Topological Considerations in DNA Structure
Topology introduces a structural feature in addition to the base-paired and helical aspects of DNA. The origin of this structure can best be understood by considering a mathematical property of two closed rings. The number of times that one ring encircles or links the other must be an integral number. It cannot be changed without physically opening
one of the rings. That is, their linking number is a topological invariant. Many types of DNA molecules found in cells are covalently closed circles because each strand is circular. Hence the concept of a linking number applies to DNA molecules obtained from many sources. The concept also applies to linear DNA if the ends are prevented from freely rotating, either because of the extreme length of the DNA or because the DNA is attached to something else.
The forces tending to hold double-stranded DNA in a right-handed helix with about 10.5 base pairs per turn add a dimension to the analysis of the structures of covalently closed circles. These forces are suffi-ciently great that the linking number, Lk, generally resolves itself into two easily distinguished components: the twist, Tw, which in DNA’s usual right-helical form has a value of 1 per each 10.5 base pairs, and the writhe, Wr. Twist is the local wrapping of one of the two strands about the other. If Lk does not equal Tw, then the discrepancy must be made up from a global writhing of the molecule as such global effects can alter the actual number of times one strand encircles the other. These global effects are called supercoiling or superhelical turns. Their computation is most difficult because the entire path of the DNA duplex must be considered. To repeat, for any covalently closed double-stranded DNA molecule, no matter how it is distorted, unless its phos-phodiester backbone is broken, Lk = Tw + Wr. This equation sometimes is written τ = α + β where τ = Lk, α = Tw, and β = Wr. It is curious that the topological invariant Lk equals the sum of two terms, each of which is not invariant.
It is convenient to normalize the deviation of the linking number from the normal, unconstrained value, Lk0. Since the normal linking is 1 per 10.5 base pairs of DNA, linking values other than this must change either the twist, the writhe, or both. Rather than just give the linking number deviation for a DNA molecule, it is more informative to give the linking number deviation per unit length of the DNA, Lk - Lk0 divided by Lk0 . This value is denoted by σ. Typical values of σ in DNA extracted from bacterial cells are -0.02 to -0.06. Colloquially σ is called supercoiling density, but it must be remembered that part of the linking number deviation goes into changing the twist of the DNA molecules and part goes into generating writhe.