The Use of Anisotropic Potentials in Modeling Water and Free Energies of Hydration

Theoretically grounded approaches to account for polarization effects in fixed-charge force fields

Non-polarizable, or fixed-charge, force fields are the workhorses of most molecular simulation studies. They attempt to describe the potential energy surface (PES) of the system by including polarization effects in an implicit way. This has historically been done in a rather empirical and ad hoc manner. Recent theoretical treatments of polarization, however, offer promise for getting the most out of fixed-charge force fields by judicious choice of parameters (most significantly the net charge or dipole moment of the model) and application of post facto polarization corrections. This Perspective describes these polarization theories, namely the "halfway-charge" theory and the molecular dynamics in electronic continuum theory, and shows that they lead to qualitatively (and often, quantitatively) similar predictions. Moreover, they can be reconciled into a unified approach to construct a force field development workflow that can yield non-polarizable models with charge/dipole values that provide an optimal description of the PES. Several applications of this approach are reviewed, and avenues for future research are proposed.

What is the Optimal Dipole Moment for Nonpolarizable Models of Liquids?

In classical nonpolarizable models, electrostatic interactions are usually described by assigning fixed partial charges to interaction sites. Despite the multitude of methods and theories proposed over the years for partial charge assignment, a fundamental question remains─what is the correct degree of polarization that a fixed-charge model should possess to provide the best balance of interactions (including induction effects) and yield the best description of the potential energy surface of a liquid phase? We address this question by approaching it from two separate and independent viewpoints: the QUantum mechanical BEspoke (QUBE) approach, which assigns bespoke force field parameters for individual molecules from ab initio calculations with minimal empirical fitting, and the Polarization-Consistent Approach (PolCA) force field, based on empirical fitting of force field parameters with an emphasis on transferability by rigorously accounting for polarization effects in the parameterization process. We show that the two approaches yield consistent answers to the above question, namely, that the dipole moment of the model should be approximately halfway between those of the gas and the liquid phase. Crucially, however, the reference liquid-phase dipole needs to be estimated using methods that explicitly consider both mean-field and local contributions to polarization. In particular, continuum dielectric models are inadequate for this purpose because they cannot account for local effects and therefore significantly underestimate the degree of polarization of the molecule. These observations have profound consequences for the development, validation, and testing of nonpolarizable models.

Comparison of the United- and All-Atom Representations of (Halo)alkanes Based on Two Condensed-Phase Force Fields Optimized against the Same Experimental Data Set

The level of accuracy that can be achieved by a force field is influenced by choices made in the interaction-function representation and in the relevant simulation parameters. These choices, referred to here as functional-form variants (FFVs), include for example the model resolution, the charge-derivation procedure, the van der Waals combination rules, the cutoff distance, and the treatment of the long-range interactions. Ideally, assessing the effect of a given FFV on the intrinsic accuracy of the force-field representation requires that only the specific FFV is changed and that this change is performed at an optimal level of parametrization, a requirement that may prove extremely challenging to achieve in practice. Here, we present a first attempt at such a comparison for one specific FFV, namely the choice of a united-atom (UA) versus an all-atom (AA) resolution in a force field for saturated acyclic (halo)alkanes. Two force-field versions (UA vs AA) are optimized in an automated way using the CombiFF approach against 961 experimental values for the pure-liquid densities ρ_liq and vaporization enthalpies ΔH_vap of 591 compounds. For the AA force field, the torsional and third-neighbor Lennard-Jones parameters are also refined based on quantum-mechanical rotational-energy profiles. The comparison between the UA and AA resolutions is also extended to properties that have not been included as parameterization targets, namely the surface-tension coefficient γ, the isothermal compressibility κ_T, the isobaric thermal-expansion coefficient α_P, the isobaric heat capacity c_P, the static relative dielectric permittivity ϵ, the self-diffusion coefficient D, the shear viscosity η, the hydration free energy ΔG_wat, and the free energy of solvation ΔG_che in cyclohexane. For the target properties ρ_liq and ΔH_vap, the UA and AA resolutions reach very similar levels of accuracy after optimization. For the nine other properties, the AA representation leads to more accurate results in terms of η; comparably accurate results in terms of γ, κ_T, α_P, ϵ, D, and ΔG_che; and less accurate results in terms of c_P and ΔG_wat. This work also represents a first step toward the calibration of a GROMOS-compatible force field at the AA resolution.

