Skip to main content

A semi-stochastic cell-based formalism to model the dynamics of migration of cells in colonies

Abstract

We consider the movement and viability of individual cells in cell colonies. Cell movement is assumed to take place as a result of sensing the strain energy density as a mechanical stimulus. The model is based on tracking the displacement and viability of each individual cell in a cell colony. Several applications are shown, such as the dynamics of filling a gap within a fibroblast colony and the invasion of a cell colony. Though based on simple principles, the model is qualitatively validated by experiments on living fibroblasts on a flat substrate.

References

  1. Boussinesq J (1885) Application des potentiels á l’ étude de l’ équilibre et du mouvementdes solides élastiques. Gauthier-Villars, Paris

    Google Scholar 

  2. Burmister D (1945) The general theory of stresses and displacements in layered systems I. J Appl Phys 16: 89–94

    Article  Google Scholar 

  3. Burmister D (1945) The general theory of stresses and displacements in layered soil systems II. J Appl Phys 16: 126–127

    Article  Google Scholar 

  4. Burmister D (1945) The general theory of stresses and displacements in layered soil systems III. J Appl Phys 16: 296–302

    Article  Google Scholar 

  5. Califano JP, Reinhart-King CA (2010) Substrate stiffness and cell area predict cellular traction stresses in single cells and cells in contact. Cell Mol Bioeng 3(1): 68–75

    Article  Google Scholar 

  6. Dallon JC, Ehrlich HP (2008) A review of fibroblast populated collagen lattices. Wound Repair Regen 16: 472–479

    Article  Google Scholar 

  7. Dallon JC (2010) Multiscale modeling of cellular systems in biology. Curr Opin Coll Interface Sci 15: 24–31

    Article  Google Scholar 

  8. Gaffney EA, Pugh K, Maini PK (2002) Investigating a simple model for cutaneous wound healing angiogenesis. J Math Biol 45(4): 337–374

    Article  MATH  MathSciNet  Google Scholar 

  9. Gefen A (2010) Effects of virus size and cell stiffness on forces, work and pressures driving membrane invagination in a receptor-mediated endocytosis. J Biomech Eng 132: 084501–1-084501-5 (To appear)

  10. Graner F, Glazier J (1992) Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys Rev Lett 69: 2013–2016

    Article  Google Scholar 

  11. Haga H, Irahara C, Kobayashi R, Nakagaki T, Kawabata K (2005) Collective movement of epithelial cells on a collagen gel substrate. Biophys J 88(3): 2250–2256

    Article  Google Scholar 

  12. Hoehme S, Drasdo D (2010) A cell-based simulation software for multi-cellular systems. Bioinformatics 26(20): 2641–2642

    Article  Google Scholar 

  13. Javierre E, Vermolen FJ, Vuik C, van der Zwaag S (2009) A mathematical analysis of physiological and morphological aspects of wound closure. J Math Biol 59: 605–630

    Article  MATH  MathSciNet  Google Scholar 

  14. Johnson KL (1985) Contact mechanics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  15. Lemmon CA, Chen CS, Romer LH (2009) Cell traction forces direct fibronectin matrix assembly. Biophys J 96: 729–738

    Article  Google Scholar 

  16. Lo CM, Wang HB, Dembo M, Wang YL (2000) Cell movement is guided by the rigidity of the substrate. Biophys J 79(1): 144–152

    Article  Google Scholar 

  17. Luding S (2008) Introduction to discrete element methods: basics of contact force models and how to perform the micro-macro transition to continuum theory. Eur J Environ Civil Eng 12(7–8(Special Issue: Alert Course, Aussois)): 785–826

    Google Scholar 

  18. Maggelakis SA (2004) Modeling the role of angiogenesis in epidermal wound healing. Discret. Cont. Syst. 4: 267–273

    Article  MATH  MathSciNet  Google Scholar 

  19. Merks MH, Koolwijk P (2009) Modeling morphogenesis in silico and in vitro: towards quantitative, predictive, cell-based modeling. Math Model Nat Phenom 4(4): 149–171

    Article  MATH  MathSciNet  Google Scholar 

  20. Merkel R, Kirchgesner N, Cesa CM, Hoffmann B (2007) Cell force microscopy on elastic layers of finite thickness. Biophys J 93: 3314–3323

    Article  Google Scholar 

  21. Murray JD (2004) Mathematical biology II: spatial models and biomedical applications. Springer, New York

    Google Scholar 

  22. Olsen L, Sherratt JA, Maini PK (1995) A mechanochemical model for adult dermal wound closure and the permanence of the contracted tissue displacement role. J Theor Biol 177: 113–128

    Article  Google Scholar 

  23. Plank MJ, Sleeman BD (2004) Lattice and non-lattice models of tumour angiogenesis. Bull Math Biol 66: 1785–1819

    Article  MathSciNet  Google Scholar 

  24. Reinhart-King CA, Dembo M, Hammer DA (2008) Cell-cell mechanical communication through compliant substrates. Biophys J 95: 6044–6051

    Article  Google Scholar 

  25. Sarvestani AS (2010) On the effect of substrate compliance on cellular mobility. J Biochip Tissue Chip 1: 101. doi:10.4172/2153-0777.1000101

    Google Scholar 

  26. Schugart RC, Friedman A, Zhao R, Sen CK (2008) Wound angiogenesis as a function of tissue oxygen tension: a mathematical model. Proc Nat Acad Sci USA 105(7): 2628–2633

    Article  Google Scholar 

  27. Schwarz US, Bischofs IB (2005) Physical determinants of cell organization in soft media. Med Eng Phys 27: 763–772

    Article  Google Scholar 

  28. Sherratt JA, Murray JD (1991) Mathematical analysis of a basic model for epidermal wound healing. J Math Biol 29: 389–404

    Article  MATH  Google Scholar 

  29. Vermolen FJ (2009) A simplified finite-element model for tissue regeneration with angiogenesis. ASCE J Eng Mech 135(5): 450–461

    Article  MathSciNet  Google Scholar 

  30. Vermolen FJ, Javierre E (2009) On the construction of analytic solutions for a diffusion-reaction equation with a discontinuous switch mechanism. J Comput Appl Math 231: 983–1003

    Article  MATH  MathSciNet  Google Scholar 

  31. Wang JHC, Lin J-S (2007) Cell traction force and measurement methods. Biomech Model Mechanobiol 6: 361–371

    Article  Google Scholar 

  32. Xue C, Friedman A, Sen CK (2009) A mathematical model of ischemic cutaneous wounds. Proc Nat Acad Sci USA 106(39): 16783–16787

    Article  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Affiliations

Authors

Corresponding author

Correspondence to F. J. Vermolen.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Vermolen, F.J., Gefen, A. A semi-stochastic cell-based formalism to model the dynamics of migration of cells in colonies. Biomech Model Mechanobiol 11, 183–195 (2012). https://doi.org/10.1007/s10237-011-0302-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10237-011-0302-6

Keywords

  • Cell migration
  • Cell-based model
  • Semi-stochastic model