Statistical topology and knotting of fluctuating filaments. (English) Zbl 1514.82245
Summary: The aim of these notes is to provide an introduction to the topic of statistical topology. With this name we refer to a combination of ideas and techniques from statistical mechanics and knot theory used to study the entanglement properties of fluctuating filaments. Some questions that we are going to address are the following: (i) What is a knot and how can we identify it? (ii) Which is the probability of finding a random curve that is knotted? (iii) How complex are these knots? (iv) How big are they? To try to partially answer these questions we will make use of few paradigmatic problems and try to investigate them by providing some “state of the art” theoretical and numerical techniques. Exercises and lists of open problems will be provided too.
Keywords:
statistical mechanics of polymers; knot theory; Monte Carlo simulations; lattice models of fluctuating filamentsReferences:
| [1] | Dean, F. B.; Stasiak, A.; Koller, T.; Cozzarelli, N. R., J. Biol. Chem., 260, 4975-4983 (1985) |
| [2] | Taylor, W. R., Nature, 406, 916-919 (2000) |
| [3] | Wassermann, S. A.; Cozzarelli, N. R., Science, 232, 951-960 (1986) |
| [4] | Sumners, D. W.; Ernst, C.; Spengler, S. J.; Cozzarelli, N. R., Q. Rev. Biophys., 28, 253-313 (1995) |
| [5] | Tkalec, U.; Ravnik, M.; Čopar, S.; Žumer, S.; Mǔevič, I., Science, 333, 62-65 (2011) · Zbl 1355.82069 |
| [6] | Dennis, M. R.; King, R. P.; Jack, B.; O’Holleran, K.; Padgett, M. J., Nat. Phys., 6, 118-121 (2010) |
| [7] | Kleckner, D.; Kauffman, L. H.; Irvine, W. T.M., Nat. Phys., 12, 650-655 (2016) |
| [8] | Sauvage, J. P.; Dietrich-Buchecker, C., Catenanes, Rotaxanes and Knots (1999), VCH-Wiley: VCH-Wiley Weinheim, Germany |
| [9] | Gil-Ramirez, G.; Leigh, D. A.; Stephens, A. J., Angew. Chem., Int. Ed. Engl., 54, 6110-6150 (2015) |
| [10] | Leigh, D. A.; Pritchard, R. G.; Stephens, A. J., Nature Chem., 6, 978-982 (2014) |
| [11] | Polles, G.; Marenduzzo, D.; Orlandini, E.; Micheletti, C., Nature Commun., 6, 6423 (2015) |
| [12] | Coluzza, I.; Van Oostrum, P. D.J.; Capone, B.; Reimhult, E.; Dellago, C., Phys. Rev. Lett., 110, 75501 (2013) |
| [13] | C.E. Soteros, D.W. Sumners, S.G. Whittington, J. Knot Theory Ramifications, 21, 1250116. · Zbl 1264.57009 |
| [14] | Rolfsen, D., (Knots and Links. Knots and Links, Mathematics Lecture Series, vol. 7 (1976), Publish or Perish, Inc.: Publish or Perish, Inc. Houston, Texas) · Zbl 0339.55004 |
| [15] | Adams, C. C., The Knot Book (1994), Freeman · Zbl 0840.57001 |
| [16] | J. Hoste, M. Thistlethwaite, J. Weeks, Springer Verlag, vol 20, 1998, p. 33. · Zbl 0916.57008 |
| [17] | Crowell, R. H.; Fox, R. H., Intoduction to Knot Theory (2012), Springer-Verlag: Springer-Verlag New York |
| [18] | Vologodskii, A. V.; Lukashin, A. V.; Frank-Kamenetskii, M. D.; Anshelevich, V. V., Sov. Phys.—JETP, 39, 1059-1063 (1974) |
| [19] | Janse van Rensburg, E. J., The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles (2015), Oxford University Press · Zbl 1314.82019 |
| [20] | Clisby, N.; Liang, R.; Slade, G., J. Phys. A, 40, 10973-11017 (2007) · Zbl 1136.82020 |
| [21] | Madras, N.; Slade, G., The Self Avoiding Walk (1999), Birkhäuser Basel |
