IN SILICO MODEL QSPR FOR PREDICTION OF STABILITY CONSTANTS OF METAL-THIOSEMICARBAZONE COMPLEXES

In the present work, the stability constants log11 and the concentration of metal ion and thiosemicarbazone in the solutions of their complex were determined by using in silico models. The 2D, 3D, physicochemical and quantum descriptors of the complexes were generated from the molecular geometric structure and semi-empirical quantum calculation PM7 and PM7/sparkle. The quantitative structure and property relationships (QSPRs) were constructed by using the ordinary linear regression (OLR) and artificial neural network (ANN). The best linear model QSPROLR (with k of 6) involved descriptors k0, core-core repulsion, xp5, xch5, valence, and SHHBd. The quality of model QSPROLR had the statistical values: R2train = 0.898, R2adj = 0.889, Q2LOO = 0.846, MSE = 1.136, and Fstat = 91.348. The neural network model QSPRANN with architecture I(6)-HL(6)-O(1) had the statistical values: R2train = 0.9768, and Q2LOO = 0.8687. The predictability of QSPR models for the complexes of the test group turned out to be in good agreement with those from the experimental data in the literature.


Introduction
Thiosemicarbazone compounds and its metal complexes have many practical applications. Thiosemicarbazones are known as analytical reagents [1,2] and have biological activities [3]. The complexes of thiosemicarbazones and metal ions have biological applications and great medicinal activities including antibacterial, antifungal, antimalarial, antitumor, and antiviral activity [4][5][6]. They are also used as a catalyst in chemical reactions [7].
For complexes, the stability constant is an important parameter. This is used to identify the complex stability in solutions. It is also a measure of the strength of the interaction between the ligand and the metal ions to form different complexes. In addition, the stability constant of complexes is the basic factor to explain such phenomena as reaction mechanism and various properties of biological systems. We can calculate the equilibrium concentration of ingredients in a solution based on the stability constant. The changes of the complex structure in solutions can be forecast by using the initial concentration of the metal ion and the ligand. Recently, the stability constant of the complexes has been estimated by incorporating the UV/VIS spectrophotometric method and the computational techniques [8,9]. Furthermore, the theoretical methods are also used for predicting the stability constant of complexes based on the relationships between the structural descriptors and the properties [9]. A few complex descriptors between the metal ions and thiosemicarbazone were determined by using quantum mechanics methods [12,13,20].
In recent years, computers have been becoming a helpful tool and an effective means of strong calculation in different areas of chemistry, such as inorganic chemistry, analytical chemistry, organic chemistry, physical chemistry, material simulation, and data mining [14][15][16].
The molecular design by means of a computer is also a tool to accelerate the discovery process for resulting knowledge of material properties. This is also a tendency to reduce the classical trial-and-error approach [17]. In this case, the development of molecular models, such as quantitative structure and property relationship (QSPR) and conformational search methodologies has also contributed greatly to the discovery and development of new molecules [18][19][20]. In this way, the multivariate analysis methods have been becoming a convenient and easy tool for supporting empirical and theoretical models. The multivariable linear relationships can be used to assess different characteristics of the systems.
In this work, we report the construction of the quantitative structure and property relationships using the structural descriptors and stability constant of complexes between the metal ions and thiosemicarbazone. The QSPROLR and QSPRANN models were successfully built based on the regression technique and neural network. The stability constant log11 of the complexes between the metal ions and thiosemicarbazone in the test set resulting from the QSPR models was validated and compared with those from experimental data in the literature.
The overall or stability constant, given the symbol β, is the constant for the formation of the complex from the reagents. The stability constant for the formation of MpLq is given by The stability constant β refers to the formation of the complex ML in one step with p = 1 and q = 1
The complexes of metal ions and ligand thiosemicarbazone were re-built and optimized by means of quantum mechanics on the MoPac 2016 system [30]. The quantum descriptors were calculated by using the semi-empirical quantum method with new version PM7 and PM7/sparkle for lanthanides [31]. The 2D and 3D topological descriptors were calculated by using the QSARIS system [10,32]. The construction of QSPROLR models was performed using the back-elimination and forward regression technique on the Regress system [33] and MS-Excel [10,14,34]. The artificial neural network model QSPRANN was constructed using the multilayer training technique on the Visual Gene Developer system [35]. The predictability of the QSPRs models was cross-validated by means of the leave-one-out method (LOO) using the statistic Q 2 LOO.

Ordinary least square regression
The ordinary least square regression (OLR) was used to model and predict the values of one or more dependent quantitative or qualitative variables by means of a linear combination of one or more explanatory quantitative and/or qualitative variables. This technique did not face the constraints of ordinary least square regression (OLR) on the number of variables versus the number of observations.
The ordinary least square regression or ordinary linear regression is more commonly named linear regression [33,34]. In this case, the regression model with k explanatory variables where Y is the dependent variable, β0 is the intercept of the model, βj is the coefficient of the j th explanatory variable, Xj corresponds to the j th explanatory variable (with j = 1 to k), and  is the random error with mean 0 and variance  2 .
In the case of k observations, the estimation of the predicted value of the dependent variable Y is given by expression (5) The OLR method corresponds to minimizing the sum of squared differences between the observed and predicted values. This minimization leads to the following estimators of the parameters of the model. The models were screened by using the values R 2 train and Q 2 LOO [10,[33][34][35][36][37][38][39][40]. These were assessed by the same formula (6) where Yi, Ŷi, and Ȳ are the experimental, predicted and average value of the response, respectively; n is the total number of observations.
Adjusted R² (R²adj) is the adjusted determination coefficient for the model. The value of R²adj can be negative if the R² is close to zero. This coefficient is only calculated if the constant of the model has not been fixed by the user. R²adj is defined by R²adj is a correction to R², which takes into account the number of variables used in the model. The error mean square (MSE) is defined by

