Incorporating Hydration Force Determined by Boundary Element Method into StochasticDynamics

Cun Xin Wang,* ,† Shun Zhou Wan,‡ Zhe Xin Xiang,‡,§ and Yun Yu Shi‡

Center for Fundamental Physics and Department of Biology, UniVersity of Science and Technology of China,Hefei, Anhui 230026, P. R. China

ReceiVed: May 29, 1996; In Final Form: October 16, 1996X

This paper describes a new method for incorporating the extra mean force terms of solvent including theCoulombic interactions with the induced surface charge and surface pressure of solvent into stochastic dynamicssimulation. The boundary element method based on the classical continuum approximation has been used toevaluate those mean force terms. Calculations with our method have been performed on a cyclic undecapeptidecyclosporin A that has been investigated by stochastic dynamics and molecular dynamics in the previoussimulations. The detailed analysis has been accomplished in terms of internal hydrogen-bonding forms,conformational fluctuations, and atomic positional correlation function. Compared with the data obtainedfrom previous simulations, our results indicate that the new simulation method presented in this work providesan obvious improvement over the conventional stochastic dynamics simulation technique. This suggests thatthe average solvent effects should produce a significant influence on the simulated structure and dynamicsbehavior of biomolecules in aqueous solution.

Introduction

Molecular dynamics (MD) simulation has become a widelyused technique for analyzing the structure and function ofchemical and biological systems.1-4 The interactions of alldegrees of freedom in the system are explicitly taken intoaccount in MD simulation. For a system of protein in water,most of the computational time is used to calculate theinteractions between solvent atoms and solute atoms as well asamong the solvent atoms. In many cases one is not interestedin the details of the solvent itself but only in its effects on themacromolecule. Therefore, one can eliminate the explicittreatments of solvent molecules in the simulation to save largeamounts of computational time on simulating water molecules.The conventional stochastic dynamics (SD) simulation yieldsan approximation with missing solvent to the dynamics ofmacromolecule in solution. Shi and co-workers suggested thatfor nonpolar solvents conventional SD simulation without usingan extra mean force term seems to be a good approximation.However, for polar solvents the omission of the mean force ofsolvent effects is not justified.5

Recently, there has been a great deal of interest in theincorporation of solvent effect into molecular mechanics or MDsimulations using the Poisson-Boltzmann equation (PBE).6-9

Davis and McCammon have pointed out that there should existthe solvent boundary pressure acting on the surface solute atomsbesides the electrostatic force on a solute atom resulting fromthe interaction of the atomic charge with the local electricalfield.10 Thus, any method for incorporating the PBE into MDor SD simulations must account for this additional pressure forceterm. Zauhar has reported on the feasibility of incorporatingcontinuum electrostatic forces including surface pressure inmolecular mechanics calculations using boundary element

method (BEM).6 Since this method includes an explicitdielectric boundary surface, it is possible to calculate directlyboundary pressure using the Maxwell stress tensor. Sharp haspresented a way of combining solvent and ion screening withconventional SD using the finite difference method for solvingthe PBE.7 More recently, Gilson and co-workers have describedan efficient method for incorporating the effects of solvent intoMD simulations using rather coarse finite difference grids tosolve the PBE.9 Up to now, it is found that no one has reportedany approach to combine the solvent effects determined by theBEM into SD simulation of biomolecules.In this article, we have applied the BEM to evaluate the

solvent effects that is regarded as the mean force of solventand combined with SD simulation for cyclosporin A (CPA).We have selected the CPA as our simulated system becausethe conventional SD and full MD simulations for CPA havebeen studied in the previous simulations5,11 with which ourmethods and results can be examined. The real goal of thepresent work is to study the methodology of SD with a potentialof mean force determined by the BEM and to test the reliabilityof the method through comparing our analysis results with theprevious simulation data.

