# QSAR Assessing the Efficiency of Antioxidants in the Termination of Radical-Chain Oxidation Processes of Organic Compounds

^{*}

## Abstract

**:**

_{7}= 0.01–6.65 (where k

_{7}is the rate constant for the reaction of antioxidants with peroxyl radicals generated upon oxidation). Based on the atomic descriptors (Quantitative Neighborhood of Atoms (QNA) and Multilevel Neighborhoods of Atoms (MNA)) and molecular (topological length, topological volume and lipophilicity) descriptors, we have developed 9 statistically significant QSAR consensus models that demonstrate high accuracy in predicting the lgk

_{7}values for the compounds of training sets and appropriately predict lgk

_{7}for the test samples. Moderate predictive power of these models is demonstrated using metrics of two categories: (1) based on the determination coefficients R

^{2}(R

^{2}

_{TSi}, R

^{2}

_{0}, Q

^{2}(

_{F1}), Q

^{2}(

_{F2}), $\overline{{\mathrm{R}}_{\mathrm{mTSi}}^{2}}$) and based on the concordance correlation coefficient (CCC)); or (2) based on the prediction lgk

_{7}errors (root mean square error (RMSEP), mean absolute error (MAE) and standard deviation (S.D.)) The RBF-SCR method has been used for selecting the descriptors. Our theoretical prognosis of the lgk

_{7}for 8-PPDA, a known antioxidant, based on the consensus models well agrees with the experimental value measure in the present work. Thus, the algorithms for calculating the descriptors implemented in the GUSAR 2013 program allow simulating kinetic parameters of the reactions underling the liquid-phase oxidation of hydrocarbons.

## 1. Introduction

- Termination of the chains by the reaction of the inhibitor with peroxyl radicals;
- Termination of the chains by the reaction of the inhibitor with alkyl radicals;
- Termination with the compounds inducing the non-radical decomposition of the organic hydroperoxide (only if the latter is the main auto-initiator).

_{50}concentration is widely used as a semi-quantitative measure of AOA (IC

_{50}relates to the intensity of the oxidation process reduced two times or the amount of malonic dialdehyde formed upon the oxidation of unsaturated fatty acids [21,22,23,24,25,26]). The reliable quantitative characterization of AOA may be based on the rate constants of the reaction of antioxidant molecule with radicals relaying the oxidation chain. These rate constants are usually designated as k

_{7}or k

_{In}in the case of compositions based on different antioxidants. Measuring these kinetic parameters may be performed by known methods of chemical kinetics [14,27].

_{HOMO}and E

_{LUMO}, difference in the heats of formation of phenol and its radical (ΔΔH

_{R}), bond dissociation enthalpies etc.) [28,29,30,31,32,33,34]. However, all these works are built on fairly narrow sets of similar compounds, e.g., the effect of para-substituents on the antioxidant activity of 10 and 27 phenol derivatives was studied in works [28] and [29], respectively. Based on quantum chemical calculations, the authors of [28] found that the smaller the BDE number, the higher the antioxidant activity of the modeled compounds. In work [29], using the model reaction of AIBN-induced styrene oxidation (T = 323 K), it was found that the inhibition rate constants k

_{7}for para-substituted phenols correlates with the Brown-Okamoto constants σ

_{p}

^{+}. The smaller its numerical value for the para-substituent (i.e., the higher the positive inductive effect of this substituent, the higher the numerical value of k

_{7}). In the work [30], as a result of modeling 14 para-substituted phenols, 4 QSAR models were built, in which the reducing potency (E

_{mid}) correlates with quantum chemical descriptors such as ionization potential of the parent molecule (IP

_{p}), spin delocalization of the intermediate radical cation (D

_{s}

^{c}), LUMO of the parent molecule (E

_{LUMOp}), difference between the heats of formation of the phenoxyl radical (ΔΔH

_{R}) and parent molecule (ΔΔH

_{p}). In work [31] it was shown that quantum chemical descriptors such as ionization potentials (IP), absolute electronegativity (χ), activation energy (ΔΔH

^{#}), difference in the heats of formation of compounds and their radicals (ΔΔH

_{R}) make a decisive contribution to the antioxidant activity of phenols. The calculation of these descriptors was performed using the AM1 method (MOPAC 6.0 software). In work [32], moderate correlations for 30 Schiff bases were obtained between antiradical activity and quantum chemistry descriptors including the bond dissociation enthalpies related to the first and second hydrogen atom transfer (BDE and BDEd), the number of OH groups (nOH); the spin density of the active OH groups (SD); and the free enthalpy of the reaction of the reactivity of phenolic Schiff bases with the DPPH radical (ΔG). Work [33] reported that the antioxidant activity of phenols correlates with electron affinity (EA) and hardness (η) calculated with AM1 and PM3. The authors of work [34] based on 15 antioxidants built a QSAR model with high values of statistical parameters using enthalpy of homolytic dissociation of OH bonds (BDE-OH) and ionization potential (IP), and two lipophilic parameters, lipophilicity (LogP) and relative lipophilicity (LogD). In this model, the most pronounced antioxidant effect was found for the compounds with electron-donor groups directly bonded with the aromatic ring. The results of these and other similar studies are fundamental. However, it is incorrect to perform virtual screening with QSAR models based on training samples with less than 30 compounds. Thus, the use of these quantum-chemically calculated descriptors and values of Taft constants, despite of their clear physical meaning, is difficult for wide sets of compounds with different structures, i.e., such approaches are applicable only to training sets of closely related compounds.

