Development And Test of Highly Accurate Endpoint Free Energy Methods. 3: Partition Coefficient Prediction Using a Poisson-Boltzmann Method Combined with Solvent Accessible Surface Area Model for SAMPL Challenges

Abstract

Accurately predicting solvation free energy is the key to predict protein-ligand binding free energy. In addition, partition coefficient (logP), which is an important physicochemical property that determines the distribution of a drug in vivo, can be derived directly from transfer free energies, i.e., the difference between solvation free energies (SFEs) in different solvents. Within the Statistical Assessment of the Modeling of Proteins and Ligands (SAMPL) 9 challenge, we applied the Poisson-Boltzmann (PB) surface area (SA) approach to predict toluene/water transfer free energy and partition coefficient (logP_{toluene/water}) from SFEs. For each solute, only a single conformation automatically generated by the free software Open Babel was used. PB calculation directly adopts our previously optimized boundary definition - a set of general AMBER force field 2 (GAFF2) atom-type based sphere radii for solute atoms. For the non-polar SA model, we newly developed the solvent-related molecular surface tension parameters $\gamma$ and offset $b$ for toluene and cyclohexane targeting experimental SFEs. This approach yielded the highest predictive accuracy in terms of root mean squared error (RMSE) of 1.52 kcal/mol in transfer free energy for 16 small drug molecules among all 18 submissions in SAMPL9 blind prediction challenge. The re-evaluation of the challenge set using multi-conformation strategies based on molecular dynamic (MD) simulations further reduces the prediction RMSE to 1.33 kcal/mol. At the same time, an additional evaluation of our PBSA method on SAMPL5 cyclohexane/water distribution coefficient (logD_{cyclohexane/water}) prediction revealed that our model outperformed COSMO-RS, the best submission model with RMSE_PBSA = 1.88 versus RMSE_COSMO-RS = 2.11 log unit. Two external logP_{toluene/water} and logP_{cyclohexane/water} datasets that contain 110 and 87 data points, respectively, are collected for extra validation and provide in-depth insight of the error source of PBSA method.

Graphical Abstract

Introduction

In this Statistical Assessment of the Modeling of Proteins and Ligands (SAMPL) 9 challenge, the organizers provided the simplified molecular-input line-entry system (SMILES) strings of 16 drug molecules as shown in Fig. 1 and solicited blind prediction of toluene-water partition coefficients (logP_{toluene/water}) on this set of molecules.¹ Unlike the distribution coefficient (logD) predictions of the previous SAPML challenge,^{2, 3} the logP predictions do not require to account for the ionization state and the tautomer of the solute molecules. Therefore, it is unnecessary to re-model or introduce external empirical corrections for the charges. This also reduces the difficulty of making predictions based on the Poisson-Boltzmann surface area (PBSA) method in this study.

Fig. 1.

Structures of the 16 molecules involved in the SAMPL9 partition coefficient challenge.

In most cases, logP_i/j is proportional to the transfer free energy of the solute molecule from solvent $j$ to solvent $i$:

$$log{P}_{i/j}= \frac{-\Delta G_{j\to i}}{RTln10}$$ (1)

where $i$, $j$ are two immiscible solvents, $R$ is gas constant (8.314 J·mol⁻¹·K⁻¹), and $T$ is thermodynamic temperature.

Transfer free energy can be derived from the difference between the solvation free energies (SFEs) of the solute in these two solvents:

$$\Delta G_{j\to i}=\Delta G_i-\Delta G_j$$ (2)

In PBSA-based SFE predictions, electrostatic interactions are usually derived from Poisson-Boltzmann (PB) equation, and the free energy associated with cavitation and dispersion is usually described by solvent accessible surface area (SASA) model.⁴

$$\Delta G_{\mathit{\text{Solv}}}=\Delta G_{PB}+\Delta G_{\mathit{\text{SASA}}}$$ (3)

$$\Delta G_{\mathit{\text{SASA}}}=\gamma \mathit{\text{SASA}}+b$$ (4)

The solute-solvent boundary has uncertainty in implicit solvent models that include the PB method. This is due to the homogenization approximation of the solvent by implicit solvent models and the fact that the solute-solvent boundaries cannot be fully defined by atomic radii based on atomic number. This also implies that it is necessary to clarify the coupled electronic structure method when discussing the definition of solute-solvent boundaries. Moreover, the separating measurements of the electrostatic and non-electrostatic contributions to the solvation effect are typically not available, hence it is difficult to optimize the electrostatic and non-electrostatic contributions individually.⁵ Modeling the solvent effect as a whole may lead to overfitting and the unbalanced contributions of the two types of solvent effect.

