Biomedical Engineering Reference
In-Depth Information
Table 3.2
Boolean regulating
functions for
Synthetic
5-Gene Network
Gene
Regulating Function
x
1
x
2
+
x
4
+
x
5
x
2
(
x
1
x
3
x
5
)
x
3
x
1
x
2
x
4
x
4
x
1
(
x
3
x
5
+
x
3
x
5
)
+
x
1
(
x
3
x
5
+
x
3
x
5
)
x
5
(
x
1
+
x
2
+
x
3
)
Table 3.3
Boolean regulating
functions for
p53
network
Gene
Regulating Function
dna_dsb
(DNA
damage is an external signal)
ATM
Wip
1(
A
TM
+
dna
_
dsb
)
p
53
Mdm
2(
ATM
+
Wip
1)
Wip
1
p
53
Mdm
2
ATM
(
p
53
+
Wip
1)
number of gene expression observations, but in our tests, each SAT operation is
less than 1 s. Accordingly, All-SAT runtime takes approximately
n
seconds for
n
satisfying solutions.
3.4.2
Method Sensitivity to Input
To investigate how the number of solutions (the number of satisfying BNs) depends
on the number of available gene expression observations, we measure the sensitivity
of our SAT algorithm in the following manner. For a GRN with
n
genes, there are 2
n
gene expression states in total, which completely determines the GRN. To test the
sensitivity of the number of obtained solutions to the number of gene expressions
observations
i
, we randomly select
i
gene expressions from the 2
n
total, and run our
algorithm to see how many surviving solutions there are. Because the number of
surviving solutions can change depending on the specific gene expressions selected,
we resample
x
times, and find the mean number of satisfying solutions among the
x
samples. We repeat this process for different values of
i
between 1 and 2
n
.
In Fig.
3.3
, we show the sensitivity of the algorithm by plotting of the average
number of satisfying solutions against
i
(the number of gene expression observations).
For each value of
i
, we re-sampled
x
100 times, and all satisfying solutions were
recorded. The mean number of solutions is plotted.
From the plot, we observe that as additional gene expressions are included in
the algorithm, the solution space reduces exponentially until only a few surviving
solutions remain. At this point, adding more gene expressions do not significantly
change the size of the solution space. In both examples, the inflection point appears
to be
i
=
16 (roughly half the total number of gene expression observations). These
plots show the importance of including additional gene expressions in reducing the
size of the solution space.
These results show that our method works well for GRNs with fewer genes.
For a network with large number of genes, a corresponding large number of gene
=
