Influence of Grid Spacing in Poisson-Boltzmann Equation Binding Energy Estimation

Abstract

Grid-based solvers of the Poisson-Boltzmann, PB, equation are routinely used to estimate electrostatic binding, ΔΔG_el, and solvation, ΔG_el, free energies. The accuracies of such estimates are subject to grid discretization errors from the finite difference approximation to the PB equation. Here, we show that the grid discretization errors in ΔΔG_el are more significant than those in ΔG_el, and can be divided into two parts: (i) errors associated with the relative positioning of the grid and (ii) systematic errors associated with grid spacing. The systematic error in particular is significant for methods, such as the molecular mechanics PB surface area, MM-PBSA, approach that predict electrostatic binding free energies by averaging over an ensemble of molecular conformations. Although averaging over multiple conformations can control for the error associated with grid placement, it will not eliminate the systematic error, which can only be controlled by reducing grid spacing. The present study indicates that the widely-used grid spacing of 0.5 Å produces unacceptable errors in ΔΔG_el, even though its predictions of ΔG_el are adequate for the cases considered here. Although both grid discretization errors generally increase with grid spacing, the relative sizes of these errors differ according to the solute-solvent dielectric boundary definition. The grid discretization errors are generally smaller on the Gaussian surface used in the present study than on either the solvent-excluded or van der Waals surfaces, which both contain more surface discontinuities (e.g., sharp edges and cusps). Additionally, all three molecular surfaces converge to very different estimates of ΔΔG_el.

INTRODUCTION

Many cellular processes, such as signal transduction, gene expression and protein synthesis, are governed by the binding of biomolecules. In pharmaceutical applications, accurate and fast predictions of the binding free energy, ΔΔG, of drugs to biomolecular targets, such as proteins and nucleic acids, are necessary, particularly in the final stages of the structure-based drug discovery. Predicting ΔΔG accurately is, however, challenging because it usually involves combining several energies, each of which is subject to numerical and model errors. Some of these energy terms, such as the Coulombic interactions between binding partners, usually favor binding, while others, such as the desolvation penalty of removing charged residues from the solvent environment upon binding, oppose it. Frequently, ΔΔG is much smaller than the contributing energy terms, so that errors judged small when compared against these individual contributions assume much greater significance when estimating ΔΔG.

The present study illustrates this general problem by examining the numerical computation of the electrostatic component, ΔΔG_el, of ΔΔG, estimated by the Poisson-Boltzmann, PB, equation^1-7. Typically, one obtains ΔΔG_el from ΔΔG_el= (ΔG_el)_c – (ΔG_el)₁ – (ΔG_el)₂ + (ΔΔG_el)_coulomb, where (ΔG_el)_c is the electrostatic solvation free energy of the complex, (ΔG_el)₁ and (ΔG_el)₂ are the corresponding electrostatic solvation free energies of the unbound components, and (ΔΔG_el)_coulomb is the electrostatic binding free energy of the two components in vacuum. Here we assume that there are no conformational changes upon binding and thus, the bound and unbound states of the binding partners are the same. This same assumption is made in the most popular single trajectory MM-PBSA protocol, where only a single all-atom molecular dynamics of the complex is performed to compute ΔΔG. Many studies assume that the differences in ΔΔG between closely-related complexes, such as the binding of similarly charged organic drugs to a single protein or nucleic acid target or the mutation of polar or charged side chains in protein-ligand complexes, are dominated by the change in ΔΔG_el,^{8, 9}. However, many of these studies have used grid spacings of 0.5 Å or larger under the justification that ΔG_el estimates computed with these grid spacings are consistent with free energy methods using explicit solvent molecular dynamics simulations¹⁰ or experimental solvation free energy data¹¹. As demonstrated below, however, such coarse grids produce estimates of ΔΔG_el with unacceptably large errors because the magnitude of ΔΔG_el is typically orders of magnitude smaller than (ΔG_el)₁, (ΔG_el)₂, and (ΔG_el)_c.

Finite difference estimates of ΔΔG_el by the PB equation contain two kinds of numerical grid discretization error. The first error, denoted by σ_GP, is the variation in ΔΔG_el due to grid positioning. This error increases with increasing grid spacing but can be controlled by averaging over different grid placements, as described in the Methods, or through averaging over many similar molecular conformations from a molecular dynamics trajectory, as done in molecular mechanics PB surface area, MM-PBSA, methods¹². However, these estimates of ΔΔG_el also contain a second, systematic error associated with grid spacing that is not eliminated by this averaging. For example, if the estimate of ΔΔG_el obtained by averaging over grid positions is denoted by <ΔΔG_el> then <ΔΔG_el> obtained at a 1.0 Å grid spacing can differ by more than 100 kcal/mol from that obtained using a 0.3 Å grid size, as shown in the Results. This error persists when averaging over multiple grid placements and can only be reduced by using finer grid spacing. For the solvent-excluded, SE, and van der Waal's, vdW, surface definitions, the Results confirm that the popular 0.5 Å grid spacing is too coarse for many applications. Recommending a single universally useful grid spacing is difficult because of the innate dependence upon the system considered, surface definition, and energy accuracy required by the application. Rather, users of PBE solvers are urged to perform calculations at multiple grid spacings to ensure that their calculations have converged.

