Calculation of the Maxwell stress tensor and the Poisson-Boltzmann ...

THE JOURNAL OF CHEMICAL PHYSICS 123, 084904 2005Calculation of the Maxwell stress tensor and the Poisson-Boltzmann forceon a solvated molecular surface using hypersingular boundaryintegralsBenzhuo Lu aDepartment of Chemistry and Biochemistry, Center for Theoretical Biological Physics,University of California at San Diego, La Jolla, California 92093-0365Xiaolin ChengCenter for Theoretical Biological Physics, Howard Hughes Medical Institute,University of California at San Diego, La Jolla, California 92093-0365Tingjun HouDepartment of Chemistry and Biochemistry, Center for Theoretical Biological Physics,University of California at San Diego, La Jolla, California 92093-0365J. Andrew McCammonDepartment of Chemistry and Biochemistry, Center for Theoretical Biological Physics,Department of Pharmacology, Howard Hughes Medical Institute, University of California at San Diego,La Jolla, California 92093-0365Received 13 May 2005; accepted 30 June 2005; published online 1 September 2005The electrostatic interaction among molecules solvated in ionic solution is governed by thePoisson-Boltzmann equation PBE. Here the hypersingular integral technique is used in a boundaryelement method BEM for the three-dimensional 3D linear PBE to calculate the Maxwell stresstensor on the solvated molecular surface, and then the PB forces and torques can be obtained fromthe stress tensor. Compared with the variational method also in a BEM frame that we proposedrecently, this method provides an even more efficient way to calculate the full intermolecularelectrostatic interaction force, especially for macromolecular systems. Thus, it may be more suitablefor the application of Brownian dynamics methods to study the dynamics of protein/protein dockingas well as the assembly of large 3D architectures involving many diffusing subunits. The method hasbeen tested on two simple cases to demonstrate its reliability and efficiency, and also compared withour previous variational method used in BEM. © 2005 American Institute of Physics.DOI: 10.1063/1.2008252I. INTRODUCTIONa Electronic mail: blu@mccammon.ucsd.eduThe electrostatic interactions among biomolecules inionic solution can be described by the Poisson-Boltzmannequation PBE. Different methods have been developed tosolve the PBE, such as finite difference FD methods, 1–3finite element FE methods, 4,5 and boundary element methodsBEMs. 6,7 However, most of these works and their applicationsin biomolecular studies are focused on the electrostaticpotential and interaction energy calculations.Relatively few studies have considered electrostatic forcecalculations. Dynamical study of biomolecular systems is ofgreat interest in chemical physics and molecular biology.Therefore, the force calculation is important for many theoreticalstudies and molecular-dynamics MD simulations.PB forces have recently been applied in several MDsimulations. 8–10 Gilson et al. 11 derived an expression for PBforce calculation that is correct for the FD approach. Theformula is compatible with the Maxwell stress tensormethod, and a smooth dielectric boundary is required in thatmethod. But for a system with two or more interacting macromolecules,as in protein assembly, the FD method is notpractical for the full PB force calculation “on the fly” due tothe tremendous CPU time required. The Brownian dynamicsBD program in the UHBD Ref. 12 actually uses a partialPB force between protein and ligand, because only the reactionfield of the fixed protein is calculated. For rigid bodyproblems, the BEM can use the molecular surface informationrepeatedly, so that it has fast performance for the full PBforce updating required in a protein-protein encounter dynamicssimulation. The force calculation in a BEM using a“polarized charge” method was first described by Zauhar; 13the force included both qE forces and boundary pressures.This idea was used in a later BEM work. 8,14 In these works,the BEM used a single-layer representation. However, theaccuracy of the boundary pressure computed by this techniquehas not yet been demonstrated. More generally, the useof only a single- or double-layer representation of the BEMto calculate the forces and torques has still not been adequatelytested. In our recent work, 15 to our knowledge, wegave the first rigorous procedure for using a variational approachembedded in the BEM frame to calculate the full PBforce and torque on a molecule in an interacting system withan arbitrary number of molecules in ionic solution. That is0021-9606/2005/1238/084904/8/$22.50123, 084904-1© 2005 American Institute of PhysicsDownloaded 02 Sep 2005 to 132.239.16.167. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

084904-3 Stress tensor on a solvated molecular surface J. Chem. Phys. 123, 084904 2005limi y,xvx − W i y,x v =0, 12→0 +S−e +s VndSxwhereW i = Gy i= 14r 2r ,i,13FIG. 1. Exclusion of the singular point y by a vanishing neighborhood o . e denotes the dotted part of the boundary.osmotic pressure of ion to the stresses at the level of linearPBE. This term is considered to be part of the hydrodynamicstress rather than the electric stress. For a system with noconvective flow as is the case in this work, the abovementionedformula gives the total stress present. Therefore,to obtain the boundary stress tensor, the derivative of thepotential, that is, the negative of E on the boundary, shouldbe known. In the integral form of the potential solution, Eqs.3 and 4, the derivative of the potential with respect toposition p, noted as ,p , is obtained directly by differentiatingon both sides of the equations, and then this value on thesurface can be obtained by taking the limit as p approachesthe boundary S. In order to use some results from the literaturein which Laplace’s equation is usually studied, here wefirst treat the interior potential solution of the PBE by subtractingthe contribution of the source point charges of themolecule, that is p = int p − 1D int q k G pk , p,k . 10kThen, the modified potential p satisfies Laplace’s equation,which is equal to the case without point charges. We will firstcalculate the derivative of p , noted as ,p , then return to thederivative of the original potential, i.e., ,p , by recoveringthe contribution from the source point charges, which is easilycalculated using Coulomb’s law.Now let us first consider the standard boundary integralequation for the harmonic function on the 3D domain ,bounded by the regular surface S see Fig. 1, 16,18lim→0 +S−e +s Gy,x x − Gy,x xdSx =0.nn11The fundamental solution G has a weak singularity of orderr −1 , when r=x−y→0, while its normal derivative has astrong singularity of order r −2 . Since in Eq. 11 both thepoints y and x lie on the surface S, a limiting process isnecessary. Actually, since Eq. 11 stems from Green’s secondidentity, it may be only formulated on a domain notincluding the singular point y. The situation is exemplified inFig. 1, where a vanishing neighborhood o of y has beenremoved from the original domain . The integration is thusperformed on the boundary S =S−e +s of the new domain =−o Fig. 1.Differentiating Eq. 11 with respect to any coordinatey i , we obtainV i =G n k x =− 1x k y i 4r 33r r,in − n i, 14where r ,i =r/x i =−r/y i , and n i is the ith component of thenormal vector n. As expected, the kernel W i shows a strongsingularity of order r −2 , while the kernel V i is hypersingularof order r −3 ,asr→0.Through the limiting process analysis as in Refs. 16 and19 the hypersingular boundary integral equation for ourmodified electrostatic potential v can be written in the followingform:c ik yv ,k y + limi y,xvx→0 +S−e V − W i y,x vxdSx + vy b iyn =0, 15where c ik and b i are bounded coefficients that only dependupon the local geometry of S at y. Ify is a smooth boundarypoint and s had a spherical shape, the coefficient c ik simplyreduces to 0.5 ik . This is then similar to Eq. 4.4.5 inRef. 