# Gene frequency problem

1/330 of white people in south Africa have a disease called Porphyri.

What is the frequency of the gene if it is inherited autosomal recessive?

This one I understand. The genotype frequency is $q^2 = 1/300$ so the gene frequency has to be $\sqrt{1/300}$ as it is inherited aa.

What is the frequency of the gene if it is inherited autosomal dominant?

Here's where I run into trouble. So if $p$ is the dominant allele and $q$ is the recessive allele the allele frequency has to be $2pq+p^2$ right? As we can either see them as $Aa$ or $AA$

But my book says it is $2pq+q^2$ and then solves it algebraically by pretending $q^2$ is so close to 0 that it's not in the calculations. I strongly disagree that $p^2$ is very small if I am right with my assumption. Which one is right and why?

I think you pretty much nailed the problem. If I understand correctly, you are just confused about the approximation that your textbook does, an approximation that is correct but not needed. Let me restate the solution to the problem first.

Resolution of the problem

As you said the frequency of sick people $\frac{1}{300} = 2pq + q^2$ in this case, where $q = 1-p$ is the frequency of the (dominant) allele causing the disease. It is a simple quadratic equation with one unknown so one doesn't even need to make any approximation to solve for $q$ (or for $p$). Rearranging the equation and expressing $p$ in terms of $q$ (p=1-q), you'll find that $q ≈ 0.00167$.

Are the results intuitive?

Note that the frequency needed to create a frequency of $\frac{1}{300}$ sick individuals is $\sqrt{\frac{1}{300}} ≈ 0.05$ if the allele causing the disease is recessive while it is only about $0.0016$ (31 times less frequent) if the allele causing the disease is dominant. It shows that indeed a lower frequency of a more dominant allele is enough to create more sick people.

Is the $q^2$ term negligible?

Given that $q ≈ 0.00167$, the term $q^2 ≈ 10^{-6}$, while the term $2pq ≈ 10^{-3}$. So, indeed the term $q^2$ is rather negligible compared to $2pq$.

• You write 1/300 = 2pq+q^2 like my book. I say that it's 1/300 = 2pq+p^2 because p is the dominant allele and 1 out of 300 people has Aa or AA. 2pq + p^2. I don't understand why you have +q^2 in the end as we're not counting the recessive aa's ? – Paze Jun 5 '15 at 19:45
• It is just a matter of what you want to call q and what you want to call p. They are totally interchangeable. In my example, I called q the frequency of the dominant allele causing the disease. But if you feel more confortable the other way around, you can call p the frequency of the dominant allele causing the disease. – Remi.b Jun 5 '15 at 21:40
• Ah, okay, I thought it was the norm in literature to denote p as the dominant allele and q as the recessive allele, but I guess I am wrong. Thank you. – Paze Jun 6 '15 at 14:01