Biology Reference
In-Depth Information
Figure
1.4
the model variables are
x
1
=
M
,
x
2
=
E
, and
x
3
=
L
. The functions
defining the systems transition will then be denoted as
f
x
1
=
f
M
,
f
x
2
=
f
E
,
f
x
3
=
f
L
.
Deciding on the actual expressions for these Boolean functions is a critical step
in the definition of a model. It should take into account all relevant parts of the
known biological mechanism of the system being modeled. In the next section we
will consider ways of doing this for the
lac
operon.
Exercise 1.2.
How many possible states does a system of
n
Boolean variables
have if:
a.
n
=
2?
b.
n
=
4?
c.
n
=
5?
d.
n
=
10?
e.
n
=
100?
Exercise 1.3.
For each of the Boolean expressions below, compute the value if
x
1
=
1. In the cases where the expressions are not
fully parenthesized, be mindful of the operation precedence.
1
,
x
2
=
1
,
x
3
=
0
,
x
4
=
0
,
x
5
=
a.
x
3
∧
x
5
b.
(
x
3
∨
x
5
)
∧
(
x
2
∨
x
1
)
∨
x
4
c.
(
x
2
∧
x
1
)
∨
(
x
2
∧
x
3
)
∧
x
5
d.
(
x
1
∨
x
2
)
∧
(
x
2
∧
x
3
)
.
Exercise 1.4.
Show that
(
0
,
0
,
0
)
is a fixed point for the system described by
Eqs. (
1.1
) and (
1.2
).
Exercise 1.5.
form a cycle of length two. That
is, show that Eqs. (
1.1
) and (
1.2
) lead to the following trajectories that alternate
between these two states:
Show that
(
0
,
1
,
0
)
and
(
1
,
0
,
0
)
(
0
,
1
,
0
)
→
(
1
,
0
,
0
)
→
(
0
,
1
,
0
)
→
...
and
(
1
,
0
,
0
)
→
(
)
→
...
Exercise 1.6.
For the system whose transition functions are given by Eqs. (
1.1
) and
(
1.2
), compute the trajectory of
0
,
1
,
0
)
→
(
1
,
0
,
0
Exercise 1.7.
Verify that the wiring diagram and the state space diagram for the
Boolean network defined by Eqs. (
1.3
) are as depicted in Figure
1.6
.
(
1
,
0
,
1
)
.
1.3.3
Creating a Boolean Model of the
Lac
Operon
Once a choice for model variables has been made and the wiring diagram has been
constructed, the Boolean transition functions for the model are determined from the
wiring diagram and additional available information or assumptions regarding the
variable interactions. When multiple quantities impact a third, we need to know if the
simultaneous presence of each of these quantities is necessary to exert the effect or
if the presence of just one of them would be enough. In the first case, the transition
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