Factors Influencing the Potency of Alzheimer Inhibitors: Computational and Docking Studies

Abstract

Density functional theory (B3LYP/6-31G [d]) is performed to study the effect of molecular and electronic structures of the investigated β-secretase 1 (BACE1) Alzheimer’s inhibitors on their biological activities and discuss the correlation between their inhibition efficiencies and quantum chemical descriptors. IC₅₀ values of the investigated compounds are mostly affected by the substituted R₂ phenyl moiety. The calculations show that the presence of electron withdrawing group increases the half maximal inhibitory concentration (IC₅₀). Structure–activity relationship studies show that the electronic descriptors, energy of high occupied molecular orbital, ΔE, lipophilicity, hardness, and ionization potential index, are the most significant descriptors for the correlation with IC₅₀. Molecular docking simulation is performed to explain the mode of interaction between the most potent drug and the binding sites of the BACE1 target. A good correlation between the experimental and the theoretical data confirms that the quantum chemical methods are successful tools for the discovery of novel BACE1 drugs.

Introduction

Alzheimer’s disease (AD) is ultimately a fatal form of dementia generally affecting people aged 60 and older. The disease progresses from mild cognitive impairment to profound dementia, loss of motor functions, and finally death. ^1,2

The formation of insoluble extracellular amyloid plaques by the accumulation of amyloid β peptide is one of the key pathological features in the brains of AD. ^3,4 Amyloid β peptide is generated by the proteolytic cleavage of the β-amyloid precursor protein (APP). β-secretase 1 (β-site APP cleaving enzyme-1, or BACE1) is one of 2 enzymes responsible for APP and is considered rate limiting in the proteolytic cleavage process. ⁵ The inhibition of BACE1 is considered as a promising therapeutic approach for the treatment of AD. ⁶

The identification of small-molecule inhibitors of BACE1 has been the focus of many pharmaceutical and academic groups worldwide for the past 20 years. ^{7 –9} Recently, potent nonpeptidomimetic BACE1 inhibitors are described. ¹⁰ However, the successful design of inhibitors has been hampered by poor pharmacokinetic properties of peptidomimetic compounds associated with inhibitors of aspartyl proteases required to attain penetration of small molecules across the blood–brain barrier. ¹¹ The protonation states of aspartyl proteases (ASP32 and ASP228) for BACE1 has been discussed using molecular dynamic simulation, which showed that the protonation states of residues are dependent on ligand binding. ^12,13

Although hydroxyethylamine (HEA) suffers from poor metabolic stability, it is found to be a potent inhibitor of BACE1. Although compound HEA exhibits a high affinity for BACE1 (2 nmol/L), it is rapidly metabolized by rat and human microsomes. Unfortunately, removal of the dipropyl amide results in a significant loss of affinity for BACE1. ^{14 –16}

The addition of computer-aided drug design (CADD) technologies to the research could lead to a reduction of up to 50% in the cost of drug design. ¹⁷ Designing a drug is the process to find or create a molecule with specific activity on a biological organism.

Quantum mechanical methods are becoming popular in computational drug design to estimate (relative) binding affinity and compute the binding interaction at the atomic level. The objective of the present work is to study the effect of substituents on the electronic structures and biological potency of the investigated inhibitors. Also, we will try to find a good correlation between quantum chemical descriptors and the efficiency of the investigated compounds.

Computational Methods

Quantum Chemical Calculations

In this study, the molecular structures of the investigated compounds are optimized initially with AM1 semiempirical method implemented in Gaussian 03 program package ¹⁸ to speed up the calculations, and the resulting optimized structures are fully reoptimized using density functional theory (DFT), ¹⁹ with the B3LYP hybrid functional, which is a combination of Becke’s 3 parameters (B3) exchange functional ²⁰ with the Lee, Yang, and Parr (LYP) correlation functional ²¹ and the 6-31G (d) basis set. The optimization of the structures together with the vibrational analysis of the optimized structures is calculated in order to determine whether they correspond to a maximum or a minimum in the potential energy curve, and no imaginary frequency is found, indicating minimal energy.

