J. Phys. Chem. 1994,98, 1328-1332
1328
Association and Dissociation of Nonpolar and Polar van der Waals Pairs in Water. Manifestation of the Hydrophobic and Hydrophilic Effect Pave1 Jungwirth’ and Rudolf Zahradnik J. Heyrovskj Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolejzkova 3, 18223 Prague 8, Czech Republic Received: July 27, 1993”
Solvent-induced free energy profiles of the association/dissociation of model van der Waals pairs were obtained via molecular dynamics simulations. For molecules of various sizes, a smooth shift from associative to dissociative interaction was observed with charging of the van der Waals pair. This effect was quantified and discussed in the context of the hydrophobic and hydrophilic phenomena.
1. Introduction The specific behavior of nonpolar solutes in a polar solvent is a key effect in many physical, chemical, and biochemical applications, such as immiscibility of polar and nonpolar species, clathrate hydrate formation, protein folding, formation of micelles, etc.14 It is usually rationalized in terms of hydrophobic interactions;5 however, there is a continuing discussion on the very nature of this effect and its relevance to the above processes.6 The main technical problem lies in the fact that the hydrophobic effect which refers to the solvent-induced interaction between two or more nonpolar molecules is hardly directly amenable to experiment because of the extremely low solubility of nonpolar solutes in water and other polar solvents. Rather, hydrophobic hydration (Le., the interaction of a single nonpolar solute molecule with water) can be subjected to experimental investigation.’ Unfortunately, it is not fully justifiable to deduce the character of the hydrophobic interaction from solubility data, and the corresponding results should be employed with caution. Great popularity has been achieved by the elegant integral equation theory of Pratt and Chandler8 based on the concept of cavity particle^.^ Their results on the hydrophobic hydration of small hydrocarbon molecules in water are in good agreement with experiment,I0 and as will be discussed later, computer simulations confirm thequalitative (if not quantitative) predictive power of the Pratt-Chandler theory for solvent-induced solutesolute interactions.’ In the field of hydrophobic interactions, computer simulations to some extent replace (hardly accessible) experiments. Their role is 2-fold: to serve as a research tool to obtain new results on specific systems and to provide more or less objective data for testing theoretical models. Although computer simulations are not “true” experiments and one should be careful in accepting their results as “final truths”, in this field they probably represent the most reliable technique and with the feedback from theory they are advancing our knowledge. The first computer simulation study of the interaction between two nonpolar molecules in water is that of Dashevsky and Sarkison,13whocalculated, using the Monte Carlo (MC) method, for a pair of methanelike molecules and for two hard spheres in water the potential of mean force (PMF), Le., the free energy profile as a function of the solute-solute separation. Unfortunately, their approach consisted of calculating the P M F from the free energy of the total system as a function of the solute-solute distance, a method which is subject to a large statistical error and has been seriously criticized.14 A much more reliable study by Pangali et al.I5 used a forcebias M C scheme to obtain the P M F of two Lennard-Jones spheres @
Abstract published in Advance ACS Abstracts, December 15, 1993.
