Direct numerical solution of the Ornstein–Zernike integral equation and spatialdistribution of water around hydrophobic moleculesMitsunori Ikeguchi and Junta Doi Citation: The Journal of Chemical Physics 103, 5011 (1995); doi: 10.1063/1.470587 View online: http://dx.doi.org/10.1063/1.470587 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/103/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Extended molecular Ornstein-Zernike integral equation for fully anisotropic solute molecules: Formulation in arectangular coordinate system J. Chem. Phys. 139, 084119 (2013); 10.1063/1.4819211 Ornstein–Zernike equation for convex molecule fluids J. Chem. Phys. 115, 925 (2001); 10.1063/1.1379762 Analytic solution of the molecular Ornstein–Zernike equation for nonspherical molecules. Spheres withanisotropic surface adhesion J. Chem. Phys. 84, 1833 (1986); 10.1063/1.450430 Generalized recursive solutions to Ornstein–Zernike integral equations J. Chem. Phys. 73, 2457 (1980); 10.1063/1.440397 Invariant Expansion. II. The OrnsteinZernike Equation for Nonspherical Molecules and an Extended Solution tothe Mean Spherical Model J. Chem. Phys. 57, 1862 (1972); 10.1063/1.1678503

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Direct numerical solution of the Ornstein–Zernike integral equationand spatial distribution of water around hydrophobic molecules

Mitsunori Ikeguchi and Junta DoiDepartment of Biotechnology, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113, Japan

~Received 13 March 1995; accepted 22 June 1995!

The Ornstein–Zernike integral equation~OZ equation! has been used to evaluate the distributionfunction of solvents around solutes, but its numerical solution is difficult for molecules with acomplicated shape. This paper proposes a numerical method to directly solve the OZ equation byintroducing the 3D lattice. The method employs no approximation the reference interaction sitemodel ~RISM! equation employed. The method enables one to obtain the spatial distribution ofspherical solvents around solutes with an arbitrary shape. Numerical accuracy is sufficient when thegrid-spacing is less than 0.5 Å for solvent water. The spatial water distribution around a propanemolecule is demonstrated as an example of a nonspherical hydrophobic molecule using iso-valuesurfaces. The water model proposed by Pratt and Chandler is used. The distribution agrees with themolecular dynamics simulation. The distribution increases offshore molecular concavities. Thespatial distribution of water around 5a-cholest-2-ene~C27H46) is visualized using computergraphics techniques and a similar trend is observed. ©1995 American Institute of Physics.

I. INTRODUCTION

The Ornstein–Zernike integral equation~OZ equation!theory has been widely used to evaluate the distributionfunction of solvents around solutes. The OZ equation issolved together with an approximate equation called the‘‘closure equation.’’ In the case of simple fluids with spheri-cally symmetric intermolecular potentials, the OZ equation isreduced to a one-dimensional integral. In such cases, the OZequation can be solved numerically on the radial grid.1,2 Fornonspherical solutes and solvents, it is difficult to solve theequation since the OZ equation includes integrals over mo-lecular orientational coordinates. Methods to solve the OZequation for nonspherical molecules using the spherical har-monic expansion have been proposed.3–12 However, thesemethods treat molecules with a simple shape such as di-atomic molecules and hard spheres with dipoles and quadru-poles. In particular, for elongated molecules, many terms ofthe spherical harmonic expansion are required and the prac-tical application is difficult.

The approximate integral equation called the ‘‘referenceinteraction site model~RISM! equation’’ was proposed fornonspherical molecules using site–site correlation func-tions.13 The closure equation was also approximated usingthe site–site correlation functions. Several applications of theRISM equation have been already reported.14–23 However,the RISM equation is limited to only estimating the site–sitedistribution, and cannot evaluate the spatial distribution ofsolvents around nonspherical solutes.

This paper proposes a numerical method to directlysolve the OZ equation by introducing the three dimensional~3D! lattice, thus avoiding any approximation employed bythe RISM equation. The closure equation is also discretelyformulated. This method makes it possible to obtain the spa-tial distribution of spherical solvents around solutes with anarbitrary shape.

As an application of our method, the spatial distributionof water around hydrophobic molecules with an arbitrary

shape is evaluated. We used a water model proposed by Prattand Chandler~PC water model!.24 The model used thesphericalized solvent-solvent correlation function. The ap-proach is a full representation of the solvent with the as-sumption that there is no direct interaction potential betweenthe hydrogen and the solute. In the Percus–Yevick~PY! clo-sure one then gets that the direct correlation function for thesolute-hydrogen is zero, and the correlation functions involv-ing hydrogen are immaterial. They calculated the radial dis-tribution of water around a spherical hydrophobic solute us-ing the model and obtained hydrophobic properties whichagrees with experiments and molecular simulations.

