Re: How are allele probabilites calculated?

Hi Cory,

Thanks for the interesting questions. You can find a previous answer (written by me) to some related questions in our archives. This answer goes into some detail about the chances of inheriting a particular allele in various pedigrees, as well as the chance of inheriting a particular allele given the frequency of that allele in the general population, so I am going to refer you to that answer as background for this answer.

Just as in that previous answer, I've drawn some pedigree figures to help explain this answer. These are labelled A, B, and C in the image below. Each of these pedigrees is labelled to show you the percent chance of inheriting ONE copy of a particular allele that the mother in generation one (G1) had. I emphasized one copy there, because the chances of inheriting TWO copies are very different.

Figure A is a pedigree that describes five generations of non-consanguinous mating, so that everyone in the pedigree mates with someone to whom they are unrelated. Person 1 in G1 has a particular allele of a particular gene that we are tracking. Her chance of having that allele is 100%. With each generation, the chance of inheriting this allele is decreased by half, so that by G5, there is only a 6.25% chance of inheriting that allele. Another way to look at this is to say that the chance of getting the allele in question is proportional to the number of great-great-grandparents (GGGPs) in G1. The person in G5 has 16 GGGPs, and 6.25% is 100% divided by 16.

Figure B describes five generations in which there are two first cousin matings. The first takes place in G3 and the second takes place in G4. I don't know what Albert Einstein's real pedigree looks like, and this one is just a made up version that matches the description you provided. So, in this figure, person 1 in G4 is the product of the first cousin mating in G3. This person has a 25% chance of having a particular allele found in his great-grand-mother (person 1 in G1) because he has a 12.25% chance of getting it from his mother, and a 12.25% chance of getting it from his father, the cousins. Note that his cousin (person 2 in G4) has half of this chance, because her pedigree looks just like the pedigree in Figure A. Their child has an 18.75% chance of inheriting her great-great-grandmother's allele; she has a 6.25% chance of getting it from her mother, and a 12.5% chance of getting it from her father. So, her chance is 3 times greater than it would have been if her father had not been related to her mother, and 1.5 times higher than it would have been if her father had still been her mother's first cousin, but not the product of a first-cousin mating himself.

Figure C describes five generations of a family in which there are four generations of full sibling matings. Lets assume that the parents in G1 are unrelated. As with the other pedigrees, the mother has a 50% chance of passing a particular allele to each of her children. Here, each generation consists of a brother-sister mating. So, each subsequent set of offspring has a 25% chance of getting that allele from the father, and a 25% chance of getting that allele from the mother, so they each again have a 50% chance of having one copy of that allele.

When we think about the chance of inheriting TWO copies of a given allele, there is an important distinction between G2 and the subsequent generations in Figure C. If we assume that the allele in question is only seen in the mother in G1, and that she has only one copy, then the people in G2, have no chance of being homozygous (having received two copies) for that allele. However, of the offspring in G3, 25% have a chance of getting the allele from one parent, and 25% of these have a chance of getting the same allele from the other parent. So each of the people in G3 (and in subsequent generations) has a 6.25% chance of having inherited two copies of that allele.

Lets compare this to Figure B, making the same assumptions about the mother in G1 having the only copy of this allele in G1. In this pedigree, only person 1 in G4 and the person in G5 have any chance of being homozygous for this allele. Person 1 in G4 has about a 1.6% chance of being homozygous, and the person in G5 has about a 0.8% chance of being homozygous.

Of course, there is no chance of homozygotes (homozygous people) in Figure A, because the allele in question is only seen in this family, and they never mate with one another. But that is not really that interesting (or realistic) a model. As I suggested in the previous answer, most alleles exist in the population at a certain (generally low) frequency, so there is always a chance of homozygosity. In the previous answer, I showed that "the offspring of any kind of first cousin mating have a base chance of homozygosity of about 1.6%." Here, you can see that the base chance of homozygosity for sibiling matings (assuming the allele is unique to that family) in G3 and higher is already 4 times greater than the chance for first cousin matings (with an allele that is at a low frequency in the population).

Finally, this entire issue is confounded by the fact that the number of alleles that are transmitted from the parents to the next generation is a function of the number of offspring that they have. While it is true that a child has a 50% chance of getting a particular allele from its parent, if it does not get that allele (which is the other 50% chance), then it has a 0% chance of transmitting it to the next generation. The more offspring that a parent has, the greater the chance of transmitting both of the alleles that it has at a given locus to the next generation. So, in Figure C, the chance of the allele in question not being transmitted to either child in G2 is 50% X 50% or 25%. If the parents in G1 had four children, this chance would decrease to (50%)^4 or 6.25%. So, when there are only two children per generation, there is only a 75% chance of a given allele being transmitted to at least one of them.

I hope that this is helpful for you. Thanks for a great set of questions!