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Statistical mechanics and topology of polymer chains

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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

  1. Flory, P. J., Principles of Polymer Chemistry (Cornell University Press, Ithaca, New York, 1953).

    Google Scholar 

  2. Vol'kenshtein, M. V., Configurational Statistics of Polymer Chains (Wiley, New York, 1963).

    Google Scholar 

  3. Birshtein, T. M., and Ptitsyn, O. B., Conformations of Macromolecules (Nauka, Moscow, 1964).

    Google Scholar 

  4. Flory, P. J., Statistical Mechanics of Chain Molecules (Wiley, New York, 1969).

    Book  Google Scholar 

  5. Schill, G., Catenanes, Rotaxanes and Knots (Academic, New York, 1971).

    Google Scholar 

  6. Hershey, A. D., (ed.), The Bacteriophage Lambda (Cold Spring Harbour, 1971).

  7. Wang, J. C., Acc. chem. Res., 6, 252–256 (1973).

    Article  CAS  Google Scholar 

  8. Delbrück, M. in Mathematical Problems in the Biological Sciences (edit. by Bellman, R. E.), 55–63 (Proc. Symp. Math., 14, 1962).

    Book  Google Scholar 

  9. Frisch, H. L., and Wasserman, E., J. Am. chem. Soc., 83, 3789–3795 (1961).

    Article  CAS  Google Scholar 

  10. Edwards, S. F., J. Phys. A, 1, 15–28 (1968).

    Article  ADS  Google Scholar 

  11. Krivoshei, I. V., Zh. Strukt. Khimii, 9, 285–289 (1968).

    CAS  Google Scholar 

  12. Shugalii, A. V., Frank-Kamenetskii, M. D., and Lazurkin, Yu. S. Molek. biol. USSR. 3, 133–145 (1969).

    CAS  Google Scholar 

  13. Ninio, J. Biochimie, 53, 485–494 (1971).

    CAS  Google Scholar 

  14. 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).

    Article  CAS  Google Scholar 

  15. Crippen, G. M. J. theor. Biol., 45, 327–338 (1974).

    Article  CAS  Google Scholar 

  16. Boeckmann, J., and Schill, G., Tetrahedron, 30, 1945–1957 (1974).

    Article  CAS  Google Scholar 

  17. Vologodskii, A. V., Lukashin, A. V., Frank-Kamenetskii, M. D., and Anshelevich, V. V., Zh. eksp. teor. Fiz., 66, 2153–2163 (1974).

    CAS  Google Scholar 

  18. Vologodskii, A. V., Lukashin, A. V., and Frank-Kamenetskii, M. D., Zh. eksp. teor. Fiz., 61, 1875–1885 (1974).

    Google Scholar 

  19. Knott, G. G., Life and Scientific Work of Peter Guthrie Tait (Cambridge University Press, 1911).

    MATH  Google Scholar 

  20. Crowell, R. H., and Fox, R. H., Introduction to Knot Theory (Ginn, 1963).

    MATH  Google Scholar 

  21. Reidemeister, K., Knotentheorie, ser. Ergebnisse der Mathematik 1, No. 1, 1932.

  22. Rosenbluth, M. N., and Rosenbluth, A. W., J. chem. Phys., 23, 356–360 (1955).

    Article  ADS  CAS  Google Scholar 

  23. El'yashevich, A. M., and Skvortsov, A. M., Molek. biol. USSR, 5, 204–213 (1971).

    CAS  Google Scholar 

  24. Landau, L., and Lifshitz, E. Statistical Physics (Nauka, Moscow, 1964).

    MATH  Google Scholar 

  25. Wang, J. C., and Schwartz, H., Biopolymers, 5, 953–966 (1967).

    Article  CAS  Google Scholar 

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Authors and Affiliations

  1. Department of Biology, I. V. Kurchatov Institute of Atomic Energy, Moscow, 123182, USSR

    M. D. Frank-Kamenetskii, A. V. Lukashin & A. V. Vologodskii

Authors
  1. M. D. Frank-Kamenetskii
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  2. A. V. Lukashin
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  3. A. V. Vologodskii
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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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