Formulation of a new and simple nonuniform size-modified Poisson-Boltzmann description

The nonlinear Poisson-Boltzmann equation (PBE) governing biomolecular electrostatics neglects ion size and ion correlation effects, and recent research activity has focused on accounting for these effects to achieve better physical modeling realism. Here, attention is focused on the comparatively simpler challenge of addressing ion size effects within a continuum-based solvent modeling framework. Prior works by Borukhov et al. (Phys. Rev. Lett. 1997, 79, 435; Electrochim. Acta 2000, 46, 221) have examined the case of uniform ion size in considerable detail. Generalizations to accommodate different species ion sizes have been performed by Li (Nonlinearity 2009, 22, 811; SIAM J. Math. Anal. 2009, 40, 2536) and Zhou et al. (Phys. Rev. E 2011, 84, 021901) using a variational principle, Chu et al. (Biophys. J. 2007, 93, 3202) using a lattice gas model, and Tresset (Phys. Rev. E 2008, 78, 061506) using a generalized Poisson-Fermi distribution. The current work provides an alternative derivation using simple statistical mechanics principles that place the ion size effects and energy distributions on a consistent statistical footing. The resulting expressions differ from the prior nonuniform ion-size developments. However, all treatments reduce to the same form in the cases of uniform ion-size and zero ion size (the PBE). Because of their importance to molecular modeling and salt-dependent behavior, expressions for the salt sensitivities and ionic forces are also derived using the nonuniform ion size description. Emphasis in this article is on formulation and numerically robust evaluation; results are presented for a simple sphere and a previously considered DNA structure for comparison and validation. More extensive application to biomolecular systems is deferred to a subsequent article.