Polyelectrolyte electrostatics: Salt dependence, entropic, and enthalpic contributions to free energy in the nonlinear Poisson–Boltzmann model

Abstract

AbstractUsing the nonlinear Poisson–Boltzmann (PB) equation, expressions for the nonspecific salt dependence of the free energy, and the entropic and enthalpic contributions to the electrostatic free energy of a polyelectrolyte are derived. The results are generalized for any number and valence of mobile ions, and now permit a more rigorous treatment of the salt dependence of polyelectrolyte–ligand binding using the PB equation. The salt dependence of the electrostatic free energy depends on an “osmotic pressure like” term identified previously [K. Sharp and B. Honig (1990), Journal of Physical Chemistry, Vol. 94, pp. 7684–7692], which in turn depends on the integrated excess/deficit of all mobile ion species around the polyelectrolyte. For the simple salt case (two ions only) this expression is equivalent to that derived previously [H. Eisenberg (1976), Biological Macromolecules and Polyelectrolytes in Solution, Clarendon. Oxford] in terms of Donnan, or preferential interaction coefficients. Moreover, since all ions enter into the PB expression for the salt dependence of the free energy in an identical way, it is equally applicable to polyelectrolytes with charge of one sign (e.g., nucleic acids), with oppositely charged groups (such as proteins), and even polyelectrolytes that have no net charge, for which the Donnan coefficient is ill‐defined. The distribution of counterions around three simple polyelectrolyte geometries—spherical, cylindrical, and planar—was examined. In each case the excess number of counterions associated with the polyelectrolyte is not constant, but increases with decreasing salt. Although a limiting value is reached at low salt for a cylinder, in agreement with the limiting law condensation models, for the spherical geometry the number of associated counterions decreases to zero, while for the planar case it increases to 100%. For the cylindrical case Donnan coefficients agree closely with those computed previously by Monte Carlo simulations [M. C. Olmstead, C. F. Anderson, and M. T. Record (1991) Biopolymers, Vol. 31, pp. 1593–1604]. An all atom B‐DNA model shows about 5% less accumulation of counterions than a cylinder, due to the helical charge arrangement and groove structure. The entropic and enthalpic contributions to the salt‐dependent free energy of a cylinder were examined in the PB model. In addition to the well known ion cratic (redistribution) entropy, there are significant contributions from electrostatic enthalpy, and from dielectric entropy (water re‐orientation due to the electrostatic field from polyelectrolyte charges and salt ions). The latter entropy contribution has not previously been taken into account in salt‐dependent polyelectrolyte phenomena, but at higher salt concentrations (> 0.01M), this can be as large as the cratic entropy contribution. © 1995 John Wiley & Sons, Inc.