QSAR Study of Antimicrobial 3-Hydroxypyridine-4-one and 3-Hydroxypyran-4-one Derivatives Using Different Chemometric Tools

Abstract

A series of 3-hydroxypyridine-4-one and 3-hydroxypyran-4-one derivatives were subjected to quantitative structure-antimicrobial activity relationships (QSAR) analysis. A collection of chemometrics methods, including factor analysis-based multiple linear regression (FA-MLR), principal component regression (PCR) and partial least squares combined with genetic algorithm for variable selection (GA-PLS) were employed to make connections between structural parameters and antimicrobial activity. The results revealed the significant role of topological parameters in the antimicrobial activity of the studied compounds against S. aureus and C. albicans. The most significant QSAR model, obtained by GA-PLS, could explain and predict 96% and 91% of variances in the pIC₅₀ data (compounds tested against S. aureus) and predict 91% and 87% of variances in the pIC₅₀ data (compounds tested against C. albicans), respectively.

1. Introduction

Quantitative structure activity relationships (QSAR) studies, as one of the most important areas in chemometrics, give information that is useful for molecular design and medicinal chemistry [1–5]. QSAR models are mathematical equations constructing a relationship between chemical structures and biological activities. These models have another ability, which is providing a deeper knowledge about the mechanism of biological activity. In the first step of a typical QSAR study one needs to find a set of molecular descriptors with the higher impact on the biological activity of interest [6–9]. A wide range of descriptors has been used in QSAR modeling. These descriptors have been classified into different categories, including constitutional, geometrical, topological, quantum chemical and so on. There are several variable selection methods including multiple linear regression (MLR), genetic algorithm (GA), partial least squares (PLS), principle component or factor analysis (PCA/FA), and so on. [7–9]. MLR yields models that are simpler and easier to interpret than PCR and PLS, because these methods perform regression on latent variables that don’t have physical meaning. Due to the co-linearity problem in MLR analysis, one may remove the collinear descriptors before MLR model development. MLR equations can describe the structure activity relationships well but some information will be discarded in MLR analysis. On the other hand, factor analysis–based methods such as PLS regression can handle the collinear descriptors and therefore better predictive models will be obtained by PLS method [10].

It is almost 120 years since physicians revealed that the coincidence of blood and bacteria in a wound may cause a life-threatening infection. It has also been shown that blood or hemoglobin enhance the lethality of intraperitoneal or subcutaneous inocula of bacteria such as Escherichia coli. The effective component of hemoglobin is iron, and various soluble iron compounds exert an equivalent effect [11]. Administration of iron compounds to the host can increase the virulence of Escherichia coli, Listeria monocytogenes, Salmonella typhimurium and other pathogens [12]. In fact, iron is an essential element required for the growth and virulence of virtually all microbial pathogens [13, 14]. The availability of iron is critically important in host-parasite interactions [15]. Vertebrate hosts withhold iron from microbial invaders as a major defence mechanism against infection [13, 15]. This task is achieved by sequestration of iron with iron-binding proteins, the most abundant, haemoproteins [16]. Some natural antibiotics, called siderophores, are low-molecular-weight chelating agents that form stable complexes with iron [17, 18]. There are many reports of the antimicrobial activity of chelating agents with different chemical structures [19–22]. Kojic acid (5-hydroxy-2-hydroxymethyl-pyran-4-one) and its 3-hydroxypyranones derivatives are examples of these compounds [19]. The bidentate chelating ligand 3-hydroxypyranone, which has a catechol-like function, forms stable complexes with several metal ions such as Fe³⁺. In vitro antibacterial and antifungal activities of 3-hydroxy-pyridinones, bioisoster derivatives of 3-hydroxypyranones with metal chelating ability have been described. They have an inhibitory effect on the growth of Escherichia coli, Listeria inocua and Staphylococcus aureus [22]. More recently antibacterial and antifungal activities of carboxamide derivatives of 3-hydroxypyranones, 5-hydroxypyranones and 5-hydroxypyridinones have been reported [23, 24].

