# Lecture 19

# Autoregulation (continued)

Can we say something quantitative about…

- Noise reduction for autorepression.
- Switching time for auto-activation.

Last lecture we came up with an expression for the steady-state probability distribution for an autoregulating protein, but the expression was unwieldy.

In this class we will use the *linear noise approximation* (LNA) to get a nicer form.

## Linear Noise Approximation

The LNA requires the following:

- Molecule number is large, .
- Noise is small,

We will treat these functions as being functions as continuous variables since for large molecule number, a change of is small and approximately continuous. So we can do a Taylor series expansion about .

After some algebra this leads to the following expression for the probability time derivatives.

The is the Fokker-Planck equation. The first term on the right-side of the equation represents *drift* in the probability density whereas the second term represents *diffusion*. We can further simplify the equality i the following way:

We know that in steady-state the time derivative of the probability function equals zero.

Now lets figure out what that constant is. We know that for our probability distribution to normalizable then it must go to zero as the number of molecules approaches infinity.

Since is a constant then it must equal zero not just out at infinity, but for all values of .

Now lets use the second approximation we stated at the very beginning, that noise is small. In order to use this approximation we will perform a change of variables so that our function are expressed in terms of instead of .

For probability to be conserved the following must be true:

We can do a series expansion since is small.

Ignoring terms in our expansion of order or higher gives the following:

We can do the same sort of thing for .

If you want to take my word for it, the solution of should look something like…

And after doing some math we can the follwoing relationship:

The denominator is a positive quantity since is negative.

Now let’s solve for the variance.

## Switching times in a Bi-stable System

The probability distribution of a bi-stable system always has two local maximums indicating separate locations of stability in the system. Let *A* represent the location of the first local maximum and *B* represent the location of the second local maximum. We also want to define the local minimum in the probability distribution betwenn *A* and *B* and we will designate that minimum as *C*.

Let’s seperate the probability distribution into the parts. One part is the probability of being below point *C* (associated with stable point *A*) and the other is the probability of being above point *C* (associated with stable point *B*).

The average time spent in each respective bi-stable region is proportional to its probability distribution.

From *Kramer’s escape rate* we can relate the rate system switching from *A* or from *B* to the energies associated with being in those states.

Let’s assume that the system starts a point *A*. Calculate the escape rate to the other stable point, *B*.

Let’s calculate $$\beta | U’’ | $$ using Kramer’s escape rate. |

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