Gojobori T, Ishii K, Nei M
J Mol Evol. 1982;18(6):414-23. doi: 10.1007/BF01840889.
A formal mathematical analysis of Kimura's (1981) six-parameter model of nucleotide substitution for the case of unequal substitution rates among different pairs of nucleotides is conducted, a new formulae for estimating the number of nucleotide substitutions and its standard error are obtained. By using computer simulation, the validities and utilities of Jukes and Cantor's (1969) one-parameter formula, Takahata and Kimura's (1981) four-parameter formula, and our six-parameter formula for estimating the number of nucleotide substitutions are examined under three different schemes of nucleotide substitution. It is shown that the one-parameter and four-parameter formulae often give underestimates when the number of nucleotide substitutions is large, whereas the six-parameter formula generally gives a good estimate for all the three substitution schemes examined. However, when the number of nucleotide substitutions is large, the six-parameter and four-parameter formulae are often inapplicable unless the number of nucleotides compared is extremely large. It is also shown that as long as the mean number of nucleotide substitutions is smaller than one per nucleotide site the three formulae give more or less the same estimate regardless of the substitution scheme used.
对木村（1981）提出的不同核苷酸对间替换率不等情况下的六参数核苷酸替换模型进行了形式化数学分析，得到了估计核苷酸替换数及其标准误差的新公式。通过计算机模拟，在三种不同的核苷酸替换方案下，检验了朱克斯和坎托（1969）的单参数公式、高桥和木村（1981）的四参数公式以及我们的六参数公式在估计核苷酸替换数方面的有效性和实用性。结果表明，当核苷酸替换数较大时，单参数和四参数公式常常给出低估结果，而六参数公式对于所检验的所有三种替换方案通常都能给出较好的估计。然而，当核苷酸替换数较大时，除非比较的核苷酸数极大，否则六参数和四参数公式常常不适用。还表明，只要每个核苷酸位点的平均核苷酸替换数小于1，无论使用何种替换方案，这三个公式给出的估计结果大致相同。
