Electrostatic free energy of the DNA double helix in counterion condensation theory

Polyelectrolyte theory based on counterion condensation is extended from the standard line charge model to helical and double helical charge arrays. The number of condensed counterions turns out to be the same as for a line charge with charge density equal to the axial charge density of the helix. Also, the logarithmic salt dependence of the electrostatic free energy is the same in the range of lower salt concentration, so that the limiting laws remain unchanged. However, the internal free energy of the condensed layer of counterions and the overall electrostatic free energy depend on the helical parameters. At higher salt, the free energies of both single and double helix are negative, indicating electrostatic stabilization of the helical charge lattices due to the mixing entropy of the condensed counterions. Except at very low salt, the free energy of a single helix is higher than the free energy of a double helix with twice the charge density. With B-DNA parameters and single strands modeled as single helices, the predicted salt dependence of the free energy of transition from double helix to separated single strands has a maximum at approximately 0.2 M salt, close to the location in the laboratory of this well-known feature of the DNA strand separation transition. We also calculate the electrostatic free energy for the transition of the DNA double helix from the B to the A conformation. The B form is electrostatically stable over most of the salt range, but there is a spontaneous electrostatic transition to A near 1 M salt. The electrostatic free energy values are close to the experimental values of the overall (electrostatic plus non-electrostatic) transition free energies for A-philic base pair sequences. We are led to suggest that the experimentally observed B-to-A transition for A-philic sequences near 1 M salt in water is governed by the polyelectrolyte properties of these two conformations of the DNA double helix. The effect of ethanol, however, cannot be attributed to lowering of the bulk dielectric constant.