An Efficient Method for Calculating Atomic Charges of Peptides and Proteins from Electronic Populations

Abstract

An efficient method is presented to calculate atomic charges of peptides and proteins derived from Mulliken electronic populations for terminally blocked amino acids (Ac–X–NHMe, X = any neutral or charged residue) calculated at the B3LYP/6-31G(d,p)//HF/6-31G(d,p) level of theory. This electronic population-derived atomic charges (EPAC) method is based on the geometry-dependent atomic charge (GDAC) method proposed by Cho et al., in which atomic charges are calculated by using the partial equalization of atomic electronegativities with electronegativity parameters and damping factors given by interatomic distances between covalently bonded atoms in a molecule. The overall mean absolute difference (mad) and root-mean-square deviation (rmsd) between dipole moments μ_EPAC and μ_B3LYP, obtained from EPAC charges and from B3LYP/6-31G(d,p) level calculations, respectively, for Ac–X–NHMe are estimated to be 0.38 and 0.59 D, respectively, for 26 representative conformations in the training set, and 0.54 and 0.79 D, respectively, for 172 representative conformations not used for parametrization. For Ac–(Ala)_n–NHMe (n = 2−6), the EPAC method reasonably predicts the increase of the dipole moment with increase of the chain length, although the deviations from the μ_B3LYP values are somewhat larger. For Ac–Ala–NHMe and Ac– (Ala)₃–NHMe, the EPAC charge for a specific type of atom does not depend on its position in the sequence or on the length of the sequence. In addition, charge neutrality holds for any Ala residue of these two peptides. Thus, these results suggest that the EPAC charges derived from B3LYP/6-31G(d,p) Mulliken populations can be used reliably for conformational analysis of peptides and proteins.

Introduction

Although atomic charges in a molecule cannot be measured experimentally, they have been used widely for conformational analysis and for interpreting molecular reactivity. For three decades, molecular force fields, such as ECEPP,¹ CHARMM,² AMBER,³ GROMOS,⁴ CVFF,⁵ OPLS,⁶ and others, have been used successfully for computing structures, and thermodynamic and kinetic properties of proteins, as well as their interactions. Electrostatic interactions, calculated from atomic charges, have been recognized as one of the essential factors for describing these properties of proteins.

Atomic charges in a molecule have been calculated by using quantum mechanical and empirical methods. The first quantum mechanical methods include orbital-based population analyses,^7–9 a topological electron density analysis (the so-called “atoms in molecules” approach),¹⁰ a population analysis based on atomic polar tensors,¹¹ and the electrostatic potential (ESP)-fitting methods,¹² with their theoretical backgrounds and applicabilities discussed in detail.^13,14 Electronic populations of organic molecules with various functional groups, computed by quantum mechanical methods with different basis sets, have been compared.^13,14 A modified formalism has been proposed to obtain atomic charges from orbital-based population analyses by parametrizing bond dipoles to reproduce experimental dipole moments.¹⁵ Although these quantum mechanical methods were applied successfully to obtain atomic charges for small molecules, they cannot be adopted practically for larger molecules such as peptides and proteins.

Most of the empirical methods have been derived on the basis of the electronegativity equalization principle proposed by Sanderson,¹⁶ which states that when two or more atoms that initially differ in electronegativity combine to form a bond, they become adjusted to the same intermediate electronegativity within the compound. Several formalisms to implement this principle, such as the partial equalization of orbital electronegativity (PEOE),¹⁷ the full equalization of orbital electronegativity, ¹⁸ the electronegativity equalization method (EEM),¹⁹ the modified partial equalization of orbital electronegativity (MPEOE),²⁰ the charge equilibrium approach (QEq),²¹ the atom-bond electronegativity equalization method (ABEEM),²² and the geometry-dependent atomic charge (GDAC) method,²³ have been proposed. Recently, the parameters for EEM²⁴ and ABEEM²⁵ were reoptimized using Mulliken atomic charges calculated at the B3LYP/6-31G(d) level. New parameters have been proposed for PEOE²⁶ and QEq²⁷ by fitting to experimental solvation free energies of organic molecules using the Poisson–Boltzmann (PB) equation. A combined EEM and ESP-fitting method has been suggested to reduce the mathematical instability of the ESP-fitting methods.²⁸

The atomic charges of the CHARMM,² GROMOS,⁴ CVFF,⁵ and OPLS⁶ force fields are empirical, and have been parametrized to reproduce thermodynamic properties of organic liquids and/or crystal structures of organic compounds and peptides, whereas Mulliken charges and ESP-fitted charges were employed for the ECEPP¹ and AMBER³ force fields, respectively. The AMBER force field with restrained ESP (RESP)-fitted charges and a scale factor of 1/1.2 for 1-4 electrostatic interactions satisfactorily reproduced observed relative conformational energies of small organic molecules, DNA base-pairing energies, and solvation free energies of methanol and N-methyacetamide. ²⁹ However, the ESP-fitted charges have inherent problems, namely, they depend strongly on molecular conformation,^30–32 on the fitting procedure,^12g,14,33,34 and on sampling the grid points on the potential surface.^12g,14,33,34 In particular, the conformational dependence of ESP-fitted charges can lead to problems of transferability of the force-field parameters and violation of charge neutrality of each neutral residue in peptides and proteins. For example, the ESP-fitted charges of the N and C^α atoms for extended conformations of Ac–Ala–NHMe and Ac–(Ala)₃–NHMe, which were calculated by using the Merz–Singh–Kollman scheme^12c,12f and by constraining charges to reproduce the dipole moment, are different depending on their positions in the sequence and on the length of the sequence, even though the backbone structures are identical (see Table S1 of the Supporting Information). In addition, charge neutrality for each residue does not hold except for the second Ala residue of Ac–(Ala)₃–NHMe (Table S1 of the Supporting Information). To avoid this non-neutrality problem in the RESP-fitted charges for Ac–X–NHMe (X = any neutral residue) in the AMBER force field, additional constraints were applied to force each residue X and the two end groups to be neutral.³⁵ Later, the effect of self-polarization was considered in the RESP charge-fitting procedure to improve the interaction energies for nucleic acid base pairs, the solvation free energies of methanol and N-methyl acetamide, and the chloroform/water partition coefficients of nucleic acid bases.³⁶

