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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

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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.

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

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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

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

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

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7 More complicated model. 7

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

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рублей. Stationary points and cycles of the system 9

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

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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.

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

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

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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.

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15 A trajectory and a limit cycle. ! !

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Trajectories and bifurcation cycles.

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17 dim = 105

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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?

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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).

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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.

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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):

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Homotopy properties. ! 22 2 cycles 2 cycles?

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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.

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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.

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

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26 Aleksei Andreevich Lyapunov, 1911 - 1973.

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27 Thank you for your patience

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

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