A model of one biological 2-cells complex
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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
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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
).()(/
);()(/
2122
1211
zzzRdtdz
zzzRdtdz
., 4
21 Rzz ., 421 Rzz
21, zzcorrespond to concentrations of species in these two cells. This model is quite hypothetical.
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negative feedbacks N ···◄ (AS-C).positive feedbacks (AS-C) → Dl ;
Two cells complex in a natural gene network:
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;)( 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.
*
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Trajectories of the system.
THEOREM 1. An unstable cycle can appear near S2.
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Either K1 or K2 becomes the Parental Cell with the Central Regulatory Contour. The other one goes to the Proneural Cluster.
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More complicated model. 7
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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
;1
914
3
1 xxdt
dx
;110
241
12 xxx
dtdx
.
110
332
23 xxx
dtdx
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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
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;)( 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.
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(A2); (A1) n=2k+1, one equilibrium point.
(A1) n=2k, “many” equilibrium points.
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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
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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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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.
!
!
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Trajectories and bifurcation cycles.
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dim = 105
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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
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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
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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
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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{...
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Homotopy properties
.
!
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1441235 \ SDDSDSQ 2 cycles
.\ 1331134 SDDSDSEQ 2 cycles?
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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
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AlekseiAndreevich Lyapunov,
1911 - 1973.
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Thank you for your patience
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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.