Abstract
The coupled Wright–Fisher diffusion is a multi-dimensional Wright–Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright–Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright–Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright–Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.
Highlights
The coupled Wright–Fisher diffusion was introduced by Aurell et al (2019) with the purpose of analysing networks of loci in recombining populations of bacteria, or more precisely, detecting couples of loci co-evolving under strong selective pressure when the linkage disequilibrium is low across the genome
In Skwark et al (2017), it is explained that the high amount of homologous recombination in populations of Streptococcus Pneumoniae, which results in low linkage disequilibrium across the genome, makes this population ideal for detecting genes that evolve under shared selection pressure
The results show that, in this model, the dual process corresponds to the block counting process of L coupled ancestral selection graph (ASG), one for each locus, evolving simultaneously
Summary
The coupled Wright–Fisher diffusion was introduced by Aurell et al (2019) with the purpose of analysing networks of loci in recombining populations of bacteria, or more precisely, detecting couples of loci co-evolving under strong selective pressure when the linkage disequilibrium is low across the genome. The coupled Wright–Fisher diffusion is obtained as the weak limit of a sequence of discrete Wright– Fisher models characterised by the assumption that the evolution of the population at one locus is conditionally independent of the other loci given that the previous generation at each locus is known, see Aurell et al (2019) for details. It is based on quasi-linkage equilibrium where the fitness coefficients, see Sect. Kimura (1955) suggests a Wright–Fisher model for multi-locus and multi-allelic genetic frequencies and conjectures that the stationary density is of the form π em, where π is the product of Dirichlet densities and m is a generic mean fitness term
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