The molecular structures of the investigated inhibitors, the charge density of high occupied molecular orbital (HOMO) and low molecular orbital (LUMO), and molecular electrostatic potential (MEP) are visualized using GaussView program, version 5. ²²

Quantum Chemical Parameters

Using HOMO and LUMO orbitals energies, the ionization energy (I) and electron affinity (A) can be expressed as follows: I = −E_HOMO, A = −E_LUMO, respectively. The following relations give the hardness (η) and chemical potential (µ), respectively:

$$\text{η}=\left(\text{I}-\text{A}\right)/2\text{ and µ}=\left(\text{I}+\text{A}\right)/2,\text{respectively}.$$

²³

Softness (σ) is a property of a molecule that measures the extent of chemical reactivity. Electronegativity, χ² = −µ. Parr and Pearson ²⁴ proposed the global electrophilicity power of a ligand (ω = µ²/2η), which measures the stabilization of the system by accepting an additional electronic charge from the environment. Electrophilicity has the ability of electrophile to accept an electronic charge and the resistance of the molecule to exchange charge with the environment. The maximal electron flow between donor and acceptor (ΔN_max) is calculated from the equation:

$$\Delta {\text{N}}_{\text{max}}=-\text{µ}/\text{η}.$$

Molecular Docking Calculations

Recently, X-ray analysis of the crystallographic structure of the BACE1 enzyme with the cocrystallized ligand was reported. ²⁵ The optimized molecular structure obtained from DFT calculations are used as input file for conformational search using systematic search method. The lowest energy conformer is used for docking calculations. Our studies begin with the examination of the known structure obtained from the protein data bank (PDB code 2xfj with resolution 1.8 Å). The native ligand was removed from the active site and then redocked into the binding site of its enzyme in order to characterize the binding pocket of BACE1 (the grid dimensions are x = 24.39, y = −0.53, z = 30.13 Å, and the radius is 15 Å). Schematic 2-dimensional representations of the docking results are generated using program for visualization (LIGPLOT). ²⁶

Results and Discussion

The Correlation of Quantum Chemical Parameters With the IC₅₀ of HEA-β Secretase Inhibitors

Experimentally, 14 compounds are found to possess a good or moderate reactivity (inhibition of HEA-BACE1). It is shown from the experimental results that the presence of different substituents on the S₂ and S₃ moieties (Table 1) may create an additional binding interaction with hydrogen bond substituents in BACE1. ²⁵

Table 1.

Molecular Structures of the Investigated Inhibitors and Their Biological Activities.

Compound	R1	R2	R3	R4	R5	BACE1 IC50(nM)
19a	Pr	Me	F	H	Et	2
19b	Pr	Me	H	H	OMe	100
19c	Pr	CONH2	H	H	OMe	100
19d	Pr	CONH2	F	H	OMe	11
19e	Pr	CN	H	H	OMe	90
19f	Pr	CH2CN	H	H	OMe	360
19g	H	CN	H	H	OMe	770
19h	H	CONH2	H	H	OMe	1900
19i	Pr	CON(Pr)2	H	H	OMe	250
19j	Pr	CONH2	F	H	Et	5
19k	Pr	CONH2	H	OH	OMe	300
19l	Pr	COOH	H	H	OMe	30
19m	Pr	CON(Me)2	H	H	OMe	70
19n	Pr	COOEt	H	H	OMe	132

Abbreviations: BACE1, β-secretase 1; CN, cyanide group.

Accordingly, quantum chemical calculations are performed using DFT/B3LYP at 6-31G (d) basis set to investigate the effect of the molecular and structural parameters on the efficiency of the investigated inhibitors and predict the mode of interaction between that inhibitor and the β-secrete enzyme. The molecular structures of the investigated drugs together with IC₅₀ values are shown in Table 1.

Quantum chemical parameters obtained from the DFT calculations, such as the energy of HOMO (E_HOMO), the energy of LUMO (E_LUMO), the energy gap ΔE, (E_LUMO − E_HOMO), the dipole moment (DM), electronegativity (χ²), chemical potential (µ), the softness (σ), the electrophilicity index (ω), the maximum electron flow from donor to the acceptor (ΔN_max), and the molecular volume, are represented in Table 2.

Table 2.

