J.
Phys. Chem. 1982, 86, 873-880
873
ARTICLES
Hydration of Inert Solutes. A Molecular Dynamics Study
D. C. Rapaporlt and H. A. Scheraga'
&kef Labolgtcyv of (%SmWy, Cornel UnhwsIty, Ithem, New York 14853 (Recelved: JU!V 17, 7981;
I n F h l Form: September 4, 198 1)
Molecular dynamics techniques have been used to study the structural changes in the "configuration interaction"
model of water resulting from the introduction of several nonpolar, essentially hard-sphere solutes. The use
of a high-speed array processor permitted very long (70 ps) simulations of systems of 343 molecules of which
4 are solute molecules. Structural reorganization within the shells surrounding the solutes and a small degree
of slowing down of the molecular motions in the shells were observed, in agreement with previous theoretical
and experimental work. The question of whether there exists a solvent-induced hydrophobic attraction between
the nonpolar solutes was also examined; the simulations failed to reveal any tendency for the solute molecules
to aggregate.
Introduction
Understanding the solvent molecular rearrangement that
occurs when a nonpolar molecule is dissolved in water is
an important step on the path to explaining the conformational preferences and folding pathways of biological
macromolecules.1-3 There have recently been several
theoretical attempts to examine both the solute and solvent behavior at a microscopic level by means of computer
simulation;"1° these studies focused on the manner in
which the local water structure and dynamics were altered
by the presence of the solute molecule, and also on the
solvent-modified mean force exerted by one solute on its
partner.
Two terms have been introduced to classify distinct
aspects of the phenomenon occurring upon the insertion
of one or more nonpolar solute molecules into water2 the
first is hydrophobic hydration, a term which covers the
solvent restructuring around a single solute; the other is
the so-called hydrophobic interaction, which is that part
of the net effective interaction between a pair of solute
molecules arising from the presence of the solvent, and
which excludes the direct solute-solute interaction (although the latter is sometimes included in the definition3).
Interpretation of thermodynamic data-in particular the
existence of a negative entropy of solution-has led to the
assumption that, when nonpolar molecules are dissolved
in water, there is an entropy-driven tendency for them to
c l ~ s t e r .At
~ the intuitive microscopic level this phenomenon is attributed to an aversion on the part of the water
to broken hydrogen bonds-forcing the nonpolar solutes
to cluster minimizes the area of the soluteaolvent interface
region, thereby reducing the number of water molecules
which are directly affected by the presence of the solute,
while at the same time reorienting these water molecules
in an attempt to replace broken hydrogen bonds by bonds
which are merely distorted.
Among the computer simulations which have been undertaken in recent years in an effort to examine the microscopic details of the hydration phenomenon, the two
On leave from the Physics Department, Bar-Ilan University,
Ramat-Gan, Israel.
O022-3654l82l2O86Q87380 1.2510
most extensive studiesel@have dealt with pairs of Lennard-Jones (LJ) type solutes dissolved in water which itself
was modeled by the empirically derived ST2 potential. In
one studf the motions of the molecules were followed by
using the molecular dynamics (MD) method; the solutes
were initially adjacent, but during the course of the run
they separated sufficiently to allow a single solvent layer
to penetrate between them. As the authors of this study
pointed out, however, the degree of sampling of the system
was insufficient to permit definite conclusions regarding
any tendency for nonadjacent solutes to associate; in other
words, it is most likely that a prolongation of the simulation run would have revealed the solutes undergoing even
further separation. A somewhat different approach@to the
same problem employed the Monte Carlo (MC) method
to analyze the potential of mean force (equivalent to the
solute-solute pair distribution function) acting between
a pair of LJ solutes. In order to control the range of
separations between the solutes, an additional direct
harmonic interaction was introduced; the effect of this
artificial force was subsequently eliminated by suitably
weighting the statistical averages. The results yielded a
highly plausible oscillatory form for the potential of mean
force, implying the existence of at least two preferred
ranges of intersolute separation; however, owing to the
nature of the sampling method used, the question of
(1) F. Franks in "Water,a Comprehensive Treatise",Vol. 2, F. Franks,
Ed., Plenum Press, New York, 1973, Chapter 1.
(2) A. Ben-Naim, "HydrophobicInteractions", Plenum Press, New
York, 1980.
(3) G. NBmethy, W. J. Peer, and H. A. Scheraga, Anno. Rev. Biophys.
Bioeng., 10, 459 (1981).
(4) J. C. Owicki and H. A. Scheraga, J. Am. Chem. SOC.,99, 7413
(1977).
.
(5jS. Swaminathan, S. W. Harrison, and D. L. Beveridge, J. Am.
Chem. SOC.,100, 5705 (1978).
(6) A. Geiger, A. Rahman, and F. H. Stillinger,J.Chem. Phys., 70,263
11979).
,--. _,.
(7) P. J. h k y add M. Karplus, J. Am. Chem. SOC.,101,1913 (1979).
(8) S. Swamhathan and D. L. Beveridge, J.Am. Chem. Soc., 101,5832
(1979).
