Ionic Solution: What Goes Right and Wrong with Continuum Solvation Modeling

Abstract

Solvent-mediated electrostatic interactions were well recognized to be important in the structure and function of molecular systems. Ionic interaction is an important component in electrostatic interactions, especially in highly charged molecules, such as nucleic acids. Here we focus on the quality of the widely used PBSA continuum models in modeling ionic interactions by comparing with both explicit solvent simulations and experiment. In this work, the molality-dependent chemical potentials for sodium chloride (NaCl) electrolyte were first simulated in the SPC/E explicit solvent. Our high-quality simulation agrees well with both previous study and experiment. Given the free energy simulations in SPC/E as the benchmark, we used the same sets of snapshots collected in the SPC/E solvent model for PBSA free energy calculations in the hope to achieve the maximum consistency between the two solvent models. Our comparative analysis shows that the molality-dependent chemical potentials of NaCl were reproduced well with both linear PB and nonlinear PB methods, though nonlinear PB agrees better with SPC/E and experiment. Our free energy simulations also show that the presence of salt increases the hydrophobic effect in a nonlinear fashion, in qualitative agreement with previous theoretical studies of Onsager and Samaras. However, the lack of molality-dependency in the non-electrostatics continuum models dramatically reduces the overall quality of PBSA methods in modeling salt-dependent energetics. These analyses points to further improvements needed for more robust modeling of solvent-mediate interactions by the continuum solvation frameworks.

Graphical Abstract

1 Introduction

A common strategy to model water molecules in molecular simulations is to treat them implicitly, offering an opportunity to reach higher computational efficiency and to model more complex systems, particularly macromolecules in biochemistry. After years of research and development, solvation models based on the Poisson-Boltzmann equation (PB) have been widely applied in implicit/continuum modeling of solvation interactions.^1–21 In this implicit/continuum solvent strategy, the solute molecule is described as a low dielectric constant region and the solvent is described as a high dielectric constant region. Fixed interior point charges are located atomic centers in solute. The charges on mobile ions or electrolyte are modeled, in a mean field manner, as a potential-dependent continuum charge density obeying the Boltzmann distribution.^22–32 The continuum representation of the solution system poses as a well-defined classical electrostatic problem that can be solved via the Poisson equation.^22–32 The nonpolar solvation interactions are empirically estimated by the surface area (SA) method. This classical surface area-dependent method is widely adopted for practical biomolecular applications to estimate the total nonpolar solvation contribution in the case of macromolecules, although its deficiencies were also reported due to overlooking solvation contributions of interior (buried) atoms.^{24, 33–52} Newer models were proposed to overcome these deficiencies, though they are yet to be widely received by the community.

The quality of solvent-mediated electrostatics by the PBSA methods was well recognized to be important issue in the past and multiple efforts have been devoted to examining this issue.^{33, 53–58} However, studies of solvent-mediated electrostatics cannot ignore ionic interactions. There are studies on the limitations of the mean field representation of ions in PBSA methods and remedies were proposed to improve the modeling quality.^59–71

Outside the field of continuum solvent modeling, the proper modeling of ionic interactions has also been well received to be an important topic.^72–80 Indeed molecular modeling can provide molecular level insights to explain many observed behaviors in experiment.^72–80 Given its importance, researchers continuously focus on developing more accurate molecular models and exploring more powerful tools in simulations of ionic solutions.^81–88 The electrolyte of sodium chloride (NaCl), one of the most common salts, has been a target in many such simulation efforts.^89–92 However some basic properties of the NaCl electrolyte, such as the molality-dependent chemical potential and solubility, are hard to simulate accurately. One of the crucial reasons is the difficulty to choose a proper water and ion force field set. Most widely used explicit water models are three-site models such as SPC/E⁹³ and TIP3P⁹⁴ due to their simplicity and wide availability in most simulation programs, though recently optimized water models, such as TIP4Pew and OPC^95–96 start to gain broad acceptance. Ion force fields are developed in the context of a water model, apparently due to the requirement to balance ion-ion and ion-water interactions accurately. Joung-Cheatham (JC) ion force field is widely accepted to model mono-valent ions, which is based on a simple additive Amber framework.^{92, 97} In the JC force field, the van der Waals radii and well depths for alkali and halide ions were all optimized to reproduce a series of experimental properties.^{92, 97} These water and ion models are additive in nature and do not model the polarizable effects, which have been found to be important for modeling certain ions, such as divalent ions, in the development of polarizable force fields like AMOEBA.^{84, 88, 98–101} Another difficulty in simulating ions is that the time required both to relax the system well and to obtain converged free energies, due to the slow time scale and the long-range nature of ionic interactions.

In this work, we first used the thermodynamic integration method to simulate the molality-dependent chemical potential of NaCl solution in the explicit water model. The high quality free energy data and trajectories were then used as a benchmark to assess how well both linear and nonlinear PBSA models in the Amber package agree with both the explicit solvent simulation and experiment. We pointed out what goes right and wrong in the current PBSA framework, in the hope to facilitate future developments of more robust continuum solvent models.

