Twisted paths in Euclidean groups: Keeping track of total orientation while traversing DNA
Seminar Room 1, Newton Institute
This talk introduces a new mathematical structure for modeling global twist in DNA. The relative rigid-body motion between reference frames attached either to a backbone curve, bi-rods, or individual bases in DNA, can be described well using elements of the Euclidean motion group, SE(n). However, the group law for Euclidean motions does not keep track of overall twist. In the planar case, the universal covering group of SE(2) identifies orientation angle as a quantity on the real line rather than on the circle, and hence keeps track of ``global'' rotations (not modulo 360 degrees). However, in the three-dimensional case, no such structure exists since the the orientational part of the universal cover of SE(3) can be identified with the quaternion sphere. In this talk a new mathematical structure for ``adding'' framed curves and extracting global twist is present. Though reminiscent of the group operation in braid theory and in homotopy theory, this structure is distinctly different, as it is geometric in nature, rather than topological. The motivation for this mathematical structure and its applications to DNA conformation will be presented.