Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 56

A

B

a

b

Fig. 2.4 Mutual inhibition between two genes. The white rectangles represent genes a and b ,
which are transcribed and then translated into the respective proteins, A and B (this is represented
by the arrows ). Each of these proteins inhibits the transcription of the gene of the other protein
(A B, B A)

a b

f 2 ( x 1 ,x 2 )=0

x 2 x 2

f 1 ( x 1 ,x 2 )=0

f 1 ( x 1 ,x 2 )=0

f 2 ( x 1 ,x 2 )=0

x ( t )

x ( t )

x 1 x 1

Fig. 2.5 Phase plane for the system of Eq. ( 2.17 ), for two different sets of parameters: ( a )
bistability, or ( b ) a single steady state. Stable steady states are marked by a black circle ,andthe
unstable steady state by an open rectangle . In each region delimited by the nullclines ( f 1 ( x )=0
in red , f 2 ( x )=0 in blue ), the sign of the vector field of each coordinate is unchanged. One
solution is shown in black . Parameter values for case ( a ): κ 10 =0 . 02 , κ 11 =1 , θ 1 =0 . 3 , γ 1 =1 ,
κ 20 =0 . 05 , κ 21 =2 , θ 2 =0 . 6 , γ 1 =1 . 3 , m 1 = m 2 =4 ; for case ( b ) the only difference is
κ 21 =1

⎧
⎨

θ m 2
2

x 1 = κ 10 + κ 11 − γ 1 x 1 ,

θ m 2
2

x m 2
2

+

(2.17)

⎩

θ m 1
1

x 2 = κ 20 + κ 21 − γ 2 x 2 .

θ m 1
1

+ x m 1
1

(Note that, in this section, the function f i denotes the full right-hand side of the x i
equation, as in Sect. 2.2.2.1 .) The dynamics of this system depend on the values
of the parameters. The nullclines and steady states for system in Eq. ( 2.17 )are
represented in Fig. 2.5 . Two distinct generic cases can be identified:

(a) m i large, θ i < ( κ i 0 + κ i 1 ) /γ i ,forall i =1 , 2. There are two stable and one
unstable steady states. The stable steady states can be intuitively understood:
one of the proteins is present at high concentration hence inhibiting transcription
of the second gene, and implying that the second protein is only present in low
concentration. This co-existence of two stable steady states is called bistability ;
(b) m i large, θ i > ( κ i 0 + κ i 1 ) /γ i ,forsome i =1 , 2. There is a unique steady state,
which is stable.