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April 2021 Chromosome painting: How recombination mixes ancestral colors
Amaury Lambert, Verónica Miró Pina, Emmanuel Schertzer
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Ann. Appl. Probab. 31(2): 826-864 (April 2021). DOI: 10.1214/20-AAP1606

Abstract

We consider a Moran model with recombination in a haploid population of size N. At each birth event, with probability 1ρNR the offspring copies one parent’s chromosome, and with probability ρNR she inherits a chromosome that is a mosaic of both parental chromosomes. We assume that at time 0 each individual has her chromosome painted in a different color and we study the color partition of the chromosome that is asymptotically fixed in a large population, when we look at a portion of the chromosome such that ρ:=limNρNN2. To do so, we follow backwards in time the ancestry of the chromosome of a randomly sampled individual. This yields a Markov process valued in the color partitions of the half-line, that was introduced by Esser, Probst and Baake (J. Math. Biol. 73 (2016) 161–197), in which blocks can merge and split, called the partitioning process. Its stationary distribution is closely related to the fixed chromosome in our Moran model with recombination. We are able to provide an approximation of this stationary distribution when ρ1 and an error bound. This allows us to show that the distribution of the (renormalised) length of the leftmost block of the partition (i.e., the region of the chromosome that carries the same color as 0) converges to an exponential distribution. In addition, the geometry of this block can be described in terms of a Poisson point process with an explicit intensity measure.

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Amaury Lambert. Verónica Miró Pina. Emmanuel Schertzer. "Chromosome painting: How recombination mixes ancestral colors." Ann. Appl. Probab. 31 (2) 826 - 864, April 2021. https://doi.org/10.1214/20-AAP1606

Information

Received: 1 April 2019; Revised: 1 June 2020; Published: April 2021
First available in Project Euclid: 1 April 2021

Digital Object Identifier: 10.1214/20-AAP1606

Subjects:
Primary: 60B12
Secondary: 05A18 , 60G55 , 60J25 , 60K35 , 92D10 , 92D20

Keywords: ancestral recombination graph , coagulation , experimental evolution , fragmentation , partition-valued process , recombination

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2021
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