Hepatitis C virus (HCV), identified in 1989 as the etiological agent of parenteral non-A non-B hepatitis, often causes the development of malignant chronic disease, including liver cirrhosis and hepatocellular carcinoma, frequently resulting in death (Alter et al., 1992[3]; Choo et al., 1989[8]; Leyssen et al., 2000[24]). With an estimated 3 % of the global population infected with HCV, including 4.1 million in the United States alone, and no protective vaccine available at present, this disease has emerged as a serious global health problem (Wasley and Alter, 2000[45]; Alter et al., 1999[2]). Although significant advances have been made in the development of treatments for chronic hepatitis C, their efficacy is not universal and only 50 % success has been reported in achieving a sustained viral response for the current combination therapy with new pegylated (PEG) forms of interferon plus ribavirin (Dillon, 2004[11]; Hügle and Cerny, 2003[19]; Walker et al., 2003[43]; Wang and Heinz, 2000[44]). Moreover, this therapy has considerable liabilities including serious adverse side effects and high cost, thus highlighting the need to develop improved therapeutic options to target HCV infections (Cornberg et al., 2003[10]).

HCV is an envelope positive-stranded RNA virus. Its single-stranded ~9.6 kb RNA genome encodes a large polyprotein of ~3010 amino acids comprising 4 structural proteins (Core, E1, E2, and p7) and 6 nonstructural proteins (NS2, -3, -4A, -4B, -5A, and -5B) (Grakoui et al., 1993[14]; Hijikata et al., 1991[17]; Lohmann et al., 1995[26]). One of the NS proteins, NS5B, an RNA-dependent RNA polymerase (RdRp) is the most studied target for anti-HCV therapy as it is a crucial and unique component of the viral replication machinery (Dillon, 2004[11]; Kaushik-Basu et al., 2007[22]; Wang and Heinz, 2000[44]). NS5B, a 68 kDa membrane-associated protein contains motifs shared by all RdRps in which the catalytic domain is arranged around a central cleft in an organization that resembles a right hand, with the “palm” “finger” and “thumb” subdomains common to polymerases (Bressanelli et al., 2002[7]; Love et al., 2003[27]). Recombinant expression of active, soluble NS5B in a variety of systems has been achieved by various C-terminal deletions between 21 and 55 amino acid residues and its biochemical properties investigated (Kaushik-Basu et al., 2007[22]). All of these reported recombinant HCV RdRps utilize a wide range of RNAs as template in vitro without preference, although they do prefer certain homo-polyribonucleotides to others and their activity is stimulated by GTP under specified conditions. Many screening assays for NS5B inhibitors utilize synthetic homopolymeric templates/primers. NS5B inhibitors thus far identified by these screening procedures can be broadly classified as either nucleoside (NI) or non-nucleoside (NNI) inhibitors (Kaushik-Basu et al., 2007[22]).

Quantitative structure-activity relationships (QSAR) studies play a key role in predicting the biological activity of new compound and provide information that is useful for molecule designing and medicinal chemistry (Karbakhsh and Sabet, 2011[21]; Noorizadeh and Farmany, 2014[31]). QSAR model establishes the mathematical relationship between chemical properties or activities of compounds with their various structural parameters (descriptors) such as topological, physicochemical, stereochemical or electronic indices (Pourbasheer et al., 2014[33]; Rathod, 2011[36]). The most important step in building QSAR models is the selection of one or more molecular descriptors that can represent the true interpretation of molecular structure with its activity or properties (Niazi et al., 2006[30]). Therefore, a validated QSAR model can provide valuable information, not only about the effect of fragments in molecular graph, but also it can predict the biological activities without performing any experimental efforts that the designing results are not clear. In this contribution, multiple linear regression (MLR) technique was employed to build QSAR models using the theoretical molecular descriptors selected by stepwise (SW) and genetic algorithm (GA) methods based on the training set compounds (Li et al., 2008[25]) in order to correlate the biological activities of taken compounds with their chemical strutures.

The primary goal of this work was to develop a new and validated QSAR model, and then investigating the molecular structural requirements for improving the biological activities based on the derived models.

