Page 153

Chapter 6—
Winding the Double Helix:
Using Geometry, Topology, and Mechanics of DNA

James H. White
University of California, Los Angeles

Crick and Watson's double helix model describes the local structure of DNA, but the global structure is more complex. The DNA double helix follows an axis that is typically curved—creating a phenomenon called supercoiling, which is crucial for a wide variety of biological processes. Understanding supercoiling requires ideas from geometry and topology. In this chapter, the author discusses three key descriptors of the geometry of supercoiled DNA molecules: linking, twisting, and writhing. These quantities are related by a fundamental theorem with important consequences for experimental biology, because it allows biologists to infer any one of the quantities from measurements of the other two.

Deoxyribonucleic acid (DNA) is usually envisioned as a pair of helices, the sugar-phosphate backbones, winding around a common linear axis. In the famous model of Crick and Watson, one turn of the double helix occurs approximately every 10.5 base pairs. However, the actual structure of DNA in a cell is typically more complex: the axis of the double helix may itself be a helix or may, in general, assume almost any configuration in space. In the late 1960s it was discovered that many DNA molecules are also closed; that is, the axis as well as the two backbone strands are closed curves. (A closed curve is a curve of finite length, the "starting point" and "endpoint" of which coincide.) In this case the DNA is called closed circular or simply closed. This chapter is



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Page 153 Chapter 6— Winding the Double Helix: Using Geometry, Topology, and Mechanics of DNA James H. White University of California, Los Angeles Crick and Watson's double helix model describes the local structure of DNA, but the global structure is more complex. The DNA double helix follows an axis that is typically curved—creating a phenomenon called supercoiling, which is crucial for a wide variety of biological processes. Understanding supercoiling requires ideas from geometry and topology. In this chapter, the author discusses three key descriptors of the geometry of supercoiled DNA molecules: linking, twisting, and writhing. These quantities are related by a fundamental theorem with important consequences for experimental biology, because it allows biologists to infer any one of the quantities from measurements of the other two. Deoxyribonucleic acid (DNA) is usually envisioned as a pair of helices, the sugar-phosphate backbones, winding around a common linear axis. In the famous model of Crick and Watson, one turn of the double helix occurs approximately every 10.5 base pairs. However, the actual structure of DNA in a cell is typically more complex: the axis of the double helix may itself be a helix or may, in general, assume almost any configuration in space. In the late 1960s it was discovered that many DNA molecules are also closed; that is, the axis as well as the two backbone strands are closed curves. (A closed curve is a curve of finite length, the "starting point" and "endpoint" of which coincide.) In this case the DNA is called closed circular or simply closed. This chapter is

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Page 154 concerned with the geometry, topology, and energetics of closed supercoiled DNA. Supercoiling of closed DNA is ubiquitous in biological systems. It can arise in two ways. First, it can result when DNA winds around proteins. Second, supercoiling can also result from topological constraints known as under- or overwinding, in which case the axis of the DNA usually assumes an interwound, or plectonemic, form. Supercoiling is important for a wide variety of biological processes. For example, supercoiling is a way of storing free energy—which can be used to assist the vital processes of replication and transcription, processes that require untwisting or separation of DNA duplex strands. Thus, supercoiling helps enzymes called helicases, polymerases, and other proteins to force apart the two strands of the DNA double helix, allowing access to the genetic information stored in the base sequence. It also promotes a variety of structural alterations that lead to DNA unwinding, such as z-DNA (left-handed double helical DNA) and cruciforms (cross-shapes). In higher organisms, supercoiling helps in cellular packaging of DNA in structures called nucleosomes, in which DNA is wound around proteins called histones. It is crucial in bringing together and aligning DNA sequences in site-specific recombination. It also changes the helical periodicity (number of base pairs per turn) of the DNA double helix; such changes can alter the binding of proteins to the DNA or the phasing of recombinant sequences. Understanding supercoiled DNA is thus essential for the understanding of these diverse processes. Numerous biological experiments—including those based on sedimentation, gel electrophoresis, electron microscopy, X-ray diffraction, nuclease digestion, and footprinting—can give information about these matters. However, mathematical methods for describing and understanding closed circular DNA are needed to explain and classify the data obtained from these experiments. This chapter defines and elucidates the major geometric descriptors of supercoiled DNA: linking, twisting, and writhing. It applies these concepts to classify the action of the major types of cutting enzymes, topoisomerases of Type I and Type II. It then develops the differential topological invariants necessary to describe the structural changes that occur in the DNA that is bound to proteins. Other chapters in this book explore applications of topology and geometry to DNA coiling. Chapter

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Page 155 7 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

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Page 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.

