Hardy-Weinberg equilibria in random mating populations

The structure of multiloci random mating populations is examined. Sufficient conditions for the existence of stable local Hardy-Weinberg equilibria for n loci and an arbitrary number of alleles per locus, are then derived for specified situations under the assumption of multiplicative gene action between loci. It is shown that a stable Hardy-Weinberg equilibrium can not be a local maximum of the mean fitness function with multiplicative gene action between loci. The stability of Hardy-Weinberg type border points and the condition for the increase of newly introduced genes are topics on which some n-loci results are also obtained for an arbitrary number of alleles per locus in systems that allow Hardy-Weinberg equilibria.     

A competitive selection model

A multi-locus model for complete positive assortative mating is discussed. For a two-locus model, if the gene frequencies for the two loci are different, as they are likely to be, it is shown that in equilibrium the population is not composed of only two homozygous types, as is usually thought. The limiting distribution will have three homozygous genotypes depending upon the initial gene frequencies. If there are m-loci such that gene frequencies at all loci are different, there will be (m+1) such homozygous genotypes present in the equilibrium population, one in each phenotypic group.     

The effect of epistasis on the excess of the additive and nonadditive variances after population bottlenecks

The effect of population bottlenecks on the components of the genetic variance generated by two neutral independent epistatic loci has been studied theoretically (VA, additive; VD, dominant; VAA, additive x additive; VAD, additive x dominant; VDD; dominant x dominant components of variance). Nonoverdominance and overdominance models were considered, covering all possible types of marginal gene action at the single locus level. The variance components in an infinitely large panmictic population (ancestral components) were compared with their expected values at equilibrium, after t consecutive bottlenecks of equal size N (derived components). Formulae were obtained in terms of allele frequencies and effects at each locus and the corresponding epistatic value. An excess of VA after bottlenecks can be assigned to two sources: (1) the spatiotemporal changes in the marginal average effects of gene substitution alpha(i), which are equal to zero only for additive gene action within and between loci; and (2) the covariance between alpha2(i) and the heterozygosity at the loci involved, which is generated by dominance, with or without epistasis. Numerical examples were analyzed, indicating that an increase in VA after bottlenecks will only occur if its ancestral value is minimal or very small. For the nonoverdominance model with weak reinforcing epistasis, that increase has been detected only for extreme frequencies of the negative allele at one or both loci. With strong epistasis, however, this result can be extended to a broad range of intermediate frequencies. With no epistasis, the same qualitative results were found, indicating that dominance can be considered as the primary cause of an increase in VA following bottlenecks. In parallel, the derived total nonadditive variance exceeded its ancestral value (V(NA) = V(D) + V(AA) + V(AD) + V(DD)) for a range of combinations of allele frequencies covering those for an excess of VA and for very large frequencies of the negative allele at both loci. For the overdominance model, an increase in V(A) and V(NA) was respectively observed for equilibrium (intermediate) frequencies at one or both loci or for extreme frequencies at both loci. For all models, the magnitude of the change of V(A) and V(NA) was inversely related to N and t. At low levels of inbreeding, the between-line variance was not affected by the type of gene action. For the models considered, the results indicate that it is unlikely that the rate of evolution may be accelerated after population bottlenecks, in spite of occasional increments of the derived V(A) over its ancestral value.     

GENETIC DIVERSITY AND POPULATION STRUCTURE IN CAMELLIA JAPONICA L. (THEACEAE)

Camellia japonica is a widespread and morphologically diverse tree native to parts of Japan and adjacent islands. Starch gel electrophoresis was used to score allelic variation at 20 loci in seeds collected from 60 populations distributed throughout the species range. In comparison with other plant species, the level of genetic diversity within C. japonica populations is very high: 66.2% of loci were polymorphic on average per population, with a mean number of 2.16 alleles per locus; the mean observed and panmictic heterozygosities were 0.230 and 0.265, respectively. Genotypic proportions at most loci in most populations fit Hardy-Weinberg expectations. However, small heterozygote deficiencies were commonly observed (mean population fixation index = 0.129). It is suggested that the most likely cause of the observed deficiencies is population subdivision into genetically divergent subpopulations. The overall level of population differentiation is greater than is typically observed in out-breeders: The mean genetic distance and identity (Nei's D and I) between pairs of populations were 0.073 and 0.930, respectively, and Wright's Fst was 0.144. Differences among populations appeared to be manifested as variation in gene frequencies at many loci rather than variation in allelic composition per se. However, the patterns of variation were not random. Reciprocal clinal variation of gene frequencies was observed for allele pairs at six loci. In addition, principal components analysis revealed that populations tended to genetically cluster into four regions representing the geographic areas Kyushu, Shikoku, western Honshu, and eastern Honshu. There was a significant relationship between genetic and geographic distance (r = 0.61; P < 0.01). Analysis of variance on allozyme frequencies showed that there was approximately four times as much differentiation among populations within regions, as among regions. It is likely that the observed patterns of population relationships result from the balance between genetic drift in small subpopulations and gene flow between them.     