Therefore, recently we conducted a series of studies on the development of high accurate PBSA model for SFE prediction,^{6, 7} which is combined with the general AMBER force field 2 (GAFF2) and our recently developed ABCG2 charge model⁸. In this new PBSA model, we developed a set of atom radii for PB calculation targeting the electrostatic (polar) contribution from thermodynamic integration (TI) calculations of hydration free energy (HFE); then the non-electrostatic (non-polar part) term was fitted targeting experimental values of HFE or SFE. We implemented this new PBSA strategy and obtained a root mean square error (RMSE) of 1.05 kcal/mol on HFEs of 544 molecules.⁶ Extending this method to solvent n-octanol yielded a prediction error of RMSE = 0.91 log units on logP_{octanol/water} calculations of 707 drug molecules in the ZINC database.⁷ We called this transfer free energy-based logP method as FELogP. Note that the PB atomic radii optimized from HFE were directly utilized for SFE calculation in organic solvent, by this way only non-polar $\Delta G_{\mathit{\text{SASA}}}$ model needs to be redeveloped electrostatic (non-polar part) term was fitted targeting experimental values of HFE or SFE. We implemented this new PBSA strategy and obtained a root mean square error (RMSE) of 1.05 kcal/mol on HFEs of 544 molecules.⁶ Extending this method to solvent n-octanol yielded a prediction error of RMSE = 0.91 log units on logP_{octanol/water} calculations of 707 drug molecules in the ZINC database.⁷ We called this transfer free energy-based logP method as FELogP. Note that the PB atomic radii optimized from HFE were directly utilized for SFE for individual organic solvents. In this study, we essentially used the previously developed PB boundary definitions,^{6, 8} and derived the solvent dependent parameters $\gamma$ and $b$ for toluene and cyclohexane solvents. The parameterization of $\gamma$ and $b$ targeted to fit experimental SFEs and using multiple conformations to avoid overfitting.

In addition to blind testing on the SAMPL9 dataset, we collected 110 molecules of toluene/water logP for additional testing. Furthermore, we tested FElogP for cyclohexane using both the SAMPL5 logD_{cyclohexane/water} dataset (110 solutes) and an additional logP_{cyclohexane/water} dataset (87 solutes) compiled by us.

Method

Data Preparation

In training sets, all the experimental data of SFE in organic solvents, in this work toluene and cyclohexane, were taken from the Minnesota Solvation Database v2012,⁹ and the experimental data of HFE were taken from the FreeSolv v0.52 database.¹⁰ All the initial structures from Minnesota Solvation Database v2012 are in xyz format, and all initial structures from FreeSolv v0.52 database are in mol2 format. All the structures were imported to Schrödinger Maestro v11.2¹¹ for visual inspection and were saved in mol2 files for further processing. In total 47 molecules have both HFE and SFE in toluene, and 83 molecules have both HFE and SFE in cyclohexane.

The initial structures of SAMPL9 molecules are converted from SMILES strings to mol2 files by Open Babel 3.1.0 with the “-gen3d” option.¹² The additional logP test set data were taken from the works done by Leo et al,¹³ Shalaeva et al,¹⁴ and Byrne et al,¹⁵ and the structures were downloaded from PubChem as sdf files and converted to mol2 files by Open Babel 3.1.0.¹²

The modified module of ANTECHAMBER¹⁶ in AMBER Tools was utilized to assign GAFF2 topologies and ABCG2 charges.

Molecular Dynamic Simulation

Selected solute molecules were solvated in explicit water molecules with at least 15 Å distance from any solute atom to the edges of cubic simulation box. The solute molecules were treated with the GAFF2 force field parameters.¹⁷ The adopted water model was TIP3P. The periodic boundary condition and the NPT ensemble were applied with P = 1.0 atm and T = 298.15 K. The time step was set to 1.0 fs and the total simulation time was 10.0 ns for each system. The software AMBER18¹⁸ was utilized for MD simulations.

