DNA Supercoil - Mathematical Description

Mathematical Description

In nature, circular DNA is always isolated as a higher-order helix-upon-a-helix, known as a superhelix. In discussions of this subject, the Watson-Crick twist is referred to as a "secondary" winding, and the superhelices as a "tertiary" winding. The sketch at right indicates a "relaxed", or "open circular" Watson-Crick double-helix, and, next to it, a right-handed superhelix. The "relaxed" structure on the left is not found unless the chromosome is nicked; the superhelix is the form usually found in nature.

For purposes of mathematical computations, a right-handed superhelix is defined as having a "positive" number of superhelical turns, and a left-handed superhelix is defined as having a "negative" number of superhelical turns. In the drawing (shown at the right), both the secondary (i.e., "Watson-Crick") winding and the superhelical winding are right-handed, hence the supertwists are positive(+3 in this example).

The superhelicity is presumed to be a result of underwinding, meaning that there is a deficiency in the number of secondary Watson-Crick twists. Such a chromosome will be strained, just as a macroscopic metal spring is strained when it is either overwound or unwound. In DNA which is thusly strained, supertwists will appear.

DNA supercoiling can be described numerically by changes in the linking number Lk. The linking number is the most descriptive property of supercoiled DNA. Lk_o, the number of turns in the relaxed (B type) 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.

Lk is merely the number of crosses a single strand makes across the other . L_k, known as the "linking number", is the number of Watson-Crick twists found in a circular chromosome in a (usually imaginary) planar projection. This number is physically "locked in" at the moment of covalent closure of the chromosome, and cannot be altered without strand breakage.

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, does not change. However, there may be complementary changes in TW and Wr without changing their sum.

Tw, called "twist", refers to the number of Watson-Crick twists in the chromosome when it is not constrained to lie in a plane. We have already seen that native DNA is usually found to be superhelical. If one goes around the superhelically twisted chromosome, counting secondary Watson-Crick twists, that number will be different from the number counted when the chromosome is constrained to lie flat. In general, the number of secondary twists in the native, supertwisted chromosome is expected to be the "normal" Watson-Crick winding number, meaning a single 10-base-pair helical twist for every 34 Å of DNA length.

Wr, called "writhe", is the number of superhelical twists. Since biological circular DNA is usually underwound, L_k will generally be less than Tw, which means that Wr will typically be negative.

Now we can see that if DNA is underwound, it will be under strain, exactly as a metal spring is strained when forcefully unwound, and that the appearance of supertwists will allow the chromosome to relieve its strain by taking on negative supertwists, which correct the secondary underwinding in accordance with the topology equation above.

The topology equation teaches further that there is a one-to-one relationship between changes in Tw and Wr. For example, if a secondary "Watson-Crick" twist is removed, then a right-handed supertwist must have been removed simultaneously (or, if the chromosome is relaxed, with no supertwists, then a left-handed supertwist must be added).

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.

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 supercoiling.

The Gibbs free energy associated with the coiling is given by the equation below

Since the linking number L of supercoiled DNA is the number of times the two strands are intertwined (and both strands remain covalently intact), L cannot change. The reference state (or parameter) L₀ of a circular DNA duplex is its relaxed state. In this state, its writhe W = 0. Since L = T + W, in a relaxed state T = L. Thus, if we have a 400 bp relaxed circular DNA duplex, L ~ 40 (assuming ~10 bp per turn in B-DNA). Then T ~ 40.

• Positively supercoiling:

  T = 0, W = 0, then L = 0

  T = +3, W = 0, then L = +3

  T = +2, W = +1, then L = +3

• Negatively supercoiling:

  T = 0, W = 0, then L = 0

  T = -3, W = 0, then L = -3

  T = -2, W = -1, then L = -3

Negative supercoils favor local unwinding of the DNA, allowing processes such as transcription, DNA replication, and recombination. Negative supercoiling is also thought to favour the transition between B-DNA and Z-DNA, and moderate the interactions of DNA binding proteins involved in gene regulation.