2 Method

2.1 Thermodynamic integration method

By definition, the chemical potential of the NaCl electrolyte, μ_NaCl, is the change of Gibbs free energy G by adding one mol of NaCl to the solution. μ_NaCl is a thermodynamic state function of system temperature T, pressure P, the mole number of Na⁺ ions N_Na+, the mole number of Cl⁻ ions N_Cl−, and the mole number of water molecules N_w. In molecular simulations, μ_NaCl can be computed starting from the general definition as⁸⁹

$$\begin{gathered} {\mu }_{\mathit{NaCl}}(T,P,N_{{Na}^{+}},N_{{Cl}^{-}},N_w)=G(T,P,N_{{Na}^{+}},N_{{Cl}^{-}},N_w,{\lambda }_{\mathit{ele}}=1,{\lambda }_{\mathit{vdw}}=1) \\ -G(T,P,N_{{Na}^{+}}-1,N_{{Cl}^{-}}-1,N_w,{\lambda }_{\mathit{ele}}=1,{\lambda }_{\mathit{vdw}}=1). \end{gathered}$$ (1)

Here all numbers represent the numbers of ions/molecules in a simulation box. To simulate the free energy change in eqn (1), two scaling parameters λ_ele and λ_vdw are introduced to control the interactions between a pair of tagged ions to be removed and rest of the solution system. The values of 1 meaning that the tagged ions are in full interaction potential with rest of the system with full charges and full van der Waals well-depth parameters; and the values of 0 meaning the tagged ions are completely decoupled from rest of the system with zero ion charges and zero van der Waals well-depth.

Given the above preparation, the Gibbs free energy G of the solution, in the grand canonical ensemble, can be expressed as an isothermal-isobaric (NPT) partition function Ξ⁸⁹

$$\beta G(T,P,N_{{Na}^{+}},N_{{Cl}^{-}},N_w,{\lambda }_{\mathit{ele}},{\lambda }_{\mathit{vdw}})=-ln\mathrm{\Xi }(T,P,N_{{Na}^{+}},N_{{Cl}^{-}},N_w,{\lambda }_{\mathit{ele}},{\lambda }_{\mathit{vdw}}).$$ (2)

Here the partition function is

$$\mathrm{\Xi }(T,P,N_{{Na}^{+}},N_{{Cl}^{-}},N_w,{\lambda }_{\mathit{ele}},{\lambda }_{\mathit{vdw}})=\frac{z_{{Na}^{+}}^{N_{{Na}^{+}}}{z}_{{Cl}^{-}}^{N_{{Cl}^{-}}}{z}_w^{N_w}}{N_{{Na}^{+}}!N_{{Cl}^{-}}!N_w!{\mathrm{\Lambda }}^{3(N_{{Na}^{+}}+N_{{Cl}^{-}}+N_w)}}\times \int {\mathit{dVe}}^{-\beta PV}\int {dr}^{N_{{Na}^{+}}}{dr}^{N_{{Cl}^{-}}}{dr}^{N_w}{e}^{-\beta U-\beta X({\lambda }_{\mathit{vdw}})-\beta Y({\lambda }_{\mathit{ele}})}.$$ (3)

In eqn (3), z_Na⁺, z_Cl⁻ and z_w are configurational partition functions of Na⁺, Cl⁻, and water, respectively. Λ is the thermal de Broglie wavelength. U is the potential energy of N_Na⁺ − 1 sodium ions, N_Na⁺ − 1 chloride ions and N_w water molecules. X(λ_vdw) is the van der Waals potential energy of the tagged NaCl ion pair with the scaling parameter λ_vdw, and Y(λ_ele) is the Coulomb potential energy of tagged NaCl ion pair with the scaling parameter λ_ele.

Substituting relationships

$$\frac{z_k}{{\mathrm{\Lambda }}_k^3}=\beta P^{0}exp(-\beta {\mu }_k^0)(k={Na}^{+},{Cl}^{-}\mathit{and}w),$$ (4)

into eqn (1)–(3), we obtain⁸⁹

$$\beta {\mu }_{\mathit{NaCl}}=\beta {\mu }_{\mathit{NaCl}}^{ig}+\beta {\mu }_{\mathit{NaCl}}^{ex},$$ (5)

$$\beta {\mu }_{\mathit{NaCl}}^{ig}=\beta {\mu }_{{Na}^{+}}^0+\beta {\mu }_{{Cl}^{-}}^0+2ln\frac{N_{\mathit{NaCl}}}{\beta P^0\langle V\rangle },$$ (6)

