Valence Topological Charge-Transfer Indices for Dipole Moments

Abstract

New valence topological charge-transfer indices are applied to the calculation of dipole moments. The algebraic and vector semisum charge-transfer indices are defined. The combination of the charge-transfer indices allows the estimation of the dipole moments. The model is generalized for molecules with heteroatoms. The ability of the indices for the description of the molecular charge distribution is established by comparing them with the dipole moments of a homologous series of phenyl alcohols. Linear and non-linear correlation models are obtained. The new charge-transfer indices improve the multivariable non-linear regression equations for the dipole moment. When comparing with previous results, the variance decreases 92%. No superposition of the corresponding G_k–J_k and – pairs is observed. This diminishes the risk of co-linearity. Inclusion of the oxygen atom in the π-electron system is beneficial for the description of the dipole moment, owing to either the role of the additional p orbitals provided by the heteroatom or the role of steric factors in the π-electron conjugation. Linear and non-linear correlations between the fractal dimension and various descriptors point not only to a homogeneous molecular structure but also to the ability to predict and tailor drug properties.

Introduction

The rationale for transdermal drug delivery needs to be carefully identified [1]. There has been great interest in developing systems for controlled delivery of drugs and other bioactive substances [2]. One technique is to attach a specialized patch on the skin with uncoated polymer matrices containing the embedded drug [3]. Other technique involves covering most of the matrix with an impermeable material [4]. The pathways that exist in a porous membrane used to deliver drugs over a continuous period form a percolating path with usual fractal structures [5,6].

Percutaneous penetration depends essentially on the lipophilicity of the substances tested [7]. The use of homologous series of compounds made it possible to establish behaviour models that allow predicting the percutaneous absorption capacity of chemically related substances [8,9]. Much of the theoretical background information on percutaneous absorption was developed from studies of nonelectrolytic permeating species [10,11]. However, the vehicle’s pH may have a profound influence upon the percutaneous delivery from topical products [12]. Depending on the pK_a of the compound and on the pH of the vehicle, an equilibrium mixture of ionized and unionized species would be present in the immediate vicinity of the skin [13]. The permeation of ionized drugs through the skin must be taken into account [14]. Penetration enhancers may act by either interacting with the highly ordered lipid structure or modifying the partitioning of the drug into the tissue [15,16].

Different phenyl alcohols are included in many pharmaceutical and cosmetic products, e.g., bacteriostatic agents and essential oils. Benzyl alcohol, which is used as a bacteriostatic agent and also as a solvent, was evaluated as a percutaneous enhancer [17]. López et al. assessed other phenyl alcohols as enhancers for 5-fluorouracil [8], which was used in in vitro diffusion experiments [18]. Other authors employed dipalmitoylphosphatidylcholine as enhancer for azone [19].

In previous papers, molecular topological charge-transfer indices were applied to the calculation of the molecular dipole moment of hydrocarbons [20]. New valence charge-transfer indices were introduced to take into account the presence of heteroatoms in the molecules and applied to valence-isoelectronic series of benzene and styrene [21]. In this paper, the method has been applied to the calculation of the dipole moment of a homologous series of phenyl alcohols. In the next section, the topological charge-transfer indices are revised. Next, the valence charge-transfer indices for heteroatoms are presented. Following that, the calculation results are discussed. The last section summarizes my conclusions.

Topological charge-transfer indices

The two most important matrices that delineate the labelled chemical graph are the adjacency () [22,23,24] and the distance () matrices, wherein D_ij = I_ij if i=j, “0” otherwise; l_ij is the shortest edge count between vertices i and j [25]. In , A_ij = 1 if vertices i and j are adjacent, “0” otherwise. The matrix is the matrix whose elements are the squares of the reciprocal distances [26]. Now, the intermediate matrix is defined as the matrix product of by :

The charge-transfer matrix is defined as , where is the transpose of [27]. By agreement, C_ii = M_ii. For i ≠ j, the C_ij terms represent a measure of the intramolecular net charge transferred from atom j to i.

