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

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

).()(/

);()(/

2122

1211

zzzRdtdz

zzzRdtdz

., 4

21 Rzz ., 421 Rzz

21, zzcorrespond to concentrations of species in these two cells. This model is quite hypothetical.

3

negative feedbacks N ···◄ (AS-C).positive feedbacks (AS-C) → Dl ;

Two cells complex in a natural gene network:

4

;)( iii xzftd

xd ;)( ii

i yxtd

yd

i=1, 2; xi (t)= [AS-C], yi (t)=[Dl], zi (t)=[N].

;)( 12*1 zytd

zd

.)( 21*2 zytd

zd

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: S1 and S3

The point S2 is unstable.

*

4

5

Trajectories of the system.

THEOREM 1. An unstable cycle can appear near S2.

6

Either K1 or K2 becomes the Parental Cell with the Central Regulatory Contour. The other one goes to the Proneural Cluster.

7

More complicated model. 7

8

Five equilibrium points in the system (DM), three of them are stable.

[AS-C] (t)

рублей.Stationary points and cycles of the system 9

;1

914

3

1 xxdt

dx

;110

241

12 xxx

dtdx

.

110

332

23 xxx

dtdx

10

APPENDIXt

.,...2;10)(;102)( nixforxLxforAxL iii

;,...2;12)(;100)( nixforAxxforx iii

“Threshold” functions describing positive feedbacks:

and negative feedbacks:

Sometimes we consider their smooth analogues.

,constAi

11

;)( 111 xxLtd

xdn

a

.,...2;)( 1 nixxLtd

xdiii

i (A1)

;)( 111 xxLtd

xdn .,...2;)( 1 nixx

td

xdiii

i (A2)

)).(max())(max( xorxLB iii

Parallelepiped ispositively 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).

],0...[],0[],0[ 21 nn BBBQ

(A2) is the Glass-Tyson (et al) dynamical system, (A1) was considered in our previous papers.

12

(A2); (A1) n=2k+1, one equilibrium point.

(A1) n=2k, “many” equilibrium points.

13

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

}001{ }011{

}110{

}100{

}001{}101{}100{}110{}010{}011{}001{

}10{}11{ cdabcdab }01{}00{ cdabcdab

;}001{}011{ F

.}001{}101{1 F

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

u1 u2 u3( ) rp1 rp2 rp3( )

;1

65x

zdt

dx

;

1

37y

xdt

dy

;7 5 ze

dt

dz y

A trajectory and a limit cycle.

!

!

Trajectories and bifurcation cycles.

17

dim = 105

18

Consider the system (A2) for n=4 and its state transition diagram (A3) (J.Tyson, L.Glass et al.)

134 \ DSEQ Trajectories of all points in (in (A4) ) do not approach E in a fixed direction.

}1110{}1100{}1000{}0000{

}1111{}0111{}0011{}0001{

(A3)Level=1

}0010{}0110{}0100{}0101{

}1010{}1011{}1001{}1101{ (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?

3P

.\ 313 DSEP

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).

4P

20

Consider the system (A1) for n=4. The state transition diagram:

}1001{}1101{}1100{}1110{

}1011{}0011{}0111{}0110{

(A5)Level=2

},0101{ }1010{ 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.

5P

224 \ DSQ

21

5D case for (A1)

.],0...[],0[],0[ nRAAAQ

Invariant piece-wise linear 2-D surfaces containing 2 cycles of corresponding system were constructed in Q. n=5:

}...11011{}11001{}11101{}11100{}11110{... Level=3 ~(A4).

Level=1 ~ (A3):

}...01011{}01001{}01101{}00101{}10101{...

Homotopy properties

.

!

22

1441235 \ SDDSDSQ 2 cycles

.\ 1331134 SDDSDSEQ 2 cycles?

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

math+biol

26

AlekseiAndreevich Lyapunov,

1911 - 1973.

27

Thank you for your patience

Trajectories of some 3-D systems right:

left:

.1

17)(,10)(,

1

10)(

33135.0

231

2

y

yyfexf

zzf x

.1

17)(,

1

10)()(

33321 y

yyf

zwfwf

16

An inverse problem N 3: to reconstruct integral manifolds inside and outside of the cycles.