Figure 1. Topology basics. (a) A doughnut and a tea cup have the same topology, as evidenced by the fact that one can be freely deformed into the other, while clearly having different geometries. (b) The DNA strands of a double helix become topologically linked when the 5¢-end of each strand is ligated to the 3¢-end of the same strand; the linking number equals the initial number of helical turns. In this illustration, the linking number between the blue and red strands is three. (c) The topology or linking number of the strands does not change, no matter how distorted, as long as neither strand is broken.

Figure 2. Reducing the number of times two strands are linked creates negatively supercoiled DNA. If the strands of a relaxed DNA circle are broken and untwisted several times before being resealed, the resulting closed circle will negatively supercoil.

Figure 3. Interwound (a) and toroid structures (b) of negatively supercoiled DNA. The DNA double helix is shown as a single line for simplicity in these illustrations. For each model, the DNA length is 3 kb and the specific linking difference is –0.06. (a) Purified, negatively supercoiled DNA is predicted to have a noncollapsed, branched structure. (b) The same DNA shown in (a); however, the DNA is packaged as nucleosomes stacked end-to-end with no linker DNA. See Cozzarelli et al. (1990) for a detailed description.

Figure 4. Titration of ethidium (shown as red lines) unto an initially negatively supercoiled DNA circle. The ethidium concentration increases from zero, as shown as a progression of molecules from (a) to (e). The increasing concentrations of ethidium have no effect on Lk of the molecule, but it does decrease Lk°. See text for details.

Figure 5. Schematic representation of six DNA samples run in agarose gels in the absence (a) and presence (b) of a subsaturating concentration of an intercalating agent. The six samples are all topoisomers of the same DNA plasmid. See text for details. ‘OC’ shows the position of open-circular or nicked DNA.

Figure 6. Knots. An illustration of the node convention is shown in (a) and described in the text. The two isomers of trefoil knots are shown in (b) and (c). (d) shows the only four-noded knot, the figure-8 knot. (e) and (f) show examples of more complex toroid and plectonemic knots, respectively.