Introduction

Oxidative stress, a disturbance in the prooxidant-antioxidant balance, is associated with the pathogenesis of many diseases associated with radical damage, including neurodegenerative diseases (Alzheimer’s disease and Parkinson’s disease), several types of diabetes, cardiovascular diseases, cancer, etc. [1]. Oxidative stress is associated with high levels of the so-called reactive species such as reactive oxygen (ROS), reactive nitrogen (RNS), and reactive sulfur species (RSS), namely free radicals (FR). The overproduction of FR may result in DNA, lipid, and protein damage [2]. Antiradical (AR) molecules can interact with FR and terminate their chain reaction by different mechanisms [3].

AR prevent radical damage by an oxidation mechanism or a reduction mechanism of FR [4]. The mechanisms of AR are primary, chain-breaking, by forming stable FR. In secondary mechanisms, preventive antiradicals do not involve direct reactions with FR and do not convert them to more stable products. Finally, tertiary mechanisms fix damaged biomolecules [5]. Moreover, some ARs may exhibit more than one mechanism; they are classified as multifunctional antiradicals. AR exist in different sources: endogenous (enzymatic or nonenzymatic compounds in the human body) and exogenous (compounds in fruits, vegetables, and plants) [6]. The effort of research groups devoted to finding chemical compounds with antiradical properties to prevent radical damage is well documented [7,8,9].

Several research groups have performed theoretical studies to explore mechanisms and antiradical properties of certain molecules to suggest new AR [10,11,12,13,14,15,16]. Such is the case of exogenous molecules produced by some plants such as lycopene, a red carotenoid present in tomatoes, watermelon, guava, etc. [14]; silybin, a medicinal plant [16]; curcumin, a yellow-orange pigment derived from the rhizome of Curcuma longa [17]; and vitamins such as vitamin C and vitamin A [18, 19]. These studies show different possible mechanisms as primary mechanisms: single electron transfer (SET) and hydrogen atom transfer (HAT). Altogether, the results describe structural properties that are important for antiradical activity. In addition, these properties help against FR. As a result, they show that some exogenous molecules can be good AR.

Some recent experimental and theoretical studies of exogenous molecules suggest that they can be good antioxidants. These molecules are curcumin [20]; chicoric acid (CA), a caffeic acid derivative present in Echinacea purpurea and basil leaves (Ocimum basilicum) [21, 22]; and caffeic acid phenethyl ester (CAPE), a polyphenolic chemical compound present in the apiproduct propolis, obtained from beehives [23]. These studies suggest that curcumin, CA, and CAPE are good antioxidants; however, their mechanism of action and their antiradical properties are still not well known. In this work, we show how to characterize them and whether they are good AR. For this purpose, we examine the antiradical properties and mechanisms of action of curcumin, CA, and CAPE. We use vitamin A and E as controls. We obtain reactivity properties, log P, solvation free energy, bioavailability, and toxicity using density functional theory (DFT).

We examine curcumin, both tautomers of curcumin (keto-enol curcumin (CKE) and diketone curcumin (CDK)), CA, CAPE, vitamin A, and vitamin E, (structures in Fig. 1). Moreover, we use Conceptual Density Functional Theory (CDFT) [24], to compute the global chemical descriptors of reactivity. Furthermore, we study three possible oxidative stress mechanisms: SET, HAT, and xanthine oxidase (XO) inhibition; the latter is an enzyme that generates reactive oxygen species [25]. Moreover, to locate the specific atoms in the molecule that can donate or accept electrons from FR, we use the local ionization potential map and the LUMO map.

Fig. 1
figure 1

Chemical structures of the antiradicals studied in this work. In blue color, H atoms selected for dehydrogenation studies, and the calculated energy of the related reaction, \({D}_{0}\), values in kcal/mol

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Benchmark and computational details

All electronic calculations were performed with the Gaussian 09 package [26]. Local minima were identified by the absence of imaginary frequencies. A small benchmark study comparing several DFT methods was carried out to validate the chosen computational procedure selected for the best cost/performance compromise. Eight levels of theory (B3LYP/6–31 + G*, B3LYP/6-311G**, B3LYP/6–311 + G**, M06-L/6-31G*, M06-L/6–31 + G*, M06/6–31 + G*, LC-ωPBE/6–31 + G*, and M06-2X/6–31 + G*) were employed to compute the vertical ionization energy (I) of Vit A, used as control. The calculated I values were compared with the experimental value, 6.95 eV, for Vit A [27]. In addition, we compared the computational time for Vit E and Vit A in every level of theory. The continuum solvation model based on density (SMD) [28] was used to include solvent effects of water (ε = 78.36) and pentyl ethanoate (ε = 4.73), chosen to mimic aqueous and lipid environments.

