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Suggested Citation:"DNA SUPERHELICITY - MATHEMATICS AND BIOLOGY." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Page 181
Suggested Citation:"DNA SUPERHELICITY - MATHEMATICS AND BIOLOGY." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Page 182
Suggested Citation:"DNA SUPERHELICITY - MATHEMATICS AND BIOLOGY." National Research Council. 1995. Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology. Washington, DC: The National Academies Press. doi: 10.17226/2121.
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Page 183

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UNWINDING THE DOUBLE HELIX: USING DIFFERENTIAL MECHANICS TO PROBE CONFORMATIONAL CHANGES IN 181 DNA 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)

UNWINDING THE DOUBLE HELIX: USING DIFFERENTIAL MECHANICS TO PROBE CONFORMATIONAL CHANGES IN 182 DNA Although Lk is fixed in a topological domain, both Tw and Wr may still vary, provided they do so in a complementary manner. Cutting one DNA strand in a domain releases the topological constraint of constant Lk, allowing it to find its most relaxed state. The two resulting ends may rotate freely, relaxing any torsional deformation imposed on the molecule. Writhing deformations can be converted to twist and then removed by this rotational relaxation. The sum of the twist and writhe in this relaxed state determines a relaxed linking number Lk0. Note that, while the linking number Lk of a circular DNA molecule must be an integer, the relaxed linking number Lk0 need not be integral. Stresses are imposed on a topological domain whenever its linking number Lk differs from the relaxed value. The resulting linking difference α = Lk − Lk0 must be accommodated by twisting and/or writhing deformations: α = ∆ Tw + ∆ Wr. (7.2) Topological domains in living systems are commonly found in a negatively superhelical state, in which the imposed linking number is smaller than its relaxed value, so α < 0. Negative superhelicity provides a mechanism for driving strand separation. Because the separated strands are less twisted than the B-form, they localize some of the linking deficiency as a decrease of twist at the transition site, thereby allowing the rest of the domain to relax a corresponding amount. Since strand separations require energy, they are disfavored in unconstrained or relaxed molecules. However, in a negatively superhelical domain, local strand separations are energetically favored to occur at equilibrium whenever the topological strain energy that is thus relieved exceeds the energetic cost of locally disrupting the base pairing between strands. The linking differences imposed on topological domains in vivo are carefully regulated. Virtually all organisms produce enzymes that alter Lk through the introduction of transient strand breaks (Gellert, 1981). The action of these molecules maintains topological domains in negatively superhelical, underlinked states (i.e., α <0). On average, bacteria and other primitive organisms maintain approximately half their domains in a superhelical state. Moreover, the amount of superhelicity imposed on DNA in vivo is known to vary with the cell division cycle in

UNWINDING THE DOUBLE HELIX: USING DIFFERENTIAL MECHANICS TO PROBE CONFORMATIONAL CHANGES IN 183 DNA a carefully regulated manner (Dorman et al., 1988). The extent of superhelicity also varies in response to environmental changes (Bhriain et al., 1989; Malkhosyan et al., 1991). In multicelled organisms, superhelicity occurs primarily within domains containing actively expressing genes. The DNA within malignant cancer cells is maintained at more extreme negative linking differences than that characterizing the corresponding DNA in normal cells (Hartwig et al., 1981). Many important regulatory events are sensitive to the degree of superhelical stress imposed on the DNA. These include the initiation of gene expression (Smith, 1981; Pruss and Drlica, 1989; Weintraub et al., 1986) and of DNA replication (Kowalski and Eddy, 1989; Mattern and Painter, 1979). Substantial evidence suggests that superhelically driven strand separations may be involved in these processes. One wellcharacterized case occurs at the origin of DNA replication of the bacterium E. coli (Kowalski and Eddy, 1989). The DNA sequence at this origin site contains a triple repeat of an A+T−rich run of 13 base pairs that is required for the initiation of DNA replication. Deletion and substitution experiments have shown that the key functional attribute of this sequence is its susceptibility to superhelical strand separation. DNA sequence changes at this site that retain this attribute preserve its ability to initiate replication in vivo; DNA sequence changes that degrade this susceptibility destroy in vivo origin function. No other sequence specificity is observed. Such sequences are called duplex unwinding elements (DUEs) and are present at origins of DNA replication in many organisms (Umek et al., 1989). Superhelicity also is known to modulate the expression of some genes. In bacteria, superhelicity regulates the expression of the so-called SOS system, a suite of genes that are activated in response to environmental stresses or DNA damage. The bacterial response to deleterious environmental changes is to increase the superhelicity of its DNA, which activates expression of the SOS genes (Bhriain et al., 1989; Malkhosyan et al., 1991). Experimental (Kowalski et al., 1988) and theoretical (Benham, 1990) results indicate that the susceptibility of some DNA molecules to superhelical strand separation is confined to sites that bracket specific genes. This suggests that there may be at least two classes of genes, distinguishable by their sensitivities to superhelical separation, whose mechanisms of operation may be different.

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As researchers have pursued biology's secrets to the molecular level, mathematical and computer sciences have played an increasingly important role—in genome mapping, population genetics, and even the controversial search for "Eve," hypothetical mother of the human race.

In this first-ever survey of the partnership between the two fields, leading experts look at how mathematical research and methods have made possible important discoveries in biology.

The volume explores how differential geometry, topology, and differential mechanics have allowed researchers to "wind" and "unwind" DNA's double helix to understand the phenomenon of supercoiling. It explains how mathematical tools are revealing the workings of enzymes and proteins. And it describes how mathematicians are detecting echoes from the origin of life by applying stochastic and statistical theory to the study of DNA sequences.

This informative and motivational book will be of interest to researchers, research administrators, and educators and students in mathematics, computer sciences, and biology.

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