Academic journal article Genetics

Genotype Probabilities at Intermediate Generations in the Construction of Recombinant Inbred Lines

Academic journal article Genetics

Genotype Probabilities at Intermediate Generations in the Construction of Recombinant Inbred Lines

Article excerpt

ABSTRACT The mouse Collaborative Cross (CC) is a panel of eight-way recombinant inbred lines: eight diverse parental strains are intermated, followed by repeated sibling mating, many times in parallel, to create a new set of inbred lines whose genomes are random mosaics of the genomes of the original eight strains. Many generations are required to reach inbreeding, and so a number of investigators have sought to make use of phenotype and genotype data on mice from intermediate generations during the formation of the CC lines (so-called pre-CC mice). The development of a hidden Markov model for genotype reconstruction in such pre-CC mice, on the basis of incompletely informative genetic markers (such as single-nucleotide polymorphisms), formally requires the two-locus genotype probabilities at an arbitrary generation along the path to inbreeding. In this article, I describe my efforts to calculate such probabilities. While closed-form solutions for the two-locus genotype probabilities could not be derived, I provide a prescription for calculating such probabilities numerically. In addition, I present a number of useful quantities, including single-locus genotype probabilities, two-locus haplotype probabilities, and the fixation probability and map expansion at each generation along the course to inbreeding.

THE mouse Collaborative Cross (CC) is a panel of eight-way recombinant inbred lines (RIL): eight diverse parental strains are intermated, followed by repeated sibling mating, many times in parallel (see Figure 1D), to create a new set of inbred lines whose genomes are random mosaics of the genomes of the original eight strains (Complex Trait Consortium 2004; Collaborative Cross Consortium 2012). There are similar efforts for Drosophila (Macdonald and Long 2007) and Arabidopsis (Kover et al. 2009); the panels will serve as important reference populations for the systemic genetic analysis of complex traits.

Many generations are required for inbreeding of such RIL, and so a number of investigators have sought to make use of phenotype and genotype data onmice from intermediate generations during the formation of the CC lines: the pre-CC mice (e.g., see Aylor et al. 2011). The mapping of quantitative trait loci (QTL) with data on pre-CC mice, whether by interval mapping (Lander and Botstein 1989) or Haley-Knott regression (Haley and Knott 1992), requires the calculation of conditional genotype probabilities given incompletely informative marker data (e.g., at single-nucleotide polymorphisms). Such probabilities are generally derived using a hidden Markov model (HMM). The construction of an HMM for pre-CC mice formally requires the calculation of two-locus diplotype probabilities at arbitrary generations along the course to inbreeding. Thus, I sought to calculate single-locus genotype probabilities and two-locus diplotype probabilities at generation G2 : Fk (see Figure 1D), with the latter being a function of the recombination fraction between the two loci.

Previous work on genotype probabilities in RIL has focused largely on the final lines (Haldane and Waddington 1931; Broman 2005; Teuscher and Broman 2007), although Haldane and Waddington (1931) did calculate a portion of the probabilities for intermediate generations in two-way RIL by selfing. More recently, Johannes and Colomé-Tatché (2011) fully derived the two-locus genotype probabilities for two-way RIL by selfing and described numerical calculations for the autosome in two-way RIL by sibling mating.

Here I extend these results to the case of four- and eight-way RIL by selfing and sibling mating, including consideration of the X chromosome. The basic problem is to calculate the k-step probabilities of a Markov chain with many states. While I was not able to obtain closed-form solutions for the two-locus diplotype probabilities at Fk in RIL by sibling mating, I do provide recipes for calculating the probabilities numerically. And I was able to obtain closed-form solutions for single-locus genotype probabilities and two-locus haplotype probabilities. …

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