Abstract
A topological inevitability of early developmental events through the use of classical topological concepts is discussed. Topological dynamics of forms and maps in embryo development are presented. Forms of a developing organism such as cell sets and closed surfaces are topological objects. Maps (or mathematical functions) are additional topological constructions in these objects and include polarization, singularities and curvature. Topological visualization allows us to analyze relationships that link local morphogenetic processes and integral devel- opmental structures and also to find stable spatio-temporal topological character- istics that are invariant for a taxonomic group. The application of topological principles reveals a topological imperative: certain topological rules define and direct embryogenesis. A topological stability of embryonic morphogenesis is pro- posed and a topological scenario of developmental and evolutionary transformations is presented.
Highlights
Local and Global OrderThe term topology was first introduced by Johann Listing (1848), who gave the following definition of topology: ‘‘By topology we mean the doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity’’
It is known that the distribution of certain sets of mRNAs in the egg ooplasm determines the spatial pattern of the future embryo (Nuccitelli 1984; Nusslein–Volhard 1991; Gilbert 2010); spatial anisotropy in distribution of gene products creates a morphogenetic field specifying the polarity of future organism
We believe that the topological approach is a powerful tool, which has not been used sufficiently in previous biological investigations
Summary
The term topology was first introduced by Johann Listing (1848), who gave the following definition of topology: ‘‘By topology we mean the doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity’’. Let X be a topological space homotopy equivalent to some cell complex (CW– complex). Topology of cell connections inside an embryo will be described The process of making a set of oriented epithelial cell layers (and transferring from 0 to 2 homotopy dimension) is called gastrulation It is known that the distribution of certain sets of mRNAs in the egg ooplasm determines the spatial pattern of the future embryo (Nuccitelli 1984; Nusslein–Volhard 1991; Gilbert 2010); spatial anisotropy in distribution of gene products creates a morphogenetic field (scalar concentration field and resulting vector polarity field) specifying the polarity of future organism Animal development is serial topological rearrangements of 2D closed surfaces (Sect. 10) This conceptual accent allows us to indicate topological simulations of onto- and phylogenesis (Sect. 11)
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