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

Recent Developments in Free Energy Calculations for Drug Discovery

The grand challenge in structure-based drug design is achieving accurate prediction of binding free energies. Molecular dynamics (MD) simulations enable modeling of conformational changes critical to the binding process, leading to calculation of thermodynamic quantities involved in estimation of binding affinities. With recent advancements in computing capability and predictive accuracy, MD based virtual screening has progressed from the domain of theoretical attempts to real application in drug development. Approaches including the Molecular Mechanics Poisson Boltzmann Surface Area (MM-PBSA), Linear Interaction Energy (LIE), and alchemical methods have been broadly applied to model molecular recognition for drug discovery and lead optimization. Here we review the varied methodology of these approaches, developments enhancing simulation efficiency and reliability, remaining challenges hindering predictive performance, and applications to problems in the fields of medicine and biochemistry.

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.

Functional Group Distributions, Partition Coefficients, and Resistance Factors in Lipid Bilayers Using Site Identification by Ligand Competitive Saturation

Small molecules such as metabolites and drugs must pass through the membrane of the cell, a barrier primarily comprising phospholipid bilayers and embedded proteins. To better understand the process of passive diffusion, knowledge of the ability of various functional groups to partition across bilayers and the associated energetics would be of utility. In the present study, the site identification by ligand competitive saturation (SILCS) methodology has been applied to sample the distributions of a diverse set of chemical solutes representing the functional groups of small molecules across phospholipid bilayers composed of 0.9:0.1 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine/cholesterol and a mixture of 0.52:0.18:0.3 1,2-dioleoyl-sn-glycero-3-phospho-l-serine/1,2-dioleoyl-sn-glycero-3-phosphocholine/cholesterol used in parallel artificial membrane permeability assay experiments. A combination of oscillating chemical potential grand canonical Monte Carlo and molecular dynamics in the SILCS simulations was applied to achieve solute sampling through the bilayers and surrounding aqueous environment from which the distribution of solutes and the functional groups they represent were obtained. Results show differential distribution of aliphatic versus aromatic groups with the former having increased sampling in the center of the bilayers versus in the region of the glycerol linker for the latter. Variations in the distribution of different polar groups are evident, with large differences between negative acetate and positive methylammonium with accumulation of the polar-neutral and acetate solutes above the bilayer head groups. Conversion of the distributions to absolute free energies allows for a detailed understanding of energetics of functional groups in different regions of the bilayers and for calculation of absolute free-energy profiles of multifunctional drug-like molecules across the bilayers from which partition coefficients and resistance factors suitable for insertion into the homogenous solubility-diffusion equation for calculation of permeability were obtained. Comparisons of the calculated bilayer/solution partition coefficients with 1-octanol/water experimental data for both drug-like molecules and the solutes show overall good agreement, validating the calculated distributions and associated absolute free-energy profiles.

Implicit Solvents for the Polarizable Atomic Multipole AMOEBA Force Field

Computational protein design, ab initio protein/RNA folding, and protein-ligand screening can be too computationally demanding for explicit treatment of solvent. For these applications, implicit solvent offers a compelling alternative, which we describe here for the polarizable atomic multipole AMOEBA force field based on three treatments of continuum electrostatics: numerical solutions to the nonlinear and linearized versions of the Poisson-Boltzmann equation (PBE), the domain-decomposition conductor-like screening model (ddCOSMO) approximation to the PBE, and the analytic generalized Kirkwood (GK) approximation. The continuum electrostatics models are combined with a nonpolar estimator based on novel cavitation and dispersion terms. Electrostatic model parameters are numerically optimized using a least-squares style target function based on a library of 103 small-molecule solvation free energy differences. Mean signed errors for the adaptive Poisson-Boltzmann solver (APBS), ddCOSMO, and GK models are 0.05, 0.00, and 0.00 kcal/mol, respectively, while the mean unsigned errors are 0.70, 0.63, and 0.58 kcal/mol, respectively. Validation of the electrostatic response of the resulting implicit solvents, which are available in the Tinker (or Tinker-HP), OpenMM, and Force Field X software packages, is based on comparisons to explicit solvent simulations for a series of proteins and nucleic acids. Overall, the emergence of performative implicit solvent models for polarizable force fields opens the door to their use for folding and design applications.

pKa Calculations with the Polarizable Drude Force Field and Poisson-Boltzmann Solvation Model

