Charge renormalization of helical macromolecules

Some time ago a theory of electrostatic interaction between helical macromolecules was proposed (Kornyshev and Leikin 1997 J. Chem. Phys. 107 3656): the Kornyshev-Leikin (KL) theory. We place this theory on a more rigorous statistical mechanical grounding, starting from the free energy that can be derived from a grand partition function. We see that the long range behaviour of the force is indeed given by the KL theory, no matter whether the distributions of 'condensed' ionic charge are at the surface of the macromolecule or extend away from it. Thus, for the limiting behaviour, we need only self-consistently calculate the distribution of the condensed fraction of ions for a single macro-ion. This distribution can be related back to interaction parameters: KL parameters. Furthermore, we are able to see within the formalism where corrections due to the hard core radius of the ion enter. For the adjustment of the 'condensed' ions, we show an expression for the leading order contribution, as well as relevant decay lengths. As a demonstration of the theoretical 'machinery', as well as a study of qualitative effects, we calculate the KL parameters in one instance. We use a DNA-like surface charge distribution, where a fraction of the ions are assumed to be bound in the grooves at the surface of a DNA molecule, whereas the rest of the charge distribution is calculated self-consistently. Also, the electrostatic contribution to the counter-ion binding potentials that ions experience within the grooves can be calculated.