The dynamic geometry of mass cell movements in animal morphogenesis.

There is an infinite number of interactions between morphogenetic processes of different time and space scales. How do these unfold in a regular series of mass morphogenetic movements to produce a basically simple and reproducible structure? I present a new morphogenetic concept -- the spatial unfolding (SU) of cell movements, whose definition rests on the correspondence between the continuous spatial series of cell shapes and the succession of changes in the shape of a single cell moving in an epithelial sheet whose shape is also subject to change. The change in the shape of moving cells is the only measure of their translocation both in space and time. The SU provides a morpho-dynamics description of mass cell movements which is completely independent of both an external coordinate system and external forces. The cell geometry of SU allows us to derive the future embryonic form from the actual one by a movement-shaping algorithm operating on the basis of positive and negative geometric feedbacks between the cell movement in the epithelial sheet plane and the epithelial sheet shaping, the feedback system providing a geometric alternative to Turing's self-organization via reaction-diffusion systems. Putting together histological, quantitative morphological and experimental data permits us to isolate four SU, each acting in morphogenesis as an irreducible whole, which seem to include all real examples of epithelial morphogenesis in multicellular animals, from Coelenterates to Chordates.