Journal of Mathematical Biology

, Volume 73, Issue 3, pp 627–664 | Cite as

Mutational pattern of a sample from a critical branching population

  • Cécile Delaporte
  • Guillaume Achaz
  • Amaury Lambert
Article

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 \(g_i:x\mapsto x^{-i}\), \(i\in \mathbb Z_+\), 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 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.

Keywords

Critical birth-death process Sampling Coalescent point process Site frequency spectrum Infinite-site model  Poisson point measure Invariance principle 

Mathematics Subject Classification

Primary 92D10 60J80 Secondary 92D25 60F17 60G55 60G57 60J85 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Cécile Delaporte
    • 1
    • 2
  • Guillaume Achaz
    • 2
    • 3
    • 4
  • Amaury Lambert
    • 1
    • 2
  1. 1.Laboratoire de Probabilités et Modèles AléatoiresUMR 7599 CNRS and UPMC Univ Paris 06ParisFrance
  2. 2.Centre Interdisciplinaire de Recherche en BiologieUMR 7241 CNRS and Collège de FranceParisFrance
  3. 3.Évolution Paris-SeineUMR 7138 CNRS and UPMC Univ Paris 06ParisFrance
  4. 4.Atelier de Bioinformatique, MNHNParisFrance

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