In our article, Glémin and Ronfot 2012, the correct form for equation (2) is:
The factor “2” is lacking in the initial version. This is only a typo error and the different figures were drawn with the correct equation. Note that there is the same typo error in equation (16) in Glémin (2007).
However, Aneil Agrawal pointed us to a potential problem with this equation. Our aim was to use a not too arbitrary function for the α term, so we used a rather crude approximation. This equation can be obtained for instance from equation (8) in Hudson and Kaplan (1995) using H + F − HF instead of H and C(1 − F) instead of C for dominance and recombination coefficient, respectively. Equation (8) of Hudson and Kaplan (1995) then becomes:
where f0(F) is the reduction of effective population size caused by background selection. Equation (1) is simply obtained as the ratio: f0(F)/f0(0). This derivation implicitly relies on the assumption that the probability of effective recombination (i.e., sites are separated by an odd number of crossing-over) maps linearly with physical distances, which is not true for large genetic map, because of the saturation due to multiple crossing-over events (see for instance Haldane's mapping function). Using linear mapping function is accurate in outcrossers because the effect of background selection mainly comes from closely linked sites for which the linear mapping function is highly accurate. However, partial selfing shrinks the map so that physically more distant sites—for which the linear approximation is much less accurate—also affect background selection; although, for complete selfing (F = 1) the linear approximation should converge to the true expression. For the size of the map used in Glémin and Ronfort 2012, especially for C = 5, the effect of selfing on background selection is thus underestimated, mainly for high-selfing rates (0.8–0.99; A. Agrawal, pers. comm.). However, the qualitative patterns are very similar, and overall this reinforces the conclusion that selfing could lower adaptation because of reduced effective population size due to background selection.
Also note another typo error in equation (4). In the right-hand term, the z2 term should simply be z:
We are very grateful to Aneil Agrawal who carefully analyzed the problem and for sharing his results with us.