PBSA Calculation

All PB calculations were performed using Delphi V4 release 1.1.^{19, 20} The salt concentration was set to 0 mol/L; the grid spacing was set to 1.2 grids/Å; the percentage of the object longest linear dimension to the lattice linear dimension was set to 80%; and the boundary condition was set as coulombic boundary with 1.4 Å probe radius. The internal dielectric constant was always set to 1.00, and the dielectric constant of solvent was set to 80.00 for water, 2.3741 for toluene, and 2.0165 for cyclohexane, respectively. Calculation mode was set as reaction field energy, which is regarded as the electrostatic component of solvation free energy $\Delta G_{PB}$. The solvent accessible surface area $\mathit{\text{SASA}}$ was generated by an internal program called MS²¹ using Bondi’s van der Waals radii²² and water probe (radius of 1.4 Å). This program is also available upon request. SASA was used to derive non-electrostatic term $\Delta G_{\mathit{\text{SASA}}}$ using Equation 4.²¹

Toluene and Cyclohexane Modeling

The same PB radius parameters derived using hydration free energies in our previous work^{6, 7} are directly applied in toluene and cyclohexane, therefore, the only parameters of toluene and cyclohexane that differ from those of water are $\gamma$ and $b$ of Equation 4 in addition to the dielectric constant. The parameterization of $\gamma$ and $b$ can be obtained directly by linear regression analysis (single data point per solute), but given the limited amount of data in organic solvents, we used the multi-conformation approach when conduct the linear regression process (multiple data points per solute). All conformations are generated by the “-conformer” option of the Open Babel software through genetic algorithm,¹² with the generation criterion being set to minimum energy and the maximum number of generated conformations being set to 20. The numbers of conformations $m$ associated with individual molecules were listed in Table S1. The advantage of generating multiple conformations through Open Babel is that the number of conformations depends on the degree of freedom of the molecule. Therefore, the modeling of toluene and cyclohexane is the fitting of the following linear equations:

$$\Delta G_{SFE,M}^{\mathit{\text{expt}}}-\Delta G_{PB}^{\mathit{\text{calc}}}\left({\mathbf{\text{R}}}_{M_k}\right)={\gamma }_s\mathit{\text{SASA}}\left({\mathbf{\text{R}}}_{M_k}\right)+b_s$$ (5)

where ${\mathbf{\text{R}}}_{M_k}$ is the kth conformation of molecule $M$, $s$ is organic solvent, here represent for either toluene or cyclohexane.

Calculate logD from logP

Only one ionization state is considered for the logD calculation from logP. The modified Henderson-Hasselbalch equation is used:

$$\text{log}D=\text{log}P-\text{log}\left(1+{10}^{p{K}_a-pH}\right)$$ (6)

$$\text{log}D=\text{log}P-\text{log}\left(1+{10}^{pH-p{K}_a}\right)$$ (7)

Equation 6 is used for basic solutes and Equation 7 is used for acidic solutes. For amphipathic molecules, acidic pK_a is adopted as the correction factor.

Thermodynamic Integration Simulation Protocol

We compared the PBSA method with thermodynamic integration (TI) method on SAMPL9 and SAMPL5 dataset, and the TI calculation details were elaborated in this section. The alchemical enhanced sampling (ACES) method,²³ proposed by Lee et al and implemented in the graphic processing unit (GPU) version^24–26 of thermodynamic integration modules in AMBER22, was employed for HFE and SFE calculations.

The TLEAP module in AMBER22 was used to generate all solute-solvent boxes. For a solute molecule being solvated in water, the minimum distance between any solute atoms and an edge of the water box was set to 15 Å. Similarly, a solute molecule was solvated in the cubic box of toluene or cyclohexane utilizing TLEAP. Note that toluene solvent box which has a dimension of 33.623 Å and cyclohexane solvent box which has a dimension of 39.418 Å were first created following the standard procedure as detailed in our previous publication.⁸

The organic solute-solvent system was first subjected to an initial equilibration for 200 ps using the CPU-TI at λ = 0.01592. A 2 ns MD simulation was conducted for each of the 9 λ windows (0.01592, 0.08198, 0.19331, 0.33787, 0.5, 0.66213, 0.80669, 0.91802, 0.98408). For the first λ window (λ = 0.01592), the initial configurations were sampled from the CPU-TI, while the initial configurations for the other eight λ windows were obtained from the preceding λ window. Following the system setup, periodic boundary condition and the isothermal-isobaric NPT ensemble were produced in all simulations. Using Langevin dynamics to maintain the temperature at 298K, with the collision frequency (gamma_ln) set to 2.0 ps⁻¹. The pressure was kept at 1.01325 bar with Monte Carlo barostat and the pressure relaxation time being set to 5.0 ps. Disable the SHAKE constrains for solute and set time step to 1fs. It is pointed out that the purpose of running GPU-TI here was to provide an equilibrium system for the ACES simulation protocol. Specifically, we enlarged the simulation boxes for the organic solvents about 15–40% from the last snapshots of the GPU TI runs for the λ = 0.5 window. The new simulation boxes have dimensions around 46.0 Å.