Conformational search

We performed a conformational search in Spartan 08 [29]. The utility “conformer distribution” was employed with the MMFF force field [30,31,32,33]. We restrict the search to 10,000 conformers, to obtain the 10 lower energy final conformers without imposing any symmetry restrictions. These 10 conformers were reoptimized at the B3LYP/6–31 + G* level of theory. Finally, we selected the minima energy conformers for each molecule studied and reoptimized with M06-2X/6–31 + G* level of theory.

pK a calculation

To mimic any molecule under physiological conditions (aqueous phase), it is important to know which would be the prevailing acid/base form. To that end, it is necessary to know the pKa value which is many times not known experimentally. Therefore, we estimated the pKa value of all the AR under study using the proton-exchange scheme [34]. In this work, we used the so-called direct method, cycle A below,

figure a

Scheme 1 pKa calculation via the direct method

with the following equations,

$$A{H}_{aq}\to {A}_{aq}^{-}+{H}_{aq}^{+}$$
(1)
$${pK}_{a}=\frac{\Delta {G}_{soln}^{*}}{RTln(10)}$$
(2)
$${\Delta G}_{soln}^{*}={\Delta G}_{gas}^{*}+\sum_{i=1}^{N\;products} {n}_{i}{\Delta G}_{solv,i}^{*}-\sum_{j=1}^{N\;reactants} {n}_{j}{\Delta G}_{solv,j}^{*}$$
(3)

Some needed values were taken from the work of Coote, M. L. [34].

Global descriptors

To understand the reactivity of our molecules towards the FR, we employed the following reactivity global descriptors of CDFT [24]. Electronegativity (χ) measures the electrons tendency to escape from the system. χ is obtained by a finite differences approximation [35] as follows:

$$\chi =\frac{I+A}{2}$$
(4)

where the vertical ionization energy (I) and electron affinity (A) have been used. I measures the vertical energy required to remove an electron from the neutral system ground state. A measures the vertical energy when a neutral system in its ground state captures one electron [36, 37]. I and A are calculated as follows:

$$I=E\left(N-1\right)-E(N)$$
(5)
$$A=E\left(N\right)-E(N+1)$$
(6)

It is necessary to compute single point energies of the cationic \(\left(N-1\right)\) and anionic \((N+1)\) molecules at the optimized neutral structure.

Chemical hardness (\(\eta\)) measures the resistance of a molecule to intramolecular charge transfer. \(\eta\) indicates the reactivity of the molecule [38, 39]. \(\eta\) is calculated as follows:

$$\eta =I-A$$
(7)

Electrophilicity index (ω) measures the stabilization energy when a system attracts electronic charge [40]. To measure the response of the system for the addition and/or removal of charge, it developed the electrodonating power index (\({\omega }^{-}\)) and the electroaccepting power index (\({\omega }^{+}\)) [41]. \({\omega }^{-}\) measures the capability of the system to donate a fractional amount of charge and \({\omega }^{+}\) measures the capability of the system to accept charge. \({\omega }^{-}\) and \({\omega }^{+}\) can be approximated as follows [41]:

$$\omega \equiv \frac{{(I+A)}^{2}}{8(I-A)}$$
(8)
$${\omega }^{+}\equiv \frac{{(I+3A)}^{2}}{16(I-A)}$$
(9)
$${\omega }^{-}\equiv \frac{{(3I+A)}^{2}}{16(I-A)}$$
(10)

Solvation free energy and log P

Solvation free energy (\(\Delta {G}_{solv}^{^\circ }\)) is a property useful for the thermodynamical description of a solution. \(\Delta {G}_{solv}^{^\circ }\) describes the relative stability of a chemical species in the solution concerning the gas phase at equilibrium. As a result, \(\Delta {G}_{solv}^{^\circ }\) indicates the preference of one phase over the other [42]. The solvation energies were obtained as follows:

$$\Delta {G}_{solv}^{^\circ }={G}_{solvent}-{G}_{gas}$$
(11)

Then, it was necessary to compute the Gibbs free energies of the optimized structure in gas (\({G}_{gas}\)), water, and pentyl ethanoate (\({G}_{solvent}\)).

Whereas the log P value is the octanol/water partition coefficient, which is traditionally used to assess the hydrophobicity of compounds and estimate their membrane permeability [43]. Log P value was calculated for every optimized structure in Spartan 18 [44] with the utility QSAR. Log P was calculated using predictions based on quantitative structure relationships (QSAR) of Ghose, Pritchet, and Crippen [45].

