J . Phys. Chem. 1987, 91, 4109-4117 of the centers of the spheres, u, can be computed. b. Computation of the Volume v:. The volume of this segment can be derived in a manner analogous to that of u,, because the volume of the segments does not depend on the orientation of the axes used, Thus, if the two spheres s and t are interchanged and all indices s and t are permuted, the two angles 8, and 4, become X, - 0, and x,- T,, respectively, where T , = X, - cos-' (hr/rr)(see Figure 4C). With these substitutions, the derivation presented in eq 17-28 is applicable, and eq 20 becomes
v,' = rI3((a/3)[1 - cos (x,- r,)12[1 + y2 cos (x,- T , ) ] y3 sin-] [sin (x,- 8,)/sin (x,- T,)] + cos (x,- ~ , ) [ 1- y3 cos2 (x,- T , ) ] sin-' [tan (x,- Or)/ tan (xr - 7 r ) I + X cos3 ( x r - Tr) tan (xr - o r ) X [tan2 (x,- T,) - tan2 (x,(29) c. Computation of the Other Volumes v, and vm', with m = s or t . By cyclical permutation of the indices r, s, and t and corresponding permutation of the definitions of the various angles in eq 20 and 29, all other volumes can be computed. The total triple overlap volume V,,, is then given by eq 16. An alternative derivation of the volume of intersection of three overlapping spheres of unequal size is given by Gibson and S ~ h e r a g a .Although ~~ the two expressions differ in form, they give identical numerical answers. For example, for three spheres with radii r, = 1.0, r, = 2.0, and r, = 3.0 A and with their centers separated by the distances rrs = 2.0, rst = 4.0, and rrt = 3.0 A, respectively, the triple overlap volume V,,, calculated from eq 16 is 0.5736 A3, in exact agreement with the result obtained by Gibson and S ~ h e r a g a . ~ ~ d . Numerical Test. Equations 16, 20, and 29 were tested by the calculation of the total volume and the overlap volume of three spheres of unit radii and unit separation distances, because these results can be compared with the calculation by R o ~ l i n s o n . ' ~ From the coordinates of the centers and the radii of the spheres, it is found that xr = 713, 4, = a/3, 0, = n/6,and r, = 0. Hence, u, = u,' = 0.1 120. The value of the triple overlap volume is urSt = 0.6718 which is exactly the same value as that given by Row1ins0n.I~Also, eq 14 gives a double overlap volume between two spheres as uSt= 1.3090. The value of the total volume of these (36).Gibson, K. D.; Scheraga, H. A. J . Pbys. Chem., fourth of four papers in this issue.
4109
three overlapping spheres is 9.31 19, and hence the volume per sphere for this example is equal to 3.1037. The latter is the same value as obtained by numerical integration of the analytical surface area function by Richmond.23 Conclusions
The hydration shell model has been improved by an exact computation of the water-accessible volume of each group in a molecule, in which the water-excluded volume of the hydration shell of each group, due to neighboring groups, is expressed in terms of double and triple overlap volumes of groups. A general expression for the triple overlap volume has been derived and tested. This expression can be used in computations of the overlap between any kind of spheres, e.g., van der Waals spheres (in the computation of volumes of molecules), or of van der Waals spheres and hydration shells, as used here. In the following papers of this series, the hydration shell model will be applied to the calculation of the free energies of hydration for neutral28and charged29organic molecules and peptides,jO and the results will be discussed in comparison with experimental data. The free energy of hydration of these molecules will be computed from the water-accessible volumes, computed as described here, and empirical free energy densities of hydration (cf. eq 9) derived from experimental thermodynamic data.
Note Added in Prooj After the work described here had been started, two papers were published by Lustig in which alternative derivations are given for the intersection of three and four spheres.37 These derivations, like that of R o ~ l i n s o n , apply '~ to spheres of equal sizes, while the derivation presented here is applicable to spheres of any size. Acknowledgment. We thank Dr. K. D. Gibson for many helpful discussions. This work was supported by research grants from the National Science Foundation (DMB84-01811) 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. thanks the Korea Science and Engineering Foundation for support. (37) Lustig, R. Mol. Phys. 1985, 55, 305. 1986, 59, 195
Free Energies of Hydration of Solute Molecules. 2. Application of the Hydration Shell Model to Nonionic Organic Molecules Young Kee Kang? George NCmethy, and Harold A. Scheraga* Baker Laboratory of Chemistry, Cornel1 University, Ithaca, New York 14853- 1301 (Received: December 19, 1986) An improved formulation of the hydration shell model is described for determining the free energies of hydration of conformationally flexible solute molecules. Twenty-seven compounds are used to obtain the hydration parameters, viz. the radius of the hydration shell and the free energy density of hydration in the shell. The model is tested by the calculation of the free energies of hydration of 130 uncharged organic molecules (not counting those used in determining the parameters), including aliphatic and aromatic hydrocarbons, ethers, alcohols, ketones, esters, acids, amines, amides, sulfides, thiols, and polyfunctional molecules containing oxygen and nitrogen atoms. The average absolute difference between the calculated and experimental values and the standard deviation are 0.46 and 0.07 kcal/mol, respectively. These correlation coefficients demonstrate t h e validity of t h e model.
As indicated in the Introduction of the first paper of this series' (to be referred to as part I), hydration plays an important role
in determining the conformation and function of molecules in aqueous solution. In Part 1, a method for computing the average free energy of hydration, AGhyd. of a flexible solute molecule has
'On leave from the Department of Chemistry, Chungbuk National University, Cheongju, Chungbuk 3 10, Korea.
(1) Part 1: Kang, Y.K.; NBmethy, G.; Scheraga, H. A. J . Pbys. Cbem., first of four papers in this issue.
Introduction
0022-3654/87/209 1-4109$01.50/0
0 1987 American Chemical Society
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The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
been proposed, based on a hydration shell model and on the use of empirical potential functions. Our earlier hydration shell mode12q3has been improved by calculating the water-accessible volume, VwaJ,of each groupj in a molecule exactly. This volume can be computed' from the double and triple overlap volumes of groups in a molecule. Monte Carlo and molecular dynamics simulations of nonionic organic molecules in aqueous solutions represent a fundamental source from which hydration numbers, energies of hydration, and the volume change of solution can be d e t e r m i ~ ~ e dNevertheless, .~~ few thermodynamic parameters for practical use (viz. free energies of hydration) have been obtained so far from such simulations, because of some problems with current simulation techniques.s The direct calculation of the free energy of hydration of only N-acetyl-L-alanyl-N'-methylamidewas reported with the Monte Carlo method by Mezei et a1.6b A molecular dynamics simulation study of the same molecule yielded the energy but not the free energy of h y d r a t i ~ n . ' ~ Thermodynamic properties associated with the transfer of nonionic organic molecules from gas to dilute aqueous solution have been described repeatedly in terms of simple schemes of group contribution^.^-'^ In such schemes, molecules are subdivided into groups, each of which is assumed to make a constant contribution to the thermodynamic quantity, independent of any conformational changes of the molecule. The concept of group additivity was applied to pairwise interactions between solute molecules in aqueous solution by Wood et aI.l3 It was assumed in these studiesg-I3 that a given kind of group in different molecules contributes a constant amount to the thermodynamic properties of the molecules. Therefore, changes in the thermodynamic parameters that correspond to different conformations cannot be obtained by these approaches. In contrast, our group contribut i o n ~ ~(including ~ ~ * ' ~ those in the work presented here) pertain to the free energy of hydration which is a product of two factors, one which is intrinsic to a given group and is independent of conformation and another which depends on the conformation. Recently, an alternative approach to the computation of the free energy of hydration of solute molecules has been pr0posed.'~3'~ The basic assumption is that the free energy of interaction of a given conformation of the solute with water can be obtained as a sum of contributions by functional groups and that these contributions are proportional to the water-accessible surface areas of the groups.16 Comparisons of this alternative and of the present approach are in progress. In this work, we assume a uniform free energy density of hydration, AghJ,around each group j , while the previous model3 ~~
(2) Gibson, K. D.; Scheraga, H. A. Proc. Natl. Acad. Sci. U.S.A. 1967, 58, 420.
