ē¬ÄRæÕŖwö_ŖwĢö Acta Medica Okayama 0474-0254ü@ 97 1 2008 Mathematical Model for the Calculation of Full and Half Sib Covariance in an Artificial Autotetraploid Population Including Aneuploids 17 24 EN Tetsuo Morisawa Kenji Kato 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 Füü0. The coefficients defined in Kempthorne's model were separated into two parts: (i) A, D, T and Q, and (ii) āė and āš. 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 ā╚ and ā╔ (ā╚ü{ā╔üü 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. ü@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. No potential conflict of interest relevant to this article was reported.
ē¬ÄRæÕŖwö_ŖwĢö Acta Medica Okayama 0474-0254ü@ 97 1 2008 Calculation of Full and Half Sib Covariances in an Artificial Autotetraploid Population Including Aneuploids, in Astragalus Sinica L. 25 31 EN Tetsuo Morisawa Kenji Kato Full and half sib covariances were investigated in an artificial autotetraploid population with random mating in Astragalus sinicus L.. Since a set of homologous chromosomes is not necessarily involved in aneuploidy, the covariances must be averaged for two cases, that is, with and without involvement. To average the covariances, the probability that a set of homologous chromosomes was involved in aneuploidy was assumed as 3/8, where üg8üh and üg3üh represent the chromosome number of a genome and the mean number of quadrivalent chromosomes formed in a euploid, respectively. The covariances were calculated under the assumption that quadrivalent chromosomes were distributed to the poles by 2-2 and 1-3 with probabilities ā╚üü 0.8 and ā╔ üü0.2 (ā╚ü{ā╔üü1) respectively, and that trisomic and pentasomic chromosomes were distributed by 1-2 and 2-3 both with a probability of 1. It was also assumed that the inbreeding coefficient of the parents was Füü 0, and that 2x and 2xü{ 1 pollens and all female gametes could fertilize equally. The covariance of a family was taken as an average of the covariance of each sib combination in a family. As a result, the covariance of a population could be obtained as an average of the covariance of each family in a population. The coefficients of variance components calculated under these assumptions were different from those calculated under the same condition except that 2xü{ 1 pollen could not fertilize. Differences in the coefficient of additive genetic variance components were about 3.3% and 7.2% for full and half sib covariances, respectively. Coefficients of the other variance components were also different between the two cases. However, 2xü{1 pollen could rarely fertilize, since their ability to fertilize in a practical population were lower than 2x pollen. Therefore, it would be valid to calculate full and half sib covariances in an artificial autotetraploid population of Astragalus sinicus L. under the condition thatonly 2x pollen could fertilize. No potential conflict of interest relevant to this article was reported.