(9) C. Pang&, M. Rao, and B. J. Berne, J. Chem. Phys., 71,2975,2982
(1979).
(10) S. Okazaki, K. Nakanishi, H. Touhara, N. Watanabe, and Y.
Adaahi, J. Chem. Phys., 74,5863 (1981).
0 1982 American Chemical Society
874
Rapaport and Scheraga
The Journal of Physical Chemistry, Vol. 86, No. 6, 7982
whether a solvent-driven association of the solutes really
occurs again went unanswered. Nevertheless, the apparent
existence of a stable solvent-separated arrangement of the
solutes runs contrary to the intuitive picture of the hydrophobic interaction mentioned earlier. The measurement of the effective mean force between a pair of methane
molecules maintained at various fmed distances in aqueous
solution formed the subject of another MC study;8 the
conclusions were similar, namely, that the mean force
oscillated in direction as the solute separation was altered.
The modification of the solvent shell structure in the
neighborhood of a nonpolar solute molecule-the hydrophobic hydration effect-was analyzed in the course of two
of the above simulation^;^^^ it has also formed the subject
of several studies involving just a single solute molecule
in water. The types of nonpolar solutes studied ranged
from a simple LJ sphere,1° through a methane molecule;'"
to a terminally blocked amino acid residue complete with
several internal degrees of freedom.' These MC and MD
simulations examined the environmental changes experienced by those water molecules close to the solute-the
hydration shell-both in regard to the thermodynamic and
structural properties and, where the MD method was used,
the alterations to the dynamical behavior of the solvent
as well.
A problem common to any attempt at simulating a dilute solution-and the above-mentioned studies are all
examples of moderately dilute solutions (typicallyonly 1 %
solute concentration)-is one of sampling. Unlike singlecomponent systems, where any statistical uncertainty in
the properties measured can be reduced by averaging over
all of the molecules of the system, in the dilute solution
there are far fewer molecules over which to average if one's
principal interest lies in the properties of the solutes and
their immediate hydration shells. In particular, for simulations employing MD techniques, the only obvious
means of compensating for the fact that the molecules of
interest constitute just a small fraction of the total system
is by undertaking runs of length considerably greater than
those considered adequate for pure systems. Only in this
way is it possible to establish, for example, whether the
two relatively long-lived separations observed during the
MD study of the pair of L J solutes in watere represent a
real effect or is merely an expected artifact of a short
simulation run.
It was with the aim of attempting to overcome this kind
of sampling problem that we undertook our own study of
the behavior of nonpolar molecules in aqueous solution.
Unlike previous studies6vgwhich employed the full LJ
potential including the attractive tail, we have attempted
to isolate the hydrophobic part of the effective solutesolute interaction from any direct attractive force between
the solutes by modeling the solutes as essentially hard
spheres and have in fact used a modified form of the LJ
potential for the solutes which generates only a shortirange
repulsive interaction. Each of the simulations involves a
set of four solute molecules dissolved in a solution consisting of 339 water molecules; the water molecules themselves were modeled by using a potential derived from
quantum-mechanical configuration interaction
calculations-the MCY-CI model.'l By way of contrast,
previous work4* involved only one or two solutes in a
solution of up to 215 water molecules, and sometimes as
few as 63.1° The results that will be described here are
based on runs of approximately 70-ps duration; these run
lengths may be compared with earlier MD work involving
(11) 0. Matsuoka, E. Clementi, and M. Yoehimine, J. Chem. Phys.,
64, 1351 (1976).
TABLE I: Summary of the MD runs
run
solute
a,A
simulated
period,
no. o f
PS
steps
temp,
"C
70.6
76.3
65060
70340
30.6
27.6
configurations
analyzed
598
434
~~
1
2
3.1
3.3
simulated time periods of only about 5.ij6and even 1.5' ps.
The considerably enhanced degree of sampling, made
possible by using both a greater number of solutes and a
longer time period, ought to yield results of greater precision than hitherto possible. Even so, the statistical uncertainty associated with the dynamical properties of the
system is still considerable, indicating that limitations
remain as to what one can realistically expect to learn from
simulations of this nature.
Molecular Dynamics Computations
In a recent article12we described an MD study of pure
liquid water based on the MCY-CI model. We pointed out
that a considerable improvement in simulational capability
could be achieved by replacing the normal Euler angle
representation of the molecular orientational coordinates
by quaternion components and by developing an optimized
computer program designed to run on an array processor.
A similar approach was employed in the work reported
here; in fact, the only change made to the computational
procedure was to replace certain water molecules by "inert"
solutes, the characteristics of which are discussed below.
The simplest kind of solute for studying the hydrophobic
phenomenon is the hard sphere. Unlike previous studies
of the hydrophobic interacti~n,".~
which employed the full
LJ interaction, simulation of the hard sphere solute-or
a continuous potential function closely approximating a
hard sphere-is capable of isolating the putative hydrophobic effect from any direct solute-solute interaction.