In Sec. II, the OZ equation, the closure equations and theRISM equation are reviewed. Our method to directly andnumerically solve the OZ equation with the closure equa-tions formulated in a discrete form is then described. In Sec.III, our method is applied to a spherical solute and its nu-merical accuracy is tested. Our method is then applied topropane as an example of a nonspherical hydrophobic solute.The result is compared with the molecular dynamics simula-tion. The spatial distribution is visualized using its iso-valuesurface. The influence of the solute shape on the spatial dis-tribution is discussed. An application to 5a-cholest-2-ene~C27H46) is presented as an example of a hydrophobic solutewith a complicated shape. The spatial distribution is visual-ized using computer graphics techniques. In Sec. IV, con-cluding remarks are given.

II. THEORY AND METHODS

A. The OZ equation for infinite dilution

The OZ equation is solved together with the closureequation to evaluate the distributionguv(r ) of solvent arounda solute. The OZ equation for infinite dilution is

huv~r !5cuv~r !1rvE cuv~r 8!hvv~r2r 8!dr 8, ~1!

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whererv is the density of the solvent,huv(r ) is the solute–solvent total correlation function,cuv(r ) is the solute–solvent direct correlation function, andhvv(r ) is thesolvent–solvent total correlation function. The coordinatesr ,r 8 include orientational coordinates. The relationship be-tween the distributionguv(r ) and the solute–solvent totalcorrelation functionhuv(r ) is

guv~r !5huv~r !11. ~2!

The closure equation to be solved together with Eq.~1! is

cuv~r !5exp@2buuv~r !#exp@ tuv~r !1Euv~r !#

2tuv~r !21, ~3!

whereb is 1/kT, k is the Boltzmann constant, andT is theabsolute temperature. The functiontuv(r ) in Eq. ~3! is de-fined by

tuv~r !5huv~r !2cuv~r !. ~4!

The functionuuv(r ) is the solute–solvent intermolecularpotential. For the aqueous solution of a spherical hydropho-bic solute, the following Lennard–Jones functionuuv

l j (r ) isused foruuv(r ):

uuvl j ~r !54«uvF S suv

r D 122S suv

r D 6G . ~5!

The termEuv(r ) in Eq. ~3! is called ‘‘the bridge function,’’which is the complex functional of the total correlation func-tion. The exact calculation of the bridge function is so diffi-cult that several approximations have been already proposed.

The bridge function is known as a short range function.If the bridge functionEuv(r ) is neglected, a closure called‘‘the hypernetted-chain~HNC! closure’’ is obtained as fol-lows:

cuv~r !5exp@2buuv~r !#exp@ tuv~r !#2tuv~r !21. ~6!

On the other hand, a closure called ‘‘the Percus–Yevick~PY! closure’’ is proposed by replacing the term exp@tuv(r )#in Eq. ~6! with @11tuv(r )#. The PY closure is as follows:

cuv~r !5exp@2buuv~r !#@11tuv~r !#2tuv~r !21. ~7!

For an aqueous solution of hydrophobic molecules, Prattand Chandler24 proposed a method to use the repulsive termof the Lennard–Jones potential instead of the full Lennard–Jones potential@Eq. ~5!# in the PY closure@Eq. ~7!#. Wedenote it ‘‘the R-PY method’’ in this paper. The repulsiveterm of the Lennard–Jones potential is written as

uuv~0!~r !5H uuvl j ~r !1«uv ~r<suv2

1/6!

0 ~r.suv21/6!,

~8!

where«uv andsuv are the parameters of the Lennard–Jonespotential@Eq. ~5!#.

B. The RISM equation method

The RISM equation is proposed as an approximate inte-gral equation to apply the OZ equation@Eq. ~1!# to non-spherical solutes and/or solvents by introducing site–site cor-relation functions. The RISM equation is given as follows:

huv5vuu* cuv* ~vvv1rvhvv!, ~9!

where the symbol ‘‘* ’’ denotes a convolution. The termhuvis a matrix having elements of the site–site total correlationfunctionhag(r ) between the interaction sitea of the soluteuand the siteg of the solventv. The termcuv is the matrix ofthe site–site direct correlation functioncag(r ). The termsvuu andvvv are the matrices of the site–site intramolecularcorrelation functions of the soluteu and the solventv, re-spectively. The termhvv is the matrix of the solvent–solventtotal correlation function.

