# QSPR Modeling and Experimental Determination of the Antioxidant Activity of Some Polycyclic Compounds in the Radical-Chain Oxidation Reaction of Organic Substrates

^{1}

^{2}

^{*}

## Abstract

**:**

_{7}, where k

_{7}is the rate constant for the oxidation chain termination by the antioxidant molecule. These results can be used to search for new potentially effective antioxidants in virtual libraries and databases and adequately predict logk

_{7}for test samples. A combination of MNA- and QNA-descriptors with three whole molecule descriptors (topological length, topological volume, and lipophilicity) was used to develop six statistically significant valid consensus QSPR models, which have a satisfactory accuracy in predicting logk

_{7}for training and test set structures: R

^{2}

_{TR}> 0.6; Q

^{2}

_{TR}> 0.5; R

^{2}

_{TS}> 0.5. Our theoretical prediction of logk

_{7}for antioxidants AO1 and AO2, based on consensus models agrees well with the experimental value of the measure in this paper. Thus, the descriptor calculation algorithms implemented in the GUSAR2019 software allowed us to model the kinetic parameters of the reactions underlying the liquid-phase oxidation of organic hydrocarbons.

## 1. Introduction

_{7}parameter in the series of phenolic antioxidants [45,46,47,48,49,50,51,52,53].

_{7}, in order to search for new potentially effective antioxidants in virtual libraries and databases. However, the training sets used in our previous models for predicting logk

_{7}for phenolic antioxidants did not contain the structures of the chromanol conjugates with lupanoic acids, 20-hydroxyecdysone, and for this reason, they could not be used for the quantitative AOA prediction for the structural analogues of these organic compounds.

_{7}for biologically active phenolic antioxidants with general formulas

**I**–

**VIII**(Figure 1) (k

_{7}is the rate constant for chain termination by the antioxidant molecule and is actually an objective quantitative characteristic of AOA) and to predict and experimentally determine this quantitative AOA parameter for two promising antioxidants, chromane derivatives. The practical significance of this study should be the applicability of these QSPR models to predict logk

_{7}for the biologically active phenolic derivatives and, hence, an objective selection of such compounds as oxidation inhibitors from virtual and synthetic libraries and databases.

## 2. Results and Discussion

#### 2.1. Prediction of the Numerical Values of the Parameter k_{7} Using the GUSAR2019 Program

_{7}for the phenolic type antioxidants, namely the sulfur-containing alkylphenols, the natural phenols, the chromane derivatives, the betulonic and betulinic acids, and 20-hydroxyecdysone. These models differ in the type of descriptors they contain and the number of partial regression relationships. The descriptive power characteristics of the M1–M6 consensus models, calculated automatically in the GUSAR2019 program by comparing the experimental logk

_{7}values with those predicted by these six models are presented in Table 1. Note that the determination coefficients, the standard deviations, and Fisher’s criterion values presented in Table 1 are the average values obtained taking into account all partial regression models included in the consensus model Mi (i = 1–6).

_{7}values for the antioxidant structures contained in test sets TR1 and TR2. As in our previous studies [67,68,69,70], in addition to the parameters calculated in the GUSAR2019 program (average R

^{2}, average Q

^{2}, average F), we used metrics based on the R

^{2}determination factors (R

^{2}, R

^{2}

_{0}, R

^{2’}

_{0}, average R

^{2}

_{m}, ΔR

^{2}

_{m}Q

^{2}

_{F1}, Q

^{2}

_{F2}, CCC); and the metrics designed to estimate the prediction errors of the logk

_{7}values (RMSE, MAE, SD) [34,35,36,37]. The metrics based on the prediction error estimates were used to determine the true prediction quality index for the parameter logk

_{7}for the compounds of both test sets. Their calculation was performed using the Xternal Validation Plus 1.2 program [71]. The same program was used to check the models for systematic errors.

^{2}, R

^{2}

_{0}, R

^{2’}

_{0}, average R

^{2}

_{m}, and CCC, evaluated by comparing the values of logk

_{7}

^{pred}and logk

_{7}

^{exp}fully met all of the the requirements, corresponding to the models with a high descriptive power listed in part 2.3. The M6 model (100% and 95% of the data) had the highest descriptive power in a number of determination coefficients (R

^{2}, R

^{2}

_{0}, CCC). At the same time, other criteria indicated the best reproducibility of the experimental data of test sets TS1 and TS2 (100% and 95% data) using the models M3 (R

^{2’}

_{0}, ∆R

^{2}

_{m}), M4 (average R

^{2}

_{m}), and M5 (R

^{2’}

_{0}, average R

^{2}

_{m}). The M3 and M6 models were characterized by the lowest values of the prediction errors of the logk

_{7}value for the structures of both training sets (RMSE, MAE, SD) at 100% and 95% of the data contained in them. At the same time, the best characteristics were in the M3 model (Table 2). The minimum SD value at 100% of the data in both training sets was shown by the M4 model. In the case of 95% of the data in these training sets, the best result was observed for the M6 model. Since the numerical values of the MAE for all of the models were in the range of 0.0599–0.0679, which is significantly lower than 0.706 (10% of the range of the simulated logk

_{7}values), and simultaneously, the numerical values of the MAE+3SD criterion were also significantly smaller than 0.706, we can conclude that almost all of the models had a high descriptive power.