All the subsequent ACES simulations were based on the new simulation boxes following the same protocol of GPU-TI except that the van der Waals and electrostatic interactions were scaled by smoothstep soft-core potential^{27, 28} with switching function $W\left(r_{ij}\right)$:

$$r_{ij}^{VDW}\left(\lambda ;{\alpha }^{VDW}\right)={\left[r_{ij}^n+W\left(r_{ij}\right)\cdot {\alpha }^{VDW}\cdot S_P\left(\lambda \right)\cdot {\sigma }_{ij}^n\right]}^{1/n}$$ (8)

$$r_{ij}^{\mathit{\text{Elec}}}\left(\lambda ;{\alpha }^{\mathit{\text{Coul}}}\right)={\left[r_{ij}^m+W\left(r_{ij}\right)\cdot {\alpha }^{\mathit{\text{Elec}}}\cdot S_P\left(\lambda \right)\cdot {\sigma }_{ij}^m\right]}^{1/m}$$ (9)

The lower boundary of the switching function $W\left(r_{ij}\right)$ was set to 8 Å and the upper boundary was set to 10 Å. Additionally, the internal VDW interactions scaling within soft-core region were disabled by setting the gti_add_sc to 5. Nine equally-spaced $\lambda$ windows (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9) were applied to decouple the endpoint states. Hamiltonian exchange between different λ windows was performed every 100000 steps under the REMD²³ framework to achieve the enhanced sampling. It is pointed out that the above ACES protocol is same as that reported by Lee et al.²³ with an aim to achieve the consistent performance. The free energies were derived from unweighted integration of the alchemical pathway as below:

$$\Delta G=G\left(\lambda =1\right)-G\left(\lambda =0\right)={\int }_0^1{\langle \frac{\partial V}{\partial \lambda }\rangle }_{\lambda }\cdot d\lambda \approx \sum 0.1\times {\langle \frac{\partial V}{\partial \lambda }\rangle }_i$$ (10)

Three independent ACES based GPU-TI runs were performed for each solute, with 2 ns MD simulations for each $\lambda$ windows. For each MD run, the beginning 0.5 ns simulation was considered as the equilibration phase and excluded from the later free energy analysis. The final HFE and SFE were then derived from the arithmetic average of the three independent TI runs, while the standard deviation of the three independent runs was calculated to measure the precision of the protocol. The corresponding logP was calculated from HFE and SFE using Equation 1, and the logD was calculated from logP using Equations 6 and 7.

Ab initio logP Calculation

We used quantum mechanics (QM) based SMD model implemented in the Gaussian 16²⁹ software to derive the logP benchmark for our model. The principle of SMD derived logP is also based on the transfer free energy as Equation 1. Geometry optimization in the liquid phase at the B3LYP/6–31G* level of theory was first performed prior to SMD calculations, with the solvent specified directly by keywords; then the optimized geometries were read out to perform single point calculations in gas phase at the same level of theory. The energy difference between the liquid and gas phase is regarded as the SFE.

Results and Discussion

Modeling of Toluene and Cyclohexane

With the multi-conformation strategy described above applied on the training sets, the descriptors ($\gamma$ and $b$) of toluene and cyclohexane for SASA model were derived: ${\gamma }_{\mathit{\text{toluene}}}=-0.023556$, $b_{\mathit{\text{toluene}}}=4.40$ and ${\gamma }_{\mathit{\text{cyclohexane}}}=-0.024237$ ,$b_{\mathit{\text{cyclohexane}}}=4.64$. $\Delta G_{SFE,M}^{\mathit{\text{expt}}}-\Delta G_{PB}^{\mathit{\text{calc}}}$ and $\mathit{\text{SASA}}$ data for molecules in training set to conduct liner regression were listed in Table S1.

SAMPL9 Toluene/Water logP Blind Prediction

As required by the SAMPL9 organizer, we submitted predicted transfer free energies ΔG_{toluene/water} of the 16 drug molecules before the deadline. Note that only a single conformation (with minimum energy) automatically generated by Open Babel for each drug molecule was used for the PBSA calculation of HFEs in water and SFEs in toluene. Based on the analysis result on all 18 submissions provided by the organizer (https://github.com/samplchallenges/SAMPL9/tree/main/logP/Analysis/prelim_analysis), our submission achieved the lowest overall RMSE of 1.52 kcal/mol. After the completion of this blind prediction contest, we also applied MD simulation conjugated with PBSA to re-calculate the transfer free energy ΔG_{toluene/water} for the 16 molecules and summarized the results in Table 1 and Fig. 2. Table 1 reports the calculated HFE, SFE in toluene and the transfer free energy derived from the difference between HFE and SFE. Fig. 2 shows the correlation between experimental and calculated transfer free energies. The re-calculated transfer free energies achieved better RMSE of 1.33 kcal/mol and the Pearson correlation coefficient (R) of 0.94.

