2942
J . Phys. Chem. 1991, 95, 2942-2944
Continuum-Based Calculations of Hydratfon Entropies and the Hydrophobic Effect Alexander A. Rashin* and Michael A. Bukatin Biosym Technologies Inc.. 1515 Route IO, Suite 1000. Parsippany, New Jersey 07054 (Received: January I I , 1991)
A mean-field approach based on continuum electrostatics is applied to the evaluation of entropies of hydration of inert gases. Calculated hydration entropies agree well with the corresponding experimental values at room temperature for all inert gases. TAS calculated with polarizabilities of inert gases and water taken into account agree with experimental values within 1 kcal/mol. The agreement is -0.4 kcal/mol worse if the polarizabilities are neglected. An interpretation of the nature of the hydration entropy of nonpolar solutes following from our results and a possible role of the approximations involved in the calculations are discussed.
Introduction A knowledge of the free energy of hydration alone is in many cases insufficient for the understanding of the thermodynamics and mechanisms of hydration effects. This is particularly true for the hydrophobic effect which is believed to play an important role in the stabilization of proteins,lv6q7and in a wide variety of chemical and biochemical p h e n ~ m e n a , ~and J * ~to be to a large extent entropic in nature (e.g., see refs 1-5 and references therein). The relative importance of the entropic contribution to the hydrophobic hydration is also expected to change significantly with temperature,6 and the curvature of the hydrophobic surface.8 Therefore, progress in the theoretical understanding of the hydration entropy can be of wide importance. At the same time, currently available molecular simulation techniques only started to be applied to the entropy evaluations and lead to error bars of the size of entropies t h e m ~ e l v e s . ~ JDiscrepancies ~ between the results of different molecular theories, and a lack of a clear theoretical understanding of an entropic nature of the hydrophobic effect, are well discussed in p ~ b l i c a t i o n s , ~and ~ ~ ~the ~ -reader ' ~ can be referred to these publications and references therein for details. Here we suggest a mean-field approximation for the calculation of hydration entropies based on the continuum representation of the solvent" that has been successful in evaluations of hydration enthalpies of polar molecules.1'J2 We demonstrate that our approach leads to a good agreement between the calculated and experimental entropies of hydration of inert gases and allows us to interpret entropic origins of the hydrophobic effect in simple terms of the changes in hydration enthalpies of water molecules near the solute relative to the bulk solvent.
where PidV, is the probability of ith configuration of the system. Within the classical continuum description of the configurational space Pi can be defined as pi = exp(-Ei(x)/kq/Sy
exp(-E(x)/kq d~
(2)
where d y is the element of volume in multidimensional configurational space ascribed to the ith configurational state of the system; x denotes Cartesian coordinates characterizing a particular configuration of the system with the energy E(x); the integration is taken over all configurational volume, V, occupied by the system under consideration; and E,(x) is the mean energy of the system in configurations with coordinates x in the element of the configurational volume d&. If each j t h molecule is rigid and is represented by the coordinates of its center of mass, x,,yj,zj, and Euler angles, $@,,a,, x in expression 2 would denote these new general coordinates, and d V (or d y ) in (2) takes the dV =
n sin 0, dxj dyj dzj d4j d e j daj J
(3)
