The chemical structures along with observed activity data of the compounds used in this study are shown in Table 1. The activity data were taken from various published studies [6,7].

A set of common molecular descriptors related to physicochemical, electronic and geometric properties of the molecules was used for this study. As all the compounds studied have a common skeleton, we found it judicious to

Table 1. Structures of molecular and their biological activities

describe the molecule by means of properties of the substituents (R₁, R₂, R₃, R₄, R₅, R₆, R₇, R₈, R₉ and X) attached to the basic skeleton (Figure 1). Determination of the pertinent properties for a given substituent may be useful for evaluating local interactions between the molecule and the receptor site.

Moreover, we tried to take into account properties of the molecule such as its molecular weight, size, height etc. This is justified by the fact that, before their possible interaction with a given receptor site, the molecules must be transported through many liquid layers and correct general dimensions for site access [8].

Molecular properties used for each substituents were:

• Size and shape described by means of van der waals volume (V) and surface (S) [9].

• Molecular dimension (length, width and height). Length (L) is the distance along the screen x-axis between the left and right most atoms plus their van der Waals radii. Width (W): is the distance along the screen y-axis between the top and bottom most atoms plus their van der Walls radii. Height (H): is the distance along the screen z-axis between the nearest and farthest atoms plus their van der Waals radii.

• Ratios V/L, V/W, W/H were also calculated.

• logP, the partition coefficient between n-octanol and water.

• Molar refractivity (MR) [10].

• Hydrogen-bonding donors (HBD) and hydrogen-bonding acceptors (HBA).

All these descriptor were calculated with the demo version of the molecular modelling program (MMP).

Since a high-correlation coefficient only indicates how well the equations fit the data, cross-validation procedure [13] was carried out in order to explore the reliability of the proposed models. In this aspect, the well-known “leave-one-out” (LOO) approach was used in which a number of models were developed with one sample ignored each time. Then, the ignored data were predicted by each model and the differences between predicted and observed activity values were evaluated. The LOO crossvalidation coefficient that is given by Equation (2) was used as an indicator of the predictive performance and stability of a model. In general, LOO cross-validated coefficient being higher than 0.5 can be considered as a statistical proof of the high-predictive ability [14]. The formulae used to calculate the aforementioned statistics are presented below [15] (Equation (2))

In the case of LMO, M represents a group of randomly selected data points which would leave out at the beginning and would be predicted by the model which was developed using the remaining data points. So, M molecules are considered as prediction set. The can be calculated by Equation (3) [16]:

where in y_exp and y_pred are the observed and predicted values for the dependent variables, respectively, and y is the average observed value [14].

The statistical quality of the equations was judged by different parameters [17] like square of correlation coefficient, explained variance, standard error of estimate and variance ratio at specified degrees of freedom [18].

ANN is a massive parallel-distributed information processing system that has certain performance characteristics, resembling biological neural networks of the human brain. ANN has been developed as a generalization of mathematical models of human cognition and neural biology [19]. The available data set is partitioned into two parts, one corresponding to training and the other corresponding to test of the model. The purpose of training is to determine the set of connection weights and nodal thresholds that cause the ANN to estimate outputs that are sufficiently close to target values. This fraction of the complete data to be employed for training should contain sufficient patterns so that the network can mimic the underlying relationship between input and output variables adequately [20].

The network consists of an input layer, an output layer and a number of hidden layers. At each node in a layer the information is received, stored, processed and communicated further to nodes in the next layer. All the weights are initialized to small random numeric values at the beginning of training. These weights are updated or modified iteratively using the generalized delta rule or steepest-gradient descent principle. The training process is stopped when no appreciable change is observed in the values associated with the connection links or some termination criterion is satisfied. Thus, the training of aback-propagation network consists of two phases: a forward pass during which the processing of information occurs from the input layer to the output and a backward pass when the error from the output layer is propagated back to the input layer and the interconnections are modified [21].

An example of a network topology is shown in Figure 2 [22].

MLR analysis was performed on the compounds described in Table 1; we have included all 65 molecules of the training set for the model generation.

After collecting the data, we submitted all parameters to regression; a few suitable models were obtained. The best model is shown in Equation (4):

^*LogP: Hydrophobic descriptor for all the molecular, LogP(R₁): Hydrophobic descriptor for the substituent 1, HBD(R₇): Hydrogen-bonding donors for the substituent 7 and MW(R₇) : Molecular weight for the substituent 7.

