TNBGGA: The No Bullshit Guide to Genetic Analysis

To start out, let's first consider an extreme case of inbreeding; namely, a brother-sister mating. The pedigree would look like this:

A useful concept to introduce here is the inbreeding coefficient ($F$), which is defined as the likelihood of homozygosity by descent at a given locusplugin-autotooltip__default plugin-autotooltip_bigLocus (plural form: loci): a physical location of a gene; often used as a synonym for a gene.. Let's take the example of the family shown in Figure 1. This family has two sets of grandparents: the paternal grandparents (individuals 1 and 2) and maternal grandparents (individuals 3 and 4). Let's imagine there is a geneplugin-autotooltip__default plugin-autotooltip_bigGene: read Chapters 02, 03, 04, 05, and 06 for a definition of gene :-) $A$, and that each grandparent carries a different alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence.: The paternal grandparents' genotypesplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. are $A1A1/A1$ (1) and $A2/A2$ (2), and the maternal granparents' genotypeplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. are $A3/A3$ (3) and $A4/A4$ (4). $F$ is the probability that the grandchild (the child of 7 and 8, marked by “?”) will have the genotypeplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. of either $A1/A1$, $A2/A2$, $A3/A3$, or $A4/A4$.

Let's first consider the question: what is the probability that the grandchild will have the genotypeplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. $A1/A1$? We can start by trying to determine if any genotypesplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. in the pedigree are known based on the information we have on the grandparents. In fact, we know the genotypesplugin-autotooltip__default plugin-autotooltip_bigGenotype: the combination of alleles within an organism or strain. When used as a verb, it means to determine the genotype experimentally. for both the parents: the father (5) must be $A1/A2$, and the mother (6) must be $A3/A4$. Given this information, the probability that the brother (7) inherits the $A1$ alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. from his father is $\frac{1}{2}$ (the alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. that he inherits from his mother (6) is irrelevant). Similarly, the probability that the sister (8) inherits the $A1$ alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. from her father is also $\frac{1}{2}$. Using the product ruleplugin-autotooltip__default plugin-autotooltip_bigProduct rule: in probability theory, the probability of two independent events both occurring (i.e., rolling snake eyes on a pair of dice) is the product of the probability of the individual events (i.e., $\frac{1}{6}\times\frac{1}{6}=\frac{1}{36}$)., the probability that both the brother and sister inherit the $A1$ alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. from their father is $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$. Finally, assuming both brother and sister are heterozygousplugin-autotooltip__default plugin-autotooltip_bigHeterozygous: a state for a diploid organism wherein the two alleles for a gene are different from each other. carriers of $A1$, the probability that their child (the grandchild) would inherit $A1$ from both their parents is $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$. Using the product ruleplugin-autotooltip__default plugin-autotooltip_bigProduct rule: in probability theory, the probability of two independent events both occurring (i.e., rolling snake eyes on a pair of dice) is the product of the probability of the individual events (i.e., $\frac{1}{6}\times\frac{1}{6}=\frac{1}{36}$). one more time, the probability that “both brother and sister inherit the $A1$ alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. from their father” and “the grandchild inhereits $A1$ from both their parents” is $\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}$.

Of course, there is nothing special about the $A1$ alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence.; the exact same logic applies to $A2$, $A3$, and $A4$. In other words:

$$p(A1/A1)=p(A2/A2)=p(A3/A3)=p(A4/A4)=\frac{1}{16}$$

Since we are interested in the probability of whether the child will be homozygousplugin-autotooltip__default plugin-autotooltip_bigHomozygous: a state for a diploid organism wherein the two alleles for a gene are identical to each other. at the $A$ geneplugin-autotooltip__default plugin-autotooltip_bigGene: read Chapters 02, 03, 04, 05, and 06 for a definition of gene :-) regardless of which alleleplugin-autotooltip__default plugin-autotooltip_bigAllele: a version of a gene. Alleles of a gene are different if they have differences in their DNA sequence. it is, we are effectively asking the question of “$p(A1/A1)$ or $p(A2/A2)$ or $p(A3/A3)$ or $p(A4/A4)$”. When solving “or” questions in probability, we use the sum rule by adding up the probabilities. Therefore:

$$p(\text{homozygous by descent})=F=\frac{1}{16}+\frac{1}{16}+\frac{1}{16}+\frac{1}{16}=\frac{1}{4}$$

A bother-sister mating is the simplest case to analyze but is of little practical consequence in human population genetics since all cultures have strong taboos against this type of consanguineous mating and the frequency of these events is extremely low.

First cousin marriages Fig. 2) do happen at an appreciable frequency. Let's calculate $F$ for offspring of 1st cousins.

As before,