Abstract.
We investigate the effects of topological constraints (entanglements) on two-dimensional polymer loops in the dense phase, and at the collapse transition (\(\Theta\)-point). Previous studies have shown that in the dilute phase the entangled region becomes tight, and is thus localised on a small portion of the polymer. We find that the entropic force favouring tightness is considerably weaker in dense polymers. While the simple figure-eight structure, created by a single crossing in the polymer loop, localises weakly, the trefoil knot and all other prime knots are loosely spread out over the entire chain. In both the dense and \(\Theta\) conditions, the uncontracted-knot configuration is the most likely shape within a scaling analysis. By contrast, a strongly localised figure-eight is the most likely shape for dilute prime knots. Our findings are compared to recent simulations.
Similar content being viewed by others
References
P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, New York, 1979).
M. Doi, S.F. Edwards, The Theory of Polymer Dynamics (Clarendon Press, Oxford, 1986).
J.D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York, 1970).
L.R.G. Treloar, The Physics of Rubber Elasticity (Clarendon Press, Oxford, 1975).
J.P. Sauvage, C. Dietrich-Buchecker (Editors), Catenanes, Rotaxanes, and Knots (VCH, Weilheim, 1999).
T.E. Creighton, Proteins: Structures and Molecular Properties\/ (W.H. Freeman, New York, 1993).
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell (Garland, New York, 2002).
V.V. Rybenkov, Science 277, 690 (1997)
W.E. Moerner, M. Orrit, Science 283, 1670 (1999)
A.D. Mehta, M. Rief, J.A. Spudich, D.A. Smith, R.M. Simmons, Science 283, 1689 (1999)
Throughout this paper we employ the terms “2D knot,” “2D knot projection,” and “2D network” to denote a polymer chain confined to two dimensions, which never intersects itself apart from explicitly imposed “crossings” (or “vertices”). The chain segments adjoining the crossings are allowed to exchange length with one another. Also the slip-linked structures considered in Sections 3 and 4 belong to this class of systems, as their equilibrium behaviour is the same. The strict non-crossing condition is relaxed only briefly near the end of Appendix A, where we consider the 2D Goldstone phase introduced in reference [28].
E. Guitter, E. Orlandini, J. Phys. A 32, 1359 (1999).
R. Metzler, A. Hanke, P.G. Dommersnes, Y. Kantor, M. Kardar, Phys. Rev. Lett. 88, 188101 (2002).
B. Duplantier, Phys. Rev. Lett. 57, 941 (1986)
K. Ohno, K. Binder, J. Phys. (Paris) 49, 1329 (1988)
M.B. Hastings, Z.A. Daya, E. Ben-Naim, R.E. Ecke, Phys. Rev. E 66, 025102 (2002).
B. Maier, J.O. Rädler, Phys. Rev. Lett. 82, 1911 (1999)
A.Yu. Grosberg, A.R. Khokhlov, Statistical Physics of Macromolecules (AIP Press, New York, 1994).
J.D. Moroz, R.D. Kamien, Nucl. Phys. B 506, 695 (1997)
E.J. Janse van Rensburg, S.G. Whittington, J. Phys. A 24, 3935 (1991)
R. Metzler, A. Hanke, P.G. Dommersnes, Y. Kantor, M. Kardar, Phys. Rev. E 65, 061103 (2002).
S.R. Quake, Phys. Rev. Lett. 73, 3317 (1994)
V.A. Bloomfield, Biopolymers 44, 269 (1997)
T. Garel, Remarks on homo- and hetero-polymeric aspects of protein folding\/, cond-mat/0305053.
B. Duplantier, J. Phys. A 19, L1009 (1986).
B. Duplantier, H. Saleur, Nucl. Phys. B 290, 291 (1987).
A.L. Owczarek, T. Prellberg, R. Brak, Phys. Rev. Lett. 70, 951 (1993)
J.L. Jacobsen, N. Read, H. Saleur, Phys. Rev. Lett. 90, 090601 (2003).
B. Duplantier, F. David, J. Stat. Phys. 51, 327 (1988).
J. Kondev, J.L. Jacobsen, Phys. Rev. Lett. 81, 2922 (1998).
M.E. Fisher, Physica D 38, 112 (1989).
E. Orlandini, F. Seno, A.L. Stella, M.C. Tesi, Phys. Rev. Lett. 68, 488 (1992)
J. Cardy, J. Phys. A 34, L665 (2001).
E. Orlandini, A.L. Stella, C. Vanderzande, Phys. Rev. E 68, 031804 (2003).
B. Duplantier, H. Saleur, Phys. Rev. Lett. 59, 539 (1987).
P. Grassberger, R. Hegger, Ann. Phys. (Leipzig) 4, 230 (1995).
Author information
Authors and Affiliations
Corresponding author
Additional information
Received: 7 October 2003, Published online: 21 November 2003
PACS:
87.15.-v Biomolecules: structure and physical properties - 82.35.-x Polymers: properties; reactions; polymerization - 02.10.Kn Knot theory
Rights and permissions
About this article
Cite this article
Hanke, A., Metzler, R., Dommersnes, P.G. et al. Tight and loose shapes in flat entangled dense polymers. Eur. Phys. J. E 12, 347–354 (2003). https://doi.org/10.1140/epje/i2003-10067-9
Published:
Issue Date:
DOI: https://doi.org/10.1140/epje/i2003-10067-9