In the classical statistical mechanics the configurational entropy, S,, is defined up to an additive ~0nstant.I~ This additive constant cancels out in differences between the entropies of different states of the same system. Only such differences are of interest here. Here we do not consider the momentum part of the entropy, S,, and thus we deal with calculated and experimental hydration entropies characterizing the transfer of a solute from the fixed position in the gas phase to the fixed position in water.I7 To greatly simplify the computations of hydration entropies we make three assumptions here: (1) only first solvation shell Methods waters contribute to the entropy change (we define a water Calculation of the Hydration Entropy. We start with a stamolecule as belonging to the first solvation shell if no other water tistical mechanical definition of the configurational e n t r ~ p y l ~ - ~ ~molecule can pass between the first water molecule and the solute); (2) correlations between the configurations of different water S, = k x P i d & In Pi (1) molecules can be neglected, so that the total entropy is the sum i of entropies of individual water molecules; (3) the influence of other water molecules and of the solute on the energetics of different configurations of a selected water molecule can be (1) Kauzmann, W. Adv. Protein Chem. 1959, 14, 1. calculated by using a mean-field approximation which represents (2) Franks, F. In Water: A Comprehensive Treatise; Franks, F., Ed.; all other water molecules as a continuum dielectric." All these Plenum: New York, 1973; Vol. 2, p 1. (3) Ben-Naim, A. Water and Aqueous Solutions: Introduction to a Moassumptions are not new, and they have been used in the literature lecular Theory; Plenum: New York, 1974. for related purposes.6."q'8-20 Our assumptions represent an ex(4) Frank, H.S.;Evans, M. W. J. Chem. Phys. 1945, 13, 507. treme of a "semicontinuum model" (see ref 21 and references (5) Pratt, L. R. Annu. Rev. Phys. Chem. 1985, 36. 433. (6) Privalov, P. L.; Gill, S . J. Adv. Prorein Chem. 1989, 39, 191. (7) Dill, K. A. Biochemistry 1990, 29, 133. (8) Rossky, P. J. Ann. N.Y.Acad. Sci. 1986, 482, 1 1 5. (9) Brooks, C. L. I11 Int. J . Quantum Chem.: Quantum Biol. Symp. 1988, I S~. . 221. (IO) Tobias, D. J.: Brooks, C. L. 111. J . Chem. Phys. 1990, 92, 2582. (11) Rashin, A . A. J. Phys. Chem. 1990, 94, 1725. (12) Rashin, A. A,; Namboodiri, K. J. Phys. Chem. 1987, 91, 6003. (1 3) McQuame, D. A. Statistical Mechanics; Harper & Row: New York, 1973. ~~~
(14) Karplus, M.; Kushick, J. N. Macromolecules 1981, 14, 325. Berendsen, H. J. C. Mol. Phys. 1984, 51, 101 1. (15) Edholm, 0.; (16) Owicki, J. C.; Scheraga, H. A. J . Am. Chem. SOC.1977, 99, 7403. (17) Ben-Naim, A. Solvation Thermodynamics; Plenum: New York, 1987. (18) Marcus, Y. Ion Solvation; Wiley: New York, 1985. (19) Rashin, A. A. J . Phys. Chem. 1989, 93, 4664. (20) Sarvazyan, A. P. Annu. Rev. Biophys. Biophys. Chem. 1990,20,321.
0022-3654/91/2095-2942%02.50/0 0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, NO. 8,1991 2943
Letters
TABLE I: Changes in Hydration Entropy of Water around Inert Cases
inert gas He Ne Ar
Kr
Xe Rn
hard-core radius: A
accessible area) A2
no. of hydration watersC
1.30 1.40 1.71 1.80 2.03 2.18
102.7 110.8 137.6 146.9 169.2 186.2
19.9 21.2 26.1 27.7 31.7 34.7
hydration entropy - T U , kcal/mol calcnC nonpolarizable polarizable atomsf atoms‘ exptd total per A2 total per A2 total per A2 2.40 3.03 4.39 4.86 5.19 5.44
(0.023) (0.027) (0.031) (0.033) (0.031) (0.029)
1.98 2.22 3.09 3.47 4.29 4.90
(0.019) (0.022) (0.022) (0.024) (0.025) (0.026)
2.20 2.46 3.40 3.83 4.69 5.34
(0.021) (0.022) (0.025) (0.026) (0.028) (0.028)
