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Contact Kinetics in Fractal Macromolecules

Maxim Dolgushev, Thomas Guérin, Alexander Blumen, Olivier Bénichou, and Raphaël Voituriez
Phys. Rev. Lett. 115, 208301 – Published 9 November 2015
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Article Text
  • Introduction.—
  • Model.—
  • Theories of first contact times.—
  • Cyclization for small capture radius.—
  • Scaling for large n.—
  • Comparison with numerical simulations.—
  • Average conformations at first contact.—
  • Conclusion.—
  • ACKNOWLEDGMENTS
    • Supplemental Material
    References

    Abstract

    We consider the kinetics of first contact between two monomers of the same macromolecule. Relying on a fractal description of the macromolecule, we develop an analytical method to compute the mean first contact time for various molecular sizes. In our theoretical description, the non-Markovian feature of monomer motion, arising from the interactions with the other monomers, is captured by accounting for the nonequilibrium conformations of the macromolecule at the very instant of first contact. This analysis reveals a simple scaling relation for the mean first contact time between two monomers, which involves only their equilibrium distance and the spectral dimension of the macromolecule, independently of its microscopic details. Our theoretical predictions are in excellent agreement with numerical stochastic simulations.

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    • Received 21 May 2015

    DOI:https://doi.org/10.1103/PhysRevLett.115.208301

    © 2015 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Fluctuation mediated interactions
    1. Physical Systems
    FractalsPolymers
    Polymers & Soft Matter

    Authors & Affiliations

    Maxim Dolgushev1, Thomas Guérin2, Alexander Blumen1, Olivier Bénichou3, and Raphaël Voituriez3

    • 1Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany
    • 2Université de Bordeaux and CNRS, Laboratoire Ondes et Matière d’Aquitaine (LOMA), UMR 5798, 33400 Talence, France
    • 3Laboratoire de Physique Théorique de la Matière Condensée, CNRS/UPMC, 4 Place Jussieu, 75005 Paris, France

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    Issue

    Vol. 115, Iss. 20 — 13 November 2015

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    Images

    • Figure 1
      Figure 1

      Structure of fractal macromolecules investigated in this Letter. (a) Vicsek fractal, here of functionality (i.e., number of nearest neighbors of the branching sites) f=4 (VF4); (b) dual Sierpiński gasket (DSG); (c) T fractal (TF). The reactive monomers for which we compute the MFCT are represented by red squares. These extended conformations show only the topology of the structures.

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    • Figure 2
      Figure 2

      TNM [Eq. (8)] and TWF [Eq. (7)] for Vicsek fractals of functionality f=4 as a function of the capture radius a. The lines represent the results of Eq. (9).

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    • Figure 3
      Figure 3

      (a) NM and (b) WF MFCT for different fractal structures as a function of n1/γ. The parameter γ is obtained from the known values [20, 31, 37] of the spectral dimension ds: for Vicsek fractals of functionality f, γ=ln(3)/ln(3f+3), for dual Sierpiński gasket γ=ln(5/3)/ln(5), and for T fractal γ=ln(2)/ln(6). The capture radius is a=l. (c) NM MFCT for the dual Sierpiński gasket for different values of a.

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    • Figure 4
      Figure 4

      Average spatial position ⟨zi⟩π of the monomers (numbered through i=1,…,n) connecting reactants at the instant cyclization in the direction of the reaction, for Vicsek fractals of different functionalities f with generation g=3, as predicted by the NM (symbols) and WF theories (solid line). Inset: same quantity determined from simulations, with the same color code.

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