Continuum model of cell adhesion and migration



The motility of cells crawling on a substratum has its origin in a thin cell organ called lamella. We present a 2-dimensional continuum model for the lamella dynamics of a slowly migrating cell, such as a human keratinocyte. The central components of the model are the dynamics of a viscous cytoskeleton capable to produce contractile and swelling stresses, and the formation of adhesive bonds in the plasma cell membrane between the lamella cytoskeleton and adhesion sites at the substratum. We will demonstrate that a simple mechanistic model, neglecting the complicated signaling pathways and regulation processes of a living cell, is able to capture the most prominent aspects of the lamella dynamics, such as quasi-periodic protrusions and retractions of the moving tip, retrograde flow of the cytoskeleton and the related accumulation of focal adhesion complexes in the leading edge of a migrating cell. The developed modeling framework consists of a nonlinearly coupled system of hyperbolic, parabolic and ordinary differential equations for the various molecular concentrations, two elliptic equations for cytoskeleton velocity and hydrodynamic pressure in a highly viscous two-phase flow, with appropriate boundary conditions including equalities and inequalities at the moving boundary. In order to analyse this hybrid continuum model by numerical simulations for different biophysical scenarios, we use suitable finite element and finite volume schemes on a fixed triangulation in combination with an adaptive level set method describing the free boundary dynamics.


Cell migration Adhesion kinetics Keratinocyte Continuum model Free boundary dynamics Level set method 

Mathematics Subject Classification (2000)


Supplementary material

285_2008_179_MOESM1_ESM.mpg (42.6 mb)
Movie 1 (related to Fig. 6 of the article): F-actin pattern formation. F-actin pattern formation starting from a constant concentration in a stationary domain. Coloring indicates the volume fraction of F-actin (red: high F-actin concentration, blue: low). Arrows represent the velocity of the F-actin field. ESM 1 (MPG 43,588 kb)
285_2008_179_MOESM2_ESM.mpg (20.9 mb)
Movie 2 (related to Fig. 7 of the article): Polarization. Polarization of an initially symmetric cell fragment. Coloring indicates the density of actin-and-surface bound integrins (red: high integrin density, blue: low). Arrows represent the velocity of the F-actin field. ESM 2 (MPG 21,385 kb)