Few reports of antimicrobial studies of 3-hydroxypyridine-4-one and 3-hydroxypyran-4-one derivatives are available [19, 21–25] and in those they were not the subject of QSAR studies. Preliminary QSAR models for a series of such derivatives have been investigated by Fassihi et al. [25]. The antimicrobial activity against C. albicans, S. aureus and P. aeroginosa was the subject of MLR analysis in this preliminary study. MLR models revealed the best relationship between the antimicrobial activity and structural properties against S. aureus and C. albicans. In the present paper, more than 600 topological, geometrical, constitutional, functional group, electrostatic, quantum and chemical descriptors were used, for the development of QSAR equations, different methods were applied for the antimicrobial activity of the studied compounds against S. aureus and C. albicans. These methods where: (i) genetic algorithm - partial least squares (GA-PLS), (ii) MLR with factor analysis as the data pre-processing step for variable selection (FA-MLR) and (iii) principal component regression analysis (PCRA). The correlation coefficient (r), standard error of regression (SE), r²cv (Q²) and RMScv (STD(r)) were employed to judge the validity of regression equation.

2. Experimental Section

2.1. Software

The two-dimensional structures of molecules were drawn using the Hyperchem 7.0 software. The final geometries were obtained with the semi-empirical AM1 method in the Hyperchem program. The molecular structures were optimized using the Polak-Ribiere algorithm until the root mean square gradient was 0.01 kcal mol⁻¹. The resulted geometry was transferred into Dragon program package, which was developed by Milano Chemometrics and QSAR Group [26]. The z-matrix of the structures was provided by the software and transferred to the Gaussian 98 program. Complete geometry optimization was performed taking the most extended conformation as starting geometries. Semi-empirical molecular orbital calculation (AM1) of the structures was preformed using Gaussian 98 program [27]. MATLAB software (version 7.1 Math Work Inc.) was used for the PLS regression method.

2.2. Data set and descriptor generation

The biological data used in this study are antimicrobial activity, (in terms of –log MIC), of a set of 3-hydroxypyridine-4-one and 3-hydroxypyran-4-one derivatives [23, 24, 25]. The structural features of these compounds are listed in

Table 1. Chemical structure of the compounds used in QSAR analysis.

Compound	X	R2	R3	R5	R6
1	NH	CH3	OH	CH2-Ra	H
2	NH	C2H5	OH	CH2-Ra	H
3	NH	CH3	OH	CH2-N(CH3)2	H
4	NH	C2H5	OH	CH2-N(CH3)2	H
5	NH	CH3	OH	CH2-N(C2H5)2	H
6	NH	C2H5	OH	CH2-N(C2H5)2	H
7	N-Ph	CH3	OH	H	H
8	N-m-OH-Ph	CH3	OH	H	H
9	N-C3H7	CH3	OH	H	H
10	N-C4H9	CH3	OH	H	H
11	O	CH2Cl	H	OH	H
12	O	CH3	H	OH	H
13	O	CH2OH	OH	H	CH3
14	O	CH2OH	OCH2Ph	H	CH3
15	O	CHO	OCH2Ph	H	CH3
16	O	COOH	OCH2Ph	H	CH3
17	O	CONHRb	OCH2Ph	H	CH3
18	O	CONHRc	OCH2Ph	H	CH3
19	O	CONHRd	OCH2Ph	H	CH3
20	O	CONHRb	OH	H	CH3
21	O	CONHRc	OH	H	CH3
22	O	CONHRd	OH	H	CH3
23	O	CH2OH	H	OCH2Ph	H
24	O	COOH	H	OCH2Ph	H
25	O	CONHPh	H	OCH2Ph	H
26	N-CH3	CONHPh	H	OCH2Ph	H
27	N-CH3	CONHPh	H	OH	H
28	O	CONH-Re	H	OCH2Ph	H
29	N-CH3	CONH-Re	H	OCH2Ph	H
30	N-CH3	CONH-Re	H	OH	H
31	O	CH2OH	H	OH	H

and then used for subsequent QSAR analysis as dependent variables. The large number of molecular descriptors was calculated using Hyperchem, Dragon package and Gaussian 98. Some chemical parameters including molecular volume (V), molecular surface area (SA), hydrophobicity (LogP), hydration energy (HE) and molecular polarizability (MP) were calculated using Hyperchem Software. Dragon software calculated different functional groups, topological, geometrical and constitutional descriptors for each molecule.

