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Relative distance between tracers as a measure of diffusivity within moving aggregates


Tracking of particles, be it a passive tracer or an actively moving bacterium in the growing bacterial colony, is a powerful technique to probe the physical properties of the environment of the particles. One of the most common measures of particle motion driven by fluctuations and random forces is its diffusivity, which is routinely obtained by measuring the mean squared displacement of the particles. However, often the tracer particles may be moving in a domain or an aggregate which itself experiences some regular or random motion and thus masks the diffusivity of tracers. Here we provide a method for assessing the diffusivity of tracer particles within mobile aggregates by measuring the so-called mean squared relative distance (MSRD) between two tracers. We provide analytical expressions for both the ensemble and time averaged MSRD allowing for direct identification of diffusivities from experimental data.


  1. 1.

    W. Pönisch, K. Eckenrode, K. Alzurqa, H. Nasrollahi, C.A. Weber, V. Zaburdaev, N. Biais, arXiv:1703.09659 (2017)

  2. 2.

    J. Taktikos, Y.T. Lin, H. Stark, N. Biais, V. Zaburdaev, PLoS ONE 10, e0137661 (2015)

  3. 3.

    S. Douezan, K. Guevorkian, R. Naouar, S. Dufour, D. Cuvelier, F. Brochard-Wyart, Proc. Natl. Acad. Sci. USA 108, 7315 (2011)

  4. 4.

    F. Pampaloni, N. Ansari, E.H. Stelzer, Cell Tissue Res. 352, 161 (2013)

  5. 5.

    F. Montel, M. Delarue, J. Elgeti, D. Vignjevic, G. Cappello, J. Prost, New J. Phys. 14, 055008 (2012)

  6. 6.

    M. Merkel, M.L. Manning, in Seminars in cell and developmental biology (Elsevier, Academic Press Inc., USA, 2016)

  7. 7.

    M.C. Munder, D. Midtvedt, T. Franzmann, E. Nüske, O. Otto, M. Herbig, E. Ulbricht, P. Müller, A. Taubenberger, S. Maharana, L. Malinovska, D. Richter, J. Guck, V. Zaburdaev, S. Alberti, eLife 5, e09347 (2016)

  8. 8.

    I.M. Tolić-N∅rrelykke, E.L. Munteanu, G. Thon, L. Oddershede, K. Berg-S∅rensen, Phys. Rev. Lett. 93, 078102 (2004)

  9. 9.

    L.F. Richardson, Proc. R. Soc. Lond. A: Contain. Pap. Math. Phys. Character 110, 709 (1926)

  10. 10.

    G. Batchelor, in Mathematical proceedings of the Cambridge Philosophical Society (Cambridge University Press, Cambridge, UK, 1952), Vol. 48, pp. 345–362

  11. 11.

    P.J. Sullivan, J. Fluid. Mech. 47, 601 (1971)

  12. 12.

    H. Qian, M.P. Sheetz, E.L. Elson, Biophys. J. 60, 910 (1991)

  13. 13.

    M. Ueda, S.I. Sasa, Phys. Rev. Lett. 115, 080605 (2015)

  14. 14.

    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

  15. 15.

    A. Caspi, R. Granek, M. Elbaum, Phys. Rev. Lett. 85, 5655 (2000)

  16. 16.

    M. Goulian, S.M. Simon, Biophys. J. 79, 2188 (2000)

  17. 17.

    I. Golding, E.C. Cox, Phys. Rev. Lett. 96, 098102 (2006)

  18. 18.

    R. Metzler, J.H. Jeon, A.G. Cherstvy, E. Barkai, Phys. Chem. Chem. Phys. 16, 24128 (2014)

  19. 19.

    Y. He, S. Burov, R. Metzler, E. Barkai, Phys. Rev. Lett. 101, 058101 (2008)

  20. 20.

    A.G. Cherstvy, A.V. Chechkin, R. Metzler, New J. Phys. 15, 083039 (2013)

  21. 21.

    G. Bel, E. Barkai, Phys. Rev. Lett. 94, 240602 (2005)

  22. 22.

    S. Burov, J.H. Jeon, R. Metzler, E. Barkai, Phys. Chem. Chem. Phys. 13, 1800 (2011)

  23. 23.

    V. Zaburdaev, S. Denisov, J. Klafter, Rev. Mod. Phys. 87, 483 (2015)

  24. 24.

    P. Beckmann, Radio Sci. J. Res. NBS/USNC-URSIs 68D, 927 (1962)

  25. 25.

    M. Siddiqui, J. Res. Natl. Bur. Stand. D 60, 167 (1962)

  26. 26.

    N. Wax, Selected papers on noise and stochastic processes (Courier Dover Publications, USA, 1954)

  27. 27.

    M. Abramowitz, I.A. Stegun et al., Handbook of mathematical functions (Courier Corporation, USA, 1966)

  28. 28.

    Z. Yücel, F. Zanlungo, T. Ikeda, T. Miyashita, N. Hagita, Sensors 13, 875 (2013)

  29. 29.

    K. Buchin, S. Sijben, T. Arseneau, E.P. Willems, in Proceedings of the 20th International Conference on Advances in Geographic Information Systems (ACM, 2012), pp. 119–128

  30. 30.

    K. Buchin, S. Sijben, E.E. van Loon, N. Sapir, S. Mercier, T.J.M. Arseneau, E.P. Willems, Mov. Ecol. 3, 18 (2015)

  31. 31.

    M. Guo, A.J. Ehrlicher, M.H. Jensen, M. Renz, J.R. Moore, R.D. Goldman, J. Lippincott-Schwartz, F.C. Mackintosh, D.A. Weitz, Cell 158, 822 (2014)

  32. 32.

    A.D. Wessel, M. Gumalla, J. Grosshans, C.F. Schmidt, Biophys. J. 108, 1899 (2015)

  33. 33.

    D.S. Banks, C. Fradin, Biophys. J. 89, 2960 (2005)

  34. 34.

    I. Bronstein, Y. Israel, E. Kepten, S. Mai, Y. Shav-Tal, E. Barkai, Y. Garini, Phys. Rev. Lett. 103, 018102 (2009)

  35. 35.

    I.N. Bronshtein, K.A. Semendyayev, Handbook of mathematics (Springer, Berlin, Heidelberg, Germany, 2015)

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Correspondence to Wolfram Pönisch.

Additional information

Contribution to the Topical Issue “Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook”, edited by Ryszard Kutner and Jaume Masoliver.

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Pönisch, W., Zaburdaev, V. Relative distance between tracers as a measure of diffusivity within moving aggregates. Eur. Phys. J. B 91, 27 (2018). https://doi.org/10.1140/epjb/e2017-80347-5

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