The actual potential form used here for both the solutesolute and solute-water interactions is a shifted and
truncated version of Lennard-Jones 6-12 interaction
r < r,
V(r)= 4c[(u/r)l2 - ( u / r ) 6+ y4]
=O
r2rc
where rc = 2lI6u. This is a purely repulsive interaction,
and both V(r)and the corresponding force are zero for r
1 rc The potential is readily seen to constitute a good
approximation for a hard sphere and is suitable for use in
the equations of motion of the system which have to be
solved numerically as part of the MD procedure.
In these calculations the energy parameter t was given
the ST2 value of 0.075 75 kcal/mol;6 the core size u was
assigned values of 3.1 and 3.3 A in different runs. The
values of u were chosen to give each solute a volume similar
to that occupied by a water molecule; the two runs employed slightly different values of u, to allow both testing
of the results for reproducibility and/or examination of
any significant u dependence.
The solvent water molecules themselves interacted
through the MCY-CI potential, the details of which have
been described elsewhere." To increase the computational
speed, we reduced the water-water interaction cutoff
distance to 8 A, as opposed to the 9-A cutoff used previously;12this change had no noticeable effect on the observed structural properties.
Two very long ( ~ 7 ps)
0 MD simulation runs were carried
out for this system. The run specifications are listed in
(12) D. C.Rapaport and H. A. Scheraga, Chem. Phys. Lett., 78,491
(1981).
Hydration of Inert Solutes
Table I. In each case, the molecules were placed in a cube
of edge length 21.7 A corresponding to the density of pure
water, and periodic boundary conditions were used. The
MD time step was the same as that of the earlier study,12
viz., approximately
s, and the equations of motion
were solved by using the same fifth-order predictor-corrector method. It should be noted that the time step is
a factor of 5 larger than that used in corresponding ST2
simulations;0as mentioned earlier, this increase is achieved
with no loss of accuracy by using quaternions rather than
Euler angles for the orientational coordinates. At the start
of each run,the solvent and solute molecules were placed
at the sites of a simple-cubic array with random orientations (for the water) and assigned random velocities; the
four solutes were positioned in such a manner as to maximize their initial separations. An equilibration period of
5OOO time steps was allowed before any measurements were
taken; during this period a series of adjustments were made
to the molecular velocities until the temperature stabilized
near the desired value. Snapshots of the molecular arrangements were stored periodically during the course of
the runs for subsequent analysis. As is usually the case
with MD calculations of this kind, the truncation of the
intermolecular potential function and the limited accuracy
of the numerical methods used lead to a slow upward drift
in temperature; to overcome this,we rescaled the velocities
a t regular intervals.
The computations were carried out with the aid of a
Floating Point Systems AP-120B array processor as were
the earlier simulations of pure water. A certain amount
of optimization of the computer program resulted in a
computational rate of approximately 900 time steps h; this
increase over the 700 steps/h rate for pure water' !/is due
principally to the reduction in the interaction cutoff distance.
Solute-Solvent Pair Distribution Functions
The averaged positional and orientational distributions
of the water molecules around each of the solutes were
computed from the accumulated configurational data. In
this and subsequent sections we discuss some of the more
significant features of these distributions and contrast
them with the results obtained for pure water.
The solute-solvent radial pair distribution function
gAO(r),where A denotes the nonpolar solute and 0 the
oxygen site of the water molecule, is defied in the usual
way to be proportional to the number of oxygen sites per
unit volume lying within a thin shell of mean radius r
The Journal of Physical Chemistty, Vol. 86, No. 6, 1982 875
centered on the solute; a second pair distribution function
gm(r), with H denoting either of the two hydrogen sites
in the water, is similarly defined. The corresponding
distribution functions for pure water are the oxygen-oxygen and oxygen-hydrogen distributions goo(r)and gOH(r).
In Figure 1we compare gAO for the case u = 3.1 8, with
goo for pure water at 31 "C taken from the earlier MD
study.I2 The fact that gA0 is considerably smoother than
the corresponding distribution in the ST2 study6 is a
consequence of the increased sample size permitted by the
longer simulation runs. Figure 2 illustrates gm together
with our pure water goH distribution (not previously
published, but practically identical with the Corresponding
MC curve13). It is quite apparent that the shell structures
surrounding the solute and regular water molecules bear
little or no resemblance to each other. In particular, the
second peak of gAO (Figure 1) occurs at approximately
twice the distance of the first (6 vs. 3 A), a distance ratio
similar to that of the simple LJ fluid,'4 whereas in the case
of gm (Figure 2) a single nearest-neighbor peak has replaced the twin peaks of
for water. A further feature
worth pointing out is that the peak in
occurs at a
slightly smaller value of r than the first peak of gAO, indicating a tendency for the water molecules adjacent to
the solute to be oriented so that their 0-H bonds are
almost tangential to an imaginary sphere surrounding the
solute.1s These changes reflect the entirely different
spatial arrangement and orientation of the solvent molecules around the solute, in contrast to the average neartetrahedral organization of the neighbors of a molecule in
pure water. A more detailed structural comparison appears
in the following section.