The total data set was separated into a training set of 59 compounds to develop the models and a test set of 13 compounds using PCA. The training and test sets are shown in Table 1(Tab. 1). After division of dataset, stepwise method was used to provide the most relevant descriptors for modeling purpose. Multiple linear regression method then was used to linearly correlate the selected descriptors based on the stepwise techniques on the biases of training set compounds, and then evaluated using group of compounds as test set. During the derivation of model, 2 compounds belonging to the test set were detected as outliers and excluded from analyses (Table 1(Tab. 1)). The derived linear equation based on SW-MLR is as follows:

pIC₅₀= 22.32 (±3.511) - 4.397 (±0.9607) EEig05x + 2.673 (±0.7931) GGI9 - 0.01958 (±0.008726) RDF065m - 0.7414 (±0.1620) Mor19m + 49.53 (±11.34) R3u+ + 0.1809 (±0.07231) C-028 (1)

N_train= 59, R²_train= 0.772, R²_test= 0.703, R²_adj= 0.745, F_train= 29.284, F_test= 0.9878, RMSE_train= 0.238, RMSE_test = 0.265, Q²_LOO=0.697, Q²_LGO= 0.720, Q²_BOOT= 0.712, CCC_train=0.871, CCC _test=0.781, r²m=0.596, r²m _average=0.433, MAE_train=0.190, MAE_test= 0.192.

In above equation, N is the number of training set compounds, R² is the squared correlation coefficient, RMSE is the root mean square error, R²_adj is adjusted R², Q²_LOO, Q²_LGO and Q²_BOOT are the squared cross-validation coefficients for leave one out, leave group out and bootstrapping respectively, and F is the Fisher F-statistic. CCC is concordance correlation coefficient and evaluates the degree to which pairs of observations fall on the 45° line through the origin (Pourbasheer et al., 2014[35]). The r²m is modified r2 value and MAE is mean absolute error. The developed model since represented lower accuracy for test set, Golbraikh and Tropsha acceptable model criteria's was employed to investigate the reliability of the derived model (Golbraikh and Tropsha, 2002[13]). Four conditions for accepting a model are as follows:

Q²_LOO > 0.5

where R is correlation coefficient between the observed and predicted values; R₀² is coefficients of calculation (correlation between predicted versus observed values with intercept of zero), and R₀′² is correlation between predicted versus observed responses for regressions through the origin; K is slope and K′ is slope of regression lines through the origin. The results of this analysis were listed in Table 2(Tab. 2). As it can be seen, the last condition for acceptance of a derived model based on SW-MLR was rejected. Therefore, the genetic algorithm as a method for variable selection was applied to the same data set (i.e. training and test set selected based on PCA) for selecting the best set of molecular descriptors. The GA-MLR analysis led to a model with six descriptors. This linear model and its statistical parameters are derived as follows:

pIC₅₀= 36.97 (±4.056) - 7.971 (±0.9724) EEig05r + 0.6368 (±0.1662) GGI4 - 0.1752 (±0.06418) SPAN - 0.5972 (±0.1320) Mor19m + 45.88 (±13.05) R3u+ - 5.624 (±1.617) R5p (2)

N_train= 59, R²_train= 0.792, R²_test= 0.713, R²_adj= 0.778, F_train= 32.985, F_test=1.3885, RMSE_train= 0.227, RMSE_test = 0.252, Q²_LOO= 0.737, Q²_LGO= 0.762, Q²_BOOT= 0.731, CCC_train=0.884, CCC_test=0.819, r²m=0.666, r²m _average=0.533, MAE_train=0.188, MAE_test= 0.213.

The PCA results were shown in Figure 1(Fig. 1). PC₁-PC₂ loadings plot using the six descriptors for the best model (GA-MLR) were shown in Figure 2(Fig. 2). In Figure 2(Fig. 2), for the loadings it is confirmed that the compounds with higher biological activity values, located on the left side which are presenting a large contribution of the R3u+ descriptor, situated on the same side in Figure 1(Fig. 1). On the other hand, compounds with lower biological activity values, on the right side, have more pronounced contributions from the other descriptors (mostly from R5p and EEig05r). Also it can be observed that the distribution of scores in Figure 1(Fig. 1) is much more in right side and upper which represent that the most of compounds in data set have higher value for descriptors that have negative values than for the descriptors with positive effects. Therefore, the selected PCs are the true representative of the molecular descriptors that can be encoded for understanding the correlation between chemical structures and biological activities.

