Abstract
The statistical–mechanical treatment of closed polymer chains based on algebraic topology is proposed. Using the Monte-Carlo method numerical results were obtained for the probability of knot formation during random closing of polymer chains of different length. For very rigid chains such as DNA double helix the probability of knot formation is rather great. Topological restrictions in a system of two polymer chains are shown to lead to a specific topological interaction between them.
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References
Flory, P. J., Principles of Polymer Chemistry (Cornell University Press, Ithaca, New York, 1953).
Vol'kenshtein, M. V., Configurational Statistics of Polymer Chains (Wiley, New York, 1963).
Birshtein, T. M., and Ptitsyn, O. B., Conformations of Macromolecules (Nauka, Moscow, 1964).
Flory, P. J., Statistical Mechanics of Chain Molecules (Wiley, New York, 1969).
Schill, G., Catenanes, Rotaxanes and Knots (Academic, New York, 1971).
Hershey, A. D., (ed.), The Bacteriophage Lambda (Cold Spring Harbour, 1971).
Wang, J. C., Acc. chem. Res., 6, 252–256 (1973).
Delbrück, M. in Mathematical Problems in the Biological Sciences (edit. by Bellman, R. E.), 55–63 (Proc. Symp. Math., 14, 1962).
Frisch, H. L., and Wasserman, E., J. Am. chem. Soc., 83, 3789–3795 (1961).
Edwards, S. F., J. Phys. A, 1, 15–28 (1968).
Krivoshei, I. V., Zh. Strukt. Khimii, 9, 285–289 (1968).
Shugalii, A. V., Frank-Kamenetskii, M. D., and Lazurkin, Yu. S. Molek. biol. USSR. 3, 133–145 (1969).
Ninio, J. Biochimie, 53, 485–494 (1971).
Vedenov, A. A., Dykhne, A. M., and Frank-Kamenetskii, M. D., Usp. Fiz. Nauk, 105, 479–519 (1971); Sov. Phys-Usp., 14, 715–736 (1972).
Crippen, G. M. J. theor. Biol., 45, 327–338 (1974).
Boeckmann, J., and Schill, G., Tetrahedron, 30, 1945–1957 (1974).
Vologodskii, A. V., Lukashin, A. V., Frank-Kamenetskii, M. D., and Anshelevich, V. V., Zh. eksp. teor. Fiz., 66, 2153–2163 (1974).
Vologodskii, A. V., Lukashin, A. V., and Frank-Kamenetskii, M. D., Zh. eksp. teor. Fiz., 61, 1875–1885 (1974).
Knott, G. G., Life and Scientific Work of Peter Guthrie Tait (Cambridge University Press, 1911).
Crowell, R. H., and Fox, R. H., Introduction to Knot Theory (Ginn, 1963).
Reidemeister, K., Knotentheorie, ser. Ergebnisse der Mathematik 1, No. 1, 1932.
Rosenbluth, M. N., and Rosenbluth, A. W., J. chem. Phys., 23, 356–360 (1955).
El'yashevich, A. M., and Skvortsov, A. M., Molek. biol. USSR, 5, 204–213 (1971).
Landau, L., and Lifshitz, E. Statistical Physics (Nauka, Moscow, 1964).
Wang, J. C., and Schwartz, H., Biopolymers, 5, 953–966 (1967).
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Frank-Kamenetskii, M., Lukashin, A. & Vologodskii, A. Statistical mechanics and topology of polymer chains. Nature 258, 398–402 (1975). https://doi.org/10.1038/258398a0
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DOI: https://doi.org/10.1038/258398a0
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