I'm having problems figuring out the logic behind two formulas related to the Hardy-Weinberg equation. Here are two relevant excerpts from my textbook, Medical Gentics by Jorde, Carey and Bamshad:
Consider, for example, a dominant disease that results in death before the person can reproduce. This is
termed a genetic lethal mutation because, even though the individual might survive for
some time, he or she contributes no genes to the next generation. Each time mutation
introduces a new copy of the lethal dominant disease allele into a population, natural
selection eliminates it. In this case, p, the gene frequency of the lethal allele in the
population, is equal to μ, the mutation rate (p = μ).
We can use the same principles to predict the relationship between mutation and selection
against recessive alleles. The Hardy-Weinberg principle showed that most copies of
harmful recessive alleles are found in heterozygotes and are thus protected from the effects
of natural selection. We would therefore expect their gene frequencies to be higher than
those of harmful dominant alleles that have the same mutation rate. Indeed, under
mutation-selection balance, the predicted frequency of a recessive allele, q, that is lethal in
homozygotes is (because μ < 1, q =√μ, resulting in a relatively higher allele frequency for lethal recessive alleles). If the allele is not lethal in homozygotes, then q=√(μ/s), where s is again the selection coefficient for those who have a homozygous affected genotype.
N.B. √ is square root in case it's not visible.
What I don't get is why are the formulas different. For a dominant lethal allele p = μ, but for a recessive lethal allele q = √μ. How did they arrive at the square root? It makes intuitive sense because there will be a higher frequency for lethal recessive alleles than dominant lethal alleles. If you have a recessive disease, AA and Aa do not have the disease so the frequency of a is higher than in the case of a dominant disease where AA and Aa have
the disease. But mathematically it doesn't make sense to me.
Does anyone have any ideas?