(3) Hodes, Z. I.; Nemethy, G.; Scheraga, H. A. Biopolymers 1979. 18, 1565. (4) (a) Owicki, J. C.; Scheraga, H. A. J . A m . Chem. Soc. 1977, 99, 7413. (b) Rapaport, D. C.; Scheraga, H. A. J . Phys. Chem. 1982.86, 873. (5) (a) Jorgensen, W. L.; Gao, J.; Ravimohan, C. J . Phys. Chem. 1985, 89,3470. (b) Jorgensen, W. L.; Madura, J. D. J. Am. Chem. SOC.1983,105, 1407. ( c ) Jorgensen, W. L.; Swenson, C. J. J . A m . Chem. SOC.1985, 107, 1489. (6) (a) Marchese, F. T.; Mehrotra, P. K.; Beveridge, D. L. J . Phys. Chem. 1984, 88, 5692. (b) Mezei, M.; Mehrotra, P. K.; Beveridge, D. L. J . Am. Chem. Sot. 1985, 107, 2239. (7) (a) Bolis, G.; Corongiu, G.; Clementi, E. Chem. Phys. Lett. 1982,86, 299. (b) Alagona, G.; Tani, A. Chem. Phys. Lett. 1982,87,337. (c) Rossky, P. J.; Karplus, M. J . A m . Chem. Soc. 1979, 101, 1913. (d) Zichi, D. A,; Rossky, P. J. J . Chem. Phys. 1986, 84, 1712. (8) Ntmethy, G.; Peer, W. J.; Scheraga, H. A. Annu. Rev. Biophys. Bioeng. 1981, 10, 459. (9) Tanford, C. J. Am. Chem. SOC.1962, 84, 4240. (10) Hine, J.; Mookerjee, P. K. J. Org. Chem. 1975, 40, 292. (1 1) Guthrie, J. P. Can. J . Chem. 1977, 55, 3700. (12) Cabani, S.; Gianni, P.; Mollica, V.; Lepori, L. J . Solution Chem. 1981, I O , 563. (13) (a) Savage, J. J.; Wood, R. H.J . Solution Chem. 1976,5, 733. (b) Suri, S . K.; Spitzer, J. J.; Wood, R. H.; Abel, E. G.; Thompson, P. T.J . Solution Chem. 1985. 14. 781. (14) Ooi, T.; Oobatakk, M.; Ntmethy, G.; Scheraga, H. A. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 3086. (15) Eisenberg, D.; McLachlan, A. D. Nature (London) 1986, 319, 199. (16) Langmuir. I Colloid Symp. Monogr. 1925, 3, 48
Kang et al. was designed to account for the interaction of polar and nonpolar groups of the solute with water in terms of qualitatively different specific (hydrogen bonding) and nonspecific contributions. The present model,' proposed in part 1, is applied here to compute the free energies of hydration for uncharged organic molecules. Based on the considerations presented in part 1 and in this paper, the free energies of hydration for ionized organic molecules such as carboxylate ions and protonated amines will be calculated in part 3 of this series."
Procedure A . Description of the Model. a. General. A general theory of the free energy of hydration of solute molecules was described in terms of the hydration shell model and empirical potential functions in part 1. The total free energy of hydration of a molecule is computed from the sums of the statistically weighted energies of each conformation of the molecule in the hydrated and unhydrated states, respectively
where AGy) is the total free energy of conformation i in the hydrated state and w iis the normalized Boltzmann factor given by wi = exp(-AG{')/RT) (2) with
Q = Cexp(-AGp)/RT) i
(3)
AEU) is the relative conformational energyls.l9 of the solute in conformationj in the unhydrated state, and w,' is its Boltzmann statistical weight. The free energy AGr) is calculated as a sum of intramolecular and intermolecular interaction terms, i.e. k
(4)
fi& is the water-accessible volume of the hydration shell around group k, and Agh,kis the free energy density of hydration for group k . The volume fld!,k of a group k may vary from one compound to another and depends on the conformation of the molecule, while is a constant for a given group k in all monofunctional compounds and all their conformations.' For polyfunctional molecules containing several N and/or 0 atoms, it has been shownI2 that a correction is necessary to account for interactions between polar groups, in order to obtain close-fitting values of free energies of hydration. In the next section of this paper, we shall describe such a correction of Aghk,in terms of the polarization of the hydration shell of a given polar group by other nearby polar groups. In the present work, the earlier assignment of a separate "specific" hydration term (corresponding to well-formed solutewater hydrogen bonds)3 has been abandoned, because of the lack of sufficient experimental data on hydrogen bonding for these interactions, especially when there is a variation of hydrogen-bond strengths as a result of stretching or bending of the hydrogen bond (occurring because hydrogen-bonded water molecules do not have completely fixed positions next to the solute). This uncertainty did not warrant the increase in computational time required for specific hydration. Overlaps between the hydration shells themselves are assumed not to influence the free energy of hydration of groups, just as in previous studies.*J It is assumed that water molecules continuously displace each other without altering the free energy of hydration of either group, when hydration shells of the two groups overlap, because the water molecules in the common region of the two hydration shells interact with both groups. (17) Part 3: Kang, Y. K.; Ntmethy, G.; Scheraga, H. A. J . Phys. Chem., third of four papers in this issue. (18) Momany, F. A,; McGuire, R. F.; Burgess, A. W.; Scheraga, H. A. J. Phys. Chem. 1975, 79, 2361. (19) Ntmethy, G.; Pottle, M. S.; Scheraga, H. A. J . Phys. Chem. 1983, 87, 1883.
Free Energies of Hydration of Solute Molecules
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4111 Le., for group 1 located inside and outside the hydration shell of group k , respectively, where rkl is the distance between groups k and 1. These equations are derived in the Appendix. In eq 5 and 6, the only new adjustable parameter is th. If there are several polar groups 1 around group k, the total polarization free energy is given by
Because of the practical difficulty in carrying out the computation of AGp,kfor a hydration shell of arbitrary shape (since the atoms are not isolated spheres but are connected by bonds), eq 5 and 6 were derived in the Appendix for a model in which group k is surrounded by a complete spherical shell with dielectric constant t h . Therefore, the polarization free energy density, Agp,k, must be. obtained by dividing AGp,kby the volume of this complete spherical shell, i.e.