Gaussian 98 was employed for calculation of different quantum chemical descriptors including, dipole moment (DM), local charges, HOMO and LOMO energies. Hardness (η), softness (S), electronegativity (χ) and electrophilicity (ω) were calculated according to the method proposed by Thanikaivelan et al. [28].

Constitutional, topological, geometrical, functional group, quantum and physicochemical indices were used in this study; brief description of some of them is listed in

Table 2. Brief description of some descriptors used in this study.

Descriptor Type	Molecular Description
Constitutional	Mean atomic van der Waals volume (Mv) (scaled on Carbon atom), no. of heteroatoms, no. of multiple bonds (nBM), no. of rings, no. of circuits, no of H-bond donors, no of H-bond acceptors, no. of Nitrogen atoms (nN), chemical composition, sum of Kier-Hall electrotopological states (Ss), mean atomic polarizability (Mp), number of rotable bonds (RBN), mean atomic Sanderson electronegativity (Me), etc.
Topological	Narumi harmonic topological index (HNar), Total structure connectivity index (Xt), information content index (IC), mean information content on the distance degree equality (IDDE), total walk count, path/walk-Randic shape indices (PW3, PW4, PW5, Zagreb indices, Schultz indices, Balaban J index (such as MSD) Wiener indices, Information content index (neighborhood symmetry of 2-order) (IC2), Ratio of multiple path count to path counts (PCR), Lovasz-Pelikan index (leading eigenvalue) (LP1), total information content index (neighborhood symmetry of 1-order) (TIC1), reciprocal hyper-detour index (Rww), Average connectivity index chi-5 (X5A), piID (conventional bond-order ID number), etc.
Geometrical	3D Petijean shape index (PJI3), Asphericity (ASP), Gravitational index, Balaban index, Wiener index, Length-to-breadth ratio by WHIM (L/Bw), etc.
Quantum	Highest occupied Molecular Orbital Energy (HOMO), Lowest Unoccupied Molecular Orbital Energy (LUMO), Most positive charge (MPC), Sum of square of positive charges (SSPC), Sum of square of negative charges (SSNC), Sum of positive charges (SUMPC), Sum of negative charges (SUMNC), Sum of absolute of charges (SAC), Standard deviation (Std), Total dipole moment (DMt), Molecular dipole moment at X-direction (DMX), Molecular dipole moment at Y-direction (DMY), Molecular dipole moment at Z-direction (DMZ), Electronegativity (χ= −0.5 (HOMO-LUMO)), Electrophilicity (ω= χ2/2 η), Hardness (η = 0.5 (HOMO+LUMO)), Softness (S=1/ η).
Functional group	Number of total secondary C(sp3) (nCs), Number of total tertiary carbons (nCt), Number of H-bond acceptor atoms (nHAcc), Number of secondary amides (aliphatic) (nCONHR), Number of unsubstituted aromatic C (nCaH), Number of ethers (aromatic) (nRORPh), Number of ketones (aliphatic) (nCO), Number of tertiary amines (aliphatic) (nNR2), Number of phenols (nOHPh), Number of total primary C(sp3) (nCp), etc.
Chemical	LogP (Octanol-water partition coefficient), Hydration Energy (HE), Polarizability (Pol), Molar refractivity (MR), Molecular volume (V), Molecular surface area (SA).

.

2.3. Data screening and model building

The calculated descriptors were collected in a data matrix whose number of rows and columns were the number of molecules and descriptors, respectively. Genetic algorithm - partial least squares (GA-PLS), MLR with factor analysis as the data pre-processing step for variable selection (FA-MLR) and principal component regression analysis (PCRA) methods were used to derive the QSAR equations and feature selection was performed by the use of genetic algorithm (GA). The genetic algorithms are efficient methods for function minimization. In descriptor selection context, the prediction error of the model built upon a set of features is optimized [29].

