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WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 177 Vinograd, 1976). As we stated above, relaxed SV40 has a linking number, Lk0, of approximately 500. Therefore minichromosomal SV40 has Lk = 475. We can now answer an important question: Is the number of base pairs per turn, the helical repeat, unchanged from the 10.5 of relaxed DNA, when DNA is wrapped on the nucleosome? The answer must be negative because of the relationship Lk = SLk + Φ. Thus, we can theoretically determine that because Lk = 475 and SLk = â45, Φ must be 520. However, we have seen above that Φ for relaxed SV40 is equal to Lk0 = 500. Because F = 520, the average helical repeat for minichromosomal SV40 equals 5,250/520 = 10.10. In this analysis, we have made a great many simplifications, but it is noteworthy that this number is in remarkably good agreement with the number 10.17 that is obtained by nuclease digestion experiments. The answer to the question is thus negative. To summarize, we have found a fundamental relationship Lk = SLk + Φ Î¿Ï ÏηÏεε Î¸Ï Î±Î½ÏιÏÎ¹ÎµÏ ÏÎ·Î±Ï Î±Ïε διÏεÏÏÎ»Ï Î±ÏÏεÏÏιβλε Ïο εξÏεÏιµενÏ, ÎÎºÎ²Ï ÎµÎ»ÎµÏÏÏοÏηοÏεÏιÏ, âÎÎºÎ²Ï ÎâÏÎ±Ï Î´Î¹ ÏαÏÏιον, ανδ Φ by digestion. If two of the three are known, one can use the relationship to predict and therefore verify the experimental evidence for finding the third. This gives a powerful use of differential topology in the field of molecular biology. REFERENCES Bauer, W.R., 1978, ''Structure and reactions of closed duplex DNA," Annu. Rev. Biophys. Bioeng. 7, 287-313. Bauer, W.R., F.H.C. Crick, and J.H. White, 1980, "Supercoiled DNA," Scientific American 243, 118-133. Boles, T.C., J.H. White, and N.R. Cozzarelli, 1990, "Structure of plectonemically supercoiled DNA," Journal of Molecular Biology 213, 931-951. Drew, H.R., and A.A. Travers, 1985, "DNA bending and its relation to nucleosome positioning," Journal of Molecular Biology 186, 773-790. Finch, J.T., L.C. Lutter, D. Rhodes, R.S. Brown, B. Rushton, M. Levitt, and A. Klug, 1977, "Structure of nucleosome core particles of chromatin," Nature 269, 29-36. Finch, J.T., R.S. Brown, D. Rhodes, T. Richmond, B. Rushton, L.C. Lutter, and A. Klug, 1981, "X-ray diffraction study of a new crystal form of the nucleosome core showing higher resolution," Journal of Molecular Biology 145, 757-770.
WINDING THE DOUBLE HELIX: USING GEOMETRY, TOPOLOGY, AND MECHANICS OF DNA 178 Richards, F.M., 1977, "Areas, volumes, packing and protein structure," Annu. Rev. Biophys. Bioeng. 6, 151-176. Richmond, T.J., J.T. Finch, B. Rushton, D. Rhodes, and A. Klug, 1984, "Structure of the nucleosome core particle at 7Ã resolution," Nature 311, 532-537. Shure, M., and J. Vinograd, 1976, "The number of superhelical turns in native virion SV40 DNA and minicol DNA determined by the band counting method," Cell 8, 215-226. Sogo, J.M., H. Stahl, T. Koller, and R. Knippers, 1986, "Structure of replicating simian virus 40 minichromosomes. The replication fork, core histone segregation and terminal structures," Journal of Molecular Biology 189, 189-204. Wang, J.C., 1985, "DNA topoisomerases," Annu. Rev. Biochem. 54, 665-697. Wang, J.C., and W.R. Bauer, 1979, "The electrophoretic mobility of individual DNA topoisomers is unaffected by denaturation and renaturation," Journal of Molecular Biology 129, 458-461. White, J.H., 1969, "Self-linking and the Gauss integral in higher dimensions," Am. J. Math. 91, 693-728. White, J.H., 1989, "An introduction to the geometry and topology of DNA structure," pp. 225-253 in Mathematical Methods for DNA Sequences, M.S. Waterman (ed.), Boca Raton, Fla: CRC Press. White, J.H., and W.R. Bauer, 1986, "Calculation of the twist and the writhe for representative models of DNA," Journal of Molecular Biology 189, 329-341. White, J.H., and W.R. Bauer, 1988, "Applications of the twist difference to DNA structural analysis," Proceedings of the National Academy of Sciences USA 85, 772-776. White, J.H., N.R. Cozzarelli, and W.R. Bauer, 1988, "Helical repeat and linking number of surface-wrapped DNA," Science 241, 323-327. White, J.H., R. Gallo, W.R. Bauer, 1989, "Effect of nucleosome distortion on the linking deficiency in relaxed minichromosomes," Journal of Molecular Biology 207, 193-199.