Fast Analytical Method for Approximate Born Radii: New Approach for GB/SA Solvation, Notas de estudo de Engenharia Elétrica

Baixe Fast Analytical Method for Approximate Born Radii: New Approach for GB/SA Solvation e outras Notas de estudo em PDF para Engenharia Elétrica, somente na Docsity! The GB/SA Continuum Model for Solvation. A Fast Analytical Method for the Calculation of Approximate Born Radii Di Qiu, Peter S. Shenkin, Frank P. Hollinger, and W. Clark Still* Department of Chemistry, Columbia UniVersity, New York, New York 10027 ReceiVed: July 8, 1996; In Final Form: October 7, 1996X Atomic Born radii (R) are used in the generalized Born (GB) equation to calculate approximations to the electrical polarization component (Gpol) of solvation free energy. We present here a simple analytical formula for calculating Born radii rapidly and with useful accuracy. The new function is based on an atomic pairwise rij-4 treatment and contains several empirically determined parameters that were established by optimization against a data set of >10 000 accurate Born radii computed numerically using the Poisson equation on a diverse group of organic molecules, molecular complexes, oligopeptides, and a small protein. Coupling this new Born radius calculation with the previously described GB/SA solvation treatment provides a fully analytical solvation model that is computationally efficient in comparison with traditional molecular solvent models and also affords first and second derivatives. Tests with the GB/SA model and Born radii calculated with our new analytical function and with the accurate but more time-consuming Poisson-Boltzmann methods indicate that comparable free energies of solventlike dielectric polarization can be obtained using either method and that the resulting GB/SA solvation free energies compare well with the experimental results on small molecules in water. I. Introduction The accurate modeling of molecules in solution using molecular mechanics is a challenging problem because solvent is an extended medium having an astronomical number of low- energy states. To treat such a medium in a molecular calcula- tion, both molecular1-3 and continuum4-10 models of solvent have been developed. Molecular solvent models employ hundreds or thousands of discrete solvent molecules and provide the most widely used method for carrying out simulations in liquid environments. Though many of the properties of solu- tions and solutes have been reproduced using calculations employing molecular solvent models, such calculations converge only slowly to precise answers because of the large number of particles and states involved. In fact, molecular solvent calculations generally require several orders of magnitude more CPU time than corresponding gas phase calculations on the same solute. Because molecular solvent models are so computation- ally demanding, we and others have a significant interest in developing more rapid continuum solvation models. Continuum models treat the solvent as a continuous medium having the average properties of the real solvent and surrounding the solute beginning at or near its van der Waals surface. In principle, such models can provide solvation effects with relatively little computational effort, because the properties of an analytical, continuum solvent are converged by nature and because the model includes no particles other than the atoms of the solute. A variety of continuum solvation models have been described over the years. Among these, treatments based on surface area or solvent accessible surface area have been recurring themes.11-13 As a method for evaluating the total solvation free energy (Gsol), however, we were concerned that area-based representations would provide poor approximations of the long range electro- static components of solvation. In particular, purely area-based treatments are problematic in that they give constant solvation energies for all arrangements of ions or other charged atoms having nonintersecting solvent-accessible surfaces. Another popular approach to continuum solvation treats a solute as a distribution of charges or electrical multipoles in a cavity in a dielectric continuum.14-17 Depending on the model, the cavity may accurately follow the van der Waals surface of the solute or it may be a simple geometrical object such as an ellipsoid that approximates the shape of the solute. These models allow one to compute approximations to a significant (in high dielectric solvents) component of solvation energy, the electrostatic solvent polarization energy (Gpol). While such continuum solvation models are computationally efficient, calculating derivatives of Gpol with respect to solute atom movement (e.g., for energy minimizations or dynamics calculations) including the effect of cavity boundary fluctuations