Inbreeding and Genetic Erosion from a Finite Model of a Synthetic Formed with Single Crosses

When a seed produced by a single-cross (SC) maize hybrid is sown, the resulting grain yield is usually lower than that of the hybrid due to the inbreeding generated. However, if a seed from a mixture of s hybrids were sown instead, the synthetic variety thus formed (Syn(SC)) would have a lower inbreeding coefficient (FSyn(SC)) and a higher grain yield. The grain yield s, the finite number of representatives of each parent SC (m) and the inbreeding coefficient of the parent lines of the SCs (F) are related to the FSyn(SC). In addition, randomness and the finite size of m can cause the loss of genes and genotypes and increase the FSyn(SC). The objectives of this study were to derive formulas for (1) expressing FSyn(SC) in terms of m, F, and s, and (2) calculating the probability of the occurrence of gene and genotype loss. It was found that for the probability of no genotype being missing from the progeny representing a parent to be at least 0.95, it is necessary that m ≥ 15. It was also found that a sample size of 7 is sufficient for FSyn(SC) to stabilize, more visibly as F is larger, and for the probability of the occurrence of erosion to be practically zero.