8
Water Inside a Hydrophobic Cavitand Molecule Jeffrey Ewell, Bruce C. Gibb, and Steven W. Rick* Department of Chemistry, UniVersity of New Orleans, New Orleans, Louisiana 70148 ReceiVed: May 19, 2008; ReVised Manuscript ReceiVed: June 18, 2008 The structure and dynamics of water inside a water-soluble, bowl-shaped cavitand molecule with a hydrophobic interior are studied using molecular dynamics computer simulations. The simulations find that the number of inside water molecules is about 4.5, but it fluctuates from being completely empty to full on a time scale of tens of nanoseconds. The transition from empty to full is energetically favorable and entropically unfavorable. The water molecules inside have fewer hydrogen bonds than the bulk and in general weaker interactions; the lower energy results from the nearest-neighbor interactions with the cavitand atoms and the water molecules at the entrance of the cavitand, interactions that are lost upon dewetting. An analysis of translational and rotational motion suggests that the lower entropy of the inside water molecules is due to decreased translational entropy, which outweighs an increased orientational entropy. The cavitand molecule acts as a host binding hydrophobic guests, and dewetting can be induced by the presence of a hydrophobic guest molecule about 3 Å above the entrance. At this position, the guest displaces the water molecules which stabilize the inside water molecules and the empty cavitand becomes more stable than the full. I. Introduction Deep-cavity cavitand 1 is a bowl-shaped host molecule that can bind a wide variety of hydrophobic guests. 1 The cavitand is water-soluble by virtue of eight carboxylic acid groups on its surface and has a large hydrophobic concave binding pocket defined by twelve aromatic rings. The interior is roughly 8 Å deep, 8 Å wide at the rim, and narrow enough at the bottom as to be closed to the passage of molecules (as represented in 1). Depending on the hydrophobicity and size of a guest, complexes of different host-guest stoichiometries can be formed by 1. If the guest is very small 2 or amphiphilic 3 the host forms 1:1 complexes, whereas with larger and/or less polar guests, the host dimerizes to form capsular complexes. These supramo- lecular species, in which one or more guest molecule is contained within the hydrophobic interior of the capsule, have been shown to bind drug molecules, 4 separate hydrocarbon gases, 2 and make excellent nanoscale reactors for bringing about novel chemistries. 5 Influencing the binding thermodynamics is the water structure in the binding site of the host. The geometric constraints near such concave surfaces leads to different water properties than water near flat or convex surfaces. 6–10 There has been signifi- cantly less studies of water near concave hydrophobic surfaces than there has been for convex or flat surfaces or pores. These studies have primarily been for water inside model hydrophobic spheres (with radii from 6 to 24 Å) 6,7 or near hydrophobic hemispheres (with radii varying from 6.5 to 12 Å). 8,9 These studies find that water molecules have fewer hydrogen bonds and a higher energy than bulk water. In contrast, the number of hydrogen bonds is the same as the bulk for water near small methane-sized hydrophobic spheres but decreases as the cur- vature increases, indicating a change in the mechanism of the hydrophobic effect as solute size increases. 11–16 The different energetics of water near concave surfaces may have other thermodynamic consequences. In the simulations of Paschek, water molecules in concave regions between hydrophobic solute pairs have an increased heat capacity relative to the bulk and to water molecules next to convex hydrophobic surfaces. 10 The water in concave hydrophobic environments may also have different dynamical properties than the bulk, although this is not clear and may depend on the details of the system. Chau finds similar translational but faster orientational dynamics for water in hydrophobic hemispheres, 8 and Brovchenko et al. find slower translational motion for water in hydrophobic spheres. 6 Faster orientational dynamics would be consistent with fewer hydrogen bonds. For water near both small and large spherical solutes, experiments 17–19 and simulations 12,16 find slower trans- lational and rotational dynamics. So the concave hydrophobic environment limits the number of neighboring water molecules resulting in a decreased number of hydrogen bonds, which allows for faster rotational motion but, maybe, slower transla- tional motion. The fewer number of hydrogen bonds for water molecules near concave surfaces may make them easier to dewet the surface. Simulations of these systems do show water densities near the surface intermediate between wet and dry 7,8 and demonstrate substantial oscillations in the number of water molecules, with the empty, or dry, state being observed, but * To whom correspondence should be addressed. E-mail: [email protected]. J. Phys. Chem. B 2008, 112, 10272–10279 10272 10.1021/jp804429n CCC: $40.75 2008 American Chemical Society Published on Web 07/29/2008

Water Inside a Hydrophobic Cavitand Molecule

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Water Inside a Hydrophobic Cavitand Molecule

Jeffrey Ewell, Bruce C. Gibb, and Steven W. Rick*Department of Chemistry, UniVersity of New Orleans, New Orleans, Louisiana 70148

ReceiVed: May 19, 2008; ReVised Manuscript ReceiVed: June 18, 2008

The structure and dynamics of water inside a water-soluble, bowl-shaped cavitand molecule with a hydrophobicinterior are studied using molecular dynamics computer simulations. The simulations find that the number ofinside water molecules is about 4.5, but it fluctuates from being completely empty to full on a time scale oftens of nanoseconds. The transition from empty to full is energetically favorable and entropically unfavorable.The water molecules inside have fewer hydrogen bonds than the bulk and in general weaker interactions; thelower energy results from the nearest-neighbor interactions with the cavitand atoms and the water moleculesat the entrance of the cavitand, interactions that are lost upon dewetting. An analysis of translational androtational motion suggests that the lower entropy of the inside water molecules is due to decreased translationalentropy, which outweighs an increased orientational entropy. The cavitand molecule acts as a host bindinghydrophobic guests, and dewetting can be induced by the presence of a hydrophobic guest molecule about 3Å above the entrance. At this position, the guest displaces the water molecules which stabilize the insidewater molecules and the empty cavitand becomes more stable than the full.

