# Noise in Gene Activation

We will continue our discussion of noise in gene activation from the previous lecture.

Last lecture we made a simple model for gene activation and the intrinsic noise in the system. Next we will consider how to derive the extrinsic noise. Recall that X is the activator molecule and Y is the protein produced from transcription / translation.

The noise in the number of Y molecules is due to both intrinsic and extrinsic noise.

Last lecture we calculated the intrinsic noise in the production Y without considering X.

Before we tackle how to compute the extrinsic noise lets look at unregulated production/degradation of Y from a different perspective.

## Generating Function

Recall from last lecture that the probability of being in a state of $n$ Y proteins is the following:

Lets perform a Fourier transform on $p_n$.

We now have an expression for the Generating Function, $G(z)$.

We can use the generating function to solve for the probabilities.

You can evaluate this integral using the Residue Theorem.

The function $G(z)$ generates moments:

We can make another function $F(z)$ from the generating function which also has useful properties.

Lets calculate the time derivative of the generating function.

Now plug in the expressions for the corresponding probability time derivatives we derived in the previous lecture.

We can simplify this expression greatly by coming some terms and simplifying some of the summation terms. Combining $-kp_0$ with the second summation term yields:

Combining $rp_1$ with the third summation term yields:

The first and fourth summation can be expressed as…

And so we go from having an expression which is a sum of infinite ordinary differential equations to one, single partial differential equation.

$z$ is a variable not necessarily equal to zero, therefore the other factor must equal zero.

Using the first moment of the generating function we can solve for the unknown coefficient $A$.

This is the same Poisson Distribution that we have derived before. Using the generating function we can verify some of our earlier results.

# Extrinsic Noise

Now after all that math, lets go back and consider how find the extrinsic noise in our earlier problem. Our system of interest look like the following:

We will simplify our expression for $k_\text{eff}(x)$ by assuming $h=1$ and $% $ and combining all the constants together.

This leads to the following rate equations for both X and Y.

Lets define the joint probability of having $m$ X proteins and $n$ Y proteins in a cell as

## Generating Function

We can calculate the mean and the noise using the following relations

After some math we get…

The term $c$ is known as the covariance.

So the noise has both intrinsic and extrinsic components.

Slow response of Y corresponds to $% $, $\ \sigma^2_\text{ext}\to 0$.

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Written on March 13, 2015