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Multi-type $Ξ$-coalescents from structured population models with bottlenecks

Marta Dai Pra, Alison Etheridge, Jere Koskela, Maite Wilke-Berenguer

TL;DR

The paper advances population-genetic theory for structured populations undergoing bottlenecks by deriving two scaling regimes, drastic and soft, in which the forward allele-frequency process converges to jump-diffusions and the backward genealogies converge to multi-type Ξ-coalescents with simultaneous mergers and migrations. A novel d_λ metric enables convergence analysis through bottleneck intervals, and the two regimes are connected by moment duality between the forward diffusion and the coalescent processes. Drastic bottlenecks yield a diffusion with jumps whose size distribution is bottleneck-driven, while soft bottlenecks produce a coalescent with an α-time window of coalescent activity and corresponding jump-diffusion dynamics. Simulations show the model can generate diverse site frequency spectra with a small set of interpretable parameters, offering a flexible framework for fitting data from structured populations such as Atlantic cod.

Abstract

We introduce an individual-based model for structured populations undergoing demographic bottlenecks, i.e. drastic reductions in population size that last many generations and can have arbitrary shapes. We first show that the (non-Markovian) allele-frequency process converges to a Markovian diffusion process with jumps in a suitable relaxation of the Skorokhod J1 topology. Backward in time we find that genealogies of samples of individuals are described by multi-type $Ξ$-coalescents presenting multiple simultaneous mergers with simultaneous migrations. These coalescents are also moment-duals of the limiting jump diffusions. We then show through a numerical study that our model is flexible and can predict various shapes for the site frequency spectrum, consistent with real data, using a small number of interpretable parameters.

Multi-type $Ξ$-coalescents from structured population models with bottlenecks

TL;DR

The paper advances population-genetic theory for structured populations undergoing bottlenecks by deriving two scaling regimes, drastic and soft, in which the forward allele-frequency process converges to jump-diffusions and the backward genealogies converge to multi-type Ξ-coalescents with simultaneous mergers and migrations. A novel d_λ metric enables convergence analysis through bottleneck intervals, and the two regimes are connected by moment duality between the forward diffusion and the coalescent processes. Drastic bottlenecks yield a diffusion with jumps whose size distribution is bottleneck-driven, while soft bottlenecks produce a coalescent with an α-time window of coalescent activity and corresponding jump-diffusion dynamics. Simulations show the model can generate diverse site frequency spectra with a small set of interpretable parameters, offering a flexible framework for fitting data from structured populations such as Atlantic cod.

Abstract

We introduce an individual-based model for structured populations undergoing demographic bottlenecks, i.e. drastic reductions in population size that last many generations and can have arbitrary shapes. We first show that the (non-Markovian) allele-frequency process converges to a Markovian diffusion process with jumps in a suitable relaxation of the Skorokhod J1 topology. Backward in time we find that genealogies of samples of individuals are described by multi-type -coalescents presenting multiple simultaneous mergers with simultaneous migrations. These coalescents are also moment-duals of the limiting jump diffusions. We then show through a numerical study that our model is flexible and can predict various shapes for the site frequency spectrum, consistent with real data, using a small number of interpretable parameters.

Paper Structure

This paper contains 12 sections, 11 theorems, 161 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Drastic bottlenecks
  4. The frequency process
  5. Duality
  6. Soft bottlenecks
  7. Coalescent process
  8. The frequency process
  9. Duality
  10. Simulations
  11. Existence and uniqueness of a strong solution for the limiting drastic bottleneck frequency process
  12. The structured Wright--Fisher model and Kingman coalescent during a soft bottleneck

Key Result

Theorem 1

Fix $N\in \mathbb{N}$, $\gamma>0$ and $L$ a probability measure on $\mathbb{N}$. Fix also a sequence $\{l_k^N\}_{k\in \mathbb{N}}$, a sequence $\{s_k^N\}_{k\in \mathbb{N}}$ and a function $F:\mathbb{N} \rightarrow \mathbb{N}$ satisfying enumerate:1, enumerate:2, enumerate:3 respectively. Let $A^\tex where $\xRightarrow{d_{\lambda}}$ denotes weak convergence in the topology induced by $d_{\lambda}$

Figures (7)

  • Figure 1: An example of a coalescent tree obtained as a limiting genealogy of our model. Different colours represent different demes, in this case they are $D=3$. We note that we can have binary mergers and single migrations as well as simultaneous events (multiple simultaneous mergers and simultaneous migrations). The latter, caused by bottlenecks, are highlighted with circles. Circles on the same level show all the events happening during a particular bottleneck and their colour corresponds to the affected deme.
  • Figure 2: Example of the effect of bottlenecks on the population size in the case D=2. The circles represent individuals, and their number is constant in between bottlenecks and decreases in a deme when a bottleneck happens there. See the main text for a full explanation of the notation.
  • Figure 3: In this picture we show the evolution of the population with $D=3$ when a bottleneck affects deme $1$ for $\tau=4$ generations and with $c_1=2$. At each step of the bottleneck two individuals come from the other demes: one from the second (white) and one from the third (grey). In $F(1)+c_1(\tau-1)=11$ different colours and shapes we highlight the families whose size we want to know at the end of the bottleneck. Their sizes at the end of the bottleneck are $\mathbf{A}^1_1(\tau)=(1,0,0),\; \mathbf{A}^1_2(\tau)=(0,0,1,1),\; \mathbf{A}^1_3(\tau)=(0,2,0,1)$.
  • Figure 4: In this figure we show a possible realisation of the discrete processes $X^N$, $V^N$, $Z^N$, $Z^N\circ (g^N)^{-1}$. In red, underlined by a square bracket, we draw what is happening during a bottleneck. In particular, we only represent the affected deme; with a red square we highlight the last generation of a bottleneck and with a blue cross the second to last generation $NT-1$ (left-hand side) which becomes $NT-\delta^N-1$ in the time scale that skips the interior of the bottlenecks (right-hand side). We highlight in a light blue rectangle the only area where $V^N$ and $Z^N \circ (g^N)^{-1}$ do not coincide.
  • Figure 5: A realization of the functions $g^N$ and $(g^N)^{-1}$ when $m^N_T=2$.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Remark 1.1
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 1
  • Remark 2.6
  • proof : Proof of Theorem \ref{['thm:drastic-diffusion']}
  • Remark 2.7
  • ...and 25 more