Quantitative Structure-Activity Relationship (QSAR) study of a series of 2-thioarylalkyl benzimidazole derivatives by The Density Functional Theory (DFT)

In this work, we used the quantum density theory (DFT), B3LYP / 6-311G (d, p) to establish a QSAR (Quantitative Structure Activity Relationships) model on a series of molecules derived from 2-thioarylalkyl-1H -Benzimidazole. This model is built with molecular descriptors and anthelmintic activities against the haemonchus contortus. The statistical indicators of this model are: the coeffi cient of determination R2, a standard deviation S, the Fisher coeffi cient F and the cross-validation coeffi cient Qcv. The statistical parameters of the model are effi cient.


Introduction
Livestock are an important source of income in developing countries and contribute to food security. In addition in Africa, it often intervenes in the gross domestic product to the tune of 10% to 20% [1]. Livestock farming in most of the African tropics is exposed to a number of factors which slow down its development, including animal diseases [2].
Among these diseases, gastrointestinal strongyliasis in cattle breeding is one of the main pathologies which causes enormous economic losses for the farmer [1].
The fi ght against infectious diseases remains a public health problem, which is explained by the high mortality and morbidity rate caused by these diseases [3].
Indeed there are three main families of anthelmintic available on the market. Unfortunately, the frequent use of its molecules has led to the appearance of resistance to its drugs.

Abstract
In this work, we used the quantum density theory (DFT), B3LYP / 6-311G (d, p) to establish a QSAR (Quantitative Structure Activity Relationships) model on a series of molecules derived from 2-thioarylalkyl-1H -Benzimidazole. This model is built with molecular descriptors and anthelmintic activities against the haemonchus contortus. The statistical indicators of this model are: the coeffi cient of determination R 2 , a standard deviation S, the Fisher coeffi cient F and the cross-validation coeffi cient Q 2 cv . The statistical parameters of the model are effi cient.
The quantum descriptors responsible for the anthelmintic activity of 2-thioarylalkyl-1H-Benzimidazole derivatives are the dipole moment (μ), the energy of the highest occupied orbital (E HOMO ), the smallest negative charge of the molecule (q -). These studies are based on the search for similarities between molecules in large databases of existing molecules whose activities are known. The discovery of such a relationship linking both activities and molecular descriptors makes it possible to predict the activities of new compounds, and therefore to guide the syntheses of new molecules.

Material and methods
All of the sixteen molecules used in our study have larvicidal concentrations ranging from 0.005 to 424 μg / ml. This concentration range does not allow a quantitative relationship to be defi ned between anthelmintic activity and theoretical descriptors.
Biological activities are generally expressed as the opposite of the base 10 logarithm of the activity so as to obtain higher mathematical values when the molecule is biologically effective.
The anthelmintic activity is then expressed by the anthelmintic potential pCL 100 defi ned by the relationship: Where M is the molecular mass (g/mol) and CL 100 the larvicidal concentration, it is the concentration necessary to eliminate 100% of the larvae of haemonchus contortus.

Theory level
The relationship between the values of the biological activity of the molecules studied and the molecular structures was highlighted by calculations of theoretical chemistry using the software Gaussian 09 [5]. The density functional theory DFT [6] was used for our calculations with its functional B3LYP with the base 6-311G (d,p) in order to determine the molecular descriptors.
Indeed, DFT is known to generate a variety of molecular properties in a QSAR study [7,8]. This method makes it possible to reduce the calculation time, increases predictability, and involves a lower cost in the design of drugs [9,10]. The model is obtained using the multilinear regression (RML) method using the XLSTAT [11] and EXCEL [12] software.

Calculated quantum descriptors
For the development of the QSAR model, several theoretical descriptors derived from the conceptual DFT were determined.
These descriptors are: the dipole moment (μ), the Energy of the Highest Occupied Molecular Orbital (E HOMO ) and the smallest negative charge (q-) of the molecule. These descriptors all determined following the optimization of the geometry of the molecules followed by the frequency calculation.
The calculation of the partial correlation coeffi cient between the descriptor pairs (a ij ) must be less than 0.70 which shows that the descriptors are independent of each other [13].

Estimation of the predictive capacity of the QSAR model
The quality of a QSAR model is determined based on the analysis of certain statistical criteria including the coeffi cient of determination R 2 , the standard deviation S, the Fisher coeffi cient F and the cross-validation coeffi cient Q 2 cv .
The statistical parameters R 2 , F and S relate to the adjustment between the experimental values and the calculated values. The cross-validation coeffi cient measures the accuracy of the model's prediction on the data from the training set [14].
The coeffi cient of determination R 2 measures the share of experimental variance explained by the model in relation to the total variance. Its value is between 0 and 1. The closer its value is to 1, the more observed and predicted values are not correlated [15,16].^2 The performance of the model according to the Erickson et al. criterion [22,23]. is characterized by the value of Q 2 cv >0.5 for a satisfactory model and for an excellent model Q 2 cv must be close to 0. 9 The cross-validation coeffi cient [21] measures the accuracy of the prediction on the data from the training set. It is calculated using the following relation:

Values of calculated molecular descriptors
In this QSAR work, three (03) pertinent molecular descriptors were calculated. These descriptors are: the dipole moment (μ), the Energy of the Highest Occupied Molecular Orbital (E HOMO ) and the smallest negative charge (q-) of the molecule. The Table 2 reports the different values of these molecular descriptors.
The partial correlation coeffi cients a ij between the descriptor pairs shows that they are less than 0.70, which demonstrates the independence of the descriptors used to develop the model.

Validation of the QSAR model
with a statistical indicators: Regarding the Fischer coeffi cient, we note a quantity which is worth F= 29.354. This refl ects our QSAR model is globally signifi cant. As for the cross validation coeffi cient, its value is Q 2 cv = 0.916 and is greater than 0.90. Likewise, the difference R 2 -Q 2 cv less than 0.3. Which means our QSAR developed model is excellent in predicting anthelmintic activity.
The external validation of the model is obtained by the ratio

Analysis of the contribution of descriptors in the model
The relative contribution of the descriptors in predicting the anthelmintic activity of the compounds is presented in Figure 2.
The energy of the highest occupied molecular orbital has the largest contribution followed by the dipole moment and the smallest negative charge in the molecule.

Conclusion
In this work, Quantitative Structure-Activity Relationship This study will play a very important role in explaining anthelmintic activity and will also provide guidance for the design of new molecules with improved anthelmintic activity.
From now on, for the design of new molecules with improved anthelmintic activity, we can simply play on the three descriptors of the QSAR developed model.