J . Phys. Chem. 1994.98, 386-389
386
Magnitude of Hydration Entropies of Nonpolar and Polar Molecules Alexander A. Rashin' BioChemComp, Inc., 543 Sagamore Ave, Teaneck, New Jersey 07666
Michael A. Bukatin Computer Science Department, Brandeis University, Waltham, Massachusetts 02254 Received: September 1, 1993; In Final Form: November 9, 1993@
A mean field model based on continuum representation of bulk water is applied to the calculation of hydration entropies of nonpolar and polar spheres. Polar spheres were assigned charges of kOSe and &le. It is found that above the radius of -3 %r hydration entropies of nonpolar and polar spheres of similar size are identical within the accuracy of our computations. Hydration entropies are similar for nonpolar and half-charged spheres of all radii. It is found that the mean field model exhibits a gaslike behavior. If entropy calculations are performed at constant pressure, the hydration entropy per %r2 of solute's accessible area does not exhibit a size dependence and stays around 30 cal/%r2. In calculations at constant volume, corresponding compression entropy terms are added to the constant-pressure results, leading to a strong size dependence. A comparison of the results to scaled particle theory suggests that the size dependence of the hydrophobic transfer energies may be closely linked to volume effects which can be absent or small in hydrophobic interactions. Introduction It is widely believed that two major factors determining molecular structure and stability in solutionare hydrophobiceffect and electrostatic interactions (including hydrogen bonds and salt bridges).'-' However, there are mounting controversiesas to the magnitude of these fact0rs.3~The hydrophobic effect is known to have mainly entropic nature at room temperature.'v3v7 It has also been suggested that entropic contribution to the free energy of hydrogen bonding can be of the order of 1 kcal/mol per bond.8 It was believed that this entropic contribution can arise from the change in the hydration of the hydrogen-bonding groups upon the formation of the bond.* The latter suggestion grew in importance due to recent results of electrostatic calculations indicating that the electrostatic contribution to the enthalpy of a hydrogen bond formation can be small for buried hydrogen bonds."' Entropic considerations have been also involved in a still controversialreevaluationof themagnitude of the hydrophobic effect.'j Recently, we suggested'2J3 an approximate method for the calculation of hydration entropies of spherical particles based on the continuum approach. The method yields results in a fair agreement with experiment for noble gasesI2 and alkali-metal and halide ions.13 Here we extend applications of the method to a wider range of particle sizes and to intermediately charged states ( f l / z e ) . Whilehydrationand bonding can becomplicated by asymmetries of the groups involved, studies of hydration entropies of simple spherical particles can provide basic insights into the physical nature and magnitudeof major structure-forming effects in solution.
Methods Following our previous work,12J3 we concentrate on the configurational contribution to the entropy, assuming that the momentum part is not relevant to the thermodynamics of the solute transfer between different phases? Because in the classical statistical mechanics the configurational entropy is defined only up to an additive constant,I2-l4 we calculate the entropy of hydration of a spherical particle as a difference between the entropies of two model configurational probability distributions. These model probability distributions represent the water with Abstract published in Advance ACS Abstracts, January 1, 1994.
