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Chapter 19. Effects of inbreeding
In this chapter we will examine how inbreeding between close relatives (also known as consanguineous matings) influences the appearance of autosomalplugin-autotooltip__default plugin-autotooltip_bigAutosome: any chromosome that is not a sex chromosome. recessiveplugin-autotooltip__default plugin-autotooltip_bigRecessive: used to describe an allele, usually in comparison to wildtype. Recessive alleles do not exhibit their phenotype when combined with a wildtype allele. traits. Inbreeding will not make a difference for dominantplugin-autotooltip__default plugin-autotooltip_bigDominant: used to describe an allele, usually in comparison to wildtype. Dominant alleles will express their phenotype when combined with a wildtype allele. traits because they need only be inherited from one parent or for X-linkedplugin-autotooltip__default plugin-autotooltip_bigLinkage: two loci are linked to each other if they are less than 50 m.u. apart. Two loci are unlinked if they are either (1) greater than 50 m.u. apart on the same chromosome, or; (2) are on separate chromosomes. traits in males since they are inherited from the mother.
Brother-sister matings and inbreeding coefficients
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 grandfather (individual 1) carries $A1$ and the paternal grandmother (2) carries $A2$; similarly, the maternal granfather 93) carries $A3$ and the maternal grandmother (4) carries $A4$. $F$ 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. $A1/A1$? In order for this to happen, the brother (7) and sister (8) must be carriers of $A1$. The brother (7) can only inherit $A1$ from the father (5), who in turn can only inherit $A1$ from the grandfather (1). The likelihood that the father (5) inherits $A1$ from the grandfather (1) is $\frac{1}{2}$, and the likelihood that the father (5) passes $A1$ to his son (the brother, 7) is also $\frac{1}{4}$. Therefore, the likelihood that the brother (7) is a carrier of $A1$ is $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$. By the exact same logic, the likelihood that the sister (8) is carrier of $A1$ is the same: $frac{1}{4}$. Therefore, the likelihood that their child is homozygousplugin-autotooltip__default plugin-autotooltip_bigHomozygous: a state for a diploid organism wherein the two alleles for a gene are identical to each other. for $A1$ 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
First cousin marriages Fig. 2) do happen at an appreciable frequency. Let's calculate $F$ for offspring of 1st cousins.
As before,