Biology Reference
In-Depth Information
1111
0111
1110
0001
0011
1011
0101
1101
1001
0110
1000
0100
1010
0010
1100
0000
FIGURE 2.8
The state space diagram for the Boolean model from Eqs. (
2.53
).
x
1
,
x
2
,...,
x
n
)
an initial condition
(
at time
t
=
0, the state of the system at time
x
1
,
x
2
,...,
x
n
)
where
x
j
x
1
,
x
2
,...,
x
n
)
t
=
1 will be
(
=
f
x
j
(
for
j
=
1
,
2
,...,
n
.
1 is given by
x
t
+
1
j
x
1
,
x
2
,...,
x
n
),
The transition between time
t
and
t
+
=
f
x
j
(
j
=
1
n
. As an example, consider the set of transition functions defined in
Eqs. (
2.53
), with the initial condition
,
2
,...,
x
1
,
x
2
,
x
3
,
x
4
)
=
(
(
0
,
0
,
1
,
1
)
. Substituting
these values into Eqs. (
2.53
) yields
x
1
=
x
1
,
x
2
,
x
3
,
x
4
)
=
f
x
1
(
f
1
(
0
,
0
,
1
,
1
)
=
1
,
x
2
=
x
1
,
x
2
,
x
3
,
x
4
)
=
f
x
2
(
f
2
(
0
,
0
,
1
,
1
)
=
1
∧
1
=
1
,
x
3
=
x
1
,
x
2
,
x
3
,
x
4
)
=
f
x
3
(
f
3
(
0
,
0
,
1
,
1
)
=
0
∧
1
=
0
,
(2.54)
x
4
=
x
1
,
x
2
,
x
3
,
x
4
)
=
f
x
4
(
f
4
(
0
,
0
,
1
,
1
)
=
0
∧
0
∧
1
=
0
.
x
1
,
x
2
,
x
3
,
x
4
)
=
(
Next, the values
(
1
,
1
,
0
,
0
)
are used to evaluate the transition
x
1
,
x
2
,
x
3
,
x
4
)
=
(
functions again, producing
(
0
,
0
,
0
,
0
)
. A subsequent iteration
x
1
,
x
2
,
x
3
,
x
4
)
=
(
of the functions
f
x
j
returns the same values
(
0
,
0
,
0
,
0
)
.Wesay
that we have computed the trajectory
(
0
,
0
,
1
,
1
)
→
(
1
,
1
,
0
,
0
)
→
(
0
,
0
,
0
,
0
)
→
(
, and that (0,0,0,0) is a
fixed point
for the Boolean network. A visual
representation of this process for all 16 different sequences
0
,
0
,
0
,
0
)
x
1
,
x
2
,
x
3
,
x
4
)
composed
of 0s and 1s as initial states is given by the directed graph in Figure
2.8
. This graph
depicts the
transition diagram
of the Boolean model. Loops of length larger than
one on the transition diagram correspond to limit cycles. Figure
2.8
shows that the
Boolean model from Eqs. (
2.53
) has two fixed points (0,0,0,0) and (1,1,1,1) and no
limit cycles.
The fixed points in a Boolean model are equivalent to the steady states
of a differential equation model.
In Chapter 1 of this volume we considered several models of the
lac
operon, specif-
ically examining their transition functions, state-space diagrams and fixed points. A
common feature of those models was the assumption that all major processes in the
(
Search WWH ::
Custom Search