An Efficient Higher-Order Fast Multipole Boundary Element Solution for Poisson–Boltzmann-Based Molecular Electrostatics

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. SCI. COMPUT. c© 2011 Society for Industrial and Applied MathematicsVol. 33, No. 2, pp. 826–848

AN EFFICIENT HIGHER-ORDER FAST MULTIPOLE BOUNDARYELEMENT SOLUTION FOR POISSON–BOLTZMANN-BASED

MOLECULAR ELECTROSTATICS∗

CHANDRAJIT BAJAJ† , SHUN-CHUAN CHEN† , AND ALEXANDER RAND†

Abstract. In order to compute polarization energy of biomolecules, we describe a boundaryelement approach to solving the linearized Poisson–Boltzmann equation. Our approach combinesseveral important features, including the derivative boundary formulation of the problem and asmooth approximation of the molecular surface based on the algebraic spline molecular surface. Stateof the art software for numerical linear algebra and the kernel independent fast multipole method isused for both simplicity and efficiency of our implementation. We perform a variety of computationalexperiments, testing our method on a number of actual proteins involved in molecular docking anddemonstrating the effectiveness of our solver for computing molecular polarization energy.

Key words. Poisson–Boltzmann equation, boundary element method, fast multipole method

AMS subject classifications. 92C40, 65N38

DOI. 10.1137/090764645

1. Introduction. Models of molecular potential energy are often used in biol-ogy to understand the structure-function relationships of proteins. Computation ofmolecular binding affinities and molecular dynamics [30, 62, 63] involves repeatedevaluation of molecular energy or forces as dynamic molecular configurations are sim-ulated and analyzed. Electrostatic interactions of a molecule within an ionic solutionare captured in the polarization term of the total potential energy. Since treatingeach solvent molecule discretely is extremely computationally expensive for a realisticnumber of molecules, a common and experimentally useful model for this polarizationinteraction is the Poisson–Boltzmann equation, which treats the solvent as a continu-ous medium [31, 34]. Finite difference, finite element, and boundary element methodshave all been used to solve the linearized Poisson–Boltzmann equation numerically[57]. Discretizing space with a regular lattice, the earliest solvers were based on fi-nite difference methods [35, 61, 70]. Later, finite difference approaches incorporatedmultigrid techniques [41, 44] and an alternate formulation [67] to improve efficiency.However, discontinuous coefficients and Dirac point charges often limit the accuracyof these methods. Finite element methods eliminate some of these challenges by al-lowing the domain to be discretized with a more geometrically accurate mesh. Finiteelement methods have been developed and analyzed for the linearized [10, 20, 27, 26]and nonlinear Poisson–Boltzmann equations [9, 10, 23, 42]. Both finite differenceand finite element methods require a discretization of three-dimensional space. If auniform mesh of size h is used, then the number of degrees of freedom is O(h−3).Boundary element methods provide an alternative in which all degrees of freedom lieon the molecular boundary and (for a uniform mesh) only O(h−2) degrees of freedomare needed.

∗Received by the editors July 10, 2009; accepted for publication (in revised form) January 14,2011; published electronically April 7, 2011. This research was supported in part by NSF grantCNS-0540033 and NIH contracts R01-EB00487, R01-GM074258, and R01-GM07308.

http://www.siam.org/journals/sisc/33-2/76464.html†Department of Computer Sciences, University of Texas at Austin, Austin, TX 78712 (bajaj@

ices.utexas.edu, [email protected], [email protected]).

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POISSON–BOLTZMANN-BASED MOLECULAR ELECTROSTATICS 827

Zauhar and Morgan [80, 81, 82] formulated the linearized Poisson–Boltzmannequation as a system of boundary integral equations (the nonderivative boundaryintegral equations, or nBIE) and solved this system numerically. The original systemhas been observed to exhibit poor conditioning for iterative linear solvers [54], butan alternative formulation (the derivative boundary integral equations, or dBIE) firststated by Juffer et al. [46] is well conditioned. Since the boundary element methodleads to a dense linear system, these and other [86] early methods suffer from the needto compute this entire matrix.

Due to the special structure of the boundary element system, the fast multipolemethod [38] can be used to efficiently approximate the necessary matrix-vector prod-ucts without creating the full matrix. This has been applied to several formulations ofthe molecular electrostatics problem: nBIE [1, 50, 55], dBIE [19, 56], models involvingonly Poisson’s equation [15, 74], and a formulation involving only single layer densities[17]. Nearly all of these codes utilize the solvent-exposed surface produced by MSMS[66], which is composed of spherical and toroidal patches but in some cases containssharp corners, and some codes approximate this surface with a flat triangulation [19].This can give hypersingular integrals which are challenging to discretize, leading to asolution error which is dominated by the geometric approximation.

For the linearized Poisson–Boltzmann equation, we have designed and imple-mented a boundary element method and additionally studied its accuracy and ef-ficiency on real protein structures. Our solver combines several key features whichproduce meaningful electrostatic calculations with modest surface mesh sizes. First,the dBIE formulation of the problem is used, providing a well-conditioned systemfor iterative methods in linear algebra. Second, by defining the molecular domainusing the C1 algebraic spline molecular surface, solutions reflect only a second-ordergeometric error from the domain approximation, and numerically problematic hy-persingular integrals are avoided. Third, a general purpose fast multipole package,KIFMM3d, is used to efficiently approximate dense matrix computations, simplifyingthe algorithm by separating the details of the fast multipole method from the restof the scheme. Our freely available solver (http://cvcweb.ices.utexas.edu/software) istested on a suite of actual proteins important in molecular docking. We show thatour software outperforms several alternative approaches (the nBIE formulation andlinear or nondifferentiable surface geometry), and we demonstrate benefits comparedto a finite difference solver. For practical examples, key parameters, including sin-gular and nonsingular quadrature orders, fast multipole approximation order, andGMRES termination tolerance, are tuned to greatly improve the method efficiencywith minimal impact on the solution error.

Motivation for computing the molecular polarization energy is contained in sec-tion 2. In section 3 the nonlinear and linearized Poisson–Boltzmann equations arestated, and then the latter equation is formulated as a pair of boundary integralequations. Our numerical scheme for solving these equations is described in section 4.Polarization energy is formulated as a postprocesses to the Poisson–Boltzmann solu-tion in section 5. Sections 6 and 7 contain implementation details and computationalexperiments, respectively.

