Probabilities of Multilocus Genotypes in SIB Recombinant Inbred Lines
Publication Type
Original research

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 2L probabilities, but we find that it is necessary to first calculate the 4L “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 4L 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 4L IBD probabilities
are determined, the 2L 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.

Statistical Genetics and Methodology
Frontiers in Genetics
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Publication Type
Both (Printed and Online)