biologia - Studia

O. RABIEIMOTLAGH, Z. AFSHARNEZHAD
itself and it is described with a Hill coefficient n ∈ N which determines the degree
of cooperativity of the ligand of p53 binding to the enzyme or receptor (Hill, 1910).
The third term represents the active process of ubiquitination of p53 by Mdm2 and
the fourth term represents the degradation of p53 independently of Mdm2.
Similarly, in the second equation above, α
2
shows the production rate of Mdm2,
and second term with coefficient α
3
represents the activation of Mdm2 by p53
with Hill coefficient 4 , and the third term represents the degradation of Mdm2
(Chicharmane et al., 2007; Vogelstein et al., 2000; McLure and Lee, 1998).
Another factor in the cell cycle process which acts as a damage sensor,
signaling presence of DNA damage, is the Atm protein (Banin et al., 1998: Kurz
and Lees-Miller, 2004), The Atm-p protein, a phosphorylated form of Atm,
associates in the cell cycle process by controlling production rate of p53. This is
the motivation for improving the model (1) by the new model
n
n
x&
= z + α1x
/( k1
+ x ) −γ
1xy
−γ
2x
(2)
4
4
y&
= α
2
+ α3x
/( k2
+ x ) − γ
3y,
where z (t)
denotes [Atm-p] (t)
. Experimental observations show that amount of
Atm and Atm-p holds for the conservative equation [Atm] (t)
+[Atm-p] (t)
= µ ,
for a positive real constant µ . Also, they are in a reaction for which the
phosphorylated form Atm-p promotes further phosphorylation of Atm, hence, once
a small amount of Atm-p is produced, it leads to further increase in Atm-p, until all
the Atm is converted into its phosphorylated form. This should occur only when a
significant DAN damage occurs. So, this process is modelled by the equation
z&
= α1s z(
µ − z)/(
k1s
+ µ − z)
−α
2sz/((
k0d
+ w)(
k2s
+ z)).
Here, w (t)
=[Dmg] (t)
denotes the DNA damage signal, representing amount of
DNA damages, and µ − z (t) = µ − [Atm-p] (t)
denotes amount of the Atm protein [2].
The first term of the above equation, with coefficient α
1s
shows phosphorylation rate of
the Atm protein and the second term, with coefficient α
2s
shows dephosphorylation
rate of the Atm protein. The parameter k 1
controls dependence of phosphorylation
rate of the Atm protein to amount of Atm. Similarly, the parameters
s
k 2 s
and k 0 d
control dependence of dephosphorylation rate of the Atm protein to amount of
Atm-p and damage signal.
In order to complete the chain of this interaction, finally, we have to take in
account that DNA is repaired by combined actions of Atm-p and p53 which can be
modelled as below .
w&
= −α wzx.
d
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