New versions of image approximations to the ionic solvent induced reaction field

Transcription

1 Computer Physics Communications wwwelseviercom/locate/cpc New versions of image approximations to the ionic solvent induced reaction field Changfeng Xue a Shaozhong Deng b a Department of undamental Sciences Yancheng Institute of Technology Yancheng Jiangsu PR China b Department of Mathematics Statistics University of North Carolina at Charlotte Charlotte NC United States Received 3 July 2007; accepted 3 August 2007 Available online 4 September 2007 Abstract A recent article by Deng Cai [Extending the fast multipole method for charges inside a dielectric sphere in an ionic solvent: High-order image approximations for reaction fields J Comput Phys 2007 doi: 006/jjcp ] introduced two fourth-order image approximations to the reaction field for a charge inside a dielectric sphere immersed in a solvent of low ionic strength To represent such a reaction field the image approximations employ a point charge at the classical Kelvin image point two line charges that extend from this Kelvin image point along the radial direction to infinity with one decaying to zero the other growing to infinity In this paper alternative versions of the fourth-order image approximations are presented using the same point charge but three different line charges all decaying to zero along the radial direction Similar discussions on how to approximate the line charges by discrete image charges how to apply the resulting multiple discrete image approximations together with the fast multipole method are also included 2007 Elsevier BV All rights reserved PACS: 420Cv; 0260-x Keywords: Method of images; Reaction field; Ionic solvent; Hybrid solvent model Introduction Electrostatic interactions play a major role in determining the structure dynamics function of biological macromolecules; as such they remain as major objects of theoretical computational studies of macromolecules Electrostatic interactions are longrange strongly dependent on the solvent the ions surrounding the biomolecule under study When modeling biological systems numerically it has been challenging however to account for such solvent environment in a manner that is computationally efficient physically accurate at the same time Explicit representation of solvent molecules [ 3] offers a detailed accurate description of a biological macromolecule yet all-atom simulations are expensive to perform due to the long-range nature of the electric forces As a result large explicitly solvated systems typically cannot be simulated for biologically relevant timescales Alternatively in implicit solvent models [45] an aqueous solvent is modeled as a continuum medium with a large dielectric constant 60 to 85 outside the macromolecule The macromolecule atoms themselves are explicitly modeled with assigned partial charges embedded in a dissimilar continuum medium of a low dielectric constant to 4 inside the macromolecule volume The neglect of explicit solvent molecules can significantly reduce the computational cost Despite the marked success of implicit solvent models however they also have fundamental limitations due to the fact that the important atomic details of how the solvent molecules interact with the surface of the macromolecule are ignored or these reasons there has been considerable recent interest in developing hybrid explicit/implicit solvent models * Corresponding author Tel: ; fax: address: shaodeng@unccedu S Deng /$ see front matter 2007 Elsevier BV All rights reserved doi:006/jcpc

