#
Recent Developments of Computational Methods for pK_{a} Prediction Based on Electronic Structure Theory with Solvation Models

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## Abstract

**:**

## 1. Introduction

_{a}values, which are based on quantum chemical electronic structure methods combined with solvation models. Here, we review the methods employed. Specifically, we review two different solvation models: the polarizable continuum model (PCM) and the reference interaction site model (RISM) theories.

_{a}values of small compounds, a number of studies using quantum chemical calculations have been reported, and the details of their accuracy are discussed in a review by Ho and Coote [3]. To estimate reliable $\mathrm{p}{K}_{\mathrm{a}}$ values, it is necessary to obtain the solvation free-energy difference of the molecules in aqueous solution, due to acid dissociation, as accurately as possible. Sprik and coworkers proposed a method for obtaining the free-energy profile due to acid dissociation from DFT-based ab initio molecular dynamics (AIMD) simulations and successfully obtained the $\mathrm{p}{K}_{\mathrm{a}}$ values of many compounds, including amino acids in aqueous solution [4]. Although this method is highly accurate for pKa prediction, long computation times are required to obtain the solvation free-energy difference, thus indicating that it is only applicable to small molecules. To avoid these high computational costs, it is, therefore, essential to employ solvation free-energy calculation methods that are based on static quantum chemical methods. Recently, combinations of various quantum chemical calculation methods and the PCM have been explored in considering the solvent effects. High-precision solvation free-energy calculations can be used to discuss deprotonation trends in aqueous solutions [5,6,7,8,9,10]. The accuracy is in good agreement with experimental values, as reported by Takano and Houk [11].

^{+}), ${G}_{\mathrm{solv}}\left({\mathrm{H}}^{+}\right)$, cannot be directly calculated because the proton has no electrons. Instead, ${G}_{\mathrm{solv}}\left({\mathrm{H}}^{+}\right)$ is sometimes calculated from the dissociation of a proton from the hydronium ion, ${\mathrm{H}}_{3}{\mathrm{O}}^{+}$, ${G}_{\mathrm{solv}}\left({\mathrm{H}}^{+}\right)={G}_{\mathrm{solv}}\left({\mathrm{H}}_{3}{\mathrm{O}}^{+}\right)-{G}_{\mathrm{solv}}\left({\mathrm{H}}_{2}\mathrm{O}\right)$. The hydronium ion is surrounded by many water molecules and forms a hydrogen-bonding network with them; hence, it is desirable to use larger proton solvation clusters and/or other proton exchange reactions involving protonated/deprotonated compounds, except for ${\mathrm{H}}_{2\mathrm{n}+1}{\mathrm{O}}_{\mathrm{n}}^{+}$ clusters. However, the computational costs increase exponentially when the size of the clusters increases because of the vast configurations of the water clusters, as in the AIMD calculations. Therefore, it is extremely difficult to obtain the solvation free energy of the proton with high accuracy, which indicates that, in many cases, it has not been possible to quantitatively reproduce $\mathrm{p}{K}_{\mathrm{a}}$ values.

_{2}, among others. As noted earlier, the ${G}_{\mathrm{solv}}\left({\mathrm{H}}^{+}\right)$ value depends strongly on the choice of methodology. Thus, the correction terms for the ${G}_{\mathrm{solv}}\left({\mathrm{H}}^{+}\right)$ value should be evaluated for each methodology and chemical group considered. Our linear scaling scheme, mentioned in Section 3.2, provided the error within 0.25 $\mathrm{p}{K}_{\mathrm{a}}$ units for standard PCM–DFT calculations. The details of this scheme are reported later.

## 2. Basics of pK_{a} Computation

## 3. Polarizable Continuum Model-Based Approach

#### 3.1. Basics of the Polarizable Continuum Model

_{k}, and finding the sum of them:

_{R}(

**s**) is similar to the SCF calculations performed, where the interaction energy, ${V}_{\sigma}\left(\mathit{r}\right)$, is obtained by iteratively calculating $\sigma \left(\mathit{s}\right)$ and then adding the energy of the vacancy formation, etc., as an empirical value, to obtain the overall energy.

