Mathematical Model for the Calculation of Full and Half Sib Covariance in an Artificial Autotetraploid Population Including Aneuploids

For the estimation of genetic variance of an artificial autotetraploid population, a mathematical model of full and half sib covariances between sibs with various chromosome numbers, which were derived from euploid or aneuploid parents, was devised for a case where the inbreeding coefficient of the parents was F＝0. The coefficients defined in Kempthorne's model were separated into two parts: (i) A, D, T and Q, and (ii) φ and ψ. The former four parameters were defined as probabilities of factor combinations, which could be compared between various sibs, for additive, digenic, trigenic, and quadrigenic effects, and were mutually independent. The latter two parameters, which were the numbers of the identical allele and the identical allele pair combinations that two sibs inherited from a parent, were defined as linear functions of the probabilities that two sibs inherited allele or allele pair from a parent, respectively. These probabilities depend on chromosome behavior during meiosis and the chromosome number of the gametes. For the estimation, it was assumed that quadrivalent chromosomes were distributed by 2-2 and 1-3 with probabilities κ and λ (κ＋λ＝ 1), respectively. The distribution of trisomic and pentasomic chromosomes to the poles was assumed to be 1-2 and 2-3. Then, the probabilities were estimated for the simple case where all male and female gametes could equally fertilize irrespective of their chromosome number, provided that tetrasomic chromosomes completely formed a quadrivalent chromosome. 　The constitution of variance components were different according to the sib combinations and family. Therefore, for the calculation of the covariance of a family, the covariances between various sibs were averaged by the combination frequency in a family, and for the calculation of the covariance of population, the family's covariances were averaged by the family's frequency in the population.