Numerical Simulation of the Dynamics of Molecular Markers Involved in Cell Polarization

  • V. Calvez
  • N. Meunier
  • N. Muller
  • R. Voituriez


In this work, we investigate the dynamics of a non-local model describing spontaneous cell polarization. It consists in a drift-diffusion equation set in the half-space, with the coupling involving the trace value on the boundary. We characterize the following behaviors in the one-dimensional case: solutions are global if the mass is below the critical mass and they blow up in finite time above the critical mass. The higher-dimensional case is also discussed. The results are reminiscent of the classical Keller–Segel system in double the dimension. In addition, in the one-dimensional case we prove quantitative convergence results using relative entropy techniques. This work is complemented with a more realistic model that takes into account dynamical exchange of molecular content at the boundary. In the one-dimensional case we prove that blow-up is prevented. Furthermore, density converges towards a non trivial stationary configuration.


Cell dynamics Cell polarization Entropy technique Exchange of molecular content 


  1. [Al07]
    Allaire, G.: Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation. Oxford University Press, Oxford (2007)MATHGoogle Scholar
  2. [AlEtAl08]
    Altschuler, S., Angenent, S., Wang, Y., Wu, L.: On the spontaneous emergence of cell polarity. Nature 454, 886–890 (2008)CrossRefGoogle Scholar
  3. [CaPeTa07]
    Calvez, V., Perthame, B., Tabar, M.S.: Modified Keller-Segel system and critical mass for the log interaction kernel. Stochastic analysis and partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 429, 45–62 (2007)Google Scholar
  4. [CaMeVo10]
    Calvez, V., Meunier, N., Voituriez, R.: A one-dimensional Keller-Segel equation with a drift issued from the boundary. C. R. Acad. Sci. Paris, Ser. 1 348, 629–634 (2010)Google Scholar
  5. [CaEtAl12]
    Calvez, V., Hawkins, R., Meunier, N., Voituriez, R.: Analysis of a nonlocal model for spontaneous cell polarization. SIAM J. Appl. Math. 72, 594–622 (2012)MathSciNetMATHCrossRefGoogle Scholar
  6. [EuEtAl07]
    Eugenio, M., Wedlich-Soldner, R., Li, R., Altschuler, S.J., Wu, L.F.: Principles for the dynamic maintenance of cortical polarity. Cell 129, 411–422 (2007)CrossRefGoogle Scholar
  7. [Ev98]
    Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)MATHGoogle Scholar
  8. [HaEtAl09]
    Hawkins, R.J., Benichou, O., Piel, M., Voituriez, R.: Rebuilding cytoskeleton roads: active transport induced polarisation of cells. Phys. Rev. E 80, 040903 (2009)CrossRefGoogle Scholar
  9. [IgDe08]
    Iglesias, P.A., Devreotes, P.N.: Navigating through models of chemotaxis. Cell Biol. 20, 35 (2008)Google Scholar
  10. [LeKeRa06]
    Levine, H., Kessler, D.A., Rappel, W.-J.: Directional sensing in eukaryotic chemotaxis: a balanced inactivation model. Proc. Natl. Acad. Sci. 103, 9761 (2006)CrossRefGoogle Scholar
  11. [Mu13]
    Muller, N.: Mathematical and numerical studies of nonlinear and nonlocal models involved in biology, Doctoral dissertation, Paris Descartes (2013)Google Scholar
  12. [OnRa07]
    Onsum, M., Rao, C.V.: A mathematical model for neutrophil gradient sensing and polarisation. PLoS Comput. Biol. 3, e36 (2007)MathSciNetCrossRefGoogle Scholar
  13. [PhKoTh09]
    Phillips, R., Kondev, J.,Theriot, J.: Physical Biology of the Cell. Garland Science, New York, (2008)Google Scholar
  14. [WeAlLi03]
    Wedlich-Soldner, L.W.R., Altschuler, S., Li, R.: Spontaneous cell polarization through actomyosin-based delivery of the cdc42 GTPase. Science 299, 1231 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • V. Calvez
    • 1
  • N. Meunier
    • 2
  • N. Muller
    • 2
  • R. Voituriez
    • 3
  1. 1.École Normale Supérieure de LyonLyon CedexFrance
  2. 2.Université Paris DescartesParisFrance
  3. 3.Université Pierre et Marie CurieParis CedexFrance

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