ON CONTROL OF CYCLES IN SOME GENE NETWORKS MODELS Golubyatnikov V.P., Sobolev Institute of Mathematics SB RAS, Kazantsev M.V., Polzunov State Technical University, Barnaul, Novosibirsk, 19 December,

Outline 1. Motivation. Simple gene networks models. Examples. 2. Topology and combinatorics of phase portraits. 3. Combinations of negative and positive feedbacks a gene network model. 2

3 1. Motivation Our current tasks: determination of conditions of regular behaviour of trajectories for PL and C ∞ cases; studies of integral manifolds, control of trajectories (and cycles), description of geometry of the phase portraits; bifurcations of the cycles, their dependence on the variations of the parameters, and possible connections of these systems with some models of mathematical biology.

4 Some simple gene networks models We study dynamical systems \eqno (n) are smooth or step monotonically decreasing functions. This corresponds to the negative feedbacks in the gene networks. Here Lemma 2. Each smooth system (2k+1) has exactly one equilibrium point in the positive octant: is an invariant parallelepiped of the system (n). Lemma 1.

5 n=2k+1, one equilibrium point. n=2k, “many” equilibrium points. (A.A.Akinshin)

6 Invariant parallelepiped Q is decomposed by n hyperplanes parallel to the coordinate ones and containing the equilibrium point. So, we obtain «2 n » (2 n-k 3 k ) disjoint parallelepipeds (domains) in Q.

7 Valency of a domain: How many (interior) faces of the domain are intersected by outgoing trajectories, or How many arrows come out of the corresponding vertex of the state transition diagram (domain graph) of the dynamical system.

8 Non-convex invariant polyhedron of the system (3) composed by six triangle prisms:. ♦

9 Theorem 1. If the equilibrium point is hyperbolic then the system (2k+1) has at least one periodic trajectory in the invariant domain Q. The following diagram shows the discrete scheme of some of the trajectories of the system (2k+1). We reduced this invariant domain to the union of 4k+2 triangle prisms in order to localize the position of the cycle. Then existence of periodic trajectories follows from the Brower’s fixed point theorem. (Here, the valency of the blocks = 1.)

10 A trajectory and a limit cycle ! ! ← equilibrium point

dim=3, biological interpretation 11/51

Let P(x,y), Q(x,y) be polynomials. To find all cycles of the system: and to describe their mutual positions on the 2D plane. 12

D.P.Furman, T.A.Bukharina. The Gene Network Determining Development of Drosophila Melanogaster Mechanoreceptors. Comp. Biol.Chemistry, 2009, v.33, pp. 231 – 234. The scheme of one «nonlinear dynamical system» T.A.B, V.G., D.P.F. A model study of the morphogenesis of D.M. mechanoreceptors: The central regulatory circuit. J. Bioinform. Comput. Biology. 2015, v. 13, N. 2.

14 Threshold (step) functions describe negative feedbacks: In some cases of 3D, 4D and 5D step-functions nonlinearity, we have constructed invariant PL-manifolds inside and outside of the corresponding cycles: Sib. Math. Journ. 2015, v. 56, N 2. J. Applied and Industr. Math. 2014, v. 8, N 2. Trajectories are piece-wise linear (PL).

15 Threshold functions describe positive feedbacks: negative feedbacks: (2 n-k 3 k ) We consider smooth cases as well. and variable feedbacks:

System (LГ). PL trajectories are linear in each domain where each equation has the form dx/dt=A – x. L.Glass, J.Pasternack, /51

In 2D there are no cycles! Otherwise there will appear invariant domains. But

18 Equilibrium points of one smooth system (LΛΛ) I II III IV V 18

Equilibrium points and cycles of that system (LΛΛ) 19/51 (A.G.Kleschev)

Iterations of the Glass-Mackey function f(x)=  x/(1+x  ),  =2,  =10. From left to right: f(x), f (3) (x), f (6) (x). Chaos can appear (V.A.Likhoshvai). 20

(A.A.Akinshin) 5D example: For PL system (5=LLLLL) two invariant PL surfaces were constructed in Q. Each of them contains a cycle; A > 3.

Two cycles in 5-D symmetric systems. (A.A.Akinshin)

One of these cycles is mentioned in Th. 1 (level = 1): Another one (level = 3): Oriented Graph of the system (5) LLLLL: 23

« 2 ». Homotopy properties cycles! (LLLLL) 2 cycles? (LΓΓΓ).

Oriented Graph of 6D system LLLLLL (levels 0, 2, 4, 6 are shown): 25/51

26 This cycle is contained in some invariant 2D surface with the vertex E=(1;1;1). We have considered also smooth cases. Stability of these cycles was studied as well. Theorem 2. If n=3, and L i are as above, then the system (3) has a unique cycle in the parallelepiped Q.

(Kleschev A.G.) ;, A trajectory convergent to the bifurcation cycle. An inverse problem of parameters identification.

