Mutation-Selection Balance at a Modifier-of-Imprinting Locus

We propose a pair of population genetic models for a modifier-of-imprinting locus for which different genotypes imprint different proportions of an imprintable target locus in their gametes. The two models examine the situations in which imprinting is advantageous or disadvantageous, and we discuss three cases for which the modifier is respectively partially dominant, dominant, or recessive. The models predict the stable equilibrium frequencies of the mutant modifier and functionally diploid individuals in a large population in terms of up to four parameters: the mutation rate at the modifier locus, V; the selection coefficient against the disadvantageous phenotype, s; the proportion of unimprinted eggs produced by homozygotes for the mutant modifier, θ, and, in the partially dominant models, the dominance parameter, k. The equilibrium frequency of the mutant phenotypes is shown to be approximately twice that of standard Mendelian models: 2V/s or 4V/s when the modifier is recessive or dominant, respectively. Mathematical equivalences between these and nonimprinting models are noted.     

A chip off the old block: a model for the evolution of genomic imprinting via selection for parental similarity

A consequence of genomic imprinting is that offspring are more similar to one parent than to the other, depending on which parent's genes are inactivated in those offspring. We hypothesize that genomic imprinting may have evolved at some loci because of selection to be similar to the parent of one sex or the other. We construct and analyze an evolutionary-genetic model of a two-locus two-deme system, in which one locus codes for a character under local selection and the second locus is a potential cis-acting modifier of imprinting. A proportion of males only migrate between demes every generation, and prebreeding males are less fit, on average, than females. We examine the conditions in which an imprinting modifier allele can invade a population fixed for a nonimprinting modifier allele and vice versa. We find that the conditions under which the imprinting modifier invades are biologically restrictive (high migration rates and high values of recombination between the two loci) and thus this hypothesis is unlikely to explain the evolution of imprinting. Our modeling also shows that, as with several other hypotheses, polymorphism of imprinting status may evolve under certain circumstances, a feature not predicted by verbal accounts.     

Population models of genomic imprinting. I. Differential viability in the sexes and the analogy with genetic dominance

Many single-locus, two-allele selection models of genomic imprinting have been shown to reduce formally to one-locus Mendelian models with a modified parameter for genetic dominance. One exception is the model where selection at the imprinted locus affects the sexes differently. We present two models of maternal inactivation with differential viability in the sexes, one with complete inactivation, and the other with a partial penetrance for inactivation. We show that, provided dominance relations at the imprintable locus are the same in both sexes, a globally stable polymorphism exists for a range of viabilities that is independent of the penetrance of imprinting. The conditions for a polymorphism are the same as in previous models with differential viability in the sexes but without imprinting and in a model of the paternal X-inactivation system in marsupials. The model with incomplete inactivation is used to illustrate the analogy between imprinting and dominance by comparing equilibrium bifurcation plots for fixed values of dominance and penetrance. We also derive a single expression for the dominance parameter that leaves the frequency and stability of equilibria unchanged for all levels of inactivation. Although an imprinting model with sex differences does not formally reduce to a nonimprinting scheme, close theoretical parallels clearly exist.     

Failure of imprinting at Igf-2: two models of mutation-selection balance.

The failure of maternal imprinting at the insulin-like growth factor II (Igf-2) locus predisposes individuals to several clinical conditions, including Wilms tumor. Having two functional Igf-2 genes, therefore, is selectively disadvantageous, and the condition is probably maintained in human populations by recurrent mutation. We propose two models that predict the expected frequency of functionally diploid individuals in a large population, in terms of a mutation rate, mu, and the selection coefficient against functionally diploid individuals, s. In the first model a mutant Igf-2 allele that cannot be imprinted arises from the standard, imprintable allele at a rate mu. Our second model hypothesizes a second modifier locus at which a recessive allele arises at rate mu. Mothers who are homozygous for this recessive modifier allele fail to imprint their eggs. Both models predict the expected frequency of affecteds to be 2 mu/s(1 + mu), approximately twice that predicted by the standard one-locus model of a recessive allele in mutation-selection balance. This frequency suggests that < or = 25% of the cases of Wilms tumor are due to the failure to imprint the maternal Igf-2 gene.     