Simple corrections for the static dielectric constant of liquid mixtures from model force fields

Pair-wise additive force fields provide fairly accurate predictions, through classical molecular simulations, for a wide range of structural, thermodynamic, and dynamical properties of many materials. However, one key property that has not been well captured is the static dielectric constant, which characterizes the response of a system to an applied electric field and is important in determining the screening of electrostatic interactions through a system. A simple correction has been found to provide a relatively robust method to improve the estimate of the static dielectric constant from molecular simulations for a broad range of compounds. This approach accounts for the electronic contribution to molecular polarizability and assumes that the charges that couple a molecule to an applied electric field are proportional to the effective force field charges. In this work, we examine how this correction performs for systems at different temperatures and for binary mixtures. Using a value for the electronic polarizability, based on the experimental index of refraction, and a charge scaling factor, determined at a single temperature, we find that the static dielectric constant can be predicted remarkably well, in comparison to the experimentally measured values. This provides good evidence that the effective charges that appear in pair-wise additive force fields developed to reproduce the potential energy surface of a system are not the same as those that determine the static dielectric constant; however, they can be captured in a relatively simple manner, which is dependent on the particular force field.

A Simple Method for Including Polarization Effects in Solvation Free Energy Calculations When Using Fixed-Charge Force Fields: Alchemically Polarized Charges

The incorporation of polarizability in classical force-field molecular simulations is an ongoing area of research. We focus here on its application to hydration free energy simulations of organic molecules. In contrast to computationally complex approaches involving the development of explicitly polarizable force fields, we present herein a simple methodology for incorporating polarization into such simulations using standard fixed-charge force fields, which we call the alchemically polarized charges (APolQ) method. APolQ employs a standard classical alchemical free energy change simulation to calculate the free energy difference between a fully polarized solute particle in a condensed phase and its unpolarized state in a vacuum. APolQ can in principle be applied to any microscopically homogeneous system (e.g., pure or mixed solvents). We applied APolQ to hydration free energy data for a test set of 45 neutral solute molecules in the FreeSolv database and compared results obtained using three different water models (SPC/E, TIP3P, and OPC3) and using minimal basis iterative Stockholder (MBIS) and restrained electrostatic potential (RESP) partial charge methodologies. In comparison with AM1-BCC, we found that APolQ outperforms it for the test set. Despite our method using default GAFF parameters, the MBIS partial charges yield absolute average deviations 1.5-1.9 kJ mol-1 lower than using AM1 bond charge correction (AM1-BCC). We conjecture that this method can be further improved by fitting the Lennard-Jones and torsional parameters to partial charges derived using MBIS or RESP methodologies.

Non-bonded force field model with advanced restrained electrostatic potential charges (RESP2)

The restrained electrostatic potential (RESP) approach is a highly regarded and widely used method of assigning partial charges to molecules for simulations. RESP uses a quantum-mechanical method that yields fortuitous overpolarization and thereby accounts only approximately for self-polarization of molecules in the condensed phase. Here we present RESP2, a next generation of this approach, where the polarity of the charges is tuned by a parameter, δ, which scales the contributions from gas- and aqueous-phase calculations. When the complete non-bonded force field model, including Lennard-Jones parameters, is optimized to liquid properties, improved accuracy is achieved, even with this reduced set of five Lennard-Jones types. We argue that RESP2 with δ≈0.6 (60% aqueous, 40% gas-phase charges) is an accurate and robust method of generating partial charges, and that a small set of Lennard-Jones types is good starting point for a systematic re-optimization of this important non-bonded term.