Table 1.

Detailed experimental and calculated transfer free energies, calculated hydration free energies in water and solvation free energies in toluene using the PBSA method. The overall Pearson correlation coefficient (R), mean signed error (MSE), mean unsigned error (MUE) and root mean square error (RMSE) were listed for 16 SAMPL9 compounds.

Compound	Experiment ΔG (kcal/mol)	Hydration ΔG (kcal/mol)	Solvation ΔG (kcal/mol)	Transfer ΔG (kcal/mol)
1	−5.11	−13.48	−15.75	−2.27
2	−3.26	−11.41	−14.38	−2.97
3	−7.49	−6.14	−12.57	−6.42
4	−7.44	−10.99	−16.07	−5.08
5	−4.91	−8.27	−13.37	−5.10
6	1.67	−18.37	−14.59	3.78
7	−5.94	−13.81	−19.83	−6.02
8	−3.79	−18.49	−22.40	−3.91
9	−6.87	−5.68	−12.39	−6.71
10	−3.36	−10.80	−14.07	−3.26
11	−1.99	−13.76	−14.10	−0.35
12	2.16	−15.42	−12.32	3.10
13	−0.49	−18.04	−17.19	0.85
14	−1.92	−14.16	−17.40	−3.24
15	1.01	−19.57	−17.75	1.82
16	−5.13	−12.52	−18.72	−6.20
R				0.94
MSE				0.68
MUE				1.03
RMSE				1.33

Fig. 2.

Experimental transfer free energy versus calculated transfer free energy using PBSA method for 16 drug molecules in SAMPL9 challenge.

In addition to the PBSA parameters and charge model that can affect the prediction accuracy of SFEs and corresponding transfer free energies, the adopted methodology and protocol for conformation generation is another factor affecting the prediction performance. The prediction error of Compound 8 significantly reduced after being treated by MD simulations compared to the value in our submission with single-conformation strategy. Also, Compound 8 has the maximum solvent accessible area, 709.35 Ų (B3LYP/6–31G* optimized geometry), and greater flexibility. Therefore, we focused on Compound 8 to investigate the conformational effect on the prediction accuracy of transfer free energies and illustrate the results in Fig. 3. The error of the calculated transfer free energies from the experimental value were evaluated using 10, 20, 50 and 100 conformations. Conformations of Compound 8 were generated through three different ways: MD simulations, genetic algorithm using Open Babel,¹² and Omega using mmff94smod_NoEstat force field parameters.³⁰ The conformations generated by MD simulation yielded the lowest computational errors among the three methods, and demonstrated a trend that the error approached to zero as shown in the panel B of Fig. 3 (from −0.76 kcal/mol on 10 conformations to −0.52 kcal/mol on 100 conformations). The magnitude of the computational error from the conformations generated by Omega also decreased as the number of conformations increases, just as the result from MD simulations, however, there was a much long way to go before the error could reduce to certain low threshold. In contrast, the computational error from the conformations generated by Open Babel fluctuated around −2.0 kcal/mol as the number of conformations changed, with the magnitude of error higher than that from MD simulation (around −0.6 kcal/mol) but lower than that from Omega (from −6.4 kcal/mol on 10 conformations to −5.0 kcal/mol on 100 conformations).

Fig. 3.

The relationship between the numbers of conformations and the prediction errors of the transfer free energies using the PBSA method. (A) prediction errors of three conformation generation methods; (B)(C)(D) are re-ranged plots for individual methods.

