Activity and toxicity modelling of some NCI selected compounds against leukemia P388ADR cell line using genetic algorithm-multiple linear regressions

2.1 Scaling of activities and descriptors data

The response variable (biological activities) and the explanatory variable (molecular descriptors) and were scaled using auto-scaling and range scaling procedure. According to Golbraikh et al. (2003), the modeling set and the evaluation set were scaled separately. Usually, variables with larger pre-scaled value have high coefficient and those with smaller pre-scaled values has a low coefficient in the regression equation (Foudah et al., 2014). Hence, the need to transform the variables data to the standard data by subtracting the mean and dividing by its standard deviation. $$x_i^{\prime }\frac{(x_i-{\widehat{\mu }}_i)}{{\widehat{\sigma }}_i}$$ where $x_i$ is the original descriptors, ${\widehat{\mu }}_i$ is the arithmetic mean of each descriptors, ${\widehat{\sigma }}_i$ is the standard deviation, and $x_i^{\prime }$ the scaled descriptors. This process removes the dependence of the regression coefficient on unit. This is good for cases where variables indicate concentrations or amounts of chemical compounds, or were variables represent measurements in unrelated units (Wehrens, 2011, Mevik et al., 2011).

Range scaling techniques or normalization usually give linear transformation that set the maximum and minimum of each scale to be [0, 1] or [−1, 1], etc. Here, the minimum value in a vector (a column representing a given variable “y”) is subtracted from every data point “y_n” of N samples and the results are divided by the range. $$N_{[-1,1]}(y_n)=2\left(\frac{y_n-y_{\mathit{min}}}{y_{\mathit{max}}-y_{\mathit{min}}}\right)-1$$ where $y_{\mathit{min}}$ and $y_{\mathit{max}}$ are, respectively, the minimum and maximum values that can be found in the data set, with respect to all the data points and the variable to normalize. The minimum and maximum value of the evaluation set was used in this normalization procedure (Tropsha, 2010, Roy et al., 2013). These range-scale descriptors have a minimums and maximum the value of −1 and 1 respectively.

These compounds were then divided into training and test sets by the Kennard-Stone algorithm (Kennard and Stone, 1969). The QSAR models were generated using the Genetic Function Approximation (GFA). The GFA technique is a collection of Genetic Algorithm used to evolve a population of equations that best fit the training set (Deb et al., 2002, Leardi et al., 1992). A unique feature of GFA is that it yields a population of models, instead of generating a single model. The developed models were then subjected to internal and external validation and Y-randomization tests so as to justify their predictability (Tropsha, 2010).

2.2 Splitting of data-set into modelling sets and evaluation test sets

The data set was divided into two sets, the modelling set, and test set. The modelling set is used in developing the model, it contains eighty percent of the entire data set. While the test set which constitutes the remaining twenty percent of the whole data set was not used in the construction of the model but to ascertain the predictive ability of the model (Tropsha, 2010).

3.3 Applicability domain study

The applicability domain of the models were evaluated using Uzairu’s plot which is novel applicability domain technique by Arthur et al. (2016a). This techniques involves plotting the standardized residuals of the activities and toxicities against the normalized mean distance between the values for the complete dataset of the molecular descriptors appearing in the model for both the toxicity and activity.

The plots shown in Figs. 5 and 6, indicates that all the compounds in both cases fell within the chemical space of the models. Fig. 5 shows the presence of one outlier with ID = 32, which is cyanomorpholinodoxorubicin, while in Fig. 6 the outlier Taxol was identified with ID = 72. These models were unable to predict the experimental values of these compounds because the molecular structure of the compound were completely different. We found out that these compounds were very large in size and they do not contain the primary molecular descriptor needed to predict their experimental values.

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Fig. 5

Uzairu’s plot: A graphical representation of Standardized residual against normalized mean distance of activity (pGI₅₀).

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Fig. 6

Uzairu’s plot: A graphical representation of Standardized residual against normalized mean distance of toxicity (pLC₅₀).

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Also, the plot of the residual against the predicted values of pLC₅₀ and pGI₅₀ for both the training and test sets shown in Figs. 2 and 4 respectively. The model did not show any proportional and systematic error because the propagation of the residuals on both sides of zero is random.