One possibility explored in this study is that the grid convergence behavior of <ΔΔG_el> is linked to the smoothness of the molecular surface. In the Results, when a smooth Gaussian surface definition¹³, which contained fewer slope discontinuities than either the SE surface, which contains sharp cusps, cones and convex edges, or the vdW surface, which contains reentrant edges, was used, <ΔΔG_el> was better converged at larger grid spacings. Unfortunately, the better numerical performance of the Gaussian surface does not necessarily imply that it should be used for PB applications.

As shown below, the three molecular surfaces choices converge to very different estimates of ΔΔG_el and, because the ability to predict changes in ΔΔG is the true test of a surface definition, more work will need to be done to determine which surface definition produces results that agree with experiment. While we do not settle this particular issue here, regardless of which surface definition is adopted, the errors associated with grid spacing will need to be accounted for to ensure that reliable estimates of ΔΔG_el are procured.

METHODS

Data Sets

Three sets of biomolecular complexes were used in this investigation:

Data Set 1: A collection of DNA-minor groove drug complexes whose experimentally-determined ΔΔG span a narrow range, thus increasing the challenge of predicting rank orderings. This data set typified the calculations required in structure-based drug design. The pdbids for Data Set 1 are as follows: 102d, 109d, 121d, 127d, 129d, 166d, 195d, 1d30, 1d63, 1d64, 1d86, 1dne, 1eel, 1fmq, 1fms, 1jtl, 1lex, 1prp, 227d, 261d, 164d, 289d, 298d, 2dbe, 302d, 311d, 328d, and 360d.

Data Set 2: Various wild-type and mutant barnase-barstar complexes were included in this study as examples of the common use of in silico mutagenesis in biophysical applications where the goal is to identify residues that make significant contributions to the total binding free energy. The structural data for all mutated complexes were available, so the effects of mutations on the three-dimensional structure can be considered without further molecular modeling. The pdbids for Data Set 2 are as follows: 1b27, 1b2s, 1b2u, 1b3s, 2aza4, 1x1w, 1x1y, 1x1u and 1x1x.

Data Set 3: RNA-peptide complexes with various biological functions, RNA folds, peptide secondary structures and binding affinities. The pdbids for Data Set 3 are as follows: 1a1t, 1a4t, 1biv, 1exy, 1g70, 1hji, 1i9f, 1mnb, 1nyb, 1qfq, 1ull, 1zbn, 2a9x and 484d.

Structure Preparation

The atomic coordinate files (pdb files) of all biomolecular complexes in this study were taken from the Research Collaboratory for Structural Bioinformatics (RCSB) database¹⁴. For the NMR structures model 1 was used. Only the chains containing the drug, nucleic acid, or protein were retained while all other unnecessary information was removed prior to further manipulation of the pdb files. For Data Set 1 the molecular modeling protocol used to create the pqr files was described elsewhere¹⁵. For the other two data sets the pdb2pqr program¹⁶ was used to prepare the pqr files, which contained the atomic coordinates, as well as atomic partial charges and van der Waals radii, assigned according to the Amber¹⁷ force field for proteins and nucleic acids. All ionizable protein residues were left in their standard ionization states at pH=7, with the histidine side chains left neutral. For the hydrogen atoms that had a zero radius, the atomic radius was taken as that of the smallest one (0.6 Å) in the Amber force field. However, these changes in atomic radii didn't change the results presented here because each of these hydrogen atoms is contained well inside another atom with finite radius (results not shown). The pqr files used in this work can be downloaded from http://www.sb.fsu.edu/~mfenley/convergence/downloads/convergence_pqr_sets.tar.gz

Poisson-Boltzmann Calculation Details

All results were obtained with the adaptive Cartesian grid (ACG) finite difference Poisson-Boltzmann equation solver described elsewhere³. This software solves the PB equation and determines the electrostatic potential on an octree structure consisting of hierarchically nested cubes, as shown in Figure 1. The hierarchical grid structure provides a variable mesh spacing capability together with an efficient means of implementing a multigrid solution scheme. In addition, the potential field is decomposed so that singularities are eliminated from the numerical solution. Specifically, inside the molecule the reaction field potential is computed, while outside the molecule the full potential is obtained. At the dielectric interface, the solutions are coupled together using the Coulombic potential, which is easily evaluated analytically. The reaction field generally varies most rapidly at the surface and becomes smooth as one moves deeper into the molecule. Similarly the full potential field calculated outside the molecule also varies most rapidly at surfaces and becomes increasingly smooth farther away. Consequently, the finest mesh spacing requirements occur near the dielectric interface. Elsewhere, coarse grid spacing is permitted, leading to substantial reductions in the overall number of grid points (and thus memory and computation time requirements) needed to compute the solution.

Figure 1.

A cut through the adaptive Cartesian grid around the B-DNA dodecamer that binds to the cationic propamidine drug (pdbid: 102d) and the surrounding high-dielectric solvent region. Note that the ACG concentrates grid points at the dielectric interface where both the interior reaction field potential and the exterior full potential solutions vary most rapidly. Deeper inside the molecule and further outside it these solutions vary more gradually, so coarser grids can be used.