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Page 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 l 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.

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Page 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 (-) 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.

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Page 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

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Page 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 -1 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

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Page 161 Figure 6.6 The writhing number of curves with one coil. The curve on the left has writhing number approximately -1 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 -2. 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.

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Page 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, nsing, 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.

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Page 163 Figure 6.8a Examples of pairs of curves  C and A  with different values of twist. (a) Simple examples in which the axis  A is a straight line and the twist is the number of times that C winds around  A, being positive for right-handed twist and negative for left-handed twist. (b) An example in which the axis  A  is a helix winding around a linear axis and the curve  C  is a superhelix winding around  A. In this case the twist of  C  around A  is the number of times that C  winds around A  (in this case approximately 3.5) plus nsing where n  is the number of times that  A winds around the linear axis and g is the pitch angle of the helix  A. Here n  is approximately 1.5 and  g  is approximately 40°. Thus, Tw  »  4.46. Applications to DNA Topoisomerase Reactions For a relaxed closed DNA that lies in a plane and has no self-crossings, we have seen that the writhing number is equal to 0. Therefore, by the fundamental formula, Tw = Lk. Thus both Tw and Lk are equal to the number of times that the backbone strands wind around the axis, or more precisely the number of times the backbone curve rises above and falls below the plane in which the axis lies. For such a DNA in the B-form, there are about 10.5 base pairs per turn of the backbone. This linking number is usually denoted Lk0. As we noted above, a relaxed DNA molecule of monkey virus SV40 with about 5,250 base pairs has Lk0 = Tw  = 500. However, the linking number of most closed circular DNA is not that of the relaxed state. The actual linking number

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Page 168 the plane and cannot link A, a curve lying entirely in the plane. Second, if the DNA axis lies on a round sphere, SLk = 0. To see this, we can assume without loss of generality that the vector field v points into the sphere. In this case the displaced curve Ae lies entirely inside the sphere and hence cannot link A. These and additional examples are illustrated in Figure 6.9. SLk is what is technically called a differential topological invariant. As such, SLk has three important properties. First, if the DNA axis-surface combined structure is deformed in such a way that no discontinuities in the vector field v occur in the neighborhood of the DNA axis A, and A itself is not broken, then SLk remains invariant. For example, if the DNA lies in a plane and that plane is deformed, SLk remains equal to 0; if it lies on a sphere that is deformed, SLk remains equal to 0. Thus, if a DNA axis lies on the surface of any type of spheroid, SLk =0. Examples of spheroids are shown in Figure 6.10a. An important example of SLk being equal to 0 is shown in Figure 6.9d, in which the DNA axis lies on the surface of a capped cylinder. A second important property of SLk is that it depends only on the surface near the axis. Hence, if the surface on which the axis lies is broken or torn apart at places not near the axis, SLk remains invariant. For example, if a DNA lies on a protein, and a portion of the protein not near the axis is broken or decomposes, SLk remains invariant. The third important property is that if a DNA lies on a surface and slides along the surface, then as long as the vector field v varies smoothly from point to point on the surface and as long as in the process of sliding the axis does not break, SLk remains invariant. Thus, if the capped cylinder in Figure 6.9d were allowed to expand and the axis curve required to remain the same length, it would have to unwind as it slid along the surface of the enlarged cylinder. However, SLk would remain equal to 0. Another class of biologically important surfaces exists for which it is possible that a DNA can have an SLk ¹ 0. These are so-called toroidal surfaces. They consist of the round circular torus and their deformations. Suppose an axis curve A traverses the entire length of a round circular torus handle once as it wraps around it a number of times, n. Suppose further that the inner radius of the torus is equal to r. For the vector field v, we choose the inward pointing surface normal. In this case, if one chooses e = r, Ae would be the central axis of the torus. Thus SLk is the linking number of the curve A with the central axis of

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Page 169 Figure 6.9 Examples of displacement curves and  SLk.  For any curve  A lying on a surface, the displacement curve Ae  is formed by moving a small distance  e along the surface normal at each point on the curve. For planar curves as in (a), all of the normal vectors can be chosen to point upward, and then Ae is above  A. The curves are unlinked, and hence  SLk =  0. For curves on a spherical surface as in (b), the surface vectors can be chosen to point inward, and hence Ae is entirely inside and therefore does not link  A. SLk  is again equal to 0. In (c) and (d) the surface normal vectors have been chosen to point inward, and  e has been set equal to the inner radii of the surfaces on which the DNA is wound. In (c), Ae becomes the central axis of the torus, and  SLk =  +4. In (d) the DNA is wrapped plectonemically around a capped cylinder. The displacement curve Ae lies entirely inside, and thus  SLk =  0.