I. Introduction

Deep-cavity cavitand 1 is a bowl-shaped host molecule thatcan bind a wide variety of hydrophobic guests.1 The cavitandis water-soluble by virtue of eight carboxylic acid groups onits surface and has a large hydrophobic concave binding pocketdefined by twelve aromatic rings. The interior is roughly 8 Ådeep, 8 Å wide at the rim, and narrow enough at the bottom asto be closed to the passage of molecules (as represented in 1).

Depending on the hydrophobicity and size of a guest,complexes of different host-guest stoichiometries can be formedby 1. If the guest is very small2 or amphiphilic3 the host forms1:1 complexes, whereas with larger and/or less polar guests,the host dimerizes to form capsular complexes. These supramo-lecular species, in which one or more guest molecule iscontained within the hydrophobic interior of the capsule, havebeen shown to bind drug molecules,4 separate hydrocarbongases,2 and make excellent nanoscale reactors for bringing aboutnovel chemistries.5

Influencing the binding thermodynamics is the water structurein the binding site of the host. The geometric constraints near

such concave surfaces leads to different water properties thanwater near flat or convex surfaces.6–10 There has been signifi-cantly less studies of water near concave hydrophobic surfacesthan there has been for convex or flat surfaces or pores. Thesestudies have primarily been for water inside model hydrophobicspheres (with radii from 6 to 24 Å)6,7 or near hydrophobichemispheres (with radii varying from 6.5 to 12 Å).8,9 Thesestudies find that water molecules have fewer hydrogen bondsand a higher energy than bulk water. In contrast, the number ofhydrogen bonds is the same as the bulk for water near smallmethane-sized hydrophobic spheres but decreases as the cur-vature increases, indicating a change in the mechanism of thehydrophobic effect as solute size increases.11–16 The differentenergetics of water near concave surfaces may have otherthermodynamic consequences. In the simulations of Paschek,water molecules in concave regions between hydrophobic solutepairs have an increased heat capacity relative to the bulk andto water molecules next to convex hydrophobic surfaces.10 Thewater in concave hydrophobic environments may also havedifferent dynamical properties than the bulk, although this isnot clear and may depend on the details of the system. Chaufinds similar translational but faster orientational dynamics forwater in hydrophobic hemispheres,8 and Brovchenko et al. findslower translational motion for water in hydrophobic spheres.6

Faster orientational dynamics would be consistent with fewerhydrogen bonds. For water near both small and large sphericalsolutes, experiments17–19 and simulations12,16 find slower trans-lational and rotational dynamics. So the concave hydrophobicenvironment limits the number of neighboring water moleculesresulting in a decreased number of hydrogen bonds, whichallows for faster rotational motion but, maybe, slower transla-tional motion.

The fewer number of hydrogen bonds for water moleculesnear concave surfaces may make them easier to dewet thesurface. Simulations of these systems do show water densitiesnear the surface intermediate between wet and dry7,8 anddemonstrate substantial oscillations in the number of watermolecules, with the empty, or dry, state being observed, but* To whom correspondence should be addressed. E-mail: [email protected].

J. Phys. Chem. B 2008, 112, 10272–1027910272

10.1021/jp804429n CCC: $40.75 2008 American Chemical SocietyPublished on Web 07/29/2008

rare, in long simulations.9 Simulations of water in modelhydrophobic pores20 and carbon nanotubes21–23 in which theconfined environment also leads to a reduced number ofhydrogen bonds show fluctuations between empty and full states.In both the concave and pore studies, the water-solute interac-tions are critical, with purely repulsive potentials leading todewetting but with weak attractive interactions, correspondingto more realistic interactions, leading to surface wetting.6,7,21,22

Dewetting fluctuations are seen in simulations of the folding ofproteins and other polymers,24–27 although it is not establishedthat folding proceeds through dewetting as opposed to a gradualwater expulsion mechanism.28 In the simulations of Miller etal. the rate-limiting event in hydrophobic polymer collapseinvolves a fluctuation in the local water density in concave bendon the polymer.27 More related to host-guest binding, the studyof Setny of the potential of mean force between a methanemolecule and a hydrophobic hemisphere find that the surfacecompletely dewets as the methane center is within 1 Å of thetop of the cavity (as indicated by the center of the cavity atoms),with partial dewetting occurring as larger distances.29

In this study, we use molecular dynamics computer simula-tions to study water in the concave hydrophobic surface of theoctaacid cavitand (1). We aim to characterize the water density,the energetic factors that lead to hydration versus dewetting,and the dynamical properties of the water molecules inside thecavitand. We also describe how the water is affected by thepresence of a guest molecule as it approaches the host.