Sex-specific viability, sex linkage and dominance in genomic imprinting

Genomic imprinting is a phenomenon by which the expression of an allele at a locus depends on the parent of origin. Two different two-locus evolutionary models are presented in which a second locus modifies the imprinting status of the primary locus, which is under differential selection in males and females. In the first model, a modifier allele that imprints the primary locus invades the population when the average dominance coefficient among females and males is >12 and selection is weak. The condition for invasion is always heavily contingent upon the extent of dominance. Imprinting is more likely in the sex experiencing weaker selection only under some parameter regimes, whereas imprinting by either sex is equally likely under other regimes. The second model shows that a modifier allele that induces imprinting will increase when imprinting has a direct selective advantage. The results are not qualitatively dependent on whether the modifier locus is autosomal or X linked.     

Somatic mutations in organisms with complex life histories.

A theoretical model is developed of the fate of mutations for organisms with such life-history characteristics as indeterminate growth and clonal reproduction. It focuses on how the fate of a particular mutant depends on whether it arises during mitotic cell division (somatic mutation) or during meiotic cell division (meiotic mutation). At gamete production, individuals carrying somatic mutations will produce some proportion of gametes reflecting the original, zygotic genotype and some proportion reflecting genotypes carrying the somatic mutation. Focusing on allele frequencies at gamete production allows the effects of growth and clonal reproduction to be summarized. The relative strengths of somatic and meiotic mutation can be determined, as well as the conditions under which the change in allele frequency due to one is greater than that due to the other. Examples from a published demographic study of clonal corals are used to compare somatic and meiotic mutation. When there is no selection acting on either type of mutation, only a few cell divisions per time unit on average are needed for the change in allele frequency due to somatic mutation to be greater, given empirically based mutation rates. When somatic selection is added, the most dramatic effect is seen with fairly strong negative selection acting against the somatic mutation within individuals. In this case, selection within organisms can effectively counteract the effects of somatic mutation, and the change in allele frequency due to somatic mutations will not be greater than that due to meiotic mutations for reasonable numbers of within-generation cell divisions. The majority of the mutation load, which would have been due to somatic mutation, is purged by selection within the individual organism.     

The evolution of genomic imprinting via variance minimization: an evolutionary genetic model

A small number of mammalian loci exhibit genomic imprinting, in which only one copy of a gene is expressed while the other is silenced. At some such loci, the maternally inherited allele is inactivated; others show paternal inactivation. Several hypotheses have been put forward to explain how this genetic system could have evolved in the face of the selective advantages of diploidy. In this study, we examine the variance-minimization hypothesis, which proposes that imprinting arose through selection for reduced variation in levels of gene expression. We present an evolutionary genetic model incorporating both this selection pressure and deleterious mutations to elucidate the conditions under which imprinting could evolve. Our analysis implies that additional mechanisms such as genetic drift are required for imprinting to evolve from an initial nonimprinting state. Other predictions of this hypothesis do not appear to fit the available data as well as predictions for two alternative hypotheses, genetic conflict and the ovarian time bomb. On the basis of this evidence, we conclude that the variance-minimization hypothesis appears less adequate to explain the evolution of genomic imprinting.     

Ancestral processes for non-neutral models of complex diseases

We consider non-neutral models for unlinked loci, where the fitness of a chromosome or individual is not multiplicative across loci. Such models are suitable for many complex diseases, where there are gene-interactions. We derive a genealogical process for such models, called the complex selection graph (CSG). This coalescent-type process is related to the ancestral selection graph, and is derived from the ancestral influence graph by considering the limit as the recombination rate between loci gets large. We analyse the CSG both theoretically and via simulation. The main results are that the gene-interactions do not produce linkage disequilibrium, but do produce dependencies in allele frequencies between loci. For small selection rates, the distributions of the genealogy and the allele frequencies at a single locus are well-approximated by their distributions under a single locus model, where the fitness of each allele is the average of the true fitnesses of that allele with respect to the distribution of alleles at other loci.     

The effect of genetic conflict on genomic imprinting and modification of expression at a sex-linked locus

We examine how genomic imprinting may have evolved at an X-linked locus, using six diallelic models of selection in which one allele is imprintable and the other is not. Selection pressures are generated by genetic conflict between mothers and their offspring. The various models describe cases of maternal and paternal inactivation, in which females may be monogamous or bigamous. When inactivation is maternal, we examine the situations in which only female offspring exhibit imprinting as well as when both sexes do. We compare our results to those previously obtained for an autosomal locus and to four models in which a dominant modifier of biallelic expression is subjected to the same selection pressures. We find that, in accord with verbal predictions, maternal inactivation of growth enhancers and paternal inactivation of growth inhibitors are more likely than imprinting in the respective opposite directions, although these latter outcomes are possible for certain parameter combinations. The expected outcomes are easier to evolve than the same outcomes for autosomal loci, contradicting the available evidence concerning the direction of imprinting on mammalian sex chromosomes. In most of our models stable polymorphism of imprinting status is possible, a behavior not predicted by verbal accounts.     

