4118
J . Phys. Chem. 1987, 91, 4118-4120
Free Energies of Hydratlon of Solute Molecules. 3. Appllcation of the Hydratlon Shell Model to Charged Organic Molecules Young Kee Kang; George NGmethy, and Harold A. Scheraga* Baker Laboratory of Chemistry, Cornel1 University, Ithaca, New York 14853-1 301 (Received: December 19, 1986)
The hydration shell model, used for determining the free energies of hydration of conformationally flexible solute molecules, has been extended to the treatment of charged organic molecules. The model is applied here to carboxylate and primary, secondary, and tertiary alkyl and aryl ammonium ions. Five compounds have been used to derive the free energy density of hydration around various charged groups in the shell, used as empirical hydration parameters. Values for the free energy densities for uncharged groups were taken over from the model for uncharged molecules. The model has been tested by computing the free energy of hydration for 14 ions that had not been used for determining the parameters. The average absolute difference between the calculated and experimental values and the standard deviation are 1.7 and 0.6 kcal/mol, respectively. The lowest individual deviations occur for those ions that serve as models for charged amino and carboxyl groups in the backbone and side chains of proteins, viz. alkylammonium ions (0.2-0.8 kcal/mol) and the butyrate ion (0.5 kcal/mol). Thus, the model is shown to be applicable to the inclusion of free energies of hydration of charged groups in conformational energy computations of peptides.
Introduction Knowledge of the hydration of ionic organic molecules such as carboxylate ions and protonated amines is important for the understanding of the hydration of peptides and proteins and of the role of water in biological structure and function. In part 1 of this series,l a general method to compute the free energy of hydration, AGhyd,of conformationally flexible solute molecules has been presented using the hydration shell model and empirical potential functions. Earlier formulations of the hydration shell mode12s3have been improved by computing the water-accessible volume, VWaj,of each group j in a molecule exactly. An application of this model to calculate the free energies of hydration of uncharged organic molecules has been presented in part 2 of this s e r i e ~ . ~ Physical properties such as hydration numbers, energies, and volume changes have been obtained from Monte Carlo (MC) and molecular dynamics simulations of ionic molecules in aqueous solution. Romano and Clement? carried out M C calculations for clusters consisting of water molecules surrounding glycine in the neutral and zwitterionic forms. In a recent work of Mezei et aL,6 the results of a M C simulation for a dilute aqueous solution of the glycine zwitterion were analyzed in terms of the quasicomponent distribution functions. Alagona et reported M C simulations for acetate and methylammonium ions in water. Jorgensen and Gao8 have performed M C simulations of dilute aqueous solutions of NH4+, CH3NH3+,(CH3)4N+,HCOO-, and CH3CO- at 25 OC. These simulations provide information on the energy and related properties of solutions but do not supply the free energy of hydration. Therefore, it is necessary to use an empirical thermodynamic approach to determine theoretical free energies of hydration. There have been several theoretical appro ache^^*'^ to compute the free energies of hydration of nonionic organic molecules, based on the group additivity concept. They are not applicable, however, to conformational dependence, because they are computed independently of any conformational changes, as discussed in part 2.4 In this work an application of the improved hydration shell model to ionic organic molecules, including carboxylate and protonated amine ions, is presented. Procedure A . Description of the Model. The free energy of hydration of a molecule is composed of additive contributions from each of +On leave from the Department of Chemistry, Chungbuk National University, Cheongju, Chungbuk 3 10, Korea.
