Tightness of slip-linked polymer chains

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 65, Issue 6, 2002, Pages

  Metzler, Ralf ^a,b   Hanke, Andreas ^a,c   Dommersnes, Paul G ^a,d   Kantor, Yacov ^a,e   Kardar, Mehran ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b NORDITA   (Denmark)

^c Theoretical Physics   (United Kingdom)

^d NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^e TEL AVIV UNIVERSITY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

FEYNMANN GRAPHS; POLYMER CHAINS; SLIDING RINGS; TOPOLOGICAL CONSTRAINTS;

CONSTRAINT THEORY; DNA; GRAPH THEORY; MATHEMATICAL MODELS; MONOMERS; PROBABILITY; PROBABILITY DENSITY FUNCTION; TOPOLOGY;

ENTROPY;

ARTICLE;

EID: 41349097244     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.65.061103     Document Type: Article

References (119)

1. P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, NY, 1953).
2. P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, NY, 1979).
3. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon Press, Oxford, 1986).
4. L. R. G. Treloar, The Physics of Rubber Elasticity (Clarendon Press, Oxford, 1975).
5. J. D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York, 1970).
6. A. Yu. Grosberg and A. R. Khokhlov, Statistical Mechanics of Macromolecules (AIP, New York, 1994).
7. K. Reidemeister, Knotentheorie (Springer, Berlin, 1931) [Knot Theory (BSC Assocs., Moscow, ID, 1983)].
8. L. H. Kauffman, Knots and Physics, Series on Knots and Everything, Vol. I (World Scientific, Singapore, 1993).
9. A. C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (Freeman, New York, 1994).
10. H. L. Frisch and E. Wassermann, J. Am. Chem. Soc. 83, 3789 (1961);
11. M. Delbrück, in Mathematical Problems in Biological Sciences, edited by R. E. Bellman [Proc. Symp. Appl. Math. 14, 55 (1962)].
12. D. W. Sumners and S. G. Whittington, J. Phys. A 21, 1689 (1988);
13. N. Pippenger, Discrete Appl. Math. 25, 273 (1989);
14. see also Y. Diao, N. Pippenger, and D. W. Sumners, J. Knot Theory Ramif. 3, 419 (1994).
15. B. Alberts, K. Roberts, D. Bray, J. Lewis, M. Raff, and J. D. Watson, The Molecular Biology of the Cell (Garland, New York, 1994);S. R. Bolsover, J. S. Hyams, S. Jones, E. A. Shephard, and H. A. White, From Genes to Cells (Wiley, New York, 1997).
16. S. A. Wassermann and N. R. Cozzarelli, Science 232, 951 (1986).
17. S. A. Wassermann and N. R. Cozzarelli, Proc. Natl. Acad. Sci. U.S.A. 82, 1079 (1984);
18. S. A. Wassermann, J. M. Dungan, and N. R. Cozzarelli, Science 229, 171 (1985).
19. T. R. Strick, V. Croquette, and D. Bensimon, Nature (London) 404, 901 (2000);
20. S. Y. Shaw and J. C. Wang, Science 260, 533 (1993);
21. T. Deguchi and K. A. Tsurusaki, J. Knot Theory Ramif. 3, 321 (1994);
22. S. A. Wasserman, J. M. Dungan, and N. R. Cozzarelli, Science 229, 171 (1985).
23. J. Yan, M. O. Magnasco, and J. F. Marko, Nature (London) 401, 932 (1999);
24. V. V. Rybenkov, C. Ullsperger, A. V. Vologodskij, and N. R. Cozzarelli, Science 277, 690 (1997).
25. F. Major, D. Gautheret, and R. Cedergren, Proc. Natl. Acad. Sci. U.S.A. 90, 9408 (1993);
26. J. D. Puglisi, J. R. Wyatt, and I. Tinoco, J. Mol. Biol. 214, 437 (1990);RNA Structure and Function, edited by R. W. Simons and M. Grunberg-Manago (Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY, 1998).
27. R. Bundschuh and T. Hwa, Phys. Rev. Lett. 83, 1479 (1999);
28. U. Gerland, R. Bundschuh, and T. Hwa, e-print cond-mat/0101250.
29. D. Poland and H. A. Scheraga, J. Chem. Phys. 45, 1464 (1966);Theory of Helix-Coil Transitions in Biopolymers (Academic Press, New York, 1970).
30. Y. Kafri, D. Mukamel, and L. Peliti, Phys. Rev. Lett. 85, 4988 (2000).
31. Y. Kafri, D. Mukamel, and L. Peliti, e-print cond-mat/0108323.
32. A. Hanke and R. Metzler, e-print cond-mat/0110164.
33. W. R. Taylor, Nature (London) 406, 916 (2000);
