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Predicting the Behaviour of Rings in Solution

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Topological Interactions in Ring Polymers

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

Polymeric systems offer an incredible richness of behaviour. Depending on the solution concentration, its temperature or its quality and the polymers length, or topology, every system made of polymers can be categorised into a “universality class”, within which it finds a physical characterisation (scaling) of its macroscopic properties.

No man is obliged to learn and know every thing; [...]; yet all persons are under some obligation to improve their understanding; otherwise it will be a barren desert, or a forest with overgrown weed and brambles.

I. Watts

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Notes

  1. 1.

    Local (short-ranged) correlations, do not affect the scaling.

  2. 2.

    “Surprisingly” because Flory’s theory actually overestimates the repulsive term by neglecting monomer-monomer correlations, but also overestimates the elastic term, thereby balancing out the errors and leading to a very accurate estimation of the scaling of the real size R (de Gennes 1979).

  3. 3.

    This means that \(M\rightarrow M/M_e\), \(\sigma \rightarrow \sigma M_e^{1/2}\) and \(L_c \rightarrow L_c/M_e^{1/2}\).

  4. 4.

    A blob being a polymer segment made of several (g) monomers where \(1 \ll g \ll M\) and assuming a size described by the scaling \(R(g)\sim g^\nu \) with \(\nu =3/5\), being not interacting with other chains (de Gennes 1979).

  5. 5.

    These are obtained by using \(1/d \le \nu \le 1\), \(1-1/d\le \beta _{c} = \beta _{g} \le 1\), in Eqs. (2.24) and (2.27).

  6. 6.

    In this case the computation can scale as \((NM)^2\), for a system of N chains M beads long, rather than \(NM^2\).

References

  • Aichele, M., Baschnagel, J.: Glassy dynamics of simulated polymer melts: coherent scattering and van Hove correlation functions. Eur. Phys. J. E 5(2), 229 (2001)

    Article  Google Scholar 

  • Berthier, L., Biroli, G.: Theoretical perspective on the glass transition and amorphous materials. Rev. Mod. Phys. 83(2), 587 (2011)

    Article  ADS  Google Scholar 

  • Brás, A., Gooßen, S., Krutyeva, M.: Compact structure and non-Gaussian dynamics of ring polymer melts. Soft Matter 10, 3649 (2014)

    Article  ADS  Google Scholar 

  • Cates, M., Deutsch, J.: Conjectures on the statistics of ring polymers. J. Phys. Paris 47, 2121 (1986)

    Article  Google Scholar 

  • Cremer, T., Cremer, C.: Chromosome territories, nuclear architecture and gene regulation in mammalian cells. Nat. Rev. Genet. 2(4), 292 (2001)

    Article  Google Scholar 

  • Daoud, M., Joanny, J.: Conformation of branched polymers. J. de phys. 42(10), 1359 (1981)

    Article  Google Scholar 

  • de Gennes, P.G.: Scaling concepts in polymer physics, Cornell University Press (1979)

    Google Scholar 

  • de Gennes, P.G.: Coherent scattering by one reptating chain. J. Phys. (Paris) 42(5), 735 (1981)

    Article  Google Scholar 

  • Doi, M., Edwards, S.: The Theory of Polymer Dynamics, Oxford University Press, Oxford (1988)

    Google Scholar 

  • Doi, Y., Matsubara, K., Ohta, Y., Nakano, T., Kawaguchi, D., Takahashi, Y., Takano, A., Matsushita, Y.: Melt Rheology of Ring Polystyrenes with Ultrahigh Purity. Macromolecules 48(9), 3140 (2015)

    Article  ADS  Google Scholar 

  • Edwards, S.: Statistical mechanics with topological constraints: I. Proc. Phys. Soc. 91, 513 (1967)

    Article  ADS  MATH  Google Scholar 

  • Edwards, S.: Statistical mechanics with topological constraints: II. J. Phys. A: Math. Gen. 1, 15 (1968)

