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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.

The living organisms inhabit and develop in the real physical space and are organized according to the properties of this space. There are general principles in physics, geometry and topology certainly is applied to many situations in biology [

Topological analysis of DNA structure and functions showed the power of the topological methodology see [

The topological approach to description of biological forms and morphogenetic processes has become a more common practice covering the fields of evolutionary and developmental biology, animal behavior and self-or- ganization, neurology, metagenomic biology and biomedicine [

The topological methodology makes it possible to consider as a whole the succession of shape transformations during metazoan development and evolution. Topological approach reveals also a certain set of topological rules constraining and directing biological morphogenesis during evolution and development.

Because an animal body is formed from the spherical egg cell, symmetry breakdown is one of the fundamental processes of development. Symmetry breaking and symmetry propagation are fundamental processes in biological morphogenesis, in metazoan evolution and development [

In most metazoans, the animal-vegetal (anterior-posterior) axis of the egg and future embryo is established during oogenesis and usually depends on the position of oocyte with respect to its cellular and non-cellular surrounding. The first symmetry breakdown is associated with the emergence of morphological and functional gradients along animal-vegetal axis of an oocyte. It was shown that spatial anisotropy in distribution of gene products in ooplasm determines the polarity of egg and embryo specifying polar axes of future organism [

The axial polarity of an egg is manifested in transcellular ion flows, which generates the extracellular electric field as signals and effectors of the egg polarity [

Egg cell polarization as the emergence of the topological singularity (or singularities) of the vector field on a sphere is inevitable according to the Poincaré-Hopf theorem: “For any smooth vector field given on two-dimen- sional sphere S^{2}, a singular point of the field exists” [

In some animals, at least in chordates, the second symmetry breaking takes place following gamete contact and fusion; this symmetry breaking process involves the asymmetrical flow of cortical actin networks [

In a cleaving zygote, the spatial organization of blastomeres is a genetically determined feature of taxonomic value. The cleavage results in a pattern of cellular contacts on the surface of an embryo that is a discrete cell

field [

Gastrulation is the first spherical surgery in development, a topological catastrophe. The completion of gastrulation is topological transition from the sphere with singularity (singularities) to the torus, which is a topologically stable object with homogeneous discrete field, without singularities [

Gastrulation is an extensive rearrangement of cells in an embryo. In embryos, cell migration pathways during gastrulation and other morphogenetic processes [

In metazoan evolution, appearance of the epithelial tissue was one of the most significant innovations [

To translate traditional morphological descriptions of a metazoan morphogenesis into topological language the external shape of an organism is modeled as a set of smooth, closed, orientable surfaces formed by epithelial layers neglecting the thickness of the cellular layers [

Metazoan morphogenesis may be represented as topological modification(s) of epithelial surfaces of spherical or toroidal shape. According to the theorem of elementary topology, any closed surface in three-dimensional space is homeomorphic (topologically equivalent) to the sphere with a certain number (p) of handles [

Topological handles in biological objects are usually realized as channels (or canals, holes), for example, the digestive tube. We consider only epithelized through channels as topological handles. Repeated addition of “handles” or creation of “holes”, “tunnels” has led to formation of double, triple, and higher order toroids [

Topological transitions in metazoan development occur due to topological surgeries realized by “cutting” and “gluing” of epithelial sheets [

We can see two kinds of topological surgeries of epithelial surfaces in biological morphogenesis. Spherical surgeries change the genus of the surface. Another kind of topological surgeries changes the connectivity of epithelial layers by the separation of additional closed epithelial surfaces from preexisting ones or by the fusion of preexisting layers. The developmental modifications of connectivity do not change the topological pattern of the outer surface of animal body, so only the spherical surgeries changing the genus of the surface are considered here.

Topological patterns in metazoan evolution, from sponges to chordates, were analyzed using the genus of the surface as a topological invariant. We considered here (

In evolution of Metazoa we can find several topological transitions [

Subsequent evolution has led to the formation of the through channels of the coelomic and respiratory systems. The surface of an organism having another additional through channel besides the digestive tube is a surface of genus 2, which is topologically equivalent to a sphere with two handles, double torus or “pretzel”. For example, echinoderms exhibit the second through channel, namely the ambulacral system. The spatial organization of the circular ambulacral system connected with the external medium by a canal is topologically equivalent to a through channel [

The next evolutionary level of topological design in animals was attained through the development of the respiratory system. The system of paired tracheal tubules with lateral connections (of ectodermal origin) in Tracheata (higher terrestrial arthropods) or paired gill slits (of entodermal origin) in a chordate archetype evolved (

There are various evolutionary modifications of the archetypical body plan in many phyla due to additional topological surgeries. Colonial cnidarians are “multi-oral” organisms, i.e., multiple tori. In Spongia and in some Cnidaria, an increase of genus p up to very high values and the shaping of topologically complicated fractal systems are evident. In some scyphomedusae and creeping ctenophores branching and anastomosing canals of gastro-vascular system result in the increasing of genus p up to indefinite high values. In some flat worms (Platyhelminthes) complicated topological and fractal patterns have arisen on the base of the branched gut and excretory system [

Hemichordata have a through gut and several pairs of gill slits (p = 1 + 2n) similar to chordate archetype; in some Appendicularia, the digestive canal has two additional lateral openings (

Thus, topological transformations of the body surface during metazoan evolution resulted in better distribution of flows to and from the external medium, used as the source of nutrients and oxygen and the sink of excreta, so ensuring greater metabolic intensity and better adaptation of organism to the environment.

An analysis of topological organization of the closed epithelial layer that covers the outer body surface as a continuous envelope is applicable to the topological organization of the body surface in metazoan development [

No modification is observed in topology of the body surface during embryonic development of those animals which in adult state have the zero genus of the surface as some solitary cnidarians and Acoelomorpha [

The surfaces of an egg, blastula as well as early gastrula in metazoan animals are surfaces of the genus 0, homeomorphic to a sphere (

The surface of an organism having another additional through channel besides the digestive tube is the surface of genus 2, topologically equivalent to a sphere with two handles, double torus or “pretzel”. For example, sea urchins and other echinoderms exhibit a second through channel, formed by the ambulacral system (

During embryonic development of the Tracheata (among Arthropoda) and some representatives of Hemichordata, the following transformations of the genus of the body surface take place: p = 0 → p = 1 → p = 1 + 2n [

The detailed analysis of topological form dynamics in ontogenesis may provide additional possibilities for metazoan description and classification. For example, we have considered the different dynamics of numerical characteristics of four types of channels (gut, ambulacral system, hydropore, and pores through the madrepore plate) during embryogenesis, metamorphosis and in adult animals in different classes of Echinodermata [

Topological surgeries of another kind modify connectivity of germ layers and result in the separation of a set of additional closed epithelial surfaces from preexisting ones, for example, enterocoelic formation of mesoderm in Deuterostomia, neurulation and eye cap formation in Chordata and so on [

At cell level, topological approach was used to describe cell membrane system and membrane dynamics [

The spatial organization of a eukaryotic cell may be represented topologically as a number of inner closed membrane surfaces of cell organelles embedded inside the outer cell membrane. Following Y. Tashiro [

Topological singularities inevitably emerging and transforming in biological morphogenesis disrupt preexisting symmetry pattern. The symmetry of a metazoan body, including the scale symmetry of fractal structures, the translational symmetry of metamerism and other variants of morphofunctional iterations is an efficient means of

modular morphogenesis. Evolutionary developmental biology includes the concept of modularity, according to which ontogenesis consists of relatively independent processes, dissociable modules; structural elements of the developing organism may also be presented as discrete dissociable units. Dissociation of developmental modules and their recombination leads to evolutionary transformations with duplications and subsequent divergence, insertions, deletions and substitutions (see [

The architecture of gene regulatory networks is modular (see [

Symmetry breaking during oogenesis and early development is a fundamental event establishing the main axial coordinates of a future organism, while in later development the scale of symmetry breaking is decreasing. Topological rehandlings are regular processes in ontogeny and in phylogeny and involve the emergence of discontinuities [

Subsequent metazoan morphogenesis may be represented as a succession of spherical surgeries of closed orientable epithelial surfaces. The local topological surgery leads to the global topological modification of biological forms. 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 increases this interface, ensuring metabolic efficiency and better adaptation of organism to the environment.

The course and outcome of biological evolution is strongly influenced by constraints; evolution and development result in discontinuities and directionality of morphological transformations [

It was supposed that some biological forms, such as branching structures, are the most functional design; these forms “are topological attractors that evolution cannot avoid” [