_{7}values for the 100% antioxidant structures contained in test sets TS1 and TS2. Thus, the coefficient of determination R

_{2}and its analogs (R

^{2}

_{0}, R

^{2’}

_{0}) were in the range of 0.4500–0.6882, the CCC criterion ranged from 0.6483 to 0.8086, which allowed us to characterize the prognostic ability of these models as low. At the same time, the most successful predictions, if we focus on these criteria, were observed for the structures of test set TS2. Meanwhile, a more reliable estimate of the predictive power of the M1–M6 models, taking into account 100% of the data in the test sets, can be obtained by analyzing the criteria based on the logk

_{7}prediction errors for the same antioxidant structures. Specifically, the MAE and MAE+3SD criteria ranged from 0.3472 (M6, TS2) to 0.4696 (M5, TS1) and from 1.4586 (M4, TS2) to 2.1112 (M5, TS1). According to these criteria, the models with moderate predictive powers are M1 (TS1), M3 (TS1), and M4–M6 (TS2). Thus, the analysis of prediction errors for antioxidant structures contained in test sets TS1 (100% data) and TS2 (100% data) did not remove the uncertainty factor in assessing the predictive power of the M1–M6 models.

_{7}prediction errors for the structures contained in TS1 and TS2.

^{2}criterion increased approximately by 30% and ranged from 0.7289 to 0.8204. The coefficient of determination R

^{2}

_{0}increased in parallel and was almost in the same range: 0.7263–0.8115. The maximum values of these criteria were found in both cases when the M1 model was used for the prediction tasks in the series of antioxidants contained in test set TS1 (95% of the data). According to the criteria mentioned in Part 2.3, the M5 and M6 models were insignificantly inferior in their predictive power. This fact was established in the prediction of logk

_{7}for the antioxidants included in test set TS2 (95% data). From the analysis of the numerical values of all of the other types of determination coefficients, which are presented in Table 3, we can conclude that in some cases, the M5 model demonstrated the greatest prognostic ability. We reached this conclusion by analyzing CCC, R

^{2’}

_{0}, and average R

^{2}

_{m}values for the compounds of test set TS2 (95% data). The highest values of the criteria Q

^{2}

_{F1}, Q

^{2}

_{F2}differed in the results of the prediction of logk

_{7}for the structures of the same test set performed using the M6 model (Table 3). When evaluating the prognostic ability of the M1-M6 models, taking into account the prediction errors of the logk

_{7}values for 95% of the data in test sets TS1, TS2, the most successful predictions were also observed for the test set TS2 structures. The M6 model showed the lowest values of the RMSEP error, the SD standard deviation, and the MAE+3SD criterion. On the same dataset, the M5 model showed the minimum MAE error.

_{7}values for the antioxidant structures contained in test sets TS1 and TS2. An obvious proof of this fact is the plot depicted in Figure 2, which shows a satisfactory correlation between the experimental and predicted values of logk

_{7}for the structures of test sets TS1 and TS2 (95% data).

_{7}values for the antioxidants can be constructed using either one particular type of descriptor (QNA or MNA descriptors) or a combination of the descriptors in a consensus approach.

_{7}for the antioxidants AO1 and AO2. The results of these calculations are summarized in Table 4.

#### 2.2. Experimental Determination of the Inhibition Rate Constants k_{7} for Compounds AO1 and AO2. Methods of the Kinetic Experiment to Determine the Antioxidant Activity of Compounds AO1 and AO2

_{7}values for compounds AO1 and AO2 were determined by the manometric method using air oxygen absorption as a model liquid-phase oxidation of 1,4-dioxane, initiated by azobis(isobutyronitrile) (AIBN). The experiments were performed according to the standard technique described earlier [72,73,74,75,76,77,78]. The model reaction was carried out in a thermostatically controlled glass reactor where the solutions of the initiator (AIBN) and the studied substances in 1,4-dioxane were loaded. The temperature of the reaction mixture was 348 K. The reaction mixture was maintained in the thermostat for 5 min. The kinetic curves was measured using a universal manometric differential unit, the design of which was reported earlier [75,76,77,78]. Subsequently, the initial rates of the oxidation of 1,4-dioxane were calculated from the initial sections of the kinetic curves recorded in the absence and in the presence of compounds AO1 and AO2 using the least-squares method. The numerical values of the effective inhibition rate constants for compounds AO1 and AO2 were calculated from the degree of the decrease in the initial oxygen uptake rate during the oxidation of 1,4-dioxane. The initiation rate of the oxidative process was constant and was V