Mulliken charges have been used widely, but they are known to have a strong dependence on the basis set.^{11,12b,12d,13,14} However, they have two good features, namely, transferability and neutrality. As seen here for Ac–Ala–NHMe and Ac–(Ala)₃–NHMe in Table S1 of the Supporting Information, the Mulliken charge for a specific type of atom does not depend on its position in the sequence or on the length of the sequence. In particular, charge neutrality holds for any Ala residue of these two peptides. Thus, these results suggest that Mulliken charges can be used successfully for conformational analysis of peptides, if the proper level of theory and basis set is adopted.

In this article, we present an efficient method to calculate atomic charges of peptides and proteins derived from Mulliken electronic populations for terminally blocked amino acids calculated at the B3LYP/6-31G(d,p)//HF/6-31G(d,p) level of theory. This electronic population-derived atomic charges (EPAC) method is based on the geometry-dependent atomic charge (GDAC) method reported previously.²³

Methods

The procedure adopted here consists of several steps. First, atomic charges and dipole moments are computed for Ac–XNHMe at the B3LYP/6-31G(d,p) level. To test the reliability of these B3LYP charges, the dipole moments, ${\mu }_{\text{B3LYP}}^{*}$, computed from these B3LYP charges are compared to the dipole moments, μ_B3LYP, computed directly at the same B3LYP level. Then, because B3LYP/6-31G(d,p) calculations cannot be carried out practically for molecules much larger than Ac–X–NHMe, an empirical method based on the use of electronegativities is used to derive (transferable) charges for larger peptides and proteins. There are several definitions of electronegativity, and we chose electronegativities based on the MPEOE method^20,23 that was found to reproduce experimental and theoretical (ab initio) dipole and/or quadrupole moments of small organic molecules. Hence, we derive a set of electronegativity parameters for each atom of Ac–X–NHMe to match the charges calculated at the B3LYP/6-31G(d,p) level and obtain empirical atomic charges from these electronegativity parameters.

Theoretical Background

According to the PEOE method,¹⁷ the atomic charge Q_A on an atom A can be calculated by an iterative procedure that converges when the charge transfer between all atoms in a molecule is zero. The degree of charge transfer, $d{q}_{\text{AB}}^{(n)}$, from atom A to atom B at the nth iteration is described by¹⁷

$$d{q}_{\text{AB}}^{(n)}=[\{{\text{χ}}_{\text{B}}^{(n-1)}-{\text{χ}}_{\text{A}}^{(n-1)}\}/{\text{χ}}_{\text{A+}}]{(f_{\text{AB}})}^n{\text{\hspace{0.25em}ifχ}}_{\text{B}}^{(n-1)}>{\text{χ}}_{\text{A}}^{(n-1)}$$ 1

where ${\text{χ}}_{\text{A}}^{(n-1)}$ and ${\text{χ}}_{\text{B}}^{(n-1)}$ are the electronegativities of atoms A and B at the (n − 1)th iteration, and χ_A+ is the electronegativity of the positive ion of atom A. f_AB is a damping factor for the bond A−B, which is 0.5 in the PEOE method¹⁷ and a parameter determined during the optimization in the MPEOE method.²⁰ In the GDAC method²³ and in the procedure used here, f_AB is expressed, however, as a function of a distance R_AB between atoms A and B and is given by

$$f_{\text{AB}}=1-R_{\text{AB}}/(R_{\text{A}}^{\text{vdw}}+R_{\text{B}}^{\text{vdw}})$$ 2

where $R_{\text{A}}^{\text{vdw}}$ and $R_{\text{B}}^{\text{vdw}}$ are the van der Waals radii of atoms A and B, respectively, for which the values of Rowland and Taylor³⁷ are used because they were obtained by analyzing more than 100 000 organic and organometallic compounds. Thus, the ratio of R_AB to $(R_{\text{A}}^{\text{vdw}}+R_{\text{B}}^{\text{vdw}})$ represents the orbital overlap between atoms A and B depending on the molecular conformation. For example, the values of f_AB vary from 0.567 to 0.641 for the extended conformation of Ac–Ala–NHMe. As the iteration proceeds, i.e., as n increases, the value of (f_AB)ⁿ decreases dramatically.

The atomic charge on atom A at the nth iteration, Q_A⁽ⁿ⁾, is obtained by^17,20,23

$$Q_{\text{A}}^{(n)}=Q_{\text{A}}^{(0)}+{\displaystyle \sum _n{\displaystyle \sum _{B}d{q}_{\text{AB}}^{(n)}}}$$ 3

where $Q_{\text{A}}^{\text{(0)}}$, which was introduced by No et al. in the MPEOE method,²⁰ represents the initial atomic charge before the iterative procedure starts. The index B sums over all the atoms covalently bonded to atom A, whereas the index n sums over the number of iterations until the convergence criterion $|Q_{\text{A}}^{(n)}-Q_{\text{A}}^{(n-1)}|$ < 10⁻⁶ e is satisfied, where e is the electronic charge. The quantity $d{q}_{\text{AB}}^{(n)}$ is obtained from eq 1.

