The pair interaction between two stiff parallel linear DNA molecules depends not only on the distance between their axes but on their azimuthal orientation. The positional and orientational order in columnar B-DNA assemblies in solution is investigated, based on the DNA-DNA electrostatic pair potential that takes into account DNA helical symmetry and the amount and distribution of adsorbed counterions. A phase diagram obtained by lattice sum calculations predicts a variety of positionally and azimuthally ordered phases and bundling transitions strongly depending on the counterion adsorption patterns.PACS numbers: 82.35. Rs, 87.14.Gg, 82.70.Dd DNA is a polyelectrolyte molecule. In aqueous electrolyte solutions, cations along its helices dissociate from it and dissolve into the mixture, leaving behind negative charges that reside on the phosphates of the DNA backbone. According to the Manning condensation theory, a fraction of the cations condenses into the Bjerrum layer near the molecular surface [1]. If some of the ions specifically adsorb onto DNA, its surface could be almost fully neutralized [2] or even overcharged [3]. Far from its axis, DNA can be apprehended as a charged cylinder. If the charge were continuously smeared, there would be only an electrostatic repulsion between two molecules, exponentially screened by the electrolyte. However, the net distribution of charge on the molecules is not homogeneous and this can dramatically alter the interaction potential at intermediate distances. Indeed, in order to condense DNA in an aggregate, one has either to apply osmotic stress [4] or use condensing agents, such as salts with Mn 2+ , Cd 2+ , spermidin, protamine or cobalt hexammine [5] cations. These cations are known to specifically adsorb on DNA, predominantly in the DNA grooves [6]. Other counterions, such as, e.g., Ca 2+ or Mg 2+ , that have strong affinity to phosphates and adsorb preferentially on the strands do not induce DNA aggregation. Obviously, one effect of these specifically adsorbing counterions is the reduction of the net charge on the DNA. However, were this to be the only effect, it would have been hard to explain the observed sensitivity to the sort of counterions of DNA condensation [5] and of the mesomorphism of resulting aggregates [7].Recently a new explanation of the features of DNA aggregation was suggested [8] resting on a Debye-Bjerrum theory of electrostatic interaction between helical macromolecules [9]. The theory offered first a formalism for a description of interaction between cylindrical molecules (with parallel axes) for arbitrary surface charge distributions on the molecules [9]. Then it explored its consequences for helical charge distributions, including those typical for double stranded B-and A-forms of DNA [9,10]. Various patterns of adsorbed counterions, including those spiraling through DNA major and minor groves, were considered. Thus, the effect of helically structured separation between negative and positive charges on each molecule was rationalized, explainin...