Theory

The normal form of SD for a system of N atoms is given bythe ordinary Langevin equation:

wherexi is the coordinate component of all atoms,γi is theatomic friction coefficient, andRi denotes the random force.The system forceFi is derived from the system potential ofmean forceVmean({xi}) and has a form

whereVmeancan be written as

* To whom correspondence should be addressed. E-mail: xin@bioser.bio.ustc.edu.cn.

† Center for Fundamental Physics.‡ Department of Biology.§ Present address: Department of Biochemistry and Molecular Biophysics

and Center for Biomolecular Simulation, Columbia University, 630 West168th St., New York, NY 10032.

X Abstract published inAdVance ACS Abstracts,December 1, 1996.

mixi ) Fi({xi}) - miγixi + Ri(t) (1)

Fi(t) ) -∂Vmean({xi})/∂xi (2)

Vmean({xi}) ) Vint({xi}) + Vsol({xi}) (3)

230 J. Phys. Chem. B1997,101,230-235

S1089-5647(96)01556-8 CCC: $14.00 © 1997 American Chemical Society

whereVint({xi}) presents the internal potential of solute atomin the system, andVsol is the potential of mean force from theenvironment solvent. Obviously, eq 2 can be rewritten as

where the internal forceFiint is due to the explicitly simulated

atoms and the external forceFisol is called hydration force

originating from the environment.In the present study, the hydration forceFi

sol is calculated byusing the classical continuum models of electrostatic interactionsand considered as the sum of the Coulombic interaction withthe induced surface charge and a purely mechanical pressureof the polarized solvent.6,8 The electrostatic component of thehydration force acting on chargeqi is calculated with theexpression

where the integration is carried out over the entire molecularsurfaceS and σ(r ) is the density of the induced polarizationcharge at an arbitrary pointr on the molecular surfaceS. Therelation between theσ(r ) and the exterior fieldEe(r ) is givenby12

whereDe andDi are the dielectric constants of the solvent andthe molecular interior, respectively. The exterior electric fieldEe(r ) can be evaluated on the solvent side of the molecularsurface, andn is the outward unit normal to the surface.The purely mechanical boundary pressure can be calculated

based on the expression developed by Gilson and co-workers8

as

whereP(r ) is a normal pressure exerted by solvent andEi isthe internal electric field. Expanding dot productEe‚Ei in eq7, it is derived as6

Thus, the force of a solvent pressure on theith atoms is givenby

where the integration is performed only over the portion of thesolvent accessible surface associated with theith atom. There-fore, both the charged atom and the uncharged atom on themolecular surface will be subject to a force exerted by theboundary pressure of the solvent.In this work, the BEM is used to calculate the electrostatic

potential inside and around the molecule. Since the detaileddiscussions of the BEM can be found in a number of otherworks,12-19 here we only give an outline of the method andmain equations used in this work.

The interior potentialφi and the exterior potentialφe aregoverned by the Poisson equation and the linearized PBE,respectively:15

whererp is a point inside or outside the molecule, andδ(rp -r k) in eq 10 is the delta function atr k at which thekth chargeqk is placed, andκ in eq 11 is the Debye inverse screeninglength. The potentialφi and φe should satisfy the followingboundary conditions:

The fundamental solutions of eqs 10 and 11 can be expressedas

whererpq is the distance between pointsp andq, andκ is theDebye inverse screening length.Gpq and µpq are the funda-mental solutions to the eqs 10 and 11, respectively. UsingGreen’s second theorem on eqs 10 and 11, we obtain thefollowing two boundary integral equations for the interior andexterior potentials:

where the integration is taken over the entire molecule surfaceS, k is a point on the surfaceS, andp is a point inside or outsidethe molecule. If the jump discontinuity of the double-layerpotential at the boundaryS is taken into account,14 eqs 16 and17 can be written as

The surface potential and their normal derivatives can beevaluated from eqs 18 and 19 plus boundary conditions eqs 12and 13 with the BEM. Then, the internal electric fieldEi andthe exterior fieldEe are obtained with following expressions:

Fi ) -∂Vint({xi})

∂xi-∂Vsol({xi})

∂xi(4)

