Chemotaxis migration and morphogenesis of living colonies

Open Access
Regular Article
Part of the following topical collections:
  1. Physical constraints of morphogenesis and evolution

Abstract.

Development of forms in living organisms is complex and fascinating. Morphogenetic theories that investigate these shapes range from discrete to continuous models, from the variational elasticity to time-dependent fluid approach. Here a mixture model is chosen to describe the mass transport in a morphogenetic gradient: it gives a mathematical description of a mixture involving several constituents in mechanical interactions. This model, which is highly flexible can incorporate many biological processes but also complex interactions between cells as well as between cells and their environment. We use this model to derive a free-boundary problem easier to handle analytically. We solve it in the simplest geometry: an infinite linear front advancing with a constant velocity. In all the cases investigated here as the 3 D diffusion, the increase of mitotic activity at the border, nonlinear laws for the uptake of morphogens or for the mobility coefficient, a planar front exists above a critical threshold for the mobility coefficient but it becomes unstable just above the threshold at long wavelengths due to the existence of a Goldstone mode. This explains why sparsely bacteria exhibit dendritic patterns experimentally in opposition to other colonies such as biofilms and epithelia which are more compact. In the most unstable situation, where all the laws: diffusion, chemotaxis driving and chemoattractant uptake are linear, we show also that the system can recover a dynamic stability. A second threshold for the mobility exists which has a lower value as the ratio between diffusion coefficients decreases. Within the framework of this model where the biomass is treated mainly as a viscous and diffusive fluid, we show that the multiplicity of independent parameters in real biologic experimental set-up may explain varieties of observed patterns.

