Nonpolar Solvation Free Energy from Proximal Distribution Functions

Abstract

Using precomputed near neighbor or proximal distribution functions (pDFs) that approximate solvent density about atoms in a chemically bonded context one can estimate the solvation structures around complex solutes and the corresponding solute–solvent energetics. In this contribution, we extend this technique to calculate the solvation free energies (ΔG) of a variety of solutes. In particular we use pDFs computed for small peptide molecules to estimate ΔG for larger peptide systems. We separately compute the non polar (ΔG_vdW) and electrostatic (ΔG_elec) components of the underlying potential model. Here we show how the former can be estimated by thermodynamic integration using pDF-reconstructed solute–solvent interaction energy. The electrostatic component can be approximated with Linear Response theory as half of the electrostatic solute–solvent interaction energy. We test the method by calculating the solvation free energies of butane, propanol, polyalanine, and polyglycine and by comparing with traditional free energy simulations. Results indicate that the pDF-reconstruction algorithm approximately reproduces ΔG_vdW calculated by benchmark free energy simulations to within ~ kcal/mol accuracy. The use of transferable pDFs for each solute atom allows for a rapid estimation of ΔG for arbitrary molecular systems.

INTRODUCTION

The accurate calculation of solvation free energies is critical to computational drug design and structure optimization as well as understanding important biophysical processes like aggregation and protein folding. Two central issues in computational free energy calculations include ensuring sufficient sampling of phase space and the accuracy of the force field used to model molecular interactions. The sampling issue remains a bottleneck in applications of free energy approaches to ever more complex systems; that is, in order to investigate thermodynamic behavior of a system, sampling of a representative number of configurations within the given ensemble is required, which often requires expensive simulations.^1–3Force fields continue advancing in accuracy often at the expense of added complexity.^4–6

Due to the transferable nature of atomic level force fields for a given chemical context, molecular mechanics force fields provide an extensible architecture to study the thermodynamic properties of a wide range of solutes.⁷ Free energies of solvation (ΔG) depend on the solute–solvent potential,⁸ and the distribution of solvent around a solute. Here, we seek a set of transferable functions to approximate solvent density distributions which will allow for rapid estimation of ΔG and its differences due to chemical processes in solution.

In assessing free energy differences it is common to consider a thermodynamic cycle for a given reaction process. For example, the relative change in free energy associated with the change or mutation of molecule A to B is given by the difference in solvation free energy (ΔΔG) of the two molecules.^9–14 In computational free energy methods, the solvation process in which a solute is transferred from the gas phase to the aqueous phase is often decomposed into two stages or contributions. First an uncharged, solute cavity is created in the solution and the free energy required to insert this cavity is defined as ΔG_vdW. This cavity is then gradually charged and the change in free energy associated with this process is defined as ΔG_elec.¹⁵ This process, though nonphysical, does allow us to calculate each component of the potential function with explicit simulations¹⁶ or individual functional theories with less computational costs.^17,18 While ΔG is a state function and path independent, the components ΔG_elec and ΔG_vdW are well-known to be defined only by a process and therefore path-dependent.¹⁹

Many approximate techniques have been developed to rapidly estimate ΔG. ΔG_elec is often estimated using dielectric continuum theory based techniques,^15,20 such as the generalized Born model,^21,22 Poisson–Boltzmann equation,^23,24 or Linear Response theory (LRT).^25–27 The nonpolar cavity contribution, ΔG_vdW, has been approximated as a linear function of the solvent accessible surface area (SASA).^28–30 However, the correlation between ΔG_vdW calculated by SASA and explicit solvent models using rigorous thermodynamic integration or free energy perturbation is far from exact for organic molecules, especially at the level of small nonpolar molecules.^16,31–33 Further decomposition of ΔG_vdW into the solute–solvent repulsive and attractive interactions has been proposed with an additional estimator accounting for solvent accessible volume.¹⁸ The use of implicit solvation models to estimate ΔG_vdW remains a challenge due to nonadditive, multibody correlations with neighboring atoms,^33–35 and the complex geometry of the solute–solvent interface.^36–39 An approach that yields estimates of solvation free energies rapidly but retains the accuracy associated with explicit solvent models would be highly desirable. Methods relying on precomputed quantities from explicit simulations of small solutes to estimate ΔG_vdW of large solutes are therefore being developed. For example, the Semi-Explicit Assembly (SEA) approach^40–42 precomputes the modified nonbonded van der Waals interactions between a series of nonpolar solute spheres and water, then assembles the total interactions based on the given configurations of large solutes.

Here, using precomputed near neighbor or proximal distribution functions (pDFs) of solvation about atoms in chemically bonded (small molecule) context we estimate the solvation structures around larger and more complex solutes. We then use these approximately transferable distributions to calculate ΔG. pDFs approximate solute–solvent pair correlation functions at the near neighbor level to describe the local solvent structure around a solute.^43–47 They have been shown to be a near universal and transferable descriptor between different globular proteins.⁴⁸ That is, pDFs constructed for small molecules can be used to predict the solvent structures around complex biological macromolecules that are composed of chemically similar components.^49–51 Average solute–solvent interaction energies, and subsequently solvation free energies, can then be calculated from these reconstructed solvent structures.

