Abstract

The aim of this study is to develop a predictive model of the antidiabetic activity of a series of twenty-four (24) 1,3,4-thiadiazole molecules using quantum chemical methods. The molecules were optimised from the B3LYP/6-31+G (d, p) level of theory. The extracted descriptors are: bond length (l(C-N)), bond angle α(S-C-N); dipole moment (µ(D)) and standard entropy of formation (∆S⁰_f). This study was carried out quantitatively and qualitatively using the Multiple Linear Regression (RLM) and Non-Multiple Linear Regression (NMR) methods. Through this study, we have developed two regression models that are accredited with good statistical indicators. The statistical indicators of the model obtained by the RLM method are: the coefficient of determination R²= 0.931, the standard deviation S = 0.045, the Fischer coefficient F = 202.657, and the correlation coefficient of the cross-validation Q²cv=0.931. Those of the second model developed by the RLNM method are: R² =0.942; S of 0.048, F of 241.887 and Q²cv= 0.942. Furthermore, the bond angle α(S-C-N) is the priority descriptor for the prediction of the biological activity of the studied compounds. The acceptance criteria of Eriksson et al. used for the test set are verified. The external validation set was also verified for all criteria of Tropsha et al and Roy et al.

1. Introduction

Diabetes is a serious chronic disease that occurs when the pancreas does not produce enough insulin (the hormone that regulates the concentration of sugar in the blood, or glucose), or when the body does not properly use the insulin it produces ¹. The International Diabetes Federation (FID) has released new figures showing that 573 million adults are now living with diabetes, an increase of 16% (74 million people) since the FID's 2019 estimates ¹. This highlights an alarming growth in the prevalence of diabetes worldwide. It is increasing in low- and middle-income countries by about 80% and prevalence is higher in urban than in rural areas, with greater reach to disadvantaged social groups. In Côte d'Ivoire, WHO estimates for the year 2010 indicate a national prevalence of 9.6%, of which 9.8% in men and 9.3% in women ². Generally, the two most common forms of diabetes are: type 1 diabetes (DT1) and type 2 diabetes (DT2). We will focus on DT2as it is the most common. However, effective approaches exist to prevent DT2 and its complications and premature death ¹. Diabetics use oral antidiabetic drugs (ADO) which have side effects including digestive disorders, hypoglycaemia, risks of urinary tract infections ¹. The development of new ADO for the treatment of diabetes based on thiadiazole derivatives could make a contribution to the management of diabetics. 1,3,4-thiadiazole and its derivatives possess a wide range of therapeutic activities such as antimicrobial ³, antifungal, diuretic, antiulcer ⁴, antimycobacterial ⁵, antioxidant/radio-protective ⁶, anti-inflammatory, anticonvulsant, antidepressant, anticancer, anti-lesion ^{7, 8}, and antidiabetic ⁹. The Quantitative Structure-Activity Relationship (QSAR) study is the method that correlates the molecular structure with a well-determined effect such as biological activity. It is increasingly used to reduce the excessive number of experiments, which are sometimes long, dangerous and costly in terms of time and money ^{10, 11}.

The overall objective of this work is to develop reliable models to explain and predict the anti-diabetic activity (blood glucose (mg/dl)) of a series of twenty-four (24) thiadiazole (DT) derivatives. All these molecules were synthesised and tested biologically in rats for blood glucose levels (blood glucose mg/dl) ^{12, 13}. Figure 1 shows the different molecules.

2. Materials and Methods

Thiadiazole derivatives were optimised using the gaussian 09 software ¹⁴. Density Functional Theory (DFT) methods are generally known to generate a variety of molecular properties in QSAR studies. These increase the predictability of QSAR models while reducing the computational time and cost implication in new drug design ^{15, 16}. The DFT method with the B3LYP/6-31+G(d,p) level of theory was used to determine the molecular descriptors. Modelling was done using two methods. The first method is multiple linear regression and the second method is non-multiple linear regression which are implemented in Excel ¹⁷ and XLSTAT ¹⁸ spreadsheets. The twenty-four (24) molecules used in this study have blood glucose levels ranging from 103,4 to 271,5 mg/dl. This biological activity was expressed by the biological potential pIC₅₀ ¹⁹ defined by equation (1)

(1)

Where M is the molecular weight of the compound in g/mol, and IC50 is the inhibitory concentration in g /l.