Except for Compound 8, other compounds which have prediction errors close to 2 kcal/mol should also be noticed. The prediction error of Compounds 1, 6 and 11 most likely arose from the formation of intramolecular hydrogen bond. As reported by Shalaeva et al,¹⁴ the difference between logP_{octanol/water} and logP_{toluene/water} is a potential descriptor to indicate the formation of intramolecular hydrogen bond. Molecular fragments that have the structural potential to form intramolecular hydrogen bonds in 6- or 7-membered rings are screened in a highly dielectric medium such as water (ε = 80) and form intermolecular hydrogen bonds with water molecules. Such molecule first undergoes desolvation during water-toluene phase transfer, and then, due to the jump in the dielectric environment, is more inclined to form intramolecular hydrogen bonds, thus decreasing the molecular polarity and increasing solubility. As such, Compounds 1 and 11 adopt different conformations in the two different solvents, and the large prediction errors of transfer free energies of the two compounds may be due to using the same set of conformations. Unfortunately, it is necessary to use the same set of conformations for the solvation free energy calculation in two different solvents to achieve the best error cancellation.³¹

Since the TI method demonstrates high accuracy in free energy calculations, we also employed TI method to calculate the logP_{toluene/water} for the 16 molecules in SAMPL9 dataset. The result of TI-calculated transfer free energies versus the experimental values was shown in Fig. 4, and the detailed data were summarized in Table S2. The overall prediction error of TI in terms of RMSE was 2.11 kcal/mol, and the Pearson correlation coefficient of TI predictions was 0.92. Note that the prediction error of TI was slightly larger than that of the COSMO-RS method, but smaller than those of the other 11 submissions in this SAMPL9 challenge.

Fig. 4.

Experimental transfer free energy versus calculated transfer free energy using the TI method for 16 drug molecules in SAMPL9 challenge. The uncertainties of calculated transfer free energy were standard deviations derived from three independent TI runs.

SAMPL5 Cyclohexane/Water logD Prediction

In addition to modeling toluene for the SAMPL9 challenge, we also modeled cyclohexane and tested the cyclohexane/water logD prediction for 53 organic molecules in SAMPL5 challenge as well as the cyclohexane/water logP prediction for 87 molecules we collected.² The prediction results of comparing our PBSA method with the best-ranked SAMPL5 submission from Klamt et al using COSMO-RS method³² (hereafter referred to as COSMO-RS) were summarized in Fig. 5 and Table 2. Panel A in Fig. 5 shows the correlation between experimental logD and PBSA calculated logD, and panel B illustrates the correlation between experimental logD value and the initial submitted logD using COSMO-RS method by Klamt et al.³² The overall RMSE prediction error of our PBSA method is 1.88 log units, which is smaller than that of COSMO-RS (RMSE = 2.11 log units). It is worth noting, however, that the logD values calculated by the PBSA method were corrected from logP values using Equations 6 and 7, and the solutes’ pK_a values were borrowed from Klamt et al. According to their report, the pK_a values were predicted using the ab initio COSMOtherm program.³³ In addition to the COSMOtherm, ab initio calculations using the Schrödinger Jaguar pK_a module³⁴ can yield comparable accurate predictions (RMSD within 0.2–0.5 pK_a units) for logD predictions. As shown in Fig. 5, the yielded large prediction errors by the PBSA method were mainly for some neutral and basic molecules, among which Compounds 74 and 82 also had large prediction errors by the COSMO-RS method. Regarding to Compound 74, based on our experience in developing the PBSA method, the conformation of polyhydroxylated compounds represented by glycerol has a significant effect on the prediction accuracy, and the use of a multi-conformation approach sampled by MD simulations usually leads to a predicted SFE of such molecules closer to the experimental value. The prediction error for SAMPL5_083 raises from using a less dominate tautomer as reported by Klamt et al.³² Similarly, we conducted TI calculations on the SAMPL5 logD_{cyclohexane/water} dataset for comparison. We also adopted the predicted pK_a (summarized in Table 2) to correct the TI calculated logP to obtain logD. The performance of TI predictions was illustrated in Fig. 6 and the detailed data were listed in Table S3. The overall prediction error of TI in terms of RMSE was 2.15 log units, which was comparable with the COSMO-RS method.

Fig. 5.

Correlation between experimental and calculated logD. (A) Calculated with PBSA method (this work); (B) Calculated using the COSMO-RS method.

Table 2.

Experimental logD, calculated logP and logD values of the PBSA and COSMO-RS methods. The pK_a values adopted to correct the ionization effect were from Klamt et al. If the molecule is an amphipathic molecule, the acidic pK_a was used to compute the correction factor.