2. Motivation. We begin with a general outline of the molecular energeticsproblem, including a description of the specific role of the polarization energy.

Molecular potential energy. The total free energy of the system G is given byG = U−TS, where U is the potential energy, T is the temperature of the system, andS is the solute entropy. The potential energy of a molecule in solution is divided into

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828 C. BAJAJ, S.-C. CHEN, AND A. RAND

two components: U = EMM +Gsol, where EMM is the molecular mechanical energy,and Gsol is the solvation energy. A common model for the molecular mechanicalenergy EMM is given in [52]. For a molecule in solution, additional potential energyresulting from interaction of the solute and solvent is called the solvation energy Gsol.The solvation energy is often modeled by three terms:

(2.1) Gsol = Gcav +Gvdw +Gpol,

where Gcav is the energy to form a cavity in the solvent, Gvdw is the van der Waalsinteraction energy between solute and solvent atoms, and the polarization energy Gpol

is the electrostatic energy due to solvation [30, 34, 40, 69, 71].

Polarization energy. The polarization energy of a molecule occupying region Ω isthe change in the electrostatic energy due to the induced polarization of the solvent,

(2.2) Gpol =1

2

∫Ω

φrxn(z)ρ(z) dz,

where ρ(z) is the charge density at position z and the reaction electrostatic potentialφrxn(z) indicates the change in electrostatic potential caused by solvation, i.e., φrxn =φsol − φgas, where φsol and φgas are the potential of the molecular in solution and ina gas, respectively.

A number of applications involve the computation of polarization energy. Forexample, the binding effect of a drug (molecule 1) and its target (molecule 2) is thedifference between the potential energy of the complex of the two molecules minusthe sum of the potential energy of the individual molecules:

ΔGbind = Gcomplex − (Gmolecule1 +Gmolecule2).

Polarization energy is an important component of each of these energy calculations.Different theoretical approaches for computing binding solvation energy can be

divided into two broad categories: explicit [13, 36, 53, 58] and implicit [30, 34, 65, 73].Explicit solvent models adopt an atomistic treatment of both solvent and solute.Explicit approaches sample the solute-solvent space by molecular dynamics or MonteCarlo techniques which involve a large number of ions, water molecules, and molecularatoms [73]. This requires considerable computational effort, and explicit solutions areoften not practical, especially for large domains [75].

Implicit solvent models treat the solvent as a featureless dielectric material andadopt a semimicroscopic representation of the solute. The effects of the solvent aremodeled in terms of dielectric and ionic physical properties. The most widely usedimplicit model for molecular electrostatics is the Poisson–Boltzmann equation: itpossesses a solid theoretical justification and has been used to explain a number ofexperimental observations [30, 31, 32, 65, 71, 75]. Since solving partial differentialequations still requires substantial computational effort, several other implicit modelshave been developed to approximate results of the Poisson–Boltzmann model. Themost common of these models is the generalized Born formula [73, 5], which hasalso been used to successfully approximate polarization energy for some applications[2, 33].

3. The Poisson–Boltzmann equation. Amolecule is defined as a stable groupof at least two atoms in a definite arrangement held together by very strong chemicalcovalent bonds. For a molecule embedded in an ionic solution, the domain (R3) is

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separated into open interior (Ω) and exterior (R3\Ω) regions divided by the molecularsurface Γ = Ω ∩ R3 \ Ω [51]; see Figure 3.1.

Two important coefficients, the dielectric coefficient ε(z) and the ion strengthI(z), are assumed to be constant over Ω and R

3 \ Ω:

ε(z) =

{εI , x ∈ Ω,

εE, x ∈ R3 − Ω,

and I(z) =

{0, z ∈ Ω,

I, z ∈ R3 − Ω.

The electrostatic potential in the interior and exterior of a molecule is governedby Poisson’s equation,

(3.1) ∇(ε(z)∇φ(z)) = ρ(z),

where ρ(z) is a variable charge density. This charge density contains two components:charged atoms belonging to the molecule itself and mobile ions as part of the solution.Atomic charges are assumed to be Dirac distributions, while mobile ions in solutionare modeled with the Boltzmann distribution,

(3.2) ρ(z) := ρc(z) + ρb(z) = −4π

nc∑k=1

qkεIδ(z− zk) + λ(z)

∑i

eczicie−ecziφ(z)/kBT .

Since ρ(z) depends on φ, (3.1) is the nonlinear Poisson–Boltzmann equation ratherthan merely Poisson’s equation. Definitions of each of the parameters in (3.2) and afew other parameters needed for the linearized equation and its numerical discretiza-tion are listed in the table of variables below.

ε(z) dielectric coefficient at zqk charge of the atom kzk location of charge qknc number of point chargesλ(z) characteristic function of the set R3 \ Ωec charge of an electronkB Boltzmann’s constantT absolute temperatureI = 1

2

∑i ciz

2i ionic strength

ci, zi concentration and charge of ith ionic species

κ(z) =√

8πe2cI(z)kBT modified Debye–Huckel parameter

nd number of basis functionsnq number of quadrature pointsnb number of molecular surface mesh patches

Selecting a linear approximation to the nonlinear term ρb produces the linearizedPoisson–Boltzmann equation,

(3.3) ∇(ε(z)∇φ(z)) = ρc(z) + ρLb (z),

where ρLb (z) = κ2(z)φ(z) is the first term of the Taylor expansion of ρb(z). In manycases the linearized Poisson–Boltzmann equation provides a sufficiently accurate ap-proximation of the nonlinear Poisson–Boltzmann equation; see [32] and the referencestherein.

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x

n(x)

n(y)

y

z1 z2 z3

z4

Γ

R3 \ Ω

Ω

Fig. 3.1. Molecular domain Ω for the boundary element formulation. Γ denotes the surface ofmolecular interior Ω. Atomic centers zk are contained inside Ω, while mobile ions in solution occuroutside Ω. x and y are used to denote points on the molecular surface and the surface normal aredenoted by �n(x) and �n(y). In the discrete system, x is typically used to identify a collocation point,while y usually represents a quadrature point.