is computationally intensive and has not been widely used. Furthermore, such dielectric con- tinuum models of the solvent do not include solvent-solvent cavity terms (Gcav) or attractive van der Waals solvent-solute interaction terms (GvdW). Because of the shortcomings of previous models and because we needed a practical solvation model for molecular mechanics and dynamics calculations requiring derivatives, several years ago we developed a new continuum solvation model (termed the GB/SA model) that provided solvation free energies (Gsol) based on a generalized Born (GB) treatment of Gpol and surface areas (SA) for approximating the cavity and van der Waals contributions to solvation.9 In the GB/SA model, the total solvation free energy (Gsol) is given as the sum of a solvent-solvent cavity term (Gcav), a solute-solvent van der Waals term (GvdW), and a solute-solvent electrostatic polarization term (Gpol): Because saturated hydrocarbons are nonpolar molecules (Gpol ∼ 0) and their Gsol in water is approximately linearly related11-13 to their solvent accessible surface areas (SA), the GB/SA model computes Gcav + GvdW together by evaluating solvent-accessible surface areas:4,5X Abstract published in AdVance ACS Abstracts, April 1, 1997. Gsol ) Gcav + GvdW + Gpol (1) 3005J. Phys. Chem. A 1997, 101, 3005-3014 S1089-5639(96)01992-5 CCC: $14.00 © 1997 American Chemical Society where SAk (Å2) is the total solvent-accessible surface area of all atoms of type k and σk (kcal/(mol Å2)) is an empirically determined atomic solvation parameter. For hydrophobic atoms in water and a solvent-accessible surface lying 1.4 Å outside the van der Waals surface, σ has the value of ∼0.01 kcal/(mol Å2). In the work described here, solvent accessible surface areas were computed numerically. For Gpol (kcal/mol), we began with the generalized Born equation and modified it to allow for application to irregularly shaped solutes: where Rij (Å) ) (RiRj)0.5 and Dij ) rij2/(2Rij)2 and the double sum runs over all pairs of atoms (i and j). Ri is the so-called Born radius of atom i (see below). Dij is the squared ratio of the i,jth atom pair separation to their mean Born diameters, and its exponential is used to force Gpol to approximate the dielectric part of Coulomb’s law rapidly as atoms i and j move beyond the contact distance of their Born radii. This model has been modified by Truhlar and co-workers and successfully used in conjunction with semiempirical molecular orbital calculations.10 Although eq 3 is a simple, pairwise expression, it requires a Born radius (R) for each atom in the solute having an atomic charge (or partial charge). For a simple spherical solute with a charge located at its center (e.g., a model for a metal ion), R can simply be taken as the van der Waals radius of the solute. But for more complex solutes, the Born radius of the ith atom (Ri) depends upon the positions and volumes of all other atoms in the solute because they displace the solventlike dielectric medium. The Born radius of a charged particle is actually not so much a radius as it is a kind of average distance from the atomic charge to the boundary of the dielectric medium. For certain simple systems, the value of R is thus obvious. For example, R for an atom at the center of a spherical macromol- ecule would be the radius of the macromolecule. For systems having irregular shapes, however, R is more complicated to evaluate. In previous descriptions of the GB/SA model, R for such solutes has been obtained by a numerical, finite difference method based on the Born equation.9 While this numerical evaluation provided well-defined and reasonably accurate evalu- ations of R’s, it was also the most time-consuming part of the GB/SA solvation calculation (eq 3). Furthermore, because of the numerical nature of the previous R evaluation, full deriva- tives of Gpol were not readily obtained. In this paper, we describe a significant enhancement to the GB/SA solvation model in the form of a fast, analytical approach to computing atomic Born radii. Though our analytical ap- proach to R is not exact, we show here that it yields Born radii that compare reasonably well with accurate R’s calculated numerically. Furthermore, in conjunction with the GB/SA model for water, experimental solvation energies are well reproduced by GB/SA calculations using R’s computed either by our rapid approximate method or by a slower but accurate numerical method. Because the new approach to computing Born radii makes the GB/SA solvation model fully analytical, we implemented it several years ago with full first and second derivatives as an unpublished feature in our molecular modeling program MacroModel/BatchMin.18 In this paper, we describe our analytical approach to Born radii in detail, reoptimize the parameters associated with the