_{i}= 1 × 10

^{−7}mol·l

^{−1}·s

^{−1}. It was determined using the equation V

_{i}= 2ek

_{p}[AIBN], where k

_{p}is the rate constant of the AIBN decay, e is the probability of the radical escape into the bulk). For k

_{p}, the value measured in cyclohexanol was taken [79]:

_{p}= 17.70 − 35/(4.575T·10

^{−3}), e = 0.5

^{−6}mol/L for AO1 or AO2, respectively, into 1,4-dioxane being oxidized, led to a decrease in the initial oxidation rate. Thus, the qualitative analysis allows us to conclude that we consider that both compounds effectively inhibit the oxidation process of the model substrate (Figure 4 and Figure 5, Table 5).

_{7}for each of the antioxidants were calculated using Equation (2). The condition for the applicability of this equation is a linear dependence of the inhibition parameter F on the concentration of the antioxidants. As can be seen from Figure 6, in the oxidation chain regime in the (0.44–3.13) × 10

^{−6}mol/L concentration range of the AO1 and AO2 compounds, the inhibition parameter F, calculated from the initial rates of the inhibited oxidation of 1,4-dioxane by formula (2) actually followed a linear dependence on the AO1 and AO2 concentrations (Figure 6):

_{0}and V are the initial rates of the oxygen uptake during the oxidation of 1,4-dioxane in the absence and in the presence of each of the antioxidants taken separately, respectively, [AO] is the concentration of the added AO, k

_{7}and 2k

_{6}are the rate constants of the oxidation chain termination by the antioxidant and the quadratic chain termination via peroxyl radicals of the substrate, respectively [1,2,3,4,5,6,7], [RH] is the concentration of 1,4-dioxane ([RH] = 11.75 mol/L), k

_{2}is the rate constant of the chain propagation for the oxidation of the model substrate (k

_{2}= 7.9 l·mol

^{−1}·s

^{−1}[2]). When calculating these values, we used the quadratic chain termination rate constant 2k

_{6}= 6.67 × 10

^{7}l·mol

^{−1}·s

^{−1}known from the literature [2]. The errors in determining the fk

_{7}and f values were calculated using the Excel 2016 word processor (Regression tab).

^{6}M

^{−1}s

^{−1}and f = (1.08 ± 0.2) × 10

^{6}M

^{−1}s

^{−1}, respectively. In addition, to determine the numerical value of the stoichiometric inhibition coefficient, we studied the dependence of the induction period, which appeared on the kinetic curves of the oxygen uptake, on the concentrations of AO1 and AO2. As can be seen from Figure 7, the dependence of the induction period τ on the concentrations of AO1 and AO2 is linear. In this case, it is correct to use Equation (3) to determine the stoichiometric inhibition coefficient f:

_{i}is the initiation rate of the oxidation. Conversion of the experimental data in the coordinates of Equation (3) gave the stoichiometric inhibition factors f for the antioxidants AO1 and AO2 to be 30 ± 4 and 40 ± 2, respectively.

_{7}

^{exp}for AO1 and AO2 was calculated by formula (4):

_{7}

^{exp}for the antioxidants AO1 and AO2 were k

_{7}

^{exp}= (4.3 ± 1.0) × 10

^{4}M

^{−1}s

^{−1}and k

_{7}

^{exp}= (2.7 ± 0.5) × 10

^{4}M

^{−1}s

^{−1}, respectively.

_{7}

^{pred}and the experimental logk

_{7}

^{exp}values for compounds AO1 and AO2 (Table 4) suggests that the M1–M6 QSPR consensus model has a moderate predictive ability and can be applied to the search and development of new antioxidants. The difference between the predicted and experimentally determined logk

_{7}values for these antioxidants does not exceed the 2RMSEP range.

_{7}values for training set structures TR1 and TR2, the external and internal test set structures TS1 and TS2, and compounds AO1 and AO2. These models can be used for the screening of virtual libraries and databases in order to search for new antioxidants in the series of some sulfur-containing alkylphenols, natural phenols, chromane and lupanoic acids, betulonic and betulinic acids, and 20-hydroxyecdysone.

_{7}parameter. Thus, this program can be recommended as an additional tool in the search for new antioxidants.

