Skip to main content

Analysis of dynamic and stationary pattern formation in the cell cortex

Journal of Mathematical Biology volume 31, pages 25–71 (1992)Cite this article

Abstract

We study a sol-gel mechanochemical model for cellular cytoplasm. Using conservation equations and a force balance equation, we derive equations for the sol-gel dynamics. Regular perturbation analysis suggests the growth of patterns which may be either dynamic or stationary, depending on parameter values. Nonlinear analysis, which indicates that these patterns remain bounded, is confirmed by numerically solving the mechanochemical equations. We use these analytical and numerical results to model two different biological problems: the dynamic formation of filopodia in nerve growth cones, and the growth of microvilli in epithelial cells.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

References

  • Abercrombie, M., Heaysman, J. E. M., Pegrum, S. M.: The locomotion of fibroblasts in culture. II. ‘Ruffling’. Exp. Cell Res. 60, 437–444 (1970)

    Google Scholar 

  • Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., Watson, J. D.: Molecular Biology of the Cell. London: Garland 1983

    Google Scholar 

  • Argiro, V., Bunge, M. B., Johnson, M. L.: A quantitative study of growth cone filopodial extension. J. Neurosci. Res. 13, 149–162 (1985)

    Google Scholar 

  • Bray, D., White, J. G.: Cortical flow in animal cells. Science 239, 883–888 (1988)

    Google Scholar 

  • Cohan, C. S., Connor, J. A., Kater, S. B.: Electrically and chemically mediated increases in intracellular calcium in neuronal growth cones. J. Neurosci. 7 (11), 3588–3599 (1987)

    Google Scholar 

  • Condeelis, J.: Rheological properties of cytoplasm: significance for the organization of spatial information and movement. In: McIntosh, J. (ed.) Modern Cell Biology, vol. 2: Spatial Organization of Eucaryotic Cells, pp. 225–240. New York: Alan R. Liss 1983

    Google Scholar 

  • Connor, J. A.: Digital imaging of free calcium changes and of spatial gradients in growing processes in single, mammalian central nervous system cells. Proc. Natl. Acad. Sci. USA 83, 6179–6183 (1986)

    Google Scholar 

  • Forscher, P.: Calcium and polyphosphinositide control of cytoskeletal dynamics. Trends Neurosci. 12 (11), 468–474 (1989)

    Google Scholar 

  • Forscher, P., Smith, S. J.: Actions of cytochalasins on the organization of actin filaments and microtubules in a neuronal growth cone. J. Cell Biol. 107, 1505–1516 (1988)

    Google Scholar 

  • Gordon-Weeks, P. R.: GAP-43 — What does it do in the growth cone? Trends Neurosci. 12 (10), 363–365 (1989)

    Google Scholar 

  • Harris, A. K.: Cell surface movements related to cell locomotion. (Ciba Found. Symp., New Ser. 14, pp. 3–26) Amsterdam: Associated Scientific Publishers 1973

    Google Scholar 

  • Hudspeth, A. J.: The hair cells of the inner ear. Sci. Am. 248, 42–52 (1983)

    Google Scholar 

  • Kater, S. B., Mattson, M. P., Cohan, C., Connor, J.: Calcium regulation of the neuronal growth cone. Trends Neurosci. 11 (7), 315–321 (1988)

    Google Scholar 

  • Landau, L. D., Lifshitz, E. M.: Theory of Elasticity, 2nd ed. London: Pergamon 1970

    Google Scholar 

  • Lewis, M. A.: Analysis of Dynamic and Stationary Biological Pattern Formation. D.Phil. Thesis, University of Oxford (1990)

  • Lewis, M. A., Murray, J. D.: Analysis of stable two-dimensional patterns in contractile cytogel. J. Nonlin. Sci. 1, 289–311 (1991)

    Google Scholar 

  • Lur'e, A. I.: Three-Dimensional Problems of the Theory of Elasticity, chap. 1. New York: John Wiley 1964

    Google Scholar 

  • Matkowski, B. J.: Nonlinear dynamic stability. SIAM J. Appl. Math. 18, 872–883 (1970)

    Google Scholar 

  • Murray, J. D.: Parameter space for Turing instability in reaction diffusion mechanisms: A comparison of models. J. Theor. Biol. 99, 161–199 (1982)