Bioavailability, ADME properties, and toxicity

Lipinski’s rules [46], Ghose’s rules [47], and Veber’s criteria [48] provide empirical values of properties useful to assess the drug’s pharmacokinetics in the human body, such as absorption, distribution, metabolism, and excretion (ADME). These rules determine if a molecule is likely to perform well as an orally active drug in humans. Lipinski’s rules state that drugs should have no more than 5 hydrogen bond donors (HBD), no more than 10 hydrogen bond acceptors (HBA), a molecular weight (MW) under 500 amu, a log P value lower than 5. On the other hand, Ghose’s rules state that drugs should have a log P value from − 0.4 to 5.6, molar refractivity (AMR) from 40 to 130 m3mol−1, MW from 160 to 480 amu, and the number of atoms from 20 to 70 (nAtom). Finally, the Veber’s criteria establish that the successful drugs should have no more than 10 rotable bonds (nRB), no more than 12 HBD, and a polar surface area (PSA) equal to or less than 140 Å2. Molecules that violate more than one rule may have problems with bioavailability. Our molecules’ ADME values were obtained using DruLiTo software [49].

We used the Toxicity Estimations Software Tool (T.E.S.T) to investigate the toxicity [50]. This program makes its predictions using quantitative structure–activity relationships (QSAR). The descriptors computed in this way were the median lethal dose (LD50) and the mutagenicity (M). LD50 is the amount of material (mg/kg), which causes the death of 50% of rats after oral ingestion. M, known as Ames test, is a bacterial bioassay to evaluate the mutagenicity caused on the DNA of Salmonella typhimurium.

Single electron transfer

To investigate the SET mechanism, Eq. 12, we used two graphical strategies. The full-electron donor–acceptor maps (FEDAM) is employed to evaluate and characterize the electron-transfer process between AR and FR. FEDAM graphs \(I\) versus \(A\). Thus, FEDAM provides information about electron-donor and electron-acceptor behaviors of a given molecule [51]. In this research, we worked with the following FR: \(O{H}^{^\bullet}\), \({NO}_{2}^{^\bullet}\), \({HOO}^{^\bullet}\), and \({CH}_{3}{O}^{^\bullet}\). These FRs were selected as representative of FRs containing oxygen, carbon, and nitrogen. The donator-acceptor map (DAM) classifies a molecule in terms of its electron donating-accepting capacity [52]. DAM is useful when comparing different molecules; they can be classified according to their electron donating-accepting capacity relative to F and Na. Then, it is necessary to graph the electron acceptance (\(Ra\)) versus the electron donation index (\(Rd\)), defined by Eqs. 13 and 14.

$$AR+{FR}^{^\bullet}\to {AR}^{^\bullet+}+{FR}^{-}$$
(12)
$$Ra=\frac{{\omega }_{L}^{+}}{{\omega }_{F}^{+}}$$
(13)
$$Rd=\frac{{\omega }_{L}^{-}}{{\omega }_{Na}^{-}}$$
(14)

Hydrogen atom transfer

Another possible mechanism for scavenging FR is HAT, represented as follows:

$$AR+{FR}^{^\bullet}\to {AR}_{(-H)}^{^\bullet}+FRH$$
(15)

For HAT, we also studied the dissociation energy of one hydrogen atom within the molecule (D0).

$${D}_{0}\equiv AR\to {AR}_{\left(-H\right)}^{^\bullet}+{H}^{^\bullet}$$
(16)

We computed the adiabatic Gibbs free energy for all reactions. We evaluated the antiradical action of all molecules under study with the following FRs: \({OH}^{^\bullet}\), \({NO}_{2}^{^\bullet}\), \({HOO}^{^\bullet}\), and \({CH}_{3}{O}^{^\bullet}\).

Inhibition of xanthine oxidase

Docking studies were based on the X-ray structure of Bos Taurus (90% homology with human XO) at 2.5 Å resolution when forming a complex with its competitive inhibitor salicylic acid (SAL), with PDB code: 1FIQ [53]. XO and ligands docking studies were performed with AutoDock Tools package version 1.5.6 and AutoDock 4.2.6 [54]. To merge nonpolar hydrogens, we calculated Kollman charges. We removed A and B chains of the protein and small molecules except molybdopterin cofactor (MTE) and Mo cofactor (MOS) in the C subunit of the protein. We used grids for docking evaluation with a spacing of 0.375 Å and 40 × 40 × 60 points centered in 26.569 × 10.228 × 113.088 with a Lamarckian Genetic Algorithm [55]. We performed flexible Docking studies. We selected the following residues as flexible: LEU873, LEU648, GLU802, SER876, ARG880, PHE914, PHE1009, THR1010, VAL1011, LEU1014, ALA1079, and GLU1261 where GLU802, THR1010, ARG880, PHE914, PHE1009, LEU873, VAL1011, and LEU648 are catalytic residues and GLU802, GLU1261, and ARG880 play key roles in the hydroxylation of substrate xanthine [56]. All molecular graphics material was prepared using the Discovery Studio 2017 R2 Client (ver. 17.2.0, Accelrys Software Inc., San Diego, CA, USA).

