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Bulletin of Mathematical Biology

, Volume 55, Issue 3, pp 585–608 | Cite as

Langevin equation, Fokker-Planck equation and cell migration

  • M. Schienbein
  • H. Gruler
Article

Abstract

Cell migration can be characterized by two independent variables: the speed,v, and the migration angle, ϕ. Each variable can be described by a stochastic differential equation—a Langevin equation. The migration behaviour of an ensemble of cells can be predicted due to the stochastic processes involved in the signal transduction/response system of each cell. Distribution functions, correlation functions, etc. are determined by using the corresponding Fokker-Planck equation. The model assumptions are verified by experimental results. The theoretical predictions are mainly compared with the galvanotactic response of human granulocytes. The coefficient characterizing the mean effect of the signal transduction/response system of the cell is experimentally determined to 0.08 mm/V sec (galvanotaxis) or 0.7 mm/sec (chemotaxis) and the characteristic time characterizing stochastic effects in the signal transduction/response system is experimentally determined as 30 sec. The temporal directed response induced by electric field pulses is investigated: the experimental cells react slower but are more sensitive than predicted by theory.

Keywords

Autocorrelation Function Directed Movement Stochastic Differential Equation Applied Electric Field Pulse Electric Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature

  1. Alt, W. 1990. Correlation analysis of two-dimensional locomotion paths. InLecture Notes in Biomathematics. Biological Motion, W. Alt and G. Hoffmann (eds), pp. 254–268. Berlin, Heidelberg: Springer Verlag.Google Scholar
  2. de Boisfleury-Chevance, A., B. Rapp and H. Gruler. 1989. Locomotion of white blood cells: A biophysical analysis.Blood Cells 15, 315–333.Google Scholar
  3. Dunn, G. A. and A. F. Brown. 1987. A unified approach to characterizing cell motility.J. Cell Sci. Suppl. 8, 81–102.Google Scholar
  4. Franke, K. and H. Gruler. 1990. Galvanotaxis of human granulocytes: electric field jump studies.Eur. Biophys. J. 18, 335–346.CrossRefGoogle Scholar
  5. Franke, K. and H. Gruler. 1992. Directed cell movement in pulsed electric fields. To be published.Google Scholar
  6. Gruler, H. 1984. Cell movement analysis in a necrotactic assay.Blood Cells 10, 107–121.Google Scholar
  7. Gruler, H. and B. Bültmann. 1984. Analysis of cell movement.Blood Cells 10, 61–77.Google Scholar
  8. Gruler, H. and Nuccitelli. 1986. New insights into galvanotaxis and other directed cell movements. InIonic Currents in Development, R. Nuccitelli (ed.), pp. 337–347. New York: A. R. Liss Inc.Google Scholar
  9. Gruler H. and A. de Boisfleury-Chevance. 1987. Chemokinesis and necrotaxis of human granulocytes: the important cellular organelles.Z. Naturforsch. 42c, 1126–1134.Google Scholar
  10. Gruler, H. 1988. Cell movement and symmetry of the cellular environment.Z. Naturforsch. 43c, 754–764.Google Scholar
  11. Gruler, H. 1989. Biophysics of leukocytes: neutrophil chemotaxis, characteristics, and mechanisms. InThe Neutrophil: Cellular Biochemistry and Physiology, I. Hellett (ed.), pp. 63–95. Boca Raton, Florida: CRC Press Inc.Google Scholar
  12. Gruler, H. 1990. Chemokinesis, chemotaxis and galvanotaxis. Dose-response curves and signal chains. InLecture Notes in Biomathematics. Biological Motion, W. Alt and G. Hoffmann (eds), pp. 396–414. Berlin, Heidelberg: Springer-Verlag.Google Scholar
  13. Gruler, H. and K. Franke. 1990 Automatic control and directed movement.Z. Naturforsch. 45c, 1241–1249.Google Scholar
  14. Gruler, H. and N. A. R. Gow. 1990. Directed growth of fungal hyphae in an electric field.Z. Naturforsch. 45c 306–313.Google Scholar
  15. Gruler, H. and R. Nuccitelli. 1991. Neural crest cell galvanotaxis: new data and novel approach to the analysis of both galvanotaxis and chemotaxis.Cell Mot. Cytoskel. 19, 121–133.CrossRefGoogle Scholar
  16. Rapp, B., A. de Boisfleury-Chevance and H. Gruler. 1988. Galvanotaxis of human granulocytes. Dose-response curve.Eur. Biophys. J. 16, 313–319.CrossRefGoogle Scholar
  17. Risken, H. 1984.The Fokker-Planck Equation. Heidelberg Springer Verlag.Google Scholar
  18. Scharstein, H. and W. Alt. 1990. The influence of discrete position measurements on the correlation analysis of 2-dimensional tracks. InLecture Notes in Biomathematics. Biological Motion, W. Alt and G. Hoffmann (eds), pp. 278–280. Berlin, Heidelberg: Springer Verlag.Google Scholar
  19. Stokes, C. L., D. A. Lauffenburger and S. K. Williams. 1990. Endothelial cell chemotaxis in angiogenesis. InLecture Notes in Biomathematics. Biological Motion, W. Alt and G. Hoffmann (eds), pp. 442–452. Berlin, Heidelberg: Springer Verlag.Google Scholar
  20. Tranquillo, R. T. and D. A. Lauffenburger. 1987. Stochastic model of leukocytes chemosensory movement.J. math. Biol. 25, 229–262.MATHMathSciNetCrossRefGoogle Scholar
  21. Tranquillo, R. T., S. H. Zigmond and D. A. Lauffenburger. 1988. Measurement of the chemotaxis coefficient for human neutrophils in the under-agarose migration assay.Cell Mot. Cytoskel. 11, 1–15.CrossRefGoogle Scholar
  22. Trinkaus, J. P. 1984.Cells into Organs. Englewood Cliffs, New Jersey: Prentice Hall Inc.Google Scholar
  23. Wiener, N. 1961.Cybernetics: Or Control and Communication in Animal and the Machine. Cambridge: MIT Press.Google Scholar
  24. Wilkinson, P. C. 1982.Chemotaxis and Inflammation. London: J. & A. Churchill.Google Scholar
  25. Zigmond, S. H. 1977. Ability of polymorphonuclear leukocytes to orient in gradients of chemotactic factors.J. cell Biol. 75, 606–616.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1993

Authors and Affiliations

  • M. Schienbein
    • 1
  • H. Gruler
    • 1
  1. 1.Abteilung für BiophysikUniversität UlmUlmGermany

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