Figure 1. Definition of geometrical parameters used in deriving the polarization of the hydration shell of a polar group k by the interacting polar group I , located inside the hydration shell of group k. Rv,kand Rh,k are the radii of the van der Waals sphere and the hydration shell of group k, respectively. ei, eh, and e, are dielectric constants in the cavity occupied by group k in the hydration shell (shaded) and in the bulk, respectively. q k and q, are net charges of groups k and I , respectively, and rkl is a distance between them. M is an arbitrary point (that can be located in any of the three regions shown) where the electric potential is measured as described in the Appendix; r and s are the distances from M to k and 1, respectively, and 8 is the angle between r and r k p
In part 1,' the method to compute vi;,$ was described in detail. The procedure to obtain the other necessary parameters for each group k , viz. the van der Waals radius Rvsk,the radius of the hydration shell Rh,k, and the free energy density will be described here in section C. b. Polarization of the Hydration Shell in Polyfunctional for a given polar group k of Molecules. The value of polyfunctional molecules may be modified by the presence of other polar groups. This effect can be described in terms of electrostatic polarization of the hydration shell of group k by the other polar groups. Let us consider a polar group k , imbedded inside concentric shells of dielectric continua, and another polar group 1 that can be located either inside or outside the hydration shell of group k (Figure 1). The central cavity, containing group k , has a net charge qk and is characterized by the van der Waals radius Rv,k and the dielectric constant e,, The hydration shell of group k, with radius Rh,+is considered to consist of a polarizable dielectric medium with the dielectric constant ' h . The quantity 6 does not appear in the expression for the polarization free energy (see Appendix). Extending beyond this hydration shell is the bulk solvent with dielectric constant eo. In the present work, eo = 78.4 (at 25 "C). For the purpose of this correction, the polar group I is assumed to be a point, with net charge qe in order to simplify the model. Following derivations by Kirkwoodm and by Beveridge and Schnuelle*l for the reaction electrostatic potential of the charged cavity in concentric dielectric continua,22the contribution to the free energy of hydration of a polar group k, resulting from the polarization of its hydration shell by the polar group 1, can be written as
and
(20) Kirkwood, J. G. J . Chem. Phys. 1934, 2, 351. (21) Beveridge, D. L.; Schnuelle, G. W. J. Phys. Chem. 1975, 79, 2562. (22) Linder, B. Adu. Chem. Phys. 1967, 12, 225.
where Vwa,kois the volume of a complete spherical shell around group k as defined in eq 1 of part 1.' Hence, eq 4 must be modified for polyfunctional molecules by adding a polarization term, so that it becomes
The quantity &h,k is a fixed parameter for each type of group (see Table I). For a specific conformation, the quantity Agp,kmust be calculated from the coordinates and charges of all atoms, by is calculated as described means of eq 5-8, and the volume in part 1;' is the corresponding intramolecular ECEPP energy for the given conformation. B. Geometry and Partial Atomic Charges. The geometry used for the nonionic organic molecules is that given in the structural literature2, for each molecule. The positions of the hydrogen atoms were chosen so as to represent refined hydrogen positions, determined from the literature on electron and neutron diffra~tion.~, In the case of molecules for which no information on bond lengths and bond angles is available, the geometry was taken to be the same as that of the similar molecules with known structure. The partial charges qk for each atom k of the molecules were determined by using the CNDO/2 (ON) method.24 These charges are listed in the supplementary material.25 In conformational energy calculations, bond lengths and bond angles were fixed and only the dihedral angles for internal rotation have been taken as the variables. C. Hydration Shell Parameters. The functional groups or atoms of the solute are assigned the van der Waals radii, R,, used in the ECEPP'8,19and UNICEPP22algorithms for conformational energy computations on peptides. Aliphatic and aromatic CH,, CH2, and C H groups are represented as united atoms26for the calculation of h y d r a t i ~ n . ~ ' A unique value of the hydration shell radius, Rhris assigned to each kind of atom (or central atom of a group), irrespective (23) (a) Sutton, L. E. Tables of Interatomic Distances and Configuration in Mofecules and rons; The Chemical Society: London, 1958. (b) Harmony, M. D.; Laurie, V. W.; Kuczkowski, R. L.; Schwendeman, R. H.; Ramsay, D. A.; Lovas, F. J.; Lafferty, W. J.; Maki, A. G.J . Phys. Chem. ReJ Data 1979, 8 , 619. (24) (a) Pople, J. A.; Sepal, 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. (25) Supplementary material (30 pages), consisting of tables containing the computed partial charges for all compounds considered in this paper and in the accompanying paper," is available on microfiche or photocopy. See NAPS document No. 04487. (26) Dunfield, L. G.;Burgess, A. W.; Scheraga, H. A. J . Phys. Chem. 1978, 82, 2609. (27) The united atom representation is used to describe the interaction of the group mentioned with the s o l ~ e n tbecause , ~ ~ ~ the hydration contributions of the C and H atoms cannot be separated easily. The computation of the intramolecular conformational energy does not depend on the presence or absence of hydration. Therefore, the present model (with united atoms used to compute the hydration contribution) can be used in conjunction with intramolecular energies computed in either the full atom (as in ECEPP)'8,19or the united atom (as in UNICEPP)26representation.
4112 The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
Kang et al.
TABLE I: Hydration Shell Parameters atom or group hvdroxvl. carboxvl, amine. or amide H polyfunctional hydroxyl, amine H thiol H aliphatic C H I aliphatic CH, aliphatic C H aliphatic C alicyclic CH, alicyclic C H alicyclic C aromatic C H aromatic C bridged aromatic C of fused rings aromatic C bonded to 0 carbonyl or carboxylic C primary amine N secondary amine N tertiary amine N alicyclic amine N aromatic N amide N ether or hydroxyl 0 ester or carboxylic 0 carbonyl 0 ester, amide, or carboxylic carbonyl 0 thiol or sulfide S S bonded to aromatic carbon I
I
RWa
Rhi
1.415 1.415 1.415 2.125 2.225 2.375 2.06 2.225 2.375 2.06 2.10 1.85 1.85 1.85 1.87 1.755 1.755 1.755 1.755 1.755 1.755 1.62 1.62 1.56 1.56 2.075 2.075
4.17 4.17 4.17 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.35 5.05 5.05 5.05 5.05 5.05 5.05 4.95e 4.95 4.95 4.95 5.37 5.37
A
A
k , b
kcal/(mol.A3) -1.057 X lo-’ -5.470 X lo-’ 6.065 X lo-’ 1.577 X lo-’ 3.533 x 10-4 -4.151 x 10-4 -1.488 X lom3 4.469 X lo4 -6.783 X -1.488 X IO-’ -2.734 X -1.991 X IO-’ -1.410 X IO-’ 6.298 X -1.488 X lo-’ -1.352 X IO-’ -9.958 X lo-’ -1.569 X IO-’ -8.656 X -9.409 X lo-’ -5.371 X lo-’ -9.450 x 10-3 -5.334 x 10-3 -1.218 X -5.334 x 10-3 -6.737 X IO-’ -3.777 x 10-3
ref compd‘ 2-propano1, 1-propanol 2-methoxyethanol, 2-methoxyethanamine ethanethiol ethane propane isobutane neopentane cyclopropane methylcyclopentane d benzene toluene naphthalene phenol
d n-propylamine diethylamine triethylamine azetidine 2-methylpyridine acetamide diethyl ether and methyl n-propyl ether n-propyl acetate 2-pen tanone f
diethyl sulfide methyl phenyl sulfide
Ovan der Waals radii of the atoms and united groups taken from ref 18, 19, and 25. *The precision of Agh corresponds to three significant figures. A fourth digit is included in order to reduce rounding errors. ‘Used in the determination of adjustable parameters. See text. dAssumed to be the same as the aliphatic C. ‘Determined as described in the text. /Assumed to be the same as the ether 0 of esters.