an added spherical solute ("main" model) and pure water ("reference" model)." The entropy in each of these models is14 S = - k J p ( i ) ln[p(i)] d V
(1)
V
whereiis a point in theconfigurational space, dVis its differential element, and p ( i ) is the probability density of a configuration i p ( i ) = exp(-E(i)/kT)/Q
(2) where E ( i ) is the energy of a configuration i and Q is the configurational integral
Q=
exp(-E(i)/kT) d V
(3)
V
To greatly simplify the computations of hydration entropies, we make two a s s u m p t i ~ n s : ~(a) ~ Jthe ~ total entropy of water in a system is the sum of entropies of individual water molecules; (b) the contributions to E ( i ) of all water molecules in the system other than the selected one can be calculated by using a mean field approximation which represents these water molecules as a continuum dielectric.I*J3Js In the main model the position of the solute particle is fixed. One water molecule is picked up, allowed to run over the space occupied by the system, and allowed to have all orientations. Each such position of the water molecule corresponds to a configuration in our model space. A total energy value, Em&), ascribed to each configuration is the sum of energies of interaction between all parts of the system under the study'3 E . ( j ) = Ewater+solutc water-wluta watar-solute main
solvation
('1
+ ECoulomb
('1
+ ELsnnard-Jons(')
(4) water+wlute where Emlvation is the energy of interaction of the molecular subsystem of a solute and one water molecule with the continuum dielectric representing the rest of water in the system; ~ ~ o and ~ Eu ~ * ~ ~are ~energies $ ~of the a Coulomb and Lennard-Jones interactions, respectively, between the water and the solute in the subsystem described in a molecular detail. In calculations of entropy of nonpolar solutes the last two terms in
0022-3654/94/2098-0386S04.50/0 0 1994 American Chemical Society
Letters
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 387
eq 4 are set to zero; in calculations for polar solutes they are
w calculated and wmoriuntal hydration uhtropion
evaluated as described in ref 13. The reference model has the same configurational space as the main model. However, the energies of all configurations are the same, E,&). The actual value of the constant Emf(?)does not influence the value of the entropy, S,f (see eqs 1-3). For convenience we define13
10
15 a
Psh
where and denote energies of hydration in a continuum solventl2J3J5 of one water molecule and a solute molecule, respectively. In calculations of entropy of nonpolar solutes the last term in eq 5 is set to zero. The totd hydration entropy of water in the volume V of configurational space is
P
\
rl
d 10 5
where N is the number of water moleculesthat can simultaneously occupy volume V (N= V/30).'2 The way to make calculations for infinite V is described in ref 13. Configurations with sterically overlapping solute and water molecule are included in the calculations as a new feature. These configurations are treated in two different ways: (1) they are assigned the energy of E,&); (2) they are assigned high positive energy. The first treatment assumes that water molecules that occupy these configurationsin the reference model (in the absence of the solute) are transferred somewhere else in the bulk water in the main model. Within our model this treatment closely correspondsto the solute transfer at constant pressure. The second treatment forces water molecules out of the solute excluded area in the main model and decreases the total volume occupied by water in the main model compared to that in the reference model. Within our model this treatment closely correspondsto the solute transfer at constant volume. It can be noted that our definition of hydration entropies in eqs 1-3 and 6, suggested in ref 12, is equivalent to eq 42 of ref 16derived later and using water-solute pair correlation function, g(r), for calculations of hydration entropies of nonpolar solutes. It follows from our eqs 1-3 and eq 12.12 of ref 14 that our p(r) = g(r)/ Vand J'g(i)dV- V. We can also write down the entropy of our reference model as S,f = In( 1/ V). Substitution of these expressions into eqs 1 and 6 leads to eq 42 of ref 16:
N J p ( i ) In p ( i ) dV- N In += f J g ( i ) In g(i) dV+
5V In I-Jg(i) dVV
N In
= f J g ( i ) In g(i)d V (7)
However, the physics of the problem in our approach is reflected in&) calculated with mean field continuum approach and in ref 16 in g(r) obtained with molecular computer simulations. A similarity of potentialsof mean force, w(r) (w(r)= -kTln[g(r)]), or probability distributions from continuum calculations and molecular simulations has been demonstrated for some systems.3J2.13.15 A detailed discussion of such similarities and differences will be presented elsewhere."