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Page 170 Figure 6.10  (a) Deformations of the round sphere into spheroids. (b) Deformation of the round circular torus into toroids. the torus. This implies that if the wrapping is right-handed, SLk = +n, and if the wrapping is left-handed, SLk = -n. By the invariant properties mentioned above, SLk remains invariant even if the round torus is deformed. Examples are shown in Figure 6.10b. The concept of SLk can also be applied to DNA that are not attached to real protein surfaces but are free in space. For example, the most common kind of free DNA, that is, DNA free of any protein attachment, is plectonemically wound DNA. Here the DNA can be considered to lie on the surface of a spheroid such as the one shown in Figure 6.9b (or a deformation of it), the exact shape of which is determined by the energyminimum DNA conformation. Then the surface may be allowed to vanish and reappear without changing the shape of the DNA superhelix. The DNA is said to be wrapped on a virtual surface (White et al., 1988). Thus, the SLk of the DNA in Figure 6.9d is equal to 0 regardless of whether the surface is virtual or is that of a real protein. More generally, these concepts can be applied to DNA wrapped on a series of proteins

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Page 171 with virtual surfaces joining them. An example of this is presented below in our discussion of the minichromosome. The Winding Number and Helical Repeat We next give a formal definition of the winding number of a DNA wrapping on a surface. Because the vector v is perpendicular to the surface, it is also perpendicular to the DNA axis A and thus lies in the perpendicular planar cross section at each point of the DNA. Therefore, at each point this vector v and the strand-axis vector vac, defined above lie in this same planar cross section. In this plane, the vector vac makes an angle f with the vector v (Figure 6. 1). As one proceeds along the DNA segment, vac spins around v, as the backbone curve C alternately rises away from and falls near to the surface, while the angle f turns through 2p radians (that is, 360°). The total change in the angle f, divided by the normalizing factor 2p (or 360°), during this passage is called the winding number of the DNA and is denoted F (White et al., 1988). This number may also be thought of as the number of times that vac rotates past v as the DNA is traversed. A related quantity called the helical repeat, denoted h, is the number of base pairs necessary for one complete 360° revolution. For closed DNA, because the beginning and ending point are the same for one complete passage of the DNA, the vectors v and vac are the same at the beginning and at the end. In this case, therefore, F must necessarily be an integer. There are equivalent formulations for the winding number of a closed DNA wrapping on a protein surface. During each 360° rotation of the vector vac in the perpendicular plane, f assumes the values of 0 (or 0°) and p (or 180°) exactly once. When f = 0, vac = v, and when f = p, vac = -v. In the latter case, the backbone strand is at maximal distance from the protein surface, and in the former it comes into contact with the protein. Thus the winding number of a closed DNA on a protein surface is the number of times one of the backbone strands contacts the protein surface, or it is the number of times the strand is at maximal distance from it. In this case, it also follows from its definition that the helical repeat is the number of base pairs between successive contact points of one of the backbones or the number of base pairs between successive

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Page 172 Figure 6.11 Definition of the surface vectors necessary to define the winding number. The duplex DNA axis  A lies on the surface. The backbone curve  C will pass above and below the surface as it winds around  A. To describe surface winding, two vectors need to be defined originating at a point  a on A, namely, the unit surface normal vector  v and the strand- axis vector  vac along the line connecting  a to the corresponding point  c on the backbone C. f is defined to be the angle between these two vectors. The winding number F measures  the number of times that f turns through 360°, or how many times  vac rotates past v. points of maximal distance from the protein surface. This latter number can be measured directly by digestion or footprinting methods, which involve probes that search for points of the backbones to cut, the easiest being those points at maximal distance from the surface. The winding number F is also a differential topological invariant and therefore has the same three properties mentioned above for SLk. In particular, it remains invariant if the DNA surface structure is deformed without any breaks in the DNA or any introduction of discontinuities in the vector field v. Under the same conditions, it also remains unchanged if the DNA is allowed to slide along the surface.