    Article  ADS  MATH  Google Scholar 

  • Flory, P.J.: Principles of polymer chemistry, Cornell University Press. Ithaca, New York (1953)

    Google Scholar 

  • Frey, S., Weysser, F., Meyer, H., Farago, J., Fuchs, M., Baschnagel, J.: Simulated glass-forming polymer melts: Dynamic scattering functions, chain length effects, and mode-coupling theory analysis. Eur. Phys. J. E 38(11), 1 (2015)

    Google Scholar 

  • Gooßen, S., Brás, A.R., Krutyeva, M., Sharp, M., Falus, P., Feoktystov, A., Gasser, U., Wischnewski, A., Richter, D.: Molecular Scale Dynamics of Large Ring Polymers. Phys. Rev. Lett. 113, 169302 (2014)

    Article  Google Scholar 

  • Grosberg, A.: Annealed lattice animal model and Flory theory for the melt of non-concatenated rings: towards the physics of crumpling. Soft Matter 10, 560 (2014)

    Article  ADS  Google Scholar 

  • Grosberg, A.Y., Rabin, Y., Havlin, S., Neer, A.: Crumpled globule model of the three-dimensional structure of DNA. Europhys. Lett. 23(5), 373 (1993)

    Article  ADS  Google Scholar 

  • Gutin, A., Grosberg, A., Shakhnovich, E.: Polymers with annealed and quenched branchings belong to different universality classes. Macromolecules 26(5), 1293 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  • Halverson, J.D., Lee, W.B., Grest, G.S., Grosberg, A.Y., Kremer, K.: Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. I. Statics. J. Chem. Phys. 134(20), 204904 (2011a)

    Article  ADS  Google Scholar 

  • Halverson, J.D., Lee, W.B., Grest, G.S., Grosberg, A.Y., Kremer, K.: Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. II. Dynamics. J. Chem. Phys. 134(20), 204905 (2011b)

    Article  ADS  Google Scholar 

  • Halverson, J.D., Kremer, K., Grosberg, A.Y.: Comparing the results of lattice and off-lattice simulations for the melt of nonconcatenated rings. J. Phys. A 46(6), 065002 (2013)

    Article  ADS  Google Scholar 

  • Halverson, J.D., Smrek, J., Kremer, K., Grosberg, A.: From a melt of rings to chromosome territories: the role of topological constraints in genome folding. Rep. Prog. Phys. 77, 022601 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  • Isaacson, J., Lubensky, T.C.: Flory exponents for generalized polymer problems. J. Phys. 41, 469 (1980)

    Google Scholar 

  • Iyer, B.V.S., Arya, G.: Lattice animal model of chromosome organization. Phys. Rev. E 86(1), 011911 (2012)

    Article  ADS  Google Scholar 

  • Kapnistos, M., Lang, M., Vlassopoulos, D., Pyckhout-Hintzen, W., Richter, D., Cho, D., Chang, T., Rubinstein, M.: Unexpected power-law stress relaxation of entangled ring polymers. Nat. Mater. 7(12), 997 (2008)

    Article  ADS  Google Scholar 

  • Klein, J.: Dynamics of entangled linear, branched, and cyclic polymers. Macromolecules 118(33), 105 (1986)

    Article  ADS  Google Scholar 

  • Kremer, K., Grest, G.S.: Dynamics of entangled linear polymer melts: A molecular-dynamics simulation. J. Chem. Phys. 92(8), 5057 (1990)

    Article  ADS  Google Scholar 

  • Lieberman-Aiden, E., van Berkum, N.L., Williams, L., Imakaev, M., Ragoczy, T., Telling, A., Amit, I., Lajoie, B.R., Sabo, P.J., Dorschner, M.O., Sandstrom, R., Bernstein, B., Bender, M.A., Groudine, M., Gnirke, A., Stamatoyannopoulos, J., Mirny, L.A., Lander, E.S., Dekker, J.: Comprehensive mapping of long-range interactions reveals folding principles of the human genome. Science 326(5950), 289 (2009)