The closures, dedicated to the RISM equation, are alsoapproximated using the site–site correlation functions. TheHNC closure for the RISM is as follows:

cag~r !5exp@2uag~r !/kT#exp@ tag~r !#2tag~r !21,~10!

whereuag(r ) is the potential function between sitesa andg. The functiontag(r ) is defined by

tag~r !5hag~r !2cag~r !. ~11!

On the other hand, the PY closure for the RISM is asfollows:

cag~r !5exp@2uag~r !/kT#@11tag~r !#2tag~r !21.~12!

For the R-PY method, the repulsive term of theLennard–Jones potentialuag

(0)(r ) is substituted into the site–site potentialuag(r ) in Eq. ~12!. The R-PY method for theRISM is as follows:

cag~r !5exp@2uag~0!~r !/kT#@11tag~r !#2tag~r !21.

~13!

The RISM equation can estimate the site–site distribu-tion. However, it cannot evaluate the spatial distribution.

C. The direct numerical solution of the OZ equationusing the 3D lattice

In our method, the OZ equation for infinite dilution isdirectly solved by introducing the 3D lattice without anyapproximation the RISM employed. Solutes are of arbitraryshape, whereas a solvent water is the PC water model. Thecorrelation functions between a solute and water are calcu-lated at each 3D grid point. The OZ equation@Eq. ~1!# isgiven in the discrete form as follows:

huv~x,y,z!5cuv~x,y,z!1rv (x8,y8,z8

cuv~x8,y8,z8!

3hvv~ ur2r 8u!DxDyDz, ~14!

wherehuv(x,y,z) is the discrete solute–solvent total corre-lation function,cuv(x,y,z) is the discrete solute–solvent di-rect correlation function,hvv(ur u) is the solvent–solvent totalcorrelation function, and the vectorsr and r 8 have elements(x,y,z) and (x8,y8,z8), respectively. The functionhvv(ur u)is sphericalized as the PC water model.

Equation~14! is written in the momentum~k! space asfollows:

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t uv~k,l ,m!5huv~k,l ,m!2 cuv~k,l ,m!

5rvcuv~k,l ,m!hvv~ uku!, ~15!

where the symbol ‘‘ ’’ indicates the Fourier transform andthe vectork has the element (k,l ,m).

The closure equation@Eq. ~3!# for the OZ equation isalso formulated in the discrete form as follows:

cuv~x,y,z!5exp@2buuv~x,y,z!#exp@ tuv~x,y,z!

1Euv~x,y,z!#2tuv~x,y,z!21 ~16!

where uuv(x,y,z) is the solute–solvent potential function.The potentialuuv(x,y,z) is calculated at each 3D grid pointby taking a summation of the Lennard–Jones potentials be-tween atoms of the soluteu and water as follows:

uuv~x,y,z!5(a

uavl j ~ ur2rau!, ~17!

where the suffixa denotes an atom of the soluteu, andra isthe coordinate of the atoma.

The HNC closure is given in the discrete form by

cuv~x,y,z!5exp@2buuv~x,y,z!#exp@ tuv~x,y,z!#

2tuv~x,y,z!21. ~18!

On the other hand, the PY closure is given in the discreteform by

cuv~x,y,z!5exp@2buuv~x,y,z!#

3@11tuv~x,y,z!#2tuv~x,y,z!21. ~19!

For the discrete R-PY method, the repulsive term of theLennard–Jones potentialuuv

(0)(x,y,z) is substituted to the po-tential uuv(x,y,z) in Eq. ~19!. The discrete R-PY method isgiven as follows:

cuv~x,y,z!5exp@2buuv~0!~x,y,z!#

3@11tuv~x,y,z!#2tuv~x,y,z!21. ~20!

The repulsive term of the Lennard–Jones potentialuuv(0)(x,y,z) is calculated at each 3D grid point, that is,

uuv~0!~x,y,z!5(

auav

~0!~ ur2rau!. ~21!

The spatial distribution of waterguv(x,y,z) at each 3Dgrid point is calculated fromtuv(x,y,z) and cuv(x,y,z) asfollows:

guv~x,y,z!5tuv~x,y,z!1cuv~x,y,z!11. ~22!