#### 3.2. The AKB Scheme

_{2}(aniline) groups, and plotted it with the experimental $\mathrm{p}{K}_{\mathrm{a}}$ values, where linear regression curves were estimated by least-squares analysis (see Figure 2). The calculations by B3LYP/6-31++G(d, p) with the PCM-SMD provide a good result, within $0.2\mathrm{p}{K}_{a}$ units (mean absolute error) in aqueous solution [32]. Note that, here, the slopes ($k)$ for the COOH, OH (phenol), and NH

_{2}(aniline) groups differ from each other, indicating that the curves also exhibit the chemical-group dependence. Once the linear regression curve is obtained, from Equation (14), one can estimate the Gibbs energy of the proton in aqueous solution using the slope and the abscissa, as follows:

_{2}(aniline) groups are 1115.71, 1065.32, and 1094.92 kJ/mol, respectively. This result indicates that, if one adopts the same ${G}^{\mathrm{solv}}\left({\mathrm{H}}^{+}\right)$ value for evaluating the $\mathrm{p}{K}_{a}$ values of these compounds, a large error is obtained when using Equation (2). The AKB scheme complements the errors arising from both a difference in the method and the chemical groups, simultaneously.

#### 3.3. Some Applications of the AKB Scheme

#### 3.3.1. Application to Salicylic Acid

^{−}and OH groups stabilizes the deprotonated compound. This detail is presumed from the structures and experimental results, but it has not yet been accurately confirmed by theory. Although salicylic acid has good analgesic action, it can simultaneously cause the perforation of stomach ulcers as one of its adverse effects because of the low $\mathrm{p}{K}_{a}$. To reduce the adverse effect while retaining the medicinal action, acetylsalicylic acid (commercially known as aspirin) was synthesized. Here, we examine both the first and second $\mathrm{p}{K}_{a}$ values of the hydroxybenzoic acids, and the $\mathrm{p}{K}_{a}$ value of the acetylsalicylic acid, to clarify the relationship between the $\mathrm{p}{K}_{a}$ and the structure of the conformers.

#### 3.3.2. Solvent Dependence

## 4. Integral Equation-Based Approach

#### 4.1. Basics of RISM-SCF and 3D-RISM-SCF

#### 4.2. First-Principles Calculation of pK_{a} and pK_{w}

#### 4.3. Data-Driven Approach for pK_{a} Prediction with 3D-RISM-SCF

## 5. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Example of linear fitting at B3LYP/6-31++G(d, p) + PCM-SMD level. Three chemical groups (COOH, phenol, and aniline) were considered. R

^{2}is the coefficient of determination.

**Figure 3.**Two conformers of salicylic acid. Conformer (

**a**) forms a hydrogen bond between a phenol OH and a COOH group (shown by the broken line), conformer (

**b**) does not form a hydrogen bond, and conformer (

**c**) is aspirin. The black, red, and white balls represent carbon, oxygen, and hydrogen atoms, respectively.

**Figure 4.**Density and temperature dependences of $\mathsf{\Delta}\mathrm{p}{K}_{\mathrm{w}}$, the difference of the $\mathrm{p}{K}_{\mathrm{w}}\left(\rho ,T\right)$ from $\mathrm{p}{K}_{\mathrm{w}}$ (1.0 g cm

^{−3}, 273.15 K). The computational and experimental values are provided in panels (

**a**) and (

**b**), respectively. These figures are reprinted with permission from [26], Copyright 2014, American Chemical Society.

**Figure 5.**pH-dependent mole fractions for (

**a**) PCBA, and (

**b**) PCBA-complex. These figures are reprinted with permission from [42], Copyright 2016, Elsevier.

**Figure 6.**Comparison of the computed $\mathrm{p}{K}_{a}$ values with the experimental values determined using (

**a**) the LFC/3D-RISM-SCF, and (

**b**) first-principles approach of the 3D-RISM-SCF. These figures are reprinted with permission from [28], Copyright 2018, Royal Society of Chemistry.

**Table 1.**Deprotonation Gibbs energy (in kJ/mol) and estimated $\mathrm{p}{K}_{a}$ in each conformer of salicylic acid. Numbers in parentheses indicate experimental values.