In these calculations, the electronegativity of an atom i at the nth iteration is described as a linear function of the atomic charge $Q_i^{(n)}$ as in the MPEOE²⁰ and GDAC²³ methods

$${\text{χ}}_i^{(n)}={\text{a}}_i+{\text{b}}_i{Q}_i^{(n)}$$ 4

where a_i is the electronegativity of the neutral atom and b_i is a coefficient describing the response of the electronegativity of the atom i to the changes in electron density. The quantity $Q_i^{(n)}$ is obtained from eq 3. For given sets of parameters a_i and b_i for each atom type, the atomic charge Q_A on an atom A can be calculated by an iterative procedure using eqs 1–4.

In the MPEOE method,²⁰ the damping factors and the sets of electronegativity parameters a_i and b_i were varied to reproduce experimental dipole moments and quadrupole moments of organic molecules with various functional groups. In the GDAC method,²³ the sets of electronegativity parameters ai and bi were, however, derived only by fitting to the three components of the dipole moment and the total dipole moment computed at the B3LYP/6-31G(d,p) level of theory for 50 molecules representing the backbone and side chains of peptides and proteins. The dipole moments calculated at the B3LYP/6-31G(d,p) level were found to be in good agreement with experimental values, as shown in Table 1 of ref 23.

TABLE 1.

Comparison of Dipole Moments for Extended Conformations of Ac–X–NHMe in the Training Set Computed by the B3LYP, EPAC, and GDAC Methods^a

X	confb	μB3LYPc	${\mu }_{\text{B3LYP}}^{*}$d	Δe	μEPACf	Δg	μGDACh	Δi
Ala	E	2.92	2.97	0.05	2.95	0.03	2.41	0.51
Val	Eg+g+	2.76	2.62	0.14	2.75	0.00	2.28	0.47
Leu	Eg−t	2.27	2.46	0.19	2.72	0.45	2.14	0.14
Ile	Ett	2.79	2.71	0.08	2.76	0.03	2.32	0.46
Gly	E	3.25	3.13	0.12	3.18	0.07	2.62	0.62
Pro	tCdj	2.65	3.06	0.41	2.80	0.14	2.78	0.13
Cys	Etg+	1.72	2.52	0.80	2.55	0.83	1.32	0.39
Met	Etg+	1.17	2.05	0.88	2.33	1.17	0.99	0.18
His(δ)k	Etg+	2.74	2.39	0.35	2.24	0.50	2.25	0.48
His(ε)k,l	Etg−	4.55	3.83	0.72	3.55	1.00	3.29	1.25
Phe	Etg+	2.71	2.75	0.03	2.68	0.03	2.21	0.50
Tyr	Eg+g+	1.62	2.23	0.61	2.31	0.69	1.37	0.24
Trp	Etg−	4.24	2.69	1.55	2.60	1.64	3.98	0.25
Asn	Etg−	0.82	0.84	0.02	0.77	0.05	0.97	0.14
Gln	Eg−g−	2.88	2.87	0.01	2.51	0.37	2.39	0.49
Ser	Etg+	1.68	1.36	0.33	1.48	0.20	1.32	0.37
Thr	Etg+	1.69	1.44	0.25	1.30	0.39	1.27	0.43
Lys	Eg+t	3.99	3.98	0.01	3.97	0.02	3.61	0.38
Arg	Etg+	5.85	4.59	1.27	4.52	1.33	4.82	1.04
Asp	Etc	2.60	2.25	0.36	2.56	0.04	2.05	0.55
Glu	Eg−g−	2.32	2.19	0.14	2.25	0.07	1.81	0.52
Lys+l	Eg−g−	11.08	10.90	0.18	11.12	0.04	16.52	5.44
Arg+l	Eg−g−	11.99	11.43	0.56	12.37	0.38	19.90	7.91
His+l	Eg−g+	9.61	8.58	1.02	9.72	0.11	12.69	3.08
Asp−l	Ett	8.70	7.98	0.72	8.76	0.06	12.57	3.88
Glu−l	Etg+	9.41	8.79	0.63	9.65	0.24	13.92	4.50
				0.44m		0.38m		1.32m
				0.60n		0.59n		2.34n

^aUnits in Debyes. The 26 representative residues of the training set are listed in Table S4 of the Supporting Information.

^bConformational letter codes. Each code represents the backbone expressed by a capital letter depending on its torsion angles Ф and ψ, and two side-chain torsion angles χ¹ and χ². See the text for definition of torsion angles.³⁸,³⁹

^cDipole moments calculated directly at the B3LYP/6-31G(d,p)//HF/6-31G(d,p) level.

^dDipole moments calculated from atomic charges obtained at the B3LYP/6-31G(d,p)//HF/6-31G(d,p) level.

^eAbsolute differences between ${\mu }_{\text{B3LYP}}^{*}$ and μ_B3LYP values.

^fDipole moments calculated by the EPAC method.

^gAbsolute differences between μ_EPAC and μ_B3LYP values.

^hDipole moments calculated by the GDAC method of Cho et al.²³

ⁱAbsolute differences between μ_GDAC and μ_B3LYP values.

^jPro with the trans prolyl peptide bond and the down puckering.