Three molecular surface definitions are considered here: the SE surface, the vdW surface, and a Gaussian surface definition. In the vdW surface definition, a molecule is represented by a union of spheres, one for each atom, centered at the coordinates of that atom in the structure file and the vdW radius of the atom. The molecular surface then consists of the non-overlapping regions of the surfaces of these spheres. As shown in Figure 2, the vdW surface contains numerous reentrant edges coinciding with the sphere intersection arcs and characterized by slope discontinuities. The SE surface defines the molecular interior as the region that cannot be covered by a solvent-sized spherical probe (here taken to have a radius of 1.4 Å) without the probe overlapping the interior of the molecule defined by the vdW surface. This definition eliminates the reentrant edges, but introduces cusps and convex edges pointing into the exterior region where the solvent probe intersects itself (i.e., can approach the surface from different directions) as observed from Figure 2. Generally, the SE surface has fewer locations with slope discontinuities than the vdW surface, and the geometries of the discontinuities are different. These factors are known to affect the local behaviors of the potential field and can give rise to singular electric fields (potential gradients). Supposing that the number and nature of surface slope discontinuities also affects binding energy predictions is therefore not unreasonable. Finally, the Gaussian surface¹³ is implicitly defined by F(r)=0, where:

$$\mathrm{F}\left(\mathbf{r}\right)=\sum \exp (-{\mid \mathbf{r}-{\rho }_{\mathrm{i}}\mid }^2∕\alpha {{\mathrm{a}}_{\mathrm{i}}}^2)-\exp (-1∕\alpha )$$ (1)

where r is the evaluation point, ρ_i and a_i are the atomic centers and radii, respectively, α is the smoothing parameter, and the summation is taken over all atoms. Unless otherwise stated, the results presented here used α = 1.4. Note that for a single sphere the Gaussian, SE and vdW definitions all coincide. Note too that the dielectric constant in the Gaussian surface used in this study does not vary gradually at the molecular surface (as in the PBE solver, ZAP¹⁸). Instead, the surface where F(r)=0 forms a sharp demarcation between the interior and exterior dielectric regions. The Gaussian surface generally contains no slope discontinuities, though regions with high curvature can occasionally arise. These regions become more pronounced as α approaches zero and the Gaussian surface converges to the vdW surface. All three surface definitions can contain cavities or regions within the molecule that do not connect with points in the exterior. Per the surface definitions above, these cavities are assigned an exterior dielectric constant in this study.

Figure 2.

Comparison of the surface electrostatic potential on a B-DNA dodecamer (pdbid: 102d) using different molecular surfaces to specify the solute-solvent dielectric boundary. a. A licorice representation of the B-DNA, with the phosphorus atom of the phosphate group colored yellow. b-d. The electrostatic potentials on the van der Waals surface, the Gaussian surface (α = 1.4), and the solvent-excluded surface, respectively. The electrostatic potentials are displayed in units of kcal/mol/e, and the scale is shown in the figure. The cationic propamidine drug binds in the AT-rich and narrow minor groove of the B-DNA, which displays a pronounced negative potential, but this drug is not shown in this figure.

All of the surface definitions are processed analytically within the ACG PBE solver. No reference to an externally-generated triangulated surface is made.

The temperature of the ionic solution was 298.15 K, the dielectric constants for solute and solvent were 1 and 80, and the ionic strength was 100 mM NaCl. No ion-exclusion region was considered in this study. All of the calculations in this study used the linear PB equation, but nonlinear PB results did not differ significantly from those presented here (data not shown). The finite-difference equations were iteratively solved until the change in dimensionless potential at any grid point was less than 10⁻⁹, which typically occurred within at most 200 multigrid iterations. The total grid size was set to 3 times the largest dimension of the solute, and the outer boundary conditions were assigned using charge conservation principles as described elsewhere². Grids with common dimensions and alignments were generated for each separate binding partner and complex. The ACG predictions of ΔG_el and ΔΔG_el were corroborated by similar computations with APBS¹⁹ and an in-house stochastic linear PB solver^5-7 (results not shown).

To quantify the error, σ_GP, due to grid placement, ΔΔG_el was computed on 30 different grids generated by shifting the original grid by a uniformly-distributed fraction of a grid spacing in each of the x, y, and z directions. The average (<ΔΔG_el>), and standard deviations (σ_GP), of these energies at each grid spacing are reported in Results.

RESULTS AND DISCUSSION

Examining the Convergence of ΔΔG_el for Biomolecules

A grid spacing of 0.5 Å is often used to predict ΔΔG_el because experience^{20, 21} has shown that it generates seemingly well-converged estimates of ΔG_el, as shown in Figure 3, where ΔG_el for the DNA-drug complexes computed with 0.5 Å and 0.2 Å grid spacings based on the SE surface are compared. The linear fit through these results produces an R² = 0.999 and a slope of 1.00 confirming converged results at the 0.5 Å spacing. Similarly well-converged results are obtained for the other complexes and surface definitions. However, note that the ΔG_el vary over a range of thousands of kcal/mol whereas the ΔΔG_el are typically several orders of magnitude smaller. Thus, the adequacy of a 0.5 Å spacing for the evaluation of ΔΔG_el is not ensured and must be determined separately. Indeed, an analogous plot for <ΔΔG_el>, produces R²=0.87, after a single outlier with <ΔΔG_el> > 60 kcal/mol (pdbid: 360d) is excluded (this case predicts a <ΔΔG_el> that is almost an order of magnitude larger than for all other complexes).

