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We model DNA as a Kirchhoff rod, making rigorous the foundations, scales, and assumptions behind the theory. DNA experiences powerful torsional forces while held in constrained structures that, on the scales of a few hundred base pairs, can be viewed as linear. As such, the structures of DNA can be studied by analysis of a twisted, intrinsically straight rod under tension. The novelty of this work lies in several points. First, this is a fully dynamic approach, including a continuous injection of twist at one domain boundary where the other is held fixed from rotation. This is a special case, most applicable to DNA, but which has not been previously examined. This work gives careful consideration to the derivation and novel incorporation of drag forces, and a rigorous treatment of the relationships between the drag model and the equations of motion. The model is examined using linear analysis about the twisted, straight rod under tension, assymtotics, and numerical analysis of the driven rod. We then apply the insights obtained through this work to examine the distribution of a inverted repeat sequences that are theoretically susceptible to supercoiling-induced structural transitions to cruciforms driven by these torsional stresses.