Figure 3 shows the cumulative numbers of neighbors as
a function of radial distance from the solute molecules and,
for comparison, the corresponding results from the pure
water simulation. The solute is seen to have more neighbors than the water molecule in the 3-5-A region. If the
number of nearest neighbors of a molecule is defined to
be the molecule count within a sphere whose radius is
determined by the distance to the first minimum of the
radial distribution function, then the solute has approximately 15 neighbors, pure water a mere 5.
Qualitatively similar conclusions were reached in earlier
simulations of dilute aqueous solutions,"1° although the
(13)G.C. Lie, E. Clementi, and M. Yoerhimine, J . Chem. Phys., 64,
2314 (1976).
(14)L.Verlet, Phys. Reu., 165, 201 (1968).
(15)F.H.Stillinger, Science, 209, 451 (1980).
Rapaport and Scheraga
The Journal of Physical Chemistry, Vol. 86, No. 6, 1982
878
I
I
'
'
-
2 4 t
1
___
(JCM-CM
I1
BCM-CM
2.01
/
I
I
0
/
1
2
3
4
5
6
7
8
r(i)
Flgure 3. Cumulative numbers of neighbors of a solute and a water
molecule, nAo(r)and noo(r). These curves are obtalned by suitably
integrating those of Figure 1.
locations of the peaks and the values of the coordination
numbers differed according to the characteristics of the
solutes used. The question has also been raised as to
whether the first peak contains a small shoulder-a feature
observed during the MD study of the ST2 solvent6but not
in the corresponding MC simulationsg (the difference may
be attributable to differing solute diameters); in any event,
thisfeature does not appear here. In the simulations which
examined pairs of solutes,6git was necessary to divide the
regions surrounding each solute into hemispherical sections
and, for each solute, to consider only the half-shell farther
from the companion solute in order to minimize any spurious disruptions that its proximity may cause. In our
analysis, the possibility of one solute influencing the solvent structure around another was minimizedby excluding
the contribution of a solute from the overall distribution
function averages if it was found to have another solute
within a distance of 5.5 A. A teat of whether this arbitrary
choice of distance was sufficient to achieve its purpose was
made by reevaluating the distribution functions with the
minimum separation set to 7 A; no significant change in
the results was detected, but fewer configurations were,
of course, available for incorporation into the final averages.
The statistical uncertainty in the distributions illustrated
in Figures 1 and 2 can be estimated by analyzing the data
for each of the solutes separately and plotting the individual solutesolvent pair distribution functions. As might
be expected, the graphs thus obtained were found to be
less smooth than the averaged distributions shown, but
each of the curves lay within an envelope whose total
vertical width was approximately 0.1; thus, the resulta can
be taken as b e i i subject to an overall uncertainty of about
5 % . The significant deviations from the mean of individual molecule distribution functions recently noted to
occur in MC simulations of water,'6 which almost certainty
reflected inadequate configuration space sampling, do not
present a problem here. The potential lack of reliability
of the individual molecule distributions would pose a serious problem for MC simulations of hydration; however,
it appears that, by making fairly substantial modifications
to the MC technique, in regard to both the method of
generating the random moves of the molecules (the use of
force-bias MC) and the scheme for selecting trial molecules
(16)P. K.Mehrotra, M. Mezei, and D. L. Beveridge, NRCC Proceedings, 'Computer Simulationeof Organic and Biological Molecules",
LBL-12979,Jan 1981,p 63.
0
1
1
2
/
3
4
r
(A)
5
6
7
8
F@ro 4. octahebai resdution of the centerof-mess pair distribution
function for pure water g d r ) (essentially goo). The actual orientatlon of the octahedron with respect to the central water molecule
Is also shown: the four type I (hydrogen bonding) faces are shaded,
and those of type I1 are not. The curves here appear slightly less
smooth than goo In Figure 1 because fewer configurations were used
in their construction.
(by means of preferential sampling), it is possible to eliminate this problem.16
The g A 0 and g m curves for u = 3.3 A are s i m i i to those
for u = 3.1 A and are not shown; the only important difference is that the features of both distributions have been
shifted out to larger distances by the expected 0.2 A.
Solvent Orientation within the Hydration Shell
In order to place the analysis of solvent orientation
within the hydration shell in context, and because our
earlier study12 did not include an analysis of the shortrange orientational order, we start by describing the corresponding situation for the shells surrounding molecules
in pure water. The results are based on the same 31 "C
MD simulation referred to earlier.
As was pointed out in one of the originalpapers on water
simulation'' (involving the BNS model, the predecessor
of ST2),a much more vivid picture of the local structure
of water can be derived if the radial distribution function
is resolved into polyhedral components; conceptually, for
a given molecule this amounts to surrounding it with a
suitably chosen and oriented polyhedron and then
grouping the molecule's neighbors into classes according
to the faces penetrated by the lines joining them to the
central molecule. Separate radial distribution functions
are then constructed for the neighbors belonging to each
class. The two polyhedra originally considered were the
octahedron and the icosahedron, both of which were suggested by the tetrahedral structure of normal hexagonal
ice, and it was abundantly clear from the results that the
majority of neighbors contributing to the first peak of goo
were positioned in a manner indicative of local tetrahedral
structure.