Results and discussion

To select a reliable level of theory, we tested the following levels: B3LYP/6–31 + G*, B3LYP/6-311G**, B3LYP/6–311 + G**, M06-L/6-31G*, M06-L/6–31 + G*, M06/6–31 + G*, LC-ωPBE/6–31 + G*, and M06-2X/6–31 + G*. These were chosen because they represent a variety of the most employed levels of theory in other works and some were selected by their low computational cost. We compared the absolute error between calculated \(I\) and experimental \(I\) (Table 1), computational time, and global descriptors; see Figs. S1 and S2 in the supporting information. According to the computational times, with B3LYP, M06-L, and M06-2X functionals and small basis sets, it is less expensive to obtain optimization and frequencies calculations (Fig. S1). We compared computed values of some global descriptors: \(I\), \(\eta\), \(\omega\), and \({\omega }^{-}\) at various levels of theory, finding low variation in the values. The range percent values are for \(I\) 14%, for \({\omega }^{-}\) 12%, for \(\eta\) 18%, and for \(\omega\) 14% (Fig S2). Finally, according to Table 1, M06-2X/6–31 + G* yields the smallest absolute error; therefore, it was selected to perform all the forthcoming studies, since it exhibits the best cost/performance compromise.

Table 1 Vit A vertical ionization energy calculated in eV at different levels of theory and % absolute error between calculated \(I\) and experimental \(I\) value
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Conformational search

We used the optimized geometry at the global minimum and the closest conformers found at local minima to examine the global reactivity descriptors, but only the global minimum conformer was used in the other studies. First, at the B3LYP/6–31 + G* level of theory, we selected the conformers closest to the global minimum conformer within a difference of ~ 0.6 kcal/mol. Thereafter, these conformers were reoptimized at the M06-2X/6–31 + G* selected level of theory. Furthermore, we used the root mean square deviation (RMSD, Eq. 17) for dihedral angles to compare the closest conformers with the global minimum conformer; see Fig. S3.

$$RMSD=\sqrt{\frac{1}{N}{\sum_{i=1}^{N}{({X}_{i,test}-{X}_{i,global})}^{2}}}$$
(17)

where \(N\) is the number of dihedral angles tested, \({X}_{test}\) is the value of the dihedral angle in the tested conformer, and \({X}_{global}\) is the value of the same dihedral angle in the reference global minimum conformer. As a result, we only found different conformers for Vit A, CDK, and CAPE. For CKE, CA, and Vit E, we only have the global minimum conformer. CA has an asymmetric carbon; our structure is the L-chicoric acid ((2R,3R)-2,3-bis[[(E)-3-(3,4-dihydroxyphenyl)prop-2-enoyl]oxy]butanedioic acid). Our CA optimized molecule is a V-shaped conformation with large separation between the tail-ends, with no hydrogen bonds. Nobela et al. have reported that the RR conformation is the most stable due to hydrogen bond interactions [57]. This conformer was obtained by molecular dynamics calculations with the AmberTools algorithm. Its conformer has several internal hydrogen bonds and is not V-shaped, so it is very different from ours.

For Vit A, we found two conformers with an RMSD value of 127.19°. The difference between conformers of Vit A is in the alkyl chain. For CDK, there are three conformers. Conformer 2 has an RMSD value of 229.18° and conformer 3 of 74.48°; they exhibit several geometry changes in the orientation of the methoxy group and the keto groups in the hepta-1,6-diene-3,5-dione structure. For CAPE, there are three different conformers. Conformer 2 has an RMSD value of 149.54° and conformer 3 of 87.08°. The principal changes observed correspond to orientation possibilities of the phenyl group; these are more significant for conformer 2. We found that conformer CKE is like the one found by Galano et al., and in the same work, they report our second CDK conformer as the minimum [17]. The second CDK conformer has an energy difference with the minimum of 0.001 kcal/mol which is very small. On the other hand, Singh et al. [58] and Puglisi et al. [59] report similar conformers with the B3LYP/6–31 + g(d, p) level of theory and molecular dynamics.

