Experimental techniques such as X-ray crystallography and nuclear magnetic resonance provide ways to infer precise distances between atoms. However, these methods are not well suited to studying the dynamic mechanism by which enzymes act. Interestingly, topology can shed light on this key issue. The topological approach to enzymology is an experimental protocol in which the descriptive and analytical powers of topology and geometry are employed in an indirect effort to determine the enzyme mechanism and the structure of active enzyme-DNA complexes in vitro (in a test tube) (Wasserman and Cozzarelli, 1986; Sumners, 1987a). Once the enzyme structure and mechanism are understood in a controlled laboratory situation, this knowledge can be extrapolated to enzyme mechanism in vivo, that is, in a living cell.
Topology is a branch of mathematics related to geometry. It is often characterized as "rubber-sheet geometry," because topological equivalence of spaces allows stretching, shrinking, and twisting of an object in order to make it congruent to another object. Topology is the study of properties of objects (spaces) that are unchanged by allowable elastic deformations. When a given topological property differs for a pair of spaces, then one can be sure that one space cannot be transformed into the other by elastic deformation. Changes that can produce nonequivalent spaces include cutting the space apart and reassembling the parts to produce another space. It is precisely this topological breakage and reassembly of DNA that characterizes the mechanism of some life-sustaining cellular enzymes, enzymes that facilitate replication, transcription, and transposition. Chapter 6 describes aspects of the geometry and topology of DNA and points out various topological transformations that must be performed on DNA by enzymes in order to carry out the life cycle of the cell. In the present chapter, we describe how recent results in three-dimensional topology (Culler et al., 1987; Ernst and Sumners, 1990; Sumners, 1990, 1992) have proven to be of use in the description and quantization of the action of these life-sustaining enzymes on DNA.
The DNA of all organisms has a complex and fascinating topology. It can be viewed as two very long curves that are intertwined millions of