Accurate Evaluation of Electrostatics for Macromolecules in Solution

Abstract

Abstract. Most biochemical processes involve macromolecules in solution. The corresponding electrostatics is of central importance for understanding their structures and functions. An accurate and efficient numerical scheme is introduced to evaluate the corresponding electrostatic potential and force by solving the governing Poisson-Boltzmann equation. This paper focuses on the following issues: (i) the point charge singularity problem, (ii) the dielectric discontinuity problem across a molecular surface, and (iii) the infinite domain problem. Green's function associated with the point charges plus a harmonic function is introduced as the zeroth order approximation to the solution to solve the point charge singularity problem. A jump condition capturing finite difference scheme is adopted to solve the discontinuity problem across molecule surfaces, where a body-fitting grid is used. The infinite domain problem is solved by mapping the outer infinite domain into a finite domain. The corresponding stiffness matrix is symmetric and positive definite, therefore, fast algorithm such as preconditioned conjugate gradient method can be applied for inner iteration. Finally, the resulting scheme is second order accurate for both the potential and its gradient.

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

“…There has been recently significant interest from applied mathematicians to develop PB solvers with interface methods that specifically deal with the continuity and accuracy issues at the molecular surfaces. Methods such as the jump condition capturing finite difference scheme, 63,64 and the matched interface and boundary 62,65,66 seem very promising and we are currently investigating ways to incorporate them in AQUASOL. Finite element methods represent a viable alternative to the finite difference methods discussed above. They allow for non-Cartesian meshes that provide better approximation of the geometry of the solutes.…”

Poisson–Boltzmann–Nernst–Planck model

“…19,34 Split ψ into the regular part ψ(r) and the singular part ψ(r), i.e., ψ = ψ + ψ, where ψ(r) is defined only in m . 19,35 Define ψ(r) = ψ * (r) + ψ 0 (r), here ψ * (r) is the Green's function which can be given analytically…”

SDPBS Web Server for Calculation of Electrostatics of Ionic Solvated Biomolecules

“…Consequently, the PBE solution can be found from calculating Ψ andΦ without involving any singular difficulty. Note that our solution decomposition differs from the ones from [11,12,67,68]. For example, in [12], the PBE solution was split within Dp only, resulting in different interface problems from ours.…”

“…So far, several PBE solution decomposition formulas were proposed [11,12,59,67,68]. The one from [12] has been adopted to the improvement of software packages MIBPB [22] and PBSA [57] for example.…”

“…So far, several PBE solution decomposition formulas were proposed [11,12,59,67,68]. The one from [12] has been adopted to the improvement of software packages MIBPB [22] and PBSA [57] for example. In SDPBS, we used the one developed by our group [59], which splits the PBE solution as a sum of three functions, G, Ψ, andΦ (see (2.3)), according to electrostatic contributions from the biomolecular charges, the boundary and interface conditions, and the ionic solvent charges, respectively.…”