Topology of DNA: When manipulation supports the lack of ‘space-filling’ imagination

Abstract

Introduction Topology, the branch of mathematics that deals with geometrical objects that remain unchanged under transformations called homeomorphisms or deformations, 1 is a powerful tool for describing the structure of DNA. 2 Among the objects studied by topologists, knots and closed circular curves have attracted the attention of organic chemists and biochemists since the sixties, when Edel Wasserman and co-workers first introduced the concept of topological bonds (bonds that, unlike any others, are maintained independent of direct interaction) and synthetized interlocked organic rings. 3 A topological approach to DNA structure gained importance when a large body of experimental data made it clear that many DNA molecules are circular. For example, the small monkey DNA virus SV40, the short chromosomes of single-stranded phages, most plasmid DNAs, and almost all bacterial chromosomes belong to the class of circular molecules. Circular shapes initially appeared incompatible with the ability of DNA to make its information accessible by unwinding the doublestranded helix during copying processes (ie replication and transcription). The unwinding puzzle disappeared when specific enzymes called topoisomerases, able to make double-strand breaks (type II topoisomerases) or singlestrand breaks (type I topoisomerases) without digesting DNA molecule into pieces, were discovered. 4'5 axis of the helix and writhe, expressed by the supercoiling phenomenon, to the pathway in space of the axis of the helix. The correlation relating the topological parameters is simply expressed by the equation L = T + W , where linking number and twist are positive since the DNA duplex is a right-hand helix, and writhe is negative if the supercoils are right-handed. In a hypothetical circular molecule of a DNA duplex, containing for example 210 base pairs (bp) and having the same number of bp per turn as the linear DNA B (10.5), the total number of turns is then equal to 20 if the molecule is in the relaxed conditions (ie with its axis lying completely on a plane). In such a molecule, in which one strand crosses the other 20 times, linking number and twist have the same value, while the number of supercoils (W) is equal to 0.