The Fractional Coalescent

Abstract

A new approach to the coalescent, the fractional coalescent ( f -coalescent), is introduced. Two derivations are presented: first, the f -coalescent is based on an extension of the discrete-time Wright-Fisher model. In this extension, for the population of size N , the probability that two randomly selected individuals have the same parent in the previous generation depends on the variable α. Second, the f -coalescent is based on an extension of the discrete-time Canning population model that the variance of the number of offspring is assumed as a random variable which depends on the variable α. In the second derivation, the f -coalescent emerges also as a continuous-time semi-Markov process. The additional parameter α affects the variability in the patterns of the waiting times; values of α < 1 lead to an increase of short time intervals, but allows occasionally for very long time intervals. When α = 1, the f -coalescent and Kingman’s n -coalescent are equivalent. The mode of the distribution of the time of the most recent common ancestor in the f -coalescent is lower than n -coalescent when the number of sample size increases, and the time which modes happen on that is smaller compare to n -coalescent. Also, this distribution showed that the f -coalescent leads to distributions with heavier tails than the n -coalescent. Also, the probability that n genes descend from m ancestral genes for f -coalescent is derived. The f -coalescent has been implemented in the population genetic model inference software M IGRATE . Simulation studies suggest that it is possible to infer the correct α values from data that was generated with known α values. When data is simulated using models with α < 1 or for three example datasets (H1N1 influenza, Malaria parasites, Humpback whales), Bayes factor comparisons show an improved model fit of the f -coalescent over the n -coalescent.

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