Supercoiling
A DNA plasmid that has no nicks (single-strand breaks) or cuts (double-strand breaks) is topologically constrained (see demonstration). The two DNA strands are interwound and cannot be separated without breaking one of them. There are three quantities that are useful for describing a two-stranded closed topologic states. These quantities are (i) the linking number, (ii) the twist, and (iii) the writhe. Note that these terms have no meaning if the plasmid is nicked or cut.
The linking number (Lk) is the number of times two closed strands are linked or, the number of times that one DNA strand winds around another. Lk is an integer that is most easily visualized when the DNA axis is restrained to lay on a plane. It is impossible to changed the linking number of a plasmid without breaking a covalent bond, i.e., without breaking one strand, passing the second strand through the break and rejoining the break.
The twist (Tw) is the number of times one strand appears to cross the other during the journey along the complete, closed helical axis. When the DNA axis is restrained to lay on a plane, Lk is equivalent to Tw.
The writhe (Wr) is the number of times that the axis is crossed during the same journey. When the DNA axis is restrained to lay on a plane, the writhe is zero.
For a plasmid with no nicks of cuts,
The linking number is fixed. Any change in twist must be compensated by a change in writhe.
If one nicks a plasmid, lets it fully relax, and then ligates it, the linking number will be Lko.
Where N is the number of base pairs in the plasmid and h is the number of base pairs per turn (10pb/turn for B-DNA). But when isolated from cells, DNA is usually underwound, so Lk<Lko (i.e., DLK = Lko-Lk is negative). The superhelical density (s) is DLK normalized to the length.