Compound	Expt	pKa		Calc logP		Calc logD
	logD	Acid	Base	COSMO-RS	PBSA	COSMO-RS	PBSA
SAMPL5_002	1.40			1.70	0.58	1.70	0.58
SAMPL5_003	1.90			2.80	1.75	2.80	1.75
SAMPL5_004	2.20		6.85	4.10	0.57	4.00	0.46
SAMPL5_005	−0.86			1.50	1.15	1.50	1.15
SAMPL5_006	−1.02			0.70	−0.14	0.70	−0.14
SAMPL5_007	1.40		7.02	1.80	1.90	1.60	1.74
SAMPL5_010	−1.70	4.86	6.03	−2.20	−0.22	−4.70	−2.76
SAMPL5_011	−2.96	4.01	4.55	1.10	1.33	−2.30	−2.06
SAMPL5_013	−1.50			0.90	0.50	0.90	0.50
SAMPL5_015	−2.20	4.35		−4.00	−0.70	−7.10	−3.74
SAMPL5_017	2.50			3.80	2.98	3.80	2.98
SAMPL5_019	1.20		6.55	4.00	−0.08	3.90	−0.13
SAMPL5_020	1.60			2.00	0.30	2.00	0.30
SAMPL5_021	1.20			2.50	1.85	2.50	1.85
SAMPL5_024	1.00			2.60	1.66	2.60	1.66
SAMPL5_026	−2.60	4.73		−0.90	0.74	−3.60	−1.93
SAMPL5_027	−1.87			−2.10	−2.13	−2.10	−2.13
SAMPL5_033	1.80			4.20	4.76	4.20	4.76
SAMPL5_037	−1.50		8.17	−1.70	−0.04	−2.60	−0.88
SAMPL5_042	−1.10			0.40	0.31	0.40	0.31
SAMPL5_044	1.00			2.80	0.10	2.80	0.10
SAMPL5_045	−2.10			−1.30	−1.08	−1.30	−1.08
SAMPL5_046	0.20			0.50	−0.29	0.50	−0.29
SAMPL5_047	−0.40			2.00	−1.95	2.00	−1.95
SAMPL5_048	0.90			1.50	0.72	1.50	0.72
SAMPL5_049	1.30			3.40	1.48	3.40	1.48
SAMPL5_050	−3.20	7.24	3.86	−6.70	0.01	−7.10	−0.38
SAMPL5_055	−1.50			−1.80	−1.56	−1.80	−1.56
SAMPL5_056	−2.50	8.09	−4.19	−4.60	−1.51	−4.70	−1.59
SAMPL5_058	0.80			1.60	1.49	1.60	1.49
SAMPL5_059	−1.30			−0.90	−1.20	−0.90	−1.20
SAMPL5_060	−3.90	4.95		−1.90	−1.55	−4.40	−4.00
SAMPL5_061	−1.45		7.03	−1.70	0.36	−1.80	0.21
SAMPL5_063	−3.00		9.05	−5.80	−1.00	−7.50	−2.66
SAMPL5_065	0.70		8.43	3.40	3.99	2.30	2.92
SAMPL5_067	−1.30		8.85	2.60	2.46	1.10	1.00
SAMPL5_068	1.40			2.20	2.22	2.20	2.22
SAMPL5_069	−1.30	8.91	7.74	1.70	−0.01	1.20	−0.02
SAMPL5_070	1.60		9.32	5.80	4.20	3.80	2.28
SAMPL5_071	−0.10			−0.20	0.34	−0.20	0.34
SAMPL5_072	0.60		8.62	4.10	3.30	2.90	2.06
SAMPL5_074	−1.90			−8.00	−7.06	−8.00	−7.06
SAMPL5_075	−2.80		8.50	1.30	2.72	0.10	1.59
SAMPL5_080	−2.20			−1.90	−2.06	−1.90	−2.06
SAMPL5_081	−2.20		8.28	−3.60	−3.90	−4.50	−4.84
SAMPL5_082	2.50		8.11	7.40	6.98	6.60	6.20
SAMPL5_083	−1.90			−2.30	3.12	−2.30	3.12
SAMPL5_084	0.00		8.18	2.00	4.13	1.20	3.29
SAMPL5_085	−2.20			−1.80	−0.50	−1.80	−0.50
SAMPL5_086	0.70		9.52	4.00	4.80	1.90	2.68
SAMPL5_088	−1.90			0.00	0.43	0.00	0.43
SAMPL5_090	0.80			1.30	2.26	1.30	2.26
SAMPL5_092	−0.40			1.30	1.98	1.30	1.98
R				0.79	0.55	0.85	0.68
MSE				1.05	1.26	0.49	0.71
MUE				1.79	1.84	1.65	1.44
RMSE				2.26	2.34	2.10	1.88

Fig. 6.

Correlation between experimental logD and TI calculated logD. Uncertainties were standard deviations from three independent TI runs.

