  We scrutinize the effect of polyvalent ions on polymer-DNA interactions. We
extend a recently developed test charge theory to the case of a stiff polymer
interacting with a DNA molecule in an electrolyte mixture. The theory accounts
for one-loop level electrostatic correlation effects such as the ionic cloud
deformation around the strongly charged DNA molecule as well as image-charge
forces induced by the low DNA permittivity. Our model can reproduce and explain
various characteristics of the experimental phase diagrams for polymer
solutions. First, the addition of polyvalent cations to the electrolyte
solution results in the attraction of the negatively charged polymer by the DNA
molecule. The glue of the like-charge attraction is the enhanced shielding of
the polymer charges by the dense counterion layer at the DNA surface. Secondly,
through the shielding of the DNA-induced electrostatic potential, mono- and
polyvalent cations of large concentration both suppress the like-charge
attraction. Within the same formalism, we also predict a new opposite-charge
repulsion effect between the DNA molecule and a positively charged polymer. In
the presence of polyvalent anions such as sulfate or phosphate, their repulsion
by the DNA charges leads to the charge screening deficiency of the region
around the DNA molecule. This translates into a repulsive force that results in
the decomplexation of the polymer from DNA. This opposite-charge repulsion
phenomenon can be verified by current experiments and the underlying mechanism
can be beneficial to gene therapeutic applications where the control over
polymer-DNA interactions is the key factor.