Figure 3.

The electrostatic solvation energy, ΔG_el, for all complexes in all three data sets computed with a solvent-excluded, SE, surface with a finest grid spacing of 0.5 Å plotted against the same energy computed with a finest grid spacing of 0.2 Å. The best-fit line has a slope of 1.00 and an R² = 0.999.

Table 1 displays <ΔΔG_el> and σ_GP for the molecules, surface definitions, and finest grid spacings used in this study. As expected, σ_GP generally increases with finest grid spacing (Figure 4), but the rate of increase differs with surface definition. The Gaussian surface, for example, shows a much slower rate of increase and smaller values of σ_GP than either the SE or vdW surfaces. For both the vdW and SE surfaces, σ_GP can exceed 10 kcal/mol at a 0.5 Å spacing, while for the Gaussian surface it is always less than 0.7 kcal/mol. Therefore, with the vdW or SE surfaces a large number of samples would be required to guarantee a reasonable estimate of <ΔΔG_el>, either using MM-PBSA methods¹² or random grid positioning, as was done in the present study.

Table 1.

Average electrostatic binding free energies, <ΔΔG_el>, and standard deviations, σ_GP, in units of kcal/mol for all of the complexes used in this study and based on three different surface definitions and four different finest grid spacings.

			0.3 Å		0.4 Å		0.5 Å		0.75 Å		1Å
		Complexes	<ΔΔGel>	σ GP	<ΔΔGel>	σ GP	<ΔΔGel>	σ GP	<ΔΔGel>	σ GP	<ΔΔGel>	σ GP
	DNA-drug	102d	9.8	0.1	9.9	0.7	12.9	2.3	32.4	8.8	55.9	19.3
		109d	3.1	0.1	3.6	0.6	5.3	4.3	18.5	9.1	43.0	14.6
		121d	23.9	0.1	23.7	1.3	27.2	4.3	58.1	14.2	90.8	19.8
		127d	29.7	0.2	29.3	1.4	32.9	4.7	52.9	16.0	85.3	18.8
		129d	40.3	0.1	40.3	0.6	41.9	2.4	56.1	9.7	95.4	19.9
		166d	15.1	0.1	15.0	0.9	15.3	1.5	25.1	6.3	51.0	10.9
		195d	3.0	0.1	3.1	0.5	3.4	2.2	21.9	8.0	54.4	16.6
		1d30	11.0	0.1	11.5	0.7	12.7	2.7	29.4	11.0	69.6	23.4
		1d63	12.9	0.1	12.7	1.1	15.3	3.7	47.1	20.4	71.1	19.3
		1d64	14.9	0.1	15.2	0.2	14.8	1.2	23.8	5.2	52.5	17.7
		1d86	25.2	0.1	25.4	1.2	30.9	4.3	71.0	17.8	115.1	28.1
		1dne	21.9	0.1	21.4	2.3	26.5	4.3	57.5	9.8	99.0	13.7
		1eel	15.6	0.2	15.6	1.5	18.7	5.2	43.2	18.1	79.3	24.7
		1fmq	14.7	0.1	14.8	0.8	16.7	2.0	42.0	15.7	63.9	17.8
		1fms	26.0	0.1	26.2	0.5	27.0	2.0	45.2	11.5	72.1	17.9
		1jtl	11.5	0.2	11.6	1.0	15.2	4.7	32.8	12.9	86.0	34.6
		1lex	11.0	0.2	11.3	1.1	16.2	4.0	49.1	9.6	87.8	29.2
		1prp	11.7	0.1	11.6	0.8	12.2	1.5	24.7	10.3	48.5	18.0
		227d	6.5	0.2	6.6	1.2	11.3	5.4	41.1	13.4	71.7	34.5
		261d	3.3	0.2	3.7	1.4	9.1	3.4	37.7	13.4	79.7	24.1