0022-3654/87/2091-4118$01.50/0
its constituent groups or atoms. These contributions, in turn, are expressed as the product of the water-accessible volume, VwaJ,and the free energy density of hydration, Aghj,of the hydration shell of group j . A detailed derivation of these quantities has been presented in part 1,' and the main features have been summarized in part 2.4 The same procedure is followed here in treating charged groups. Each ionized groupj is assigned a hydration shell, with radius RhJand a uniform free energy density AghJ,as described below in section C. An earlier formulation, in terms of two kinds of water molecules in the hydration shell," has not been used here because of the lack of experimental data that must be used to determine the parameters of the model for the charged groups of interest here. Instead, the same free energy density is assumed for all parts of the hydration shell. This assumption corresponds to that introduced in the first formulation of the hydration shell model.* B. Geometry and Partial Atomic Charges. The procedure described in part 24 has been followed; Le., bond lengths and bond angles have been based on structural inf~rmation,'~,'~ and partial charges for each atom (including ionized groups) were determined by using the CNDO/2 (ON) method.14 C. Hydration Shell Parameters. The van der Waals radii, R,, the radii of the hydration shell, Rh,and the free energy densities of hydration, Agh, of all uncharged groups and atoms are those (1) Part 1: Kang, Y. K.; NCmethy, G.; Scheraga, H. A. J . Phys. Chem., first of four papers in this issue. ( 2 ) Gibson, K. D.; Scheraga, H. A. Proc. Natl. Acad. Sci. U.S.A. 1967, 58, 420. (3) Hodes, Z. I.; Ndmethy, G.; Scheraga, H. A. Biopolymers 1979, 18, 1565. (4) Part 2: Kang, Y. K.; NCmethy, G.; Scheraga, H. A. J . Phys. Chem., second of four papers in this issue. (5) Romano, S.; Clementi, E. Int. J . Quantum Chem. 1978, 14, 839. (6) Mezei, M.; Mehrotra, P. K.; Beveridge, D. L. J . Biomol. Struct. Dyn.
1984, 2, 1. (7) Alagona, G.; Ghio, C.; Kollman, P. J. Am. Chem. SOC.1986, 108, 185. (8) Jorgensen, W. L.; Gao, J. J . Phys. Chem. 1986, 90, 2174. (9) Cabani, S.; Gianni, P.; Mollica, V.; Lepori, L. J . Solution Chem. 1981, 10, 563. (10) Savage, J. J.; Wood, R. H. J . Solution Chem. 1976, 5, 733. ( 1 1) Paterson, Y.; NBmethy, G.;Scheraga, H. A. J. Solution Chem. 1982, 11, 831. (12) Sutton, L. E. Tables of Interatomic Distances and configuration in Molecules and Ions;The Chemical Society: London, 1958.
(13) Harmony, M. D.; Laurie, V. W.; Kuczkowski, R. L.; Schwendeman, R. H.; Ramsay, D. A,; Lovas, F. J.; Lafferty, W. J.; Ma Ki, A. G. J. Phys. Chem. Re$ Data 1979, 8, 619. (14) (a) Pople, J. A,; Segal, G. A. J . Chem. Phys. 1966, 44, 3289. (b) Momany, F. A.; McGuire, R. F.: Yan, J. F.; Scheraga, H. A. J . Phys. Chem. 1971, 75, 2286.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4119
Free Energies of Hydration of Solute Molecules TABLE I: Hydration Shell Parameters for Ionic Organic Molecules R"?b
Rkb
Agh,c kcal/ (m01.A~)
1.415 1.87 1.755 1.755 1.755 1.755 1.56
4.17 5.35 5.05 5.05 5.05 5.05 4.95
-2.95 X IO-'' -1.488 X lo-' -9.83 X lo-' -1.21 x 10-1 -1.32 X IO-' -1.14 X 10-I -9.29 X lo-'
A
atom or group protonated amine H carboxylate C protonated primary amine N protonated secondary amine N protonated tertiary amine N protonated aromatic amine N carboxylate 0
ref iond ethyl-, diethyl-, and triethylammonium
f
ethylammonium diethylammonium triethylammonium pyridinium propionate
"van der Waals radii of the atoms and united atoms taken from ref 15 and 16. bIdentical with the radii given for the corresponding uncharged atoms or groups in part 2.4 cThe precision of Agh corresponds to three significant figures. A fourth digit is added in order to reduce rounding errors. dUsed in the determination of Agh as an adjustable parameter. See text. eThis free energy density is applicable to protonated amines in monofunctional compounds only. In compounds containing two or more polar groups, Agh must be corrected for polarization effects, as discussed in ref 4. See also footnote 17. ?Value of Agh taken from part 2.4 1
I
I
AGhyd(AH)
I
I
I
Figure 1. A thermodynamic cycle used to determine the free energies of hydration for carboxylate ions, using experimental data obtained in the gas phase and in solution, as described in the text.