3.1. Boundary integral formulation. Potential theory [47, 72] provides thetools needed to derive a boundary integral formulation of the linearized Poisson–Boltzmann equation. We begin by separating (3.3) into the interior and exterior re-gions and explicitly stating interface conditions which must hold on molecular bound-ary Γ:

∇ (εI∇φ(z)) = −nc∑k=1

qkδ(z− zk), z ∈ Ω,(3.4)

∇ (εE∇φ(w)) = κ2φ(w), w ∈ R3 \ Ω,(3.5)

φ(z)|z=x = φ(w)|w=x , x ∈ Γ,(3.6)

∂φ

∂n(z)

∣∣∣∣z=x

=εEεI

∂φ

∂n(w)

∣∣∣∣w=x

, x ∈ Γ.(3.7)

Carefully applying Green’s second identity to the interior and exterior regions andtaking limits approaching Γ yields the nBIE,

1

2φ(x) +

∫Γ

[∂G0(x,y)

∂n(y)φ(y) −G0(x,y)

∂φ

∂n(y)

]dy =

nc∑k=1

qkεIG0(x, zk),(3.8)

1

2φ(x) +

∫Γ

[∂Gκ(x,y)

∂n(y)φ(y) − εI

εEGκ(x,y)

∂φ

∂n(y)

]dy = 0,(3.9)

where G0 and Gκ denote the fundamental solutions of the Poisson–Boltzmann equa-tions,

G0(x,y) =1

4π ||x− y|| and Gκ(x,y) =e−κ||x−y||

4π ||x− y|| .

Recall Figure 3.1 for an example domain including normal vectors at labeled boundarypoints x and y.

An alternative boundary element formulation of the linearized Poisson–Boltzmannequation was proposed by Juffer et al. [46]. This system (dBIE) is produced by taking

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linear combinations of the original boundary integral equations and their derivatives:

(3.10)

1

2

(1 +

εEεI

)φ(x) +

∫Γ

(∂G0(x,y)

∂n(y)− εEεI

∂Gκ(x,y)

∂n(y)

)φ(y) dy

−∫Γ

(G0(x,y) −Gκ(x,y))∂φ(y)

∂n(y)dy =

nc∑k=1

qkεIG0(x, zk),

(3.11)

1

2

(1 +

εIεE

)∂φ(x)

∂n(x)+

∫Γ

(∂2G0(x,y)

∂n(x)∂n(y)− ∂2Gκ(x,y)

∂n(x)∂n(y)

)φ(y) dy

−∫Γ

(∂G0(x,y)

∂n(x)− εIεE

∂Gκ(x,y)

∂n(x)

)∂φ(y)

∂n(y)=

nc∑k=1

qkεI

∂G0(x, zk)

∂n(x)dy.

This combination of the derivatives of (3.8) and (3.9) has been selected so the kernel∂2G0(x,y)∂�n(x)∂�n(y) − ∂2Gκ(x,y)

∂�n(x)∂�n(y) in (3.11) is not hypersingular. For certain numerical schemes,

this reformulation has been observed to produce a better well-conditioned linear sys-tem and faster convergence of iterative linear solvers when compared to the originalboundary integral equations [54].

3.2. Discretization by the collocation method. The boundary integral equa-tions (either nBIE or dBIE) are discretized by selecting a finite-dimensional functionspace and a set of collocation points. Each unknown function is required to belongto the selected function space, and the integral equations are required to hold ex-actly at the collocation points. The most commonly selected pairs of function spacesand collocation points are piecewise constant functions with triangle centroid collo-cation points and piecewise linear functions with mesh vertex collocation points. Let{ψi}nd

i=1 be a basis for the finite-dimensional function space, i.e., φ(x) =∑nd

i=1 φiψi(x)

and ∂φ∂�n(x) =

∑nd

i=1 ∂φiψi(x), and let xi denote the collocation points. Then the nBIEformulation becomes a linear system of equations,

1

2

nd∑j=1

φjψj(xi) +

∫Γ

∂G0(xi,y)

∂n(y)

nd∑j=1

φjψj(y) dy

−∫Γ

G0(xi,y)

nd∑j=1

∂φjψj(y) dy =

nc∑k=1

qkεIG0(xi, zk),

i = 1..nd,(3.12)

1

2

nd∑j=1

φjψj(xi) +

∫Γ

∂Gκ(x,y)

∂n(y)

nd∑j=1

φjψj(y) dy

−∫Γ

εIεEGκ(x,y)

nd∑j=1

∂φjψj(y) dy = 0,

i = 1..nd.(3.13)

A similar system can be derived for the dBIE system. Solving this dense linear system(for unknowns φi and ∂φi) involves a number of complications and simplifications.We briefly outline the general issues here and in the next section describe our specificapproaches as applied to realistic proteins.

The integrals in (3.12) and (3.13) must be discretized by some quadrature rules,but the singular kernels prevent the use of a fixed quadrature rule over a triangulation

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(or similar discretization) of the boundary. For a boundary subdivided into patches{Γb}nb

b=1, the integral is usually broken into three parts: nonsingular, nearly singular,and singular components. A different quadrature rule is used for each type of bound-ary patch based on which component of the integral it belongs to. The singular andnonsingular integrals are usually performed only in a small neighborhood of the singu-larity xi. The remaining integrals are evaluated using a fixed nonsingular quadraturerule, and due to the rapid decay of the kernels, the simultaneous computation of theseintegrals for each collocation point can be accelerated with the fast multipole method[38]. For example, if the first integral in (3.12) is discretized using a quadrature rule{(yq, ww)}nq

q=1, then the resulting summations,

nq∑q=1

∂G0(xi,yq)

∂n(yq)wq

nd∑j=1

φjψj(yq), i = 1..nd,(3.14)

can be accurately approximated via the fast multipole in O(max(nq, nd)) operations,assuming that the support of each basis function intersects a bounded number ofboundary patches, i.e., the sum in j in (3.14) involves a bounded number of nonzeroterms.

Following the fast multipole calculation, each of the values is corrected to includeaccurate singular and nearly singular quadrature rules for the appropriate boundarypatches. Singular integration is usually performed with quadrature rules tailored tothe position of the singularity [29, 39, 45], while nearly singular integration usuallyinvolves (possibly adaptive) refinement of the boundary patches [18, 39, 45, 68]. Insome cases, singular and nearly singular integration has been studied with respect tocertain specific surfaces associated with the linearized Poisson–Boltzmann equation[12, 76].

