Download Dynamical robustness of the Mammalian cell cycle network: a

Dynamical robustness of the Mammalian cell cycle
network: a mathematical approach
Marco Montalva, Eric Goles and Gonzalo A. Ruz
Facultad de Ingenierı́a y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile
ECCS’13 - Barcelona, September 16-20, 2013
Summary
Fixed points in Case 1 (Main Theorem)
A logical model for the control of the mammalian cell cycle was presented in [1] where
the simulations performed to obtain the dynamic of the network, in parallel mode,
showed that in the presence of Cyclin D protein (CycD), any trajectory converges to a
unique limit cycle. In this work, we prove mathematically that, in fact, the presence of
CycD is a necessary and sufficient condition to assure that the network dynamics has no
fixed point, regardless of the deterministic updating mode considered (in particular, the
parallel one). Finally, we present a detailed study of all the deterministic dynamics
including the other case (absence of CycD).
Mammalian cell cycle model
The logical model for the control of the Mammalian cell cycle
[1,3]
CycD
CycE
p27
Rb
E2F
CycA
Cdc20
UbcH10
Cdh1
CycB
Theorem: Let S be the set of all the deterministic updating
schemes of the Mammalian network. Then,
CycD = 1 ⇔ ∀s ∈ S, the mammalian dynamics, generated by
the nine remaining nodes, has no fixed point.
Proof: ⇐) If CycD = 0, then by [1], the Mammalian dynamic
has one fixed point for the parallel update.
⇒) Let CycD = 1 and s ∈ S. We know that in at most one
iteration step, Rb = 0 and p27 = 0. Therefore, the Mammalian
dynamic generated by the seven remaining nodes can be
described by the dynamic of the above subnetwork. Let N be
such subnetwork, we will prove (by contradiction) that N has
no fixed point.
Let us consider that
y 0 = (E2F, CycE, CycA, Cdc20, Cdh1, UbcH10, CycB) ∈ {0, 1}7 is
a fixed point of N. We can consider two possibilities;
Cdc20 = 1 or Cdc20 = 0 and we will have the following
implications for each:
I If Cdc20 = 1, then CycB = 1 in (14) but, also, if Cdc20 = 1,
then CycB = 0 in (17), which is a contradiction.
I If Cdc20 = 0, then CycB = 0 in (14). If (Cdc20 = 0 ∧ CycB = 0),
then Cdh1 = 1 in (17). If (Cdc20 = 0 ∧ CycB = 0 ∧ Cdh1 = 1),
then CycA = 0 in (15). If (Cdc20 = 0 ∧ Cdh1 = 1 ∧ CycA = 0),
then (E2F = 0 ∨ UbcH10 = 1) in (13) but, on the other hand, if
we consider the above values of; CycA = 0, CycB = 0,
Cdc20 = 0 and Cdh1 = 1, we will have that E2F = 1 and
UbcH10 = 0 in (11) and (16) respectively, which is a
contradiction.
Therefore, N cannot have fixed points.
Local boolean functions
Mammalian cell cycle network dynamic with the following
updating scheme:
(CycD, Rb, Cdc20, Cdh1, CycA)(p27, UbcH10, CycB)(E2F)(CycE).
The red circle represents the fixed point, the 12 green circles
represent the states that belong to two limit cycles, one of
length 8 and another of length 4:
Pajek
Example 2: Dynamic with 3 limit cycles
Mammalian cell cycle network dynamic with the following
updating scheme:
(CycD, p27, Cdc20, Cdh1, UbcH10, CycB)(E2F)(CycE)(Rb, CycA).
The red circle represents the fixed point, the 14 green circles
represent the states that belong to three limit cycles, two of
length 6 and one of length 2:
Dynamics summary for Case 1
(1)
Rb =(CycD ∧ CycB) ∧ ([CycE ∧ CycA] ∨ p27)
(2)
E2F =(Rb ∧ CycA ∧ CycB) ∨ (p27 ∧ Rb ∧ CycB)
(3)
CycE =(E2F ∧ Rb)
(4)
CycA =(Rb ∧ Cdc20 ∧ Cdh1 ∧ UbcH10) ∧ (E2F ∨ CycA)
(5)
p27 =(CycD ∧ CycB) ∧ ([CycE ∧ CycA] ∨ [p27 ∧ CycE ∧ CycA]) (6)
Cdc20 =CycB
(7)
Cdh1 =(CycA ∧ CycB) ∨ Cdc20 ∨ (p27 ∧ CycB)
(8)
UbcH10 =Cdh1 ∨ (Cdh1 ∧ UbcH10 ∧ [Cdc20 ∨ CycA ∨ CycB])
(9)
CycB =Cdc20 ∧ Cdh1
(10)
CycD =CycD
Our analysis is separated in two cases; CycD = 1 and CycD = 0.