II. Methods

The molecular dynamics simulations were performed usingthe AMBER 7 simulation package,30 with the TIP4P-Ewpotential for water.31 The TIP4P-Ew model accurately repro-duces many of the important properties of water over a rangeof temperatures31 and also accurately reproduces the solvationproperties for simple nonpolar solutes.32 Charges for the octaacidmolecule were generated from a RESP charge fitting procedurewith input from Hartree-Fock calculations at the 6-31G* levelusing the Gaussian03 program.33 Additional parameters for theoctaacid and ethane molecules were generated using the gaffparameter set.34 The initial coordinates for the octaacid 1 werefrom an unpublished X-ray structure. Of the eight carboxylicacids, the four at the top rim of the structure, which are extendedout into the solvent, were assumed to be unprotonated. Of thefour at the bottom, which are in relatively close proximity, two,taken to be diagonally across from each other, were assumedto be protonated. This gives a total net charge of -6. Chargeneutrality of the system was created by adding 6 sodium ions,using the Amber 99 parameter set.35 The system contained 1178water molecules. Simulations were carried out primarily in theisothermal-isobaric (T,P,N) ensemble, for lengths of 10 ns,except for microcanonical (E,V,N) simulations performed fordetermining dynamical properties. A 1-fs time step was used,and long-ranged electrostatics were treated with particle meshEwald. A cutoff for the Lennard-Jones and the real-space partof the Ewald interactions equal to 9.0 Å is used. All analysesof the simulations were done with our own programs. Theoctaacid molecule was free to translate and rotate, as well aschange conformation, so the spatial analysis of water structurewere done by rotating the simulation coordinates onto theoriginal octaacid structure, which has its C4 axis along the zaxis. The x axis is defined to be along the direction connectingtwo opposite carbon atoms connected to the carboxylic acidson the bottom of the molecule. The atoms at the top of themolecule roughly form a square; the x axis connects across the

middle of two sides. By symmetry, the y axis would beequivalent to the x axis. All error estimates represent onestandard deviation of the data. Simulations were also carriedout which excluded water from the interior of the cavitand.Water were kept out of the interior using a short ranged repulsivepotential inside the cavity, E(z) ) Rz12, where z is the distancefrom a plane perpendicular to the C4 axis, with R equal to 2345Å12 kcal/mol. The plane is positioned at a distance d from thebottom of the cavitand.

III. Results

Water Density Structure in the Cavitand. Figure 1 showsthe water density as a function of x and z (see Methods). Thedensity shows the average position of the water oxygenpositions. The 4-fold symmetry of the molecule was used togenerate the density. The origin of z corresponds to the positionof the four methine carbon atoms from which the -CH2-CH2-CO2 groups connect. A number of conclusions can bemade from this figure. First, water does enter the cavitand to asignificant amount. There is no water density at the bottom ofthe cavitand, indicating that the bottom of the cavitand is closedto the passage of water. Inside the cavitand, two separate regionsof water density are apparent. The two regions are separatedby a region of low density right below z ) 4 Å, correspondingto the positions of four benzal hydrogen atoms. The lowerregions contains a single, considerably localized, water molecule.The upper region contains more water molecules, the numberof which depends on the definition of the interior region. Thechoice of value of z that separates interior water molecules fromthe others is arbitrary. We chose to use the positions of the 8oxygen atom on the 32-member ring of the rim, which are theuppermost heavy atoms. These have an average z position equalto 7.3 Å. Notice that the excluded volume of the cavitandmolecule extends beyond this distance, to almost 10 Å. Thatvalue of z was chosen because it is in the region where thecavitand-water surface changes from convex to concave, thisbeing one way to define interior rather than exterior solutesurfaces, and also because above this position there are alwayswater molecules, whereas below there are configurations forwhich there are no water molecules. In addition, because thisis the position of oxygen atoms, this could be considered theupper limit of the hydrophobic part of the cavitand. With thisdefinition, there are on average 4.34 ( 0.04 water molecules,although this number fluctuates considerably.

For the 10-ns simulation, the cavitand is empty of watermolecules for only a small fraction of the time, 3 out of 10000stored configurations. (These represent distinct events, separatedby configurations in which there are water molecules in thecavitand.) From this, the free energy of the filled relative toempty cavitand can be estimated from ∆G ) -kT ln[(10000 -3)/3) or -4.8 ( 0.6 kcal/mol. Because the empty configurationsare so rare, there is a good deal of uncertainty in this number.This is in addition to its arbitrariness, due to the ambiguity ofthe definition of interior. However, the simulations do showthat a dewetting transition or a liquid-vapor oscillation takesplace on a time scale of approximately 5 ns. Less ambiguous isthe presence of a lower-most water molecule, between 0 and4.0 Å. A water molecule is present in this position a fraction0.88 ( 0.04 of the time, giving a free energy difference(occupied relative to empty) equal to -kT ln[0.88/(1 - 0.88)]) -1.2 ( 0.1 kcal/mol. From the probabilities of having Nwater molecules in the cavitand, P(N), a free energy relative tothe empty state can be found from -kT ln[P(N)/P(0)]. Theprobabilities and free energies are shown in Figure 2. The most

Water Inside a Hydrophobic Cavitand Molecule J. Phys. Chem. B, Vol. 112, No. 33, 2008 10273

probable, or lowest free energy, number of water molecules is4, with 5 being very close. The highest number of watermolecules seen inside is 7.

One measure of the effect of the confined environment onthe water structure is the number of hydrogen bonds betweenwater molecules. We use the Mancera and Buckingham defini-tion of a hydrogen bond: a hydrogen bond exists if theoxygen-oxygen distance is less than 3.6 Å and the anglebetween the O-H vector on the hydrogen bonding donor andthe O-O vector is between 130 and 180°.37 The most confinedwater is in the lowest position; this water has on average 1.33( 0.02 hydrogen bonds. The lower water tends to form hydrogenbonds as a hydrogen bond acceptor. The other water moleculesinside the cavitand (from around 4 to 7.3 Å) have on average2.65 ( 0.01 hydrogen bonds. In the region at the top of thecavitand, from about 7.3 to 10.3 Å, the number of hydrogenbonds is increased to 3.33 ( 0.01, closer to the bulk value of3.64 ( 0.02. (Bulk properties are calculated using the ten watermolecules furthest from the center of the cavitand.) Becausethe water molecules in the concave part of the cavitand haveone less hydrogen bond than bulk water, the energies of thesemolecules must be significantly different.

