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Allele trees for the mother-dependent neutral mutations model and their scaling limits in the rare mutations regime

Airam Blancas, Maria Clara Fittipaldi, Sarai Hernandez-Torres

TL;DR

The paper analyzes the genealogy of a population evolving under mother-dependent neutral mutations with a finite number of alleles. By developing a clone-mutant Markov chain and constructing a multitype allele tree, it derives a scaling limit in which rescaled allele trees converge to a tree-indexed continuous-state branching process (CSBP) with reproduction measure $\nu(dz)$ and inverse-Gaussian initial condition for the first generation. The full limiting object retains type information and mutant descendants, and is expressible through Bertoin’s universal allele tree, thereby extending Bertoin's infinite-alleles framework to multidimensional types. The results establish a rigorous link between discrete multitype genealogies and continuous-state tree-indexed limits under a rare-mutations regime, providing explicit MGFs for transition laws and a clear path for applications to structured populations.

Abstract

The mother-dependent neutral mutations model describes the evolution of a population across discrete generations, where neutral mutations occur among a finite set of possible alleles. In this model, each mutant child acquires a type different from that of its mother, chosen uniformly at random. In this work, we define a multitype allele tree associated with this model and analyze its scaling limit through a Markov chain that tracks the sizes of allelic subfamilies and their mutant descendants. We show that this Markov chain converges to a continuous-state Markov process, whose transition probabilities depend on the sizes of the initial allelic populations and those of their mutant offspring in the first allelic generation. As a result, the allele tree converges to a multidimensional limiting object, which can be described in terms of the universal allele tree introduced by Bertoin (2010).

Allele trees for the mother-dependent neutral mutations model and their scaling limits in the rare mutations regime

TL;DR

The paper analyzes the genealogy of a population evolving under mother-dependent neutral mutations with a finite number of alleles. By developing a clone-mutant Markov chain and constructing a multitype allele tree, it derives a scaling limit in which rescaled allele trees converge to a tree-indexed continuous-state branching process (CSBP) with reproduction measure and inverse-Gaussian initial condition for the first generation. The full limiting object retains type information and mutant descendants, and is expressible through Bertoin’s universal allele tree, thereby extending Bertoin's infinite-alleles framework to multidimensional types. The results establish a rigorous link between discrete multitype genealogies and continuous-state tree-indexed limits under a rare-mutations regime, providing explicit MGFs for transition laws and a clear path for applications to structured populations.

Abstract

The mother-dependent neutral mutations model describes the evolution of a population across discrete generations, where neutral mutations occur among a finite set of possible alleles. In this model, each mutant child acquires a type different from that of its mother, chosen uniformly at random. In this work, we define a multitype allele tree associated with this model and analyze its scaling limit through a Markov chain that tracks the sizes of allelic subfamilies and their mutant descendants. We show that this Markov chain converges to a continuous-state Markov process, whose transition probabilities depend on the sizes of the initial allelic populations and those of their mutant offspring in the first allelic generation. As a result, the allele tree converges to a multidimensional limiting object, which can be described in terms of the universal allele tree introduced by Bertoin (2010).
Paper Structure (25 sections, 8 theorems, 115 equations, 3 figures)

This paper contains 25 sections, 8 theorems, 115 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mother-dependent neutral mutations model
  3. Main result on multiallele trees
  4. Relation to previous work
  5. Some words about our notation
  6. Organization of this work
  7. Background on multitype BGW processes
  8. Plane trees
  9. Labelling a multitype BGW tree
  10. Colored trees and subtrees
  11. General branching property
  12. From trees to forests
  13. Clone-mutant Markov chain
  14. Definition of the clone-mutant Markov chain
  15. Transition probabilities of the clone-mutant Markov chain
  16. ...and 10 more sections

Key Result

Theorem 1.1

Consider a sequence of rescaled allele trees where each $\mathscr{A}^{(n)}$ is associated with a mother-dependent neutral mutations model following the law $\mathbb{P}_ { n^{-1}\mathbf{e}_j }^{r(n)}$, starting with one indivivual of type $j \in [d]$, under the hypothesis (Hhyp:mutation) and (Hhyp:size). Then, we have the following convergence where $\left(\mathcal{Y}_u : u \in \mathbbm{U}\right)

Figures (3)

  • Figure 1: Left: A realization of the mother-dependent neutral mutation model with $d = 3$ types. The types $1$, $2$ and $3$ are represented in green, orange, and purple, respectively. Mutations are indicated by marks on the edges, and the representative of each allelic subfamily is shown in a bolder shade. Right: Multitype allele tree associated with the colored genealogical tree on the left.
  • Figure 2: Below, in \ref{['eq:linemut']}, we define the set $L_n$ of mutant individuals in the $n$-th allelic generation. Each $L_n$ forms a stopping line. In this figure, we show the stopping lines $L_1$ and $L_2$ are highlighted in green and pink, respectively.
  • Figure 3: Multitype allele tree associated with the colored genealogical tree in Figure \ref{['fig:stoppingline']}. The green-shaded nodes in level one are the children of the mother with the highest number of mutant children. The remaining two nodes descend from different mothers who each have the same number of mutant children; therefore, we order them in ascending order by type.

Theorems & Definitions (14)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2: Jagers1989
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Corollary 3.4
  • Lemma 3.5
  • Remark 4.1
  • ...and 4 more