		264d	31.7	0.1	32.1	1.0	34.7	3.1	54.0	10.2	79.9	20.0
		289d	16.8	0.1	16.7	1.8	18.2	4.4	49.0	18.2	72.2	18.3
		298d	15.5	0.1	15.4	0.6	17.6	4.5	38.2	8.6	71.8	21.5
SE		2dbe	5.5	0.1	5.5	0.5	7.1	2.6	24.6	12.7	59.4	18.7
		302d	24.9	0.2	25.7	1.3	29.3	6.6	58.2	12.6	81.3	17.2
		311d	10.3	0.2	9.8	2.7	13.5	6.1	42.5	13.8	80.7	28.0
		328d	19.1	0.1	19.1	0.3	18.7	1.4	26.0	3.7	58.6	15.2
		360d	55.5	0.1	55.6	1.6	57.2	3.0	82.3	18.0	109.8	19.7
	Barnase-Barstar	1b27	85.9	0.2	86.2	2.2	96.7	4.8	172.9	24.1	273.8	46.4
		1b2s	72.5	0.2	72.0	2.0	81.8	6.2	145.2	20.0	221.7	49.2
		1b2u	78.9	0.2	79.0	1.9	87.4	5.5	146.2	17.4	199.8	36.2
		1b3s	49.8	0.2	49.5	1.7	59.0	7.7	120.0	16.3	223.4	28.2
		1x1u	75.6	0.3	75.8	2.3	85.9	6.0	167.9	24.7	257.8	26.5
		1x1w	93.7	0.3	93.9	1.5	106.1	7.0	181.6	29.1	287.9	40.7
		1x1x	117.8	0.4	118.2	2.9	133.5	9.1	214.1	31.5	309.2	41.9
		1x1y	88.8	0.2	88.9	2.5	102.0	8.5	187.6	20.3	271.7	33.2
		2za4	75.6	0.3	76.1	2.3	85.2	6.7	135.7	25.9	222.5	40.4
	RNA-peptide	1a1t	114.9	0.3	115.2	3.2	131.3	6.7	237.5	21.3	351.1	47.2
		1a4t	136.3	0.2	136.2	2.3	148.4	7.3	255.2	32.2	338.0	38.3
		1biv	137.2	0.4	138.1	5.1	162.2	15.1	337.1	40.8	468.7	46.5
		1exy	249.3	0.5	251.2	3.9	276.0	7.6	416.7	45.6	565.8	46.2
		1g70	174.2	0.3	173.2	3.2	193.5	8.8	292.0	19.1	390.9	43.0
		1hji	61.1	0.2	60.5	2.1	69.6	5.6	120.3	26.2	181.5	34.8
		1i9f	18.3	0.3	19.5	3.6	43.9	7.3	172.8	32.8	255.8	39.2
		1mnb	171.3	0.4	171.1	2.7	193.4	10.7	305.4	46.3	420.2	47.9
		1nyb	35.4	0.2	35.3	2.1	47.9	12.4	124.5	28.8	228.9	30.9
		1qfq	56.3	0.4	56.4	2.7	72.0	16.9	136.4	37.6	239.0	46.0
		1ull	84.4	0.3	82.5	4.4	118.8	11.5	319.0	59.3	516.5	60.4
		1zbn	297.9	0.3	298.3	2.4	318.2	11.1	439.2	35.6	572.5	37.3
		2a9x	414.0	0.2	414.3	1.4	421.5	5.0	480.0	31.7	551.9	27.8
		484d	232.3	0.3	232.6	2.5	254.6	8.1	423.6	38.8	635.2	74.3
	DNA-drug	102d	−5.8	0.1	−5.0	0.3	−2.7	1.3	6.9	5.3	15.8	6.2
		109d	−4.5	0.1	−3.6	0.4	−1.7	1.5	9.5	7.8	32.1	13.4
		121d	1.1	0.2	2.6	0.6	4.6	2.3	24.3	10.7	45.3	16.4
		127d	0.6	0.1	1.6	0.7	4.1	2.1	16.4	10.3	37.0	17.2
		129d	4.3	0.3	5.8	0.5	8.7	2.3	19.2	8.5	37.3	13.3
		166d	−2.8	0.1	−2.1	0.9	0.6	2.3	5.4	4.4	23.4	8.5
		195d	−8.5	0.2	−7.8	0.8	−6.7	3.4	9.4	9.0	27.3	10.8
		1d30	−1.1	0.2	−0.1	0.6	1.5	2.2	10.2	5.9	31.1	12.7
		1d63	−4.0	0.1	−3.1	0.6	−2.1	2.2	15.9	10.7	40.0	18.3
		1d64	0.2	0.1	0.9	0.4	1.9	1.4	6.6	3.6	20.0	8.1
		1d86	−7.8	0.1	−7.1	1.2	−4.5	3.1	11.0	6.6	41.3	18.1
		1dne	−3.4	0.2	−2.0	1.5	2.4	3.2	26.5	12.1	62.2	16.9
		1eel	−3.5	0.2	−2.9	1.5	0.3	5.3	18.9	17.4	46.6	26.8
		1fmq	−3.9	0.2	−3.1	0.8	−0.5	3.0	16.5	15.0	30.2	18.6