| [22] | Clisby, N., Phys. Rev. Lett., 104, 055702 (2010) |
| [23] | Marko, J. F.; Siggia, E. D., Macromolecules, 28, 8759-8770 (1995) |
| [24] | Rybenkov, V. V.; Cozzarelli, N. R.; Vologodskii, A. V., Proc. Natl. Acad. Sci. USA, 90, 5307-5311 (1993) |
| [25] | Chen, Y. D., J. Chem. Phys., 74, 2034-2038 (1981) |
| [26] | Koniaris, K.; Muthukumar, M., Phys. Rev. Lett., 66, 2211-2214 (1991) |
| [27] | Kremer, K.; Grest, G. S., J. Chem. Phys., 92, 5057 (1990) |
| [28] | Weeks, J. D.; Chandler, D.; Andersen, H. C., J. Chem. Phys., 54, 5237 (1971) |
| [29] | Bates, A.; Maxwell, A., DNA Topology (2005), Oxford Univ. Press: Oxford Univ. Press New York |
| [30] | Frisch, H. L.; Wasserman, E., J. Am. Chem. Soc., 83, 3789-3795 (1961) |
| [31] | Delbruck, M., (Bellman, R. E., Mathematical Problems in the Biological Sciences. Mathematical Problems in the Biological Sciences, Proceedings of Symposia in Applied Mathematics, vol. 14 (1962), American Mathematical Society: American Mathematical Society Providence, Rhode Island), 55-63 · Zbl 0116.00205 |
| [32] | Sumners, D. W.; Whittington, S. G., J. Phys. A: Math. Gen., 21, 1689-1694 (1988) · Zbl 0659.57003 |
| [33] | Diao, Y.; Pippenger, N.; Sumners, D. W., J. Knot Theory Ramifications, 3, 419-429 (1994) · Zbl 0846.57003 |
| [34] | Diao, Y., J. Knot Theory Ramifications, 4, 189-196 (1995) · Zbl 0846.57004 |
| [35] | Hammersley, J., Proc. Cambridge Philos. Soc., 57, 516-523 (1961) · Zbl 0122.36501 |
| [36] | Kesten, H., J. Math. Phys., 4, 960 (1963) · Zbl 0122.36502 |
| [37] | Fox, R. H., A quick trip through knot theory, (Fort, M. K., Topology of Three-Manifolds and Related Topics (1962), Prentice-Hall: Prentice-Hall New York) · Zbl 1246.57002 |
| [38] | Janse van Rensburg, E. J.; Sumners, D. W.; Wasserman, E.; Whittington, S. G., J. Phys. A: Math. Gen., 25, 6557-6566 (1992) · Zbl 0770.60099 |
| [39] | Hammersley, J.; Handscomb, Monte Carlo Methods (1964), Methuen & Co. London and John Wiley & Sons: Methuen & Co. London and John Wiley & Sons New York · Zbl 0121.35503 |
| [40] | Millet, K., Knotting of regular polygons in 3-space, (Millett, K. C.; Sumners, D. W., Random Knotting and Linking, Vol. 7 (1994), World Scientific: World Scientific Singapore), 31-46 |
| [41] | Micheletti, C.; Marenduzzo, D.; Orlandini, E.; Sumners, D. W., J. Chem. Phys., 124, 064903 (2006) |
| [42] | Baiesi, M.; Orlandini, E.; Stella, A. L., Phys. Rev. Lett., 99, 058301 (2007) |
| [43] | M. Baiesi, E. Orlandini, S.G. Whittington, J. Chem. Phys., 131, 154902. |
| [44] | Micheletti, C.; Marenduzzo, D.; Orlandini, E., Phys. Rep., 504, 1-73 (2011) |
| [45] | Berg, B.; Foerster, D., Phys. Lett. B, 106, 323-326 (1981) |
| [46] | Aragao de Carvalho, C.; Caracciolo, S., J. Phys. (France), 44, 323-331 (1983) |
| [47] | Janse van Rensburg, E. J.; Whittington, S. G., J. Phys. A: Math. Gen., 24, 5553-5567 (1991) · Zbl 0765.65005 |
| [48] | Janse van Rensburg, E. J.; Rechnitzer, A., J. Phys. A, 44, 165001 (2011) · Zbl 1216.82019 |
| [49] | Janse van Rensburg, E. J.; Whittington, S. G., J. Phys. A: Math. Gen., 24, 3935 (1991) · Zbl 0956.82503 |
| [50] | Baiesi, M.; Orlandini, E.; Stella, A. L., J. Stat. Mech., P06012 (2010) |
| [51] | Janse van Rensburg, E. J.; Whittington, S. G., J. Phys. A: Math. Gen., 23, 3573-3590 (1991) · Zbl 0709.57001 |
