New Methods of Esterification of Nanodiamonds in Fighting Breast Cancer—A Density Functional Theory Approach

The use of nanodiamonds as anticancer drug delivery vehicles has received much attention in recent years. In this theoretical paper, we propose using different esterification methods for nanodiamonds. The monomers proposed are 2-hydroxypropanal, polyethylene glycol, and polyglicolic acid. Specifically, the hydrogen bonds, infrared (IR) spectra, molecular polar surface area, and reactivity parameters are analyzed. The monomers proposed for use in esterification follow Lipinski’s rule of five, meaning permeability is good, they have good permeation, and their bioactivity is high. The results show that the complex formed between tamoxifen and nanodiamond esterified with polyglicolic acid presents the greatest number of hydrogen bonds and a good amount of molecular polar surface area. Calculations concerning the esterified nanodiamond and reactivity parameters were performed using Density Functional Theory with the M06 functional and the basis set 6–31G (d); for the esterified nanodiamond–Tamoxifen complexes, the semi-empirical method PM6 was used. The solvent effect has been taken into account by using implicit modelling and the conductor-like polarizable continuum model.


Introduction
Breast cancer is a leading cause of death worldwide. This disease is mainly characterized by high levels of estrogen receptor (ER) and progesterone receptor (PR) expressions [1]. One of the treatments prescribed for patients with breast cancer is tamoxifen (TAM) [2], which belongs to the family of selective estrogen receptor modulators (SERMs) [3]. Nonetheless, prolonged use of TAM causes adverse effects such as endometrial cancer, thromboembolisms, and menopause symptoms [4,5].
Fortunately, nanomedicine has demonstrated exceptional promise for improving the delivery of therapeutic agents for the treatment of breast cancer through nanomaterials and, in particular, with nanodiamonds (ND). These nanomolecules are being studied due to their unique characteristics that make them suitable as a platform for diagnosis and therapeutics [6]. In addition, ND are stable and dispersible in water, making them a promising and clinically important modality for improving the efficacy of the treatment of diseases, and even some cancers, at the molecular level [7]. This carbon-based material is medically relevant because of its biological applications and its physical and chemical properties including stability, scalability, small size, good adsorption, and biocompatibility [6,8,9]. ND are known for easily forming complexes with biopolymers [10]

Esterified Nanodiamond Complexes (NDE-TAM)
The optimization of the esterified nanodiamond complex (NDE-TAM) was performed using the semi-empirical method PM6 [38] (to achieve convergence of the multi-atom complex), taking into account the good past results on the structural parameters of the method [39][40][41]. Water was used as the solvent with the conductor-like polarizable continuum model.

Reactivity Parameters
The electronic properties of NDE and the NDE-TAM complex were calculated by means of M06/6-31G(d). The solvation model was a conductor-like polarizable continuum model, with water as the solvent. The DFT approach has been successfully used for chemical descriptors such as electron affinity (EA), ionization potential (IP), chemical hardness (η) [42], electronegativity (χ) [43,44], chemical potential (µ) [45], and electrophilicity (ω) [46]. These reactivity descriptors were obtained by single point calculations, considering the energy (E) as a function of the number of electrons, N: E(N − 1), E(N), and E(N + 1). Approximations were also done of IP and AE to HOMO and LUMO energies obtained from the Kohn-Sham scheme.

Molecular Polar Surface Area
The molecular polar surface area (PSA) was obtained using a commercial package based on Bayesian statistics (Molinspiration miscreen toolkit) [47]. To obtain the PSA, the NDE and the NDE-TAM complex were encoded with SMILE (Simplified Molecular Input Line System), which is a chemical notation system designed for modem chemical information processing [48].