Electrostatic interaction energies estimated using pDFs in combination with LRT have been shown to approximate ΔG_elec well for a variety of solutes compared to simulation.^52–54 We have recently shown that van der Waals solute–solvent interaction energies can be approximated with good accuracy using precomputed pDFs, but requires a finer spatial resolution than what is used to estimate the more smoothly varying electrostatic interactions.⁵⁵ Here, we extend the pDF-approach to calculate ΔG_vdW of a variety of solutes and assess the accuracy of this approach by a comparison with the more expensive explicit solvent thermodynamic integration.

Below we review the generation of pDFs from small solutes and how they can be used to reconstruct the distribution of solvent density around an arbitrary solute to estimate average solute–solvent interaction energies. Then we show how to calculate ΔG_vdW by constructing a set of pDFs as a function of a coupling parameter that gradually scales the van der Waals interactions between a small solute and solvent (i.e., we define a pathway along which to calculate ΔG_vdW). To benchmark this approach, we calculate ΔG_vdW of butane, and propanol and compare with more rigorous estimates obtained from alternative approaches. We then generate transferable pDFs from monomers of alanine and glycine and use those to calculate ΔG_vdW for various triglycine and deca-alanine systems. We study the effects of flexibility, or disorder, on ΔG_vdW by comparing ΔG_vdW of a fixed, extended conformation of triglycine to that obtained for a large ensemble of triglycine conformers generated from a molecular dynamics (MD) simulation. For completeness, we include linear response estimates of ΔG_elec and thus construct an approximate total solvation free energy.

THEORY

In this section, we briefly review nearest neighbor proximal distribution functions (pDFs) and how they are used to reconstruct solute–solvent density distributions and subsequently estimate solute–solvent energetics. Then we discuss how the pDF formalism can be modified to estimate ΔG_vdW using a thermodynamic integration scheme.

Distribution Functions and Solute–Solvent Energetics

Proximal distribution functions (pDFs, or g_⊥(r)) are average density distributions calculated from solvent molecules to their nearest solute atom.^43,56,57 The solvent probability distribution most nearly perpendicular to the k_th solute atom ${\mathrm{g}}_{\perp }^k(r)$ can be written as

$${\mathrm{g}}_{\perp }^k(r)=\sum _{t=0}^T\sum _{j=1}^n\frac{\delta \left(\text{Inf}{[\mid {\overrightarrow{r}}_{\mathit{ij}}(t)\mid ]}_{i=1,m}-r\right)}{\delta \tau \left({\overrightarrow{r}}_j(t),k\right)}$$ (1)

where ${\overrightarrow{r}}_{\text{ij}}(t)$ represents the position vector from the i_th solute atom to the j_th solvent atom at time t, m is the number of solute atoms and n is the total number of solvent molecules in the system. $\text{Inf}{[\mid {\overrightarrow{r}}_{\text{ij}}(t)\mid ]}_{i=1,m}$ returns the distance between a particular solvent atom and the nearest solute atom k, i.e., k is the assigned solute atom with the closest distance to the j_th solvent atom. In this way, the solvent space is effectively divided into Voronoi polyhedra⁵⁸ and the resulting distribution functions are defined essentially perpendicularly to the exposed surface of the respective solute atoms⁵² (illustrated in Supporting Information). $\delta \tau ({\overrightarrow{r}}_j(t),k)$ is the volume element around the j_th solvent molecule, which is defined by all $\overrightarrow{r}$ vectors, where $\mid {\overrightarrow{r}}_k-\overrightarrow{r}\mid \le \mid {\overrightarrow{r}}_i-\overrightarrow{r}\mid$. A graphical illustration for this process can be found in the Supporting Information. In practice, a pDF can be computed from a MD simulation mapped to a three-dimensional grid where each individual grid point along with its corresponding time-averaged solvent density is assigned to the closest solute atom for ${\mathrm{g}}_{\perp }^k(r)$ calculations. The detailed theory and computational procedure can be found elsewhere.^43,45

A variety of solute atom sets or groupings can be used to define the pDFs. For example, at one extreme one could take each atom of a molecule as having a unique g_⊥(r). Alternatively, pDFs can be constructed for each force field atom type or, using an even coarser grouping, by chemical identity or similarity.⁴⁹ In general, we refer to a particular grouping or atom set by χ and refer to a particular pDF defined by this grouping with χ_k. Previous work shows that the pDFs classified using force field atom types for proteins and nucleic acids yields nearly universal functions which are thus transferable to chemically similar solute molecules.^45,48,50

We can reconstruct the solvation density around a given solute configuration with a three-dimensional grid and assign solvent density to the grid point (x, y, z) with the precomputed ${\mathrm{g}}_{\perp }^{{\chi }_k}(r^{\prime })$:

$$\mathrm{g}(x,y,z)={\mathrm{g}}_{\perp }^{{\chi }_k}(r^{\prime })$$ (2)

where $r^{\prime }=\mid {\overrightarrow{r}}_i-(x,y,z)\mid$ is the minimum distance between the grid point and all solute atoms i, χ_k is the assigned atom set type of the closest solute atom and (x, y, z) denotes the center of the grid volume. In this framework, the solvent density at grid point (x, y, z) is therefore

$$\rho (x,y,z)=\rho {\mathrm{g}}_{\perp }^{{\chi }_k}(r^{\prime })$$ (3)

where ρ is the bulk solvent number density. In our pDF calculations (both pDF-collections and solvent density reconstructions), we used 0.2 and 0.01 Å as the reconstruction grid space resolution and g_⊥(r) collection resolution, respectively. The selection of g_⊥(r) resolution is more important in the van der Waals reconstruction process and a resolution of 0.01 Å is suggested based on previous work.⁵⁵

