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Limit theorems for the site frequency spectrum of neutral mutations in an exponentially growing population

Einar Bjarki Gunnarsson, Kevin Leder, Xuanming Zhang

TL;DR

It is shown that the SFS of a Galton-Watson process evaluated at a fixed time converges almost surely to a random limit and the results can be used to construct consistent estimators for the extinction probability and the effective mutation rate of a birth-death process.

Abstract

The site frequency spectrum (SFS) is a widely used summary statistic of genomic data. Motivated by recent evidence for the role of neutral evolution in cancer, we investigate the SFS of neutral mutations in an exponentially growing population. Using branching process techniques, we establish (first-order) almost sure convergence results for the SFS of a Galton-Watson process, evaluated either at a fixed time or at the stochastic time at which the population first reaches a certain size. We finally use our results to construct consistent estimators for the extinction probability and the effective mutation rate of a birth-death process.

Limit theorems for the site frequency spectrum of neutral mutations in an exponentially growing population

TL;DR

It is shown that the SFS of a Galton-Watson process evaluated at a fixed time converges almost surely to a random limit and the results can be used to construct consistent estimators for the extinction probability and the effective mutation rate of a birth-death process.

Abstract

The site frequency spectrum (SFS) is a widely used summary statistic of genomic data. Motivated by recent evidence for the role of neutral evolution in cancer, we investigate the SFS of neutral mutations in an exponentially growing population. Using branching process techniques, we establish (first-order) almost sure convergence results for the SFS of a Galton-Watson process, evaluated either at a fixed time or at the stochastic time at which the population first reaches a certain size. We finally use our results to construct consistent estimators for the extinction probability and the effective mutation rate of a birth-death process.
Paper Structure (29 sections, 10 theorems, 153 equations)

This paper contains 29 sections, 10 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Branching process model with neutral mutations
  4. Special case: Birth-death process
  5. Asymptotic behavior
  6. Site frequency spectrum
  7. Results
  8. General case
  9. Special case: Birth-death process
  10. Connection to previous results
  11. Proof of Theorem \ref{['thm:mainresult']}
  12. Decomposition into increasing processes $S_{j,+}(t)$ and $S_{j,-}(t)$
  13. Approximation results for $S_{j,+}^k(t)$ and $S_{j,-}^k(t)$
  14. Limit theorems for $S_{j,+}(t)$ and $S_{j,-}(t)$
  15. Proof of the fixed-time result \ref{['eq:fixedtimemainresult']}
  16. ...and 14 more sections

Key Result

Theorem 1

Theorems & Definitions (27)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • ...and 17 more