The number of hydrogen bonds with neighboring watersdepends both on the geometric constraints of the cavitand, whichlimits the number of water molecules and the density of thewater molecules within the available volume. From Figure 1 itis apparent that the density in the lower part of the cavity isstructured, with regions of high density (around x ) 0 Å, z )2 Å, and x ) (1 Å, z ) 5 Å) and regions of low density (x )0 Å, z ) 6 Å). Above z ) 7 Å, the density appears similar tothe bulk value (above z ) 11 Å). The structure of the watercan be further characterized by F(z), the number of watermolecules that are inside the cavitand and have an oxygenposition between z and z +δz [Figure 3A]. Three sections ofwater density can be seen. A section from 0 to 4 Å integratesto give 0.88 ( 0.04 water molecules (as reported above). Amiddle section from 4 to 6.6 Å has 2.82 ( 0.04 water molecules,and the a top section from 6.6 to 9 Å contains 3.21 ( 0.02molecules. To make comparisons to the bulk density, someestimates of the volume of the cavitand must be made. For aparticular value of z, a volume can be found from the area, A(z),of the plane tangential to the z axis, which is within the solvent-accessible surface38 of the cavitand. To calculate the solvent-accessible surface, we use a water radius of 1.4 Å and radii for

Figure 1. Two-dimensional density of water molecules in the vicinity of the octaacid cavity. The density is in units relative to the bulk. The plotwas prepared with the XFARBE program.36

10274 J. Phys. Chem. B, Vol. 112, No. 33, 2008 Ewell et al.

the cavitand atoms were calculated using the method of Hummeret al. in which the radii correspond to the distance where thewater-solute interaction equals 1 kT.39 This gives radii equal to1.1, 1.5, and 1.6 Å for hydrogen, oxygen, and carbon atoms,respectively. The resulting density (the dotted line in Figure3A) shows that the lowest and second lowest position havedensities slightly higher than the bulk density, and near the topthe density is about equal to the bulk value (1 g/cm3).

Energetics of the Cavitand Water Molecules. The energyof a molecule depends on its position in the cavitand, asillustrated in Figure 3B. The bulk energy is -22.26 ( 0.03 kcal/mol, -21.96 ( 0.03 kcal/mol from the water-water interactionsand -0.31 ( 0.02 kcal/mol from the water-octaacid interac-tions. The energy is less than the bulk value inside the cavitandand only reaches the bulk value at positions outside the cavitand,above 10 Å. The energy is highest for the lower-most water,where the water-water and water-octaacid energies have

similar values. The average energy for a water molecule in thisposition is -15.31 ( 0.03 kcal/mol (the water-water contribu-tion is -8.57 ( 0.04, and the water-octaacid contribution is-6.74 ( 0.03). For the other water molecules in the concavesection of the cavitand from around 4 to 7.3 Å, the energy isalso noticeably higher than the bulk, with an average value of-18.99 ( 0.02 kcal/mol (the water-water energy is -16.16( 0.02, and the water-octaacid energy is -2.83 ( 0.02 kcal/mol). The energetically stable position in the cavitand around4.4 Å is stabilized by favorable water-water interactionsbetween a water molecule at that position and the water localizedin the bottom position of the cavitand. This analysis shows thatthe energy lost from having fewer water neighbors inside thecavitand is not completely compensated for by the interactionwith the cavitand atoms, which is not surprising given thenonpolar nature on the cavitand interior. It is also apparent thatthe regions of high water density (near 2 and 5 Å) do notcorrespond to regions where the energy is relatively low. Clearly,water molecules are in places that are not low in energythemselves, so they must lower the free energy of the entiresystem by being in these positions.

Given that the energy of the water molecules inside thecavitand is higher than the bulk water, the question is whatfactors favor hydration of cavity. To fully understand theenergetic contribution to the hydration process, the energy ofthe entire system with the empty and full cavitand should becompared. However, the difference from the movement of afew water molecules would be very small relative to the totalenergy of the system, which contains over a thousand watermolecules. For this reason, we use a method which only includesthe small subset of water molecules in the vicinity of the cavity.We split up the water molecules into three types. There are then water molecules inside the cavitand, the m water moleculesat the boundary between the inside and bulk water molecules,and the N - n - m remaining or bulk (even though these includemany water molecules near the solute) waters. The potentialenergy as a function of n is

⟨E(n) ⟩ ) ⟨Einsideww (n) ⟩ + ⟨Einside

ow (n) ⟩ + ⟨Eboundaryww (n,m) ⟩ +

⟨Eboundaryow (n, m) ⟩ + ⟨Ebulk

ww (N- n-m) ⟩ +

⟨Ebulkow (N- n-m) ⟩ + ⟨Eoo(n)⟩ (1)

where the superscripts ww, ow, and oo indicate water-water,octaacid-water, and octaacid-octaacid energies, respectively.The energy ⟨Einside

ww (n)⟩ is the sum of all the water-waterinteractions involving the inside water molecules, ⟨Eboundary

ww (n,m)⟩is the sum of the water-water interactions between the boundaryand bulk water molecules, and ⟨Ebulk

ww (N-n-m)⟩ is the sum ofinteractions only between the bulk water molecules, all averagedover those configurations with n inside water molecules. Oursimulations find that both ⟨Eboundary

ow (n, m)⟩ and ⟨Eoo(n)⟩ areindependent of n, with this assumption the energy differencebetween the hydrated and empty cavitand is

⟨∆E(n) ⟩ ) ⟨E(n) ⟩ -⟨E(0) ⟩ ) ⟨Einsideww (n) ⟩ + ⟨Einside

ow (n) ⟩ +

⟨Eboundaryww (n, m) ⟩ -⟨Eboundary

ww (0, m) ⟩ -⟨Ebulkww /Nbulk ⟩ n-

⟨Ebulkow /Nbulk ⟩ n (2)

The change in energy involves the gain in energy from thewater-octaacid interactions and the water-water interactionsinvolving the inside water molecules, a loss of the bulkwater-octaacid and water-water interactions, and a modifica-tion of the water-water interactions in the boundary. The loss

Figure 2. (A) The probability of having N water molecules inside thecavity and the (B) the free energy for N water molecules relative tozero water molecules. The line in part B is a fourth-order polynomialfit to the free energy data, and the line in part A is from the samepolynomial.

