DNA minicircles, where the length of DNA is below the persistence length, are highly effective, preferred, ligands for HMG-box proteins. The proteins bind to them "structure-specifically" with affinities in the nanomolar range, presumably to an exposed widened minor groove. To understand better the basis of this preference, we have studied the binding of HMG1 (which has two tandem HMG boxes linked by a basic extension to a long acidic tail) and Drosophila HMG-D (one HMG box linked by a basic region to a short and less acidic tail), and their HMG-box domains, to 88 bp and 75 bp DNA minicircles. In some cases we see cooperative binding of two molecules to the circles. The requirements for strong cooperativity are two HMG boxes and the basic extension; the latter also appears to stabilize and constrain the complex, preventing binding of further protein molecules. HMG-D, with a single HMG box, does not bind cooperatively. In the case of HMG1, the acidic tail is not required for cooperativity and does not affect binding significantly, in contrast to a much greater effect with linear DNA, or even four-way junctions (another distorted DNA substrate). Such effects could be relevant in the hierarchy of binding of HMG-box proteins to DNA distortions in vivo, where both single-box and two-box proteins might co-exist, with or without basic extensions and acidic tails.