In this study, to model the structure-antimicrobial activity relationships better, genetic algorithm-partial least square (GA-PLS) was employed [30, 31]. Partial least squares (PLS) linear regression is a recent technique that generalizes and combines features from principal component analysis and multiple regressions. PLS is a method suitable for overcoming the problems in MLR related to multicollinear or over-abundant descriptors [10].

Application of PLS method thus allows the construction of larger QSAR equations while still avoiding over-fitting and eliminating most variables. This method is normally used in combination with cross-validation to obtain the optimum number of components [32, 33]. The PLS regression method used was the NIPALS-based algorithm existed in the chemometrics toolbox of MATLAB software (version 7.1 Math Work Inc.). In order to obtain the optimum number of factors based on the Haaland and Thomas F-ratio criterion, leave-one-out cross-validation procedure was used [34].

In our previous study the classical approach of multiple regression technique was used for developing QSAR relation [25]. Here, FA-MLR was also performed on the dataset. Factor analysis (FA) was used to reduce the number of variables and to detect structure in the relationships between them. This data-processing step is applied to identify the important predictor variables and to avoid collinearities among them [35]. Principle component regression analysis, PCRA, was also tried for the dataset along with FA-MLR. With PCRA collinearities among X variables are not a disturbing factor and the number of variables included in the analysis may exceed the number of observations [36]. In this method, factor scores, as obtained from FA, are used as the predictor variables [35]. In PCRA, all descriptors are assumed to be important while the aim of factor analysis is to identify relevant descriptors.

3. Results and Discussion

3.1. GA-PLS

In PLS analysis, the descriptors data matrix is decomposed to orthogonal matrices with an inner relationship between the dependent and independent variables. Therefore, unlike MLR analysis, the multicolinearity problem in the descriptors is omitted by PLS analysis. Because a minimal number of latent variables are used for modeling in PLS; this modeling method coincides with noisy data better than MLR. In order to find the more convenient set of descriptors in PLS modeling, genetic algorithm was used. To do so, many different GA-PLS runs were conducted using different initial set of populations. The data set (compounds tested against S. aureus, n = 31) was divided into two groups: calibration set (n = 25) and prediction set (n = 6). Given 25 calibration samples; the leave-one-out cross-validation procedure was used to find the optimum number of latent variables for each PLS model. The most convenient GA-PLS model that resulted in the best fitness contained 17 indices, 5 of them being those obtained by MLR. The PLS estimate of coefficients for these descriptors are given in Figure 1.

As it is observed, a combination of quantum, topological, geometrical, constitutional, and functional group descriptors have been selected by GA-PLS to account the antimicrobial activity of the studied compounds. The majority of these descriptors are topological indices. The resulted GA-PLS model possessed very high statistical quality R² = 0.96 and Q² = 0.91. The values of pMIC using PLS model (refined from cross-validation or external prediction set) along with the corresponding relative errors of prediction (REP) are shown in

Table 3. Experimental and predicted activity of compounds against Staphylococcus aureus.

Compound	Experimental pMICa	Predicted pMIC	REP b (%)
1	3.29	3.3205	0.9173
2	3.29	3.3007	0.3242
3	3.29	3.2266	−1.9664
4*	3.29	3.3976	3.1675
5	4.19	3.7498	−11.740
6	3.29	3.3205	0.9173
7	3.89	3.8255	−1.6850
8	3.29	3.2698	−0.6172
9	3.29	3.2886	−0.0440
10*	3.89	3.9283	0.9738
11	3.59	3.6207	0.8470
12	3.59	3.7254	3.6340
13	3.59	3.5063	−2.3883
14	3.59	3.6212	0.8627
15*	4.19	4.1563	−0.8119
16	3.59	3.5611	−0.8123
17	3.59	3.6177	0.7647
18	3.59	3.5548	−0.9915
19*	3.89	3.8950	0.1293
20	4.19	4.0995	−2.2079
21	3.59	3.7117	3.2787
22	5.10	5.0840	−0.3141
23	3.59	3.5533	−1.0318
24*	3.59	3.7223	3.5534
25	3.89	3.9222	0.8214
26	3.89	3.9779	2.2092
27	4.80	4.8022	0.0453
28	3.89	3.8591	−0.8011
29	3.59	3.4907	−2.8470
30*	4.49	4.5105	0.4549
31	3.59	3.4728	−3.3746

^a pMIC= −log (MIC),

^b REP = Relative Error Prediction

*Compounds used as prediction set

. Very small values of relative errors confirm the accuracy of the proposed GA-PLS model for modeling antimicrobial activity of the studied compounds.

