Calculation of the twist and the writhe for representative models of DNA

Abstract

The geometric quantities twist (Tw) and writhe (Wr) are of primary importance for a complete description of the structure of DNA. In the case of a closed circular DNA, the sum of Tw and Wr is constant and equal to the linking number, Lk. In this paper we present a general method for calculation of the twist in terms of a pair of curves and a correspondence surface that joins them. The twist of any pair of curves (C1 and C2) may take on different values depending upon their ordering, and in general Tw(C1, C2) is not equal to Tw(C2, C1). We describe four models that may be taken to represent the structure of DNA and compute the twist for both orderings in each case. The four models examined are: I, a regular helix about a linear axis; II, a toroidal helix about a closed circular axis; III, a superhelix about a regular helical axis; and IV, a superhelix about a closed toroidal helix. In cases II and IV these results are also used to calculate Lk and Wr. Case III is used to analyze the winding of DNA in a nucleosome.