Electrostatics of charged dielectric spheres with application to biological systems. III. Rigorous ionic screening at the Debye-Hückel level.

The unequivocal role of electrostatic forces in biological (and colloidal) systems underscores the importance of attaining accurate and rapid calculations of electrostatic forces if one wishes to faithfully simulate the electrostatic aspect of a biological system. This paper makes significant progress toward this aspect as it rigorously incorporates ionic screening at the Debye-Hückel level for an electrolyte system containing dielectric spheres of finite radii. We investigated earlier this system without mobile ions via a surface charge method. However, the need for computing a large number of Wigner rotation matrix elements per configuration can significantly slow down the numerical calculations. This difficulty was recently overcome by our Wigner-matrix-free formalism. Unfortunately, in that method ions can only be included individually, making it impractical to investigate, for example, ionic screening in a system modeled by charged dielectric spheres immersed in a solution of mobile ions. Here, we overcome this difficulty by extending the surface charge method to treat ions implicitly. Previous treatments of charged dielectric spheres in a solution of mobile ions did not emphasize the energy reciprocity of electrostatics and are largely limited to a few spheres and/or special symmetries. Our new formalism respects reciprocity and accommodates arbitrarily many dielectric spheres of different dielectric constants and sizes while being rigorous at the Debye-Hückel level. The differences, and the relationship, between our new implicit ion treatment and our previous ion-free (or explicit ion) approach are described. A closed form for the electrostatic energy with implicit ions is also provided. This new formalism speeds up the computation of the electrostatic energy in the presence of ions, and accommodates permanent and induced multipoles that are very important when the polarization effect needs to be correctly included. We also mention how the proposed method can be transformed to a numerical method for use with arbitrary nonspherical surfaces.