“Hard-core radii: He, Ne, Ar, Kr, Xe;27Rn;280 and bThe surface area of the sphere with the radius equal to the van der Waals radius, r, (see the text), increased by 1.4 A. (Calculated for the first hydration shell described in the text. dFrom ref 17; values in parentheses are entropies per A* of the accessible surface. eThis work; the entropy of the formation of a cavity with zero radiusZ8J6(-25 kcal/mol) was added to the calculated results; values in parentheses are entropies per A2. fSPC model of water’O and nonpolarizable inert gases. #The water molecule has on its atoms partial charges reproducing the vacuum dipole moment (1.855 D), and additive polarizabilitiesreproducing experimental molecular polarizability;Il polarizabilitiesof inert gases are taken from ref 24; induced dipoles on atoms do not interact with partial charges or induced dipoles on other atoms of the same molecule.” therein), in which only water molecules in the first hydration shell are treated explicitly, and the rest of the solvent is represented by a continuum. Extensions of our approach to semicontinuum models with more than one explicit water molecule in the first hydration shell require significantly more extensive computations and will be considered elsewhere. In our formulation expressions 1 and 2 correspond to the entropy and configurational probability for a single water molecule in the configurational space that can be occupied by this water molecule in the first solvation shell around the solute. To obtain the total loss of water entropy around the solute we have to multiply the entropy loss of a single water molecule by their number, N . Experimental data suggest that water densities around different solutes are the same to within 10% of the bulk value (see ref 20 and references therein). To check this we calculated the volume per water molecule, Vo,from the volumes assigned to the first coordination shells of nonpolar solutes, and spherical cations, and the water coordination shells of nonpolar solutes, and spherical cations, and the water coordination numbers around them obtained in molecular simulation^.^^,^^ VOthus obtained equals the bulk value of 30 A3 to within -5%. Therefore we define N as N = V/Vo- 1, where Vis the total volume of water around the solute. As only entropy differences are uniquely defined in the classical statistical mechanics, we need to calculate the reference entropy value for no solute present in water (“water in water”). Such reference value can be useful in the processes of folding and association, when part of the hydration water is released into the bulk. In the simplest model of the reference state used here we just fill the cavity formed by the solute by the “continuum water” and perform the summation and integration in (1) and (2) over the same configurational volume as in the presence of the solute. The energies in (2) are the same for all configurations of the water molecule in this reference model (because it is always surrounded by the homogeneous dielectric). The volume of the system should be sufficiently large to ensure the convergence of the calculated total loss of the entropy. The Mean-Field Approximation. As our method for the computation of hydration enthalpies of polar molecules based on the continuum electrostatics has been reviewed in detail recently, here we give only a sketch of the ideas and the formalism involved. The reader is referred to publications”J2 for details. In our calculations of hydration energies we represent the dielectric interface between the continuum solvent and the solute with an adjacent water molecule by the “molecular surface”.11J2~26
-
(21) Tanger. J. C. IV; Pitzer, K. S. J . Phys. Chem. 1989, 93,4941. (22) Jorgensen, W.L.; Ciao, J.; Ravimohan, C. J . Phys. Chem. 1985,89, 3470. (23) Impey, R. W.; Madden, P. A,; McDonald, 1. R. J. Phys. Chem. 1983, 87, 5071. (24) CRC Handbook of Chemistry and Physics, 66th ed.; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 1985. (25) Edsall, J. T.; McKenzie, H. A. Ado. Biophys. 1983, 16, 53. (26) Connolly, M.L. Science 1983, 221, 709.
