Multiallelic selection polymorphism

The existence and stability of an internal (i.e., completely polymorphic) equilibrium for viability selection at a single multiallelic locus is investigated. Generations are discrete and nonoverlapping; the population is panmictic, monoecious, and diploid. Various necessary and sufficient conditions for the existence of an internal equilibrium are established and applied to the loss of alleles. Some necessary conditions for the existence of an asymptotically stable internal equilibrium are also established. All these conditions are simpler and yield general biological conclusions more easily than the classical necessary and sufficient conditions.     

Evolution under multiallelic migration-selection models

The loss of a specified allele and the convergence of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection are investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Sufficient conditions are established for global fixation and for global loss of a particular allele. When migration is either sufficiently weak or sufficiently strong relative to selection, the equilibria are described, convergence of the gene frequencies is demonstrated, and sufficient conditions for the increase of a suitably defined mean fitness are offered. If the selection pattern is the same in every colony and such that in a panmictic population there is a globally asymptotically stable, internal (i.e., completely polymorphic) equilibrium point, then under certain weak assumptions on migration, the gene frequencies in the subdivided population converge globally to that equilibrium point. Thus, in this case, the ultimate state of the population is unaffected by geographical structure.     

Selective sweeps in multilocus models of quantitative traits

We study the trajectory of an allele that affects a polygenic trait selected toward a phenotypic optimum. Furthermore, conditioning on this trajectory we analyze the effect of the selected mutation on linked neutral variation. We examine the well-characterized two-locus two-allele model but we also provide results for diallelic models with up to eight loci. First, when the optimum phenotype is that of the double heterozygote in a two-locus model, and there is no dominance or epistasis of effects on the trait, the trajectories of selected mutations rarely reach fixation; instead, a polymorphic equilibrium at both loci is approached. Whether a polymorphic equilibrium is reached (rather than fixation at both loci) depends on the intensity of selection and the relative distances to the optimum of the homozygotes at each locus. Furthermore, if both loci have similar effects on the trait, fixation of an allele at a given locus is less likely when it starts at low frequency and the other locus is polymorphic (with alleles at intermediate frequencies). Weaker selection increases the probability of fixation of the studied allele, as the polymorphic equilibrium is less stable in this case. When we do not require the double heterozygote to be at the optimum we find that the polymorphic equilibrium is more difficult to reach, and fixation becomes more likely. Second, increasing the number of loci decreases the probability of fixation, because adaptation to the optimum is possible by various combinations of alleles. Summaries of the genealogy (height, total length, and imbalance) and of sequence polymorphism (number of polymorphisms, frequency spectrum, and haplotype structure) next to a selected locus depend on the frequency that the selected mutation approaches at equilibrium. We conclude that multilocus response to selection may in some cases prevent selective sweeps from being completed, as described in previous studies, but that conditions causing this to happen strongly depend on the genetic architecture of the trait, and that fixation of selected mutations is likely in many instances.     

Random temporal variation in selection intensities: One-locus two-allele model

Summary The paper formulates a one locus two allele diploid model influenced by temporal random selection intensities. The concept of stochastic local stability for equilibria states is formulated. Necessary and sufficient conditions are derived for the stochastic local stability of the fixation states for non-dominant and dominant traits. A general model, where a fixed polymorphic equilibrium is maintained is investigated. The complete local evolutionary picture is derived and some cases of global convergence are established.Supported in part by N. I. H. Grant GM 10452-09 and N. S. F. Grant 7507129.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

Epistatic selection opposed by immigration in multiple locus genetic systems

A model of “complete” epistatis is considered in which all “plus” alleles must be present in an individual before the adaptive phenotype is expressed. The conditions under which the plus alleles and hence the adaptive phenotype can increase and reach a stable equilibrium in the presence of immigration of gametes carrying minus alleles are found. In haploids and diploids in which the plus alleles are recessive, frequencies of the plus alleles are the same at all loci, regardless of the linkage relationships. Tight linkage favors the existence of a locally stable polymorphic equilibrium, but the equilibrium with only minus alleles is locally stable unless there is very tight linkage or very strong selection. Thus, this kind of epistasis, which provides a simple model for a character that requires several components to be present at the same time, is very sensitive to even a small amount of immigration. Hence, the evolution of such characters is likely only in completely rather than partially isolated populations.     