Figure 3. (A) The density of water molecules inside the cavity,expressed in terms of F(z), the number of water molecules which havean oxygen position between z and z +δz (solid line) and F(z) scaledby the area within the solvent-accessible surface, A(z) (dashed line).(B) The total (solid line), water-water (dashed line), and water-octaacid(dotted line) interaction energy of a water molecule as a function of itsoxygen position along the z axis inside the cavitand. The constantdot-dashed line is the bulk total energy.

Water Inside a Hydrophobic Cavitand Molecule J. Phys. Chem. B, Vol. 112, No. 33, 2008 10275

in the bulk water-water energy is only half the water-waterenergy (-21.94 kcal/mol) to avoid overcounting. This factorof a half is consistent with the fact that, as a water moleculeleaves the liquid phase, the neighboring water molecules willadjust to fill the space left by the exiting water molecule andremake their interactions. This is in contrast to the watermolecules inside the cavitand. When they exit the cavitand, thewater-octaacid interactions and the water-water interactions,most significantly the interactions between the inside andboundary water molecules, are lost. On average, there are 3.15( 0.03 hydrogen bonds between the inside and boundary watermolecules. How the boundary water molecules respond whenthese interactions are lost will affect the energy change throughthe terms ⟨Eboundary

ww (n, m)⟩ - ⟨Eboundaryww (0, m)⟩ .

To answer that question, we have to define the boundaryregion. Figure 4 shows the total energy of a water molecule asa function of its z position within the cavitand averaged for theconfigurations with n ) 0 (the dashed line) and n ) 4 (the solidline). To generate more configurations with n ) 0, a plane insidethe cavitand (at a position z ) 6 Å) with a short-ranged repulsiveinteraction on the water molecules was used to eliminate watersfrom the cavitand, as described in Methods. We started withthe three n ) 0 configurations and four additional configurationswith water molecules all above 6 Å (equilibrating for 3 ps firstto empty the cavitand of waters), taken from the unbiasedsimulations, and ran for 0.4 ns, to give a variety of startingconfigurations and 1.4 ns of total simulation time. The energiesbecome independent of the number of inside water moleculesonce a distance over 10 Å is reached, and so the boundary regionis taken to be from 7.3 to 10.3 Å. This region contains onaverage 8 water molecules, so we set m ) 8. Also shown is theenergy of water molecules with n ) 4 less the interaction withthose four inside water molecules (the dotted line). The dottedline then shows what the energy of the boundary watermolecules would be if there were no changes in structurebetween the full and empty cavitands. The fact that the dottedline is slightly higher in energy than the dashed line indicatesthat the water molecules do reorientate to partially make up forthe interactions lost by the removal of the interior watermolecules. This same effect is seen in the hydrogen bondaverages for water molecules in boundary region. With n ) 4,a water molecule at the bottom of the boundary region, and so

closest to the inside, has an average of 3.34 ( 0.02 hydrogenbonds, 2.38 ( 0.03 with other boundary water molecules and0.96 ( 0.03 with the inside water molecules. With n ) 0, thewater molecule at the bottom of the boundary region has 2.63( 0.03 hydrogen bonds, all, obviously, with other boundaryregion water molecules. So the one hydrogen bond lost withthe absence of an inside water molecule is partially made up(with only 0.25 of a hydrogen bond) through more hydrogenbonds with waters in the boundary region.

The energies needed as input to eq 2 are given in Table 1.The energies for each value of n are given, along with theweighted average for each energy

⟨A ⟩ )∑n)1

7

A(n)e-∆G(n)⁄kT

∑n)1

7

e-∆G(n)⁄kT

(3)

For the free-energy change, the average given is the to-tal free energy change between the empty and hydratedcavitand as given earlier. The energy ⟨∆Eboundary

ww ⟩ is equal to⟨Eboundary

ww (n)⟩ - ⟨Eboundaryww (n ) 0)⟩ , with ⟨Eboundary

ww (n ) 0)⟩ )-137.3 ( 1.2 kcal/mol. This energy only contributes a smallamount to the overall energy change, about 1 kcal/mol,indicating that there is not a lot reorganization of watermolecules in the boundary region. The amount of the totalwater-water energy involving the n inside water molecules,⟨Einside

ww ⟩ , that comes from interactions with the m boundarywater molecules, ⟨Einside-boundary

ww ⟩ , is also shown. The energychange, ⟨∆E(n)⟩ , for all n except for n ) 1 (but here theerror bars are large) is negative, so the process of hydratingthe cavitand is exothermic, even though the interactions ofwater molecules inside the cavitand are less than they are inbulk. There is also a PV contribution to the enthalpy change,arising from the larger volume of the n ) 0 state from thelarger number of water molecules in the bulk. The molecularvolume of liquid water is 29.9 Å3, and the PV energy permolecule is only 4 × 10-4 kcal/mol.

