Genotype probabilities at intermediate generations in the construction of recombinant inbred lines

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.

Figure 1

(A–D) The generation of two-way RIL by selfing (A), two-way RIL by sibling mating (B), four-way RIL by sibling mating (C), and eight-way RIL by sibling mating (D). A single autosome is shown. In A and B, the generation of multiple RIL in parallel is shown, while C and D illustrate the generation of a single RIL.

Figure 2

Fixation probability at generation F_k for an arbitrary locus as a function of k. (A) The cumulative probability. (B) The probability of fixation precisely at F_k. The results for RIL by selfing are in green. The results for two-way and four-way RIL by sibling mating are in blue and red, respectively, with the solid curves corresponding to an autosomal locus and the dashed curves corresponding to an X chromosome locus.

Figure 3

Map expansion at generation F_k as a function of k, for two-way (blue), four-way (red), and eight-way (black) RIL by selfing (dashed curves) and by sibling mating (solid curves). The displayed results for RIL by sibling mating are for the autosomes; values for the X chromosome are exactly two-thirds those for the autosomes.