determined as described in part 2.4 They are listed in Table I of ref 4. For the atoms contained in the charged groups considered here, the three parameters are listed in Table I. The same values were used for the radiiIsJ6 R, and Rh as for the corresponding atoms in uncharged groups.4 The values of Agh for each atom have been determined in this work, by fitting the computed free energy of hydration of the compounds listed in the fifth column of Table I to the experimental free energy of hydration, as described beiow.l7J8 All free energy changes refer to isothermal transfers of the molecule from the ideal gas state at 1 M concentration to the hypothetical ideal 1 M aqueous solution at 25 O C . a. Carboxylate Zons. The experimental free energies of hydration of carboxylate ions were determined by means of the thermodynamic cycle shown in Figure 1. The free energy change of acid dissociation in water, AG:&(AH), can be obtained from the experimental value of pKa.19 The free energy of hydration of the neutral acid AH, &,,(AH) (corrected for the free energy of dissociation in water), was obtained from solubility data.9 The free energy change of acid dissociation in the gas phase, AGiCid(AH),was derived from high-pressure mass spectrometric measurements.20-22 The free energy of hydration of H+, AGhYd(H+),is based on the measured ionization potential of the (15) Ntmethy, G.; Pottle, M. S.; Scheraga, H. A. J . Phys. Chem. 1983, 87, 1883. (16) Dunfield, L. G.; Burgess, A. W.; Scheraga, H. A. J . Phys. Chem. 1978, 82, 2609. (17) The value of Agh derived here for protonated amine hydrogens is applicable only to monofunctional amines. As discussed in part 2,4 a correction for polarization of the hydration shell around the H atom must be made in bifunctional compounds (containing two or more polar groups). This correction will be applied in part 4 of this series,l*dealing with amino acids and peptides. (18) Kang, Y.K.; N h e t h y , G.; Scheraga, H. A,, in preparation. (19) IUPAC: Ionization Constants of Organic Acids in Aqueous Solution; IUPAC CDS-No. 23; Pergamon: Oxford, 1979. (20) The free energy change of acid dissociation in the gas phase, AGiCid(AH),is calculated as AGicid(AH) = Aqe,(AH) + AGf,,(HCl). Here, A@,,(AH) is the relative value referred to the free energy change for gaseous HCl.*' The values of AGfeI(AH)were taken from ref 21 and corrected to 298 K. (21) Kebarle, P. Annu. Reu. Phys. Chem. 1977, 28, 445. (22) Aecid(HCl)is obtained by using the data from: JANAF Thermochemical Tables, 2nd ed.; U S . National Bureau of Standards: Washington, DC, 1971; NSRDS-NBS 37.
