## Inbreeding in Infinite Populations and Identity by Descent

In finite populations random mating will occasionally produce mating between relatives. Here we consider mating between relatives in infinite populations. Two individuals are related if they have a common ancestor. Siblings and half-siblings have at least one common parent; first cousins have common grandparents, etc. If two individuals have a common ancestor, then it is possible that at a given locus they have inherited the same allele from that ancestor. Such an allele is said to be inherited identical by descent (IBD). The coefficient of relatedness of two individuals is defined to be the probability that at a given locus a randomly selected allele from one of the individuals is identical by descent with a randomly selected allele at the same locus in the other individual. For example, two siblings, whose parents are unrelated, have two common ancestors, their mother and their father. If we select a random allele from one of the siblings, there is probability 1/2 it was inherited from their mother and probability 1/2 it was inherited from their father. In either case, if we select a random allele from the other sibling, there is a 1/2 chance it was inherited from the same parent, and if so, there is then a 1/2 chance it is the same allele. Hence the coefficient of relatedness of the siblings is (1/2) x (1/2) = 1/4. If two relatives mate, the coefficient of inbreeding of their offspring is by definition the probability of relatedness of the parents, i.e., the probability that at a given locus the two alleles in that offspring are identical by descent.

The following equations modify Hardy-Weinberg equilibrium to accommodate inbreeding. Assume that in a population that is otherwise in Hardy-Weinberg equilibrium, mating occurs between two relatives having a coefficient of relatedness F. Then at a locus with alleles A and a, a child can have a genotype AA because (i) it inherits the A allele from one parent and the same allele IBD from the other parent, which occurs with probability FpA, or (ii) it inherits the allele A independently (not IBD) from both parents, which happens with probability (1 — F)pA. Adding these two terms together, we find that the probability of the genotype AA is pAA = pA + FpA(1 — pA). A similar formula holds for the genotype aa. For the genotype Aa, the alleles cannot be inherited IBD, since they are different, so pAa = 2pA(1 — pA)(1 — F). A consequence of inbreeding is an increase in homozygosity compared to random mating.

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