of its bonding state. Thus, the same value of Rh is used for all C atoms and C-containing groups, viz. CH3, CH2,CH, and C in both aliphatic and aromatic compounds. Similarly, all N atoms have the same Rhras do all 0 atoms, all s atoms, and all ”polar” H atoms (Le., hydrogens attached to N or 0),as shown in Table I. Table I lists the parameters of the hydration shell. For each type of atom or group listed, Agh has been obtained by fitting the computed free energy of hydration of only one compound (listed in the fifth column of Table I) to the experimental free energy,28 while Rh is determined either from some structural consideration or by fitting some experimental free energies of hydration. The procedures used to obtain each of these adjustable parameters are described below in detail for each class of compounds. In this study and in subsequent work,” all free energy changes refer to the isothermal transfer of the molecule from the ideal gas state at 1 M concentration to the hypothetical ideal 1 M aqueous solution at 25 “C. a. Aliphatic Hydrocarbons. The radius of the hydration shell of all groups containing carbon atoms was assigned as Rh,C= 5.35 A. This value corresponds to the location of the first minimum in the C-.O radial distribution functions of the Monte Carlo simulation for methane and other alkanes in water at 25 ‘CSa Then, iigh,CH, for the aliphatic CH, group was obtained by using the experimental value of ethane in the following way.2s Because ethane has only one stable rotameric conformation (the fully staggered one) and the computed water-accessible volume of each CH3group is calculated to be Vw/wa,CH3 = 580.4 A3 (computed by the method of ref l ) , &h,CH3 is obtained by using the relation Substituting the value of AGh,CH, and AE = 0.0 into eq 1 and 4 gives Agh,CH, = 1.577 x 1 0 - ~kcal/(mol.A3). Next, Agh,CH2is computed from the experimental AGhjd for propane. The relation 2 A G h . c ~-t~ AGh,CH2 = AGg;$Lropane = 1.96 kcal/mol ~
~~
(1 1)
~
(28) All the experimentalvalues of the free energy of hydration were taken from ref I 2 and references cited herein, except as noted
is used, with 2AGhcH, = 1.76 kcal/m01.~~Using the computed v ~= 557.6 ~ Ai ,and substituting ~ ~ into ~ eq 1 and 4 give Agh,CH2 = 3.533 X lo4 kcal/(mol.A3). Application of the same procedure to the experimental AGhydof isobutane and neopentane, respectively, provides the values of Agh listed in Table I for the aliphatic CH group and the C atom with no attached hydrogen. The values of Agh for alicyclic CH2 and CH were obtained from the experimental free energies of hydration of cyclopropane and methylcyclopentane, respectively. b. Aromatic Hydrocarbons. Rh was assigned the same value as for aliphatic hydrocarbons. Benzene was used to obtain Agh,CH of the aromatic CH group. The Agh,c of the aromatic C (with no hydrogen attached) was calculated by using the data for toluene. For carbon atoms bonded to three other carbons in fused aromatic rings, Agh,c was obtained from naphthalene. The numerical values are listed in Table I. c. Ethers. Experimental data for two ethers were used to determine the value of both parameters of the hydration shell, viz. the radius and the free energy density. Rh,Ohad to be treated as an adjustable parameter, because use of the position of the first minimum in the O-.O radial distribution function in a Monte Carlo simulation of methanol in watersb (in analogy with the proceudre described for C in hydrocarbons) gives poor agreement for the free energies of hydration of ethers. Therefore, Rh,Owas varied so as to optimize the fit for the free energy of hydration of methyl n-propyl ether. For each choice of Rh,O,a corresponding value of Agh,o was calculated by fitting diethyl ether alone. Best fitting for both ethers was obtained with the choice of Rh,o= 4.95 8, and i i g h , O = -9.450 X kcal/(mol.A3). The same values were also used for cyclic ethers, because experimental data are available only for a few cyclic compounds. d . Alcohols. Rh and Agh of the 0 atom in aliphatic and aromatic alcohols were assigned to be the same as those of the ether oxygen. The two parameters for the hydroxyl H were (29) This value of AGh,CH3was obtained as the product of Vw.,cH3calculated for the hydration shell around the methyl groups in propane and of Agh,CH,derived for ethane. The numerical values of Vwa,cHare different in ethane and propane because the volume excluded by neighboring groups is different for the compounds, as explained in connection with eq 9 of paper I .’
