Maintenance of genetic variability under the pressure of neutral and deleterious mutations in a finite population

In order to assess the effect of deleterious mutations on various measures of genic variation, approximate formulas have been developed for the frequency spectrum, the mean number of alleles in a sample, and the mean homozygosity; in some particular cases, exact formulas have been obtained. The assumptions made are that two classes of mutations exist, neutral and deleterious, and that selection is strong enough to keep deleterious alleles in low frequencies, the mode of selection being either genic or recessive. The main findings are: (1) If the expected value (q) of the sum of the frequencies of deleterious alleles is about 10% or less, then the presence of deleterious alleles causes only a minor reduction in the mean number of neutral alleles in a sample, as compared to the case of q = 0. Also, the low- and intermediate-frequency parts of the frequency spectrum of neutral alleles are little affected by the presence of deleterious alleles, though the high-frequency part may be changed drastically. (2) The contribution of deleterious mutations to the expected total number of alleles in a sample can be quite large even if q is only 1 or 2%. (3) The mean homozygosity is roughly equal to (1--2q)/(1 + theta 1), where theta 1 is twice the number of new neutral mutations occurring in each generation in the total population. Thus, deleterious mutations increase the mean heterozygosity by about 2q/(1 + theta 1). The present results have been applied to study the controversial problem of how deleterious mutations may affect the testing of the neutral mutation hypothesis.

Effect of changes in population size on the correlation between mutation rate and heterozygosity

The effect of changes in population size on the correlation between mutation rate and heterozygosity was studied by using two models: sudden change in population size and gradual change. It was shown that the results for the two models are close to each other, unless the rate of change for the latter is exceedingly slow. Thus, in many cases, the former model, which is much simpler than the latter, can be used to treat the present problem. Numerical computations showed that the correlation in a population that is expanding or has expanded in the recent past is stronger while the correlation in a population that is decreasing or has experienced a population reduction or bottleneck in the recent past is weaker than that for an equilibrium population with the same mean heterozygosity. However, regardless of whether the population is at equilibrium or not, the proportion of variation in heterozygosity that is attributable to variation in molecular weight over loci is rather small if the mean heterozygosity of the population is low, say of the order 0.05 or smaller.

Persistence of common alleles in two related populations or species

Mathematical studies are conducted on three problems that arise in molecular population genetics. (1) The time required for a particular allele to become extinct in a population under the effects of mutation, selection, and random genetic drift is studied. In the absence of selection, the mean extinction time of an allele with an initial frequency close to 1 is of the order of the reciprocal of the mutation rate when 4Nv less than 1, where N is the effective population size and v is the mutation rate per generation. Advantageous mutations reduce the extinction time considerably, whereas deleterious mutations increase it tremendously even if the effect on fitness is very slight. (2) Mathematical formulae are derived for the distribution and the moments of extinction time of a particular allele from one or both of two related populations or species under the assumption of no selection. When 4Nv less than 1, the mean extinction time is about half that for a single population, if the two populations are descended from a common original stock. (3) The expected number as well as the proportion of common neutral alleles shared by two related species at the tth generation after their separation are studied. It is shown that if 4Nv is small, the two species are expected to share a high proportion of common alleles even 4N generations after separation. In addition to the above mathematical studies, the implications of our results for the common alleles at protein loci in related Drosophila species and for the degeneration of unused characters in cave animals are discussed.

Maintenance of genetic variability under mutation and selection pressures in a finite population

Formulas are developed for the distribution of allele frequencies and the mean and variance of heterozygosity under mutation and selection pressures. In large populations, even slight selection drastically changes the shape of the distribution of allele frequencies and reduces heterozygosity. On the other hand, the number of rare alleles in a sample is much less affected by selection. Under genic selection, heterozygosity may decrease with increasing population size. As a test statistic, the variance of heterozygosity can be used to detect the presence of selection, though it is not efficient when selection is very slight.

Distribution of nucleotide differences between two randomly chosen cistrons in a finite population

Watterson's (1975) formula for the steady-state distribution of the number of nucleotide differences between two randomly chosen cistrons in a finite population has been extended to transient states. The rate for the mean of this distribution to approach its equilibrium value is 1/2N and independent of mutation rate, but that for the variance is dependent on mutation rate, where N denotes the effective population size. Numerical computations show that if the heterozygosity (i.e., the probability that two cistrons are different) is low, say of the order of 0.1 or less, the probability that two cistrons differ at two or more nucleotide sites is less than 10 percent of the heterozygosity, whereas this probability may be as high as 50 percent of the heterozygosity if the heterozygosity is 0.5. A simple estimate for the mean number (-d) of site differences between cistrons is d = h/(1 - h) where h is the heterozygosity. At equilibrium, the probability that two cistrons differ by more than one site is equal to h2, the square of heterozygosity.

The transient distribution of allele frequencies under mutation pressure

The transient distribution of allele frequencies in a finite population is derived under the assumption that there arekpossible 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 whenk= ∞. The rate of approach to the steady state distribution is generally very slow, the asymptotic rate being 2v+ 1/(2N), wherevandNare 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.