^kHis(δ) and His(ε) are isomers of the His residue depending on its protonation at N^δ and N^ε, respectively.

^lInitial atomic charges were obtained by a trial-and-error procedure starting from the corresponding values in ref 23 to reproduce the B3LYP/6-31G(d,p) atomic charges for only these residues; see Figure S1 of the Supporting Information.

^mMean values over all residues.

ⁿRms deviations over all residues.

Training and Test Sets

Although the MPEOE²⁰ and GDAC²³ charges satisfactorily reproduced experimental dipole moments and quadrupole moments, and quantum-mechanically computed dipole moments of small organic molecules, respectively, it was found that these two methods occasionally produced unacceptable atomic charges and/or dipole moments for larger molecules such as terminally blocked amino acids (see Table 1 for comparison of dipole moments for Ac–X–NHMe, X = any neutral or charged residue, computed at the B3LYP/6-31G(d,p) level and by the GDAC method, and Table S2 of the Supporting Information for Ac–Ala–NHMe). In addition, whereas the Mulliken charges depend on the molecular conformation, the degree of dependence is relatively smaller than that found for ESP-fitted charges (see Table S3 of the Supporting Information). To avoid these two problems, we chose extended conformations of Ac–X–NHMe (X = any neutral or charged residue, except for Pro), shown in Table 1 and in boldface in Table S4 of the Supporting Information as a training set to derive the electronegativity parameters a_i and b_i. Because the Pro residue does not have an extended conformation, the tCd conformation was chosen as a representative one.^1c

Several representative conformations were selected for each residue X of Ac–X–NHMe, not used for parametrization, and their dipole moments, computed with the EPAC charges, were compared with those calculated directly at the B3LYP/6-31G-(d,p) level in order to demonstrate the reliability of the EPAC method. In particular, the dipole moments of representative conformations of Ac–(Ala)_n–NHMe (n = 2−6) computed with both the EPAC charges and also directly at the B3LYP/6-31G-(d,p) level are compared to examine the transferability of the present method for longer peptide chains (Table S5 of the Supporting Information).

The initial representative conformations for each residue were taken from the local minima for Ac–X–NHMe (X = any neutral residue)^1c,38 optimized by the ECEPP2^1b and ECEPP3^1c force fields. All local minima for Ala, Gly, and Pro residues found by the ECEPP force fields were taken as initial conformations. For other neutral residues, however, local minima were chosen if they have C or E backbone conformations³⁹ and different χ¹ or χ² side-chain conformations. Each conformation is represented by a letter code composed of a capital letter plus two side-chain codes depending on its backbone torsion angles ϕ and ψ, and side-chain torsion angles χ¹ or χ¹, according to the definition suggested by Zimmermann et al.³⁹ All selected conformations of neutral His, Lys, Arg, Asp, and Glu residues were used as starting points for optimizations of charged His⁺, Lys⁺, Arg⁺, Asp⁻, and Glu⁻ residues, respectively. For Ac–(Ala)₂–NHMe and Ac–(Ala)₃–NHMe, the initial representative conformations were generated by a combination of C, E, and A conformations for Ac–Ala–NHMe. Three representative conformations, for which every residue has the same A (or C or E) backbone conformation, were constructed for each peptide of Ac–(Ala)_n–NHMe (n = 4−6). Then, these initial conformations were optimized using the ECEPP3 force field^1c. In addition, mutants of extended Ac–Ala–NHMe with different N- and C-terminal groups were included in the training set only to derive the parameters for these terminal groups.

Quantum Mechanical Calculations

All representative conformations for Ac–X–NHMe, Ac–(Ala)n–NHMe (n = 2−6), and Ac–Ala–NHMe mutants with different N- and C-terminal groups obtained from the ECEPP3 force field were optimized at the HF/6-31G(d,p) level of theory using the Gaussian 98⁴⁰ and 03⁴¹ packages. This level has previously been usedsuccessfullyforconformationalanalysesofAlaoligopeptides^42–44 and in developing potential parameters of force fields.^6b,45 B3LYP/6-31G(d,p) single-point calculations were carried out for all representative conformations optimized at the HF/6-31G-(d,p) level. The dipole moments for small molecules representing the backbone and side chains of peptides and proteins calculated at the B3LYP/6-31G(d,p) level were found to be in good agreement with experimental values.²³ The Mulliken charges and dipole moments for these representative conformations computed at the B3LYP/6-31G(d,p)//HF/6-31G(d,p) level were used for deriving the electronegativity parameters a_i and b_i in eq 4 and in testing the quality of the method and its parameters. All representative conformations for Ac–X–NHMe and Ac–(Ala)_n–NHMe (n = 2−6), optimized at the HF/6-31G(d,p) level, are listed in Tables S4 and S5, respectively, of the Supporting Information.

Parametrization Procedure

For given electronegativity parameters a_i and b_i for each type of atom in a molecule, the conformation-dependent atomic charges were calculated by using eqs 1–4. For most residues and end groups, the initial atomic charges Q⁽⁰⁾ in eq 3 were chosen to be zero. However, for some residues such as His(Є) and charged residues (Lys⁺, Arg⁺, His⁺, Asp⁻, and Glu⁻) and for some N- and C-terminal groups, which are listed in Figure S1 of the Supporting Information, finite initial atomic charges were assigned. This was necessary in order to avoid large, sometimes oscillatory, changes in the electronegativity function, often preventing convergence of the procedure, as in the MPEOE^20b and GDAC²³ methods. The choice of these initial atomic charges was arbitrary, but their values (listed in Figure S1 of the Supporting Information) were obtained by a trial-and-error procedure, starting from the corresponding values in ref 23, to reproduce the B3LYP/6-31G(d,p) atomic charges during the optimization described below.

