A Quantum-chemical Study of the Relationships Between Electronic Structure and Anti-proliferative Activity of Quinoxaline Derivatives on the HeLa Cell Line

A study of the relationships between electronic structure and anti-proliferative activity of quinoxaline derivatives on the HeLa cell line was carried out. For this QSAR study the technique employed is the Klopman-Peradejordi-Gómez (KPG) method. We obtain a statistically significant equation (R= 0.97 R= 0.94 adj-R= 0.91 F (8, 15)=29.50 p<0.000001 and SD=0.06). The results showed that the variation of the activity depends on the variation of the values of eight local atomic reactivity indices. The process seems to be charge and orbital-controlled. Based on the analysis of the result, a partial twodimensional pharmacophore was built. The results should be useful to propose new molecules which higher activity.


Introduction
Hela is the first immortalized cell line [1]. This cell line originates from a cervical cancer tumor of a patient named Henrietta Lacks, who later died of her cancer in 1951 [1]. One of the earliest uses of HeLa cells was to develop the vaccine against the polio virus [2]. The genomic and transcriptomic resource for a HeLa cell line based on deep DNA and RNA sequencing was created in 2013 [3]. Several studies are performed to found molecules that inhibit the proliferation of this cell line . Theoretical studies were also done and are useful to explain the mechanism, the affinity and the activities of different compounds [26][27][28][29]. This work presents the results of the use of the KPG method [30] to obtain quantitative relationships between the electronic structure of quinoxaline derivatives and their antiproliferative activities on the HeLa cell line.

Methods and Models
For this study we use the Klopman-Peradejordi-Gomez (KPG) method. In 1967, Klopman and Hudson presented a general perturbation model for chemical reactivity including ionic interactions and not restricted only to π electron [31][32][33]. In their model, the electronic energy change, ∆E, associated with the interaction of atom i of molecule A with atom j of molecule B is given by: where Q i is the net charge of atom I, F mi is the Fukui index of OM m of atom i, β ij is the resonance integral (assumed to be independent of the kind of atomic orbitals (OA) because the A-B complex does not involve covalent bonds), E m (E m ') Electronic Structure and Anti-proliferative Activity of Quinoxaline Derivatives on the HeLa Cell Line is the energy of the m-th occupied MO (m' for the empty MOs) of molecule A. n and n' refer to molecule B. The summation on p is over all interacting atom pairs. The first term of the right side of Equation 1 represents the electrostatic interaction between atom with net charges Q i and Q j . The next two terms introduce the interactions between occupied MOs of one molecule with the empty MOs of the other molecule and vice versa. As this model represents the interaction energy in terms of atom-atom interactions, it was only a matter of time that someone applied it for pharmacological/biological problems. Then, in 1971, Peradejordi et al published an article where they presented the results of a quantum-chemical study of the structure-activity relationships of tetracycline antibiotics [34]. The authors proposed that the inhibitory rate constants, I i K , can be expressed as: log tan log where c i K is the ribosome-tetracycline equilibrium constant. Now, let us consider the state of thermodynamic equilibrium and a 1:1 stoichiometry in the formation of the drug-receptor complex: where i D is the drug, R the receptor and i D R the drugreceptor complex. According to statistical thermodynamics the equilibrium constant i K is written as: Peradejordi et al consider that the partition function terms and the solvation energy are constant. After overs considerations and approximations (for details see [34]), the linear equations is obtained: where a, b and c are constants, parameters of the substituents. Throughout this paper HOMO j * refers to the highest occupied molecular orbital localized on atom j and LUMO j * to the lowest empty MO localized on atom j.
where the summation over t is over the different substituents of the molecule, , i t m is the mass of the i-th atom belonging to the t-th substituent, , i t R being its distance to the atom to which the substituent is attached. This approximation allows him to transform a molecular property into a sum of substituent properties. He proposed that these terms represent the fraction of molecules attaining the proper orientation to interact with a given site. He called them Orientational Parameters (OP). The new local atomic reactivity indices (LARIs) of Eq. 7 are defined as follows: Local atomic electronic chemical potential: Local atomic hardness: Local electrophilic superdelocalizability of the HOMO* of atom i and local nucleophilic superdelocalizability of the LUMO* of atom i: Local atomic softness of atom i: Local atomic electrophilicity of atom i: The maximal amount of charge atom i may receive: The physical meaning of these indices is summarized in Table 1.  The Klopman-Peradejordi-Gómez (KPG) method is also discussed in many previous papers [30,35,36,38,39,42,[44][45][46]. From a conceptual perspective, the work presented here is a test of the hypothesis stating that the KPG model can provide a quantitative and formal relationship between the molecular structure and any biological activity. Nowadays, the KPG model produced excellent results in all its applications [35,44,[46][47][48][49][50][51][52][53].

Selection of Molecules
For this study, a series of quinoxline derivatives were selected [23]. These molecules have an anti-proliferative activity on the HeLa cell line. The experimental data was taken from a recent study [23]. The structures of the Electronic Structure and Anti-proliferative Activity of Quinoxaline Derivatives on the HeLa Cell Line compounds are shown in Figure 1 and Table 2 which also summarizes the values of their median inhibitory concentrations expressed as log(IC 50 ).