4. BEM on molecular surfaces. We describe the details of our boundaryelement method: how the molecular surface is defined and discretized, what basisfunctions are selected, and how quadrature is performed.

4.1. Construction of the molecular surface. To define the molecular surfaceΓ, we begin with an experimentally derived protein structure from the RCSB ProteinData Bank (PDB) [14], a worldwide data repository containing thousands of largebiomolecules. Each PDB structure contains of list of spacial locations for each of theatoms in a molecule. The molecular model for electrostatic calculations is obtainedfrom a PDB file by assigning charge and radius parameters derived from one of avariety of force fields, e.g., AMBER [63], CHARMM [21], etc. For example, theadaptive Poisson–Boltzmann solver, APBS, applies the all-atom AMBER 99 force field[28] by default.

From a configuration of atomic positions and radii a molecular surface can bedefined. The simplest surfaces, the van der Waals and solvent accessible surfaces, aremerely the boundary of a union of balls [51]; see Figure 4.1(a). Alternatively, thesolvent excluded surface [25, 64] is defined to be the boundary of the region outsidethis union of balls which is accessible by a probe sphere. The solvent excluded surfaceeliminates many, but not all, of the nondifferentiable cusps which occur in the union-of-balls surfaces. For a smooth surface, the level set of a sum of Gaussian functionsassociated with each atom is often considered; see [16, 37], for example. The mostcommon molecular surface used for solving the Poisson–Boltzmann equation is thesolvent excluded surface; see, e.g., [1, 19, 46, 61, 70, 74]. In some cases, solvent

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(a) (b) (c)

(d) (e) (f)

Fig. 4.1. Molecular model of a protein (PDB id:1PPE, 436 atoms). (a) The van der Waalssurface of the protein which models the molecule as a union of balls. (b) The variational molecularsurface gives a smooth approximation of the van der Waals surface. (c) The variational surfaceis then triangulated and then decimated to produce a smaller mesh. This decimated mesh contains1,000 triangles. (d) The algebraic spline molecular surface (ASMS) fits a smooth surface over thetriangular mesh. (e) Electrostatic potential computed using the 1,000 patch ASMS. (f) Electrostaticpotential using an ASMS with 74,812 patches. The surfaces in (e) and (f) are colored by the elec-trostatic potential, ranging from −3.8 kbT/ec (red) to +3.8 kbT/ec (blue). Color is available onlyin the online version.

accessible [54], Gaussian [43], and slightly modified solvent excluded [82] surfaceshave also been considered.

We utilize the molecular surface constructed in [7, 8]; the surface is generatedby constructing a Gaussian density function for the atom based on atomic positionsand radii, evolving this function according to a variational formulation and then con-sidering a level set of this function; see Figure 4.1(b). For the resulting surface, atriangular mesh with surface normal vectors at the vertices is constructed using adual contouring method [83]. If the surface mesh generated contains too many tri-angles, it is decimated following the approach in [4], and further mesh smoothing isperformed as necessary; see Figure 4.1(c).

4.2. Surface parametrization. To provide a smooth surface which interpo-lates mesh vertices and prescribed surface normals, we utilize the algebraic splinemolecular surface (ASMS) [84]. This surface is constructed from algebraic patchesor A-patches which are a kind of low degree algebraic surface with dual implicit andrational parametric representations [3]. The result is a molecular surface depicted inFigure 4.1(d) which can be parametrized in terms of the barycentric coordinates ofthe triangles, allowing for easy construction of basis functions as described in the nextsection. We give a brief overview of this construction; complete details can be foundin [84, 85].

For some triangular element Γj with vertices v1, v2, and v3 and normals n1, n2,

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v1 v2

v3n1 n2

n3

λ = 1

λ = −1Γj

Γj

(a)

v1

v2

v3

v4Γ1 Γ2

(b)

Fig. 4.2. (a) A single prismatic scaffold region for the triangle with vertices v1, v2, and v3

and associated surface normals �n1, �n2, and �n3. The surface patch Γi interpolates these normals.(b) The ASMS is smooth between two scaffold patches Γ1 and Γ2.

and n3, the A-patch Γj is defined on the prism,

D(Γj) := {y : y = b1v1(λ) + b2v2(λ) + b3v3(λ), −1 ≤ λ ≤ 1},where vi(λ) = vi+λni and (b1, b2, b3) are the barycentric coordinates of the triangle;see Figure 4.2. We define a function over the prism D(Γj) in Bernstein–Bezier splineform by

Fd(b1, b2, b3, λ) =∑

i+j+k=d

bijk(λ)Bdijk(b1, b2, b3),

where Bdijk(b1, b2, b3) =

d!i!j!k! b

i1b

j2b

k3 . For d ≥ 3, coefficients bijk(λ) can be selected so

that Fd is continuous between adjacent patches and for each vertex Fd(vi) = 0 and∇Fd(vi) = ni.

The molecular surface Γj is the zero level set of Fd,

(4.1) Γj = {y : y = b1v1(λ) + b2v2(λ) + b3v3(λ), Fd(b1, b2, b3, λ) = 0}.This can be viewed as a parametric representation in two parameters b1 and b2. Thethird barycentric coordinate can be computed from the first two, b3 = 1− b1− b2, andunder some mild restrictions on the mesh shape and vertex normals, Fd(b1, b2, b3, λ) =0 can be solved for λ in terms of b1 and b2. In practice this nonlinear equation is solvednumerically with Newton’s method; see, e.g., [22, pp. 611–620] and [48, pp. 81–92].

4.3. Selection of basis functions. We consider two different types of basisfunctions for the solution space and associated collocation points: piecewise constantbasis functions with triangle centroids as collocation points and piecewise linear basisfunctions with mesh vertices as collocation points. In both cases, these functions aredefined based on the barycentric coordinates of an underlying triangular mesh. Sincethe A-patches can be parametrized by the barycentric coordinates, this constructioncan be applied to the ASMS directly.