The negative energy change can be attributed to two factors.The water-water interactions for the inside water moleculeswhich average to -54.5 ( 0.2 kcal/mol are less than thewater-water interactions the water molecules have in the bulk,-47.70 ( 0.04 kcal/mol. This is largely due to the fact that thewater molecules in the boundary region do not make newinteractions when the interior water molecules are absent andthe energy between the inside and boundary water molecules(-11.3 ( 0.2 kcal/mol) is lost. The boundary water moleculeschange structure to make up only about 1 kcal/mol of thatenergy. The empty cavitand creates a small vapor-liquidinterface costing about -10 kcal/mol in energy (⟨Einside-boundary

ww ⟩- ⟨∆Eboundary

ww ⟩). Some (about 4.2 kcal/mol) of that energy mustbe compensated for by the stronger interactions in the bulk sothe total contribution to the energy change from the water-waterinteractions is -5.7 ( 1.3 kcal/mol (⟨Einside

ww ⟩ - ⟨∆Eboundaryww ⟩ -

n⟨Ebulkww /Nbulk⟩). The other factor is the lost interactions between

the interior water molecules and the octaacid molecule (-15.4( 0.2 kcal/mol), which is significantly greater than the interac-tion between bulk water molecules and the octaacid (-1.35 (0.05 kcal/mol). This factor then contributes about -14 kcal/mol, which together with the water-water contribution makesup the total energy change of about -20 kcal/mol.

If the enthalpy change is as large as -20 kcal/mol, then theremust be an entropy change, -T∆S, of about 15 kcal/mol togive ∆G ) -5 kcal/mol. The large entropy change would mean

Figure 4. The total interaction energy for a water molecule with four(solid line) and no (dashed line) water molecules inside the cavitand.The dotted line shows the total interaction energy with four inside watermolecules minus the interaction with the four inside water molecules.

10276 J. Phys. Chem. B, Vol. 112, No. 33, 2008 Ewell et al.

that there is a tendency for the cavitand to be less occupiedwith water as temperature increases. To test this, we ranadditional simulations at 313 and 328 K. The simulations diddemonstrate a trend for the cavitand to be empty with a greaterprobability as temperature increased. The probability of beingempty increases from 0.0003 ( 0.0002 at 298 K to 0.003 (0.002 at 313 K to 0.007 ( 0.005 at 328 K. The entropy change,⟨∆S(n)⟩ , can be found from the slope (as determined from alinear fit) of ⟨∆G(n)⟩ . An estimate of the enthalpy change canthen be found from ⟨∆G(n)⟩ + T⟨∆S(n)⟩. This is shown in Table1, as ∆HT. The agreement between the energy change from eq2 and from the temperature dependent data is very good. Bothgive -20 kcal/mol. The agreement is not as good for some ofthe less common hydration states, particularly n ) 1, but bothmethods agree that not only is the average energy changenegative but also that it tends to decrease as n increases.

Water Structure in the Presence of a Guest Molecule.Because the inside water molecules are stabilized throughinteractions with water molecules just outside the cavitand,disrupting the water structure outside the cavitand could leadto less water molecules inside. To test how a guest moleculeinfluences the water structure inside the cavitand, we performedsimulations with using ethane as a potential guest. Ethane isthe one (purely) hydrocarbon guest that has been shown to forma stable 1:1 complex with cavitand 1.2 (Methane does not bindto the pocket, whereas propane triggers dimerization of the hostto form a 2:2 capsular complex.)2 The center-of-mass of theethane molecule in constrained at various distances along theC4 (z) axis from the bottom of the cavitand, using a harmonicrestraint, with force constant 2.0 kcal/mol/Å2. Four separatesimulations of 10 ns each were performed, at distances of 10,11, 12, and 14 Å. From these simulations, the fraction that thecavitand was empty as a function of ethane distance wascalculated (Figure 5). The ethane molecule does induce dew-etting of the cavitand when it is in a position to displace theboundary region water molecules. The hydration of the cavitandis not changed until the ethane gets to a point around 11 Å.Beyond 12 Å the probability of finding an empty cavitand istoo low for it to show in the probability histograms. Forcomparison, the probability of the empty cavitand without anethane molecules (0.0003) is shown; this is essentially the sameprobability as that when the ethane is constrained to be 12 Åor farther away. At a distance of 11 Å, the number of insidewater molecules oscillates from being fully occupied, with 4or more molecules, to empty, on a time scale of 1.1 ( 0.4 ns.When the ethane molecule is closer than 10 Å to the bottom ofthe cavitand, the probability to be empty increases to almostone. When the ethane molecule is actually inside the cavitand(unconstrained), there is not much room for water molecules,

and there is on average less than one (0.3) water molecule inside.Inside the cavitand, the ethane molecule is positioned with anaverage center-of-mass position equal to z ) 4.26 ( 0.06, withone carbon atom in the lowermost position (less than 4 Å) alarge fraction (0.77 ( 0.04) of the time. When the ethane is inthe cavitand, a water molecule was never in the lower position.