Free Energies of Hydration of Solute Molecules determined by fitting the experimental free energy of hydration of 2-propanol and 1-propanol, using a procedure similar to that described above for the ether oxygen. Rh,H was varied so as to optimize the agreement for 1-propanol. For each choice of the radius, Agh,Hwas determined by fitting the free energy of hydration for 2-propanol alone. N o choice of Rh,H allowed perfect fitting for both alcohols, but the best fitting was obtained with the value of Rh,H = 4.17 A, with an error of 0.32 kcal/mol for 1-propanol. Subsequently, the same values of the hydration shell radius and free energy density were used for all polar hydrogens (attached to 0 or N). The same 0 and H parameters were used for aliphatic and aromatic alcohols. For the aromatic C atom bonded to the oxygen, Agh,c was fitted to the experimental data for phenol. e. Ketones. The hydration shell radii determined above were used for the carbonyl C and 0,together with the Agh,c for C with no hydrogen attached. Then, Agh,Owas determined from the experimental free energy for 2-pentanone. f. Esters. The value of Agh,o for both oxygen atoms of the ester was determined by using the data of propyl acetate and assuming that the Aghvalues of both 0 atoms of the COO group in esters are the same. g. Carboxylic Acids. No new parameters had to be introduced for this class of compounds. The parameters for the C and 0 atoms were those used in esters, while the parameters for the H atoms were those of hydrogens in hydroxyl groups. h. Amines. The values of Rh and Agh of the H atom in amines were assumed to be equal to the values of the H atom in alcohols. This assumption was used in order to reduce the number of adjustable parameters in the model. The thickness of the hydration shell around nitrogen atoms was assumed to be the average of the thicknesses of the shells around c and 0 atoms.30 This gives R h , N = 5.05 8,as the radius of the hydration shell. For the N atom in primary amines, Agh,N was determined from the free energy of n-propylamine. Agh,N of the N atom in secondary amines was determined from the data on diethylamine. For the N atom of the tertiary amines, was computed from experimental data for triethylamine. Azetidine was used in the calculation of Agh,N of the N atom in alicyclic amines. For the aromatic N atom, Agh,N was assigned from the experimental free energy of hydration of 2-methylpyridine. i. Amides. The CO group of amides was considered to be identical with the CO group in esters. The values of the parameters for the H atom in the amide group are identical with the values of the amine H atom. Agh,Nof the N atom is obtained from the experimental free energy of hydration of acetamide. j . Sulfides and Thiols. Because there are few experimental data, it was assumed that the thickness of the hydration shell of the S atom in sulfides is equal to that of the N atom. This choice was made because S forms weaker hydrogen bonds than 0. The results are not sensitive to the choice of Rh,s, The value of Rh,S is thus determined to be 5.37 A. The value of Agh,s of the S atom in aliphatic compounds was obtained from the experimental free energy of hydration of diethyl sulfide. For the S atom bonded to an aromatic C atom, Agh,s was calculated from the data on methyl phenyl sulfide. Ethanethiol was used to obtain the value of Agh,H of the H atom in the SH group. k . Polyfunctional Molecules Containing 0 and N Atoms. It is knownI2 that the solution thermodynamic parameters of bifunctional molecules containing two or more 0 and N atoms, in which two functional groups are usually separated by two or three C atoms, are not additive; Le., the free energy of hydration of a bifunctional molecule cannot be expressed as a sum of the group contributions obtained from monofunctional compounds. The perturbation of the additivity in the free energy may arise from interactions either between neighboring polar groups or between (30) Calculations with various choices of Rh,Nas an adjustable parameter (as descnbed for Rh.0 of ethers above) have shown that the computed results are not sensitive to the choice of Rh,N, and therefore the fitting procedure cannot be used to determine the choice of Rh,N. Consequently, it does not matter what value of Rh,Nis used.
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4113 TABLE II: Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Aliphatic Hydrocarbons" moleculeb calcd exptlC Ad8 ethane* propane* n-butane isobutane* n-pentane isopentane neopentane* n-hexane isohexane 3-methylpentane 2,2-dimethylbutane cyclopropane* cyclopentane methylcyclopentane* cyclohexane methylc yclohexane
1.83 1.96 2.07 2.32 2.22 2.41 2.50 2.39 2.55 2.51 2.60 0.75 1.16 1.36 1.35 1.54
1.83 1.96 2.08 2.32 2.33 2.38 2.50 2.49 2.52 2.5 1 2.59 0.75 1.20 1.36 1.23 1.71
0.00 0.00 -0.01 0.00 -0.1 1 0.03 0.00 -0.01 0.03 0.00 0.01 0.00 -0.04 0.00 0.12 -0.17
"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 OC. *The starred molecules were used to obtain the numerical values of Agh. See the text for details. CExperimental values taken from ref 12 and references therein. d A = calcd - exptl. eAverage absolute difference AAD = 0.05 kcal/mol and standard deviation SD = 0.02 kcal/mol, on the basis of 10 molecules not used to obtain hydration parameters.
these groups and the water molecules surrounding the two groups. In order to correct for these interactions, eq 9 is used instead of eq 4 to calculate the free energy of hydration of polyfunctional molecules. The dielectric constant of the hydration shell, th, was used as an adjustable parameter. Its value of 2.78 was determined from the best fit of the free energy of hydration of 1,2-dimethoxyethane to its experimental value. The same value of t h is then used for all polar groups. Another correction is necessary for bifunctional compounds in which one of the polar groups contains a hydrogen atom (i.e., OH, N H , N H 2 groups), in order to fit the observed free energies. For this case, the value of Agh,H= -5.470 X kcal/(mol.A3) was obtained as the average of fitting the observed free energies of hydration of 2-methoxyethanol and 2-methoxyethanamine. This value is more positive than the Agh,H = -1.057 X kcal/ (mol.A3) obtained above for monofunctional alcohol and amine hydrogens; Le., hydration appears to be weaker in the bifunctional compounds. This may be caused by the perturbation of tight hydration around the small polar H atom by a nearby polar group.
Results and Discussion Experimental and computed free energies of hydration for the ten classes of compounds considered here are compared in Tables 11-XI. The tables also include those compounds (marked by asterisks) that were used to determine the parameters of the hydration shell model. 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 molecules, the average absolute difference (AAD) and the standard deviation (SD) respectively are defined as AAD = N-IxIAGCaICd hyd - AGexPtl hyd I
(12)
and SD = N-'[x(AGi$d
- AG$$t')2]1/2
(13)
where N is the number of molecules used in testing, Le., not used in obtaining hydration parameters. A . Aliphatic Hydrocarbons. Experimental and calculated free energies of hydration of 16 aliphatic hydrocarbons, including six alicyclic molecules, are shown in Table 11. For the 10 molecules used to test the model, AAD = 0.05 kcal/mol and S D = 0.02 kcal/mol. The largest differences (A), occurring for the two cyclic molecules, do not exceed 0.2 kcal/mol. Thus, the hydration shell model presented here gives very good results for aliphatic hydrocarbons.
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TABLE 111: Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Aromatic Hydrocarbons“ moleculeb calcd exptl‘ Ad,e benzene* -0.87 -0.87 0.00 toluene* -0.89 -0.89 0.00 o-xylene m-xylene p-xylene ethylbenzene 1,2,4-trimethyIbenzene biphenyl naphthalene* a-methylnaphthalene 1,3-dimethylnaphthalene 1,4-dimethylnaphthalene 2,3-dimethylnaphthalene 2,6-dimethylnaphthalene fluorene anthracene phenanthrene pyrene
-0.91 -0.87 -0.88 -0.66 -0.82 -3.07 -2.39 -2.35 -2.25 -2.28 -2.28 -2.38 -3.36 -3.77 -3.71 -4.89
-0.90 -0.84 -0.81 -0.80 -0.86 -2.64 -2.39 -2.37 -2.47 -2.82 -2.78 -2.63 -3.44 -4.23 -3.95 -4.46
-0.01 -0.03 -0.07 0.14 0.04 -0.43 0.00 0.02 0.22 0.54 0.50 0.25 0.08 0.46 0.24 -0.43
“dSee corresponding footnotes in Table 11. e A A D = 0.23 kcal/mol and SD = 0.08 kcal/mol on the basis of 15 molecules not used to determine the parameters.
TABLE IV: Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Ethers“ moleculeb dimethyl ether diethyl ether* methyl n-propyl etherf methyl isopropyl ether ethyl n-propyl ether methyl tert-butyl ether di-n-propyl ether diisopropyl ether di-n-butyl ether tetrahydrofuran 2-methyltetrahydrofuran
2,5-dimethyltetrahydrofuran methyl phenyl ether
calcd -2.33 -1.64 -1.66 -1.44 -1.30 -1.08 -0.96 -0.60 -0.59 -2.80 -2.40 -2.00 -3.07
exutlc
Adse
-1.90 -1.64 -1.66 -2.01 -1.81 -2.21 -1.15 -0.53 -0.83 -3.47 -3.30 -2.92 -1.04
-0.43 0.00 0.00 0.57 0.51 1.13 0.19 -0.07 0.24 0.67 0.90 0.92 -2.03
a-dSee corresponding footnotes in Table 11. ‘ A A D = 0.70 kcal/mol and SD = 0.26 kcal/mol on the basis of 11 molecules not used to determine the parameters. fMolecule used to fit the value of Rh,o. See the text for details.
