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Sampling probabilities, diffusions, ancestral graphs, and duality under strong selection

Martina Favero, Paul A. Jenkins

TL;DR

The paper analyzes Wright--Fisher diffusion with mutation and an exploding first-selective allele, establishing three diffusion limits as $\sigma\to\infty$: a deterministic logistic trajectory, Gaussian fluctuations around this trajectory, and non-Gaussian fluctuations governed by independent continuous-state branching processes with immigration near the boundary equilibrium. It then derives a Gamma-driven leading-term approximation for sampling probabilities and a full inverse-$\sigma$ asymptotic expansion via a double-recursive coefficient scheme, applicable to general mutation models; for the two-allele PIM case, explicit closed forms via confluent hypergeometric functions are provided. The ancestral processes are studied through the reduced conditional ASG, revealing a two-timescale dynamic with fast unfit-lineage coalescence and slow fit-lineage coalescence, and an asymptotic duality is established between the limiting CBI diffusion and the fast ancestral process. Collectively, the results yield practical asymptotic approximations for sampling probabilities under strong selection and illuminate the genealogical structure via a robust asymptotic duality framework, with implications for inference and exact-simulation approaches in multi-allele contexts.

Abstract

Wright-Fisher diffusions and their dual ancestral graphs occupy a central role in the study of allele frequency change and genealogical structure, and they provide expressions, explicit in some special cases but generally implicit, for the sampling probability, a crucial quantity in inference. Under a finite-allele mutation model, with possibly parent-dependent mutation, we consider the asymptotic regime where the selective advantage of one allele grows to infinity, while the other parameters remain fixed. In this regime, we show that the Wright-Fisher diffusion can be approximated either by a Gaussian process or by a process whose components are independent continuous-state branching processes with immigration, aligning with analogous results for Wright-Fisher models but employing different methods. While the first process becomes degenerate at stationarity, the latter does not and provides a simple, analytic approximation for the leading term of the sampling probability. Furthermore, using another approach based on a recursion formula, we characterise all remaining terms to provide a full asymptotic expansion for the sampling probability. Finally, we study the asymptotic behaviour of the rates of the block-counting process of the conditional ancestral selection graph and establish an asymptotic duality relationship between this and the diffusion.

Sampling probabilities, diffusions, ancestral graphs, and duality under strong selection

TL;DR

The paper analyzes Wright--Fisher diffusion with mutation and an exploding first-selective allele, establishing three diffusion limits as : a deterministic logistic trajectory, Gaussian fluctuations around this trajectory, and non-Gaussian fluctuations governed by independent continuous-state branching processes with immigration near the boundary equilibrium. It then derives a Gamma-driven leading-term approximation for sampling probabilities and a full inverse- asymptotic expansion via a double-recursive coefficient scheme, applicable to general mutation models; for the two-allele PIM case, explicit closed forms via confluent hypergeometric functions are provided. The ancestral processes are studied through the reduced conditional ASG, revealing a two-timescale dynamic with fast unfit-lineage coalescence and slow fit-lineage coalescence, and an asymptotic duality is established between the limiting CBI diffusion and the fast ancestral process. Collectively, the results yield practical asymptotic approximations for sampling probabilities under strong selection and illuminate the genealogical structure via a robust asymptotic duality framework, with implications for inference and exact-simulation approaches in multi-allele contexts.

Abstract

Wright-Fisher diffusions and their dual ancestral graphs occupy a central role in the study of allele frequency change and genealogical structure, and they provide expressions, explicit in some special cases but generally implicit, for the sampling probability, a crucial quantity in inference. Under a finite-allele mutation model, with possibly parent-dependent mutation, we consider the asymptotic regime where the selective advantage of one allele grows to infinity, while the other parameters remain fixed. In this regime, we show that the Wright-Fisher diffusion can be approximated either by a Gaussian process or by a process whose components are independent continuous-state branching processes with immigration, aligning with analogous results for Wright-Fisher models but employing different methods. While the first process becomes degenerate at stationarity, the latter does not and provides a simple, analytic approximation for the leading term of the sampling probability. Furthermore, using another approach based on a recursion formula, we characterise all remaining terms to provide a full asymptotic expansion for the sampling probability. Finally, we study the asymptotic behaviour of the rates of the block-counting process of the conditional ancestral selection graph and establish an asymptotic duality relationship between this and the diffusion.
Paper Structure (20 sections, 7 theorems, 132 equations)

This paper contains 20 sections, 7 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Two different diffusion limits
  3. Gaussian fluctuations around the logistic trajectory
  4. CBI fluctuations around the equilibrium point
  5. Stationary distributions
  6. Sampling probabilities
  7. Two alleles and parent independent mutation
  8. Gamma approximation
  9. Full asymptotic expansion for general mutation models
  10. Ancestral processes
  11. Asymptotic analysis of the rates
  12. Interpretation of the ancestral process under strong selection
  13. Duality
  14. Asymptotic duality
  15. Appendix $\cdot$ Limits of Wright--Fisher models
  16. ...and 5 more sections

Key Result

proposition 1

Let $\boldsymbol{X}^{(\sigma)}$ be the WF diffusion defined as the solution to eq:WF_SDE. Then the following convergence results hold in the strong selection limit. (a) $\boldsymbol{X}^{(\sigma)}$ converges to a deterministic logistic trajectory. That is, assuming $\boldsymbol{X}^{(\sigma)}(0)\xrigh where $\boldsymbol{\xi}$ is the logistic trajectory defined by the following ODE or, more explicit

Theorems & Definitions (19)

  • proposition 1
  • proof
  • remark 1: degenerate Gaussian stationary distribution
  • remark 2: Gamma stationary distribution
  • proposition 2
  • proof
  • lemma 1
  • proof
  • remark 3
  • theorem 1
  • ...and 9 more