19. This is the value we take in our case, because weconsider that our molecular boundary has been smoothedbefore discretization.The derivative of the potential appears in Eq. 15, butthere still is a limiting process to be treated. We rearrange theequation in the hypersingular integral form on a discretizedboundary for v in our study as12 v ,iy =−v o ,i − lim V i y,x vx→0 +s e −e n dSx+ vy b iy + lim W i y,xvxdSx,→0 +s e −e 16where s e denotes the boundary elements involving the singularpoint y, and v o ,i denotes the value from integration inEq. 15 on all other elements S−s e , which can be obtainedby regular numerical integration. The term in brackets in Eq.16, defined in terms of a limit, is called the “Hadamard” 20or “finite part” of the unregularized integral, which is thehypersingular integral part. Guiggiani et al. 16 presented aprocedure using Laurent series expansion to treat the generalform of hypersingular integral and finally transformed it intoa sum of a double integral and a one-dimensional regularintegral. Then the standard quadrature formulas can be used.The procedure was improved a little bit later and expressedin a more complete form, 17 which does not affect the formulasfor our case. Similar results can be obtained by usingother approaches see the appendixes of Ref. 17. The strongDownloaded 02 Sep 2005 to 132.239.16.167. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

084904-4 Lu et al. J. Chem. Phys. 123, 084904 2005r i /4r 2 N v J. J is the Jacobian from the coordinatetransformation, and here simply doubles the element area,From the coordinate transformation from r to ,, wefindx i − y i = A i ,20FIG. 2. A flat triangle in the three-dimensional space is mapped to a rightisosceles triangle on the plane. The singular point y, its surroundingcircular patch e , and their images in the transformed triangle are alsoshown.singular integral part, i.e., the last term on the right-hand sideof Eq. 16, can be treated following a similar procedure.For our specific case, because we use a flat triangularelement and linear shape functions, the derivation of hypersingularintegral formulation can be more simple andstraightforward. Here we give the detailed derivation. Let usfirst consider the case that the singular point y locates in theelement, and suppose in the limiting process the patch e iscircular and of radius centered at point y see Fig. 2. Inthis case, we can then show as following that part of theintegral the circular patch-related part is solved analyticallyand the remainder numerically, which was also discussed byAllison. 21 In the implementation of the boundary elementintegral, the surface triangles in the physical threedimensionalspace are mapped to an isosceles triangle on the parametric plane, as shown in Fig. 2, where the integrationon each element is actually performed.The values of the coordinates, the potential, and its normalderivative at any position in the element are obtained bylinear interpolation from the corresponding values on threenodes, respectively. For example, for a point , in parametricspace, its coordinates in the original space is x=N 1 ,x 1 +N 2 ,x 2 +N 3 ,x 3 , where N i i=1,2,3 arethe shape functions, and x 1 ,x 2 ,x 3 are the original coordinatesof the three nodes of the element. Here, N 1 =1−−, N 2=, and N 3 =.Now, we use polar coordinate , in the parametricspace , to perform the integral. Denoting Eq. 16 as12 v ,i =−v o ,i − I + I W ,17we find that the hypersingular part I and the strong singularpart I W areI = lim→0 +s e −e V i y,xN v n Jdd + vy b iy, 18Jdd I W = lim W i y,xN v ,→0 +s e −e 19where v and v n ,=1,2,3, denote the value v and vx/nat node , respectively, and the doubly appeared superscriptsor subscripts implicitly mean a summation, i.e., N v n= N v n . Note that r/n=0 on the flat surface, the hypersingularintegrand in Eq. 18 is thus reduced ton i /4r 3 N v n J, and the strong singular integral is given bywherer = A,A i = x 2 i − x 1 i cos + x 3 i − x 1 i sin ,21223A = A i 21/2, 23i=1andN = N o + N 1 , = 1,2,3. 