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concerned with the geometry, topology, and energetics of closed supercoiled DNA.

Supercoiling of closed DNA is ubiquitous in biological systems. It can arise in two ways. First, it can result when DNA winds around proteins. Second, supercoiling can also result from topological constraints known as under- or overwinding, in which case the axis of the DNA usually assumes an interwound, or plectonemic, form.

Supercoiling is important for a wide variety of biological processes. For example, supercoiling is a way of storing free energy—which can be used to assist the vital processes of replication and transcription, processes that require untwisting or separation of DNA duplex strands. Thus, supercoiling helps enzymes called helicases, polymerases, and other proteins to force apart the two strands of the DNA double helix, allowing access to the genetic information stored in the base sequence. It also promotes a variety of structural alterations that lead to DNA unwinding, such as z-DNA (left-handed double helical DNA) and cruciforms (cross-shapes). In higher organisms, supercoiling helps in cellular packaging of DNA in structures called nucleosomes, in which DNA is wound around proteins called histones. It is crucial in bringing together and aligning DNA sequences in site-specific recombination. It also changes the helical periodicity (number of base pairs per turn) of the DNA double helix; such changes can alter the binding of proteins to the DNA or the phasing of recombinant sequences.

Understanding supercoiled DNA is thus essential for the understanding of these diverse processes. Numerous biological experiments—including those based on sedimentation, gel electrophoresis, electron microscopy, X-ray diffraction, nuclease digestion, and footprinting—can give information about these matters. However, mathematical methods for describing and understanding closed circular DNA are needed to explain and classify the data obtained from these experiments.

This chapter defines and elucidates the major geometric descriptors of supercoiled DNA: linking, twisting, and writhing. It applies these concepts to classify the action of the major types of cutting enzymes, topoisomerases of Type I and Type II. It then develops the differential topological invariants necessary to describe the structural changes that occur in the DNA that is bound to proteins. Other chapters in this book explore applications of topology and geometry to DNA coiling. Chapter

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