28 Dim = 18 (A.A.Akinshin)

29 Dim = 7 (Akinshin A.A.)

J.Tyson (1977), L.Glass (1978) et al considered similar systems (LГГГ…) with monotonically increasing functions f i, i ≥2, and f 1 as above. An example: J.Tyson et al obtained conditions of existence of cycles for similar smooth systems. In this PL case, L.Glass and J.S.Pasternack described conditions of existence of a stable cycle C... 30

31 a (n)(n) (GT) The domain is positive invariant for both systems (n), (GT) and contains exactly one «equilibrium» point E of the system (GT) for all n. For odd n it contains exactly one «equilibrium» point E of the system (n). (GT) = Glass-Tyson system, (n) was considered above.

32 4-dimensional symmetric system (LLLL) Theorem 3. If A > 2, then symmetric dynamical system (4) with threshold functions has exactly one symmetric cycle. It is contained in an invariant piece-wise linear 2-D surface. It travels in the cube Q according to the diagram (d4). The valency of these blocks = 2, see the next slide: (4) (d4)

33 Consider the system (n) (LLLL) for n=4. State transition diagram (d4) valency=2 have zero valency. For symmetric case, the union of the domains of the diagram (d4) contains a cycle. We have obtained conditions of its existence and uniqueness. There is an 1-D invariant manifold Δ: two straight trajectories which approach E. Each of them contains a stable equilibrium point. Δ={x 1 =x 2 =x 3 =x 4 }.

34 For 4D-system (T) (LΓΓΓ) consider the state transition diagram (T2 ) (J.Tyson, L.Glass et al.) Trajectories of (T) of the points in do not approach E by straight lines. (T2) valency=1 (T3) valency=3 The union of the domains in the diagram (T2) can contain a cycle C. Conditions of its existence were obtained by Glass and Tyson. What about its uniqueness?

35 Theorem 4. The union of domains in the diagram (T3) for a symmetric dynamical system contains one-parameter family of trajectories which remain there for all t >0. These trajectories fill a PL-surface M 2 with vertex E composed by 8 plane domains, contained in the domains of (T3). M 2 does not contain cycles. Theorem 5 holds for smooth analogs of the system (T) as well. Let M 1 be the surface spanned by the cycle C. Consider intersections of M 1 and M 2 with the boundary of 4D cube Q 4.

These intersections have non-trivial link on ∂Q 4 36 Hopf link

37 Theorem 5. Domain graphs of arbitrary PL-systems of the same dimension with functions L and Γ are isomorphic if the parities of these systems coincide. i.e., quantities of Γ-functions in their right-hand sides coincide (mod 2). Definition: Parity of a PL- system: how many (mod 2) functions Γ are there in its right-hand sides. Odd and even systems.

38 Theorem 4, n=4. a) ΓLΓΓ, b) LLΓL. b) a) See slide 34. Valency=k Valency=(k – 2) Valency=k

39 Theorem 4, n=5. a) LΓLΓL, b) LLLLL. Arrows as above. a) b) a)

« 3 » Cycles of the system LГП : (000)→(100) →(110) →(111) 40 (001) ←(011) ←(021) ←(121) o

Phase portrait of the system LГП: О 41

42 Construction of a cycle in phase portrait of the system LГП, step 1.

43 Step 2.

44 A cycle in phase portrait of the system LГП.

45 Summary 1.For some low-dimensional PL dynamical systems (n), (LГП) existence of cycles and 2D-invariant surfaces is shown. 2. In the phase portrait of 4-dimensional system of the type (LLLL) we construct explicitly a hyper-surface which separates attraction basins of stable equilibrium points of this system. 3. For the 4D system (LΓΓΓ) we have shown existence of infinitely many trajectories which are not attracted to the stable cycle discovered by Glass and Pasternack. Non-trivial (Hopf) link. 4. For similar PL systems of arbitrary dimensions we prove conditions of equivalency of their domain graphs.

46 Some recent publications: N.B.Ayupova, V.G. On the uniqueness of a cycle in an asymmetric three-dimensional model of molecular repressilator. J. Applied and Industr. Math. 2014, v. 8, N 2. Yu.A. 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. N.B.Ayupova, V.G. On two classes of nonlinear dynamical systems: the 4-D case. Sib. Math. Journ. 2015, v. 56, N 2. V.G., A.E.Kalenykh On structure of phase portraits of some nonlinear dynamical systems. Bulletin of Novosibirsk State University, 2015, v.15, N 1. M.V.Kazantzev Some properties of the domain graphs of dynamical systems. Sib. Zh. Ind.Appl. Math., 2015, v. 18, N 4. T.A.Bukharina, V.G., D.P.Furman A model study of the morphogenesis of D. melanogaster mechanoreceptors: The central regulatory circuit. Journal of Bioinformatics and Computational Biology. 2015, v. 13, N. 2.

Projections of trajectories of one 4-D system onto the plane OXYZ. Reparation of DNA damage by the p53 gene. (2009, with Eric Mjolsness, UCI), 2012!! 47

Maxim Kazantsev, Andrey Akinshin. RFBR

49 Aleksei Andreevich Lyapunov

Thank you for your attention