    Google Scholar 

  • Murray, J. D.: Mathematical Biology. Berlin Heidelberg New York: Springer 1989

    Google Scholar 

  • Odell, G. M.: A mathematically modelled cytogel cortex exhibits periodic Ca++-modulated contraction cycles seen in Physarum shuttle streaming. J. Embryol. Exp. Morphol. 83, 261–287 (1984)

    Google Scholar 

  • Oster, G. F.: On the crawling of cells. J. Embryol. Exp. Morphol. 83, 329–364 (1984)

    Google Scholar 

  • Oster, G. F., Murray, J. D., Harris, A. K.: Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morphol. 78, 83–125 (1983)

    Google Scholar 

  • Oster, G. F., Murray, J. D., Odell, G. M.: The formation of microvilli. In: Molecular Determinants of Animal Form, pp. 365–384. New York: Alan R. Liss 1985

    Google Scholar 

  • Oster, G. F., Odell, G. M.: Mechanics of cytogels I: Oscillations in Physarum. Cell Motil. 4, 469–503 (1984a)

    Google Scholar 

  • Oster, G. F., Odell, G. M.: The mechanochemistry of cytogels. Physica 12D, 333–350 (1984b)

    Google Scholar 

  • Purcell, E.: Life at Low Reynolds Number. Am. J. Phys. 45, 1–11 (1977)

    Google Scholar 

  • Silver, R. A., Lamb, A. G., Bolsover, S. R.: Calcium hotspots caused by L-channel clustering promote morphological changes in neuronal growth cones. Nature 343, 751–754 (1990)

    Google Scholar 

  • Stossel, T. P.: The spatial organization of cortical cytoplasm in macrophages. In: McIntosh, J. (ed.) Modern Cell Biology, vol. 2: Spatial Organization of Eucaryotic Cells, pp. 203–223. New York: Alan R. Liss 1983

    Google Scholar 

  • Tanaka, T.: Gels. Sci. Am. 244 (1), 124–138 (1981)

    Google Scholar 

  • Taylor D., Hellewell, S., Virgin, H., Heiple, J.: The solation-contraction coupling hypothesis of cell movements. In: Hatano, S. (ed.) Cell Motility, pp. 363–377. Baltimore: University Park Press 1979

    Google Scholar 

  • Tilney, L. G.: The role of actin in nonmuscle cell motility. In Inoue, S., Stephens, R. E. (eds.) Molecules and Cell Movement, pp. 339–388. New York: Raven 1975

    Google Scholar 

  • Tilney, L. G., De Rosier, D. J.: Actin filaments, stereocilia and hair cells of the bird cochlea IV. How the actin filaments become organized in developing stereocilia and in the cuticular plate. Dev. Biol. 116, 119–129 (1986)

    Google Scholar 

  • Tilney, L. G., Inoue, S.: Acrosomal reaction of Thyrone sperm. II. The kinetics and possible mechanism of acrosomal process elongation. J. Cell Biol. 93, 820–827 (1982)

    Google Scholar 

  • Tosney, K. W., Wessells, N. K.: Neuronal motility: The ultrastructure of veils and microspikes correlates with their motile activities. J. Cell Sci. 61, 389–411 (1983)

    Google Scholar 

  • Weeds, A.: Actin-binding proteins — regulators of cell architecture and motility. Nature 296, 811–816 (1982)

    Google Scholar 

  • Zuber, M. X., Goodman, D. W., Karns, L. R., Fishman, M. C.: The neuronal growth-associated protein GAP-43 induces filopodia in non-neuronal cells. Science 244, 1193–1195 (1989)

    Google Scholar 

Download references

Author information

Author notes

  1. M. A. Lewis

    Present address: Department of Mathematics, University of Utah, 84112, Salt Lake City, UT, USA

Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, FS-20, 98195, Seattle, Washington, USA

    M. A. Lewis & J. D. Murray

  2. Department of Zoology, University of Washington, NJ-15, 98195, Seattle, Washington, USA

    M. A. Lewis

Authors
  1. M. A. Lewis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. D. Murray
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lewis, M.A., Murray, J.D. Analysis of dynamic and stationary pattern formation in the cell cortex. J. Math. Biol. 31, 25–71 (1992). https://doi.org/10.1007/BF00163842

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00163842

Key words

  • Cytoplasm
  • Mechanochemical
  • Sol-gel
  • Microvilli
  • Filopodia