2 72 C Xue S Deng / Computer Physics Communications [6 8] in which the macromolecule together with a boundary layer or shell of the solvent molecules are considered explicitly within a cavity outside the cavity the solvent is treated implicitly as a dielectric continuum Electric charges within the cavity will polarize the surrounding solvent medium which in turn makes a contribution called the reaction field to the electric field throughout the cavity The electric potential inside the cavity is thus expressed as Φ = Φ S + Φ R Φ S is the potential due to direct Coulomb interactions between source charges within the cavity Φ R is the reaction field ast accurate calculation of such a reaction field will have a far-reaching impact on computational simulations for chemical biological systems involving electrostatic interactions within a solvent In case of a spherical cavity a popular approach is the method of images in which the reaction field is represented in terms of potentials of discrete image charges or the pure water solvent namely with no ions present in the solvent a variety of approaches exist for calculating the reaction field for charges inside the spherical cavity for example the high-order accurate multiple image approximation [9] references therein or an ionic solvent in [0] by assuming that the ionic strength of the solvent is low enough so the product of the inverse Debye screening length of the solvent the radius of the spherical cavity is less than one a first- a second-order image approximation to the ionic solvent induced reaction field were developed Both approaches employ the same point line charges as those obtained for the pure water solvent while the ionic strength effect is included in additional correction terms Two fourthorder image approximations were further developed in [] using a point charge at the classical Kelvin image point two line charges that extend from this Kelvin image point along the radial direction to infinity with one decaying to zero the other growing to infinity In this paper new versions of the fourth-order image approximations to the ionic solvent induced reaction field shall be developed so that only line charges decaying to zero are utilized The paper is organized as follows In Section 2 after briefly reviewing the series solution to the ionic solvent induced reaction field due to a point charge inside a dielectric sphere we present the new versions of the fourth-order image approximations to such the reaction field Then in Section 3 the discretization of the line charges by point image charges is summarized Numerical results are given next in Section 4 to validate the convergence property investigate the efficiency of the new versions of the fourth-order image approximations In addition how to apply the proposed multiple discrete image approximations together with the fast multipole methods is discussed in Appendix A 2 Image approximations to ionic solvent induced reaction fields By linear superposition the reaction field due to a single source charge q inside a spherical cavity centered at the origin only needs to be considered Without loss of generality let us consider a dielectric sphere of radius a immersed in an ionic solvent The sphere has the dielectric constant ɛ i the surrounding ionic solvent is represented as a homogeneous dielectric continuum of the dielectric constant ɛ o The point charge q is located on the x-axis inside the sphere at a distance r S <afrom the center of the sphere as shown in ig Inside the sphere the electrostatic potential Φ is given by the solution to the Poisson s equation ɛ i Φr = qδ r r S δr is the Dirac delta function Outside the sphere on the other h by assuming that the mobile ion concentration in the ionic solvent is given by the Debye Hückel theory for a solvent of low ionic strength the linearized Poisson Boltzmann equation [23] 2 Φr λ 2 Φr = 0 2 ig A point charge a dielectric sphere immersed in an ionic solvent

3 C Xue S Deng / Computer Physics Communications ig 2 Illustration of the reaction field inside a spherical cavity of radius in the case that r S = 08 λ = 08 ɛ i = 2 ɛ o = 80 can be used to approximate the screened Coulomb potential in the solution Here λ is the inverse Debye screening length defined by λ 2 = 8πN Ae 2 I 000ɛ o k B T N A is Avogadro s number e is the electron charge k B is the Boltzmann constant T is the absolute temperature I is the ionic strength of the solvent Using the classical electrostatic theory the reaction field of the spherical dielectric can be solved analytically [0] Moreprecisely with respect to a spherical coordinate system r θ φ originating in the center of the sphere the pole is denoted by the x-axis in this paper due to the azimuthal symmetry the reaction field at an observation point r = rθφinside the sphere is Φ R r = A n r n P n cos θ n=0 P n x are the Legendre polynomials A n are the expansion coefficients given by A n = q r n K ɛ i n + k n u + ɛ o uk n u ɛ i nk n u ɛ o uk n u n 0 Here u = λa = a 2 /r S with = 0 0 denoting the conventional Kelvin image point k n r are the modified spherical Hankel functions defined as [45] k n r = π n n + k! 5 2r e r n 0 k!n k! 2r k k=0 In theory any desired degree of accuracy can be obtained using the direct series expansion 3 to calculate the ionic solvent induced reaction field In the case that the point charge is close to the spherical boundary when calculating the reaction field at an observation point also close to the boundary however the convergence by the series expansion is slow due to r/ = rr S /a 2 requiring a great number of terms in the series expansion to achieve high accuracy in the reaction field ig 2 plots the reaction field on a disk inside a spherical cavity of radius in the case that r S = 08 λ = 08 ɛ i = 2 ɛ o = 80 Let us now consider the problem of finding images outside the spherical region giving rise to the reaction potential inside the sphere As mentioned earlier in [] by assuming that the ionic strength of the solvent is low enough so the product of the inverse Debye screening length the radius of the spherical cavity is less than one u = λa < two fourth-order image approximations in terms of u = λa to the ionic solvent induced reaction field were developed Both approximations employ a point charge two line charges with one decaying to zero the other growing to infinity Next we shall follow a similar methodology to develop alternative versions of the fourth-order image approximations As discussed in detail in [] the assumption of u = λa < is physically justifiable since the condition of low ionic strength is required for the linearization of the Poisson Boltzmann equation [67] to be meaningful In [] the construction of the fourth-order image approximations is based on the fact that the modified spherical Hankel functions their derivatives can be exped respectively in terms of /r as k n r = π 2n 2! n! 22n 2r n+ 42r n + O r n 3 n