**I**–

**V**(Figure 1) using the GUSAR 2013 program and constructed the corresponding statistically significant QSAR models for predicting the k

_{7}values.

## 2. Computational Details

#### 2.1. Computational Methodology

_{7}values is presented in Supplementary Materials (Table S2). The experimental data k

_{7}of phenol, aminophenol and uracil derivatives (in L·mol

^{−1}·s

^{−1}) were selected from the literature [27,48,49] and converted to logarithmic values (lgk

_{7}) for QSAR analysis.

#### Formation of Training and Test Sets

_{7}values are known [48,49,50]. Thus, the k

_{7}values are used as a quantitative parameter for assessing target AOA.

_{7}. The TR2 training set is designed to develop QSAR models M4–M6 and includes 59 antioxidant structures. The TS1 set was used to test the predictabilities of models M4–M6. Both of these sets were obtained by the separation of the TR1 set in the ratio 4:1, i.e., each fifth compound was transferred to TS1 from TR1. Before this, all the structures of TR1 were ranked according to the increase in lgk7 parameter. The features of sets TR1–TR2 and TS1 are presented in Table 1 and Table 2. The statistical characteristics presented in these tables indicate a fairly uniform distribution of data in all training and test sets. The average values of the lgk7 parameter and the range of its variation in all training and test sets are numerically close. The observed response values lgk7 of training sets TRi and test sets TSi are similarly distributed around training mean. These facts indicate the correctness of the formation of training and test sets.

_{7}values within the training sets are Δlgk

_{7}> 6. Thus, the condition for development of reliable QSAR models is fulfilled [53].

#### 2.2. QSAR Model Development

- zero-level MNA descriptor for each atom is the mark A of the atom itself;
- any next-level MNA descriptor for the atom is the substructure notation A (D
_{1}D_{2}…D_{i}…), where Di is the previous-level MNA descriptor for i–th immediate neighbor of the atom A.

_{1}D

_{2}…D

_{i}… are arranged in a unique manner. This may be, for example, a lexicographic sequence. MNA descriptors are generated using an iterative procedure, which results in the formation of structural descriptors that include the first, second, etc. neighborhoods of each atom. The label contains not only information about the type of atom, but also additional information about its belonging to a cyclic or acyclic system, etc. For example, an atom that does not enter a ring is marked with a “―“.

^{3}, where R is the atomic radius [45].

- (1)
- Selecting descriptors using the SCR method. This is a regularized method of the least squares. Independent parameters a are calculated in this method according to the Equation (4) [43]:$$\mathrm{a}=\mathrm{ArgMin}\left[{\displaystyle \sum}_{\mathrm{i}=1}^{n}{\mathrm{y}}_{\mathrm{i}}-{\displaystyle \sum}_{\mathrm{k}=0}^{\mathrm{m}}{\mathrm{x}}_{\mathrm{i}\mathrm{k}}{\mathrm{a}}_{\mathrm{k}}{)}^{2}+{\displaystyle \sum}_{\mathrm{k}=1}^{\mathrm{m}}{\mathrm{v}}_{\mathrm{k}}{\mathrm{a}}_{\mathrm{k}}^{2}\right]$$where a is the regression coefficient, n is the number of objects, y
_{i}is the response value of the i-th object, m is the number of independent variables, x_{ik}is the value of the k-th independent variable of the i-th object, a_{k}is the k-th value of the regression coefficients, and v_{k}is the k-th value of the regularization parameters. Equation (4) has the following solution:$$\mathrm{a}={\mathrm{TX}}^{\mathrm{T}}\mathrm{y}{,}_{}\mathrm{T}={({\mathrm{X}}^{\mathrm{T}}\mathrm{X}+\mathrm{V})}^{-1}$$where X^{T}is the transposed regression matrix X, and V is the diagonal matrix of the regularization parameters. The regression coefficients obtained from the SCR reflect the contribution of each particular descriptor (variable) to the final equation. The higher the absolute value of the coefficient, the greater its contribution. Thus, the regression coefficients obtained after the SCR can be used to weight the descriptors (variables) depending on their importance. - (2)
- Calculating the radial basis functions using the weighted coefficient of SCR as a criterion of similarity. The RBF-SCR method can be expressed as [39]:$$\mathrm{y}\left(\mathrm{x}\right)={\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{w}}_{\mathrm{i}}\mathrm{\phi}\left(\Vert \mathrm{a}\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}\Vert \right)=\mathrm{\Phi}\mathrm{w}$$where a is taken from Equation (4).The weights w are calculated as:$$\mathrm{w}={\mathrm{\Phi}}^{-1}\mathrm{y}$$
- (3)
- Calculating the weighting coefficients RBF by the least squares.

_{7}values for a particular compound is formed on the averaged predicted lgk7 values of the partial QSAR regression models.