Test of the PBSA method on Additional logP Datasets

Finally, to further validate the developed PBSA models for toluene and cyclohexane, additional test molecules were collected to predict the logP_{toluene/water} and logP_{cyclohexane/water} values. For 110 organic molecules in toluene, the PBSA method achieved an RMSE of 1.83 log units. In contrast, the QM-based SMD method calculated at the B3LYP/6–31G* level of theory had a prediction error of 2.31 log units. The comparison results were shown in a scatter plot between the experimental logP and calculated logP (Fig. 7), and the raw data were listed in Table S4.

Fig. 7.

Correlation between experimental and calculated logP_{toluene/water}. (A) Calculated logP_tol/wat using PBSA method; (B) Calculated logP_{toluene/water} using SMD method.

Interestingly, there was a strong agreement between the PBSA method and the SMD method for molecules with large prediction errors, which are: 8-Hydroxyquinoline, 2-Methyl-8-Quinolinol, Bromothymol blue, and Schiff base. Some others with larger errors by the PBSA method are phosphorus-containing molecules, for which the phosphorus-related bond charge correction parameters were not adequately adjusted for the ABCG2 charge model. Still other six molecules with experimental logP values between 3.0 – 4.0 have systematic errors in the PBSA calculations, but not in SMD calculations. Examination on their structures revealed that most of them are halogen-substituted benzenes except for cyclohexene. This systematic error is probably due to the inability of the implicit solvent model described by the dielectric constant to adequately model the π-π interactions arising from the benzene rings in the toluene and solute molecules. Of course, the systematic error may also come from the inadequate description of the σ-hole effect by the ABCG2 charge model. This systematic error in structure-dependent SFE calculations recurs in the PBSA model and has attracted our attention to deal with those “difficult” molecules in the future.

As to the 87 organic molecules in the additional cyclohexane test set, the PBSA method achieved an RMSE of 1.11 log units, which is slightly larger than that of the SMD method (RMSE=0.99) as shown in Fig. 8. Nevertheless, the prediction error is much lower than the RMSE of logD prediction in SAMPL5 challenge.

Fig. 8.

Correlation between the experimental and calculated logP_{cyclohexane/water}. (A) Calculated logP_{cyclohexane/water} using the PBSA method; (B) Calculated logP_{cyclohexane/water} using the SMD method.

Conclusion

In this study, we extended the scope of our PBSA method for predicting solvation free energies in toluene and cyclohexane for organic molecules by parameterizing the nonpolar part and successfully applied this model to predict toluene-water partition coefficients in the SAMPL9 challenge. The PBSA method performed the best out of a total of 18 submissions in terms of RMSE. The RMSE error of our submission, 1.52 kcal/mol, was further reduced to 1.33 kcal/mol after using the multi-conformations generated through MD simulations. The distribution coefficient dataset from SAMPL5 challenge was adopted to test the performance of the PBSA solvation free energy model for cyclohexane, and the prediction error of our model, RMSE = 1.88 log units, was better than that of COSMO-RS, which had the lowest prediction error (RMSE = 2.11 log units) among the 63 submissions of the SAMPL5 challenge. The ACES TI was conducted to calculate toluene-water transfer free energy in SAMPL9 dataset and cyclohexane-water logD in SAMPL5 dataset. The RMSE of TI were 2.11 kcal/mol on SAMPL9 dataset and 2.15 log units on SAMPL5 dataset. This further proved the reliability of our PBSA-based approach for partition coefficient prediction. In addition, we discussed the potential sources of errors for some poor predictions. More excitingly, we found the prediction error of our models can be further reduced when using multiple conformations. Among the three conformational ensemble generation methods, MD simulation achieved the best performance. We further evaluated our two PBSA solvation free energy models using two larger molecule sets. Overall, our FElogP model performance is comparable or better to that of quantum mechanics based SMD method.

Associated Content

Supporting information includes the calculated nonpolar free energies and SASA for each conformation (Table S1), SMILES strings for each molecule in training set (Table S1), ACES based TI calculation results for SAMPL9 dataset (Table S2) and SAMPL5 dataset (Table S3), additional logP experimental data and corresponding molecules’ SMILES strings (Table S4). AMBER topology/coordinates, additional force field parameters (in frcmod format) and molecular structures with ABCG2.1 charges (in mol2 format) for training set molecules are accessible from https://mulan.pharmacy.pitt.edu/publication/supplementary/pccp_2023/Trainingset.tar.gz. The internal program for solvent-accessible surface area calculation, ms, is available upon request.