Modelling Collective Cell Motion in Biology

  • P. K. MainiEmail author
  • R. E. Baker
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 87)


This paper reviews three mathematical modelling approaches that have recently been used to understand three different modes of collective cell motion in biology. Firstly, a cell-based model is presented for the study of cell motion in epithelial sheets, then a hybrid discrete cell-based model is described for neural crest cell invasion and, finally, a traditional partial differential equation model is described for tumour cell invasion. It is shown that the behaviour of all of these models can, in limiting cases, be recapitulated by nonlinear diffusion equations where the particular nonlinearity of the diffusion coefficient captures, on the global scale, the inherent interactions on the local scale.


Cell-based model Vertex-based model Travelling waves Volume exclusion 


  1. 1.
    Odell, G.M., Oster, G., Alberch, P., Burnside, B.: The mechanical basis of morphogenesis. Dev. Biol. 85, 446–462 (1981)CrossRefGoogle Scholar
  2. 2.
    Honda, H.: Description of cellular patterns by Dirichlet domains: the two-dimensional case. J. Theor. Biol. 72, 523–543 (1978)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Vincent, J.-P., Fletcher, A.G., Baena-Lopez, L.A.: Mechanisms and mechanics of cell competition in epithelia. Nat. Rev. Mol. Cell Biol. 14, 581–591 (2013)CrossRefGoogle Scholar
  4. 4.
    Fletcher, A.G., Osterfield, M., Baker, R.E., Shvartsman, S.Y.: Vertex models of epithelial morphogenesis. Biophys. J. 106(11), 2291–2304 (2014)CrossRefGoogle Scholar
  5. 5.
    Blankenship, J.T., Backovic, S.T., Sanny, J.S., Weitz, O., Zallen, J.A.: Multicellular rosette formation links planar cell polarity to tissue morphogenesis. Dev. Cell 11, 459–470 (2006)CrossRefGoogle Scholar
  6. 6.
    Trichas, G., Smith, A.M., White, N., Wilkins, V., Watanabe, T., Moore, A., Joyce, B., Sugnaseelan, J., Rodriguez, T.A., Kay, D., Baker, R.E., Maini, P.K., Srinivas, S.: Multi-cellular rosettes in the mouse visceral endoderm facilitate the ordered migration of anterior visceral endoderm cells. PLoS Biol. 10(2), e1001256 (2012)CrossRefGoogle Scholar
  7. 7.
    Farhadifar, R., Roper, J.C., Aigouy, B., Eaton, S., Julicher, F.: The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing. Curr. Biol. 17, 2095–2104 (2007)CrossRefGoogle Scholar
  8. 8.
    Meineke, F., Potten, C.S., Loeffler, M.: Cell migration and organization in the intestinal crypt using a lattice-free model. Cell Prolif. 34(4), 253–266 (2001)CrossRefGoogle Scholar
  9. 9.
    Graner, F., Glazier, J.A.: Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys. Rev. Lett. 69(13), 2013–2016 (1992)CrossRefGoogle Scholar
  10. 10.
    Newman, T.J.: Modeling multi-cellular systems using sub-cellular elements. Math. Biosci. Eng. 2, 611–622 (2005)CrossRefGoogle Scholar
  11. 11.
    Murray, P.J., Edwards, C.M., Tindall, M.J., Maini, P.K.: From a discrete to a continuum model of cell dynamics in one dimension. Phys. Rev. E 80, 031912 (2009)CrossRefGoogle Scholar
  12. 12.
    Lushnikov, P.M., Chen, N., Alber, M.: Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact. Phys. Rev. E 78, 061904 (2009)CrossRefGoogle Scholar
  13. 13.
    Murray, P.J., Edwards, C.M., Tindall, M.J., Maini, P.K.: Classifying general nonlinear force laws in cell-based models via the continuum limit. Phys. Rev. E 85, 021921 (2012)CrossRefGoogle Scholar
  14. 14.
    Pitt-Francis, J., Pathmanathan, P., Bernabeu, M.O., Bordas, R., Cooper, J., Fletcher, A.G., Mirams, G.R., Murray, P., Osborne, J.M., Walter, A., Chapman, S.J., Garny, A., van Leeuwen, I.M., Maini, P.K., Rodriguez, B., Waters, S.L., Whiteley, J.P., Byrne, H.M., Gavaghan, D.J.: Chaste: a test-driven approach to software development for biological modelling. Comput. Phys. Comm. 180, 2542–2471 (2009)CrossRefGoogle Scholar
  15. 15.
    Murray, P.J., Walter, A., Fletcher, A.G., Edwards, C.M., Tindall, M.J., Maini, P.K.: Comparing a discrete and continuum model of the intestinal crypt. Phys. Biol. 8, 026011 (2011)CrossRefGoogle Scholar
  16. 16.
    Murray, P.J., Kang, J.-W., Mirams, G.R., Shin, S.-Y., Byrne, H.M., Maini, P.K., Cho, K.-H.: Modelling spatially regulated β-catenin dynamics and invasion in intestinal crypts. Biophys. J. 99, 716–725 (2010)CrossRefGoogle Scholar
  17. 17.
    McLennan, R., Dyson, L., Prather, K.W., Morrison, J.A., Baker, R.E., Maini, P.K., Kulesa, P.M.: Multiscale mechanisms of cell migration during development: Theory and experiment. Development 139, 2935–2944 (2012)CrossRefGoogle Scholar
  18. 18.
    Dyson, L.: Models of cranial neural crest cell migration. D.Phil. thesis, University of Oxford (2013)Google Scholar
  19. 19.
    Dyson, L., Maini, P.K., Baker, R.E.: Macroscopic limits of individual-based models for motile cell populations with volume exclusion. Phys. Rev. E 86, 031903 (2012)CrossRefGoogle Scholar
  20. 20.
    Gatenby, R.A., Gawlinski, E.T.: A reaction-diffusion model of cancer invasion. Cancer Res. 56, 5745–5753 (1996)Google Scholar
  21. 21.
    Warbug, O.: The Metabolism of Tumors. Arnold Constable, London (1930)Google Scholar
  22. 22.
    Fasano, A., Herrero, M.A., Rodrigo, M.R.: Slow and fast invasion waves in a model of acid-mediated tumour growth. Math. Biosci. 220, 45–56 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    McGillen, J.B., Gaffney, E.A., Martin, N.K., Maini, P.K.: A general reaction-diffusion model of acidity in cancer invasion. J. Math. Biol. 68, 1199–1224 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, New York (2003)Google Scholar
  25. 25.
    Robey, I.F., Baggett, B.K., Kirkpatrick, N.D., Roe, D.J., Dosescu, J., Sloane, B.F., Gatenby, R.A., Raghunand, N., Gillies, R.J.: Bicarbonate increases tumor pH and inhibits spontaneous metastases. Cancer Res. 69, 2260–2268 (2009)CrossRefGoogle Scholar
  26. 26.
    Martin, N.K., Gaffney, E.A., Gatenby, R.A., Gillies, R.J., Robey, I.F., Maini, P.K.: A mathematical model of tumour and blood pHe regulation: the HCO3/CO2 buffering system. Math. Biosci. 230, 1–11 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Wolfson Centre for Mathematical BiologyUniversity of OxfordOxfordUK

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