Biology Reference
In-Depth Information
external lactose, while
L
e
=
0
;
L
e
high
=
0 and
L
e
=
1
;
L
e
high
=
1 represent low and
high external lactose, respectively.
As in the case of the 3 variable differential equation model from Table
2.1
,itwas
shown in [
9
] that the delays in the equations do not affect the bistable behavior of
the system. With this motivation, our first Boolean model for the 5-variable system
ignores the delays. We also choose to introduce a single “old” variable for each of the
model variables (see Section
2.7
for more on this). The transition functions for this
Boolean model are:
f
M
=
A
∨
(
M
∧
M
old
)
f
B
=
M
∨
(
B
∧
B
old
)
f
M
old
=
A
∧
M
f
B
old
=
M
∧
B
f
A
=
(
B
∧
L
)
∨
(
L
∧
L
e
h
igh
)
∨
((
A
∧
A
old
)
∧
B
)
f
P
=
M
∨
(
P
∧
P
old
)
f
A
old
=
((
B
∨
L
)
∧
(
L
∨
L
e
high
)
∧
A
f
P
old
=
M
∧
P
f
L
=
((
P
∧
L
e
)
∨
L
e
high
)
∨
((
L
∧
L
old
)
∧
(
B
∧
P
))
f
L
old
=
((
∨
L
e
)
∧
L
e
high
)
∧
P
L
(2.59)
The justification for the transition functions of the “old” variables is the same as
before: no new quantities are made and previously available amounts have not already
been reduced once by dilution and degradation. Also note that the justifications for
the transition functions for
M
and
B
are as those for the Boolean models from the
previous section.
Transition Equation for A
: There are three possible ways for
A
to be available at
time
t
-galactosidase is available together with at least medium
concentration of lactose. Under those conditions
+
1: (i) At time
t
,
β
β
-galactosidase will convert lactose
into allolactose by time
t
1. (ii) At time
t
, internal lactose is already present and the
concentration of extracellular lactose is high; this will ensure that by time
t
+
+
1, the
intracellular lactose concentration is high enough to find available trace amounts of
β
-galactosidase and initiate enough conversions into allolactose to bring the concen-
tration of allolactose above the discretization threshold. (iii) At time
t
the concentration
of
A
is high enough not to be reduced below the threshold level at time
t
1 due
to dilution or degradation and that it will not be lost to conversion into glucose and
galactose (mediated by
+
1.
Transition Equation for L
: There are three possible ways for
L
to be available at
time
t
β
-galactosidase) by time
t
+
1: (i) At time
t
, permease above the discretization threshold is available
together with at least medium levels of external lactose. Under those conditions, the
permease will facilitate the transport of external lactose into the cell. (ii) At time
t
, the concentration of
P
may be below the discretization threshold but the high
concentration of external lactose at time
t
will guarantee that the external lactose
enters the cell and brings, by time
t
+
1, internal lactose above the threshold. (iii)
At time
t
there is still high enough concentration of
L
that will not be lost due to
the reversible nature of permease-facilitated lactose transport and also not lost to
conversion into allolactose mediated by
B
.
+
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