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DNA Superhelicity—
Mathematics and Biology

DNA in living cells is held in topological domains whose linking numbers can be individually regulated. In practice there are two types of domains. Small DNA molecules can occur as closed circles, whereas larger DNA molecules are formed into a series of loops by periodic attachments to a protein scaffold in a way that precludes local rotations at the attachment site. This arrangement constrains the portion of DNA between adjacent attachment sites to be a topological domain analogous to a closed circle.

For simplicity we consider a closed circular duplex DNA molecule as the paradigm of the topological domain. (Closed circles are also the molecules of choice for experiments in this field.) The two strands that make up the DNA duplex each have a chemical orientation induced by the directionality of the bonds that join neighbor bases. This is called the 5'-3' orientation because each phosphate group in a strand joins the 5' carbon of one sugar to the 3' carbon of the next. This orientation must be the same for every phosphate group within a strand, which imparts a directionality to the strand as a whole. The two strands of the B-form duplex are oriented so their 5'-3' directions are antiparallel. In consequence, a duplex DNA molecule can be closed into a circle only by joining together the ends of each individual strand. Circularization by joining the ends of one strand to those of the other to form a Möbius strip is forbidden because the bonds required would violate the conservation of 5'-3' directionality. Hence a closed circular DNA molecule is composed of two interlinked, circular (antiparallel) strands.

Circularization fixes the linking number of the resulting molecule; the linking number is the number of times that either strand links through the closed circle formed by the other strand. (Topological domains formed by periodic attachments have a functionally equivalent constraint.) The fixing of the linking number Lk within a topological domain provides a global constraint that topologically couples its secondary and tertiary structures according to White's (1988) formula

Lk = Tw+ Wr.                              (7.1)



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