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

Abstract

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 programmed 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.

An accelerated nonlocal Poisson‐Boltzmann equation solver for electrostatics of biomolecule

“…We then set D = ( − 2,2) 3 , τ = 1, and η = 4 to yield Ω 7 = ( − 3,3) 3 and Ω = ( − 6,6) 3 . We calculated the series solutions U ( r ) and W ( r ) using their first 20 terms, which were known to be accurate enough . The overrelaxation parameter ω of our overlapped box iterative method was set as 1.25 in solving for (Ψ,Ψ w ) or for ( p k , q k ).…”

“…Both U and W are known functions, which are represented in the simple series expressions in terms of Legendre polynomials and modified spherical Bessel functions. 15 Clearly, they are the analytical solution of the test model (27). Hence, they can be used to validate our new NMPBE program package.…”

“…We calculated the series solutions U(r) and W(r) using their first 20 terms, which were known to be accurate enough. 15 The overrelaxation parameter of our overlapped box iterative method was set as 1.25 in solving (16) for (Ψ, Ψ w ) or (19) for (p k , q k ). We also constructed four nested meshes using the mesh size h = 1.0, 0.5, 0.25, 0.125 for testing the convergence of the new NMPBE solver.…”

“…Using the new NMPBE software package, we made numerical experiments on 12 proteins and compared the performance of our new solver with that of the finite element solver reported in Xie and Jiang . We also validated this new solver and program package using a NMPBE test model with an analytical solution for a protein domain being a ball containing multiple charges . In these experiments, our new solver reduced the total CPU runtime of the finite element solver up to 79 % without compromising solution accuracy.…”

“…11 We also validated this new solver and program package using a NMPBE test model with an analytical solution for a protein domain being a ball containing multiple charges. 15 In these experiments, our new solver reduced the total CPU runtime of the finite element solver up to 79% without compromising solution accuracy. For example, the new solver sharply reduced the CPU time of the finite element solver from 512 seconds to 114 seconds for a protein (with the PDB ID 4PTI) on a mesh with 541 329 mesh nodes on one 3.7 GHz processor of the Mac Pro Workstation with 64 GB memory (see Tables 2 and 3).…”