pK a calculation

According to our computed pKa values (Table 2) at physiological pH (7.4), all the molecules are in neutral form. Therefore, the neutral form was used in this work to study the free radical scavenging activity. To assess which H is involved in the deprotonation mechanism, we chose H atoms from -NH, -NH2, -COOH, and -OH groups. For Vit A, we studied the hydrogen in the -OH group and for Vit E, the hydrogen at -OH group. For CDK, we selected hydrogens H7a and H7a’ at -OH groups (Fig. 1). For CKE, according to the lower \({D}_{0}\) value in Fig. 1, we selected H7b from the -OH group. For CA, we selected H7a and H7a’ from -OH groups; these are the hydrogens with the lower \({D}_{0}\) value (Fig. 1). For CAPE, we selected H7a from the -OH group. Our results were compared with experimental results. Experimental previous work shows that CAPE [60], CA [61, 62], and curcumin [63] are not ionized at physiological pH 7.4. The experimental pKa reported for curcumin is 8.38 ± 0.04 [63]. It is known that cycle A, for the proton exchange scheme, is not very accurate, but our results agree with experimental results, which is enough for this work’s scope.

Table 2 First pKa values and molar fraction calculated from the neutral (mfneutral) and anionic (mfanion) species at pH = 7.4 for Vit A, Vit E, CKE, CDK, CA, and CAPE at 1 M standard state
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Variation in global descriptors

To study the electron transfer between AR and FR, we analyzed the global chemical reactivity descriptors of CAPE, CA, curcumin, Vit A, and Vit E. We found that CDK, CKE, CA, and CAPE behave as antireductants while vitamins as antioxidants. Furthermore, we found that all of them are more reactive in water than in pentyl ethanoate (Fig. 2), where we present the values of the global chemical reactivity descriptors. In Fig. 2a and c, we see those molecules with the lower values of I have the following order Vit A < Vit E < CKE < CAPE < CDK < CA; a lower I means a higher probability of losing an electron, i.e., Vit A and Vit E have a higher likelihood of donating an electron to a FR. We saw that the molecules with higher values of A might be arranged in the following order CKE > CDK > CAPE > CA > Vit A > Vit E; a higher value of A means a higher probability of gaining an electron from a FR, i.e., CKE, CDK, and CAPE can more easily accept an electron from a FR.

Fig. 2
figure 2

Global descriptors computed for CAPE, CA, CDK, CKE, Vit A, and Vit E at M06-2X/6–31 + G* level of theory. a and b show values in water and c and d show values in pentyl ethanoate

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On the other hand, molecules with the lower values of \(\eta\) follow the order CKE < Vit A < CDK < CAPE < Vit E < CA; a lower value of \(\eta\) means higher reactivity (higher values of \(\eta\) for more stable molecules). Additionally, the molecules with a lower value of \(\chi\) are Vit A and Vit E; they may share electrons more easily. In Fig. 2b, we see that the molecules with the lower values of \(\omega\) have the following order Vit E < Vit A < CA < CDK < CKE < CAPE; molecules with a lower value of \(\omega\) are expected to be efficient for scavenging free radicals via electron transfer, then Vit A and Vit E should be more reactive. Also, we see those molecules with the highest values of \({\omega }^{+}\) have the following order CKE > CDK > CAPE > Vit A > CA > Vit E; a higher value of \({\omega }^{+}\) means a higher probability to accept charge from a FR, then CKE and CDK should accept charge more easily. Moreover, we see that the molecules with the lower value of \({\omega }^{-}\) follow the order Vit E < Vit A < CA < CAPE < CDK < CKE; the lower value of \({\omega }^{-}\) means higher probability to donate charge to a FR, then Vit E and Vit A would donate charge more easily to a FR. This trend was observed in the pentyl ethanoate phase. We observe that there are no differences in behavior between the different conformers of each molecule.

Solvation energy and log P

Calculations of \(\Delta {G}_{solv}^{^\circ }\) and log P help to assess the solubility of a molecule in each solvent. We found that CKE, CDK, and CAPE are lipophilic molecules according to their log P and \(\Delta {G}_{solv}^{^\circ }\) values; see Table 3. We compare the results of CDK, CKE, CAPE, and CA with Vit A and Vit E. It is clear that Vit A and Vit E are lipophilic, in agreement with the literature [64, 65]. Hence, our methodology can reproduce the known experimental results. We added Vit C, a known hydrophilic molecule, as control. Our results also show that Vit C is hydrophilic. Thus, CDK, CKE, and CAPE are more energetically stable in pentyl ethanoate solvent than in water, but they are also stable. There is only a small difference between energies in water and pentyl ethanoate. This little difference would indicate the capacity to cross the cell membrane given this amphiphilic character. On the other hand, for CA, its log P value shows CA as a lipophilic molecule, while its \(\Delta {G}_{solv}^{^\circ }\) shows CA as hydrophilic. However, the difference in energies is small. According to its computed pKa value, CA is mostly lipophilic, in agreement with the known fact that phenolic compounds have low solubility in water.