34. M. L. Mansfield, Nat. Struct. Biol. 1, 213 (1994);
35. Nat. Struct. Biol.M. L. Mansfield4, 116 (1997);
36. F. Takusagawa and K. Kamitori, J. Am. Chem. Soc. 118, 8945 (1996).
37. T. E. Creighton, Proteins: Structures and Molecular Properties (W. H. Freeman, New York, 1993).
38. S. Y. Shaw and J. C. Wang, Science 260, 533 (1993);
39. V. V. Rybenkov, N. R. Cozzarelli, and A. V. Vologoskij, Proc. Natl. Acad. Sci. U.S.A. 90, 5307 (1993);
40. A. Stasiak, V. Katritch, J. Bednar, D. Michoud, and J. Dubochet, Nature (London) 384, 122 (1996);
41. A. V. Vologodskij, N. J. Crisona, B. L. P. Pieranski, V. Katritch, J. Dubochet, and A. Stasiak, J. Mol. Biol. 278, 1 (1998);
42. S. Trigueros, J. Arsuaga, M. E. Vazquez, D. W. Sumners, and J. Roca, Nucleic Acids Res. 29, e67 (2001) (on the journal web page).
43. C. W. Pouton and L. W. Seymour, Adv. Drug Delivery Rev. 46, 187 (2001);Protein Targeting and Translocation, edited by D. A. Phoenix (Princeton University Press, Princeton, NJ, 1998).
44. H. Fujisawa and M. Morita, Genes Cells 2, 537 (1997);
45. C. E. Catalano, D. Cue, and M. Feiss, Mol. Microbiol. 16, 1075 (1995).
46. G. Schill, Catenanes, Rotaxanes and Knots (Academic Press, New York, 1971).
47. J.-M. Lehn, Supramolecular Chemistry (VCH, Weinheim, 1995).
48. Catenanes, Rotaxanes and Knots, edited by J.-P. Sauvage and C. Dietrich-Buchecker (VCH, Weilheim, 1999).
49. Single molecule spectroscopy: W. E. Moerner and M. Orrit, Science 283, 1670 (1999);
50. W. E. Moerner and L. Kador, Phys. Rev. Lett. 62, 2535 (1989).
51. Optomicroscopical imaging: A. van Oudenaarden and J. A. Theriot, Nat. Cell Biol. 1, 493 (1999);
52. J. A. Theriot, T. J. Mitchison, L. G. Tilnet, and D. A. Portnou, Nature (London) 37, 257 (1992).
53. Atomic force microscopy: G. Binnig, C. F. Quate, and C. Gerber, Phys. Rev. Lett. 56, 930 (1986).
54. Optical tweezers: A. Ashkin, Phys. Rev. Lett. 24, 156 (1970);
55. A. Ashkin, J. M. Dziedi, and T. Yamane, Nature (London) 330, 769 (1987);
56. M. D. Wang, H. Yin, R. Landeck, J. Gelles, and S. M. Block, Biophys. J. 72, 1335 (1997);
57. J.-C. Meiners and S. R. Quake, Phys. Rev. Lett. 84, 5014 (2000).
58. Y. Arai, R. Yasuda, K.-I. Akashi, Y. Harada, H. Miyata, K. Kinosita, Jr., and H. Itoh, Nature (London) 399, 446 (1999).
59. M. Rief, M. Gautel, F. Oesterhelt, J. M. Fernandez, and H. E. Gaub, Science 276, 1109 (1997);
60. U. Bockelmann, B. Evassez-Roulet, and F. Heslot, Phys. Rev. Lett. 79, 4489 (1997);
61. U. BockelmannB. Evassez-RouletF. HeslotPhys. Rev. E 58, 2386 (1998);
62. R. E. Thomson and E. D. Siggia, Europhys. Lett. 31, 335 (1995).
63. B. Maier and J. O. Rädler, Phys. Rev. Lett. 82, 1911 (1999).
64. J. Kepler, Johannes Kepler, Gesammelte Werke, edited by W. von Dyck and M. Caspar (Beck, München, 1937).
65. L. Euler, Acad. Sci. Imp. Petropolitanae 8, 128 (1736) [Sci. Am. 189, 66 (1953)].
66. J. B. Listing, Vorstudien zur Topologie, Göttinger Studien (Vandenhoeck und Ruprecht, Göttingen, 1848).
67. W. Thomson, Lord Kelvin, Philos. Mag. 34, 15 (1867);
68. W. ThomsonLord KelvinProc. R. Soc. Edinburgh 9, 59 (1875–1876).
69. http://www-groups.dcs.st-and.ac.uk/(Formula presented)istory/HistTopics/Knots_ and_physics.html
70. P. G. Tait, Trans.-R. Soc. Edinburgh 28, 145 (1876–1877);
71. Trans.-R. Soc. EdinburghP. G. Tait32, 327 (1883–1884);
72. Trans.-R. Soc. EdinburghP. G. Tait32, 493 (1884–1885);Scientific Papers (Cambridge University Press, London, 1898).
73. T. P. Kirkman, Proc. R. Soc. Edinburgh 13, 363 (1884–1885);
74. T. P. KirkmanTrans. R. Soc. Edinburg 32, 483 (1884–1885).
75. C. N. Little, Trans. Conn. Acad. Arts Sci. 7, 27 (1888);
76. C. N. LittleTrans.-R. Soc. Edinburgh 35, 771 (1890);
77. Trans.-R. Soc. EdinburghC. N. Little36, 253 (1890–1891).
78. T. A. Vilgis, Phys. Rep. 336, 167 (2000).
79. A. L. Kholodenko and T. A. Vilgis, Phys. Rep. 298, 251 (1998).