Resdta and Discussion Results of our calculations of TAS at constant pressure along with the curve from scaled particle theory (SPT)17 and experimental data are shown in Figure 1. The dependence of calculated entropies on the solute size is qualitatively different for nonpolar and half-charged solutes on one side and ions on the other. While for the first group of solutes hydration entropy increases monotonically with the solute size, for ions it has a minimum corresponding to a change from typically ionic to 'hydrophobic" hydration. In ionic hydration Emin(?) exhibits ion-dipole attraction at short separations in solution, which is absent in
0
a
j 3 4 5 particlo-oxygon minimum noparation in A
6
Figure 1. Hydration entropies of spherical solutes: ( 0 )calculatedvalues
for nonpolar solutes with van der Waals radii 1,2,3,4,5 A and for noble (t)calculatedvalues gases;12 ( 0 )experimentalvalucsfor n~blegases;~J~ for cations with ionic radii of alkali-metal ions and with radii 1,2, 3,4, 5 A; (A)experimental values for alkali-metal ions;13*26( 0 )calculated values for anions with radii of halide ions13and with radii 2,3,4, A; ( 0 ) experimentalvalues for halideiom;13* @) calculatedvalues for 'cationic" sphereswith charge45e and ionic radii of 1,2,3,4,5 A; (B) calculated values for 'anionic" spheres with charge 0.5e and ionic radii of 1, 2, 3, 4, A; (w ) calculated for nonpolar solutes with SF'T.16 Only polarizable water moleculesl~*~ were used in calculations. Minimum solutewater separation is taken as their equilibrium separation minus 0.25 A, same as in ref 13 for ions but slightly different than in ref 12 for noble gases. For Rn and smaller solutes cavity radii were corrected as dcacribed in ref 19; for all solutes larger than Rn the correction was assumed to be the same as for Rn. "hydrophobic" hydration. For nonpolar solutes calculated TAS closely agrees with SPT for small solutes, agrees within 1 kcal/ mol with experimental results for noble gases,7J2 and become progressivelysmaller in absolutevalue than SPT results for larger solutes. For the largest nonpolar solute shown (van der Waals radius of 5 A) the difference between SPT and our results is =7 kcal/mol. It is even larger if parameters from ref 18 are used in SPT. Calculated TAS for cations closely agree with experimental results for alkali-metal ionsI3 and for larger cations practically coincide with the curve for nonpolar solutes. Calculated TAS for solutes with positive charge 4.5e agree within 1 kcal/mol with the calculated values for nonpolar solutes in range of the solute sized studied. (Results for van der Waals radii from 1 to 5 A are shown in Figure 1.) Calculated TAS for anions agree with experimentalresults for halide ions,l3 are similar to values calculated for nonpolar solutes at anionic radii around 2 A, and then drop below the curve for nonpolar solutes. The latter result is likely to be an artifact of the rule defining cavity radii for anions in continuum calculations.1~~~9~20 For cations and nonpolar solutes the cavity radius equals ionic or van der Waals radius plus -0.6 A.15.19120 For anions the cavity radii are approximately equal to ionic radii because of strongly oriented water molecules hydrogen bonded to small anions. Calculations show that for halide ions in solution the energy minimum correspondingto anion-water hydrogen bond is lower than anionwater interaction energy in solution at any other separations.13 Our calculations show that for anions with ionic radii of 4 A or larger the water-separated minimum becomes deeper than the 'hydrogen-bonded" one, and thus a well-defined hydrogen bond is unlikely to form in solution. Orientational preference for water around large anions is also largely lost, thus displacing hydrogens
388 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 from the line between the anion and water oxygen. This makes water surroundingsof large anions much more like that of cations or nonpolar solutes, suggesting that the same rule defining cavity radii should be used for all these solutes. If the cavity radius for anion with 4-A radius is taken to be the same as for cation or nonpolar solute with the same ionic or van der Waals radius, calculated TAS for such three solutes are indistinguishable from each other. This behavior is repeated for solutes with negative charge of OSe, for which calculated TAS are similar to thevalues for nonpolar solutes at smaller solute radii (see Figure 1). Thus, our results strongly suggest that in the range of solute charges between e and -e TAS are similar for all solutes with the same radii except