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WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 155 introduces the concepts necessary to describe the mechanical equilibria of closed circular DNA and gives an analysis of transitions of superhelical transitions, dealing specifically with strand separation. Chapter 8 applies the topology of knot theory to explain the action of enzymes in carrying out the fundamental process of site-specific recombination. DNA GEOMETRY AND TOPOLOGY: LINKING, TWISTING, AND WRITHING To understand supercoiling in DNA, we model DNA (Bauer et al., 1980; White and Bauer, 1986) in the simplest possible way that will be useful for both ''open" linear DNA and closed circular supercoiled DNA wrapped around a series of proteins. Linear DNA is best modeled by a pair of cylindrical helices, C and W, representing the backbones winding right-handedly around a finite cylinder whose central axis, A, is a straight line (Figure 6.1a). Such DNA has a "starting point" and an endpoint. Relaxed closed circular DNA is modeled by bending the cylinder to form a closed toroidal surface in such a way that the axis, A, is a closed planar curve and the ends of the curves C and W are also joined (Figure 6.1b). Finally, closed supercoiled DNA can be modeled by supercoiling the toroidal surface itself (Figure 6.1c). (Closed DNA can be used to model "open" linear DNA because the reference frame is fixed at the starting point and the endpoint of open DNA even during biological changes.) We first wish to describe the fundamental geometric and topological quantities that can be used to characterize supercoiling, namely, the three quantities linking, writhing, and twisting (White, 1989). These are quantities that can be used to measure the interwinding of the backbone strands and the compacting of the DNA into a relatively small volume. The linking number is a mathematical quantity associated with two closed oriented curves. This important property is unchanged even if the two curves are distorted, as long as there is no break in either curve. For closed DNA the linking number is that of the two curves C and W. This number can therefore be changed only by single-or double-stranded breaks in the DNA. We assume that the two strands are oriented in a parallel fashion. This assumption is not consistent with the bond polarity
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 156 Figure 6. 1 (a) The linear form of the double helical model of DNA. (b) The relaxed closed circular form of DNA. (c) The plectonemically interwound form of supercoiled closed DNA.
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 157 but greatly facilitates the mathematics necessary for the description of supercoiling. Because either backbone curve can be deformed into the axis curve A without passing through the other, the linking number of a closed DNA is also the linking number of either backbone curve C or W with the central axis curve A. Therefore, we describe the linking number of a DNA in terms of the linking number of C with A. To define the linking number, the simplest technique is to use the so-called modified projection method. The pair of curves, A and C, when viewed from a given point distant from the two curves appear to be projected into a plane perpendicular to the line of sight except that the relative overlay of crossing segments is clearly observable. Such a view gives a modified projection of the pair of curves. In any such projection, there may be a number of apparent crossings. To each such crossing is attached a signed number ±1 according to the sign convention described in Figure 6.2. If one adds all of the signed numbers associated with this projection and divides by 2, one obtains the linking number of the curves A and C, Lk(A,C), which we denote for simplicity by Lk. In Figure 6.3 a number of simple cases are illustrated. An important fact about the linking number of a pair of curves is that it does not depend on the projection or view of the pair; that is, the total of the signed numbers for crossings corresponding to any projection is always the same. For socalled relaxed closed DNA the average number of base pairs per turn of C around W or C around A is 10.5. Thus Lk can be quite large for closed DNA. For a relaxed circular DNA molecule of the monkey virus SV40, which has approximately 5,250 base pairs, Lk is about 500, and for bacteriophage λ of about 48,510 base pairs, Lk is about 4,620. The linking number of a DNA, though a topological quantity, can be decomposed into two geometric quantities: writhe Wr and twist Tw, which can be used to describe supercoiling (White, 1969). The linking number is a measure of the crossings seen in any view. These crossings can be divided into two categories, distant crossings, which occur because the DNA axis is seen to cross itself, and local crossings, which occur because of the helical winding of the backbone curve around the axis. In the former, the backbone curve of one crossing segment is seen to cross the axis of the other segment. Distant crossings are measured by writhe, and local crossings by twist. We now give precise definitions of these two quantities.
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 158 Figure 6.2 Sign convention for the crossing of two curves in a modified projection. The arrows indicate the orientation of the two crossing curves. To determine the sign of the crossing, the arrow on top is rotated by an angle less than 180° onto the arrow on the bottom. If the rotation required is clockwise as in (a), the crossing is given a (â) sign. If the rotation required is counterclockwise as in (b), the crossing is given a (+) sign. Figure 6.3 Examples of pairs of curves with various linking numbers, using the convention described in Figure 6.2 and the method described in the text.
