Biology Reference
In-Depth Information
lac
operon incorporating inducer inclusion and catabolite repression. The networks
for those models are then reduced by this method to smaller models that no longer
include those regulatory mechanisms but preserve the bistable behavior of the system.
In the Boolean models proposed in this chapter we use “old” variables to track
the dilution and degradation of concentrations that require multiple steps of time for
reduction below the discretization threshold. In the context of the notation used in
Section
2.5
, if a variable
Y
regulates the production of
X
, which (if no new amounts
are produced) would degrade below threshold levels in
n
steps, the Boolean model
will include the following motif:
f
X
=
Y
∨
(
X
∧
X
old
(
n
)
)
. . .
(2.62)
f
X
old
(
1
)
=
Y
∧
X
f
X
old
(
n
−
1
)
=
Y
∧
X
old
(
n
−
2
)
f
X
old
(
2
)
=
Y
∧
X
old
(
1
)
f
X
old
(
n
)
=
Y
∧
X
old
(
n
−
1
)
...
f
Y
=
...,
We can reduce the network by eliminating
X
old
(
n
)
. To do this, substitute its tran-
sition function
in place of
X
old
(
n
)
in the transition function of X:
f
X
=
Y
∨
(
X
∧
Y
, resulting in exactly the
same structural motif with one less “old” variable. Variables
X
ol
d
(
n
−
1
)
,...,
∧
X
old
(
n
−
1
)
)
, which simplifies to
f
X
=
Y
∨
(
X
∧
X
old
(
n
−
1
)
)
X
old
(
2
)
ca
n be eliminated similarly, leading to the reduction
f
X
=
Y
∨
(
X
∧
X
old
(
1
)
),
f
X
old
(
1
)
=
Y
. When applied to the Boolean models in this chapter, this reduc-
tion process indicates that the bistability property of the system does not depend
on the number of “old” variables used in the model. We finally note that if we
choose to also eliminate
X
old
(
1
)
∧
X
,
f
Y
=
...
from the model, this leads to the transition equations
, reflecting a situation in which
X
is considered completely
stable (a situation in which the combined degradation and dilution rate for
X
is
infinitely small and thus, the growth rate is negligible), which is biologically unreal-
istic. Thus, at least one “old” variable is
needed.
Exercise 2.12.
Confirm that
Y
f
X
=
Y
∨
X
,
f
Y
=
...
The use and treatment of “old” variables here differ from the approach introduced
in [
16
]. We stipulate that an “old” variable has value 1 at time
t
∨
(
X
∧
Y
∧
X
old
(
n
−
1
)
)
=
Y
∨
(
X
∧
X
old
(
n
−
1
)
)
.
1 only when
conditions for new production are not met at time
t
and
when previously produced
amounts available at time
t
have not already been reduced by a certain factor due
to dilution and degradation. In [
16
], an “old” variable has value 1 at time
t
+
1
when conditions for new production are not met at time
t
. Our approach provides a
mechanism to track the level of reduction of
X
: since only one
X
old
(
k
)
+
could be equal
to 1 at each time step,
X
old
(
k
)
(
1 means that at time
t
the concentration of
X
has
already been reduced exactly
k
times
t
)
=
.
Thus, when delay variables are added to our Boolean model from Eqs. (
2.59
) (see
Exercise
2.9
) the resulting model provides a Boolean approximation of the 5-variable
differential equation model from Table
2.57
that differs from the Boolean approxima-
tion of the same model presented in [
16
]. Both models capture the bistable behavior
of the lactose operon but the model in [
16
] generates several fixed points that are not
(
k
=
1
,
2
,...,
n
)
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