Table 3 Log P value and Solvation free energy (\(\Delta {G}_{solv}^{^\circ }\)) values in water (\(\Delta {G}_{solv, \mathrm{w}}\)) and pentyl ethanoate (\(\Delta {G}_{solv,\mathrm{p}}\)) of each molecule at the M06-2X/6–31 + G*/SMD level of theory
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Bioavailability and toxicity

To investigate the use of our molecules as drugs with potential therapeutic applications, we consider several molecular descriptors from the Lipinski’s rules, Ghose’s rules, and Veber’s criteria and descriptors for toxicity (Table 4). As a result, CKE, CDK, and CAPE do not violate more than one of these rules; then, they are predicted to have good bioavailability. It is important to note that these rules are empirical, so they are a guideline, not rigorous law. On the other hand, we consider the toxicity descriptors, the lower value of M, and the larger value of LD50 means low toxicity for molecules. Henceforth, CDK, CKE, CA, and CAPE are not toxic due to a lower and negative value of M and a larger value of LD50. Our results were compared with the values of the non-toxic Vit C and the toxic DDT and pipobroman, values in Table 4. Thus, we can confirm that our molecules should be non-toxic. As can be seen, CKE, CDK, and CAPE seem to be the most promising for being further tested as drugs with potential therapeutic use.

Table 4 Molecular descriptors (log P, PSA, HBD, HBA, MW, nAtom, nRB, and AMR) from Lipinski’s rules, Ghose’s rules, and Veber’s criteria and descriptors for toxicity, oral rat 50% lethal dose (LD50), and Ames mutagenicity (M) for CAPE, CA, CDK, CKE, and controls Vit A, Vit E, Vit C, DDT, and pipobroman
Full size table

Single electron transfer

To evaluate the SET mechanism, we used simple strategies as DAM and FEDAM maps. In the DAM map, we found that CDK and CKE are good antireductants in water, as shown in Fig. 3. CDK and CKE are much weaker electron donors and better electron acceptors. Therefore, CDK and CKE have better antiradical properties in water than Vit A and Vit E. This result may be due to their phenolic structure. Hence, CKE and CDK have more -OH groups than Vit A and Vit E. However, CDK and CKE have fewer -OH groups than CA, but CA has difficulties donating or accepting electrons, as we saw in the global chemical reactivity descriptors. On the other hand, in Fig. 4, we used the FEDAM map to evaluate fast transfer processes and analyze the feasibility of donating or accepting an electron from FR. In this map, the molecules down and left will transfer electrons more easily to the FR molecules located up and right, and molecules near FR can accept electrons from FR. CDK and CKE are up and right to \({HOO}^{^\bullet}\); they have higher A than the other molecules in both solvents. CDK and CKE are expected to be able to accept an electron from \({HOO}^{^\bullet}\). Consequently, only CDK and CKE would undergo the SET mechanism and mostly in water.

Fig. 3
figure 3

DAM map of the molecules studied. a Molecules in water and b molecules in pentyl ethanoate

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Fig. 4
figure 4

FEDAM map of the molecules studied. a Molecules in water and b molecules in pentyl ethanoate

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To identify which atoms can more easily donate or accept charge from a FR, we propose to use the LUMO map and local ionization potential map (local chemical descriptors). The LUMO map shows the absolute value of the LUMO mapped onto an electron density isosurface with a color code. Blue indicates a high concentration of the LUMO, while red suggests low LUMO. With high values in the LUMO map, a nucleophilic attack is predicted to occur more likely; then, atoms with blue color can more easily accept an electron from a FR molecule. Local ionization potential map indicates the ease or difficulty for local electron removal, and this map shows the value of the local ionization potential mapped onto an electron density isosurface with a color code. Red indicates low ionization potential, while blue indicates high ionization potential. Therefore, atoms in red can more easily donate an electron to a FR molecule. In this sense, in Fig. 5, we can see the atoms, marked with a black arrow, in every molecule that can donate or accept an electron from a FR. In both solvents, we can see the same atoms behave similarly. From the values of the global chemical reactivity descriptors, we can see that CDK, CKE, CAPE, and CA have difficulties donating an electron; thus, acceptance of electrons occurs. Hence, the atoms shown in the LUMO map with blue color can accept an electron from a FR more readily. Electrophilic or nucleophilic agents can easily attack these sites in real biological systems. The results for the vitamins are in the supporting information (Fig. S4).

