biologia - Studia

O. RABIEIMOTLAGH, Z. AFSHARNEZHAD
Especially they can not have negative initial values. Therefore, for verifying validity of
the model, we have to consider solutions beginning from positive initial values and
we show that their positive trajectories remain non negative. Furthermore, we
assume that amount of the p53 protein and the Mdm2 protein is initially nonzero,
however, because of biological facts, amount of the [Atm-p] protein or the DNA
damage signal may be initially zero.
Definition 1: A solution of (3) with the initial values x (0) = x0
> 0 ,
y (0) = y0
> 0 , 0 ≤ z (0) = z0
≤ µ , w (0) = w ≥ 0
0 is called a compatible solution.
It will be shown later that positive trajectories of compatible solutions remain non
negative, so they have such adequacy to be called compatible. It is easy to check
that the sets (manifolds) N = R
3 × {0} and N w
= R× {0} × { w}
, w∈ R , are
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invariant for (3), that is, any solution of (3) which intersects N ( or N
w
), is
contained in N ( in N
w
). A solution which is contained in N
w
, for some w∈ R ,
implies that the Atm-p is constantly zero. Similarly, a solution which is contained
in N , implies that amount of damage signal is permanently zero. Although, these
invariant sets may be theorically important, but, we are interested to situations
which damage signal and the Atm-p protein coexist simultaneously. This is why
we do not study these invariant sets here. What makes these sets interesting for us
is that each of them can be the ω -limit set for some compatible solutions.
Therefore, interaction between the DNA damage signal and other proteins is
bifurcated when the ω -limit set of a compatible solution changes from N to N
or conversely from N
w
to N . This change of ω -limit set is biologically important
because, as we will see later, it can be interpreted as switching from DNA healing
process to DNA permanent damage, or conversely from DNA permanent damage
to DNA healing process. This is the reason for introducing bifurcation diagram of
the DNA damage signal by change of ω -limit set of compatible solutions. This is
summarized in the theorem below
Theorem 2: Suppose that Λ = µ ( µ + k ) − 1 s
k1s
k2s
, U = k ( )
1s
µ + k1s
+ k2s
and define
∆ = ( µ + k1 s
) α
2s
− µ k0d
k2sα1s,
Ω = α
2s
U /( α1s
( U − U )( U − k1s
)) − k0d
(1) For Λ ≤ 0 , change of sign of ∆ makes a bifurcation for the DNA
damage signal. Indeed, for ∆ ≤ 0 , compatible solutions converge to N and for
∆ > 0 , compatible solutions converge to N
w
, for some w∈ R .
(2) For Λ > 0 , change of sign of Ω makes a bifurcation for the DNA
damage signal. Indeed, for Ω ≤ 0 , some compatible solutions converge to N and
for Ω > 0 , all compatible solutions converge to N
w
, for some w∈ R .
The proof of the theorem will be done through a few lemmas. The theorem shows that
the bifurcation occurs with respect to the parameters α , i = 1,2 and k js
, j = 0,1,2 ;
is
w