Figure 1. Relationship of linking number, twist and writhe of closed circular forms of DNA. Closed DNA circles can be made by formation of covalent 5¢–3¢ phosphodiester bonds on each strand of a linear molecule. For a linear molecule with 36 helical turns, the linking number of this unconstrained state (Lk⁰) is 36. Closure into an unconstrained planar circle, as shown on the left side of the figure, produces a molecule with twist (Tw) = 36 and writhe (Wr) = 0. If the number of helical turns is altered before closure, the DNA molecule adopts a supercoiled conformation. On the right side of the figure, four helical turns are removed from the molecule, reducing the linking number (Lk) to 32. For simplicity, the figure shows all unwinding partitioned as Wr, although such changes are usually partitioned between Tw and Wr. Unwinding of helical turns produces negatively supercoiled DNA (or –Lk) as shown, whereas the inclusion of additional turns produces positively supercoiled DNA. For DNA with –Lk in the interwound form, the superhelical turns are right-handed. Note that separation of DNA strands removes negative supercoils (equivalent to the addition of positive supercoils).

Figure 2. Measurement of linking number by gel electrophoresis. (a) Schematic illustration of a DNA sample separated by electrophoresis through an agarose gel with and without an intercalator. DNA isolated from bacterial cells contains molecules with different topology: some have their backbones unbroken and are negatively supercoiled (SC), some have one strand broken or ‘nicked’ (N) and some have both strands broken to produce a linear molecule (L). Note that the supercoiled DNA consists of a Gaussian distribution of different topoisomers. Upon addition of intercalator, the migration of intact molecules is altered, but that of nicked and linear molecules is not changed. (b) Enzymatic relaxation of plasmid DNA in the presence of varying concentrations of intercalator produces samples containing topoisomers at different levels of supercoiling. Utilization of multiple gels with different concentrations of intercalator allows measurement of Lk. For each sample, average superhelical density () is shown above the lane. Note that in each gel, samples can have positively or negatively supercoiled topoisomers. The inclusion of intercalator in the running buffer alters the electrophoretic mobility of all topoisomers equivalently. Superhelical density can be measured for experimental samples (‘native’) by comparison with those of known . (c) Two-dimensional agarose gel electrophoresis of topoisomers ranging from high negative to moderate positive . A DNA sample is loaded in a single well in a large agarose gel and electrophoresis is performed under specific conditions (usually without intercalator) in direction D1. After soaking of the gel in buffer containing intercalator, electrophoresis is continued in direction D2 (90° to D1). The gel shown contained 20 g mL^–1 chloroquine during the second electrophoresis, resulting in all topoisomers having positive . Deviation of topoisomers from a smooth curve indicates that structural transitions reduced their negative during the first direction of electrophoresis. Spots marked ‘N’ and ‘L’ indicate the position of migration of ‘nicked’ and ‘linear’ DNA molecules, respectively.

Figure 3. Representation of knots and catenanes. (a) Topological knots may be formed in closed circles of DNA. The simplest knot that can be formed is called a trefoil because there are three lobes to the structure when it is laid flat. Two isomers of the trefoil knot are shown. Many other more complex knots may be formed within cccDNA molecules. (b) Catenanes are formed when two circular DNA molecules are interlocked. Catenanes containing complex, multiple links and involving many DNA circles have been observed in naturally occurring DNA. The arrows indicate that the polarity of a knot or catenane is influenced by the directionality of the DNA sequence.

Figure 4. Twin domains of supercoiling are generated during transcription. (a) The shaded cylinders flank a closed domain of DNA containing eight helical turns. (b) To accommodate the transcriptional complex, some unwinding of the DNA helix occurs producing slight overwinding of the remaining DNA within each closed domain. (c) During transcription elongation, rotation of the large transcriptional complex around the DNA is hindered and positive and negative supercoiling are generated ahead and behind the polymerase, respectively. In this diagram, positive and negative supercoiling is represented by the presence of the same number of helical turns over a shorter and longer distance of DNA, respectively. Several mechanisms exist to remove these supercoils.