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Journal of Electrostatics 70 (2012) 126e135

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Journal of Electrostatics

journal homepage: www.elsevier .com/locate/elstat

Electrical double layer: A numerical treatment of stern layer in biomolecularelectrostatics

Osman Goni*

Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong

a r t i c l e i n f o

Article history:Received 5 July 2011Received in revised form1 November 2011Accepted 25 November 2011Available online 13 December 2011

Keywords:Linearized PoissoneBoltzmann equationElectrostatic solvation free energyReaction potentialStern layerDebye-HückelDiffuse layer

* Tel.: þ852 2241 5683.E-mail address: osman_goni@yahoo.com.

0304-3886/$ e see front matter � 2011 Elsevier B.V.doi:10.1016/j.elstat.2011.11.006

a b s t r a c t

PoissoneBoltzmann equation (PBE) is widely used in the context of deriving the electrostatic energy ofmacromolecular systems and assemblies in aqueous salt solution. Macromolecules and their ion pene-trability with the presence of stern layer have been discussed theoretically and analytically. Whilenumerous numerical solvers for the 3D PBE have been developed, the integral equation formulation forthe boundary treatments used in these methods has only been loosely addressed, especially in the ionexclusion stern layer. The de facto standard in current linear PBE implementations is to estimate thepotential at the outer boundaries using the (linear) Debye-Hückel (DH) approximation. However, asassessment of how these outer boundary treatments affect the overall solution accuracy in the stern layerdoes not appear to have been previously made. As will be demonstrated here, this DH approximation canunder certain conditions, produce completely erroneous estimates of the potential and energy saltdependencies. In this work, the sets of boundary conditions are invoked that take into account theimpenetrability of the ions to the macromolecule. Using surface integral equation, this new treatment isable to give an accurate description of the electrostatic potential distribution, electrostatic solvation freeenergy etc. not only in a macromolecular system by means of continuum model but also focus on physicsof the ion impenetrable stern layer. The accuracy of the results obtained by using the boundary elementmethod (BEM) is tested by comparison with analytical TanfordeKirkwood results for a model sphericalsolute system. Finally, the author also examined how the general ion exclusion layers would tend toincrease the surface electrostatic potential under physiological salt conditions. To facilitate presentationand computational domain, attention is restricted here to the 3D spherically symmetric linear PBE.Though geometrically limited, the modeling principles nevertheless extend to general linear PBE solvers.The 3D linear PBE model can also be used to benchmark and validate the salt effect prediction capa-bilities of existing PBE solvers. This choice promises to be particularly useful in the context of biologicalapplications, where the solvation energy, arising from medium polarization, has a prime role.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Electrostatic interactions are important factors in determiningthe native structures of both proteins and nucleic acids as well astheir complexes with low-molecular-weight drugs [1,2,3]. Thelongrange nature of electrostatic interactions, even in aqueoussolution, is one reason why their theoretical treatment is difficult.In order to circumvent the considerable and often prohibitivecomputational expense of microscopic (explicit) solvent modelswhich, in principle, afford an exact treatment of electrostaticinteractions in solution, there has been much renewed interest inthe use of simpler continuum models [1e9]. In one class of

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continuum models [5e9] the explicit structural features of thesolvent are replaced by a linear high dielectric constant continuumsurrounding the solute, which is modeled as a low dielectricconstant charge-containing cavity. For ionic solutions, the iondistribution is modeled as a mean field, determined from statisticalmechanics according to a Boltzmann distribution. The solute chargedistribution and, at nonzero ionic strength, the mobile ion distri-bution polarize the solvent, giving rise to a solvent reactionpotential. The calculation of the polarization of the solvent iscarried out by solving the Poisson equation or, when ionic strengtheffects are to be included, by solving the more general Pois-soneBoltzmann (PB) equation.

The interaction of the solvent reaction potential with the solutecharge distribution determines the free energy of solvation of thesystem. Although very simple, such continuum models have been

O. Goni / Journal of Electrostatics 70 (2012) 126e135 127

useful for making predictions concerning electrostatic effects inproteins [10e13] which show reasonable agreement with exper-imental observations. A set of very elegant calculations hasrecently shown that continuum models reproduce solute-solventfree energies obtained by using a microscopic treatment of thesolvent [37]. It is likely that the success of such continuummodelsis in part due to cancellation effects in the behavior of water at themolecular level [14,15]. The use of such continuum models isespecially suitable for aqueous systems because of the uniquebehavior of the local dielectric constant in the region of thedielectric boundary. At the boundary, the dielectric constant variesvery rapidly over a microscopic distance to the value of thedielectric constant for bulk water. This is consistent with theassumption in the continuum model that there are only two(discontinuous) dielectrics separated by a molecular interface. Inaddition, calculations of the potential of mean force between ionsin aqueous solution using integral equation theories [16] haveshown that water completely screens vacuum Coulombic inter-actions within one hydration shell.

