7 introduces the concepts necessary to describe the mechanical equilibria of closed circular DNA and gives an analysis of transitions of superhelical transitions, dealing specifically with strand separation. Chapter 8 applies the topology of knot theory to explain the action of enzymes in carrying out the fundamental process of site-specific recombination.
To understand supercoiling in DNA, we model DNA (Bauer et al., 1980; White and Bauer, 1986) in the simplest possible way that will be useful for both ''open" linear DNA and closed circular supercoiled DNA wrapped around a series of proteins. Linear DNA is best modeled by a pair of cylindrical helices, C and W, representing the backbones winding right-handedly around a finite cylinder whose central axis, A, is a straight line (Figure 6.1a). Such DNA has a "starting point" and an endpoint. Relaxed closed circular DNA is modeled by bending the cylinder to form a closed toroidal surface in such a way that the axis, A, is a closed planar curve and the ends of the curves C and W are also joined (Figure 6.1b). Finally, closed supercoiled DNA can be modeled by supercoiling the toroidal surface itself (Figure 6.1c). (Closed DNA can be used to model "open" linear DNA because the reference frame is fixed at the starting point and the endpoint of open DNA even during biological changes.)
We first wish to describe the fundamental geometric and topological quantities that can be used to characterize supercoiling, namely, the three quantities linking, writhing, and twisting (White, 1989). These are quantities that can be used to measure the interwinding of the backbone strands and the compacting of the DNA into a relatively small volume.
The linking number is a mathematical quantity associated with two closed oriented curves. This important property is unchanged even if the two curves are distorted, as long as there is no break in either curve. For closed DNA the linking number is that of the two curves C and W. This number can therefore be changed only by single- or double-stranded breaks in the DNA. We assume that the two strands are oriented in a parallel fashion. This assumption is not consistent with the bond polarity