Gene Frequency Clines in the Presence of Selection Opposed by Gene Flow

Suppose we have a cline in gene frequency which results from spatially varying selection forces (which tend to establish the cline) opposed by gene flow (which tends to blur the cline). The characteristic width, W, of this gene frequency cline can be assessed qualitatively in terms of the parameters characterizing the selection and gene flow processes. We define Δ as the transition distance in the selection gradient, s as the maximum selection intensity (see fig. 1), l as the root-mean-square gene flow distance (see eq. [3]), and l_c = l/ $$\sqrt s$$ (see eq. [5]): then for Δ < l_c we have W ∼ l_c, and consequently W > Δ (as discussed fully by Slatkin 1973); conversely for Δ > l_c we have W ∼ (l_c² Δ)^1/3, and consequently l_c < W < Δ. The former circumstance may be thought of as relatively strong gene flow and a relatively blurred cline (W > Δ); the latter circumstance as relatively weak gene flow and a relatively sharp cline (W < Δ). The above qualitative results pertain to all situations where the fitness of the heterozygotes, Aa, is intermediate between (or, in the event of a dominant allele A or a, coincident with one of) the homozygotes AA and aa. In particular, the detailed results of table 1 apply if the heterozygotes are of precisely intermediate fitness and table 2 applies if there is dominance, so that the heterozygote fitness coincides with one of the homozygotes. More generally, if the heterozygotes are at a strong disadvantage, measured by the selection parameter h of equation (15), the gene frequency cline is steepened and has typical width W ∼ l/ $$\sqrt{-h}$$ . Conversely, strong heterozygous advantage leads to a very smooth cline and eventually to essentially equal frequencies of A and a at all points. The presence of asymmetry in the gene flow, as may easily arise from a prevailing wind or other environmental effect, can substantially modify the slope and position of a gene frequency cline. This can happen even when the characteristic scale length of the asymmetry (see eq. [22]) is relatively small. These results unify a large number of numerical simulations of such situations, some of which have been done for a stepped environmental gradient (effectively Δ < l_c) and others for a continuous selection gradient (effectively Δ > l_c). Some studies on real clines produced by the tension between selection and gene flow contain enough data to permit rough comparison with theory. The agreement between nature and qualitative theory is surprisingly good in several cases (see table 3). Field data of this sort are rare, as quantitative information about gene flow and about selective forces is difficult to get. The main aim of this paper is to present an orderly framework within which to assemble such data.