Construction of the Proposed Systems
Nanodiamond is often described as a crystalline diamond core with a perfect diamond lattice surrounded by an amorphous shell with a combination of sp2 and sp3 bonds [49]. The constructed ND for this work is octahedral with 351 atoms with 111 facets and a lattice constant of 3.56 Å. The size of the nanodiamond used is 1.5 nm due to the stability of the NDs in the range between 1.5 and 2.0 nm [50,51].
The functionalization of the ND begins with the creation of carboxyl functional groups on its surface, simulating acid treatment [52,53]. Three different molecules were used to create the esterification: 2-hydroxypropanal (2HP), polyethylene glycol (PEG), and polyglicolic acid (PGA) monomers. The construction of the esterified structures was carried out following the reaction protocol established by Landeros-Martinez et al. [25]. This process involves bonding the carboxyl group on the surface of the nanodiamond in a covalent bond with the different monomers proposed in this work. This generates an available extended linear chain on the surface of the nanodiamond.
Also, the analysis of nucleophilicity index of the alcohols and the electrophilicity index of the ND was taken into account in the reactivity section. The simplest approach was used: considering that high (low) nucleophilicities are associated with low (high) ionization potentials, nucleophilicity was related with a negative IP value. The nucleophilicity index was calculated in the manner introduced by Domingo et al. [54]. Tetracyanoethylene (TCE) is taken as a reference for this scale of nucleophilicity because it presents the lowest HOMO energy in a large series of molecules already considered.
The alcohols showed low nucleophilicity index values and high values of electrophilicity, characterizing them as marginal nucleophiles and strong electrophiles according to Domingo et al. [55]. However, the ND has a strong nucleophile character with N > 3.0 eV. Results are in Table 1. After the construction of the systems, they were optimized in aqueous phase and named ND2HP, NDPEG, and NDPGA. Once the optimization was reached, frequency calculations were performed to ensure that the structures were at their lowest energy level. The structures are shown in Figure 1 where the view is from the top of the ND. We observed that the structure of the ND did not change before or after optimization. However, the linear chain of esterification had significant changes in its dihedral angles. These changes will outline the position of the TAM drug for its loading in the nanoparticle. The alcohols showed low nucleophilicity index values and high values of electrophilicity, characterizing them as marginal nucleophiles and strong electrophiles according to Domingo et al. [55]. However, the ND has a strong nucleophile character with N > 3.0 eV. Results are in Table 1. After the construction of the systems, they were optimized in aqueous phase and named ND2HP, NDPEG, and NDPGA. Once the optimization was reached, frequency calculations were performed to ensure that the structures were at their lowest energy level. The structures are shown in Figure 1 where the view is from the top of the ND. We observed that the structure of the ND did not change before or after optimization. However, the linear chain of esterification had significant changes in its dihedral angles. These changes will outline the position of the TAM drug for its loading in the nanoparticle. After optimization in 2HP the D1 (O352-C351-O353-C354) changed from 3.

Reactivity Parameters of the NDE
The following reactivity parameters were obtained: electron affinity (EA), ionization potential (IP), electronegativity (χ), chemical potential (μ), chemical hardness (η), and electrophilicity (ω). The reactivity parameters mentioned above were obtained using two different approximations. As stated in the computational details section, the first considered the energy (E) as a function of the number of electrons, and the second one used the approximation of IP and EA to HOMO and LUMO energies. Results are in Table 2, where the second approximation values are in cursive font. The values in Table 2 suggest greater reactivity for ND2HP. It has the lowest hardness in both approximations, and has the highest value for electrophilicity index even when the three of the esterified systems are classified as moderate electrophiles, in accordance with Domingo et al. [55] with ω < 1.5 eV. In this work, the main area of interest is that the reaction with TAM and ND2HP showed the highest tendency to attract electrons and presented better stability with the electrons that were in the surrounding area.
Another important feature that quantifies permeation is polar surface area (PSA). This property is defined as the surface sum of all the polar atoms, including the oxygen, nitrogen, and hydrogen attached to these atoms [56]. PSA is a useful parameter for the prediction of drug transport properties. Molecules with value greater than 140 Å 2 are poorly absorbed at cell membranes, while those with a value less than 90 Å 2 are completely absorbed [57]. The results were 63.60 Å 2 for ND2HP and NDPGA, and 55.70 Å 2 for NDPEG, meaning that all the proposed systems have good cell permeability.

Construction of the NDE-TAM Complex
To build the NDE-TAM complex, it was necessary to find the equilibrium distance between the NDE and the drug. With the optimized structure obtained in the previous section, an equilibrium distance analysis was performed. This exploration involved searching for the distance requiring the least energy to bind a pair of molecules-in this case, NDE and TAM. The analysis consisted of singlepoint energy calculations across a rectangular grid involving a scan of previously selected internal

