[Paper Review] Dynamic sampling bias and overdispersion induced by skewed offspring distributions

Takashi Okada, Oskar Hallatschek|arXiv (Cornell University)|Mar 9, 2021

Evolution and Genetic Dynamics60 references11 citations

TL;DR

This paper introduces a dynamic sampling bias that arises in populations with skewed offspring distributions (power-law with exponent 1 < α < 2), which suppresses minor alleles over time and leads to overdispersed allele frequency trajectories. Using exact asymptotic analysis and scaling hypotheses, the authors derive universal scaling laws for fixation probabilities, extinction times, and site frequency spectra, showing that this bias—driven by time-dependent sampling of large families—dominates over traditional diffusion processes and explains non-Wright-Fisher dynamics in systems like range expansions and epidemics.

ABSTRACT

Natural populations often show enhanced genetic drift consistent with a strong skew in their offspring number distribution. The skew arises because the variability of family sizes is either inherently strong or amplified by population expansions, leading to so-called `jackpot' events. The resulting allele frequency fluctuations are large and, therefore, challenge standard models of population genetics, which assume sufficiently narrow offspring distributions. While the neutral dynamics backward in time can be readily analyzed using coalescent approaches, we still know little about the effect of broad offspring distributions on the dynamics forward in time, especially with selection. Here, we employ an exact asymptotic analysis combined with a scaling hypothesis to demonstrate that over-dispersed frequency trajectories emerge from the competition of conventional forces, such as selection or mutations, with an emerging time-dependent sampling bias against the minor allele. The sampling bias arises from the characteristic time-dependence of the largest sampled family size within each allelic type. Using this insight, we establish simple scaling relations for allele frequency fluctuations, fixation probabilities, extinction times, and the site frequency spectra that arise when offspring numbers are distributed according to a power law $~n^{-(1+\alpha)}$. To demonstrate that this coarse-grained model captures a wide variety of non-equilibrium dynamics, we validate our results in traveling waves, where the phenomenon of 'gene surfing' can produce any exponent $1<\alpha <2$. We argue that the concept of a dynamic sampling bias is useful generally to develop both intuition and statistical tests for the unusual dynamics of populations with skewed offspring distributions, which can confound commonly used tests for selection or demographic history.

Motivation & Objective

• To understand forward-time allele frequency dynamics under broad, power-law offspring distributions (n−(1+α)) with 1 < α < 2.
• To identify and characterize a time-dependent sampling bias that suppresses minor alleles due to preferential sampling of large families.
• To develop a coarse-grained model that captures overdispersion and non-diffusive fluctuations in allele frequency trajectories.
• To validate the model in non-equilibrium systems like traveling waves and gene surfing, where such distributions naturally arise.
• To provide a framework for detecting selection and demographic history in systems where standard Wright-Fisher models fail.

Proposed method

• Employs exact asymptotic analysis to study allele frequency dynamics under power-law offspring distributions with 1 < α < 2.
• Introduces a scaling hypothesis to relate the time-dependent sampling bias to the size of the largest family in each allelic type.
• Derives scaling relations for fixation probability, extinction time, stationary distribution, and site frequency spectrum using the emergent bias.
• Uses backward Fokker-Planck equations and Laplace transforms to analyze area under frequency trajectories, particularly for rare mutants.
• Validates results via numerical simulations in the Eldon-Wakeley model and in traveling wave systems with gene surfing.
• Compares results to the Wright-Fisher diffusion and the α = 1 case (Bolthausen-Sznitman coalescent) to establish the intermediate regime.

Experimental results

Research questions

• RQ1 How does a power-law offspring distribution with 1 < α < 2 affect forward-time allele frequency dynamics in the presence of selection or drift?
• RQ2 What is the origin and time-dependence of the sampling bias that suppresses minor alleles in such systems?
• RQ3 How do fixation probabilities, extinction times, and site frequency spectra scale under broad offspring distributions compared to the Wright-Fisher model?
• RQ4 Can the dynamic sampling bias explain overdispersion and non-diffusive fluctuations observed in real populations like expanding microbes or epidemics?
• RQ5 To what extent can this model replace or improve upon standard coalescent and diffusion approximations in demographic inference?

Key findings

• A time-dependent sampling bias emerges that suppresses minor alleles by preferentially sampling large families, leading to overdispersed allele frequency trajectories.
• The bias fades over time as larger families are sampled more thoroughly, leading to a transient but strong influence on dynamics.
• Fixation probabilities and extinction times scale with N−(1−α/2) for 1 < α < 2, deviating significantly from Wright-Fisher predictions.
• The site frequency spectrum exhibits a power-law tail with exponent 1−α, differing from the neutral Wright-Fisher spectrum.
• Numerical simulations confirm that the model captures non-diffusive behavior in systems like traveling waves and gene surfing, where α ∈ (1,2) is naturally generated.
• The model explains why standard tests for selection or demography fail in populations with skewed offspring distributions, due to confounding sampling bias.