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 159 Writhe can be defined in a manner analogous to the linking number. It is a property of a single curve, in this case the central axis A. In any modified projection of the single curve A, there will be a number of apparent crossings. To each such crossing is attached a sign as in the case of the linking number. If one adds all these signed numbers, one obtains the projected writhing number. Unlike the linking number, projected writhe may change depending on the view that one takes. This is demonstrated by the different views of the same curve in Figure 6.4. The writhing number is defined as the average over all possible views of the projected writhing number. If two distant segments of a DNA axis are brought very close together, then this proximity will contribute approximately ±1 to the writhing number because in almost all views this proximity will be seen as a crossing. If the DNA axis lies in a plane and has no self-crossings, then Wr must be equal to zero, because in all views (except along the plane itself) there will be no apparent crossings. If the DNA lies in a plane except for a few places where the curve crosses itself, then the writhe is the total of the signed numbers attached to the self-crossings. Figure 6.5 gives the approximate writhe of some examples of tightly coiled DNA axes. Note that for consistently coiled curves of uniform handedness the larger the absolute value of the writhe the more compact the curve is. An important fact about the writhe of a space curve is that if the curve is passed through itself, at the moment of self-passage the writhe changes by ±2. This is because, at the moment of self-passage, no change takes place except at the point of passage, and the interchange of the under- and oversegments at the point of passage changes the writhe by precisely ±2. This is illustrated in Figure 6.6. The orientation of the axis curve is not important because the writhe does not change if the orientation is reversed. This fact enables one to choose the orientation of the axis curve, A, to be consistent with that of the backbone strand orientation. We next define the twist of the DNA. For closed DNA the twist will usually refer to the twist of one of the backbone curves, say C around the axis A, which is denoted Tw(C,A) or simply Tw. To define the twist, we need the use of vectors (White and Bauer, 1986). Any local cross section of the DNA perpendicular to the DNA axis contains a unique point a of the axis A and a unique point c of the backbone C
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 160 Figure 6.4 Illustration of the dependence of projected writhing number on projection. The axis of the same nonplanar closed DNA is shown in two different projections obtained by rotating the molecule around the dashed line. The points Q and R on the axis help illustrate the rotation. The segment QR crosses in front in (a) but is in the upper rear in (b). The projected writhing number in (a) is â1 and 0 in (b). Figure 6.5 Examples of closed curves with different writhing numbers. (Figure 6.7). We designate by vac a unit vector along the line joining the point a to the point c. Then as one proceeds along the DNA, since the backbone curve C turns around the axis A, the vector vac turns around the axis, or more precisely around T, the unit vector tangent to the curve A. The twist is a certain measure of this turning. As the point a moves along the axis A, the vector vac may change. The infinitesimal change in vac, denoted dvac, will have a component tangent to the axis and a component perpendicular to the axis. The twist is the measure of the total perpendicular component of the change of the vector vac, as the point a traverses the entire length of the DNA. It is therefore given by the line
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 161 Figure 6.6 The writhing number of curves with one coil. The curve on the left has writhing number approximately â1 and on the right approximately +1. One curve can be obtained from the other by a self-passage at the crossing, which changes the writhing number by +2 or â2. Figure 6.7 Cross-section of DNA. The plane perpendicular to the DNA axis A intersects the axis at the point a and intersects the backbone curve C at the point c. The unit vector along the line joining a to c is denoted vac. Note that as the intersection plane moves along the DNA, this vector turns about the axis.
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 162 integral expression: When A is a straight line segment or planar curve, dvac always is perpendicular to the curve A, so that in these cases, Tw reduces to the number of times that vac turns around the axis. Examples are shown in Figure 6.8a. Furthermore, if the DNA axis is planar and is also closed, Tw must necessarily be an integer, because the initial vector vac and the final vector vac are the same. Tw is not always the number of times that C winds around A. Indeed, Tw is usually not the number of times that C winds around A if the axis is supercoiled; Figure 6.8b gives an example in which A itself is a helix and C a superhelix winding around A. In this case, the twist is the number of times that C winds around A plus a term, nsinγ, which depends on the geometry of the helix A. In addition, in most cases where the DNA is closed and supercoiled, the twist is not integral (White and Bauer, 1986). The linking number, writhe, and twist of a closed DNA are related by the well-known equation (White, 1969): Lk = Tw + Wr. Thus for a closed strand of DNA of constant linking number, any change in Wr must be compensated by an equal in magnitude but opposite in sign change in Tw. This interchange is most easily seen by taking a rubber band or some simple elastic ribbon-like material that has two edges and while holding it fixed with one hand, twisting it with the other. After some time, much of the twisting will be seen to introduce writhing of the axis of the elastic material. The linking number of the two edges will stay the same because no breaks occur in the twisting. Because the model is held fixed by one hand, the twisting must be compensated by writhing. Though more complicated to explain, it is the same phenomenon that accounts for the supercoiling of most heavily used telephone cords. The constant twisting of the cord is eventually compensated by writhing.
