Biology Reference
In-Depth Information
The
lac
permease will then bring the extracellular lactose inside the cell, ensuring the
presence of intracellular lactose at time
t
+
1; or (ii) Internal lactose is available at
time
t
but no
-galactosidase (as r
ep
resented again by the polypeptide
E
) is available
at time
t
to metabolize it
β
(
L
(
t
)
∧
E
(
t
)
=
1
)
. Thus, the internal lactose will still be
present at time
t
+
1.
1.3.4
Initial Testing of the Boolean Model of the
Lac
Operon from
Eqs. (
1.4
)
Now that we have defined a model of the
lac
operon, it must be analyzed and validated.
Since a model can never be shown to be correct in an absolute sense and is always just
an approximation of the actual system, its validation is only appropriate within the
context of the questions that the model is developed to help answer. In our case, the
simple model we have created should be able to describe the basic qualitative dynamic
properties of the
lac
operon. Thus, at a minimum, our model should show that the
operon has two steady states, On and Off. When extracellular glucose is available,
the operon should be Off. When extracellular glucose is not present and extracellular
lactose is, the operon must be On. We next demonstrate that our model satisfies these
conditions.
Recall that the operon is On when mRNA is being produced
(
M
=
1
)
. When
mRNA is present, the production of
lac
permease, and
β
-galactosidase is also turned
(
,
,
)
=
(
,
,
)
on. This corresponds to the fixed-point state
. On the other hand,
when mRNA is not made, the operon is Off. This also means no production of
lac
permease, and
M
E
L
1
1
1
β
-galactosidase. This corresponds to the fixed-point state
(
M
,
E
,
L
)
=
(
.
For the Boolean model of the
lac
operon from Eqs. (
1.4
), there are four possible
combinations for the values
L
e
and
G
e
of the model parameters:
L
e
0
,
0
,
0
)
=
0
,
G
e
=
0
;
L
e
1. For each one of these
pairs of values we can determine the state space transition diagram of the model
from the update functions in Eqs. (
1.4
). The results are shown in Figure
1.7
. Notice
that according to the model, the operon is On only when external glucose is unavail-
able and external lactose is present. In all other cases, the operon is Off. These model
predictions reflect exactly the expected behavior of the
lac
operon based on the under-
lying regulatory mechanisms described earlier. This means that our initial model is
capable of describing the most fundamental behavior of the
lac
operon system and
captures the main qualitative properties of
lac
operon regulation.
Exercise 1.8.
Verify that the space state diagram for the Booleanmodel described by
Eqs. (
1.4
) is as presented in Figure
1.7
b. Notice that for some values of the parameters,
the transition functions simplify significantly when we apply short-circuit evaluation
for the appropriate Boolean expressions. For instance, when
G
e
=
0
,
G
e
=
1
;
L
e
=
1
,
G
e
=
0; and
L
e
=
1
,
G
e
=
=
1, the equations
for the transition functions will be:
x
M
(
t
+
1
)
=
f
M
(
t
+
1
)
=
1
∧
(
L
(
t
)
∨
L
e
(
t
))
=
0,
regardless of the
va
lues of
L
and
L
e
and
x
L
(
t
+
1
)
=
f
L
(
t
+
1
)
=
1
∧
((
E
(
t
)
∧
L
e
(
t
))
∨
(
L
(
t
)
∧
E
(
t
)))
=
0, regardless of the values of
L
,
E
, and
L
e
.
Search WWH ::
Custom Search