Reactivity Parameters of the NDE
The following reactivity parameters were obtained: electron affinity (EA), ionization potential (IP), electronegativity (χ), chemical potential (µ), chemical hardness (η), and electrophilicity (ω). The reactivity parameters mentioned above were obtained using two different approximations. As stated in the computational details section, the first considered the energy (E) as a function of the number of electrons, and the second one used the approximation of IP and EA to HOMO and LUMO energies. Results are in Table 2, where the second approximation values are in cursive font. The values in Table 2 suggest greater reactivity for ND2HP. It has the lowest hardness in both approximations, and has the highest value for electrophilicity index even when the three of the esterified systems are classified as moderate electrophiles, in accordance with Domingo et al. [55] with ω < 1.5 eV. In this work, the main area of interest is that the reaction with TAM and ND2HP showed the highest tendency to attract electrons and presented better stability with the electrons that were in the surrounding area.
Another important feature that quantifies permeation is polar surface area (PSA). This property is defined as the surface sum of all the polar atoms, including the oxygen, nitrogen, and hydrogen attached to these atoms [56]. PSA is a useful parameter for the prediction of drug transport properties. Molecules with value greater than 140 Å 2 are poorly absorbed at cell membranes, while those with a value less than 90 Å 2 are completely absorbed [57]. The results were 63.60 Å 2 for ND2HP and NDPGA, and 55.70 Å 2 for NDPEG, meaning that all the proposed systems have good cell permeability.

Construction of the NDE-TAM Complex
To build the NDE-TAM complex, it was necessary to find the equilibrium distance between the NDE and the drug. With the optimized structure obtained in the previous section, an equilibrium distance analysis was performed. This exploration involved searching for the distance requiring the least energy to bind a pair of molecules-in this case, NDE and TAM. The analysis consisted of single-point energy calculations across a rectangular grid involving a scan of previously selected internal coordinates. When a scan was requested, the variables in the molecular structure, and the range of values which they could take, had to be specified. These variables were defined according to the structure systems and the intended bond type. The scan was performed in 15 steps with approximations of 1 Å. The atoms involved in the approximation are a hydrogen of the NDE, an oxygen in TAM in ND2HP and NDPEG, and two oxygens in NDPGA. The rest of the atoms remained fixed. The energy curve results were 4 Å for ND2HP-TAM and NDPEG-TAM, and 6 Å for NDPGA-TAM. The three different curves obtained are shown in Figure 3. coordinates. When a scan was requested, the variables in the molecular structure, and the range of values which they could take, had to be specified. These variables were defined according to the structure systems and the intended bond type. The scan was performed in 15 steps with approximations of 1 Å. The atoms involved in the approximation are a hydrogen of the NDE, an oxygen in TAM in ND2HP and NDPEG, and two oxygens in NDPGA. The rest of the atoms remained fixed. The energy curve results were 4 Å for ND2HP-TAM and NDPEG-TAM, and 6 Å for NDPGA-TAM. The three different curves obtained are shown in Figure 3. Considering that the scan helps to determine the approximate location of the minimum energy structure, but does not include optimized geometry [58], all the NDE-TAM complexes were optimized. In this case, the geometry optimization and the frequency calculation were developed using the PM6 semi-empirical method, using water as solvent due to the size of the system. The results on the binding bonds are 7.79 Å for ND2HP-TAM, 4.21 Å for NDPEG-TAM, and 2.18 Å for NDPGA-TAM.

Hydrogen Bonds
The hydrogen bond is also one of the most important intermolecular interactions because it determines molecular conformation, molecular aggregation, and the function of a vast number of chemical systems ranging from inorganic to biological [59]. The distance and angles observed in the HBs are classified as weak, according to Jeffrey classification [60]. The results in Table 3 show the type of HBs and the interaction type, where the groups C-H and O-H act as proton donors. These kinds of interactions are possible, even when the groups have slight polarity, as stated by Steiner [59]. N and O act as acceptors. Considering that the scan helps to determine the approximate location of the minimum energy structure, but does not include optimized geometry [58], all the NDE-TAM complexes were optimized. In this case, the geometry optimization and the frequency calculation were developed using the PM6 semi-empirical method, using water as solvent due to the size of the system. The results on the binding bonds are 7.79 Å for ND2HP-TAM, 4.21 Å for NDPEG-TAM, and 2.18 Å for NDPGA-TAM.

Hydrogen Bonds
The hydrogen bond is also one of the most important intermolecular interactions because it determines molecular conformation, molecular aggregation, and the function of a vast number of chemical systems ranging from inorganic to biological [59]. The distance and angles observed in the HBs are classified as weak, according to Jeffrey classification [60]. The results in Table 3 show the type of HBs and the interaction type, where the groups C-H and O-H act as proton donors. These kinds of interactions are possible, even when the groups have slight polarity, as stated by Steiner [59]. N and O act as acceptors. Table 3. Hydrogen bonding in NDE-TAM complexes.