In order to develop our QSAR model, some theoretical descriptors were determined. In particular, the dipole moment (μ(D)), the bond length (C-N), the bond angle (α(S-C-N)) and the standard entropy of formation (Δ_fS⁰), The dipole moment is based on the existence of electrostatic dipoles. It is a global distribution of electric charges in a molecular system, such that the barycentre of positive charges does not coincide with that of negative charges. The dipole moment is a vector quantity. For a distribution of charges at distance each, the total dipole moment is established as follows:

(2)

The dipole moment is used to describe the overall polarity as well as the existence of interactions of molecular systems such as Van der Waals forces, and also to predict their solubility in polar solvents. The dipole moment is an important property that gives an idea of the reactivity of the molecule ²⁰. Furthermore, it indicates the stability of a molecule in water. Thus, a high dipole moment will reflect a low solubility in organic solvents and a high solubility in water.

The geometric descriptors used are the bond length l (C-N) in Armstrong (A°) and the valence angle α(S-C-N) in degrees (°) (Figure 2). These geometric descriptors are of great importance in modelling ²¹. These descriptors are illustrated in the figure below on the thiadiazole moiety side.

The thermodynamic quantity of formation, i.e. the standard entropy of formation, was determined according to J.W. Otchersky et al ¹⁴.

(3)

x: Number of atoms of X in the molecule

The quality of a model is determined on the basis of various statistical analysis criteria including the coefficient of determination R², the standard deviation S, Fischer's F and the cross-validation correlation coefficients Q²_CV. R², S and F relate to the fit of the calculated and experimental values. They describe the predictive capacity within the limits of the model, and allow the accuracy of the calculated values to be estimated on the test set ^{21, 22}. The cross-validation coefficient Q²_CV provides information on the predictive power of the model. This predictive power is said to be "internal" because it is calculated from the structures used to build the model. The correlation coefficient R² gives an evaluation of the dispersion of the theoretical values around the experimental values. The quality of the modelling is better when the points are close to the fit line ²³. The fit of the points to this line can be evaluated by the coefficient of determination determined by the following expression:

(4)

Where:

: Experimental value of anti-diabetic activity

: Theoretical value of anti-diabetic activity and

: Average value of the experimental values of the anti-diabetic activity.

The closer the R² value will be to 1 the more the theoretical and experimental values are correlated

Furthermore, the variance is determined by relation 5:

(5)

Where k is the number of independent variables (descriptors), n is the number of molecules in the test or training set and n-k-1 is the degree of freedom.

Another statistical indicator used is the standard deviation or RMCE. It is used to assess the reliability and accuracy of a model according to the relationship:

(6)

The Fisher F test is also used to measure the statistical significance of the model, i.e. the quality of the choice of descriptors making up the model. It is determined by the following relationship:

(7)

The coefficient of determination of the cross-validation, Q²_CV is used to evaluate the accuracy of the prediction on the test set. It is calculated using the following relationship :

(8)

Model acceptance Criteria

The performance of a mathematical model, according to Eriksson et al ²⁴, is characterised by a value of Q²cv > 0.5 for a satisfactory model when for the excellent model Q²cv > 0.9. According to them, given a test set, a model will perform well if the acceptance criterion R²- Q²cv< 0.3 is met.

Multiple Linear Regression and Non-Multiple Linear Regression (RLM and NMR)

The statistical technique of multiple linear regression (RLM) is used to study the relationship between a dependent variable (Property) and several independent variables (Descriptors). This statistical method minimises the differences between the actual and predicted values. It was also used to select the descriptors used as input parameters in the non-multiple linear regression (NMRL). As for the NLRM analysis, it also allows for the improvement of the structure-activity relationship in order to quantitatively assess the Activity. It is the most common tool for studying multidimensional data. It is based on the following pre-programmed XLSTAT functions:

(9)

Where a, b, c, d,... represent the parameters and, x1, x2, x3, x4,... represent the variables

