Analytical solutions of nonlocal Poisson dielectric models with multiple point charges inside a dielectric sphere

The nonlocal dielectric approach has led to new models and solvers for predicting electrostatics of proteins (or other biomolecules), but how to validate and compare them remains a challenge. To promote such a study, in this paper, two typical nonlocal dielectric models are revisited. Their analytical solutions are then found in the expressions of simple series for a dielectric sphere containing any number of point charges. As a special case, the analytical solution of the corresponding Poisson dielectric model is also derived in simple series, which significantly improves the well known Kirkwood's double series expansion. Furthermore, a convolution of one nonlocal dielectric solution with a commonly used nonlocal kernel function is obtained, along with the reaction parts of these local and nonlocal solutions. To turn these new series solutions into a valuable research tool, they are programed as a free fortran software package, which can input point charge data directly from a protein data bank file. Consequently, different validation tests can be quickly done on different proteins. Finally, a test example for a protein with 488 atomic charges is reported to demonstrate the differences between the local and nonlocal models as well as the importance of using the reaction parts to develop local and nonlocal dielectric solvers.

Figure 1

A unit spherical solute region $D_p$ containing 488 point charges from a protein (2LZX) in a cubic domain ${[-2,2]}^3$.

Figure 2

A comparison of model 1 with model 2 and the local Poisson model in their reaction functions of $\mathrm{\Psi }$ defined on a uniform mesh of ${[-2,2]}^3$ with $h=0.1$ for the unit spherical solute region $D_p$ containing 488 point charges from a protein (2LXZ).                           

Figure 3

A comparison of three linear interpolation functions of the analytical solution $\mathrm{\Phi }$ for model 1 defined on three uniform meshes of ${[-2,2]}^3$ with $h=0.1$, 0.05, and 0.01. Here, $D_P$ contains 488 point charges, and the figures are plotted by setting $y$ coordinates to be zero.