Question

Connection to quantitative traits:

SNPs are inherited in a Mendelian fashion and are often polygenic in nature. We can think of SNPs in terms of either contributing or non-contributing alleles. A study of SNPs correlated with heart disease has shown that heart probles are severe if nine OR MORE of the alleles at 6 loci are of the contributing variety. What is the probability the following parents will have a child that is susceptible heart disease?

AaBbccDDEEFf x AaBbCCDdEeff

Answer 1

I know it has something to do with binomials

Answer 2

Which alleles are considered contributing factors? The dominant or recessive?

Answer 3

dominant

Answer 4

Thanks. I assume all the traits encoded by the different alleles occur independently of one another.

First notice that every child will be heterozygous for the \(\mathrm C\) genotype, so you know that every child will have at least one of the contributing factors. This reduces the problem to finding how many have at least eight other alleles in the dominant configuration.

At each locus, you have, respectively 3, 3, 1, 2, 2, and 2 possible genotypes yielded from the given cross, so there is a total of \(3^2\times1\times2^3=48\) possible genotypes among the offspring.

How to quickly find out how many combinations contain at least nine dominant alleles is escaping me at the moment, but there is a systematic way of figuring it out. (It works out rather quickly, but that's thanks to the fortunate setup of the question.)

Consider the extreme case where the number of dominant alleles is maximized. This occurs among offspring with the genotype \(\mathrm{AABBCcDDEEFf}\), which has 10 dominant alleles.

So we know that we can swap at most one of the homozygous loci for a hetrozygous one and still have at least nine dominant factors. This leaves us with four additional possibilities,
\[\begin{matrix}
\mathrm{AaBBCcDDEEFf}&\mathrm{AABbCcDDEEFf}\\[1ex]
\mathrm{AABBCcDdEEFf}&\mathrm{AABBCcDDEeFf}
\end{matrix}\]
This means you have a \(\dfrac{5}{48}\) probability of having a child with increased susceptibility to developing heart disease.