$$\beta {\mu }_{\mathit{NaCl}}^{ex}=-ln\frac{\int {\mathit{dVe}}^{-\beta PV}\int {dr}^{N_{{Na}^{+}}}{dr}^{N_{{Cl}^{-}}}{dr}^{N_w}{e}^{-\beta U-\beta X({\lambda }_{\mathit{vdw}}=1)-\beta Y({\lambda }_{\mathit{ele}}=1)}}{\int {\mathit{dVe}}^{-\beta PV}\int {dr}^{N_{{Na}^{+}}}{dr}^{N_{{Cl}^{-}}}{dr}^{N_w}{e}^{-\beta U}},$$ (7)

where ${\mu }_{\mathit{NaCl}}^{ig}$ is the chemical potential of adding the ideal gas atoms to the solution, ${\mu }_{\mathit{NaCl}}^{ex}$ is the excess chemical potential from the Coulomb and van der Waals interactions from adding the tagged NaCl pair. P⁰ = 1 bar is the standard state pressure, and 〈V〉 is the ensemble average volume of the solution system without the tagged ion pair (N_Na⁺ − 1, N_Cl⁻ − 1, N_w) and N_w ). Here N_NaCl is equal to N_Na⁺ or N_Cl⁻. Finally ${\mu }_{{Na}^{+}}^0$ and ${\mu }_{{Cl}^{-}}^0$ are standard molar chemical potentials, and can be retrieved from NIST-JANAF thermochemical tables, ${\mu }_{{Na}^{+}}^0=574.317\text{kJ}/\text{mol}$ and ${\mu }_{{Cl}^{-}}^0=-240.167\text{kJ}/\text{mol}$.¹⁰²

The excess chemical potential ${\mu }_{\mathit{NaCl}}^{ex}$ is obtained by the Bennett acceptance ratio (BAR) method. We used 10 windows for the electrostatic part with λ_ele chosen as 0, 0.1, 0.2, …, 1.0, to make the charges of the NaCl pair disappear. The charges are changed as q_Na⁺,λele = λ_ele · q_Na⁺,_{λele = 1} and q_Cl⁻,λele = λ_ele · q_Cl⁻,_{λele = 1}. Because the van der Waals part is harder to converge, we used 15 windows with λ_vdW chosen as 0, 0.05, 0.1, 0.15, …, 0.45, 0.5, 0.6, …, 1.0. The K-6 perturbation scheme was used, i.e. the well depth is changed as ${\epsilon }_{ij}={\lambda }_{\mathit{vdW}}^6{\epsilon }_{ij}$. Following these strategies, summation of 10 electrostatic free energy intervals and 15 van der Waals free energy intervals together leads to ${\mu }_{\mathit{NaCl}}^{ex}$ as

$${\mu }_{\mathit{NaCl}}^{ex}={\mathrm{\sum }}_{k=1}^{10}{\mu }_{\mathit{NaCl},k}^{\mathit{ele}}+{\mathrm{\sum }}_{l=1}^{15}{\mu }_{\mathit{NaCl},l}^{\mathit{vdW}}$$ (8)

2.2 Poisson-Boltzmann Surface Area method

The Poisson-Boltzmann equation (PBE) is based on the fundamental Poisson equation. To model the electrostatic interaction in a salt water solution, electrostatic potential ϕ(r→) can be described by the PBE as³²

$$\nabla ·\epsilon (\overrightarrow{r})\nabla \varphi (\overrightarrow{r})+\lambda (\overrightarrow{r})f(\varphi (\overrightarrow{r}))=0,$$ (9)

with

$$f\left(\varphi (\overrightarrow{r})\right)=4\pi {\mathrm{\sum }}_i^N{z}_{i}e{c}_{i}exp\left(-\frac{z_{i}e\varphi (\overrightarrow{r})}{k_{B}T}\right)+4\pi {\rho }^f(\overrightarrow{r}).$$ (10)

Here ε(r→) is a predefined dielectric distribution function for a solvation system, λ(r→) is the ion-exclusion function with values of 0 within the Stern layer and the molecular interior and 1 outside the Stern layer, z_i is the valence of ion type i, ρ^f(r→) is the fixed charge density, and c_i is the bulk concentration of ion type i.

In cases where both the ionic strength and solvent potential are low, and when symmetric electrolytes are considered, the PBE can be linearized to

$$\nabla ·\epsilon (\overrightarrow{r})\nabla \varphi (\overrightarrow{r})=-4\pi {\rho }^f(\overrightarrow{r})+\lambda (\overrightarrow{r}){\epsilon }_{\mathit{out}}{\kappa }^2\varphi (\overrightarrow{r}),$$ (11)

where ${\kappa }^2=\frac{8\pi e^{2}I}{{\epsilon }_{\mathit{out}}{k}_{B}T}$. Here ε_out denotes the dielectric constant of outside solvent, I represents the ionic strength of the solution, and is defined as I = z²c.