Golbraikh and Tropsha acceptable model criteria's was employed for evaluating the prediction capability of the built GA-MLR model. The results are listed in Table 2(Tab. 2). As it can be seen, the all conditions were accepted for GA-MLR and therefore, it was used as a main model for prediction purpose. The experimental and predicted activities based on this model were given in Table 1(Tab. 1). The plot of the predicted pIC₅₀ versus the experimental pIC₅₀ is demonstrated in Figure 3(Fig. 3). As can be seen from Table 1(Fig. 1) and Figure 3(Fig. 3), the calculated activity values are in good agreement with experimental activity values.

The inter-correlation between the six selected descriptors was inspected by calculating their variance inflation factor (VIF), which are also given in Table 3(Tab. 3). The VIF values, calculated as 1/1- r², where r² is the multiple correlation coefficient of one descriptor's effect regressed on the remaining molecular descriptors. If VIF equals to 1, then no inter-correlation exists for each variable; if VIF falls into the range between 1 and 5, the related model is acceptable; and if VIF is larger than 10, the related model is unstable and a recheck is necessary (Maryam et al., 2012[28]). As it can be seen by the given information of Table 3(Tab. 3), most of the variables had VIF values of less than 5, indicating that the GA-MLR model has statistic significance.

The built GA-MLR model was validated using the leave-one-out and leave-group-out cross-validated correlation coefficients (Q²_LOO and Q²_LGO). The robustness of the GA-MLR model and its predictive ability was confirmed by the high Q²_BOOT source based on bootstrapping repeated 5000 times (Hadizadeh et al., 2013[15]). The results produced by the Q²_LOO, Q²_LGO and Q²_BOOT parameters along with other validation parameters showed the higher quality of the developed GA-MLR model. Therefore, this model can be used to predict the inhibition activity of the compounds.

The robustness of the QSAR model was further assessed by applying Y-randomization test. The dependent variable vector (inhibitory activity) was shuffled randomly and the new QSAR models (after several repetitions) would be anticipated to have low R² and Q²_LOO values (Figure 4(Fig. 4)) (Asadollahi et al., 2011[4]). As it can be seen from Figure 4(Fig. 4), after 200 times shuffling the biological response for compounds, all of the derived new models were less than that of obtained in real response.

The Williams plot, the plot of the standardized residuals versus the leverage (h), is used to visualize the applicability domain (AD) of QSAR models (Vahdani and Bayat, 2011[42]). From the Williams plot (Figure 5(Fig. 5)), it is obvious that there are only two compounds (No. 1 and No. 6 belonging to the training set) have the leverage higher than the warning h∗ value of 0.356, thus they can be considered as structural outliers. From Figure 4(Fig. 4), it is obvious that the standardized residuals observed for all the compounds in the training and test sets are smaller than three standard deviation units (3δ). Thus, the generated model is acceptable for prediction purpose.

A robust QSAR model was developed based on PCA-GA-MLR for a dataset consisting of 72 HCV NS5B polymerase inhibitors. The derived models were validated based on several validation techniques, and it was observed that GA-MLR is more accurate than the derived SW-MLR model. Based on the obtained results of GA-MLR, it was observed that the capability of having more resonances in molecular graph is not appropriate and since most of the functional groups belonging to polar groups can represent the presence, therefore, the replacing of more polar groups should be avoided addressing to the negative effect of EEig05r and R5p descriptors. It was also seen that distance of substituents from mass center would cause negative impact over biological activities. However, a good biological activity can be presented if the charge transfer between bonding network and steric contributions to molecules shape increase. In this work, the proposed models could identify and provide better insights about the chemical structure requirements for increasing the pIC₅₀ values.