Such behavior is well represented by simple continuummodels.Analytical solutions of the PB equation can be obtained for onlya very few, simple cavity shapes. Hence, in order to study macro-molecular systems in aqueous ionic solution using cavity-basedcontinuum models, efficient methods for obtaining approximatenumerical solutions to the PB equation have been developed,although some drawbacks remain with each method. There arebroadly two different approaches in seeking approximate numer-ical solutions of the PB equation. One such approach is the finite-difference (FD) method, first used to study bio-macromolecularsystems by Warwicker and Watson [5], with several very impor-tant algorithmic advances being added later by Gilson et al. [6] andNicholls and Honig [18]. The finite-difference method is verygeneral and has been used to obtain solutions of the full nonlinearPoissoneBoltzmann (NLPB) equation [19]. In this method, thesolute and solvent are mapped onto a cubic lattice. Each of thesmall cubes defining the lattice is assigned an appropriate value ofthe charge density, dielectric constant, and ionic strength param-eters that appear in the PB equation. The method of finite-differences is then used to obtain the electrostatic potential overthe entire grid iteratively. This technique involves N3 variables (thetotal number of lattice sites), where N is the number of points peredge of the lattice. There are some difficulties encountered whenusing finite-difference techniques. One concerns the necessarychoice of boundary conditions. These can be obtained at sufficientlylarge distance with respect to the dielectric boundary from eitherCoulomb’s law or Debye-Hückel theory. To achieve a high degree ofaccuracy, it is necessary to consider the continuum solvent that isfar from the solute; this entails increasing the lattice size (relativeto themolecule) and hence the expense of the calculation. A secondproblem associated with the finite-difference technique arisesbecause of the necessity to map the molecular charge distributiononto lattice points. The resulting error arising from this disturbanceof the optimal charge distribution is a function of the lattice spacing(although it is in general small); however, when the molecularcharge distribution is approximated by a set of distributed multi-ples, it is likely that mapping onto the lattice would have to beachieved by use of a limiting monopole distribution. Faerman andPrice [20] have recently demonstrated the utility of using sucha distributed multipole description to obtain very accuratedescriptions of the electrostatic field/potential at the molecularsurface, for peptide molecules, a prerequisite for the success ofelectrostatic continuum solvent models. An alternative approachfor obtaining solutions of the Poisson equation is the boundaryelement method, first developed for macromolecules by Zauharand Morgan [21], with different algorithmic improvements

proposed by Rashin and Namboodiri [9] and Zauhar and Morgan[21,22,59].

The key feature of the boundary element method is the reduc-tion of the problem to the solution of an integral equation overa two-dimensional surface. The polarization of the solvent by thesolute induces a field throughout the volume of the surroundingdielectric medium. Calculation of the polarization field is equiva-lent to the calculation of induced polarization charge density at thedielectric boundary [7,23].

The boundary element method is a function of S independentvariables, where S is the number of elements covering the two-dimensional surface, which serves as the dielectric interface.There is no requirement to displace atomic charge distributionswhen using this method, and in general the method allows fora more accurate description of the molecular surface than thefinite-difference method. The boundary element method has thusfar been used to calculate the total electrostatic potential and theassociated electrostatic component of the free energy of solvation[7,9]. Rashin [24] has described a combined iterative boundaryelement method to obtain solutions of the general PB equation, buthas not presented details for carrying out accurate volume inte-grations, or about the convergence properties of the scheme. Theinclusion of ionic strength effects in continuum models is oftenachieved by using a linearized version of the PoissoneBoltzmannequation [1,17,25,26]. Unlike the full nonlinear version, the linearPoissoneBoltzmann (LPB) equation is formally correct in the limitof low ionic strength and can be derived within a statisticalmechanical framework from a partition function [26]. However, useof the LPB equation is unlikely to be suitable for all investigationsconcerning macromolecular structure. This is because, even at lowionic strength, the main condition for linearization, i.e., .qif(r)/kT << 1.(where f(r) is the electrostatic potential, qi is the ioncharge, T is temperature, and k is the Boltzmann constant), appearsto break down at room temperature in aqueous solution if thedistance between an ion and an exposed polar atom is smaller than5 Å, according to our calculations. The ion charge density predictedby the LPB equation is generally too low and leads to incorrectestimations of ion screening between charged atoms. A detaileddiscussion concerning the validity of the NLPB equation, as well asderivations of various forms of the associated total electrostaticenergy, has been given recently by Sharp and Honig [27].

The purpose of the present paper is to develop a procedure toobtain solutions to the NLPB equation within the framework ofthe previously described boundary element method. The under-lying physical basis of our method is our observation that, atrelatively low ionic strength (�1 M), the distribution of mobileions around the solute molecule is determined primarily by thepotential due to the solute charge distribution and the reactionsolvent potential (viz. the potential due to the surface chargesobtained in the boundary element method). This makes possiblethe calculation of the mobile ion distribution around the moleculein a way that the polarization of the solvent by the solute chargedistribution is calculated by using a boundary element method.The accuracy of the results obtained by using the boundaryelement method (BEM) is tested by comparison with analyticalTanfordeKirkwood results for a model spherical solute system.Finally, the author also consider how the general ion exclusionlayers tend to increase the surface electrostatic potential underphysiological salt conditions. To facilitate presentation andcomputational domain, attention is restricted here to the 3Dspherically symmetric linear PBE. Though geometrically limited,the modeling principles nevertheless extend to general linear PBEsolvers. The 3D linear PBE model can also be used to benchmarkand validate the salt effect prediction capabilities of existing PBEsolvers.

