The concordance of gene trees and species trees at two linked loci

The gene genealogies of two linked loci in three species are analyzed using a series of Markov chain models. We calculate the probability that the gene tree of one locus is concordant with the species tree, given that the gene tree of the other locus is concordant. We define a threshold value of the recombination rate, r*, to be the rate for which the difference between the conditional probability of concordance and its asymptotic value is reduced to 5% of the initial difference. We find that, although r* depends in a complicated way on the times of speciation and effective population sizes, it is always relatively small, <10/N4, where N4 is the effective size of the species represented by the internal branch of the species tree. Consequently, the concordance of gene trees of neutral loci with the species tree is expected to be on roughly the same length scale on the chromosome as the extent of significant linkage disequilibrium within species unless the effective size of contemporary populations is very different from the effective sizes of their ancestral populations. Both balancing selection and selective sweeps can result in much longer genomic regions having concordant gene trees.