) Fiint + Fi

sol

Fipol(R) ) I

S

qiσ(r )(R - r )

|R - r |3d2r (5)

σ(r ) ) 14π(Di - De

Di)Ee(r )‚n(r ) (6)

P(r ) ) 18π(De - Di)Ee‚Ei (7)

P(ri) ) 18π(De - Di)[(Ee‚n)

2(De - Di

Di) + |E0|2] (8)

Fipress)∫S-np(r ) d2r (9)

∇2φpi ) -1

Di∑k

qkδ(rp - r k) (10)

∇2φpe ) κ2φe(rp) (11)

φi ) φ

e (12)

Di∂φ

i

∂n) De

∂φe

∂n(13)

Gpq ) 14πrpq

(14)

µpq ) exp(-κrpq)/4πrpq (15)

φpi ) I

S[Gpk

∂φki

∂n- φk

i∂Gpk

∂n ] dAk +1

Di∑k

qkGpk (16)

φpe ) I

S[-µpk

∂φke

∂n+ φk

e∂µpk

∂n ] dAk (17)

1

2φpi ) I

S[Gpk

∂φki

∂n- φk

i∂Gpk

∂n ] dAk +1

Di∑k

qkGpk (18)

1

2φpe ) I

S[-µpk

∂φke

∂n+ φk

e∂µpk

∂n ] dAk (19)

Ei ) -∇φi (20)

Ee ) -∇φe (21)

Stochastic Dynamics Simulation J. Phys. Chem. B, Vol. 101, No. 2, 1997231

Therefore, the density of the induced polarization charge canbe calculated with eq 6, and the hydration force on charge andthe boundary pressure can be obtained from eqs 5 and 9,respectively.

Methods

In order to incorporate the total hydration force defined byeqs 5 and 9 into SD simulation, we first linked our BEMprograms (MACBEM)18 with the GROMOS package.20 It iswell-known that the most difficult part in BEM is to generatea proper triangulation. In the present work, we followed thetriangulation method developed by Juffer and co-workers.15 Thetriangulation procedure contained three steps: (1) A high-densitydot surface of the molecule was generated using the Connollyprogram.21 (2) After the triangulated grids on a sphere surfacewere defined from a regular polyhedron, the center of the spherewas translated and coincided with the center of mass of themolecule. (3) The triangulation vertices on the molecularsurface were assigned by selecting the surface point with theshortest distance to the vectors between the center of mass andthe vertices on the sphere, in which each vector of the verticesmust intersect the molecular surface only once. Figure 1 showsa triangulation of the solvent accessible surface for X-raystructure of CPA. The probe radius is 0.14 nm. The surface isdefined at the center of probe rolling around the molecule. Thenumber of total triangles is 760.If the coordinates and the charges of the system are given,

the potentials in the interior or exterior region can be determinedby integrating over all surface triangle elements with eq 18 oreq 19, respectively. The distribution of induced polarizationcharge on the molecular surface and the corresponding hydrationforce on charge were evaluated from eqs 6 and 5. The solventboundary pressure acting on the surface atoms was computed

with eqs 8 and 9, in which the pressure was distributed over allsurface atoms associated with the solvent accessible surfaceelements. The direction of the pressure is toward the inside ofthe molecule and perpendicular to the molecular surface.The subroutine programs for calculating the hydration force

on charge and the solvent pressure based on the BEM werewritten with FORTRAN 77 and jointed with the subroutineprogram FORCE in GROMOS backage. The relevant inputand output and the corresponding subroutine programs inGROMOS were modified to set up the SDBEM programpackage with which the following simulation was carried out.The initial structure of CPA molecule was used from the