The multi-collinearity amid the descriptors existing in the models was spotted by calculating their variation inflation factors (VIF), which can be calculated as follows: $$\mathit{VIF}=\frac{1}{1-R^2}$$ where R² is the correlation coefficient of the multiple regression between the variables within the model (Shapiro et al., 2002). The corresponding VIF values of the descriptors were presented in Table 3.1 and 3.2. The tables show, all the variables have VIF values of less than five except for two descriptors, indicating that the descriptors were reasonably orthogonal.

In order to assess the strength of the model, the Y-randomization test was used in this study (Golbraikh et al., 2003, Tropsha et al., 2003). Y-randomization test settles whether the model is gotten through coincidental correlation, and is a true structure–activity relationship to validate the capability of the training set molecules.

The new QSAR models (after several repetitions) would be expected to have low R² and Q²_LOO values (Table 2.1 and 2.2). If the opposite happens, then an acceptable QSAR model cannot be obtained for the specific modeling method and data. The results of Tables 2.1 and 2.2 indicate that an acceptable model is obtained by GA–MLR method and the model developed is statistically significant and robust.

3.4 Interpretation of descriptors

By interpreting the descriptors contained in the QSAR model, it is plausible to increase a few bits of information into factors, which are identified with the anti-leukemia action. Hence, a satisfactory understanding of the chosen descriptors is given below. The brief representations of descriptors shown in Table 3.1 and 3.2. To look at the comparative meaning and also the importance of every descriptor in the model, the assessment of the mean effect (MF) was established for every descriptor (Pourbasheer et al., 2009, Riahi et al., 2009); This was achieved by using an MF mathematical statement which is given as $${\mathit{MF}}_j=\frac{{\beta }_j{\sum }_{i=1}^{i=n}{d}_{\mathit{ij}}}{{\sum }_j^m{\beta }_j{\sum }_i^n{d}_{\mathit{ij}}}$$ ${\mathit{MF}}_j$ is given as the mean effect for the considered molecular descriptor j, while ${\beta }_j$ is the coefficient of the descriptor j, $d_{\mathit{ij}}$ represents the values for the target descriptors of each molecule, and m is the total number of descriptors in the model. The MF values proves the relative implication of a descriptor, associated with other descriptors in the model. Its sign shows the variation direction in the estimations of the model as an effect of the descriptor values.

The dipole moment is an electric polarization descriptor; it encodes information about charge distribution in molecules. They are also important in modelling solvation properties of the compounds which depend on solute/solvent interactions since the mean effect of the dipole moment was found to be negative hence a reduction in the polarity of these compounds was found to steadily decrease the toxicity of anti-leukemia compounds. The dipole moment is given as $$\mu =-{\displaystyle \sum _{i=1}^{\mathit{occ}}}{\int }_{(V)}{\varphi }_i\widehat{r}{\varphi }_i\mathit{dv}+{\displaystyle \sum _{a=1}^M}{Z}_a{\overrightarrow{R}}_a$$

• ${\varphi }_i$ – molecular orbitals

• $\widehat{r}$ – electron position operator

• Z_a – a-th atomic nuclear charge

• ${\overrightarrow{R}}_a$ – position vector of a-th atomic nucleus

Methanal as a functional group count descriptor, whose mean effect was also found to negatively affect the toxicity of the compounds, while the shadow length Lx is the maximum dimensions of the molecular surface projections and it also negatively affects the toxicities of these compounds, this was confirmed by the negative values of the mean effect.

AATSC4i, MATS3e, and AATS8e, ATSC6C, ATSC6i, AATSC6v, AATSC1p, GATS7v are 2D Autocorrelation descriptors developed by (Todeschini and Consonni, 2009), The 2D autocorrelation descriptors have been successfully employed by Fernandez et al. (Fernandez-Lozano et al., 2015) Caballero (Caballero, 2010, Fernández et al., 2005, Vilar et al., 2009).

In these descriptors, the molecule atoms represent a set of discrete points in space, and the atomic property and function are evaluated at those points. The sign on the mean effects influences their behaviors in whatever model they are found in. theses descriptors as defined on Table 3.1 and 3.2 describes the weight by first ionization potential, weighted by Sanderson electro-negativities, weighted by charges, by van der Waals volumes and by polarizabilities of the molecules used determines the potency of anti-leukemia compounds.