positive supercoiling activity is activated in vitro by Cdk1 phosphorylation, suggesting that it is likely one of the physiological activities directly involved in mitotic chromosome assembly. It is postulated that this activity of condensin I helps fold DNA and promotes topoisomerase II-mediated resolution of sister chromatids. Early single-DNA-molecule experiments also demonstrated in real time that condensin I is able to compact DNA in an ATP-hydrolysis dependent manner. Most recently, single-molecule experiments have demonstrated that budding yeast condensin I is able to translocate along dsDNA (motor activity) and to "extrude" DNA loops (loop extrusion activity) in an ATP hydrolysis-dependent manner. In the latter experiments, the activity of individual condensin complexes on DNA was visualized by real-time fluorescence imaging, revealing that condensin I indeed is a fast loop-extruding motor and that a single condensin I complex can extrude 1,500 bp of DNA per second in a strictly ATP-dependent manner. It has been proposed that condensin I anchors DNA between Ycg1-Brn1 subunits and pulls DNA asymmetrically to form large loops. Moreover, it has been shown that condensin complexes can traverse each other, forming dynamic loop structures and changing their sizes. It is unknown how condensins might act on nucleosomal DNA. Recent development of a reconstitution system has identified the histone chaperone FACT as an essential component of condensin I-mediated chromosome assembly in vitro, providing an important clue to this problem. It has also been shown that condensins can assemble chromosome-like structures in cell-free extracts even under the condition where nucleosome assembly is largely suppressed. This observation indicates that condensins can work at least in part on non-nucleosomal DNA in a physiological setting. Only limited information is currently available as to the functional contribution of individual subunits of condensins to their activities. An SMC2-SMC4 dimer has an ability to reanneal complementary single-stranded DNA. This activity does not require ATP. For eukaryotic complexes, it has been reported that HEAT repeat subunits contribute to part of DNA binding and to the assembly of chromosome axes. Flexible and extendable nature of HEAT repeats could underlie the dynamic action of condensins and the architecture of mitotic chromosomes. Mathematical modeling Several attempts on mathematical modeling and computer simulation of mitotic chromosome assembly, based on molecular activities of condensins, have been reported. Representative ones include modeling based on loop extrusion, stochastic pairwise contacts and a combination of looping and inter-condensin attractions. Functions in chromosome assembly and segregation Mitosis In human