Calculating the Secrets of Life: Contributions of the Mathematical Sciences to Molecular Biology (1995)

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WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 174 the spinning of the vector field vac around the curve A. Thus, the difference Tw−STw measures the spinning of vac around v. But this is exactly the winding number Φ. Hence Tw − STw = Φ. We recall the fundamental formula Wr(A) + Tw(C,A) = Lk(C,A) or Wr + Tw = Lk. A similar formula relating STw, Wr, and SLk holds: WrA + STw = LkA,Aε or Wr + STw = SLk because STw is the twist of Aε around A and SLk is the linking number of Aε and A. Combining the two formulas and using the result that Tw − STw = Φ, we obtain Lk = SLk + Φ. The biological importance of this relationship is that all three of these quantities are experimentally measurable. Thus, having determined any two of them, one can calculate the other and then compare with the experimental value. In the next section, we show by a classical example from molecular biology, the minichromosome, the power of this theorem. APPLICATION TO THE STUDY OF THE MINICHROMOSOME A minichromosome is a structure that consists of a closed DNA bound to a series of core nucleosomes. Such a structure allows the compaction of a very long DNA into a small volume, in the same way that a long piece of thread is compacted by wrapping it on a spool. Understanding such structures is essential to a knowledge of how DNA 

WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 175 is packaged in the cell. In this section, we study the geometry and topology of DNA in such a structure. Each nucleosome may best be described as a cylinder, the histone octamer, around which the DNA wraps approximately 1.8 times in a left-handed manner. The DNA segments between successive nucleosomes are called linker regions. Thus, the DNA divides between linker DNA and core-associated DNA. An example of such a structure is shown in Figure 6.12. Such a compound structure consists of a toroidal surface, part of which is the real surface of the nucleosome cores and part of which is virtual linker surfaces joining successive cylinders. These virtual pieces are deformed cylindrical sections, all of the same radius, on which the linker DNA are constrained to lie. The specification of each of these surfaces is arbitrary as long as it takes into account the coiling of the linker. The linker DNA can thus be thought of as a generating curve for the cylindrical section. An important condition to be imposed is that the linker DNA does not wind around the piece on which it lies. This condition will ensure that all contributions to SLk due to winding around the torus handle will come only from the intranucleosome winding. Any additional contribution to SLk must therefore come from the coiling of the linker DNA. To simplify our example, we will assume that the minichromosome is relaxed. This means that the linker regions are planar and that all contributions to SLk come from the winding of the DNA around the histone octamers. Such a relaxed state can be achieved by the introduction into the minichromosome of topoisomerases, which relax the linker DNA but leave unaffected the DNA on the nucleosome cores. In this case, SLk can be directly measured by X-ray diffraction and found to be −1.8 m, where m is the number of nucleosomes. An example with 5 nucleosomes is shown in Figure 6.13, for which SLk = −9. For SV40 DNA, there are about 25 nucleosomes (Sogo et al., 1986). Therefore, SLk = −45. The linking number of the DNA on the relaxed SV40 minichromosome is measured in an indirect way. First, the DNA is stripped of the nucleosome particles, becoming in the process a plectonemically interwound free DNA. By means of gel electrophoresis, its linking number can be experimentally measured. In actuality, what is measured is the difference of its linking number and the linking number of the same DNA totally relaxed, ∆Lk, as defined above. ∆Lk is found to be about −1 per nucleosome core; that is, ∆Lk =−25 (Shure and 

WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 176 Figure 6.12 Cartoon of a minichromosome. Three cylinders representing histone octamers are wound by DNA so as to form three nucleosomes. The nucleosomes are connected by linker DNA segments. Successive real nucleosomes are connected by virtual deformed cylindrical pieces, the deformations of which are determined by the coiling of the linker. Reprinted, by permission, from White et al. (1988). Copyright 1988 by American Association for the Advancement of Science. Figure 6.13 Diagram of a relaxed minichromosome with five cylindrical nucleosomes. The DNA wraps left-handedly 1.8 times around each nucleosome. The contribution to SLk is −1.8 for each nucleosome and 0 for each linker region. For the entire structure, SLk = − 9. Reprinted, by permission, from White et al. (1989). Copyright 1989 by Academic Press Limited.