Any solute atom may exclude solvent from any grid point when there is overlap or volume exclusion. In solvent density reconstruction, a grid point is assigned zero solvent density if any ${\mathrm{g}}_{\perp }^{{\chi }_k}(r^{\prime })\approx 0$. As an example we consider a grid point density assignment near a hydrogen covalently bonded to a heavy atom whose van der Waals radius effectively eclipses that of the hydrogen’s. While the hydrogen atom is the nearest grid point and ${\mathrm{g}}_{\perp }^{\mathrm{H}}(r^{\prime })>0$, the pDF of the bonded heavy atom (i.e., next nearest neighbor) could dictate that solvent should be excluded at the grid point with the given distance, r′. This becomes increasingly important at small distances when using a soft-core van der Waals potential as is common in free energy calculations (addressed in next section, Supporting Information, and ref 55).

We use a two step process to ensure that grid points are assigned the appropriate solvent density during reconstruction. First we eliminate grid points at distances in which the pDFs of heavy atoms in the set χ predict essentially zero solvent occupation (referred to as the cavity rule). Conceptually, solvent space is divided into Vornoi polyhedra with heavy atoms at this step. Then, we exclude or zero out any grid points in a similar manner using ${\mathrm{g}}_{\perp }^{\chi }$ of the remaining atoms (referred to as the exclusion rule). The whole reconstruction process including the cavity and exclusion rules can therefore be formulated as

$$\begin{array}{cc}\hfill \rho & (x,y,z)=\{\prod _{i\ne {\chi }_k}^{m_{\mathrm{H}}}{\delta }_{\text{cav}}^i(\mid (x,y,z)-{\overrightarrow{r}}_i\mid )\}\cdot \rho {\mathrm{g}}_{\perp }^{{\chi }_k}(r^{\prime }).\hfill \\ \hfill & \{\prod _{j\ne {\chi }_k}^m{\delta }_{\text{excl}}^j(\mid (x,y,z)-{\overrightarrow{r}}_j\mid )\}\hfill \end{array}$$ (4)

Here m_H is the number of solute heavy atoms, and m is the number of all solute atoms. The cavity factor ${\delta }_{\text{cav}}^i$ is defined as

$${\delta }_{\text{cav}}^i=\{\begin{array}{cc}1,\hfill & \text{if}{\mathrm{g}}_{\perp }^{{\chi }_i}(\mid (x,y,z)-{\overrightarrow{r}}_i\mid )>0\hfill \\ 0,\hfill & \text{if}{\mathrm{g}}_{\perp }^{{\chi }_i}(\mid (x,y,z)-{\overrightarrow{r}}_i\mid )=0\hfill \end{array}\phantom{\}}$$ (5)

and notice that i loops over solute heavy atoms only. Similarly, the exclusion factor ${\delta }_{\text{excl}}^j$ is written as

$${\delta }_{\text{excl}}^j=\{\begin{array}{cc}1,\hfill & \text{if}{\mathrm{g}}_{\perp }^{{\chi }_j}(\mid (x,y,z)-{\overrightarrow{r}}_j\mid )>0\hfill \\ 0,\hfill & \text{if}{\mathrm{g}}_{\perp }^{{\chi }_j}(\mid (x,y,z)-{\overrightarrow{r}}_j\mid )=0\hfill \end{array}\phantom{\}}$$ (6)

where j loops over the remaining atoms in the set χ.

Once the solvent density is reconstructed around the solute, the average solute–solvent interaction energy can then be written as

$$\langle U_{\text{solu}-\text{solv}}\rangle \approx \sum ^{x,y,z}\sum _{i=1}^m\Delta \nu \rho (x,y,z)U(\mid (x,y,z)-{\overrightarrow{r}}_i\mid )$$ (7)

where $U(\mid (x,y,z)-{\overrightarrow{r}}_i\mid )$ is the solute–solvent interaction energy for solute atom i, Δv is the unit volume which depends on the spatial resolution and the outer sum is taken over all grid points. Depending on the potential energy function, U may be decomposed into the sum of electrostatic and van der Waals interaction energies. The average electrostatic and van der Waals solute–solvent interaction energies can then be calculated using eq 7.

We have previously shown that the solute–solvent electrostatic interaction energy and electrostatic solvation free energy can be reasonably approximated from the pDF-reconstructed solvent density. Here, we extend the pDF approach to estimate the change in free energy (ΔG_vdW) upon inserting uncharged solutes in solution using a soft-core van der Waals potential and a thermodynamic integration scheme.