TABLE 11: Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Carboxylate Ions' ionb acetate propionate* butyrate
calcdC -82.29 -78.30 -76.96
exptld -79.9 -78.3 -77.5
At
-2.4 0.0 0.5
"The free energy of hydration refers to the isothermal transfer of the molecule from the ideal 1 M gas state to the hypothetical ideal 1 M aqueous solution at 25 O C . bThe starred ions were used to obtain the parameters of See the text for details. cAverage absolute difference AAD = 1.5 kcal/mol and standard deviation S D = 1.2 kcal/ mol, on the basis of two ions. "Experimental values were obtained from the thermodynamic cycle shown in Figure 1 by using the experimental data for each step. See the text for details. A = calcd - exptl. TABLE 111: Comparison of Calculated and Experimental Free Energies of Hydration ( k c a l h o l ) for Protonated Amines" ionb
calcdc
exptl"
methvlammonium ethyl&nonium* n-propylammonium isopropylammonium n-butylammonium rerr-butylammonium dimethylammonium diethylammonium* di-n-propylammonium pyrrolidinium piperidinium trimethylammonium triethylammonium* 1-methylpyrrolidinium pyridinium* 4-methylpyridinium
-72.34 -68.40 -66.51 -64.68 -65.97 -61.59 -67.09 -58.89 -53.62 -63.20 -59.06 -61.03 -50.21 -56.23 -56.1 1 -55.35
-71.3 -68.4 -66.7 -65.5 -66.2 -63.1 -63.9 -58.9 -57.7 -61.6 -60.0 -56.6 -50.2 -54.6 -56.1 -54.0
Ae -1.0 0.0 0.2 0.8 0.2 1.5 -3.2 0.0 4.1 -1.6 0.9 -4.4 0.0 -1.6 0.0 -1.4
n,b,e See corresponding footnotes in Table 11. AAD = 1.7 kcal/mol and S D = 0.6 kcal/mol on the basis of 12 ions. "Free energy changes of hydration taken from ref 24 and converted to the standard state used in this paper.
free proton.23 Thus, by use of the above data, the free energy of hydration, AGhyd(A-), for the carboxylate ion (A-) can be calculated from the thermodynamic cycle of Figure 1. The values of Agh for the aliphatic CH, and CH2 groups and of the carboxylate C atom were taken from part 2:4 they are 1.577 X 3.533 X and -1.488 X lo-, kcal/(mol.A3), respectively. The free energy density of hydration of the 0 atoms in the carboxylate ion was determined from the computed wateraccessible volume (VWa)for each atom and the experimental AGhyd of the propionate ion
where the sum of the first three terms equals 0.2 kcal/mol. Thus, (23) Friedman, H. L.; Krishnan, C. V. In Water, A Comprehensiue Treatise; Franks, F., Ed.; Plenum: New York, 1973; Vol. 3, Chapter 1.
4120 The Journal of Physical Chemistry, Vol, 91, No. 15, 1987
AGh,O= -39.3 kcal/mol and Agh,o = -9.29 X lo-' kcal/(mol.A3) because the computed Vwa,o = 422.9 A3. b. Protonated Amines. Published values of the experimental values of AGhydwere sed.'^-'^ For the determination of the parameters, it was assumed (i) that the N atoms in the ethylammonium, diethylammonium, and triethylammonium ions make equal contributions to the free energy of hydration of the ions and (ii) that the free energy density of hydration, Agh,H, around the nitrogen-bound H atoms is the same in all ions. With these assumptions, Agh,H was found to be -2.95 X lo-' kcal/ (mol.A3), which is about 3 times greater than that of a nonionic polar H atom.4 For the N atom in protonated primary amines, Agh,N was computed from the calculated water-accessible volume and the experimental AGhyd of the ethylammonium ion. The diethylammonium ion was used to obtain the &h,N of the N atom in protonated secondary amines. The Agh,Nof the N atom in protonated tertiary amines was determined from the value of Agh," computed for the N atom in the trietylammonium ion. Agh,N of the protonated aromatic N atom was determined from experimental data on the pyridinium ion.