4 74 C Xue S Deng / Computer Physics Communications k n 2n 2! 42n n + r = π n! 2r n+2 n 22r n + O r n 2 n 2 Correspondingly we have k n r 2 + r 2 + 4n = r r r 2 n + 4n 2 + O r 4 n 2 rk n Inserting this approximation in 4 leads to A n = q we denote r n K a + a 2 n + a 3 n 2 b + b 2 n + b 3 n 2 + O u 4 n 2 a = ɛ i + ɛ o u 2 2ɛ i ɛ o a 2 = ɛ i ɛ o 2 u 2 a 3 = 4ɛ i ɛ o b = ɛ o 2 u 2 b 2 = ɛ i + ɛ o u 2 2ɛ i ɛ o b 3 = 4ɛ i + ɛ o After some algebraic manipulation the expansion coefficients A n can be further expressed as γ + + O u 4 n 2 A n = q r n K γ = ɛ i ɛ o ɛ i + ɛ o r n K α + α 2 n β + β 2 n + n 2 a α = γ b a2 α 2 = γ b 2 β = b β 2 = b 2 a 3 b 3 a 3 b 3 b 3 b 3 Note that β < 0 when 0 u< Then using partial fractions gives us A n = q γ + δ + δ 2 + O u 4 n 2 n + σ n σ 2 β2 2 σ = 4β + β 2 2 β2 2 σ 2 = 4β β 2 2 δ = α 2σ α σ + σ 2 δ 2 = α 2σ 2 + α σ + σ 2 It can be shown that 0 <σ < 0 <σ 2 < Moreover /2 <σ when ɛ i <ɛ o /5 σ 2 < /2 when ɛ i <ɛ o σ 2 = /2 when ɛ i = ɛ o σ 2 > /2 when ɛ i >ɛ o Therefore for our target applications in hybrid solvent models for biomolecular simulations the typical dielectric constants are to 4 for nd 60 to 85 for ɛ o respectively we have /2 <σ < 0 <σ 2 < /2 On the other h for n applying the exact expressions of k 0 r k r in fact we can arrive at A 0 = C 0 u q C 0 u = ɛ i + uɛ o + uɛ o A = C u q C u = 2 + uɛ i 2 + 2u + u 2 ɛ o + uɛ i u + u 2 ɛ o Inserting now the approximation of A n given by 7 into 3 the reaction field inside the sphere is taken as 7 Φ R r = A 0 + A r cos θ + S + S 2 + S 3 + O u 4 with S S 2 S 3 representing three series defined respectively by S = γq r n P n cos θ 8

5 S 2 = δ q S 3 = δ 2q r n P n cos θ n + σ r n P n cos θ n σ 2 C Xue S Deng / Computer Physics Communications To find image representations of the three series we first recall that the Coulomb potential Φ S r the potential at r due to a point charge at r S can be exped in terms of the Legendre polynomials of cos θ as follows [8] q n rs P q n cos θ r S r a Φ S r = 9 ɛ i r r S = ɛ i r r n=0 q r n P n cos θ 0 r r S ɛ i r S r S n=0 Then the first series S can be written as S = γq a r n P n cos θ γq + r cos θ ɛ i r S n=0 the first part of the right-h side is exactly the expansion given by 9 for a point charge of magnitude q K = γaq/r S outside the sphere at the classical Kelvin image point = 0 0 namely q K S = 0 ɛ i r γq + r cos θ Next using the integral identity n + σ = rn+σ K x n+σ + dx which is valid for all n 0 when σ>0 the second series S 2 can be written as [ q L x r n S 2 = P n cos θ] dx δ q + r cos θ ɛ i x x σ + σ q L x = δ q a n=0 x σ x Note that the integr of the above integral again is the expansion given by 9 for a charge of magnitude q L x outside the sphere at the point x = x 0 0 Therefore q L x can be regarded as a continuous line charge that extends from the classical Kelvin image point along the radial direction to infinity Thus the second series S 2 becomes S 2 = q L x ɛ i r x dx δ q + r cos θ σ + σ However an image representation similar to 3 in which the line charge decays to zero along the radial direction cannot be obtained for the third series S 3 by following the same methodology Instead in [] this series is represented using a line charge that grows to infinity In this paper we shall develop a new approach in which only line charges that decay to zero are involved To this end we first use the recursion formula P n x = 2n xp n x n n n P n 2x n 2 to rewrite the third series as S 3 = S 4 S 5 with the two series S 4 S 5 being defined as S 4 = q 2n δ 2 nn σ 2 r n cos θp n cos θ S 5 = q n δ 2 r nn σ 2 n P n 2cos θ 2 3

