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Nature © Macmillan Publishers Ltd 1998 8 bounded above and below across each family. No p<3/4 can work, because CአA and there is a universal lower bound 12 for L/A 3/4 . Furthermore, rope-length minimizers in a family with L~C 3/4 must satisfy C~A; thus these tight curves have L~A 3/4 . Any knot or link arises from a 'braid' , a collection of N ascending arcs in a cylinder Z, joining N points on the bottom of Z to the same N points on the top. Z is bent into a solid torus S, the top and bottom disks are identified, and the arcs are joined to give a link in S. For a braid with bounds on its slope, curvature and horizontal strand separation, this bending fixes rope length within a uniform factor (depending on the shape of Z and S).For a family of N-component Hopf links with L~C 3/4 , we took a cylinder Z of height h and radius r. N points are distributed on the bottom disk of Z, and rotated one full turn as they ascend. The resulting braid (Fig. 1a) has N helical strands of bounded slope and curvature, provided h≈r; to separate strands, we needed r~N 1/2 . Thus the braid (and the corresponding link, Fig. 1b) has rope length L~hN~N 3/2 . As each component links every other component exactly once , CኑN(N-1); the standard projection has N(N-1)crossings, so C = N(N-1)~N 2 .For (N,N-1)-torus knots in S with LC 3/4 , we repeated this construction in the lower half of Z. To define the rest of the braid, we chose a circuit joining the N points (Fig. 1c) and slid each point along this circuit to the next while ascending through the upper half of Z. The horizontal distance a point travels is comparable to, at most, r. So, as before, we needed h~r~N 1/2 and rope length L~N 3/2 . The minimum crossing number of this knot 15 is C=N(N-2)~N 2 . (Lattice knots with the same growth of crossing number have been 10.

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Type II topoisomerases resolve problematic DNA topologies such as knots, catenanes, and supercoils that arise as a consequence of DNA replication and recombination. Failure to remove problematic DNA topologies prohibits cell division and can result in cell death or genetic mutation. Such catastrophic consequences make topoisomerases an effective target for antibiotics and anticancer agents. Despite their biological and clinical importance, little is understood about how a topoisomerase differentiates DNA topologies in a molecule that is significantly larger than the topoisomerase itself. It has been proposed that type II topoisomerases recognize angle and curvature between two DNA helices characteristic of knotted and catenated DNA to account for the enzyme's preference to unlink instead of link DNA. Here we consider the electrostatic potential of DNA juxtapositions to determine the possibility of juxtapositions occurring through Brownian diffusion. We found that despite the large negative electrostatic potential formed between two juxtaposed DNA helices, a bulk counterion concentration as small as 50 mM provides sufficient electrostatic screening to prohibit significant interaction beyond an interhelical separation of 3 nm in both hooked and free juxtapositions. This suggests that instead of electrostatics, mechanical forces such as those occurring in anaphase, knots, catenanes, or the writhe of supercoiled DNA may be responsible for the formation of DNA juxtapositions.