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Minimal clade size and external branch length under the neutral coalescent

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Michael G. B. Blum  and
Olivier François
Michael G. B. Blum*
Affiliation:
Institut National Polytechnique de Grenoble
Olivier François*
Affiliation:
Institut National Polytechnique de Grenoble
*
∗ Postal address: Laboratoire TIMC-TIMB, Institute for Health and Information Engineering, Faculty of Medicine, F38706 La Tronche cedex, France.
∗ Postal address: Laboratoire TIMC-TIMB, Institute for Health and Information Engineering, Faculty of Medicine, F38706 La Tronche cedex, France.
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Abstract

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Given a sample of genes taken from a large population, we consider the neutral coalescent genealogy and study the theoretical and empirical distributions of the size of the smallest clade containing a fixed gene. We show that the theoretical distribution is strongly related to a Yule distribution of parameter 2, and that the empirical count statistics are asymptotically Gaussian as the number of genes grows to infinity. Then we consider external branches of the coalescent tree, and describe their lengths. Using the infinitely many sites model of mutation, we also describe the conditional distribution of the external branch lengths, given the number of pairwise differences between a reference DNA sequence and the sequence of one closest relative in the sample.

Keywords

Coalescent genealogiestree shape statisticsstatistical genetics

MSC classification

Primary: 92D10: Genetics 92D20: Protein sequences, DNA sequences
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 647 - 662
Copyright
Copyright © Applied Probability Trust 2005 

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