D. Numerical procedures

The numerical procedures to calculate the spatial distri-bution of water are as follows:

~1! The solute-solvent potential functionuuv(x,y,z) is cal-culated at each 3D grid point using Eq.~17! for the PYor for the HNC closure. The repulsive term of theLennard–Jones potential is calculated using Eq.~21! forthe R-PY method.

~2! The functiontuv(x,y,z) is initialized to zero.

~3! The functioncuv(x,y,z) is calculated from the functionstuv(x,y,z) anduuv(x,y,z) using the HNC@Eq. ~18!#, thePY @Eq. ~19!#, or the R-PY method@Eq. ~20!#.

~4! The functioncuv(x,y,z) is transformed to the functioncuv(k,l ,m) using the 3D fast Fourier transform~3D-FFT!.

~5! The functiont uv(k,l ,m) is calculated from the functioncuv(k,l ,m) using the discrete OZ equation@Eq. ~15!#.

~6! The functiont uv(k,l ,m) is transformed totuv8 (x,y,z) us-ing the inverse 3D-FFT. Steps~3!–~6! are iterated untilthe difference betweentuv8 (x,y,z) obtained here andtuv(x,y,z) in step ~3! becomes smaller than the giventhreshold value«.

~7! The distribution of waterguv(x,y,z) in Eq. ~22! is ob-tained from the functionstuv(x,y,z) andcuv(x,y,z).

The convolution integral using the 3D-FFT in steps~4!,~5!, and~6! assumes that the input functions are periodic. Inour case, the error arising from the periodicity is negligiblebecausecuv(x,y,z) and hvv(r ) are short-ranged comparedwith the system size. Our calculation agrees with the spheri-cal calculation as discussed in Sec. III A.

E. The spatial distribution of water

The visualization of the spatial distribution of water isessential to analyze it in the 3D space. In this paper, it isvisualized using the Marching Cube algorithm.25 The arbi-trary iso-value surface of the distribution at each 3D gridpoint can be generated. The spatial distribution can be com-prehensively displayed by transforming the iso-value surfaceinto the solid model. For example, the boolean operationssuch as the plane-cut and the translation are applied to themodel. A detailed observation becomes possible in this man-ner.

III. RESULTS AND DISCUSSION

The PC water model is used as the solvent in ourmethod. The experimentally determined oxygen–oxygencorrelation function for pure water26 at 25 °C is used for thewater–water correlation function.

A. Calculations for a spherical solute

The OZ equation has been solved for spherical mol-ecules using the spherical coordinate, because there is noorientational problem. Pratt and Chandler24 solved the OZequation for the aqueous solution of a hard sphere of CH4.We also solved the OZ equation for a spherical Lennard–Jones CH4 solute to test the numerical accuracy. The param-eters of the Lennard–Jones potential@Eq. ~5!# are as samevalues as Pratt and Chandler used. The two closures, HNCand PY, and the R-PY method are used. The grid resolutionis 64364364. The grid-spacings of 0.5 Å , 1.0 Å, and 1.5 Åfor our calculations and the radial grid-spacing, 0.038 Å, forthe spherical solution are used. Figure 1 shows the radialdistribution of water around a Lennard–Jones sphericalCH4 solute, where the distance is normalized by the potentialparametersuv for CH4 and water. In the figure, our methodwith a grid-spacing of 0.5 Å is compared with the sphericalsolution using the HNC closure. Comparisons of these cal-

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culations are summarized in Table I. When the grid-spacingis 0.5 Å , the maximum deviations are 0.028 for the HNC,0.019 for the PY, and 0.011 for the R-PY. These values aresmall enough in comparison with the water distributionguvon the order of 1~Fig. 1!. The deviations for the grid-spacingof 1.0 Å and 1.5 Å are not negligible. It suggests that thegrid-spacing must be set at less than 0.5 Å~1/3 of the radiusof a water molecule!. The maximum deviations are similarwhen the solute is 0.25 Å shifted~Table I!. The grid-spacingis set at less than 0.5 Å throughout this paper.

B. Spatial distribution of water around a propanemolecule

Our method is applied to propane as an example of anonspherical hydrophobic solute. The spatial distribution iscompared with the molecular dynamics simulation. TheOPLS parameters27,28 are used for the potential. The grid-spacing is 0.2 Å and the grid resolution is 12831283128.This means that the distribution inside a cube with an edge of25.6 Å is calculated. In Fig. 2, the iso-value surface of thedistribution guv51.5 is represented using the solid model.The plane-cut operation is applied to remove the upper partof the distribution to show the space-fill model of the pro-pane molecule. Figure 3 shows a contour map of the distri-bution on the cutting-plane shown in Fig. 2. These figures arediscussed in Sec. III B 3.