#### 2.3. Assessment of the Descriptive and Predictive Ability of QSAR Models

_{7}values for the structures of TR1–TR3 using a metrics based on the determination coefficients of R

^{2}(R

^{2}

_{TSi}, R

^{2}

_{0}, $\overline{{\mathrm{R}}_{\mathrm{mTSi}}^{2}}$) and based on the concordance correlation coefficient (CCC). The prognostic ability of these models was assessed with the predicted lgk

_{7}values for the structures of test sets TS1 and TS2 using the metrics of two categories: (1) based on the determination coefficients R

^{2}(R

^{2}

_{TSi}, R

^{2}

_{0}, Q

^{2}(

_{F1}), Q

^{2}(

_{F2}), $\overline{{\mathrm{R}}_{\mathrm{mTSi}}^{2}}$ and CCC); or (2) based on the prediction lgk

_{7}errors (root mean square error (RMSEP), mean absolute error (MAE) and standard deviation (S.D.)) [55,56,57,58,59]. The calculations of these statistical parameters were performed using the Xternal Validation Plus 1.2 program [60].

_{7}values for N-2-ethylhexyl-N′-phenyl-p-phenylenediamine (8-PPDA), a promising industrial antioxidant, which was not included in the data array MD1. The experimental lgk

_{7}values for this compound were measured by a manometric method based on the absorption of atmospheric oxygen using the model liquid-phase ethylbenzene oxidation initiated by azodiisobutyronitrile (AIBN) at 348 K. The kinetic curves were recorded using a universal manometric differential device [61,62,63,64,65,66].

_{7}was a quantitative parameter of AOA, where f is the inhibitor capacity, equal to the number of radical intermediates decaying in the interaction with one inhibitor molecule (8-PPDA) [14]. This kinetic parameter was determined by the concentration effect of 8-PPDA on the oxidation rate of ethylbenzene, a model substrate. When analyzing the experimental data, we used the basic mechanism of the inhibited radical-chain oxidation of organic compounds (Scheme 1) [14,27].

## 3. Results and Discussion

_{7}values for these nine models are presented in Table 3. The experimental and predicted lgk

_{7}values used for calculating the statistical parameters of models M1–M9 are collected in Supplementary Materials (Tables S3–S7).

_{i}.

_{7}values based on models M3, M6 and M9 versus experimental ones is shown in Figure 4. Dependencies depicted in this figure clearly indicate a fairly high descriptive and predictive ability of models M3, M6 and M9.

^{2}(R

^{2}

_{TSi}, R

^{2}

_{0}, Q

^{2}(

_{F1}), Q

^{2}(

_{F2}), $\overline{{\mathrm{R}}_{\mathrm{mTSi}}^{2}}$, CCC); and (2) metrics that allow estimating prediction errors of lgk

_{7}values (RMSEP, MAE, S.D.) [53,55,56,57,58,59]. These statistical parameters were calculated using the Xternal Validation Plus 1.2 program [60]. Additionally, in this program, we estimated the systematic errors of the models.

_{i}possesses a high descriptive ability if its determination coefficients of different types for 95% of the data TR

_{i}are close to each other and tend to unity.

_{i}possesses a high predictive ability if the following four conditions are simultaneously fulfilled for 95% of the test sets:

- (1)
- determination coefficients R
^{2}, R^{2}_{0}, R^{2}′_{0}, Q^{2}_{F1}, Q^{2}_{F2}and CCC criterion are close to each other and tend to unity; - (2)
- R
^{2}_{m}> 0.5 if ΔR^{2}_{m}< 0.2; - (3)
- MAE value does not exceed 10% of the Δlgk
_{7}interval of the compounds from TR_{i}; - (4)
- the sum MAE + 3·S.D. does not exceed 20% of the Δlgk
_{7}interval of the compounds from TR_{i}.

_{i}has a low predictive ability if the following four conditions are simultaneously fulfilled for 95% of the test sets:

- (1)
- determination coefficients R
^{2}, R^{2}_{0}, R^{2}′_{0}, Q^{2}_{F1}, Q^{2}_{F2}and CCC criterion do not exceed the threshold value equal to 0.6; - (2)
- R
^{2}_{m}≤ 0.5 if ΔR^{2}_{m}≤ 0.2; - (3)
- MAE value is higher than 15% of the Δlgk
_{7}interval of the compounds from TR_{i}; - (4)
- the sum MAE + 3·S.D. is larger than 25% of the Δlgk
_{7}interval of the compounds from TR_{i}.

_{7}are presented as diagrams (Figure 5 and Figure 6). The full set of the calculated statistical parameters for the TR1–TR3 and TS1–TS2 structures is available as Supplementary Materials (Tables S3–S7). The diagrams demonstrate that models M1–M9 possess high descriptive and predictive abilities.

_{7}values. Figure 5 shows the structures of 7 compounds with the general structural formula I used for the comparative analysis. More detailed information on the effect of functional groups on the antioxidant activity of compounds with the general structural formulas I-V is presented in Supplementary Materials (Figures S1–S10, Tables S8–S9).

_{7}parameter. At the same time, electron-acceptor substituents in the same positions lead to the opposite effect. In compounds V, the effect of electron-donor substituents is ambiguous. Methyl groups in positions R

_{1}, R

_{2}, R

_{3}increase the nominal value of lgk

_{7}but replacing the hydrogen atom from positions R

_{1}and R

_{2}by the hydroxyl or amino group, in contrast, reduces the antioxidant properties. A detailed description of this fact is presented in Supplementary Materials (Figure S6). Uracil derivatives with electron-acceptor substituents in R

_{1}, R

_{2}, R

_{3}and R

_{4}positions are not used in the modeling, and for this reason not discussed [61].

