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 gi:x↦x−ig\_i:x{\textbackslash}mapsto x{\textasciicircum}\{-i\

@article{delaporte_mutational_2016,

title = {Mutational pattern of a sample from a critical branching population},

issn = {0303-6812, 1432-1416},

url = {http://link.springer.com/article/10.1007/s00285-015-0964-2},

doi = {10.1007/s00285-015-0964-2},

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 gi:x↦x−ig\_i:x{\textbackslash}mapsto x{\textasciicircum}\{-i\}, i∈ℤ+i{\textbackslash}in {\textbackslash}mathbb Z\_+, including the so-called uniform (i=0i=0) and log-uniform (i=1i=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 gig\_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.},

language = {en},

urldate = {2016-02-03},

journal = {Journal of Mathematical Biology},

author = {Delaporte, Cécile and Achaz, Guillaume and Lambert, Amaury},

month = jan,

year = {2016},

keywords = {60F17, 60G55, 60G57, 60J80, 60J85, Applications of Mathematics, Coalescent point process, Critical birth-death process, Infinite-site model, Invariance principle, Mathematical and Computational Biology, Poisson point measure, Primary 92D10, Sampling, Secondary 92D25, Site frequency spectrum},

pages = {1--38},

file = {Snapshot:/Users/amaury/Library/Application Support/Firefox/Profiles/x9yjaxzc.default/zotero/storage/8BBHV33J/10.html:text/html}

}