4.4. Quadrature. Let {(bq, wq)}nq

q=1 be a (generic) quadrature rule for a ref-erence triangle T , where bq denote the barycentric coordinates. Using a change ofvariables, this rule can be transferred to an arbitrary A-patch using the parametriza-tion (4.1). The resulting quadrature rule on Γj is {(y(bq), J(bq)wq)}nq

q=1, where J(bq)denotes the Jacobian of the parametrization.

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x

y

x′

y′x′ = x

y′ = y/(1− x)

(a) (b)

Fig. 4.3. Singular quadrature rules. (a) Quadrature rule for a triangle with a weak singularityat a triangle vertex. (b) When singularity occurs in the triangle interior, the triangle is dividedinto three subtriangles at the singularity, and then the scheme depicted in (a) can be applied to eachsubtriangle.

Next we outline the quadrature rules for nonsingular, singular, and nearly singularintegrals to be computed. In each case, quadrature rules on a reference triangle canbe transferred to the curved molecular surface using the aforementioned change ofvariables.

Nonsingular quadrature and the fast multipole method. Nonsingular quadrature isperformed using a fixed Gaussian quadrature rule. This gives a single quadrature rulefor the entire surface producing integrals of the form of (3.14). The source densitywq

∑nd

j=1 φjψj(yq) must be computed at each quadrature point yq. Since the basisfunctions are locally supported, the summation over j involves only a bounded numberof terms for any particular quadrature point. Then for all collocation points xi thesummation in q can be approximated by the fast multipole method in O(nq · nb)operations.

Singular quadrature. For smooth surfaces, the kernels in (3.8)–(3.11) are all in-tegrable. By performing a change of variables to polar coordinates around the sin-gularity, a smooth integrand is produced. For singularities occurring at a vertex ofa triangle, a more computationally useful change of variables is described clearly in[29]. This coordinate change maps the triangle into a square where a tensor-productGaussian quadrature rule can be applied; see Figure 4.3(a).

When a triangle centroid is selected to be a collocation point, the integrandsingularity occurs in the interior of the triangle. Suitable quadrature rules are formedby subdividing the triangle into three new triangles with the singularity as a newvertex; see Figure 4.3(b). Then the previous quadrature rule (which was designedfor triangles with singularities at a vertex) can be applied to each of the three newtriangles.

Nearly singular quadrature. Nearly singular quadrature is performed by subdi-vision. On each subdivided triangle a Gaussian quadrature rule is applied. Preciseconvergence analysis imposes many restrictions on how this refinement should be per-formed and which integrals must be considered nearly singular; see, e.g., [45, 78]. Insection 7 we demonstrate that nearly singular quadrature has a very limited impacton our results for practical molecular structures and thus have avoided implementinga more complex (and computationally demanding) quadrature procedure.

5. Polarization energy computation. After solving for the electrostatic po-tential φ and its normal derivative ∂φ

∂�n over the molecular surface, the total polarizationenergy can be computed. Combining the expressions for the polarization energy (2.2)

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and the charge density (3.2) gives

(5.1) Gpol =

∫Ω

φrxn(z)

nc∑k=1

qkδ(z− zk)dz =1

2

nc∑k=1

φrxn(zk)qk,

where φrxn(x) = φ(x)−φgas(x) is the difference between the potential induced by themolecule in solution and the molecule in a gas.

Using Green’s second identity as in the derivation of the boundary integral equa-tions, formulas for the potential both inside and outside the molecule can be obtained;see [46] for complete details. For a point z ∈ R

3 \ Γ,ε(z)

εIφ(z) =

∫Γ

(εEεI

∂Gκ(z,y)

∂n(y)− ∂G0(z,y)

∂n(y)

)φ(y) dy

+

∫Γ

(G0(z,y) −Gκ(z,y))∂φ(y)

∂n(y)dy +

nc∑k=1

qkεIG0(z, zk).

(5.2)

The potential of the molecule in a gas is the solution to Poisson’s equation (3.1)with constant dielectric ε(z) := εI and no charge density due to mobile ions ρ(z) =ρc(z). As the right-hand side contains only a sum of Dirac functions, φgas is the sumof fundamental solutions to Poisson’s equation,

(5.3) φgas(z) =

nc∑k=1

qkεIG0(z, zk).

Subtracting (5.3) from (5.2) yields

φrxn(z) =

∫Γ

(εEεI

∂Gκ(z,y)

∂n(y)− ∂G0(z,y)

∂n(y)

)φ(y) + (G0(z,y) −Gκ(z,y))

∂φ(y)

∂n(y)dy

for all z ∈ Ω. The fast multipole method is then used to simultaneously evaluate φrxnat each atomic position zk for the energy computation (5.1).

6. Implementation details. Here we outline the steps in our software pipelinefollowed by a description of the key parameters to the algorithm.

6.1. Data pipeline and software architecture. Given a molecular structure,a force field, and the concentrations of ions in solution, our code computes polarizationenergy in the following steps.

1. Molecular structure preparation. Molecular structures contained in the PDB[14] contain the types and positions of most of the atoms in a molecule. The softwarepackage PDB2PQR [28] places missing hydrogen atoms in the original structure andassigns partial charges and atomic radii based on the force field selected.

2. Molecular surface and triangular surface mesh construction. Based on the po-sitions and radii of the atoms, a molecular surface is constructed through a level-setformulation with software described in [7, 8]. The level-set surface is approximatedas a quality triangular mesh with surface normal directions specified at the verticesusing a dual contouring method [83]. If necessary, this triangular mesh is decimated,and a geometric flow algorithm is applied to improve mesh quality [4].

3. Surface parametrization. The molecular surface is locally parametrized usingthe algebraic spline construction described in section 4.2. Quadrature points arecomputed for each type of integral listed in section 4.4.

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4. Numerical solution. The linear system ((3.12) and (3.13) or the equivalentsystem for the dBIE formulation) is solved using the GMRES routine provided byPETSc (portable, extensible toolkit for scientific computation) [11]. Matrix-vectorproducts are implemented manually using PETSc’s shell matrix construction. Insideeach matrix-vector product, KIFMM3d (kernel-independent fast multipole 3d method)[77] is used to efficiently perform summations for a fixed quadrature rule, and thensingular and near field quadrature rules are used to provide a local correction to theleast accurate portions of the integrals.

5. Energy computation. The polarization energy is computed using the formula-tion in section 5. Numerical integration is again performed using KIFMM3d with localquadrature corrections to singular or nearly singular integrals.

