What is the probability of an offspring sharing identical HLA typing as one of their parents?

Question

DISCLAIMER: I have yet to thoroughly study HLA 100% to the bone, and hence I won't know everything about it at the back of the hand.

Recently I came across this information on a San Francisco hospital website claiming the chance of a full HLA match between 2 randomly selected people is 1/1,000,000 and the chance of a full HLA match between a child and one of their parents is 1/200.$^{[1]}$

What struck me as odd is the 1/200 chance of a parent matching their offspring's HLA, which seemed too high of an estimate. The immediate intuition is for a child to have identical HLA typing as one of their parents, their parents need to share some HLA haplotypes, which on its own should be a rare occurrence. Hence I decided to attempt verification of this number to confirm if it was within the realm of possibility

According to Stanford$^{[2]}$, the HLA cell surface receptors that mix during inheritance are HLA-A, HLA-B in MHC I, and HLA-DR in MHC II. A set of these 3 forms a haplotype. [HLA-C and HLA-DQ are probably involved too, but we'll omit them for simplicity's sake.]

For our estimation, we use data from the same Stanford source$^{[2]}$ and find there are 59 HLA-A serotypes, 118 HLA-B serotypes, and 124 HLA-DR serotypes. This gives us $59\times118\times124=863288$ different possible haplotypes.

To get the probability of 2 parents sharing at least 1 common haplotype, we do some math: $\frac{1}{863288} \times \frac{863287}{863288} + \frac{1}{863288} \times \frac{863287}{863288} + 2 \times \frac{1}{863288} \times \frac{1}{863288} = 1/431644$

The above is assuming the parents have met randomly in the world and no inbreeding has taken place. Inbreeding will pretty much ensure 2 parents share common HLA haplotypes.

This means the chance of an offspring to even possibly have the same HLA typing as one of their parents is 1 in 431,644, which is already way lower than the 1 in 200 chance stated on the hospital's website.

Just for calculation sake, the precise chance of an offspring sharing the same HLA typing with either one of their parents after accounting for different possible haplotype combinations produced by the parents would be: $100\% \times \frac{1}{863288}^2 \times 2 + 25\% \times \frac{1}{863288} \times \frac{863286}{863288} \times 2 + 50\% \times 2 \times \frac{1}{863288}^2 = 5.792 \times 10^{-7}$

An exceedingly small chance.

Issues with my estimation:

There were 2 assumptions made during this estimatey computation. I didn't take into account the weighted and uneven distribution of the haplotypes, and just assumed they are all statistically evenly distributed. Some haplotypes are more common than others, and this factor was not considered during the estimation. If existing data on haplotype occurrence is used, it will likely affect the estimation quite a bit, however, this is highly difficult as the data varies by region and ancestry.

I also did not take Genetic matchmaking into account, which is a factor that will lower the chance of 2 parents sharing common HLA haplotypes.

If exact computation for the probability proves to be difficult, what would be a reasonable estimation on the order of magnitude of said probability? Is 1/200 within the realm of possibility once you take into account the uneven distribution of haplotypes caused by certain more common ones?

Sources:

$^{[1]}$https://www.ucsfbenioffchildrens.org/education/types_of_bmt_donors/#2

$^{[2]}$http://web.stanford.edu/dept/HPST/transplant/html/hla.html

Answer 1

If you follow your math, you will definitely not find a 1/1,000,000 chance of two random people being an HLA match, so it seems clear that the independence and equal weighting of types is not a great assumption.

We can make a different set of back-of-envelope assumptions, and just say that the 1/1000000 number comes from having 6 combined matches. In that case, you would interpret 1/1000000 as due to 6 independent matchings of probability 1/10. In that case, with a parent you only have to randomly match the other three. 1/10^3 = 1/1000, much closer to the 1/200. Further non-independence, unequal frequencies, and homozygosity makes 1/200 seem reasonable in a whole population (though individuals with more rare HLA would be less likely to match a parent or a random individual), and I presume the 1/200 number comes from actual observed frequencies rather than theoretical calculations.

Perhaps someone with more HLA expertise (I have basically none) will provide a better answer based on more specifics, but I just meant this as a quick napkin calculation for whether the numbers given are plausible, and I think that they are.