The parameters a_i and b_i for the training set were obtained by minimizing the following function S with the secant unconstrained minimization solver (SUMSL)⁴⁶ with given sets of initial atomic charges

$$S={\displaystyle \sum _k{{\displaystyle \sum _i{w}_{ik}(Q_{ik}^{\text{EPAC}}-Q_{ik}^{\text{EPAC}})}}^2}$$ 5

where the index k refers to a molecule in the training set, and i to an atom in molecule k; w_ik, $Q_{ik}^{\text{EPAC}}$, and $Q_{ik}^{\text{B3LYP}}$ are the statistical weight, EPAC charge, and B3LYP/6-31G(d,p) charge for atom i in molecule k, respectively. wik was taken as 1.000 during minimization. For those residues and end groups having nonzero initial atomic charges, the assignment of initial atomic charges and the optimization of parameters a_i and b_i were repeated until the difference between the computed charge and the corresponding B3LYP/6-31G(d,p) charge for each atom becomes less than 0.01 e.

The minimization of the function S, however, encounters the multiple-minima problem. It is not easy to find the best set of parameters if a large number of parameters are involved in the minimization procedure because the procedure is likely to be trapped in local minima. To overcome this problem and to obtain the best set of parameters depending on the chemical structure of each side chain, we first derived the parameters for aliphatic (Ala, Val, Leu, and Ile), Gly, and Pro residues, which correspond to the parameters for backbone and aliphatic groups of side chains. The sets of parameters for side chains of the other residues were optimized with the parameters from these six residues fixed. In this procedure, the residues not employed in the first optimization were classified into seven groups: (1) Asn, Gln, Ser, and Thr; (2) Cys and Met; (3) Asp and Glu; (4) Lys and Arg; (5) aromatic His(δ), His(Є), Phe, Tyr, and Trp; (6) positively charged Lys⁺, Arg⁺, and His⁺; (7) negatively charged Asp⁻, and Glu⁻. Then, all parameters a_i and b_i were reoptimized simultaneously to avoid accumulation of errors and to provide consistency in the parameters with initial atomic charges obtained in the previous optimization. The parameters a_i and b_i obtained from the GDAC method²³ were used as initial values for optimization in the present EPAC method. In addition, the parameters for the N- and C-terminal groups were derived by a similar procedure by using the extended Ac–Ala–NHMe mutants with the parameters for the Ala residue fixed. Thus, we employed a total of 69 sets of atomic parameters a_i and b_i to reproduce 954 B3LYP/6-31G(d,p) atomic charges for Ac–X–NHMe (X = 21 neutral and 5 charged residues) and 10 mutants of Ac–Ala–NHMe with different end groups used in the training set.

Results and Discussion

Torsion angles for backbone and side chain, relative conformational energies, and dipole moments of representative conformations for Ac–X–NHMe and Ac–(Ala)_n–NHMe (n = 2−6) calculated at the B3LYP/6-31G(d,p)//HF/6-31G(d,p) level, and their dipole moments computed by the EPAC method, are listed in Tables S4 and S5, respectively, of the Supporting Information.

B3LYP Charges and Dipole Moments

First, we compared the B3LYP/6-31G(d,p) directly computed dipole moments μ_B3LYP with the dipole moments ${\mu }_{\text{B3LYP}}^{*}$ computed by using B3LYP/6-31G(d,p) atomic charges for 21 neutral and 5 charged representative conformations of Ac–X–NHMe in the training set, which are shown in Table 1 and Figure 1a. The mean absolute difference (mad) between μ_B3LYP and ${\mu }_{\text{B3LYP}}^{*}$ values is 0.44 D and its rmsd is 0.60 D, although the absolute differences (ad's) for Trp, Arg, and His⁺ residues are larger than 1.00 D. These results suggest that the B3LYP/6-31G(d,p) Mulliken charges reproduce the dipole moment calculated at the B3LYP/6-31G(d,p) level well and can be used in deriving the EPAC parameters. Comparison of panels b and c in Figure 1 shows the improvement of the EPAC parameters over those of GDAC.

Figure 1.

Comparison of dipole moments μ_B3LYP computed directly at the B3LYP/6-31G(d,p) level with dipole moments ${\mu }_{\text{B3LYP}}^{*}$, μ_EPAC, and μ_GDAC calculated with B3LYP/6-31G(d,p) Mulliken, EPAC, and GDAC charges, respectively, for representative conformations of Ac–X–NHMe (X = 21 neutral and 5 charged residues) in the training set: (a) ${\mu }_{\text{B3LYP}}^{*}$ vs μ_B3LYP; (b) μ_EPAC vs μ_B3LYP; (c) μ_GDAC vs μ_B3LYP. Units in Debyes. The same conformations shown in boldface in Table S4 of the Supporting Information were used in panels a–c.