Benchmarking Electronic Structure Methods for Accurate Fixed-Charge Electrostatic Models

The accuracy of classical molecular mechanics (MM) force fields used for condensed phase molecular simulations depends strongly on the accuracy of modeling nonbonded interactions between atoms, such as electrostatic interactions. Some popular fixed-charge MM force fields use partial atomic charges derived from gas phase electronic structure calculations using the Hartree-Fock (HF) method with the relatively small 6-31G* basis set (HF/6-31G*). It is generally believed that HF/6-31G* generates fortuitously overpolarized electron distributions, as would be expected in the higher dielectric environment of the condensed phase. Using a benchmark set of 47 molecules, we show that HF/6-31G* overpolarizes molecules by just under 10% on average with respect to experimental gas phase dipole moments. The overpolarization of this method/basis set combination varies significantly though and, in some cases, even leads to molecular dipole moments that are lower than experimental gas phase measurements. We further demonstrate that using computationally inexpensive density functional theory (DFT) methods, together with appropriate augmented basis sets and a continuum solvent model, can yield molecular dipole moments that are both more strongly and more uniformly overpolarized. These data suggest that these methods-or ones similar to them-should be adopted for the derivation of accurate partial atomic charges for next-generation MM force fields.

Development and Validation of the Quantum Mechanical Bespoke Protein Force Field

Molecular mechanics force field parameters for macromolecules, such as proteins, are traditionally fit to reproduce experimental properties of small molecules, and thus, they neglect system-specific polarization. In this paper, we introduce a complete protein force field that is designed to be compatible with the quantum mechanical bespoke (QUBE) force field by deriving nonbonded parameters directly from the electron density of the specific protein under study. The main backbone and sidechain protein torsional parameters are rederived in this work by fitting to quantum mechanical dihedral scans for compatibility with QUBE nonbonded parameters. Software is provided for the preparation of QUBE input files. The accuracy of the new force field, and the derived torsional parameters, is tested by comparing the conformational preferences of a range of peptides and proteins with experimental measurements. Accurate backbone and sidechain conformations are obtained in molecular dynamics simulations of dipeptides, with NMR J coupling errors comparable to the widely used OPLS force field. In simulations of five folded proteins, the secondary structure is generally retained, and the NMR J coupling errors are similar to standard transferable force fields, although some loss of the experimental structure is observed in certain regions of the proteins. With several avenues for further development, the use of system-specific nonbonded force field parameters is a promising approach for next-generation simulations of biological molecules.

Solvation Free Energy Calculation Using a Fixed-Charge Model: Implicit and Explicit Treatments of the Polarization Effect

In this work, IPolQ-Mod charges and the reference potential scheme are used to calculate the solvation free energies of a set of organic molecules. Both methods could capture the phase transfer of a solute with accompanying polarization cost utilizing a fixed-charge model. The IPolQ-Mod charges, which are the average of two charge sets fitted in a vacuum state and a condensed phase, take account of the polarization effect implicitly. For the reference potential method, the quantum mechanics polarization corrections are calculated explicitly by thermodynamic perturbation. The polarization effect captured by the IPolQ-Mod charges is an approximation to that of the reference potential method theoretically. In the present study, the reference potential method shows a slight improvement over the classical restrained electrostatic potential (RESP) charges, which perform pretty well in predicting the solvation free energy. However, IPolQ-Mod(MP2) shows a poor agreement with the experimental data. Compared with IPolQ-Mod(MP2), IPolQ-Mod(M06-2X) or IPolQ-Mod(ωB97X) is found to give more appropriate prediction of the molecule's dipole and the solvation free energies calculated by IPolQ-Mod(M06-2X) or IPolQ-Mod(ωB97X) are more compatible with those of the RESP charges. If the other force field parameters remain unchanged, M06-2X or ωB97X is recommended to derive the IPolQ-Mod charges.

