Asymptotic solution of the cylindrical nonlinear Poisson–Boltzmann equation at low salt concentration: Analytic expressions for surface potential and preferential interaction coefficient

The analytic solution to the nonlinear Poisson–Boltzmann equation describing the ion distributions surrounding a nucleic acid or other cylindrical polyions as a function of polyion structural quantities and salt concentration ([salt]) has been sought for more than 80 years to predict the effect of these quantities on the thermodynamics of polyion processes. Here we report an accurate asymptotic solution of the cylindrical nonlinear Poisson–Boltzmann equation at low to moderate concentration of a symmetrical electrolyte (≤0.1 M 1:1 salt). The approximate solution for the potential is derived as an asymptotic series in the small parameter ɛ(−1), where ɛ ≡ κ(−1)/a, the ratio of the Debye length (κ(−1)) to the polyion radius (a). From the potential at the polyion surface, we obtain the coulombic contribution to the salt–polyelectrolyte preferential interaction (Donnan) coefficient (Γ [Formula: see text]) per polyion charge at any reduced axial charge density ξ. Γ [Formula: see text] is the sum of the previously recognized low-salt limiting value and a salt-dependent contribution, analytically derived here in the range of low-salt concentrations. As an example of the application of this solution, we obtain an analytic expression for the derivative of the midpoint temperature of a nucleic acid conformational transition with respect to the logarithm of salt concentration (dT(m)/d ln[salt]) in terms of [salt] and nucleic acid structural quantities. This expression explains the experimental observation that this derivative is relatively independent of salt concentration but deviates significantly from its low-salt limiting value in the range 0.01–0.1 M.