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.     

Some circumstances assuring monomorphism in subdivided populations

It is well known that in a subdivided population subject to soft selection with two alleles at one locus, instability of both fixation states (a “protected polymorphism”) entails at least one stable polymorphic equilibrium. Although stable polymorphic and monomorphic equilibria can coexist in general, a stable fixation state (monomorphic equilibrium) precludes the existence of any polymorphic equilibrium under the circumstances of haploid or submultiplicative diploid viabilities. This provides that a stable monomorphism is robust against random fluctuations in allele frequencies. It also increases the known circumstances where there is a unique globally attracting stable equilibrium, i.e., where allele frequencies are determined by the selection-migration structure independent of the history of the system.     

A deterministic analysis of self-incompatibility alleles

A number of plant species have a self-incompatibility locus that prevents self-fertilization. We 'analyse a deterministic model with an arbitrary number of alleles. We prove that the only polymorphic equilibrium is the one for which all (heterozygous) genotypes are equally frequent, and we prove that all (initially) polymorphic populations converge to this equilibrium.     

A model of migration modification

Two subpopulations whose different sizes are in a constant ratio interact via migration. The fitness of the diploid organisms is determined by two alleles at a single locus and by the niche the organism is in. The rates of migration depend upon two neutral modifier genes at a second locus. The second modifying allele is introduced into an equilibrium where the first modifying allele is fixed, and where the other two alleles are already polymorphic. It is shown that the new migration modifier is selected for when it reduces migration. The similarity between this result and some recombination modifier models is noted.     

Normal variation at the myotonic dystrophy locus in global human populations.

Myotonic dystrophy (DM) is a dominant neuromuscular disease that results from an unstable CTG-repeat expansion in the 3' UTR of the myotonin kinase gene at 19q13.3. This repeat is normally polymorphic with a trimodal distribution reflecting 5-, 11-17-, and 19-30-repeat-length alleles. An absolute association between expanded CTG alleles and the 1-kb insertion allele of an intragenic polymorphism in Caucasians has led to the proposal that the 5-repeat allele gives rise to alleles of 19-30 repeats, from which expanded alleles are derived, a transition not involving the 11-17-repeat alleles. A survey of eight global populations confirms the stability of the 11-17-repeat alleles but shows disociation between the 1-kb insertion polymorphism and both the 5- and 19-30-repeat-length alleles. These data indicate more than one ancestral allele from which expanded alleles are derived and suggest that widely variable population frequencies of DM may reflect distinct frequencies of such predisposed alleles.     

Evolution and polymorphism in the multilocus Levene model with no or weak epistasis

Evolution and the maintenance of polymorphism under the multilocus Levene model with soft selection are studied. The number of loci and alleles, the number of demes, the linkage map, and the degree of dominance are arbitrary, but epistasis is absent or weak. We prove that, without epistasis and under mild, generic conditions, every trajectory converges to a stationary point in linkage equilibrium. Consequently, the equilibrium and stability structure can be determined by investigating the much simpler gene-frequency dynamics on the linkage-equilibrium manifold. For a haploid species an analogous result is shown. For weak epistasis, global convergence to quasi-linkage equilibrium is established. As an application, the maintenance of multilocus polymorphism is explored if the degree of dominance is intermediate at every locus and epistasis is absent or weak. If there are at least two demes, then arbitrarily many multiallelic loci can be maintained polymorphic at a globally asymptotically stable equilibrium. Because this holds for an open set of parameters, such equilibria are structurally stable. If the degree of dominance is not only intermediate but also deme independent, and loci are diallelic, an open set of parameters yielding an internal equilibrium exists only if the number of loci is strictly less than the number of demes. Otherwise, a fully polymorphic equilibrium exists only nongenerically, and if it exists, it consists of a manifold of equilibria. Its dimension is determined. In the absence of genotype-by-environment interaction, however, a manifold of equilibria occurs for an open set of parameters. In this case, the equilibrium structure is not robust to small deviations from no genotype-by-environment interaction. In a quantitative-genetic setting, the assumptions of no epistasis and intermediate dominance are equivalent to assuming that in every deme directional selection acts on a trait that is determined additively, i.e., by nonepistatic loci with dominance. Some of our results are exemplified in this quantitative-genetic context.     