model based on Poisson equation derived Gpol energies, and show how the model performs in reproducing accurate Born radii and experimental solvation energies. We also compare its performance to a different approach to Born radii recently described by Hawkins et al.19 II. Methods To define our approach to computing Born radii (R), we begin with the original Born expression (eq 4) for a monoatomic spherical ion surrounded by a continuum dielectric medium representing a solvent which relates the total dielectric polariza- tion energy of the system (Gpol, kcal/mol) to the charge (q, electrons), the dielectric constant () of the medium, and R, the ion’s effective or Born radius (or, more precisely, the distance from the center of the ion to the boundary of the dielectric, Å). For such a spherical system in a solventlike, continuum dielectric medium, the effective dielectric boundary will be found at some fixed distance (previously defined9 as the dielectric offset distance φ) from the van der Waals surface of the solute. Thus, for a spherical monoatomic solute, there is a simple relationship between R and the distance from the ion center to its van der Waals surface (R ) φ + RvdW). For a polyatomic solute, however, the corresponding distance from an atomic charge to the molecular van der Waals surface will vary depending on which part of the molecular surface is being considered. To avoid the mathematical complexities associated with such an angularly dependent R, we sought an appropriate way to average the various distances from a given charge to all points on the dielectric boundary to produce a single value of R for use in eq 3. The approach we developed begins with eq 4 and the following idea. Imagine a polyatomic solute whose atoms are all electrically neutral but displace the dielectric solvent medium to create a solute-shaped cavity in the medium. For such a system, Gpol ) 0. Now, choose an atom (i) and place an electrical charge (qi) on it. The resulting system will now have some nonzero Gpol. If we could compute this Gpol, we could then use eq 4 to calculate Ri, a value corresponding to a spherically averaged, effective Born radius of atom i. Thus, given a method to calculate Gpol for a system consisting of a continuum dielectric and an irregularly shaped solute with a single charge located at any position within the solute, Born radii for each atom in the solute could be readily calculated. This general approach to Born radii was introduced as part of the original GB/SA solvation model and assumes that the Born radius of a given atom does not depend upon the charge distribution in the system.9 The following paragraphs describe a method for carrying out such Gpol calculations in the context of a solute having atom-centered charges in a continuum dielectric medium. We begin by describing our analytical method for the rapid calculation of such Gpol’s and thus Born radii. An Analytical Approach to Born Radii (r). In order to compute R efficiently, we sought an analytical function leading to usefully accurate Born radii via a simple pairwise evaluation of the atoms in a molecular solute. The idea we developed is best described with the aid of Figure 1. Imagine that we wish to compute Ri in a polyatomic solute in a solvent represented by a continuum dielectric. All of the atoms of the solute may be considered to displace any dielectric within their van der Waals surfaces to create a solute-shaped cavity in the solventlike Gcav + GvdW ) ∑ k)1 N σkSAk (2) Gpol ) -166.0(1 - 1  )∑ i)1 n ∑ j)1 n qiqj (rij 2 + Rij 2e-Dij)0.5 (3) Gpol ) -166.0(1 - 1) q2 R (4) 3006 J. Phys. Chem. A, Vol. 101, No. 16, 1997 Qiu et al. While the work described here was underway, a different pairwise approach for the evaluation of Born radii was reported by Hawkins et al.19 Their method was based on our original finite difference Born shell approach to Born radii but employed a pairwise descreening approximation27 to make the approach both rapid and analytical. We term this method the PDA method. The PDA method is similar in spirit to our V/r4 method (eq 5a) in that it computes an atom’s Born radius by summing the effects of dielectric displacement by all other atoms in the molecule. Though both methods approach the Born radius problem using an atomic pairwise algorithm, the basic underly- ing models differ in that the PDA method is an analytical approximation to our original Born shell model,9 whereas our new work is based on the V/r4 model of Figure 1. To see how the PDA method performs relative to eq 5a and accurate FDPB results, we programmed and tested the Hawkins formula for R-1. Like our V/r4 model, the PDA approach is simplistic in that it does not explicitly deal with voids between atoms or atomic overlaps. Instead, empirical scaling factors (Sx) are used to adjust the atomic radii