Biomedical Engineering Reference
In-Depth Information
will be referred to as one of balanced inactivation, for reasons that will soon
become apparent. The goal of the model is to explain the fluorescent response
data, specifically how the external signal leads to a roughly symmetric tran-
sient followed by a robust decision. It assumes that the cell does not have
perfect adaptation.
The basic idea of our model relies on the general notion of rapid local
activation followed by non-local inhibition, originally proposed by Parent and
Devreotes [38] and explicitly modeled by Iglesias and Levchenko [39, 40, 41].
The key is the mode of inhibition; we will assume that the inhibitor molecule is
produced in exactly the same numbers as the activator, and that the inhibitor
acts by binding to the activator and sequestering it. In more explicit detail,
we will first ignore the specifics of the binding process of the chemoattractant
to the receptors and will assume that the concentration of activated receptors,
S
, is directly related to the chemoattractant concentration. These activated
receptors produce a membrane-bound species
A
and a cytosolic species
B
at
equal rates
k
a
; this is absolutely crucial. The cytosolic species diffuses inside
the cell and can attach itself to the membrane at a rate
k
b
, where we will
label it
B
m
. There, it can inactivate
A
with rate
k
i
, a process that will be
assumed to be irreversible. Thus,
A
plays the role of activator and
B
plays
the role of inhibitor in our model. Finally, we will allow for the spontaneous
degradation of
A
and
B
m
at rates
k
−a
and
k
−b
, respectively. These rates will
be taken to be small compared to both the activation and the recombination.
In mathematical terms, these reactions are written as:
∂A
∂t
=
k
a
S
−
k
−a
A
−
k
i
AB
m
at the membrane
∂B
m
∂t
=
k
b
B
−
k
−b
B
m
−
k
i
AB
m
at the membrane
(3.5)
∂B
∂t
2
B
=
D
∇
in the cytosol
with a boundary condition for the outward pointing normal derivative of the
cytosolic component:
D
∂B
∂n
=
k
a
S
−
k
b
B
(3.6)
Let us first look at the steady state solution resulting from a uniform
stimulus,
S
0
. In this case, the inhibitor concentration
B
becomes uniform and
equals:
B
0
=
k
a
S
0
k
b
(3.7)
Substituting into the remaining two equations, we find
B
m,
0
=
k
b
B
0
/
(
k
i
A
0
+
k
−b
) where the concentration of the activator
A
0
is given by
k
−
a
k
−
b
+
(
k
−
a
k
−
b
)
2
+4
k
a
k
i
k
−
a
k
−
b
S
0
2
k
i
k
−a
A
0
=
−
(3.8)
(
k
a
k
i
S
)
/
(
k
−a
k
−b
)
is large, with
S
equaling the average value of
S
along the cell wall (here being
We always imagine that the dimensionless parameter
K
≡
