Chromosomal Elements:- DNA Underwinding Is Defined by Topological Linking Number
The field of topology provides a number of ideas that are useful to this discussion, particularly the concept of linking number. Linking number is a topological prop erty of double-stranded DNA, because it does not vary when the DNA is bent or deformed, as long as both DNA strands remain intact. Linking number (Lk) is illustrated in Figure 1. Let’s begin by visualizing the separation of the two strands of a double-stranded circular DNA. If the two strands are linked as shown in Figure 1 a, they are effectively joined by what can be described as a topological bond. Even if all hydrogen bonds and base stacking interactions were abolished such that the strands were not in physical contact, this topological bond would still link the two strands. Visualize one of the circular strands as the boundary of a surface (such as a soap film spanning the space framed by a circular wire before you blow a soap bubble). The linking number can be defined as the number of times the second strand pierces this surface. For the molecule in Figure 1a, Lk=1; for that in Figure 1 b, Lk=6. The linking number for a closed-circular DNA is always an integer. By convention, if the links between two DNA strands are arranged so that the strands are interwound in a right-handed helix, the linking number is defined as positive (+); for strands interwound in a left-handed helix, the linking number is negative (-). Negative linking numbers are, for all practical purposes, not en countered in DNA. We can now extend these ideas to a closed-circular DNA with 2,100 bp (Fig. 2 a). When the molecule is relaxed, the linking number is simply the number of base pairs divided by the number of base pairs per turn, which is close to 10.5; so, in this case, Lk 200. For a circular DNA molecule to have a topological property such as linking number, neither strand may contain a break. If there is a break in either strand, the strands can, in principle, be unraveled and separated completely. In this case, no topological bond exists and Lk is undefined (Fig. 2 b). We can now describe DNA underwinding in terms of changes in the linking number. The linking number in relaxed DNA, Lk0, is used as a reference. For the molecule shown in Figure 2a, Lk0=200; if two turns are removed from this molecule, Lk=198. The change can be described by the equation ΔLk=Lk-Lk0=198-200=-2 It is often convenient to express the change in linking number in terms of a quantity that is independent of the length of the DNA molecule. This quantity, called the specific linking difference (σ), or superhelical density, is a measure of the number of turns removed relative to the number present in relaxed DNA:
In the example in Figure 24–16c, σ=-0.01, which means that 1% (2 of 200) of the helical turns present in the DNA (in its B form) have been removed. The degree of underwinding in cellular DNAs generally falls in the range of 5% to 7%; that is, σ =-0.05 to -0.07. The negative sign indicates that the change in linking number is due to underwinding of the DNA. The supercoiling induced by underwinding is therefore defined as negative supercoiling. Conversely, under some conditions DNA can be overwound, resulting in positive supercoiling. Note that the twisting path taken by the axis of the DNA helix when the DNA is underwound (negative supercoiling) is the mirror image of that taken when the DNA is overwound (positive supercoiling) (Fig. 3). Supercoiling is not a random process; the path of the supercoiling is largely prescribed by the torsional strain imparted to the DNA by decreasing or increasing the linking number relative to B-DNA.
Linking number can be changed by 1 by breaking one DNA strand, rotating one of the ends 360 about the unbroken strand, and rejoining the broken ends. This change has no effect on the number of base pairs or the number of atoms in the circular DNA molecule. Two forms of a circular DNA that differ only in a topological property such as linking number are referred to as topoisomers.
Linking number can be broken down into two structural components called writhe (Wr) and twist (Tw) (Fig. 3). These are more difficult to describe than linking number, but writhe may be thought of as a measure of the coiling of the helix axis and twist as deter mining the local twisting or spatial relationship of neigh boring base pairs. When the linking number changes, some of the resulting strain is usually compensated for by writhe (supercoiling) and some by changes in twist, giving rise to the equation Lk=Tw+ Wr Tw and Wr need not be integers. Twist and writhe are geometric rather than topological properties, because they may be changed by deformation of a closed-circular DNA molecule. In addition to causing supercoiling and making strand separation somewhat easier, the underwinding of DNA facilitates a number of structural changes in the molecule. These are of less physiological importance but help illustrate the effects of underwinding. Recall that a cruciform generally contains a few un paired bases; DNA underwinding helps to maintain the required strand separation . Underwinding of a right-handed DNA helix also facilitates the formation of short stretches of left-handed Z-DNA in regions where the base sequence is consistent with the Z .
FIGURE 1 Linking number, Lk. Here, as usual, each blue ribbon represents one strand of a double-stranded DNA molecule. For the molecule in (a), Lk=1. For the molecule in (b), Lk 6. One of the strands in (b) is kept untwisted for illustrative purposes, to define the border of an imaginary surface (shaded blue). The number of times the twisting strand penetrates this surface provides a rigorous definition of linking number.
FIGURE 2 Linking number applied to closed-circular DNA molecules. A 2,100 bp circular DNA is shown in three forms: (a) relaxed, Lk=200; (b) relaxed with a nick (break) in one strand, Lk undefined; and (c) underwound by two turns, Lk=198. The underwound molecule generally exists as a supercoiled molecule, but underwinding also facilitates the separation of DNA strands.
FIGURE 3 Negative and positive supercoils. For the relaxed DNA molecule of Figure 24–16a, underwinding or overwinding by two helical turns (Lk=198 or 202) will produce negative or positive supercoiling, respectively. Note that the DNA axis twists in opposite directions in the two cases.