## 3. Research Methods

#### 3.1. The Methodology of the Computational Experiment

**I**–

**VIII**(Figure 1) was performed using the GUSAR2019 (General Unrestricted Structure Activity Relationships) software [54,55,56,57,58,59,60,61,62].

#### 3.2. Formation of the Training and Test Sets

_{7}values.

_{7}was obtained by taking logarithms of the numerical values of the inhibition rate constant k

_{7}for the simulated antioxidants, which were measured experimentally and reported in the literature [67,68,69,70,76,77,78,79,80,81,82,83,84]. In fact, the inhibition rate constant k

_{7}, which we chose as the simulated parameter, reflects the specific rate of the inhibition of the liquid-phase oxidation of the organic substrates similar in oxidative capacity by the antioxidants. In modeling, it was assumed that the oxidation of the organic substrates in the presence of antioxidants, proceeds in several steps and can be schematically described by the following key steps, which have been studied in detail and described in the literature [1,2,3,4,5,6,7] (Figure 10).

_{2}

^{•}and via the antioxidant molecule, respectively. The antioxidant effect of the antioxidants included in the S1 data array is implemented through their reaction with the peroxyl radical of the RO

^{2-}oxidation substrate RO

_{2}

^{•}. As a result, the peroxyl radical active in the chain propagation reaction is replaced by an inactive antioxidant radical. This is the AOA mechanism of the simulated compounds. Obviously, the higher the numerical value of the inhibition rate constant k

_{7}, the more pronounced the antioxidant properties of the organic compound.

_{7}values. To test the predictive power of the M1–M3 models, we used test set TS1, which contained 25 antioxidant structures with their corresponding logk

_{7}values. Both of these sets were derived by a 5:1 split of the S1 data set by transferring every sixth compound from S1 to TS1. The remaining 123 antioxidant structures were used to form the training set TR1. Preliminarily, all structures of the data array S1 were ranked in ascending order of the numerical value of logk

_{7}.

_{7}values and was designed to build the M4–M6 QSPR models. The validity of the M4–M6 QSPR models was tested using test set TS2. Both TR2 and TS2 sets were formed on the basis of the training set TR1. In this case, the TR1 set was subjected to a 5:1 split, with the transfer of every sixth compound from TR1 to TS2. The characteristics of the training sets TR1 and TR2 and test sets TS1 and TS2 are presented in Table 6 and Table 7, respectively. The data of these tables indicate that the compounds of the training and test sets are fairly evenly distributed over the entire range of the logk

_{7}variability. At the same time, the AOA of the compounds of the TR1 and TR2 sets varies over a wide range (∆logk

_{7}= 7.06). The range of variability of logk

_{7}for the compounds of the test sets does not go beyond the range Δlogk

_{7}= 7.06. In addition, as can be seen from Figure 1, the training sets are characterized by a high degree of molecular diversity. These conditions are important for building high-quality QSPR models and the correct forecasts based on them [34].

#### 3.3. Building QSPR Models

- 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 D_{i}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. The 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.

_{7}for a particular compound using a particular model was formed based on the results of averaging the predicted logk

_{7}values of the single regression QSPR models included in this consensus model.

#### 3.4. Assessment of the Descriptive and Predictive Powers of the QSPR Models

^{2}(R

^{2}, R

^{2}

_{0}, R

^{2’}, average R

^{2}

_{m}, CCC) and the metrics evaluating the prediction errors of the logk

_{7}values (root mean square error (RMSE), mean absolute error (MAE), standard deviation (SD)) [34,35,36,37]. These statistical parameters were calculated using Xternal Validation Plus 1.2 for 100% and 95% of the data (to account for the outliers) in the training and test sets [88]. The Supplementary Material provides the formulas by which these criteria are calculated in this program. The internal validation of the M1–M6 models was performed using LMO cross-validation (Q

^{2}

_{LMO}) with a 20-fold exclusion of 20% of the compounds from the training sets.

_{7}values with the experimental values of the same parameter for the new promising antioxidants AO1 and AO2, which were not included in the S1 data set (Figure 10).

- For 95% of the data of the training set TRi, the numerical values of the determination coefficients R
^{2}, R^{2}_{0}, R^{2’}_{0}, and the CCC criterion should be close to each other and tend to unite; - Numerical value of the criterion R
^{2}_{m}> 0.85 with ΔR^{2}_{m}< 0.15; - Numerical value of the average absolute error MAE should not exceed 10% of the activity range Δlogk
_{7}of the simulated training set TRi; - MAE+3SD parameter value (where SD is standard deviation) should not exceed 10% of the activity range Δlogk
_{7}of the simulated training set TRi; - Numerical values of the determination coefficients Q
^{2}_{F1}, Q^{2}_{F2}(calculated for the test sets) should be close to each other and tend to unite.

