# Network Motifs

We have already examined the one node motif of auto-regulation, let’s now explore 3 node motifs.

Let $n$ be the number of nodes and $g$ be the number of edges. For a 3 node motif $n=3$ and the number of edges can vary from 2 all the way to 6.

## Random Network

What are our expectations for a random network with $N$ nodes and $E$ edges? Let’s try to figure out the expected number of sub-graphs of a particular type, $\bar{s}$.

### Calculate $\bar{s}$

First, we need to choose $n$ nodes from a total of $N$.

Knowing this, what is the probability of choosing the desired edge placements?

Assuming as $N$ increases the ratio $E/N$ remains fixed.

For a 3 node, $n=3$ motif this yields the following results.

g $\ \ \ \ \ \bar{s}$
2 $\sim N$
3 $\sim N^0$
6 $\sim N^{-3}$

In the case of $g=3$ the number of sub-graphs does not change $N$ 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.

# Feedforward Loop

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).

## C1 Loop

Assume the concentration of X is quickly turned on at time $t=0$.

### AND Logic

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.

### OR Logic

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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Written on April 1, 2015