  1. 1.
    Adalsteinsson D., Sethian J.: The fast construction of extension velocities in level set methods. J. Comput. Phys. 148(1), 2–22 (1999)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alt W.: Biomechanics of actomyosin-dependent mobility of keratinocytes. Biophysics 41(1), 181 (1996)MathSciNetGoogle Scholar
  3. 3.
    Alt W., Dembo M.: Cytoplasm dynamics and cell motion: two-phase flow models. Math. Biosci. 156, 207–228 (1999)MATHCrossRefGoogle Scholar
  4. 4.
    Alt W., Tranquillo R.T.: Protrusion-retraction dynamics of an annular lamellipodial seam. In: Alt, W. et al. (eds) Dynamics of Cell and Tissue Motion, pp. 73–81. Birkhäuser, Basel (1997)Google Scholar
  5. 5.
    Balaban N., Schwarz U., Riveline D., Goichberg P., Tzur G., Sabanay I., Mahalu D., Safran S., Bershadsky A., Addadi L., Geiger B.: Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates. Nature Cell Biol. 3, 466–472 (2001)CrossRefGoogle Scholar
  6. 6.
    Beningo K., Dembo M., Kaverina I., Small J., Wang Y.: Nascent focal adhesions are responsible for the generation of strong propulsive forces in migrating fibroblasts. J. Cell Biol. 153(4), 881–888 (2001)CrossRefGoogle Scholar
  7. 7.
    Bertolazzi E., Manzini G.: A cell-centered second-order accurate finite volume method for convection-diffusion problems on unstructured meshes. Math. Models Methods Appl. Sci 14(8), 1235–1260 (2004)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Borm B., Requardt R., Herzog V., Kirfel G.: Membrane ruffles in cell migration: indicators of inefficient lamellipodia adhesion and compartments of actin filament reorganization. Exp. Cell Res. 302(1), 83–95 (2005)CrossRefGoogle Scholar
  9. 9.
    Bottino D., Mogilner A., Roberts T., Stewart M., Oster G.: How nematode sperm crawl. J. Cell Sci. 115(2), 367–384 (2002)Google Scholar
  10. 10.
    Brenner S., Scott L.: The Mathematical Theory of Finite Element Methods. Springer, Heidelberg (2002)MATHGoogle Scholar
  11. 11.
    Bretschneider T., Diez S., Anderson K., Heuser J., Clarke M., Müller-Taubenberger A., Köhler J., Gerisch G.: Dynamic actin patterns and Arp2/3 assembly at the substrate-attached surface of motile cells. Curr. Biol. 14(1), 1–10 (2004)CrossRefGoogle Scholar
  12. 12.
    Bruinsma R.: Theory of force regulation by nascent adhesion sites. Biophys. J. 89(1), 87–94 (2005)CrossRefGoogle Scholar
  13. 13.
    Burridge K., Chrzanowska-Wodnicka M., Zhong C.: Focal adhesion assembly. Trends Cell Biol. 7(9), 342–347 (1997)CrossRefGoogle Scholar
  14. 14.
    Chung C., Funamoto S., Firtel R.: Signaling pathways controlling cell polarity and chemotaxis. Trends Biochem. Sci. 26(9), 557–566 (2001)CrossRefGoogle Scholar
  15. 15.
    Dembo M.: Mechanics and control of the cytoskeleton in Amoeba proteus. Biophys. J. 55(6), 1053–1080 (1989)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Frank D., Carter W.: Laminin 5 deposition regulates keratinocyte polarization and persistent migration. J. Cell Sci. 117(8), 1351–1363 (2004)CrossRefGoogle Scholar
  17. 17.
    Gracheva M., Othmer H.: A continuum model of motility in ameboid cells. Bull. Math. Biol. 66(1), 167–193 (2004)CrossRefMathSciNetGoogle Scholar
  18. 18.
    He X., Dembo M.: On the mechanics of the first cleavage division of the sea urchin egg. Exp. Cell Res. 233(2), 252–273 (1997)CrossRefGoogle Scholar
  19. 19.
    Herant M., Marganski W., Dembo M.: The mechanics of neutrophils: synthetic modeling of three experiments. Biophys. J. 84(5), 3389–3413 (2003)CrossRefGoogle Scholar
  20. 20.
    Hinz B., Alt W., Johnen C., Herzog V., Kaiser H.: Quantifying lamella dynamics of cultured cells by SACED, a new computer-assisted motion analysis. Exp. Cell Res. 251, 234–243 (1999)CrossRefGoogle Scholar
  21. 21.
    Hood P., Taylor C.: A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73–100 (1973)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hou L., Ravindran S.: A penalized neumann control approach for solving an optimal Dirichlet control problem for the Navier-Stokes equations. SIAM J. Control Optim. 36(5), 1795–1814 (1998)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Jiang G., Giannone G., Critchley D., Fukumoto E.: Two-piconewton slip bond between fibronectin and the cytoskeleton depends on talin. Nature 424, 334–337 (2003)CrossRefGoogle Scholar
  24. 24.
    Jiang G., Peng D.: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (1999)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Landau L., Lifshitz E.: Fluid Mechanics, 2nd edn. Butterworth Heinemann, London (1999)Google Scholar
  26. 26.
    Lauffenburger D., Horwitz A.: Cell migration: a physically integrated molecular process. Cell 84(3), 359–69 (1996)CrossRefGoogle Scholar
  27. 27.
    Lo C., Wang H., Dembo M., Wang Y.: Cell movement is guided by the rigidity of the substrate. Biophys. J. 79(1), 144–152 (2000)CrossRefGoogle Scholar