		1fms	0.8	0.3	1.9	0.6	3.4	1.9	14.7	5.3	27.5	10.7
		1jtl	−4.9	0.1	−4.0	1.0	0.9	3.1	19.7	9.8	54.6	18.2
		1lex	−6.8	0.1	−5.4	0.8	−1.5	2.7	20.3	8.8	49.7	19.2
		1prp	−1.7	0.1	−0.9	0.2	0.1	0.5	6.1	3.9	19.3	8.0
		227d	−5.3	0.3	−5.1	1.4	0.1	5.6	18.3	8.8	31.8	16.5
		261d	−7.5	0.1	−6.7	1.3	−2.9	2.6	17.4	9.5	51.1	14.3
		264d	4.7	0.2	6.1	0.7	8.8	1.5	23.1	7.9	44.9	15.4
		289d	−3.3	0.2	−2.9	1.8	−0.2	4.0	21.3	19.9	32.9	24.0
		298d	−2.3	0.2	−1.5	0.8	0.9	3.3	16.0	9.4	38.3	22.8
		2dbe	−5.7	0.2	−5.3	0.7	−3.7	2.8	8.3	9.4	24.1	14.8
		302d	−0.7	0.1	0.6	0.4	2.3	1.7	14.9	6.1	33.7	9.1
		311d	−2.9	0.2	−2.3	0.8	−0.6	2.7	17.3	9.6	49.6	19.8
		328d	−1.8	0.1	−1.1	0.2	−0.2	0.2	2.5	1.2	15.4	5.0
		360d	61.4	0.2	61.6	1.7	65.2	3.7	96.6	20.2	121.3	24.4
	Barnase-Barstar	1b27	−0.3	0.3	2.0	1.9	13.1	6.6	79.3	26.7	142.1	29.7
		1b2s	7.7	0.3	9.6	2.3	23.2	6.4	91.1	18.3	140.6	34.3
		1b2u	13.9	0.2	15.4	1.9	25.5	5.3	77.3	17.2	126.6	33.0
		1b3s	−19.3	0.2	−17.8	1.5	−8.4	4.5	48.3	18.9	108.7	25.4
vdW		1x1u	−9.8	0.4	−8.9	2.7	3.6	6.2	61.6	17.5	128.0	33.4
		1x1w	3.2	0.3	6.1	1.5	17.8	6.5	81.8	25.2	151.8	33.2
		1x1x	16.0	0.4	18.4	1.8	27.3	4.2	90.9	21.5	152.2	46.9
		1x1y	1.3	0.4	3.3	1.7	14.2	7.3	77.3	24.4	121.8	26.1
		2za4	10.0	0.3	12.1	1.4	17.8	5.8	53.1	24.7	100.3	26.6
	RNA-peptide	1a1t	8.3	0.4	10.8	2.2	23.6	6.1	85.4	20.4	167.1	22.0
		1a4t	−0.1	0.2	3.6	1.2	10.1	3.5	67.2	18.9	138.2	34.3
		1biv	−3.2	0.5	3.1	3.3	21.9	12.0	159.3	36.0	282.5	68.7
		lexy	35.5	0.6	41.9	2.0	57.2	7.3	146.6	24.7	234.1	47.2
		1g70	3.8	0.2	7.3	0.6	11.6	1.7	32.2	8.1	123.5	52.2
		1hji	−9.1	0.2	−8.2	0.9	−3.9	2.9	21.3	13.7	50.9	22.2
		1i9f	−42.3	0.3	−39.6	3.0	−24.5	5.3	59.6	27.1	139.5	41.4
		1mnb	8.2	0.3	14.4	2.1	25.0	4.0	78.8	19.8	162.2	26.3
		1nyb	−21.0	0.4	−20.0	2.0	−8.9	8.8	41.5	28.4	114.3	28.3
		1qfq	−6.1	0.4	−4.2	2.3	8.5	12.8	66.8	28.4	155.0	49.8
		1ull	−69.3	0.5	−65.5	2.9	−39.1	9.4	122.3	53.6	323.3	56.3
		1zbn	84.7	1.1	93.5	3.3	108.4	4.2	175.2	26.7	276.0	34.8
		2a9x	329.2	0.4	331.9	1.5	337.6	4.7	373.7	22.8	439.7	30.0
		484d	15.6	0.7	21.8	3.3	42.5	7.2	145.0	33.0	304.6	59.5
	DNA-drug	102d	−52.0	0.0	−52.3	0.1	−52.5	0.1	−53.8	0.5	−51.7	4.0
		109d	−59.3	0.0	−59.4	0.0	−59.6	0.1	−60.1	0.3	−60.4	1.2
		121d	−56.3	0.1	−56.4	0.1	−56.4	0.1	−57.5	0.6	−53.5	3.9
		127d	−20.5	0.0	−20.6	0.1	−20.5	0.2	−21.3	0.7	−20.5	2.1
		129d	−16.0	0.0	−16.2	0.0	−16.2	0.1	−16.9	0.2	−17.2	0.8
		166d	−43.2	0.0	−43.4	0.1	−43.5	0.1	−44.7	0.6	−42.0	2.1
		195d	−68.1	0.0	−68.2	0.0	−68.4	0.1	−69.4	0.5	−65.2	2.5
		1d30	−49.8	0.0	−50.0	0.0	−50.3	0.1	−51.8	1.4	−46.2	5.2
