Gene Activation

In the previous lecture we discussed a simple model of protein production with an activator.

\[\text{X}+\text{D} \rightleftharpoons^{k_+}_{k_-} \text{C} \xrightarrow{k}\text{Y}+\text{X}+\text{D}\]

Now lets consider a more involved model.

Additional Considerations

X has its own dynamics

\[\text{D} \xrightarrow{g}\text{D}+\text{X} \\ \text{X} \xrightarrow{s}\emptyset\]

We can write down a rate equation describing the dynamics.

\[\dot{x} = \frac{g}{V}-sx\]

In steady state, the concentration of X is constant.

\[x_\text{ss}=\frac{g}{sV} \\ \therefore \ y_\text{ss}=\frac{f_+(x_\text{ss})}{r}\]

Transcription Factors

Transcription factors (in this case it is X) often oligomerize before binding to DNA. Therefore the dynamics must be modified in order to take this into account.

\[\underbrace{X+X+X+... +X}_h \rightleftharpoons^{k'_+}_{k'_-} X' \\ X'+\text{D} \rightleftharpoons^{k_+}_{k_-} \text{C} \xrightarrow{k}\text{Y}+X'+\text{D} \\\]

So the rate equation needs to take into account the many X oligomerize and then bind to DNA.

\[\dot{x}'=k'_+x^h-k'_-x'\]

Oligomerization is faster than transcription/translation. So we can safely assume

\[\dot{x}'\approx0 \\ \Rightarrow x'=\frac{k'_+}{k'_-}x^h\sim x^h \\ \Rightarrow f_+(x) \to \frac{k}{V}\frac{x^h}{x^h+K_d^h}\]

The above function is known as a Hill function. The function sharpens as \(h\) increases. Note the following limiting case:

\[x<<K_d \ \rightarrow \ f_+(x)\approx\frac{k}{V}\frac{x^h}{K_d^h}\]

• Oligomerization (also referred to as cooperativity) acts to sharpen gene regulation functions.
• However, when comparing with experiments our model does not agree exactly with experimental results.
  • Experiments show that biological systems are very noisy.
• Theoretically there are two possible explanations in regards to gene expression to account for the noise seen in biological systems.
  1. X is noisy only, and that noise propagates to Y.
  2. X is noisy and Y is also noisy on its own.
  • Turns out that statement 2 is correct.

Noise

We will describe the noise in Y as having two contributions. One due to the intrinsic noisiness of Y and the other due to the propagation of noise from X (extrinsic noise).

\[\sigma_y^2 = \sigma_\text{int}^2+\sigma_\text{ext}^2\]

The noise from X is denoted with subscript “ext” for extrinsic noise. The noise due to Y itself is labelled “int” for intrinsic noise.

Intrinsic Noise

Lets first consider only intrinsic noise without the presence of X.

\[\text{D} \xrightarrow{k}\text{D}+\text{Y} \\ \text{Y} \xrightarrow{r}\emptyset\]

We already wrote down the simple rate equation for this model in the previous lecture. Instead of thinking in terms of rate equations lets now look at this in terms of probabilities. Let \(n\) denote the number of Y proteins in the cell. Lets formulate the probabilities of going to state with one more or one less Y protein.

Transition Probabilities and Master Equation

\[p(n\to n+1 \text{ in }\Delta t)=p_{n+1}=k\Delta t \\ p(n\to n-1 \text{ in }\Delta t)=p_{n-1}=k\Delta t\]

Then the master equation is the following:

\[\begin{align*} \dot{p_0}&=p_1r-p_0k \\ \dot{p_n}&=p_{n-1}+p_{n+1}r(n+1)-p_nk-p_nrn \end{align*}\]

In stationary state all time derivatives are equal to zero. For n = 0 :

\[0 = p_1r-p_0k \\ \Rightarrow p_1=\frac{k}{r}p_0=\lambda p_0 \\ \lambda \equiv \frac{k}{r}\]

For n = 1 :

\[0 = p_{0}+p_{2}r(n+1)-p_1k-p_1r \\ \Rightarrow p_2 = \frac{\lambda^2}{2}p_0\]

A pattern emerges (take my word for it)…

\[p_n = \frac{\lambda^n}{n!}p_0\]

This is a Poisson distribution! We can normalize our distribution in order to solve for \(p_0\) which yields:

\[p_n = e^{-\lambda}\frac{\lambda^n}{n!}\]

Like any Poisson distribution the mean and variance are equal.

\[\bar{n}=\lambda, \ \ \ \sigma^2=\lambda.\]

We now have an expression for the intrinsic noise!

\[\sigma^2_\text{int}=\bar{n}=\lambda=\frac{k}{r}\]

Relative Noise

Relative noise is the ratio of the noise to the mean number squared.

\[\eta^2 \equiv \frac{\sigma^2}{\bar{n}^2} \\ \therefore \eta_\text{int}=\frac{1}{\sqrt{\bar{n}}}\]

So the relative intrinsic noise decreases as the average number of proteins increases.

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