Topology in Biology: Singularities and Surgery Transformations in Metazoan Development and Evolution

Abstract

The review presents a topological description and interpretation (analysis) of some events in metazoan development and evolution through the use of well-known mathematical concepts and theorems (using topological approach). It is the topological language that can provide strict and adequate description of various phenomena in developmental and evolutionary transformations. Topological singularities inevitably arising and transforming during early development destroy the preexisting pattern of symmetry. The symmetry breaking of preexisting spatial pattern plays a critical role in biological morphogenesis in development and evolution. Some events of early development are interpreted in terms of symmetry breakdown and related to well-known mathematical theorems. A topological inevitability of some developmental events through the use of classical topological concepts is discussed. The topological approach makes it possible to consider the succession of spherical surgeries, which change the topological genus of an animal body surface. We model the biological shape as a set of smooth, closed, oriented surfaces—membrane or epithelial layers. Membrane and epithelial surfaces are boundary layers, interfaces between a living structure and its environment, ensuring metabolism. Toroid forms as well as fractal structures in metazoans can be considered as functionally optimized biological design and attractors in biological morphogenesis. The epithelial surface is an interface between the internal medium of an organism and the outside environmental medium; topological and fractal transformations during metazoan evolution and development increase this interface, ensuring better adaptation of organism to the environment. Fractal structures as well as toroid forms can be considered as a functionally optimized design in Metazoa. Topological methodology reveals a certain set of topological rules constraining and directing biological morphogenesis during evolution and development.