_{7}parameter of 8-PPDA, an aromatic amine antioxidant. We choose these models for this purpose as they are based on the descriptors of different types and, therefore, expected to provide more accurate estimates. The results of the lgk7 calculation for 8-PPDA within the M3, M6 and M9 models are shown in Table 6.

_{7}.

^{−4}mol·L

^{−1}:

_{0}are the rates of the oxygen absorption with and without 8-PPDA in the reaction system, respectively.

^{exp}

_{7}= (4.8 ± 0.2)·10

^{5}mol·L

^{−1}·s

^{−1}. The induction period (τ) on the oxygen absorption curves corresponding the 8-PPDA-inhibited ethylbenzene oxidation (Figure 6) linearly depends on the inhibitor concentration. This dependence (Figure 8) allows assessing the stoichiometric inhibition coefficient via Equation (8):

_{i}is the initiation rate of the oxidation process. The calculation according to Equation (8) gives f = 2. Thus, the inhibition rate constant k

_{7}

^{exp}for 8-PPDA can be calculated as:

_{7}

^{pred}) with the experimental one (lgk

_{7}

^{obs}) demonstrates a high predictability of the QSAR consensus models M3, M6 and M9. Hence, they can be applied to screening novel antioxidant structures.

_{7}values for the TR1–TR3 structures, external and internal TS1 and TS2 test sets, and 8-PPDA. These models can be used for screening on virtual libraries and databases to find new antioxidants among substituted phenols, polyphenols, aminophenols, aromatic amines and uracils.

_{7}) with high accuracy and this program could be recommended as an auxiliary tool when searching for new antioxidants.

## 4. Conclusions

_{7}parameter (lgk

_{7}0.01 ÷ 6.65 for these compounds). Operating with MNA/QNA descriptors, descriptors corresponding to the whole molecule (topological length, topological volume and lipophilicity), RBF-SCR method, we have developed 9 statistically significant QSAR consensus models. The QSAR models demonstrate high accuracy in predicting the lgk

_{7}values for the compounds of training sets and appropriately predict lgk

_{7}for the test sets (R

^{2}

_{TR}> 0.6; Q

^{2}

_{TR}> 0.5; R

^{2}

_{TS}> 0.5). We recommend using the QSAR models M3, M6 and M9 for virtual screening of new antioxidants because they are based on a combination of the descriptors different types. This guarantees reliability of the predicted lgk

_{7}values.

_{7}parameter only for antioxidants with general structural formulas I–V. At the same time, the range of correct prediction of the lgk

_{7}parameter using all the QSAR models developed by us, including the M3, M6, M9 models, is quite wide. The experimental values of the lgk

_{7}parameter for the modeled antioxidants, on the basis of which all QSAR models were developed and their validity was assessed, varied from 0.01 to 6.65.

_{7}parameter. Electron-acceptor substituents in the same positions reduce the antioxidant activity of phenols. In uracils with the general structural formulas V, the effect of electron-donor substituents is ambiguous. The introduction of methyl groups in the positions R

_{1}, R

_{2}, R

_{4}increases the nominal value of the parameter lgk

_{7}but replacing the hydrogen atom from R

_{3}by the hydroxyl or amino group, in contrast, leads to a decrease in the antioxidant properties (see Supplementary Materials, Figures S1–S6).

_{7}values for the compounds of the test sets and 8-PPDA, a ”young” member of the inhibitors family, indicates that the GUSAR 2013 algorithms are applicable to simulating the kinetic parameters of the model liquid-phase oxidation reactions of organic hydrocarbons. Thus, the GUSAR 2013 program allows modeling kinetic parameters relating to non-specific activity in addition to the ADMET properties and diverse biological activities mentioned in the introduction. Previously, we have reported the first results of a successful QSPR simulation of nonspecific activity using the GUSAR 2013 program. In the previous study [68], we have demonstrated the efficiency of the GUSAR descriptors for simulating the photovoltaic performances of the methanofullerene derivatives (we built six statistically significant QSPR consensus models for predicting the power conversion efficiencies of the methanofullerene-based organic solar cells). This QSPR study develops the idea of successfully applying the GUSAR approaches to describing nonspecific activities of the compounds. Our studies in this area will be continued.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**The effect of structural features on the antioxidant activity of compounds with general structural formula I.

**Figure 6.**The kinetic curves of the oxygen absorption upon the ethylbenzene oxidation (V

_{i}= 2.4·10

^{−7}mol·L

^{−1}·s

^{−1}, 348 K) without (

**×**) and with 8-PPDA: (

**a**) 2 × 10

^{−5}mol·L

^{−1}(

`▲`) and 4.03 × 10

^{−5}mol·L

^{−1}(

`■`); (

**b**) 1 × 10

^{−4}mol·L

^{−1}(●) and 1.61 × 10

^{−4}mol L

^{−1}(♦).

**Figure 7.**Dependence of the inhibition parameter on the 8-PPDA concentrations (V

_{i}=2.4·10

^{−7}mol·L

^{−1}·s

^{−1}, 348 K); the correlation coefficient R = 0.95.