Gálvez et al. defined the topological charge-transfer indices G_k as the sum, in absolute value, of the C_ij terms defined for the vertices i,j placed at a topological distance D_ij equal to k:

(1)

where N is the number of vertices in the graph, D_ij are the entries of the matrix, and δ is the Kronecker δ, being δ = 1 for i = j and δ = 0 for i ≠ j. G_k represents the sum of all the C_ij terms, for every pair of vertices i and j at topological distance k. Gálvez et al. also introduced other topological charge-transfer index, J_k, as

(2)

This index represents the mean value of the charge transfer for each edge, since the number of edges for acyclic compounds is N-1. The G_k and J_k indices were applied to the calculation of the molecular dipole moment of hydrocarbons, the boiling temperature of hydrocarbons and alcohols, the vaporization enthalpy of alkanes, the activity of drugs [28,29,30], the capacity and separation factors in chromatographic chiral separations of biomolecules [31], the resonance energy of hydrocarbons, the log IC₅₀ of the D₂ dopamine receptor and the σ receptor of piperidines and the sedative character of chiral barbiturates [32].

The algebraic semisum charge-transfer index μ_alg is defined as

(3)

where C^e is the C_ij index for vertices i and j connected by edge e. In the first equation, the sum extends for all pairs of adjacent vertices in the molecular graph. In the second expression, the sum extends for the m edges in the molecular graph. Therefore, μ_alg is intended as a measure of the molecular dipole moment calculated by algebraic semisum of C^e.

An edge-to-edge analysis of μ suggests that each edge dipole moment μ^e connecting vertices i and j can be evaluated from the corresponding edge C^e index as

Each edge dipole can be associated with a vector in space. This vector has magnitude μ^e, lies in the edge e connecting vertices i and j, and its direction is from j to i. The molecular dipole moment vector results the vector sum of the edge dipole moments as

summed for all the m edges in the molecular graph. The vector semisum charge-transfer index μ_vec is defined as the norm of :

(4)

Therefore, μ_vec is intended as a measure of the molecular dipole moment calculated by vector semisum of edge .

As μ_vec is associated with a vector in space, the three-dimensional structure of the molecule is needed. The method followed to get it is outlined as follows.

• 1.Read the Cartesian coordinates of the atoms.
• 2.Determine which atoms are bonded to which other atoms. The distance between two atoms is calculated and, if it is less than a certain value, a bond is assumed to exist between the two atoms.
• 3.Build the charge transfer terms C_ij.
• 4.Calculate the vector semisum dipole moment μ_vec.

Therefore, μ_vec is not a pure topological index. It is rather a linear combination of topological indices, but the coefficients are geometric descriptors.

Valence charge-transfer indices for heteroatoms

In valence charge-transfer indices terms, the presence of each heteroatom is taken into account by introducing its electronegativity value in the corresponding entry of the main diagonal of the adjacency matrix . For each heteroatom X, its entry A_ii is redefined as

(5)

to give a modified matrix, where x_x and x_c are the electronegativities of heteroatom X and carbon, respectively, in Pauling units. Notice that the subtractive term keeps for the carbon atom (Equation 5). Moreover, the multiplicative factor reproduces for oxygen, which was taken as standard in previous works. From the modified matrix, the valence and matrices, , and topological charge-transfer indices and can be calculated by following the former procedure with the matrix. The , and descriptors are graph invariants. The main difference between μ_vec and is that μ_vec is sensitive only to the steric effect of the heteroatoms while is sensitive to both electronic and steric effects.

Calculation results and discussion

The molecular charge-transfer indices G_k, J_k, and (with k < 6) for a series of 13 phenyl alcohols (8 form a homologous series and 5 are congeneric) are reported in Table 1. In the homologous series, G₁ and G₂ are sensitive to the presence of the alkyl chain. G₃, G₄ and G₅ indicate the presence of at least 2, 3 and 4 carbon atoms in the alkyl chain, respectively. is influenced by the presence of the alkyl chain. – point out the presence of at least 2–5 carbon atoms in the alkyl chain.

Table 1.

Values of the G_k and J_k charge-transfer indices up to the fifth order for a homologous series of phenyl alcohols.