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Page 173 Relationship between Linking, Surface Linking, and Winding It is remarkable that the three quantities Lk, SLk, and F, although very different in definition, are related by a theorem from differential topology. In fact, for a closed DNA on a surface, the linking number is the sum of the surface linking number and the winding number (White et al., 1988); that is, Lk = SLk + F. Before we outline the proof of this result, we first give some simple examples. We then give the proof and conclude with the example of the minichromosome. For DNA that lies in a plane or on a spheroid, SLk =0. Therefore, Lk = F, and if there are N base pairs in the DNA, the helical repeat is given by h = N / Lk. These two cases include relaxed circular DNA, for which Lk Lk0, and plectonemically interwound DNA, the most common form of supercoiled DNA. For DNA that traverses the handle of a round circular torus while wrapping n times around the handle, Lk = n + F if the wrapping is right-handed, and Lk = -n + F if the wrapping is left-handed. In both cases, Lk is unchanged if the torus is smoothly deformed. We now outline the proof of the main result. To do this, we first define the surface twist, STw, of the vector field v along the axis curve A (White and Bauer, 1988; White et al., 1988). This is basically defined the same as the twist of the DNA except that the vector field v is used and not the vector field vac, Hence, STw is given by the equation Thus, STw measures the perpendicular component of the change of the vector v as one proceeds along the axis A, and thus is a measure of the spinning of the vector field v around the curve A. It can also be considered to be the twist of Ae around A. We recall that Tw measures

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Page 174 the spinning of the vector field vac around the curve A. Thus, the difference Tw- STw measures the spinning of vac around v. But this is exactly the winding number F. Hence Tw - STw = F. We recall the fundamental formula Wr(A) + Tw(C,A) = Lk(C,A) or Wr + Tw = Lk. A similar formula relating STw, Wr, and SLk holds: WrA + STw = LkA,Ae or Wr + STw = SLk because STw is the twist of Ae around A and SLk is the linking number of Ae and A. Combining the two formulas and using the result that Tw - STw = F, we obtain Lk = SLk + F. The biological importance of this relationship is that all three of these quantities are experimentally measurable. Thus, having determined any two of them, one can calculate the other and then compare with the experimental value. In the next section, we show by a classical example from molecular biology, the minichromosome, the power of this theorem. Application to the Study of the Minichromosome A minichromosome is a structure that consists of a closed DNA bound to a series of core nucleosomes. Such a structure allows the compaction of a very long DNA into a small volume, in the same way that a long piece of thread is compacted by wrapping it on a spool. Understanding such structures is essential to a knowledge of how DNA

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Page 175 is packaged in the cell. In this section, we study the geometry and topology of DNA in such a structure. Each nucleosome may best be described as a cylinder, the histone octamer, around which the DNA wraps approximately 1.8 times in a left-handed manner. The DNA segments between successive nucleosomes are called linker regions. Thus, the DNA divides between linker DNA and core-associated DNA. An example of such a structure is shown in Figure 6.12. Such a compound structure consists of a toroidal surface, part of which is the real surface of the nucleosome cores and part of which is virtual linker surfaces joining successive cylinders. These virtual pieces are deformed cylindrical sections, all of the same radius, on which the linker DNA are constrained to lie. The specification of each of these surfaces is arbitrary as long as it takes into account the coiling of the linker. The linker DNA can thus be thought of as a generating curve for the cylindrical section. An important condition to be imposed is that the linker DNA does not wind around the piece on which it lies. This condition will ensure that all contributions to SLk due to winding around the torus handle will come only from the intranucleosome winding. Any additional contribution to SLk must therefore come from the coiling of the linker DNA. To simplify our example, we will assume that the minichromosome is relaxed. This means that the linker regions are planar and that all contributions to SLk come from the winding of the DNA around the histone octamers. Such a relaxed state can be achieved by the introduction into the minichromosome of topoisomerases, which relax the linker DNA but leave unaffected the DNA on the nucleosome cores. In this case, SLk can be directly measured by X-ray diffraction and found to be -1.8 m, where m is the number of nucleosomes. An example with 5 nucleosomes is shown in Figure 6.13, for which SLk = -9. For SV40 DNA, there are about 25 nucleosomes (Sogo et al., 1986). Therefore, SLk = -45. The linking number of the DNA on the relaxed SV40 minichromosome is measured in an indirect way. First, the DNA is stripped of the nucleosome particles, becoming in the process a plectonemically interwound free DNA. By means of gel electrophoresis, its linking number can be experimentally measured. In actuality, what is measured is the difference of its linking number and the linking number of the same DNA totally relaxed, DLk, as defined above. DLk is found to be about -1 per nucleosome core; that is, DLk =-25 (Shure and

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Page 176 Figure 6.12 Cartoon of a minichromosome. Three cylinders representing histone octamers are wound by DNA so as to form three nucleosomes. The nucleosomes  are connected by linker DNA segments. Successive real nucleosomes are connected by virtual deformed cylindrical pieces, the deformations of which are determined by the coiling of the linker. Reprinted, by permission, from White et al. (1988). Copyright © 1988 by American Association for  the Advancement of Science. Figure  6.13 Diagram  of a relaxed  minichromosome with five cylindrical nucleosomes. The DNA wraps left-handedly 1.8 times around each nucleosome. The contribution to  SLk  is -1.8 for each nucleosome and 0 for each linker region. For the entire structure,  SLk =  - 9. Reprinted, by permission, from White et al. (1989). Copyright © 1989 by Academic Press Limited.

