Quantitative Structure-Activity Relationship Study of a Benzimidazole-Derived Series Inhibiting Mycobacterium tuberculosis H 37 Rv

This work was carried out on a series of twenty-two (22) benzimidazole derivatives with inhibitory activities against Mycobacterium tuberculosis H37Rv by applying the Quantitative Structure-Activity Relationship (QSAR) method. The molecules were optimized at the level DFT/B3LYP/6−31 + G (d, p), to obtain the molecular descriptors. We used three statistical learning tools namely, the linear multiple regression (LMR) method, the nonlinear regression (NLMR) and the artificial neural network (ANN) method. These methods allowed us to obtain three (3) quantitative models from the quantum descriptors that are, chemical potential (μ), polarizability (α), bond length l (C = N), and lipophilicity. These models showed good statistical performance. Among these, the ANN has a significantly better predictive ability R 2 = 0.9995; RMSE = 0.0149; F = 31879.0548. The external validation tests verify all the criteria of Tropsha et al. and Roy et al. Also, the internal validation tests show that the model has a very satisfactory internal predictive character and can be considered as robust. Moreover, the applicability range of this model determined from the levers shows that a prediction of the pMIC of the new benzimidazole derivatives is acceptable when its lever value is lower than 1.


Introduction
Tuberculosis is an infectious and contagious disease caused by Koch's bacillus (strains of the Mycobacterium tuberculosis complex). This infectious agent is transmitted by air, via droplets containing the bacteria and expectorated by the cough of the patients. Tuberculosis is present in all regions of the world [1]. In Computational Chemistry capability of QSAR models while shortening the computational time and cost implication in the design of new drugs [18] [19]. The twenty-two (22) compounds used to conduct this work have minimum inhibitory concentration (MIC) values that range from 0.08 to 100 μg/mL. Minimum inhibitory concentration (MIC) is an indicator of the effectiveness of a given compound in inhibiting a specific biological or biochemical function. Concentration values are usually expressed as the inverse of the logarithm of activity based on decimal places ( ) ( ) 10 log C − to obtain better mathematical values when structures are biologically active [20] [21]. The anti-tuberculosis activity will be expressed by the potential inhibitory concentration pMIC defined by Equation (1): 3 10 MIC pMIC log 10 M where MIC, the minimum inhibitory concentration in μg/mL. The data modeling was developed using three statistical learning methods. These are the Linear Multiple Regression (LMR) and Non-Linear Multiple Regression (NLMR) methods that are integrated in Excel [22] and XLSTAT [23]. The last method is that of artificial neurons which is included in the JMP Pro software [24].

Molecular Descriptors Used
In the development of our QSAR model, quantum descriptors have been calculated. In particular, chemical potential (μ), polarizability (α), bond length (l (C = With: The polarizability designates a phenomenon caused by the moment of the electric charges of the atom. A molecule placed in an electric field E undergoes a deformation and acquires an induced dipole electric moment proportional to the field E, the polarizabilities are expressed in Å 3 . They have the dimension of a volume. The atomic polarizability increases with the size of the atoms [25]. Lipophilicity reflects the ability of a molecule to adhere to a lipidic environment, oil, cell membrane, lipidic solvent, etc. [26]. This physico-chemical descriptor is generally evaluated by the distribution of the molecule, neutral, soluble, between water and another immiscible solvent: most often n-octanol (or octan-1-ol) [26] [27] [28] [29]. This parameter is estimated from the logP value. The logP is equal to the logarithm of the ratio of the concentrations of the test substance in octanol and in water logP = log(C oct /C Water ). Indeed, if the logP is positive and very high, it expresses the fact that the molecule considered is much more soluble in octanol than in water, which reflects its lipophilic character, and conversely, if the logP is negative it means that the molecule considered is hydrophilic. A zero logP means that the molecule is as soluble in one solvent as in the other. In this work, the software Chemsketch [30] allowed us to determine the values of logP. In practice we express the lipophilicity by the decimal logarithm of the partition coefficient logP. Thus:

Estimation of the Predictive Capacity of a QSAR Model
The quality of a model is determined based on different statistical criteria of analysis including the coefficient of determination R 2 , standard deviation (S) or Root Mean Square Error (RMSE), cross-validation correlation coefficients 2 CV Q and Fischer F. R 2 , S and F refer to the fit of the simulated and experimental values. They represent the predictive capacity within the limits of the model, and allow us to estimate the accuracy of the values calculated on the test set [31] [32]. As for the cross-validation coefficient 2 CV Q , it provides information on the predictive character of the model. This predictive potential is said to be "internal" because it is calculated from the structures used to build the model. The coefficient of determination R 2 gives an evaluation of the dispersion of the theoretical values around the experimental values. The quality of the modeling is better when the points are close to the fit line [33]. The fit of the points to this line can be evaluated by the coefficient of determination.
where: The closer the R 2 value is to 1, the more the theoretical and experimental values are correlated The Root Mean Square Error RMSE is another statistical indicator used. It allows to evaluate the reliability and the precision of a model: The Fisher F test is also used to measure the level of statistical significance of the model, i.e. the quality of the choice of descriptors constituting the model The coefficient of determination of the cross-validation 2 CV Q allows to evaluate the accuracy of the prediction on the training set. It is calculated using the following relation:

Acceptance Criteria of a Model
The performance of a mathematical model, according Eriksson et al. [34], is characterized by a value of 2 0.5 cv Q > for a satisfactory model, when for the excellent model 2 0.9 cv Q > . According to these authors, given a set of tests, a model will perform well if the acceptance criterion 2 According to Tropsha et al. [35] [36] [37], or the external validation set, the predictive power of a model can be obtained from five criteria. These criteria are the following: 3)

Linear and Non-Linear Multiple Regressions (LMR and NLMR)
The statistical method of multiple linear regression (MRL) is used to examine the relationship between a dependent variable (Property) and various independent variables (Descriptors). This statistical approach limits the differences between the actual and predicted values. It was also used to select the descriptors used as input parameters in the multiple non-linear regression (NLMR). As for the NLMR analysis, it is also used to refine the structure-property relationship in order to quantitatively assess the property. It is the most common tool for studying multidimensional data. It is based on the following pre-programmed functions of XLSTAT:  [40]. The most commonly used networks are multilayer perceptrons (MLPs) whose neurons are usually divided into layers [41]. In this paper, the artificial neural network was made from the 4-3-1 multilayer perceptron network, i.e., the network consists of five (4) neurons in the input layer, three (3) neurons in the hidden layer and one (1) neuron in the output layer. The output layer consists of a sigmoid function. The architecture of the ANN models used is described in Figure 3 below.

Area of Applicability
The domain of applicability of a QSAR model is the physicochemical, structural or biological field in which the model equation can be used to make predictions about new drugs [42]. It corresponds to the area of the chemical space including the compounds of the training set and similar compounds, which are close in this same space [43]. In particular, the model, which is built on the basis of a limited number of compounds, by relevant descriptors cannot be a universal tool to predict the activity of any other molecule with certainty. It appears necessary, to determine the DA of any QSAR model. This is recommended by the Organization (OECD) in the development of a QSAR model of economic cooperation and development [44]. There are different methods to establish the domain of applicability of a model [43]. Among these, the method used in this work is the leverage approach. This method is based on the variation of the standardized residuals of the dependent variable with the distance between the values of the descriptors and their mean, called leverage [45].   (13) H is defined by the expression (12): The field of applicability is delimited by a threshold value of the lever noted h*. In general, it is fixed at where n is the number of compounds in the training set, and p is the number of descriptors in the model [46] [47]. For standardized residuals, the two limit values generally used are ±3σ, σ being the standard deviation of the experimental values of the quantity to be explained [48]: it is "the rule of the three sigmas" [49].

Results and Discussion
In our QSAR study, we used a series of twenty-two (22) benzimidazole derivatives. These compounds were synthesized and tested on Mycobacterium tuberculosis H37Rv. The compounds were split into two groups, fifteen (15) were used for the learning set and seven (7) for the validation set. This part of the work will concern the modeling of the antitubercular activity of the benzimidazole derivatives based on the descriptors presented in Table 1. The values of the descriptors as well as those of the experimental biological activities of the molecules are recorded in Table 1.

Interdependence of Descriptors
In order to better understand the interdependence of the descriptors used, we present the values of the partial correlation coefficients aij of these descriptors in Table 2.