The non-polar part, ΔG_{solv,non–polar}, is typically estimated by the surface area (SA) method. Recent Amber molecular dynamic packages have implemented two options for nonpolar solvation modeling, the original and the revised approaches. The revised approach uses separate terms to model cavity and van der Waals dispersion contributions to overcome reported limitations of the original approach.^{24, 33–52} In our previous study, protein-ligand binding affinities computed with the revised method were found to agree better with experiment than the original method in all six tested groups of receptors.²⁴

The original approach uses the solvent accessible surface area (SAS) to correlate the total nonpolar solvation free energy as

$$\mathrm{\Delta }{G}_{\mathit{solv},\mathit{non}-\mathit{polar}}=\mathit{COEF}\ast \mathit{SAS}+\mathit{OFFSET}.$$ (12)

The revised approach employs the solvent accessible surface area/or its enclosed volume (SAV) to correlate the repulsive (hydrophobic/cavity) term, and utilizes a surface-integration approach to calculate the van der Waals (dispersion) term as³³

$$\mathrm{\Delta }{G}_{\mathit{solv},\mathit{non}-\mathit{polar}}=\mathrm{\Delta }{G}_{\mathit{dispersion}}+\mathrm{\Delta }{G}_{\mathit{cavity}}=\mathrm{\Delta }{G}_{\mathit{dispersion}}+\mathit{COEF}\ast \mathit{SAV}+\mathit{OFFSET}$$ (13)

2.3 Simulation details

We designed 10 different NaCl concentrations in this study, by solvating 1, 2, 3, 6, 10, 20, 30, 40, 50 and 60 pairs NaCl into cubic boxes respectively with 540 SPC/E⁹³ water molecules in each box, corresponding to NaCl molalities of 0.10, 0.21, 0.31, 0.62, 1.03, 2.06, 3.08, 4.10, 5.14, 6.17 mol per 1 kilogram of water. The Na+ and Cl− ions parameters are for Ewald and SPC/E water from Joung & Cheatham.⁹⁷ For explicit MD simulations, 100ns NPT equilibrium runs were first conducted with 1 bar standard state pressure and 298.15K room temperature for all of 10 concentrations, employing the pmemd-cuda module in the Amber16 package.¹⁰³ The simulation speed is around 650ns per day. After the long equilibration simulation, the simulation box volumes and densities were found to all converge well.

Next, free energy was simulated by decoupling one pair of NaCl in each simulation system. For each MD simulation at a given λ_ele we started with a 10ns NVT trajectory, saving 2000 snapshot (one snapshot per 5ps). The length of each MD simulation was doubled until the free energy change was less than 0.5 kJ/mol. Here we used the sequential approach in all free energy simulations, i.e. the final restart file of the λ_ele =1.0 run is the input restart file of the λ_ele =0.9 run and so on. Thus, the charges of tagged ions are decrease step by step with λ_ele changed from 1.0 to 0.0, sequentially with a total of 110ns MD simulated for 10ns windows, and 220ns for 20ns windows and so on. Here we chose the sequential approach instead of the parallel approach, i.e. all λ_ele MDs start from the same final equilibrated restart because the free energies were found to converge faster in our previous analysis. Next the vdW terms were decoupled in a similar strategy, in 15 windows from 1.0 to 0.0. The MD runs were also doubled from 10ns to 20ns and so on until the free energy changes were less than 0.5 kJ/mol. To further improve the throughput of the free energy simulations, the NVT ensemble was utilized instead of NPT because pmemd-cuda runs faster in NVT.

To calculate the free energy change for each window, the BAR method was applied for each of electrostatics and vdW parts.^104–109 For each window, only the second half of the MD trajectories were used in the BAR calculation, with the first part treated as equilibration at the current lambda values. Details can be found in our previous publication.¹¹⁰

PBSA calculation was conducted with the sander module in the Amber16 package,¹⁰³ and with the same snapshots (the tagged ions only as it is an implicit solvent model) as above BAR calculations in the explicit solvent. In PBSA calculations, the grid space was 0.5 Å, inner side dielectric constant was 1 and outer side dielectric constant was 72.2.¹¹¹ Nonperiodic boundary condition was used and the box size was set to be twice as large as the solute dimension. The incomplete Cholesky conjugate gradient was chosen as the numerical PBE solver.^{32, 112–113} Iteration convergence criterion was set as 10⁻³. Atomic radii were initially set as the default mbondi radii as in the Amber package.¹⁰³ The solvent probe radius for the molecular surface is defined as the default 1.4 Å and the mobile ion probe radius for the ion accessible surface is the default value of 2.0 Å. The atom-based cutoff distance for van der Waals interactions is 15 Å and the cutoff distance of adding short-range pairwise charge-based interactions is 7 Å, while the long-range interactions were directly from the PBE numerical solution.¹¹⁴ Ionic strength was set according to the molality of each input trajectory. The linear Poisson-Boltzmann (LPB) and Non-linear Poisson-Boltzmann (NPB) methods were both used to calculate ${\mu }_{\mathit{NaCl}}^{\mathit{ele}}$, respectively; and original and revised nonpolar solvent model options were employed for modeling ${\mu }_{\mathit{NaCl}}^{\mathit{vdW}}$, respectively. After obtaining the solvation free energy from PBSA calculation for each single snapshot, the BAR method was also utilized to calculate the final ${\mu }_{\mathit{NaCl}}^{\mathit{ele}}$ and ${\mu }_{\mathit{NaCl}}^{\mathit{vdW}}$ as in the explicit solvent model.

