Ancestral reproductive bias in branching processes

TL;DR

This paper investigates how sampling a single cell from a branching process introduces a bias in the observed ancestral lineage, revealing increasing reproductive output over time due to sampling bias, with implications for understanding mutation rate variation.

Contribution

It provides an explicit characterization of the evolution of reproduction rates along sampled lineages as a mixture of Poisson processes, accounting for sampling bias in branching processes.

Findings

Reproductive output along sampled lineages increases over time due to bias.

Sampling bias causes cells with more offspring to be overrepresented.

The model explains variation in mutation rates in human embryonic development.

Abstract

Consider a branching process with a homogeneous reproduction law. Sampling a single cell uniformly from the population at a time $T > 0$ and looking along the sampled cell's ancestral lineage, we find that the reproduction law is heterogeneous - the expected reproductive output of ancestral cells on the lineage from time $0$ to time $T$ continuously increases. This `inspection paradox' is due to sampling bias, that cells with a larger number of offspring are more likely to have one of their descendants sampled by virtue of their prolificity, and the bias's strength grows with the random population size and/or the sampling time $T$. Our main result explicitly characterises the evolution of reproduction rates and sizes along the sampled ancestral lineage as a mixture of Poisson processes, which simplifies in special cases. The ancestral bias helps to explain recently observed variation in mutation rates along lineages of the developing human embryo.