Molecule	N	G1		G2		G3		G4		G5
Phenol	7	2.0000		0.8889		0.3750		0.2222		0.0000
benzyl alcohol	8	1.2500		6.7778		0.8125		0.4133		0.1250
2-phenyl-1-ethanol	9	1.2500		6.7778		0.8750		0.5644		0.2431
3-phenyl-1-propanol	10	1.2500		6.7778		0.8750		0.6044		0.3333
4-phenyl-1-butanol	11	1.2500		6.7778		0.8750		0.6044		0.3611
5-phenyl-1-pentanol	12	1.2500		6.7778		0.8750		0.6044		0.3611
6-phenyl-1-hexanol	13	1.2500		6.7778		0.8750		0.6044		0.3611
7-phenyl-1-heptanol	14	1.2500		6.7778		0.8750		0.6044		0.3611
1-phenyl-2-propanol	10	2.2500		6.7778		0.9375		0.7156		0.3611
2-phenyl-2-propanol	10	2.7500		7.2222		1.5625		0.7956		0.3750
3-phenyl-2-propen-1-ol	10	1.2500		8.2222		0.8750		0.4844		0.2708
1-phenyl-1-pentanol	12	1.7500		7.1111		1.3125		0.8756		0.4861
1-phenyl-2-pentanol	12	2.2500		7.0000		1.0625		0.7556		0.4792
Molecule	J1		J2		J3		J4		J5
phenol	0.3333		0.1481		0.0625		0.0370		0.0000
benzyl alcohol	0.1786		0.9683		0.1161		0.0590		0.0179
2-phenyl-1-ethanol	0.1563		0.8472		0.1094		0.0706		0.0304
3-phenyl-1-propanol	0.1389		0.7531		0.0972		0.0672		0.0370
4-phenyl-1-butanol	0.1250		0.6778		0.0875		0.0604		0.0361
5-phenyl-1-pentanol	0.1136		0.6162		0.0795		0.0549		0.0328
6-phenyl-1-hexanol	0.1042		0.5648		0.0729		0.0504		0.0301
7-phenyl-1-heptanol	0.0962		0.5214		0.0673		0.0465		0.0278
1-phenyl-2-propanol	0.2500		0.7531		0.1042		0.0795		0.0401
2-phenyl-2-propanol	0.3056		0.8025		0.1736		0.0884		0.0417
3-phenyl-2-propen-1-ol	0.1389		0.9136		0.0972		0.0538		0.0301
1-phenyl-1-pentanol	0.1591		0.6465		0.1193		0.0796		0.0442
Molecule
phenol	2.2000		1.1000		0.3639		0.0847		0.0000
benzyl alcohol	2.9500		6.6611		0.5625		0.2533		0.0370
2-phenyl-1-ethanol	2.9500		7.1056		0.7444		0.4044		0.1319
3-phenyl-1-propanol	2.9500		7.1056		0.9944		0.5019		0.2222
4-phenyl-1-butanol	2.9500		7.1056		0.9944		0.6619		0.2824
5-phenyl-1-pentanol	2.9500		7.1056		0.9944		0.6619		0.3936
6-phenyl-1-hexanol	2.9500		7.1056		0.9944		0.6619		0.3936
7-phenyl-1-heptanol	2.9500		7.1056		0.9944		0.6619		0.3936
1-phenyl-2-propanol	3.4500		7.6556		0.8069		0.5556		0.2500
2-phenyl-2-propanol	3.4500		8.2056		1.3125		0.6356		0.2870
3-phenyl-2-propen-1-ol	2.9500		8.3278		0.6306		0.3819		0.1597
1-phenyl-1-pentanol	2.9500		7.3222		1.1819		0.7731		0.4861
Molecule
phenol	0.3637		0.1833		0.0606		0.0141		0.0000
benzyl alcohol	0.4214		0.9516		0.0804		0.0362		0.0053
2-phenyl-1-ethanol	0.3688		0.8882		0.0931		0.0506		0.0165
3-phenyl-1-propanol	0.3278		0.7895		0.1105		0.0558		0.0247
4-phenyl-1-butanol	0.2950		0.7106		0.0994		0.0662		0.0282
5-phenyl-1-pentanol	0.2682		0.6460		0.0904		0.0602		0.0358
6-phenyl-1-hexanol	0.2458		0.5921		0.0829		0.0552		0.0328
7-phenyl-1-heptanol	0.2269		0.5466		0.0765		0.0509		0.0303
1-phenyl-2-propanol	0.3833		0.8506		0.0897		0.0617		0.0278
2-phenyl-2-propanol	0.3833		0.9117		0.1458		0.0706		0.0319
3-phenyl-2-propen-1-ol	0.3278		0.9253		0.0701		0.0424		0.0177
1-phenyl-1-pentanol	0.2682		0.6657		0.1074		0.0703		0.0442
1-phenyl-2-pentanol	0.3136		0.6960		0.0956		0.0666		0.0335