The Calculated Quantum Chemical Parameters Obtained From DFT/B3LYP/6-31G(d) for the Investigated Inhibitors.

Molecule	EHOMOeV	ELUMO eV	ΔE eV	DM D	LogP	η eV	σ eV−1	µ eV	Χ eV	ω	ΔNmax e	MV m3/mol	TNC (e)
19a	−6.260	−1.035	5.225	6.937	6.76	2.613	0.383	3.648	−3.648	2.546	−1.396	414.200	−9.873
19b	−5.990	−0.928	5.062	5.671	5.37	2.531	0.395	3.459	−3.459	2.363	−1.367	417.746	−9.168
19c	−6.073	−1.272	4.800	7.264	3.71	2.400	0.417	3.672	−3.672	2.809	−1.529	361.131	−9.902
19d	−6.145	−1.360	4.785	7.629	3.99	2.392	0.418	3.752	−3.752	2.943	−1.568	381.476	−10.602
19e	−5.767	−1.772	3.994	2.919	4.87	1.997	0.501	3.769	−3.769	3.557	−1.888	458.882	−8.953
19f	−5.959	−1.326	4.633	3.349	5.19	2.317	0.432	3.642	−3.642	2.863	−1.572	425.722	−9.623
19g	−6.024	−1.721	4.303	3.312	3.81	2.152	0.465	3.872	−3.872	3.484	−1.799	357.759	−8.371
19h	−6.059	−1.378	4.682	5.856	2.66	2.341	0.427	3.719	−3.719	2.954	−1.589	406.474	−9.253
19i	−5.772	−1.236	4.535	8.006	5.83	2.268	0.441	3.504	−3.504	2.707	−1.545	581.901	−11.191
19j	−6.331	−1.354	4.978	6.425	5.11	2.489	0.402	3.843	−3.843	2.966	−1.544	429.846	−10.583
19k	−5.770	−1.273	4.498	7.592	3.43	2.249	0.445	3.522	−3.522	2.757	−1.566	446.448	−10.501
19l	−6.070	−1.627	4.443	6.341	4.60	2.222	0.450	3.848	−3.848	3.333	−1.732	375.788	−9.659
19m	−6.061	−1.121	4.940	6.681	4.21	2.470	0.405	3.591	−3.591	2.611	−1.454	422.278	−10.209
19n	−6.045	−1.411	4.634	4.601	4.97	2.317	0.432	3.728	−3.728	2.999	−1.609	404.819	−9.989

Abbreviations: DFT, density functional theory; DM, dipole moment; E_HOMO, energy of high occupied molecular orbital; E_LUMO, energy of low molecular orbital; MV, molecular volume.

It is found experimentally that the compound 19d with fluoro atom at the position R₃ possesses a higher BACE1 affinity than 19c (Table 2). This could be explained from the calculated quantum chemical parameters which show that the presence of fluoro group (electron withdrawing group) decreases the LUMO energy of 19d (−1.360 eV) compared to that of 19c (−1.272 eV) and accordingly increases its electron affinity  (EA) (Table 2). This means that compound 19d could react as an electrophile with the BACE1 enzyme.

Electrophilicity is the descriptor of reactivity and is sufficient enough to describe the toxicity of the molecule. It is also provided with the direct relationship between the rates of reactions and the ability to identify the function of the capacity of an electrophile. It is observed from the calculations that the electron withdrawing substituent increases the ω and accordingly increases the potency of 19d.

The maximum amount of electronic charge ΔN_max acquired from the environment (donor) by an inhibitor (acceptor) is a reactivity index measured by the stabilization in the energy of the complex. The inhibitor 19d has higher ΔN_max (−1.568 e) than 19c (−1.529 e; Table 2), which could allow the transfer and exchange of electrons and accordingly increases the reactivity of 19d as an electrophile.

ΔE is a function of reactivity, and decreasing the value of ΔE increases the reactivity of the inhibitors. The calculated small energy gap between HOMO and LUMO of 19d compound facilitates the transfer and exchange of electron, which leads to an increase its reactivity. This is in a good agreement with the experimental observation.