Thermodynamic Integration with Soft-Core Potentials

To calculate the solvation free energy of cavity formation (ΔG_vdW) we use a thermodynamic integration (TI) approach. We construct a pathway between the initial gas phase and final solvated state by means of a coupling parameter, λ, which varies between 0 and 1 such that when λ = 0 the interaction energy (U_ij) between solute atom i and solvent atom j is zero and when λ = 1, $U_{\mathit{ij}}=U_{\mathit{ij}}^{\text{vdW}}$, or the typical van der Waals potential energy function. To avoid singularities and numerical instabilities at the λ = 0 end-point, a soft-core potential is commonly used to scale $U_{\mathit{ij}}^{\text{vdW}}$ along the pathway:^59,60

$$U_{\mathit{ij}}^{\text{vdW}}(r;\lambda )=4{ϵ}_{\mathit{ij}}\lambda \left[{\left(\frac{{\sigma }_{\mathit{ij}}^2}{r_{\mathit{ij}}^2+\delta (1-\lambda )}\right)}^6-{\left(\frac{{\sigma }_{\mathit{ij}}^2}{r_{\mathit{ij}}^2+\delta (1-\lambda )}\right)}^3\right]$$ (8)

where r_ij is the distance between solute and solvent atoms, ϵ is the van der Waals well depth, and σ is the contact distance. δ is a radius-shifting coefficient of λ and here is taken to be 5.0 throughout the simulations and pDF-reconstructions. Trivially at λ = 1 this is the usual Lennard-Jones form. We note that the effective σ_ij is reduced as λ decreases, which leads to water residing at closer distances to the solute atoms.

The total solute–solvent van der Waals interaction energy (U^vdW) is the sum of $U_{\mathit{ij}}^{\text{vdW}}$ over all solute–solvent atom pairs. The van der Waals solvation free energy is calculated as the integral of the average derivative of U^vdW with respect to λ:

$$\Delta G_{\text{vdW}}={\int }_0^1{\langle \frac{\partial U^{\text{vdW}}(\lambda )}{\partial \lambda }\rangle }_{\lambda }d\lambda$$ (9)

One can approximately calculate ΔG_vdW using numerical integration methods with the ensemble averages of ∂U^vdW/∂λ at various values of λ. The derivative of U^vdW with respect to λ is readily obtained from the analytical expression:

$$\begin{array}{cc}\hfill \frac{\partial U_{\mathit{ij}}^{\text{vdW}}(r;\lambda )}{\partial \lambda }=& 4{ϵ}_{\mathit{ij}}\{{\left[\frac{{\sigma }_{\mathit{ij}}^2}{r_{\mathit{ij}}^2+\delta (1-\lambda )}\right]}^6\left[1+\frac{6\lambda {\sigma }_{\mathit{ij}}}{r_{\mathit{ij}}^2+\delta (1-\lambda )}\right]\phantom{\}}\hfill \\ \hfill & -\phantom{\{}{\left[\frac{{\sigma }_{\mathit{ij}}^2}{r_{\mathit{ij}}^2+\delta (1-\lambda )}\right]}^3\left[1+\frac{3\lambda {\sigma }_{\mathit{ij}}}{r_{\mathit{ij}}^2+\delta (1-\lambda )}\right]\}\hfill \\ \hfill & \frac{\partial U^{\text{vdW}}}{\partial \lambda }=\sum _{i,j}\frac{\partial U_{\mathit{ij}}^{\text{vdW}}}{\partial \lambda }\hfill \end{array}$$ (10)

In our pDF approach, we first construct ${\mathrm{g}}_{\perp }^{{\chi }_k}$ for the atom type set of a small representative chemical group (e.g., an alkane or peptide substituent) from MD simulations performed at regularly spaced values of λ. For each λ, we obtain unique ${\mathrm{g}}_{\perp }^{{\chi }_k,}\mathrm{s}$. Then using these precomputed pDFs, we reconstruct the average, λ-dependent solvent density distributions on a 3D grid around a given solute configuration (e.g., a long alkane or polypeptide) for each λ value. We directly estimate ⟨∂U^vdW/∂λ⟩_λ using a form like eq 7 but with U_ij replaced with $\partial U_{\mathit{ij}}^{\text{vdW}}∕\partial \lambda$. Finally, numerically integrating ⟨∂U^vdW/∂λ⟩_λ along λ yields ΔG_vdW. Note that the λ-dependent pDFs only need to be constructed once but may be used to predict ΔG_vdW of more complex solutes that are composed of similar atom type sets for which the pDFs were generated.

METHODS

Simulation

The solutes butane, propanol, ala₁, ala₁₀, gly₁, gly₃ were chosen for this study. Molecular dynamics simulations were performed with NAMD 2.9 and 2.10^61,62 with the CHARMM36 force field parameters⁶³ to generate the pDFs and calculate free energy benchmarks. pDFs were generated for butane and propanol as a control to compare with more rigorous free energy calculations. The pDFs generated for ala₁ and gly₁ were used in the analysis of longer peptides. Each solute was solvated with TIP3P water⁶⁴ in a volume with at least 10 Å from each boundary to the solute atoms. Peptides were capped with neutral acetyl (ACE) and N-methyl amide (NME) chemical groups. The initial conformations of decaalanines and triglycine peptides are from previous studies^33,52,55,65 and presented in the Supporting Information. Three-dimensional periodic boundary conditions were applied. A rigid water geometry is enforced using SHAKE.⁶⁶ Particle Mesh Ewald (PME)⁶⁷ was used to treat electrostatic interactions using a grid of 1.0 Å resolution. The Lennard-Jones (L-J) interactions were gradually switched off over the range 10 Å to 11 Å. The temperature was fixed at 300 K via a Langevin thermostat with damping coefficient of 5 ps⁻¹. A time step of 2.0 fs was used to integrate the equations of motion.

