Modeling and Analysis of Gene Regulatory Networks - Modeling in Computational Biology and Biomedicine - page 54

If no other reactions take place, there is conservation of mass D + C = D T (the
total amount of promoter sites, free or bound, remains constant), which can be used
to reduce the number of variables. One can view A as an external variable. Using the
observation that binding processes are typically faster than transcription, the system
in Eq. ( 2.8 ) can be simplified through a process similar to Tikhonov's method, by
setting the equations for C and D at “quasi steady state” ( C ≈ 0), to obtain:

⎧
⎨

A m

C = D T

θ A + A m ,
D = D T −

(2.9)

θ A

⎩

C = D T

θ A + A m ,

with θ A =( k 2 /k 1 ) 1 /m . The amount of mRNA molecules (denoted M ) produced
may depend both on the concentration of DNA sites bound to an activator or
repressor, and on the amount of free DNA sites. For simplicity, it will be assumed
that the effect of activators and repressors can be modeled independently. Since
binding of A to D is fast, the most natural form for the production of mRNA is a
linear dependence on D and C . In addition, mRNA decays at a constant rate γ M ,
which yields the equation:

M = α 0 D + α 1 C

− γ M M. (2.10)

In the case of an activator, the contribution of C to mRNA production is much larger
than that of D , which can be represented by specifying α 1 α 0 . Defining κ 0 =

α 0 D T and κ 1 =( α 1 − α 0 ) D T , substitution of the quasi-steady state expressions

for C and D in Eq. ( 2.9 ) into Eq. ( 2.10 ) yields:

A m
θ A + A m −

M = κ 0 + κ 1

γ M M. (2.11)

In the case of a repressor, the contribution of C to mRNA production is much smaller
than that of D ( α 1 α 0 ). Now define κ 0 = α 1 D T and κ 1 =( α 0 − α 1 ) D T ,to

obtain:

θ A
θ A + A m −

M = κ 0 + κ 1

γ M M (2.12)

In either case, the parameter κ 0 ( κ 1 ) denotes a residual or basal activity. For

further modeling aspects see also [ 3 ].
The next step is translation, or protein synthesis from mRNA. This is itself a
sequence of several intermediate steps, and can be modeled as a linear function of
the mRNA, together with a natural decay term:

P = κ 2 M

− γ P P. (2.13)

The translation and transcription system in Eqs. ( 2.10 )-( 2.13 ) can be further sim-
plified by using the fact that mRNA degrades faster than protein, or γ M γ P .