We have already examined the one node motif of auto-regulation, let’s now explore 3 node motifs.
Let be the number of nodes and be the number of edges. For a 3 node motif and the number of edges can vary from 2 all the way to 6.
What are our expectations for a random network with nodes and edges? Let’s try to figure out the expected number of sub-graphs of a particular type, .
First, we need to choose nodes from a total of .
Knowing this, what is the probability of choosing the desired edge placements?
Assuming as increases the ratio remains fixed.
For a 3 node, motif this yields the following results.
In the case of the number of sub-graphs does not change varies. Two different sub-graphs in this category are the feedback loop and the feedforward loop. In the picture below, the sub-graphs in the center represents a feedback loop and the sub-graph on the right is a feedforward loop.
Compared to the expected number of sub-graphs in a random network, feedforward loops are very much over-represented in E.coli.
There are 8 distinct types of feedforward loops. Half of them are incoherent and the other half are coherent.
Let’s examine the behavior of the Coherent type 1 loop (C1).
Assume the concentration of X is quickly turned on at time .
Assume that the signals from X and Y combine in an AND logic fashion to create output Z.
- There is a time delay $\tau$ between X turning on and Z turning on.
- There is no time delay between X turning off and Z turning off.
Assume that the signals from X and Y combine in an OR logic fashion to create output Z.
- There is no time delay between X turning on and Z turning on.
- There is a time delay $\tau$ between X turning off and Z turning off.
Time delays between changes in X and changes in Z may be useful for suppressing noise.
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