Biology Reference
In-Depth Information
this criterion. It might, at first glance, seem intuitive to use
the Erd˝s
e
R´nyi networks as the grounds for comparison.
However, a more careful look shows that this is not suffi-
cient. The reason is that the expected frequency of a given
sub-graph is dictated by the degree distribution of the graph
as a whole. Consider, for instance, the tetrahedral module
discussed above. This module consists of four nodes and six
links. In order for such a module to emerge, first we need to
have a node with a degree of at least 3. Then there must be
at least three additional links among this node's nearest
neighbors. The likelihood of the first condition is dictated
by P
function
[101]
. To demonstrate the functional importance
of this motif, we focus on two different versions of the
motif. The first is a coherent feedforward loop, observed in
the arabinose utilization and in the flagella systems of E.
coli
[102
e
103]
, and the second is an incoherent feedfor-
ward loop, which appears in the galactose system of E. coli
[104]
.
In the coherent feedforward loop all the directed links
represent the process of activation. Thus gene x activates
both genes y and z, and yet gene y itself activates z once
again. This might seem redundant, but can be shown to have
important functional implications. Consider the case where
the target gene, z, can only be activated if it receives a signal
from both x and y. Using a computational analogy, we say
that it serves as an AND gate, as it yields a positive output
only when both of its inputs are positive. The motif will
feature a time lag from when x is activated to when z
responds. This is because z will be activated only after
a sufficient concentration of y products has been produced.
The result is that short sporadic expressions of the x gene
will die off before z is ever activated. This motif, therefore
functions as a filter, ignoring stochastic short-term pertur-
bations and responding only to persistent ones. The
complementary feature rises when the target gene serves as
an OR gate. In this case, z is activated by either x or y. Here
the delayed response will appear if x suddenly ceases to be
expressed, in which case z will still remain active for some
time, as long as a sufficient abundance of y's product
persists. Thus the stability of z's expression is assured
against sudden short-term drops in the production of x. This
type of behavior is observed in the flagella system of E. coli,
where a persistent activation of the flagella is maintained
even under transient loss of the input signal
[105]
.
A surprising, but nevertheless prevalent, version of the
feedforward motif is the incoherent feedforward loop.
Here, while x activates both y and z, the link between y and
z is inhibitory. This seemingly contradictory wiring leads to
an interesting functional feature. Consider a sudden acti-
vation of the gene x, due to, say, an external signal. As
a result both y and z will be activated too. For a short time
after x's activation, the expression levels of z will be
constantly rising, owing to its activation by x. However,
after a sufficient amount of y products has been produced,
the expression of z will be suppressed, due to its inhibition
by y. This version of the motif therefore translates
a persistent signal induced by x into a spike of activation of
the target gene, z.
ð
k
Þ
, and the likelihood of the second is determined by
C
. In the broader sense what this means is that the
macroscopic features of the network, given by P
ð
k
Þ
ð
k
Þ
and
C
, are in close relationship with the detailed structure of
its modules. In the context of the current discussion, it
states that the abundance of a given sub-graph is not
independent of P
ð
k
Þ
ð
Þ
. Thus, in order to deem a certain
module as over-represented in a particular network, we
must compare its abundance to that of a randomized
network with the same degree distribution
[94]
. Such
a randomized network can be constructed by randomly
rewiring all the links in the original network, preserving
each node's degree, and hence P
ð
k
Þ
k
, but deleting fine
structure, such as the recurrence of motifs.
Autoregulation and the Feedforward Loop
We now briefly discuss two noted examples of highly
recurring motifs found in transcriptional regulatory
networks. The first motif is the negative autoregulator,
which is one of the simplest and most abundant network
motifs found in E. coli
[95
e
96]
. It includes a single tran-
scription factor, which represses its own transcription.
Graphically, this motif, shown in
Figure 9.4
(b), is simply
a single node loop. It was shown to have two important
functions. The first function is response acceleration.
Compared to alternative regulating processes, such as
protein degradation, the process of autoregulation allows
for a faster response to signals. This was shown both
theoretically and experimentally by employing synthetic
gene circuits in E. coli
[97]
. The second advantage is that
the motif increases the stability of the gene product
concentration against stochastic noise. It therefore reduces
the variations in protein levels between different cells
[98
e
99]
.
Another motif frequently encountered in regulatory
networks is the feedforward loop
[100]
. This motif consists
of three nodes, x, y and z, where x is directly linked to both y
and z, and in addition y is also directly linked to z
(
Figure 9.4
(c),(d)). The direct links can symbolize the
activation or the inhibition of the target gene, or any
combination thereof. Thus eight different versions of this
motif can be constructed, each with a different biological
GOING BEYOND TOPOLOGY
Despite its success, the purely topological approach
possesses inherent limitations in the race to understand
cellular networks. In focusing on topology alone, we have
neglected the fact that not all edges are created equal.
