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The transient distribution of allele frequencies in a finite population is derived under the assumption that there are k possible alleic states at a locus and mutation occurs in all directions. At steady state this distribution becomes identical with the distribution obtained by Wright, Kimura and Crow when k = ∞. The rate of approach to the steady state distribution is generally very slow, the asymptotic rate being 2v + 1/(2N), where v and N are the mutation rate and effective population size, respectively. Using this distribution it is shown that when population size is suddenly increased, the expected number of alleles increases more rapidly than the expected heterozygosity. Implications of the present study on testing hypotheses for the maintenance of genetic variability in populations are discussed.
The accumulation of beneficial and harmful mutations in a genome is studied by using analytical methods as well as computer simulation for different modes of reproduction. The modes of reproduction examined are biparental (bisexual, hermaphroditic), uniparental (selfing, automictic, asexual) and mixed (partial selfing, mixture of hermaphroditism and parthenogenesis). It is shown that the rates of accumulation of both beneficial and harmful mutations with weak selection depend on the within-population variance of the number of mutant genes per genome. Analytical formulae for this variance are derived for neutral mutant genes for hermaphroditic, selfing and asexual populations; the neutral variance is largest in a selfing population and smallest in an asexual population. Directional selection reduces the population variance in most cases, whereas recombination partially restores the reduced variance. Therefore, biparental organisms accumulate beneficial mutations at the highest rate and harmful mutations at the lowest rate. Selfing organisms are intermediate between biparental and asexual organisms. Even a limited amount of outcrossing in largely selfing and parthenogenetic organisms markedly affects the accumulation rates. The accumulation of mutations is likely to affect the mean population fitness only in long-term evolution.
A mathematical method for evaluating the probability that a locus is monomorphic for the same allele in related species is developed under the neutral mutation hypothesis. A formula for the proportion of identically monomorphic loci in related species is also worked out. The results of the application of this method to Drosophila data do not support Prakash & Lewontin's (1968) contention that the strong association between gene arrangements (inversion chromosomes) and alleles at protein loci is evidence of coadaptation of genes in the inverted segment of chromosomes. Similarly, unlike Haigh & Maynard Smith's (1972) contention, the monomorphism of the haemoglobin α chain locus in man can be accommodated with the neutral mutation hypothesis without invoking the bottleneck effect.
Wehrhahn (1975) introduced the method of probability generating function to study the distribution of charge differences between homologous proteins in a population but considered only the special case where the population starts with a single allele. Some of his results, however, contained errors. In this paper, all the formulae are presented in general, correct yet much simpler forms. It is also shown that the method of diffusion equations (Ohta & Kimura, 1973) can produce the same results. Numerical computations show that the difference between the one-step and two-step models of charge changes is practically negligible. The results obtained have also been applied to study Nei's genetic distance. Numerical computations indicate that the genetic distance computed from electrophoretic data is about 10% smaller than the expected number of amino acid substitutions involving charge changes in the early stage of divergence of populations and may give a serious underestimate in comparisons between species.
With the aim of knowing the probable magnitude of non-random association between inversion chromosomes and electromorphs, both deterministic and stochastic studies are conducted on the evolutionary change of non-random association, which is defined as the difference in the frequency of a given allele between inversion and non-inversion chromosomes. In these studies inversion chromosomes are assumed to be subject to selection but electromorphs are selectively neutral, and recombination is allowed to occur between inversion and non-inversion chromosomes with a low frequency. The deterministic study has shown that in a variety of selective schemes for inversion chromosomes the non-random association decays at a rate equal to the recombination value in every generation. Thus, if the recombination value is of the order of 10−5 ˜ 10−4, it would take a long time for the non-random association to disappear. Furthermore, the stochastic study has indicated that random genetic drift generates non-random association of inversions and electromorphs in finite populations and the standard error of non-random association often becomes larger than the mean. In addition to these problems the time required for the electromorph frequencies in the inversion and noninversion chromosomes to become equal in a finite population and the probability that the population of inversion chromosomes remains monomorphic for the allele which existed in the initial inversion introduced are studied. Considering all these quantities, it is concluded that data on the non-random association between electromorphs and inversions are not very informative for the study of the maintenance of protein polymorphism. It is also indicated that in the study of association between electromorphs and inversion chromosomes non-random association or Yule's coefficient of association has a better property than the usual linkage disequilibrium measure or correlation coefficient. Implications of this study on some experimental observations are discussed.
Using the moments of gene frequencies, the drift variances of heterozygosity and genetic distance in transient states have been studied under the assumption that all mutations are selectively neutral. Interestingly, this approach provides a simple derivation of Stewart's formula for the variance of heterozygosity at steady state. The results obtained indicate that if all alleles in the initial population are equally frequent, the standard derivation of heterozygosity is very small and increases linearly with time in the early generations. On the other hand, if the initial allele frequencies deviate appreciably from equality, then the standard deviation in the early generations is much larger but increases linearly with the square root of time. Under certain conditions, the standard deviation of genetic distance also increases linearly with time. Numerical computations have shown that the standard deviations of heterozygosity and genetic distance relative to their means are so large that a large number of loci must be used in estimating the average heterozygosity and genetic distance per locus.
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