# Autoregulation

In a transcrption network with $N$ nodes and $E$ edges there are $N$ possible ways to draw an arrow from one gene to itself or the other $N-1$ genes. This possibilities represent a microcanonical ensemble. A gene regulates its own production is called an autoregualating gene (auto=self).
The probability of autoregulation in a random transcription network is \begin{align} p_S=\frac{1}{N} \end{align} The probaility of having $k$ self edges can be described by a binomial distribution \begin{align} P(k)=\begin{pmatrix} E \\ k \\ \end{pmatrix} {p_s}^{k} \cdot (1-p_s)^{E-k} \end{align}
Thus the mean number of network motifs with autoregulation $N_S$ ($N_{Self}$) is simply \begin{align} N_S=\frac{1}{N} \cdot E \end{align} For the well known part of the E. coli transcription network with $N=519$ and $E=424$ we get \begin{align} N_S=\frac{519}{424} \approx 1,2 \end{align} with a standard deviation of \begin{align} \sigma=\sqrt{\frac{E}{N}} \approx \sqrt{1,2} \approx 1,1 \end{align} In the part of the real E. coli network this autogenous control occurs $40$ times. This is more than $30$ standard deviations more often than in the random network. Thus autoregulation is a network motif.
There are two types of autoregulation, one is positive and the other negative. In the real subnetwork of E. coli $34$ self-edges perform negative autoregulation, which means that a gene e.g. Y represses its own production.