The development of a higher organism is controlled by a complex network of biochemical reactions. We have developed models for different developmental situations. Many of these models have found meanwhile direct support by molecular-genetic experiments. By ON-LINE computer simulations it will be shown that the models describe many regulatory phenomena as they have been experimentally observed. The following processes are proposed to play a key role: (i) Primary pattern formation is accomplished by autocatalysis and long range inhibition. Graded, periodic or stripe-like distributions can be generated. (ii) Cells obtain a stable state of differentiation by direct or indirect autoregulation of genes accompanied by a mutual competition among alternative genes. In this way, only one of several alternative genes can remain active within a particular cell. The robustness of development is proposed to result from the self-regulating generation of pattern (i) combined with the self-regulating response of the cells toward these signals. (iii) By mutual long range stabilization of cell states, a controlled neighbourhood of structures can be achieved. Insect segmentation is proposed to result by a cyclic mutual activation of such locally self-stabilizing cell states. (iv) Boundaries between regions generated by these mechanisms can obtain organizing properties for the finer subdivision of an organism. Substructures such as eyes, legs or wings are proposed to be initiated around the intersection of two borders. This mechanism accounts for the pair-wise initiation of these structures at the correct position and with the correct handedness. Pattern on the shells of tropical mollusks are especially instructive since they are generated by a one-dimensional pattern forming process and bear, in the second dimension, a complete historical record. Since they are free of functional constraints, nature could vary these patterns without endangering the species. These pattern show unusual behaviour, for instance, travelling waves that can penetrate each other without annihilation and waves that form spontaneously backwards running waves. By ON-LINE computer simulations it will be shown that the models account for these patterns in fine details.

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1998-1999 Mathematics in Biology