Biomedical Engineering Reference
In-Depth Information
We provide below a textual description of the main reactions occurring and
illustrated in Fig.
4
. VEGF binds to the VEGF receptor (VEGFR2) located at the
surface of endothelial cells, which leads to the formation of homodimers of
receptors and the autophosphorylation of their intracellular kinase domains [
46
]
that then activates molecules at the top of signaling cascades. We can distinguish
three main pathways:
• The phosphorylation of PLCc leads to the activation of PKC and then the Raf/
MEK/ERK pathway [
49
] which stimulates cell proliferation [
47
].
• The MAPKAPK/hsp27 pathway is activated through the phophorylation of p38.
This molecular network triggers actin reorganization and cell migration [
51
].
• The phosphorylation of PI3K permits the transformation of PIP2 into PIP3
which can bind to Akt and permit its bi-phosphorylation. Akt- PI-PP will then
phosphorylate Casp9. Casp9 P is then unable to play its role in cellular apop-
tosis, which leads to an improvement of the cell survival [
52
,
53
].
It is then possible to translate the molecular system into a system of ODEs. It is
important to note, however, that this step implies the following hypothesis: the
molecular concentrations are continuous, reactions happen in a homogeneous
medium of a volume large enough, and reactions are deterministic.
The dynamics of reactions can then be modeled using mass action kinetics,
which describes reactions rates as proportional to the concentrations. For example
in the reaction A
þ
B
$
AB
;
where compounds A and B bind to each other to form
a stable complex AB at the rate k
1
and the product AB dissociates itself into A and
B at the rate k
1
:
The speed of the reaction is then given by:
v
¼
k
1
½½
k
1
½
AB
:
For some complex multi-step reactions, like phosphorylations, the Michaelis-
Menten approximation can be used to reduce the complexity of the mathematical
model and conserves the dynamic properties of the reaction. For instance, A
P
þ
B
!
A
P
B
!
A
P
þ
B
P
where A
P
is a catalyzer of the activation of B into B
P
;
can be
simplified to A
P
þ
B
!
B
P
with the corresponding rate equation: v
¼
VA
½ ½
B
K
½
B
:
When a unique catalyzer triggers several reactions, writing the rate equations is
more complicated (the reader may refer to [
50
,
54
] for further details).
In doing so, a differential equation can be associated with each reaction as
follows:
dt
¼
X
v
pro
;
c
i
X
v
con
;
c
i
;
dc
i
where v
pro
;
c
i
and v
con
;
c
i
are respectively the velocities of the reaction producing and
consuming the concentration c of the molecule i.
Writing all equations for the reaction depicted in Fig.
4
leads to a huge system
composed by 39 equations with 78 parameters (reaction rates and Michaelis-
Menten constants). Simulation of the model in principle reproduces the time
evolution of the different elements composing the network following stimulation
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