With respect to circular DNAs and supercoiling, students should note the following:

1. Circular DNAs have no free ends.

2. Enzymes called topoisomerases can take apart a circular DNA, introduce additional twists into it, and then reseal the structure (Figure 4.24). Adding twists to circular DNA introduces tension into the molecule. This kind of tension would not stay in linear DNA for very long because the force could be dissipated through the ends, and the DNA would "relax". The extra tension in circular DNA (or in linear DNA whose ends are anchored to prevent tension from being released) usually causes the molecule to writhe to alleviate the tension. Like an overwound rubber band, the circular DNA assumes a new shape, called a supercoil.

3. Supercoiling can be positive (additional twists added beyond the normal amount for linear DNA) or negative (reduced numbers of twists compared to linear DNA).

Figure 4.24 illustrates three kinds of circular DNA, unstrained circle, strained circle, and supercoil. Similarly, Figure 4.18 shows an electron micrograph of a relaxed (unstrained) circle and two supercoiled circles. The unstrained circle contains the same number of twists as linear DNA. It is under no superhelical tension. To make the strained circle, one twist was removed (compared to linear DNA) and the resulting circular DNA is strained because it has the same number of base pairs (105), but fewer numbers of turns (twists). Thus, the strained circle has a higher number of base pairs per turn than the unstrained circle. To relieve the strain, the strained molecule can introduce another superhelical turn within itself, called a writhe.

After the writhe, the number of twists (turns) is 10 again so the number of base pairs per turn is 10.5 again, too. However, the three-dimensional shape of the molecule has changed in response to the initial change in the number of twists. The change in shape of the molecule can be observed as an alteration in the electrophoretic mobility. As noted above, the change in shape is called supercoiling. Supercoiling can come about by either adding or subtracting twists relative to unstrained circular DNA.

The linking number (L) is simply the algebraic sum of the number of twists (T) and writhes (W) of a molecule:

L = T + W

Consequently, the change in the linking number is also equal to the change in the twists and writhes for a molecule:

L = T + W.

The superhelical density is defined as L/L0, where L0 is the linking number of the DNA in its unstrained (relaxed state).

Many naturally occurring DNA molecules have superhelical densities of about -0.06. To get an idea of what this means, consider a hypothetical DNA molecule of 10,000 bp, which is in the "classical" B form, with 10.0 bp/turn. Then L0 is 10,000 bp/(10.0 bp/turn), or 1000 turns. Each DNA strand crosses the other 1000 times in the relaxed circle. If the topoisomerase gyrase twisted the molecule to a superhelical density of -0.06, then L = -0.06 L0, or L = -60. This change could be accommodated, for example, by the helix axis writhing about itself 60 times in a left-hand sense, which would correspond to W = -60, T = 0; the molecule would have 60 left-hand superhelical turns.

Alternatively, the twist of the molecule could change so that it had 940 turns in 10,000 bp (T = 940) or 10,000/940 = 10.64 bp/turn. This would correspond to W = 0, T = -60. Although any combination of T and W that sums to -60 could occur, real molecules release strain mainly by writhing into superhelical turns, because it is easier to bend long DNA than it is to untwist it.

Besides writhing, unwinding DNA, cruciform formation (via palindromes), triple helix formation, and Z-DNA formation, can all reduce superhelical tension, too. For example, unwinding one repeat of the DNA helix (10 base pairs) is equivalent to T = -1. If one superhelical turn were removed at the same time (W = +1), then L would remain unchanged. Similarly, converting from B- to Z-DNA causes a change of T = -2. Thus, two superhelical turns could be removed in this manner. There are likely physiological consequences of superhelicity alteration. For example, unwinding of duplex DNAs is a factor in both transcription and in DNA replication. Moreover, the A-T rich sequence regions near promoters of DNA (regions of DNA where copying of RNA occurs) allow for easier separation of strands of DNA and may well be easily opened by changes in superhelical density.