Directional selection in a population yields reduced genetic variance due to the Bulmer effect. While this effect has been thoroughly investigated in mammals, it is poorly studied in social insects with biological peculiarities such as haplo-diploidy or the collective expression of traits. In addition to natural adaptation to climate change, parasites, and pesticides, honeybees increasingly experience artificial selection pressure through modern breeding programs. Besides selection, many honeybee breeding schemes introduce controlled mating. We investigated which individual effects selection and controlled mating have on genetic variance. We derived formulas to describe short-term changes of genetic variance in honeybee populations and conducted computer simulations to confirm them. Thereby, we found that the changes in genetic variance depend on whether variance is measured between queens (inheritance criterion), worker groups (selection criterion) or both (performance criterion). All three criteria showed reduced genetic variance under selection. In the selection and performance criteria, our formulas and simulations showed an increased genetic variance through controlled mating. This newly described effect counterbalanced and occasionally outweighed the Bulmer effect. It could not be observed in the inheritance criterion. A good understanding of the different notions of genetic variance in honeybees therefore appears crucial to interpret population parameters correctly.

Additional file 6. Genetic gain, $${R}{GS}$$ R GS , in the initial selection cycle of different breeding schemes applying CBS and GPS. The table presents the configuration of the breeding schemes shown in Fig. 4, as well as 9 reruns of the optimal breeding scheme with ssGBLUPBQ, and 1765 other runs chosen at even spacing to represent the remaining schemes. Equations (18), (19), and (20) were used to calculate $${R}{GS}$$ R GS in the scenarios described in Table 1 for different ratios of preselected BQ and DPQ, with the prediction accuracy adjusted to the proportion on preselected BQ in all scenarios. Genetic gain is given in the units of the selection criterion. The parameter setting MOD was used. The standard deviations of the true breeding values $${\sigma }{pW}$$ σ pW (worker groups from year 8) and $${\sigma }{uQ}$$ σ uQ (queens from year 9) are shown in Additional file 5. The 1765 remaining schemes were picked by the numbers of dams of DPQ chosen for GPS, ( $${N}{DPQ}^{GPS}$$ N DPQ GPS ), dams of BQ chosen for GPS ( $${N}{BQ}^{GPS}$$ N BQ GPS ), candidate DPQ per dam for GPS ( $${n}{DPQ}^{GPS}$$ n DPQ GPS ), and candidate BQ per dam for GPS ( $${n}{BQ}^{GPS}$$ n BQ GPS ) with the step widths 10, 20, 8, and 8, respectively.

Additional file 6. Genetic gain, $${R}{GS}$$ R GS , in the initial selection cycle of different breeding schemes applying CBS and GPS. The table presents the configuration of the breeding schemes shown in Fig. 4, as well as 9 reruns of the optimal breeding scheme with ssGBLUPBQ, and 1765 other runs chosen at even spacing to represent the remaining schemes. Equations (18), (19), and (20) were used to calculate $${R}{GS}$$ R GS in the scenarios described in Table 1 for different ratios of preselected BQ and DPQ, with the prediction accuracy adjusted to the proportion on preselected BQ in all scenarios. Genetic gain is given in the units of the selection criterion. The parameter setting MOD was used. The standard deviations of the true breeding values $${\sigma }{pW}$$ σ pW (worker groups from year 8) and $${\sigma }{uQ}$$ σ uQ (queens from year 9) are shown in Additional file 5. The 1765 remaining schemes were picked by the numbers of dams of DPQ chosen for GPS, ( $${N}{DPQ}^{GPS}$$ N DPQ GPS ), dams of BQ chosen for GPS ( $${N}{BQ}^{GPS}$$ N BQ GPS ), candidate DPQ per dam for GPS ( $${n}{DPQ}^{GPS}$$ n DPQ GPS ), and candidate BQ per dam for GPS ( $${n}{BQ}^{GPS}$$ n BQ GPS ) with the step widths 10, 20, 8, and 8, respectively.

Results of the individual parameter informations. All necessary info can be found in the README file.

Results of the individual parameter informations. All necessary info can be found in the README file.