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Mutational pattern of a sample from a critical branching population

Abstract : 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 nearly critical branching populations with rare 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 [Formula: see text], [Formula: see text], including the so-called uniform ([Formula: see text]) and log-uniform ([Formula: see text]) 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 with prior [Formula: see text] 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.
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Submitted on : Thursday, March 16, 2017 - 11:08:16 AM
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Cécile Delaporte, Amaury Lambert, Guillaume Achaz. Mutational pattern of a sample from a critical branching population. Journal of Mathematical Biology, Springer Verlag (Germany), 2016, 73 (3), pp.627-664. ⟨10.1007/s00285-015-0964-2⟩. ⟨hal-01490969⟩



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