The polar molecules are distorted better than nonpolar molecules, so the polarity of the molecule is generally expressed in terms of DM. It is a physicochemical property of a drug candidate, which is widely used in medicinal chemistry as an index of lipophilicity and membranes. In general, the solubility of a drug substance in water increases with increasing DM. ²⁷ It is obvious from the calculations that the presence of electron withdrawing group (fluoro group), 19d inhibitor, leads to a slight increase in its DM (7.629 D) and becomes more polarizable than inhibitor 19c (7.264 D; Table 2). This is also confirmed by the increasing value of log P, which may be responsible for increasing the IC₅₀ of compound 19d.

It is concluded from the abovementioned discussion that the calculated quantum chemical parameters confirm that 19d inhibitor has a higher affinity to BACE1 comparable to 19c inhibitor, which agrees well with the experimental observations.

The replacement of carboxamide group (CONH₂), in position R₂, 19h inhibitor, with cyanide group (CN) as in the case of 19g inhibitor, leads to a high increase in the IC₅₀. It is confirmed from the calculations that the presence of CN increases the HOMO energy of 19g inhibitor by about 0.035 eV, which leads to increase in the electron-donating ability and accordingly increases its IC₅₀. Also, the LUMO energy is decreased by 0.343 eV, which means that the 19g inhibitor reacts as an electrophile. This leads to a high decrease in the value of ΔE of 19g by about 0.379 eV and accordingly increases its reactivity to interact with theBACE1 enzyme. Meanwhile, the presence of CN leads to a decrease in the DM (3.312 D) and show the hydrophobic nature of 19g.

It is found from the experimental results that the presence of propyl group instead of H-atom in position R₁ leads to a high increase in the IC₅₀ in case of 19e versus 19 g inhibitors (Table 2). This could be explained from the calculated quantum chemical parameters which show that the highest reactivity of the 19e inhibitor compared to 19g could correspond to the increase in HOMO energy and decrease ionization potential (IP) of 19e inhibitor by about (0.257 eV) and leads to a decrease in ΔE value of 19e to 3.994 eV, more than in the case of 19g compound, which indicates the increase in the reactivity of 19e compound.

The experimental data showed that the effect of substituent CN group instead of CH₂CN group at R₂ position increases the potency of 19e with respect to 19f (Table 1). This could be explained from the calculated quantum chemical parameters which show that the presence of CN group leads to a decrease in IP and LUMO energy (Table 2), which means that 19e inhibitor could accept the electron from the enzyme. The DM decreases to be 2.919 D, which probably increase the lipophilic character of the 19e compound. This is confirmed from the calculated logP (4.87). Also, the increase in the electrophilicity and softness probably increases the IC₅₀ of 19e with respect to 19f compound (Table 2). According to the abovementioned discussion, we can order the potency as follows: 19a >19f >19g > 19h, which agrees well with the experimental expectations.

The presence of electron withdrawing substituent such as a carboxyl group as in the case of compound 19l instead of (CON[Me]₂, COOEt) groups as in the case of compounds 19m and 19n, respectively, leads to decrease in the IC₅₀ of inhibitors 19m and 19n. This could be explained by increasing the reactivity of 19l with lower ΔE value than those of 19m and 19n. Also, the presence of electron withdrawing group increases the LUMO energy and softness of 19l, which leads to the enhancement of the inhibition efficiency of the19l compound. It is shown that 19l has a lower DM which probably increases its lipophilic character. We conclude from the above that the potency is in order 19l > 19m > 19n, which is in a good agreement with the experimental results.

Frontier Molecular Orbital Densities

Frontier molecular orbitals (FMOs) are the most important orbitals that play an important role in the interaction. The charge density distribution of the HOMO level is highly localized on the substituted phenyl ring, m-methoxy group, and nitrogen atom for all investigated compounds except for 19k molecule, where the charge density is highly localized on the moiety that contains a phenyl ring with a substituted hydroxyl group, carbon atom of methylene group, and nitrogen atom. This means that these moieties could be considered as a nucleophilic site (electron donor) in the investigated compounds (Figure 1; Supplemental Figure S1). Also, the calculations show that the charge density of the LUMO level is highly delocalized on the phenyl substituent in position R₂, carbonyl group, and nitrogen atom for all investigated compounds which mean that these moieties could also interact with the enzyme as an electrophile (electron acceptor; Figure 1 and Supplemental Figure S2). It is concluded from FMOs that the substituents play an important role to enhance the affinity of the tested compounds toward the BACE1.