The data set (compounds tested against C. albicans, n = 28) was again divided into two groups: calibration set (n = 23) and prediction set (n = 5). Given 23 calibration samples; the leave-one-out cross-validation procedure was used to find the optimum number of latent variables for each PLS model. Here, the most convenient GA-PLS model contained 15 indices, five of them being those obtained by MLR. The PLS estimate of coefficients for these descriptors are given in Figure 2. As it is observed, a combination of quantum, topological, geometrical and functional group descriptors have been selected by GA-PLS to account the antimicrobial activity of the compounds. The majority of these descriptors are topological indices again. The resulted GA-PLS model possessed very high statistical quality R² = 0.91 and Q² = 0.87. The values of pMIC using PLS model along with the corresponding REPs are shown in

Table 4. Experimental and predicted activity of compounds against Candida albicans.

Compd.	Experimental pMIC	Predicted pMIC	REP(%)
2	3.29	3.4139	3.6304
4*	3.29	3.3893	2.9303
5	3.89	3.8920	0.0514
6	3.29	3.3591	2.0577
7	3.29	3.3835	2.7631
8	3.59	3.6477	1.5813
9	3.29	3.3208	0.9272
10*	3.59	3.6196	0.8175
11	3.89	3.9567	1.6857
12	3.89	3.7481	−3.7870
13	3.89	3.9092	0.4922
14	3.89	3.7076	−4.9191
15	3.89	3.8892	−0.0203
16	3.89	3.8422	−1.2433
17*	4.49	4.3961	−2.1360
18	4.49	4.4476	−0.9524
19	3.89	3.7076	−4.9191
20	3.89	3.8014	−2.3296
21	3.89	3.9525	1.5813
23	3.89	3.7450	−3.8727
24*	3.89	3.9056	0.3994
25	3.89	3.9969	2.6755
26	3.89	3.8489	−1.0691
27	3.89	3.7573	−3.5304
28	3.89	3.9503	1.5262
29*	3.89	3.9964	2.6619
30	3.89	3.8978	0.2006
31	3.89	3.8732	−0.4333

*Compounds used as prediction set

. Very small values of relative errors confirm the accuracy of the proposed GA-PLS model for modeling antimicrobial activity of the studied compounds.

3.2. FA-MLR and PCRA

Table 5. Numerical values of factor loading numbers 1–4 for some descriptors after VARIMAX rotation (against S. aureus).

	1	2	3	4	Commonality
MPC	0.588	−0.105	0.587	−0.313	0.799
DMy	0.195	−0.054	0.762	0.071	0.627
HOMO	0.059	0.637	−0.013	0.620	0.794
Electonegativity	−0.643	−0.206	−0.199	−0.496	0.741
Mv	0.751	−0.413	0.362	−0.259	0.934
Me	0.001	−0.781	0.097	−0.298	0.708
RBN	0.087	0.902	0.068	0.003	0.826
HNar	0.866	0.051	0.217	−0.252	0.863
Xt	−0.645	−0.505	−0.307	0.081	0.772
IDDE	0.746	0.359	0.215	0.324	0.837
LP1	0.667	0.460	0.368	0.292	0.877
TIC1	0.714	0.413	0.175	0.127	0.726
PJI3	0.375	0.611	−0.315	−0.276	0.689
nCS	−0.559	0.578	−0.411	0.199	0.855
nCaH	0.894	−0.140	−0.143	−0.079	0.845
nCONHR	0.261	0.220	0.695	−0.906	0.765
nCO	−0.082	0.081	−0.214	0.853	0.787
pMIC S. aureus	0.041	−0.116	0.898	−0.051	0.824
%variance	29.87	20.10	17.15	12.12	79.24

shows the five factor loadings of the variables (after VARIMAX rotation) for the compounds tested against S. aureus. As it is observed, about 79% of variances in the original data matrix could be explained by selected four factors.

