Mutations on a random binary tree with measured boundary

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 (infinitelymany-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 supercritical, 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. K: coalescent point process; branching process; random point measure; allelic partition; regenerative set; tree-valued process. MSC2000 subject classifications: primary 05C05, 60J80; secondary 54E45; 60G51; 60G55; 60G57; 60K15; 92D10.

Convergence of genealogies through spinal decomposition with an application to population genetics

“…Let (d ij ; i, j ≥ 1) and (Y i ; i ≥ 1) be an exchangeable marked ultrametric matrix. There exists a random marked probability UMS [U, d, U , µ] (that is, µ(U ) = 1 a.s.) such that the exchangeable marked ultrametric matrix obtained by sampling from it as in (11) is distributed as (d ij ; i, j ≥ 1) and (Y i ; i ≥ 1). Moreover this marked UMS is unique in distribution.…”

Random ultrametric trees and applications

“…It can be shown [8] that the boundary of (τ f ,d f ) is indeed (Ī,d f ), which explains why we keep the same notation for the two distances. Also that for each t ∈ I, L t = ρ, α t , where α t ∈Ī is equal to (t, r).…”

“…Proof. We know from [8] that ϕ(t) is the genealogy of a reversed (i.e. time flows from T to 0) pure-birth tree with birth rate β = β • ϕ −1 .…”

“…In particular, we can define µ(t) := µ([0, t]). Then it can be seen [8] that if 0 µ(x)ν(dx) < ∞ the total number of mutations is finite a.s. whereas if 0 µ(x)ν(dx) = ∞ then the number of mutations in any clade (set of all descendants of a point) is infinite a.s.…”

“…where T is here to recall the dependence on the height of the CPP, and the measure Λ T is defined on N when ν is finite [20] and on (0, ∞) when ν is infinite [8]. In addition, the following convergence holds weakly on [0, ∞)…”