Biology Reference
In-Depth Information
function will use the operator AND, and in the second, it will use the operator OR.
We illustrate this by defining a set of transition functions consistent with the wiring
diagram in Figure
1.4
for the
lac
operon.
We make the following additional model assumptions. Although some of them
may appear too restrictive, keep in mind that they can always be modified or removed
after the initial testing and analysis of the model:
Transcription and translation require one unit of time. This means that when all
necessary conditions for the activation of the molecular mechanism are present at
time
t
, the protein production will occur and the product will be available at time
t
1.
Degradation of all mRNA and proteins occurs in one time step. When conditions
for making new mRNA or proteins are not met at time
t
, all previously available
amounts of mRNA and proteins have already fallen below the threshold levels and
will be absent at time
t
+
+
1.
β
In the presence of
-galactosidase, lactose metabolism occurs in one time step. If
β
lactose and
-galactosidase are available at time
t
but the conditions for bringing
new lactose through the cell membrane from the external medium are not met at
time
t
, by time
t
1 all of the lactose is converted to glucose and galactose, and
lactose is not available at time
t
+
+
1.
With this, we define the transition functions (update rules) for the “minimal”
lac
operon model based on the wiring diagram presented in Figure
1.4
as:
x
M
(
t
+
1
)
=
f
M
(
t
+
1
)
=
G
e
∧
(
L
(
t
)
∨
L
e
(
t
))
x
E
(
t
+
1
)
=
f
E
(
t
+
1
)
=
M
(
t
)
x
L
(
t
+
1
)
=
f
L
(
t
+
1
)
=
G
e
∧
((
E
(
t
)
∧
L
e
(
t
))
∨
(
L
(
t
)
∧
E
(
t
))).
(1.4)
We explain these equations next.
Boolean functions for M:
The update rule states that for mRNA to be present at
time
t
1, there should be no external glucose at time
t
, and either internal or external
lactose should be present. When external glucose is present at time
t
+
(
G
e
(
t
)
=
1
)
,no
mRNA will be available at time
t
+
1
(
M
(
t
+
1
)
=
0
)
. Also, when there is no external
glucose at time
t
(
G
e
(
t
)
=
0
)
and there are high concentrations of lactose inside the
cell at time
t
, lactose concentrations in the
cell will be sufficiently high (above threshold levels) to cause mRNA production and
make mRNA available at time
t
(
L
(
t
)
=
1
)
or outside the cell
(
L
e
(
t
)
=
1
)
.
Boolean function for E:
With mRNA available at time
t
+
1
(
M
(
t
+
1
)
=
1
)
(
M
(
t
)
=
1
)
, the LacZ
polypeptide will be produced and available at time
t
.
Boolean function for L:
When external glucose is available at time
t
+
1
(
E
(
t
)
=
1
)
,
no lactose will be brought into the cell from the medium and, thus, no internal lactose
will be available at time
t
(
G
e
(
t
)
=
1
)
+
1
(
L
(
t
+
1
)
=
0
)
. When external glucose is absent from the
medium at time
t
1 when at
least one of the following conditions is satisfied: (i) External lactose and
lac
permease
(as represented by the polypeptide
E
) are both present at time
t
(
G
e
(
t
)
=
0
)
, external lactose will be available at time
t
+
(
E
(
t
)
∧
L
e
(
t
)
=
1
)
.
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