Fig. 5
figure 5

LUMO map (left) and local ionization potential map (right) for CKE, CDK, CAPE, and CA. Black arrow points to atoms that can accept or donate an electron more readily to a FR. For the LUMO map, blue color identifies atoms that can accept charge easily, while in the local ionization potential map, red color identifies the atom that can donate charge more easily

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Hydrogen atom transfer

Furthermore, we evaluated the hydrogen atom transfer mechanism, where we found that CA and CAPE should be good antiradicals following this mechanism. We calculated the dissociation energy of one hydrogen atom (D0) shown in Fig. 1 and adiabatic Gibbs free energy (\({\Delta G}\)) for the FR: \({OH}^{^\bullet}\), \({NO}_{2}^{^\bullet}\), \({HOO}^{^\bullet}\), and \({CH}_{3}{O}^{^\bullet}\). To find the best antiradical following this mechanism, we must consider the number of reactive hydrogen atoms in the given molecule and the Gibbs free energies for the hydrogen atom transfer. Negative values for Gibbs free energies indicate that the reaction is exergonic and energetically possible. The dissociated hydrogen atom which produces exergonic reactions is considered an active hydrogen. Molecules with many active hydrogen atoms are more reactive; therefore, they should behave as better antiradicals. The total number of active hydrogens is shown in Table 5 (see supporting information Tables S1-S4 for more detailed information). We studied the bond dissociation energy of the active hydrogens of each molecule shown in blue in Fig. 1. For example, for CKE, we studied eight bonds and for CAPE, nine bonds. On the other hand, we only considered the side of the molecule of CDK and CA that has symmetry. Thus, for CDK, we studied four bonds, but it has eight bonds susceptible to donate H, while for CA, we reviewed eight, but it has 16 bonds in total. These can be seen in Fig. 1.

Table 5 Atoms studied in hydrogen atom transfer mechanism for each molecule This table shows the number of reactive atoms within each FR; in parenthesis, the more exergonic reactions are shown
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The more reactive antiradical was found to be CA because it is reactive with all studied FRs in both solvents. However, it is more reactive in pentyl ethanoate than in water, as can be seen in Table S4; we show this result with bold letters in Table 5. Also, CA reacts with all the FR in the order shown in parenthesis. As we can see, the most favorable reaction of CA should be with \({OH}^{^\bullet}\). The second more reactive antiradical is CAPE with nine active hydrogen atoms. CAPE is more reactive in water with all the evaluated FR, especially with \({OH}^{^\bullet}\). The third is CKE and CDK with eight reactive hydrogens; both molecules were more reactive in water. In our previous results, we found that all these molecules are lipophilic; however, they can donate hydrogen atoms more easily in water. This result indicates that they should be able to transfer hydrogen atoms in both solvents. The hydrogen atom that more easily transfers is an acidic hydrogen in all cases.

We also studied the formation of bond hydrogens. A strong intermolecular hydrogen bonding is thermodynamically more favorable; this may cause the highest value of \({D}_{0}\) and \({\Delta G}\), and in consequence, the lowest antioxidant potential. We found moderate and weak hydrogen bonds in CDK, CKE, and CAPE, as shown in Fig. S5. CKE and CDK have three and two hydrogen bonds, respectively, while CAPE has one. CKE has two weak hydrogen bonds between the -OH groups of the phenol and the methoxy group, and it also has one moderate hydrogen bond in the keto-enol species. CDK has two moderate hydrogen bonds between the -OH groups of the phenol and the methoxy. Finally, CAPE has one moderate hydrogen bond between the -OH groups. These hydrogen bonds may indicate why it is more difficult for CDK and CKE to transfer a hydrogen atom than for CA and CAPE.

Inhibition of xanthine oxidase

XO is an enzyme involved in the catabolism of purines in humans. XO generates \({O}_{2}^{^\bullet-}\) and \({H}_{2}{O}_{2}\) during its mechanism of reaction [66]. Then, XO generates significant amounts of FRs; it represents an important source of free radicals [67]. To inhibit XO activity, we used molecular docking to explore the binding affinity, binding mode, molecular interactions, and inhibition constant of CDK, CKE, CAPE, CA, Vit A, and Vit E. To validate the docking methodology, we computed the RMSD value for bond lengths to assess the overall quality of obtained geometries when compared with a known species; the method is effective if a value of RMSD is smaller than 3.5 Å when comparing the bonds in a calculated system with the same bonds in the reference. The docking method validation was done by redocking the natural ligand receptor on the active site, the salicylic acid (SAL). Also, we used the allopurinol molecule, a well-known competitive inhibitor of the XO enzyme [68].