6 76 C Xue S Deng / Computer Physics Communications Now using partial fractions again leads to 2n δ 2 nn σ 2 = δ 2 n + δ 22 n σ 2 n δ 2 nn σ 2 = δ 2 n + δ 23 n σ 2 we denote δ 2 = δ 2 σ 2 δ 22 = 2σ 2 δ 2 σ 2 δ 23 = σ 2 δ 2 σ 2 Then the series S 4 can be written as S 4 = S 6 + S 7 S 6 = q δ 2 r n cos θp n cos θ= q r cos θ δ 2 r m P m cos θ n m + S 7 = q δ 22 n σ 2 r n cos θp n cos θ= q r cos θ δ 22 r m P m cos θ m + σ 2 Now applying the integral identity with σ = σ = σ 2 respectively gives us r q L2 x S 6 = ɛ i r x dx δ 2q cos θ Here S 7 = q L22 x ɛ i r x dx δ 22 q σ 2 q L2 x = δ 2q x x a r cos θ q L22 x = δ 22q x σ2 x a Combining 4 5 yields r q L2 x S 4 = ɛ i r x dx q δ 2 cos θ σ 2 q L2 x = q L2 x + q L22 x = δ 2q a [ x x σ2 ] + 2σ 2 Analogously the series S 5 can be written as S 5 = S 8 + S 9 S 8 = q δ 2 r n P n 2cos θ= q r 2 δ 2 r n m + 2 S 9 = q δ 23 n σ 2 r m=0 m P m cos θ n P n 2cos θ= q r 2 δ 23 r m P m cos θ m + 2 σ 2 m=

7 C Xue S Deng / Computer Physics Communications Applying the integral identity with σ = 2 σ = 2 σ 2 respectively gives us S 8 = S 9 = q L3 x r ɛ i r x dx q L32 x r ɛ i r x dx 2 2 the functions q L3 x q L32 x are 7 8 q L3 x = δ 2q a x 2 x q L32 x = δ 23q x 2 σ2 x a Combining 7 8 yields q L3 x r S 5 = ɛ i r x dx 2 q L3 x = q L3 x + q L32 x = δ 2q a [ x 2 x 2 σ2 ] + σ 2 Combining we finally have a fourth-order image approximation to the ionic solvent induced reaction field as follows Φ R r = q K ɛ i r + q L x ɛ i r x dx + + Φ C + Φ C2 r + O u 4 q L2 x r ɛ i r x dx Φ C is a position-independent correction potential given by Φ C = q C 0 u γ δ σ on the other h Φ C2 r defines a position-dependent correction potential Φ C2 r = q C u γ δ δ 2 r cos θ + σ σ 2 q L3 x r cos θ ɛ i r x dx Likewise q L2 x q L3 x can be regarded as continuous line charges extending from the classical Kelvin image point along the radial direction to infinity Asymptotically all three line charges q L x q L2 x q L3 x decay to zero faster than x /2 at infinity More precisely asymptotically q L x decays to zero as rapidly as x σ q L2 x as x σ 2 q L3 x as x 2 σ 2 respectively ig 3 shows the density distributions of the image line charges q L x q L2 x q L3 x for a source point charge q = atr S = 08 onthex-axis in the case of ɛ i = 2 ɛ o = 80 a = λ = 08 As pointed out in [] the accuracy of the fourth-order image approximation 20 can be improved by including more correction terms In particular for n = 2 from the exact expression of k 2 r in fact we can arrive at A 2 = C 2 u q r 2 K C 2 u = u + u2 ɛ i 9 + 9u + 4u 2 + u 3 ɛ o u + u 2 ɛ i u + 4u 2 + u 3 ɛ o