3 Result and discussion

3.1 Benchmark data collection

Density equilibration

NaCl solutions of all ten different molalities (mol of NaCl/kg water) were first equilibrated for 100ns in the isothermal-isobaric NPT ensemble (298.15K and 1 bar). Detailed volume data from all simulations are listed in Table S-1 of Supplementary Materials. Overall the simulation model with the JC force field^{92, 97} of NaCl and the SPC/E water model⁹³ agrees well with experiment as shown in Figure 1, with the molality-dependent density trend reproduced very well, consistent with a previous study in the Monte Carlo simulation setup.⁸⁹ Nevertheless, it is also clear that our simulation mode slightly overestimates the electrolyte solution density at all 10 settings. The largest deviation from experiment appears to be at the highest molality of 6.17mol/kg: the simulation yields a density of 1.206 g/cm³, and the experimental density is 1.197 g/cm³, with a relative difference of less than 1%.

Figure 1.

Simulated density of NaCl solution in SPC/E versus molality at T=298.15K and P=1 bar after 100ns NPT equilibration simulation. Here SPC/E water model and JC force field for Na⁺ and Cl⁻ were used. Experimental data are form Rogers and Pitzer¹¹⁵ and are fitted to a second order polynomial.

Convergence analysis of excess chemical potentials

After NPT equilibration, NVT simulations with different λ_ele and λ_vdW values were conducted sequentially, first by decharging step by step, then by decoupling van der Waals interactions step by step between ions and water as mentioned in the Method section. Due to the challenge of reaching convergence in the free energy simulations, we doubled the simulation length for each lambda from 10ns to 20ns and so on until the free energy change is less than 0.5 kJ/mol. Table 1 collects all simulated excess chemical potentials, for both the electrostatic and van der Waals components. With our extensive equilibration and sequential simulation procedure, the van der Waals component appears to converge relatively easier: we only see one case where 40ns simulation is necessary to observe free energy change less than 0.5 kJ/mol. On the other hand, the electrostatic component is more difficult to converge, particularly for solutions of lower molality. This is reasonable because the long-range electrostatic interactions are less screened by unperturbed ions. It is worth pointing out that the trajectories utilized here are already much longer than most previous free energy simulations. However, it is possible that these simulated free energies are still not “truly” converged.

Table 1.

Convergence check of simulated excess chemical potential (both electrostatic, ele, and van der Waals, vdW, components). Molecular dynamics simulation of 10ns, 20ns and 40ns for each lambda was conducted to study the convergence of excess chemical potentials. All free energies are in the unit of kJ/mol. The molality is in the unit of mol/kg.

Molality	10ns		20ns		40ns
	ele	vdW	ele	vdW	ele	vdW
0.10	−773.13	30.37	−773.66	30.28	−774.08	NA
0.21	−773.35	29.95	−773.61	30.62	−773.74	30.28
0.31	−774.22	31.66	−774.60	31.23	−774.69	NA
0.62	−775.44	32.64	−776.35	32.31	−776.04	NA
1.03	−775.72	33.41	−776.00	33.45	NA	NA
2.06	−777.09	34.64	−776.12	34.38	−776.17	NA
3.08	−777.35	37.91	−777.53	35.50	NA	NA
4.10	−776.23	36.83	−776.95	36.53	−777.16	NA
5.14	−777.79	38.81	−777.59	38.48	NA	NA
6.17	−777.15	39.58	−777.30	39.47	NA	NA

Molality-dependent chemical potential

Once excess chemical potentials are ready, the total chemical potential (μ_NaCl) can be readily computed according to Eqn (5) and (6) by summing up both electrostatic and van der Waals excess chemical potentials, standard state chemical potentials, i.e. ${\mu }_{{Na}^{+}}^0=574.317\text{kJ}/\text{mol},{\mu }_{{Cl}^{-}}^0=-240.167\text{kJ}/\text{mol}$,¹⁰² and the volume correction term. Table S-2 in the Supplementary Material lists these detailed data used to ensemble the total chemical potentials that can be compared with experiment.¹¹⁶ Previous simulation results from Mester and Panagiotopoulos⁸⁹ are also presented in Table S-2 as reference. The final molality-dependent chemical potentials are shown in Figure 2. This shows that our current simulation data agrees well with those of Mester and Panagiotopoulos,⁸⁹ even though a different simulation protocol was used. It is apparent that our computer model tends to give chemical potentials higher than experimental values, and the differences are larger at higher molality. At the highest molality of 6.17 mol/kg, the difference is around 4.7 kJ/mol. This indicates that our computer model tends to overestimate the unfavorable interactions among ions when the concentration is increased. It is possible that the systematic overestimation is attributed by the tested water model and/or the lack of polarization effect. The later was known to lead to overestimation of electrostatic interactions. Thus, an interesting direction to pursue in the future is to study the system with more recently optimized water models^95–96 or with a reasonable polarizable computer model for both water and ions of interest.