to minimize such effects. These scaling factors were optimized to best reproduce known solvation energies of more than 100 organic molecules using a GB/SA- like solvation model with SM2 atomic radii and AM1-derived partial charges. Their values as given by Hawkins et al. are shown below under the Sx column: In comparing the results from the PDA method with those from our V/r4 method (eq 5a) and FDPB, we used Hawkins et al.’s original scaling factors (Sx) as well as modified scaling parameters (Sx*) that we optimized to minimize the same error function (eq 7) and data set used to optimize P1 through P5. These modified scaling parameters should be more appropriate for the OPLS atomic radii and charges used here. We term the results of the PDA method with our modified Sx* scaling factors as the PDA* results in the discussion below. For compounds containing sulfur, phosphorus, or halogens, no PDA or PDA* calculations were carried out because scaling parameters were unavailable for those atom types. III. Results and Discussion In the following sections, we test the ability of our V/r4 model (eq 5a) to reproduce solventlike, continuum dielectric polariza- tion energies (atomic G′pol,i and molecular Gpol) from FDPB calculations. We also use V/r4 and FDPB-derived Born radii in our GB/SA solvation model to compute the total solvation energies (Gsol) of small molecules in water where experimental data are available. Finally, we provide similar comparisons using the PDA approach to Born radii. IIIa. Comparison with Finite Difference Poisson-Boltz- mann (FDPB) Results. Because solution of the Poisson equation provides polarization energies for charge-bearing objects in a continuum dielectric medium, it provides a convenient source of accurate G′pol,i data for comparison with corresponding energies calculated using the V/r4 model (eq 5a). This comparison for all 10 034 atoms in our data base is shown graphically in Figure 2A. To make the correlation graphs easier to read, we have plotted G′pol,i for small molecules and large molecules (various nonapeptide conformations and crambin) separately. As these graphs show, the V/r4 model does the best job at reproducing FDPB atomic polarization energies for those atoms having the largest (most negative) solvent polarization energies. Those atoms are the ones having the smallest Born radii. Thus, the atoms making the largest contributions to solvent polarization energies are most accurately treated. While most atoms had solvent polarization energies that were within a few percent of the correct FDPB results, a significant number of atoms having larger Born radii (less negative G′pol,i) had Born radii that were systematically smaller than those from the FDPB calculations. These outliers occurred primarily in the larger molecules and tended to be those atoms most deeply buried within a solute. In the FDPB method, a Connolly surface is used to define the solute boundary and to exclude dielectric from small voids in regions of densely packed solute atoms. In contrast, the V/r4 model uses atomic van der Waals surfaces to define those volumes from which the dielectric is excluded. This approach effectively leaves small, interstitial void volumes containing dielectric within the solute. While the parameterization de- scribed above minimizes such differences in an average way, differences still remain, especially for deeply buried atoms in large molecules. The net effect is that these atoms often experience a higher microdielectric environment in the V/r4 model than in the FDPB treatment. Table 2, column eq 5a*, provides statistical data on the correlation between eq 5a and FDPB calculations of atomic medium polarization energies. Given there are correlation coefficients (r) for a linear fit between G′pol,i computed by the two methods. Atomic polarization energies for small molecules are better reproduced (r ) 0.96) than those of large molecules (r ) 0.92), as expected given our systematic overestimation of G′pol,i for buried atoms. The average unsigned error in G′pol,i for all 10 034 atoms in our data set is 4.27 kcal/(mol atom) (just under 6% of the mean atomic polarization energy). TABLE 1: Average Errors in Born Radii (r) Based on Atom Type av error ) R - R(FDPB), Å atom MacroModel atom type eq 5aa eq 5a*b no. atomsc C(sp) C1 -0.028 0.002 1 C(sp2) C2 0.024 0.070 1668 C(sp3) C3 -0.135 -0.102 723 CH(sp3) CA -0.043 0.015 727 CH2(sp3) CB 0.062 0.096 597 CH3(sp3) CC 0.145 0.180 393 CH(sp2) CD 0.096 0.132 245 O(sp2) O2 -0.099 -0.066 940 O(sp3) O3 -0.138 -0.097 331 O- OM 0.001 0.029 209 N(sp) N1 -0.011 0.015 1 N(sp2) N2 -0.249 0.014 876 N(sp3) N3 -0.406 -0.225 14 N+(sp2) N4 0.119 0.153 69 N+(sp3) N5 -0.048 -0.018 86 H(C) H1 -0.069 -0.051 1638 H(O) H2 -0.187 -0.072 232 H(N) H3 -0.564 -0.410 857 H(N+) H4 -0.023 0.000 365 S S1 -0.574 -0.241 