8 78 C Xue S Deng / Computer Physics Communications ig 3 Density distributions of the image line charges q L x q L2 x q L3 x for a source point charge q = atr S = 08 in the case of ɛ i = 2 ɛ o = 80 a = λ = 08 Now instead of 8 we express the reaction field inside the sphere as Φ R r = A 0 + A r cos θ + A 2 r 2 P 2 cos θ+ n=3 q r n K γ + δ + δ 2 r n P n cos θ+ O u 4 n + σ n σ 2 Note that the only difference between 8 23 is in the calculation of the r 2 P 2 cos θ-terms Therefore the fourth-order image approximation 20 can be improved simply by including another position-dependent correction potential resulting in an improved fourth-order image approximation as follows Φ R r = q K ɛ i r + q L x ɛ i r x dx + + Φ C + Φ C2 r + Φ C3 r + O u 4 q L2 x r ɛ i r x dx the second position-dependent correction potential is Φ C3 r = q C 2 u γ δ δ 2 r 2 P 2 cos θ 2 + σ 2 σ 2 3 ourth-order multiple discrete image approximations q L3 x r cos θ ɛ i r x dx Now let us consider how to represent approximately each image line charge involved in the image approximations by a set of discrete image charges This can be achieved by approximating the involved integrals by numerical quadratures Without losing any generality let us consider I = x σ fx dx σ>0 irst by introducing the change of variables /x = s/2 τ with τ>0 we have I = 2 τσ τ s α hs τ ds α = τσ hs τ = 2τ 2 τ s τ f s τ 26 27

9 C Xue S Deng / Computer Physics Communications Note that α> because σ>0 τ>0 Also s = corresponds to the Kelvin image point x = Therefore we can choose either Jacobi Gauss or Jacobi Gauss Radau quadrature to approximate the integral in 27 More precisely let s m ω m m = 2M be the Jacobi Gauss or Jacobi Gauss Radau points weights on the interval [ ] with α = τσ β = 0s = if Jacobi Gauss Radau quadrature is used which can be obtained with the program ORTHPOL [9] Then we have M I 2 τσ τ ω m x m fx m 28 for m = 2M 2 τ x m = s m The parameter τ>0 in the change of variables /x = s/2 τ can be used as a parameter to control the accuracy of numerical approximations When τ = /σ we have α = 0 in this case the quadrature given by 28 simply reduces to the usual Gauss or Gauss Radau quadrature 3 Discretization of q L x 29 Note that q L x ɛ i r x dx = δ q x r x σ dx Recall that 0 <σ < furthermore when ɛ i <ɛ o /5 we have σ > /2 Then using 28 with σ = σ leads to q L x M ɛ i r x dx qm L ɛ i r x m for m = 2M q L m = 2 τσ τδ ω m x m a q Discretization of q L2 x Discretization of q L2 x can be carried out by discretizing q L2 x q L22 x separately in the same way as how q L x is discretized which in general will introduce a total of 2M discrete charges even when the same τ valueisusedtodiscretizeq L2 x q L22 x In order to obtain an approach that uses only M discrete charges we reformulate q L2 x as q L2 x = δ 2q a [ + 2σ 2 x σ2 ] x or q L2 x = δ [ 2q x σ2 ] x σ2 + 2σ 2 a Then using 28 with σ = orσ = σ 2 >/2 leads to q L2 x M ɛ i r x dx qm L2 ɛ i r x m for m = 2M 32 or q L2 m = 2 τ τδ 2 ω m [ + 2σ 2 q L2 m xm σ2 ] xm a q [ σ2 ] = 2 τ σ2 xm xm τδ 2 ω m + 2σ 2 a q 33 34