- For 95% of the data of the training sample Tri, the numerical values of the determination coefficients R
^{2}, R^{2}_{0}, R^{2’}_{0}, and the CCC criterion should not exceed the threshold value 0.6; - Numerical value of R
^{2}_{m}≤ 0.5 with ΔR^{2}_{m}≤ 0.2; - Numerical value of the mean absolute error of the MAE exceeded 20% of the activity interval of the Δlgk
_{7}compounds simulated by the training sample TRi; - The value of the MAE+3SD parameter exceeded 25% of the activity interval of the lgk
_{7}compounds simulated by the training sample TRi; - Numerical values of the determination coefficients Q
^{2}_{F1}< 0.70, Q^{2}_{F2}< 0.70 (calculated for the test sets) should be less than 0.70.

## 4. Conclusions

**I**–

**VIII**. Six statistically significant valid QSPR consensus models were built. The models demonstrated a satisfactory predictive accuracy in predicting the parameter logk7 for training and test set structures: R

^{2}TR > 0.6; Q2TR > 0.5; R2TS > 0.5. All models showed a high performance, as they reproduced the known experimental data for the training sets with a high degree of accuracy. The cross-validation with a 20-fold exclusion of 20% of the training set data also showed good results. The validation of the prediction of logk7 by the estimation of these parameters for the compounds of two test sets and two compounds that were subsequently studied, experimentally demonstrated a moderate predictive power of the M1–M6 QSPR models. Despite the high performance and satisfactory external validation results found for all of the models, we recommend using the M3 and M6 QSPR models for the virtual screening and search for new antioxidants. The M3 and M6 models are based on the combination of the different types of descriptors, which ensures the most objective prognostic estimates of logk

_{7}.

_{7}

^{pred}values and the experimentally determined logk

_{7}

^{exp}values for the compounds of the test sets TS1, TS2 and antioxidants AO1 and AO2, provides the conclusion that the calculation and selection algorithms for the descriptors, the algorithms of the generation of the regression equations, and their consensus combination implemented in the GUSAR 2019 program allow the correct modeling of the kinetic parameter logk

_{7}, which is determined experimentally in the model liquid-phase oxidation reactions of organic hydrocarbons.

## 5. Patents

## Supplementary Materials

^{2}and MAE metrics; Table S2: The validation parameters of the QSPR models estimated using the Xternal Validation Plus 1.2 program based on the experimental and predicted lgk

_{7}values of the compounds form internal training sets TR1 and TR2; Table S3: The validation parameters of the QSPR 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; Table S4: Prediction of the lgk

_{7}values for the TR1 compounds using models M1-M3; Table S5: Prediction of the lgk

_{7}values for the TR2 compounds using models M4-M6; Table S6: Prediction of the lgk

_{7}values for the TS1 compounds using models M1-M3; Table S7: Prediction of the lgk

_{7}values for the TS1 compounds using models M4-M6; Table S8: Prediction of the lgk

_{7}values for the TS2 compounds using models M4-M6.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

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**Figure 1.**General structural of the formulas of the modeled antioxidant inhibitors (Pht=CH

_{2}[CH

_{2}CH

_{2}CH(CH

_{3})CH

_{2}]

_{3}H).

**I, VIII**(a phenol derivative),

**II–V**(chromone derivatives),

**VI**(20-hydroxyecdysone derivatives with chroman-2-yl moiety),

**VII**(triterpenoids derivatives with chroman-2-yl moiety).

**Figure 4.**Typical kinetic curves of the oxygen uptake during the oxidation of 1,4-dioxane in the absence (1) and in presence of AO1 taken in concentrations, mol/L: 0.44 × 10

^{−6}(2); 1.24 × 10

^{−6}(3); 1.88 × 10

^{−6}(4); 2.50 × 10

^{−6}(5); 3.13 × 10

^{−6}(6). T = 348 K, V

_{i}= 1 × 10

^{−7}M/s.

**Figure 5.**Typical kinetic curves of the oxygen uptake during the oxidation of 1,4-dioxane in the absence (1) and in the presence of AO2 taken in concentrations, mol/L: 0.44 × 10

^{−6}(2); 0.94 × 10

^{−6}(3); 1.25 × 10

^{−6}(4); 1.88 × 10

^{−6}(5); 3.13 × 10

^{−6}(6). T = 348 K, V

_{i}= 1 × 10

^{−7}M/s.

**Figure 6.**Dependence of the inhibition efficiency parameter on the concentration of AO1 and AO2, V

_{i}= 1 × 10

^{−7}M/s, T = 348 K.

**Figure 7.**Dependence of the induction period on the injected initial concentration of the inhibitor. T = 348 K, V

_{i}= 1 × 10

^{−7}M/s.