Fig. 1. The continuum model of a solvated protein. The molecular interior (Region I),containing point charges qi, surrounded by a salt solution with high dielectric constantand inverse Debye length, k (Region II).

O. Goni / Journal of Electrostatics 70 (2012) 126e135128

2. Numerical analysis of biomolecular electrostatics

Biomolecular structure and interactions in an aqueous envi-ronment are determined by a complicated interplay betweenphysical and chemical forces including solvation, electrostatics, vander Waals forces, the hydrophobic effect, and covalent bonding.Electrostatic forces have received a great deal of study due to theirlongrange nature and the tradeoff between desolvation and inter-action effects [2,3,6,28]. In addition, electrostatic interactions playa significant role within a biomolecule as well as betweenbiomolecules, making the balance between the two vital to theunderstanding of macromolecular systems. As a result, much efforthas been devoted to accurate modeling and simulation of biomol-ecule electrostatics. One important application of this work is tocompute the strength of electrostatic interactions for a biomoleculein an electrolyte solution, as well as the potential that the moleculegenerates in space. There are two valuable uses for these simula-tions. First, it provides a full picture of the electrostatic energetics ofa biomolecular system, improving our understanding of howelectrostatics contribute to stability, function, and molecularinteractions [29]. Second, these simulations serve as a tool formolecular design, since electrostatic complementarity is animportant feature of interacting molecules [30]. Through exami-nation of the electrostatics and potential field generated bya protein molecule, for example, it may be possible to suggestimprovements to other proteins or drug molecules that interactwith it, or perhaps even design new interacting molecules de novo[31e33].

There are two approaches to simulating a protein macromole-cule in an aqueous solution with nonzero ionic strength. Discrete,atomistic approaches based onMonto-Carlo or molecular dynamicssimulations treat the macromolecule and solvent explicitly at theatomic level [27,34e38]. An enormous number of solvent mole-cules are often required to provide reasonable accuracy, particularlywhen the electric fields of interest are far away from the macro-scopic surface. In addition, free ions within the solvent are difficultto model with this approach. In this paper, we adopt insteada mixed discreteecontinuum approach based on combininga continuum description of the macromolecule and solvent witha discrete description of the atomic charges [5,27,39,40,41,96].

We will briefly describe a widely used approximate model ofbiomolecule electrostatics, and then we will derive a coupledintegral formulation for the problem [60]. The numerical schemesused for computing solutions to the coupled integral formulationwill be described.

2.1. Mixed discreteecontinuum formulation

One commonly used simplified model for biomolecule elec-trostatics was introduced by Tanford and Kirkwood in 1957 [96].In this model the interior of a protein molecule is approximated asa collection of point charges in a uniform dielectric material,where the dielectric constant is typically two to four times largerthan the permittivity of free space. Any surrounding solvent ismodeled as a much higher permittivity electrolyte whosebehavior is described by the Debye-Hückel theory. The interfacebetween the protein and the solvent is defined by determininghow close the solvent molecules can approach the biomolecule[77,78].

The Tanford and Kirkwood model for a single protein ina solvent is depicted in Fig. 1, where Region I corresponds to theinterior of the protein and Region II corresponds to the surroundingsolvent. The electrostatic behavior in Region I, the protein interior,is governed by a Poisson equation

V2f1ð r!Þ ¼ �Xnc

i¼1

qiεIdð r!� r!iÞ ðRegion IÞ (1)

where fI is the electrostatic potential, r! is an evaluation position,r!i is the location of the ith protein point charge, qi is the pointcharge strength, nc is the number of point charges, and 3I is thedielectric constant in the protein interior. Note also that d is thestandard DiraceDelta function.

To determine the electrostatic potential in the solvent, Debye-Hückel theory suggests that the electrostatic potential shouldsatisfy a nonlinear PoissoneBoltzmann equation, but the nonline-arity generates an unnecessarily complicated model. Instead, thesimpler linearized PoissoneBoltzmann equation, which is alsoa Helmholtz equation, is more commonly used, and has been testedextensively and shown to accurately predict biomolecular proper-ties under a variety of conditions. Therefore, the electrostaticpotential in the solvent, Region II of Fig. 1, is presumed to satisfy theHelmholtz equation

V2f2ð r!Þ ¼ k2f2ð r!Þ ðRegion IIÞ (2)

where k is the inverse Debye screening length (or Debye-Hückelparameter).

A wide variety of numerical techniques can be used to computesolutions to the combination of Eqs. (1) and (2). For the biomoleculeapplication, the most commonly used approach is based on thefinite-difference method for discretizing partial differential equa-tions, with researchers frequently making use of the DelPhi soft-ware package [6,11,18,42,43]. Although finite-difference methodshave proven to be effective, there are several characteristics of thebiomolecule application which are problematic for such methods.Inaccuracies can be generated when projecting the discretecharges, which appear in Eq. (1), on to finite-difference grids. Theproblem is particularly troublesome when attempting to computereaction forces at those point charge locations [44]. In addition, thelarge jump in dielectric constant across the irregularly shapedprotein-solvent boundary must be treated carefully. Finally, thesolvent region is unbounded, at least formally, and must besomehow truncated before applying a finite-difference method.Modifications of the basic finite-difference method have beendeveloped to resolvemany of these difficulties [11,43,45,46,54e56],though often at considerable computational cost.

