National Institute of Genetics, Mishima, 411 Japan.
Proc Natl Acad Sci U S A. 1983 Oct;80(20):6317-21. doi: 10.1073/pnas.80.20.6317.
Assume a diploid species consisting of an infinite number of competing demes, each having N(e) reproducing members and in which mating is at random. Then consider a locus at which a pair of alleles A and A' are segregating, where A' is the "altruistic allele," which has selective disadvantage s' relative to A with respect to individual selection, but which is beneficial for a deme in competition with other demes; namely, a deme having A' with frequency x has the advantage c(x - x) relative to the average deme, where c is a positive constant and x is the average of x over the species. Let varphi = varphi(x;t) be the distribution function of x among demes in the species at time t. Then, we have partial differentialvarphi/ partial differentialt = L(varphi) + c(x - x)varphi, where L is the Kolmogorov forward differential operator commonly used in population genetics [i.e., L = (1/2) ( partial differential(2)/ partial differentialx(2))V(deltax) - ( partial differential/ partial differentialx)M(deltax)], and M(deltax) and V(deltax) stand for the mean and variance of the change in x per generation within demes. As to migration, assume Wright's island model and denote by m the migration rate per deme per generation. By investigating the steady state, in which mutation, migration, random drift, and intra- and interdeme selection balance each other, it is shown that the index D = c/m - 4N(e)s' serves as a good indicator for predicting which of the two forces (i.e., group selection or individual selection) prevails; if D > 0, the altruistic allele predominates, but if D < 0, it becomes rare and cannot be established in the species.
假设一个由无数相互竞争的群体组成的二倍体物种，每个群体有 N(e)个繁殖成员，并且交配是随机的。然后考虑一个位于其中一对等位基因 A 和 A'分离的基因座，其中 A'是“利他等位基因”，与个体选择相比，A'相对于 A 具有选择性劣势 s'，但对于与其他群体竞争的群体是有益的；即，一个具有频率 x 的 A'的群体相对于平均群体具有优势 c(x - x)，其中 c 是一个正常数，x 是物种中 x 的平均值。令 varphi = varphi(x;t)为物种中群体在时间 t 时的 x 分布函数。那么，我们有偏微分varphi/偏微分t = L(varphi) + c(x - x)varphi，其中 L 是群体遗传学中常用的 Kolmogorov 前向微分算子[即，L = (1/2) ( partial differential(2)/ partial differentialx(2))V(deltax) - ( partial differential/ partial differentialx)M(deltax)]，其中 M(deltax)和 V(deltax)分别表示群体内 x 的变化的均值和方差。至于迁移，假设 Wright 的岛屿模型，并记 m 为每个群体每代的迁移率。通过研究稳态，其中突变、迁移、随机漂移以及群体内和群体间选择相互平衡，可以表明指标 D = c/m - 4N(e)s'是预测两种力量（即群体选择或个体选择）哪个占主导地位的一个很好的指标；如果 D > 0，则利他等位基因占主导地位，但如果 D < 0，则它变得稀有，无法在物种中建立。
