SAMPL6: calculation of macroscopic pKa values from ab initio quantum mechanical free energies

Abstract

Macroscopic pK_a values were calculated for all compounds in the SAMPL6 blind prediction challenge, based on quantum chemical calculations with a continuum solvation model and a linear correction derived from a small training set. Microscopic pK_a values were derived from the gas-phase free energy difference between protonated and deprotonated forms together with the Conductor-like Polarizable Continuum Solvation Model and the experimental solvation free energy of the proton. pH-dependent microstate free energies were obtained from the microscopic pK_as with a maximum likelihood estimator and appropriately summed to yield macroscopic pK_a values or microstate populations as function of pH. We assessed the accuracy of three approaches to calculate the microscopic pK_as: direct use of the quantum mechanical free energy differences and correction of the direct values for short-comings in the QM solvation model with two different linear models that we independently derived from a small training set of 38 compounds with known pK_a. The predictions that were corrected with the linear models had much better accuracy [root-mean-square error (RMSE) 2.04 and 1.95 pK_a units] than the direct calculation (RMSE 3.74). Statistical measures indicate that some systematic errors remain, likely due to differences in the SAMPL6 data set and the small training set with respect to their interactions with water. Overall, the current approach provides a viable physics-based route to estimate macroscopic pK_a values for novel compounds with reasonable accuracy.

Keywords: Quantum chemistry; SAMPL challenge; pH; pK a.

Fig. 1

Chemical structures of the SAMPL6 data set. SM20 is the only compound that contains a single titratable proton; all other compounds contain multiple titratable protons and, in some cases, tautomers.

Fig. 2

Chemical structures of the QM1 training data set (neutral acids); see also Table 1.

Fig. 3

Chemical structures of the QM2 training data set (cationic acids); see also Table 1.

Fig. 4

Training data set. The pK_a of the training data set compounds are used to derive a simple linear model that relates the free energy correction $\Delta G_{\text{corr}}^{\text{*}}$ to the experimental pK_a. Two linear models were derived: a global linear model (black dashed line), utilizing all data, and a piecewise linear model that applies to either neutral acids (subset QM1, blue) or to positively charged acids (subset QM2, green). a: Correlation between experimental and calculated pK_a of the training data set. The dashed line indicates ideal correlation with the gray band indicating 1 pK_a unit deviation. b: Global linear fit of the calculated $\Delta G_{\text{corr}}^{\text{*}}$ to the experimental pK_a. c: Linear fits of the calculated $\Delta G_{\text{corr}}^{\text{*}}$ to the experimental pK_a, split between the QM1 and the QM2 subsets. In (b) and (c) the dashed lines are linear models to the data, with shaded bands indicating 95% confidence intervals from 1000 bootstrap samples.

Fig. 5

Signed error Δ_id of individual predictions. The calculated pK_a was matched to the experimental pK_a for each compound (indicated by the SAMPL6 pKa ID) and the deviation from the experimental value represented as a bar. Observations for the same compound have the same color. a: pK_a were directly estimated from the quantum mechanical free energy differences. b: The quantum mechanical pK_a were corrected with the global linear model. c: compounds were corrected depending on their membership in subsets 1 or 2 with the piecewise linear model.

Fig. 6

Correlation between experimental and calculated pK_a values for the SAMPL6 compounds. a: pK_a were directly estimated from the quantum mechanical free energy differences. b: The quantum mechanical pK_a were corrected with the global linear model. c: compounds were corrected depending on their membership in subsets 1 or 2. The black dashed line indicates ideal correlation, the shaded green bars show 0.5 and 1 pK_a units deviation from ideal. Blue lines are linear regression fits to the data, with the blue shaded area indicating the 95% confidence interval from 1000 bootstrap samples.

Fig. 7

Comparison of chemical properties of the training (light blue) and SAMPL6 (orange) data sets. a: normalized histograms of the number of rotatable bonds; b: normalized histograms of the number of hydrogen bond acceptors; c: correlation between the number of heavy atoms and the number of acceptors with linear regressions shown as solid lines and their 95% confidence interval from 1000 bootstraps indicated by shaded areas.

Fig. 8

Microstate probabilities p_i for SM14. a: Computed microstate probabilities (for the piecewise linear fit) are shown as heavy solid lines and experimentally derived probabilities as thin dashed lines. The experimental p_i were calculated in the same way as the calculated ones (Eq. 19) by directly using the experimental microstate pK_as. b: Microstate diagram with arrows indicating deprotonation. Bold numbers near solid arrows are the calculated microstate pK_a (from (a)) and italic numbers near dashed arrows are the experimental numbers, assigned to the experimentally identified microstate transitions. The gray solid arrows with gray bold numbers indicate the calculated macroscopic pK_a from N = 3 protons (microstate SM14_micro003) to N = 2 protons (mixture of SM14_micro002 and SM14_micro004, indicated by the orange box) to N = 1 proton in SM14_micro001 (and SM14_micro005, which is not shown because computation and experiment indicate that it is suppressed relative to SM14_micro001).

Fig. 9

RMSE of all SAMPL6 submissions (blue), including our new calculations for all SAMPL6 compounds (red) and for completeness our original submissions (gray), which only included predictions for SM15, SM20, and SM22 and is only of limited statistical validity because of the large variance of the RMSE itself for only three samples [37]. The submission IDs p0jba and xxxc correspond to the piecewise linear model, 35bdm and xxxb to the global linear model, and xxxa to directly using the quantum chemical free energies. Other IDs belong to other regular SAMPL6 submissions. The error bars indicate 95% confidence intervals from 1000 bootstrap samples.