10 80 C Xue S Deng / Computer Physics Communications Discretization of q L3 x Discretization of q L3 x can be carried out in the same way as how q L2 x is discretized or instance to obtain an approach that uses only M discrete charges we reformulate q L3 x as q L3 x = δ [ 2q x σ2 ] x 2 + σ 2 a or q L3 x = δ [ 2q x σ2 ] x 2 σ2 + σ 2 a Then using 28 with σ = 2orσ = 2 σ 2 >3/2 leads to q L3 x M ɛ i r x dx qm L3 ɛ i r x m for m = 2M 35 or q L3 m = 2 2τ τδ 2 ω m [ + σ 2 xm σ2 ] xm a q 36 q L3 m [ σ2 ] = 2 τ2 σ2 xm xm τδ 2 ω m + σ 2 a q In conclusion in general we have the following fourth-order multiple discrete image approximations to the reaction potential inside the sphere a A fourth-order multiple discrete image approximation: Φ R r q M K ɛ i r + qm L ɛ i r x m + + Φ C + Φ C2 r M q L2 m ɛ i r x m r M cos θ b An improved fourth-order multiple discrete image approximation: q M K Φ R r ɛ i r + qm L M ɛ i r x m + qm L2 r ɛ i r x m cos θ 4 Numerical results + Φ C + Φ C2 r + Φ C3 r M qm L3 r 2 ɛ i r x m qm L3 r 2 ɛ i r x m or demonstration purpose let us consider a unit dielectric sphere The dielectric constants of the sphere its surrounding medium are assumed to be ɛ i = 2 ɛ o = 80 respectively The results obtained by the direct series expansion with 400 terms are treated as the exact reaction fields to calculate the errors of various image approximations Unless otherwise specified 3 with τ = /σ 34 with τ = / σ 2 37 with τ = /2 σ 2 are used to approximate the image line charges q L x q L2 x q L3 x respectively for simplicity we always choose the same M value so that the same Gauss quadrature points weights s m ω m are involved We begin by verifying the convergence property of the proposed line image approximations To this end let us consider a single point charge located on the x-axis inside the sphere at a distance r S = 095 from the center of the sphere A large M value M = 50 is used to discretize the line charges so that the error arising in the discretization of the image line charges by discrete charges is negligible compared to the error arising in the approximation of the reaction field by the point image line charges or each selected value of u = λa we calculate the relative error of the image approximations in the reaction field respectively at 0000 observation points uniformly distributed under the polar coordinates within the unit disk that contains the x-axis The maximal relative error E at the 0000 observation points for various u values the corresponding order of convergence are shown in Table As can be observed the results clearly demonstrate the Ou 4 convergence property of the new versions of the fourth-order image approximations One natural concern with the proposed multiple discrete image approximations is the final number of discrete image charges required to achieve certain order of degree of accuracy or a desired accuracy this number depends on both the locations of the

11 C Xue S Deng / Computer Physics Communications Table Convergence property of the proposed image approximations u 4th-order Improved 4th-order E Order E Order E 4 68E E E E E E E E E ig 4 Accuracy of the improved fourth-order image approximation to the ionic solvent induced reaction field due to a source charge inside the unit sphere at distance r S from the center The total number of discrete image charges is 3M + source charge the ionic strength It should be small if compared to the number of terms needed to achieve the same order of degree of accuracy in the direct series expansion to make the image approximations useful in the practice In this test we shall confine ourselves to the improved fourth-order image approximation we consider four different source locations r S = as well as different u = λa values ranging from 005 to 095 or each selected source position u value we approximate the reaction fields at the same 0000 points within the sphere first by the direct series expansion with various numbers of terms then by the improved fourth-order image approximation with various numbers of discrete charges The results are compared to the exact ones obtained by the direct series expansion with 400 terms the maximal relative errors are plotted in ig 4 irst of all as can be seen the approximation error increases as the source charge moves to the spherical boundary Second as pointed out earlier in the case that the source charge is close to the spherical boundary the Kelvin image point is close to the boundary as well Therefore when calculating the reaction field at an observation point also close to the spherical boundary the

