1. Optimization of studies of gene networks models phase portraits
Golubyatnikov V.P.
Sobolev institiute of mathematics
Joined work with Akinshin A.A.,
Ayupova N.B., and Kazantsev M.V.
13 December

2. Our goal is to give mathematical explanations, and predictions to some numerical experiments with periodic trajectories of nonlinear dynamical systems considered as gene networks models (existence, stability, non-uniqueness). A.N.Kolmogorov, I.G.Petrovskii, N.S.Piskunov Moscow University Herald, 1937.

3. Content Molecular repressilators model.
Modeling of 3 cells of D.melanogaster’s
  mechanoreceptors morphogenesis.

4. I Molecular repressilator’s model:
Elowitz M.B., Leibler S. A synthetic oscillatory network
of transcriptional regulators. Nature
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5. EL (2)
pj(t) are concentrations of 3 proteins, mj (t) are concentrations of corresponding mRNA.
The system is symmetric
with respect to
f0(p)=α0+α(1+pγ)-1 .
1→2→3→1
Glyzin, S.D., Kolesov, A.Y. & Rozov, N.Kh.
Theor Math Phys (2016) 187: Issue 3, 935–951.

6. f0(p)=α0+α(1+pγ)-1 . (EL=2):
pj(t) – концентрации трех белков, mj (t) – концентрации соответствующих им мРНК.
Симметричная система.
f0(p)=α0+α(1+pγ)-1 .
1→2→3→1

7. Symmetric system
Akinshin A.A.

8. Positive and negative feedbacks:
“f0(p)=α0+α(1+pγ)-1 “

9. Dynamical system (1): 1 3 5 2 4 6
pj(t) are concentrations of 3 proteins, mj (t) are concentrations of corresponding mRNA.
Smooth functions fj decrease monotonically
(negative feedbacks, repressilator).

10. Dynamical system(1): f0(p)=α0+α(1+pγ)-1 .
pj(t) are concentrations of 3 proteins, mj (t) are concentrations of corresponding mRNA.
f0(p)=α0+α(1+pγ)-1 .

11. dim = 105
f0(p)=α0+α(1+pγ)-1

12. Trajectories of some 3-D systems right: left:12
Trajectories of some 3-D systems right: left:
An inverse problem: to reconstruct integral manifolds inside and outside of the cycles.

13. Dynamical system (1): Smooth functions fj decrease monotonically3 5 2 4 6
Smooth functions fj decrease monotonically
(negative feedbacks, repressilator). m1

14. Invariant domain Aj:=fj(0)/kj; for the system (1) we get: Aj:=fj(0).
Q:=[0, A1]×[0, A1]×[0, A2]×[0, A2]×[0, A3]×[0, A3].
Russian Journal of Numerical Analysis and Mathematical Modeling. 2011, v. 28, N 4.
Contemporary mathematics. 2011, v. 553.
Numerical analysis and applications. 2010, т. 13, N 4.
The journal of three dimensional images 3D Forum. Japan, 2004, v. 18, N 4.

15. Decomposition of the phase portrait:
, является инвариантной областью системы (1).
Lemma 1. 1) Polyhedron Q in the positive octant
is an invariant domain of the system (1).
2) The system (1) has exactly one equilibrium point S0 , it is contained in Q.
Let be its coordinates.
Consider decomposition of Q by hyperplanes
to 64 blocks, and denote them
by binary indices:
E={ε1ε2ε3ε4ε5ε6}.

16. . Q S0 о
3D analogy:
"

17. Directions of trajectories
means ≤ for ε(2j-1) =0, and ≥ for ε(2j-1) =1;
means ≤ for ε(2j) =0, and ≥ for ε(2j) =1.
Lemma 2. For any two adjacent blocks E1, E2 of this partition, all trajectories of (1) intersect their common 5D face in one direction only: either E1→ E2, or E2→ E1.

18. Valency of blocks:
Valency of a block E is the number of adjacent blocks where trajectories of the points of E can pass from E .
The next diagram shows the blocks of the valency 1 which can cntain a cycle
of the system (1).

