Superhelical DNA with local substructures. A generalization of the topological constraint in terms of the intersection number and the ladder-like correspondence surface

The presence of certain local structural elements in superhelical DNA, such as cruciforms and denatured loops, complicates the topological and geometric analysis of these molecules. In particular, the duplex axis is often difficult to define. In consequence, the usual conservation condition, Lk = Tw + Wr, is often inapplicable as formulated in terms of the winding of either strand of the DNA about the duplex axis. We present here a more general formulation of the topological conservation condition in terms of a model in which the two strands of DNA are regarded as twisting about one another, and in which one of the two strands is considered to writhe. We define a ladder-like correspondence surface, which connects the two strands nd is independent of whether or not a unique duplex axis is locally available. These considerations lead to the definition of a new topological property of superhelical DNA, the intersection number, In. This quantity describes the complexity of a local structural element; in the case of a cruciform, for example, the intersection number is a measure of the number of duplex turns removed from the major segment of the DNA by the cruciform formation. In more general terms, the topological constraint applicable to closed circular DNA is given by Lk(W,C) + In(S,C) = Tw(W,C) + Wr (C), where W and C represent the two strands of the DNA and S is the ladder-like correspondence surface that connects the two strands.