A theoretical derivation of response to selection with and without controlled mating in honeybees

Abstract

Background: In recent years, the breeding of honeybees has gained significant scientific interest, and numerous theoretical and practical improvements have been made regarding the collection and processing of their performance data. It is now known that the selection of high-quality drone material is crucial for mid to long-term breeding success. However, there has been no conclusive mathematical theory to explain these findings.

Methods: We derived mathematical formulas to describe the response to selection of a breeding population and an unselected passive population of honeybees that benefits indirectly from genetic improvement in the breeding population via migration. This was done under the assumption of either controlled or uncontrolled mating of queens in the breeding population.

Results: Our model equations confirm what has been observed in simulation studies. In particular, we have proven that the breeding population and the passive population will show parallel genetic gain after some years and we were able to assess the responses to selection for different breeding strategies. Thus, we confirmed the crucial importance of controlled mating for successful honeybee breeding. When compared with data from simulation studies, the derived formulas showed high coefficients of determination [Formula: see text] in cases where many passive queens had dams from the breeding population. For self-sufficient passive populations, the coefficients of determination were lower ([Formula: see text]) if the breeding population was under controlled mating. This can be explained by the limited simulated time-frame and lower convergence rates.

Conclusion: The presented theoretical derivations allow extrapolation of honeybee-specific simulation results for breeding programs to a wide range of population parameters. Furthermore, they provide general insights into the genetic dynamics of interdependent populations, not only for honeybees but also in a broader context.

Fig. 1

Illustration of the genetic contributions of the breeding and passive colonies with uncontrolled mating. This figure motivates the recursion formulas Eq. 1 and 3. Worker groups in year t receive their breeding values in equal parts from their queens and the drones that are mating partners of the queens. Passive queens in year t receive their breeding values either from unselected breeding colonies of year $t-2$ (probability $q_t$) or from passive colonies of year $t-2$ (probability $1-q_t$). In the former case, the average inherited breeding value is $B_{t-2}$; in the latter case, it is $P_{t-2}$. Breeding queens inherit their breeding values from selected breeding colonies of year $t-2$, with average breeding values equal to $B_{t-2}+S_{1,t-2}$. Drones which mate with breeding or passive queens in year t may be offspring (and thus carry the breeding values) of either unselected breeding queens (probability $p_t$) or passive queens (probability $1-p_t$) of year $t-2$. For the breeding values of unselected breeding and passive queens of year $t-2$, the same considerations apply as for those of year t

Fig. 2

Illustration of the genetic contributions to the breeding colonies with controlled mating. This figure motivates the recursion formula Eq. 5. Breeding colonies in year t receive half of their genetic material from their queens which as in the case of uncontrolled mating have an average breeding value equal to $B_{t-2}+S_{1,t-2}$. The other half is inherited from the drones on a mating station whose common granddam on average has a breeding value equal to $B_{t-3}+S_{2,t-3}$

Fig. 3

Illustration of the generation intervals in the simulations with uncontrolled or controlled mating of queens

Fig. 4

Comparison of predicted (horizontal axis) and simulated values (vertical axis) for genetic progress. a: Annual genetic improvement in the breeding population, $\mathrm{\Delta }{B}_t$, in individual years 8 to 17. b: Average annual genetic improvement in the breeding population, $\mathrm{\Delta }B$, over the years 8 to 17. c: Average annual genetic improvement in the passive population, $\mathrm{\Delta }P$, over the years 8 to 17. d: Average genetic lag, D, over the years 8 to 17. Values are highlighted for maternal, direct, and total breeding values; diagonal equality lines are drawn for orientation

Fig. 5

Comparison of predicted (horizontal axis) and simulated values (vertical axis) for genetic progress. a: Annual genetic improvement in the breeding population, $\mathrm{\Delta }{B}_t$, in individual years 8 to 17. b: Average annual genetic improvement in the breeding population, $\mathrm{\Delta }B$, over the years 8 to 17. Values are highlighted for maternal, direct, and total breeding values; diagonal equality lines are drawn for orientation

Fig. 6

Comparison of predicted (horizontal axis) and simulated values (vertical axis) for genetic progress. a: Average annual genetic improvement in the passive population, $\mathrm{\Delta }P$, over the years 8 to 17. b: Average genetic lag, D, over the years 8 to 17. Values are highlighted for maternal, direct, and total breeding values; diagonal equality lines are drawn for orientation