2.2. Integral equation formulation

As this section will make clear, numerical methods based onsolving an integral formulation of Eqs. (1) and (2) can treat point

O. Goni / Journal of Electrostatics 70 (2012) 126e135 129

charges, irregularly shaped regions with large jumps in parameters,unbounded domains, and the reaction force computation muchmore naturally than finite-difference methods. For this reason,a number of researchers have developed integral formulations[21,60,61e63], but most efforts have only addressed systems withzero ionic strength (k ¼ 0 in Eq. (2)). In Juffer et al. [61], an integralformulation was presented which allows for a general, but theformulation uses integrals with hypersingular kernels, and thoseintegrals are challenging to evaluate accurately. In this work wefollowed the approach of Yoon and Lenhoff [60], as their approachallows for a general k and avoids hypersingular kernels. Eventhough integral formulations have many advantages for thisapplication, they are not often used; the available numerical tech-niques for solving integral equations were too computationallyexpensive to use on complicated problems.

To begin the formulation derivation, first consider that thewellknown fundamental solutions to Eqs. (1) and (2) are,respectively,

GIð r!; r!0Þ ¼ 1

4pj r!� r!0j; (3)

GIIð r!; r!0Þ ¼ e�kj r!� r!0j

4pj r!� r!0j: (4)

The two fundamental solutions can be combined with Green’ssecond theorem to generate an integral equations for the potentialand its normal derivative. In particular, the integral equation forRegion I and II are [60,61,80e87]:

fIð r!Þ ¼ZU

�GIð r!; r!0Þ vfI

vnð r!0Þ � fIð r!

0Þ vGI

vnð r!; r!0Þ

�dr!0

þXnc

i¼1

qiεIGIð r!; r!iÞ; ð5Þ

and

fIIð r!Þ ¼ZU

�� GIIð r!; r!0Þ vfII

vnð r!0Þ þ fIIð r!

0Þ vGII

vnð r!; r!0Þ

�dr!0

;

(6)

where n! is the outward pointing normal as shown in Fig. 1, and thedomain of integration for the integrals, U, is the boundary surfaceseparating the low permittivity protein interior from the highpermittivity solvent. The potentials fI and fII must satisfy a pair ofmatching conditions on the boundary surface U. In particular, theelectric potential is continuous and the normal derivative of thepotential jumps by an amount related to the ratio of the dielectricconstants,

fIð r!0Þ ¼ fIIð r!0Þ; 3IvfIvn

ð r!0Þ ¼ 3IIvfIIvn

ð r!0Þ; (7)

where r!0˛U, and 3I and 3II are the dielectric constant of the region Iand II respectively. To enforce these matching boundary conditions,take the limit of Eq. (5) as r!/U from the inside, and use the limitof Eq. (6) as r!/U from the outside. In this limit, GI, GII, vGI/vn, andvGII/vn are kernels with integrable singularities, so care must betaken in carrying out the integrations. Note that the potential due toa monopole layer is continuous across the layer, while the potentialdue to a dipole layer is discontinuous across the layer.

The results generated by applying the limiting processes to Eqs.(5) and (6) along with the boundary conditions yields

fIð r!0Þ þZU

�fIð r!

0Þ vGI

vnð r!0; r

!0Þ � GIð r!0; r!0Þ vfI

vnð r!0Þ

�dr!0

¼Xnc

i¼1

qi3IGIð r!0; r

!iÞ ð8Þ

and

fIð r!0Þ þZU

�� fIð r!

0Þ vGII

vnð r!0; r

!0Þ þ GIIð r!0; r!0Þ

� 3I

3II

vfIvn

ð r!0Þ�dr!0 ¼ 0 ð9Þ

Eqs. (8) and (9) can be used to compute fI and vfI/vn on U. Thenthose surface potentials and their normal derivatives can be used inEqs. (5)e(7) to compute the potentials anywhere. Therefore, tocompute the reaction potentials at the charge locations, which areneeded to determine energy changes, one need only evaluate

fREACð r!iÞ ¼ZU

�GIð r!i; r

!0Þ vfIvn

ð r!0Þ � fIð r!0Þ vGI

vnð r!i; r

!0Þ�dr!0

(10)

3. Numerical solution

3.1. Discretization

A standard piecewise constant centroidecollocation scheme(Point Matching) or Galerkin method can be used to discretize Eqs.(8) and (9). In bothmethods, the surface is first discretized into a setof panels, and a piecewise constant (pulse) basis function, fn, isassociated with each panel. Then, the potentials are represented asa weighted combination of the panel basis functions. That is,

fIð r!0ÞzXNn

anfnð r!0Þ;vfIvn

ð r!0ÞzXNn

bnfnð r!0Þ (11)

where n is the panel index, N is the total number of panels and anand bn are weights of individual basis functions.

The basis function weights are determined by insisting thatwhen (11) is substituted for the potential and its normal derivativein (8) and (9), the resulting equations are exactly satisfied for thosevalues of r0 which correspond to centroids or quadrature points onthe panels. The resulting system of equations can be denoted asa matrix of the form [81]:2664I þ

ZvGI

vnds �

ZGIds

I �Z

vGII

vnds þ 3I

3II

ZGIIds

3775"fvf

vn

#¼

"PnciqiεIGI

0

#(12)

where I denotes the identity operator.