Genetic and ecological study of aboriginal populations of northeastern Siberia. II. Polymorphic blood systems, immunoglobulin allotypes and other genetic markers in asian eskimos. Genetic structure of Bering sea eskimos

Blood groups, immunoglobulin allotypes, serum proteins and red cell enzymes were tested in four populations of Asiatic Eskimos. 12 of 16 gene loci studied were found to be polymorphic. Analysis of covariance and variance matrices for gene frequencies of 8 Bering Sea populations revealed major aspects of regional genetic structure. Regression of average heterozygosity on genetic distance from centroid permitted us to interpret it in the light of counterbalancing action of systematic and non-systematic pressure. Analysis of R matrix containing gene frequencies for both Eskimo and Chukchi populations revealed two different clusters - more heterogeneous Eskimo cluster and less heterogeneous Chukchi cluster. Good fit was observed between regional genetic structure, history and geography.     

Distribution of Gene Frequency as a Test of the Theory of the Selective Neutrality of Polymorphisms

The variation in gene frequency among populations or between generations within a population is a result of breeding structure and selection. But breeding structure should affect all loci and alleles in the same way. If there is significant heterogeneity between loci in their apparent inbreeding coefficients F=s_p²/p(1-p), this heterogeneity may be taken as evidence for selection. We have given the statistical properties of F and shown how tests of heterogeneity can be made. Using data from human populations we have shown highly significant heterogeneity in F values for human polymorphic genes over the world, thus demonstrating that a significant fraction of human polymorphisms owe their current gene frequencies to the action of natural selection. We have also applied the method to temporal variation within a population for data on Dacus oleae and have found no significant evidence of selection.     

The deviation from linkage equilibrium with multiple loci varying in a stepping-stone cline

The balance between the creation of associations between alleles at different loci by immigration and the convergence to linkage equilibrium due to the recombination process is studied in a theoretical model. The geographical structure of the model is a stepping-stone chain of populations linking two genetically constant source populations. The model assumes an arbitrary number of autosomal loci and considers genetic variation (two alleles at each locus) that is not subject to natural selection. The gene frequencies at each locus will then show a linear cline through the stepping-stone chain of populations. The deviation from linkage equilibrium through the stepping-stone cline is characterized by an equation for linear measures that provide the linkage disequilibrium measures for a given set of loci in terms of the gene frequencies and the linkage disequilibria in the source populations and in terms of the linkage disequilibrium measures through the cline for lower numbers of loci. Numerical examples of this iterative solution are given, and it is shown that the build-up of the higher order Bennett-disequilibria through the cline is considerably more pronounced than the build-up of two-locus disequilibria.     

Two-locus,fourth-order gene frequency moments: Implications for the variance of squared linkage disequilibrium and the variance of homozygosity

Identity coefficients are used to construct a sufficient set of equations to determine the fourth-order moments of gene frequencies for two linked loci. This allows the variance of the expected squared linkage disequilibrium to be found. It is shown that the coefficient of variation is generally greater than one and if the mutation rate is small, the standard deviation is more than four times the size of the mean. This demonstrates that squared linkage disequilibrium is a highly variable quantity. The variance of homozygosity for a gene which consists of two sites can also be obtained. Recombination between these sites increases the variance of homozygosity, suggesting that intragenic recombination significantly changes all the expected moments of gene frequencies if 4Nμ > 1.0 and r > μ (where N is the population size, μ is the mutation rate of the gene to neutral alleles, and r is the recombination rate between two sites within the gene).     

Structure of Genetic Variation within and between Populations of Mycophagous Drosophila

Patterns of genetic variation within and between populations of five species of mycophagous Drosophila were examined by gel electrophoresis of several polymorphic loci. Populations of the five species could not be shown to be subdivided into sympatric host-adapted races. Statistically significant, but small, between-host differences in gene frequencies were observed at three of 15 loci. Mean gene frequencies at all loci were similar in New York and Tennessee, and, with one exception, relatively little genetic differentiation was observed among study sites within those two regions. Gene frequencies generally were stable over several years of collecting as well. The unpredictable nature of the fungal hosts may preclude the site fidelity and continuity of diversifying selection necessary for adaptive divergence of populations.     