**Figure 8.**Dependence of the induction period on the initial concentration of the inhibitor (348 K, V

_{i}= 2.4·10

^{−7}mol·L

^{−1}·s

^{−1}).

Value | Designation of TR_{i} | Code of the Set | ||
---|---|---|---|---|

TR1 | TR2 | TR3 | ||

Number of compounds | N | 74 | 59 | 62 |

Mean value of lgk_{7} | $\overline{{\mathrm{lgk}}_{7}}$ | 3.3750 | ||

Spread of lgk_{7} | Δlgk_{7} | 6.5500 | ||

Distribution of the observed response values of training sets TRi around training mean (in %) | $\overline{{\mathrm{lgk}}_{7}}\pm 0.5$ (%) | 20.2703 | 20.0000 | 16.6667 |

$\overline{{\mathrm{lgk}}_{7}}\pm 1.0$ (%) | 50.0000 | 46.6667 | 50.0000 | |

$\overline{{\mathrm{lgk}}_{7}}\pm 1.5$ (%) | 72.9730 | 73.3333 | 75.0000 | |

$\overline{{\mathrm{lgk}}_{7}}\pm 2.0$ (%) | 83.7838 | 80.0000 | 83.3333 | |

0.10 × Δlgk_{7} | 0.6550 | |||

0.15 × Δlgk_{7} | 0.9825 | |||

0.20 × Δlgk_{7} | 1.3100 | |||

0.25 × Δlgk_{7} | 1.6375 |

Value | Designation of TS_{i} | Code of the Set | |
---|---|---|---|

TS1 | TS2 | ||

Number of compounds | N | 15 | 12 |

Mean value of lgk_{7} | $\overline{{\mathrm{lgk}}_{7}}$ | 4.1727 | 4.0821 |

Spread of lgk_{7} | Δlgk_{7} | 6.0800 | 5.8000 |

Distribution of the observed response values of test sets TSi around test mean (in %) | $\overline{{\mathrm{lgk}}_{7}}\pm 0.5$ (%) | 53.3333 | 50.0000 |

$\overline{{\mathrm{lgk}}_{7}}\pm 1.0$ (%) | 73.3333 | 75.0000 | |

$\overline{{\mathrm{lgk}}_{7}}\pm 1.5$ (%) | 86.6667 | 83.3333 | |

$\overline{{\mathrm{lgk}}_{7}}\pm 2.0$ (%) | 86.6667 | 91.6667 | |

Distribution of the observed response values of test sets TSi around train mean (in %) | $\overline{{\mathrm{lgk}}_{7}}\pm 0.5$ (%) | 20.0000 | 16.6667 |

$\overline{{\mathrm{lgk}}_{7}}\pm 1.0$ (%) | 46.6667 | 50.0000 | |

$\overline{{\mathrm{lgk}}_{7}}\pm 1.5$ (%) | 73.3333 | 75.0000 | |

$\overline{{\mathrm{lgk}}_{7}}\pm 2.0$ (%) | 80.0000 | 83.3333 |

**Table 3.**Statistical parameters and accuracy of the predicted lgk

_{7}values of the compounds from training sets TR1–TR3 within the M1–M9 consensus models (using RBF-SCR). Δlgk

_{7 (TR1)}= Δlgk

_{7 (TR2)}= Δlgk

_{7 (TR3)}= 6.55.

^{1}

Training Set | Model | N | N_{PM} | $\overline{{\mathbf{R}}^{2}}$ | $\overline{\mathbf{F}}$ | $\overline{\mathbf{S}.\mathbf{D}.}$ | $\overline{{\mathbf{Q}}^{2}}$ | V |
---|---|---|---|---|---|---|---|---|

QSAR Models Based on the QNA Descriptors | ||||||||

TR1 | M1 | 74 | 20 | 0.999 | 10.457 | 0.525 | 0.843 | 22 |

TR2 | M4 | 59 | 20 | 0.999 | 7.676 | 0.587 | 0.799 | 18 |

TR3 | M7 | 62 | 20 | 0.999 | 6.059 | 0.544 | 0.829 | 24 |

QSAR Models Based on the MNA Descriptors | ||||||||

TR1 | M2 | 74 | 20 | 0.999 | 18.207 | 0.478 | 0.867 | 18 |

TR2 | M5 | 59 | 20 | 0.998 | 11.554 | 0.556 | 0.819 | 15 |

TR3 | M8 | 62 | 20 | 0.999 | 9.864 | 0.567 | 0.810 | 17 |

QSAR Models Based on Both QNA and MNA Descriptors | ||||||||

TR1 | M3 | 74 | 100 | 0.999 | 10.744 | 0.490 | 0.872 | 22 |

TR2 | M6 | 59 | 100 | 0.999 | 7.768 | 0.559 | 0.830 | 18 |

TR3 | M9 | 62 | 100 | 0.999 | 7.011 | 0.535 | 0.845 | 21 |

^{1}N is the number of structures in the training set; N

_{PM}is the number of regression equations used for the consensus model; $\overline{{\mathrm{R}}_{}^{2}}$ is the determination coefficient calculated for the compounds of TR

_{i}; $\overline{{\mathrm{Q}}_{}^{2}}$ is the correlation coefficient calculated for the training set with the by cross-validation with exception of one; $\overline{\mathrm{F}}$ is the Fisher criterion; $\overline{\mathrm{S}.\mathrm{D}{.}_{}}$—standard deviation; V is the number of variables in the final regression equation.