1. Comparison with the molecular dynamicssimulation

In previous RISM reports, the distributions of wateraround solutes are compared with the molecular dynamicssimulation.18,20,22 We compared our spatial distribution ofwater with the molecular dynamics simulation. The molecu-lar dynamics simulation is made using AMBER 4.0.29,30TheOPLS parameters are also used. The TIP3P model is used forwater. The number of water molecules involved in the calcu-lation is 210 and the periodic boundary is applied. The cutofflength of the potential is 9.0 Å. The bond length and theangle of a propane molecule are fixed using the SHAKEmethod.31 The simulation is made for 100 ps with a time stepof 2 fs after an equilibration period of 4 ps. Coordinates ofthe water molecules during the simulation are recorded every

FIG. 1. The distribution of waterguv around a Lennard–Jones sphericalCH4 solute. The grid-spacing of 0.5 Å is compared to the spherical solutionwith the radial grid-spacing of 0.038 Å . The HNC closure is used.

TABLE I. The maximum deviations of our distributions of waterguv arounda Lennard–Jones spherical solute to the spherical solution. Calculations withgrid-spacings of 0.5 Å, 1.0 Å, and 1.5 Å are listed. The radial grid-spacingfor the spherical solution is 0.038 Å . The two closures, HNC and PY, andthe R-PY method are compared.

Grid-spacing HNC PY R-PY

0.5 Å 0.028 0.019 0.0110.5 Åa 0.035 0.015 0.0071.0 Å 0.349 0.488 0.2491.5 Å 0.411 0.388 0.257

aThe maximum deviation when the solute is 0.25 Å shifted with respect tothe 0.5 Å grid coordinate.

FIG. 2. Internal and external iso-value surfacesguv51.5 of the distributionof water around a propane molecule. The upper part of the surface is re-moved for a more comprehensive observation. The space–fill model in thecentral part is the propane molecule.

FIG. 3. The contour map of the distribution on the cutting-plane in Fig. 2.Dashed line,guv50.8; solid line,guv51.0; hatched region,guv51.5–2.1;double hatched region,guv52.1;; dotted region in the central part, a pro-pane molecule shown in the space-fill model.

5014 M. Ikeguchi and J. Doi: Distribution of water around hydrophobic molecules

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20 fs. The distributions around the methyl group~CH3) andaround the methylene group~CH2) of the propane moleculeare calculated from the coordinates of the water moleculesduring the simulation. We compared them with the results ofour method using the closures, HNC and PY, and the R-PYmethod.

Figure 4 shows the distributions of water around the me-thyl group~CH3) of the propane molecule. The result of ourmolecular dynamics simulation is similar to that of Ichie andChandler,20 although the used parameters are somewhat dif-ferent. Our molecular dynamics simulation is compared withour method. The distribution of the R-PY method is the clos-est to the molecular dynamics simulation. The first peak at3–4 Å is close to the molecular dynamics simulation, al-though the peaks of the PY and HNC closures are different.

Figure 5 shows the distribution of water around the me-thylene group~CH2) of the propane molecule. The R-PYdistribution agrees fairly well with the molecular dynamicssimulation. The first peaks of the PY and the HNC areshifted to the distance shorter than that of the molecular dy-namics simulation. Figures 4 and 5 indicate that our method

combined with the R-PY method agrees with the moleculardynamics simulation.

2. The PC and three-site water models in the RISMequation method

The three-site water model is proposed to explicitly in-clude the hydrogens of water in the RISM method.18 Wecompared the PC water model in the RISM with the three–site water model. The water–water correlation function iscalculated from the TIP3P water potential27 using theXRISM-HNC method17 for the three-site water model. Forthe PC water model, the experimentally determined oxygen–oxygen correlation function for pure water is used. The R-PYmethod is used for both calculations. Figure 6 shows thesite–site distributions between the methylene group CH2 ofthe propane molecule and water. The result of the PC watermodel is similar to that of the three-site water model in theRISM method. The first peaks of both distributions at 3–4 Åare commonly observed. The first peak becomes higher if theHNC or the PY closure is used although they are not shownhere. The same trend is also observed in the distribution withthe three-site model reported by Ichie and Chandler20 usingthe RISM-HNC, although their potential parameters aresomewhat different. The distributions with the three-sitemodel reported by Lue and Blankschtein22 using the RISMwith the Martynov–Sarkisov closure and with the Ballone–Pastore–Galli–Gazzillo closure are similar to that of ourRISM-R-PY shown in Fig. 6. The first peaks observed in theRISM with various closures do not appear in the distributionof the molecular dynamics simulation. The disagreement be-tween the RISM and the molecular dynamics simulation isestimated due to the approximation which the RISM em-ployed. Our method, avoiding such an approximation of theRISM employed, agrees with the molecular dynamics simu-lation. Consequently, the PC water model is valid as judgedfrom these results.

