During the random cyclization of long polymer chains, knots of different types are formed. We investigated experimentally the distribution ofknot types produced by random cyclization of phage P4 DNA via its long cohesive ends. The simplest knots (trefoils) predominated, but more complex knots were also detected. The fraction of knots greatly diminished with decreasing solution Na+ concentration. By comparing these experimental results with computer simulations of knotting probability, we calculated the effective diameter ofthe DNA double helix. This important excluded-volume parameter is a measure of the electrostatic repulsion between segments of DNA molecules. The calculated effective DNA diameter is a sensitive function of electrolyte concentration and is several times larger than the geometric diameter in solutions of low monovalent cation concentration.A classic problem in polymer physics is the probability that the random cyclization of a polymer produces a knot of a particular topology. The issues were formulated more than 30 years ago by Frisch and Wasserman (1) and by Delbruck (2). The initial solution was provided by a Monte Carlo simulation in 1974 (3), and since then several computational analyses of the knotting probability of polymer chains have appeared (reviewed in refs. 4 and 5).Experimental measures of knotting frequency have not been made, however, despite the considerable interest in these forms. Ever since the production oflinked hydrocarbon rings, organic chemists have striven to synthesize knotted molecules. These attempts have recently come to fruition with the elegant and complete synthesis of a hydrocarbon knot by Dietrich-Buchecker and Sauvage (6). A distinct synthetic route based on the properties of nucleic acids has been used by Seeman's group to synthesize different types of knots (7).Knots made of DNA, first identified in 1976 (8), are much easier to synthesize and to analyze than other knotted polymers. Knotted DNA has been identified in numerous in vitro and in vivo experiments (9, 10). The topology of the knots is diagnostic of the mechanism of the enzymes that produce them and of the structure of the DNA substrates (9). The experimental studies of DNA knots have been guided by the development of mathematical methods for analyzing DNA topology (11,12).DNA is thus the most suitable polymer for the experimental investigation of the probability of knotting. The measurement of the probability of knotting duplex DNA is of special interest because it provides an excellent means for determining the effective diameter of DNA. This important parameter characterizes the excluded volume of DNA. The effective diameter of DNA is the diameter of an uncharged polymer chain that mimics the conformational properties of actualThe publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. electrically charged DNA. Because DNA is a highly charge...

Type II DNA topoisomerases catalyze the interconversion of DNA topoisomers by transporting one DNA segment through another. The steady-state fraction of knotted or catenated DNA molecules produced by prokaryotic and eukaryotic type II topoisomerases was found to be as much as 80 times lower than at thermodynamic equilibrium. These enzymes also yielded a tighter distribution of linking number topoisomers than at equilibrium. Thus, topoisomerases do not merely catalyze passage of randomly juxtaposed DNA segments but control a global property of DNA, its topology. The results imply that type II topoisomerases use the energy of adenosine triphosphate hydrolysis to preferentially remove the topological links that provide barriers to DNA segregation.

Xenopus 13S condensin converts interphase chromatin into mitotic-like chromosomes, and, in the presence of ATP and a type I topoisomerase, introduces (+) supercoils into DNA. The specific production of (+) trefoil knots in the presence of condensin and a type II topoisomerase shows that condensin reconfigures DNA by introducing an ordered, global, (+) writhe. Knotting required ATP hydrolysis and cell cycle-specific phosphorylation of condensin. Condensin bound preferentially to (+) supercoiled DNA in the presence of ATP but not in its absence. Our results suggest a mechanism for the compaction of chromatin by condensin during mitosis.

Type II DNA topoisomerases actively reduce the fractions of knotted and catenated circular DNA below thermodynamic equilibrium values. To explain this surprising finding, we designed a model in which topoisomerases introduce a sharp bend in DNA. Because the enzymes have a specific orientation relative to the bend, they act like Maxwell's demon, providing unidirectional strand passage. Quantitative analysis of the model by computer simulations proved that it can explain much of the experimental data. The required sharp DNA bend was demonstrated by a greatly increased cyclization of short DNA fragments from topoisomerase binding and by direct visualization with electron microscopy.T ype II topoisomerases are essential enzymes that pass one DNA through another and thereby remove DNA entanglements. They make a transient double-stranded break in a gate segment (G segment) that allows passage by another segment (T segment) of the same or another DNA molecule (reviewed in refs. 1 and 2). Thus, these enzymes have the potential to convert real DNA molecules into phantom chains that freely pass through themselves to generate an equilibrium distribution of knots, catenanes, and supercoils.The actual picture is more complex and more interesting. The observed steady-state fractions of knotted, catenated, and supercoiled DNAs produced by type II topoisomerases are up to two orders of magnitude lower than at equilibrium (3). Thermodynamically, there is no contradiction in this finding because the enzymes use the energy of ATP hydrolysis. Active topology simplification by topoisomerases has an important biological consequence. It helps explain how topoisomerases can remove all DNA entanglements under the crowded cellular conditions which favor the opposite outcome. The challenge, though, is to understand how type II topoisomerases actively simplify DNA topology. Topology is a global property of circular DNA molecules, and yet it is determined by the much smaller topoisomerases, which can act only locally.Two models have been suggested to explain active simplification of DNA topology. First, if type II topoisomerases corral the T segment within a small loop of DNA containing the G segment, active disentanglement would result (3). However, it was pointed out when this model was suggested (3) that to account for the large effects observed, the loop trapping would need substantial energy input from ATP hydrolysis for the transport of the DNA along the enzymes, and these enzymes are energetically efficient (4). Moreover, no direct experimental data supporting the model have been presented.Second, a kinetic proofreading model proposed that two successive bindings of T segments are required for strand passage (5). The first binding event converts the enzyme bound with a G segment to an activated state. An assumption of the model is that segment collision in the knotted state occurs about 1͞P k times more often than in the unknotted state, where P k is the equilibrium probability of knotting. Our computer simulations below show that th...