DNA supercoil

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In a "relaxed" double-helical segment of DNA, the two strands twist around the helical axis once every 10.4 base pairs of sequence. To add or subtract twists, as some enzymes can do, is to impose a strain. If a DNA segment under twist strain were to be closed into a circle by joining its two ends and then allowed to move freely, the circular DNA would contort into new shape, such as a simple figure-eight. Such a contortion is a supercoil.

The simple figure eight is the simplest supercoil, and is the shape a circular DNA assumes to accommodate one too many or one too few helical twists. The two lobes of the figure eight will appear rotated either clockwise or counterclockwise with respect to one another, depending on whether the helix is over or underwound. For each additional helical twist being accommodated, the lobes will show one more rotation about their axis.

The noun form "supercoil" is rarely used in the context of DNA topology. Instead, global contortions of a circular DNA, such as the rotation of the figure-eight lobes above, are referred to as writhe. The above example illustrates that twist and writhe are interconvertable. "Supercoiling" is an abstract mathematical property, and represents the sum of twist and writhe. The relationship of twist, writhe and supercoiling is expressed as the equation:

S = T + W.

Extra helical twists are positive and lead to positive supercoiling, while subtractive twisting causes negative supercoiling. Many topoisomerase enzymes sense supercoiling and either generate or dissipate it as they change DNA topology. DNA of most organisms is negatively supercoiled.

In part because chromosomes may be very large, segments in the middle may act as if their ends are anchored. As a result, they may be unable to distribute excess twist to the rest of the chromosome or to absorb twist to recover from underwinding--the segments may become supercoiled, in other words. In response to supercoiling, they will assume an amount of writhe, just as if their ends were joined.

[edit] Modeling using mathematics

DNA supercoiling can be described numerically by changes in the 'linking number' Lk. The linking number is the most descriptive property of supercoiled DNA. Lkο, the number of turns in the relaxed DNA plasmid/molecule, is determined by dividing the total base pairs of the molecule by the relaxed bp/turn which, depending on reference is 10.4-10.5.

L k o = b p / 10.4

Lk is merely the number of crosses a single strand makes across the other in a planar projection. The topology of the DNA is described by the equation below in which the linking number is equivalent to the sum of Tw, which is the number of twists or turns of the double helix, and Wr which is the number of coils or 'writhes'. If there is a closed DNA molecule, the sum of Tw and Wr, or the linking number, do not change. However, there may be complementary changes in Tw and Wr without changing their sums.

L k = T w + W r

The change in the linking number, ΔLk, is the actual number of turns in the plasmid/molecule, Lk, minus the number of turns in the relaxed plasmid/molecule Lk_o.

Δ L k = L k - L k o

If the DNA is negatively supercoiled ΔLk < 0. The negative supercoiling implies that the DNA is underwound.

A standard expression independent of the molecule size is the "specific linking difference" or "superhelical density" denoted σ. σ represents the number of turns added or removed relative to the total number of turns in the relaxed molecule/plasmid, indicating the level of supercoling.

σ = Δ L k / L k o

The Gibbs free energy associated with the coiling is given by the equation below. Here, R is the universal gas constant 8.314472 J · K^-1 · mol^-1, T is the tempeature of the DNA in Kelvin, and N is the number of base pairs in the DNA molecule.