for small ions with charge f e which are larger in absolute value than TAS for half-charged or nonpolar solutes. As large negative hydration entropy is considered to be the cause of the hydrophobic effect,3-7 our results ask for some clarification of the usually accepted distinction between “hydrophobic” (“dislikingto bedissolved in water”) and “hydrophilic” solutes. All solutes are “hydrophobic” entropically. Polar nonionic (modeled here by charge f0.5e) solutes and nonpolar solutes of the same size are equally “hydrophobic”in this sense. Larger ions (radii of more than 3 A) have the same “entropic hydrophobicity”as nonpolar solutes of the same size. Small ions are even more “entropically hydrophobic” than nonpolar solutes. Thus “hydrophilicity”can come only from enthalpic component of the free energy of transfer. This component is well expressed in transfer of polar solutes from nonpolar (vacuum or nonpolar liquid) environment to a polar one (water). The situation becomes less clear in transfer of polar groups from polar water to polar hydrogen-bonding environment. In such cases the enthalpy of transfer can becomevery small,gJl and the transfer may bedriven by negative entropies of hydration as for nonpolar solutes. Thus, the free energy of hydrogen bonding in proteins may become mainly entropy driven. These conclusions, of course, can be so far valid only at room temperature for which the calculations were performed in this work. They currently do not address differences in temperature dependencies of the transfer thermodynamics of different solutes.2J1 Another interesting result of our computations is the size dependence of the hydration entropy. Our model intrinsically possesses gaslike properties. This comes from inclusion of correlations only in the mean field which is effectively felt as an external field by each individual water molecule. In uniformly distributed gas (at constant temperature) the entropy is defined by its volume14
Letters with radius of 5 A, shown in Figure 1, and it is -200% for solutes with radius of 10 A (not shown). Our calculations at constant pressure show that TAS per unit of accessible surface area of a nonpolar solute is 18 cal/AZ for a solute with radius of 1 A, 26 cal/A2 for a solute of 2-A radius, and 31 cal/AZ for a solute of 3-A radius, and changes further only to 35 cal/A2 for a solute of lo-A radius. Thus, after 3-A radius entropic contribution to the free energy of hydration practically does not exhibit any size dependence in transfer at constant pressure (which corresponds to usual experimental conditions). Calculated TAS per unit of accessible surface area of a polar solute and the average surface area of 50 A2 buried upon formation of a hydrogen bond2 lead to an estimate of entropic hydrogen bond stabilization of 1-1.5 kcal/mol (compare to ref 8). However, the corresponding values become strongly size-dependent in transfer at constant volume (see eq 9). It can be noted that in SPT the free energy (which is mainly entropic17)of transfer of a solute excluding centers of solvent molecules from the sphere of radius r = aX is22
-
AG(X) = 4kT~pa’S,”(X’)~c(X’) dX’
(10)
and if G(X’) = 1 , AG(X) = &Tno
(11) with defined in exactly the same way as in eq 9. For small and moderate size solutes the real c(X) has a maximum of -2 at r = 2 and then decreases, staying close to 1.22923 Thus, a comparison of eqs 9 and 1 1 suggests that in SPT the size dependence of the free energy of hydration can be mainly due to a volume change upon transfer. According to our calculations, the effect should not be present in the entropy of transfer at constant pressure. Recent analysis of experimental data for methane solutions also suggests a difference of 1.5 kcal/mol between entropies of transfer at constant pressure and at constant volume.24 Equation 9 leads to the value of 1.8 kcal/mol for the same transfer if an equilibrium water-methane separation is 3.4 A. Volume changes in protein association and folding can be much smaller than in transfer of molecules of similar sizes. Therefore, size-dependent effects may be irrelevant for these processes. Currently accepted view of the thermodynamics of transfer7 holds, however, that free energiesof transfer at constant pressure, AG, and constant volume, AA, are equal. It is argued that for real liquids expanding the volume of the solute-solvent system by the volume of a molecular solute requires negligible work against the atmospheric pres~ure.~ This is not the case for gaslike Ssas = N k In V,, or hard-sphere liquids where a pressure of thousands of atmoand for VI - VZ= &VO