12 82 C Xue S Deng / Computer Physics Communications convergence by the direct series expansion Φ R r = q ɛ i n + k n u + ɛ o uk n u r n ɛ i nk n u ɛ o uk n u P n cos θ n=0 shall be slow due to r/ = rr S /a 2 requiring a great number of terms to achieve high accuracy in the reaction field As indicated to achieve an accuracy with the error being less than 0 2 around terms have to be included in the direct series expansion for the cases of r S = respectively To achieve the same accuracy however for all source locations all u 095 only 4 discrete charges including the point charge at the Kelvin image point M = are needed in the multiple discrete image approximation Similarly as can be seen to achieve an accuracy with the error being less than 0 3 around terms have to be included in the direct series expansion for the cases of r S = respectively On the other h to achieve the same accuracy for relatively small relatively large u values u<05 u> discrete charges M = 2 M = 3 are needed in the multiple discrete image approximation respectively Therefore to calculate reaction fields at points close to the spherical boundary due to source charges also close to the boundary the improved fourth-order image approximation is clearly much more efficient than the method of direct series expansion 5 Conclusions In this paper we have presented new versions of the fourth-order image approximations to the reaction field due to a point charge inside a dielectric sphere immersed in an ionic solvent Compared to the original ones which employ two image line charges with one decaying to zero the other growing to infinity the new versions utilize three image line charges all decaying to zero along the radial direction Numerical results have demonstrated the efficiency of the new versions of the fourth-order image approximations compared to the direct series expansion Acknowledgements C Xue thanks the support of the National Natural Science oundation of China grant number: S Deng thanks the support of the National Institutes of Health grant number: R0GM for the work reported in this paper The authors also thank Dr Wei Cai for many interesting discussions during this work Appendix A ast calculation of the image approximations The main purpose for discrete image approximations to reaction fields is to apply existing fast algorithms such as the precorrected T [20] or the fast multipole methods MMs [2 25] directly in calculating the electrostatic interactions among N charges inside the spherical cavity in ON log N or even ON operations In particular below we give a straightforward but far from optimal MM-based ON implementation of the discrete image approximations or convenience let r i = xi y i z i i = 2NbeN observation points rs j = xs j ys j zs j j = 2Nbethe locations of N source charges with charge strengths q q 2 q N By linear superposition the reaction field at an observation point r i in the case that the improved fourth-order image approximation 39 is employed becomes N [ Φ R r q Kj M i ɛ i r i j + qmj L r ɛ i r i x mj + M i cos θ ij j N + Φ Cj + Φ C2j r i + Φ C3j r i qmj L2 r 2 ɛ i r i x mj M i j qmj L3 ] ɛ i r i x mj θ ij is the angle between r i r S j a quantity with a second subscript j designates a quantity associated with the source charge r S j such as A q Kj = γ a rj S q j j = a2 2 τ rj S x mj = j s m A Calculation of the correction potentials in ON operations Obviously the position-independent correction potential q j Φ Cj = C 0 u γ δ σ

13 C Xue S Deng / Computer Physics Communications can be evaluated in ON operations The evaluation of the second correction potential N Φ C2j r q j i = C u γ δ δ 2 r i cos θ ij + σ σ 2 j in ON operations however needs special treatment due to its position-dependence Expressing the cosine of the angle θ ij between r i r S j in terms of their rectangular coordinates we get N Φ C2j r i = c q j r i j x i x S j + y i ys j + z i zs j r i rs j c = C u γ δ δ 2 + σ σ 2 Rearranging terms in A2 leads to Φ C2j r i = d xi + d 2 yi + d 3 zi d = c a 2 q j x S j d 2 = c a 2 i = 2N q j y S j d 3 = c a 2 q j zj S Note that d d 2 d 3 depend just on the source charges each can be calculated in ON operations Then it is clear that after obtaining d d 2 d 3 the calculation of A3 for all observation points can be evaluated in ON operations Analogously the third correction term can also be evaluated in ON operations In fact using P 2 cos θ= 3 cos 2 θ /2 we get Φ C3j r i = e x 2 i + e2 y 2 i + e3 z 2 i + e4 xi y i + e 5 xi z i + e 6yi z i + e 7 r 2 i i = N e k k = 7 dependent just on the source charges are gievn by e = 3c 2 2a 4 e 4 = 3c 2 a 4 e 7 = c 2 2a 4 q j x S j 2 e2 = 3c 2 2a 4 q j x S j ys j e 5 = 3c 2 a 4 q j r S 2 j c 2 = C 2 u γ δ δ σ 2 σ 2 q j y S j 2 e3 = 3c 2 2a 4 q j x S j zs j e 6 = 3c 2 a 4 A2 Calculation of the potentials of the image charges in ON operations q j z S 2 j q j yj S zs j The MMs are known to be extremely efficient in the evaluation of pairwise interactions in large ensembles of particles such as expressions of the form Φr i = j i q j i = 2N r i r j for the electrostatic potential r r 2 r N are points in R 3 q q 2 q N are the corresponding charge strengths or instance the adaptive MM of [25] requires ON work breaks even with the direct calculation at about N = 750 for three-digit precision N = 500 for six-digit precision N = 2500 for nine-digit precision respectively [26] Using such an adaptive MM A2 A3