The domain of applicability of a QSAR model is the physico-chemical, structural or biological space, in which the model equation is applicable to make predictions for new compounds ²⁵. It corresponds to the region of chemical space including the compounds in the training set and similar compounds, which are close in the same space ²⁶. Indeed, the model, which is built on the basis of a limited number of compounds, by relevant descriptors, chosen among many others, cannot be a universal tool to predict the activity of any other molecule with confidence. It appears necessary, even mandatory, to determine the DA of any QSAR model. This is recommended by the Organisation for Economic Co-operation and Development (OECD) in the development of a QSAR model ²⁷. There are several methods for determining the domain of applicability of a model ²⁶. Among these, the approach used in this work is the leverage approach. This method is based on the variation of the standardised residuals of the dependent variable with the distance between the values of the descriptors and their mean, called leverage ²⁸. The h_ii are the diagonal elements of a matrix H called hat matrix. H is the projection matrix of the experimental values of the explained variable into the space of the predicted values of the explicated variable Y_pred such that:

(10)

H is defined by the expression:

(11)

The applicability domain is delimited by a threshold value of the lever noted h*. In general, it is set at 3 (p+1)/n, where n is the number of compounds in the training set, and p is the number of descriptors in the model ^{29, 30}. For standardised residuals, the two limit values generally used are ±3σ, σ being the standard deviation of the experimental values of the quantity to be explained ³¹: this is the "three sigma rule" ³².

3. Results and Discussion

The learning set of the seventeen (17) molecules and the seven (7) molecules of the test set for anti-diabetic activity are presented in Table 1.

The values of the partial correlation coefficients aij of the descriptors are presented in Table 2.

The partial correlation coefficients aij contained in Table 2 between the descriptor pairs (α(S-C-N), l(C-N)), (α(S-C-N) (μ(D)), (α(S-C-N), (ΔfS0)) (l(C-N)) (ΔfS0)), (l(C-N) (μ(D)) and (μ(D)) (ΔfS0)) This indicates the non-dependence of the descriptors used to develop the models as (aij <0.7) ^{33, 34}.

In a model equation, the negative or positive sign of the coefficient of a descriptor reflects the effect of proportionality between the evolution of the inhibitory concentration IC and this physico-chemical parameter of the regression equation. Thus, the negative sign indicates that when the value of the descriptor is high, the inhibitory concentration IC decreases, whereas the positive sign reflects the opposite effect. In this work, two statistical analysis tools were used, namely Multiple Linear Regression (RLM) and Non-Multiple Linear Regression (NMR).

The equation of the QSAR model is presented below. The statistical indicators are given in the table below (Table 3).

(12)

The negative sign of the coefficient of the bond length and the standard entropy of formation reflects that the antidiabetic activity will be enhanced for low values of these descriptors. In contrast, the positive sign of the coefficient of the dipole moment (μ(D)) and the bond angle α(S-C-N) indicates that high values of these descriptors improve the anti-diabetic activity.

The coefficient of determination R² indicates a 93.1% inclusion of the molecular descriptors that define this model with a standard deviation of 0.043. The significance of the model descriptors is given by the Fischer Test F= 202.657 > 2.9. This high value reflects a strong relationship between the blood glucose level and the model parameters. The cross-validation correlation coefficient Q ²_CV= 0.931>0.9 indicates that this model has excellent predictive power according to Erikson et al. ²⁴. Also, the acceptability of this model was proven by calculating R²-Q²_CV= 0.931-0.931= 0.000<0.3. These different observations are corroborated by the corresponding regression line of this model (Figure 3). The blue points correspond to the test set and the red points to the validation set.

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 (S=0.043) between the values of pIC^exp and pIC^th, so there is a good similarity in these values. This similarity is illustrated in Figure 4.

The internal validation of the RLM model was carried out using the randomisation test.

The randomisation test of the RLM model was carried out on the molecules of the training set by randomly swapping the values of the activities while keeping the descriptors for the construction of the model. We stopped at ten (10) iterations. The randomised coefficients of determination (R²r) for each iteration are listed in Table 4.

The value of Roy's parameter (R²p=0.5357) is greater than 0.5, hence the model really exists and is not due to chance ³³.

The values of the external validation test were checked by applying the Tropsha and Roy criteria. These criteria are shown in Table 5 and Table 6 below.

values met Tropsha's criteria, so the model is acceptable for predicting the All antidiabetic activity of thiadiazole derivatives.

