Transformed Poisson-Boltzmann relations and ionic distributions

For many applications, charge distributions around macromolecules in aqueou s solution are of greater interest than the electrical potential. We show t hat it is possible to use the Poisson-Boltzmann (PB) relation to develop di fferential equations for the ionic distributions. The solutions to these eq uations are the integral distribution functions whose derivatives give the charge density functions for counterions and colons. In this formalism the salt-free atmosphere of a cylindrical polyelectrolyte is very easily solvab le for the counterion. Quantities such as the "condensation radius" (Le Bre t, M.; Zimm, B. Il. Biopolymers 1984, 23, 287-312) and the "Bjerrum associa tion radius" (Bjerrum, N. Investigations on Association of Ions, I. In Niel s Bjerrum Selected Papers; Munksgaard: Copenhagen, 1926; pp 108-19) appear naturally as inflection points in curves of the counterion distribution fun ctions. Moreover, a number of the properties of condensation theory arise a s scaling limits of the transformed PB equation. In the presence of added s alt separate equations can be derived for the excess charge distributions o f counterions and colons. In this case the total excesses of counterion phi (ct) and of colon phi (co) are simply related to experiment. Various combi nations of these two quantities lead to formulas for (1) the total charge, (2) Donnan exclusion, (3) counterion release (Record, M. T.; Lohman, T. M.; de Haseth, P. J. Mel. Biol. 1976, 107, 145-158), and (4) fraction of "cond ensed" ions. Bjerrum's theory of ion assocition and Manning's theory of cou nterion condensation are discussed in the context of the transformed Poisso n-Boltzmann theory.