We take away the one molecule having higher than as defined in Equation (5):

With is the mean of the observed activity. Consequently, a new regression model was derived using 64 molecules Equation (6):

^*LogP: Hydrophobic descriptor for all the molecular, Log P(R₁): Hydrophobic descriptor for the substituent 1, HBD(R₇): Hydrogen-bonding donors for the substituent 7 and MW(R₇): Molecular weight for the substituent 7.

In QSAR equations, n is the number of data points, r is the correlation coefficient between observed values of the dependent and the values calculated from the equation, r² is the square of the correlation coefficient and represents the goodness of fit, q² is the cross-validated r² (a measure of the quality of the QSAR model), and s is the standard deviation [23].

The 64 active compounds were randomly divided into a training set of 43 compounds and a test set of 21 compounds. The training set compounds were used to develop a QSAR model, and the test set compounds were used to validate the reliability and the predictive ability of the model (Table 2).

In the above results, the most significant variable is Hydrogen-bonding donors descriptor, following by the contribution of the and, respectively. Molecular weights contribute poorly. This result is justified by calculating the descriptors contribution (Table 3), according to the method of Gore (1952). [24,25] (Equation (7)):

With: Regression coefficient for descriptor

Standard deviation for descriptor.

• Fraction of the variance: It is believed that the closer the value of r² to unity, the better the QSAR model. The value of r² for this QSAR model is 0.8364 which suggest that these QSAR model explain 83.64% of the variance in the data. According to the literature, the predictive QSAR model must have.

• Cross-validation test: The values of for these QSAR models are 0.72 with multiple regressions in the other hand; the value is 0.724 with artificial neural network. The high values of q² validate these QSAR models. According to the literature, the predictive QSAR model must have [26].

• Standard deviation(s): s is the standard deviation

• about the regression line. The smaller the value of s the better the QSAR model. The value of s for this QSAR model is 0.4168.

• Fischer statistics (F): Fischer statistics (F) is the ratio between explained and unexplained variance for a given number of degree of freedom. The larger the value of F the greater the probability that the QSAR model is significant.

The result of the credibility test of 64 molecules shows that the descriptors used express the activity studied very well because the statistical quality of the model decreases dramatically. The correlation coefficient, standard deviation and Fischer statistics pass are respectively to. Finally, the plot of experimental and predicted values of activity (Figure 3) from multiple linear regressions showed a good fitting function.

As a second step, we were interested to investigate the non-linear characteristics of the activity parameter. Therefore, a back propagation artificial neural network [19] was developed using the descriptors appearing in the MLR model as its inputs.

The optimal architecture of the selected NN model was [4-3-1] after optimization study (Figure 4), which means that the model had 4 input neurons in the input layer (the selected descriptors), 3 hidden neurons in the hidden layer, and one neuron in the output layer. Data set was separated into two groups: training and test sets.

The training set, consisted of 43 molecules, was used

Table 2. Validation tests of the reliability and predictive ability of model.

Table 3. Descriptors contribution in Equation (6).

for the model generation. However, the test set, consisted of 21 molecules, was used to take care of the overtraining.

The standard deviation between calculated and observed activity was 0.345 by using Levenberg-Marquardt Method, which was found to be superior that obtained using MLR. In addition, the correlation coefficient square between observed and calculated value 0.8724 by using Levenberg-Marquardt Methods.

Finally, the plot of experimental and predicted values of activity (Figure 5) from artificial neural network showed a good fitting function.

The contribution of descriptors was estimated from the [4-3-1] neural network architecture. The descriptor under study was removed from the [4-3-1] calculated the output of each molecule as usual. The mean of the deviations absolute values between the observed activity and the estimated one for all compounds was calculated. This process was reiterated for each descriptor. Finally, the contribution of each descriptor is given by Equation (8) [27]:

With: Mean value of absolutes deviations between predicted and calculated activity.

Sum of means values of absolutes deviations between predicted and calculated activity for 4 descriptors.

Table 4. Property of the physicochemical descriptors and their contributions.

According to the results above (Figure 6), it appears that LogP have the same classification in the two methods (MLR and ANN) used but the other descriptors have changed order. These results indicate the existence of non-linear relationships between activity and molecular descriptors that appeared pertinent for the linear model.