing the GAFF force field and AM1-BCC atomic partial charges model, is parametrized and tested against the experimental hydration free energies of small molecules from the FreeSolv database. The resulting accuracy on the test set (RMSE ∼ 0.9 kcal/mol) is 12% better than that of the explicit solvation (TIP3P) treatment, which is orders of magnitude slower. We also find that the coupling between polar and nonpolar parts of the solvation free energy is essential to ensuring that several features of the IWM-GB model are physically meaningful, including the sign of the nonpolar contributions.

A Closed-Form, Analytical Approximation for Apparent Surface Charge and Electric Field of Molecules

Closed-form, analytical approximations for electrostatic properties of molecules are of unique value as these can provide computational speed, versatility, and physical insight. Here, we have derived a simple, closed-form formula for the apparent surface charge (ASC) as well as for the electric field generated by a molecular charge distribution in aqueous solution. The approximation, with no fitted parameters, was tested against numerical solutions of the Poisson equation, where it has produced a significant speed-up. For neutral small molecules, the hydration free energies estimated from the closed-form ASC formula are within 0.8 kcal/mol RMSD from the numerical Poisson reference; the electric field at the surface is in quantitative agreement with the reference. Performance of the approximation was also tested on larger structures, including a protein, a DNA fragment, and a viral receptor-target complex. For all structures tested, a near-quantitative agreement with the numerical Poisson reference was achieved, except in regions of high negative curvature, where the new approximation is still qualitatively correct. A unique efficiency feature of the proposed "source-based″ closed-form approximation is that the ASC and electric field can be estimated individually at any point or surface patch, without the need to obtain the full global solution. An open-source software implementation of the method is available: https://people.cs.vt.edu/~onufriev/CODES/aasc.zip.

Optimized Halogen Atomic Radii for PBSA Calculations Using Off-Center Point Charges

In force-field methods, the usage of off-center point charges, also called extra points (EPs), is a common strategy to tackle the anisotropy of the electrostatic potential of covalently bonded halogens (X), thus allowing the description of halogen bonds (XBs) at the molecular mechanics/molecular dynamics (MM/MD) level. Diverse EP implementations exist in the literature differing on the charge sets and/or the X-EP distances. Poisson-Boltzmann and surface area (PBSA) calculations can be used to obtain solvation free energies (ΔG_solv) of small molecules, often to compute binding free energies (ΔG_bind) at the MM-PBSA level. This method depends, among other parameters, on the empirical assignment of atomic radii (PB radii). Given the multiplicity of off-center point-charge models and the lack of specific PB radii for halogens compatible with such implementations, in this work, we assessed the performance of PBSA calculations for the estimation of ΔG_solv values in water (ΔG_hyd), also conducting an optimization of the halogen PB radii (Cl, Br, and I) for each EP model. We not only expand the usage of EP models in the scope of the general AMBER force field (GAFF) but also provide the first optimized halogen PB radii in the context of the CHARMM general force field (CGenFF), thus contributing to improving the description of halogenated compounds in PBSA calculations.

An Effective MM/GBSA Protocol for Absolute Binding Free Energy Calculations: A Case Study on SARS-CoV-2 Spike Protein and the Human ACE2 Receptor

The binding free energy calculation of protein-ligand complexes is necessary for research into virus-host interactions and the relevant applications in drug discovery. However, many current computational methods of such calculations are either inefficient or inaccurate in practice. Utilizing implicit solvent models in the molecular mechanics generalized Born surface area (MM/GBSA) framework allows for efficient calculations without significant loss of accuracy. Here, GBNSR6, a new flavor of the generalized Born model, is employed in the MM/GBSA framework for measuring the binding affinity between SARS-CoV-2 spike protein and the human ACE2 receptor. A computational protocol is developed based on the widely studied Ras-Raf complex, which has similar binding free energy to SARS-CoV-2/ACE2. Two options for representing the dielectric boundary of the complexes are evaluated: one based on the standard Bondi radii and the other based on a newly developed set of atomic radii (OPT1), optimized specifically for protein-ligand binding. Predictions based on the two radii sets provide upper and lower bounds on the experimental references: -14.7(ΔGbindBondi)<-10.6(ΔGbindExp.)<-4.1(ΔGbindOPT1) kcal/mol. The consensus estimates of the two bounds show quantitative agreement with the experiment values. This work also presents a novel truncation method and computational strategies for efficient entropy calculations with normal mode analysis. Interestingly, it is observed that a significant decrease in the number of snapshots does not affect the accuracy of entropy calculation, while it does lower computation time appreciably. The proposed MM/GBSA protocol can be used to study the binding mechanism of new variants of SARS-CoV-2, as well as other relevant structures.

