biologia - Studia

Atm PROTEIN CAN SWITCH OFF THE DNA DAMAGE SIGNAL IN A p53 MODEL
+
Proof: Suppose that ~γ (0)∈ A and
~ γ ( t ) dose not intersects G (g)
. From
lemma 7,
~ γ ( t ) is bounded and defined for t ≥ 0 . Since ~γ +
( t ) = ( z(
t),
w(
t))
∈ A ,
so z (t)
and w (t)
are monotone functions, in fact, z (t)
is increasing and w (t)
is
decreasing. Let lim ( z ( t ), w ( t )) = ( z 1,
w
t → +∞
1)
, then from the mean value theorem
we have
z(
t + 1) − z(
t)
= z&
( c ) = z(
c ) h(
z(
c ), w(
c )),
w(
t + 1) − w(
t)
= w&
( c2)
= −α d
w(
c2)
z(
c2)
x(
c2),
where | c i
− t | ≤ 1 , i = 1,2 . We can let t → +∞ and obtain
0 = lim z(
t + 1) − z(
t)
= z h(
z , w ),
t→+∞
0 = lim w(
t + 1) − w(
t)
= −α
w z
t→+∞
1
1
1
d
1
1
1
1
lim x(
t).
1 1
t→+∞
( z , w 1 1)
The first equation shows that h = 0 which means that
( z1,
w1
) ∈ G(
g)
. Now, we claim that limt
→+∞
x(
t)
≠ 0 , if not, one more time
applying of the mean value theorem yields
n
n
x( t + 1) − x(
t)
= x&
( c)
= z(
c)
+ α1x
( c)/(
k1
+ x ( c))
− γ
1x(
c)
y(
c)
− γ
2x(
c),
where | c − t |≤ 1. Since y (t)
is bounded, so if t → +∞ we obtain 0 = z
1
, which is a
contradiction. Therefore, from the second equation of (3) we obtain that w
1
= 0 ,
hence
~ γ ( t ) converges to a intersection point of G (g)
and z axis. (see figure 4).
(4)
~ t
Fig. 4. Typical behavior of γ ( ) .
Lemma 9: If the curve
~ γ ( t ) enters
−
A then it converges to a intersection
point of G (g)
and z axis, or it is converges to a point (0, w ) on w axis with
w ≤ g(0) .
75