Figure 1.

HOMO, LUMO, and MEP for 19g and 19k compounds. MEP, molecular electronic potential. HOMO, high occupied molecular orbital; LUMO, low molecular orbital.

Molecular Electrostatic Potentials

Molecular electrostatic potential is a real physical property related to the electronic density and is used as a highly beneficial descriptor for the determination of electrophilic and nucleophilic sites as well as hydrogen-bonding interactions. Also, it is well suited for analyzing processes based on the recognition of one molecule from another as is in drug–receptor interaction. It is clear from Figure 1 and Supplemental Figure S3 that the negative region for the electrophilic attack is localized at oxygen atoms, which means that they are H-bond acceptor from the active site of the BACE1 enzyme. The positive region for the nucleophilic attack is at hydrogen atoms and most rest of the molecule, which means that they are H-bond donor from the active site of BACE1. This is important to generate an ideal docking pose of the drug within the binding pocket of BACE1 and to describe the IC₅₀ of the drugs.

Molecular Docking Analysis

Docking computations are carried out to explore the probable binding conformations of a potential inhibitor to the receipt and to inspect significant interactions with the protein. Root mean square deviation value is calculated between the cocrystal (2xfj) and the redocked one to validate the docking reliability which is 0.72 Å. The value shows a very high reliability of the docking method to reproduce the experimentally binding mode of the investigated inhibitor (Figure 2). In an effort to elucidate the possible mechanism by which the most potent compound 19d can induce anti-Alzheimer’s activity, molecular docking computations are performed. The catalytic sites of BACE1 Alzheimer’s are Thr 292, Asp 93, Asp 289, Thr 293, Gln 134, Asn 294, and Thr 133. If the inhibitor 19d interacts with the catalytic site of the BACE1 target (Figure 3), it will reduce its activity and change the protein conformation. The calculations show that the inhibitor 19d form 13 hydrogen bond interactions with the catalytic amino acid residues of the receptor target (Table 3). The quantities of hydrogen bond interactions indicate the ability of a tested compound to inhibit the protein target.

Figure 2.

Superimposition of the native ligand found within the crystal structure and the redocked pose of the same ligand.

Figure 3.

Three-dimensional scheme of interactions between 19d and BACE1 active site. Hydrogen bond interactions are indicated by dotted lines. Protein is represented by sticks (thin), and 19d is represented by sticks. BACE1 indicates β-secretase 1.

Table 3.

H-Bond Distances (Å) and Angles for Cognate Inhibitor (19d) with BACE1 Residues.

Participant Amino Acidsa	Optimized H-bond Distances (Å) State	H-Bond Angle (°)
Thr 292	2.82	19.47
Asp 93	3.22b	15.72
Asp 93	3.16c	4.55
Asp 93	3.37d	14.41
Asp 289	3.24e	16.54
Asp 289	3.09f	17.07
Thr 293	3.13g	12.42
Thr 293	3.17h	18.08
Gln 134	2.83i	18.28
Gln 134	2.95j	12.75
Asn 294	3.31	11.60
Thr 133	2.85k	5.92
Thr 133	3.23l	18.04

Abbreviation: BACE1, β-secretase 1.

^aFor detailed participant atoms, readers are referred to Figure 9.

^bAsp 93-N (3) H.

^c,dAsp 93-O(1)H.

^e,fAsp 289-O(1)H.

^gThr 293-O (2).

^hThr 293-O(3).

^IGln 134-N(4)H₂.

^jGln 134-O(4).

^k,lThr 133-O(4).

Ligand–Receptor Interactions

Our model system including the interacted amino acids and a bound ligand (19d) is shown in Figure 4. H-bond geometries are reported as H-donor-acceptor angles. It should be noted that H-bond lengths are obtained considering H-acceptor distances. ²⁸

Figure 4.