Allele Frequencies at Microsatellite Loci: The Stepwise Mutation Model Revisited

We summarize available data on the frequencies of alleles at microsatellite loci in human populations and compare observed distributions of allele frequencies to those generated by a simulation of the stepwise mutation model. We show that observed frequency distributions at 108 loci are consistent with the results of the model under the assumption that mutations cause an increase or decrease in repeat number by one and under the condition that the product Nu, where N is the effective population size and u is the mutation rate, is larger than one. We show that the variance of the distribution of allele sizes is a useful estimator of Nu and performs much better than previously suggested estimators for the stepwise mutation model. In the data, there is no correlation between the mean and variance in allele size at a locus or between the number of alleles and mean allele size, which suggests that the mutation rate at these loci is independent of allele size.     

Likelihoods and simulation methods for a class of nonneutral population genetics models

Methods for simulating samples and sample statistics, under mutation-selection-drift equilibrium for a class of nonneutral population genetics models, and for evaluating the likelihood surface, in selection and mutation parameters, are developed and applied for observed data. The methods apply to large populations in settings in which selection is weak, in the sense that selection intensities, like mutation rates, are of the order of the inverse of the population size. General diploid selection is allowed, but the approach is currently restricted to models, such as the infinite alleles model and certain K-models, in which the type of a mutant allele does not depend on the type of its progenitor allele. The simulation methods have considerable advantages over available alternatives. No other methods currently seem practicable for approximating likelihood surfaces.     

Population Genetic Models of Genomic Imprinting

The phenomenon of genomic imprinting has recently excited much interest among experimental biologists. The population genetic consequences of imprinting, however, have remained largely unexplored. Several population genetic models are presented and the following conclusions drawn: (i) systems with genomic imprinting need not behave similarly to otherwise identical systems without imprinting; (ii) nevertheless, many of the models investigated can be shown to be formally equivalent to models without imprinting; (iii) consequently, imprinting often cannot be discovered by following allele frequency changes or examining equilibrium values; (iv) the formal equivalences fail to preserve some well known properties. For example, for populations incorporating genomic imprinting, parameter values exist that cause these populations to behave like populations without imprinting, but with heterozygote advantage, even though no such advantage is present in these imprinting populations. We call this last phenomenon "pseudoheterosis." The imprinting systems that fail to be formally equivalent to nonimprinting systems are those in which males and females are not equivalent, i.e., two-sex viability systems and sex-chromosome inactivation.     

Exact moment calculations for genetic models with migration,mutation, and drift

Using properties of moment stationarity we develop exact expressions for the mean and covariance of allele frequencies at a single locus for a set of populations subject to drift, mutation, and migration. Some general results can be obtained even for arbitrary mutation and migration matrices, for example: (1) Under quite general conditions, the mean vector depends only on mutation rates, not on migration rates or the number of populations. (2) Allele frequencies covary among all pairs of populations connected by migration. As a result, the drift, mutation, migration process is not ergodic when any finite number of populations is exchanging genes. In addition, we provide closed-form expressions for the mean and covariance of allele frequencies in Wright's finite-island model of migration under several simple models of mutation, and we show that the correlation in allele frequencies among populations can be very large for realistic rates of mutation unless an enormous number of populations are exchanging genes. As a result, the traditional diffusion approximation provides a poor approximation of the stationary distribution of allele frequencies among populations. Finally, we discuss some implications of our results for measures of population structure based on Wright's F-statistics.     

A method for detecting recent selection in the human genome from allele age estimates

Mutations that have recently increased in frequency by positive natural selection are an important component of naturally occurring variation that affects fitness. To identify such variants, we developed a method to test for recent selection by estimating the age of an allele from the extent of haplotype sharing at linked sites. Neutral coalescent simulations are then used to determine the likelihood of this age given the allele's observed frequency. We applied this method to a common disease allele, the hemochromatosis-associated HFE C282Y mutation. Our results allow us to reject neutral models incorporating plausible human demographic histories for HFE C282Y and one other young but common allele, indicating positive selection at HFE or a linked locus. This method will be useful for scanning the human genome for alleles under selection using the haplotype map now being constructed.     