24Here, A, A i , N 0 , and N 1 have the same definitions as in Ref.16. N 0 is a constant that depends on the position of the singularpoint, namely, the position where we want to calculatethe derivative of the potential. The contour of the neighborhoode given by the expression =r now in the parameterplane is expressed in polar coordinates by = A i .25By using the reversion of the above expression, we thenobtain the expression in of the equation in polar coordinatesof the contour e the image of e , see Fig. 2 = /A i .26Now, using the above formulas expansion Eqs.20–26 and performing the hypersingular and strong singularintegrals, we obtainI = lim→0 +02 A−3Jn i N 1lnˆ −ln +lnA4− A−3 Jn i N 04 1ˆ + 1Ad + vy b iy , 27I W = lim→0 +02 A−3A i JN 1ˆ4+ A−3 A i JN 0lnˆ −ln4+lnAd .28Note that A=A+, A i =−A i +, and N 1 =−N 1 +, the integral of the terms involving ln in bothEqs. 27 and 28 is equal to zero. Because the final value Iin Eq. 27 is finite, in the limiting process, the integral resultof the term involving 1/ must be canceled by the termvyb i y/. Then, the singular integrals are transformedwithout any approximation into regular integrals one dimensionin our case, they areI =012F −1 lnˆA − F −2 ˆd,29Downloaded 02 Sep 2005 to 132.239.16.167. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

084904-5 Stress tensor on a solvated molecular surface J. Chem. Phys. 123, 084904 2005I W =02F W 0 ˆ + F W −1 lnˆAd,30withF −1 = A−3 Jn i N 1, 314F −2 = A−3 Jn i N 0, 324F W 0 = A−3 A i JN 1, 334F W −1 = A−3 A i JN 0. 344The above two formulas hold for linear triangularboundary elements employed when the singular point is locatedin the element. A general form for hypersingular integralfor any kind of boundary element employed can befound in Refs. 16 and 17. If the singular point is chosen to benodal points on the bounding surface, a similar circular patchcan be selected, but subdivided into parts belonging to differentneighboring elements. Then a similar procedure presentedabove can be used, and the singular integral can beanalytically resolved. The results become 2I = mF m −1 lnˆ mA m mm 1− F m −2 1ˆ md ,mI W 2= F Wm 0 ˆ m + F Wm −1 mm 1lnˆ mA m d ,3536where the index m refers to the mth element around the collocationpoint, F m −1 , F m −2 , F Wm −1 , F Wm 0 , and A m are the similar coefficient as in Eqs. 29 and 30 but correspondto the mth element, and m 1 m 2 on the element. Itshould be noted here that, when the adjacent elements are ona plane, the sum of all the sectional angle ranges is equal to2. This was restated in the work of Huber et al. 22 However,in a real discretized surface such as the triangulated molecularsurface in our case, the adjacent elements sharing a commonnode may deviate somewhat from a plane, and thus thesum of the angles might not be 2. Nevertheless, in ourpresent stress calculations on the node, we suppose that themolecular surface is smoothed enough and the adjacent elementsaround a node are nearly on a plane; Eqs. 35 and36 are still used for the boundary integral on each relatedelement. Because the integral on each element is also actuallytransformed onto an independent parametric triangle asis routinely done, the angle interval in the integral is /2 iftaking the singular node as the first node of each element inthe transformation.In this work, we implement two options to select thecollocation point, a singular point where the v ,i is to be calculated.One is to select the points in the element not onthe edge, which isare just the Gauss quadrature points forthe integration in Eqs. 29 and 30. We name this hypersingularintegral method type I, denoted as HS1. The otheroption is to put the collocation point at each corner node,and Eqs. 35 and 36 will be used, for which case the derivativeof the potential at any point in the element will beobtained by interpolation. This second method is denoted asHS2.For the first case where the singular point is selected asinside the triangular element, we just show the integral onthe parametric triangle where the boundary element integralis actually performed. Figure 3 shows the singular point in aparametric triangle.From the figure, the needed function ˆ can be calculated,1−a − bsin/4ˆ 1 = , − 1 − 2 ,sin/4 + ˆ 2 =ˆ 3 =acos − , − 2 3 2 − 3, 37bcos − 3/2 , 32 − 3 2 − 1 ,where 1 =arctan b/1−a, 2 =arctan1−b/a, 3=arctan a/b, and a,b are the parametric coordinates of thesingular point in the parametric triangle; e.g., if the singularpoint is in the center of the triangle, then a=1/3 and b=1/3.The values in N in Eqs. 31–34 are=1−a − b, =1N 0a, =2b, =3,andFIG. 3. A singular point in a parametric triangle.=− cos − sin , =1N 1 cos , =238sin , =3.Downloaded 02 Sep 2005 to 132.239.16.167. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