Two-dimensional scheme of interactions between 19d and BACE1 active site generated by LIGPLOT, PDB deposition code: 2xfj. BACE1 indicates β-secretase 1; PDB, protein data bank.

Ligand–residue binding energies are calculated using Molegro Virtual Docker program. ²⁹ Binding energies of inhibitor (19d) with individual amino acid residues surrounding the BACE1 binding site obtained from calculations are summarized in Figure 5.

Figure 5.

Binding energies for the interacted BACE1 amino acids and 19d.

H-bonds are detected between Gln 134 and Thr 293 residues and amide nitrogen (N₄-H) and carboxamide oxygen (O₂) of the ligand, respectively. Moreover, the oxygen atom of a hydroxyl group (O₁) is stabilized by 2 residues Asp 93 and Asp 289, which act as hydrogen bond acceptor. The stabilization of amide nitrogen (N₂) is due to a hydrogen bond interaction with Thr 292 residue which acts as hydrogen bond donor. The carbonyl oxygen (O₃) of the ligand is stabilized by Asn 294 residue. Also, the carbonyl oxygen (O₄) is stabilized by 2 residues; Gln 134 acts as hydrogen bond donor and Thr 133. In addition, amide nitrogen of Thr 293 contributed to 2 key H-bonds involving oxygen atom of a hydroxyl group (O₃) and carboxamide oxygen (O₂) of the ligand (Figure 3).

The molecular docking analysis shows 5 attractive hydrophobic contacts between the inhibitor 19d and Phe 169, Thr 292, Thr 293, Leu 91, and Ile 179 catalytic residues (Figure 6). It is shown that the phenyl group in S₅ moiety is close to the side chain of Leu 91 and Ile 179 with distances of 2.44 and 2.33 Å, respectively. This phenyl moiety of inhibitor also provided a π–π stacking interaction with Phe 169 with distance of 2.8 Å. Meanwhile, the phenyl group in S₂ moiety of the inhibitor is close to the side chain of Thr 292 and Thr 293 residues with distances 2.74 and 2.89 Å, respectively. It is concluded from the abovementioned discussion that these kinds of interactions are responsible for the stabilization of protein–drug complex.

Figure 6.

The hydrophobic binding pattern of 19d in the BACE1 active site, PDB deposition code: 2xfj. BACE1 indicates β-secretase 1; PDB, protein data bank.

Structure–Activity Relationship

The ultimate aim of structure–activity relationship (SAR) study is to correlate the IC₅₀ of a series of compounds with some appropriate quantum chemical descriptors. ³⁰ Different descriptors such as E_HOMO, energy gap, ionization energy, hardness, softness, and lipophilicity (log p) are tested for SAR analysis. Figure 7 shows some representative plots of the correlations between the experimental IC₅₀ and some of the quantum chemical parameters. Herein, a good correlation with a correlation coefficient (R = .69) is found between 1\log IC₅₀ and E_HOMO. The IC₅₀ of the studied compounds increases as the energy of HOMO decreases. Lipophilicity descriptor (usually expressed as log P) represents the ability of a molecule to enter the cell membrane and contact with the interacting sites. Lipophilicity is related to the free energy change associated with the desolvation of a compound, as it moves from an aqueous phase to the biological part. This property is correlated well with 1\log IC₅₀ values of the studied compounds with an R value of .61. The equation contains HOMO energy with a high correlation coefficient that is used to estimate the predictive correlation coefficient (R_pred) by plotting a graph between the experimental and the predictive IC₅₀ values (Figure 8). R_pred is 0.69, which indicates that this model describes 69% of the variance of IC₅₀ of the investigated molecules.

Figure 7.

The correlations between experimental 1\log IC₅₀ and different quantum chemical parameters.

Figure 8.

The correlation between calculated 1\log IC₅₀ and predicted biological activity.

Quantitative Structure–Activity Relationship Using Multiple Linear Regression Analysis

Quantitative structure–activity relationship is a set of methods that try to find a mathematical relationship between IC₅₀ of a molecular system and a set of descriptors of molecules based on its geometric and chemical characteristics.

Objective feature selection uses the independent variables alone to filter out the nonuseful descriptors without using the dependent variables, and this procedure involves:

• All descriptors with the same values for the molecules are omitted.