Electrophoretic identity of proteins in a finite population and genetic distance between taxa

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.

A note on the arrival probability, first arrival time and age of a mutant gene in a finite population

It is pointed out that the arrival or fixation probability of a mutant gene can be easily inferred analytically. The mean first arrival time for a single overdominant mutation to reach frequency y attains its maximum when x is close to but still slightly less than y/2, where x is the equilibrium frequency of the mutant gene in an infinitely large population. For an advantageous mutation, the mean first arrival time decreases with an increasing degree of dominance if selection is strong, but it first increases, after reaching a maximum, then decreases as the degree of dominance increases, if selection is weak. Contrary to our intuition, the mean age of an advantageous mutant gene increases with increasing degree of dominance, except when selection is very strong. A simple explanation is given in terms of the sojourn time at a particular gene frequency.

Growth of deleterious mutant genes in a large population

Formulae for the arrival probability and first arrival time for a single mutant gene to reach a certain number have been obtained by using a continuous branching process. If the mean of the progeny number of heterozygotes is less than one the arrival probability increases with increasing variance of the progeny distribution whereas if the mean is greater than one the contrary is true. Since most human populations are growing at fairly high rates, the result indicates that the probability for a single mutation to grow to a large number is quite high. Numerical computations show that the mean first arrival time decreases with increasing variance of the progeny distribution of the mutant carriers. The results have been applied to investigate the case of the acheiropodia gene in Brazil. The hypothesis that all acheiropodia genes in Brazil were derived from a single mutation seems to be tenable.

Genetics of acheiropodia (the handless and footless families of Brazil). VII. Population dynamics

Since carriers of the acheiropodia gene cannot be distinguished from noncarriers, parents and normal sibs of affected individuals have been used to estimate the fitness of heterozygotes. No significant difference in biologic fitness (viability and fertility) between normal sibs and the general population could be detected. A comparison between acheiropods and their normal sibs showed the following: (1) a nonsignificant difference in stillbirth rate; (2) a higher mortality rate of acheiropods in the first 5 years of life; (3) a relative viability not larger than .7; (4) a relative fertility no greater than .14, due to "cosmetic effects"; and (5) a fitness of .10 or lower. The total number of acheiropodia genes in Brazil has been calculated as 25,000 in the 1970s. The data are compatible with an extremely low mutation rate and a very stable locus. It is suggested that all Brazilian acheiropods can be traced to a single mutation. A conservative estimate of the number of acheiropods to appear in the future in Brazil is 14,000 with an extinction time of no less than 2,300 generations or almost 70,000 years. A variety of other parameters have been calculated.

Probability of identical monomorphism in related species

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 toDrosophiladata 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.

Drift variances of heterozygosity and genetic distance in transient states

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.

The first arrival time and mean age of a deleterious mutant gene in a finite population

The mean and standard deviation of the first arrival time for a single mutant to reach a certain frequency and the mean age of a mutant persisting in a population have been studied using diffusion methods. These quantities are shown to be highly dependent on both the heterozygous effect and the population size. For partially recessive deleterious mutations, both the mean first arrival time and the mean age decrease with increasing selection coefficient against heterozygotes. For overdominant mutations, the mean age always increases very rapidly with increasing heterozygous advantage, while the mean first arrival time first increases rapidly with increasing heterozygous advantage to a maximum and then decreases rapidly with increasing heterozygous advantage. The standard deviation of the first arrival time is small while that of the age is large. The results of this study have been applied to study the case of the sickle cell anemia mutant in Africa. It is argued that the present prevalence may be explained without the necessity of quite so great a heterozygous advantage as .25 or higher as proposed by some workers. A reasonable range for the heterozygous advantage seems to be from .05 to .18.

Linkage disequilibrium in subdivided populations

The linkage disequilibrium in a subdivided populaton is shown to be equal to the sum of the average linkage disequilibrium for all subpopulations and the covariance between gene frequencies of the loci concerned. Thus, in a subdivided population the linkage disequilibrium may not be 0 even if the linkage disequilibrium in each subpopulation is 0. If a population is divided into two subpopulations between which migration occurs, the asymptotic rate of approach to linkage equilibrium is equal to either r or 2(m(1) + m(2)) - (m(1) + m(2))(2), whichever is smaller, where r is the recombination value and m(1) and m(2) are the proportions of immigrants in subpopulations 1 and 2, respectively. Thus, if migration rate is high compared with recombination value, the change of linkage disequilibrium in subdivided populations is similar to that of a single random mating population. On the other hand, if migration rate is low, the approach to lnkage equilibrium may be retarded in subdivided populations. If isolated populations begin to exchange genes by migration, linkage disequilibrium may increase temporarily even for neutral loci. If overdominant selection operates and the equilibrium gene frequencies are different in the two subpopulations, a permanent linkage disequilibrium may be produced without epistasis in each subpopulation.