Twist and Writhe DNA

The linking number is sum of two geometric components called twist and writhe. Let twist first. Twist is simply the number of helical turns of one strand about other, the number of times one strand completely wraps around the other strand. Consider a cccDNA that is lying flat on a plane. In this flat conformation, the linking number is fully composed of twist. Indeed, the twist can be easily determined by counting the number of times the two strands cross each other. The helical crossovers (twist) in a right-handed helix are defined as positive such that the linking number of DNA will have a positive value.

But cccDNA is generally not lying flat on plane. Rather, it is usually torsionally stressed such that the long axis of the double helix crosses over itself, often repeatedly, in three-dimensional space. This is called writhe. To visualize the distortions caused by torsional stress, think of the coiling of a telephone cord that has been overtwisted. Writhe can take two forms. One form is the interwound or plectonemic writhe, in which the long axis is twisted around itself. The other form of writhe is a toroid or spiral in which the long axis is wound in a cylindrical manner, as often occurs when DNA wraps around protein. The writhing number (Wr) is the total number of interwound and/or spiral writhes in cccDNA. For example, the molecule shown in  has a writhe of 4 from 4 interwound writhes.Interwound writhe and spiral writhe are  topologically equivalent to each other and are readily interconvertible geometric properties of cccDNA. Also, twist and writhe are interconvertible. A molecule of cccDNA can readily undergo distortions that convert some of its twist  to writhe or some of its writhe to twist without the breakage of any covalent bonds. The only constraint is that the sum of the twist number (Tw) and the writhing number (Wr) must remain equal to the linking number (Lk). This constraint is described by the equation: Lk = Tw + Wr.