Experimental atomic polarizability can be assigned to each atom.” The dielectric constant inside of the cavities, Di, equals 1. Each atom is also assigned a hard-core radius, so that two atoms (one from the solute, and another from the water molecule) cannot be closer than the sum of their hard-core radii. The values of the hard-core radii, rhc, are taken from the literature (see legend to Table I). The van der Waals radii, rvw,are calculated as r , = 2‘16 rhc,26 and the cavity radii are calculated from r, as described in ref 12. The solvent outside the cavity is represented only by its dielectric constant, Do = 78. We ignore the van der Waals attraction here because the magnitude of its configurational variations is about the same (and often compensating) for the solute-water and water-water interactions and is much smaller than the magnitude of such variations in the electrostatic interaction between water molecules. Therefore, variations in the energy of different water-solute configurations are evaluated as variations in the electrostatic part of the hydration energy of the single-water + solute complex. The electrostatic part of the hydration energy includes two terms: (1) half of the energy of interactions of partial charges on atoms with the dipoles, wi,induced on atoms, and (2) half of the interaction energy of the same partial charges on atoms with the surface polarization charges on the dielectric boundary.”J2 The surface polarization charges and the induced dipoles on atoms can be calculated from the simultaneous system of linear algebraic equations derived and explained in detail in refs 11 and 12. It has been demonstrated” that calculations of hydration enthalpies thus performed lead to excellent agreement with experimental values. Therefore, we can expect that they may be sufficiently accurate for the evaluations of expressions 1 and 2 and thus for the evaluations of hydration entropies undertaken here. All computations were performed on Personal Iris workstations.
Results and Discussion Table I shows that hydration entropies calculated with our method are in good agreement with experimental data. The agreement is better for hydration entropies of all inert gases when polarizabilities are taken into account in the calculations. A better agreement for the model accounting for atomic polarizabilities is satisfying as such a model is expected to be more realistic than a nonpolarizable one. It is interesting that our calculations reproduce the experimentally observed range of variations in TAS/AZ(Table I). We have not attempted fitting of the parameters to achieve better agreement with experimental data. To save the computational effort, the calculations were performed with two rather crude grids in the configurational space (30°,0.25 A; and 1 8 O , (27) Hirshfelder, J. 0.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. (28) Pierotti, R. A. J . Phys. Chem. 1965, 69, 281. (29) Weiner, S. J.; Kollman, P. A.; Nguyen, D. T.; Case, D. A. J . Comput. Chem. 1986, 7, 230.
2944
J. Phys. Chem. 1991, 95, 2944-2946 0.7
.
4
0.6 0.5
0.4
0.3 0.2
I
I
-1
0 CO.
1
e
Figure 1. Relative probabilities of different orientations of a water molecule near Ne (curves for other inert gases are similar). 0 is the Ne-0-H angle. The probabilities for cos 8 = A1 are not very accurate
because of a crude configurational grid employed in the computations (see the text).
0.15 A). The evaluation of expressions 1 and 2 was performed with two schemes, one of them underestimating and the other overestimating their values. We assumed that the converged values equal to the mean of the corresponding results obtained with the two schemes, after checking that they converge toward each other at similar rates. The estimated error due to numerical inaccuracies is less than 0.3 kcal/mol. Some details of the results obtained with our method are given in Figure 1 that shows relative probabilities of different orientations of water molecules near the solute. The shape of the curve in Figure 1 is very similar to that of the corresponding curves from the molecular s i m ~ l a t i o n . ~ , ~The ~ * 'ratio ~ of the maximum to minimum probablities in the middle of our curve is similar to the corresponding ratio obtained in the simulation36 with the same water model, but is about half that obtained in the other simul a t i ~ n . * This * ~ ~ means that the corresponding difference in energies for different orientations of a water molecule near a nonpolar solute is about 0.4 kcal/mol larger in molecular simulationssJ3 than in our computations and in ref 36. It may reflect the difference in water models used in the computations or the contribution of orientational correlations that are neglected in this study and will be analyzed elsewhere. The interplay of different factors contributing to hydration entropy is rather complex, and the magnitude of the spread in energies of different configurations of hydration water is a major factor. It may be worth noting that