Hydration Free Energies in the FreeSolv Database Calculated with Polarized Iterative Hirshfeld Charges

Computer simulations of biomolecular systems often use force fields, which are combinations of simple empirical atom-based functions to describe the molecular interactions. Even though polarizable force fields give a more detailed description of intermolecular interactions, nonpolarizable force fields, developed several decades ago, are often still preferred because of their reduced computation cost. Electrostatic interactions play a major role in biomolecular systems and are therein described by atomic point charges. In this work, we address the performance of different atomic charges to reproduce experimental hydration free energies in the FreeSolv database in combination with the GAFF force field. Atomic charges were calculated by two atoms-in-molecules approaches, Hirshfeld-I and Minimal Basis Iterative Stockholder (MBIS). To account for polarization effects, the charges were derived from the solute's electron density computed with an implicit solvent model, and the energy required to polarize the solute was added to the free energy cycle. The calculated hydration free energies were analyzed with an error model, revealing systematic errors associated with specific functional groups or chemical elements. The best agreement with the experimental data is observed for the AM1-BCC and the MBIS atomic charge methods. The latter includes the solvent polarization and presents a root-mean-square error of 2.0 kcal mol-1 for the 613 organic molecules studied. The largest deviation was observed for phosphorus-containing molecules and the molecules with amide, ester and amine functional groups.

Comparison of RESP and IPolQ-Mod Partial Charges for Solvation Free Energy Calculations of Various Solute/Solvent Pairs

The calculation of solvation free energies ΔG_solv by molecular simulations is of great interest as they are linked to other physical properties such as relative solubility, partition coefficient, and activity coefficient. However, shortcomings in molecular models can lead to ΔG_solv deviations from experimental data. Various studies have demonstrated the impact of partial charges on free energy results. Consequently, calculation methods for partial charges aimed at more accurate ΔG_solv predictions are the subject of various studies in the literature. Here we compare two methods to derive partial charges for the general AMBER force field (GAFF), i.e. the default RESP as well as the physically motivated IPolQ-Mod method that implicitly accounts for polarization costs. We study 29 solutes which include characteristic functional groups of drug-like molecules, and 12 diverse solvents were examined. In total, we consider 107 solute/solvent pairs including two water models TIP3P and TIP4P/2005. Comparison with experimental results yields better agreement for TIP3P, regardless of the partial charge scheme. The overall performance of GAFF/RESP and GAFF/IPolQ-Mod is similar, though specific shortcomings in the description of ΔG_solv for both RESP and IPolQ-Mod can be identified. However, the high correlation between free energies obtained with GAFF/RESP and GAFF/IPolQ-Mod demonstrates the compatibility between the modified charges and remaining GAFF parameters.

Further along the Road Less Traveled: AMBER ff15ipq, an Original Protein Force Field Built on a Self-Consistent Physical Model

We present the AMBER ff15ipq force field for proteins, the second-generation force field developed using the Implicitly Polarized Q (IPolQ) scheme for deriving implicitly polarized atomic charges in the presence of explicit solvent. The ff15ipq force field is a complete rederivation including more than 300 unique atomic charges, 900 unique torsion terms, 60 new angle parameters, and new atomic radii for polar hydrogens. The atomic charges were derived in the context of the SPC/E_b water model, which yields more-accurate rotational diffusion of proteins and enables direct calculation of nuclear magnetic resonance (NMR) relaxation parameters from molecular dynamics simulations. The atomic radii improve the accuracy of modeling salt bridge interactions relative to contemporary fixed-charge force fields, rectifying a limitation of ff14ipq that resulted from its use of pair-specific Lennard-Jones radii. In addition, ff15ipq reproduces penta-alanine J-coupling constants exceptionally well, gives reasonable agreement with NMR relaxation rates, and maintains the expected conformational propensities of structured proteins/peptides, as well as disordered peptides—all on the microsecond (μs) time scale, which is a critical regime for drug design applications. These encouraging results demonstrate the power and robustness of our automated methods for deriving new force fields. All parameters described here and the mdgx program used to fit them are included in the AmberTools16 distribution.

