Information Technology Reference
In-Depth Information
Network
Shorthand
y
1
y
2
y
1
y
2
Genes:
f
2
f
1
Promoters
x
1
x
2
x
1
x
2
Products:
Fig. 7.2: Example 1, simple network with two genes. With feedforward and feed-
back connections included (
left
). Equivalent configuration redrawn with bidirec-
tional connections (
right
).
simple example, Example 1 (Fig. 7.2), if genes
y
1
and
y
2
lead to product
x
1
, then by
definition the
x
1
promoter
f
1
affects genes
y
1
and
y
2
. Similarly if gene
y
2
leads to
product
x
2
, gene
y
2
is additionally regulated by product
x
2
. Thus gene
y
1
is regulated
by product
x
1
and gene
y
2
is regulated by products
x
1
and
x
2
. Due to the feedback
model, the activation of gene
y
1
can depend on the level of product
x
2
, because
if products
x
1
and
x
2
are consumed equally, then gene
y
2
will be promoted at the
expense of gene
y
1
.
The network is evaluated until it settles onto a steady state. The solutions are pre-
sented as (
products consumed
)
(
genes expressed
). Since there are two products
and two genes in Example 1, the solution is written in the form
→
(
x
1
,
x
2
)
→
(
y
1
,
y
2
)
.
The steady-state solution for Example 1 is
(
x
1
,
x
2
)
→
(
y
1
=
x
1
−
x
2
,
y
2
=
x
2
)
.The
mathematical equations and their derivation follow:
y
1
(
t
)
x
1
y
2
(
t
)
x
1
x
2
y
2
(
y
1
(
t
+
dt
)=
)
,
y
2
(
t
+
dt
)=
)
+
.
(7.5)
y
1
(
t
)+
y
2
(
t
2
y
1
(
t
)+
y
2
(
t
t
)
The network solution at steady state is derived by setting
y
1
(
t
+
dt
)=
y
1
(
t
)
and
y
2
(
t
+
dt
)=
y
2
(
t
)
and solving these equations. The solutions are
y
1
=
x
1
−
x
2
and
x
1
+
x
2
.
This solution demonstrates efficient outcomes where minimal products are wasted.
Neither
x
1
nor
x
2
is produced if they are not needed. For example, when products
x
1
and
x
2
are equally consumed
y
2
=
x
2
.If
x
1
≤
x
2
then
y
1
=
0 and the equation for
y
2
becomes
y
2
=
(
=
)
then gene
y
2
is expressed and gene
y
1
is
silenced. This occurs because
x
1
and
x
2
equally usurp promoter
f
1
. From the per-
spective of the genes, gene
y
1
has all of its promoters reduced when
f
1
is usurped,
while gene
y
2
still has an independent promoter
f
2
. Gene
y
2
expression becomes
preferred and in the process inhibits gene
y
1
. The final result is that if product
x
2
is
not consumed, gene
y
2
is not expressed.
If only product
x
1
is consumed
x
1
x
2
then only gene
y
1
is expressed avoiding
extraneous products. There are consumption patterns where this configuration is
not efficient. For example, if only product
x
2
is consumed
(
x
2
=
0
)
(
x
1
=
0
)
then only
y
2
is
expressed but extraneous product
x
1
is produced.