**Table 4.**The validation parameters of the QSAR models estimated using the Xternal Validation Plus 1.2 program based on the experimental and predicted lgk

_{7}values of the compounds form test sets TS1 and TS2. Δlgk

_{7(TR1)}= Δlgk

_{7(TR2)}= Δlgk

_{7(TR3)}= 6.55; Δlgk

_{7(TS1)}= 6.08; Δlgk

_{7(TS2)}= 5.80.

^{1}

Comments | Prediction Parameters | QSAR Model Used for Predicting lgk_{7} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

TR1 | TR2 | TR3 | ||||||||

M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9 | ||

Classical Metrics (100% data) | R^{2} | 0.9894 | 0.9922 | 0.9923 | 0.9878 | 0.9897 | 0.9899 | 0.9888 | 0.9902 | 0.9893 |

R^{2}_{0} | 0.9887 | 0.9906 | 0.9894 | 0.9869 | 0.9888 | 0.9876 | 0.9876 | 0.9890 | 0.9871 | |

R^{2′}_{0} | 0.9878 | 0.9899 | 0.9881 | 0.9860 | 0.9881 | 0.9863 | 0.9866 | 0.9882 | 0.9856 | |

$\overline{{\mathrm{R}}_{\mathrm{m}}^{2}}$ | 0.9538 | 0.9568 | 0.9387 | 0.9520 | 0.9550 | 0.9369 | 0.9493 | 0.9514 | 0.9357 | |

ΔR^{2}_{m} | 0.0120 | 0.0101 | 0.0115 | 0.0128 | 0.0109 | 0.0123 | 0.0122 | 0.0108 | 0.0129 | |

CCC | 0.9947 | 0.996 | 0.9952 | 0.9929 | 0.9942 | 0.9934 | 0.9932 | 0.9942 | 0.9930 | |

Classical Metrics (after removing 5% data with high residuals) | R^{2} | 0.9917 | 0.9939 | 0.9936 | 0.9899 | 0.9921 | 0.9918 | 0.9922 | 0.9936 | 0.9928 |

R^{2}_{0} | 0.9911 | 0.9932 | 0.9918 | 0.9893 | 0.9914 | 0.9900 | 0.9917 | 0.9931 | 0.9913 | |

R^{2′}_{0} | 0.9571 | 0.9589 | 0.9420 | 0.9553 | 0.9571 | 0.9402 | 0.9628 | 0.9648 | 0.9467 | |

$\overline{{\mathrm{R}}_{\mathrm{m}}^{2}}$ | 0.9624 | 0.9632 | 0.9470 | 0.9606 | 0.9614 | 0.9452 | 0.9670 | 0.9683 | 0.9511 | |

ΔR^{2}_{m} | 0.0094 | 0.0074 | 0.0089 | 0.0104 | 0.0084 | 0.0099 | 0.0080 | 0.0067 | 0.0085 | |

CCC | 0.9962 | 0.9973 | 0.9965 | 0.9944 | 0.9955 | 0.9947 | 0.9956 | 0.9962 | 0.9954 | |

Mean absolute error and standard deviation for test set (100% data) | RMSE | 0.1341 | 0.1195 | 0.1277 | 0.1518 | 0.1372 | 0.1454 | 0.1466 | 0.1364 | 0.1486 |

MAE | 0.1014 | 0.0894 | 0.0997 | 0.1132 | 0.1012 | 0.1115 | 0.1060 | 0.0976 | 0.1103 | |

S.D. | 0.0920 | 0.0834 | 0.0841 | 0.1020 | 0.0934 | 0.0941 | 0.1021 | 0.0960 | 0.1004 | |

MAE+3·S.D. | 0.3774 | 0.3396 | 0.3520 | 0.4192 | 0.3814 | 0.3938 | 0.4123 | 0.3856 | 0.4115 | |

Mean absolute error and standard deviation for test set (after removing 5% data with high residuals) | RMSE | 0.1284 | 0.1192 | 0.1247 | 0.1326 | 0.1234 | 0.1289 | 0.1180 | 0.1095 | 0.1203 |

MAE | 0.0982 | 0.0889 | 0.099 | 0.1003 | 0.0910 | 0.1003 | 0.0887 | 0.0812 | 0.0936 | |

S.D. | 0.0853 | 0.0821 | 0.0795 | 0.0874 | 0.0842 | 0.0816 | 0.0785 | 0.0741 | 0.0762 | |

MAE+3·S.D. | 0.3541 | 0.3352 | 0.3375 | 0.3627 | 0.3436 | 0.3453 | 0.3241 | 0.3036 | 0.3222 | |

Distribution of prediction errors (in %) | ωN in range 0.10 × Δlgk_{7 (TR)} | 0.000 ^{a} | 0.000 ^{a} | 0.000 ^{a} | 0.000 ^{b} | 0.000 ^{b} | 0.000 ^{b} | 0.000 ^{c} | 0.000 ^{c} | 0.000 ^{c} |