in water, which they showed to be an oscillating function with two well-developed minima, from which the distant (solventseparated) one was more populated. The preferenceof the solventseparated minimum was also observed by Geiger et a1.16 within a (short) molecular dynamics (MD) simulation. The importance of solvent-separated structures has been confirmed by more recent M C studies on the methane dimer in water by Jorgensen et a1.I’ and Ravishanker et a1.18 Watanabe and Andersen19 investigated krypton-like molecules in water and found a tendency for the solute species to dissociate rather than to associate. In this context they introduced the term “hydrophobic repulsion”. The lack of hydrophobic attraction for various hydrocarbon pairs in water was also very recently reported by Herman.20 Contrary to the above findings, WallqvistZ1in his recent MD study of 107 water molecules and 18 methane-like molecules clearly observed hydrophobic association of thenonpolar solutes. In light of previous contradictory studies on the methane dimer in water, he concluded that it is the many-body part of the potential of mean force which drives the solute particles together. Finally, Smith et aL2*reinvestigated the methane-methane P M F in water with special emphasis on the entropy contribution. They found that the attractive hydrophobic effect is an entropy-driven process, which was also most recently confirmed in a MD study by Skipper.23 The main goal of our study was to obtain the solvent-induced potential of mean force (SIPMF) of a series of van der Waals pairs of various sizes (diameter of each Lennard-Jones sphere of 4-6 A) and polarity (modeled by placing zero or small and opposite charges on the spheres) in water, and thereby get a rather comprehensive picture of hydrophobic/hydrophilic interactions in a polar solvent. Our findings as well as critical compilation of the literature encouraged us also to make a poor man’s contribution to the discussion on the nature of the hydrophobic effect. In section 2 we provide the reader with the theory necessary for calculating the S I P M F from the MD simulation, details of which are described in section 3. Section 4 contains the results of the computer experiments. Discussion and main conclusions of our study are summarized in section 5 . 2. Theory
The direct measure of the solvent effect on the association/ dissociation of a pair of solute molecules is the SIPMF which is given as the difference between the total P M F and the gas-phase solute-solute interaction energy.’ Although most of the earlier works discuss the influence of the solvent in terms of the total PMF, for our comparable study the SIPMF is a more suitable quantity, as it allows us to elucidate the general features of the
0022-365419412098-1328$04.50/0 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 1329
Nonpolar and Polar van der Waals Pairs
TABLE 1: SoluteSolvent Lennard-Jones Interaction Parameters Used in This Study uky
vdW sphere (R = 2 A) 0 vdW sphere (R = 2 A) H vdW sphere (R = 2.5 A) 0 vdW sphere (R = 2.5 A) H vdW sphere (R = 3 A) 0 vdW sphere (R = 3 A) H
(A)
3.80 3.00 4.30 3.50 4.80 4.00
CAX (kcal/mol)
0.10 0.01 0.10 0.01 0.10 0.01
interaction with the liquid environment independently of thedirect solute-solute interactions. For a specific solute pair, the P M F can easily be obtained by summing up the (gas-phase) solutesolute energy with the SIPMF. Our strategy in obtaining the SIPMFwas based on the following relation between the solute-solute pair correlation function g A A ( r ) and the S I P M F Gsol(r),valid for the canonical ensemble:Is
where r is the solute-solute distance, k is the Boltzmann constant, and T is the absolute temperature. In principle, g A A ( r ) can be obtained directly from the computer simulation with “switched off“ gas-phase solute-solute interaction:
where Cis a normalization constant, 6 denotes the delta function, r a b is the instantaneous solute-solute separation and ( ) means ensemble averaging. However, such an approach would be very inefficient as we would extensively sample uninteresting regions of the configurational space (large solute-solute separations). Therefore, analogously to Pangali et al.,l5 we introduced an artificial weak harmonic attraction U&) between the two solutes to preferentially sample-relevant portions of the configurational space. Such a procedure (details of which will be described in the next section) is the MD parallel to the well-known “umbrella sampling” used in M C simulation^.^^ The solvent-induced potential of mean force (normalized to then zero for the largest studied solute-solute separation I,,,) follows from eq 1 by subtracting the artificial solute-solute potential uH(r):
(3) Equation 3 is the key to the method used in our work. A similar preferential sampling M D approach has already been successfully applied to a study of two nonpolar solutes in waterk5as well as to the solvent-induced torsional dynamics of alkane^.^^^^^ 3. Computer Simulation A series of constant temperature (298 K) and pressure (1 atm) M D simulations was performed for pairs of van der Waals (vdW) spheres with radii of 2,2.5, and 3 8,dissolved in 209-214 TIP3P2’ water molecules (the TIP3P model was chosen mainly for comparison with earlier studies). The vdW spheres were either nonpolar or polar; in the latter case they bore small partial charges of the same size and opposite sign (+0.1/-0.1 and +0.3/-0.3 e). Given the vdW radius U A of the sphere, the solute oxygen (UAO) and solute hydrogen (u A ~ Lennard-Jones ) parameters were evaluated as UAO = UA + uo and UAH = UA + U H . This is the convention used in AMBER2* (which is the program of use in this work) where U A X (X = 0 or H) corresponds to the position of the Lennard-Jones minimum. In other conventions (e.g., in CHARMMZ9)(TAX is the zero crossing point on the LennardJones curve and u(AMBER) = 2Il6u(CHARMM). The (shallow) depths EAX of the Lennard-Jones curves were chosen so as to roughly correspond t o a typical small hydrocarbon-water interaction. Concrete values are given in Table 1. The mass of
I
4
2.53 2.01
1
O.Of 1
4.5j -1 .o
3
4
5
6 r (Angstrom)
7
8
9
Figure 1. Solvent-induced potential of mean force of two van der Waals spheres with a diameter of 4 A. X, nonpolar species; partial charges of 0.1/-0.1 e; partial charges of 0.3/-0).3 e.
*,
+,
the vdW sphere which is an arbitrary parameter with respect to the SIPMF was set to 20 au. In the course of the M D simulation, no ”realistic” solutesolute interaction was introduced; rather we mildly constrained the solute-solute separation to relevant values by an artificial harmonic potential. In ref 15, Pangali et al. performed a whole set of simulations with several harmonic potentials (differing in the position of the minimum), through which they obtained the pair correlation function in a piecewise manner and then “glued” the pieces together. By trial and error we found that in our case it is ideal to introduce two very weak solutesolute harmonic potentials (force constant of 0.1 kcal mol-’ A-2). Thus, for each size and polarity of the vdW pair, we performed two M D simulations. Each of these computer experiments consisted of 50 ps of equilibration (providing an ensemble with relative energy drift per step < lo4 and temperature fluctuations 7.1 A and finally becomes practically constant for r > 8.7 A. The free energy difference AG;;; between r = 2R (tight
Jungwirth and Zahradnfk
1330 The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 I
-
g
0.8-
\
1I .o 5j
."
0.6
0.4-
0.5
i5
(3
0.0
-0.5 -1.01 -1.5-1
,I: y
&& A.
-1.o
, , , , , , , , , , , ~, , , , , , , , , , , , , / ,
/ ,
, , ,, ,,, , , , ( , , , , , , , , , , , , ,
,I
5
6 7 8 9 r (Angstrom) Figure 4. Solvent-induced potential of mean force of two van der Waals spheres of a diameter of 2 A as a function of the length of the MD simulation. 100 000 steps; S, 500 000 steps; (plain) full line, 800 000
steps.
3
4
+,
L . "
TABLE 2 SIPMF (in kcal/mol) of Polar vdW Pairs with Partial Charges 0.1/-0.1 e and 0.3/-0.3 e and Sizes R = 2 and 3 A following from MD Simulations (MD) and from Simple Continuum Model (CM)
1.5: 1.o:
MD MD r(A) (R=2A) (R=3A)
E 1 - 0.0 (3
4 5 6 7 8 9
-0.5 -1.0
-1.5 4
0.3/-0.3 e
0.1/-0.1 e
0.51 ~
5
6
7
8
9
10
r (Angstrom)
Figure 3. Solvent-induced potential of mean force of two van der Waals spheres with a diameter of 6 A. X, nonpolar species; +, partial charges of 0.1/-0.1 e; %, partial charges of 0.3/-0.3 e. geometry) and r = m is approximately -0.3 kcal/mol. A shallow (-0.2 kcal/mol) solvent-separated minimum occurs a t about r = 7.1 A. the solvent-induced barrier between the tight and solventseparated geometry lies some 0.5 kcal/mol above the former and 0.4 kcal/mol above the latter structure and corresponds to r = 5.1 A. On polarizing the 2-A pair to +O. 1/-0. 1 e the SIPMF becomes mildly repulsive for r > 4 A and remains attractive only for r C 4 A. The free energy difference AG;; is now positive and equals + O S kcal/mol. An even more stronglyrepulsive SIPMF without any attractive part corresponds to the +0.3/-0.3 polar pair. In this case, AG:;; N 2.0 kcal/mol. No solvent-separated minimum can be found on the S I P M F of polar vdW spheres with radius of 2 A. For nonpolar spheres of R = 2.5, the SIPMF is attractive for r C 6.9 A, mildly repulsive for r E (6.9, 8.1 A) and attractive again for r > 8.1 A. Finally, it becomes constant for r > 9.1 A. The barrier between the tight geometry (the corresponding AG;;; = -0.7 kcal/mol) and the shallow solvent-separated minimum (at 8.1 A) lies a t 6.9 A and barely touches the zero line on t h e y axis. Under polarization to 0.1/-0.1 e, the S I P M F becomes practically constant for r > 5 A (but remains attractive for r C 5 A) with zero value of AG;;;. The larger polarization of 0.3/ -0.3 e results in a repulsive SIPMF with AG:;; = +1.3 kcal/mol. The S I P M F of nonpolar spheres with R = 3 A is strongly attractive for r < 7.7 A, marginally repulsive for r E (7.7, 8.6
0.55 0.49 0.38 0.29 0.17 0.00
-0.76 -0.28 -0.13 -0.32 -0.27
CM
MD MD (R=2A) (R=3A)
0.45 0.29 0.18 0.11 0.05
1.83 1.37 0.95 0.87 0.60
0.00
0.00
0.18 0.06 0.12 0.21 0.07
CM
4.08 2.61 1.63 0.93 0.41 0.00
A), mildly attractive again for r > 8.6 A and becomes more or less constant for r > 9.5 A. The free energy difference, AI'$;, is approximately -1 .O kcal/mol. A very shallow solventseparated minimum lies at r = 8.6 8,and is separated by a 0.10 kcal/mol barrier from the attractive part of the curve. Interestingly, this barrier (at r = 7.7 A) lies some0.25 kcal/mol below the large separation limit of the SIPMF (Gsol(-)). The S I P M F of polar +O. 1/-0.l e spheres of R = 3 A is similar tothatforthenonpolarspeciesforr> 8 A. However,theattractive part of the curve is less steep (Gf;; = -0.4 kcal/mol) and the barrier between the tight and solvent-separated structure lies at 6.9 A and equals G,,I(-). For the +0.3/-0.3 e polar pair, the SIPMF is practically constant with AG;;; = +0.20 kcal/mol. Comparison of the P M F of polar vdW pairs of diameter R = 2 and 3 A following from our simulations, with the free energy profiles obtained from a simple continuum model is presented in Table 2. While in the former case we extracted data from Figures O 1 and 3, in the latter case we put Gsol = -zz'(l - l / ~ ) / r , ~where z and z' are the two partial charges and e = 70 is the relative permittivity of the TIP3P water. The dependence of the shape and "smoothness" of the SIPMF of two nonpolar spheres of R = 2 A on the total length of the two corresponding M D simulations is shown in Figure 4. A 0.1-ns run already gives the correct shape of the curves, however, with large fluctuations. No significant deviations from the 0.5-ns SIPMF curve were observed for longer simulation times, indicating that our sampling was sufficiently extensive. Two snapshots (Figure 5) from the course of the M D run demonstrate the typical water structure around the tight and solvent-separated nonpolar vdW pairs ( R = 2 A). To put more flesh into our results we present in Figure 6 the total potentials of mean force of "real" nonpolar vdW pairs. The P M F is obtained simply by summing up the S I P M F with a
The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 1331
Nonpolar and Polar van der Waals Pairs a)
b)
Figure5. Snapshotshowing thewater structurearoundthe two (a) closely packed ( r = 4.3 A) and (b) solvent-separated( r = 7.2 A) non-polar van der Waals spheres with a diameter of 4 A. The stereo stick model shows
all the water molecules closer than 4.5 A, while the space-fillingmodel all the water molecules closer than 3.3 A to the two solutes and their center of mass. (Arrows indicate the solutes.) L.”
1.5 1.o
- 1:: 0
-0.5 -1.o
-1.5
-2.0 3
4
5
6 7 r (Angstrom)
8
9
10
Figure 6. Total potential of mean force of two nonpolar van der Waals spheres with a diameter of (X) 4 A, (+) 5 A, and (%) 6 A.
direct (gas phase) Lennard-Jones solute-solute energy profile (UAA = 4,5, and 6 A, and CAA = 0.1 kcal/mol). The introduction of a mildly attractive solute-solute potential does not significantly alter the shape of the free energy curves except for the repulsive short separation region. 5. Discussion and Conclusions The principal goal of our investigation was to obtain a complex picture of the influence of a polar solvent on the association/ dissociation of nonpolar and polar vdW pairs. We aimed to get a flavor on how the solvent effect depends on the size and polarity of the vdW pair and, as a result, how (associative) hydrophobic interactions smoothly change into (dissociative) hydrophilic interactions. The solvent effect was quantified in terms of the S I P M F of two vdW molecules and the direct solutesolute interaction was eventually added (see Figure 6). Such an approach is justifiable for dilute solutions within the pairwise additive intermolecular interactions Ansatz. Let us first consider the nonpolar species. The parameters of the small ( R = 2 A) vdW pair were chosen so as to approximately
model the solvent-induced interaction between two methane molecules, a system extensively studied in the literat~re.~3-18.20-22 Our S I P M F is in excellent agreement with a recent M D study by Smith et a1.,22although we used a different method for the evaluation of the free energy curve, slightly different water model and parameters of the attractive part of the solute-solvent interaction potential. It is a well-known fact that the SIPMF is sensitive to the strength of the vdW molecule-oxygen attractive interaction1’ (the attraction with water hydrogens is negligible); however, our model should be at least qualitatively applicable to “real” systems with ~ A OE (0.05, 0.20 kcal/mol), which covers most “common” small hydrocarbon solutes. It is worthmentioning that the positionof themethanemethane solvent-separated minimum following from our simulation (7.1 A) is in good agreement with that predicted by the theory of Pratt and Chandler (6.8 A).lo However, our minimum is more than 1 A farther than that resulting from the considerations of Ben-Naim, based on a simple water models6 It seems that the dynamic water structure around the vdW pair is rather complicated and that more than a single water molecule effectively bridges the solvent-separated pair (see also Figure 5 and ref 16). The only relation of our results to the experiment is through the solubility data, as mentioned in the Introduction. A simple way to estimate the strength of the hydrophobic effect between two methane molecules at a very short separation ( r N 1.54 8, is based on the relation AG?i4,- = AG,(ethane) = ZAG,(methane)
N
-2.0 kcal/mol
(4) where AG:i4,- is the total (Le., gas-phase methane-methane plus solvent induced) free energy stabilization and AGs represents the (experimental) solvation free energ y of the given m o l e ~ u l e . ~ , ~ ~ It is clear that this value greatly overestimates the real methanemethane stabilization at the optimal van der Waals distance of approximately 4 A, which was extrapolated to thevalue of -0.78 kcal/mol.18 If we take the direct (gas phase) methane-methane interaction to be -0.3 kcal/mol,6 we can conclude that the experiment “predicts” AG;;; = -0.48 kcal/mol. Given the crude way in which this number was obtained, the agreement between M D simulations and the experiment is reasonably good. For larger nonpolar vdW spheres (roughly corresponding to molecules of the size of neopentane) the attractive interaction becomes stronger and operates at larger separations, in agreement with both common sense and the integral equation theory of Pratt and Chandler.Io The solvent-separated minimum is shifted toward larger separations and becomes more shallow with increasing size of the vdW pair; the barrier between the tight and solventseparated structures moves below the dissociation limit free energy. Upon polarization of thevdW pair the associative hydrophobic interaction smoothly changes into the dissociative hydrophilic interaction. The dissociatve effect is much stronger for smaller spheres, which can be qualitatively traced to the originally (before polarization) weaker hydrophobic attraction, if compared with larger vdW pairs. The direct “size proportionality” and the inverse “polarity-proportionality” of the associative effect has an interesting consequence-for each size of the vdW pair a value of polarity can be found, where the P M F is roughly constant for r > 2R due to cancellation between hydrophobic and hydrophilic forces. From the experimentalists point of view it is interesting to see how the solvent induced “packing force”, given quantitatively by the value of AGkY-, depends on the excluded solute surface area at the contact. Figure 7 shows this dependence, which is roughly (within the statistical error) linear, for both nonpolar and polar vdW pairs. The effect of polarization of vdW pairs for R = 2 and 3 A is quantized in Table 2 and compared with predictions of a simple
Jungwirth and Zahradnfk
1332 The Journal of Physical Chemistry, Vol. 98, No. 4, 1994
\
0
1
1
-0.5 -1
\ I 150
100
s (A”2)
230
Figure 7. Solvent-induced potential of mean force at the vdW contact as a function of the excluded solute surface. X, nonpolar species; +, charges of 0.1/-0.1 e; charges of 0.3/-0.3 e.
*,
continuum model, which does not depend on R . It is obvious that the continuum model, which works qualitatively well for small ionic species, fails for large (especially slightly polar) pairs. Only in the case of small ( R = 2 A) polar pairs under study the continuum model provides at least a first estimate of the PMF. From the experimentalists point of view a question may arise, what is the physical relevance of simulation of the association/ dissociation of polar vdW spheres if such species (dissociated dimers bearing partial charges) do not, in fact, exist in nature. It probably does not suffice to say that our study is a theoretical model for the investigation of competing hydrophobic and hydrophilic interactions. However, simulations of polar pairs serve directly as a model of predissociation of polar species and give at least a qualitative picture of the interaction between polar sites in large molecules, an effect of great relevance in biochemistry. Information on the entropic and enthalpic contributions to the hydrophobic/hydrophilic interaction would be very valuable. Unfortunately, with the method used in this study we are not able to give reliable values of the above quantities, as they are subject to a much larger statistical error than the free energy profiles themselves. The statistical error of our S I P M F curves, which reflects itself directly in the “unsmoothness” of the profiles, slowly decreases with increasing length of the simulation (see Figure 4); however, it remains too large to permit precise entropy evaluation, even for rather long simulation times. Another question concerns systematic errors which can possibly affect our results. We are mainly aware of the fact that we use, together with other authors, pairwise additive intermolecular potentials instead of exact many-body potentials. It was recently shown by van Belle and Wodak3I that explicit consideration of polarizable water molecules has a nonnegligible influence on the shape of the PMF. Further work in this direction is certainly necessary and is in progress also in our laboratory. One remarkable problem which can be traced throughout the literature remains to beclarified. Almost all computer simulations as well as the Pratt-Chandler theory predict for nonpolar vdW pairs in a polar solvent an associative hydrophobic solute-solute i n t e r a ~ t i o n . ~ On ~ - ~the ~ ~other ~ ~ ~hand, * ~ if the two solutes are allowed to move freely in the solvent, no tendency to associate is seen in the computer simulation, rather the vdW pair tends to dissociate. 15-1619-20 Recently, WallqvistZ1obtained a different result from a M D simulation of 18 methane-like particles in 107 water molecules. As mentioned in the Introduction, he observed a strong tendency
of the Lennard-Jones particles to form a cluster, which could be “dissolved” only under very high pressure (for discussion of the pressure dependence of the P M F see also ref 32). For two molecules, if no intermolecular interaction is “switched on”, the probability P(r) of finding the two particles at separation r is proportional to the spherical surface element, i.e., to 9.This is basically, as mentioned by the referees, the ideal entropy for mixing, which also prevents the hydrophobic association of vdW pairs at infinite dilution. Watanabe and Andersen19 found via a computer experiment that the ideal infinite dilution behavior persists to concentrations of at least 3%and that at these concentrations of nonpolar solutes in water no hydrophobic clusterization but rather the ‘hydrophobic repulsion” can be observed. However, it follows from the simulation of Wallqvistz’ that at concentrations around 15% one canclearly see the formation of methaneclustersin water. Similar concentration dependence of clusterization of ion pairs in bulk and even more evidently in a cluster of a polar liquid was recently reported in a theoretical and M D study by Makov and N i t ~ a n . ~ ~ So, the main lesson from this exercise is that the P M F of a vdW pair at infinitedilution cannot fully explain the hydrophobiceffect and that most probably the concentration-dependent many-body terms in the P M F are largely responsible for the hydrophobic associative effect observed in the nature.
Acknowledgment. We are grateful to Drs. Z. Havlas and J. Vondrdiek from the Institute of Organic Chemistry and Biochemistry for generous allocation of computer time on IBM Risc6000 stations. We also wish to thank the referees for valuable comments. References and Notes (1) Ben-Naim, A. Hydrophobic Interactions;Plenum: New York, 1980. (2) Chan, D. Y.C.; Mitchell, D. J.; Ninham, B. W.; Pailthorpe, B. A. In Water, a Comprehensive Treatise; Frank, F., Ed.; Plenum: New York, 1979; Vol. 6, Chapter 5 . (3) Sloan, E. D. Clathrate Hydrates of Natural Gases; Dekker: New York, 1990. (4) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley: New York, 1980. ( 5 ) Kauzmann, W. Adu. Protein Chem. 1959, 14, 1. (6) Ben-Naim, A. J . Chem. Phys. 1989, 90, 7412. (7) (a) Ben-Naim, A. WaterandAqueousSo1utio~;Plenum:New York, 1974. (b) Laaksonen, A.; Stilbs, P. Mol. Phys. 1991, 74, 747. (8) Pratt, L. R.; Chandler, D. J. Chem. Phys. 1976, 66, 2925. (9) Chandler, D. Farraday Discuss. Chem. Soc. 1978, 66, 184. (10) Pratt, L. R.; Chandler, D. J. Chem. Phys. 1977, 67, 3683. (11) Pratt, L. R.; Chandler, D. J . Chem. Phys. 1980, 73, 3434. (12) Tani, A. Mol. Phys. 1984, 51, 161. (13) Dashevsky, V. G.; Sarkison, G . N. Mol. Phys. 1974, 27, 1271. (14) Owicki, J. C.; Scheraga, H. A. J . Am. Chem. SOC.1977,99, 7403. (15) Pangali, C.; Rao, M.; Berne, B. J. J . Chem. Phys. 1979, 71, 2975. (16) Geiger, A.; Rahman, A.; Stillinger, F. H. J . Chem. Phys. 1979, 70, 263. (17) Jorgensen, W. L.; Buckner, J. K.; Boudon, S.; Tirado-Rives, J. J . Chem. Phys. 1988, 89, 3742. (18) Ravishanker, G.;Mezei, M.; Beveridge, D. L. Faraday Symp. Chem. SOC.1982, 17, 79. (19) Watanabe, K.; Andersen, H. C. J. Phys. Chem. 1986, 90,795. (20) Herman, R. B. J. Compul. Chem. 1993, 14, 741. (21) Wallqvist, A. J . Phys. Chem. 1991, 95, 8921. (22) Smith, D. E.; Zhang, L.; Haymet, A. D. J. J. Am. Chem. Soc. 1992, 114, 5875. (23) Skipper, N. T. Chem. Phys. Lett. 1993, 207, 424. (24) Torrie, G . M.; Valleau, J. P. J . Comput. Phys. 1977, 23, 187. (25) Robertus, D. W.; Berne, B. J.; Chandler, D. J . Chem. Phys. 1979, 70, 3395. (26) Jungwirth, P.; Zahradnfk, R. Chem. Phys. Lett. 1993,212, 211. (27) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D. J. Chem. Phys. 1983, 79, 926. (28) Weiner, P. K.; Kollman, P. A. J . Comput. Chem. 1981, 2, 287. (29) Brooks, B. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. J.; Swaminathan, S.; Karplus, M. J . Comput. Chem. 1983, 4, 187. (30) Bader, J. S.; Chandler, D. J. Phys. Chem. 1992, 96, 6423. (31) van Belle, D.; Wodak, S. S . J. Am. Chem. SOC.1993, 115, 647. (32) Wallqvist, A. J. Chem. Phys. 1992, 96, 1655. (33) Makov, G.;Nitzan, A. J. Phys. Chem. 1992, 96, 2965.