80. For comparison, self-avoidance in 3D is usually treated as a perturbation, i.e., as a “soft constraint,” in analytical studies 2.
81. S. F. Edwards, Proc. Phys. Soc. London 91, 513 (1967);
82. S. F. EdwardsJ. Phys. A 1, 15 (1968).
83. P.-G. de Gennes, Macromolecules 17, 703 (1984).
84. M. Otto and T. A. Vilgis, Phys. Rev. Lett. 80, 881 (1998).
85. A. Yu. Grosberg, Phys. Rev. Lett. 85, 3858 (2000).
86. S. R. Quake, Phys. Rev. Lett. 73, 3317 (1994);
87. S. R. QuakePhys. Rev. E 52, 1176 (1986).
88. A. Yu. Grosberg, A. Feigel, and Y. Rabin, Phys. Rev. E 54, 6618 (1996).
89. M. E. Cates and J. M. Deutsch, J. Phys. (Paris) 47, 2121 (1986);
90. M. Müller, J. P. Wittmer, and M. E. Cates, Phys. Rev. E 53, 5063 (1996);
91. Phys. Rev. EM. MüllerJ. P. WittmerM. E. Cates61, 4078 (2000).
92. A. Grosberg and S. Nechaev, J. Phys. A 25, 4659 (1991).
93. S. Nechaev, e-print cond-mat/9812205.
94. M. Doi and S. F. Edwards, J. Chem. Soc., Faraday Trans. 2 74, 1802 (1978);
95. R. C. Ball, M. Doi, S. F. Edwards, and M. Warner, Polymer 22, 1010 (1981);
96. P. G. Higgs and R. C. Ball, Europhys. Lett. 8, 357 (1989);
97. S. F. Edwards and T. A. Vilgis, Polymer 27, 483 (1986).
98. B. Erman and J. E. Mark, Structures and Properties of Rubberlike Networks (Oxford University Press, New York, 1997).
99. B. D. Hughes, Random Walks and Random Environments, Volume 1: Random Walks (Oxford University Press, Oxford, 1995).
100. R. Metzler, A. Hanke, P. G. Dommersnes, Y. Kantor, and M. Kardar, Phys. Rev. Lett. 88, 188101 (2002).
101. E. J. Janse van Rensburg and S. G. Whittington, J. Phys. A 24, 3935 (1991).
102. E. Orlandini, M. C. Tesi, E. J. Janse van Rensburg, and S. G. Whittington, J. Phys. A 31, 5953 (1998);
103. J. Phys. AE. OrlandiniM. C. TesiE. J. Janse van RensburgS. G. Whittingtoncompare 29, L299 (1996).
104. E. Guitter and E. Orlandini, J. Phys. A 32, 1359 (1999).
105. V. Katritch, W. K. Olson, A. Vologodskii, J. Dubochet, and A. Stasiak, Phys. Rev. E 61, 5545 (2000).
106. M. K. Shimamura and T. Deguchi, Phys. Rev. E 64, 020801 (2001).
107. Here and in the following we consider two configurations of a polymer chain different if they cannot be matched by translation. In addition, the origin of a given structure is fixed by a vertex point, i.e., a point where several legs of the polymer chain are joint. For a simply connected ring polymer, such a vertex is a two-vertex anywhere along the chain.
108. Note that the above factorization does not take place in structures where additional links are introduced, such as connecting the two corner loops in the necklace configuration sketched in Fig. 22(h); this creates conditional probabilities, and the associated arc diagrams no longer correspond to Hartree graphs.
109. R. Metzler, Y. Kantor, and M. Kardar (unpublished).
110. B. Duplantier, Phys. Rev. Lett. 57, 941 (1986);
111. B. DuplantierJ. Stat. Phys. 54, 581 (1989).
112. L. Schäfer, C. von Ferber, U. Lehr, and B. Duplantier, Nucl. Phys. B374, 473 (1992).
113. K. Ohno and K. Binder, J. Phys. (Paris) 49, 1329 (1988).
114. B. Nienhuis, Phys. Rev. Lett. 49, 1062 (1982);
115. B. NienhuisJ. Stat. Phys. 34, 731 (1984).
116. M. B. Hastings, Z. A. Daya, E. Ben-Naim, and R. E. Ecke, e-print cond-mat/0110612.
117. J. des Cloizeaux, J. Phys. (Paris) 41, 223 (1980).
118. If the segments (Formula presented) are loose, the cutoff a is no longer relevant and characteristic lengths can be calculated by multiple integrals of the form (Formula presented)whose Laplace transform (Formula presented) (Formula presented)yields a simple product form by the integration theorem for the first integral and the convolution theorem for the remaining factors, and therefore (Formula presented)With similar expressions for the normalization, one obtains the results quoted in the text.
119. J. F. Marko and E. D. Siggia, Macromolecules 28, 8759 (1995).