Environmental gradients and the establishment of chromosomal inversions

Chromosomal inversions are thought to promote local adaptation by capturing combinations of locally beneficial alleles and shielding them from recombination. Many of the best-documented examples are inversions that form clines along environmental gradients, yet analytical theory for this mechanism has remained confined to continent-island and two-deme models. We combine classical cline theory with quasi-linkage equilibrium approximations and spatial branching processes to derive the growth rate, establishment probability, and total establishment rate of a rare inversion along continuous environmental gradients of varying shape. After rescaling space by the characteristic cline width, all three quantities follow universal profiles and simple scaling laws. At a sharp environmental transition, the local growth rate of an inversion equals the selection acting on the captured alleles, independently of dispersal. Yet its establishment probability is only about one tenth of that value, again independently of dispersal, because a founding lineage disperses over the same spatial scale on which its fitness changes. In contrast to two-deme models, where the total establishment rate peaks at intermediate migration, along a cline it increases monotonically with dispersal, as successful inversions originate over an increasingly broad region on the flanks of the cline. Because the local growth rate depends only on the resident single-locus clines and the dispersal rate, the potential for inversion establishment can be estimated from observable allele-frequency clines without knowledge of the underlying fitness landscape. Stochastic simulations confirm these predictions. Our results provide a quantitative link between clinal patterns of genetic variation and the evolution of chromosomal inversions.