Biomedical Engineering Reference
In-Depth Information
Thus, the situation with regard to models of the signaling system is some-
what mixed. On the one hand, there are several good phenomenological mod-
els available for use in constructing theories of wave pattern dynamics and
streaming response; these will be presented in subsequent chapters. On the
other hand, there is no one model that starts from known biochemistry and
recovers all the desired features. This remains a challenge for future work.
3.4 The cAMP Wavefield
The cAMP signaling system studied in the last Section forms the basis upon
which we can understand the cAMP waves that are formed during the early
hours of the aggregation phase and thereafter guide the cells to the nascent
mounds. To get from the signaling models to models for the spatially-extended
system, we need to add transport of cAMP from cell-to-cell. We will do this
within the context of one of the simpler forms of the MG model, a four variable
reduction that keeps the ATP concentration
α
as a dynamical variable in
addition to the three given above in Equation 3.1,
d
ρ
T
d
t
=
−
f
1
(
γ
)
ρ
T
+
f
2
(
γ
)(
G
−
ρ
T
)
d
α
d
t
k
α
=
ν
−
−
Φ
(
G, ρ
T
,γ,α
)
d
β
d
t
(
k
i
G
3
+
k
t
)
β
=
qΦ
(
G, ρ
T
,γ,α
)
−
d
γ
d
t
k
t
h
β
k
e
G
2
γ
+
D
2
γ
=
−
∇
(3.2)
where
k
1
+
k
2
γ
1+
γ
f
1
(
γ
)=
k
1
L
1
+
k
2
L
2
cγ
1+
cγ
f
2
(
γ
)=
σαG
2
(
λθ
+
G
2
Y
2
)
1+
αθ
+
G
2
Y
2
(1 +
α
)
.
Φ
(
G, ρ
T
,γ,α
)=
(3.3)
ρ
T
γ
1+
γ
.ForATP,
ν
and
k
are the production and decay rates,
respectively. All the various constants appearing in Equation 3.3 are given in
Table 3.1. At the moment, we will imagine
G
to be fixed; the dynamical system
is excitable (i.e., can propagate waves) for 1
.
0
<G<
1
.
11, self-oscillatory
for 1
.
11
<G<
1
.
65, and again excitable for 1
.
65
<G<
2
.
24. Later, we
will introduce a dynamical equation for
G
that changes in time due to the
feedback between the waves and the genetic development program. Finally,
in the MG model, density changes are incorporated by scaling the ratio of
extracellular to intracellular volume by the parameter
δ
, which then affects
and
Y
=