Complex
Interaction Type

Bond Angles (A-H----B) (°) Type
ND2HP  The presence of the hydrogen bonds found was corroborated qualitatively with an analysis of the IR spectra of the free A-H group with no HB group. According to Desiraju, in a comparison of the IR spectrum of a free A-H group and another one with a HB, in the HB, the A-H stretching frequency is reduced. This indicates a hydrogen bonding [61]. For our study, we compare the TAM's interaction with the A-H----B hydrogen bond group in the NDE-TAM complexes. The bonding of the atoms to H decreases the vibration frequency. Figure 5 shows the IR spectrum of TAM at the M06/6-31G(d) level [25] versus the IR spectra of the ND2HP-TAM, NDPEG-TAM, and NDPGA-TAM complexes. The latter three use the PM6 semi-empirical method. The comparison shows the energy reduction on the frequencies as follows:  The presence of the hydrogen bonds found was corroborated qualitatively with an analysis of the IR spectra of the free A-H group with no HB group. According to Desiraju, in a comparison of the IR spectrum of a free A-H group and another one with a HB, in the HB, the A-H stretching frequency is reduced. This indicates a hydrogen bonding [61]. For our study, we compare the TAM's interaction with the A-H--B hydrogen bond group in the NDE-TAM complexes. The bonding of the atoms to H decreases the vibration frequency. Figure 5 shows the IR spectrum of TAM at the M06/6-31G(d) level [25] versus the IR spectra of the ND2HP-TAM, NDPEG-TAM, and NDPGA-TAM complexes. The latter three use the PM6 semi-empirical method. The comparison shows the energy reduction on the frequencies as follows:  According to Lipinski´s rule of five, which states that poor permeation is more likely when there are more than five HB, the number of these bonds plays a crucial role in determining if a drug will be more active [62]. The three analyzed NDE-TAM complexes fulfill this rule.
The PSA was calculated at 12.47 Å 2 for the three complexes. This is due to the number of polar oxygen and nitrogen atoms and number of hydrogen atoms attached to the oxygen. The three systems have the same number of these atoms, creating good cell permeability.

Chemical Reactivity Descriptors of NDE-TAM
A quantitative analysis of the reactivity of the NDE-TAM complexes was performed by means of the global reactivity descriptors. The values of the global reactivity descriptors, calculated for each of the NDE-TAM complexes using the vertical electron affinity and ionization potential, are shown in Table 4.  According to Lipinski´s rule of five, which states that poor permeation is more likely when there are more than five HB, the number of these bonds plays a crucial role in determining if a drug will be more active [62]. The three analyzed NDE-TAM complexes fulfill this rule.
The PSA was calculated at 12.47 Å 2 for the three complexes. This is due to the number of polar oxygen and nitrogen atoms and number of hydrogen atoms attached to the oxygen. The three systems have the same number of these atoms, creating good cell permeability.

Chemical Reactivity Descriptors of NDE-TAM
A quantitative analysis of the reactivity of the NDE-TAM complexes was performed by means of the global reactivity descriptors. The values of the global reactivity descriptors, calculated for each of the NDE-TAM complexes using the vertical electron affinity and ionization potential, are shown in Table 4. The results of chemical reactivity show similar values for each of the complexes studied. ND2HP-TAM shows the highest value for EA and IP, therefore, the highest electrophilicity index with 1.53/1.42 eV. The value calculated with the energies approximation places ND2HP-TAM as strong, while the result for the HOMO-LUMO approximation places it as a moderate electrophile, and therefore close to the level of strong electrophiles. The Nucleophilicity indices calculated for these systems are 6.06, 6.14, and 6.16 eV for NDE2HP, NDEPEG, and NDEPGA, respectively. All three are classified as strong nucleophiles.

Conclusions
The esterification of nanodiamond with 2-hydroxypropanal, polyethylene glycol, and polyglicolic acid allowed the generation of a longer linear chain in the carrier vehicle that interacts electrostatically with the linear chain of the drug tamoxifen. The nanodiamond esterified with 2-hydroxipropanal showed the highest reactivity. The three molecular systems proposed as new esterification models have good permeability in cells according to their PSA. The hydrogen bond analysis showed that NDPGA-TAM generates a greater number of hydrogen bonds. The presence of HBs was corroborated with the vibrational frequencies analysis. The reactivity parameters show ND2HP-TAM as the only strong electrophile. NDPEG-TAM and NDPGA-TAM are classified as moderate electrophiles, in accordance with classification criteria by Domingo et al. The three proposed complexes have good biodistribution, permeability, and pharmacological action on cancer cells, according to Lipinski's Law.