Appendix 2—figure 3. The long-term phenotypic contribution of minor alleles is amplified if they start above a critical initial frequency, which depends on their magnitude, and it is diminished if they start below this critical MAF.

In the linear approximation, we found that

$${\displaystyle \mathrm{\Delta }{z}_{\mathrm{\infty }}\left(a,x_0\right)\approx (1+A)\cdot \mathrm{\Delta }{z}_{\mathrm{\infty }}^L\left(a,x_0\right)=(1+A)\cdot \mathrm{\partial }\pi /\mathrm{\partial }x(a,x_0)\cdot \mathrm{\Delta }{z}_d^L\left(a,x_0\right)}$$

(Equation 24 in the main text). Further assuming that ${\displaystyle \mathrm{\Delta }{z}_{t_1}\left(a,x_0\right)\approx \mathrm{\Delta }{z}_d^L\left(a,x_0\right)}$ (which holds if the shift is not miniscule, that is, ${\displaystyle \mathrm{\Lambda }\gg \delta }$ ), we find that the (multiplicative) amplification/reduction of the phenotypic contribution is approximated by ${\displaystyle (1+A)\cdot \mathrm{\partial }\pi /\mathrm{\partial }x(a,x_0)}$ . In this approximation, the critical MAF for alleles with magnitude satisfies

$${\displaystyle (1+A)\cdot \mathrm{\partial }\pi /\mathrm{\partial }x(a,{\mathit{x}}_{\mathit{c}})=1.}$$

(A) The amplification/reduction for alleles with small (left), moderate (middle), and large (right) effects as a function of initial MAF, in the Lande (red) and non-Lande (blue) case. The curves and critical MAFs are calculated from the above equations. Given a factor ${\displaystyle A>0}$ , the contribution of alleles with sufficiently small effect sizes are amplified for any initial MAF ${\displaystyle x_0}$ (Figure 6), because ${\displaystyle \mathrm{\partial }\pi /\mathrm{\partial }x(a,x_0)\approx 1}$ and thus ${\displaystyle \mathrm{\Delta }{z}_{\mathrm{\infty }}^L\left(a,x_0\right)\approx \mathrm{\Delta }{z}_d^L\left(a,x_0\right)}$ and ${\displaystyle \mathrm{\Delta }{z}_{\mathrm{\infty }}^v\left(a,x_0\right)>\mathrm{\Delta }{z}_{t_1}\left(a,x_0\right)}$ . In turn, for sufficiently large effect sizes, the curves for ${\displaystyle \mathrm{\Delta }{z}_{\mathrm{\infty }}^v\left(a,x_0\right)}$ and ${\displaystyle \mathrm{\Delta }{z}_d^L\left(a,x_0\right)}$ ( ${\displaystyle \approx \mathrm{\Delta }{z}_{t_1}\left(a,x_0\right)}$ ) intersect (Figure 6B) and thus a critical MAF exists (i.e., ${\displaystyle 0<x_c<1/2}$ ). These considerations explain why, for sufficiently large effect sizes, the long-term contribution of alleles with low initial MAFs is diminished relative to their short-term contribution. (B) The critical MAF as a function of effect size in the Lande and non-Lande case (based on ${\displaystyle (1+A)\cdot \mathrm{\partial }\pi /\mathrm{\partial }x(a,{\mathit{x}}_{\mathit{c}})=1}$ ). The critical MAF is lower in the non-Lande case, because ${\displaystyle A>0}$ . This proportion declined with increasing allele magnitude both because initial MAFs at equilibrium decrease and because the critical MAF increases (panel B). It is greater in the non-Lande case because the critical MAF is lower (panel B). (C) The proportion of segregating sites at equilibrium with MAF exceeding the critical MAF.