The molecular dipole moments μ (experimental and calculated as vector semisums of C_ij or ) are listed in Table 2. As experimental values are not available for all the series, some reference values are calculated with program MOPAC-AM1. The reliability of the results has been tested with the first four entries in Table 2, for which experimental data are available. AM1 calculations adequately reproduce the oscillatory behaviour of the experimental data, mimicking two minima for phenol and 2-phenyl-1-ethanol and two maxima for benzyl alcohol and 3-phenyl-1-propanol. This test suggests that AM1 allows a good approximation, at least for the general performance of the homologous series as a whole, and that the error is sufficiently constant throughout the homologous series. The number of compounds in the homologous series has not been increased because longer phenyl alcohols are not percutaneous enhancers owing to their lower transdermal penetration. In particular, 3-phenyl-2-propen-1-ol has been chosen in order to compare the influence of a double bound in the alkyl chain region of a phenyl alcohol. The experimental μ decreases ca. 3% from that of 3-phenyl-1-propanol due to the presence of the double bond. The presence of an enol group (conjugated double bond in β position with respect to the -OH group) lends 3-phenyl-2-propen-1-ol to particular structural characteristics (greater rigidity than for 3-phenyl-1-propanol) with a lower μ. For the three phenyl propanols, μ_experiment decreases 5% from 3-phenyl-1-propanol (primary alcohol) to 1-phenyl-2-propanol (secondary alcohol) and 11% to 2-phenyl-2-propanol (tertiary alcohol). For the three phenyl pentanols, μ_experiment increases 7% from 5-phenyl-1-pentanol (primary alcohol) to 1-phenyl-1-pentanol (secondary alcohol with both –phenyl and –OH groups in terminal position) and decreases 8% to 1-phenyl-2-pentanol (secondary alcohol).

Table 2.

Molecular dipole moment values, μ (D), for a series of phenyl alcohols with charge-transfer indices.

Molecule	Number of carbon atoms in alkyl chain	Vector semisum	Valence vector semisum	Experimenta
phenol	0	0.737	2.431	1.400 (1.233b)
benzyl alcohol	1	0.589	2.487	1.700 (1.568b)
2-phenyl-1-ethanol	2	0.700	2.257	1.590 (1.497b)
3-phenyl-1-propanol	3	0.573	2.519	1.640 (1.597b)
4-phenyl-1-butanol	4	0.702	2.249	1.345b
5-phenyl-1-pentanol	5	0.573	2.519	1.626b
6-phenyl-1-hexanol	6	0.702	2.250	1.346b
7-phenyl-1-heptanol	7	0.573	2.518	1.634b
1-phenyl-2-propanol	3	0.923	2.347	1.564b
2-phenyl-2-propanol	3	0.780	2.821	1.463b
3-phenyl-2-propen-1-ol	3	0.561	2.495	1.591b
1-phenyl-1-pentanol	5	0.426	2.794	1.746b
1-phenyl-2-pentanol	5	0.895	2.367	1.496b

^a Taken from Reference 60.

^b Calculations carried out with program MOPAC-AM1.

