Computing likelihoods for coalescents with multiple collisions in the infinitely many sites model

Abstract

One of the central problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of the classical Fleming-Viot process, such as the Wright-Fisher- or the Moran model, then the standard means to describe its genealogy is Kingman's coalescent. For this coalescent process, powerful inference methods are well-established. An important feature of the above class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size, and hence all ancestral collisions are binary only. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised Lambda-Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called Lambda-coalescents, which allow multiple collisions. Moreover, Eldon and Wakeley (Genetics 172:2621-2633, 2006) provide evidence that such more general coalescents might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier and Griffiths (Ann Probab 15(2):515-545, 1987) and Griffiths and Tavaré (Theor Pop Biol 46:131-159, 1994a, Stat Sci 9:307-319, 1994b, Philos Trans Roy Soc Lond Ser B 344:403-410, 1994c, Math Biosci 12:77-98, 1995) to obtain a likelihood based inference method for general Lambda-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of Lambda-coalescents, the parametrisable sub-family of Beta(2 - alpha, alpha)-coalescents, where alpha in (1, 2], are of particular relevance. We illustrate our method using simulated datasets, thus obtaining maximum-likelihood estimators of mutation and demographic parameters.

Coalescence 2.0: a multiple branching of recent theoretical developments and their applications

“…The properties of the beta-coalescent are well studied and various aspects of its genealogies computed such as length of coalescent trees, the expected sitefrequency spectrum (SFS) and number of segregating sites (Fig. 2, (Berestycki et al 2007(Berestycki et al , 2008Birkner & Blath 2008). These results are key for the potential application to analysis of polymorphism data.…”

“…For the Beta-coalescent, the ratio R 1 tends to the value 2-α for large sample size (n→∞; Berestycki et al 2007Berestycki et al , 2008. In practice, inference of α is possible based on the SFS and the total number of segregating sites of a large enough sample n of individuals (Birkner and Blath 2008;Birkner et al 2011;Steinrücken et al 2013).…”

“…It is defined as the Beta(α, 2-α) coalescent with a specific rate of multiple coalescent events α (see Appendix). In mathematical terms, multiple merger events occur on the time scale of O(1/N α ) (with α < 2) as described in Birkner and Blath (2008). The Beta-coalescent is obtained as a specific case of Λ-coalescent assuming a beta distribution, with parameter α, of multiple merger probabilities (Schweinsberg 2003).…”

“…The aim is here to infer from SNP data the rate of sweepstake events, that is how often they occur and which percentage of the population is affected. Likelihood methods have thus been developed for the Λ-coalescent (Birkner et al 2011), Beta-coalescent (Birkner and Blath 2008;Steinrücken et al 2013) or ψ-coalescent (Eldon & Wakeley 2006;Cenik & Wakeley 2010;Eldon 2011) in order to infer the parameters of interest based on the site frequency-spectrum.…”

“…The Beta-coalescent simulations are computed using the recursion (R-code) from Birkner et al (2008).…”