FIG. 4. The site–site distributions between the methyl group~CH3) of pro-pane and water. The results of our method using the three closures arecompared with the molecular dynamics simulation.

FIG. 5. The site–site distributions between the methylene group~CH2) ofpropane and water. The results of our method using the three closures arecompared with the molecular dynamics simulation.

FIG. 6. The site–site distributions between the methylene group CH2 of apropane molecule and water. The PC water model is compared with thethree-site water model in the both RISM calculations. The distributions ofour method and of the molecular dynamics simulation are also shown. TheR-PY method is used for all calculations.

5015M. Ikeguchi and J. Doi: Distribution of water around hydrophobic molecules

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3. Spatial distribution of water around propane

Figure 2 shows that the iso-value surface of the distribu-tion of waterguv51.5 around propane. There are two sur-faces, internal and external. The upper half part of the distri-bution is removed to show three carbon atoms of the propanemolecule. Figure 3 shows the contour map of the distributionon the cutting-plane in Fig. 2. The large values of the distri-bution are observed offshore concavities of the propane mo-lecular surface. The large values are also observed in ourmolecular dynamics simulation. It is reported that the mo-lecular concavities contribute to the positive hydrophobicsolvation free energy.32,33 On the other hand, it is reportedthat the value greater than 1 of the distribution contributes tothe decrease in the solvation entropy.34,35Our result suggeststhat the large values of the distribution offshore molecularconcavities contribute to the large positive hydrophobic sol-vation free energy.

C. Spatial distribution of water around5a-cholest-2-ene

The computer graphics techniques, such as the MarchingCube method, visualize the spatial distribution at each 3Dgrid point in combination with the solid modeling represen-tation. Our method is applied to 5a-cholest-2-ene~C27H46)as an example of a hydrophobic molecule with a complicatedshape. The atom coordinates are from the Cambridge Struc-tural Database.36,37The OPLS potential parameters are used.The grid-spacing is 0.3 Å and the grid resolution is 12831283128. The distribution inside a cube with 38.4 Å edges iscalculated. Figure 7 shows the iso-value surface of the dis-

tribution guv51.5 around the 5a-cholest-2-ene molecule. Apart with a 4 Å high band between the top hat part and thecentral part is removed to observe the internal atoms. Thecentral part is cut by a vertical plane. The right half is dis-played using the semitransparent representation, and the lefthalf is translated to the left by 5 Å to observe both the dis-tribution and the molecule. A concavity of the iso-value sur-face locates offshore the molecular concavity as observed inthe propane distribution.

The calculation is done using a Workstation HP712~122SPECfp92!. It takes about 3 cpu h for the 12831283128grid resolution.

IV. CONCLUDING REMARKS

This paper proposes a numerical method to directlysolve the Ornstein–Zernike integral equation by introducingthe 3D lattice. The method employs no approximation theRISM used. The method enables one to obtain the quantita-tive spatial distribution of solvents around solutes of an ar-bitrary shape. Numerical accuracy due to the grid-spacingand the solute placement is tested. A spacing of less than 0.5Å is recommended. The spatial distribution of water is cal-culated around a propane molecule using the PC watermodel. The distribution agrees with the molecular dynamicssimulation. The distribution increases offshore molecularconcavities of the solute. The spatial distribution of wateraround 5a-cholest-2-ene is visualized and the similar trend isobserved.

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FIG. 7. The iso-value surface of the distributionguv51.5 around a5a-cholest-2-ene molecule. A part with a 4 Åhigh band between the top hatpart and the central part is removed. The right half is displayed using asemitransparent representation, and the left half is translated to the left sideby 5 Å to observe the space-fill model of the 5a-cholest-2-ene molecule.

5016 M. Ikeguchi and J. Doi: Distribution of water around hydrophobic molecules

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