Figure 1 illustrates how the dipole moment of the homologous phenyl alcohols fluctuates with the number of C atoms in the alkyl chain, n. Experimental μ and calculated vary in an alternate fashion: for odd n, μ is greater than for even n. However, μ_vec presents the opposite tendency. As μ_vec is not sensitive to the electronic effect of the oxygen atom, it is clear that the steric (μ_vec) and electronic () factors are antagonistic, and that the electronic effect dominates over the steric one. The μ_experiment decreases ca. 4% either from to 6 or from to 7.

Figure 1.

Dipole moment of homologous phenyl alcohols as a function of the number of C atoms in alkyl chain.

The experimental dipole moments and charge-transfer indices have been correlated for the homologous series. The best linear regression results:

(6)

The mean absolute percentage error (MAPE) is 4.53% and the approximation error variance (AEV) is 0.5087. All other models with greater MAPE and AEV have been discarded. The presence of charge-transfer (G₃) and valence () indices is representative of charge-transfer processes and clearly conditions the polar character of the correlated compounds. The best non-linear model for μ does not improve the results.

If the congeneric set is included in the model of phenyl alcohols, the best linear regression for all μ results:

and AEV increases 21%. The best non-linear model for μ does not improve the results.

If all μ are calculated with AM1, the best linear regression results:

and AEV augments 12%. The best non-linear model for μ does not improve the results.

If μ_vec is included in the model of phenyl alcohols, the best linear regression for μ results:

(7)

and AEV decreases 72%. Notice that the negative sign of the μ_vec coefficient is due to the opposite trend of μ_vec and μ_experiment. The best non-linear model for μ results:

(8)

and AEV diminishes 86%.

If is included in the model, the best linear regression for μ results:

(9)

and AEV drops 73%. The non-linear model with results:

(10)

and AEV decreases 92%. Due to the complexity of Equations 8 and 10, a Fortran program has been written to calculate μ. The program is available from the author at Internet (Francisco.Torrens@uv.es).

No superposition of the corresponding G_k–J_k and – pairs is observed in Equations 6–10. This diminishes the risk of co-linearity in the fit, given the close relationship between each pair G_k, J_k in Equation 2 [33].

The correlation coefficient found between cross-validated representatives and the property values R_cv has been calculated with the leave-n-out procedure for Equations 6–10 [34]. The procedure furnishes a new method for selecting the best set of descriptors according to the criterion of maximization of the value of R_cv. The R_cv calculations for a homologous series of phenyl alcohols are given in Table 3 for 1 ≤ n ≤ 5. In general, R_cv decreases with n. In particular, Equations 9-10 give greater R_cv than Equations 7-8 (2%) and than Equation 6 (86%) for the whole range of n given in Table 3, although the results for Equations 7 and 9 are rather similar. The corresponding interpretation is that Equations 9-10 are more predictive than Equations 7-8 or 6.

Table 3.

Cross-validation correlation coefficient in a leave-n-out procedure for a homologous series of phenyl alcohols.

n	Equation 6	Equation 7	Equation 8	Equation 9	Equation 10
1	0.477	0.834	0.915	0.823	0.956
2	–	0.827	0.916	0.821	0.954
3	–	0.819	0.916	0.821	0.951
4	–	0.806	0.915	0.820	0.935
5	–	–	0.890	0.823	-0.951

In the rest of this section, new parameters are calculated to illustrate the importance of linear and non-linear models in pharmacology. Another intramolecular property related with charge-transfer is the quadrupole moment tensor Θ, defined by

where is the i-th element of charge at the point relative to an origin fixed at the centre of mass in the molecule [35]. The δab stands for the Kronecker δ. The subscripts a, b… denote vector or tensor components and can be equal to the Cartesian components x, y, z. Tensor Θ is calculated with our program POLAR [36,37]. The fractal dimension D of the solvent-accessible surface A_s of the phenyl alcohols may then be obtained according to Lewis and Rees as

where R is the radius of a probe representing the solvent [38]. It is calculated with our program TOPO. The fractal dimension D provides a quantitative indication of the degree of surface accessibility toward different solvents [39,40,41,42,43,44,45,46]. In previous papers, the fractal dimension of transdermal-delivery drug models (phenyl alcohols [47] and 4-alkylanilines [48]) was calculated. A linear model for the molecular quadrupole moment of the homologous phenyl alcohols vs. fractal dimension is shown in Figure 2.