X-ray structure.22 The schematic structure of the cyclic un-decapeptide CPA can be found in Figure 1b. The GROMOSforce field was used for all simulations. Nonpolar hydrogenatoms were included in the carbon atoms (united atom approach)while polar hydrogen atoms were treated explicitly. The systemcontained 90 atoms in total. In the beginning, 100 steps ofenergy minimization with the steepest descent method wereperformed. Then 10 ps conventional SD simulation withoutthe extra mean force terms was carried out for system equilib-rium. After that time, 50 ps SDBEM simulation was performed,and the configurations of the trajectories were saved every 25time steps for further analysis. The time step, 2 fs, was takenfor integrating the equations of motion. No cutoff radius wasapplied to the nonbonded interactions. All bond lengths werekept rigid using the SHAKE algorithm with a relative toleranceof 10-4.23 Coupling to a temperature bath and a pressure bathwas applied to keep the system at 300 K and 1 atm.24 Allsimulations and analysis were carried out on an IRIS 4D/50GTworkstation and a Micro VAX 3500 computer.

Results and Discussion

Convergence Behavior of the Simulation.To verify theconvergence behavior of the SDBEM simulation, we analyzedthe root mean square (rms) deviation of atomic positions versusthe simulation time relative to two reference structures, i.e., thestarting structure (t ) 0) after 10 ps SD equilibration (Figure2a) and the X-ray crystal structure (Figure 2b). As shown inFigure 2a, the RMS shift in position reaches a stable value of0.08-0.12 nm for all atoms and 0.04-0.08 nm for CR atomsaround 10-50 ps. The magnitudes of the RMS values relativeto the X-ray structure (Figure 2b) also have a stable values of0.08-0.17 nm for all atoms and 0.04-0.12 nm for CR atomsfrom 10 to 50 ps of the simulation. It is clear from Figure 2that our SDBEM simulation appears to reach an equilibriumstate after 10 ps SDBEM simulation. After this initial period,the trajectories obtained from final 40 ps simulation are usedfor the following conformation and dynamics analyses.Hydrogen-Bonding Analysis. Table 1 lists the results of

hydrogen-bond (H bond) analysis in CPA for SDBEM trajec-tories data of the last 40 ps. In order to compare our resultsobtained from the SDBEM simulation with previous simulationsusing different approaches, we have also reported the resultsobtained from MD11and conventional SD5 simulations in Tables1-4.The criteria used to determine an H bond are purely

geometric: for each coordinate set, every potential donor-acceptor is tested and considered to form an H bond if thehydrogen to acceptor distance is less than 0.25 nm and thedonor-acceptor angle is larger than 90°. The frequencies ofH bonds are determined from the occurrences registered on thesimulation trajectory frames.As seen in Table 1, the occupancies of H bonds among

2Abu-11Meval, 5Val-2Abu, and 5Val-3Sar obtained from

Figure 1. (a) A complete triangulation of the solvent accessible surfacefor X-ray structure of CPA. The number of total triangles is 760. (b)The X-ray structure of CPA and its solvent accessible surface that wereplotted with points. The probe radius is 0.14 nm.

232 J. Phys. Chem. B, Vol. 101, No. 2, 1997 Wang et al.

our SDBEM simulation are higher than that from SD simulationand close to the results obtained from MD simulation. More-over, the H bond between 1MeBmt side-chain hydroxyl groupand its own carbonyl is not found in our SDBEM simulation,which is exactly the same as that in MD simulation due to thecompetition of the water molecules for H-bond donor andacceptor for 1MeBmt carbonyl (see Table VII of ref 11). Forthe same reason, some other H bonds in our SDBEM simulationexcept three H-bonding patterns among 1MeBmt-10MeLeu,5Val-3Sar, and 5Val-5Val have lower frequencies comparedwith that from SD simulation. This means that our SDBEMsimulation decreases the formation of those H bonds since we

have added the hydration force on charge and solvent surfacepressure into the mean force of solvent in our simulation.Conformation and Structure Analysis. Table 2 shows the

averaged backbone dihedrals (φ, ψ, ω) and their fluctuations inparentheses for three various simulations. Since the influenceof solvent effects on the conformation of backbone is not somuch, most of backbone dihedrals for both SD and SDBEMsimulated structure are quite similar within their RMS fluctua-tions. However, the values of several backbone dihedral anglesobtained from the SDBEM simulation, such as dihedrals 2Abuφ, ψ, ω; 3Sarω; 4MeLeuφ; and 5Valω, are much closer tothat from MD simulation compared with the data in SDsimulation (see Table 2).The averaged side-chain dihedrals and their rms fluctuations

