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WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 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 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 turns through 2Ï radians (that is, 360°). The total change in the angle , divided by the normalizing factor 2Ï (or 360°), during this passage is called the winding number of the DNA and is denoted Φ (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, Φ 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, assumes the values of 0 (or 0°) and Ï (or 180°) exactly once. When = 0, vac = v, and when = Ï, vac = âv. 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
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 172 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. 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. is defined to be the angle between these two vectors. The winding number Φ measures the number of times that turns through 360°, or how many times vac rotates past v. The winding number Φ 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.
