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Duality plays an important role in population genetics. It can relate resultsfrom forwards-in-time models of allele frequency evolution with those ofbackwards-in-time genealogical models; a well known example is the dualitybetween the Wright-Fisher diffusion for genetic drift and its genealogicalcounterpart, the coalescent. There have been a number of articles extendingthis relationship to include other evolutionary processes such as mutation andselection, but little has been explored for models also incorporating crossoverrecombination. Here, we derive from first principles a new genealogical processwhich is dual to a Wright-Fisher diffusion model of drift, mutation, andrecombination. Our approach is based on expressing a putative dualityrelationship between two models via their infinitesimal generators, and thenseeking an appropriate test function to ensure the validity of the dualityequation. This approach is quite general, and we use it to find dualities forseveral important variants, including both a discrete L-locus model of a geneand a continuous model in which mutation and recombination events are scatteredalong the gene according to continuous distributions. As an application of ourresults, we derive a series expansion for the transition function of thediffusion. Finally, we study in further detail the case in which mutation isabsent. Then the dual process describes the dispersal of ancestral geneticmaterial across the ancestors of a sample. The stationary distribution of thisprocess is of particular interest; we show how duality relates thisdistribution to haplotype fixation probabilities. We develop an efficientmethod for computing such probabilities in multilocus models.

A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning