Topological Models in Developmental and Evolutionary Biology

1. Topological Models in Developmental and Evolutionary Biology Eugene Presnov and Valeria Isaeva Volcani Center, Israel; Far East State University and Institute of Marine Biology, Vladivostok, Russia

2. Influence and Attraction • In 1969 the brilliant French mathematician René Thom (Fields medal - 1958) published his famous article “Topological Models in Biology” in journal “Topology” . • Following, R.Thom, we applied formal topological concepts to the description of dynamics of forms in ontogeny and evolution.

3. Embryogenesis

4. Previous Biological Ideas

5. Classification of living organisms by topology of definitive forms Extension of this classification by topology of developmental forms Description of cellular mechanisms of topological transformations in development Heuristic view: Topological laws and topology constraints of development Aims

6. Short History(in Russian & in English)

7. Methodology • Epithelial cell layers are characterized by morphological and functional connectivity, closeness of the intact surface and apical-basal polarity, i.e. outer-inner anisotropy. • To translate anatomical descriptions into topological language the external shape of an organism is modeled by smooth closed surfaces. • Metazoan morphogenesis may be represented as topological modifications of the epithelial surfaces.Maresin & Presnov, 1985 - Journal Theoretical Biology; Jockusch & Dress, 2003 - Bulletin of Mathematical Biology.

8. I.1 Illustration of the Methodology on the example of sea urchin

9. I.2 Smooth deformation (ambulacral system in echinoderms)

10. I.3 Irrigative & Gastrovascular Systems(Isaeva, Presnov & Chernyshev, in preparation) Gastrovascular System i-l - CTENOPHORA: i,k -two openings (pores) l - multiple anastomoses Gastrovascular System d-h - CNIDARIANS: d - solitary polip e - colonial polip f-h - Medusoid forms: f - hydromeduse g - hydromeduse with branching gastrovascular canals h - scyphomeduse with branching and anastomosing gastrovascular canals Irrigation System a-c - SPONGLES: a,b - ascon, leucon c - budding sponge

11. I.4 Digestive Systems(Isaeva, Presnov & Chernyshev, in preparation) k, l - HEMICHORDATES and lower CHORDATES: k - digestive system with paired branchial clefts (lower CHORDATES) l - multiple openings of branchial system (some Enteropneusta and Tunicata) a-i - PLATHELMINTHES: a - blind digestive system (Acoela) b, h - digestive system with oral and anal openings c - g - digestive system in different representatives of flatworm: e - Trematoda c, f - Turbellaria Triclada d, g - Turbellaria Polyclada i - digestive system with two additional lateral canals (some Gastrotriсha)

12. The first important topological surgery in evolution of Bilateria (Triploblastica) is the appearance of the through intestinal tube instead of the blind gut resulting in progressive differentiation of the digestive system. Mollusks and echinoderms exhibit, besides the digestive tube, a second through channel – the coelomic system. The next evolutionary level of topological organization in animals is attained due to the development of the respiratory system. I.5 Evolutionary modifications of main body plan

13. I.6 Scheme of main body plan using genus of surface of definitive forms

14. I.7 Classification of Definitive Forms

15. Classification of living organisms by topology of definitive forms Extension of this classification by topology of developmental forms Description of cellular mechanisms of topological transformations in development Heuristic view: Topological laws and topology constraints of development Aims

16. II.1 Variability of echinodermsusing genus modifications in development

17. II.2 Cladogram of echinoderms

18. II.3 Observation that has never been done • The pigment band on the embryo’s surface of sea urchin. • Question: What might have happened to a canonical loop during development? • This dynamics may provide an additional marker of classification.

19. Classification of living organisms by topology of definitive forms Extension of this classification by topology of developmental forms Description of cellular mechanisms of topological transformations in development Heuristic view: Topological laws and topology constraints of development Aims

20. III.1 Mechanisms of topological transformations in development • Topological surgeries in epithelial morhogenesis occur locally by involving complex processes at the cellular level: sheet disintegration followed by cell adhesion and cytoskeletal cooperation results in newly formed cell sheet or sheets. Topological transitions in metazoan development occur due to topological surgeries realizing by “cutting” and “gluing” of epithelial sheets (Maresin & Presnov, 1985), as “self-wounding and healing” (Jockusch & Dress, 2003).

21. III.2 Neurulation • The local topological surgeries lead to global topological modifications of biological forms. • This is a positional information

22. III.3 Topological surgeries in development of chordates

23. Classification of living organisms by topology of definitive forms Extension of this classification by topology of developmental forms Description of cellular mechanisms of topological transformations in development Heuristic view: Topological laws and topology constraints of development Aims

24. IV.1 Topological laws and topology constraints.Polarization: Poincaré-Hopf’s theorem • For any smooth vector field given on two-dimensional sphere a singular point of the field exists. • It says that field singularity (or singularities) on the sphere are inevitable. • This singularity is breaking the spherical symmetry of the cortical vector field of microfilaments, polarizing the egg (positional information)

25. IV.2 Cortical changes following fertilization(gray crescent in amphibians) • Poincaré-Hopf’s theoremalso regulates the appearance of new singularities of cortex after fertilization: • New singularity initiates the appearance of another one (gray crescent).

26. IV.3 Cortical reactions • The Brouwer fixed point theorem is the statement that a self map of a convex subset of Rn always has a fixed point. • This indicates the existence of the ovum’s biological pole.

27. IV.4Cortical Flows:Model for the Establishment and Maintenance of PAR Domains • (A) A network of interactions among PAR proteins, the cortical actomyosin cytoskeleton, and the sperm MTOC in the one-cell C. elegans embryo. • (B and C) Dynamic consequences of these network interactions during the establishment and the maintenance of AP polarity. • A signal from the sperm MTOC weakens the cortex locally causing a posterior-directed cortical flow that transports PAR-3, PAR-6, and PKC-3 to the anterior. • Munro E. at al, 2004, Developmental Cell, v.7(3), 413-424.

28. IV.5 Discrete Morphogenetic Field • On the surface of fertilized and cleaving egg the discrete morphogenetic field mi emerges as a pattern of blastomere contacts. • Here, miis the number of neighboring cells; • For every cell with pattern mi, a value kican be defined. • ki is called a discrete cell curvature

29. IV.6 Gauss-Bonnet theorem (Euler’s theorem for polyhedron) With the consequence of Euler’s theorem for structurally stable graphs on the sphere (whose each apex order is equal to 3): kiis the discrete analogue of the continuous value of Gaussian curvature. ~

30. IV.7 Symmetry of the discrete morphogenetic field is breaking. • There are indeed only five homogeneous discrete fields on sphere corresponding to five regular (convex) polyhedra. • And only three of them - structurally stable (left column). • During synchronous cleavage divisions only first four blastomeres can create a homogeneous field on the embryo surface, later the field pattern of cellular contacts on the embryo surface inevitably becomes topologically inhomogeneous.

31. IV.8 Negative curvature – a reason of gastrulation k > 0 – exogastrulation (mi<6) Nematoda; Annelida; Echiurida; Mollusca; Crustacea; Phoronidea k < 0 - gastrulation (mi>6)

32. Last Chapter • We have used mathematical consepts in biology • Another question: • How can we transfer biological abstractions back to mathematics?

33. Systematics of morphogenetic fields

34. Chaos and Slow Processes in Modulating the Cell-Cycle Clock:by cyclin and MPF levels (Presnov & Agur, in preparation) Local order Integral order Limit cycle Existence of an integral order is the positional information

35. Synchronization of Cell Division(Presnov, 1999) Local order: One part of cells passes from i stage to i+1 stage. Another part of cells does not transit to the next developmental stage. • Positional Information: • The system of difference equations has an ergodic property: any initial condition gives a solution like a normal distribution if all coefficients biare equal and near 1. • Moreover, for unequal coefficients each initial distribution approaches a stationary distribution. Integral order

36. Global Decomposition of Vector Fields (Presnov, 2002) • Local order: The vector field F(x) • Global order: Decomposition • Positional information: the extraction of irrotational, solenoidal and rotational fields

37. Nikolas Bourbaki about … From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms – the mathematical structures; and it so happens – without our knowing why – that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation. Of course, it can not be denied that most of these forms had originally a very definite intuitive content; but, it is exactly by deliberately throwing out this content, that it has been possible to give these forms all the power which they were capable of displaying and to prepare them for new interpretations and for the development of their full power.