Detecting Attractors in Biological Models with Uncertain Parameters 51d[dtX]=k 1 Kyn 11
Kny 11 +[Y]n 1 ·Knz 12
Knz 12 +[Z]n 2 −φX[X]
d[dtY]=k 2 Knx^32
Knx 23 +[X]n 3 ·Kzn 24
Knz 24 +[Z]n 4 −φY[Y]
d[dtZ]=k 3 Knx^53
Knx 35 +[X]n^5 ·Kyn 36
Kny 36 +[Y]n^6 −φZ[Z]
∀i, j;ki=1,
Kxi=Kyi=Kzi=5,nj=5,
φX=φY=φZ=0. 1ki φI
(0. 1 ,10)(0,1)Fig. 5.The tri-stable toggle switch regulatory network (left) and its ODE model (mid-
dle). The parameter value intervals we have considered for alli∈{ 1 , 2 , 3 },I∈{X, Y, Z}
(right).
Fig. 6.The parameter space and the corresponding number of terminal components
(one in white, two in green, three in blue). Thanks to the symmetry of the model, there
are only 3 pairs of allowed parameters (Color figure online).
Regulation of theG 1 /SCell Cycle Transition.As the last case, we have
investigated a well-known model representing the central module of the genetic
regulatory network governing theG 1 /S cell cycle transition in mammalian
cells [ 24 ]. In particular, the model explains the mechanism behind the irreversible
decision for cell division described by a two-gene regulatory network of interac-
tions between the tumour suppressor proteinpRBand the central transcription
factorE 2 F1 (Fig. 7 left). In high concentration levels,E 2 F1 activates theG 1 /S
transition mechanism. In low concentration ofE 2 F1, committing toS-phase is
refused and that way the cell avoids DNA replication. For suitable parameter
values, two distinct stable attractors exist. A numerical bifurcation analysis of
E 2 F1 stable concentration depending on the degradation parameter ofpRB
(φpRB) has been provided in [ 24 ].
A bistability region has been discovered forφpRB∈[0. 012 , 0 .0145] in [ 5 ]. Our
algorithm has found a bistability region in (0. 010 , 0 .0146) for parametrisedφpRB.
This extension of parameter interval has the same reason as in the first case
study—the presence of a non-trivial terminal component, instead of asink[ 5 ].
Additionally, we have managed to analyse this model for all pairs of parame-
ters allowed for the implementation (Fig. 8 ).