Appendix 3—figure 9. A comparison of the linear and nonlinear approximations with simulation results.

The simulation results were generated using the single allele simulation, as described in Section 2.1; errors are not visible because they are smaller than the points. The analytic predictions were based on the linear (Equations A3-40 and A3-45) and nonlinear (Equations A3-40 and A3-48) approximations averaged over the initial MAF distribution (Equation A3-27) with the corresponding parameters. The nonlinear approximation is quite accurate throughout and substantially more accurate than the linear approximation for larger effect sizes and low polygenicity (see, e.g., A, B, and D). In the linear approximation, frequency changes with directional selection alone scale with $\mathrm{\Lambda }/V_A(0)\cdot (1-D_L(t_1)/\mathrm{\Lambda })$ (Equations A3-39, A3-40 and A3-45). We therefore normalized the frequency changes by this factor to make the results for the two shift sizes and three extents of polygenicity comparable.