Roy's criteria are verified as the is greater than 0.5 and the is less than 0.2. This model is robust and has good predictive power

The equation of this QSAR model is a polynomial equation of degree 2 presented below. The statistical indicators are also given in Table 7.

(13)

The coefficient of determination R² = 0.942 justifies the good correlation between the predicted and observed values. The model therefore has excellent explanatory power. The low value of the standard deviation (S= 0.048) shows the good statistical fit. The significance of the model descriptor is expressed by the Ficher-Snedecor coefficient whose value is: 241.887. This high value, well above the threshold value (=2.9), indicates that there is a strong relationship between the blood glucose level and the descriptors of this model. The cross-validation correlation coefficient (Q²cv) is equal to 0.942 ˃ 0.9 indicating an excellent model according to Eriksson et al ²⁴. R²-Q²cv= 0.942- 0.942= 0.000 < 0.3 means that the model is acceptable. All these statistical indicators show that the model developed explains the antidiabetic activity in a significant and satisfactory way. It can be used to predict the anti-diabetic activity of other molecules.

The regression line of the RLNM model between the experimental and predicted anti-diabetic activities of the training set and the validation set is shown in Figure 5.

The low value of the standard deviation (S) of 0.048 for the RLNM model attests to the good regression between the predicted and experimental values. The curves (Figure 6) show the similar evolution of the data from this model for the prediction of the inhibitory concentration (IC₅₀) of 1,3,4-thiadiazole derivatives.

The internal validation of the model by the randomisation method, ten (10) iterations were carried out. The R² values obtained after these iterations are shown in Table 8.

The value of Roy's parameter (R²p=0.5755) is greater than 0.5, hence the model really exists and is not due to chance ³³.

The values of the external validation test were checked by applying the Tropsha and Roy criteria. These criteria are shown in Table 9 and Table 10 below.

All values meet the Tropsha criteria, so the model is acceptable for predicting the antidiabetic activity of thiadiazole derivatives

The analysis of Table 10 shows that, the is greater than 0.5 and the is less than 0.2. This result reflects that the model the Roy criteria. It can therefore be stated that the model is robust and has good predictive power.

The study of the relative contribution of the descriptors in the prediction of the antidiabetic activity of the compounds was carried out. The different contributions are presented in Figure 7.

The bond angle α(S-C-N) shows a significant contribution compared to the bond length or the dipole moment or the standard entropy of formation. Thus, the bond angle turns out to be the priority descriptor in predicting the antidiabetic activity of the 1,3,4-thiadiazole derivatives studied.

Models cannot be used to predict the biological activity of all molecules. It is therefore necessary to define a range in which the formulae can be used. Figure 8 shows the range of applicability of the models using the lever method.

The Williams diagram shows that the standardised residue values of the compounds are between -3δ and + 3δ. Moreover, the leverage values of the molecules are lower than the threshold leverage value h*(h*=0.882). Also, all the molecules are within the applicability range of the model.

4. Conclusion

The dipole moment (µ(D)), bond angle α(S-C-N), bond length l(C-N) and standard entropy standard of formation ((Δ_fS⁰)) allowed us to predict the antidiabetic activity of 1, 3, 4-thiadiazole derivatives. This study revealed strong correlations between the predicted and experimental values of the blood glucose potentials (pIC₅₀) and descriptors. These proposed models identify the angle of the angle as a priority descriptor for improving antidiabetic activity. Furthermore, the negative signs of the binding length and standard entropy of formation indicate that anti-diabetic activities will be enhanced for a low value of the binding length. Both models are accredited with good statistical indicators. Thus, for the RLM model, we have: (R² = 0.931, S = 0.043, F = 202.657> 2.9, Q²cv = 0.931) and for the RLNM model we have: (R² = 0.942, S = 0.048, F = 241.887 > 2.9, ( = 0.942). Both models show that the bond angle α(S-C-N) is the primary descriptor in predicting the antidiabetic activity of the 1, 3, 4-thiadiazole derivatives studied. The two models obtained were validated using a validation set comprising seven molecules by applying the Tropsha criteria.

References