Calculations
The electronic structure of each fully optimized molecule was obtained using the Density Functional Theory (DFT) at the B3LYP/6-31G (d, p) level with the Gaussian software [54]. The local atomic reactivity indices were calculated from the single point results of Gaussian03 using the D-Cent-QSAR software [55] with a correction for Mulliken populations [56]. All populations of electrons less than or equal to 0.01e are considered null [56]. The orientational parameters of the substituents are calculated in the usual manner [57,58]. We have used the concept of common skeleton defined as a set of atoms common to all the molecules analyzed. We hypothesize that the variation of the numerical values of the local atomic reactivity indices (LARIs) of the atoms of this common skeleton accounts for almost all the variation of the biological activity. As the number of LARIs involved is greater that the number of molecules, the solving of the linear systems of equations is not possible. For this reason we employed the technique of multiple linear regression analysis (LMRA) to determine the atoms that are directly involved in the variation of the biological activity. The data matrix contains log (IC 50 ) as a dependent variable, and the local indices of atomic reactivity of all the atoms of the common skeleton as independent variables. The Statistica 10 software was used to perform LMRA studies [59]. The numbering of the common skeleton atoms is shown in Figure. 2.

Results
The best statistically significant equation obtained is the following:  Table 3 shows the beta coefficients and the t-test results for the significance of coefficients of equation 1. Concerning independent variables, Table 4 shows that the highest internal correlation is r 2 [F 20 (HOMO)*, F 21 (HOMO)*]=0.43. Figure 3 shows the plot of observed values vs. calculated values of log(IC 50 ). The associated statistical parameters of Eq.16 show that this equation is statistically significant and that the variation of the numerical values of eight LARIs explains about 91% of the variation of the variation of the biological activity.    Tables 5 and 6

Discussion
The HeLa inhibition mechanism is unknown. We have stated that "it is important to stress that our hypothesis covers multi-step (for example, in the n-th step molecules must cross a pore) and multimechanistic (for example, to cross the pore molecules must interact consecutively with j unknown sites) processes. Therefore it seems logical to state that a necessary condition to obtain good structure-activity relationships is that all the steps and all the mechanisms inside each step must be the same for all the group of molecules under study" [44]. If the molecules studied here employ multi-step and/or multimechanistic action mechanisms that are not exactly the same for all, we may expect that the linear multiple regression results contain sometimes variables whose interpretation seems contradictory.
The beta values shows that the importance of variables is Atom 21 is a carbon atom in the lateral chain of ring C (Figure 2). Table 6 shows that all local MO have σ nature. A low value of 21 E S indicates that atom 21 should interact with a sigma electron rich center through its empty sigma local MOs. Note that the local HOMO* and the local LUMO* do not coincide with the molecule's frontier MOs. The interactions can be of the σ-π or σ-σ kind. The σ-σ interaction may occur with the sigma MOs of the -CH 2 -groups of some amino acids. This coincides with the requirement of a low value for 21  Low numerical values are obtained by shifting upwards the energy of the empty MOs, making this atom a bad electron acceptor. Therefore, we suggest that atom 16 is interacting with an electron deficient center. On the other hand, Eq. 16 shows that a high inhibitory activity is related with a positive value for Q 16 , fact that seems to be contradictory with the interaction with an electron deficient center. Examining Table   II we may see that 16 N S is more significant than Q 16 . Therefore, and as a first approximation, we shall not consider Q 16 . Atom 20 is a carbon atom of the side chain of ring C ( Figure 2). All local MOs have σ nature (Table 6). A low value for 20 ( ) * F HOMO suggests that atom 20 is probably interacting with a center rich in sigma electrons. Note that this condition is the same that the one for atom 21. Atom 22 is the carbon atom of the carboxylate moiety of the side chain of ring C (Figure 2). (LUMO) 22 * is a π MO in all molecules (  ( ) * LUMO eigenvalue and making the MO less reactive. So, we suggest that atom 22 is interacting with a π electron deficient center. Atom 15 is a carbon atom in ring C (Figure 2). 15 ( ) * LUMO is a MO of π nature (Table   5). A high value for 15 ( ) * F LUMO suggests that this lowest unoccupied local MO is interacting with an electron rich center. Atom 23 is an oxygen atom of the carboxylate moiety in the side chain of ring C (Figure 2). A high value for 23 ( ) * F HOMO suggests that the highest occupied local is interacting with an electron deficient center. All the above suggestions are shown in the partial 2D pharmacophore of Figure 4.

Conclusion
We obtained a statistically significant relationship between the variation of the anti-proliferative activity of some quinoxaline derivatives and the variation of the numerical values of a set of local atomic reactivity indices. This allowed us to build the associated pharmacophore that should serve as a starting point for chemical modifications producing more active compounds. According to the obtained pharmacophore, it is not necessary to modify the indices of the atoms of the quinoxaline cycle. But the indices which would be modified to improve the anti-proliferative activity are those from the side chain.