14 84 C Xue S Deng / Computer Physics Communications with ON computational complexity the calculation of the potentials of the discrete image charges for all observation points can be evaluated in ON operations in a straightforward way Such evaluation can be carried out with five MM runs The first run includes all point charges q Kj at the corresponding Kelvin image points j all discrete image point charges qmj L of q Lx at x mj to calculate q Kj ɛ i r i j + M qmj L ɛ i r i x mj In the case that the total potential is to be calculated all original source charges are also included in the MM computational box And all charges are taken as acting in a homogeneous medium of the dielectric constant ɛ i To calculate the third term in A we write it as [ r i = j [ r i = x i cos θ ij j M M q L2 mj ɛ i r i x mj r i rs j ] x i xj S + y i ys j + z i zs M j qmj L2 ] ɛ i r i x mj q L2x mj ɛ i r i x mj + y i M q L2y mj ɛ i r i x mj + z i M q L2z mj ɛ i r i x mj q L2x mj = xs j mj a 2 ql2 ql2y mj = ys j a 2 ql2 mj Then each double summation involved such as M qmj L2x ɛ i r i x mj i = 2N ql2z mj = zs j a 2 ql2 mj can be evaluated in ON operations with a MM run by including in the MM box all discrete image charges qmj L2x ql2y mj orql2z mj at x mj That being done the third term can then be evaluated in another ON operations Similarly the fourth term in A can be written as [ r i j 2 M r S 2 qmj L3r = j a 2 qmj L3 q L3 mj ɛ i r i x mj ] = ri N 2 M q L3r mj ɛ i r i x mj The double summation for i = 2N can be calculated in ON operations with the last MM run by including all discrete image charges qmj L3r at x mj in the MM box After that the fourth term can be evaluated simply in an additional ON operations References [] P Koehl Electrostatics calculations: latest methodological advances Curr Opin Struct Biol [2] RM Levy E Gallicchio Computer simulations with explicit solvent: Recent progress in the thermodynamic decomposition of free energies in modeling electrostatic effects Annu Rev Phys Chem [3] C Sagui TA Darden Molecular dynamics simulation of biomolecules: Long-range electrostatic effects Annu Rev Biophys Biomol Struct [4] M eig CL Brooks III Recent advances in the development application of implicit solvent models in biomolecule simulations Curr Opin Struct Biol [5] NA Baker Improving implicit solvent simulations: A Poisson-centric view Curr Opin Struct Biol [6] A Okur C Simmerling Hybrid explicit/implicit solvation methods in: D Spellmeyer Ed Annu Rep Comput Chem vol Chapter 6 [7] MS Lee R Salsbury Jr MA Olson An efficient hybrid explicit/implicit solvent method for biomolecular simulations J Comput Chem [8] MS Lee MA Olson Evaluation of Poisson solvation models using a hybrid explicit/implicit solvent method J Phys Chem B [9] W Cai S Deng D Jacobs Extending the fast multipole method to charges inside or outside a dielectric sphere J Comput Phys

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