The pDFs for this free energy study were generated in 11 windows with a λ-spacing of 0.1 between 0 and 1. Solvent reconstruction and calculation of ΔG_vdW were performed with these λ values. In general, it is necessary to sample more extensively at certain λ-windows depending on the chosen soft-core potential function (near end points in our case) to accurately depict the repulsive van der Waals forces when inserting the solute into the system.¹¹ More detail on the analysis can be found in the Results Section below. In this article we show how to use pDFs to reproduce ⟨∂U^vdW/∂λ⟩ at all λ-windows with our pDF-reconstruction algorithm, which is subsequently integrated to estimate ΔG_vdW.

Throughout the simulations, the solute molecules were rigid except as noted below for gly₃. The pDF sampling simulations were initially equilibrated in the NVT ensemble for 1 ns. The simulations were then switched to NPT at 1 atm pressure, with the first 1 ns excluded as equilibration. The pDFs and ⟨∂U^vdW/∂λ⟩ were calculated from production simulations that ranged between 4 and 10 ns at each λ. Trajectory snapshots were saved every 0.2 ps for analyses. The convergence of ⟨∂U^vdW/∂λ⟩ at each λ window is discussed in the Supporting Information. For each λ window, the uncertainty of ⟨∂U^vdW/∂λ⟩ is estimated as the block standard error.⁶⁸ The final uncertainty of ΔG_vdW is calculated by the propagation of errors across all λ.

In addition to the fixed solute conformations, we also examined ΔG_vdW of gly₃ without positional constraints (i.e., allowed to be completely flexible during simulations). The simulation parameters and λ-windows are identical to those of simulations with gly₃ fixed in an extended conformation. We used 10000 solute configurations from a typical 50 ns MD simulation (λ = 1) of flexible gly₃ for calculating ΔG_vdW from pDF-reconstructions.

In the following sections, we also provide estimates of ΔG_elec based on LRT. That is, ΔG_elec can be approximated as half of the solute–solvent electrostatic interaction energy instead of integrating the average derivative of U^elec with respect to a coupling parameter that gradually scales solute–solvent electrostatic interactions. Thus, a set of simulations with charged solutes is employed to achieve full electrostatic solute–solvent interactions. In our previous work, we have reported the averaged solute–solvent electrostatic interaction energies for configurations of butane, propanol, and ala₁₀,⁵⁵ along with the corresponding pDFs used for solvent density reconstruction. In this contribution, with a goal of calculating total solvation free energy, we further include estimates of ΔG_elec for fixed ala₁, gly₁, and flexible gly₃ from free energy simulations to compare with pDF-reconstructions. The simulation details are the same as the above free energy simulations except for having charges on solute atoms and the total sampling time for each system is 100 ns with a 2 ps save frequency.

RESULTS AND DISCUSSIONS

Butane and Propanol

We compare the ⟨∂U^vdW/∂λ⟩ of a single butane/propanol molecule obtained from TI simulations and pDF-reconstruction using the same solute configuration and same number of λ points. In general, the more extensive the set of atom types we include, the higher the accuracy in solvent structure and solute–solvent energetics we can expect during the reconstructions,^51,55 especially for the nonpolar contributions. Here, we consider all six force field atom types (i.e., the set χ) for butane and propanol: methylene carbon (CT2), methyl carbon (CT3), methylene hydrogen (HA2), methyl hydrogen (HA3), the hydroxyl oxygen (O1), and the bonded polar hydrogen (HO1). The pDFs of each atom type with respect to water oxygen atom (OT) at selected λ windows are shown in Figure 1. The pDFs of each atom type with water hydrogen atoms (HT) are included in the Supporting Information.

Figure 1.

pDF of the butane/propanol atoms to water oxygen atoms at different λ. Numbers above each pDF indicate the corresponding λ of this atom type.

Ideally, the pDFs at λ = 0 should be 1 (i.e., bulk solvent density) at all interatomic distances. Therefore, we set the pDFs at λ = 0 to 1 for all distances. Essentially, all atom types are uncharged and considered as pure covalently bonded L-J spheres, and thus show similar behaviors; as λ increases, the position of the peaks are shifted to larger distances with stronger magnitudes in the first peak.