Multidimensional Global Optimization and Robustness Analysis in the Context of Protein-Ligand Binding

Accuracy of protein-ligand binding free energy calculations utilizing implicit solvent models is critically affected by parameters of the underlying dielectric boundary, specifically, the atomic and water probe radii. Here, a global multidimensional optimization pipeline is developed to find optimal atomic radii specifically for protein-ligand binding calculations in implicit solvent. The computational pipeline has these three key components: (1) a massively parallel implementation of a deterministic global optimization algorithm (VTDIRECT95), (2) an accurate yet reasonably fast generalized Born implicit solvent model (GBNSR6), and (3) a novel robustness metric that helps distinguish between nearly degenerate local minima via a postprocessing step of the optimization. A graph-based "kT-connectivity" approach to explore and visualize the multidimensional energy landscape is proposed: local minima that can be reached from the global minimum without exceeding a given energy threshold (kT) are considered to be connected. As an illustration of the capabilities of the optimization pipeline, we apply it to find a global optimum in the space of just five radii: four atomic (O, H, N, and C) radii and water probe radius. The optimized radii, ρW = 1.37 Å, ρC = 1.40 Å, ρH = 1.55 Å, ρN = 2.35 Å, and ρO = 1.28 Å, lead to a closer agreement of electrostatic binding free energies with the explicit solvent reference than two commonly used sets of radii previously optimized for small molecules. At the same time, the ability of the optimizer to find the global optimum reveals fundamental limits of the common two-dielectric implicit solvation model: the computed electrostatic binding free energies are still almost 4 kcal/mol away from the explicit solvent reference. The proposed computational approach opens the possibility to further improve the accuracy of practical computational protocols for binding free energy calculations.

Tackling Halogenated Species with PBSA: Effect of Emulating the σ-Hole

To model halogen-bond phenomena using classical force fields, an extra point (EP) of charge is frequently introduced at a given distance from the halogen (X) to emulate the σ-hole. The resulting molecular dynamics (MD) trajectories can be used in subsequent molecular mechanics (MM) combined with Poisson-Boltzmann and surface area calculations (PBSA) to estimate protein-ligand binding free energies (Δ G_bind). While EP addition improves the MM/MD description of halogen-containing systems, its effect on the calculation of solvation free energies (Δ G_solv) using the PBSA approach is yet to be assessed. As the PBSA calculations depend, among other parameters, on the empirical assignment of radii (PB radii), a problematic issue arises, since standard halogen radii are smaller than the typical X···EP distances, thus placing the EP within the solvent dielectric. Herein, we took a common literature EP parametrization scheme, which uses X···EP = R_min and RESP charges in the context of GAFF, and performed a comprehensive study on the performance of PBSA (using three different setups) in the calculation of Δ G_solv values for 142 halogenated compounds (bearing Cl, Br, or I) for which the experimental values are known. By conducting an optimization (minimizing the error against experimental values), we provide a new optimized set of halogen PB radii, for each PBSA setup, that should be used in the context of the aforementioned scenario. A simultaneous optimization of PB radii and X···EP distances shows that a wide range of distance/radius pairs can be used without significant loss of accuracy, therefore laying the basis for expanding this halogen radii optimization strategy to other force fields and EP implementations. As ligand Δ G_solv estimation is an important term in the determination of protein-ligand Δ G_bind, this work is particularly relevant in the framework of structure-based virtual screening and related computer-aided drug design routines.

Generalized Born Implicit Solvent Models for Biomolecules

It would often be useful in computer simulations to use an implicit description of solvation effects, instead of explicitly representing the individual solvent molecules. Continuum dielectric models often work well in describing the thermodynamic aspects of aqueous solvation and can be very efficient compared to the explicit treatment of the solvent. Here, we review a particular class of so-called fast implicit solvent models, generalized Born (GB) models, which are widely used for molecular dynamics (MD) simulations of proteins and nucleic acids. These approaches model hydration effects and provide solvent-dependent forces with efficiencies comparable to molecular-mechanics calculations on the solute alone; as such, they can be incorporated into MD or other conformational searching strategies in a straightforward manner. The foundations of the GB model are reviewed, followed by examples of newer, emerging models and examples of important applications. We discuss their strengths and weaknesses, both for fidelity to the underlying continuum model and for the ability to replace explicit consideration of solvent molecules in macromolecular simulations.