3.2. Computational results

A spherical molecule of unit radius, in aqueous salt solution,with a single charge located at the center was calculated. Adielectric constant of 2 was used inside the molecule, anda dielectric constant of 20was used externally; k¼ 5 correspondingto the ionic concentration of 2.3 M in this example. No stern layerwas used, allowing the ionic strength to reach the molecularsurface. Fig. 2 shows the normalized electrostatic potentials at thediffuse layer region starting from the surface of the sphere ob-tained with numerical method (Eq. (6)), theoretical and/or

Fig. 2. Electrostatic potentials at the diffuse layer region starting from the surface ofthe sphere obtained with numerical Eq. (23), theoretical [Hunter, Ohshima, YoonLenhoff], and asymptotic form [Ohshima, R Tuinier].

Fig. 4. Percentage of relative error due to difference in analytical [Shaozhong Deng]and numerical [Eq. (27)] results of reaction potential as a function of the number ofpanels (surface elements) with different inverse Debye length [�1].

O. Goni / Journal of Electrostatics 70 (2012) 126e135130

analytical [50,51,60,76,79,88e95], and asymptotic form[47e49,90]. The electrostatic theory of surfaces originates fromGouy and Chapman. It describes how the electrostatic potential onsurfaces depends on the charge density and on the ionic strength ofthe salt solution. For membranes one finds that the electrostaticpotential decays fast as a function of the distance from themembrane. The characteristic decay length is called the Debyelength. Ionic distribution governed by the Boltzmann distributionwith the function of distance from the surface to the diffuse layerregion is shown in Fig. 3 which depends on the potential presentedin Fig. 3. The reaction potential calculated at the charge locationwas compared with the analytical result [50e53]. The percentageof relative error due to the difference in analytical and numerical(Eq. (10)) results of reaction potential as a function of the numberof panels(surface elements) with different inverse Debye length(IDL) is presented in Fig. 4.

4. Extension to multiple dielectrics, solvent cavities, and ionexclusion layers (stern layer)

Continuum electrostatic models of biomolecular systems can bedefined by multiple embedded regions of differing homogeneousdielectric constant and salt treatment. Integral equation formula-tions that can solve these problems often possess a complicated

Fig. 3. Ionic distribution (Boltzmann distribution) given by Eq. (4) with the function ofdistance in diffuse layer region with the potential presented in Fig. 3.

block structure because there exist numerous operators that couplevariables on one surface to conditions on other surfaces. To illus-trate this block structure, we next present Green’s theoremformulations for two-surface problems.

Fig. 5 is a schematic of a two-surface problem in molecularelectrostatics; salt ions are not permitted to directly reach themolecular surface a, but instead are bounded by an accessiblesurface b a specified distance outside the molecule. The enclosedvolume between the surfaces is termed the ion exclusion layer.Region I, again representing the molecular interior, has dielectricconstant εI and nc point charges. The ion exclusion layer, region II,has dielectric constant 3II, and in this region, the Laplace equationgoverns the electrostatic potential. Region III represents solventwith mobile ions and has dielectric constant 3III (usually the sameas 3II) but contains a Debye-Hückel salt treatment; the potential inthis region is governed by the linearized PoissoneBoltzmannequation.

Recalling from the previous section, the potential can also becomputed directly as

V2fIð r!Þ ¼ �Xnc

i¼1

qi3Idð r!� r!iÞ ðRegion IÞ (13)

In region II, the charge density function is given by the r2ð r!Þ ¼0 due to the absence of fixed charge and mobile ions. Potential inregion II is then

Fig. 5. A two-surface problem in molecular electrostatics. The molecular interior(Region I), containing point charges qi, is surrounded by an ion exclusion layer withsolvent dielectric and no salt (Region II), which in turn is surrounded by solvent witha salt treatment (Region III)[73e75].

O. Goni / Journal of Electrostatics 70 (2012) 126e135 131

V2fIIð r!Þ ¼ 0 ðRegion IIÞ (14)

and the equation for Region III, similar to previous section,

V2f3ð r!Þ ¼ k2f3ð r!Þ ðRegion IIIÞ (15)

Integral equation for the potential for interior region I boundedby a

fIð r!aÞ þZa

fIð r!0Þ vGI

vnð r!a; r

!0ÞdA0 �Za

vfI

vnð r!0ÞGIð r!a; r

!0ÞdA0

¼Xi

qiεiGIð r!a; r

!iÞ ð16Þ

Exterior region II of the boundary awithout considering anotherboundary b:

fIIð r!aÞ�Za

fIIð r!0ÞvGII

vnð r!a; r

!0ÞdA0

þZa

vfII

vnð r!0ÞGIIð r!a; r

!0ÞdA0 ¼ 0 ð17Þ

Now the contribution of the second boundary b to the potentialterm, as region II is interior to the boundary b

