Implicit solvation based on generalized Born theory in different dielectric environments

In this paper we are investigating the effect of the dielectric environment on atomic Born radii used in generalized Born (GB) methods. Motivated by the Kirkwood expression for the reaction field of a single off-center charge in a spherical cavity, we are proposing extended formalisms for the calculation of Born radii as a function of external and internal dielectric constants. We demonstrate that reaction field energies calculated from environmentally dependent Born radii lead to much improved agreement with Poisson-Boltzmann solutions for low dielectric external environments, such as biological membranes or organic solvent, compared to previous methods where the calculation of Born radii does not depend on the environment. We also examine how this new approach can be applied for the calculation of transfer free energies from vacuum to a given external dielectric for a system with an internal dielectric larger than one. This has not been possible with standard GB theory but is relevant when scoring minimized or average structures with implicit solvent.     

The treatment of solvation by a generalized Born model and a self-consistent charge-density functional theory-based tight-binding method

We present a model to calculate the free energies of solvation of small organic compounds as well as large biomolecules. This model is based on a generalized Born (GB) model and a self-consistent charge-density functional theory-based tight-binding (SCC-DFTB) method with the nonelectrostatic contributions to the free energy of solvation modeled in terms of solvent-accessible surface areas (SA). The parametrization of the SCC-DFTB/GBSA model has been based on 60 neutral and six ionic molecules composed of H, C, N, O, and S, and spanning a wide range of chemical groups. Effective atomic radii as parameters have been obtained through Monte Carlo Simulated Annealing optimization in the parameter space to minimize the differences between the calculated and experimental free energies of solvation. The standard error in the free energies of solvation calculated by the final model is 1.11 kcal mol(-1). We also calculated the free energies of solvation for these molecules using a conductor-like screening model (COSMO) in combination with different levels of theory (AM1, SCC-DFTB, and B3LYP/6-31G*) and compared the results with SCC-DFTB/GBSA. To assess the efficiency of our model for large biomolecules, we calculated the free energy of solvation for a HIV protease-inhibitor complex containing 3,204 atoms using the SCC-DFTB/GBSA and the SCC-DFTB/COSMO models, separately. The computed relative free energies of solvation are comparable, while the SCC-DFTB/GBSA model is three to four times more efficient, in terms of computational cost.     

Ion solvation thermodynamics from simulation with a polarizable force field

Thermodynamic measurements of the solvation of salts and electrolytes are relatively straightforward, but it is not possible to separate total solvation free energies into distinct cation and anion contributions without reference to an additional extrathermodynamic assumption. The present work attempts to resolve this difficulty using molecular dynamics simulations with the AMOEBA polarizable force field and perturbation techniques to directly compute absolute solvation free energies for potassium, sodium, and chloride ions in liquid water and formamide. Corresponding calculations are also performed with two widely used nonpolarizable force fields. The simulations with the polarizable force field accurately reproduce in vacuo quantum mechanical results, experimental ion-cluster solvation enthalpies, and experimental solvation free energies for whole salts, while the other force fields do not. The results indicate that calculations with a polarizable force field can capture the thermodynamics of ion solvation and that the solvation free energies of the individual ions differ by several kilocalories from commonly cited values.     

GBr6NL: a generalized Born method for accurately reproducing solvation energy of the nonlinear Poisson-Boltzmann equation

The nonlinear Poisson-Boltzmann (NLPB) equation can provide accurate modeling of electrostatic effects for nucleic acids and highly charged proteins. Generalized Born methods have been developed to mimic the linearized Poisson-Boltzmann (LPB) equation at substantially reduced cost. The computer time for solving the NLPB equation is approximately fivefold longer than for the LPB equation, thus presenting an even greater obstacle. Here we present the first generalized Born method, GBr(6)NL, for mimicking the NLPB equation. GBr(6)NL is adapted from GBr(6), a generalized Born method recently developed to reproduce the solvation energy of the LPB equation [Tjong and Zhou, J. Phys. Chem. B 111, 3055 (2007)]. Salt effects predicted by GBr(6)NL on 55 proteins overall deviate from NLPB counterparts by 0.5 kcal/mol from ionic strengths from 10 to 1000 mM, which is approximately 10% of the average magnitudes of the salt effects. GBr(6)NL predictions for the salts effects on the electrostatic interaction energies of two protein:RNA complexes are very promising.     