Optimized Sets of Parameters

The sets of parameters a_i and b_i for the EPAC method, optimized for 26 representative conformations of Ac–X–NHMe in the training set, are listed in Table 2, in which the sets of parameters for N- and C-terminal groups are also listed. The dipole moments μ_EPAC calculated by using the EPAC charges for these representative conformations are compared with the B3LYP/6-31G(d,p) dipole moments μ_B3LYP in Table 1 and Figure 1b. The mad and rmsd between μ_EPAC and μ_B3LYP values are 0.38 and 0.59 D, respectively, although the ad's for Met, Trp, and Arg residues are larger than 1.00 D. This shows that the EPAC charges provide somewhat better dipole moments than the B3LYP/6-31G(d,p) charges, with the better results for the Val, Pro, Asp, Lys⁺, Arg⁺, His⁺, Asp⁻, and Glu⁻ residues making the main contribution to the mad and rmsd. In addition, the dipole moments μ_GDAC, calculated using the atomic charges from the GDAC method²³ for these 26 representative conformations, are compared with the B3LYP/6-31G(d,p) dipole moments μ_B3LYP in Table 1 and Figure 1c. The mad and rmsd between μ_GDAC and μ_B3LYP values are 1.32 and 2.34 D, respectively, to which the charged residues contribute the larger deviations.

TABLE 2.

Atomic Types and Optimized Parameters for Their Atomic Electronegativities

atomic type	description	aia	bia
H1	aliphatic H	2.1917	0.0489
H2	Hα of the peptide backbone	2.5062	−1.2773
H3	H of the Pro ring	0.8253	12.7055
H4	aromatic H	1.0280	13.4508
H5	amine H	2.4868	1.7882
H6	amide H	1.1118	5.7519
H7	H of OH	−0.5001	10.5861
H8	H of COOH	2.8982	1.6472
H9	H of SH	1.9730	0.4021
H10	H of H−N in His and Trp side chains	−2.1730	16.8150
H11	H attached to H2N+	1.7926	0.8722
H12	H of His+ side chain	2.6350	0.7970
H13	H of the formyl amino end group	3.1946	−0.5623
H14	Hα when the N-terminal group is H2N+	2.2773	−1.0843
H15	Hα when the C-terminal group is COO−	1.6636	4.0451
C1	sp3 C of CH3	2.5925	1.2129
C2	sp3 C of CH2	2.4955	1.5058
C3	sp3 C of CH	2.3457	1.3918
C4	sp3 Cα for non-Pro residues	2.4504	0.6371
C5	sp3 C attached to C═O	2.4158	0.7327
C6	sp3 C attached to amide N−H	2.5234	1.4027
C7	sp3 C of the Pro ring	2.6156	0.4805
C8	sp3 Cβ attached to the aromatic ring	2.6586	1.6877
C9	sp3 C of CH2 attached to OH of Ser	2.3773	0.5901
C10	sp3 C of CH attached to OH of Thr	2.3507	0.5820
C11	sp3 C of CH2 or CH3 attached to S	2.4492	0.2428
C12	sp3 C attached to C═O of the side chain in Asn, Gln, Asp, and Glu	2.3983	0.7665
C13	sp2 C of C═O	−5.1345	14.1482
C14	aromatic C	2.4449	0.4126
C15	fused aromatic C	2.3480	0.1250
C16	aromatic C attached to sp3 C	2.2973	0.6777
C17	aromatic C attached to OH of Tyr	2.0084	1.9595
C18	Cδ of His and Cδ1 of Trp	2.1780	4.8280
C19	Cε of His	3.8312	−3.1644
C20	Cε of Arg	2.3846	1.5047
C21	Cζ of Arg	−3.4851	9.6585
C22	sp3 C attached to COO−	2.7051	1.8741
C23	sp3 C attached to N+	3.5982	11.6359
C24	sp2 C of COO−	−1.7286	7.7191
C25	Cζ of Arg+	2.4697	1.0926
C26	Cζ of Arg+	−14.8197	21.5072
C27	Cγ of His+	0.2430	8.0356
C28	Cδ and Cε of His+	2.7851	0.1440
C29	sp3 tertiary C of the t-Boc amino end group	0.7942	5.5220
C30	sp3 C of CH2 or CH3 in OMe and OEt carboxylic end groups	2.3756	−0.0071
C31	sp2 C of C═O for the first residue when the N-terminal group is t-Boc	−5.1838	8.9183
C32	sp2 C of C═O in OMe and OEt carboxylic end groups	−5.0094	11.3808
C33	Cα when the N-terminal group is H2N+	2.2800	0.8454
C34	Cβ when the N-terminal group is H2N+	2.4330	−1.5951
C35	C of C═O for the first residue when the N-terminal group is H2N+	−4.1541	11.3556
C36	Cα when the C-terminal group is COO−	2.2838	5.2758
N1	sp3 amine N	2.4955	−8.9472
N2	sp2 amide N	2.9703	−0.1802
N3	sp2 amide N of the Pro ring	3.0683	-0.3431
N4	sp2 Nε and Nη2 of Arg	2.8267	−0.8363
N5	sp2 Nη1 of Arg	2.5473	0.0262
N6	aromatic Nη of His and Trp	3.2388	1.3470
N7	aromatic N of His and Trp	4.4553	6.1260
N8	N+ of Lys+	3.4576	2.4448
N9	N+ of Arg+	7.4806	10.3003
N10	N+ of His+	9.4721	13.8330
N11	N of the NH amino end group	2.6867	−2.6812
O1	sp3 O of OH	2.3877	−2.5413
O2	sp3 O of COOH	4.0066	−1.5831
O3	sp3 O of the t-Boc amino end group	3.3914	−0.6520
O4	sp2 O of C═O	4.2182	−1.6118
O5	sp2 O of COO−	2.7582	0.4253
O6	sp3 O of OMe and OEt carboxylic end groups	2.9383	0.0708
S1	S in thiol, sulfide, and disulfide	2.2551	2.4403

^aElectronegativity parameters of each atom in the molecule are expressed in eq 4. According to the definitions of Mulliken⁴⁷ and Parr and Pearson,⁴⁸ the electronegativity χ has the dimension of energy. Therefore, the dimensions of a_i and b_i are energy and energy per electronic charge, respectively.