Compound | $\Delta {\mathit{G}}_{0,\mathit{a}1}$ | $\mathit{p}{\mathit{K}}_{\mathit{a}1}$ | $\Delta {\mathit{G}}_{0,\mathit{a}2}$ | $\mathit{p}{\mathit{K}}_{\mathit{a}2}$ |
---|---|---|---|---|

(a) | 1157.9 | 2.69 (2.97) | 1271.4 | 13.29 (13.40) |

(b) | 1180.0 | 4.10 (2.97) | 1232.1 | 10.76 (13.40) |

(c) | 1174.6 | 3.76 (3.49) | - | - |

**Table 2.**Solvent dependence on the scaling factor, $k$, Gibbs energy of a proton, and mean absolute error (MAE) in computing $\mathrm{p}{K}_{a}$ for aniline derivatives. $N$ indicates the samples of aniline derivatives that have experimental $\mathrm{p}{K}_{a}$ values for the solvents.

Solvent | ε | N | Ref. | k | G (H^{+}) | MAE |
---|---|---|---|---|---|---|

Water | 78.4 | 14 | [35] | 0.09641 | −1094.9 | 0.38 |

Methanol | 32.7 | 6 | [35] | 0.09454 | −1080.5 | 0.19 |

DMSO | 46.7 | 4 | [35] | 0.11580 | −1112.2 | 0.22 |

Acetonitrile | 36.0 | 9 | [35] | 0.10487 | −1046.3 | 0.50 |

THF ^{a} | 7.58 | 4 | [36] | 0.11712 | −1066.4 | 0.19 |

THF ^{b} | 7.58 | 7 | [37] | 0.05938 | −988.7 | 0.42 |

Acetone ^{b} | 20.7 | 5 | [37] | 0.06186 | −1042.7 | 0.25 |

Nitromethane ^{b} | 35.9 | 6 | [37] | 0.09395 | −1048.2 | 0.23 |

^{a}$\mathrm{p}{K}_{a}$ is listed.

^{b}$\mathrm{p}{K}_{b}$ is listed.

**Table 3.**Reaction free energies and $\mathrm{p}{K}_{a}$ values for PCBA and its complex. This table is reprinted with permission from [42], Copyright 2016, Elsevier.

Reaction | $\Delta \mathit{G}\left[\mathbf{kcal}\text{}{\mathbf{mol}}^{-1}\right]$ | $\mathit{p}{\mathit{K}}_{\mathit{a}\mathit{c}\mathit{o}\mathit{m}\mathit{p}}$ | $\mathit{p}{{\mathit{K}}_{\mathit{a}}}^{\mathbf{b}}$ | |
---|---|---|---|---|

PCBA + H_{2}O→PCBA^{−} + H_{3}O^{+} | 28.74 | 21.1 | 4.7 | |

PCBA^{−} + 2H_{2}O→PCBA^{2−} + H_{3}O^{+} | 36.76 | 26.9 | (8.7 ^{a}) | 10.6 |

PCBA-complex + H_{2}O→PCBA-complex^{−} + H_{3}O^{+} | 28.13 | 20.6 | 4.3 | |

PCBA-complex^{−} + 2H_{2}O→PCBA-complex^{2−} + H_{3}O^{+} | 31.22 | 22.9 | 6.5 |

^{a}Experimental value taken from [41].

^{b}Adjusted $\mathrm{p}{K}_{a}$ values.

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**MDPI and ACS Style**

Fujiki, R.; Matsui, T.; Shigeta, Y.; Nakano, H.; Yoshida, N.
Recent Developments of Computational Methods for p*K*_{a} Prediction Based on Electronic Structure Theory with Solvation Models. *J* **2021**, *4*, 849-864.
https://doi.org/10.3390/j4040058

**AMA Style**

Fujiki R, Matsui T, Shigeta Y, Nakano H, Yoshida N.
Recent Developments of Computational Methods for p*K*_{a} Prediction Based on Electronic Structure Theory with Solvation Models. *J*. 2021; 4(4):849-864.
https://doi.org/10.3390/j4040058

**Chicago/Turabian Style**

Fujiki, Ryo, Toru Matsui, Yasuteru Shigeta, Haruyuki Nakano, and Norio Yoshida.
2021. "Recent Developments of Computational Methods for p*K*_{a} Prediction Based on Electronic Structure Theory with Solvation Models" *J* 4, no. 4: 849-864.
https://doi.org/10.3390/j4040058