A mechanical model for biological pattern formation: A nonlinear bifurcation analysis
We present a mechanical model for cell aggregation in embryonic development. The model is based on the large traction forces exerted by fibroblast cells which deform the extracellular matrix (ECM) on which they move. It is shown that the subsequent changes in the cell environment can combine to produce pattern. A linear analysis is carried out for this model. This reveals a wide spectrum of different types of dispersion relations. A non-linear bifurcation analysis is presented for a simple version of the field equations: a non-standard element is required. Biological applications are briefly discussed.
KeywordsDispersion Relation Homogeneous Steady State Uniform Steady State Biological Pattern Formation Chick Wing
Unable to display preview. Download preview PDF.
- Davidson, D. (1983). The mechanism of feather pattern development in the chick. 1. The time determination of feather position. J.Embryol.Exp. Morph., 74, 245–259.Google Scholar
- Meinhardt, H. (1982). Models of biological pattern formation. Academic Press, London.Google Scholar
- Murray, J.D. (1977). Lectures on non-linear differential equation models in biology. Oxford University Press: Oxford.Google Scholar
- Murray, J.D. & Oster, G.F. (1984). Generation of biological pattern and form. I.M.A. J. Maths. Appl. to Biol. & Med., 1Google Scholar
- Rawles, M. (1963). Tissue interactions in scale and feather development as studied in dermal-epidermal recombinations. J. Embryol. Exp. Morph., 11, 765–789.Google Scholar
- Saunders, J.W. & Gasseling, M.T. (1968). Ectodermal-mesenchymal interactions in the origin of limb symmetry. In epithelial-mesenchymal interactions (ed. R. Fleischmajer & R.E. Billingham), pp. 78–97.Google Scholar
- Smith, J.C. & Wolpert, L. (1981). Pattern formation along the anteroposterior axis of the chick wing: the increase in width following a polarizing region graft and the effect of X-irradiation. J. Embryol. Exp. Morph., 63, 127–144.Google Scholar
- Thomas, D. (1975). In analysis and control of immobilized enzyme systems, (Thomas, D. & Kernevez, J.P. eds.) New York: Springer-Verlag, pp. 115–150.Google Scholar
- Trinkaus, J.P. (1984). Cells into organs: The forces that shape the embryo. (Prentice-Hall).Google Scholar
- Wolpert, L. & Hornbruch, A. (1981). Positional signalling along the anteroposterior axis of the chick wing. The effect of multiple polarizing region grafts. J. Embryo. Exp. Morph., 63, 145–159.Google Scholar