Explicit ions/implicit water generalized Born model for nucleic acids

The ion atmosphere around highly charged nucleic acid molecules plays a significant role in their dynamics, structure, and interactions. Here we utilized the implicit solvent framework to develop a model for the explicit treatment of ions interacting with nucleic acid molecules. The proposed explicit ions/implicit water model is based on a significantly modified generalized Born (GB) model and utilizes a non-standard approach to define the solute/solvent dielectric boundary. Specifically, the model includes modifications to the GB interaction terms for the case of multiple interacting solutes-disconnected dielectric boundary around the solute-ion or ion-ion pairs. A fully analytical description of all energy components for charge-charge interactions is provided. The effectiveness of the approach is demonstrated by calculating the potential of mean force for Na⁺-Cl^- ion pair and by carrying out a set of Monte Carlo (MC) simulations of mono- and trivalent ions interacting with DNA and RNA duplexes. The monovalent (Na⁺) and trivalent (CoHex³⁺) counterion distributions predicted by the model are in close quantitative agreement with all-atom explicit water molecular dynamics simulations used as reference. Expressed in the units of energy, the maximum deviations of local ion concentrations from the reference are within k _B T. The proposed explicit ions/implicit water GB model is able to resolve subtle features and differences of CoHex distributions around DNA and RNA duplexes. These features include preferential CoHex binding inside the major groove of the RNA duplex, in contrast to CoHex biding at the "external" surface of the sugar-phosphate backbone of the DNA duplex; these differences in the counterion binding patters were earlier shown to be responsible for the observed drastic differences in condensation propensities between short DNA and RNA duplexes. MC simulations of CoHex ions interacting with the homopolymeric poly(dA·dT) DNA duplex with modified (de-methylated) and native thymine bases are used to explore the physics behind CoHex-thymine interactions. The simulations suggest that the ion desolvation penalty due to proximity to the low dielectric volume of the methyl group can contribute significantly to CoHex-thymine interactions. Compared to the steric repulsion between the ion and the methyl group, the desolvation penalty interaction has a longer range and may be important to consider in the context of methylation effects on DNA condensation.

Accuracy Comparison of Generalized Born Models in the Calculation of Electrostatic Binding Free Energies

The need for accurate yet efficient representation of the aqueous environment in biomolecular modeling has led to the development of a variety of generalized Born (GB) implicit solvent models. While many studies have focused on the accuracy of available GB models in predicting solvation free energies, a systematic assessment of the quality of these models in binding free energy calculations, crucial for rational drug design, has not been undertaken. Here, we evaluate the accuracies of eight common GB flavors (GB-HCT, GB-OBC, GB-neck2, GBNSR6, GBSW, GBMV1, GBMV2, and GBMV3), available in major molecular dynamics packages, in predicting the electrostatic binding free energies ( ΔΔ G_el) for a diverse set of 60 biomolecular complexes belonging to four main classes: protein-protein, protein-drug, RNA-peptide, and small complexes. The GB flavors are examined in terms of their ability to reproduce the results from the Poisson-Boltzmann (PB) model, commonly used as accuracy reference in this context. We show that the agreement with the PB of ΔΔ G_el estimates varies widely between different GB models and also across different types of biomolecular complexes, with R² correlations ranging from 0.3772 to 0.9986. A surface-based "R6" GB model recently implemented in AMBER shows the closest overall agreement with reference PB ( R² = 0.9949, RMSD = 8.75 kcal/mol). The RNA-peptide and protein-drug complex sets appear to be most challenging for all but one model, as indicated by the large deviations from the PB in ΔΔ G_el. Small neutral complexes present the least challenge for most of the GB models tested. The quantitative demonstration of the strengths and weaknesses of the GB models across the diverse complex types provided here can be used as a guide for practical computations and future development efforts.

Reproducing the Ensemble Average Polar Solvation Energy of a Protein from a Single Structure: Gaussian-Based Smooth Dielectric Function for Macromolecular Modeling

Typically, the ensemble average polar component of solvation energy (ΔG_polar^solv) of a macromolecule is computed using molecular dynamics (MD) or Monte Carlo (MC) simulations to generate conformational ensemble and then single/rigid conformation solvation energy calculation is performed on each snapshot. The primary objective of this work is to demonstrate that Poisson-Boltzmann (PB)-based approach using a Gaussian-based smooth dielectric function for macromolecular modeling previously developed by us (Li et al. J. Chem. Theory Comput. 2013, 9 (4), 2126-2136) can reproduce that ensemble average (ΔG_polar^solv) of a protein from a single structure. We show that the Gaussian-based dielectric model reproduces the ensemble average ΔG_polar^solv(⟨ΔG_polar^solv⟩) from an energy-minimized structure of a protein regardless of the minimization environment (structure minimized in vacuo, implicit or explicit waters, or crystal structure); the best case, however, is when it is paired with an in vacuo-minimized structure. In other minimization environments (implicit or explicit waters or crystal structure), the traditional two-dielectric model can still be selected with which the model produces correct solvation energies. Our observations from this work reflect how the ability to appropriately mimic the motion of residues, especially the salt bridge residues, influences a dielectric model's ability to reproduce the ensemble average value of polar solvation free energy from a single in vacuo-minimized structure.

