Studying models of balancing selection using phase-type theory

Balancing selection (BLS) is the evolutionary force that maintains high levels of genetic variability in many important genes. To further our understanding of its evolutionary significance, we analyse models with BLS acting on a biallelic locus: an equilibrium model with long-term BLS, a model with long-term BLS and recent changes in population size, and a model of recent BLS. Using phase-type theory, a mathematical tool for analysing continuous time Markov chains with an absorbing state, we examine how BLS affects polymorphism patterns in linked neutral regions, as summarised by nucleotide diversity, the expected number of segregating sites, the site frequency spectrum, and the level of linkage disequilibrium (LD). Long-term BLS affects polymorphism patterns in a relatively small genomic neighbourhood, and such selection targets are easier to detect when the equilibrium frequencies of the selected variants are close to 50%, or when there has been a population size reduction. For a new mutation subject to BLS, its initial increase in frequency in the population causes linked neutral regions to have reduced diversity, an excess of both high and low frequency derived variants, and elevated LD with the selected locus. These patterns are similar to those produced by selective sweeps, but the effects of recent BLS are weaker. Nonetheless, compared to selective sweeps, non-equilibrium polymorphism and LD patterns persist for a much longer period under recent BLS, which may increase the chance of detecting such selection targets. An R package for analysing these models, among others (e.g., isolation with migration), is available.