Figure 2.

Chemical potential μ_NaCl versus molality at T=298.15K and P=1 bar computed in this work (SPC/E, solid circles) and in the literature⁸⁹ (Mester and Panagiotopoulos, open triangles) versus experiment (solid line).¹¹⁶

3.2 PBSA simulations

Linear PB modeling of electrostatic interactions

An important issue that must be addressed in PBSA modeling is the ionic radii to be used in these calculations because atomic sizes crucially influence solvation free energies. The default radii of Cl⁻ and Na⁺ are 1.7 Å and 1.5 Å, respectively in Amber, that were revised from the Bondi radius set, but may not reproduce the ionic solvation free energy in the SPC/E water. Thus, the first step in our PBSA calculation is to use one data point in the molality-dependent curve to calibrate the PBSA model. Here the lowest molality system, i.e. one pair of NaCl, was used in the calibration.

The calibration was conducted with the linear PB (LPB) model as the salt concentration was very low and the salt contribution to the overall solvation free energy is also very small. Figure 3 compares the three sets of BAR calculations in the calibration, showing LPB/BAR calculations with the default Amber radii and the LPB/BAR calculations with scaled Amber radii along with the SPCE/BAR calculation. The radius scaling factor was found to be 1.12, i.e. 12% larger than the default values to achieve the minimal root mean squared deviations (RMSD) from the SPCE/BAR data. Here all ten lambda-dependent ΔG’s were used in the RMSD calculation. Indeed, the LPB μ_ele value was −868.63 kJ/mol before optimization, which is apparently larger than μ_ele value of −774.28 kJ/mol in SPC/E. The LPB μ_ele value was −774.08 kJ/mol after optimization. It is worth pointing out that the SPC/E charging free energy is not linear. Here we used a second-order polynomial to fit its charging curve versus λ_ele. In contrast, the LPB charging free energy is apparently linear, consistent with the way the charging free energy is computed in LPB.

Figure 3.

Optimization of the ionic radii via detailed analysis of electrostatic excess chemical potential μ_BAR versus λ_ele for the lowest molality at 0.10 mol/kg. The benchmark data from the SPC/E BAR calculation are shown as red dots. LPB before and LPB after the optimization are shown as blue squares and open triangles, respectively. The ion-solvent surface is defined as classical solvent excluded surface using given ionic radii starting from the default Amber radii (before) or using those scaled to minimize the RMSD with respect to the SPC/E BAR values (after).

Nonlinear PB modeling of electrostatic interactions

For nonlinear PB (NPB) calculations, another key factor to watch is the size of ion probe, which is needed to define how close the continuum ion can approach a molecular solute, i.e. how far away the Stern layer would be placed. The default ion probe size is 2.0 Å in Amber, which may also need further calibration.

Here to highlight the nonlinear effect, we used the salt solution of the highest molality of 6.17 mol/kg to calibrate the ion probe. Our calibration shows that the default ion probe size of 2.0 Å is already good enough and it best reproduces the SPCE/BAR data. Figure 4 shows the agreement of NPB/BAR data with SPCE/BAR data for the tested salt solution. Here LPB/BAR data is also enclosed as reference. It is interesting to note that the differences of NPB and LPB free energies per BAR window are very small for human eyes to catch. However, the cumulative μ_ele data by NPB and LPB are noticeably different, as given in Table 2, which list these data for all models and all molality’s tested.

Figure 4.

Optimization of Stern layer definition in NPB via detailed analysis of electrostatic excess chemical potential μ_BAR versus λ_ele for the highest molality at 6.17 mol/kg. The benchmark data from the SPC/E BAR calculation are shown as red dots. LPB and NPB BAR calculations using the same snapshots from the SPC/E BAR calculation are shown as open triangles and blue squares, respectively. The Stern layer is defined with the default two Angstrom ion exclusion radius.

Table 2.