B. Aromatic Hydrocarbons. Table 111 shows similar comparisons with experiment for aromatic hydrocarbons. Free energies of hydration for benzene and its derivatives are reproduced well by the present model, while the errors for highly fused polycyclic compounds are somewhat larger. The A A D = 0.23 kcal/mol and S D = 0.08 kcal/mol are somewhat higher than for the aliphatic hydrocarbons, but the agreement is good. C. Ethers. The calculated results are shown in Table IV. For the 11 ethers used in testing, A A D = 0.70 kcal/mol and S D = 0.26 kcal/mol. Major deviations occur for the cyclic ethers, for dimethyl ether, which is the lowest member of the homologous series, and for methyl tert-butyl ether and methyl phenyl ether, two compounds with bulky branching next to the oxygen atom. Usually, the latter two types of compounds show the largest deviations in most classes considered below. It is generally known that many physical properties of the lowest members of homologous series deviate from linear trends observed for the higher members. The present results are thus consistent with the general behavior. Bulky groups (especially aromatic ones) next to a polar atom (such as the ether oxygen) probably cause deviations from the linear dependence of AG, on the accessible volume (cf. eq 4) by perturbing water-water interactions in the small remaining volume of the shell around the oxygen atom. This effect is not considered in the hydration shell model. D. Alcohols. Table V lists the results for 22 alcohols. The A A D and S D are 0.36 and 0.09 kcal/mol, respectively. The largest deviations occur for methanol, ethanol, and the highly branched alcohols such as 2-methyl-2-butano1, 2,3-dimethyl-2-
Kang et al. TABLE V Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Alcohols’ moleculeb methanol ethanol 1-propanof 2-propanol* 1-butanol 2-methyl- I-propanol 2-butanol tert-butyl alcohol 1-pentanol 3-methyl- 1-butanol 2-pentanol 3-pentanol 2-methyl-2-butanol I-hexanol 2,3-dimethyl-2-butanol 3-hexanol 4-methyl-2-pentanol 2-methyl-3-pentanol 2-methyl-2-pentanol cyclopentanol cyclohexanol phenol* 2-methylphenol 4-methylphenol 4-tert-butylphenol
calcd
exptlc
-6.08 -5.57 -5.15 -4.76 -5.04 -4.65 -4.48 -4.21 -4.86 -4.61 -4.30 -4.04 -3.93 -4.71 -3.21 -3.90 -3.89 -3.50 -3.76 -5.28 -4.94 -6.62 -6.51 -6.65 -5.69
-5.12 -5.01 -4.83 -4.76 -4.72 -4.52 -4.58 -4.51 -4.47 -4.42 -4.39 -4.35 -4.43 -4.36 -3.91 -4.08 -3.74 -3.89 -3.93 -5.49 -5.48 -6.62 -5.87 -6.14 -5.92
Ad.e
-0.96 -0.56 -0.32 0.00 -0.32 -0.13 0.10 0.30 -0.39 -0.19 0.09 0.31 0.50 -0.35 0.70 0.18 -0.15 0.39 0.17 0.21 0.54 0.00 -0.64 -0.51 0.23
4-dSee corresponding footnotes in Table 11. ‘ A A D = 0.36 kcal/mol and SD = 0.09 kcal/mol on the basis of 22 molecules not used to determine the parameters. fMolecule used to fit the value of Rh,”. See the text for details.
TABLE VI: Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Ketonesa moleculeb acetone 2-butanone 2-pentanone* 3-pentanone 3-methyl-2-butanone 2-hexanone 4-methyl-2-pentanone 2-heptanone 4-heptanone
2,4-dimethyl-3-pentanone acetophenone
calcd -4.25 -3.81 -3.53 -3.34 -3.22 -3.30 -2.91 -3.09 -2.73 -2.21 -6.32
exptl‘ -3.85 -3.64 -3.53 -3.41 -3.24 -3.29 -3.06 -3.04 -2.93 -2.74 -4.58
Adse
-0.40 -0.17 0.00 0.07 0.02 -0.01 0.15 -0.05 0.20 0.53 -1.74
a-dSee corresponding footnotes in Table 11. ‘ A A D = 0.33 kcal/mol and SD = 0.19 kcal/mol on the basis of I O molecules not used to determine the parameters.
butanol, and 2-methyl-3-pentanol. The free energies of hydration are well reproduced, however, for most of the aliphatic alcohols. The errors for phenol derivatives are similar to those in aliphatic alcohols. E. Ketones. The calculated results for ketones are shown in Table VI, with A A D = 0.33 kcal/mol and SD = 0.19 kcal/mol. The overall agreement between the calculated and experimental values is similar to that for the alcohols. The larger deviations for acetone, 2,4-dimethyl-3-pentanone,and acetophenone are similar to those discussed above for ethers. F. Esters. Table VI1 lists results for 13 esters. The A A D and S D are 0.20 and 0.10 kcal/mol, respectively. All the data, except those for methyl benzoate, are reproduced well with a small absolute difference. G. Carboxylic Acids. Table VI11 shows the comparisons with experiment of the free energies of hydration calculated for three carboxylic acids for which experimental data are available. The A A D and SD are low, being comparable to those of the alcohols and ketones, although no new parameter was introduced for the acids. H . Amines and Amides. The calculated results for 22 compounds are shown in Table IX, for which the A A D and S D are 0.37 and 0.10 kcal/mol, respectively. The individual AAD’s for
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4115
Free Energies of Hydration of Solute Molecules TABLE VII: Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Esters‘
TABLE X: Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Sulfides and Thiols’
moleculeb
calcd
exutlC
Ad,e
molecule’
calcd
exPtlC
methyl acetate ethyl acetate n-propyl acetate* isopropyl acetate methyl propionate ethyl propionate n-propyl propionate isopropyl propionate methyl butanoate ethyl butanoate n-propyl butanoate methyl pentanoate ethyl pentanoate methyl benzoate
-3.50 -3.14 -2.86 -2.53 -3.10 -2.73 -2.42 -2.1 1 -2.83 -2.44 -2.15 -2.62 -2.24 -5.57
-3.32 -3.10 -2.86 -2.65 -2.93 -2.80 -2.46 -2.22 -2.83 -2.50 -2.28 -2.57 -2.52 -4.28
-0.18 -0.04 0.00 0.12 -0.17 0.07 0.04 0.11
dimethyl sulfide diethyl sulfide* methyl phenyl sulfide* methanethiol ethanethiol* benzenethiol
-2.04 -1.43 -2.73 -1.57 -1.30 -2.32
-1.54 -1.43 -2.73 -1.24 -1.30 -2.55
Adre
~~
0.00
’
0.06 0.13 -0.05 0.28 -1.29
“-dSee corresponding footnotes in Table 11. ‘AAD = 0.20 kcal/mol and S D = 0.10 kcal/mol on the basis of 13 molecules.