Tests for Other Conformations of Ac–X–NHMe Not Used for Parametrization

The optimized sets of parameters a_i and b_i for the EPAC method were tested for their applicability to other conformations of Ac–X–NHMe not used for parametrization. The mad's and rmsd's between the μ_EPAC and μ_B3LYP values for a total of 172 representative conformations of Ac–X–NHMe not used for parametrization are listed in Table 3, and their dipole moments are plotted in Figure 2. The overall mad and rmsd are calculated to be 0.54 and 0.79 D, respectively, which are similar in magnitude to those estimated for the training set. The calculated mad's and rmsd's for 38 aliphatic, 31 nonpolar, 27 aromatic, 13 polar, 40 charged-neutral, and 23 charged-ionized conformations are 0.19 and 0.25 D, 0.38 and 0.49 D, 0.87 and 1.05 D, 0.30 and 0.40 D, 0.55 and 0.72 D, and 1.03 and 1.41 D, respectively. Large deviations are found for aromatic and charged-ionized residues.

TABLE 3.

Mean Absolute Deviations (MADs) and Root-Mean-Square Deviations (RMSDs) of Dipole Moments for Ac–X–NHMe, Not Used for Parametrization, Calculated by the EPAC Method

group	X	no.a		madb	rmsdc
aliphatic	Ala	5		0.41	0.44
	Val	4		0.17	0.20
	Leu	15		0.14	0.20
	Ile	14		0.18	0.22
			av.d	0.19	0.25
nonpolar	Gly	2		0.58	0.58
	Pro	4		0.47	0.52
	Cys	10		0.27	0.38
	Met	15		0.41	0.54
			av.d	0.38	0.49
aromatic	His(δ)k	2		1.07	1.09
	His(ε)k	8		0.58	0.71
	Phe	5		0.27	0.34
	Tyr	4		1.22	1.23
	Trp	8		1.30	1.45
			av.d	0.87	1.05
polar	Asn	3		0.28	0.36
	Gln	4		0.36	0.54
	Ser	4		0.28	0.30
	Thr	2		0.27	0.28
			av.d	0.30	0.40
charged-neutral	Lys	15		0.53	0.64
	Arg	5		1.15	1.44
	Asp	8		0.42	0.49
	Glu	12		0.40	0.45
			av.d	0.55	0.72
charged-ionized	Lys+	3		2.16	2.64
	Arg+	5		0.91	1.08
	His+	5		1.49	1.70
	Asp−	5		0.45	0.50
	Glu−	5		0.60	0.79
			av.d	1.03	1.41
			overall av.e	0.54	0.79

^aNumber of conformations considered.

^bMean absolute deviations from dipole moments calculated at the B3LYP/6-31G(d,p)//HF/6-31G(d,p) level.

^cRoot-mean-square deviations from dipole moments calculated at the B3LYP/6-31G(d,p)//HF/6-31G(d,p) level.

^dAverage values for each group.

^eOverall mean values for all 172 conformations.

Figure 2.

Comparison of dipole moments μ_EPAC, calculated with EPAC charges, with dipole moments μ_B3LYP computed directly at the B3LYP/6-31G(d,p) level for 172 representative conformations of Ac–X–NHMe not used for parametrization; units in Debyes: (a) aliphatic residues Ala (red), Val (blue), Leu (green), and Ile (open); (b) nonpolar residues Gly (red), Pro (blue), Cys (green), and Met (open); (c) aromatic residues His(δ) (red), His(ε) (blue), Phe (green), Tyr (open), and Trp (yellow); (d) polar residues Asn (red), Gln (blue), Ser (green), and Thr (open); (e) charged-neutral residues Lys (red), Arg (blue), Asp (green), and Glu (open); (f) charged-ionized residues Lys⁺ (red), Arg⁺ (blue), His⁺ (green), Asp⁻ (open), and Glu⁻ (yellow).

Tests for Longer Peptides

The dipole moments of representative conformations for Ac–(Ala)_n–NHMe (n = 2−6), computed with the EPAC charges and at the B3LYP/6-31G-(d,p) level, are compared to assess the transferability of the present method for longer peptide chains. The mad's and rmsd's between μ_EPAC and μ_B3LYP values for these representative conformations are listed in Table 4 and their dipole moments are plotted in Figure 3. The overall mad and rmsd between μ_EPAC and μ_B3LYP values for these 23 representative conformations are estimated to be 0.95 and 1.40 D, respectively, to which the large deviations of Ac–(Ala)_n–NHMe (n = 4–6) contribute dominantly. However, the mad's and rmsd's for Ac–(Ala)₂–NHMe and Ac–(Ala)₃–NHMe are only 0.19 and 0.20 D and 0.36 and 0.52 D, respectively. It should be noted that the EPAC method reasonably predicts the increase of dipole moment with the increase of chain length, although there are increasing errors with increasing n. In particular, the smaller deviations are found for extended conformations with the same E backbone conformation for every residue than for folded conformations with the same C or A backbone conformations for every residue for Ac–(Ala)_n–NHMe (n = 4−6), which may imply that nonadditive effects, not considered here, would operate on folded structures of longer peptides.

TABLE 4.