Artificial neural network
A neural network as a function of a set of derived inputs is called hidden nodes. The hidden nodes are nonlinear functions of the original inputs. The neural network can specify many layers of hidden nodes [41,42].
The functions applied at the nodes of the hidden layers are called activation functions.
The activation function is a transformation of a linear combination of the X variables. The function applied at the response is a linear combination of continuous responses, or a logistic transformation for nominal or ordinal responses [43,44]. There are three transfer functions, namely sigmoid, hyperbolic tangent, and Gaussian transfer function.
The main advantage of the neural network is that it can efficiently model different response surfaces. Neural networks are very snappy models and tend to overfit data. When that happens, the forecast of the model is very good but predicts future observations poorly. The weakness of the neural network model is that the results are not easily explainable, since there are intermediate layers rather than a direct path from the X variables to the Y variables, as in the case of regular regression [45,46]. To alleviate overtraining, the neural network is validated by use of an independent data set to evaluate the predictive ability of the model [41].
Validation is a process of using a part of the data set to estimate the model parameters and using the other part to assess the predictability of the neural network. The first part is the training set used to estimate the model parameters. The second part is the validation set used to validate the predictability of the model. The test set is the final, independent assessment of the model predictability [42].
In this work, we used a typical feed-forward neural network, which was trained by using an error back-propagation learning algorithm. This neural network style propagates information in the feed-forward direction using equation (9) where ai is the input factor, bj is the output factor, wij is the weight factor between two nodes, Tj is the internal threshold, and  is the transfer function.
There exist many transfer functions that are used in neural networks such as hyperbolic tangent, Gaussian, sigmoid… In this study, we used the hyperbolic tangent function. The hyperbolic tangent learning algorithm is based on a generalized delta rule accelerated by a momentum term. To increase the efficiency of the neural network, both the weight factors and the internal threshold values were adjusted using equations (10) and (11) [41,42] , = , + · ∑ δ , · , + α · ∆ , = + · ∑ δ , + α · ∆ (11) where  is the learning rate;  is the momentum coefficient; W is the previous weight factor change; T is the previous threshold value change; O is the output -the gradient-descent correction term; and k stands for the pattern. The performance of the trained network was verified by determining the error between the predicted value and the real value. All the data of the patterns were normalized to be less than 1 before training the neural network; the initial weight factors were randomly generated from -0.2 to 0.2, and the initial internal threshold values were set to zero.

Constructing models QSPROLR
The QSPROLR model was constructed from the database of complexes between metal ions and the ligands including the 2D and 3D molecular descriptors, and the quantum parameters.
The general complex structure is shown in Fig. 1a and 1b, and the stability constant logβ11 is given in Table 1.  Table 2.

Constructing models QSPRANN
In addition to model QSPROLR, the QSPRANN model was also developed with the neural network technique on the Visual Gene Developer system [35] upon the molecular descriptors of model QSPROLR. The architecture of the neural network comprising three layers is I(6)-HL(6)-O(1) (Fig. 3a); the input layer I(6) includes 6 neurons (k0, core-core repulsion, xp5, xch5, valence, and SHHBd); the output layer O(1) includes 1 neuron, that is, logβ11; the hidden layer includes 6 neurons. The error back-propagation algorithm was used to train the network. The hyperbolic tangent transfer function was set on each node of the layers; the training network parameters included the learning rate of 0.01, the momentum coefficient of 0.1, and the sum of error of 0.000016 with 1,000,000 loops. The results of the training process are given in Table 4.

Predictability of QSPR models
The predictability of the QSPROLR and QSPRANN model was carefully evaluated by means of the phasing-each-case technique. The predicted results received for 9 randomly chosen substances with the experimental values [27-29, 45, 46] are presented in Table 5.
The average absolute values of the relative error MARE used to assess the overall error of the QSPR models were calculated according to formula (14) , where , % = |log β 11, − log β 11, | log β 11, , n is the number of test substances; and β11,exp and β11,cal are the experimental and calculated stability constants.

Conclusion
This work successfully built the quantitative structure and property relationships incorporating ordinary linear regression and artificial neural network. The QSPR models were constructed by using the data set of structural descriptors resulting from the semi-empirical quantum calculation and molecular mechanics. The models were cross-validated carefully using the leave-one-out method upon statistical values R 2 train, Q 2 LOO, MARE, %, and the single factor ANOVA method. The QSPRANN model I(6)-HL(6)-O(1) turned out to be satisfactory for actual applicability. The results from this work could serve for designing new thiosemicarbazone derivatives that are helpful in the fields of analytical chemistry, pharmacy, and environment.