TABLE VIII: Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Carboxylic Acids’ moleculeb
calcd
exptlc
acetic acid propionic acid butyric acid
-7.20 -6.76 -6.52
-6.70 -6.48 -6.36
Ad,‘
-0.50 -0.28 -0.16
“*cdSeecorresponding footnotes in Table 11. ’None of these molecules were used in the parametrization. See text. eAAD = 0.31 kcal/mol and S D = 0.20 kcal/mol on the basis of three molecules.
TABLE IX: Comparison of Calculated and Experimental Free Energies of Hvdration (kcal/mol) for Amines and Amides“ moleculeb
calcd
exptle
methylamine ethylamine n-propylamine* n-butylamine n-pentylamine n-hexylamine dimethylamine diethylamine* di-n-propylamine di-n-butylamine trimethylamine triethylamine* aziridine azetidine* pyrrolidine piperidine N-methylpyrrolidine N-methylpiperidine pyridine 2-methylpyridine* 3-methylpyridine 2-ethylpyridine 3-ethylpyridine 2,3-dimethylpyridine 3,5-dimethylpyridine acetamide* propionamide 3-methylindole
-5.23 -4.61 -4.39 -4.24 -4.04 -3.90 -5.22 -4.07 -3.31 -2.86 -4.35 -3.02 -6.1 1 -5.56 -5.13 -4.63 -4.65 -4.13 -4.78 -4.63 -4.68 -4.20 -4.44 -4.53 -4.56 -9.71 -9.25 -6.43
-4.56 -4.50 -4.39 -4.30 -4.10 -4.03 -4.29 -4.07 -3.66 -3.33 -3.24 -3.02 -5.41 -5.56 -5.48 -5.11 -3.98 -3.89 -4.70 -4.63 -4.77 -4.33 -4.60 -4.83 -4.84 -9.71 -9.41f -5.91/
Ad.e
-0.67 -0.11 0.00 0.06 0.06 0.13 -0.93 0.00 0.35 0.47 -1.11 0.00 -0.70
0.00 0.35 0.48 -0.61 -0.24 -0.08
0.00 0.09 0.13 0.16 0.30 0.28 0.00 0.16 -0.52
“dSee corresponding footnotes in Table 11. eAAD = 0.37 kcal/mol and SD = 0.10 kcal/mol on the basis of 22 molecules. /Taken from ref 32.
primary, secondary, cyclic amines, and pyridines are 0.21, 0.58, 0.49, and 0.22 kcal/mol, respectively. The small homologues, such as methylamine, dimethylamine, trimethylamine, aziridine, and N-methylpyridine, show the highest deviations, just as in the other classes. Because of the lack of experimental values for free energies of hydration for amides, only acetamide and propionamide were considered in this work. Only the parameter for the amide N atom was derived from acetamide; those for all other atoms of the amide group were transferred from other classes of compounds. The
-0.50 0.00 0.00 -0.33 0.00 0.23
‘-dSee corresponding footnotes in Table 11. ‘AAD = 0.35 kcal/mol and SD = 0.21 kcal/mol on the basis of three molecules.
TABLE XI: Comparison of Calculated and Experimental Free Energies of Hydration (kcal/mol) for Polyfunctional Molecules of N and 0 Atoms‘ molecule’
calcd
exptl‘
Ad8
dimethoxymethane 1,2-dimethoxyethand 1,l-diethoxyethane 1,2-diethoxyethane 1,3-dioxolane 1,4-dioxane 2-methoxyethanol* 2-ethoxyethanol 2-propox yet hano 2-butoxyethanol ethylene glycol glycerol 1,2-ethanediamine piperazine N-methylpiperazine N,N’-dimethylpiperazine 2-methylpyrazine 2-ethylpyrazine 2-methoxyethanamine*g 3-methoxy-1-propanamine morpholine 4-methylmorpholine n-propylguanidine 4-methylimidazole
-3.88 -4.84 -2.30 -4.20 -4.98 -5.20 -7.33 -6.95 -6.66 -6.41 -9.63 -13.94 -5.66 -7.74 -8.31 -8.90 -8.00 -7.55 -5.42 -5.46 -6.46 -7.15 -9.67 -10.61
-2.93 -4.84 -3.27 -3.53 -4.10 -5.06 -6.77 -6.61 -6.42 -6.27 -7.66 -9.22 -7.60 -7.38 -7.78 -7.58 -5.52 -5.46 -6.55 -6.93 -7.18 -6.34 -10.92h -10.25h
-0.95 0.00 0.97 -0.67 -0.88 -0.14 -0.56 -0.34 -0.24 -0.20 -1.97 -4.72 1.94 -0.36 -0.53 -1.32 -2.48 -2.09 1.13 1.47 0.72 -0.81 1.25 -0.36
‘-dSee corresponding footnotes in Table 11. eAAD = 1.16 kcal/mol and SD = 0.34 kcal/mol on the basis of 21 molecules not used to determine the parameters. /Molecule used to obtain the dielectric constant within the hydration shell, ch = 2.78. See the text for details. EMolecules used to obtain Agh of the hydroxyl or amine hydrogen atom of polyfunctional molecules. See the text for details. Taken from ref 32.
good agreement for propionamide (A = 0.16 kcal/mol) suggests that the present model can be used to compute reliable free energies of hydration for peptides3’ 3-Methylindole was included32 in the testing because it serves as a model compound for the side chain of tryptophan. The agreement is fairly good ( A = -0.52 kcal/mol), indicating that the hydration behavior of this side chain can be represented well by the model. I. S u y d e s and Thiols. The data in Table X show an AAD = 0.35 kcal/mol and SD = 0.21 kcal/mol for three compounds. It can be expected that the hydration of cysteine side chains and of disulfide bonds in peptides will be reproduced with a similar accuracy. J. Polyfunctional Molecules Containing 0 and N Atoms. In order to check and extend the present model, several polyfunctional molecules containing 0 and N atoms were considered, including diethers, cyclic diethers, ether-alcohols, bi- and trialcohols, diamines, azines, and ether-amines. The calculated results are shown in Table XI, in which the AAD and SD are 1.16 and 0.34 kcal/mol, respectively, for a wide variety of molecular structures. The errors for the polyfunctional molecles are somewhat larger than those of the other molecules considered previously. The overall agreement between experimental and computed free en(31) Kang, Y . K.; NEmethy, G.; Scheraga, H. A,, to be published. (32) Wolfenden, R.; Andersson, L.; Cullis, P.M.; Southgate, C. C. B. Biochemistry 1981, 20, 849.