		1d63	−50.8	0.0	−51.1	0.0	−51.4	0.1	−53.0	1.5	−45.9	5.4
		1d64	−43.4	0.0	−43.5	0.1	−43.6	0.2	−44.5	0.6	−40.5	3.0
		1d86	−58.3	0.1	−58.6	0.1	−58.8	0.1	−59.8	0.8	−55.2	3.3
		1dne	−71.2	0.1	−71.5	0.1	−71.7	0.1	−73.0	0.8	−68.9	2.5
		1eel	−54.9	0.0	−55.2	0.0	−55.4	0.1	−56.2	0.3	−56.9	1.7
		1fmq	−53.3	0.0	−53.6	0.0	−53.8	0.1	−54.4	0.2	−55.0	1.4
		1fms	−47.7	0.0	−48.0	0.0	−48.2	0.1	−49.1	0.3	−50.2	2.4
		1jtl	−44.5	0.0	−44.6	0.1	−44.7	0.1	−45.3	0.9	−42.4	4.5
		1lex	−74.3	0.0	−74.6	0.0	−74.7	0.1	−75.8	0.9	−68.9	5.4
		lprp	−45.4	0.0	−45.6	0.0	−45.7	0.1	−46.7	0.7	−44.3	2.1
		227d	−49.8	0.0	−50.1	0.0	−50.1	0.1	−51.8	1.8	−44.8	5.1
		261d	−86.9	0.0	−87.1	0.0	−87.3	0.1	−88.1	1.0	−82.2	3.0
		264d	−26.3	0.0	−26.4	0.0	−26.4	0.1	−27.0	0.3	−27.6	1.7
		289d	−53.0	0.0	−53.3	0.0	−53.5	0.0	−54.3	0.2	−54.2	2.3
		298d	−47.0	0.0	−47.1	0.0	−47.3	0.0	−48.3	0.2	−49.0	2.0
		2dbe	−52.4	0.0	−52.6	0.0	−52.8	0.1	−54.1	1.4	−50.0	6.0
		302d	−31.4	0.1	−31.5	0.1	−31.5	0.1	−32.3	0.4	−30.8	2.0
		311d	−68.6	0.3	−68.3	0.0	−68.6	0.2	−69.7	0.9	−62.0	6.5
		328d	−39.5	0.0	−39.6	0.0	−39.8	0.1	−40.5	0.4	−41.0	1.2
		360d	−11.2	0.0	−11.4	0.0	−11.6	0.0	−12.4	0.3	−13.2	1.5
	Barnase-Barstar	1b27	−78.9	0.1	−79.2	0.1	−79.6	0.1	−81.2	1.3	−75.0	4.5
		1b2s	−52.6	0.0	−52.6	0.1	−52.9	0.1	−53.9	1.3	−48.0	3.3
		1b2u	−36.7	0.1	−36.8	0.1	−37.2	0.2	−38.7	1.6	−33.3	6.1
		1b3s	−112.9	0.0	−113.1	0.0	−113.7	0.2	−115.0	1.3	−110.2	3.9
Gaussian		1x1u	−108.4	0.1	−108.8	0.1	−109.4	0.1	−110.9	1.1	−105.1	5.9
		1x1w	−89.9	0.0	−84.3	0.1	−84.7	0.1	−86.3	1.4	−81.8	5.0
		1x1x	−42.1	0.1	−42.3	0.1	−42.8	0.2	−44.5	1.6	−37.9	5.0
		1x1y	−65.5	0.0	−65.6	0.0	−66.2	0.1	−67.6	1.2	−61.8	6.5
		2za4	−39.4	0.1	−39.6	0.1	−39.9	0.1	−40.8	1.5	−36.9	3.6
	RNA-peptide	1a1t	−104.3	0.1	−105.0	0.1	−105.7	0.2	−108.8	2.6	−96.1	5.2
		1a4t	−140.1	0.2	−140.8	0.2	−141.7	0.2	−144.8	3.1	−124.8	9.3
		1biv	−267.4	0.1	−268.3	0.1	−269.6	0.2	−273.5	3.0	−256.5	10.2
		1exy	−105.5	0.1	−106.8	0.2	−107.7	0.2	−112.4	3.0	−93.3	8.4
		1g70	−63.1	0.1	−63.6	0.1	−64.3	0.2	−67.9	1.1	−58.5	7.7
		1hji	−76.0	0.1	−76.6	0.1	−77.2	0.6	−79.7	1.3	−68.3	6.5
		1i9f	−164.5	0.1	−165.0	0.1	−165.4	0.6	−167.7	2.9	−148.6	4.8
		1mnb	−88.4	0.1	−88.6	0.1	−89.4	0.2	−92.8	2.1	−76.8	8.8
		1nyb	−147.0	0.1	−147.5	0.1	−148.0	0.2	−149.7	1.7	−132.4	12.3
		1qfq	−173.3	0.1	−173.4	0.1	−174.1	0.2	−176.3	2.0	−169.5	9.8
		1ull	−508.9	0.1	−509.9	0.2	−511.8	0.3	−518.8	2.0	−498.9	9.2
		1zbn	−81.0	0.1	−81.6	0.1	−82.8	0.3	−86.0	2.3	−71.4	7.5
		2a9x	245.5	0.1	245.0	0.1	244.5	0.1	242.4	1.7	245.7	7.1
		484d	−178.2	0.1	−178.9	0.2	−180.4	0.3	−184.2	3.4	−162.0	9.1

