Prediction of Stability Constants of Zinc(II) Complexes with 2-aminobenzamide and Amino Acids

We developed a model for the stability (log βZnLB) of Zn 2+ mixed complexes (N = 16) with 2-aminobenzamide (L) and four amino acids (B) glycine, alanine, valine, and phenylalanine at 300, 310, 320, and 330 K. The model based on the quadratic regression function of the molecular valence connectivity index of the third order, χ, yielded S.E. = 0.02. We also developed an overall model for K1, K2 and βZnLB of the same system at all of the four temperatures (N = 48). This model yielded S.E. = 0.05.


Introduction
Although models based on graph theoretical indices rely on somewhat vague concepts, as do other models based on topological approaches in theoretical chemistry, 1,2 they have proved quite successful in many applications. Graph theoretical indices correlate well with many physicochemical parameters [3][4][5][6] and biological activities (QSAR). 7,8 Our efforts to use topological indices, especially the valence connectivity index of the 3 rd order, 3 χ v , in order to build regression models for the prediction of stability constants of coordination compounds 9 ended with varying success, depending on the nature of coordination compounds as well as on the quality of experimental data. Stability constants of copper(II) complexes with α-amino acids were reproduced with such a precision that it was even possible to evaluate the results of two electroanalytical methods (GEP and SWV) used for their measurement. 10 For copper(II) complexes with derivatives of thioflavin T and clioqiunol, used as model compounds for the study of Alzheimer's disease, we obtained even better results 11 than those achieved by the more demanding DFT method. 12 In some cases, our method, despite being strictly empirical, enabled the analysis of coordination, i.e. it gave insight into the structure of the complex. 13,14 However, in other cases additional variables 15 and unusual forms of regression functions 16 had to be introduced in or-der to obtain an acceptable agreement between theory and experiment.
The topic of this contribution are mixed zinc(II) complexes of 2-aminobenzamide (2-AB) with amino acids. These complexes have already been studied by pH-metric and spectrophotometric methods, 17 because 2-aminobenzamide and its derivatives have desirable pharmacological properties, 18 and are also used as analytical reagents. 19 On the other hand, zinc(II) participates in many biological processes, 20 so the study of the above mentioned complexes should lead to a better understanding of the pharmacokinetics of 2-aminobenzamide and its derivatives.

1. Calculation of Topological Indices
We calculated topological indices using the E-DRAG-ON program system, developed by R. Todeschini and coworkers, capable of yielding 119 topological indices in a single run, along with many other molecular descriptors. 21,22 Connectivity matrices were constructed with the aid of the Online SMILES Translator and Structure File Generator. 23 The valence molecular connectivity index of the 3 rd order, 3 χ v , was defined as: 24-27 3 χ where δ(i), δ(j), δ(k), and δ(l) are weights (valence values) of vertices (atoms) i, j, k, and l making up the path of length 3 (three consecutive chemical bonds) in a vertexweighted molecular graph. The valence value, δ(i), of a vertex i is defined by: where Z v (i) is the number of valence electrons belonging to the atom corresponding to vertex i, Z(i) is its atomic number, and H(i) is the number of hydrogen atoms attached to it. The 3 χ v index for all monoand mixed complexes was calculated from the graph representations of aquacomplexes (Fig. 1), assuming that the metal in the monocomplexes is tetracoordinated and in mixed complexes hexacoordinated.

Regression Calculations
Regression calculations, including the leave-one-out procedure (LOO) of cross validation were done using the CROMRsel program. 28 The standard error of the cross-validation estimate was defined as: (6) where ΔX and N denotes cv residuals and the number of reference points, respectively.

Results
In order to model the logarithm of stability constant, log β ZnLB , (Eq. 1, Table 1) of mixed Zn 2+ complexes with 2-aminobenzamide and amino acids (ZnLB), we used the quadratic function of 3 χ v (Figure 2): because it proved better than the linear function (e.g. at 300 K, S.E. = 0.04 and 0.06 for quadratic and linear function, respectively; N = 4).
ding stability constants of a referent complexes (either with glycine, alanine, valine, or phenylalanine); 3 χ v denotes 3 χ v (ZnB) in the case of K 1 constant and for K 2 and β ZnLB it corresponds to 3 χ v (ZnLB) normalized on 3 χ v (Zn-Gly):  As the dependence of log β ZnLB on temperature in the range 300-330 K is linear, 17 we proposed the model for all four of the temperatures (N = 16) simply by including temperature into Eq. (7): where T is the absolute temperature at which constants, log β ZnLB , were measured. The model yielded S.E. cv = 0.03 (Model 1, Table 2). The values of log K 1 and log K 2 (Eqs. 2 and 3) show the same quadratic dependence on 3 χ v (Figure 3). This enabled the development of the model (N = 12) for all of the constants at a given temperature: where K denotes K 1 , K 2 and β ZnLB ; K 0 are the correspon- Taking glycine complexes as referent, the model gave S.E. cv = 0.09 (Model 2, Table 2).
The same function, Eq. (9), can be applied to K 1 , K 2 and β ZnLB at all four of the temperatures (N = 48). In this case, K 0 denotes the corresponding stability constants of a referent complexes at all of the temperatures. Taking glycine complexes as referent, the model gave S.E. cv = 0.06 (Model 3, Table 2). Furthermore, we have found that it is possible to make the model more predictive. To be more precise, it is enough to know the K 0 , i.e. K 1 , K 2 and β ZnLB constants of a referent complex at only two temperatures. The K 0 values at the other two temperatures can be then easily calculated from the linear dependence of K 1 , K 2 and β ZnLB on T. This way, fewer experimental constants are needed for calibration. Such a model, calibrated on K 1 , K 2 and β of glycine complexes at 300 and 310 K, yielded S.E. cv = 0.06 (Figure 4).

Conclusion
We developed two kinds of models. A model for the prediction of stability (log β ZnLB ) of Zn 2+ mixed complexes with 2-aminobenzamide (L) and four amino acids (B) at four temperatures (N = 16), and a model for the prediction of K 1 , K 2 and β ZnLB of the same system (N = 48). In both cases, the theoretical results showed excellent agreement with the experimental ones, yielding errors commensurable with the errors of measurements. Specifically, the maximal differences between the experimental and theoretical (cross-validated) values were 0.06 and 0.15 for the two models (Models 1 and 3, Table 2), respectively, while the declared standard error of measurements were 0.03-0.08 for log K 1 and 0.03-0.09 for log β ZnLB .
It should also be pointed out that the log K 1 function has the same form as the function for copper(II) monocomplexes with amino acids. 29