Maruyama T
Genetics. 1974 Feb;76(2):367-77. doi: 10.1093/genetics/76.2.367.
A Markov process (chain) of gene frequency change is derived for a geographically-structured model of a population. The population consists of colonies which are connected by migration. Selection operates in each colony independently. It is shown that there exists a stochastic clock that transforms the originally complicated process of gene frequency change to a random walk which is independent of the geographical structure of the population. The time parameter is a local random time that is dependent on the sample path. In fact, if the alleles are selectively neutral, the time parameter is exactly equal to the sum of the average local genetic variation appearing in the population, and otherwise they are approximately equal. The Kolmogorov forward and backward equations of the process are obtained. As a limit of large population size, a diffusion process is derived. The transition probabilities of the Markov chain and of the diffusion process are obtained explicitly. Certain quantities of biological interest are shown to be independent of the population structure. The quantities are the fixation probability of a mutant, the sum of the average local genetic variation and the variation summed over the generations in which the gene frequency in the whole population assumes a specified value.
针对一个种群的地理结构模型，推导了基因频率变化的马尔可夫过程（链）。该种群由通过迁移相连的群落组成。选择在每个群落中独立进行。结果表明，存在一个随机时钟，它将原本复杂的基因频率变化过程转化为一个与种群地理结构无关的随机游走。时间参数是一个依赖于样本路径的局部随机时间。事实上，如果等位基因是选择中性的，时间参数恰好等于种群中出现的平均局部遗传变异之和，否则它们近似相等。得到了该过程的柯尔莫哥洛夫向前和向后方程。作为大种群规模的极限，推导了一个扩散过程。明确得到了马尔可夫链和扩散过程的转移概率。一些具有生物学意义的量被证明与种群结构无关。这些量包括突变体的固定概率、平均局部遗传变异之和以及在整个种群基因频率取指定值的那些世代中的变异之和。