084904-6 Lu et al. J. Chem. Phys. 123, 084904 2005In method HS2 for the singular point at the node of atriangular element, e.g., the first node 0,0, the functions inˆ reduce to only one type,1ˆ =cos + sin , 0 2 , 39and the functions in N are=1, =1N 00, =20, =3,and=− cos − sin , =1N 1 cos , =240sin , =3.Now, all the essential elements are ready for the calculationof the derivatives of the potential on the molecularsurface. The PB force calculation takes the following steps:First, we use the BEM to get the PBE solution potentialsand their normal derivatives on the boundary elements foran arbitrary number of solvated molecules, as described inour former work. 15 Second, the singular integrals as describedabove are used to calculate the derivatives of themodified potential v ,i , then the derivatives of the potential ,iare obtained by adding the contribution of the source pointcharges to v ,i , and the stress tensor T at any point on thesurface can be computed from Eq. 9. Finally, the PB forceF and torque M acting on each molecule are calculated byintegrationsF =STx · dSx, 41M r c x Tx · dSx,42=Swhere r c x is a vector from the center of mass of the targetmolecule to the surface point x, and the dot and cross vectormultiplication are applied to the vector and tensor quantities.IV. TESTS FOR SOME SIMPLE CASESSince we have previously compared the potentials calculatedfrom a variational approach with the analytical resultson some cases, 15 here we just calculate and compare theelectrostatic interaction forces between solvated moleculescomputed by all these methods. For a test model, we choosetwo point charges in vacuum, in which each charge is surroundedby a unit sphere discretized by 320 flat triangularelements 162 nodes. Both charges are put on the x axis,therefore, the nonzero force component is along the x direction,i.e., F x , and the other two components F y and F z arezero in theory. Figure 4 shows the forces F x along the xdirection as a function of the distance between the two pointcharges calculated using the two hypersingular integralmethods HS1 and HS2, the variational approach, and theanalytical formula.FIG. 4. The force calculation comparison using different approaches: analyticalresults solid line, hypersingular integral type I HS1 with singularpoint in the center of each element dotted line, hypersingular integral typeII HS2 with singular point on the nodes dot-dashed line, and variationalapproach dashed line.It is found that the forces F x calculated by both HS1 andthe variational approach are very close to the analytical onesin all the distance range shown in Fig. 4. The results from thehypersingular integral method HS1 are even more accuratethan those from the variational approach; the HS1 curve isbetween those obtained by analytical solution and the variationalapproach over all the range. For HS2, the results atclose distance are even more accurate than those frommethod HS1, but HS2 is less accurate at a large distance. It isalso found that the more separated the two charges become,the smaller the relative errors of HS1 and the variationalmethods are. For example, the relative errors given by thevariational approach and the HS1 method at a distance of 4Å are 5.6% and 3.1%, respectively, and 2.8% and 0.3% at adistance of 10 Å, respectively. HS2 gives a more accurateresult with a relative error of 2.7% at distance of 4 Å, but abigger relative error 5.6% at a distance of 10 Å. One reasonmay be the approximation of treating the adjacent elementsas on a plane when using Eqs. 35 and 36, as mentionedabove. Another possible reason will be discussed later. Inaddition, note that as mentioned in our previous work theforce calculations at moderate to large distances are evenmore accurate than the potential itself on the boundary. Thisis quite promising for BD trajectory