| [52] | Yao, A.; Matsuda, H.; Tsukahara, H.; Shimamura, M. K.; Deguchi, T., J. Phys. A: Math. Gen., 34, 7563-7577 (2001) · Zbl 0982.82015 |
| [53] | Saitta, A. M.; Soper, P. D.; Wasserman, E.; Klein, M. L., Nature, 399, 46-48 (1999) |
| [54] | Farago, O.; Kantor, Y.; Kardar, M., Europhys. Lett., 60, 53-59 (2002) |
| [55] | Caraglio, M.; Orlandini, E.; Micheletti, C., Phys. Rev. Lett., 115, 188301 (2015) |
| [56] | Taylor, W. R.; Xiao, B.; Gamblin, S. J.; Lin, K., Comput. Biol. Chem., 27, 11-15 (2003) |
| [57] | Millett, K.; Dobay, A.; Stasiak, A., Macromolecules, 38, 601-606 (2005) |
| [58] | Tubiana, L.; Orlandini, E.; Micheletti, C., Progr. Theoret. Phys., 191, 192-204 (2011) |
| [59] | Tubiana, L.; Rosa, A.; Fragiacomo, F.; Micheletti, C., Macromolecules, 46, 3669-3678 (2013) |
| [60] | Virnau, P.; Kantor, Y.; Kardar, M., J. Am. Chem. Soc., 127, 15102-15106 (2005) |
| [61] | Kauffman, L. H., European J. Combin., 20, 663-691 (1999) · Zbl 0938.57006 |
| [62] | Silver, D. S.; Williams, S. G., Notices Amer. Math. Soc., 64, 461-464 (2017) · Zbl 1366.57007 |
| [63] | Turaev, V., Osaka J. Math., 49, 195-223 (2012) · Zbl 1271.57030 |
| [64] | Alexander, K.; Taylor, A. J.; Dennis, M. R., Sci. Rep., 7, 42300 (2017) |
| [65] | Goundaroulis, D.; Dorier, J.; Benedetti, F.; Stasiak, A., Sci. Rep., 7, 6309 (2017) |
| [66] | Rawdon, E. J.; Millett, K. C.; Stasiak, A., Sci. Rep., 5, 8928 (2015) |
| [67] | King, N. P.; Yeates, E. O.; Yeates, T. O., J. Mol. Biol., 12, 153-166 (2007) |
| [68] | Marcone, B.; Orlandini, E.; Stella, A. L.; Zonta, F., J. Phys. A: Math. Gen., 38, L15-L21 (2005) · Zbl 1067.82531 |
| [69] | Sulkowska, J. I.; Rawdon, E. J.; Millett, K. C.; Onuchic, J. A., Proc. Natl. Acad. Sci. USA, 109, E1715-E1723 (2012) |
| [70] | Jamroz, M.; Niemyska, W.; Rawdon, E. J.; Stasiak, A.; Millett, K. C.; Sulkowski, P.; Sulkowska, J. I., Nucl. Acid. Res., 43, D306-D314 (2015) |
| [71] | Millett, K. C., J. Knot Theory Ramifications, 19, 601-615 (2010) · Zbl 1191.57009 |
| [72] | Tubiana, L.; Orlandini, E.; Micheletti, C., Phys. Rev. Lett., 107, 188302 (2011) |
| [73] | Katritch, V.; Olson, W. K.; Vologodskii, A.; Dubochet, J.; Stasiak, A., Phys. Rev. E, 61, 5545-5549 (2000) |
| [74] | Mansfield, M. L.; Douglas, J. F., J. Chem. Phys., 133, 044903 (2010) |
| [75] | Mansfield, M. L., Macromolecules, 31, 4030-4032 (1998) |
| [76] | Marcone, B.; Orlandini, E.; Stella, A. L.; Zonta, F., Phys. Rev. E, 75, 041105-041111 (2007) |
| [77] | Marenduzzo, D.; Orlandini, E.; Stasiak, A.; Sumners, d. W.; Tubiana, L.; Micheletti, C., Proc. Natl. Acad. Sci. USA, 106, 22269-22274 (2009) |
| [78] | Marenduzzo, D.; Micheletti, C.; Orlandini, E.; Sumners, D. W., Proc. Natl. Acad. Sci. USA, 110, 20081-20086 (2013) |
| [79] | Metzler, R.; Hanke, A.; Dommersnes, P. G.; Kantor, Y.; Kardar, M., Phys. Rev. E, 65, 061103 (2002) |
| [80] | Orlandini, E.; Stella, A. L.; Vanderzande, C., J. Stat. Phys., 115, 681-700 (2004) · Zbl 1157.82422 |
| [81] | Baiesi, M.; Orlandini, E.; Stella, A. L.; Zonta, Z., Phys. Rev. Lett., 106, 258301 (2011) |
| [82] | Baiesi, M.; Orlandini, E.; Stella, A. L., Macromolecules, 47, 8466-8476 (2014) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