**Figure 9.**Construction of the training and test sets for the M1–M6 models in the design of the QSPR consensus models (S is set, TR and TS are training and test sets, M is the model, N is the number of compounds included in the corresponding sets and arrays). Designations: (1) S1 is the overall data set; (2) S2 is the training set TR1 for the M1–M3 models; (3) S3 is the external test set TS1 for the M1–M6 models; (4) S4 is the training set TR2 for the M4–M6 models; (5) S5 is the internal test set TS2 for the M4–M6 models.

**Figure 10.**Mechanism of the inhibited radical chain oxidation of organic compounds (I, RH and InH are the initiator, oxidized substrate, and inhibitor, respectively), where I is the initiator of the oxidation process, r

^{•}is the radical that was formed upon the decay of the initiator I, RH is the oxidation substrate, R

^{•}is the radical that was formed upon the elimination of a hydrogen atom from the substrate molecule by the initiator radical r

^{•}, RO

_{2}

^{•}is the peroxyl radical formed upon the reaction of the substrate radical R

^{•}with an oxygen molecule, InH is antioxidant, In

^{•}is the radical formed as a result of the hydrogen atom elimination from the antioxidant molecule by the substrate peroxyl radical RO

_{2}

^{•}.

**Table 1.**Statistical parameters and the accuracy of the predicted logk

_{7}values of the compounds included in the training sets TR1, TR2 within the M1–M6 consensus models (using Both). ∆logk

_{7(TR1)}= ∆logk

_{7(TR2)}= 7.057

^{1}.

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

QSPR models based on the QNA descriptors | ||||||||

TR1 | M1 | 123 | 20 | 0.968 | 7.675 | 0.548 | 0.760 | 29 |

TR2 | M4 | 103 | 20 | 0.962 | 7.337 | 0.587 | 0.740 | 24 |

QSPR models based on the MNA descriptors | ||||||||

TR1 | M2 | 123 | 20 | 0.968 | 7.008 | 0.550 | 0.763 | 29 |

TR2 | M5 | 103 | 20 | 0.964 | 7.891 | 0.578 | 0.756 | 22 |

QSPR models based on both QNA and MNA descriptors | ||||||||

TR1 | M3 | 123 | 320 | 0.976 | 8.708 | 0.512 | 0.802 | 28 |

TR2 | M6 | 103 | 320 | 0.973 | 8.057 | 0.551 | 0.787 | 23 |

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

**Table 2.**Validation parameters of the QSPR models estimated using the Xternal Validation Plus 1.2 program based on the experimental and predicted logk

_{7}values of the compounds of the internal training sets TR1 and TR2. Δlogk

_{7(TR1)}= ∆logk

_{7(TR2)}= 7.057

^{1}.

Comments | Prediction Parameters | QSPR Model Used for Predicting logk_{7} | |||||
---|---|---|---|---|---|---|---|

TR1 | TR2 | ||||||

M1 | M2 | M3 | M4 | M5 | M6 | ||

Classical metrics (after removing 5% of the data with high residuals) | R^{2} | 0.9868 | 0.9849 | 0.9896 | 0.9887 | 0.9850 | 0.9925 |

R^{2}_{0} | 0.9845 | 0.9837 | 0.9876 | 0.9870 | 0.9839 | 0.9903 | |

R^{2’}_{0} | 0.9236 | 0.9338 | 0.9317 | 0.9353 | 0.9366 | 0.9364 | |

$\overline{{\mathrm{R}}_{\mathrm{m}}^{2}}$ | 0.9384 | 0.9496 | 0.9454 | 0.9419 | 0.9532 | 0.9414 | |

∆R^{2}_{m} | 0.0141 | 0.0149 | 0.0113 | 0.0132 | 0.0146 | 0.0099 | |

CCC | 0.9916 | 0.9912 | 0.9932 | 0.9932 | 0.9912 | 0.9947 | |

Mean absolute error and standard deviation for the test set (after removing 5% of the data with high residuals) | RMSE | 0.1090 | 0.1107 | 0.0975 | 0.1128 | 0.1098 | 0.0976 |

MAE | 0.0855 | 0.0894 | 0.0765 | 0.0924 | 0.0879 | 0.0773 | |

SD | 0.0679 | 0.0656 | 0.0607 | 0.0650 | 0.0661 | 0.0599 | |

MAE+3SD | 0.2892 | 0.2862 | 0.2586 | 0.2873 | 0.2861 | 0.2570 | |

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

Presence of systematic errors | - | Absent |

^{1}R

^{2}, R

^{2}

_{0}, and R’

^{2}are the determination coefficients calculated with and without taking into account the origin; average R

^{2}

_{m}is the averaged determination coefficient of the regression function calculated using the values of determination coefficients on the ordinate axis (R

^{2}

_{m}) and on the abscissa axis (R’

^{2}

_{m}), respectively; ΔR

^{2}

_{m}is the difference between R

^{2}

_{m}and R’

^{2}

_{m}; CCC is the concordance correlation coefficient; MAE is the mean absolute error; SD is the standard deviation.