The simulated and pDF-reconstructed estimates ⟨∂U^vdW/∂λ⟩ for each λ are presented in Figure 2. Differences between ⟨∂U^vdW/∂λ⟩ calculated by TI simulations and pDF reconstruction at each λ are shown in the insets. Overall ⟨∂U^vdW/∂λ⟩ calculated by pDF reconstruction for both solutes are consistent with the simulations with deviations for most windows less than 1 kcal/mol. The largest deviation (~2 kcal/mol) of ⟨∂U^vdW/∂λ⟩ between both methods occurs at λ = 0.1. The deviations are diluted upon integration when computing ΔG_vdW. ΔG_vdW calculated for both solutes and methods are listed in Table 1. Differences in simulated and pDF-reconstructed ΔG_vdW for butane and propanol are −0.33 and 0.18 kcal/mol, respectively. The uncertainty for each λ window is estimated using block standard errors,⁶⁸ and the final uncertainty of ΔG_vdW is computed using error propagation. We include the estimated ΔG_elec for butane and propanol from TI simulations and pDF-reconstructions in Table 1. These values are obtained as half of solute-water electrostatic interaction energies based on standard LRT^48,52 and are adapted from our previous work.⁵⁵ Essentially all free energy components are in good accord with those obtained from TI simulations.

Figure 2.

Simulated and pDF-reconstructed ⟨∂U^vdW/∂λ⟩ at each λ window for (a) butane (b) propanol. The inset of each panel shows the difference between the simulated and pDF-reconstructed ⟨∂U^vdW/∂λ⟩. Panel (c) illustrates the thermodynamic cycle of mutating a methyl group of a butane into hydroxyl group to form a propanol.

Table 1.

Comparison of Solvation Free Energies of Butane and Propanol Using Thermodynamic Integration with Simulations and pDF-Reconstructions^a

Δ G	simulated T.I.	pDF T.I.
butane
Δ G vdW	2.67 (0.14)	2.34
Δ G elec	−0.13 (0.01)	−0.14
Δ G total	2.54 (0.14)	2.20
propanol
Δ G vdW	1.88 (0.28)	2.06
Δ G elec	−6.84 (0.03)	−6.42
Δ G total	−4.96 (0.28)	−4.36

^aΔG_vdW is obtained using eq 9. ΔG_elec is approximated as half of electrostatic solute–solvent interaction energy using LRT and is adapted from ref 55. Uncertainties obtained as the block standard errors⁶⁸ are denoted in parentheses. Uncertainties for ΔG_vdW and ΔG_total are calculated using error propagation. The units are all in kcal/mol.

To demonstrate the utility of using this pDF approach to estimate solvation free energies, we consider a free energy cycle of mutating the methyl group in butane to a hydroxyl group to form propanol (illustrated in Figure 2). Since free energy is a state function, ΔΔG = ΔG₄ – ΔG₃ = ΔG₂ – ΔG₁. ΔG₃ and ΔG₄ are the free energies of mutating butane to propanol in vacuum and solution, respectively, and their difference can be calculated by the difference in solvation free energies of butane and propanol. Using the values reported in Table 1, we obtain the pDF-reconstructed ΔΔG = −6.56 kcal/mol, which is within 1 kcal/mol of the simulated ΔΔG = −7.50 kcal/mol.

Deca-alanine

We next estimate ΔG_vdW from precomputed pDFs for a deca-peptide. pDFs were constructed from a single alanine monomer (ala₁) and used to compute ΔG_vdW of four different conformers of deca-alanine, ala₁₀ (referred to as d, d1, d2, d3 in Supporting Information). The set of atoms chosen for pDF generation and solvent density reconstruction include {CT3, C, O, NH1, CT1, HA3, HN, HB1}, where C and O represent the carbon and oxygen atoms of the carbonyl group, NH1 is the peptide nitrogen, HN is the polar hydrogen bonded to the nitrogen, CT1 is the backbone carbon, and HB1 is the backbone hydrogen. We use the pDFs for CT3 and HA3 from our analysis of butane and propanol (Figure 1). The pDFs for the rest of the atom types at several values of λ are presented in Figure 3.

Figure 3.

pDF of alanine/glycine atoms to water oxygen atoms at different λ. Numbers above each pDF indicate the corresponding λ.

We start with comparing the simulated and pDF-reconstructed estimates of ⟨∂U^vdW/∂λ⟩ for ala₁, as shown in Figure 4a, using the same solute configuration. From the inset we observe that the deviations of ⟨∂U^vdW/∂λ⟩ between both methods are slightly larger than those observed for butane and propanol for all λ windows, but still within 2 kcal/mol. The simulated and the pDF-reconstructed ΔG_vdW and ΔG_elec are listed in Table 2. The difference in ΔG_vdW between simulation and pDF-reconstruction is 0.60 kcal/mol, while the difference in ΔG_elec is −0.51 kcal/mol. These differences cancel and yield a pDF reconstructed ΔG within the uncertainty of ΔG obtained by TI simulations. We note ΔG_elec is generally the dominating term in ΔG. Estimated uncertainties of ΔG_elec are smaller than the uncertainties of ΔG_vdW in most of our studies.

Figure 4.

Simulated and pDF-reconstructed ⟨∂U^vdW/∂λ⟩ at each λ window for (a) ala₁ (b) ala₁₀ (the configuration referred to as d in Supporting Information). The inset of each panel shows the difference between the simulated and pDF-reconstructed ⟨∂U^vdW/∂λ⟩.

Table 2.

