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WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 167 components that relate directly to the surface and surface-related experiments. The linking number of a closed DNA constrained to lie on the surface divides into two integral quantities, the surface linking number, which measures the wrapping of the DNA around the surface, and the winding number, which is a measure of the number of times that the backbone contacts or rises away from the surface (White et al., 1988). Experimentally, the first quantity can be measured by X-ray diffraction, and the second can be measured by digestion or footprinting methods. In particular for the nucleosome, the partial contribution to the surface linking number due to the left-handed wrapping around the cylindrical core is â1.85 (Finch et al., 1981; Richmond et al., 1984). Furthermore, the winding number has been measured to be the number of base pairs of the DNA on the nucleosome divided by approximately 10.17 (Drew and Travers, 1985). THE SURFACE LINKING NUMBER We now give a formal definition of the two quantities, surface linking number SLk and winding number, for a closed DNA on a protein surface. We assume that the surface involved has the property that at each point near the axis of the DNA, there is a well-defined surface normal vector. The unit vector along this vector will be denoted by v. (We assume that the surfaces are orientable. In this case, there are two possible choices for the vector field v depending on the side of the surface to which the vector field points.) If the DNA axis A is displaced a small distance ε â 0 along this vector field at each point, a new curve Aε is created. ε should be chosen small enough so that during the displacement of A to Aε no crossings of one curve with the other take place. The curve Aε is also closed and can be oriented in a manner consistent with the orientation of A. The surface linking number is defined to be the linking number of the original curve A with the curve Aε (White and Bauer, 1988). Simple examples of the surface linking number occur for DNA whose axes lie on planar surfaces or spheroidal surfaces. First, for a DNA whose axis lies in a plane, SLk = 0. This is easy to see, for in this case the vector field v is a constant field perpendicular to the plane. Hence the curve Aε lies entirely to one side of
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 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 Aε 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 â 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 ε = r, Aε would be the central axis of the torus. Thus SLk is the linking number of the curve A with the central axis of
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 169 Figure 6.9 Examples of displacement curves and SLk. For any curve A lying on a surface, the displacement curve Aε is formed by moving a small distance ε 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 Aε 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 Aε 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 ε has been set equal to the inner radii of the surfaces on which the DNA is wound. In (c), Aε becomes the central axis of the torus, and SLk = +4. In (d) the DNA is wrapped plectonemically around a capped cylinder. The displacement curve Aε lies entirely inside, and thus SLk = 0.
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 170 the torus. This implies that if the wrapping is right-handed, SLk = +n, and if the wrapping is left-handed, SLk = ân. By the invariant properties mentioned above, SLk remains invariant even if the round torus is deformed. Examples are shown in Figure 6.10b. Figure 6.10 (a) Deformations of the round sphere into spheroids. (b) Deformation of the round circular torus into toroids. 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