Molality dependence of excess chemical potential components. Electrostatic part μ_ele by LPB, NPB, and SPC/E models, van der Waals part μ_vdW (SPC/E only), and total excess chemical potentials μ^ex.

molality	μele			μvdW	μex
	SPC/E	LPB	NPB		SPC/E	LPB	NPB
0.10	−774.08	−774.06	−774.06	30.28	−743.80	−743.78	−743.78
0.21	−773.74	−774.07	−774.85	30.28	−743.46	−743.79	−744.57
0.31	−774.69	−774.07	−775.18	31.23	−743.46	−742.84	−743.95
0.62	−776.04	−774.06	−775.37	32.31	−743.73	−741.75	−743.06
1.03	−776.00	−774.05	−775.64	33.45	−742.55	−740.60	−742.19
2.06	−776.17	−774.06	−775.83	34.38	−741.79	−739.68	−741.45
3.08	−777.53	−774.05	−776.13	35.50	−742.03	−738.55	−740.63
4.11	−777.16	−774.06	−776.36	36.53	−740.63	−737.53	−739.83
5.14	−777.59	−774.05	−776.50	38.48	−739.11	−735.57	−738.02
6.17	−777.30	−774.05	−776.57	39.47	−737.83	−734.58	−737.10

LPB and NPB modeling of molality-dependent chemical potential

We are now ready to assess the quality of LPB and NPB modeling of the NaCl salt water to see how well our models reproduce its molality-dependent chemical potentials. Table 2 lists all final excess electrostatic chemical potentials modeled by LPB and NPB along with SPC/E. Here we have kept the van der Waals excess chemical potentials from SPC/E as we want to focus on the quality in modeling electrostatic interactions.

Table 2 shows that μ_ele by LPB does not change much with increasing molality. This is different from both NPB and SPC/E μ_ele values. When combined with van der Waals excess chemical potential from SPC/E, the molality-dependent chemical potential is better reproduced by NPB as clearly demonstrated in Figure 5. Here both SPC/E and NPB trend lines trace each other very closely. The comparison demonstrates that the NPB method leads to more accurate result due to its consideration of the non-linear effect.

Figure 5.

Chemical potential μ_NaCl calculated with LPB, NPB, and SPC/E models, respectively, versus experimental data for molality in the range of of 0.10 to 6.17 mol/kg. Trend lines are second-order polynomials best-fitted to the data. See Table 2 for detailed data breakdown.

Modeling of non-electrostatics/van der Waals solvation free energies by PBSA

An interesting finding of the detailed breakdown of molality dependent chemical potentials in the NaCl solution was the pronounced role that the van der Waals component plays in the overall trend of excess chemical potential over molality. Table 1 shows that with increasing molality, the electrostatic excess chemical potentials become more negative and the van der Waals excess chemical potentials become more positive. Although the absolute values of electrostatic components are dramatically larger than those of the van der Waals components, the changes of the two components from the lowest to the highest molality are similar: −3.02 kJ/mol and +9.19 kJ/mol, respectively. This shows accurate modeling of both components are important to reproduce the overall molality-dependent trend in the chemical potential.

Now the question is how well the nonpolar component in PBSA performs. Figure 6 shows the overall trends of two widely used models in modeling nonpolar solvation free energies with the SPCE/E data as benchmark. As described in the Methods, the original method uses a single surface dependent term to correlate the nonpolar solvation free energies, and the revised approach uses two separate terms to correlate van der Waals/dispersion and cavity/hydrophobic terms, respectively, to better model the nonpolar solvation interactions. In both cases, there is no concentration-dependence built in, as such efforts are yet to be seen for molecular applications in the literature. This deficiency would dramatically reduce the quality of PBSA in modeling salt-dependent energetics. Nevertheless, the improvement of the revised model over the original model is also apparent. The smaller deviation from the SPCE/E data can be in part attributed to the different solvent models and different free energy simulations used in the calibration of the revised model.³³

Figure 6.

Excess chemical potential – the van der Waals part (μ_vdW) calculated with original (INP=1) and revised (INP=2) models versus SPC/E for molality in the range of 0.10 to 6.17 mol/kg. The solid trend line is the fitted molality-dependent model of eqn (15).

Some literature study shows that Onsager and Samaras thought about this issue as early as 1934 in their theoretical studies of electrolytes. They showed that the electrolyte surface tension (σ) would follow a limiting law at temperature T as¹¹⁷

$$\sigma ={\sigma }_0+\left(\frac{79.517}{{\epsilon }_{\mathit{out}}}\right)m\times ln\left(1.143\times {10}^{-13}\frac{{({\epsilon }_{\mathit{out}}T)}^3}{m}\right).$$ (14)

where σ₀ is the pure water surface tension and m is the salt concentration in moles per liter. Since the dispersion/van der Waals term for the nonpolar solvation free energy does not change much over different concentrations of salt, eqn (14) would in principle explain the trend of simulated data in Figure 6. Substitution of the water dielectric constant gives a scaling constant about −1.0 for the m × ln(m) term, and the scaling constant is close to zero at +0.2 for the linear m term.

