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Original Research ARTICLE Provisionally accepted The full-text will be published soon. Notify me

Front. Genet. | doi: 10.3389/fgene.2019.00833

Probabilities of multilocus genotypes in SIB recombinant inbred lines

 Kamel A. Jebreen1, 2, Marianyela S. Petrizzelli1 and Olivier C. Martin1*
  • 1Université Paris-Saclay, France
  • 2Department of Mathematics, Faculty of Science, An-Najah National University, Palestine

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.

Keywords: RIL population, Sibling mating, Crossover, Self consistency, structure population

Received: 21 Jun 2019; Accepted: 13 Aug 2019.

Copyright: © 2019 Jebreen, Petrizzelli and Martin. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Prof. Olivier C. Martin, Université Paris-Saclay, Gif-sur-Yvette, Île-de-France, France, olivier.c.martin@inra.fr