ZAP

Zap is a PB solver that has implemented the idea of smooth dielectric boundary conditions based on radial, atom-centered, Gaussians. Figure 1 illustrates the nature of the dielectric at the boundary with this approach compared to a discontinous, "molecular" boundary.

The A parameter controls the switch between internal and external dielectrics and the work of Christine Kitchen, University of Sheffield, showed good correspondence for small molecule solvation between molecular and gaussian functions if A is chosen carefully.  One reason, illustrated in the next picture, is that, at a given cutoff, an isocontour of the gaussian-based dielectric function looks very like a molecular surface.

In addition to reproducing small molecule calculations the results for proteins are also very similar, as shown in figures 3 and 4:

The only significant difference we could find between protein electrostatics and small molecule electrostatics with the gaussian model is that the former would give numerical results similar to a molecular dielectric model but with an internal dielectric doubled, i.e.
gaussian + internal dielectric of 4.0= molecular + internal dielectric of 2.0
As many studies have suggested a higher internal dielectric for proteins than of its constituent amino-acids (i.e. 2.0), this result was not unpleasing.

The smooth dielectric approach gives much better numerical stability with respect to displacements on the grid used to solve the PB equation, as illustrated in figure 5:

This comes about because the induced surface charge is much more evenly distributed radially, figures 6 and 7:

This leads to the first rigorous error analysis for PB (work, again, of Christine Kitchen), Figure 8:

It's a bit fuzzy but figure 8 shows the dependence on the standard deviation of error compared to the result at very fine grid scale.

The smooth dielectric approach also helps control error when calculating binding energies.  Figure 9 illustrates the approach to a fast method:

The dependence upon the distance from the final focussing box and the ligand and the final answer shows good behaviour with the gaussian model. Figure 10
shows this for two proteins, CBS.pdb and COY.pdb

Finally, we can also extract the gradients with respect to position, or forces, from the calculation because the derivatives of the dielectric function are straight-forward.