Electrical double layer: A numerical treatment of stern layer in biomolecular electrostatics

Poisson–Boltzmann equation (PBE) is widely used in the context of deriving the electrostatic energy of macromolecular systems and assemblies in aqueous salt solution. Macromolecules and their ion penetrability with the presence of stern layer have been discussed theoretically and analytically. While numerous numerical solvers for the 3D PBE have been developed, the integral equation formulation for the boundary treatments used in these methods has only been loosely addressed, especially in the ion exclusion stern layer. The de facto standard in current linear PBE implementations is to estimate the potential at the outer boundaries using the (linear) Debye-Hückel (DH) approximation. However, as assessment of how these outer boundary treatments affect the overall solution accuracy in the stern layer does not appear to have been previously made. As will be demonstrated here, this DH approximation can under certain conditions, produce completely erroneous estimates of the potential and energy salt dependencies. In this work, the sets of boundary conditions are invoked that take into account the impenetrability of the ions to the macromolecule. Using surface integral equation, this new treatment is able to give an accurate description of the electrostatic potential distribution, electrostatic solvation free energy etc. not only in a macromolecular system by means of continuum model but also focus on physics of the ion impenetrable stern layer. The accuracy of the results obtained by using the boundary element method (BEM) is tested by comparison with analytical Tanford–Kirkwood results for a model spherical solute system. Finally, the author also examined how the general ion exclusion layers would tend to increase the surface electrostatic potential under physiological salt conditions. To facilitate presentation and computational domain, attention is restricted here to the 3D spherically symmetric linear PBE. Though geometrically limited, the modeling principles nevertheless extend to general linear PBE solvers. The 3D linear PBE model can also be used to benchmark and validate the salt effect prediction capabilities of existing PBE solvers. This choice promises to be particularly useful in the context of biological applications, where the solvation energy, arising from medium polarization, has a prime role.