Modeling of Mycobacterium tuberculosis H37Rv Activity
In the model formula, the negative or positive sign of the coefficient of a descriptor reflects the effect of proportionality between the evolution of the inhibitory concentration MIC and this physicochemical parameter of the regression equation. Thus, the negative sign indicates that when the value of the descriptor is high, the MIC inhibitory concentration decreases, whereas the positive sign expresses the opposite effect. In this work, three statistical analysis tools were used: Multiple Linear Regression (MLR), Multiple Nonlinear Regression (MNLR) and Artificial Neural Network (ANN).

Multiple Linear Regression (MLR)
The equation of the QSAR model is presented below. The statistical indicators are given in Table 3.   shows that the error committed is lower than what the model explains [50]. The  Table 4.    The results show that the models constructed, after the removal of one of the compounds from the training set (first column of the table), have statistical parameters (R 2 and RMSE) of the same order as those of the initial model, overall.
The average values of these parameters are R 2 = 0.9219, RMSE = 0.2647. We find values almost identical to those of the initial model. The cross-correlation coefficient 2 cv Q , is equal to 0.7936. This value is higher than the minimum required value of 0.5 according to Tropsha et al. [51] [52]. In addition, we note that 2 2 0.1283 0.3 cv R Q − = < [53]. All this shows that the RML model has a very satisfactory internal predictive character and can be considered as robust [54].

b) Randomization test
The randomization test of the MLR model was performed on the molecules of the training set by randomly permuting the values of the activities while keeping the descriptors for model building. We stopped at ten (10) iterations. The randomized coefficients of determination ( 2 r R ) for each iteration are listed in Table  5.
From the values in Table 5

2) External validation
The external validation of the RML model was performed on the molecules of the validation set (Table 1) using the Tropsha criteria [51] [52] and Roy [38].
The Tropsha and Roy criteria checks are recorded in Table 6 and Table 7 respectively.  The values in Table 6 show that all Tropsha criteria are met, so the model is acceptable for predicting the antitubercular activity of benzimidazole derivatives.
The analysis in Table 7 shows that the 2 m r is greater than 0.5 and the 2 m r ∆ is less than 0.2. This result reflects that the model meets Roy's criteria. We can therefore affirm that the model is robust and has a good predictive power.

Non-Linear Multiple Regression NLMR
The equation of the QSAR model is presented below. The statistical indicators are given in Table 8.   [50]. This value 356.324 of the Fisher test, higher than the critical value, shows that the error committed is lower than what the model explains [50]. The standard deviation (RMSE = 0.2396) expresses the small deviation between the predicted values and the experimental mean. This model presents a correlation coefficient of the cross-validation 2 cv Q equal to 2 0.9648 cv Q = . This value, higher than 0.9, reflects a so-called excellent model according to Erikson et al. [34]. This model is acceptable because it is in agreement with the acceptance criteria of these authors 2 2 0.9648 0.9248 0.000 0.3 cv R Q − = − = < . All these statistical indicators show that the model developed explains the TB activity in a statistically significant and satisfactory way. These different results are confirmed by the regression plot of the NLMR model presenting the theoretical anti-tuberculosis activity as a function of the experimental activity represented in Figure 6.
The regression curve of the RML model shows that all points are around the regression line. This result indicates that there is a small difference (RMSE = 0.2396) between the values of pMICexp and pMIC th, thus a good similarity in these values. This similarity is illustrated in Figure 7.

1) Internal validation a) Randomization test
The NLMR model randomization test was performed on the molecules in the training set by randomly permuting the activity values while retaining the descriptors for model building. We stopped at ten (10) iterations. The randomized coefficients of determination ( 2 r R ) for each iteration are listed in Table 9.     [52] and Roy [38]. The Tropsha and Roy criteria checks are recorded in Table 10 and Table 11, respectively.
The values in Table 10 show that all Tropsha criteria are met, so the model is acceptable for predicting the antitubercular activity of benzimidazole derivatives.
The analysis in Table 11 shows that 2 m r is greater than 0.5 and the 2 m r ∆ is less than 0.2. This result reflects that the model meets Roy's criteria. We can therefore affirm that the model is robust and has good predictive power.

Contribution of the Descriptors of the MLR and NLMR Models
The study of the relative contribution of the descriptors in predicting the antitubercular activity of benzimidazole derivatives was performed. The different contributions are presented in the pie chart in Figure 8.   From Figure 5, we can see that polarizability (α) is the descriptor with the highest contribution compared to the other descriptors. Thus, polarizability (α) is the priority descriptor in predicting anti-tuberculosis activity.