6. User interface and visualization. The Poisson–Boltzmann solver is available aspart of the MolEnergy package [60], which contains a variety of software for molecularenergetics computations [5, 24]. The molecular visualization tool TexMol [6] providesa graphical interface for the algorithm parameters as well as immediate visualizationof the results.

6.2. Algorithm parameters. When running the algorithm, a particular for-mulation must be selected, and a number of parameters must be set. One boundaryintegral formulation (nBIE or dBIE) must be selected, and either piecewise constantor piecewise linear basis functions can be used. Additionally, the following parametersmust be selected:

Ng number of points in the triangular Gaussian quadrature ruleNs number of points in the triangular singular quadrature ruleNns number of subdivisions for the nearly singular quadrature ruleDns depth of triangles for the nearly singular quadrature ruleεtol tolerance for terminating the PETSc GMRES routineNfmm KIFMM3d accuracy parameter

The parameterNfmm is the number of points used by KIFMM3d to represent equiva-lent densities and affects the accuracy of the fast multipole evaluations. KIFMM3d runs

in O(N32

fmm) time.

7. Experimental results. Two types of experiments are considered: a simpleexample with a known solution and realistic protein complexes from the ZDOCK bench-mark [59]. Results are compared to solutions of the linearized Poisson–Boltzmannequations produced by the multigrid finite difference method provided in APBS ver-sion 1.2.1 [10, 44].

7.1. Single ion model. We begin by studying the simplest molecule: a singleatom with radius r and charge q. In this case an explicit solution to the linearizedPoisson–Boltzmann equation is known [49]:

(7.1) φ∗(x) =

⎧⎨⎩

q4πεI |x| +

q4πr

[1

εE(1+κ) − 1εI

], x ∈ Ω,

qe−κ(|x|−r)

εE(1+κr)|x| , x /∈ Ω.

The resulting polarization energy is

(7.2) G∗pol =

q2

8πr

[1

εE(1 + κ)− 1

εI

].

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Table 7.1

Comparison of solution error under several quadrature procedures on the single ion example.Each quadrature scheme is listed as Ng/Ns/Dns. In all cases Nns = 6 and εtol = 10−7. (Ng =3/Ns = 9/Dns = 3): These quadrature rules are sufficiently high order such that quadrature errordoes not impact the expected convergence rates. (3/9/0): Near field quadrature is not necessary topreserve convergence rates. (3/4/0): Errors are slightly larger, but second-order convergence is stillobserved. (1/9/0) and (1/4/0): First-order Gaussian quadrature for far field integration does notyield second-order convergence.

Mesh Quadrature schemeh vertices 3/9/3 3/9/0 3/4/0 1/9/0 1/4/0

4.9e-1 42 5.88e-2 5.91e-2 6.05e-2 6.15e-2 6.29e-22.5e-1 82 1.93e-2 1.94e-2 1.98e-2 2.13e-2 2.18e-21.2e-1 162 5.36e-3 5.38e-3 5.55e-3 6.47e-3 6.64e-36.2e-2 322 1.41e-3 1.41e-3 1.49e-3 1.98e-3 2.06e-33.1e-2 642 3.62e-4 3.65e-4 4.06e-4 6.50e-4 6.90e-41.6e-2 1282 9.54e-5 9.70e-5 1.17e-4 2.38e-4 2.58e-4

Table 7.2

Comparison of the nBIE and dBIE formulations for the single ion model.

εtol 10−5 10−8

nBIE dBIE nBIE dBIEh Error It. Error It. Error It. Error It.

4.9e-1 1.63e-1 5 5.91e-2 6 1.63e-1 10 5.91e-2 82.5e-1 4.47e-2 11 1.93e-2 9 4.47e-2 26 1.93e-2 151.2e-1 1.15e-2 6 5.38e-3 6 1.15e-2 28 5.38e-3 136.2e-2 2.95e-3 4 1.41e-3 6 2.95e-3 34 1.41e-3 133.1e-2 7.49e-4 3 3.65e-4 6 7.58e-4 43 3.65e-4 12

This example is used to test the various parameter settings for MolEnergy. Relativeerror between the exact solution φ∗ and the numerical solution φ is measured in theL2-norm,

(7.3) Error =

√√√√√√∫Γ (φ

∗(y) − φ(y))2+(

∂φ∗(y)∂�n − ∂φ(y)

∂�n

)2

dy∫Γ (φ

∗(y))2 +(

∂φ∗(y)∂�n

)2

dy

.

While derivatives of φ suggest that this expression is more closely related to the H1-norm, φ and ∂φ

∂�n are independent unknowns in the boundary integral formulation. So

(7.3) is the L2-norm of the unknown vector (φ, ∂φ∂�n ).We begin by selecting acceptable quadrature rules. Table 7.1 contains a compar-

ison of the solution error under different quadrature configurations. The simplifiednature of our nearly singular quadrature scheme means that eventually (i.e., when thesize of the triangles in the surface mesh becomes small enough) the quadratic conver-gence rate of the method will be lost. The table demonstrates that if the nonsingularand singular quadrature rules are of high enough order, nearly singular quadraturecan be avoided. In practice we see that the three-point Gaussian quadrature rule fornonsingular integrals and the nine-point (i.e., three by three) rule for singular integralsare sufficient to preserve the convergence rate in the typical ranges that we consider.Higher degree integration rules do not reduce the solution error for the mesh sizeslisted.

Table 7.2 contains a comparison of the nonderivative ((3.8) and (3.9)) and deriva-tive ((3.10) and ((3.11)) boundary integral formulations. In [54] it is reported that

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Table 7.3

Comparing the performance of the algebraic spline molecular surface (ASMS) to a linear ap-proximation of the domain. The exact energy value is −81.450 kcal/mol.

A-spline Linearh L2 Error Energy It. L2 Error Energy It.

4.9e-1 5.90e-2 −75.56 6 5.81e-1 −137.67 42.5e-1 1.94-2 −80.08 9 3.60e-1 −100.73 91.2e-1 5.38e-3 −81.61 6 1.92e-1 −89.37 106.2e-2 1.41e-3 −81.43 6 9.80e-2 −85.01 93.1e-2 3.65-4 −81.44 4 1.65e-3 −83.14 9

Table 7.4

Comparison of APBS and MolEnergy for the single ion example. The exact polarization energyis −81.450 kcal/mol with the interior and exterior dielectric constants 2 and 80.