calculations.It has to be noted that in our calculations the other twoforce components F y and F z that should be zero are verysmall less than 10 −5 in the variational approach, while inhypersingular integral method HS1, these numbers are not assmall, although they are still small relative to the nonzerocomponent F x e.g., F y =0.14 kcal/mol Å and F z=0.28 kcal/mol Å at 4-Å distance. At larger distances, thesetwo force components do not decrease much, thus the relativeerrors of these force components will be increased at along distance. This is mostly due to the numerical error inherentlyexisting in the algorithms. Comparing the two approachesrefer to previous work 15 , we find that the interactionbetween molecules calculated using the variationalapproach depends on , which is the difference of the potentialon the molecular surface in the interacting molecularDownloaded 02 Sep 2005 to 132.239.16.167. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

084904-7 Stress tensor on a solvated molecular surface J. Chem. Phys. 123, 084904 2005system relative to that in the isolated case. Therefore ,and consequently the force calculation, directly depend onthe distance between the molecules. This keeps the relativeaccuracy more constant over almost all the distance rangeswhere the BEM is valid. But for the hypersingular integralmethod, as described in Sec. III the calculation of the derivativeof the potential v ,i depends on the total potential including due to the interaction, as well as the potential fromthe isolated molecule isol that is fixed independent of therelative positions of the two molecules. Therefore, at a largedistance, even if is small, isol stays the same; the numericalerror due to this part is maintained, and this makesthe relative error increase. However, as illustrated by theabove example, the relative error does not increase for themain force component, e.g., F x . So, it seems that the relativeerrors of the variational approach are more stable and independentof the relative position of the molecules, while formolecules that are closer together, where the interaction isexpected to be strong, the hypersingular methods give moreaccurate results.An interesting thing is to check how much the singularintegral parts including strong and hypersingular contributeto the total calculated electrostatic field. In the HS1 method,the singular integral contribution is just from the integrationon one element where the singular point is located, and theother part is from the integral on all of the other elements.However, it is found that the singular integral contribution islarge enough to offset the contributions from all the otherelementary integrals. For example, at 4-Å distance in theabove case, at the center of the first element, the singularintegral part HS1 gives the contributions to the electrostaticfield −47.86, −34.84, and −347.17 kcal/mol Å e on x, y, andz directions, respectively, and the regular integral part gives59.59, 36.55, and 364.69 kcal/mol Å e on x, y, and z directions,respectively. Therefore, the singular integral part cannotbe omitted in the stress tensor calculation in the BEM.The main advantage of the hypersingular integralmethod is its calculation efficiency, which is also the mainmotivation to develop this method, as mentioned above. Inthe above test case, in which 30 calculation points were selectedas shown in Fig. 4, the CPU time on an Intel PentiumIV2 GHz for the whole BEM calculation is 28.3 s for thevariational approach, 23.0 s for HS1, and 19.3 s for HS2.Moreover, as shown in the methods sections, the CPU timespent on the force calculation for the hypersingular integralis only dependent on the boundary elements on the target2molecule, i.e., N elem , while that in the variational approachis dependent on all the boundary elements of the molecularsystems. Therefore, for a large biomolecular system, the hypersingularintegral methods should display much more efficiencythan in the above simple system.The second testing model is a point charge interactingwith a dipole. The point charge surrounded by a unitspherical boundary is located