Electrophoretic variability in natural populations of Drosophila melanogaster and Drosophila simulans

Genetic variation at 59 gene loci coding for enzymes (50) and larval proteins (9) has been studied in sympatric populations of Drosophila melanogaster and D. simulans from insular and continental origin. The average number of alleles per locus, the mean proportion of polymorphic loci and the mean heterozygosity are similar both within and between species. There are however some significant differences between D. simulans populations in the genotypic frequencies for four polymorphic loci.     

Towards a theory of the evolution of modifier genes

The main findings of a study of the evolution of modifier gene frequencies in models of deterministic population genetics are presented. A wide variety of random mating systems are subject to selection with modifiers operating, in different cases, on mutation rates, migration between subpopulations, and linkage between other loci. In all these instances, the modifier frequencies evolve in such a way as to maximize the mean fitness of the population at equilibrium. This is remarkable since, the modifier genes are selectively neutral in the sense that they do not affect the fitness of their individual carriers. In nonrandom mating systems, the mean fitness concept is not well-defined, and there does not appear to be such a simple principle governing the evolution of modifier frequencies. In assortative mating systems, modifiers favoring reduced assortment propensities tend to increase. In contrast, for selfing-outcrossing systems, modifiers favoring increased selfing tend to increase.     

The consequences of fluctuating selection for isozyme polymorphisms in Daphnia

Temporal sequences of allele frequencies in natural populations of Daphnia are analyzed to obtain the mean and variance of the selection coefficient for both asexual and sexual phases. In general, the alleles at enzyme loci appear to be quasi-neutral. Although significant variation exists for the estimated selection coefficients, the means are in all cases close to zero. Estimates of the variance of selection intensity are applied to existing models to demonstrate the implications of fluctuating selection for the spatial and temporal distribution of gene frequencies in Daphnia. The empirical and analytical results are shown to provide a possible solution to some previously puzzling aspects of Daphnia population genetic surveys. Neither genetic drift nor diversifying selection are necessary conditions for the local diversification of gene frequencies.     

Population Genetics of Euphydryas Butterflies. I. Genetic Variation and the Neutrality Hypothesis

Twenty-one populations of the checkerspot butterfly, Euphydryas editha, and ten populations of Euphydryas chalcedona were sampled for genetic variation at eight polymorphic enzyme loci. Both species possessed loci that were highly variable from population to population and loci that were virtually identical across all populations sampled. Our data indicate that the neutrality hypothesis is untenable for the loci studied, and therefore selection is indicated as the major factor responsible for producing these patterns. Thorough ecological work allowed gene flow to be ruled out (in almost all instances) as a factor maintaining similar gene frequencies across populations. The Lewontin-Krakauer test indicated magnitudes of heterogeneity among standardized variances of gene frequencies inconsistent with the neutrality hypothesis. The question of whether or not to correct this statistic for sample size is discussed. Observed equitability of gene frequencies of multiple allelic loci was found to be greater than that predicted under the neutrality hypothesis. Genetic differentiation persisting through two generations was found between the one pair of populations known to exchange significant numbers of individuals per generation. Two matrices of genetic distance between populations, based on the eight loci sampled, were found to be significantly correlated with a matrix of environmental distance, based on measures of fourteen environmental parameters. Correlations between gene frequencies and environmental parameters, results of multiple regression analysis, and results of principle component analysis showed strong patterns of association and of "explained" variation. The correlation analyses suggest which factors might be further investigated as proximate selective agents.     

Use of restriction fragment length polymorphisms for genetic counseling: Population genetic considerations

Two-locus population genetic models are analyzed to evaluate the utility of restriction fragment length polymorphisms for purposes of genetic counseling. It is shown that the linkage disequilibrium between a neutral marker and a tightly linked overdominant mutant will increase rapidly as the mutant moves to its polymorphic equilibrium. The linkage disequilibrium decays for deleterious recessive mutants. Two measures involving the linkage disequilibrium are investigated to determine how much information the transmission of the neutral marker provides about the transmission of the selected gene. In certain kinds of matings, where the parental two-locus genotypes and linkage phases are known, it is possible to determine whether or not a progeny is homozygous for the selected gene on the basis of the fetal genotype at the marker locus. A quantity of primary interest is the fraction of matings between individuals heterozygous for the selected gene in which exact diagnosis can be made in this way. The expected proportion of such matings, taken over all two-locus matings involving heterozygotes at the selected locus, is calculated as a function of the gene frequencies at the two loci and the linkage disequilibrium between them. This expected value is maximized when the linkage disequilibrium is at its maximum in absolute value. Fewer than half of all matings are informative if the linkage disequilibrium is small in magnitude or if the gene frequencies at the two loci are quite different. Consideration is also given to various conditional measures of association that may be useful when the parental two-locus genotypes are unknown. The results suggest that the utility of tightly linked neutral marker genes in predicting the transmission of a selected gene is generally less when selection acts against a recessive gene than for overdominant selection.     