35 P P0 -0.128 -0.099 17 F F0 -0.112 -0.099 17 Cl Cl -0.285 -0.120 17 Br Br -0.030 0.001 1 I I0 -0.024 0.007 4 a By V/r4 model (eq 5) using OPLS radii and 1.15 Å for H. b V/r4 model (eq 5) using OPLS radii and 1.15 Å for H except that N(sp2), N(sp3), H(O), H(N), and Cl are enlarged by 5% and S is enlarged by 10% (see text). c Number of atoms in data set of designated atom type. atom Sx Sx* H 0.82 0.78 C 0.70 0.77 O 0.54 0.64 N 0.66 0.66 GB/SA Continuum Model for Solvation J. Phys. Chem. A, Vol. 101, No. 16, 1997 3009 Also given in Table 2 (eq 5a* column) are Born radius data computed from G′pol,i (eq 5b) for the same 10 034 atoms. The correlation between these V/r4- and FDPB-derived Born radii is weaker (r ) 0.85) than with the energetic equivalent G′pol,i, and the average unsigned error in Born radii over the entire set is 0.17 Å/atom. As discussed above, the greatest differences in Born radii are found with those atoms having the largest Born radii. These effectively larger atoms contribute the least to the solvation energies so that the overall V/r4-derived and FDPB energetic results still agree reasonably well, as we show below. We next investigated the agreement between GB and FDPB in computing solventlike dielectric medium () 80) polarization energies (Gpol) for molecules bearing complete sets of atomic charges. The same charge sets were used in both calculations and came from the AMBER* force field.25 The molecules were the same 189 organic and biological molecules used in the above G′pol,i and Ri calculations. The Gpol comparison in which the GB calculations employed V/r4-derived Born radii is shown in Figure 3A. There is a strong linear correlation (slope ) 1.000, correlation coefficient ) 0.999) between the two Gpol calculation results. The differences that do exist are generally small relative to the total solvation energies involved and are explicitly plotted next to the correlation graph. The molecule having the largest difference in Gpol energy as calculated by the GB method with Figure 2. Comparison of atomic dielectric medium polarization energies (G′pol,i, kcal/(mol atom)) computed by various methods. Panel A: GB radii from eq 5a with optimal atomic radii. Panel B: GB radii using the method of ref 19. Panel C: GB radii using method of ref 19 with reoptimized Sx parameters. 3010 J. Phys. Chem. A, Vol. 101, No. 16, 1997 Qiu et al. V/r4 Born radii is the protein crambin, the largest molecule in the data set. Even here the +16 kcal/mol GB/FDPB difference is relatively small: about 7% of crambin’s total dielectric polarization energy. The average unsigned error of GB(V/r4) Gpol relative to FDPB for all 189 molecules in the data set is 1.93 kcal/mol. Because the above molecular GB Gpol calculations depend both upon the validity of the GB equation (eq 3) and of the V/r4 model (eq 5), we carried out one further test to probe the sensitivity of these Gpol calculations to the above-noted errors in V/r4-derived Born radii. This test involved using accurate, FDPB-derived Born radii in the GB equation. The results are plotted in Figure 3B and indicate the best Gpol results that the GB equation (3) can provide relative to full Gpol calculations by the FDPB method. While the errors in molecular Gpol are generally smaller with FDPB-derived Born radii, the improve- ment is not dramatic and the average unsigned error for the entire data set is now 1.74 kcal/mol. IIIb. Comparison with Experimental Free Energies of Solvation in Water. While the above comparisons with Poisson-Boltzmann calculations support the utility of the GB and V/r4 approximations in computing polarization energies (Gpol) for a molecular solute in a dielectric continuum medium, the medium in real-world applications of the method is not a dielectric continuum but a real molecular solvent. We therefore ask how well GB calculations using various sources of Born radii reproduce actual solvation free energies for the polar sol- vent water. Here, accurate experimental data are more limited and are available only for certain small molecules. To compare our calculations with experiment, we chose 36 small organic molecules having diverse functional groups and whose hydration free energies appeared accurately known.28 As in the original GB/SA work,9 we also limited the molecules to those bearing functional groups having atomic partial charges defined by Jorgensen’s OPLS force field21 or derived from electrostatic potential fitting to minimally HF/6-31G* ab initio wave functions. To calculate the total solvation free energies of these molecules, we used the GB/SA solvation model (eqs 1-3) using Born radii calculated using the methods described above. In the original GB/SA work,9 all atoms were treated as hydrophobic and used the same atomic solvation parameter (σ ) 7 cal/(mol Å2)) in the solvent-accessible