A strategy for reducing gross errors in the generalized Born models of implicit solvation

The "canonical" generalized Born (GB) formula [C. Still, A. Tempczyk, R. C. Hawley, and T. Hendrickson, J. Am. Chem. Soc. 112, 6127 (1990)] is known to provide accurate estimates for total electrostatic solvation energies ΔG(el) of biomolecules if the corresponding effective Born radii are accurate. Here we show that even if the effective Born radii are perfectly accurate, the canonical formula still exhibits significant number of gross errors (errors larger than 2k(B)T relative to numerical Poisson equation reference) in pairwise interactions between individual atomic charges. Analysis of exact analytical solutions of the Poisson equation (PE) for several idealized nonspherical geometries reveals two distinct spatial modes of the PE solution; these modes are also found in realistic biomolecular shapes. The canonical GB Green function misses one of two modes seen in the exact PE solution, which explains the observed gross errors. To address the problem and reduce gross errors of the GB formalism, we have used exact PE solutions for idealized nonspherical geometries to suggest an alternative analytical Green function to replace the canonical GB formula. The proposed functional form is mathematically nearly as simple as the original, but depends not only on the effective Born radii but also on their gradients, which allows for better representation of details of nonspherical molecular shapes. In particular, the proposed functional form captures both modes of the PE solution seen in nonspherical geometries. Tests on realistic biomolecular structures ranging from small peptides to medium size proteins show that the proposed functional form reduces gross pairwise errors in all cases, with the amount of reduction varying from more than an order of magnitude for small structures to a factor of 2 for the largest ones.     

Carbohydrate solution simulations: producing a force field with experimentally consistent primary alcohol rotational frequencies and populations

We present a CHARMM Carbohydrate Solution Force Field (CSFF) suitable for nanosecond molecular dynamics computer simulations. The force field was derived from a recently published sugar parameter set.1 Dihedral angle parameters for the primary alcohol as well as the secondary hydroxyl groups were adjusted. Free energy profiles of the hydroxymethyl group for two monosaccharides (beta-D-glucose and beta-D-galactose) were calculated using the new parameter set and compared with similar force fields. Equilibrium rotamer populations obtained from the CSFF are in excellent agreement with NMR data (glucose gg:gt:tg approximately 66:33:1 and galactose gg:gt:tg approximately 4:75:21). In addition, the primary alcohol rotational frequency is on the nanosecond time scale, which conforms to experimental observations. Equilibrium population distributions of the primary alcohol conformers for glucose and galactose are reached within 10 nanoseconds of molecular dynamics simulations. In addition, gas phase vibrational frequencies computed for beta-D-glucose using this force field compare well with experimental frequencies. Carbohydrate parameter sets that produce both conformational energies and rotational frequencies for the pyranose primary alcohol group that are consistent with experimental observations should allow for increased accuracy in modeling the flexibility of biologically important (1-6)-linked saccharides in solution.     

The consistent force field: Representation of molecular structure by potential energy functions

The principles are outlined, but the presentation concentrates on: the choice of structures on which to optimise; accuracy of post-dicted structure; reliability of pre-dicted structure; model building from graphic data.     

Shells theory of solvation and the long-range Born correction

A new solvation model, named shells theory of solvation, is proposed. In this approach, the solvent is divided in two regions, the S₁ shell, close to the solute and describing specific solute–solvent interactions, and the S₂ shell, representing the remain solvent and accounting for the long-range interaction contribution. A simple theoretical equation can be derived which allows the computation of the solvation free energy using two-point thermodynamic integration and configurations generated from molecular dynamics simulation. The discrete/continuum version of this theory provides rigorous theoretical foundations for the popular long-range Born correction and presents a new reliable expression for including this contribution. Further, it converges to the full discrete representation of the solvent when the number of solvent molecules goes to infinity. The method can be easily applied when the solute–solvent interaction (S₁ shell) is treated by full quantum mechanics, while the S₂ shell is described by a dielectric continuum solvation method. A simple test of the theory was done for solvation of fluoride ion in benzene solution. The S₁ shell was composed of the fluoride ion plus 32 benzene molecules, and the interaction with the S₂ shell was calculated at Hartree–Fock level with the MINI basis set and using the polarizable continuum model.     

Ionic force field optimization based on single-ion and ion-pair solvation properties: going beyond standard mixing rules

Using molecular dynamics (MD) simulations in conjunction with the SPC/E water model, we optimize ionic force-field parameters for seven different halide and alkali ions, considering a total of eight ion-pairs. Our strategy is based on simultaneous optimizing single-ion and ion-pair properties, i.e., we first fix ion-water parameters based on single-ion solvation free energies, and in a second step determine the cation-anion interaction parameters (traditionally given by mixing or combination rules) based on the Kirkwood-Buff theory without modification of the ion-water interaction parameters. In doing so, we have introduced scaling factors for the cation-anion Lennard-Jones (LJ) interaction that quantify deviations from the standard mixing rules. For the rather size-symmetric salt solutions involving bromide and chloride ions, the standard mixing rules work fine. On the other hand, for the iodide and fluoride solutions, corresponding to the largest and smallest anion considered in this work, a rescaling of the mixing rules was necessary. For iodide, the experimental activities suggest more tightly bound ion pairing than given by the standard mixing rules, which is achieved in simulations by reducing the scaling factor of the cation-anion LJ energy. For fluoride, the situation is different and the simulations show too large attraction between fluoride and cations when compared with experimental data. For NaF, the situation can be rectified by increasing the cation-anion LJ energy. For KF, it proves necessary to increase the effective cation-anion Lennard-Jones diameter. The optimization strategy outlined in this work can be easily adapted to different kinds of ions.     

