Biology Reference
In-Depth Information
framework,
L
=
1 means that the concentration of
L
is at least medium, and
when
L
=
1 and
L
high
=
0, the concentration of
L
is only medium.
L
=
0 and
L
high
=
0 stands for low concentration of
L
;
L
=
1 and
L
high
=
1 represents
high concentration of
L
.
3.
In a Boolean model, delays can be incorporated by introducing additional vari-
ables. The number of additional variables depends on the magnitudes of the
delays and on the choice of the time step for the Boolean model. Assume a
variable
R
regulates the production of
X
but the effect that
R
exerts on
X
is
delayed by time
.
2
is commensurable with
n
time steps in the
model,
n
additional Boolean variables
R
i
,
τ
If the delay
τ
n
, can be introduced
to represent the delayed regulation. The following motif will then be present
in the set of transition functions:
R
1
(
i
=
1
,
2
,...,
t
+
1
)
=
R
(
t
),
R
2
(
t
+
1
)
=
R
1
(
t
)
,…,
R
n
(
t
+
1
)
=
R
n
−
1
(
t
),
X
(
t
+
1
)
=
R
n
(
t
)
. Expressed in terms of the transition
functions, the same can be stated as
f
X
=
R
n
,
f
R
1
=
R
2
,...,
f
R
n
=
R
n
−
1
.
We now use these techniques to create Boolean models that approximate the
Yildirim-Mackey differential equation models from Section
2.4
.
2.6
BOOLEAN APPROXIMATIONS OF THE
YILDIRIM-MACKEY MODELS
2.6.1
Boolean Variants of the 3-Variable Model
We want to build a Boolean model based on the assumptions used in the 3-variable
differential equation model from Table
2.1
. Recall that
A
τ
M
=
A
(
t
−
τ
M
),
M
τ
B
=
M
(
t
−
τ
B
)
. The delays are estimated in [
8
]tobe
τ
M
=
0
.
10 min
,τ
B
=
2
.
00 min.
Various estimates are available from the literature for the loss rates
γ
A
and in some cases the range for the estimates may be rather wide. For instance in
Yildirim and Mackey [
9
], the degradation rate of
A
is estimated at
γ
M
,γ
B
, and
52 min
−
1
,
γ
A
=
0
.
10
−
2
min
−
1
, and in Wong et al. [
17
]
in Yildirim et al. [
8
] the estimate is
γ
A
=
1
.
35
×
10
−
4
min
−
1
is considered. The degradation rates
an estimate as low as
γ
A
=
1
.
8
×
γ
M
10
−
4
min
−
1
.
The effective loss in the concentrations due to dilution is proportional to the growth
rate
411 min
−
1
and
γ
B
are estimated in both [
8
,
9
]tobe
γ
M
=
0
.
,γ
B
=
8
.
3
×
10
−
3
min
−
1
μ
μ
min
=
.
×
μ
max
=
and is estimated to be between
4
5
and
10
−
2
min
−
1
.
As Boolean models are qualitative in nature, the actual values of these constants
are not essential and the only consideration of importance is the comparative order
of their magnitudes. For our first Boolean model below, we have selected the follow-
ing values by considering middle-of-the-range estimates:
3
.
47
×
10
−
2
min
−
1
μ
=
3
×
10
−
2
min
−
1
. Thus the loss terms in the 3-variable model in
Table
2.1
are estimated to be
and
γ
A
=
1
.
35
×
γ
M
=
γ
M
+
μ
=
0
.
441
,γ
B
=
γ
B
+
μ
=
0
.
031, and
γ
A
=
γ
A
+
μ
=
0
.
044. From here the times needed to reduce the concentrations of
2
In the models we consider below,
R
will usually be just one of the regulating factors for
X
but the idea
remains the same.
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