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.

人為同質４倍体集団の遺伝分散を求めるため，両親が近交系数Ｆ＝０の同質４倍体家族の全兄弟と半兄弟の共分散を検討した．Kempthorne のモデルにおいて定義された分散の係数を①Ａ，Ｄ，Ｔ，Ｑと②φ，ψの２つに分割し た．①は互いに独立な相加，２遺伝子，３遺伝子，４遺伝子効果の組み合わせの確率である．②は兄弟が片親から受け取る同一対立遺伝子の数と対立遺伝子ペアの数である．これは兄弟が片親から対立遺伝子と対立遺伝子の組を受け取る確率の関数であり，この確率は減数分裂での染色体行動と配偶子の染色体数によって決まる.この確率を推定するため，Ⅳ価染色体は確率κ，λ（κ＋λ＝１）で２-２と１-３で分配され，Ⅲ価染色体とⅤ価染色体は１-２と２-３に分配されると仮定した．本報告では，四染色体が完全にⅣ価染色体を形成するとして，全ての雌雄の配偶子がその染色体数に関係なく受精できる単純な場合について検討した．共分散の分散成分の構造は兄弟の組み合わせと家族によって異なる．したがって，家族の共分散は各兄弟の共分散 とその組み合わせ頻度を用い平均すれば求めることができ，集団の平均の共分散は家族の共分散と集団での家族の頻度を用い平均すれば求めることができる．