**Table 3.**Validation parameters of the QSPR models estimated using the Xternal Validation Plus 1.2 program based on the experimental and predicted logk

_{7}values of the compounds of test sets TS1 and TS2. ∆logk

_{7(TR1)}= ∆logk

_{7(TR2)}= 7.057; Δlogk

_{7(TS1)}= 4.009; ∆logk

_{7(TS2)}= 3.148

^{1}.

Comments | Prediction Parameters | QSPR Model Used for Predicting logk_{7} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

TS1 | TS2 | |||||||||

M1 | M2 | M3 | M4 | M5 | M6 | M4 | M5 | M6 | ||

Classical metrics (after removing 5% of the data with high residuals) | R^{2} | 0.8204 | 0.7364 | 0.7715 | 0.7696 | 0.7289 | 0.7807 | 0.7765 | 0.8125 | 0.8071 |

R^{2}_{0} | 0.8115 | 0.7342 | 0.7701 | 0.7621 | 0.7263 | 0.7741 | 0.7739 | 0.7936 | 0.8013 | |

R^{2’}_{0} | 0.5555 | 0.4652 | 0.5346 | 0.4750 | 0.4466 | 0.5005 | 0.6304 | 0.7650 | 0.7064 | |

Q^{2}_{F1} | 0.9538 | 0.9390 | 0.9525 | 0.4750 | 0.9367 | 0.9468 | 0.9567 | 0.9608 | 0.9621 | |

Q^{2}_{F2} | 0.7966 | 0.7312 | 0.7627 | 0.7473 | 0.7212 | 0.7656 | 0.7679 | 0.7896 | 0.7969 | |

$\overline{{\mathrm{R}}_{\mathrm{m}}^{2}}$ | 0.6798 | 0.6010 | 0.6726 | 0.6191 | 0.5892 | 0.6332 | 0.6934 | 0.7434 | 0.7353 | |

∆R^{2}_{m} | 0.1673 | 0.2191 | 0.1803 | 0.2046 | 0.2249 | 0.1964 | 0.1316 | 0.0319 | 0.0653 | |

CCC | 0.8763 | 0.8371 | 0.8600 | 0.8433 | 0.8293 | 0.8563 | 0.8775 | 0.8998 | 0.8970 | |

Mean absolute error and standard deviation for the test set (after removing 5% of the data with high residuals) | RMSE | 0.4133 | 0.4750 | 0.4186 | 0.4606 | 0.4838 | 0.4436 | 0.3870 | 0.3685 | 0.3620 |

MAE | 0.3146 | 0.3309 | 0.2986 | 0.3296 | 0.3442 | 0.3129 | 0.2945 | 0.2719 | 0.2740 | |

SD | 0.2740 | 0.3485 | 0.3000 | 0.3289 | 0.3476 | 0.3215 | 0.2580 | 0.2555 | 0.2431 | |

MAE+3SD | 1.1367 | 1.3763 | 1.1985 | 1.3164 | 1.3871 | 1.2773 | 1.0684 | 1.0383 | 1.0032 | |

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

Presence of systematic errors | - | Absent |

^{1}R

^{2}, R

^{2}

_{0}, and R’

^{2}are the determination coefficients calculated with and without taking into account the origin; average R

^{2}

_{m}is the averaged determination coefficient of the regression function calculated using the determination coefficients on the ordinate axis (R

^{2}

_{m}) and on the abscissa axis (R’

^{2}

_{m}), respectively; ∆R

^{2}

_{m}is the difference between R

^{2}

_{m}and R’

^{2}

_{m}; CCC is the concordance correlation coefficient; MAE is the mean absolute error; SD is the standard deviation.