On the other hand, the fitting of the simulated nonpolar solvation free energy in Figure 6 shows a trend as

$${\mu }_{\mathit{vdW}}=29.8+3.0\times \mathit{molality}-0.8\times \mathit{molality}\times ln(\mathit{molality}).$$ (15)

Comparison with the coefficients of eqn (14) at the simulation condition shows that the m × ln(m) term agrees very well, even after considering the different concentration units used in the two equations. However, the linear term agrees poorly: the fitted trend line slope is over ten times larger than that in the limiting law of Onsager and Samaras. This is consistent with previous finding that the Onsager-Samaras limiting law underestimates the hydrophobic effect over increasing salt concentration at finite concentrations.¹¹⁸

Thus, what remains to be done is to revise the limiting law proposed by Onsager and Samaras by considering the finite concentration effect¹¹⁸ and incorporate it into both nonpolar solvent models via refitting nonpolar solvation free energies of model compounds in the context of non-trivial amount of salt so these revised models can be applied to highly charge systems, such as nucleic acids.

3.3 How to conduct PBSA calculations without explicit solvent MD?

Given its reasonable good performance in modeling electrostatic excess chemical potentials in the NaCl solution, it is natural to ask how to use PBSA for typical application settings where no explicit solvent MD is used. A natural setup is to use the minimum energy configuration for the NaCl pair to estimate the system chemical potential. This would correspond to the structure in the first shell (2.80 Å) of the Na⁺-Cl⁻ radial distribution function (Figure 7). However, the radial distribution function apparently shows that the most probably separation between the ion pair is at the second shell (4.95 Å), which is a much higher peak (Figure 7). It is worth pointing out that our current simulations yield radial distribution functions highly consistent with published studies.^{89, 119} This analysis shows that the ions do not strip their solvation shells when approaching each other. We explored to compute electrostatic excess free energies with both configurations to see which setup provides the best agreement with detailed explicit solvent/explicit ion simulations.

Figure 7.

Radial distribution functions g(r) between Na⁺ and Cl⁻ for selected molality of 0.62, 1.03, 2.06, 4.11, and 6.17 mol/kg, from top to bottom, respectively.

The calculations of electrostatic chemical potentials modeled by NPB with both setups are shown in Figure 8 after incorporating all chemical potential components. It is clear that the better strategy is to use the most probable peak in the radial distribution function, i.e. the second shell position for estimating the excess chemical potentials. Use of the first shell position would lead to estimation about 20 kJ/mol too negative. Nevertheless, it is also worth pointing out that even with the most probable configuration, the trend line is still slightly different from that computed with snapshots simulated in SPC/E, with deviations mostly in the low salt concentration cases. Overall, the single snapshot run still produces a trend line a bit too negative than that from full MD trajectories.

Figure 8.

Chemical potential μ_NaCl computed by the NPB model with ion pair separation distances of 4.95 Å and 2.8 Å, respectively, chosen from the two peaks of g(r). The trends are compared with the SPC/E model and experiment for molality in the range of 0.10 to 6.17 mol/kg.

4 Conclusions and future directions

The quality of solvent-mediated electrostatics by the PBSA methods was well recognized to be an important issue. A high-quality model offers an opportunity to use the implicit water to reach higher computational accuracy in modeling more complex systems without much loss of accuracy. In this work, we first conducted extensive free energy simulations of NaCl solutions to compute molality-dependent chemical potentials and found overall agreement with experiment. It is also apparent that our all-atom computer model tends to give chemical potentials higher values than experimental values, and the differences are larger at higher molality. This indicates that the model tends to overestimate the unfavorable interactions among ions when the concentration is increased.

Our analyses of implicit solvents show that the molality-dependent chemical potential is better reproduced by NPB while the molality dependence in LPB values is too weak. The comparison demonstrates that the NPB method leads to more accurate result due to its consideration of non-linear effects. Even with the good agreement between PBSA and explicit solvent/explicit ion models, it should be pointed out the robust PBSA performance may not always be possible when ions of higher valence present in the electrolytes, where the mean field approximation would break down due to the higher concentration of charges on such ions. Nevertheless, improved theoretical models have emerged. For example, Song and co-workers proposed a molecular Poisson-Boltzmann theory based upon a rigorous statistical mechanics foundation that goes beyond the mean-field treatment of electrolytes.^120–123 It will be interesting to explore this and other improvements with the more challenging electrolytes, for example with electrolytes of divalent ions.

In the analysis of non-electrostatics solvation free energies, an interesting finding is that the van der Waals component plays an important role in the overall trend of excess chemical potential over molality. Thus, accurate modeling of both electrostatic and van der Waals components is important to reproduce the overall molality-dependent trend of chemical potential in the tested salt solutions. In contrast, the continuum non-electrostatics solvent models do not model chemical potentials as being dependent on concentration. This deficiency dramatically reduces the quality of PBSA in modeling salt-dependent energetics. Nevertheless, the improvement of the revised model over the original one is also apparent. What remains to be done is to incorporate a revised Onsager and Samaras model into nonpolar solvent models to better capture the molality-dependent trend of nonpolar solvation free energies in the context of non-trivial amount of salt.