AUTHOR=Jebreen Kamel , Petrizzelli Marianyela , Martin Olivier C. 

TITLE=Probabilities of Multilocus Genotypes in SIB Recombinant Inbred Lines

JOURNAL=Frontiers in Genetics

VOLUME=Volume 10 - 2019

YEAR=2019

URL=https://www.frontiersin.org/journals/genetics/articles/10.3389/fgene.2019.00833

DOI=10.3389/fgene.2019.00833

ISSN=1664-8021

ABSTRACT=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 \emph{via} summations of these quantities. \emph{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.