Biomolecular Force Field Parameterization via Atoms-in-Molecule Electron Density Partitioning

Molecular mechanics force fields, which are commonly used in biomolecular modeling and computer-aided drug design, typically treat nonbonded interactions using a limited library of empirical parameters that are developed for small molecules. This approach does not account for polarization in larger molecules or proteins, and the parametrization process is labor-intensive. Using linear-scaling density functional theory and atoms-in-molecule electron density partitioning, environment-specific charges and Lennard-Jones parameters are derived directly from quantum mechanical calculations for use in biomolecular modeling of organic and biomolecular systems. The proposed methods significantly reduce the number of empirical parameters needed to construct molecular mechanics force fields, naturally include polarization effects in charge and Lennard-Jones parameters, and scale well to systems comprised of thousands of atoms, including proteins. The feasibility and benefits of this approach are demonstrated by computing free energies of hydration, properties of pure liquids, and the relative binding free energies of indole and benzofuran to the L99A mutant of T4 lysozyme.

ff14ipq: A Self-Consistent Force Field for Condensed-Phase Simulations of Proteins

We present the ff14ipq force field, implementing the previously published IPolQ charge set for simulations of complete proteins. Minor modifications to the charge derivation scheme and van der Waals interactions between polar atoms are introduced. Torsion parameters are developed through a generational learning approach, based on gas-phase MP2/cc-pVTZ single-point energies computed of structures optimized by the force field itself rather than the quantum benchmark. In this manner, we sacrifice information about the true quantum minima in order to ensure that the force field maintains optimal agreement with the MP2/cc-pVTZ benchmark for the ensembles it will actually produce in simulations. A means of making the gas-phase torsion parameters compatible with solution-phase IPolQ charges is presented. The ff14ipq model is an alternative to ff99SB and other Amber force fields for protein simulations in programs that accommodate pair-specific Lennard-Jones combining rules. The force field gives strong performance on α-helical and β-sheet oligopeptides as well as globular proteins over microsecond time scale simulations, although it has not yet been tested in conjunction with lipid and nucleic acid models. We show how our choices in parameter development influence the resulting force field and how other choices that may have appeared reasonable would actually have led to poorer results. The tools we developed may also aid in the development of future fixed-charge and even polarizable biomolecular force fields.

Implementing electrostatic polarization cannot fill the gap between experimental and theoretical measurements for the ultrafast fluorescence decay of myoglobin

Over the past few years, time-dependent ultrafast fluorescence spectroscopy method has been applied to the study of protein dynamics. However, observations from these experiments are in a controversy with other experimental studies. Participating of theoretical methods in this debate has not reconciled the contradiction, because the predicted initial relaxation from computer simulations is one-order faster than the ultrafast fluorescence spectroscopy experiment. In those simulations, pairwise force fields are employed, which have been shown to underestimate the roughness of the free energy landscape. Therefore, the relaxation rate of protein and water molecules under pairwise force fields is falsely exaggerated. In this work, we compared the relaxations of tryptophan/environment interaction under linear response approximation employing pairwise, polarized, and polarizable force fields. Results show that although the relaxation can be slowed down to a certain extent, the large gap between experiment and theory still cannot be filled.

Dipeptide Aggregation in Aqueous Solution from Fixed Point-Charge Force Fields

The description of aggregation processes with molecular dynamics simulations is a playground for testing biomolecular force fields, including a new generation of force fields that explicitly describe electronic polarization. In this work, we study a system consisting of 50 glycyl-l-alanine (Gly-Ala) dipeptides in solution with 1001 water molecules. Neutron diffraction experiments have shown that at this concentration, Gly-Ala aggregates into large clusters. However, general-purpose force fields in combination with established water models can fail to correctly describe this aggregation process, highlighting important deficiencies in how solute-solute and solute-solvent interactions are parametrized in these force fields. We found that even for the fully polarizable AMOEBA force field, the degree of association is considerably underestimated. Instead, a fixed point-charge approach based on the newly developed IPolQ scheme [Cerutti et al. J. Phys. Chem.2013, 117, 2328] allows for the correct modeling of the dipeptide aggregation in aqueous solution. This result should stimulate interest in novel fitting schemes that aim to improve the description of the solvent polarization effect within both explicitly polarizable and fixed point-charge frameworks.