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Page 177 Vinograd, 1976). As we stated above, relaxed SV40 has a linking number, Lk0, of approximately 500. Therefore minichromosomal SV40 has Lk = 475. We can now answer an important question: Is the number of base pairs per turn, the helical repeat, unchanged from the 10.5 of relaxed DNA, when DNA is wrapped on the nucleosome? The answer must be negative because of the relationship Lk = SLk + F. Thus, we can theoretically determine that because Lk = 475 and SLk = -45, F must be 520. However, we have seen above that F for relaxed SV40 is equal to Lk0 =  500. Because F = 520, the average helical repeat for minichromosomal SV40 equals 5,250/520 = 10.10. In this analysis, we have made a great many simplifications, but it is noteworthy that this number is in remarkably good agreement with the number 10.17 that is obtained by nuclease digestion experiments. The answer to the question is thus negative. To summarize, we have found a fundamental relationship Lk = SLk + Ffor three quantities that are directly accessible to experiment, Lk by electrophoresis, SLk by X-ray diffraction, and F by digestion. If two of the three are known, one can use the relationship to predict and therefore verify the experimental evidence for finding the third. This gives a powerful use of differential topology in the field of molecular biology. References Bauer, W.R., 1978, ''Structure and reactions of closed duplex DNA," Annu. Rev. Biophys. Bioeng. 7, 287-313. Bauer, W.R., F.H.C. Crick, and J.H. White, 1980, "Supercoiled DNA," Scientific American 243, 118-133. Boles, T.C., J.H. White, and N.R. Cozzarelli, 1990, "Structure of plectonemically supercoiled DNA," Journal of Molecular Biology 213, 931-951. Drew, H.R., and A.A. Travers, 1985, "DNA bending and its relation to nucleosome positioning," Journal of Molecular Biology 186, 773-790. Finch, J.T., L.C. Lutter, D. Rhodes, R.S. Brown, B. Rushton, M. Levitt, and A. Klug, 1977, "Structure of nucleosome core particles of chromatin," Nature 269, 29-36. Finch, J.T., R.S. Brown, D. Rhodes, T. Richmond, B. Rushton, L.C. Lutter, and A. Klug, 1981, "X-ray diffraction study of a new crystal form of the nucleosome core showing higher resolution," Journal of Molecular Biology 145, 757-770.

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Page 178 Richards, F.M., 1977, "Areas, volumes, packing and protein structure," Annu. Rev. Biophys. Bioeng. 6, 151-176. Richmond, T.J., J.T. Finch, B. Rushton, D. Rhodes, and A. Klug, 1984, "Structure of the nucleosome core particle at 7Å resolution," Nature 311, 532-537. Shure, M., and J. Vinograd, 1976, "The number of superhelical turns in native virion SV40 DNA and minicol DNA determined by the band counting method," Cell 8, 215-226. Sogo, J.M., H. Stahl, T. Koller, and R. Knippers, 1986, "Structure of replicating simian virus 40 minichromosomes. The replication fork, core histone segregation and terminal structures," Journal of Molecular Biology 189, 189-204. Wang, J.C., 1985, "DNA topoisomerases," ,nnu. Rev. Biochem. 54, 665-697. Wang, J.C., and W.R. Bauer, 1979, "The electrophoretic mobility of individual DNA topoisomers is unaffected by denaturation and renaturation," Journal of Molecular Biology 129, 458-461. White, J.H., 1969, "Self-linking and the Gauss integral in higher dimensions," Am. J. Math. 91, 693-728. White, J.H., 1989, "An introduction to the geometry and topology of DNA structure," pp. 225-253 in Mathematical Methods for DNA Sequences, M.S. Waterman (ed.), Boca Raton, Fla: CRC Press. White, J.H., and W.R. Bauer, 1986, "Calculation of the twist and the writhe for representative models of DNA," Journal of Molecular Biology 189, 329-341. White, J.H., and W.R. Bauer, 1988, "Applications of the twist difference to DNA structural analysis," Proceedings of the National Academy of Sciences USA 85, 772-776. White, J.H., N.R. Cozzarelli, and W.R. Bauer, 1988, "Helical repeat and linking number of surface-wrapped DNA," Science 241, 323-327. White, J.H., R. Gallo, W.R. Bauer, 1989, "Effect of nucleosome distortion on the linking deficiency in relaxed minichromosomes," Journal of Molecular Biology 207, 193-199.