Comparison of Solvation Free Energies of Single Alanine and Different Conformers of Deca-alanines Using Thermodynamic Integration with Simulations and pDF-Reconstructions^a

Δ G	simulated T.I.	pDF T.I.
ala1
Δ G vdW	1.08 (0.36)	1.78
Δ G elec	−20.21 (0.02)	−20.72
Δ G total	−19.13 (0.36)	−18.94
d
Δ G vdW	5.75 (1.77)	5.92
Δ G elec	−78.19 (0.08)	−74.26
Δ G total	−72.44 (1.77)	−68.34
d1
Δ G vdW	5.83 (2.20)	8.70
Δ G elec	−67.75 (0.08)	−60.70
Δ G total	−61.92 (2.20)	−52.00
d2
Δ G vdW	5.42 (1.81)	6.27
Δ G elec	−89.97 (0.10)	−89.00
Δ G total	−84.55 (1.81)	−82.73
d3
Δ G vdW	5.72 (1.69)	5.67
Δ G elec	−86.88 (0.08)	−85.86
Δ G total	−81.16 (1.69)	−80.19

^aΔG_vdW is obtained using eq 9. ΔG_elec is approximated as half of electrostatic solute–solvent interaction energy using LRT and is adapted from ref 55. The units are all in kcal/mol.

We use the pDFs calculated from ala₁ to estimate the ΔG_vdW of ala₁₀. The results of simulated and pDF-reconstructed ⟨∂U^vdW/∂λ⟩ at each λ for conformer d are shown in Figure 4b. The results for the other three conformers are similar and included in the Supporting Information. The inset shows the deviations of ⟨∂U^vdW/∂λ⟩ between simulations and pDF-reconstructions at each λ for all four conformers. While the deviations are large (10–15 kcal/mol/window) and increase with system size, they are suppressed upon integration when calculating ΔG_vdw in a manner similar to what we observed for butane and propanol. The pDF reconstructed estimates of ΔG_vdW of all ala₁₀ conformers are consistent with those obtained from TI simulations (Table 2). The differences between simulated and reconstructed ΔG_vdW for the different conformers in order are 0.17, 2.88, 0.84, and −0.05 kcal/mol. Differences of ΔG_elec between methods (adapted from our previous work⁵⁵) are larger than those observed for ΔG_vdW. Owing to the extremely twisted configuration of conformer d1, we find the largest deviation between simulation and reconstruction for both ΔG_vdW and ΔG_elec. We note that Poisson–Boltzmann calculations fared worse in reproducing the ΔG_elec of d1.⁵² This highly twisted conformation is a low probability structure with peculiar internal interatomic correlations. Thus, we consider this conformer to be a most challenging case for this theory based on pairwise near neighbor solvent correlations for the estimation of ΔG.

Flexible and Fixed Triglycine

Finally, we extend our approach to the calculation of ΔG_vdW for a large, diverse ensemble of gly₃ conformations and assess the effects of flexibility, or disorder, on these calculations by comparing to a single fixed/extended conformation of gly₃. We chose to calculate the pDFs from a single fixed glycine (gly₁) to evaluate the ΔG_vdW of fixed and flexible triglycines (gly₃). The configuration of fixed gly₃ is in the Supporting Information. The atom set used to generate the pDFs used here include {CT3, C, O, NH1, CT2, HA3, HN, HB2}. pDFs for these atom types have been presented in the previous sections except for HB2, which represents the aliphatic backbone hydrogen atoms bonded to the CT2 atoms. Results of simulated and pDF-reconstructed ΔG_vdW and ΔG_elec for fixed gly₁ are presented in Table 3. Differences of both components between methods are 0.06 kcal/mol. We next use these precomputed pDFs to estimate ΔG_vdW of fixed and flexible gly₃.

Table 3.

Comparison of Solvation Free Energies of Glycine and Tri-glycine Using Thermodynamic Integration with Simulations, pDF-Reconstructions, and Benchmark Free Energy Perturbation Calculations^a

Δ G	simulated TI	pDF TI	FEPb
gly1
Δ G vdW	0.39 (0.34)	0.45
Δ G elec	−16.31 (0.12)	−16.25
Δ G total	−15.92 (0.36)	−15.80
gly3, flexible
$\Delta G_{\text{vdW}}^{\text{flex}}$	−0.41 (0.32)	−0.33	−0.03 (0.05)
$\Delta G_{\text{elec}}^{\text{flex}}$		−28.75	−24.01 (0.50)
$\Delta G_{\text{total}}^{\text{flex}}$		−29.08	−24.04 (0.50)
gly3, fixed
$\Delta G_{\text{vdW}}^{\text{fix}}$	−1.29 (0.38)	−1.10	−0.84 (0.04)
$\Delta G_{\text{elec}}^{\text{fix}}$	−25.62 (0.12)	−24.13	−22.08 (0.03)
$\Delta G_{\text{total}}^{\text{fix}}$	−26.91 (0.40)	−25.23	−22.16 (0.05)

^aΔG_vdW is obtained using eq 9. For fixed configurations, ΔG_elec are approximated as half of electrostatic solute–solvent interaction energies using LRT. An average is taken over configurations for flexible solutes. The units are all in kcal/mol.

^bFree energy perturbation results are adapted from ref 65.