  28. 28.
    Machacek M., Danuser G.: Morphodynamic profiling of protrusion phenotypes. Biophys. J. 90(4), 1439–1452 (2006)CrossRefGoogle Scholar
  29. 29.
    Mitchison T., Cramer L.: Actin-based cell motility and cell locomotion. Cell 84(3), 371–9 (1996)CrossRefGoogle Scholar
  30. 30.
    Mittal R., Iaccarino G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37(1), 239–261 (2005)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Mogilner A., Edelstein-Keshet L.: Regulation of actin dynamics in rapidly moving cells: a quantitative analysis. Biophys. J. 83(3), 1237–1258 (2002)CrossRefGoogle Scholar
  32. 32.
    Mogilner A., Oster G.: Cell motility driven by actin polymerization. Biophys. J. 71(6), 3030–3045 (1996)CrossRefGoogle Scholar
  33. 33.
    Mogilner A., Oster G.: Polymer motors: pushing out the front and pulling up the back. Current Biol. 13(18), 721–733 (2003)CrossRefGoogle Scholar
  34. 34.
    Mogilner A., Verzi D.: A simple 1-D physical model for the crawling nematode sperm cell. J. Stat. Phys. 110(3), 1169–1189 (2003)MATHCrossRefGoogle Scholar
  35. 35.
    Möhl, C.: Modellierung von Adhäsions- und Cytoskelett-Dynamik in Lamellipodien migratorischer Zellen. Diploma thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2005)Google Scholar
  36. 36.
    Oliver J., King J., McKinlay K., Brown P., Grant D., Scotchford C., Wood J.: Thin-film theories for two-phase reactive flow models of active cell motion. Math. Med. Biol. 22(1), 53 (2005)MATHCrossRefGoogle Scholar
  37. 37.
    Osher S., Shu C.: High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal. 28(4), 907–922 (1991)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Palecek S.P., Horwitz A.F., Lauffenburger D.A.: Kinetic model for integrin-mediated adhesion release during cell migration. Ann. Biomed. Eng. 27(2), 219–235 (1999)CrossRefGoogle Scholar
  39. 39.
    Parent C.: A cell’s sense of direction. Science 284(5415), 765–770 (1999)CrossRefGoogle Scholar
  40. 40.
    Parsons J.: Focal adhesion kinase: the first ten years. J. Cell Sci. 116, 1409–1416 (2003)CrossRefGoogle Scholar
  41. 41.
    Plow E., Haas T., Zhang L., Loftus J., Smith J.: Ligand binding to integrins. J. Biol. Chem. 275(29), 21785–21788 (2000)CrossRefGoogle Scholar
  42. 42.
    Pollard T., Borisy G.: Cellular motility driven by assembly and disassembly of actin filaments. Cell 112(4), 453–465 (2003)CrossRefGoogle Scholar
  43. 43.
    Ponti A., Machacek M., Gupton S., Waterman-Storer C., Danuser G.: Two distinct actin networks drive the protrusion of migrating cells. Science 305(5691), 1782–6 (2004)CrossRefGoogle Scholar
  44. 44.
    Rubinstein B., Jacobson K., Mogilner A.: Multiscale two-dimensional modeling of a motile simple-shaped cell. Multiscale Model. Simul. 3, 413 (2005)MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Seifert U.: Rupture of multiple parallel molecular bonds under dynamic loading. Phys. Rev. Lett. 84(12), 2750–2753 (2000)CrossRefGoogle Scholar
  46. 46.
    Sethian J.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. USA 93(4), 1591–5 (1996)MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Sethian J.: Level Set Methods. Cambridge University Press, Cambridge (1996)MATHGoogle Scholar
  48. 48.
    Small J., Herzog M., Anderson K.: Actin filament organization in the fish keratocyte lamellipodium. J. Cell Biol. 129(5), 1275–1286 (1995)CrossRefGoogle Scholar
  49. 49.
    Small J., Stradal T., Vignal E., Rottner K.: The lamellipodium: where motility begins. Trends Cell Biol. 12(3), 112–120 (2002)CrossRefGoogle Scholar
  50. 50.
    Thoumine O., Meister J.J.: A probabilistic model for ligand–cytoskeleton transmembrane adhesion: predicting the behavior of microspheres on the surface of migrating cells. J. Theor. Biol. 204(3), 381–392 (2000)CrossRefGoogle Scholar
  51. 51.
    Turner C.: Paxillin and focal adhesion signalling. Nature Cell Biol. 2, E231–E236 (2000)CrossRefGoogle Scholar
  52. 52.
    Verkhovsky A., Svitkina T., Borisy G.: Self-polarization and directional motility of cytoplasm. Curr. Biol. 9(1), 11–20 (1999)CrossRefGoogle Scholar
  53. 53.
    Xian W., Tang J., Janmey P., Braunlin W.: The polyelectrolyte behavior of actin filaments: a 25Mg NMR study. Biochemistry 38(22), 7219–7226 (1999)CrossRefGoogle Scholar
  54. 54.
    Ye T., Mittal R., Udaykumar H., Shyy W.: An accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries. J. Comput. Phys. 156(2), 209–240 (1999)MATHCrossRefGoogle Scholar
  55. 55.
    Zhu C., Skalak R.: A continuum model of protrusion of pseudopod in leukocytes. Biophys. J. 54(6), 1115–1137 (1988)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Engineering PhysicsHelsinki University of TechnologyEspooFinland
  2. 2.Universität Bonn, Abteilung Theoretische BiologieBonnGermany

Personalised recommendations