Nearly identical allelic distributions of xanthine dehydrogenase in two populations of Drosophila pseudoobscura

In a previous study, Keith (1983) showed by sequential gel electrophoresis of the esterase-5 protein in Drosophila pseudoobscura that a highly polymorphic locus with many alleles can have very similar frequency distributions in populations separated by 500 km. The present work studies another highly polymorphic locus, xanthine dehydrogenase, in the same California population samples, using the same technique to distinguish allelic classes. Twelve electromorphs were found in one population and 15 in the other. Both populations shared a single very frequent (approximately 60%) allele, as well as five other alleles in low but similar frequencies. In addition, each population had an array of unique alleles present only once in one population sample but absent in the other. A statistical test against the stationary distribution for neutral alleles shows that, if the populations are at equilibrium, then purifying selection is operating on xanthine dehydrogenase. The extremely close similarity in frequency distributions of the alleles between populations for both the xanthine dehydrogenase and esterase-5 loci, despite differences in allele frequency distribution between loci, strongly emphasizes the importance of migration in influencing genic diversity in these populations.     

Genetic polymorphism of MJD1 alleles and molecular analysis of SCA3 patients from Rio de Janeiro, Brazil

Spinocerebellar ataxia type 3 is the most common form of autosomal dominant cerebellar ataxia. It is a severe progressive neurological disorder caused by an expansion of an exonic CAG repeat of the MJD1 gene. The repeated sequence is polymorphic among both normal individuals and patients. In general, expanded alleles are paternally inherited and the disorder exhibits anticipation. We performed a PCR-based study to determine polymorphisms of the number of CAG repeats of the MJD1 gene in an anonymous sample of normal Brazilian individuals. We also analyzed DNA samples from 9 patients with ataxia. We identified 29 different allele sizes ranging from 12 to 40 CAG repeats, with heterozygosity of 79%. The distribution of allele sizes showed two major peaks of 16 (7%) and 26 (10.1%) CAG repeats. When grouping normal alleles by size, we observed that the distribution varies between males and females, and a significant deviation from the Hardy-Weinberg equilibrium was observed with an excess of normal large alleles among males. We also detected expanded alleles with 68-73 CAG repeats in 3 out of 9 ataxic patients.     

Polymorphisms in the large subunit of human RNA polymerase II as target for allele-specific inhibition

The lack of specificity of cancer treatment causes damage to normal cells as well, which limits the therapeutic range. To circumvent this problem one would need to use an absolute difference between normal cells and cancer cells as therapeutic target. Such a difference exists in the genome of all individuals suffering from a tumor that is characterized by loss of genetic material [loss of heterozygosity (LOH)]. Due to LOH, the tumor is hemizygous for a number of genes, whereas the normal cells of the individual are heterozygous for these genes. Theoretically, polymorphic sites in these genes can be utilized to selectively target the cancer cells with an antisense oligonucleotide, provided that it can discriminate the alleles and inhibit gene expression. Furthermore, the targeted gene should be essential for cell survival, and 50% gene expression sufficient for the cell to survive. This will allow selective killing of cancer cells without concomitant toxicity to normal cells. As an initial step in the experimental test of this putative selective cancer cell therapy, we have developed a set of antisense phosphorothioate oligonucleotides which can discriminate the two alleles of a polymorphic site in the gene encoding the large subunit of RNA polymerase II. Our data show that the exact position of the antisense oligonucleotide on the mRNA is of essential importance for the oligonucleotide to be an effective inhibitor of gene expression. Shifting the oligonucleotide position only a few bases along the mRNA sequence will completely abolish the inhibitory activity of the antisense oligonucleotide. Reducing the length of the oligonucleotides to 16 bases increases the allele specificity. This study shows that it is possible to design oligonucleotides that selectively target the matched allele, whereas the expression level of the mismatched allele, that differs by one nucleotide, is only slightly affected.     