We have carried out a similar analysis for the MCY-CI
water model. The polyhedron used is the octahedron, and
ita eight faces are divided into two classes as shown in
Figure 4. The octahedron is oriented with respect to the
water molecule under consideration in such a manner that
neighboring molecules located in or near the expected
tetrahedral arrangement, and therefore capable of forming
relatively undistorted hydrogen bonds, can be joined to
the central molecule by lines passing through each of the
four type I faces; the remaining four faces are of type I1
(17) F. H. Stillinger and A. Rahman, J. Chem. Phys., 57,1281(1972).
Hydration of Inert Solutes
The Journal of Physicel Chemistry, Vol. 86, No. 6, 1982 877
r = 5.1 A
O
'
I
0
r.4.1 A
O
'\.,
I
-1.0
-0.5
0
0
r =3.l A
r=2.9%
I
I
0
0.5
cos
I
I .o
(5)
Flgwr 5. Orientatkmal distributions within shells for pure waty gt 31
'C. The dlstrlbutlons n(S;r)of cos S; = S,.PI and cos lp= 8,.81for
both face types are shown; the histograms have a bin width of 0.2,
and the spherical shells are of thickness 1 A with midpoint radii r as
shown.
and are unlikely to contain hydrogen-bonded neighbors.
The resulting decomposed center-of-mass (essentially the
oxygen-oxygen) radial distribution functions are shown
in Figure 4. The first peak near 3 A is seen to be almost
entirely due to neighbors in type I positions, and hence
to molecules that fit in with the tetrahedral pattern. The
second peak a t 4.2 A, which is a feature of the water distribution not shared by simple W fluids,14is principally
due to type I1 neighbors that are not located near the
correct tetrahedral positions. By the time the 6.8-A peak
has been reached, there is no significant difference between
the contributions arising from the two types of face and,
at least at this level of angular resolution, the distribution
has become essentially isotropic.
This kind of analysis can be extended to describe the
distribution of relative molecular orientations, divided
according to face, as a function of distance from a particular water molecule. For convenience, the neighbors are
grouped in spherical shells around the central molecule,
the shells themselves being further subdivided according
to the face type. Two quantities characterizing the relative
orientations of neighbors with respect to the central
molecule are employed, and normalized histograms showing the distributions of these quantities for various shell
radii are shown in Figure 5. One of the distributions is
for the quantity cos S; = Pc.fii, where Pc is the unit dipole
vector of the central water molecule, and Pi that of any of
its neighbors; its value 'eggs betwee-n -1 and +l. The
other quantity is cos lp= &&, where & is the unit vector
between thp two hydrogen atom sites of the central molecule, and Bi the corresponding vector for the neighbor;
because of the equivalence of the two hydrogen sites, the
range of values is only 0-1. The behavior shown in Figure
5 implies a strong degree of tetrahedral orientation in the
portion of the innermost (r = 2.9 A) shell corresponding
C"S
I
I
0
0.5
I
I
.o
(5)
Figure 8. Ollentatkmal dbtrlbutbns n({,r) within shells surrounding the
nonpolar solute (u = 3.1 A). The quantities shown are the distributions
of cos {,, = FJii and cos CY = id.+,.
The other details are as h Figure
5.
to the type I faces; this is indicated by the preference
toward parallel alignment of the p's (cos C,, = 1)and perpendicular 0's (cos l6= 0). In contrast, those misplaced
molecules occupying type I1 positions in the innermost
shell barely show any orientational preferences. In more
distant (r = 3.9 and 4.9 A) shells there is a weak tendency
toward oppositely oriented dipoles (cos C,, = -1); this
tendency is slightly more pronounced for the type I1
positions than for type I.
A different set of quantities must be defined in order
to specify the orientations within the shells surrounding
a nonpolar solute because, unlike the case of water, the
central solute molecule does not exhibit any tetrahedral
(or other) directional preferences. Following the approach
of the earlier MD work on hydration,6 we examine the
distribution of the angles whose cosines are given by cos Cg
= iCi&
and cos Cy = f&, where fciis the unit vector from
the solute to a neighboring water molecule i, & is the water
dipole vector direction as before, and Ti is the unit vector
along either of the 0-H bonds of the water molecule.
(Note that these angles are the complements of those in
ref 6.)
