Probabilities of Multilocus Genotypes in SIB Recombinant Inbred Lines

Recombinant Inbred Lines (RILs) are obtained through successive generations of inbreeding. In 1931 Haldane and Waddington published a landmark paper where they provided the probabilities of achieving any combination of alleles in 2-way RILs for 2 and 3 loci. In the case of sibling RILs where sisters and brothers are crossed at each generation, there has been no progress in treating 4 or more loci, a limitation we overcome here without much increase in complexity. In the general situation of L loci, the task is to determine 2 probabilities, but we find that it is necessary to first calculate the 4 "identical by descent" (IBD) probabilities that a RIL inherits at each locus its DNA from one of the four originating chromosomes. We show that these 4 probabilities satisfy a system of linear equations that follow from self-consistency. In the absence of genetic interference-crossovers arising independently-the associated matrix can be written explicitly in terms of the recombination rates between the different loci. We provide the matrices for L up to 4 and also include a computer program to automatically generate the matrices for higher values of L. Furthermore, our framework can be generalized to recombination rates that are different in female and male meiosis which allows us to show that the Haldane and Waddington 2-locus formula is valid in that more subtle case if the meiotic recombination rate is taken as the average rate across female and male. Once the 4 IBD probabilities are determined, the 2 probabilities of RIL genotypes are obtained via summations of these quantities. In fine, our computer program allows to determine the probabilities of all the multilocus genotypes produced in such sibling-based RILs for L<=10, a huge leap beyond the L = 3 restriction of Haldane and Waddington.