Population Bottlenecks and Nonequilibrium Models in Population Genetics. II. Number of Alleles in a Small Population That Was Formed by a Recent Bottleneck

A model is presented in which a large population in mutation/drift equilibrium undergoes a severe restriction in size and subsequently remains at the small size. The rate of loss of genetic variability has been studied. Allelic loss occurs more rapidly than loss of genic heterozygosity. Rare alleles are lost especially rapidly. The result is a transient deficiency in the total number of alleles observed in samples taken from the reduced population when compared with the number expected in a sample from a steady-state population having the same observed heterozygosity. Alternatively, the population can be considered to possess excess gene diversity if the number of alleles is used as the statistical estimator of mutation rate. The deficit in allele number arises principally from a lack of those alleles that are expected to appear only once or twice in the sample. The magnitude of the allelic deficiency is less, however, than the excess that an earlier study predicted to follow a rapid population expansion. This suggests that populations that have undergone a single bottleneck event, followed by rapid population growth, should have an apparent excess number of alleles, given the observed level of genic heterozygosity and provided that the bottleneck has not occurred very recently. Conversely, such populations will be deficient for observed heterozygosity if allele number is used as the sufficient statistic for the estimation of 4Nev. Populations that have undergone very recent restrictions in size should show the opposite tendencies.     

Patterns of multiallelic polymorphism maintained by migration and selection

Evolution at a multiallelic locus under the joint action of migration and viability selection is investigated. Generations are discrete and nonoverlapping. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. The forward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). Several examples of globally attracting multiallelic equilibria are presented. Migration can cause global fixation even if, without migration, there is a globally attracting multiallelic equilibrium in every colony. Migration can also cause the global fixation of an allele that, without migration, is eliminated in every colony. Without dominance, generically, the number of alleles present at equilibrium cannot exceed the number of colonies. Some general properties and examples of the Levene model are studied in detail. If in each colony there is either no dominance or, without migration, a globally attracting internal equilibrium, then there exists a globally attracting equilibrium with migration. Therefore, if an internal equilibrium exists, it is the global attractor.     

A theoretical and numerical assessment of genetic variability

The equilibrium behavior of one-locus viability selection models is studied numerically. The selection schemes include randomly chosen viabilities, viabilities chosen to measure a hypothetical distance between the alleles making up the genotype and viabilities that obey various allelic dominance relations. From 3 to 8 alleles are considered. Among the key conclusions are (1) equilibria that are most polymorphic do not usually have the highest mean fitness, (2) the more structure there is in the choice of the viability model, the greater is the level of polymorphism at equilibrium, and (3) for the numbers of alleles chosen here, the equilibrium reached by iteration from the centroid of the allele frequency simplex is the best predictor of the equilibrium attainable from randomly chosen starting vectors. Preliminary evidence shows that this is not the case for 16 alleles.     

Population dynamics of sex-determining alleles in honey bees and self-incompatibility alleles in plants

Mathematical theories of the population dynamics of sex-determining alleles in honey bees are developed. It is shown that in an infinitely large population the equilibrium frequency of a sex allele is 1/n, where n is the number of alleles in the population, and the asymptotic rate of approach to this equilibrium is 2/(3n) per generation. Formulae for the distribution of allele frequencies and the effective and actual numbers of alleles that can be maintained in a finite population are derived by taking into account the population size and mutation rate. It is shown that the allele frequencies in a finite population may deviate considerably from 1/n. Using these results, available data on the number of sex alleles in honey bee populations are discussed. It is also shown that the number of self-incompatibility alleles in plants can be studied in a much simpler way by the method used in this paper. A brief discussion about general overdominant selection is presented.     

Enzyme polymorphism in Schistosoma mattheei from cattle in the Eastern Transvaal Lowveld

Enzyme electrophoresis was conducted on 10 Schistosoma mattheei adult worm samples, comprising 270 individuals, collected from cattle in the Eastern Transvaal Lowveld. Glucose-6-phosphate dehydrogenase (G6PDH) was studied in all the samples and phosphoglucomutase (PGM) and malate dehydrogenase (MDH) in five populations each. Only one population was polymorphic for G6PDH. In this population, in addition to the allele found in all the other samples, a second allele occurred with a similar Rf value to S. haematobium. The two alleles were in Hardy-Weinberg equilibrium. MDH-1 exhibited two alleles. However, these alleles were not in equilibrium. In certain populations, heterozygotes occurred together with homozygotes of one of the alleles only. PGM was monomorphic in all the populations studied.