Solver # degrees Gpol Memory Timename h of freedom (kcal/mol) (mb) (seconds)

APBS

4.0e-1 173 −87.663 1.4 0.562.0e-1 333 −84.476 8.2 1.181.0e-1 653 −82.178 59 8.835.0e-2 1293 −81.831 448 57.732.5e-2 2573 −81.594 3510 426.30

MolEnergy

2.5e-1 82 −80.077 38 5.901.2e-1 162 −81.358 68 13.606.2e-2 322 −81.428 125 46.563.1e-2 642 −81.444 275 203.601.6e-2 1282 −81.449 995 830.09

matrices corresponding to the dBIE formulation are better conditioned for iterativesolvers. We observe this when performing the computation for small εtol. However,for modest εtol values, we find that both formulations terminate in many fewer iter-ations without an impact on the solution error. Since the dBIE formulation requiresfour times as many fast multipole calls as the nBIE formulation (and thus typicallyfour times the runtime), it can be desirable to use the nBIE formulation in certainsituations. This likely explains how the nBIE formulation has been used successfullyby some research groups; see, e.g., [1].

Table 7.3 contains a comparison of a curved A-spline molecular surface and alinear approximation of the geometry. Polarization energy converges at the expectedquadratic rate for the curved geometry and at a linear rate for the linear geometry.Even for very coarse meshes (i.e., before the faster convergence rate has taken effect)the curved geometry performs much better. This is likely due to the hypersingular in-tegrals associated with corners of the polygonal domain. Since the A-spline molecularsurface is differentiable, it produces no hypersingular integrals and thus no associatednumerical problems.

Table 7.4 contains a comparison of our solver with APBS for the single ion example.While the computational time for each method is linear in the number of degrees offreedom, the number of degrees of freedom grows at O(h−3) for the finite differencesolver compared to O(h−2) for the boundary element solver. The finite differencesolver is much more efficient per degree of freedom: this is expected because thelinearity of the fast multipole method involves a larger constant than the local finitedifference computations. The boundary element method gives a more accurate resultwhen compared to finite difference grids with the same length scale.

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Table 7.5

Comparison of MolEnergy and APBS on 212 molecules from the ZDOCK benchmark. Error in theenergy value is computed with respect to the finest mesh using the same solver and reported as apercentage.

Solver MolEnergy APBS

# of DOF 2000 8000 32000 653 1293 2573

Median energy error % 2.72 0.44 - 5.36 3.94 -Max energy error % 32.68 3.58 - 44.06 7.4 -

Median # of iterations 19 22 24 - - -Max # of iterations 78 53 46 - - -Median compute time 37.14 173.45 801.94 13.84 80.29 524.64Median time per iter 1.92 8.12 32.08 - - -Median memory usage 65 150 469 126 535 3577

7.2. Protein binding examples. We focus our experiments on a set of 212ligand-receptor protein complexes from the ZDOCK benchmark [59]. Based on ourexperiments on the single ion model, we choose a conservative parameter set forMolEnergy: Ng = 3, Ns = 9, Dns = 0, εtol = 10−5, and Nfmm = 6. These parametersettings were seen to preserve the expected convergence rates for the single ion modelat small length scales. Atomic charge and radius information is generated using theAMBER 99 force field.

Table 7.5 contains a summary of the results of running MolEnergy and APBS onthe set of test molecules. The runtime of both solvers is observed to be linear in thenumber of degrees of freedom as expected. Error in the energy values is computed withrespect to the energy computed at the finest level. We see that MolEnergy-computedenergy values are more consistent than those computed with APBS.

Note that the median difference between the finest scale MolEnergy and APBS

results is 3.15%. This appears to be much higher than the error in the MolEnergy

computations. Some of this discrepancy is due to the differences in the molecularsurfaces used by the two solvers since the surfaces given to MolEnergy involve somepreprocessing; recall section 4.1. Figure 7.1 contains plots of the energy values com-puted under the different solvers and mesh sizes.

For a more detailed look at the results, we consider the per-atom energy values(i.e., individual terms in the summation (5.1)) for a particular molecule, nuclear trans-port factor 2 (PDB id: 1A2K). Figure 7.2 contains plots of the per-atom energy valuesfor different mesh resolutions. The per-atom energies are consistent, especially be-tween the highest resolution meshes. The median error over all atoms is 0.03 kcal/mol,while the maximum error is 3.29 kcal/mol. Of the 3, 179 atoms, 46 have errors largerthan 1 kcal/mol, and only two atoms have an error larger than 2 kcal/mol. Figure 7.3contains comparisons of per-atom energies resulting from the APBS solver.

Figure 7.4 demonstrates an electric potential computation for a typical proteincomplex. Figures 7.4(a)–(c) depict electric potential of a molecule using differentresolution surfaces meshes. These results can be compared to those produced by APBS

shown in Figure 7.4(d). Figures 7.4(e)–(f) depict the potential computed separatelyfor the two components. Finally, Figure 7.4(g) contains the surface potential for theentire complex. An example of a pentamer (a complex of five identical proteins)from the cucumber mosaic virus is given in Figure 7.5. Each of the five monomersin the pentamer contains 2,486 atoms. The full viral capsid contains 180 monomershighlighting the need for scalable methods to process molecules with millions of atomsand surfaces discretized with a comparable number of triangles.

We demonstrate the need for the derivative boundary formulation and a curved

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(a) MolEnergy, 2,000 DOF vs. 32,000 DOF (b) MolEnergy, 8,000 DOF vs. 32,000 DOF

(c) APBS, 653 DOF vs. MolEnergy, 2, 000 DOF (d) APBS, 2573 DOF vs. MolEnergy, 32, 000 DOF

Fig. 7.1. Scatter plots of polarization energy values computed for 212 proteins using differentsolvers and mesh sizes.

(a) Comparison of per-atom energy with 2,000and 32,000 vertex meshes.

(b) Comparison of per-atom energy with 8,000and 32,000 vertex meshes.

