Acta Applicandae Mathematicae

, Volume 145, Issue 1, pp 15–45 | Cite as

Quantitative Convergence Towards a Self-Similar Profile in an Age-Structured Renewal Equation for Subdiffusion

  • Hugues Berry
  • Thomas Lepoutre
  • Álvaro Mateos González
Article

Abstract

Continuous-time random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion, i.e. subdiffusion. Here, we describe the subdiffusive continuous-time random walk using age-structured partial differential equations with age renewal upon each walker jump, where the age of a walker is the time elapsed since its last jump. In the spatially-homogeneous (zero-dimensional) case, we follow the evolution in time of the age distribution. An approach inspired by relative entropy techniques allows us to obtain quantitative explicit rates for the convergence of the age distribution to a self-similar profile, which corresponds to convergence to a stationary profile for the rescaled variables. An important difficulty arises from the fact that the equation in self-similar variables is not autonomous and we do not have a specific analytical solution. Therefore, in order to quantify the latter convergence, we estimate attraction to a time-dependent “pseudo-equilibrium”, which in turn converges to the stationary profile.

Keywords

Age-structured PDE Renewal equation Anomalous diffusion Relative entropy estimates 

Mathematics Subject Classification

35Q92 92D25 60J75 35B40 

References

  1. 1.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation (1987) CrossRefMATHGoogle Scholar
  2. 2.
    Bronstein, I., Israel, Y., Kepten, E., Mai, S., Shav-Tal, Y., Barkai, E., Garini, Y.: Transient anomalous diffusion of telomeres in the nucleus of mammalian cells. Phys. Rev. Lett. 103(018102), 1–4 (2009) Google Scholar
  3. 3.
    Di Rienzo, C., Piazza, V., Gratton, E., Beltram, F., Cardarelli, F.: Probing short-range protein Brownian motion in the cytoplasm of living cells. Nat. Commun. 5, 5891 (2014) CrossRefGoogle Scholar
  4. 4.
    Fedotov, S., Falconer, S.: Subdiffusive master equation with space-dependent anomalous exponent and structural instability. Phys. Rev. E 85, 031132 (2012) CrossRefGoogle Scholar
  5. 5.
    Fedotov, S., Falconer, S.: Nonlinear degradation-enhanced transport of morphogens performing subdiffusion. Phys. Rev. E 89, 012107 (2014) CrossRefGoogle Scholar
  6. 6.
    Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York (1966) MATHGoogle Scholar
  7. 7.
    Golding, I., Cox, E.C.: Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96, 098102 (2006) CrossRefGoogle Scholar
  8. 8.
    Henry, B.I., Langlands, T.A.M., Wearne, S.L.: Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 74(3 Pt 1), 031116 (2006) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Höfling, F., Franosch, T.: Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys. 76(4), 046602 (2013). doi:10.1088/0034-4885/76/4/046602 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Melbourne, I., Terhesiu, D.: Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 189, 61–110 (2012) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mendez, V., Fedotov, S., Horsthemke, W.: Reaction-Transport Systems (2010) CrossRefGoogle Scholar
  12. 12.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Michel, P., Mischler, S., Perthame, B.: General relative entropy inequality: an illustration on growth models. J. Math. Pures Appl. 84(9), 1235–1260 (2005) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Montroll, E., Weiss, G.: Random walks on lattices. II. J. Math. Phys. 6, 167–181 (1965) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Parry, B.R., Surovtsev, I.V., Cabeen, M.T., O’Hern, C.S., Dufresne, E.R., Jacobs-Wagner, C.: The bacterial cytoplasm has glass-like properties and is fluidized by metabolic activity. Cell 156(1–2), 183–194 (2014) CrossRefGoogle Scholar
  16. 16.
    Perthame, B.: Transport Equations in Biology. Birkäuser, Basel (2007). ISBN 978-3-7643-7842-4 MATHGoogle Scholar
  17. 17.
    Terhesiu, D.: Error rates in the Darling–Kac law. Stud. Math. 220, 101–117 (2014) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Yuste, S.B., Lindenberg, K., Ruiz-Lorenzo, J.J.: Subdiffusion-limited reactions. In: Anomalous Transport, pp. 367–395 (2008). Wiley-VCH Verlag GmbH & Co. KGaA CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Hugues Berry
    • 1
  • Thomas Lepoutre
    • 1
    • 2
  • Álvaro Mateos González
    • 1
    • 3
  1. 1.InriaVilleurbanneFrance
  2. 2.Institut Camille Jordan, CNRS UMR 5208Université de LyonVilleurbanne cedexFrance
  3. 3.Unité de Mathématiques Pures et Appliquées, CNRS UMR 5669Université de LyonLyonFrance

Personalised recommendations