## Inbreeding

Inbreeding plays two important, somewhat distinct roles in experimental genetics: first as the starting place for the breeding experiments leading to back crosses and intercrosses, and second as a resource for genetic mapping through recombinant inbreds.

An inbred strain can be created by repeated inbreeding of a small population. The most extreme examples are self-fertilizing plants such as Arabidopsis that can reproduce by themselves. A population of size one can be maintained by selfing for many generations. A heterozygous locus, say Aa, will eventually become either AA or aa and remain the same (fixed) thereafter. In mammals selfing is not an option and the population size must be at least of size two - a male and a female. However, by repeated brother-sister mating it is in principle possible to maintain a population of this minimal size. Again, each locus will eventually reach homozygosity throughout the population and will remain in that state thereafter. In much of the rest of this chapter we will assume that inbreeding is taking place in a population of size two. In practice, the stock size during inbreeding is kept larger in order to reduce the chance of the stock becoming extinct. The study of inbreeding in a larger population will be left as an exercise.

Recombinant inbred strains (RI) are a special type of inbred strains. They are constructed by an outcross of two established inbred parental strains. From the second generation of F2 animals, which are not genetically identical, a set of pairs is selected. Each selected pair is used in order to create a new inbred strain via repeated brother-sister mating. Eventually, a set of inbred strains is created, which is called a recombinant inbred set. Each strain in a given set is genetically homogeneous within, but is genetically different from the other strains in the set and from the original parental strains. Thus, at each locus for which the original strains differ, the new recombinant inbred strain is homozygous for one of the two alleles. However, the parental source of the allele may vary from one strain to the next within the recombinant inbred set, and at different loci the source of the allele may be different parental strains. For example, if the original parental strains are A/B and a/b homozygous at two loci (i.e., one strain is AA at one locus and BB at another, while the other strain is aa at the first locus and bb at the other), recombinant inbreds can be A/B, A/b, a/B, or a/b homozygous at those two loci. There are 23 = 8 possibilities with three loci, 24 = 16 with four, etc.

The name of the RI set is formed by combining the names of the two parental inbred strains. For example, if the Fi mice are created by mating an A female and a C57BL/6J (B6) male, then the resulting recombinant inbred set is denoted AXB. If, on the other hand, the female is a B6 and the male is A then the set is called BXA.

Recombinant inbred sets can be used for mapping traits. At each locus the allele is equally likely to have come from each of the parental strains. In the notation of the previous chapter, the genotype is either AA or aa with probability 1/2 each. The representation (2.3) can be re-written (Prob. 3.1) in the simpler form y = m + a(x — 1) + e, where x is 0 or 2 with probability 1/2. Hence the genetic variance component for a recombinant inbred set at the given locus is a2. As we will see in later chapters, mapping is conducted by correlating expressed phenotype with genotypes. A major advantage of recombinant inbred sets, and an important motivation for their establishment, is that genotypes are fixed for the entire set. Consequently, the experiment for mapping a trait with recombinant inbreds is conducted by phenotyping mice from the different strains that form the set. Genotyping is unnecessary for the commercially available sets, since their genotypes have already been determined. In addition, since mice from the same strain are genetically identical, one can average the phenotypes of a number of different mice from the same strain, which has the effect of decreasing the environmental variance relative to the genetic variance.

In order to understand the dynamics of inbreeding, in the following subsection we will investigate the process of forming a recombinant inbred strain and expand the discussion of the segregation of the phenotype to include the case of a recombinant inbred strain. In the next subsection, we take a closer look at a mathematical analysis of inbreeding.

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