arXiv subjects

Verónica Miró Pina

Publications and source records attributed to Verónica Miró Pina.

3 recordsLinked to original sources

arXiv2022

The Symmetric Coalescent and Wright-Fisher models with bottlenecks

We define a new class of $Ξ$-coalescents characterized by a possibly infinite measure over the non negative integers. We call them symmetric coalescents since they are the unique family of exchangeable coalescents satisfying a symmetry property on their coagulation rates: they are invariant under any transformation that consists in moving one element from one block to another without changing the total number of blocks. We illustrate the diversity of behaviors of this family of processes by introducing and studying a one parameter subclass, the $(β,S)$-coalescents. We also embed this family in a larger class of $Ξ$-coalescents arising as the limit genealogies of Wright-Fisher models with bottlenecks. Some convergence results rely on a new Skorokhod type metric, that induces the Meyer-Zheng topology, which allows to study the scaling limit of non-markovian processes using standard techniques.

math.PR↗

arXiv2020

Chromosome Painting: how recombination mixes ancestral colors

We consider a Moran model with recombination in a haploid population of size $N$. At each birth event, with probability $1-ρ_N R$ the offspring copies one parent's chromosome, and with probability $ρ_N R$ 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 $ρ:= \lim_{N\to \infty} \frac{ρ_N N}{2}\to \infty$. 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 et al. (2016), 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 $ρ\gg 1$ 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.

math.PR↗

arXiv2018

How does geographical distance translate into genetic distance?

Geographic structure can affect patterns of genetic differentiation and speciation rates. In this article, we investigate the dynamics of genetic distances in a geographically structured metapopulation. We model the metapopulation as a weighted directed graph, with d vertices corresponding to d subpopulations that evolve according to an individual based model. The dynamics of the genetic distances is then controlled by two types of transitions -mutation and migration events. We show that, under a rare mutation - rare migration regime, intra subpopulation diversity can be neglected and our model can be approximated by a population based model. We show that under a large population - large number of loci limit, the genetic distance between two subpopulations converges to a deterministic quantity that can asymptotically be expressed in terms of the hitting time between two random walks in the metapopulation graph. Our result shows that the genetic distance between two subpopulations does not only depend on the direct migration rates between them but on the whole metapopulation structure.

q-bio.PE↗