The normalized shell orientation histograms for the u
= 3.1 A solute are shown in Figure 6. The results for u
= 3.3 A are essentially the same. The shell radii in Figure
6 were chosen to be slightly larger than those of Figure 5
to allow for the fact that the first g A O peak occurs at
somewhat greater r than goo. In the case of the innermost
(r = 3.1 A) shell, the histograms reveal a clear tendency
on the part of the dipole vector to point inward toward the
solute (cos {, = -l),an arrangement consistent with the
clathrate type of organization generally believed to describe
the time-averaged water molecule arrangement around the
solute. Furthermore, the preferred alignment of the 0-H
bonds is such that they point either inward but at an angle
distributed around 60° from the radial vector (cos CY =
878
Rapaport and Scheraga
The Journal of Physical Chemlstty, Vol. 86, No. 6, 1982
-0.5) or radially outward (cos f7 = 1). There is a marked
lack of shell molecules having either an 0-H bond pointing
directly toward the solute (cos f7 = -1) or a dipole vector
pointing radially outward from the solute (cos f, = 1). The
orientational anisotropy in the second (r = 4.1 A) shell,
which includes the remaining solvent molecules up to the
fust minimum of gAo,is considerably less pronounced than
in the first shell. A residual anisotropy persists into the
third (t = 5.1 A) and higher shells (the runs for both solute
sizes showed this effect), indicating that, as might have
been anticipated, the solute-induced structural modifications are by no means confined to the nearest neighbors
of the solute.
The results are consistent with those obtained in the
earlier MD study of solutes in ST2 water: although in that
case the dipole distribution for the innermost shell was
peaked rather than monotonic as in Figure 6; this might,
however, be due to the choice of shell size. In the case of
ST2, an attempt was made to specify the orientational
ordering within the first shell to a higher degree of precision, and the results could indeed be interpreted as
supporting a more detailed averaged clathrate picture;
however, those conclusions must be tempered by the fact
that over the entire sampling period the dipole orientation
autocorrelation function for the shell molecules had yet
to decay to half ita initial value, and thus the molecular
coordinates remained strongly correlated over the entire
set of configurations used in computing the orientational
distributions. The other, much shorter-and therefore
subject to even stronger correlations over the entire sampling period-MD study' considered the orientations
within those regions of a shell bordering on the nonpolar
portions of a terminally blocked amino acid residue; once
again the tendency for all four of the tetrahedral hydrogen-bonding directions to avoid pointing directly at the
nonpolar groups was apparent.
The overall picture emerging from these three systems
is that there is a clear tendency to avoid the wastage of
a hydrogen bond which would occur if an 0-H bond were
allowed to point directly toward a nonpolar solute; by
adopting a straddling configuration with three of the
tetrahedral hydrogen-bonding directions nearly tangential
to a sphere surrounding the solute, the water molecules
in the shell are still able to form their full complement of
possibly distorted, but nevertheless unbroken, hydrogen
bonds. In a sense the results can be regarded as being
consistent with a clathrate type of picture; however, as the
results for the dynamical behavior will show, this structure
can in no sense be taken as being more "stable" than the
tetrahedzal structure observed in the case of pure
water-the time-averaged molecular correlation functions
are clathratelike, but that is as far as the similarity extends.
A more detailed analysis of the shell organization, analogous to the octahedral decomposition of the structure of
pure water described earlier, ought to be possible, but we
have not attempted such a calculation.
Hydrophobic Association
A feature of the MD approach not shared by MC is its
ability to probe the true microscopic dynamical behavior
of the system. A particularly interesting question in this
context is whether there exista a solvent-induced force
whose effect is to drive the solutes together-the hydrophobic interacti0n.l
In Figure 7 we plot the time variation of the distances
between each of the six possible solute pairs for the two
runs. While it is apparent that the solute molecules are
highly mobile (see also the diffusion results of the next
section) there is no indication of any tendency for the
I
0 . .
0
5
ll,21,
, ,
0 20
20 40 60
'
11.31
'
<
'
S
40 60
I
11,2)
I
(l,3)
Fbum 7. Separation of solute ks as a functkn of time for both m s :
(a) u = 3.1 A, (b) u = 3.3
The separatlons are measured at
approximately 0.5-ps Intervals.
P
solutes to aggregate. There was only one pair which approached to nearest-neighbor separation (pair 2,4 for the
u = 3.3 A solutes), but the duration of the period of close
approach is not of sufficient length to hint at any special
mutual affinity. Thus, we are compelled to conclude,
subject to the assumption that the sampling has been
adequate, and within the framework of the model, that
there is no indication of any "hydrophobic attraction"
between the nonpolar solutes.
The earlier studies of U solutes in ST2 water6>@
were
not really designed to attempt to address this aspect of the
problem. In the course of the MD simulation,6the initially
adjacent solute molecules became separated by a single
intermediate water layer, and it is most probable that
further separation would have been observed had the run
been extended to longer. times. On the other hand, the MC
analysis@
found that the form of the potential of mean force
between the solutes suggested that there were at least two
preferred distances within the overall range of solute
separations considered. These correspond to the same
adjacent and single-solvent-layer-separated solute arrangements observed in the MD simulations. The existence of more than one preferred solute-solute separation
has also been predicted on the basis of an approximate
integral equation approach1* inspired by the OrnsteinZernicke theory of simple fluids.