Comparative electrophoretic examination of the two European species of angler-fish (Lophiidae): Lophius piscatorius (L.) and Lophius budegassa (Spinola) and assessment of their genetic relationship

1. A multilocus electrophoretic examination of the genetic relationships between the two European species of angler-fish was carried out using starch gel electrophoresis. 2. At 4 gene loci fixation for different alleles separated Lophius piscatorius and L. budegassa while at an additional 5 loci, frequencies of shared alleles were significantly heterogeneous. 3. A mean genetic distance of I = 0.771 was calculated, corresponding to a mean genetic distance of D = 0.260 between these species. 4. This research identifies 7 polymorphic loci in L. piscatorius and 5 in L. budegassa which will potentially serve as genetic markers for examination of population structure.     

Hierarchical gene diversity and genetic structure of tribal populations of Andhra Pradesh,India

Gene diversity and genetic structure of tribal populations of Andhra Pradesh, India, have been analyzed under a hierarchical model consisting of five regions of the state, tribes within the regions, and local subpopulations within the tribes. Average gene diversity has been estimated from gene frequency data for 15 polymorphic loci by using nested gene diversity analysis of G_ST. The intralocation coefficient of gene diversity was estimated at 96% of the total, whereas the intertribal, within—and between—regional gene diversities were found to be only 1.90, 0.95, and 1.43%, respectively. The estimate of gene diversity was higher for loci with higher degrees of polymorphism such as ABO, MN, ESD, and PTC and lower for loci with low-level polymorphism and extreme gene frequencies such as Hb, Tf, PHI, 6PGD, and Hp. The nature of selective preference or neutrality at the loci seems to be important in this respect. Tribes of the plains exhibit the least gene diversity, apparently because of higher gene flow among them. The contribution of loci with intermediate gene frequencies in intertribal and regional gene diversity was found to be higher than for loci with extreme allelic frequencies. These results suggest that the most significant component of variation is between individuals within locations and that variation between local subpopulations is negligible in the genetic structure of a population. Forces like selection, gene flow and drift also influence the diversity depending upon the nature of the locus. © 1993 Wiley-Liss, Inc.     

Gene Flow and Selection in a Two-Locus System

A model of gene flow and selection in two linked loci is analyzed. The problems considered are the effects of linkage on the clines in frequencies at the two loci and the role of gene flow in producing linkage disequilibrium between the loci. Also, the possible significance of linkage as a mechanism for permitting a population of "track" spatial changes in the environment is considered. The results are that when the recombination fraction between the loci is of the same order of magnitude as the selection coefficients or smaller, then linkage is important in determining the gene frequencies and a substantial amount of linkage disequilibrium is present in the cline. Depending on the spatial pattern of selection on the two loci, linkage can either decrease or increase a population's response to local selection.     

Enzyme polymorphism and species discrimination in fruit flies of the genus Dacus (Tephritidae).

Sympatric populations of D. tryoni and D. neohumeralis are difficult to completely distinguish taxonomically. Using five pigmentation characters, each of some taxonomic value, a small proportion of individuals cannot be assigned to either species nor definitely classified as hybrids. To aid in species discrimination and hybrid identification gene frequencies in natural populations were estimated at three polymorphic protein loci, an alcohol dehydrogenase (Adh), an octanol dehydrogenase (Odh) and an esterase (E-2). Samples of flies were taken from four sites spread over 1200 miles along the Australian eastern coast. Within each species allelic frequencies at each locus were largely the same at all localities. Consistent differences in gene frequencies between species occurred at all three loci, strongly supporting the hypothesis of two distinct gene pools. The Adh locus best discriminated between species with a unique allele occurring in D. neohumeralis at a frequency of 0-85. None of the loci showed complete differentiation and hence it was not possible to find a quick and easy method to distinguish the species nor to detect field hybrids. Directional selection of laboratory populations for a change in callus colour (the best pigmentation character for separating the species) indicated that at the Adh and E-2 loci frequencies of major alleles were not genetically associated with major genes for callus colour. Thus genotype determination at these loci when considered together with pigmentation characters may be valuable taxonomically for further distinguishing between the species.     

A Complex Model for Gene Effects taking into Account Selection and Crossing Over

A general model for gene effects including the influence of selection and crossing over is presented. It is shown, how gene frequencies of alleles in two loci system are changed in selected populations in the presence of crossing over. This is used for estimation of gene effects.