Mean Absolute Deviations (MADs) and Root-Mean-Square Deviations (RMSDs) of Dipole Moments for Ac–(Ala)_n–NHMe (n = 2−6) Not Used for Parametrization, Calculated by the EPAC Method

n	no.a		madb	rmsdc
2	5		0.19	0.20
3	9		0.36	0.52
4	3		1.27	1.29
5	3		2.13	2.18
6	3		2.71	2.80
		overall av.d	0.95	1.40

^aNumber of conformations considered.

^bSee footnotes b and c of Table 3.

^cSee footnotes b and c of Table 3.

^dOverall mean values for all 23 conformations.

Figure 3.

Comparison of dipole moments μ_EPAC calculated with EPAC charges with dipole moments μ_B3LYP computed directly at the B3LYP/6-31G(d,p) level for representative conformations of Ac–(Ala)_n–NHMe (n = 2−6) not used for parametrization, units in Debyes: n = 2, ●; n = 3, ○; n = 4, ▲; n = 5, ∆; n = 6, ■.

Transferability and Neutrality of EPAC Charges

As seen for representative conformations of Ac–X–NHMe in the training set and for those not for used for parametrization, the EPAC charges reproduced the B3LYP/6-31G(d,p) dipole moments well, which suggests transferability of the charges obtained with the EPAC method. In particular, the EPAC charge for a specific type of atom does not depend on its position in the sequence or on the length of the sequence, as seen for Ac–Ala–NHMe and Ac–(Ala)₃–NHMe in Table S1 of the Supporting Information. In addition, charge neutrality holds for any Ala residue of these two peptides. Thus, these results suggest that the EPAC charges derived from B3LYP/6-31G(d,p) Mulliken populations can be used reliably for conformational analysis of peptides and proteins.

For another test of transferability of the EPAC method to small molecules, we calculated the atomic charges and dipole moment of the water molecule optimized at the HF/6-31(d,p) level using the parameters of atom types H₇ and O₁ for the OH group (Table 2). The calculated EPAC charges are −0.7136 and +0.3568 e for O and H of water, respectively, and its dipole moment is computed to be 1.95 D. At the same B3LYP/6-31-(d,p)//HF/6-31(d,p) level, the corresponding ESP-fitted charges are computed to be −0.7340 and +0.3670 e, respectively, by using the Merz–Singh–Kollman scheme^12c,12f and by constraining the charges to reproduce the dipole moment, and its dipole moment is computed to be 2.00 D. The EPAC charges of the water molecule are quite similar to the ESP-fitted charges, but the EPAC dipole moment is a little better than that from ESP-fitted charges because the experimental dipole moment is 1.84 D.

Conformational Variations of Atomic Charges

As seen for Ac–Ala–NHMe and Ac–(Ala)₃–NHMe with the extended backbone conformation in Table S1 of the Supporting Information, the atomic charge for a specific type of atom does not depend on its position in the sequence or on the length of the sequence. Although atomic charges could vary from conformation to conformation, the deviations are quite small. For example, for local minima (C, E, D, C*, A*, and F*, shown in Table S4 of the Supporting Information) of Ac–Ala–NHMe, the mean atomic charges for O and C of the acetyl end group, N and H of the Ala residue, and N and H of the NHMe end group (refer to Table S2 of the Supporting Information for atomic definitions) are computed as −0.5265, +0.5671, −0.5110, +0.2757, −0.5263, and +0.2755 e, respectively. Although these atoms are involved in the formation of intramolecular hydrogen bonds for some conformations, their deviations are calculated to be ±0.008, ±0.005, ±0.004, ±0.0002, ±0.002, and ±0.0005 e, respectively. The EPAC dipole moments for these local minima C, E, D, C*, A*, and F* are calculated to be 1.96, 2.95, 4.60, 2.68, 5.93, and 2.63 D, respectively, shown in Table S4 of the Supporting Information. Thus, it can be said that the different dipole moments for the different conformations are caused by the configuration of atoms in a molecule and not by the different atomic charges depending on conformation.

Computation Time

The mean CPU time required for the calculation of the atomic charges and dipole moment by the EPAC method for a single conformation of Ac–(Ala)₆–NHMe is less than 1 second on a PC with an Intel Core2Duo T7200 (2.0 GHz) processor.

Conclusions

An EPAC method is presented to calculate atomic charges of peptides and proteins derived from Mulliken electronic populations for terminally blocked amino acids (Ac–X–NHMe) calculated at the B3LYP/6-31G(d,p)//HF/6-31G(d,p) level of theory. The EPAC atomic charges are calculated by using the partial-equalization-of-atomic electronegativities method with electronegativity parameters and damping factors given by interatomic distances between covalently bonded atoms in a molecule.

For Ac–X–NHMe, the dipole moments μ_EPAC obtained with EPAC charges agree well with the dipole moments μ_B3LYP calculated directly at the B3LYP/6-31G(d,p) level for both the conformations in the training set and the conformations not used for parametrization. For Ac–(Ala)_n–NHMe (n = 2−6), the EPAC method predicts the increase of dipole moment with the increase of chain length reasonably well, although the deviations from the μ_B3LYP values are somewhat larger.

For Ac–Ala–NHMe and Ac–(Ala)₃–NHMe, it was found that the EPAC charge for a specific type of atom does not depend on its position in the sequence or on the length of the sequence, and that charge neutrality holds for any Ala residue of these two peptides. Thus, these results suggest that the EPAC charges derived from B3LYP/6-31G(d,p) Mulliken populations can be used reliably for conformational analysis of peptides and proteins.