The two, and almost certainly more, preferred solute
separations are a consequence of a generalized packing
problem; in simple fluids packing is determined by the
repulsive cores of the spherical intermolecular potentials,
whereas in water it is the strong, highly directional hydrogen bonds which dictate molecular arrangement. As
might be expected, for both simple fluids and water, pairs
of neighboring solutes are forced into certain well-defined
positions (aneffect whoee details are very much dependent
on the characteristics of both solute and solvent); water
differs from simple model fluids in that strong anisotropic
interactions are involved, and hence a rearrangement of
the solvent in the shells surrounding the solutes is also to
be expected. The fact that certain solute-solute separations are favored on free-energy gro~ndSeJ*is therefore not
strictly relevant to the question of whether hydrophobic
association of initially well-separated solutes occurs in
dilute solution; indeed the present simulations show no
(la) L.R.Pratt and D.Chandler, J. Chem. Phya., 67,3683(1977);79,
3434 (1980).
Hydratlon of Inert Solutes
The Journal of Physlcal Chemlstty, Vol. 86, No. 6, 1982 87g
evidence of its manifestation.
-
I
1
1
-
-..-
Dynamics
There are a number of experimental studies of dilute
aqueous solutions which indicate that the water molecules
experience a slowing down of their motions when in the
vicinity of a nonpolar s01ute.I~ The earlier MD study of
hydratione examined this phenomenon and found that the
model system did indeed show a reduction in both the
rates of diffusion and reorientation of the shell molecules
as compared to the more distant bulk solvent molecules.
In the present study of the dynamical properties, the
measurements were able to cover time periods longer than
previously attsmptedeJ and, as will be shown below, there
was a considerable degree of solvent molecular exchange
between the shell and the bulk over the course of the
simulation. In order to distinguish between shell and bulk
solvent molecules and to avoid considering those shell
molecules which succeeded in drifting away from the solute, the earlier MD worke examined the motion of those
shell molecules that initially lay within a distance of the
solute somewhat less than the actual shell width; thus, only
that fraction of the set of shell molecules which actually
occupied this inner shell at the start of the measurement
period were treated as shell members, and the remainder
were consigned to the bulk. Over the extended time intervals considered here, this restriction is no longer adequate to ensure that the solute molecules remain in the
shell for the duration of the measurement; the mean time
that a molecule spends in the shell is much shorter than
the length of the simulation run.
A direct measure of the rate of molecular exchange between the shell and bulk components of the solvent may
be obtained by studying the shell occupancy lifetimes.
T w o different definitions of this quantity have been con.
sidered, both of which focus attention on the set of shell
occupants a t some given initial instant and characterize
the elapsed time until a particular event occurs. The
"continuous" occupancy time rc describes the average
elapsed time until a shell member first departa from the
shell. The "intermittent" occupancy time q is the average
time elapsed before a solvent molecule leaves the shell for
the last time during the observation period. In the case
of rc, molecules which return to the shell subsequent to
their initial exit make no further contribution to the results; T~on the other hand allows for all exits and returns
during the period under examination. Clearly the function
n,(t),which describes the mean fraction of shell molecules
which have not yet made their initial exit from the shell
after a time interval t, must decay monotonicaUy with time,
whereas ni(t),the fraction of original occupants actually
present in the shell after time t, may well show a certain
amount of oscillation superimposed on the overall decay.
Both functions supply estimates of the time-dependent
correlationswithin the system; the behavior of nc(t)reflects
local short-term coordinate oscillations, whereas ni(t)is a
measure of the longer-term molecular rearrangement.
The graphs of n&) and ni(t) vs. t for the two runs are
shown in Figure 8. The results represent averages over
several independent time intervals. In order to determine
whether the behavior in the vicinity of the solute is any
different from that in pure water, the figure also includes
the corresponding results for the shells surrounding several
reference water molecules chosen from the same system.
In each case the shell decay around the solute appears to
be slower than that around the reference water. For
(19) K. Hallenga, J. R. Grigera, and H. J. C. Berendsen, J. Phys.
Chem., 84, 2381 (1980),and references therein.
I
1
ni
ni
nc
nc
---
I
1
solute
water
-
- 1.0
- 0.8
solute
water
I
I
0
I
2
I
1
4
I
1
6
1
8
t(psec)
Flgun 8. Shell occupancy a8 a function of tlme for both runs. Both
the continuous and lntermlttent occupancy decay curves, n,(t) and
ndt), are shown, for both solute and reference water shells.
purposes of calculation the shell radius was taken to be
6 A, roughly corresponding to the first minimum of gA0
and just below the second minimum of goo; in both cases
the shells contain an average of approximately 16 water
molecules (from Figure 3), though the spatial distributions
and orientations within the shells are of course very different. In order to prevent interference between the solutes, the contribution of the shell surrounding an individual solute molecule to the overall results was not included if, at any stage during the particular time interval,
the solute approached to within 6.5 A of another solute;
it is apparent from Figure 7 that such close approaches
were infrequent events. A similar criterion was applied
to the reference water molecules. In all cases, the coordinates of a water molecule were taken to be those of its
center of mass.
