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The moleculein (a) has a linking number of 1. The molecule in(b) has a linking number of 6. One of the strandsin (b) is kept untwisted for illustrative purposes todefine the border of an imaginary surface (shadedblue). The number of times the twisting strandpenetrates this surface provides one definition oflinking number.Lk = 20 = LkQ(a)StrandbreakALk = - 2Even if all hydrogen bonds and base-stacking interactions are abolished such that the strands are not in physical contact, this topologicalbond will still link the two strands. If one of the circular strands isthought of as the boundary of an imaginary surface (much as a soapfilm might span the space framed by a circular wire), the linking number can be defined rigorously as the number of times the second strandpierces this surface.
For the molecule in Figure 23-14aL£ = 1; for thatin Figure 23-14b Lk = 6. The linking number for a closed-circularDNA is always an integer. By convention, if the links between twoDNA strands are arranged so that the strands are interwound in aright-handed helix, the linking number is defined as positive ( + ). Conversely, for strands interwound as a left-handed helix the linking number is negative (-). Given that left-handed Z-DNA occurs only rarely,negative linking numbers are not encountered in studies of DNA for allpractical purposes.We can now extend these ideas to a closed-circular DNA with 210base pairs (Fig. 23-15). For a closed-circular DNA molecule that isrelaxed, the linking number is simply the number of base pairs dividedby 10.5; in this case, Lk = 20. For a circular DNA molecule to have atopological property such as linking number, neither strand may contain a break.
If there is a break in either strand, it is possible in principle to unravel the strands and separate them completely (Fig. 23-15b).Clearly, no topological bond exists in this case, and Lk is undefined.We can now describe DNA underwinding in terms of changes in thelinking number. The linking number in relaxed DNA is used as a reference and called Lk0. In the molecule shown in Figure 23-15a, Lk0 =20; if two turns are removed from this molecule, Lk will equal 18.
Thechange can be described by the equationALk =Lk-Lk0 = 18 - 20 = - 2NickLk undefined(b)Figure 23-15 Linking number applied to closedcircular DNA molecules. A 210 base pair circularDNA is shown in three forms: (a) relaxed, Lk = 20;(b) relaxed with a nick (break) in one strand, Lkundefined; (c) underwound by two turns, Lk = 18.The underwound molecule can occur as a supercoiled (left) or strand-separated (right) structure.Chapter 23 Genes and Chromosomes803It is often convenient to express the change in linking number in termsof a length-independent quantity called the specific linking difference O), which is a measure of the turns removed relative to thosepresent in relaxed DNA.
The term a is also called the superhelicaldensity and is defined asALkLk0In the example in Figure 23-15c, a = -0.10, which means that 10% ofthe helical turns present in the DNA (in its B form) have been removed. The degree of underwinding in cellular DNAs generally fallsinto the range of 5 to 7%; that is, a = -0.05 to -0.07. The negative signof a denotes that the change in linking number comes about as a resultof underwinding the DNA.
The supercoiling induced by underwindingis therefore defined as negative supercoiling. Conversely, under someconditions DNA can be overwound, and the resulting supercoiling isdefined as positive. Note that the twisting path taken by the axis of theDNA helix when the DNA is underwound (negative supercoiling) is themirror image of that taken when the DNA is overwound (positive supercoiling) (Fig. 23-16).
Supercoiling is not a random process; the pathof the supercoiling is largely prescribed by the torsional strain imparted to the DNA by decreasing or increasing the linking numberrelative to B-DNA.The linking number can be changed by ±1 by breaking one DNAstrand, rotating one of the ends 360° about the unbroken strand, andrejoining the broken ends. This change has no effect on the number ofbase pairs, or indeed on the number of atoms in the circular DNAmolecule. Two forms of a given 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.
23-17). These are moredifficult to describe intuitively than linking number, but to a first approximation Wr may be thought of as a measure of the coiling of thehelix axis and Tw as determining the local twisting or spatial relationship of neighboring base pairs. When a change in linking number occurs, some of the resulting strain is usually compensated by writhe(supercoiling) and some by changes in twist, giving rise to the equationRelaxed DNALk = 20ALk = - 2ALk = +2\NegativesupercoilPositivesupercoilLk = 22Figure 23-16 For the relaxed DNA molecule ofFigure 23-15a, underwinding or overwinding bytwo helical turns (Lk = 18 or 22) will produce negative or positive supercoiling as shown.