Electronic polarization effects have been suggested to play an important role in proton binding to titratable residues in proteins. In this work, we describe a new computational method for pK_a calculations, using Monte Carlo (MC) simulations to sample protein protonation states with the Drude polarizable force field and Poisson-Boltzmann (PB) continuum electrostatic solvent model. While the most populated protonation states at the selected pH, corresponding to residues that are half-protonated at that pH, are sampled using the exact relative free energies computed with Drude particles optimized in the field of the PB implicit solvation model, we introduce an approximation for the protein polarization of low-populated protonation states to reduce the computational cost. The highly populated protonation states used to compute the polarization and pK_a's are then iteratively improved until convergence. It is shown that for lysozyme, when considering 9 of the 18 titratable residues, the new method converged within two iterations with computed pK_a's differing only by 0.02 pH units from pK_a's estimated with the exact approach. Application of the method to predict pK_a's of 94 titratable side chains in 8 proteins shows the Drude-PB model to produce physically more correct results as compared to the additive CHARMM36 (C36) force field (FF). With a dielectric constant of two assigned to the protein interior the Root Mean Square (RMS) deviation between computed and experimental pK_a's is 2.07 and 3.19 pH units with the Drude and C36 models, respectively, and the RMS deviation using the Drude-PB model is relatively insensitive to the choice of the internal dielectric constant in contrast to the additive C36 model. At the higher internal dielectric constant of 20, pK_a's computed with the additive C36 model converge to the results obtained with the Drude polarizable force field, indicating the need to artificially overestimate electrostatic screening in a nonphysical way with the additive FF. In addition, inclusion of both syn and anti orientations of the proton in the neutral state of acidic groups is shown to yield improved agreement with experiment. The present work, which is the first example of the use of a polarizable model for the prediction of pK_a's in proteins, shows that the use of a polarizable model represents a more physically correct model for the treatment of electrostatic contributions to pK_a shifts in proteins.

Further Optimization and Validation of the Classical Drude Polarizable Protein Force Field

The CHARMM Drude-2013 polarizable force field (FF) was developed to include the explicit treatment of induced electronic polarizability, resulting in a more accurate description of the electrostatic interactions in molecular dynamics (MD) simulations. While the Drude-2013 protein FF has shown success in improving the folding properties of α-helical peptides and to reproduce experimental observables in simulations up to 1 μs, some limitations were noted regarding the stability of β-sheet structures in simulations longer than 100 ns as well as larger deviations from crystal structures in simulations of a number of proteins compared to the additive CHARMM36 protein FF. The origin of the instability has been identified and appears to be primarily due to overestimated atomic polarizabilities and induced dipole-dipole interactions on the Cβ, Cγ, and Cδ side chain atoms. To resolve this and other issues, a number of aspects of the model were revisited, resulting in Drude-2019 protein FF. Backbone parameters were optimized targeting the conformational properties of the (Ala)5 peptide in solution along with gas phase properties of the alanine dipeptide. Dipeptides that contain N-acetylated and N'-methylamidated termini, excluding Gly, Pro, and Ala, were used as models to optimize the atomic polarizabilities and Thole screening factors on selected Cβ, Cγ, and Cδ carbons by targeting quantum mechanical (QM) dipole moments and molecular polarizabilities. In addition, to obtain better conformational properties, side chain χ1 and χ2 dihedral parameters were optimized targeting QM data for the respective side chain dipeptide conformations as well as Protein Data Bank survey data based on the χ1, χ2 sampling from Hamiltonian replica-exchange MD simulations of (Ala)4-X-(Ala)4 in solution, where X is the amino acid of interest. Further improvements include optimizing nonbonded interactions between charged residues to reproduce QM interaction energies of the charged-protein model compounds and experimental osmotic pressures. Validation of the optimized Drude protein FF includes MD simulations of a collection of peptides and proteins including β-sheet structures, as well as transmembrane ion channels. Results showed that the updated Drude-2019 protein FF yields smaller overall root-mean-square differences of proteins as compared to the additive CHARMM36m and Drude-2013 FFs as well as similar or improved agreement with experimental NMR properties, allowing for long time scale simulation studies of proteins and more complex biomolecular systems in conjunction with the remainder of the Drude polarizable FF.