19. Diagram: ↓ ↑ ↓ ↑ Valency = 1 ↑ ↓ {110011}→{010011}→{000011}→{001011}
{110010}
{001111} ↑ ↓
Valency = 1
{001101}
{110000} ↑ ↓
{110100}←{111100}←{101100}←{001100}

20. Part of the state transition diagram of the system (1)
Valency=5:
Valency=1.

21. Linearization of the system (1) at S0
for
We consider the case «a >> 1».

22. Lemma 3. If a is sufficiently large, then
the characteristic polynomial P(λ) of the linearization matrix at the point S0
has two roots with positive real parts.
The rest 4 roots have negative real parts. M0

23. Theorem 1.
If the linearization matrix M0 of the system (1) at the point S0 has two eigenvalues with positive real parts, and the real parts of the rest eigenvalues are negative, then
the system (1) has at least one cycle which travels through the blocks of the partition (3) according to the diagram.

24. Numerical experiments, А.А.Akinshin and М.V.Каzantsev

25. Symmetric system (2) projections of a trajectory and its limit cycle
А.А.Аkinshin

26. А.А.Аkinshin
Projection of one trajectory of (1) and its limit cycle onto the plane of the variables

27. M.V.Каzantsev
Projection of one trajectory of (1) and its limit cycle onto the plane of the variables

28. Projection of one trajectory of (1) and its limit cycle onto the plane of the variables m1, m2, m3.
M.V.Каzantsev

29. Concentrations of the proteins Left for (1), right for (2).
А.А.Аkinshin

30. Projection of one trajectory of (1) and its limit cycle onto 2 D plane. S0
M.V.Каzantsev

31. II. Three cells complex Two cells interactionK1 K2
Two cells interaction K3
Positive feedbacks (AS-C) → Dl, Dl → N;
Negative feedback N ···◄ (AS-C).

32. We use notations: N ···◄ (AS-C). (AS-C) → Dl.
Concentrations of the proteins in the cells Kj :
N ···◄ (AS-C).
(AS-C) → Dl.
j, k = 1,2,3..
Dl → N, between the cells.
System Computational Biology,
IC&G SB RAS, 2008.

33. (9D) j,k,l = 1,2,3. and we obtain 9-dimensional dynamical system
при всех
and we obtain 9-dimensional dynamical system
(9D) j,k,l = 1,2,3.
The function f decreases monotonically,
this describes negative feedback (Notch)···◄ (AS-C).
The functions σ and ζ grow monotonically, this corresponds to positive feedbacks in the cells
(AS-C)→(Delta),
and (Delta)→(Notch)←(Delta) between the cells.
Equilibrium points:

34. Equilibrium points: or
a, α, γ > 0.

35. Partially symmetric equilibrium points
is determined by the equation z=2g(z).
Left hand side grows, right hand side decreases.
This equation has a unique solution.
Theorem 2. The equilibrium points with
do not exist.
Partially symmetric equilibrium points
etc. appear from the system

36. For sufficiently large values of α, this system has 3 solutions, otherwise just one = S0.
Similarly, we get equilibrium points

37. Linearization of the system (9D) at its equilibrium points

38. S0
This point is unstable for large values of α.
are unstable for large values of α.
are stable for large values of α.

39. If trajectory of the system (9D) is attracted
to the point then the cell Kj becomes
the parental one.
In the 3-dimensional plane ,
is an invariant unstable manifold of the
system (9D).
Theorem 3. For large values of α it contains
an unstable cycle.

40. S.Smale (1974) has constructed one very hypothetical model of interaction of two cells.
It was not related to any biological process, and was represented in the form of 4-D nonlinear dynamical system.
For appropriate choice of parameters of this system, its phase portrait contains a stable cycle. In absence of interaction there are no periodic trajectories in this two-cells complex dynamics, just stable equilibrium points.

41. 39/42
Our current tasks are connected with: determination of conditions of regular behaviour of trajectories; studies of integral manifolds and non-uniqueness of the cycles, bifurcations of the cycles; their dependence on the variations of the parameters, and connections of these models with discrete models of the Gene Networks.
"

42. Some recent publications:
N.B.Ayupova, V.G. On the uniqueness of a cycle in an asymmetric three-dimensional model of molecular repressilator. J. Appl. and Industr. Math. 2014, v. 8, N 2.
N.B.Ayupova, V.G. On two classes of nonlinear dynamical systems: the 4-D case. Sib. Math. Journ. 2015, v. 56, N 2.
M.V.Kazantzev Some properties of the domain graphs of dynamical systems. Sib.Zh.Ind.Appl.M., 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.
T.A.Bukharina, V.G., D.P.Furman. Gene Network Controlling Morphogenesis of D.melanogaster mechanoreceptors. Russ. J. Development Biology. 2016, v.47, N 5.

43. Aleksei
Andreevich
Lyapunov.

44. 1958

45. Thank you for attention and
RFBR for support: grant 