at −2,0,0; the dipole is alsosurrounded by a unit spherical boundary with its two pointcharges located at 2, 0, 0.1 and 2,0,−0.1, respectively. Inthis model, there are both force and torque acting on thedipole, but only the z direction force component F z andthe y direction torque component M y are in principle notzero. The analytical solutions are F z analy =−1.04 kcal/mol Å,and M y analy =−4.15 kcal/mol. The variational approachgives F z =−1.02 kcal/mol Å, and M y =−4.08 kcal/mol,and the hypersingular integral method HS1 givesF z =−1.01 kcal/mol Å, and M y =−4.01 kcal/mol, and theyare all close to the correct values. HS2 givesF z =−1.15 kcal/mol Å, and M y =−4.58 kcal/mol. The resultsof HS2 have a relatively large deviation from the analyticalvalues, and show a less accuracy than HS1 again.V. CONCLUSIONS AND DISCUSSIONWe use the hypersingular integral technique in a BEMframe to directly calculate the derivatives of the potential onthe molecular boundary, and then the Maxwell stress is obtainedand used to compute the PB force on a molecule in aninteracting solvated molecular system. The computationalaccuracy and performance for force and torque are demonstratedin the sample tests and compared with that of theprevious variational approach. The accuracy of the hypersingularintegral method HS1 is found to be even somewhathigher than that of the variational method for the main forcecomponent computation. Compared with HS2, the first kindof hypersingular integral method HS1 is rigorous in principlewithout the plane approximation for the adjacentboundary elements, and it has a stable computational performancein terms of accuracy in different cases. Overall, themajor advantage of this hypersingular integral method is itsfaster performance compared with the variational approach,especially for biomolecular systems, and this makes it potentiallyuseful for the BD simulation of protein-proteindocking.ACKNOWLEDGMENTSThis work was supported in part by the NIH, NSF, theHoward Hughes Medical Institute, National BiomedicalComputing Resource, the NSF Center for Theoretical BiologicalPhysics, SDSC, the W. M. Keck Foundation, and Accelrys,Inc.1 M. E. Davis and J. A. McCammon, J. Comput. Chem. 10, 386 1989.2 M. Holst, R. E. Kozack, F. Saied, and S. Subramaniam, J. Biomol. Struct.Dyn. 11, 1437 1994.3 A. Nicholls and B. Honig, J. Comput. Chem. 12, 435 1991.4 T. J. You and S. C. Harvey, J. Comput. Chem. 14, 4841993.5 M. Holst, N. Baker, and F. Wang, J. Comput. Chem. 21, 1319 2000.6 R. J. Zauhar and R. S. Morgan, J. Mol. Biol. 186, 815 1985.7 H. X. Zhou, Biophys. J. 65, 955 1993.8 B. Z. Lu, C. X. Wang, W. Z. Chen, S. Z. Wan, and Y. Y. Shi, J. Phys.Chem. B 104, 6877 2000.9 B. Z. Lu, W. Z. Chen, C. X. Wang, and X. J. Xu, Proteins 48, 4972002.10 R. Luo, L. David, and M. K. Gilson, J. Comput. Chem. 23, 12442002.11 M. K. Gilson, M. E. Davis, B. A. Luty, and J. A. McCammon, J. Phys.Chem. 97, 3591 1993.12 J. D. Madura, J. M. Briggs, R. C. Wade et al., Comput. Phys. Commun.91, 571995.13 R. J. Zauhar, J. Comput. Chem. 12, 575 1991.14 A. J. Bordner and G. A. Huber, J. Comput. Chem. 24, 3532003.15 B. Z. Lu, D. Q. Zhang, and J. A. McCammon, J. Chem. Phys. 122,214102 2005.16 M. Guiggiani, G. Krishnasamy, T. J. Rudolphi, and F. J. Rizzo, Trans.ASME, J. Appl. Mech. 59, 604 1992.17 M. Guiggiani, in Singular Integrals in Boundary Element Methods,Downloaded 02 Sep 2005 to 132.239.16.167. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

084904-8 Lu et al. J. Chem. Phys. 123, 084904 2005edited by V. Sladek and J. Sladek Computational Mechanics, Southampton,UK, 1998, Chap. 3, pp. 85–124.18 O. D. Kellog, Foundations of Potential Theory Frederick Ungar, NewYork, 1929.19 C. Pozrikidis, A Practical Guide to Boundary-Element Methods with theSoftware Library BEMLIB Chapman and Hall, London, 2002.20 J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial DifferentialEquations Dover, New York, 1952.21 S. A. Allison, Macromolecules 29, 7391 1996.22 O. Huber, A. Lang, and G. Kuhn, Comput. Mech. 12, 391993.Downloaded 02 Sep 2005 to 132.239.16.167. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp