Biology Reference
In-Depth Information
x1
0 0 0
0 0 1
1 0 1
0 1 0
x2
0 1 1
1 1 0
1 0 0
x3
1 1 1
FIGURE 1.5
Wiring diagram and the state space transition diagram for the Boolean dynamical system
in Example
1.2
given by Eqs. (
1.2
). Graphs produced with DVD [
12
].
of prece
de
nce, we
co
mpute
A
=
1
;
(
D
∨
B
)
∧
A
=
1;
(
D
∨
B
)
∧
A
∧
C
=
1; and
finally
1.
Exercise 1.1.
Consider the Boolean variables
w, x , y, z
and the Bo
olean
function
f
of
those variables defined by the expression
f
(
D
∨
B
)
∧
A
∧
C
∨
B
=
(w,
x
,
y
,
z
)
=
(
x
∨
y
)
∧
(w
∧
z
)
. Determine
the values of: (1)
f
Modeling the dynamics of a Boolean network: Transition functions.
Assume the
wiring diagram contains
n
Boolean model variables (also called
nodes
) denoted by
x
1
,
(
1
,
1
,
0
,
1
)
;
(
2
)
f
(
0
,
1
,
0
,
1
)
;
(
3
)
f
(
1
,
1
,
1
,
1
)
;
(
4
)
f
(
0
,
0
,
0
,
0
)
.
x
2
,...,
x
n
. Each of these
n
variables can take a value 0 or 1, resulting in a set of
n
n
-tuples
V
, containing
2
n
elements, representing all possible states for the model variables. Time is dis-
crete and can only take values
t
={
0
,
1
}
={
(
x
1
,
x
2
,...,
x
n
)
|
x
i
∈{
0
,
1
}
,
i
=
1
,
2
,...,
n
}
=
0
,
1
,
2
,...
The values of the nodes
x
1
,
x
2
,...,
x
n
=
x
i
(
)
change with time and we write
x
i
for the value of the variable
x
i
at time
t
.
Thus, at each time step
t
, the system is represented by a binary
n
-tuple from the set
V
,
where each component stands for the value of the respective Boolean variable at time
t
. The rules for transitioning between the states at each time step are described by
n
functions
f
x
i
,
t
n
, one function for each model variable. The Boolean
expression defining the function
f
x
i
, written in terms of the Boolean operations AND,
OR, and NOT, describes in what way the values of the variables
x
1
,
i
=
1
,
2
,...,
x
2
,...,
x
n
at time
t
affect the value of the variable
x
i
at time
t
+
1. Thus, for any value of
t
=
0
,
1
,
2
,...,
the system “update” for variable
x
i
from time
t
to time
t
+
1 is determined by
x
i
(
t
+
1
)
=
f
x
i
(
x
1
(
t
),
x
2
(
t
), . . .
x
n
(
t
)),
i
=
1
,
2
,...,
n
. The functions
f
x
i
,
i
=
1
n
,arethe
transition functions
(also called
update rules
or
rules
)forthe
model. The updates we will be using here are synchronous, meaning that all variables
x
i
are computed first for time
t
and then used to evaluate the functions
f
xi
.Ifwe
write
x
,
2
,...,
=
(
x
1
,
x
2
,...,
x
n
)
and
f
(
x
)
=
(
f
x
1
(
x
), . . . ,
f
x
n
(
x
))
,the
state space
of the
model is defined by the directed graph
{
V
,
T
}
, where the set
T
={
(
x
,
f
(
x
))
|
x
∈
V
}
represents the set of edges.
Example 1.2.
Assume we have a Boolean network containing three Boolean vari-
ables
x
1
,
x
2
and
x
3
, assume that the wiring diagram for the network is depicted in
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