Probabilities of Multilocus Genotypes in SIB Recombinant Inbred Lines - Jebreen, Kamel; Petrizzelli, Marianyela; Martin, Olivier C. (Frontiers in Genetics, 2019)
Recombinant Inbred Lines (RILs) are obtained through generations of inbreeding until all alleles are fixed. 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 SIB 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 $2^L$ probabilities, but we find that it is necessary to first calculate the $4^L$ ``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 $4^L$ 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 $4^L$ IBD probabilities are determined, the $2^L$ 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 $L \le 10$, a huge leap beyond the $L=3$ restriction of Haldane and Waddington.