Electrostatic Interactions in Protein Structure, Folding, Binding, and Condensation

Charged and polar groups, through forming ion pairs, hydrogen bonds, and other less specific electrostatic interactions, impart important properties to proteins. Modulation of the charges on the amino acids, e.g., by pH and by phosphorylation and dephosphorylation, have significant effects such as protein denaturation and switch-like response of signal transduction networks. This review aims to present a unifying theme among the various effects of protein charges and polar groups. Simple models will be used to illustrate basic ideas about electrostatic interactions in proteins, and these ideas in turn will be used to elucidate the roles of electrostatic interactions in protein structure, folding, binding, condensation, and related biological functions. In particular, we will examine how charged side chains are spatially distributed in various types of proteins and how electrostatic interactions affect thermodynamic and kinetic properties of proteins. Our hope is to capture both important historical developments and recent experimental and theoretical advances in quantifying electrostatic contributions of proteins.

How Water's Properties Are Encoded in Its Molecular Structure and Energies

How are water's material properties encoded within the structure of the water molecule? This is pertinent to understanding Earth's living systems, its materials, its geochemistry and geophysics, and a broad spectrum of its industrial chemistry. Water has distinctive liquid and solid properties: It is highly cohesive. It has volumetric anomalies-water's solid (ice) floats on its liquid; pressure can melt the solid rather than freezing the liquid; heating can shrink the liquid. It has more solid phases than other materials. Its supercooled liquid has divergent thermodynamic response functions. Its glassy state is neither fragile nor strong. Its component ions-hydroxide and protons-diffuse much faster than other ions. Aqueous solvation of ions or oils entails large entropies and heat capacities. We review how these properties are encoded within water's molecular structure and energies, as understood from theories, simulations, and experiments. Like simpler liquids, water molecules are nearly spherical and interact with each other through van der Waals forces. Unlike simpler liquids, water's orientation-dependent hydrogen bonding leads to open tetrahedral cage-like structuring that contributes to its remarkable volumetric and thermal properties.

Explicit Solvent Hydration Benchmark for Proteins with Application to the PBSA Method

Explicit and implicit solvent models have a proven record of delivering hydration free energies of small, druglike solutes in reasonable agreement with experiment. Hydration of macromolecules, such as proteins, is to a large extent uncharted territory, with few results shedding light on quantitative consistency between different solvent models, let alone their ability to reproduce real water. In this work, based on extensive explicit solvent simulations employing TIP3P and SPC/E water models we analyze hydration free energy changes between fixed conformations of 5 diverse proteins, including large multidomain structures. For the two solvent models we find better agreement in electrostatic rather than nonpolar contributions (RMSE of 2.3 and 2.7 kcal/mol, respectively), even though absolute values of the latter are typically an order of magnitude smaller. We also highlight the importance of finite size corrections to relative protein hydration free energies, which turn out to be rather large, on the order of several kcal/mol, and are necessary for proper interpretation of results obtained under periodic boundary conditions. We further compare gathered data with predictions of the implicit solvent approach based on the Poisson equation and the surface or volume based nonpolar term. We find definitely lesser consistency than between the two explicit models (RMSE between implicit and TIP3 results of 11.3 and 8.4 kcal/mol for electrostatic and nonpolar contributions, respectively). In the process we determine the value of the protein dielectric constant and the geometric model for the dielectric boundary that provide for the best agreement. Finally, we evaluate the usefulness of surface and volume based models of nonpolar contributions to hydration free energy of large biomolecules.