Figure 2.

Quadrupole moment of homologous phenyl alcohols vs. the fractal dimension.

The 1-octanol–water partition coefficient of a substance soluble in 1-octanol and water is defined as

where a_1-octanol and a_water are the activities of the solute in 1-octanol and water, respectively [49]. P is calculated with the equation:

at a given T, which is taken as 298K. R is the gas constant, and and (in kJ·mol^-1) are the standard-state free energies of solvation of a given solute considered in 1-octanol and water, respectively. The solvation energies are calculated with our program SCAP [50,51,52,53,54,55]. A non-linear model for of the homologous phenyl alcohols vs. fractal dimension is illustrated in Figure 3.

Figure 3.

1-Octanol–water partition coefficient of homologous phenyl alcohols vs. the fractal dimension.

The molar concentration necessary to produce a 1:1 complex with bovine serum albumin (BSA) via equilibrium dialysis, C, provides information on the binding and conformational perturbation of macromolecules by smaller compounds [56]. It is calculated with SCAP. A non-linear model of for the homologous phenyl alcohols vs. fractal dimension is illustrated in Figure 4.

Figure 4.

Molar concentration of homologous phenyl alcohols necessary to produce a 1:1 complex with BSA via equilibrium dialysis vs. the fractal dimension.

The hydrophile–lipophile balance (HLB) is a scale that characterizes surfactants in emulsions as solubilizers, emulsifiers, wetting agents or antifoaming agents [57,58]. It is calculated with SCAP. A non-linear model of HLB for the homologous phenyl alcohols vs. fractal dimension is shown in Figure 5.

Figure 5.

Hydrophile–lipophile balance of homologous phenyl alcohols vs. the fractal dimension.

The best linear model for the fractal dimension of the phenyl alcohols vs. a series of physicochemical parameters results:

where M_W is the molecular weight, G is the molecular globularity and G’ is the molecular rugosity. Index G is calculated as G = S_e/S, where S_e is the surface area of a sphere whose volume is equal to the molecular volume V, and S is the molecular surface area. Index G’ is calculated as G’ = S/V. All the descriptors are calculated with TOPO. Notice that Equation 15 contains five parameters versus only eight data, allowing three degrees of freedom. The statistical significance of the fit is that the error introduced by fitting is much lower than the error due to the individual points and can be disregarded. Therefore, the data points can be replaced by the fit without adding an appreciable error. The very question is the predictive power of the fit for points out of the fitting set [59]. However, it is not the case because longer phenyl alcohols in the aromatic series are not percutaneous enhancers owing to their lower transdermal penetration. On the other hand, the best non-linear model for the fractal dimension does not improve the results.

Conclusions

From the preceding results the following conclusions can be drawn:

• 1Inclusion of the oxygen atom in the π-electron system is beneficial for the description of the dipole moment, owing to either the role of the additional p orbitals provided by the heteroatom or the role of steric factors in the π-electron conjugation. The analysis of both electronic and steric factors in μ caused by the presence of the oxygen atom shows that the two factors are antagonistic, and that the electronic factor dominates over the steric one.
• 2In 3-phenyl-2-propen-1-ol, the double bond in the side chain lends to a more rigid structure with a lower dipole moment.
• 3Linear and non-linear correlation models have been obtained for the molecular dipole moments of phenyl alcohols. The new μ_vec and charge-transfer indices have improved the multivariable regression equations for μ, diminishing the risk of co-linearity in the fit.
• 4The linear correlation between D and Θ, and various non-linear correlations between D, log P, log 1/C and HLB point not only to a homogeneous molecular structure of the phenyl alcohols but also to the ability to predict and tailor drug properties. The latter is nontrivial in pharmacology.

Work is in progress on the calculation of the dipole moments of a homologous series of 4-alkylanilines, which are also percutaneous enhancers.