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WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 166 will be distributed to give a change of +1.44 in average writhe and a change of +0.56 in average twist. Continuing with the same example of SV40 DNA introduced in the paragraph above, if a topoisomerase of Type II were introduced into a population of SV40 with linking number difference â25, the result would be a collection of such DNA with linking number differences â25, â23, â21, . . ., â3, â1. Such enzymes in fact were discovered because of this striking difference from topoisomerase of Type I in which DNA with all negative differences, not just the odd ones, were found. Other functions of topoisomerases are to pass single-stranded DNA through itself or to pass nicked DNA-that is, DNA in which one of the backbone strands has been cut by an enzyme-through itself. This aspect of topoisomerases is dealt with in more detail in Chapter 8. DNA ON PROTEIN COMPLEXES We now turn our attention to the geometric and topological analysis of DNA whose axes are constrained to lie on surfaces (White et al., 1988). The most well-characterized example of a protein surface is the nucleosome core (Finch et al., 1977), a cylinder of height 5.04 nanometers (nm) and radius 4.3 nm. In this case the axis A of the DNA wraps nearly twice around the core as a left-handed helix of pitch 2.8 nm. The surface on which the DNA molecule lies is the so-called solvent-accessible surface (Richards, 1977). This is the surface generated by moving a water-sized spherical probe around the atomic surface of the protein at the van der Waals distance of all external atoms and is the continuous sheet defined by the locus of the center of the probe. (In general, the surface of a protein is defined in this manner.) It is this surface that comes into contact with the DNA backbone chain. Because the DNA is approximately 1 nm in radius, the DNA axis lies on a surface that is 1 nm outside of the solvent- accessible surface to account for the separation of the backbone from the axis. This latter surface is the one to which we shall refer in the rest of this section as the surface on which the DNA, meaning the DNA axis A, lies or wraps. For DNA that lies on a surface, the geometric and topological analyses are best served by dividing the linking number not into twist and writhe, which relate only to spatial properties of the DNA, but into