Based on the procedure explained in the experimental section, the following three-parametric equation was derived.

$$\begin{array}{c}\text{pMIC}=4.786 (\pm 0.484)+ 0.196(\pm 0.063) \text{DMy}+0.1666 (\pm 0.063) \text{nCONHR}-0.130(\pm 0.058) \text{PJ13}\\ {\text{R}}^{\text{2}}=0.73 \text{S.}\text{E.}=0.31 \text{F}=11.41 {\text{Q}}^{\text{2}}=0.68 \text{RMScv}=0.34 \text{N}=31\end{array}$$

(1)

Equation 1 could explain 73% of the variance and predict 68% of the variance in pMIC data. This equation describes the effect of geometrical (PJI3), functional group (nCONHR) and quantum (DMy) indices on antimicrobial activity.

When factor scores were used as the predictor parameters in a multiple regression equation using forward selection method (PCRA), the following equation was obtained:

$$\begin{array}{c}\text{pMIC}=3.756 (\pm 0.036)+ 0.4000 (\pm 0.036) f_3\\ {\text{R}}^{\text{2}}=0.81 \text{S.}\text{E.}=0.19 \text{F}=35.05 {\text{Q}}^{\text{2}}=0.79 \text{RMScv}=0.20 \text{N}=31\end{array}$$

(2)

Equation 2 also shows high equation statistics (81% explained variance and 79% predict variance in pMIC data). Since factor scores are used instead of selected descriptors, and any factor-score contains information from different descriptors, loss of information is thus avoided and the quality of PCRA equation is better than those derived from FA-MLR.

As it is observed from Table 5, in the case of each factor, the loading values for some descriptors are much higher than those of the others. These high values for each factor indicate that this factor contains higher information about which descriptors. It should be noted that all factors have information from all descriptors but the contribution of descriptor in different factors are not equal. For example, factors 1 and 2 have higher loadings for topological, constitutional and functional group indices, whereas information about quantum and functional group descriptors is highly incorporated in factors 3 and 4. Therefore, from the factor scores used by equation E₂, significance of the original variables for modeling the activity can be obtained. Factor score 1 indicates importance of Mv, HNar, nCaH and IDDE (topological, constitutional and functional group descriptors, respectively). Factor score 2 indicates importance of RBN and Me (constitutional descriptors), Factor score 3 and 4 signify the importance of DMy, and nCONHR (quantum and functional group descriptors, respectively).

Table 6. Numerical values of factor loading numbers 1–5 for some descriptors after VARIMAX rotation (against C. albicans).

	1	2	3	4	5	Commonality
Std	−0.491	−0.431	−0.459	−0.107	0.095	0.657
DMz	−0.007	0.102	−0.209	0.860	0.322	0.898
HOMO	0.240	0.811	−0.156	−0.349	0.014	0.861
Electonegativity	−0.706	−0.389	0.142	0.323	−0.310	0.871
X5A	−0.627	−0.664	−0.134	−0.102	0.129	0.879
PW3	−0.166	0.594	−0.377	−0.158	0.893	0.584
PW5	0.913	−0.079	0.055	0.135	−0.132	0.879
IC2	0.579	0.272	−0.164	0.210	0.584	0.820
piID	0.750	−0.070	−0.333	−0.190	−0.208	0.758
ASP	−0.075	0.087	0.866	−0.198	0.322	0.905
L/Bw	0.064	0.117	0.926	−0.023	0.164	0.902
nCp	−0.206	0.754	−0.224	−0.097	−0.325	0.777
nNR2	−0.366	0.722	0.148	0.287	−0.234	0.814
nOHPh	−0.191	−0.415	−0.165	−0.447	0.356	0.562
nRORPh	0.571	−0.522	0.379	0.002	−0.341	0.858
pMIC C. albicans	0.628	−0.627	−0.277	−0.107	0.602	0.872
%variance	22.58	20.58	14.71	14.02	8.71	80.60

shows the five factor loadings of the variables (after VARIMAX rotation) for the compounds tested against C. albicans. As it is observed, about 80% of variances in the original data matrix could be explained by selected five factors.

