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

Einar Bjarki Gunnarsson, Kevin Leder, Jasmine Foo

Research output: Contribution to journalArticlepeer-review

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/j2 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.

Original languageEnglish (US)
Pages (from-to)67-90
Number of pages24
JournalTheoretical Population Biology
Volume142
Early online dateSep 21 2021
DOIs
StatePublished - Dec 2021

Bibliographical note

Funding Information:
The authors would like to thank Anna Kraut, the three anonymous reviewers and the editor for their valuable comments and suggestions. EBG and KL were supported in part by NSF, USA grant CMMI-1552764. JF was supported in part by NSF, USA grant DMS-1349724. KL and JF were supported in part by Research Council of Norway R&D Grant 309273. EBG was supported in part with funds from the Norwegian Centennial Chair Program, USA. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding Information:
The authors would like to thank Anna Kraut, the three anonymous reviewers and the editor for their valuable comments and suggestions. EBG and KL were supported in part by NSF, USA grant CMMI-1552764 . JF was supported in part by NSF, USA grant DMS-1349724 . KL and JF were supported in part by Research Council of Norway R&D Grant 309273 . EBG was supported in part with funds from the Norwegian Centennial Chair Program, USA .

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

  • Branching processes
  • Cancer evolution
  • Exponentially growing populations
  • Infinite sites model
  • Mathematical modeling
  • Site frequency spectrum

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