ωN in range 0.15 × Δlgk_{7 (TR)} | 0.000 ^{a} | 0.000 ^{a} | 0.000 ^{a} | 0.000 ^{b} | 0.000 ^{b} | 0.000 ^{b} | 0.000 ^{c} | 0.000 ^{c} | 0.000 ^{c} | |

ωN in range 0.20 × Δlgk_{7 (TR)} | 0.000 ^{a} | 0.000 ^{a} | 0.000 ^{a} | 0.000 ^{b} | 0.000 ^{b} | 0.000 ^{b} | 0.000 ^{c} | 0.000 ^{c} | 0.000 ^{c} | |

ωN in range 0.25 × Δlgk_{7 (TR)} | 0.000 ^{a} | 0.000 ^{a} | 0.000 ^{a} | 0.000 ^{b} | 0.000 ^{b} | 0.000 ^{b} | 0.000 ^{c} | 0.000 ^{c} | 0.000 ^{c} | |

Prediction quality | - | Good | ||||||||

Systematic error presence | - | Absent |

^{1}Where R

^{2}, R

^{2}

_{0}, and R′

^{2}are determination coefficients calculated with and without taking into account the origin; $\overline{{\mathrm{R}}_{\mathrm{m}}^{2}}$ is the averaged determination coefficient of the regression function, calculated using values of determination coefficients on the ordinate axis (R

^{2}

_{m}) and using them on the abscissa (R′

^{2}

_{m}) respectively; $\mathsf{\Delta}{\mathrm{R}}_{\mathrm{m}}^{2}$ is the difference between R

^{2}

_{m}and R′

^{2}

_{m}; CCC is the concordance correlation coefficient; MAE is the mean absolute error; S.D. is the standard deviation; ωΝis the percentage of training sets TR1–TR3, for which the prediction error is less than the interval proportional to 0.1, 0.15, 0.20, and 0.25 of Δlgk

_{7}of training sets TR1 (a), TR2 (b) and TR4 (c).

**Table 5.**The validation parameters of the QSAR models estimated using the Xternal Validation Plus 1.2 program based on the experimental and predicted lgk

_{7}values of the compounds form training sets TrS1–TrS3

^{1}; Δlgk

_{7(TR1)}= Δlgk

_{7(TR2)}= Δlgk

_{7(TR3)}= 6.55.

^{1}

Comments | Prediction Parameters | QSAR Model Used for Predicting lgk_{7} | |||||
---|---|---|---|---|---|---|---|

TS1 | TS2 | ||||||

M4 | M5 | M6 | M7 | M8 | M9 | ||

Classical Metrics (100% data) | R^{2} | 0.9469 | 0.9461 | 0.9388 | 0.8639 | 0.8797 | 0.8876 |

R^{2}_{0} | 0.9360 | 0.9423 | 0.9260 | 0.8638 | 0.8769 | 0.8737 | |

R^{2′}_{0} | 0.9164 | 0.9312 | 0.9005 | 0.8413 | 0.8454 | 0.8184 | |

Q^{2}_{F1} | 0.9531 | 0.9579 | 0.9454 | 0.8931 | 0.9031 | 0.9006 | |

Q^{2}_{F2} | 0.9357 | 0.9423 | 0.9251 | 0.8638 | 0.8766 | 0.8734 | |

$\overline{{\mathrm{R}}_{\mathrm{m}}^{2}}$ | 0.8152 | 0.8592 | 0.7941 | 0.7951 | 0.7754 | 0.7188 | |

ΔR^{2}_{m} | 0.0663 | 0.0570 | 0.0771 | 0.1208 | 0.1155 | 0.1285 | |

CCC | 0.9634 | 0.9686 | 0.9566 | 0.9263 | 0.9306 | 0.9254 | |

Classical Metrics (after removing 5% data with high residuals) | R^{2} | 0.9710 | 0.9715 | 0.9659 | 0.8846 | 0.9106 | 0.9046 |

R^{2}_{0} | 0.9643 | 0.9700 | 0.9575 | 0.8840 | 0.9100 | 0.8963 | |

R^{2′}_{0} | 0.8550 | 0.9051 | 0.8334 | 0.8045 | 0.8043 | 0.7186 | |

Q^{2}_{F1} | 0.9731 | 0.9784 | 0.9670 | 0.8959 | 0.9197 | 0.9038 | |

Q^{2}_{F2} | 0.9610 | 0.9687 | 0.9521 | 0.8778 | 0.9057 | 0.8870 | |

$\overline{{\mathrm{R}}_{\mathrm{m}}^{2}}$ | 0.8739 | 0.9202 | 0.8562 | 0.8342 | 0.8455 | 0.7694 | |

ΔR^{2}_{m} | 0.0364 | 0.0292 | 0.0437 | 0.0607 | 0.0835 | 0.1035 | |

CCC | 0.9783 | 0.9834 | 0.9729 | 0.9382 | 0.9505 | 0.9374 | |

Mean absolute error and standard deviation for test set (100% data) | RMSE | 0.3318 | 0.3145 | 0.3581 | 0.4984 | 0.4744 | 0.4805 |