surface area (SA) part of the model. Since that time, we have adopted Cramer and Truhlar’s approach19 of using different σ’s for different atom types though we distinguish only a few atom types in this way. Thus, we treat atoms based on their approximate hydrophobicity/hydro- philicity and use the following three area-based atomic solvation parameters for the common atom types: σ(C(sp3), S) ) 10 cal/ (mol Å2), σ(C(sp2), C(sp), P) ) 7 cal/(mol Å2), σ(O, N) ) 0 cal/(mol Å2). To speed our surface area calculations, we employ the united atom approximation in the SA part of the GB/SA model, and thus, σ(H) is zero. The results of our GB/SA hydration free energy (Gsol) calculations are summarized along with experimental data in Table 3. Results using our V/r4 Born radius model are given in column eq 5a* and plotted in Figure 4. There it can be seen that GB/SA(R from V/r4) solvation energies are strongly correlated with experiment (average unsigned error 0.9 kcal/ mol, linear fit slope ) 0.96, r ) 0.94). We also carried out analogous GB/SA calculations using accurate, FDPB-derived Born radii. Those data are shown in the FDPB column of Table 3 and have an average unsigned error of 0.8 kcal/mol. Thus, comparable accuracy relative to experiment is obtained in our GB/SA Gsol calculations regardless of the source of the Born radii. The utility of the V/r4 model for computing Born radii for use in the GB/SA solvation model would therefore seem to be validated. Finally, we tested the basic GB equation (eq 3) of our GB/ SA model for solvation by replacing the entire GB equation with a full FDPB calculation of Gpol. Those results are given in Table 3 under the FDPB/SA column and yield an average unsigned error of 0.9 kcal/mol relative to experiment. Thus, to the extent that experiment is the best yardstick for assessing the usefulness of a model in real applications, we find no evidence that the FDPB method is any better than the simpler GB approximation. Indeed, we think it likely that the ap- proximation of a polar molecular solvent by a simple continuum dielectric is more significant than the difference between FDPB and GB approaches to Gpol. IIIc. Comparison with the Pairwise Descreening Method. We also carried out similar tests on the related PDA approach to Born radii recently reported by Hawkins et al.19 As noted in the Methods section, we found it advantageous to reoptimize their Sx based on our van der Waals radii and FDPB-derived G′pol,i. We have compared the results with both the original (PDA, Sx parameters) and the revised (PDA*, Sx* parameters) parameter sets. The PDA and PDA* comparisons with FDPB G′pol,i are given in parts B and C of Figure 2. GB(PDA or PDA*) molecular solvation energies are compared with FDPB results and experiment in Figure 3C and Table 3. For atomic solvent polarization energies and Born radii, the original PDA method provided a reasonable correlation with FDPB-derived results. However, using the reoptimized scaling factors (Sx*), PDA* performed significantly better relative to FDPB, especially with large molecules (Figure 2C). Statistics for these PDA and PDA* calculations are given in Table 2 and quantify the improvement in G′pol,i and Ri with the PDA* parameter set. We then compared the Gpol from FDPB and GB(PDA) for 139 molecules in our data set that did not contain sulfur, phosphorus, or halogen. For the small molecules, PDA Born radii in conjunction with the GB equation yielded Gpol that are in good agreement with molecular FDPB results but systemati- TABLE 2: Comparison of Atomic Polarization Energies (G′pol,i, kcal/mol) and Born Radii (ri, Å) Calculated by Various Methods and Compared to Poisson-Boltzmann Resultsa polarization energies FDPBb eq 5a*c PDAd PDA*e mean energy per atom -76.7 -77.6 -85.7 -78.4 slope (small mols)f 0.93 1.01 0.97 corr (small mols)g 0.96 0.97 0.95 slope (large mols)f 0.83 0.74 0.78 corr (large mols)g 0.92 0.83 0.90 av unsigned error 4.27 9.40 5.89 Born radii FDPBb eq 5*c PDAd PDA*e mean R 2.33 2.27 2.01 2.18 slopef 0.64 0.46 0.75 corrg 0.85 0.58 0.82 av unsigned error 0.17 0.31 0.21 a Data computed for 10 034 atoms in 189 molecules bearing single- unit charges (qi ) 1) and in an external dielectric continuum with  ) 80 except for PDA and PDA* results which are based on 7407 atoms in 139 molecules (see text). b Accurate result given by finite difference Poisson-Boltzmann calculation. c V/r4 model (eq 5). d Pairwise de- screening approximation method of Hawkins et al.19 (G′pol,i ) -166Ri-1). e Pairwise descreening approximation method with Sx*-modified scaling parameters (see text). f Slope of linear fit between designated method and FDPB. g Correlation coefficient for linear fit of points between designated method and FDPB. GB/SA Continuum Model for Solvation J. Phys. Chem. A, Vol. 101, No. 16, 1997 3011