the only attempt to calculate hydration entropy (for methane) with the semiempirical molecular theory3' led to an absolute value of the computed hydration entropy 1.5 kcal/mol too large. An attempt to reproduce the temperature dependence of hydration free energies of inert gases with full molecular simulation32led to rather inconclusive results (see ref 10 for a discussion). Large uncertainties in the water-water interaction energies around nonpolar solutes obtained in molecular simulation,22along with significant discrepancies in changes of these energies and their interpretations found in different molecular s t ~ d i e s , ~ preclude -~**~~ a definite interpretation of relatively small differences between our results and those of molecular theories a t this time. Thus our results, which correctly reproduce experimentally observed decrease in entropies upon a dissolution of inert gases, suggest that it can be quantitatively explained by a larger (than in the bulk water) spread of configurational energies of water molecules around nonpolar solutes. This is determined by the loss of hydration enthalpy by water molecules in some configurations around a nonpolar solute. This loss of the hydration enthalpy, and not an "enhanced hydrogen b ~ n d i n g " ~(or~ ,"structure ~~ making") already disputed in the literature,22constitutes the basis of the hydrophobic effect according to our results. Of course, the existence of any preferred configurations can be interpreted as formation of a structure, but this is quite different from the meaning usually assigned to the structure making. The importance of the hydrophobic effect and rapidly increasing popularity of the continuum a p p r ~ a c h " ~in~ our ~ - ~opinion, ~ an exposure of the new possibilities of its application presented here. Further development and analysis of our approach are, of course, desirable, and they are under way. They may lead to a progress in our understanding of the thermodynamics of hydration, as well as of the limits of applicability of the continuum approach. Acknowledgment. We thank R . M. Fine, B. K. Lee, and H. Meirovich for productive and illuminating discussions and P. J. Rossky and J. Malinsky for helpful suggestions. This work has been supported by NIH grant GM-38144. (30)Jorgensen, W.L.;Chadrasekhar, J.; Madura, J. D. J. Chem. Phys.
1983, 79, 926.
(31) Pratt, L. R.; Chandler, D. J. Chem. Phys. 1977,67, 3683. (32)Swope, W.C.; Andersen, H. C. J. Phys. Chem. 1984, 88, 6548. (33)Zichi, D.A.; Rossky, P . J. J . Chem. Phys. 1985, 83, 797. (34)Sharp, K. A.; Honig, B. Annu. Rev. Eiophys. Eiophys. Chem. 1990,
19, 301.
(35)Davis, M. E.;McCammon, J. A. Chem. Reo. 1990,90, 509. (36)Postma, J. P. M.; Berendsen, H. J. C.; Haak, J. R. Faraday Symp. Chem. SOC.1982, 17, 55.
Heats of Sublimation from a Polycrystalline Mixture of Cg0and C,o C. Pan, M. P. Sampson, Y. Chai, R. H. Hauge,* and J. L. Margrave* Department of Chemistry and Rice Quantum Institute, Rice Unioersity. Houston, Texas 77251 (Received: January 14, 1991) Knudsen effusion mass spectrometric measurements of vapors in equilibrium with a polycrystalline mixture of C , and C ~ O were carried out over the temperature range 640-800 K. From the second-law method, average values obtained for the heats of sublimation of Cm and C70from a polycrystalline C , matrix were found to be respectively 40.1 1.3 and 43.0 2.2 kcal mol-', at the average temperatures 707 and 739 K. It was also noted that it was necessary to heat treat the samples at temperatures of at least 170 O C for greater than 12 h to achieve stable vaporization. This was consistent with the sample becoming more crystalline.
*
Introduction Cb0 has been of interest since smalley and co-workersl sueceeded in producing a remarkably stable C , cluster ion by laser
vaporization of graphite in a high-pressure supersonic nozzle in 1985. However, it was not practical to investigate traditional thermodynamic properties until Kratschmer, Huffman, and coworkers2 recently reported a procedure that for the first time
H. W.;Heath, J. R.; O'Brien, S. C.; Curl, R. F.; Smalley, R.
(2)Kriitschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman, D . R. Nature 1990, 347, 354.
( I ) Kroto,
E.Nature 1985, 318, 162.
0022-3654/91/2095-2944$02.50/0
0 1991 American Chemical Society
