Schematic representation of DNA. The double-helical axis is given by curve and one of the helical strands by curve . For the purpose of calculation of the twist of about , is to be thought of as being traced out by the head of a vector everywhere normal to the tangent vector .
The vectors associated with a base-pair plane: an origin and a mutually orthogonal triad of unit vectors, the short axis , the long axis , and the normal .
The passage from a smooth curve with circular segments of curvature to the entirely linear segments that connect the origins of the base-pair planes consistent with a high-resolution structure of DNA.
The vectors involved in the calculation of the twist of supercoiling of the DNA base-pair step bounded by the and the planes. Shown in (a), the four base-pair plane origins needed for the determination of the three vectors depicted in (b), and , which, in turn, are needed for specifying and . These last two are normal to the two planes seen in (c). Each of these planes contains a vector and a vector. Two of the angles needed for the twist calculation as given by Eq. (16) are denoted in (d), an enlargement of (c).
The vectors needed for the calculation of the step-parameter twist of the same step shown in the previous figure. Here knowledge of the direction of the two normals allows the determination of the single vector , which lies in each of the two base-pair planes. Also indicated are the two angles needed for the use of Eq. (18) for the twist calculation.
Construction of a model DNA structure characterized by a chiral deformation. Image labeled (a) shows four equally spaced and parallel base-pair planes having their origins lying on a line. The sequence of base-pair planes in (b) depicts the structure after the bend described in the text is introduced. The four origins are still coplanar, and the viewing direction is chosen to be normal to this plane. A translation of base pairs 3 and 4 as a single unit along the viewing direction, depending on the direction of the motion, results either in (c), a structure with a right-handed jog, or (d), one with a left-handed jog.
Structural deformation of DNA leading to a twist of supercoiling change dependent on the method used to determine the position of the origin, the method used here, or that in which the origin lies on a line connecting two carbon atoms on the bases. In structure (a), the same as structure (d) in Fig. 6, both methods lead to an origin for all four base pairs located in the same position. However when base pair 3 is buckled as shown to form structure (b), the origin determined by our method (blue dot) moves, but that of the other method (red dot) does not. One then observes a method-dependent twist of supercoiling for the middle step.
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