    Article  ADS  Google Scholar 

  • Lubensky, T., Isaacson, J.: Statistics of lattice animals and dilute branched polymers. Phys. Rev. A 20(5), 2130 (1979)

    Article  ADS  Google Scholar 

  • Milner, S., Newhall, J.: Stress Relaxation in Entangled Melts of Unlinked Ring Polymers. Phys. Rev. Lett. 105(20), 208302 (2010)

    Article  ADS  Google Scholar 

  • Mirny, L.A.: The fractal globule as a model of chromatin architecture in the cell. Chromosome Res. 19(1), 37 (2011)

    Article  Google Scholar 

  • Muller, M., Wittmer, J., Cates, M.: Topological effects in ring polymers. II. Influence Of persistence length. Phys. Rev. E 61(4), 4078 (2000)

    Article  ADS  Google Scholar 

  • Müller, M., Wittmer, J.P., Cates, M.E.: Topological effects in ring polymers: A computer simulation study. Phys. Rev. E 53(5), 5063 (1996)

    Article  ADS  Google Scholar 

  • Obukhov, S., Rubinstein, M.: Dynamics of a ring polymer in a gel. Phys. Rev. Lett. 73(9), 1263 (1994)

    Article  ADS  Google Scholar 

  • Parisi, G., Sourlas, N.: Critical behavior of branched polymers and the Lee-Yang edge singularity. Phys. Rev. Lett. 46(14), 871 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Pasquino, R., Vasilakopoulos, T., Jeong, C., Lee, H., Rogers, S., Sakellariou, G., Allgaier, J., Takano, A., Bras, A., Chang, T., Goossen, S., Pyckhout-Hintzen, W., Wischnewski, A., Hadjichristidis, N., Richter, D., Rubinstein, M., Vlassopoulos, D.: Viscosity of Ring Polymer Melts. ACS Macro Lett. 2, 874 (2013)

    Article  Google Scholar 

  • Raphael, E., Gay, C., de Gennes, P.G.: Progressive construction of an Olympic gel. J. Stat. Phys. 89, 111 (1997)

    Article  ADS  Google Scholar 

  • Rosa, A., Everaers, R.: Structure and dynamics of interphase chromosomes. PLoS Comput. Biol. 4(8), 1 (2008)

    Article  MathSciNet  Google Scholar 

  • Rosa, A., Everaers, R.: Ring polymers in the melt state: The physics of crumpling. Phys. Rev. Lett. 112, 118302 (2014)

    Article  ADS  Google Scholar 

  • Rubinstein, M.: Dynamics of ring polymers in the presence of fixed obstacles. Phys. Rev. Lett. 57(24), 3023 (1986)

    Article  ADS  Google Scholar 

  • Rubinstein, M., Colby, H.R.: Polymer Physics, Oxford University Press, Oxford (2003)

    Google Scholar 

  • Smrek, J., Grosberg, A.Y.: Understanding the dynamics of rings in the melt in terms of annealed tree model. J. Phys.: Condens. Matter 27, 064117 (2015)

    ADS  Google Scholar 

  • Zhang, Y., McCord, R.P., Ho, Y.-J., Lajoie, B.R., Hildebrand, D.G., Simon, A.C., Becker, M.S., Alt, F.W., Dekker, J.: Spatial organization of the mouse genome and its role in recurrent chromosomal translocations. Cell 148(5), 908 (2012)

    Article  Google Scholar 

Download references

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  1. School of Physics and Astronomy, University of Edinburgh, Edinburgh, UK

    Davide Michieletto

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  1. Davide Michieletto
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Michieletto, D. (2016). Predicting the Behaviour of Rings in Solution. In: Topological Interactions in Ring Polymers. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-41042-5_2

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