The frequency distribution of lethal chromosomes in finite populations.

The statistical techniques for studying the population dynamics of recessive lethal genes were developed mainly by Wright,'-' who studied the distribution of gene frequencies in populations. In reality, however, the frequencies of individual lethal genes cannot be observed but the frequency of lethal-bearing chromosomes is determined. In large populations the random fluctuation of the frequency of lethal chromosomes is negligibly small, but it becomes appreciably large in intermediate or small populations. Thus, it is desirable to find the frequency distribution of lethal chromosomes rather than lethal genes. To obtain a general distribution function of the chromosome frequencies is not easy, but the author obtained several formulas for some important special cases. As we shall see, these formulas give some useful relations among mutation rate, population size, allelic rate, etc., which may be used for the study of genetic structure of populations or the heterozygous effects of lethal genes. In recent years much attention has been focused on the latter problem in view of the fact that the genetic load due to these genes is profoundly affected by a slight positive or negative effect in the heterozygotes.45 Partially Recessive Lethals.-Consider a randomly mating population of size N, in which a lethal gene a and its allelic normal gene A are segregating. Let the frequency of a be q and the fitnesses of the three possible genotypes AA, Aa, and aa be 1, 1 h, and 1 s, respectively, where s is 1 for a completely lethal gene and 0.5 < s < 1 for a semilethal gene. The mean fitness, 1pu, then becomes 1 2hq (s 2h)q2. We assume that the gene A mutates to a at the rate of u per generation, the reverse mutation being negligible. Inserting these quantities into Wright's' general formula, we find the frequency distribution of lethal gene in an equilibrium population as follows: 4(q) -e4NhQ2N( ..2h)Q'q4Nul (1 q) -1. (1)