Skip to main content

A universal route to pattern formation in multicellular systems

Abstract

A general framework for the generation of long wavelength patterns in multi-cellular (discrete) systems is proposed, which extends beyond conventional reaction-diffusion (continuum) paradigms. The standard partial differential equations of reaction-diffusion framework can be considered as a mean-field like ansatz which corresponds, in the biological setting, to sending to zero the size (or volume) of each individual cell. By relaxing this approximation and, provided a directionality in the flux is allowed for, we demonstrate here that instability leading to spatial pattern formation can always develop if the (discrete) system is large enough, namely, composed of sufficiently many cells, the units of spatial patchiness. The macroscopic patterns that follow the onset of the instability are robust and show oscillatory or steady state behavior.

Graphical abstract

This is a preview of subscription content, access via your institution.

References

  1. S. Kondo, R. Asai, Nature 376, 765 (1995)

    ADS  Article  Google Scholar 

  2. J.D. Murray, Sci. Am. 258, 80 (1988)

    Article  Google Scholar 

  3. C. Klausmeier, Science 284, 1826 (1999)

    Article  Google Scholar 

  4. J.L. Maron, S. Harrison, Science 278, 1619 (1997)

    ADS  Article  Google Scholar 

  5. M. Rietkerk, J. van de Koppel, Trends Ecol. Evol. 23, 169 (2008)

    Article  Google Scholar 

  6. A.M. Turing, Phil. Trans. R. Soc. B 237, 37 (1952)

    ADS  Google Scholar 

  7. P. Ball,The self-made tapestry: Pattern formation in Nature, 1st edn. (Oxford University Press, Oxford, 1999)

  8. J.D. Murray,Mathematical biology II: Spatial models and biomedical applications (Springer-Verlag, Berlin, 2001)

  9. G. Nicolis, I. Prigogine,Self-organization in nonequilibrium systems: From dissipative structures to order through fluctuations (John Wiley & Sons, Chichester, 1977)

  10. A. Gierer, H. Meinhardt, Kybernetik 12, 30 (1972)

    Article  Google Scholar 

  11. H.G. Othmer, L.E. Scriven, J. Theor. Biol. 32, 507 (1971)

    Article  Google Scholar 

  12. H. Nakao, A.S. Mikhailov, Nat. Phys. 6, 544 (2010)

    Article  Google Scholar 

  13. R.S. Shaw, N. Packard, M. Schroter, H.L. Swinney, PNAS 104, 9580 (2007)

    ADS  Article  Google Scholar 

  14. K.S. Kim, I.S. Davis, P.A. Macpherson, T.J. Pedley, A.E. Hill, Proc. R. Soc. A 461, 273 (2005)

    ADS  Article  Google Scholar 

  15. J.C. Dallon, H.G. Othmer, Phil. Trans. R. Soc. Lond. B 352, 391 (1997)

    ADS  Article  Google Scholar 

  16. M. Asllani, R. Lambiotte, T. Carletti, Sci. Adv. 4, eaau9403 (2018)

    ADS  Article  Google Scholar 

  17. M. Postma, J. Roelofs, J. Goedhart, T.W.J. Gadella, A.J.W.G. Visser, P.J.M.V. Haastert, Mol. Biol. Cell 14, 5019 (2003)

    Article  Google Scholar 

  18. A.B. Rovinsky, M. Menzinger, Phys. Rev. Lett. 96, 1193 (1992)

    ADS  Article  Google Scholar 

  19. A.B. Rovinsky, M. Menzinger, Phys. Rev. Lett. 70, 778 (1993)

    ADS  Article  Google Scholar 

  20. M. Asllani, J.D. Challenger, F.S. Pavone, L. Sacconi, D. Fanelli, Nat. Commun. 5, 4517 (2014)

    ADS  Article  Google Scholar 

  21. R. Schnabel, M. Bischoff, A. Hintze, A.-K. Schulz, A. Hejnola, H. Meinhardt, H. Hutter, Dev. Biol. 294, 418 (2006)

    Article  Google Scholar 

  22. M. Asllani, D.M. Busiello, T. Carletti, D. Fanelli, G. Planchon, Sci. Rep. 5, 12927 (2015)

    ADS  Article  Google Scholar 

  23. P. Ciarletta, V. Balbi, E. Kuhl, Phys. Rev. Lett. 113, 248101 (2014)

    ADS  Article  Google Scholar 

  24. J.D. Murray, J. Theor. Biol. 88, 161 (1981)

    Article  Google Scholar 

  25. J.M. Pringle, A.M.H. Blakeslee, J.E. Byers, J. Roman, PNAS 108, 15288 (2011)

    ADS  Article  Google Scholar 

  26. O. Sporns,Networks of the Brain (MIT Press, Cambridge 2010)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Malbor Asllani.

Additional information

Publisher’s Note

The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Asllani, M., Carletti, T., Fanelli, D. et al. A universal route to pattern formation in multicellular systems. Eur. Phys. J. B 93, 135 (2020). https://doi.org/10.1140/epjb/e2020-10206-3

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2020-10206-3

Keywords

  • Statistical and Nonlinear Physics