Dynamics of the Interior Water Molecules. The transla-tional velocity autocorrelation function is given by

CV(t)) 1N∑

i)1

N

m⟨ViCM(t)Vi

CM(0)⟩ (4)

where m is the mass of a water molecule and ViCM(t) is the center

of mass velocity of molecule i. Rotational times scales can befound from the angular velocity autocorrelation function

cω(t)) 1N∑

i)1

N

∑j)1

3

Ij⟨ωij(t)ωij(0)⟩ (5)

where Iij is the jth principle moment of inertia of molecule iand ωij is angular velocity around the principal axis j of moleculei. These correlation functions are shown in Figure 6 for bulkwater, the lowermost water, and the inside water molecules(excluding the lowermost water). From these functions thetranslational diffusion constant can be found from40

DT ) 1/3m∫0

∞cT(t)dt (6)

and the rotational diffusion constant can be found from

TABLE 1: Energetic Contributions to Cavitand Hydration (in kcal/mol) as a Function of the Number of Inside WaterMolecules na

n

1 2 3 4 5 6 7 average

⟨Einsideww ⟩ -12.5(2.1) -21.0(0.6) -34.9(0.4) -49.5(0.2) -63.6(0.4) -79.8(0.9) -101.0(2.0) -54.5(0.2)

⟨Einside-boundaryww ⟩ -7.4(0.7) -6.5(0.5) -9.2(0.4) -11.0(0.3) -12.3(0.3) -13.4(0.3) -17.2(1.5) -11.3(0.2)

⟨Einsideow ⟩ -2.3(1.9) -7.2(0.4) -11.5(0.4) -14.4(0.2) -17.5(0.4) -19.3(0.7) -17.6(1.4) -15.4(0.2)

⟨∆Eboundaryww ⟩ 5.9(4.9) 1.6(3.0) 1.1(1.3) 0.8(1.5) 1.3(1.4) 1.4(1.5) 1.2(3.0) 1.0(0.8)

n⟨Ebulkww /Nbulk⟩ -10.97(0.02) -21.94(0.03) -32.91(0.05) -43.88(0.06) -54.85(0.08) -65.82(0.09) -76.79(0.11) -47.70(0.04)

n⟨Ebulkow /Nbulk⟩ -0.31(0.02) -0.62(0.04) -0.93(0.06) -1.24(0.08) -1.55(0.10) -1.86(0.12) -2.17(0.14) -1.35(0.05)

⟨∆E(n)⟩ 2.5(5.6) -4.1(3.0) -11.4(1.3) -18.0(1.5) -23.4(1.4) -30.2(1.5) -38.5(3.5) -19.8(0.8)⟨∆G(n)⟩ -1.1(0.5) -2.5(0.4) -3.6(0.4) -4.3(0.4) -4.2(0.4) -3.3(0.4) -1.5(0.5) -4.8(0.6)⟨∆HT(n)⟩ -14.6(7.8) -15.9(6.2) -18.1(6.0) -20.3(6.0) -21.5(6.0) -22.4(6.0) -21.9(6.7) -20.5(3.4)

a Numbers in parentheses give error estimates.

Figure 5. The probability of an empty cavitand as a function of ethanecenter-of-mass distance along the z axis of the cavitand molecules. Thedotted line shows the probability in the absence of an ethane molecule.

Water Inside a Hydrophobic Cavitand Molecule J. Phys. Chem. B, Vol. 112, No. 33, 2008 10277

DR )1

∑j)1

3

Ij

∫0

∞cwtdt (7)

The values for the translational diffusion constants are 2.6 (0.1 × 10 -5 cm2/s (bulk), 1.9 ( 0.2 × 10 -5 cm2/s (inside), and0.6 ( 0.3 × 10 -5 cm2/s (lowermost). The bulk diffusionconstant is in good agreement with the previously reported valuefor TIP4P-Ew (2.4 ( 0.00 × 10 -5 cm2/s).31 The rotationaldiffusion constants are 2.1 ( 0.1 × 1011 s -1 (bulk), 3.1 ( 0.1× 10 11 s -1 (inside), and 33. ( 2 × 10 11 s -1 (lowermost).

It is evident that the lower water has much different dynamicsthan both the bulk and the other inside water molecules. Boththe translational and rotational motion resemble that of a dampedharmonic oscillator with a period of about 0.5 ps for translationsand 0.1 ps for rotations. The lower water moves in a relativelywide potential minimum with a low frequency until it leavesthis site. The average lifetime in this site is about 0.8 ps. Therotations are relatively fast as would be consistent with amolecule that forms 1.33 ( 0.02 hydrogen bond, mostly as ahydrogen bond acceptor so the hydrogens are free to rotate.

The bulk and inside water time correlation functions bothshow the same time scales, with peaks and minima at similarpoints. The main difference between the translational correlationfunctions is the negative region for the inside water moleculesat 0.25 ps indicating that, after this amount of time, the watermolecules have had collisions with their nearest neighbors,turned around and are on average moving in the oppositedirection from the original point. This negative correlation ismuch more pronounced for the inside water molecules than thebulk. The turning round leads to a decreased translationaldiffusion constant. Some of this turning around must be due tocollisions with the cavitand surface. For rotational dynamics,the main difference is the stronger negative correlation for bulkrelative to the inside water, as given by the deeper minimum at0.025 ps. This indicates that the rotations are more hindered,or librational, for the bulk waters.

Conclusion

The simulation results show that the interior of the cavitandsare occupied with water molecules, which average about 4 or 5,

but fluctuate between 0 and 7. The water molecules have a higherenergy than bulk waters (Figure 3B), forming fewer hydrogenbonds than bulk water. The higher energy and few hydrogen bondsis in agreement with previous studies of water in concavehydrophobic environments.6–9 Despite the higher energy of thewater molecules inside, the process of adding all the watermolecules to the interior is exothermic for two reasons. First, eventhough the water-octaacid interactions are weak individually, theysum up to about -15 kcal/mol over the 4 or 5 water molecules,and these interactions are lost when the cavitand is empty (seeTable 1). The octaacid-water interactions for the bulk water isonly -1.35 kcal/mol, so emptying the cavity results in a loss of-14 kcal/mol in the water-octaacid interactions. Second, theinteractions between the water molecules in cavitand and watermolecules in the boundary between the interior and bulk regionsare also lost, as a small liquid-vapor interface is created. Thisenergy is about -11 kcal/mol. Only 1 kcal/mol of this energy isrecovered by the water molecules in the boundary reorganizing inthe absence of interior water molecules to form more hydrogenbonds with other boundary region water molecules. The smalldegree of reorganization means there is only a small entropy changeassociated with the creation of the interface. An additional amountof energy is gained in moving from the interior to bulk water fromstronger water-water interaction in the bulk, which compensatesfor the energy lost in creating the liquid-vapor interface. In total,the ∆H is about -20 kcal/mol, ∆G is about -5 kcal/mol, and-T∆S is 15 kcal/mol. These values, found from an analysis ofwater populations and energetics at 298 K are consistent withenthalpy changes found from the temperature dependence of ∆G.

