1. A model of one biological 2-cells complex Akinshin A.A., Golubyatnikov V.P. Sobolev Institute of Mathematics SB RAS, Bukharina T.A., Furman D.P. Institute of Cytology and Genetics SB RAS Novosibirsk, 24 September, Geometry Days -2014 1

2. S.Smale “A mathematical model of two cells via Turing’s equation”, AMS, Lectures in Applied Mathematics, v. 6, 1977. Each of these 4-dim variables describes one of two cells in a cell complex. Smale has shown that for some nice values of parameters this system can have non-trivial cycles, though its restriction to any cell has just a stable equilibrium point... 2 correspond to concentrations of species in these two cells. This model is quite hypothetical.

3. 3 negative feedbacks N ···◄ (AS-C). positive feedbacks (AS-C) → Dl ; Two cells complex in a natural gene network:

4. 4 i=1, 2; x i (t)= [AS-C], y i (t)=[Dl], z i (t)=[N]. Here f is monotonically decreasing, it corresponds to negative feedbacks N ···◄ (AS-C). Sigmoid functions σ and describe positive feedbacks (AS-C) → Dl. 2 stable equilibrium points: S 1 and S 3 The point S 2 is unstable. 4

5. 5 Trajectories of the system. THEOREM 1. An unstable cycle can appear near S 2.

6. 6 Either K 1 or K 2 becomes the Parental Cell with the Central Regulatory Contour. The other one goes to the Proneural Cluster.

7. 7 More complicated model. 7

8. 8 Five equilibrium points in the system (DM), three of them are stable. [AS-C] (t)

9. рублей. Stationary points and cycles of the system 9

10. 10 APPENDIX t “Threshold” functions describing positive feedbacks: and negative feedbacks: Sometimes we consider their smooth analogues.

11. 11 a (A1) (A2) Parallelepiped is positively invariant for both systems (A1) and (A2) and contains a unique «equilibrium» point E of (A2) for all n. For odd n it contains a unique «equilibrium» point E of (A1). (A2) is the Glass-Tyson (et al) dynamical system, (A1) was considered in our previous papers.

12. 12 (A2); (A1) n=2k+1, one equilibrium point. (A1) n=2k, “many” equilibrium points.

13. 13 Non-convex invariant domain of the 3-D system (A1) composed by six triangle prisms..

14. 14 Potential level of a block: How many faces of the block are intersected by outgoing trajectories, or How many arrows come out of the corresponding vertex of the state transition diagram of the dynamical system.

15. 15 A trajectory and a limit cycle. ! !

16. Trajectories and bifurcation cycles.

17. 17 dim = 105

18. 18 Consider the system (A2) for n=4 and its state transition diagram (A3) (J.Tyson, L.Glass et al.) Trajectories of all points in (in (A4) ) do not approach E in a fixed direction. (A3) Level=1 (A4) Level=3 It was shown that the union of the blocks listed in (A3) can contain a cycle, and conditions of its existence were established. What about its uniqueness?

19. 19 THEOREM 2. The union of the blocks listed in (A4) contains a trajectory which remains there for all t >0. This theorem holds for smooth analogues of the system (A2) as well. In the PL-case, there are infinitely many geometrically distinct trajectories in the diagram (A4).

20. 20 Consider the system (A1) for n=4. The state transition diagram: (A5) Level=2 have zero potential level. We show that in symmetric cases the union of the blocks listed in (A5) contains a cycle, conditions of its existence were established. It is unique in this union. There is an invariant 1-D manifold Δ which approaches E in the fixed direction.

21. 21 5D case for (A1) Invariant piece-wise linear 2-D surfaces containing 2 cycles of corresponding system were constructed in Q. n=5: Level=3 ~(A4). Level=1 ~ (A3):

22. Homotopy properties. ! 22 2 cycles 2 cycles?

23. 23 Motivation. Our current tasks are connected with: determination of conditions of regular behaviour of trajectories; studies of integral manifolds non- uniqueness of the cycles, and description of geometry of the phase portraits; bifurcations of the cycles, their dependence on the variations of the parameters, and connections of these models with other models of the Gene Networks.

24. 24 Some recent publications: Yu.Gaidov, V.G. On cycles and other geometric phenomena in phase portraits of some nonlinear dynamical systems. Springer Proc. in Math. &Statistics, 2014, v.72, 225 – 233. N.B.Ayupova, V.G. On the uniqueness of a cycle in an asymmetric 3-D model of a molecular repressilator. Journ.Appl.Industr. Math., 2014, v.8(2), 1 – 6. A.Akinshin, V.G. On cycles in symmetric dynamical systems. Bulletin of Novosibirsk State University, 2012, v.2(2), 3 – 12. T.Bukharina, V.G., I.Golubyatnikov, D.Furman. Model investigation of central regulatory contour of gene net of D.melanogaster machrohaete morphogenesis. Russian journal of development biology. 2012, v.43(1), 49 – 53. Yu.Gaidov, V.G. On the existence and stability of cycles in gene networks with variable feedbacks. Contemporary Mathematics. 2011, v. 553, 61 – 74.

25. 25 Acknowledgments: RFBR grant 12-01-00074, grant 80 of SB RAS, RAS VI.61.1.2, 6.6 and. math+biol

26. 26 Aleksei Andreevich Lyapunov, 1911 - 1973.

27. 27 Thank you for your patience

28. Trajectories of some 3-D systems right: left: 16 An inverse problem N 3: to reconstruct integral manifolds inside and outside of the cycles.