Different
dynamics
1696
|D2|
|D3|
|D4|
|D5|
|D6|
|D7|
16
298
812
432
132
6
(0.9%) (17.6%) (47.9%) (25.5%) (7.8%) (0.4%)
Attractor analysis for all the dynamics of the subnetwork of
seven nodes. In this case, there is only one limit cycle by
dynamic, with attraction basin of size 512 (including the limit
Some previous simplifications:
I If CycD = 1, then Rb = 0 in (2) and p27 = 0 in (6).
I If (Rb = 0 ∧ p27 = 0), then E2F = CycA ∧ CycB in (3).
I If Rb = 0, then CycE = E2F in (4).
I If Rb = 0, then
CycA = (Cdc20 ∧ Cdh1 ∧ UbcH10) ∧ (E2F ∨ CycA) in (5).
I If p27 = 0, then Cdh1 = (CycA ∧ CycB) ∨ Cdc20 in (8).
i.e., the Mammalian network can be reduced to the following
subnetwork:
CycE
E2F
CycA
only limit cycle has length i. We are use the algorithm
I We are presented a mathematical approach for the
Mammalian network [1] that consist in to separate its
The following table show the general attractor analysis for all
the dynamics of the subnetwork of nine nodes obtained when
CycD = 0:
CycE =E2F
CycA =(Cdc20 ∧ Cdh1 ∧ UbcH10) ∧ (E2F ∨ CycA)
Cdc20 =CycB
Cdh1 =(CycA ∧ CycB) ∨ Cdc20
UbcH10 =Cdh1 ∨ (Cdh1 ∧ UbcH10 ∧ [Cdc20 ∨ CycA ∨ CycB])
be divided into two components; one containing half of the
Different
130297
2-8
CycD = 1), the other one, with a single fixed point and,
with 1 lim. cycle
19719 (15.1%)
2-8
possibly, with at most two limit cycles (case CycD = 0).
with 2 lim. cycles
163 (0.1%)
2-6
without lim. cycles 110415 (84.7%)
–
I Note that most of the dynamics (≈ 85%) have a single
attractor; a fixed point.
I In the rest of the dynamics (≈ 15%), only one limit cycle
appears except for a few ones (≈ 0.1%) where two additional
limits cycles appears.
A more detailed analysis for the dynamics having a limit
cycle is showed in the next table:
Cdh1
(11)
(12)
(13)
(14)
(15)
(16)
(17)
I Every deterministic dynamic of the Mammalian network can
Range of lim.
cycles length
UbcH10
E2F =CycA ∧ CycB
dynamic according to the possible values of node CycD.
Number of
dynamics
Dynamics with
Average
at least 1 Basin Size
and, with the following local functions on its nodes:
Conclusion
developed in [2] for the classification of identical dynamics.
Cdc20
CycB
Pajek
cycle configurations). |Di| is the number of dynamics whose
Case 2: CycD = 0
Case 1: CycD = 1
CycB =Cdc20 ∧ Cdh1
Example 1: Dynamic with 2 limit cycles
lim. cycle of length 2
6587
140.5
lim. cycle of length 3
3383
321.7
lim. cycle of length 4
6148
273.4
lim. cycle of length 5
2892
298.1
lim. cycle of length 6
919
325
lim. cycle of length 7
21
363.3
lim. cycle of length 8
63
372.9
I The most frequent limit cycle lengths are 2 and 4 whereas the
least frequent are 7 and 8, however, these latest have the
maximum average basin sizes whereas the first ones have the
minimum average basin sizes
configurations converging to a single limit cycle (case
I When CycD = 1, the range of limit cycles lengths is between
2 and 7, being 4 the most frequent and with basin size of
29 = 512.
I When CycD = 0, the maximum length of limit cycles grows
up 8, being the most frequent the length 2 (but with average
basin size less than other longer limit cycles).
References
[1] A. Fauré, A. Naldi, C. Chaouiya, D. Thieffry, Dynamical
analysis of a generic boolean model for the control of the
mammalian cell cycle, Bioinformatics 22 (14) (2006), 124–131.
[2] J. Aracena, J. Demongeot, E. Fanchon, M. Montalva, On
the number of different dynamics in boolean networks with
deterministic update schedules, Mathematical Biosciences
242 (2) (2013), 188–194.
[3] G. A. Ruz, E. Goles, Reconstruction and update robustness
of the mammalian cell cycle network, in: The 2012 IEEE
Symposium on Computational Intelligence in Bioinformatics
and Computational Biology (CIBCB 2012), San Diego,
California, USA, 2012, 397–403.
Acknowledgment: Fondecyt 3130466 (M.M.), Fondecyt 1100003 (E.G.), Fondecyt 11110088 (G.A.R.), Basal (Conicyt)-CMM and Anillo Act-88