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Transition in the ancestral reproduction rate and its implications for the site frequency spectrum

Yubo Shuai

Abstract

Consider a supercritical birth and death process where the children acquire mutations. We study the mutation rates along the ancestral lineages in a sample of size $n$ from the population at time $T$. The mutation rate is time-inhomogenous and has a natural probabilistic interpretation. We use these results to obtain asymptotic results for the site frequency spectrum associated with the sample.

Transition in the ancestral reproduction rate and its implications for the site frequency spectrum

Abstract

Consider a supercritical birth and death process where the children acquire mutations. We study the mutation rates along the ancestral lineages in a sample of size from the population at time . The mutation rate is time-inhomogenous and has a natural probabilistic interpretation. We use these results to obtain asymptotic results for the site frequency spectrum associated with the sample.
Paper Structure (14 sections, 14 theorems, 84 equations, 5 figures)

This paper contains 14 sections, 14 theorems, 84 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Ancestral reproduction rate
  3. The site frequency spectrum
  4. The contour process of the genealogical tree
  5. Mutations in a sample from a population
  6. Bernoulli$(y)$ sampling
  7. Uniform$(n)$ sampling
  8. Asymptotics for large $n$ and $T$
  9. Approximation for the sampling probability and branch length
  10. Error bounds
  11. Proof of Theorems \ref{['Theorem: LLN']} and \ref{['Theorem']}
  12. Moment computations
  13. Proof of Theorem \ref{['Theorem: LLN']}
  14. Proof of Theorem \ref{['Theorem']} and Corollary \ref{['Corollary: number of mutations']}

Key Result

Theorem 1.1

There exists a random variable $S$ with density $F'_T(s)/(1-F_T(0))$ on [0,1] such that conditional on $S=s$, independently for each $l$, size $l$ reproduction events occur along the uniform ancestral lineage according to a time inhomogeneous Poisson point process with intensity function

Figures (5)

  • Figure 2: The planar representation of a genealogical tree
  • Figure 3: The contour process of the genealogical tree in Figure \ref{['Fig: the planar representation of a genealogical tree']}. For $i=1,2,...,8$, $t_i$ is the total length of the branches searched before we reach the $i$th individual, and the size of the jump is the lifespan of the $i$th individual.
  • Figure 4: The coalescent tree (in red) where individuals 2, 5, and 7 are sampled. This tree is obtained by tracing back the lineages of the sampled individuals from time $T$.
  • Figure 5: The same coalescent tree as in Figure \ref{['Fig: the coalescent tree']}, except that the parent mutates when there is a birth event that is not at one of the branchpoints in the coalescent tree.
  • Figure 6: A coalescent tree with sample size $n=9$. The blue mutation on the segment $AB$ supports three individuals because $\max\{H_{6,n,T}, H_{7,n,T}\}\le \min\{H_{5,n,T}, H_{8,n,T}\}$, the red mutation associated with $H_{3,n,T}$ supports two individuals because $H_{4,n,T}<H_{3,n,T}<H_{5,n,T}$.

Theorems & Definitions (22)

  • Theorem 1.1: Theorem 2.2 of cheek2023ancestral
  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Theorem 1.3
  • Corollary 1.2
  • Remark 1.1
  • Theorem 1.4: Theorem 1.3 of schweinsberg2023asymptotics
  • Theorem 1.5: Corollary 2 of johnson2023clonerate
  • Remark 3.1
  • ...and 12 more