Note thatthe twisting of the DNA axis is opposite in signin the two cases.Lk = Tw + WrTwist and writhe are geometric rather than topological properties, because they may be changed by deformation of a closed-circular DNAmolecule. In addition, Tw and Wr need not be integers.Figure 23-17 A ribbon model for illustrating twistand writhe. The ribbon in (a) represents the axisof a relaxed DNA molecule. Strain introduced bytwisting the ribbon (underwinding the DNA) can bemanifested as a change in writhe (b) or a changein twist (c).
Changes in linking number are usuallyaccompanied by changes in both writhe and twist.Straight ribbon (relaxed DNA)(a)Large writhe, small change in twist(b)Zero writhe, large change in twist(c)804Part IV Information PathwaysRelaxed DNAUnderwound DNAThe concepts outlined above can be summarized by considering thesupercoiling of a typical bacterial plasmid DNA. Plasmids are generally closed-circular DNA molecules. Because DNA is a right-handedhelix, a plasmid will have a positive linking number.
When the DNA isrelaxed, the linking number or Lk0 is simply the number of base pairsdivided by 10.5. A typical plasmid, however, is generally underwoundin the cell. Therefore, Lk is less than Lk0, cr is negative, and the plasmid is negatively supercoiled.
Typically for a bacterial plasmid, a =-0.05 to -0.07.Underwinding DNA facilitates a number of structural changes inthe molecule. Strand separation occurs more readily in underwoundDNA. This is critical to the processes of replication and transcription,and represents a major reason why DNA is maintained in an underwound state. Other structural changes are of less physiological importance but help illustrate the effects of underwinding. A cruciform (seeFig. 12-21) generally contains a few unpaired bases, and DNA underwinding helps to maintain the required strand separation (Fig. 23-18).In addition, underwinding a right-handed DNA helix facilitates theformation of short regions of left-handed Z-DNA, where the DNA sequence is consistent with Z-DNA formation (Chapter 12).Topoisomerases Catalyze Changes in theLinking Number of DNACruciformFigure 23-18 DNA underwinding promotes cruciform structures.
In relaxed DNA, cruciforms seldomoccur because the linear DNA accommodates morepaired bases than does the cruciform structure.Underwinding the DNA facilitates the partialstrand separation needed to promote cruciform formation at appropriate sequences (palindromes).In every cell, DNA supercoiling is a precisely regulated process thatinfluences many aspects of DNA metabolism. Not surprisingly, thereare enzymes in every cell whose sole purpose is to underwind and/orrelax DNA.
The enzymes that increase or decrease the extent of DNAunderwinding are called topoisomerases, and the property of DNAthey affect is the linking number. These enzymes play an especiallyimportant role in processes such as replication and DNA packaging.There are two classes of topoisomerases. Type 1 topoisomerases act bytransiently breaking one of the two DNA strands, rotating one of theends about the unbroken strand, and rejoining the broken ends; theychange Lk in increments of 1.
Type 2 topoisomerases break both DNAstrands and change Lk in increments of 2.The effects of these enzymes can be demonstrated using agarosegel electrophoresis (Fig. 23-19). A population of identical plasmidDNAs with the same linking number will migrate as a discrete bandduring electrophoresis. Topoisomers with Lk values differing by as little as 1 can be separated by this method. In this way changes in linkingnumber induced by topoisomerases can readily be observed.There are at least four different topoisomerases in E.
coli, distinguished by Roman numerals I through IV. The type 1 topoisomerases(topoisomerases I and III) generally relax DNA by removing negativesupercoils (they increase Lk). One bacterial type 2 enzyme, called topoisomerase II or, alternatively, DNA gyrase, can introduce negativesupercoils (decrease Lk).
It uses the energy of ATP and a surprisingmechanism to accomplish this (Fig. 23-20). The superhelical density ofbacterial DNA is balanced by regulation of the net activity of topoisomerases I and II.Eukaryotic cells also have type 1 and type 2 topoisomerases; inmost eukaryotes there is one known example of each type, called topoisomerase I and II, respectively.
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