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Statistical properties of pairwise distances between leaves on a random Yule tree

MPS-Authors
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Sheinman,  Michel
Evolutionary Genomics (Peter Arndt), Dept. of Computational Molecular Biology (Head: Martin Vingron), Max Planck Institute for Molecular Genetics, Max Planck Society;

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Massip,  Florian
Evolutionary Genomics (Peter Arndt), Dept. of Computational Molecular Biology (Head: Martin Vingron), Max Planck Institute for Molecular Genetics, Max Planck Society;
INRA, UR1077 Unite Mathematique Informatique et Genome, Jouy-en-Josas, France ;

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Arndt,  Peter F.
Evolutionary Genomics (Peter Arndt), Dept. of Computational Molecular Biology (Head: Martin Vingron), Max Planck Institute for Molecular Genetics, Max Planck Society;

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Sheinman.PDF
(Publisher version), 908KB

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Citation

Sheinman, M., Massip, F., & Arndt, P. F. (2015). Statistical properties of pairwise distances between leaves on a random Yule tree. PLoS One, 10(3): e0120206. doi:10.1371/journal.pone.0120206.


Cite as: http://hdl.handle.net/11858/00-001M-0000-002A-392A-7
Abstract
A Yule tree is the result of a branching process with constant birth and death rates. Such a process serves as an instructive null model of many empirical systems, for instance, the evolution of species leading to a phylogenetic tree. However, often in phylogeny the only available information is the pairwise distances between a small fraction of extant species representing the leaves of the tree. In this article we study statistical properties of the pairwise distances in a Yule tree. Using a method based on a recursion, we derive an exact, analytic and compact formula for the expected number of pairs separated by a certain time distance. This number turns out to follow a increasing exponential function. This property of a Yule tree can serve as a simple test for empirical data to be well described by a Yule process. We further use this recursive method to calculate the expected number of the n-most closely related pairs of leaves and the number of cherries separated by a certain time distance. To make our results more useful for realistic scenarios, we explicitly take into account that the leaves of a tree may be incompletely sampled and derive a criterion for poorly sampled phylogenies. We show that our result can account for empirical data, using two families of birds species.