Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 55

Consider the case of an activator as in Eq. ( 2.11 ) and change the time variable to
τ = γ P t , to obtain:

d dτ

A m
θ A +

γ γ P

= κ 0

γ P + κ 1 A m −

M,

γ P

(2.14)

dP
dτ

= κ 2
γ P

M − P.

For a fixed value of A , Tikhonov's theorem can now be applied with y = M , x = P ,
ε = γ P /γ M and with f ( x, y, ε )= κ 2
γ P

A m
θ A + A m

x , g ( x, y, ε )= κ 0

γ M + κ 1

y .
Substituting the quasi steady state expression for mRNA into the protein Eq. ( 2.14 ),
and rewriting the system in the original time variable, obtains the reduced system:

y − −

γ M

A m
θ A + A m −

P = κ 0 + κ 1

γ P P, (2.15)

where κ 0 = κ 2 κ 0 /γ M and κ 1 = κ 2 κ 1 /γ M . This yields a dynamical equation for
the protein concentration, directly dependent on the amount of activator ( A ). From
now on, all the intermediate steps (the binding of A to the promoter and synthesis
of mRNA) can be left out of the model.
The expression h + ( x, θ, m )= x m / ( θ m + x m ) (or Hill function ) is known to
fit well to synthesis and activity rates. Similarly, the inhibition function can be
represented as: h − ( x, θ, m )=1 −

h + ( x, θ, m )= θ m / ( θ m + x m ). For gene
regulatory networks, the exponent m is considered to be “large” ( m

≥ 2), according
to experimental data [ 40 ]. Note that the qualitative form of h + ( x, θ, m ) remains
essentially unchanged for m ≥ 2, with the same maximal and half-maximal values

(max( h − )=1and h ± ( θ,θ,m )=1 / 2), the only difference being the steepness
of the function around the value θ .Forlarge m , the parameter θ has therefore a
special meaning: it is a threshold value below which there is practically no activity
and above which activity is (almost) maximal. In the limit as m tends to infinity, the
Hill function becomes a step function, as described in Sect. 2.2.3.3 .

2.2.3.2 Continuous Differential Systems for Genetic Network Models

To illustrate the modeling and analysis of complex GRN, consider a regulatory motif
that appears frequently in genetic networks: two genes that mutually inhibit them-
selves or, more precisely, the protein A encoded by gene a represses transcription of
gene b , and vice-versa (Fig. 2.4 ). The concentration of each protein can be described
by x j = κ j M j −

γ j x j , and each mRNA by an expression as in Eq. ( 2.12 ):

M j = κ j 0 + κ j 1 h − ( x i ,θ i ,m i ) −

γ Mj M j , for j, i ∈{ 1 , 2 } and j = i. (2.16)

Using the quasi-steady state assumption for the protein and mRNA equations,
the system can be reduced to the dynamics of the protein concentrations, x i =
f i ( x 1 ,x 2 ) with (renaming constants):