Consider the ΔG_vdW term. The comparisons of simulated and pDF-reconstructed ⟨∂U^vdW/∂λ⟩, along with the deviations (in the insets) at each λ for flexible and fixed gly₃ are presented in Figure 5a,b. The deviations are all within 2 kcal/mol. The values of ΔG_vdW are listed in Table 3. To estimate ΔG_vdW of flexible gly₃, we extracted 10000 solute configurations from a molecular dynamics simulation and, for each one, calculated ⟨∂U^vdW/∂λ⟩ at all λ values. The ⟨∂U^vdW/∂λ⟩ values are then configurationally averaged. We show the distribution over all conformations of ⟨∂U^vdW/∂λ⟩ at λ = 0 from simulation in Figure 5c. This shows the deviation among the solute configurations. We expect the distribution of ⟨∂U^vdW/∂λ⟩ to be different depending on the force field used given that we have previously shown that CHARMM36 and Amber ff12SB generate different ensembles of structures.⁶⁹ It would be an interesting comparison to test the distributions of ΔG_vdW using different force fields as they can display different distributions of solute configurations and flexibility.

Figure 5.

Simulated and pDF-reconstructed ⟨∂U^vdW/∂λ⟩ at each λ window for (a) flexible gly₃ and (b) fixed gly₃. The inset of each panel shows the difference between the simulated and pDF-reconstruced ⟨∂U^vdW/∂λ⟩. Panel (c) shows the distribution of ⟨∂U^vdW/∂λ⟩ for flexible gly₃ at λ = 0. The average of pDF-reconstructed ⟨∂U^vdW/∂λ⟩ corresponds closely with the average of the simulated ⟨∂U^vdW/∂λ⟩. Panel (d) shows the distribution of pDF-reconstructed ΔG_elec using the solute configurations from flexible gly₃ simulations, with the red dashed line representing the ensemble average.

The differences between simulated and pDF-reconstructed ΔG_vdW for flexible and fixed gly₃ are within 0.2 kcal/mol. In addition to the 11 λ windows used for thermodynamic integration, we include the results using the free energy perturbation (FEP) method with 50 equally spaced λ windows between 0 and 1 from recently published work.⁶⁵ Due to the nature of the soft-core potential used in this contribution, the integrand is changing more rapidly (as shown in all ⟨∂U^vdW/∂λ⟩ figures) at small λ. This is a likely source of error in the ΔG_vdW reported here. The comparison of using different numbers of λ windows for calculating the ΔG_vdW of the same solute can be found in Supporting Information. We observe that $\Delta \Delta G_{\text{vdW}}=(\Delta G_{\text{vdW}}^{\text{flex}}-\Delta G_{\text{vdW}}^{\text{fix}})$ calculated from the current free energy simulations, pDF-reconstructions, and FEP are 0.88, 0.77, and 0.81, respectively. The fact that the ΔG_vdW for flexible gly₃ is less favorable (or more repulsive) than the fixed gly₃ is not surprising. With solute flexibility, gly₃ can form configurations that prevent water from accessing parts of the solute, consequently increasing ΔG_vdW. The striking part is that all three methods eventually lead to a consistent ΔΔG_vdW, which is not trivial. More extensive tests of our approximate ΔΔG_vdW are certainly warranted.

To complete the estimation of the total solvation free energy, results for ΔG_elec are presented in Table 3. Here we do not include the ΔG_elec of flexible gly₃ calculated by TI simulations since the noise of ΔG_elec computed from the ensemble of solute and solvent configurations instead of from only the ensemble of solute configurations (like in continuum models or the current concept using pDF-reconstruction) is much larger.^70–72 For the fixed gly₃, the differences of ΔG_elec between these three methods are within 4 kcal/mol. For flexible gly₃, the difference of ΔG_elec between pDF-reconstruction and FEP increases slightly and is within 5 kcal/mol considering a 0.5 kcal/mol uncertainty for FEP reported ΔG_elec. Figure 5 d shows the distribution of pDF-reconstructed ΔG_elec using 10000 solute configurations generated from simulation. This distribution has a standard deviation of 7.42 and is larger than the distribution from pDF-reconstructed ΔG_vdW. $\Delta \Delta G_{\text{elec}}=(\Delta G_{\text{elec}}^{\text{flex}}-\Delta G_{\text{elec}}^{\text{fix}})$ calculated from pDF-reconstruction and FEP are 4.62 and 1.93 kcal/mol, respectively.

DISCUSSION AND CONCLUSION

In this contribution, we used precomputed solvent density distributions at multiple alchemical windows with free energy methods to estimate the van der Waals and electrostatic solvation free energies for a variety of solutes. By computing solute–solvent pDFs from small molecules, we can reconstruct the solvent densities around arbitrary solutes with similar chemical composition and, subsequently, estimate solvation free energies without performing lengthy simulations. Using this framework, we have shown that ΔG_vdW can be reproduced to within useful accuracy. In the future, we will extend our work to the calculation of solvation free energies of complex macromolecules.

The examples of calculating ΔΔG encourage applications to different biological systems. Combinations of solvation free energies with appropriate free energy cycles may allow rapid estimations for molecular associations,¹⁰ alanine scanning,⁷³ and the predictions of pK_a values of proteins and amino acid side chains.⁷⁴