are reported in Table 3 for the MD, SD and SDBEM simula-tions, respectively. It is found that the obvious conformationalchange for three various simulations occurs in the 1MeBmt sidechain. For instance, the dihedralø2 in 1MeBmt is 72° for MDsimulation but-132° for SD simulation. This means that theside-chain dihedralø2 in 1MeBmt yields a big rotation betweenMD and SD simulations. However, a very consistent value ofdihedralø2 in 1MeBmt (68°) to that in the MD simulation isgiven in our SDBEM simulation. In addition, there also existsthe dihedral rotation for side-chain dihedralø3 in 1MeBmtbetween MD and SD simulations. Although the value ofø3 in1MeBmt in our SDBEM simulation has a near 20° differencefrom MD simulation, our result forø3 is much better than thatfrom SD simulation (see Table 3). For the side-chain dihedralsthat have big deviation between MD and SD simulation, suchas dihedrals 1MeBmtø1, ø2, ø3, ø4; 2Abu ø1; 4MeLeuø2; and10MeLeuø2, our SDBEM results have a distinct modificationcompared with SD results. On the other hand, for the remaining

Figure 2. Root mean square deviation of atomic positions versusSDBEM simulation time relative to two reference structures: (a) thestarting structure after 10 ps SD equilibration; (b) the X-ray crystalstructure.

TABLE 1: Frequencies of Intramolecular Hydrogen BondsObtained from Different Simulation Data in CPA (in %)

donor acceptor MD in watera SDb SDBEM

1MeBmt O-H 1MeBmt O 0 11 010MeLeu O 0 8 18

2Abu N-H 5Val O 82 68 5711MeVal O 22 40 26

5Val N-H 2Abu O 85 96 873Sar O 15 1 95Val O 18c 3c 7c

7Ala N-H 5Val O 13 20 411MeVal O 92 67 59

8Ala N-H 6MeLeu O 44 78 73

a Values from ref 11.b Values from ref 5.c This hydrogen bond hasthe unrealistic value for the donor-hydrogen-acceptor angleθ < 100°;all otherθ values are larger than 127°.

TABLE 2: Averaged Backbone Dihedral Angles (in deg)and Their Rms Fluctuations (in Parentheses) in CPA

angle MDa SDb SDBEM

1MeBmt φ -99 (8) -107 (9) -110 (10)ψ 96 (14) 88 (14) 92 (13)ω 180 (8) -175 (7) 176 (8)

2Abu φ -106 (17) -91(14) -109 (14)ψ 106 (11) 102 (10) 106 (13)ω -176 (10) -168 (10) 179 (11)

3Sar φ 56 (12) 57 (10) 57 (10)ψ -121(12) -114 (10) -115 (11)ω 173 (8) 166 (7) 172 (8)

4MeLeu φ -107 (10) -114 (10) -106 (11)ψ 36 (18) 32 (20) 31(26)ω -177 (6) 178 (7) -178 (8)

5Val φ -118 (19) -94 (20) -108 (22)ψ 121 (12) 113 (9) 116 (9)ω 167 (9) -176 (7) 172 (7)

6MeLeu φ -94 (11) -102 (9) -107 (7)ψ 98 (14) 90 (10) 90 (10)ω -171 (7) -175 (6) -175 (7)

7Ala φ -97 (20) -85 (12) -87 (12)ψ 87 (16) 73 (13) 71 (15)ω 175 (7) 177 (6) 178 (6)

8Ala φ 68 (17) 75 (14) 75 (15)ψ -122 (11) -126 (9) -124 (9)ω -178 (8) -177 (7) -176 (7)

9MeLeu φ -128 (9) -129 (9) -128 (8)ψ 119 (12) 116 (13) 115 (9)ω -18 (13) -16 (14) -17 (10)

10MeLeu φ -119 (9) -120 (10) -120 (8)ψ 98 (8) 98 (9) 99 (9)ω -159 (7) -158 (8) -159 (9)

11MeVal φ -129 (8) -121 (8) -118 (8)ψ 114 (9) 93 (10) 92 (8)ω 176 (7) -159 (9) -154 (10)

a Values from ref 11.b Values from ref 5.