Selection and mutation at a diallelic X-linked locus

Equilibria and convergence of gene frequencies are studied in the case of a diallelic X-linked locus under the influence of selection and mutation. The model used is that of an infinite diploid population with nonoverlapping discrete generations and random mating. It is proved that if the mutation rates and fitnesses are constant and the mutation rates are less than one-third, then global convergence of gene frequencies to equilibria occurs. The phase portraits of the dynamical system describing the change of allelic frequencies from one generation to the next are determined. Convergence of gene frequencies is monotone from a certain generation on if every other generation is skipped. In the case without mutation, our proof of this monotone convergence simplifies G. Palm's original proof [37].     

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.     

A two-locus mutation—Selection model and some of its evolutionary implications

The deterministic properties of a two-locus model with mutation and selection have been investigated. The mutation process is unidirectional, and the model is so constructed that the genetic variation at one locus is selectively neutral in the absence of a mutant allele at the other locus. All genotypes with three or four mutant alleles are deleterious, while the double heterozygotes may have the same fitness as the standard genotype. If one of the mutant alleles becomes fixed in the population, then the other locus will show a regular one-locus mutation-selection balance. Such a boundary equilibrium may be unstable or stable in the full two-locus setting. In the symmetric case, which is analyzed in details, the population will either go to one of the two boundary equilibria, or to a fully polymorphic equilibrium at which both the mutant alleles are rare. The origin of reproductive separation between two populations via the fixation of complementary deleterious mutants at different loci, and the fixation of nonfunctional alleles at duplicated loci, are two biological processes which both can be studied with the present model. In the last part of the paper we show how the results from the deterministic analysis can be used to predict how different factors will influence the rates of evolution in these systems.     

Gene interactions in the evolution of genomic imprinting

Numerous evolutionary theories have been developed to explain the epigenetic phenomenon of genomic imprinting. Here, we explore a subset of theories wherein non-additive genetic interactions can favour imprinting. In the simplest genic interaction—the case of underdominance—imprinting can be favoured to hide effectively low-fitness heterozygous genotypes; however, as there is no asymmetry between maternally and paternally inherited alleles in this model, other means of enforcing monoallelic expression may be more plausible evolutionary outcomes than genomic imprinting. By contrast, more successful interaction models of imprinting rely on an asymmetry between the maternally and paternally inherited alleles at a locus that favours the silencing of one allele as a means of coordinating the expression of high-fitness allelic combinations. For example, with interactions between autosomal loci, imprinting functionally preserves high-fitness genotypes that were favoured by selection in the previous generation. In this scenario, once a focal locus becomes imprinted, selection at interacting loci favours a matching imprint. Uniparental transmission generates similar asymmetries for sex chromosomes and cytoplasmic factors interacting with autosomal loci, with selection favouring the expression of either maternal or paternally derived autosomal alleles depending on the pattern of transmission of the uniparentally inherited factor. In a final class of models, asymmetries arise when genes expressed in offspring interact with genes expressed in one of its parents. Under such a scenario, a locus evolves to have imprinted expression in offspring to coordinate the interaction with its parent''s genome. We illustrate these models and explore key links and differences using a unified framework.     

Analysis of somatic mutations at human minisatellite loci in tumors and cell lines

Hypervariable human minisatellite loci show a substantial level of germline instability, and spontaneous mutation rates to new length alleles have been measured directly by pedigree analysis. We now show that mutation events altering the number of minisatellite repeat units are not restricted to the germline, but also arise in other tissues. Mutant alleles can be detected at a very low frequency in lymphoblastoid cell lines and at much higher frequencies in clonal tumor cell populations, most particularly in gastrointestinal adenocarcinomas. Mutant alleles in these tumors are usually present at a dosage equal to or greater than that of the progenitor allele, indicating that most or all of the tumor cells carry the same clonally derived mutant allele. As with germline mutation, the incidence of somatic mutations in tumors varies from locus to locus, with the same locus showing the highest level of germline and somatic instability. Most length changes, as those in the germline, are of only a few repeat units; however, very large changes are also observed, implying that such mutations can occur in the absence of meiosis.