Matrix inversions for chromosomal inversions: A method to construct summary statistics in complex coalescent models

Chromosomal inversions allow genetic divergence of locally adapted populations by reducing recombination between chromosomes with different arrangements. While patterns of genetic variation within inverted regions are increasingly documented, inferential methods are largely missing to analyze such data. Previous work has provided expectations for coalescence patterns of neutral sites linked to an inversion polymorphism in two locally adapted populations. Here, we define a method to construct summary statistics in such complex population structure models. Under a scenario of selection on the inversion breakpoints, we first construct estimators of the migration rate between the two habitats, and of the recombination rate of a nucleotide site between the two inversion backgrounds. Next, we analyze the disequilibrium between two sites within an inversion and provide an estimator of the distinct recombination rate between these two sites in homokaryotypes and heterokaryotypes. These estimators should be suitable summary statistics for simulation-based methods that can handle the complex dependences in the data.