Charge Scaling Manifesto: A Way of Reconciling the Inherently Macroscopic and Microscopic Natures of Molecular Simulations

Electronic polarization effects play an important role in the interactions of charged species in biologically relevant aqueous solutions, such as those involving salt ions, proteins, nucleic acids, or phospholipid membranes. Explicit inclusion of electronic polarization in molecular modeling is tedious both from the point of view of force field parametrization and actual performance of the simulations. Therefore, the vast majority of biomolecular simulations is performed using nonpolarizable force fields, which can lead to artifacts such as dramatically overestimated ion pairing, particularly when polyvalent ions are involved. Here, we show that many of these issues can be remedied without extra computational costs by including electronic polarization in a mean field way via charge rescaling. We also lay the solid physical foundations of this approach and reconcile from this perspective the microscopic versus macroscopic natures of nonpolarizable force fields.

Structural Insights into Hearing Loss Genetics from Polarizable Protein Repacking

Hearing loss is associated with ∼8100 mutations in 152 genes, and within the coding regions of these genes are over 60,000 missense variants. The majority of these variants are classified as "variants of uncertain significance" to reflect our inability to ascribe a phenotypic effect to the observed amino acid change. A promising source of pathogenicity information is biophysical simulation, although input protein structures often contain defects because of limitations in experimental data and/or only distant homology to a template. Here, we combine the polarizable atomic multipole optimized energetics for biomolecular applications force field, many-body optimization theory, and graphical processing unit acceleration to repack all deafness-associated proteins and thereby improve average structure MolProbity score from 2.2 to 1.0. We then used these optimized wild-type models to create over 60,000 structures for missense variants in the Deafness Variation Database, which are being incorporated into the Deafness Variation Database to inform deafness pathogenicity prediction. Finally, this work demonstrates that advanced polarizable atomic multipole force fields are efficient enough to repack the entire human proteome.

A Boundary-Integral Approach for the Poisson-Boltzmann Equation with Polarizable Force Fields

Implicit-solvent models are widely used to study the electrostatics in dissolved biomolecules, which are parameterized using force fields. Standard force fields treat the charge distribution with point charges; however, other force fields have emerged which offer a more realistic description by considering polarizability. In this work, we present the implementation of the polarizable and multipolar force field atomic multipole optimized energetics for biomolecular applications (AMOEBA), in the boundary integral Poisson-Boltzmann solver PyGBe. Previous work from other researchers coupled AMOEBA with the finite-difference solver APBS, and found difficulties to effectively transfer the multipolar charge description to the mesh. A boundary integral formulation treats the charge distribution analytically, overlooking such limitations. This becomes particularly important in simulations that need high accuracy, for example, when the quantity of interest is the difference between solvation energies obtained from separate calculations, like happens for binding energy. We present verification and validation results of our software, compare it with the implementation on APBS, and assess the efficiency of AMOEBA and classical point-charge force fields in a Poisson-Boltzmann solver. We found that a boundary integral approach performs similarly to a volumetric method on CPU. Also, we present a GPU implementation of our solver. Moreover, with a boundary element method, the mesh density to correctly resolve the electrostatic potential is the same for standard point-charge and multipolar force fields. Finally, we saw that for binding energy calculations, a boundary integral approach presents more consistent results than a finite difference approximation for multipolar force fields. © 2019 Wiley Periodicals, Inc.

Discrete and Continuum Models for the Salt in Crowded Environments of Suspended Charged Particles

Electrostatic forces greatly affect the overall dynamics and diffusional activities of suspended charged particles in crowded environments. Accordingly, the concentration of counter- or co-ions in a fluid-''the salt"-determines the range, strength, and order of electrostatic interactions between particles. This environment fosters engineering routes for controlling directed assembly of particles at both the micro- and nanoscale. Here, we analyzed two computational modeling schemes that considered salt within suspensions of charged particles, or polyelectrolytes: discrete and continuum. Electrostatic interactions were included through a Green's function formalism, where the confined fundamental solution for Poisson's equation is resolved by the general geometry Ewald-like method. For the discrete model, the salt was considered as regularized point-charges with a specific valence and size, while concentration fields were defined for each ionic species for the continuum model. These considerations were evolved using Brownian dynamics of the suspended charged particles and the discrete salt ions, while a convection-diffusion transport equation, including the Nernst-Planck diffusion mechanism, accounted for the dynamics of the concentration fields. The salt/particle models were considered as suspensions under slit-confinement conditions for creating crowded "macro-ions", where density distributions and radial distribution functions were used to compare and differentiate computational models. Importantly, our analysis shows that disparate length scales or increased system size presented by the salt and suspended particles are best dealt with using concentration fields to model the ions. These findings were then validated by novel simulations of a semipermeable polyelectrolyte membrane, at the mesoscale, from which ionic channels emerged and enable ion conduction.