Stochastic Dynamics Simulation J. Phys. Chem. B, Vol. 101, No. 2, 1997233

side-chain dihedrals in which there exist small deviationsbetween MD and SD simulations, our SDBEM simulation dataare consistent with the SD results. As mentioned above, wesuggest that the side-chain conformation is obviously affectedby the hydration force on charge and solvent pressure that aretaken into account in our SDBEM simulation.Table 4 represents the rms atomic positional fluctuations and

the mean anisotropy for atomic motion for the MD, SD andSDBEM simulations. As seen in Table 4, the rms atomicpositional fluctuation of our SDBEM simulation is 0.059 nmfor all atoms, which is smaller than that of SD simulation (0.072nm) and comparable to that obtained in MD simulation (0.052nm). For the averages over CR atom and other side-chain carbonatoms, the rms fluctuations of atomic positions for our SDBEMsimulation are agreement with the values of the MD simulation.In addition, the mean anisotropy in the atomic motion isprovided by evaluating the ratio of the smallest to the longestprincipal axis of the anisotropic fluctuation ellipsoids. Thevalues for anisotropy are comparable for all three simulations(see Table 4).Dynamics Behavior. The solvent environment of a biomol-

ecule not only influences its equilibrium conformation but alsoaffects its dynamics behavior. Therefore, the time evolutionof the atomic positional fluctuation has been examined usingautocorrelation function that is defined as

where the brackets⟨ ⟩ indicate an average over the simulationtime,∆r(t) ) r(t) - ⟨r⟩, and⟨∆r2⟩ ) ⟨(r - ⟨r⟩)2⟩.In the present study, we have selected two backbone atoms

and two side-chain atoms of CPA for analyzing their positional

TABLE 3: Averaged Side-Chain Dihedral Angles (in deg)and Their Rms Fluctuations (in Parentheses) in CPA

angle MDa SDb SDBEM

1MeBmt ø1 -167 (8) -148 (38) -175 (8)ø2 72 (10) -132 (42) 68 (12)ø3 173 (12) -77 (27) -169 (12)ø4 -153 (27) -139 (57) -165 (45)ø5 179 (8) -179 (8) 178 (8)

2Abu ø1 -114 (46) -131 (57) -114 (50)4MeLeu ø1 -80 (15) -76 (11) -73 (12)

ø2 -103 (43) -74 (12) -90 (33)5Val ø1 -61 (12) -62 (12) -61 (14)6MeLeu ø1 -170 (9) -166 (10) -166 (10)

ø2 -141 (31) -105 (20) -107 (27)9MeLeu ø1 -70 (13) -75 (14) -75 (14)

ø2 -77 (18) -88 (26) -88 (28)10MeLeu ø1 -169 (10) -164 (9) -166 (10)

ø2 -164 (13) -130 (40) -145 (33)11MeVal ø1 -61 (15) -59 (10) -59 (10)a Values from ref 11.b Values from ref 5.

TABLE 4: Rms Atomic Positional Fluctuationsa and theMean Anisotropy for Atomic Motion in CPA

atom type MDb SDc SDBEM

all atoms 0.052 0.072 0.059CR atoms 0.036 0.033 0.036MeLeu Câ 0.052 0.046 0.053MeLeu Cγ 0.065 0.059 0.072MeLeu Cδ 0.096 0.089 0.099anisotropyd

all atoms 0.44 0.42 0.41CR atoms 0.53 0.49 0.42

a Fluctuations are in nm.b Values from ref 11.c Values from ref 5.dRatio of the shortest to the longest axis of the anisotropic fluctuationellipsoids.