Area of Applicability of the MLR and NLMR Models
The applicability domain of the MLR and NLMR models was determined by the leverage method. The h ii -lever values of the molecules in the training set calculated from the MINTAB software are listed in Table 12.
The values of the h ii levers in Table 12 and the standardized residues of the molecules were used to plot the graph of standardized residues versus h ii levers in Figure 9.  For the 15 molecules of the training set and the 4 descriptors of the model, the threshold value of the h* levers is 1. The extreme values of the standardized residuals are ±3 according to the "three sigma rule" [49]. These different values delimit the field of applicability [55] of the model as indicated on the graph in Figure 6. Figure 6 shows us that all the molecules have levers lower than the threshold lever (h* = 1) and values of the standardized residues between +3 and −3. This result thus translates that all the molecules belong to the applicability domain.

Artificial Neural Network (ANN)
The values of the descriptors as well as those of the experimental biological activities of the molecules used for the development of the ANN model are listed in Table 13. The equation of the QSAR model is presented below. The statistical indicators are given in Table 14.   [34]. Ce modèle est acceptable car il is in agreement with the acceptance criteria of these authors    Table 15.
The results show that the models constructed, after the removal of one of the compounds from the training set (first column of the table), have statistical parameters (R 2 and RMSE) of the same order as those of the initial model, overall.
The average values of these parameters are 2 LOO R = 0.9661, RMSE LOO = 0.1216.
We find values almost identical to those of the initial model. The cross-correlation , is equal to 0.7936. This value is higher than the minimum required value of 0.5 according to Tropsha et al. [51] [52]. In addition, we note that 2 2 LOO LOO 0.1283 0.3 cv R Q − = < [53]. All this shows that the ANN model has a very satisfactory internal predictive character and can be considered as robust [54].

b) Randomization Test
The randomization test of the ANN model was performed on the molecules of the training set by randomly permuting the values of the activities while keeping the descriptors for model building. We stopped at ten (10) iterations. The randomized coefficients of determination ( 2 r R ) for each iteration are listed in Table  16.
From the values in Table 16 [38]. The Tropsha and Roy criteria checks are recorded in Table 17 and Table 18 respectively.
The values in Table 17 show that all Tropsha criteria are met, so the model is acceptable for predicting the antitubercular activity of benzimidazole derivatives. The analysis in Table 18 shows that 2 m r is greater than 0.5 and the 2 m r ∆ is less than 0.2. This result reflects that the model meets Roy's criteria. We can therefore affirm that the model is robust and has good predictive power.

Domain of Applicability of the ANN Model
The applicability domain of the model was determined by the leverage method. The hii-lever values of the molecules in the training set calculated from the MINTAB software are listed in Table 19.
The values of the h ii levers in Table 19 and the standardized residues of the molecules were used to plot the graph of standardized residues versus h ii levers in Figure 12.   For the 15 molecules of the training set and the 4 descriptors of the model, the threshold value of the h* levers is 1. The extreme values of the standardized residuals are ±3 according to the "three sigma rule" [49]. These different values delimit the field of applicability [55] of the model as shown on the graph in Figure 12. Figure 12 shows us that all the molecules have levers lower than the threshold lever (h* = 1) and standardized residue values between +3 and −3. This result means that all the molecules belong to the applicability domain.

Conclusion
At the end of this work we determined three (3) mathematical relationships between the potential minimum inhibitory concentration (pMIC) of benzimidazole derivatives against Mycobacterium tuberculosis strain H37Rv and their physicochemical descriptors. Chemical potential (μ), polarizability (α), bond length l (C = N), and lipophilicity (logP) are the parameters that explain the antitubercular activity of benzimidazole derivatives significantly. Statistical data learning methods such as multilinear regression (MLR), nonlinear multiple regression (NLMR) as well as artificial neural network (ANN) methods were used. The statistical indicators of these models (MLR, NLMR, ANN) show that they are acceptable, robust and have good predictive power. In this work, due to its statistical indicators, the artificial neuron method (ANN) (R 2 = 0.9995; RMSE = 0.0149; F = 31879.0548) proved to be the best statistical learning method for predicting the anti-tuberculosis activity of benzimidazole derivatives. Moreover, the appli- cability range of this model determined from the levers shows that a prediction of the pMIC of the new benzimidazole derivatives is acceptable when its lever value is less than 1. In perspective we intend to use the artificial neural network model to predict the biological activity of new benzimidazole derivatives.