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The Journal of Physical Chemistry, Vol. 91, No. 15, 1987
ergies of hydration is good, except for two classes of molecules, viz. the polyhydroxy and polyamino compounds (ethylene glycol, glycerol, 1,2-ethanediamine) and pyrazine derivatives. For ethylene glycol and related compounds, it is known that, as solvents, they behave similarly to water in many r e s p e c t ~ , 3because ~*~~ they can form waterlike hydrogen-bonded networks.35 Therefore, the interactions of these compounds with neighboring water molecules imply waterlike structural effects, and thus the present assumption of group additivity is not applicable. The reason for the large deviations in the case of pyrazine derivatives is not clear. n-Propylguanidine and 4-methylimidazole were included32in the testing as model compounds for the arginyl and histidyl side chains. There is good agreement\for both compounds ( A = 1.25 and -0.36 kcal/mol, respectively), suggesting that the model will reproduce well the hydration behavior of arginine and histidine. Conclusions
The hydration shell model can be used to represent satisfactorily the free energies of hydration of uncharged organic molecules containing a great variety of functional groups. The parameters describing hydration, viz. the radius of the hydration shell and the free energy density of hydration, were obtained by fitting the observed free energies of hydration of only 27 compounds. The model was tested by computing the free energy of hydration of 130 compounds (none of which were used in deriving the parameters). The average absolute difference (AAD) of the observed and computed free energies of hydration is 0.46 kcal/mol, and the standard deviation (SD) is 0.07 kcal/mol, indicating that free energies of hydration can be predicted reliably with this model. Deviations for model compounds of the peptide group and of several amino acid side chains are even lower (with an AAD = 0.23 kcal/mol) than the average deviation. This is encouraging, because one of the main goals of this work is an improved estimate of the free energies of hydration of peptides and proteins. Testing on amino acid derivatives and oligopeptides is now being carried
Kang et al.
where ti is the dielectric constant inside the cavity, P,"(cos 0) are the associated Legendre polynomials, and A,,, B,, and C,, are constants to be determined by applying the boundary conditions described below. The constant C,, is identified with a multipolar expansion of the central charge distribution and is given by
The remaining terms in A,, and B,, represent the polarization of the hydration shell and of the outside bulk dielectric continuum, respectively, and are called, by definition, the reaction potential inside the cavity. The reaction potential ar(r) is written, therefore, as @'r(r)= e;IZ(A,,, n,m
-
-. .
(35) Ndmethy, G.Comment. Pontif. Acad. Sci. 1972, 2, No. 51, 1. ( 3 6 ) The coefficients D,, and E,, can be derived from the boundary conditions2*in a manner similar to A,, and Bn,. No explicit expression is given here for them, because they are not required for the present purpose.
(AS)
> rkl
(A6)
qlrk?
O)eim@ for r
Hence a h ( r ) becomes
:I
The polarization free energy AG$ is derived for a polar group
1974 27 521 .. -
+
4r (r/rkJ2 - 2(r/rkl) cos e]'/*
= Z-Pnm(cos n,m r"+'
k contained in a spherical cavity with dielectric constant el, surrounded by a spherical shell with dielectric constant t h and by a
( 3 3 ) Ray, A. Nature (London) 1971, 231, 313. (34) Podo,F.; NCmethy, G.; Indovina, P. L.; Radics, L.; Viti, V. Mol. Phys.
(A3)
where the terms in D,, come from the electrostatic potential of the central charge distribution and are analogous to the terms in C,,. Using the definitions shown in Figure 1, we can expand q l / s in terms of the associated Legendre polynomials as follows:
Appendix. Free Energy of Polarization
bulk medium with dielectric constant to (Figure l), in the presence of a group 1 with charge 4,. The center of group 1 may be located either inside or outside the hydration shell of group k . Beveridge and Schnuelle2' derived the free energy for a similar model in the absence of qr,Le., for the effect of the polarization of the hydration shell (due to the central charge alone) on the free energy of the central charge. The derivation presented here follows theirs. Case a: qr Located inside the Hydration Shell of Group k . From the general solution of Laplace's equation in polar coordinates,2w22the potential is derived for any point (i) inside the cavity, (ii) in the hydration shell, and (iii) in the bulk. (i) The potential @l(r)at any point inside the cavity (Le., the van der Waals sphere of group k ) at a distance r from the center is given by2'
O)e"@
(ii) The potential a h ( r ) at point M in the hydration shell of group k (analogous to point M shown in Figure 1) is given by
rkJl Acknowledgment. We thank Drs. K. D.Gibson and D. Ripoll 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. thanks the Korea Science and Engineering Foundation for support.
+ B,,)r"Pnm(cos
- Pnm(cosO)e"@
for r
< rkl (A7)
or
=
(AS) (iii) The potential a,,(r) in the bulk, Le., outside the hydration shell of group k, becomes2'
because the coefficients of terms in r with positive powers of n must be zero for the potential to vanish properly at infinity. The coefficients A,,,,,,B,,,,,, C,,,,D, and E,, are related by the boundary conditions and can be expressed21in terms of C, defined in eq A2. The potential and its first derivative must be continuous across the boundary between the cavity and the hydration shell and across the boundary between the hydration shell and the bulk
The Journal of Physical Chemistry, Vol. 91, No. 15, 1987 4117
Free Energies of Hydration of Solute Molecules region. By following the similar derivations of Beveridge and Schnuelle?' we obtain36
(n + 1 ) ( 1 -e:)( = (n +
I)€;
+n
) +-
cnm 'cq, ( A 1 O ) Rv,k~n+~ rkr"+l
The first two terms on the right side of eq A22 represent the polarization free energy of the hydration shell induced by the group k itself. This contribution is already included in the empirically derived free energy of hydration for each group. The last term of eq A22, written as AGb?k, is the contribution to the free energy of hydration of a polar group k , resulting from the polarization of its hydration shell by a group I , Le.
where
(A23) If ti = 1 and qi = 0, eq A22 becomes identical with the equation derived by Beveridge and Schnuelle.21 Case b: qr Located outside the Hydration Shell of Group k . The electrostatic potential ai(r) and the reaction potential a r ( r ) inside the cavity are the same as in eq A1 and A3, respectively. But the potential in the hydration shell of group k becomes2'
And the potential outside of the hydration shell is given by
By definition, the free energy of polarization for a set of discrete charges q,, located at ri, is given by20,21
Gg1.k = %Eqi@r(ri)
( A17)
where aPr(ri) is the reaction potential at a r i n t ri inside the cavity. We can obtain the final expression for Ghl,kafter substituting A , from eq A10, B,, from eq A16, and Cnmfrom eq A 2 into eq A 3 as follows
where
=
to-'c[c rkP+I + $)Pnm(cos n,m
8)eim+
for r C rki ( A 2 6 )
where q , / s is expanded in terms of the associated Legendre polynomials as was done in the previous section. By using the same boundary conditions as in the previous section, we find that
and
In eq A18
After substituting A,, from eq A27 and B from eq A29 into eq A3, we obtain the final expression for G$,; in this second case for n = 0 to be
or
Q, = EqiqjrinrjnP,(cos0,) i,i
(A21)
where P,(cos 0,) is a simple Legendre polynomial. Each term in eq A18 represents the polarization free energy induced by an electric moment of order n of the charge distribution. Since we use a monopole approximation for the electrostatic interaction in our empirical p ~ t e n t i a l , ' ~ Jonly ~ * *monopoles ~ (Le., n = 0) are included in this work, and eq A18 becomes
So we find that