Fig. 7.2. Per-atom polarization energy values are compared for nuclear transport factor 2 (PDBid: 1A2K). Polarization energy is computed using surface meshes with 2,000, 8,000, and 32,000 meshvertices.

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(a) Per-atom energy comparison between a 333-grid cell APBS computation and a 2,000 vertexMolEnergy computation.

(b) Per-atom energy comparison between a2573-grid cell APBS computation and a 32,000vertex MolEnergy computation.

Fig. 7.3. Per-atom polarization energy values are compared for nuclear transport factor 2 (PDBid: 1A2K). Polarization energy is compared between MolEnergy and APBS.

(a) 8,000 DOF (b) 32,000 DOF (c) 113,998 DOF (d) APBS, 2573 DOF

(e) Nuclear Transport Factor 2(NTF), PDB id: 1A2K, chainsA and B.

(f) GTPase Ran, PDB id:1A2K, chains C, D, and E.

(g) GTPase Ran-NTF2 com-plex, PDB id: 1A2K.

Fig. 7.4. The electrostatic potential on the molecular surface for the complex between nucleartransport factor 2 and GTPase Ran (PDB id: 1A2K). In all cases, the potential is between −3.8kbT/ec (red) and +3.8 kbT/ec (blue). Color is available only in the online version. (a)–(c) Electricpotential of the nuclear transport factor 2 molecule using surface meshes containing 8,000, 32,000,and 113,998 vertices. (d) The surface potential computed by APBS. (e)–(f) Electric potential of thetwo component molecules. (g) Electric potential of the molecular complex.D

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(a) (b)

Vertices Energy Runtime11,596 −7, 783 22723,854 −14, 303 31944,644 −13, 265 71371,154 −12, 818 1,133108,868 −12, 554 1,536189,784 −12, 334 4,140

(c)

Fig. 7.5. Pentamer of the cucumber mosaic virus (PDB id: 1F15). Electrostatic potential foreach of the monomers is computed separately (a) and as a single subunit (b). Energy (in kcal/mol)and runtime (in seconds) for the combined subunit is given for several different resolution surfacemeshes.

Table 7.6

Comparison of nBIE and dBIE formulations on 20 example proteins. Error is computed withrespect to the numerical solutions on the same mesh using a much lower GMRES tolerance (10−7).*Computation was halted after 100 GMRES iterations: each computation involving linear geometryreached 100 iterations.

Geometry A-spline A-spline Linearformulation nBIE dBIE dBIE

Median # iterations 40 17 *Max # iterations 47 26 *

Median energy error 11.28 0.12 50.65Max energy error 17.55 0.46 61.92

Table 7.7

Results of polarization energy computation on 20 example proteins when varying Nfmm. Errorin the energy computation is reported as a percentage.

Nfmm 2 4 6 8

Median energy error 0.77 4.21× 10−3 1.24 × 10−4 -Median compute time 316 493 737 1151

approximation of the geometry by comparison to simpler alternatives. For this task weconsidered a set of 20 proteins from the ZDOCK benchmark. The derivative boundaryintegral formulation requires fewer iterations to terminate and for a fixed tolerance εtolgives a more accurate solution. Specifically we compared the dBIE and nBIE formu-lations for a very modest GMRES tolerance, εtol = 10−3. The results are tabulatedin Table 7.6. The dBIE formulation requires noticeably fewer GMRES iterations,but the dBIE formulation requires more computation time because it requires 16 fastmultipole calls per iteration compared to only four required in the nBIE formulation.However, the advantage of the dBIE formulation is seen by looking at the error in theenergy value computed after termination: the dBIE energy values are very near thefinal value for a large εtol, while the nBIE energy values contain substantial error.

The curved representation of the geometry yields a similar, yet more dramatic,impact on the energy computation: the linear geometry computation requires moreGMRES iterations while yielding much poorer energy values.

Table 7.7 contains results of our algorithm on the 20-protein test set when vary-

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ing the fast multipole accuracy parameter Nfmm. Meaningful differences in the finalenergy computation are apparent only for the lowest Nfmm value, and even then thesedifferences are small. In practice, we observe that polarization energy computationsare not very sensitive to the fast multipole accuracy, especially when compared to theeffects of the problem formulation and the molecular surface selection.

8. Final remarks. We have described a complete software pipeline for com-puting the electrostatic potential and polarization energy of biomolecules based onatomic descriptions. Our software is based on general purpose scientific computingcodes PETSc and KIFMM3d and performs favorably against a specialized linearizedPoisson–Boltzmann solver. Our experiments demonstrate the benefits of the dBIEformulation of the Poisson–Boltzmann equation and a smooth representation of themolecular surface when simulating actual proteins.

In a fashion similar to the polarization energy, interior and exterior electrostaticpotential and per-atom forces can also be computed as a postprocess to the Poisson–Boltzmann solver. Integral formulations of the interior and exterior electrostatic po-tential are given in [46], while a derivation of the atomic forces can be found in[34]. Also worth consideration are more detailed models of molecular electrostatics,including an ion exclusion layer surrounding the molecule and regions with differingdielectric constants. Altman et al. [1] formulate a system including these features withrespect to the nBIE formulation, and a similar extension should apply to the dBIEsystem. Moreover, the construction of the ASMS [85] should be useful in generatingparallel surfaces required by the ion exclusion layer by picking different level sets of asingle function over the prismatic scaffold region.

Both PETSc and KIFMM3d are designed for parallel computation [79] and can beapplied to our solution approach. However, for a number of problems, the Poisson–Boltzmann equation must be solved many times; for example, the molecular dockingproblem requires polarization energy to be computed over many potential docked con-figurations. In such cases it is often more natural to find separate Poisson–Boltzmannsolutions in parallel rather than to parallelize the individual computations.

Acknowledgments. The authors wish to thank Prof. Lexing Ying of the Dept.of Mathematics at the University of Texas-Austin for the use of KIFMM3d and severaluseful discussions. Additionally, the authors thank the members of the ComputationalVisualization Center at the University of Texas-Austin for their help in maintain-ing the software environment in which the code was developed (http://cvcweb.ices.utexas.edu/software) and the anonymous referees for insightful and thorough reviewsof the paper.

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An Efficient Higher-Order Fast Multipole Boundary Element Solution for Poisson–Boltzmann-Based Molecular Electrostatics