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WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 173 RELATIONSHIP BETWEEN LINKING, SURFACE LINKING, AND WINDING It is remarkable that the three quantities Lk, SLk, and Φ, 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 + Φ. 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 = Φ, 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 + Φ if the wrapping is right- handed, and Lk = ân + Φ 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 Aε around A. We recall that Tw measures
