Features of CPB: a Poisson-Boltzmann solver that uses an adaptive Cartesian grid

Abstract

The capabilities of an adaptive Cartesian grid (ACG)-based Poisson-Boltzmann (PB) solver (CPB) are demonstrated. CPB solves various PB equations with an ACG, built from a hierarchical octree decomposition of the computational domain. This procedure decreases the number of points required, thereby reducing computational demands. Inside the molecule, CPB solves for the reaction-field component (ϕrf ) of the electrostatic potential (ϕ), eliminating the charge-induced singularities in ϕ. CPB can also use a least-squares reconstruction method to improve estimates of ϕ at the molecular surface. All surfaces, which include solvent excluded, Gaussians, and others, are created analytically, eliminating errors associated with triangulated surfaces. These features allow CPB to produce detailed surface maps of ϕ and compute polar solvation and binding free energies for large biomolecular assemblies, such as ribosomes and viruses, with reduced computational demands compared to other Poisson-Boltzmann equation solvers. The reader is referred to http://www.continuum-dynamics.com/solution-mm.html for how to obtain the CPB software.

Keywords: Poisson-Boltzmann equation; adaptive Cartesian grid; electrostatic potential; electrostatics; implicit solvent model; surface.

Figure 1

A slice through the computational domain, containing a barnase (PDBid: 1B3S) and ionic solution dielectric regions, showing the adaptive Cartesian grid of CPB. This figure shows how CPB assigns more grid points to regions where the solution potential varies most rapidly (i.e., near the molecular surface).

Figure 2

(a) An electrostatic potential surface map (EPSM) of the isolated 16S rRNA (PDBid: 1J7T) computed without its cationic aminoglycosidic paromomycin (net charge=+4e) binding partner at 0.1 M NaCl. The paromomycin is shown as a gray surface. This cationic drug binds in the deep enlarged major groove created by a distorted phosphate backbone containing unpaired and bulging adenines. (b) A close-up view of the paromomycin binding site in the full 16S rRNA of the 30S ribosomal subunit (PDBid: 1FJG) at 0.1 M NaCl. The paromomycin is buried when bound to the A-site of the 16S rRNA in the 30S ribosomal subunit. (c) An EPSM of barnase (PDBid: 1B3S) immersed in a 0.1 M NaCl solution. (d) An EPSM of barstar (PDBid: 1B3S) embedded in a 0.1 M NaCl solution. (e) An EPSM of the barnase-barstar complex (PDBid: 1B3S) at 0.1 M NaCl. The electrostatic potential (ϕ) coloring scales (in kT/e) are as follows: green (G) = 5, blue (B) = 2.5, white (W) = 0, red (R) = −2.5, and yellow (Y) = −5 in panel a. For panel b, (G,B,W,R,Y)=(4,2,0,−2,−4). For panels c, d, and e, (G,B,W,R,Y)=(3,1.5, 0,−1.5,−3). In views (c) and (d), which show the binding interface, one observes that barnase has a large region of positive ϕ and barstar has a complementary negative ϕ.

Figure 3

Electrostatic binding free energies (ΔΔG_el) calculated with and without LSR as functions of the finest grid spacing. The data points show the average of 30 calculations for each grid spacing, and the error bars depict their standard deviations, as outlined in Methods. (a) Paromomycin bound to a 16S rRNA fragment (PDBid: 1J7T). (b) Paromomycin bound to a 30S ribosomal subunit (PDBid: 1FJG). (c) Barnase-barstar complex (PDBid: 1B3S). The NaCl concentration was set to 0.1 M.

Figure 4

Electrostatic potential (ϕ) surface maps generated with the nonlinear Poisson-Boltzmann equation (PBE) for a 40 base pair ideal A-RNA helix embedded in a NaCl solution. Using the notation in Figure 2, the ϕ scale in kT/e is (G,B,W,R,Y) = (+5,+2.5,0,−2.5, −5). The gray meshes represent isocontours where the difference between the concentration of counterions given by the size-modified PBE and that given by the nonlinear PBE is (a) 0.1 M and (b) 1.0 M NaCl.

Figure 5

(a), (b), and (c) show three views of the electrostatic potential (ϕ) surface map of the Host factor for Q beta (Hfq) protein with its companion RNA shown in black (PDBid: 1KQ2). (b) shows a close-up view of (a), highlighting the RNA-binding region. (c) shows the back view of (a). (d), (e), and (f) show isopotential contours around the Hfq protein. The blue contour has a ϕ of +1 kT/e, and the red contour has a ϕ of −1 kT/e. The dipolar nature of the Hfq protein is clearest in (e), which is a side view of (d). The ϕ scale is as follows: (G,B,W,R,Y) = (3,1.5,0, −1.5, −3) kT/e.

Figure 6

Two views of the electrostatic potential (ϕ) surface map of the rRNA (PDBid: 1HC8) obtained with the nonlinear Poisson-Boltzmann equation and a finest grid spacing of 0.1 Å. The ϕ scale is as follows: (G,B,W,R,Y) = (5.5,2.75,0, −2.75, −5.5) kT/e. The Mg²⁺ binding sites are displayed as gray spheres. Panel (a) highlights the two Mg²⁺-binding sites (MG1163 and MG1167) with the largest negative ϕ’s.

Figure 7

(a) shows the electrostatic potential (ϕ) surface map (EPSM) for the 30S ribosomal subunit immersed in a 0.1 M NaCl solution (PDBid: 1FJG). The black arrow points to the binding site of the paromomycin, which is represented as a translucent gray surface. A close-up view of the binding site showing the high quality of the generated EPSM is portrayed in Figure 2b (b) shows a similar EPSM for the satellite panicum mosaic viral capsid (PDBid: 1STM) at 0.1 M NaCl. The ϕ scale is as follows: (G,B,W,R,Y) = (4,2,0, −2, −4) kT/e. The nonlinear PBE was solved using a finest grid spacing of 0.3 Å.