• The input variables in multiple linear regressions (MLR) must not be highly correlated. Subjective feature selection searches for an information-rich subset of descriptors. Here, the dependent variable is -log IC50 (pIC₅₀) values that are considered in descriptor selection, and the subset of descriptors is used to map the set of molecular structure to the activity.

In order to explore more relevant descriptors contributing to the IC₅₀ of BACE1 inhibitors, MLR runs for several times by using molecular descriptors using material studio v.4.3 software. ³¹ One model was generated using a combination of different descriptors. We check the “intervariable correlation matrix” for the equation in the model. This parameter was used to filter off the equations that are showing intercorrelation among the descriptors, even though this equation shows a good statistical data. The best statistical significant data obtained from the model is represented in Table 4.

Table 4.

Correlation Matrix Showing Correlation Among Various Physicochemical Descriptors and Inhibitory Activity in the Model.

Parameters	EHOMO	ELUMO	DM	LogP	pIC50
E HOMO	1
ELUMO	0.28	1
DM	−0.18	0.66	1
LogP	−0.17	0.28	−0.05	1
pIC50	−0.63	−0.31	−0.36	−0.59	1

Abbreviations: DM, dipole moment; E_HOMO, energy of high occupied molecular orbital; E_LUMO, energy of low molecular orbital.

The correlation matrix containing the descriptors derived from this model shows that there are no colinearity problems between descriptors (Table 4). The equation describing the IC₅₀ for this model contains the E_HOMO, E_LUMO, DM (electronic descriptors), and logp. The obtained statistical model has a correlation coefficient R (0.88), which supports the reliability and goodness of the model.

$$pI{C}_{50}=2.66\times \left[E_{HOMO}\right]+1.32\times \left[E_{LUMO}\right]-0.23\times \left[DM\right]-0.48\times \left[LogP\right]+23.32$$

$$n=13R=0.88R^2=0.77.$$

The main points of interest from the abovementioned equation are as follows:

• E_HOMO indicates the importance of electrostatic interactions of the ligand with an enzyme when a molecule acts as electron pair donor in bond formation; the electrons are supplied from the HOMO of the molecule. HOMO descriptor denotes the nucleophilicity of the molecules, and it has the highest positive coefficient in the equation, which indicates that the increase in the electron energy of the molecules increases the activity.

• Energy of low molecular orbital measures the electrophilicity of the molecule. The high positive correlation of LUMO with the IC₅₀ indicates that the electron-donating groups are favorable for the activity.

• Dipole moment and Logp are negatively correlated with the inhibition activity with a small coefficient.

The R_pred is estimated by plotting a graph between the observed and the calculated pIC₅₀ values. Such correlation is shown in Figure 9. R_pred value is 0.88, and this is fairly high indicating the quality of this model.

Figure 9.

Graph for the actual versus predicted activities of the investigated compounds.

Conclusion

It is concluded from the DFT calculations that the calculated quantum chemical parameters could explain the effect of substituents on the IC₅₀ of the investigated compounds, which is in a good agreement with the experimental observations.

The following conclusions are obtained:

1. It is shown from the calculations that inhibitors with good IC₅₀ have a variation in LUMO energies. Meanwhile, the charge density distribution of the LUMO level explains the role of the R₂ substituent for future drug design of new potent inhibitors for the BACE1 enzyme.

2. The charge density of the HOMO is highly localized on the substituted phenyl moieties which show that these moieties could be considered as a nucleophile.

3. The calculations show the importance of the presence of electron-donating substituents on the potency of the inhibitors.

4. The presence of electron withdrawing substituent at position R₂ increases the inhibitory potency.

5. Molecular docking results show that the inhibitor 19d forms hydrogen bond and hydrophobic interactions with some active amino acids residues. These interactions enhanced the stabilization of the protein–inhibitor complex.

6. Structure–activity relationship study shows that E_HOMO, ΔE, lipophilicity, hardness, and ionization potential are the most significant descriptors for the correlation with the IC₅₀.

7. Finally, the present study shows that the quantum chemical parameters and molecular docking analysis are successful tools for better description of IC₅₀.