A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion

  • K. AnguigeEmail author
  • C. Schmeiser


We develop and analyse a discrete model of cell motility in one dimension which incorporates the effects of volume filling and cell-to-cell adhesion. The formal continuum limit of the model is a nonlinear diffusion equation with a diffusivity which can become negative if the adhesion coefficient is sufficiently large. This appears to be related to the presence of spatial oscillations and the development of plateaus (pattern formation) in numerical solutions of the discrete model. A combination of stability analysis of the discrete equations and steady-state analysis of the limiting PDE (and a higher-order correction thereof) can be used to shed light on these and other qualitative predictions of the model.


Cell-to-cell adhesion Continuous and discrete models of cell motility Nonlinear diffusion equations Ill-posed problems Modified equations 

Mathematics Subject Classification (2000)

92C17 35K55 


  1. 1.
    Armstrong N., Painter K., Sherratt J.: A continuum approach to modelling cell–cell adhesion. J. Theor. Biol. 243(1), 98–113 (2006)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Crosby C., Fleming P., Argraves S., Corada M., Zanetta L., Dejana E., Drake C.: VE-cadherin is not required for the formation of nascent blood vessels but acts to prevent their disassembly. Blood. 105(7), 2771–2776 (2005)CrossRefGoogle Scholar
  3. 3.
    Dolak Y., Schmeiser C.: The Keller-Segel model with logistic sensitivity function and small diffusivity. SIAM J. Appl. Math. 66(1), 286–308 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Enmon R., O’Connor K., Lacks D., Schwartz D., Dotson R.: Dynamics of spheroid self-assembly in liquid-overlay culture of DU 145 human prostate-cancer cells. Biotechnol. Bioeng. 72(6), 579–591 (2001)CrossRefGoogle Scholar
  5. 5.
    Glazier J., Graner F.: Simulation of the differential-adhesion driven rearrangement of biological cells. Phys. Rev. E. 47(3), 2128–2154 (1993)CrossRefGoogle Scholar
  6. 6.
    Kadmon G., Kowitz A., Altevogt P., Schachner M.: The neural cell-adhesion molecule N-CAM enhances L1-dependent cell–cell interactions. J. Cell. Biol. 110, 193–208 (1990)CrossRefGoogle Scholar
  7. 7.
    Murray J.: Mathematical Biology I: An Introduction. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  8. 8.
    Neelamegham S., Munn L., Zygourakis K.: A model for the kinetics of homotypic cellular aggregation under static conditions. Biophys J. 72(1), 5164 (1997)CrossRefGoogle Scholar
  9. 9.
    Othmer H., Stevens A.: Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57(4), 1044–1081 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Painter K., Hillen T.: Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Quart. 10(4), 501–543 (2002)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Potapov A., Hillen T.: Metastability in chemotaxis models. J. Dynam. Diff. Eq. 17(2), 293–330 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Serini G., Ambrosi D., Giraudo E., Gamba A., Preziosi L., Bussolino F.: Modelling the early stages of vascular network assembly. EMBO J. 22(8), 1771–1779 (2003)CrossRefGoogle Scholar
  13. 13.
    Sun X., Ward M.: The dynamics and coarsening of interfaces for the viscous Cahn-Hilliard equation in one spatial dimension. Stud. Appl. Math. 105, 203–234 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Taylor M.: Partial Differential Equations I. Springer, Heidelberg (1996)Google Scholar
  15. 15.
    Taylor M.: Partial Differential Equations III. Springer, Heidelberg (1996)Google Scholar
  16. 16.
    Vazquez J.L.: The Porous-Medium Equation: Mathematical Theory. Oxford Science Publications, Oxford (2007)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.RICAM, Austrian Academy of SciencesLinzAustria
  2. 2.Faculty of MathematicsUniversity of ViennaViennaAustria

Personalised recommendations