Accuracy comparison of several common implicit solvent models and their implementations in the context of protein-ligand binding

In this study several commonly used implicit solvent models are compared with respect to their accuracy of estimating solvation energies of small molecules and proteins, as well as desolvation penalty in protein-ligand binding. The test set consists of 19 small proteins, 104 small molecules, and 15 protein-ligand complexes. We compared predicted hydration energies of small molecules with their experimental values; the results of the solvation and desolvation energy calculations for small molecules, proteins and protein-ligand complexes in water were also compared with Thermodynamic Integration calculations based on TIP3P water model and Amber12 force field. The following implicit solvent (water) models considered here are: PCM (Polarized Continuum Model implemented in DISOLV and MCBHSOLV programs), GB (Generalized Born method implemented in DISOLV program, S-GB, and GBNSR6 stand-alone version), COSMO (COnductor-like Screening Model implemented in the DISOLV program and the MOPAC package) and the Poisson-Boltzmann model (implemented in the APBS program). Different parameterizations of the molecules were examined: we compared MMFF94 force field, Amber12 force field and the quantum-chemical semi-empirical PM7 method implemented in the MOPAC package. For small molecules, all of the implicit solvent models tested here yield high correlation coefficients (0.87-0.93) between the calculated solvation energies and the experimental values of hydration energies. For small molecules high correlation (0.82-0.97) with the explicit solvent energies is seen as well. On the other hand, estimated protein solvation energies and protein-ligand binding desolvation energies show substantial discrepancy (up to 10kcal/mol) with the explicit solvent reference. The correlation of polar protein solvation energies and protein-ligand desolvation energies with the corresponding explicit solvent results is 0.65-0.99 and 0.76-0.96 respectively, though this difference in correlations is caused more by different parameterization and less by methods and indicates the need for further improvement of implicit solvent models parameterization. Within the same parameterization, various implicit methods give practically the same correlation with results obtained in explicit solvent model for ligands and proteins: e.g. correlation values of polar ligand solvation energies and the corresponding energies in the frame of explicit solvent were 0.953-0.966 for the APBS program, the GBNSR6 program and all models used in the DISOLV program. The DISOLV program proved to be on a par with the other used programs in the case of proteins and ligands solvation energy calculation. However, the solution of the Poisson-Boltzmann equation (APBS program) and Generalized Born method (implemented in the GBNSR6 program) proved to be the most accurate in calculating the desolvation energies of complexes.

Protein-Ligand Electrostatic Binding Free Energies from Explicit and Implicit Solvation

Accurate yet efficient computational models of solvent environment are central for most calculations that rely on atomistic modeling, such as prediction of protein-ligand binding affinities. In this study, we evaluate the accuracy of a recently developed generalized Born implicit solvent model, GBNSR6 (Aguilar et al. J. Chem. Theory Comput. 2010, 6, 3613-3639), in estimating the electrostatic solvation free energies (ΔG(pol)) and binding free energies (ΔΔG(pol)) for small protein-ligand complexes. We also compare estimates based on three different explicit solvent models (TIP3P, TIP4PEw, and OPC). The two main findings are as follows. First, the deviation (RMSD = 7.04 kcal/mol) of GBNSR6 binding affinities from commonly used TIP3P reference values is comparable to the deviations between explicit models themselves, e.g. TIP4PEw vs TIP3P (RMSD = 5.30 kcal/mol). A simple uniform adjustment of the atomic radii by a single scaling factor reduces the RMS deviation of GBNSR6 from TIP3P to within the above "error margin" - differences between ΔΔG(pol) estimated by different common explicit solvent models. The simple radii scaling virtually eliminates the systematic deviation (ΔΔG(pol)) between GBNSR6 and two out of the three explicit water models and significantly reduces the deviation from the third explicit model. Second, the differences between electrostatic binding energy estimates from different explicit models is disturbingly large; for example, the deviation between TIP4PEw and TIP3P estimates of ΔΔG(pol) values can be up to ∼50% or ∼9 kcal/mol, which is significantly larger than the "chemical accuracy" goal of ∼1 kcal/mol. The absolute ΔG(pol) calculated with different explicit models could differ by tens of kcal/mol. These discrepancies point to unacceptably high sensitivity of binding affinity estimates to the choice of common explicit water models. The absence of a clear "gold standard" among these models strengthens the case for the use of accurate implicit solvation models for binding energetics, which may be orders of magnitude faster.

On the energy components governing molecular recognition in the framework of continuum approaches

Molecular recognition is a process that brings together several biological macromolecules to form a complex and one of the most important characteristics of the process is the binding free energy. Various approaches exist to model the binding free energy, provided the knowledge of the 3D structures of bound and unbound molecules. Among them, continuum approaches are quite appealing due to their computational efficiency while at the same time providing predictions with reasonable accuracy. Here we review recent developments in the field emphasizing on the importance of adopting adequate description of physical processes taking place upon the binding. In particular, we focus on the efforts aiming at capturing some of the atomistic details of the binding phenomena into the continuum framework. When possible, the energy components are reviewed independently of each other. However, it is pointed out that rigorous approaches should consider all energy contributions on the same footage. The two major schemes for utilizing the individual energy components to predict binding affinity are outlined as well.