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.     

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.     

Effective Born radii in the generalized Born approximation: the importance of being perfect

Generalized Born (GB) models provide, for many applications, an accurate and computationally facile estimate of the electrostatic contribution to aqueous solvation. The GB models involve two main types of approximations relative to the Poisson equation (PE) theory on which they are based. First, the self-energy contributions of individual atoms are estimated and expressed as "effective Born radii." Next, the atom-pair contributions are estimated by an analytical function f(GB) that depends upon the effective Born radii and interatomic distance of the atom pairs. Here, the relative impacts of these approximations are investigated by calculating "perfect" effective Born radii from PE theory, and enquiring as to how well the atom-pairwise energy terms from a GB model using these perfect radii in the standard f(GB) function duplicate the equivalent terms from PE theory. In tests on several biological macromolecules, the use of these perfect radii greatly increases the accuracy of the atom-pair terms; that is, the standard form of f(GB) performs quite well. The remaining small error has a systematic and a random component. The latter cannot be removed without significantly increasing the complexity of the GB model, but an alternative choice of f(GB) can reduce the systematic part. A molecular dynamics simulation using a perfect-radii GB model compares favorably with simulations using conventional GB, even though the radii remain fixed in the former. These results quantify, for the GB field, the importance of getting the effective Born radii right; indeed, with perfect radii, the GB model gives a very good approximation to the underlying PE theory for a variety of biomacromolecular types and conformations.     

Evaluation of solvation free energies for small molecules with the AMOEBA polarizable force field

The effects of electronic polarization in biomolecular interactions will differ depending on the local dielectric constant of the environment, such as in solvent, DNA, proteins, and membranes. Here the performance of the AMOEBA polarizable force field is evaluated under nonaqueous conditions by calculating the solvation free energies of small molecules in four common organic solvents. Results are compared with experimental data and equivalent simulations performed with the GAFF pairwise‐additive force field. Although AMOEBA results give mean errors close to “chemical accuracy,” GAFF performs surprisingly well, with statistically significantly more accurate results than AMOEBA in some solvents. However, for both models, free energies calculated in chloroform show worst agreement to experiment and individual solutes are consistently poor performers, suggesting non‐potential‐specific errors also contribute to inaccuracy. Scope for the improvement of both potentials remains limited by the lack of high quality experimental data across multiple solvents, particularly those of high dielectric constant. © 2016 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Small imines and oximes as model compounds in the optimization of a consistent force field potential energy function

Parameters have been optimized for a potential energy function to be used in molecular mechanics calculations of small imines as a preliminary step to calculations on larger systems. A consistent force field (CFF) program was used, and a new damping algorithm due to Sundius was introduced in the optimization procedure. Optimization of parameters has been done on structural and vibrational data of five small imines and one oxime. The quality of the derived potential energy function is examined by calculations on larger oximes.     

On a semiclassical interpretation of inter- and intramolecular interactions

The semiclassical models considered here are composed by charge distributions coming from ab initio quantum-mechanical calculations on actual molecular systems. These charge distributions interact with one another according to the laws of classical electrostatics. This article describes some results of a systematic examination of the performances of this model in a variety of cases, with the aim of putting in evidence the usefulness and the limits of this inherently approximate representation of chemical interactions. Intermolecular interactions are examined first; the test cases are interactions of neutral molecules with H⁺, Li⁺, and C1^?, and the formation of H-bonded complexes. Attention is paid mainly to the energetics of the processes; each interacting molecule is considered as a unique entity and classical molecular reactivity indexes (electrostatic potential V, polarization term P) are introduced to compute the interaction energy, to interpret the details of the interaction process, and then to elaborate on less expensive computational procedures. Intramolecular interactions are considered. Attention is paid to the question of defining chemical groups starting from SCF molecular wavefunctions. The transferability and conservation degree of groups derived from localized orbitals of actual molecules is examined in detail, taking as tests their ability to reproduce charge distribution, one-electron observables, and energy. The effect of classical fields on these groups is then examined, taking into consideration external fields originated either by a point charge or by a solvent, and internal fields deriving from substitution of chemical groups. The intergroup analysis is then extended to the case of bimolecular reaction acts by considering the whole system as a supermolecule. Approximate computational procedures able to reproduce the main features of these interactions are proposed and tested. All through the article the performances of the classical models are compared with ab initio SCF calculations (mainly of low or intermediate quality).