Based on the procedure explained in the experimental section, the following four-parametric equation was derived.

$$\begin{array}{c}\text{pMIC}=5.980 (\pm 0.695)+0.182 (\pm 0.022) \text{piID}-0.167(\pm 0.024) \text{nCp}-0.085(\pm 0.023) \text{ASP}\\ -0.058(\pm 0.023) \text{PW3}\\ {\text{R}}^{\text{2}}=0.81 \text{S.}\text{E.}=0.17 \text{F}=34.76 {\text{Q}}^{\text{2}}=0.79 \text{RMScv}=0.18 \text{N}=28\end{array}$$

(3)

Equation 3 could explain and predict 85% and 81% of the variance in pMIC data, respectively. This equation describes the effect of topological (piID and PW3), functional group (nCp) and geometrical (ASP) indices on the antimicrobial activity.

When factor scores were used as the predictor parameters in a multiple regression equation using forward selection method (PCRA), the following equation was obtained:

$$\begin{array}{c}\text{pMIC}=3.806(\pm 0.023)+0.237(\pm 0.024) f_1-0.114((\pm 0.024) f_3+0.081(\pm 0.024) f_2-0.065(\pm 0.024) f_4\\ {\text{R}}^{\text{2}}=0.83 \text{S.E.}=0.12 \text{F}=38.05 {\text{Q}}^{\text{2}}=0.81 \text{RMScv}=0.12 \text{N}=28\end{array}$$

(4)

Equation 4 shows also high equation statistics (88% explained variance and 83% predicted variance in pMIC data). It should be noted that the variables (factor scores) used in Equation 4 are perfectly orthogonal to each other. Since factor scores are used instead of selected descriptors, and any factor-score contains information from different descriptors, loss of information is thus avoided and the quality of PCRA equation is better than those derived from FA-MLR.

As it is observed from Table 6, in the case of each factor, the loading values for some descriptors are much higher than those of the others. Factors 1 and 2 have higher loadings for topological, quantum and functional group indices, whereas information about geometrical, quantum and topological descriptors is highly incorporated in factors 3, 4 and 5. Therefore, from the factor scores used by equation E₄, significance of the original variables for modeling the activity can be obtained. Factor score 1 indicates importance of PW5, piID and electronegativity (topological and quantum descriptors). Factor score 2 indicates importance of HOMO nCp and nNR₂ (quantum and functional group descriptor). Factor score 3 signifies the importance of ASP and L/Bw (geometrical descriptors) and factor score 4 and 5 signify the importance of quantum and topological descriptors (DMz and PW3).

Comparison between the results obtained by GA-PLS and the other employed regression methods indicates higher accuracy of this method in describing antimicrobial activity of the studied compounds.

Difference in accuracy of the different regression methods used in this study is visualized in Figures 3 and 4 by plotting the predicted activity (by cross-validation) against the experimental values. Obviously, all linear models represented scattering of data around a straight line with slope and intercept close to one and zero, respectively. As it is observed, the plot of data resulted by GA-PLS represents the lowest scattering and that obtained by FA-MLR and PCR analysis have lower accuracy. It should be mentioned that the model which GA-PLS method provides is better than that MLR analysis provided in our previous study [25]. In fact, MLR analysis could explain and predict 55% and 35% of variances in the pMIC data (compounds tested against S. aureus) and predict 82% and 73% of variances in the pMIC data (compounds tested against C. albicans).

4. Conclusions

Quantitative relationships between molecular structure and inhibitory activity of a series of 3-hydroxypyridine-4-one and 3-hydroxypyran-4-one derivatives were discovered by a collection of chemometrics methods including GA-PLS, FA-MLR and PCRA. The results revealed the significant role of topological parameters in the antimicrobial activity of the studied compounds against S. aureus and C. albicans. A comparison between the different statistical methods employed indicated that GA-PLS represented superior results and it could explain and predict 96% and 91% of variances in the pMIC data (compounds tested against S. aureus) and predict 91% and 87% of variances in the pMIC data (compounds tested against C. albicans). As it is observed, the plot of data resulted by GA-PLS represents the lowest scattering, and the impact of topological descriptors was the most.