MAE | 0.2664 | 0.2417 | 0.2843 | 0.4167 | 0.3748 | 0.4150 | |

S.D. | 0.2048 | 0.2083 | 0.2254 | 0.2856 | 0.3038 | 0.2529 | |

MAE + 3·S.D. | 0.8808 | 0.8666 | 0.9605 | 1.2735 | 1.2862 | 1.1737 | |

Mean absolute error and standard deviation for test set (after removing 5% data with high residuals) | RMSE | 0.2598 | 0.2326 | 0.2878 | 0.4457 | 0.3914 | 0.4285 |

MAE | 0.2254 | 0.1981 | 0.2422 | 0.3735 | 0.3172 | 0.3740 | |

S.D. | 0.1342 | 0.1266 | 0.1613 | 0.2550 | 0.2404 | 0.2194 | |

MAE + 3·S.D. | 0.6279 | 0.5778 | 0.7261 | 1.1385 | 1.0385 | 1.0322 | |

Distribution of prediction errors (in %) | ωN in range 0.10 × Δlgk_{7 (TR)} | 6.6667 ^{a} | 6.6667 ^{a} | 6.6667 ^{a} | 25.000 ^{b} | 25.000 ^{b} | 16.667 ^{b} |

ωN in range 0.15 × Δlgk_{7 (TR)} | 0.0000 ^{a} | 0.0000 ^{a} | 0.0000 ^{a} | 0.0000 ^{b} | 8.3333 ^{b} | 0.0000 ^{b} | |

ωN in range 0.20 × Δlgk_{7 (TR)} | 0.0000 ^{a} | 0.0000 ^{a} | 0.0000 ^{a} | 0.0000 ^{b} | 0.0000 ^{b} | 0.0000 ^{b} | |

ωN in range 0.25 × Δlgk_{7 (TR)} | 0.0000 ^{a} | 0.0000 ^{a} | 0.0000 ^{a} | 0.0000 ^{b} | 0.0000 ^{b} | 0.0000 ^{b} | |

Prediction quality | - | Good | |||||

Systematic error presence | - | Absent |

^{1}Where R

^{2}, R

^{2}

_{0}, and R′

^{2}are determination coefficients calculated with and without taking into account the origin; $\overline{{\mathrm{R}}_{\mathrm{m}}^{2}}$ is the averaged determination coefficient of the regression function, calculated using values of determination coefficients on the ordinate axis (R

^{2}

_{m}) and using them on the abscissa (R′

^{2}

_{m}) respectively; $\mathsf{\Delta}{\mathrm{R}}_{\mathrm{m}}^{2}$ is the difference between R

^{2}

_{m}and R′

^{2}

_{m}; Q

^{2}

_{F1}and Q

^{2}

_{F2,}are determination coefficients calculated for the compounds of test sets TS1 and TS2 taking into account the average lgk

_{7}value of the compounds from training and test sets, respectively; CCC is the concordance correlation coefficient; MAE is the mean absolute error; S.D. is the standard deviation; ωΝ is the percentage of test sets TS1 and TS2, for which the prediction error is less than the interval proportional to 0.1, 0.15, 0.20, and 0.25 of Δlgk

_{7}of training sets TR2 (a) and TR3 (b).

**Table 6.**The lgk

_{7}values of N-2-ethylhexyl-N’-phenyl-n-phenylendiamine (8-PPDA) predicted with the M3, M6 and M9 models.

Model | Applicability (AD) | Predicted Value lgk_{7}^{pred} | Predicted value k_{7}^{pred}·10^{−5}(L·mol ^{−1}·s^{−1}) |
---|---|---|---|

M3 | in AD | 5.3258 | 2.12 |

M6 | in AD | 5.3252 | 2.12 |

M9 | in AD | 5.5896 | 3.88 |

**Table 7.**The dependence of the initial rate of the ethylbenzene oxidation on the 8-PPDA concentration (V

_{i}= 2.4·10

^{−7}mol·L

^{−1}·s

^{−1}, T = 348 K).

[8-PPDA]·10^{4} (M) | V_{0}·10^{6} (mol·L^{−1}·s^{−1}) |
---|---|

0.000 | 4.700 |

0.200 | 2.480 |

0.403 | 0.801 |

0.805 | 0.481 |

1.610 | 0.191 |

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**MDPI and ACS Style**

Khairullina, V.; Safarova, I.; Sharipova, G.; Martynova, Y.; Gerchikov, A.
QSAR Assessing the Efficiency of Antioxidants in the Termination of Radical-Chain Oxidation Processes of Organic Compounds. *Molecules* **2021**, *26*, 421.
https://doi.org/10.3390/molecules26020421

**AMA Style**

Khairullina V, Safarova I, Sharipova G, Martynova Y, Gerchikov A.
QSAR Assessing the Efficiency of Antioxidants in the Termination of Radical-Chain Oxidation Processes of Organic Compounds. *Molecules*. 2021; 26(2):421.
https://doi.org/10.3390/molecules26020421

**Chicago/Turabian Style**

Khairullina, Veronika, Irina Safarova, Gulnaz Sharipova, Yuliya Martynova, and Anatoly Gerchikov.
2021. "QSAR Assessing the Efficiency of Antioxidants in the Termination of Radical-Chain Oxidation Processes of Organic Compounds" *Molecules* 26, no. 2: 421.
https://doi.org/10.3390/molecules26020421