Exact site frequency spectra of neutrally evolving tumors: a transition between power laws reveals a signature of cell viability

TL;DR

This paper derives exact formulas for the site frequency spectrum in neutrally evolving tumors, revealing how cell viability influences the spectrum's shape and enabling estimation of key parameters from genomic data.

Contribution

It provides the first exact expressions for the SFS in stochastic branching models of tumor growth, highlighting the impact of cell death and birth rates on the spectrum's shape.

Findings

The SFS transitions from a 1/j^2 to a 1/j power law as cell viability decreases.

Mutation rate scales the SFS linearly.

The shape of the SFS can estimate cell death and birth rate ratios.

Abstract

The site frequency spectrum (SFS) is a popular summary statistic of genomic data. While the SFS of a constant-sized population undergoing neutral mutations has been extensively studied in population genetics, the rapidly growing amount of cancer genomic data has attracted interest in the spectrum of an exponentially growing population. Recent theoretical results have generally dealt with special or limiting cases, such as considering only cells with an infinite line of descent, assuming deterministic tumor growth, or taking large-time or large-population limits. In this work, we derive exact expressions for the expected SFS of a cell population that evolves according to a stochastic branching process, first for cells with an infinite line of descent and then for the total population, evaluated either at a fixed time (fixed-time spectrum) or at the stochastic time at which the population reaches a certain size (fixed-size spectrum). We find that while the rate of mutation scales the SFS of the total population linearly, the rates of cell birth and cell death change the shape of the spectrum at the small-frequency end, inducing a transition between a $1/j^2$ power-law spectrum and a $1/j$ spectrum as cell viability decreases. We show that this insight can in principle be used to estimate the ratio between the rate of cell death and cell birth, as well as the mutation rate, using the site frequency spectrum alone. Although the discussion is framed in terms of tumor dynamics, our results apply to any exponentially growing population of individuals undergoing neutral mutations.