Cr(t) )⟨∆r(τ)∆r(t + τ)⟩τ

⟨∆r2⟩)

⟨∆r(0)∆r(t)⟩⟨∆r2⟩

(22)

Figure 3. Atomic positional fluctuation autocorrelation functions forfour atoms of CPA: (a) 5-Val-CR; (b) 9-MeLeu-CR; (c) 4-MeLeu-Cδ1;(d) 9-MeLeu-Cδ1. Here the dotted lines show the SDBEM simulation,the dashed lines show the SD simulation (see ref 5), and the solid linesrepresent the MD simulation (see ref 11).

234 J. Phys. Chem. B, Vol. 101, No. 2, 1997 Wang et al.

correlation function and compared with that obtained from theprevious SD and MD simulations. Figure 3a-d shows thecorrelation functions for those four atoms of CPA, in whichbackbone atom 5-Val-CR and the side-chain atom 4-MeLeu-Cδ1 are in theâ-pleated sheet, and the backbone atom 9-MeLeu-CR and the side-chain atom 9-MeLeu-Cδ1 are in the loop region.The correlation functions for 5-Val-CR atom (Figure 3a) inSDBEM and SD simulations have a similar behavior. Theyshow a rapid initial decay followed by slow oscillations with alow amplitude. For this case, both SDBEM and SD simulationsare comparable with MD simulation. For 9-MeLeu-CR atomin the loop region (Figure 3b), all the correlation functions showa slower decay than that in Figure 3a. This indicates that the5-Val-CR atom is much more fixed in its position in theâ-sheetthan the 9-MeLeu-CR atom in the loop region. For the side-chain atom 4-MeLeu-Cδ1 in the â-sheet (Figure 3c), thecorrelation functions show the much slower decay followed bya linear decrease, in which our SDBEM simulation is muchcloser to the MD simulation compared with SD simulation.However, the correlation functions for side-chain atom 9-MeLeu-Cδ1 in loop region (Figure 3d) display different features,in which the results for both SDBEM and SD simulations arevery similar and a shorter relaxation time can be found comparedwith Figure 3c. This means that the sidechain of 4-MeLeu hasmuch more flexibility than the sidechain of 9-MeLeu. Throughthe above analysis, we suggest that in general both SDBEMand SD simulation generate a similar correlation function. Butfor the side chain in theâ-sheet of CPA, the SDBEM simulateddata are better than the SD data compared with MD simulation.

Conclusions

In this work, a methodology of combining BEM with SDsimulation technique offers a possible tool to study the structureand dynamic properties of biomolecules. The two extra meanforce terms of hydration force on charge and surface pressureof solvent determined by the BEM have been added into SDsimulation to represent the average solvent effect on the soluteatoms. To examine the reliability of the new approach, we haveselected the CPA molecule as our study target to carry out 50ps SDBEM simulation. The analysis results for H-bondingnetwork, molecular conformation, and dynamic behavior havebeen compared with the previous SD and MD simulation. Ourresults show that the SDBEM simulation approach has anobvious improvement relative to the conventional SD simulationtechnique without use of the extra mean force of solvent.The further study of this method will focus on the increasing

calculation speed and precision of algorithms for solving thePoisson-Boltzmann equation using the BEM. For the CPAmolecule, 0.3 h CPU time for each picosecond was taken for

the conventional SD simulation, but 0.7 h CPU time was takenfor each picosecond for our SDBEM simulation on an IRIS 4D/50GT workstation computer. Although this method requiresmore computation time than the conventional SD simulationtechnique, we believe that with further development of algo-rithms, it should be possible to use this method for studyinglarger biomolecular systems without use of the cutoff radius inelectrostatic interaction calculation.

Acknowledgment. We are grateful to Professor W. F. vanGunsteren for kindly providing us the GROMOS package. Thiswork was supported in part by Grant No. 863-103-13-03-01for Protein Engineering of the Chinese National High Technol-ogy Project, the National Foundation Commission of Scienceand Technology of China (No. 8512-II-12), and the ChineseNatural Science Foundation.

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