Continuous approximations for optimizing allele trajectories.

The incorporation of genetic information such as quantitative trait loci (QTL) data into breeding schemes has become feasible as DNA technologies have advanced. Such strategies allow the frequency of desirable QTL to be controlled over a predefined time frame, allowing the allele trajectory for QTL to be manipulated. A continuous approximation to changes in allele frequency was developed to approximate the selection procedure as a continuous rather than a discrete process, and analytical solutions were obtained, which shed light on how allele trajectories behave under different objective functions. Three different objectives were considered: (1) minimizing the total selection intensity, (2) minimizing the sum of squared selection intensities and (3) equalizing the selection intensity applied over time. Simulations and genetic algorithms were performed to test the accuracy and robustness of the continuous approximation. Theory shows firstly that the total selection intensity required for moving an allele from a starting frequency to another frequency point can be predicted independent of its trajectory, and secondly that objectives (2) and (3) are equivalent as the number of selection opportunities (T) becomes large. The prediction of total selection intensity provides a good fit for these two objectives, with the accuracy of prediction improving as T increases. However, for (1) the continuous approximation does not fit due to the existence of a discontinuous solution in which the continuous approximation is applied before the frequency of the selected allele reaches 0.5 followed by rapid fixation.