Graphical abstract

Keywords

Topical issue: Physical constraints of morphogenesis and evolution 

References

  1. 1.
    K.W. Rogers, A.F. Schier, Annu. Rev. Cell Dev. Biol. 27, 377 (2011)CrossRefGoogle Scholar
  2. 2.
    V. Sourjik, J.P. Armitage, EMBO j. 29, 2724 (2010)CrossRefGoogle Scholar
  3. 3.
    E. Ben-Jacob, H. Shmueli, O. Shochet, A. Tenenbaum, Physica A 187, 378 (1992)ADSCrossRefGoogle Scholar
  4. 4.
    J.S. Langer, Rev. Mod. Phys. 52, 1 (1980)ADSCrossRefGoogle Scholar
  5. 5.
    M. Ben Amar, Journ. Phys. I 3, 353 (1993)Google Scholar
  6. 6.
    M. Ben Amar, E. Brener, Phys. Rev. Lett. 71, 589 (1993)ADSCrossRefGoogle Scholar
  7. 7.
    Y. Couder, Growth patterns: from stable curved fronts to fractal structures, in Chaos, Order and Patterns, edited by R. Artuso, P. Cvitanovic, G. Casati (Plenum Press, 1991)Google Scholar
  8. 8.
    M. Ben Amar, Phys. Rev. A 44, 3673 (1991)ADSCrossRefGoogle Scholar
  9. 9.
    F. Argoul, E. Freysz, A. Kuhn, C. Lger, L. Potin, Phys. Rev. E 53, 1777 (1996)ADSCrossRefGoogle Scholar
  10. 10.
    F. Argoul, E. Freysz, A. Kuhn, C. Lger, L. Potin, Phys. Rev. A 44, 3673 (1991)CrossRefGoogle Scholar
  11. 11.
    E. Ben-Jacob, H. Levine, J. R. Soc. Interface 3, 197 (2006)CrossRefGoogle Scholar
  12. 12.
    M. Reffay, L. Petitjean, S. Coscoy, E. Grasland-Mongrain, F. Amblard, A. Buguin, P. Silberzan, Biophys. J. 100, 2566 (2011)ADSCrossRefGoogle Scholar
  13. 13.
    J.R. King, S.J. Franks, Mathematical Modeling of Biological Systems, Vol. 1, edited by A. Deutsch, L. Brusch, H.M. Byrne, G. Vries, H. Herzel (Birkhauser, 2007) p. 175Google Scholar
  14. 14.
    J.S. Lowengrub et al., Nonlinearity 23, R1 (2010)MathSciNetADSCrossRefMATHGoogle Scholar
  15. 15.
    M. Eisenbach, Bacterial Chemotaxis Encyclopedia of life Sciences (Nature Publishing Group, 2001) pp. 1-14Google Scholar
  16. 16.
    A. Puliafito, L. Hufnagel, P. Neveu, S. Streichan, A. Sigal, D.K Fygenson, B.I. Shraiman, Proc. Nat. Acad. Sci. U.S.A. 109, 739 (2012)ADSCrossRefGoogle Scholar
  17. 17.
    E.F. Keller, L.A. Segel, J. Theor. Biol. 26, 399 (1970)CrossRefMATHGoogle Scholar
  18. 18.
    L. Corrias, B. Perthame, H. Zaag, Milan J. Math. 72, 1 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    J. Dolbeault, B. Perthame, C. R. Math. Acad. Sci. Paris 339, 611 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    M.A. Herrero, J. Velazquez, Journ. Math. Biol. 35, 177 (1996)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    L. Menten, M.I. Michaelis, Biochem. Z. 49, 333 (1913)Google Scholar
  22. 22.
    J.B. Xavier, E. Martinez-Garcia, K.R. Foster, Amer. Naturalist 174, 1 (2009)CrossRefGoogle Scholar
  23. 23.
    P. Ciarletta, L. Foret, M. Ben Amar, J. R. Soc. Interface 9, 305 (2011)Google Scholar
  24. 24.
    A. Saez, E. Anon, M. Ghibaudo, M.O. du Roure, J.-M. Di Meglio, P. Hersen, P. Silberzan, A. Buguin, B. Ladoux, J. Phys.: Condens. Matter 22, 19 (2010)CrossRefGoogle Scholar
  25. 25.
    S. Mark, R. Shlomovitz, N. Gov, S. Nir, M. Poujade, E. Grasland-Mongrain, P. Silberzan, Biophys. J. 98, 361 (2010)CrossRefGoogle Scholar
  26. 26.
    M. Doi, A. Onuki, J. Phys. II 2, 1631 (1992)CrossRefGoogle Scholar
  27. 27.
    G. Caginalp, P. Fife, Phys. Rev. B 33, 7792 (1986)MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    G. Caginalp, Arch. Rach. Mech. Anal. 92, 205 (1986)MathSciNetMATHGoogle Scholar
  29. 29.
    J.W. Cahn, J.E. Hilliard, J. Chem. Phys. 28, 258 (1958)ADSCrossRefGoogle Scholar
  30. 30.
    J.W. Cahn, J.E. Taylor, Acta Mettal. 42, 1045 (1994)CrossRefGoogle Scholar
  31. 31.
    M. Ben Amar, Phys. Fluids A 4, 2641 (1992)MathSciNetADSCrossRefMATHGoogle Scholar
  32. 32.
    N. Kudryashov, Phys. Lett. A 342, 99 (2005)MathSciNetADSCrossRefMATHGoogle Scholar
  33. 33.
    M. Hayek, Appl. Math. Comput. 218, 2407 (2011)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    O. Cochet-Escartin Dynamique de fermeture de blessures circulaires modèles: aspects expérimentaux et théoriques, PhD thesis of University Pierre et Marie Curie, Paris (Avril 2013)MathSciNetCrossRefGoogle Scholar
  35. 35.
    X. Trepat, Mi.R. Wasserman, T.E. Angelini, E. Millet, D.A. Weitz, J.P. Butler, Jeffrey J. Fredberg, Nat. Phys. 5, 426 (2009)CrossRefGoogle Scholar
  36. 36.
    X. Wang, T. Long, R.M. Ford, Biotech. Bioengin. 109, 1622 (2012)CrossRefGoogle Scholar
  37. 37.
    J. Dervaux, J.C. Magniez, A. Libchaber Growth and form of Bacillus subtilis biofilms, preprint (2012)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2013

Authors and Affiliations

  1. 1.Laboratoire de Physique Statistique, Ecole Normale SupérieureUPMC Univ Paris 6, Université Paris Diderot, CNRSParisFrance
  2. 2.Institut Universitaire de Cancérologie, Faculté de MédecineUniversité Pierre et Marie Curie-Paris 6ParisFrance

Personalised recommendations