biologia - Studia

Atm PROTEIN CAN SWITCH OFF THE DNA DAMAGE SIGNAL IN A p53 MODEL
Lemma 4: let γ ( t)
= ( x(
t),
y(
t),
z(
t),
w(
t))
be a compatible solution and
consider the curve
~ γ ( t)
= ( z(
t),
w(
t))
in two dimensional plane of z and w . The
curve
~ γ ( t ) dose not intersect z -axis or it is contained in it. Similarly,
~ γ ( t)
dose
not intersect w -axis or it is a single point on it.
In the rest of this section we use notations γ and ~ γ as they used in lemma
4. Moreover, we assume that initial amounts of the Atm-p protein and the DNA
damage signal are nonzero, that is, the compatible solution γ (t)
has initial
conditions z (0) = z0
> 0 and w (0) = w0
> 0 .
Lemma 5: Let T > 0 and suppose that {γ ~
( t):0 < t < T}
⊂ A . Then,
x ( t)
> 0, y(
t)
> 0 for all 0 < t < T .
Proof: Suppose that there exists t0 ∈ (0, T ) such that for all t ∈ (0 , t 0
) ,
x (t) > 0 and x ( t 0
) = 0 . Then, x (t)
must be non increasing function in a left interval of
t ; specially we have x& ( t 0
) ≤ 0 . But, from the first equation of the model (3) , we have
0
( t0)
= z(
t0
x& ) > 0 which is a contradiction. This means that for all 0 < t < T we have
x (t) > 0 . Similarly, if for all t ∈ (0 , t 0
) , y (t) > 0 and y ( t 0
) = 0 then y (t)
must be
non increasing in a left interval of t 0
; specially we have y& ( t 0
) ≤ 0 . But, from the second
4
4
equation of the model (3) , we have y &( t0)
= α
2
+α3x
( t0)/(
k2
+ x ( t0))
> 0 which is
a contradiction. This shows that for all 0 < t < T we have y (t) > 0 .
Now we study path of ~ γ in A . The forth equation of the model (3) and
the lemma 5 imply that w& < 0 in region A ; therefore, w (t)
is a decreasing map in
A . In order to know behavior of z (t)
, we study the curve z& =0. From the third
equation of the model (3) we have
⎛
=
1
( )/(
1
)
2
/((
0
)(
2
)) = 0.
⎟ ⎟ ⎞
z&
z⎜α
− + − − + +
⎜ 14s
µ z k
444444444444444
s
µ z α
24444444444444444
s
k
d
w k
s
z
3
⎝
h(
z,
w)
⎠
We can solve the equation h ( z,
w)
= 0 with respect to w and find w = g(
z)
, where
g( z)
= − k0d + α
2s
( µ + k1s
− z)/(
α1s
( µ − z)(
k2s
+ z)).
It is easy to see that limz →
+
µ g(
z)
= +∞ . Furthermore, for all
~γ
~γ
( t)
= ( z(
t),
w(
t))
∈ A with w ( t)
< g(
z(
t )) we have z& (t) < 0 . Similarly, for all
( t)
= ( z(
t),
w(
t))
∈ A with w ( t)
> g(
z(
t )) we have z& (t) > 0 . On the other hand,
from lemma 5, if ~γ ( t)
∈ A , then x& > 0 , hence w& < 0 . This shows that if the curve
~γ ( t)
∈ A is close enough to the line z = µ , then z& < 0 . Since
~ γ ( t ) dose not
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