Zb

fIIð r!0Þ vGII

vnð r!a; r

!0Þ�dA0 �

Zb

vfII

vnð r!0ÞGIIð r!a; r

!0ÞdA0

So add these terms to the above equation to get

fIIð r!aÞ�Za

fIIð r!0ÞvGII

vnð r!a; r

!0Þ dA0 þZa

vfII

vnð r!0ÞGIIð r!a; r

!0Þ

�dA0 þZb

fIIð r!0ÞvGII

vnð r!a; r

!0ÞdA0 �Zb

vfII

vnð r!0ÞGIIð r!a; r

!0Þ

�dA0 ¼ 0 ð18ÞSimilarly, consider only boundary b, so we have the integral

equation for the potential interior to b as region II ignoring a,

fII

�rb!� þ

Zb

fIIð r!0Þ vGII

vnð r!b; r

!0ÞdA0

�Zb

vfII

vnð r!0ÞGIIð r!b; r

!0ÞdA0 ¼ 0 ð19Þ

Now add the contribution of boundary a to the integral equationfor the potential. So the contributing terms are

�Za

fIIðr0ÞvGII

vnðrb; r0ÞdA0 þ

Za

vfII

vnðr0ÞGIIðrb; r0ÞdA0

So add these terms to the above equation to get

fIIð r!bÞ þZb

fIIð r!0Þ vGII

vnð r!b; r

!0Þ dA0

�Zb

vfII

vnð r!0ÞGIIð r!b; r

!0Þ dA0 �Za

fIIð r!0Þ vGII

vnð r!b; r

!0ÞdA0

þZa

vfII

vnð r!0ÞGIIð r!b; r

!0ÞdA0 ¼ 0 ð20Þ

Finally, integral equation exterior to boundary b i.e. for theregion III becomes

fIIIð r!bÞ �Zb

fIIIð r!0Þ vGIII

vnð r!b; r

!0ÞdA0

þZb

vfIII

vnð r!0ÞGIIIð r!b; r

!0ÞdA0 ¼ 0 ð21Þ

Eqs. (16), (18), (20) and (21) are the required integralequations for the potential and charge distribution on the two-surface object.

5. Interface continuity conditions

5.1. Classical boundary condition

Physically, we expect that the function f(r) to be continuousat the interface of the regions, as well as the dielectric timesthe normal derivative of the function, 3VfðrÞ$ n!, where n! is theunit outward normal vector. The discontinuous dielectricinterface then implies a discontinuous normal derivative offð r!Þ at the interfaces. In particular, on Ga ¼ UIXUII, it must betrue that:

fIð r!aÞ ¼ fIIð r!aÞ; 3IvfI

vnð r!aÞ ¼ 3II

vfII

vnð r!aÞ (22)

while on Gb ¼ UIIXUIII, we must have

fIIð r!bÞ ¼ fIIIð r!bÞ; 3IIvfII

vnð r!bÞ ¼ 3III

vfIII

vnð r!bÞ (23)

The appropriate boundary conditions for the infinite domainU ¼ UIWUIIWUIII.

The associated integral equations have four surface variables,which are the potential and normal derivative on both surfaces:fa, vfa/vn, fb, and vfb/vn. The free space Green’s functions ineach region are again denoted by G with the region label assubscript:

GIðr; r0Þ ¼ GIIðr; r0Þ ¼ 14pkr � r0k Region I; II

GIIIðr; r0Þ ¼ e�kkr�r0k

4pkr � r0k Region III

Using Eqs. (22) and (23), Eqs. (16), (18), (20) and (21) can be re-written as

fa þZ

favGI

vnðra; r0ÞdA0 �

Zvfa

vnGIðra; r0ÞdA0 ¼

Xi

qi3iGIðra; riÞ

(24)

fa �Z

favGII

vnðra; r0Þ dA0 þ 3I

3II

Zvfa

vnGIIðra; r0Þ dA0

þZ

fbvGII

vnðra; r0ÞdA0 �

Zvfbvn

ðr0ÞGIIðra; r0ÞdA0 ¼ 0 ð25Þ

fb þZ

fbvGII

vnðrb; r0Þ dA0 �

Zvfbvn

GIIðrb; r0Þ dA0

�Z

favGII

vnðrb; r0Þ dA0 þ 3I

3II

Zvfa

vnGIIðrb; r0ÞdA0 ¼ 0 ð26Þ

O. Goni / Journal of Electrostatics 70 (2012) 126e135132

fb �Z

fbvGIII

vnðrb; r0Þ dA0 þ 3II

3III

Zvfbvn

GIIIðrb; r0ÞdA0 ¼ 0

(27)

Introducing an abbreviated notation allows the equations to bewritten as [81]

266664

I þ DaI;a �SaI;a 0 0

I � DaII;a þ 3I;IISaII;a Da

II;b �SaII;b

�DbII;a þ 3I;IISbII;a I þ Db

II;b �SbII;b

0 0 I � DbIII;b þ 3II;IIISbIII;b

377775

2666664

favfa

vnfbvfb

vn

3777775

¼

266664Pi

qi3IGaI;i

000

377775

(28)

where I denotes the identity operator, 3I,II and 3II,III abbreviate 3I/ 3IIand 3II/ 3III respectively. Sui;y and Du

i;y denote the single and doublelayer operators defined as

Sui;yvfy

vn¼

Zy

GIðru; r0Þ vfy

vnðr0ÞdA0

and

Dui;yfy ¼

Zy

vGI

vnðru; r0Þfyðr0ÞdA0:

5.2. Zero source function boundary conditions

The zero source functions are a results of placing the charge onthe molecule surface. Since the charge q is assumed to be evenlydistributed over the molecular surface of area 4pR2, (where R is theradius of the spherical molecule) the molecule has the uniformcharge density:

s ¼ q4pR2

:

For boundary conditions on Ga ¼ UIXUII (at r ¼ ra), we musthave:

fIðraÞ ¼ fIIðraÞ; 3IIvfII

vnðraÞ � 3I

vfI

vnðraÞ ¼ �4ps ¼ �q

R2:

(29)

whereas on Gb ¼ UIIXUIII (at r ¼ rb), we must have:

fIIðrbÞ ¼ fIIIðrbÞ; 3IIvfII

vnðrbÞ ¼ 3III

vfIII

vnðrbÞ (30)

With these new boundary condition, in spherical coordinateswith spherical symmetry the linearized form of the Pois-soneBoltzmann equation is easily seen to reduce to:

�1r2

ddr

�r2

ddr

fIðrÞ�

¼ 0 ðRegion IÞ (31)

In region II, the charge density function is given by the r2ð r!Þ ¼0 due to the absence of fixed charge and mobile ions. Potential inregion II is then

�1r2

ddr

�r2

ddr

f2ðrÞ�

¼ 0 ðRegion IIÞ (32)

and the equation for Region III, similar to previous section,

�1r2

ddr

�r2

ddr

f3ðrÞ�þ k2f3ðrÞ ¼ 0 ðRegion IIIÞ (33)

Using Eqs. (29) and (30), Eqs. (16), (18), (20) and (21) can be re-written as in order to comply with Eqs. (31)e(33)

fa þZ

favGI

vnðra; r0ÞdA0 �

Zvfa

vnGIðra; r0ÞdA0 ¼ 0 (34)

fa �Z

favGII

vnðra; r0Þ dA0 þ 3I

3II

Zvfa

vnGIIðra; r0Þ dA0

þZ

fbvGII

vnðra; r0ÞdA0 �

Zvfb

vnðr0ÞGIIðra; r0ÞdA0

¼Z

qi3IIR2

GIIðra; r0ÞdA0 (35)

fb þZ

fbvGII

vnðrb; r0Þ dA0 �

Zvfb

vnGIIðrb; r0Þ dA0

�Z

favGII

vnðrb; r0Þ dA0 þ 3I

3II

Zvfa

vnGIIðrb; r0ÞdA0

¼Z

qi3IIR2

GIIðrb; r0ÞdA0 (36)

fb �Z

fbvGIII

vnðrb; r0Þ dA0 þ 3II

3III

Zvfb

vnGIIIðrb; r0ÞdA0 ¼ 0

(37)

Hence the new system equation incorporating the above-mentioned boundary conditions becomes in a matrix form as:

266664

I þ DaI;a �SaI;a 0 0

I � DaII;a þ 3I;IISaII;a Da

II;b �SaII;b

�DbII;a þ 3I;IISbII;a I þ Db

II;b �SbII;b

0 0 I � DbIII;b þ 3II;IIISbIII;b

377775

2666664

favfa

vnfbvfbvn

3777775

¼

2666664

0Zqi

3I 3IIR2GIIðra; r0ÞdA0Z

qi3I 3IIR2

GIIðrb; r0ÞdA0

0

3777775

(38)

5.3. Charge balance condition

For boundary conditions on Ga ¼ UIXUII (at r ¼ ra), we musthave:

fIðraÞ ¼ fIIðraÞ; 3IIvfII

vnðraÞ � 3I

vfI

vnðraÞ ¼ �4ps; (39)

whereas on Gb ¼ UIIXUIII (at r ¼ rb), we must have:

fIIðrbÞ ¼ fIIIðrbÞ; 3IIIvfIII

vnðrbÞ � 3II

vfII

vnðrbÞ ¼ �4psOHP

(40)

where sOHP is the charge density in the outer Helmholtz plane(surface b in Fig. 5

Hence the system equation becomes in terms of matrix form as:

O. Goni / Journal of Electrostatics 70 (2012) 126e135 133

266664

I þ DaI;a �SaI;a 0 0

I � DaII;a þ 3I;IISaII;a Da

II;b �SaII;b

�DbII;a þ 3I;IISbII;a I þ Db

II;b �SbII;b

0 0 I � DbIII;b þ 3II;IIISbIII;b

377775

2666664

favfa

vnfbvfb

vn

3777775

¼

266666664

0Zqi

3I 3IIR2GIIðra; r0ÞdA0Z

qi3I 3IIR2

GIIðrb; r0ÞdA0

�Z

qi3I 3IIIR2

GIIIðrb; r0ÞdA0

377777775

(41)

5.4. Computational results

A spherical molecule of unit radius, in aqueous salt solution,with a single charge located at the center was calculated. A sternlayer of thickness 0.2a (a is radius of inner boundary) was used,forming an ion exclusion layer. A dielectric constant of 2 wasagain used inside the molecule, and a dielectric constant of 20was used in the stern layer and in the solvent for the zero sourcefunction boundary condition. Fig. 6 shows the electrostaticpotentials in the diffuse layer region starting from the interface ofthe stern layer and the diffusion layer of the sphere obtainednumerically using Eqs. (21) and (28) for different values of Debyelength (DL) 1.0 Å, 0.5 Å, 0.2 Å and 0.1 Å which correspond to ionicstrength of 93 mM, 370 mM, 2.3 M and 9.25 M solution of NaClrespectively [64e67]. Since under thin layer approximation, theion exclusion layer is typically considered to be about 1.0 Å, manyresearchers adopt this approximation to simplify the computa-tional domain [64]. But more general case, this approximation isnot valid for physical system and hence this Debye length hasa great impact on the electrostatic potential for many biophysicalsystems.