Estimates of the lifetimes q and 7, for both solute and
reference water shells were obtained by assuming exponential decay for both n&) and ni(t). The results are 7,
= 4.8 pa, ri = 16 ps, for both kinds of shell; it hardly bears
mentioning that these values are very sensitive to the entirely arbitrary choice of hydration shell radius. Even
though Figure 8 reveals that solute shell occupants diffuse
away from the central molecule more slowly than the
members of a shell surrounding an ordinary water molecule, the numerical uncertainty of the decay curves does
not permit lifetime estimates to be made with sufficient
confidence to differentiate between the two cases. The exit
rate from the solute shell nevertheless seems to be some
20% lower than from the reference water shell and indicates a slowing down of the solvent motion relative to the
solute within the hydration shell; such behavior is in
qualitative agreement with experimental obser~ation,'~
although there the slowing down is by a factor of 2-3
(depending on the size assumed for the hydration shell
when interpreting the the data).
Figure 9 illustrates the mean-square displacement as a
function of time for both solute and solvent molecules. By
averging over several independent portions of the run,the
uncertainty of the solvent results was estimated at approximately 10% but, because of the smaller sample size,
that of the solute approached 50%; the irregularity of the
solute graphs is a clear indication of this shortcoming.
There is no obvious difference in self-diffusion rates between solute and solvent. The sampling errors inherent
in the data do, however, reflect on the self-diffusion estimates for hydration shell molecules obtained in the
880
The Journal of Physical Chemistry, Vol. 86, No. 6, 1982
0 = 3I A
/,,’
$!L-
4
6
b
t lpsec
Figure g. Time dependence of the mean-square displacement of
solute and solvents molecules for the two runs.
earlier MD study:s specifically,in Figure 19 of ref 6 there
is a probable sampling error large enough to bridge the gap
between the shell and bulk solvent self-diffusion data,since
the number of displacement measurements (for each value
of t ) is almost the same, a similar 50% uncertainty is
present in the results. Furthermore, as Figure 9 of this
paper reveals, the mean-square displacement rate observed
over a period of less than 2 ps is not necessarily a reliable
guide to longer-term behavior. The object lesson from the
diffusion analysis is that the extraction of reliable data for
solute and shell dynamical properties requires very long
simulation runs in order to ensure a sufficient number of
independent measurements-a great deal longer than that
deemed necessary for pure systems.12i20
Conclusions
The molecular dynamics runs reported here involve
larger systems and span time periods more than 1 order
of magnitude greater than previously attempted. The
enhanced degree of c o n f i a t i o n space sampling achieved
here can also be compared with existing MC studies: using
a criterion based on both translational and rotational
self-diffusion it has been shown2Ithat a single attempted
MC move for each molecule in the system corresponds, at
s; thus,
best, to a single MD time step of about 0.2 X
the runs that we have carried out would be equivalent to
normal MC simulations involving 100 million stepsapproximately 1 order of magnitude greater than any reported to date.
The long runs reported here have shown no indication
of any solute-solute attraction that might be attributed
(20) F. H. Stillinger and A. Rahman, J. Chem. Phys., 60,1545 (1974).
(21) M.Rao,C. Pangali, and B. J. Berne, Mol. Phys., 37,1773 (1979).
Rapaport and Scheraga
to hydrophobic effects. There is clearly a reorganization
of the water molecules around each solute which persists
at least to second-neighbor distance, but no pronounced
tendency for the solutes to approach one another or, if they
do, for them to remain associated. Hard spheres are obviously not experimentally relevant entities, and the simplified model employed in calculations of this kind cannot
be expected to completely mirror reality (in particular, we
have not attempted an examination of the thermodynamics of hydration3), but our goal here has been to attempt
to examine a possible hydrophobic effect directly, rather
than attempt to deduce its existence indirectly from solutes
which also attract each other directly, e.g., by means of the
U interacti~n.~
There is also the possibility that, despite
their length, the simulations are still of insufficient duration; however, this claim may be countered by the observation that a significant amount of solute motion is
observed to occur during the runs and, if there were a real
effective interaction leading to association, it would have
had ample opportunity to manifest itself. Yet another
possibility is that solute aggregation might be a cooperative
phenomenon, requiring the participation of many more
solute molecules than available here; to examine this hypothesis would require simulation of considerably larger
systems.
A question which had been studied previously: but
which could not be addressed in the present work, is the
form of the solute-solute pair distribution at short distances. Since there was no tendency for the solutes to
approach each other, and in other exploratory runs (not
described here) pairs of solutes which were initially adjacent showed a tendency to drift apart, no data could be
collected to determine this distribution. On the basis of
the observed molecular distribution within the hydration
shell of a single solute, it is however entirely reasonable
to expect that there will be certain preferred solute separations, corresponding to those time-averaged water
structures which favor hydrogen-bond formation among
the (multisolute) shell water molecules. This result has
J solutes,Sg but, while it is almost
been demonstrated for L
certainly of significance for solutes which are forced into
proximity for other reasons, for example, by virtue of the
fact that they are neighboring nonpolar groups on the same
polymer chain, the existence of preferred separations does
not in itself imply the presence of any longer-range hydrophobic attraction, at least within the framework of the
present model.
Acknowledgment. This work was supported by research
grants from the National Science Foundation (PCM7918336 and PCM79-20279) and the National Institute of
General Medical Sciences, National Institutes of Health
(GM-14312 and GM-25138).