The SAMPL4 hydration challenge: evaluation of partial charge sets with explicit-water molecular dynamics simulations

We used blind predictions of the 47 hydration free energies in the SAMPL4 challenge to test multiple partial charge models in the context of explicit solvent free energy simulations with the general AMBER force field. One of the partial charge models, IPolQ-Mod, is a fast continuum solvent-based implementation of the IPolQ approach. The AM1-BCC, restrained electrostatic potential (RESP) and IpolQ-Mod approaches all perform reasonably well (R(2) > 0.8), while VCharge, though faster, gives less accurate results (R(2) of 0.5). The AM1-BCC results are more accurate than those of RESP for tertiary amines and nitrates, but the overall difference in accuracy between these methods is not statistically significant. Interestingly, the IPolQ-Mod method is found to yield partial charges in very close agreement with RESP. This observation suggests that the success of RESP may be attributed to its fortuitously approximating the arguably more rigorous IPolQ approach.

Derivation of fixed partial charges for amino acids accommodating a specific water model and implicit polarization

We have developed the IPolQ method for fitting nonpolarizable point charges to implicitly represent the energy of polarization for systems in pure water. The method involves iterative cycles of molecular dynamics simulations to estimate the water charge density around the solute of interest, followed by quantum mechanical calculations at the MP2/cc-pV(T+d)Z level to determine updated solute charges. Lennard-Jones parameters are updated starting from the Amber FF99SB nonbonded parameter set to accommodate the new charge model, guided by the comparisons to experimental hydration free energies (HFEs) of neutral amino acid side chain analogs and assumptions about the computed HFEs for charged side chains. These Lennard-Jones parameter adjustments for side-chain analogs are assumed to be transferable to amino acids generally, and new charges for all standard amino acids are then derived in the presence of water modeled by TIP4P-Ew. Overall, the new charges depict substantially more polarized amino acids, particularly in the backbone moieties, than previous Amber charge sets. Efforts to complete a new force field with appropriate torsion parameters for this charge model are underway. The IPolQ method is general and applicable to arbitrary solutes.

Alchemical prediction of hydration free energies for SAMPL

Hydration free energy calculations have become important tests of force fields. Alchemical free energy calculations based on molecular dynamics simulations provide a rigorous way to calculate these free energies for a particular force field, given sufficient sampling. Here, we report results of alchemical hydration free energy calculations for the set of small molecules comprising the 2011 Statistical Assessment of Modeling of Proteins and Ligands challenge. Our calculations are largely based on the Generalized Amber Force Field with several different charge models, and we achieved RMS errors in the 1.4-2.2 kcal/mol range depending on charge model, marginally higher than what we typically observed in previous studies (Mobley et al. in J Phys Chem B 111(9):2242-2254, 2007, J Chem Theory Comput 5(2):350-358, 2009, J Phys Chem B 115:1329-1332, 2011; Nicholls et al. in J Med Chem 51:769-779, 2008; Klimovich and Mobley in J Comput Aided Mol Design 24(4):307-316, 2010). The test set consists of ethane, biphenyl, and a dibenzyl dioxin, as well as a series of chlorinated derivatives of each. We found that, for this set, using high-quality partial charges from MP2/cc-PVTZ SCRF RESP fits provided marginally improved agreement with experiment over using AM1-BCC partial charges as we have more typically done, in keeping with our recent findings (Mobley et al. in J Phys Chem B 115:1329-1332, 2011). Switching to OPLS Lennard-Jones parameters with AM1-BCC charges also improves agreement with experiment. We also find a number of chemical trends within each molecular series which we can explain, but there are also some surprises, including some that are captured by the calculations and some that are not.