The obtained docking poses for the docked inhibitor were within 1.33 Å for the RMSD of the docked ligand crystal structure. Significant interactions were depicted as hydrogen bonding, \(\pi\) \(\pi\) bonding, and \(\pi\)-alkyl bonding, in Fig. 6. The inhibitors were docked near the binding site; however, CDK, CKE, CAPE, CA, and vitamins A and E were not able to interact directly with the Mo cofactor in XO. In Fig. 6, it is noteworthy that SAL and allopurinol exhibit strong molecular interactions with XO; the binding energy and ligand efficiency reflect this. A high value of binding energy and ligand efficiency means a stronger interaction with the target. Ligand efficiency is used to compare the activity of different molecules regardless of different sizes. On the other hand, the inhibition constant, \({K}_{i}\), indicates how potent an inhibitor is; it is the concentration required to produce half-maximum inhibition.

Fig. 6
figure 6

2D poses and molecular interactions of CKE, CKD, CAPE, CA, and molecules used as controls (SAL and allopurinol) in docking calculations into the active site of XO

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It is noteworthy that SAL and allopurinol, according to their binding energy and ligand efficiency, inhibit XO more efficiently than our molecules; see Table 6. Also, we note that Vit A, CKE, and CA have high values of \({K}_{i}\); this means that high quantities of these molecules are necessary to inhibit the activity of XO. On the other hand, the molecules with a low value of \({K}_{i}\) and high value of binding energy and ligand efficiency are the best to inhibit XO. CDK and CAPE could inhibit XO; however, they are not better than SAL and allopurinol.

Table 6 Binding energy, ligand efficiency, and \({K}_{i}\) values for each molecule
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Conclusions

The antiradical properties of CDK, CKE, CAPE, and CA were investigated at the M06-2X/6–31 + G* DFT level of theory. This level was selected after we performed a benchmark study comparing results from several levels of theory with the experimental value of the vertical ionization energy of vitamin A as reference. Our results show that CDK, CKE, CAPE, and CA should behave as good antiradicals. We find that these molecules have good antiradical properties such as good bioavailability and low toxicity. Some properties were evaluated to provide a good quantitative characterization and description of the most plausible mechanisms of action of the suggested antiradicals. First, the pKa was estimated. We found that at physiological pH, all molecules are in neutral form; in consequence, neutral forms were the only studied since they are the most important species for a biological study. Thereafter, the solubility of these molecules was estimated computing the log P and \(\Delta {G}_{solv}^{^\circ }\). Our results show that CKE, CDK, and CAPE are mainly lipophilic molecules. Our methodology was validated with the vitamin A and vitamin E (lipophilic molecules) and vitamin C (hydrophilic molecule) used as references or controls. Furthermore, the ADME properties and toxicity were assessed by calculating the Lipinski’s rules, Ghose’s rules, Veber’s criteria, LD50, and mutagenicity. Our results indicate that CKE, CDK, and CAPE should be useful drugs with potential therapeutic use due to the excellent values obtained for these properties and their non-toxic nature. Finally, we evaluated some possible antiradical mechanisms of action: SET, HAT, and XO inhibition.

For the study of electron transfer, several global chemical reactivity descriptors were computed; and we found that CDK, CKE, CA, and CAPE behave as antireductants while vitamin A and E as antioxidants. In the study of the SET mechanism, we used two simple graphical techniques to accommodate in a simple manner our numerical results: the DAM map and FEDAM map. These devices allow a rapid quantitative characterization of the antiradical properties of the studied molecules. CDK and CKE are found to be weaker electron donors and better electron acceptors; they are good antireductants according to the SET mechanism. Also, we noticed that CDK and CKE can accept an electron from •OOH. However, this mechanism implies a higher energy cost.

On the other hand, the HAT mechanism was more energetically favorable. The best antiradicals turned out to react following the HAT mechanism, CA, and CAPE with more active hydrogen atoms to be transferred to the free radicals. CA and CAPE were found to be the best antiradicals to react with \({OH}^{^\bullet}\). Although the best molecules to inhibit XO are predicted to be CDK and CAPE, they are not better than the controls SAL and allopurinol.

Our DFT global descriptors of reactivity results can anticipate other mechanisms of action, not studied here, such as sequential electron-proton transfer (SET-PT). The obtained high values of I indicate that our molecules will hardly present SET-PT mechanisms in both the physiological and lipophilic medium. The same should happen with the sequential proton loss hydrogen atom transfer (SPLHAT) mechanism. Consequently, this research may motivate further studies on the subject, to include these other mechanisms not studied so far. In conclusion, we suggest that CDK, CKE, CAPE, and CA may have a potential role as protectors against oxidative stress scavengers and the associated health issues.