Acta Biotheoretica

, Volume 63, Issue 4, pp 381–395 | Cite as

Travelling Waves of Cell Differentiation

Regular Article

Abstract

The paper is devoted to modelling of cell differentiation in an initially homogeneous cell population. The mechanism which provides coexistence of two cell lineages in the initially homogeneous cell population is suggested. If cell differentiation is initiated locally in space in the population of undifferentiated cells, it can propagate as a travelling wave converting undifferentiated cells into differentiated ones. We suggest a model of this process which takes into account intracellular regulation, extracellular regulation and different cell types. They include undifferentiated cells and two types of differentiated cells. When a cell differentiates, its choice between two types of differentiated cells is determined by the concentrations of intracellular proteins. Differentiated cells can either stimulate differentiation into their own cell lineage or into another cell lineage. In the case of the positive feedback, only one lineage of differentiated cells will finally appear. In the case of negative feedback, both of them can coexist. In this case a periodic spatial pattern emerges behind the wave.

Keywords

Cell differentiation Lineage choice Reaction-diffusion waves Pattern formation 

Mathematics Subject Classification

92C15 35K57 

References

  1. Anderson ARA, Chaplain M, Rejniak KA (2007) Single cell based models in biology and medicine. Birkhäuser, BaselCrossRefGoogle Scholar
  2. Bernard S (2013) Modélisation multi-échelles en biologie. In: Le vivant discret et continu. N. Glade, A. Stephanou, Editeurs, Editions Materiologiques, pp 65-89Google Scholar
  3. Bessonov N, Crauste F, Demin I, Volpert V (2009) Dynamics of erythroid progenitors and erythroleukemia. Math Model Nat Phenom 4(3):210–232CrossRefGoogle Scholar
  4. Bessonov N, Crauste F, Fischer S, Kurbatova P, Volpert V (2011) Application of hybrid models to blood cell production in the bone marrow. Math Model Nat Phenom 6(7):2–12CrossRefGoogle Scholar
  5. Bessonov N, Kurbatova P, Volpert V (2012) Pattern formation in hybrid models of cell populations. In: Capasso V, Gromov M, Harel-Bellan A, Morozova N, Pritchard L (eds) Pattern formation in morphogenesis. Springer, Berlin, pp 107–119Google Scholar
  6. Bessonov N, Eymard N, Kurbatova P, Volpert V (2012) Mathematical modeling of erythropoiesis in vivo with multiple erythroblastic islands. Appl Math Lett 25:1217–1221CrossRefGoogle Scholar
  7. Bessonov N, Kurbatova P, Volpert V (2010) Particle dynamics modelling of cell populations. In: Proceedings of Conferences on JANO Mohhamadia. Math Model Nat Phenom 5(7):42–47Google Scholar
  8. Crauste F, Demin I, Gandrillon O, Volpert V (2010) Mathematical study of feedback control roles and relevance in stress erythropoiesis. J Theor Biol 263:303–316CrossRefGoogle Scholar
  9. Cristini V, Lowengrub J (2010) Multiscale modeling of cancer: an integrated experimental and mathematical modeling approach. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  10. Demin I, Crauste F, Gandrillon O, Volpert V (2010) A multi-scale model of erythropoiesis. J Biol Dyn 4:59–70CrossRefGoogle Scholar
  11. Deutsch A, Dormann S (2005) Cellular automaton modeling of biological pattern formation. Birkhäuser, BostonGoogle Scholar
  12. El Khatib N, Genieys S, Volpert V (2007) Atherosclerosis initiation modeled as an inflammatory process. Math Model Nat Phenom 2(2):126–141CrossRefGoogle Scholar
  13. El Khatib N, Genieys S, Kazmierczak B, Volpert V (2009) Mathematical modelling of atherosclerosis as an inflammatory disease Phil. Trans R Soc A 367:4877–4886CrossRefGoogle Scholar
  14. El Khatib N, Genieys S, Kazmierczak B, Volpert V (2012) Reaction-diffusion model of atherosclerosis development. J Math Biol 65(2):349–374CrossRefGoogle Scholar
  15. Eymard N, Bessonov N, Gandrillon O, Koury MJ, Volpert V (2014) The role of spatial organisation of cells in erythropoiesis. J Math Biol in pressGoogle Scholar
  16. Fischer S, Kurbatova P, Bessonov N, Gandrillon O, Volpert V, Crauste F (2012) Modelling erythroblastic islands : using a hybrid model to assess the function of central macrophage. J Theor Biol 298:92–106CrossRefGoogle Scholar
  17. Glade N, Stephanou A, Editeurs (2013) Le vivant discret et continu. Editions MateriologiquesGoogle Scholar
  18. Karttunen M, Vattulainen I, Lukkarinen A (2004) A novel methods in soft matter simulations. Springer, BerlinCrossRefGoogle Scholar
  19. Kurbatova P, Bernard S, Bessonov N, Crauste N, Demin I, Dumontet C, Fischer S, Volpert V (2011) Hybrid model of erythropoiesis and leukemia treatment with cytosine arabinoside. SIAM J Appl Math 71(6):2246–2268CrossRefGoogle Scholar
  20. Kurbatova P, Panasenko G, Volpert V (2012) Asymptotic numerical analysis of the diffusion-discrete absorption equation. Math Methods Appl Sci 35(4):438–444CrossRefGoogle Scholar
  21. Kurbatova P, Eymard N, Volpert V (2013) Hybrid model of erythropoiesis. Acta Biotheor 61(3):305–315CrossRefGoogle Scholar
  22. Osborne JM, Walter A, Kershaw SK, Mirams GR, Fletcher AG, Pathmanathan P, Gavaghan D, Jensen OE, Maini PK, Byrne HM (2010) A hybrid approach to multi-scale modelling of cancer. Phil Trans R Soc A 368:5013–5028CrossRefGoogle Scholar
  23. Patel AA, Gawlinsky ET, Lemieux SK, Gatenby RA (2001) A cellular automaton model of early tumor growth and invasion: the effects of native tissue vascularity and increased anaerobic tumor metabolism. J Theor Biol 213:315–331CrossRefGoogle Scholar
  24. Satoh A (2011) Introduction to practice of molecular simulation. Elsevier, AmsterdamGoogle Scholar
  25. Trewenack AJ, Landman KA (2009) A traveling wave model for invasion by precursor and differentiated cells. Bull Math Biol 71:291–317CrossRefGoogle Scholar
  26. Trofimov SY (2003) Thermodynamic consistency in dissipative particle dynamics. Eindhoven University Press, EindhovenGoogle Scholar
  27. Volpert V (2014) Elliptic partial differential equations, vol 2. Reaction-diffusion equations, BirkhäuserCrossRefGoogle Scholar
  28. Volpert V, Bessonov N, Eymard N, Tosenberger A (2013) Modèle multi-échelle de la dynamique cellulaire. In: Le vivant discret et continu. N. Glade, A. Stephanou, Editeurs, Editions Materiologiques, pp 91–111Google Scholar
  29. Volpert A, Volpert Vit, Volpert Vl (1994) Traveling wave solutions of parabolic systems. Translation of mathematical monographs, vol 140, Am Math Soc, ProvidenceGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • M. Benmir
    • 1
  • N. Bessonov
    • 2
  • S. Boujena
    • 1
  • V. Volpert
    • 3
  1. 1.Faculté des SciencesUniversity Hassan II of CasablancaCasablancaMaroc
  2. 2.Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSaint PetersburgRussia
  3. 3.Institut Camille Jordan, UMR 5208 CNRSUniversity Lyon 1VilleurbanneFrance

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