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Decomposing the site frequency spectrum: the impact of tree topology on neutrality tests

Alice Ledda, Guillaume Achaz, Thomas Wiehe, Luca Ferretti
doi: https://doi.org/10.1101/042044
Alice Ledda
1Department of Infectious Disease Epidemiology, Imperial College, London.
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Guillaume Achaz
2Evolution Paris-Seine (UMR CNRS 7138), UPMC, Paris.
3Atelier de Bio-Informatique, UPMC, Paris.
4Stochastic Models for the Inference of Life Evolution, CIRB (UMR INSERM 7241), Collège de France, Paris.
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Thomas Wiehe
5Institute of Genetics, University of Cologne.
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Luca Ferretti
2Evolution Paris-Seine (UMR CNRS 7138), UPMC, Paris.
3Atelier de Bio-Informatique, UPMC, Paris.
4Stochastic Models for the Inference of Life Evolution, CIRB (UMR INSERM 7241), Collège de France, Paris.
6The Pirbright Institute, Woking, UK.
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Abstract

We investigate the dependence of the site frequency spectrum (SFS) on the topological structure of genealogical trees. We show that basic population genetic statistics − for instance estimators of θ or neutrality tests such as Tajima’s D – can be decomposed into components of waiting times between coalescent events and of tree topology. Our results clarify the relative impact of the two components on these statistics. We provide a rigorous interpretation of positive or negative values of neutrality tests in terms of the underlying tree shape. In particular, we show that values of Tajima’s D and Fay and Wu’s H depend in a direct way on a measure of tree balance which is mostly determined by the root balance of the tree. We also compute the maximum and minimum values for neutrality tests as a function of sample size.

Focusing on the standard coalescent model of neutral evolution, we discuss how waiting times between coalescent events are related to derived allele frequencies and thereby to the frequency spectrum. Finally, we show how tree balance affects the frequency spectrum. In particular, we derive the complete SFS conditioned on the root imbalance. We show that the conditional spectrum is peaked at frequencies corresponding to the root imbalance and strongly biased towards rare alleles.

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The copyright holder for this preprint is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY-NC 4.0 International license.
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Posted March 01, 2016.
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Decomposing the site frequency spectrum: the impact of tree topology on neutrality tests
Alice Ledda, Guillaume Achaz, Thomas Wiehe, Luca Ferretti
bioRxiv 042044; doi: https://doi.org/10.1101/042044
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Decomposing the site frequency spectrum: the impact of tree topology on neutrality tests
Alice Ledda, Guillaume Achaz, Thomas Wiehe, Luca Ferretti
bioRxiv 042044; doi: https://doi.org/10.1101/042044

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