# Autoregulation

We will continue our discussion of autoregulating genetic networks. Specifically, we will continue talking about autorepression.

## Autorepression

Recall that there are two advantages of autorepression.

1. Reduces response time.
2. Buffers fluctuations.

### Noise Reduction (buffer fluctuations)

Consider a small fluctuation $\epsilon$ from the steady-state concentration $x^*$.

Let’s make a Taylor series expansion of the time derivative…

We neglect terms of order $\epsilon^2$.

In our case $% $. So fluctuations in the number of proteins will decay exponentially leading to a stable stead-state concentration of protein.

Now lets derive the stochastic model for autorepression gene regulation.

# Stochastic Model of Autorepression

In steady the time derivative of the probabilities will be equal to zero.

### For $\ n=2$

A pattern emerges…

So we now have an exact expression for the probability distribution. However, this form is very unwieldy.

# Autoactivation

This is the opposite of autorepression we can hypothesize that it has the following features:

1. Increases the response time.
2. Increases fluctuations due to noise.