Model | Applicability (AD) | Predicted Value of logk_{7}^{pred} | Experimental Value of logk_{7}^{exp 1} | Δlogk_{7} ^{2} | 2RMSEP (95%) ^{3} | |||
---|---|---|---|---|---|---|---|---|

AO1 | AO2 | AO1 | AO2 | AO1 | AO2 | |||

M1 | in AD | 5.21 | 5.10 | 4.64 | 4.43 | 0.57 | 0.67 | 0.83 |

M2 | in AD | 4.79 | 5.32 | 0.15 | 0.89 | 0.95 | ||

M3 | in AD | 5.17 | 5.21 | 0.53 | 0.78 | 0.84 | ||

M4 | in AD | 5.25 | 5.23 | 0.61 | 0.80 | 0.92 | ||

M5 | in AD | 5.07 | 5.20 | 0.43 | 0.77 | 0.97 | ||

M6 | in AD | 5.19 | 5.15 | 0.55 | 0.72 | 0.89 |

^{1}The experimental determination of logk

_{7}for compounds AO1 and AO2 is decribed in Section 3;

^{2}∆logk

_{7}= logk

_{7}

^{pred}− logk

_{7}

^{exp};

^{3}The maximum values of the RMSEP were taken; multiplying this criterion by two gives the confidence interval with 95% probability (relative to the predicted value of logk

_{7}, if the model is correct and the errors are normally distributed, which was observed in our computational experiments) [32].

**Table 5.**Dependence of the initial oxidation rate of ethylbenzene on the concentration of AO1 and AO2; V

_{i}= 1·10

^{−7}M/s, T = 348 K.

[AO1]·10^{6}, mol/L | V_{0}·10^{6}, M/s | [AO2]·10^{6}, mol/L | V_{0}·10^{6}, M/s |
---|---|---|---|

0.00 | 2.30 | 0.00 | 2.36 |

0.44 | 1.86 | 0.44 | 1.95 |

1.24 | 1.66 | 0.94 | 1.89 |

1.88 | 1.53 | 1.25 | 1.77 |

2.50 | 1.20 | 1.88 | 1.53 |

3.13 | 1.13 | 3.13 | 1.44 |

Designation of TRi | Code of the Training Set | |
---|---|---|

TR1 | TR2 | |

N | 123 | 103 |

logk_{7} | 3.529 | |

∆logk_{7} | 7.057 | |

Thresholds used to evaluate the model’s forecast | ||

0.10 × ∆logk_{7} | 0.706 | |

0.15 × ∆logk_{7} | 1.059 | |

0.20 × ∆logk_{7} | 1.411 | |

0.25 × ∆logk_{7} | 1.764 |

Designation of TSi | Code of the Test Set | |
---|---|---|

TS1 | TS2 | |

N | 25 | 20 |

$\overline{{\mathrm{lgk}}_{7}}$ | 5.106 | 5.117 |

∆logk_{7} | 4.009 | 3.148 |

Distribution of the observed response values of test sets TSi around the test mean (in %) | ||

$\overline{{\mathrm{lgk}}_{7}}$ ± 0.5, % | 32.000 | 35.000 |

$\overline{{\mathrm{lgk}}_{7}}$ ± 1.0, % | 64.000 | 70.000 |

$\overline{{\mathrm{lgk}}_{7}}$ ± 1.5, % | 88.000 | 95.000 |

$\overline{{\mathrm{lgk}}_{7}}$ ± 2.0, % | 96.000 | 100.000 |

Distribution of the observed response values of test sets TSi around the training mean (in %) | ||

$\overline{{\mathrm{lgk}}_{7}}$ ± 0.5, % | 8.000 | 10.000 |

$\overline{{\mathrm{lgk}}_{7}}$ ± 1.0, % | 32.000 | 30.000 |

$\overline{{\mathrm{lgk}}_{7}}$ ± 1.5, % | 44.000 | 45.000 |

$\overline{{\mathrm{lgk}}_{7}}$ ± 2.0, % | 68.000 | 70.000 |

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**MDPI and ACS Style**

Khairullina, V.; Martynova, Y.; Safarova, I.; Sharipova, G.; Gerchikov, A.; Limantseva, R.; Savchenko, R. QSPR Modeling and Experimental Determination of the Antioxidant Activity of Some Polycyclic Compounds in the Radical-Chain Oxidation Reaction of Organic Substrates. *Molecules* **2022**, *27*, 6511.
https://doi.org/10.3390/molecules27196511

**AMA Style**

Khairullina V, Martynova Y, Safarova I, Sharipova G, Gerchikov A, Limantseva R, Savchenko R. QSPR Modeling and Experimental Determination of the Antioxidant Activity of Some Polycyclic Compounds in the Radical-Chain Oxidation Reaction of Organic Substrates. *Molecules*. 2022; 27(19):6511.
https://doi.org/10.3390/molecules27196511

**Chicago/Turabian Style**

Khairullina, Veronika, Yuliya Martynova, Irina Safarova, Gulnaz Sharipova, Anatoly Gerchikov, Regina Limantseva, and Rimma Savchenko. 2022. "QSPR Modeling and Experimental Determination of the Antioxidant Activity of Some Polycyclic Compounds in the Radical-Chain Oxidation Reaction of Organic Substrates" *Molecules* 27, no. 19: 6511.
https://doi.org/10.3390/molecules27196511