Therefore, it is energetically unfavorable for the water moleculesthat transfer from the bulk to the cavitand interior but energeticallyfavorable for the neighboring water and octaacid atoms, resultingin a total energy change that is favorable. On the other hand,because there is not much structural rearrangement of either theboundary water molecules or the octaacid molecule, the entropychange must be mostly due to the entropy difference of thetransferring water molecules. Because the overall entropy changeis negative, the entropy of the cavitand water molecules must belower than that of bulk water. An analysis of the translational andorientational velocity time correlation functions reveals thattranslational motion is slower and that orientational motion is fasterfor water molecules in the cavitand relative to the bulk. Slowertranslational motion makes sense for a molecule in a concaveenvironment in which movement is restricted along some directionsand the water density also shows increased structure inside thecavitand (Figure 1). Faster orientational motion is consistent withfewer hydrogen bonds because the hydrogens not involved in ahydrogen bond should be freer to rotate. The dynamical resultssuggest that the decreased entropy of the interior water moleculesis due to decreased translational entropy, which must overcom-pensate any gain in rotational entropy. A correlation between theconfigurational entropy and the translational diffusion constant, assuggested by this data, is consistent with a number of previousstudies.40–43

That the contribution from the octaacid-water interactionsare so important for the stability of the inside water moleculesagrees with earlier studies of water in concave surface andhydrophobic pores. As mentioned in the Introduction, thesestudies find that water will only not be found near these surfaceif the solvent-solute interactions are purely repulsive, addingweak attractions brings the water molecules in contact with thesurface.7,6,21,22 The other contribution to the negative energy offilling the cavitand, the interaction with the water molecules atthe boundary between the interior and the bulk, can be

Figure 6. The translational velocity autocorrelation function (A) androtational translational function (B) for bulk (solid line), inside (dashedline), and bottom most water (dotted line).

10278 J. Phys. Chem. B, Vol. 112, No. 33, 2008 Ewell et al.

eliminated as a potential guest molecule approaches. Oursimulations find that as an ethane molecule gets to about 11 Åfrom the bottom of the cavity, or 4 Å from the top, the waterstructure is sufficiently disrupted, inducing dewetting of thecavitand (Figure 5). With the ethane molecule in the boundaryregion, the probability of having an empty cavitand is close to1. This is similar to the Setny study of a methane moleculenear a hydrophobic concave hemisphere, in which the surfacebecomes empty of water as the methane molecule get close totop of the cavity.29 For the octaacid/ethane system, with theethane near the entry to the cavitand, transitions between emptyand filled, or dry and wet, take place on a 1-ns time scale. Theseresults suggest a mechanism for host-guest binding, in whichthe guest approaches the guest binding site, promoting fluctua-tions in which the cavitand becomes empty of water. Once thishappens the guest could presumably move to the binding sitequickly, without a large energetic barrier.

Our results can tell us about the role of water in thethermodynamics of host-guest binding process

host{water}(aq)+ guest(aq)h host{guest}(aq) (8)

where the occupant of the host binding site (water or guest) isindicated inside the curly brackets. The process can be split upinto steps

host{water}(aq)h host{}(aq) (9)

guest(aq)h guest(gp) (10)

host{}(aq)+ guest(gp)h host{guest}(aq) (11)

where eq 9 removes the water from the guest binding site, eq10 removes the guest from the solvent, and eq 11 places thegas-phase guest into the empty host. For eq 9, our calculationsfind ∆G ) 5, ∆H ) 20, and -T∆S ) -15 kcal/mol. For thedesolvation step for the guest, in the case of ethane, ∆G ) -2,∆H ) 4, and -T∆S ) -6 kcal/mol.44 For larger hydrophobicguests, the entropy change will be larger. By assumption thatthe guest occupies the same volume as that created by the waterthat left in eq 9, the final step should not involve substantialrearrangement of the water and so the contribution to ∆S fromthe solvent is about zero. There will be solvent contribution to∆H from the interactions between the guest and the watermolecules bordering the binding site. Overall, for the host-guestassembly process there should be a large contribution to theentropy from the solvent, -T∆S ) -21 kcal/mol or ∆S ) 70cal/mol/K. The enthalpic contribution from the solvent shouldbe positive. These calculations provide valuable informationregarding the 1:1 complexes formed by cavitand 1. In theformation of capsular complexes formed by 1, for example, 2:2host-guest complexes, this is only the first step toward the finalassembly. The capular complex would have different thermo-dynamics from the 1:1 complex because it eliminates the waterin contact with the guest. The formation of this complex wouldmake for some interesting future studies.

Acknowledgment. S.W.R. gratefully acknowledges the fi-nancial support of the National Science Foundation (CHE-0611679), and B.C.G. gratefully acknowledges the financialsupport of the National Science Foundation (CHE-0718461).

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