Walsh J B
Department of Ecology and Evolutionary Biology, University of Arizona, Tucson 85721.
J Math Biol. 1990;28(1):21-31. doi: 10.1007/BF00171516.
Inconsistencies exist in the standard expansions used to approximate selection coefficients for alleles at a locus underlying a quantitative character. Allelic (marginal) fitnesses obtained from expansions based on average excesses differ from allelic fitnesses obtained from expansions based on genotypic values. Similarly, W, the mean population fitness based on summing over either allelic or genotypic fitnesses usually differs mean population fitness obtained by averaging over the unrestricted phenotypic distribution. A consistent value of W requires no variation in genotypic values. If, as suggested by Nagylaki (1984), expansions are corrected for the decrease in phenotypic variance resulting from conditioning on the presence of a particular allele or genotype, inconsistencies still exist. Unless integral of W(z)[Vzp"(z) + zp'(z) + p(z)] dz = 0, where p(z) is the phenotypic probability density function, Vz the phenotypic variance, W(z) the fitness of phenotypic value z, the primes denote differentiation with respect to z, allelic fitnesses based on average effects differ from allelic fitnesses based on genotypic values. This condition must also be satisfied in order for either expansion to give a consistent W, as first shown by Nagylaki. For arbitrary W(z), this is satisfied if and only if phenotypes are normally distributed.
在用于近似数量性状位点上等位基因选择系数的标准展开式中存在不一致性。从基于平均超额的展开式中获得的等位基因（边际）适合度与从基于基因型值的展开式中获得的等位基因适合度不同。同样，基于等位基因适合度或基因型适合度求和得到的总体平均适合度W，通常与通过对无限制表型分布求平均得到的总体平均适合度不同。W的一致值要求基因型值无变化。正如Nagylaki（1984年）所指出的，如果对因特定等位基因或基因型的存在而导致的表型方差减小进行展开式校正，不一致性仍然存在。除非∫W(z)[Vzp"(z) + zp'(z) + p(z)]dz = 0，其中p(z)是表型概率密度函数，Vz是表型方差，W(z)是表型值z的适合度，撇号表示对z的求导，否则基于平均效应的等位基因适合度与基于基因型值的等位基因适合度不同。正如Nagylaki首次表明的那样，为了使任何一个展开式给出一致的W，这个条件也必须得到满足。对于任意的W(z)，当且仅当表型呈正态分布时，这个条件才会得到满足。