Results and Discussion Experimental and computed free energies of hydration for carboxylate ions and protonated amines are compared in Tables I1 and 111. The fourth column of each table lists the difference between the observed and calculated free energies of hydration. For each class and for the entire set of ions, the average absolute difference (AAD) and the standard deviation (SD) respectively are defined4 as
where N is the number of moelcules used in testing, Le., not used in obtaining hydration parameters. A . Carboxylate Ions. The results are shown in Table 11. Experimental data were available only for the three carboxylates listed. Propionate was used to determine the free energy parameter. The observed free energy is not matched well for the acetate ion because of its small size. This parallels the behavior of other classes of compounds for which it has been pointed out4 that the lowest homologues often show deviations from the regular trends. For this reason, acetate was not used in the parametri(24) Taft, R. W. Prog. Phys. Org. Chem. 1983,14,247. In this reference, relative free energy changes of hydration are given, referred to AGhyd(NH4+) = -78.9 kcal/mol (corrected for the standard states used here). (25) In ref 24, the free energy of hydration of each ion BH+ (as listed in Table 111) was evaluated by using (i) the free energies for the process NH4+ B BH++ NH, in the gas phase and in aqueous solution, respectively, calculated from basicities determined in the gas phase by ion cyclotron resonance s p e c t r o ~ c o p yand ~ ~ in ~ ~solution ~ from pK, measurements, (ii) free energies of hydration26of NH, and the neutral molecule B, and (iii) the free AGh d(NH4+) = -78.9 kcal/mol energy of hydration of the NH4+ (corrected from the value cited in ref 24 for tge standard state used here). (26) Taft, R. W.; Wolf, J. F.; Beauchamp, J. L.; Scorrano, G.; Arnett, E. M. J. Am. Chem. SOC.1978, 100, 1240.
+
Kang et al. zation. On the other hand, the close fitting of the free energy of hydration for the butyrate ion suggests that the model works well for carboxylate ions attached to larger molecular fragments (ethyl or larger alkyl groups in the present examples). B . Protonated Amines. Computed and experimental values of AGhydare compared in Table 111 for several protonated amines. The AAD and S D are 1.7 and 0.6 kcal/mol, respectively. In the case of the aliphatic alkylammonium ions R,NH,-,+ (with m = 1, 2, 3), the largest deviations occur when R = CH3, Le., for the lowest homologues of each series, in agreement with the results for other classes of compounds: mentioned above. Relatively large deviations occur also for the tert-butyl- and di-n-propylammonium ions, Le., for bulky substitutions next to the N atom. Similar effects of large substituents have been seen for other classes of compounds as wek4 On the other hand, the agreement is very good for primary alkylammonium ions with large alkyl substituents (isopropyl and n-butyl). Deviations for cyclic molecules fall into an intermediate range, up to 2 kcal/mol.
Conclusions The form of the hydration shell model used here results in a satisfactory representation of the hydration free energies of carboxylate ions and alkylammonium ions, with errors of about 2 kcal/mol or less for many ions, except for those with one or more methyl groups or bulky substituents next to the charged group. Deviations are small when the charged group is attached to only one large (branched or unbranched) alkyl chain. This is significant, because it suggests that the method is applicable to treating charged amino and carboxyl groups in peptides and proteins, viz. the terminal a-amino and carboxyl groups of the backbone and the side chains of aspartic acid, glutamic acid, and lysine,27as well as probably other side chains. This conclusion is supported also by the relatively low deviations computed for cyclic molecules with a charged nitrogen modeling ionizable amino acid side chains. Applications of the model to the study of hydration free energies of peptides are in progress. Acknowledgment. We thank Dr. K. D. Gibson for many helpful discussions. This work was supported by research grants from the National Science Foundation (DMB84-018 11) and from the National Institute of General Medical Sciences (GM-143 12) and the National Institute on Aging (AG-00322) of the National Institutes of Health, U S . Public Health Service. Support was also received from the National Foundation for Cancer Research. Y.K.K. thansk the Korea Science and Engineering Foundation for support. Supplementary Material Available. Supplementary material (30 pages), consisting of tables containing the computed partial charges for all compounds considered in this paper and in the accompanying paper,4 is available on microfiche or photocopy. See NAPS document No. 04487. (27) The method is applicable to lysine, but a numerical correction for polarization effects must be added for this multifunctional compound, as described in footnote 17.
