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<publisher>HAL CCSD</publisher>
<title lang=en>Sample genealogy and mutational patterns for critical branching populations</title>
<creator>Achaz, G.</creator>
<creator>Delaporte, C.</creator>
<creator>Lambert, A.</creator>
<contributor>Centre interdisciplinaire de recherche en biologie (CIRB) ; Collège de France (CdF) - Institut National de la Santé et de la Recherche Médicale (INSERM) - Centre National de la Recherche Scientifique (CNRS)</contributor>
<contributor>Evolution Paris Seine ; Université Nice Sophia Antipolis (UNS) ; Université Côte d'Azur (UCA) - Université Côte d'Azur (UCA) - Centre National de la Recherche Scientifique (CNRS) - Université des Antilles et de la Guyane (UAG) - Université Pierre et Marie Curie - Paris 6 (UPMC)</contributor>
<contributor>Atelier de BioInformatique (ABI) ; Université Pierre et Marie Curie - Paris 6 (UPMC)</contributor>
<contributor>Laboratoire de Probabilités et Modèles Aléatoires (LPMA) ; Université Pierre et Marie Curie - Paris 6 (UPMC) - Université Paris Diderot - Paris 7 (UPD7) - Centre National de la Recherche Scientifique (CNRS)</contributor>
<description>28 pages, 6 figures</description>
<identifier>hal-01053343</identifier>
<identifier>https://hal.archives-ouvertes.fr/hal-01053343</identifier>
<source>https://hal.archives-ouvertes.fr/hal-01053343</source>
<source>28 pages, 6 figures. 2014</source>
<identifier>ARXIV : 1407.7720</identifier>
<relation>info:eu-repo/semantics/altIdentifier/arxiv/1407.7720</relation>
<language>en</language>
<subject lang=en>critical birth-death process</subject>
<subject lang=en>sampling</subject>
<subject lang=en>coalescent point process</subject>
<subject lang=en>site frequency spectrum</subject>
<subject lang=en>infinite-site model</subject>
<subject lang=en>Poisson point measure</subject>
<subject lang=en>invariance principle</subject>
<subject>92D10, 60J80 (Primary), 92D25, 60F17, 60G55, 60G57, 60J85 (Secondary)</subject>
<subject>[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]</subject>
<subject>[SDV.BID.EVO] Life Sciences [q-bio]/Biodiversity/Populations and Evolution [q-bio.PE]</subject>
<type>info:eu-repo/semantics/preprint</type>
<type>Preprints, Working Papers, ...</type>
<description lang=en>We study a universal object for the genealogy of a sample in populations with mutations: the critical birth-death process with Poissonian mutations, conditioned on its population size at a fixed time horizon. We show how this process arises as the law of the genealogy of a sample in a large class of critical branching populations with mutations at birth, namely populations converging, in a large population asymptotic, towards the continuum random tree. We extend this model to populations with random foundation times, with (potentially improper) prior distributions g_i: xmapsto x^{-i}, iinZ_+, including the so-called uniform (i=0) and log-uniform (i=1) priors. We first investigate the mutational patterns arising from these models, by studying the site frequency spectrum of a sample with fixed size, i.e. the number of mutations carried by k individuals in the sample. Explicit formulae for the expected frequency spectrum of a sample are provided, in the cases of a fixed foundation time, and of a uniform and log-uniform prior on the foundation time. Second, we establish the convergence in distribution, for large sample sizes, of the (suitably renormalized) tree spanned by the sample genealogy with prior g_i on the time of origin. We finally prove that the limiting genealogies with different priors can all be embedded in the same realization of a given Poisson point measure.</description>
<date>2014-07-29</date>
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