Fig. 7 shows the normalized electrostatic potentials at differentradial positions from the center (location of charge) obtained fromclosed form solutions for linear [57,58,60,68,69,90,93] andnonlinear [47,90] along with the numerical results. Up to certainrange of surface potential (z25 mV at room temp.) the linearizednumerical results agree satisfactorily well with the theoretical oranalytical results. Hence, DH approximation ðsinhjzjÞ holds truefor the low potential energy. The existence of the compact layer

Fig. 6. Numerical results of electrostatic potential in the diffuse double layer accordingto the Gouy-Chapman model for different values of Debye length (Å) as a function ofdimensionless distance from the surface.

(stern layer) can be comparedwith the results presented in [70e72]with no stern layer. Normalized electrostatic potentials at differentradial positions starting from the location of point charge (center)of a unit sphere model obtained numerically with and withoutstern layer cases and in the diffuse layer region are plotted in Fig. 8.Discontinuity of potentials can be seen near the surface and at theinterface of stern layer and diffuse layer. There are sharp rise inpotential at the interface between stern layer and diffuse layerelsewhere the exponential decay of potentials are observed clearly.Normalized electrostatic potentials obtained numerically atdifferent radial positions from the location of point charge (center)and in diffuse layer region of a unit sphere model are plotted withand without stern layer cases. The increase in surface potential isobserved clearly from Fig. 9 due to the presence of stern layer whichhas a great impact on membrane potential for many biological andbiophysical system.

6. Discussion

In this paper, by using the classical electrostatic theory andGreen’s theorem, the electric potential of point charge insidea dielectric spheroid are obtained depending on whether thesurrounding dissimilar dielectric medium of the spheroid is ionic ornonionic (k ¼ 0). Numerical experiments have also demonstratedfor the presence of stern layer (compact inner layer) and shows thevalidity of the solutionwith conventional analytical and asymptoticsolution without the stern layer. Using surface integral equation,this new treatment is able to give an accurate description of theelectrostatic potential distribution, electrostatic solvation freeenergy etc. not only in a macromolecular system by means ofcontinuummodel but also focus on physics of the ion impenetrablestern layer. The ion exclusion layers tend to increase the surfaceelectrostatic potential under physiological salt conditions.

Although continuum models have been proven useful in recentstudies, it is important to recall that they are very simplifiedmodelsof the liquid state. Their failings are generally wellknown and areprincipally connected to the considerable information about thesolvent structure that is discarded in assuming a linear dielectriccontinuum.

In summary, we have described a method for computing thetotal electrostatic potential obtained from the linear PB equation,based upon the boundary element method. The method makes useof the advantageous features of boundary element techniques tosolve the Poisson equation and PoissoneBoltzmann equation, such

Fig. 7. Electrostatic potentials at the diffuse layer region starting from the surfaceof the sphere obtained with numerical (Eq. (23)), theoretical, and asymptoticform (nonlinear). Linearized condition is maintained with low surface potential, i.e.f << kT/e.

Fig. 8. Normalized electrostatic potentials at different radial positions from the surfaceof the unit sphere model obtained from numerical results with three differentboundary conditions and without stern layer case.

O. Goni / Journal of Electrostatics 70 (2012) 126e135134

as the accurate description of the molecular surface, and makes useof the boundary element description of the molecular surface in thedesign of an efficient procedure to evaluate integrals numericallyover the volume external to the molecular cavity. We havedemonstrated that the BEM method for obtaining approximatenumerical solutions to the linear PB equation can be used toreproduce analytical Tanford-Kirkwood solutions for modelspherical systems at different ionic strengths.

Fig. 9. Normalized electrostatic potentials at different radial positions (a) from thelocation of point charge (center) (b) in diffuse layer region of a unit sphere modelobtained numerically with and without stern layer cases.

This 3D linear PBE formulation can also be used to benchmarkand validate the salt effect prediction capabilities of existing PBEsolvers. The most biophysical problem can be restricted to lowpotential and low charge density for which these approach wouldproduce great impact for different surface conditions. As thenumerical solution by the surface integral equation proposed forthe stern layer model of unit dielectric sphere can also be equiva-lently considered for the analysis of the behavior of biologicalmembrane of a cell and protein.

Acknowledgments

The author acknowledge Prof. Weng Cho Chew and Dr. LijunJiang for continuous guidance, helpful suggestions and discussions.Financial support from the Research Grants Council of Hong Kong(GRF 711609 and 711508), in part from the University GrantsCouncil of Hong Kong (Contract No. AoE/P-04/08) is acknowledged.

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