1. Mutations on a Random Binary Tree with Measured Boundary
- Author
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Duchamps, Jean-Jil, Lambert, Amaury, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), Centre interdisciplinaire de recherche en biologie (CIRB), Labex MemoLife, École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Collège de France (CdF (institution))-Ecole Superieure de Physique et de Chimie Industrielles de la Ville de Paris (ESPCI Paris), Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Paris (ENS-PSL), and Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL)
- Subjects
allelic partition ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,tree-valued process ,coalescent point process ,Quantitative Biology::Populations and Evolution ,MSC2000 subject classifications: primary 05C05, 60J80 ,secondary 54E45, 60G51, 60G55, 60G57, 60K15, 92D10 ,regenerative set ,random point measure ,branching process - Abstract
Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure (infinitely-many-allele model) on the skeleton of the tree. The partition of the boundary into distinct alleles is the so-called allelic partition. In this paper, we are interested in the infinite trees generated by super-critical, possibly time-inhomogeneous, binary branching processes, and in their boundary, which is the set of particles 'co-existing at infinity'. We prove that any such tree can be mapped to a random, compact ultrametric tree called coalescent point process, endowed with a 'uniform' measure on its boundary which is the limit as t → ∞ of the properly rescaled counting measure of the population at time t. We prove that the clonal (i.e., carrying the same allele as the root) part of the boundary is a regenerative set that we characterize. We then study the allelic partition of the boundary through the measures of its blocks. We also study the dynamics of the clonal subtree, which is a Markovian increasing tree process as mutations are removed.
- Published
- 2017