Figure 4.

The standard deviations, σGP, of 30 different measurements of the electrostatic binding free energy, ΔΔG_el, on randomly-shifted grids at finest grid spacings of 0.3 Å, 0.4 Å, and 0.5 Å, as described in Methods. a. The solvent-excluded surface. b. The van der Waal's surface. No figure was made for the Gaussian surface because σ_GP was too small to represent on this scale, as can be seen from Table 1.

More concerning though, is the trend in <ΔΔG_el> apparent in the data for the SE and vdW surfaces. For these surfaces, <ΔΔG_el> increases with grid spacing, by more than 100 kcal/mol for some complexes, and furthermore, the results at the largest grid spacings are not even correlated with those at the finest grid spacings. When R² is computed between <ΔΔG_el> at 0.3 Å and those at 1 Å for the SE surface, the results are as follows: for the DNA-drug complexes, R² = 0.41, for the barnase-barstar complexes, R² = 0.61, and for the RNA-peptide complexes, R² = 0.62. For the vdW surface, the corresponding numbers are R² = 0.59, R² = 0.12, and R² = 0.40. Clearly, rank ordering these complexes by <ΔΔG_el> with a grid spacing of 1 Å would produce misleading results. These problems are less pronounced on the Gaussian surface, where the corresponding numbers are R² = 0.98, R² = 1.0, and R² = 0.91. Indeed, <ΔΔG_el> may not even be converged with finest grid spacings of 0.3 Å, as can be seen from Figure 5, where ΔΔG_el is computed for the different surface definitions for the barnase-barstar complex (pdbid: 1b27) at very fine mesh spacings on all three surface definitions. As can be seen from Table 1 and Figure 5, although σ_GP is very small on these fine grids, <ΔΔG_el> is still changing at some of the finest grid spacings.

Figure 5.

The difference, δΔΔG_el, between ΔΔG_el computed at a finest grid spacing of 0.1 and at larger grid spacings for the barnase-barstar complex (pdbid: 1b27) using the vdW, SE, and Gaussian (α =1.4) surface definitions.

Examining the Effect of Surface Definition on ΔΔG_el

One option for improving the convergence of estimates of ΔΔG_el by grid-based solvers of the PB equation is to use a more numerically-tractable surface definition. As seen from Table 1, the Gaussian surface definition examined in the present study yielded generally smaller σ_GP and produced <ΔΔG_el> estimates that depended less on grid spacing than those produced by either the SE or vdW surfaces, indicating that the Gaussian surface may be such a numerically-tractable surface. However, ultimately a surface definition should be selected because of its ability to match experimental data. To date no single surface definition appears to offer consistently superior agreement with different experimental data sets. While the SE surface is the most popular, some groups have advocated for either the Guassian or vdW surface^{18, 22-24}, claiming that their predictions of ΔG_el, ΔΔG_el, or pKa are in better agreement with experimental results. The different surface definitions considered here give very different estimates of <ΔΔG_el>. The correlations between ΔΔGel computed based on the SE surface and those computed with the vdW surface were not very strong for the complexes considered here. R² = 0.58, 0.72 and 0.45 for the DNA-drug, RNA-peptide and barnase-barstar complexes, respectively. Similarly, when the predictions of ΔΔG_el by the Gaussian surface are compared to those with the SE surface, R² = 0.56, 0.37 and 0.20 for the DNA-drug, RNA-peptide and barnase-barstar complexes, respectively. Consequently, the different surface definitions would yield very different rankings by <ΔΔG_el>. Additionally, the attractive numerical properties of the Gaussian surface will vanish at small α because in the limit of α = 0, the Gaussian surface will be identical to the vdW surface, which has much larger σ_GP and <ΔΔG_el> estimates that depend more on grid spacing.

CONCLUSIONS

Estimates of ΔΔG_el from grid-based solvers of the PB equation inevitably contain numerical grid-discretization errors. Generally, the larger the grid spacing, the more sensitive is ΔΔG_el to the placement of the grid. Eliminating this error at larger grid spacings requires either averaging over several PB calculations on slightly shifted grids or averaging over several related molecular-dynamics snapshots as is done in MM-PBSA. However, even when this averaging is performed, the resulting best estimate of ΔΔG_el, <ΔΔG_el>, is also dependent on grid spacing. For the vdW and SE surfaces errors in <ΔΔG_el> at a grid spacing of 0.5Å can exceed 30 kcal/mol, while a 1 Å grid spacing yields errors in excess of 100 kcal/mol. Additionally, because the systematic error in <ΔΔG_el> at a particular grid spacing is very different for different complexes, ranking complexes by <ΔΔG_el> yields varying rank orders at different grid spacings. Therefore users of grid-based solvers of the PB equation that attempt to rank complexes by <ΔΔG_el>, such as in structure-based drug-design studies, must be careful to assess and control this error. We advocate that calculations at different grid spacings should be performed to verify that PB predictions have converged with respect to grid spacing.

The data presented here indicate that the commonly-used grid spacing of 0.5 Å may not be sufficiently fine to produce converged estimates of <ΔΔG_el>, at least for the vdW and SE surfaces, even though the predictions of ΔG_el appear to be well converged. Indeed, for some of the complexes, even a 0.3 Å grid appears inadequate for estimating <ΔΔG_el> in some applications.

One option for circumventing these convergence issues in grid-based solvers of the PB equation is to employ a more numerically tractable molecular surface definition. The Results show that a Gaussian surface definition with adequate α achieves much faster convergence with grid spacing, as both σ_GP and <ΔΔG_el> converge much more rapidly than for either the vdW or SE surfaces. Presumably, this numerical stability is due to the elimination of the small crevices, reentrant edges and cusps present in the SE and vdW surfaces. However, a suitable choice of surface definition must be able to match experimental data and, unfortunately, this aspect of surface selection has not been settled in the literature^{24, 25} and is not addressed in the present study. As shown in the Results, the predictions of <ΔΔG_el> produced by different surface definitions are very different. While determining an optimal surface definition remains a challenging problem, the results developed here underscore the importance of establishing convergence and selecting adequate grid size as prerequisites in any attempt to find such a surface.