Observations of metastable states of the free swelling knots and directional motion of tensioned knots in vibrated bead chains

  • Y. C. ChouEmail author
Regular Article


The free swelling of knots and the directional motion of knots under tension were studied in vertically vibrated bead chains. A metastable state of swelling was observed in the strongly vibrated two-end-free bead chains, as predicted by Grosberg and Rabin (Phys. Rev. Lett. 99, 217801 (2007)). Knots in the two-end-fixed chains were found to move directionally. The direction of motion could be changed by flipping the knot over. The velocity of motion depended on the tension in the bead chain. The effects of tension on the motion of knots were studied in one-end-fixed chains. The directional reptation might have been influenced by the random motion of the leading arc of the knot. The knots might move in a forced-reptation manner under the interaction with a simulated translocase.

Graphical abstract


Flowing Matter: Granular Materials 

Supplementary material

10189_2019_11841_MOESM1_ESM.mp4 (10.6 mb)
Swelling and untying of a $5_{1}$ knot in a both-end-free chain, $\Gamma = 2.5$ G.
10189_2019_11841_MOESM2_ESM.mp4 (29.7 mb)
A metastable $3_{1}$ knot in a both-end-free chain for $\Gamma = 4.5$ G.
10189_2019_11841_MOESM3_ESM.mp4 (19.8 mb)
Motion of a $3_{1}$ knot in a 90-bead chain with both ends fixed at the vibrational acceleration $\Gamma = 3.0$ G. The leading arc is to the left of the images.
10189_2019_11841_MOESM4_ESM.mp4 (16.6 mb)
Motion of a $3_{1}$ knot in a bead chain under a tension of 140 mgw for $\Gamma = 4.0$ G. The leading arc is to the free end.
10189_2019_11841_MOESM5_ESM.mp4 (3.4 mb)
Pushing and untying of a $3_{1}$ knot in a linear chain by a simulated helicase for $\Gamma = 3.0$ G.
10189_2019_11841_MOESM6_ESM.mp4 (10.3 mb)
Tightening of a $4_{1}$ knot in a circular chain by a simulated helicase at $\Gamma = 4.0$ G.


  1. 1.
    L.F. Liu, R.E. Depew, J.C. Wang, J. Mol. Biol. 106, 439 (1976)CrossRefGoogle Scholar
  2. 2.
    K. Koniaris, M. Muthukumar, Phys. Rev. Lett. 66, 2211 (1991)ADSCrossRefGoogle Scholar
  3. 3.
    S.A. Wasserman, N. Cozzarelli, Science 232, 951 (1986)ADSCrossRefGoogle Scholar
  4. 4.
    R. Matthews, A.A. Louis, J.M. Yeomans, EPL 89, 20001 (2010)ADSCrossRefGoogle Scholar
  5. 5.
    X.R. Bao, H.J. Lee, S.R. Quake, Phys. Rev. Lett. 91, 265506 (2003)ADSCrossRefGoogle Scholar
  6. 6.
    L. Huang, D.E. Makarov, J. Phys. Chem. A 111, 10338 (2007)CrossRefGoogle Scholar
  7. 7.
    A.R. Klotz, B.W. Soh, P.S. Doyle, Phys. Rev. Lett. 120, 188003 (2018)ADSCrossRefGoogle Scholar
  8. 8.
    V. Narsimhan, C.B. Renner, P.S. Doyle, Soft Matter 12, 5041 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    P. Szymczak, Sci. Rep. 6, 21702 (2016)ADSCrossRefGoogle Scholar
  10. 10.
    A.Y. Grosberg, Y. Rabin, Phys. Rev. Lett. 99, 217801 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    E. Ben-Naim, Z.A. Daya, P. Vorobieff, R.E. Ecke, Phys. Rev. Lett. 86, 1414 (2001)ADSCrossRefGoogle Scholar
  12. 12.
    K. Safford, Y. Kantor, M. Kardar, A. Kudrolli, Phys. Rev. E 79, 061304 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    Y.C. Chou, J. Phys. D: Appl. Phys. 51, 135401 (2018)ADSCrossRefGoogle Scholar
  14. 14.
    J.M. Sogo, A. Stasiak, M.L. Martinez-Robles, D.B. Krimer, P. Hernandez, J.B. Schvartzman, J. Mol. Biol. 286, 637 (1999)CrossRefGoogle Scholar

Copyright information

© EDP Sciences, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsNational Tsing-Hua UniversityHsinchuTaiwan

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