Estimating the age of alleles by use of intraallelic variability.

A method is presented for estimating the age of an allele by use of its frequency and the extent of variation among different copies. The method uses the joint distribution of the number of copies in a population sample and the coalescence times of the intraallelic gene genealogy conditioned on the number of copies. The linear birth-death process is used to approximate the dynamics of a rare allele in a finite population. A maximum-likelihood estimate of the age of the allele is obtained by Monte Carlo integration over the coalescence times. The method is applied to two alleles at the cystic fibrosis (CFTR) locus, deltaF508 and G542X, for which intraallelic variability at three intronic microsatellite loci has been examined. Our results indicate that G542X is somewhat older than deltaF508. Although absolute estimates depend on the mutation rates at the microsatellite loci, our results support the hypothesis that deltaF508 arose < 500 generations (approximately 10,000 years) ago.     

Fixation probability in a two-locus model by the ancestral recombination-selection graph

We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.     

A Bayesian method for jointly estimating allele age and selection intensity.

The problem of jointly estimating the intensity of past selection affecting an allele and the allele's age is formulated in a Bayesian framework. The prior distribution of allele age given its frequency is obtained from existing population genetics theory. The prior distribution of selection intensity is assumed to reflect the fact that positive selection on a new mutant is more likely to be weak than strong. The general approach is illustrated by the development of an importance sampling method applicable to low-frequency alleles. This method can be used either when the haplotypes of closely linked marker loci are known or when the lengths of linked ancestral chromosomal segments can be inferred. The method is illustrated with an application to the A-allele of G6PD in Africa. Because changes in allele frequency and recombination are both intrinsically stochastic, there are limits to the accuracy achievable with any method.     

Non-equilibrium theory of the allele frequency spectrum

A forward diffusion equation describing the evolution of the allele frequency spectrum is presented. The influx of mutations is accounted for by imposing a suitable boundary condition. For a Wright-Fisher diffusion with or without selection and varying population size, the boundary condition is lim(x downward arrow0)xf(x,t)=thetarho(t), where f(.,t) is the frequency spectrum of derived alleles at independent loci at time t and rho(t) is the relative population size at time t. When population size and selection intensity are independent of time, the forward equation is equivalent to the backwards diffusion usually used to derive the frequency spectrum, but this approach allows computation of the time dependence of the spectrum both before an equilibrium is attained and when population size and selection intensity vary with time. From the diffusion equation, a set of ordinary differential equations for the moments of f(.,t) is derived and the expected spectrum of a finite sample is expressed in terms of those moments. The use of the forward equation is illustrated by considering neutral and selected alleles in a highly simplified model of human history. For example, it is shown that approximately 30% of the expected total heterozygosity of neutral loci is attributable to mutations that arose since the onset of population growth in roughly the last 150,000 years.     

The Hill-Robertson effect and the evolution of recombination

In finite populations, genetic drift generates interference between selected loci, causing advantageous alleles to be found more often on different chromosomes than on the same chromosome, which reduces the rate of adaptation. This "Hill-Robertson effect" generates indirect selection to increase recombination rates. We present a new method to quantify the strength of this selection. Our model represents a new beneficial allele (A) entering a population as a single copy, while another beneficial allele (B) is sweeping at another locus. A third locus affects the recombination rate between selected loci. Using a branching process model, we calculate the probability distribution of the number of copies of A on the different genetic backgrounds, after it is established but while it is still rare. Then, we use a deterministic model to express the change in frequency of the recombination modifier, due to hitchhiking, as A goes to fixation. We show that this method can give good estimates of selection for recombination. Moreover, it shows that recombination is selected through two different effects: it increases the fixation probability of new alleles, and it accelerates selective sweeps. The relative importance of these two effects depends on the relative times of occurrence of the beneficial alleles.     

Disequilibrium Pattern Analysis. I. Theory

We have developed a method, disequilibrium pattern analysis, for examining the disequilibrium distribution of the entire array of two locus multiallelic haplotypes in a population. It is shown that a selected haplotype will produce a distinct pattern of linkage disequilibrium values for all generations while the selection is acting. This pattern will also presumably be maintained for many generations after the selection event, until the disequilibrium pattern is eventually broken down by genetic drift and recombination. Related haplotypes, sharing an allele with a selected haplotype, assume a value of linkage disequilibrium proportional to the frequency of the unshared allele and have a single negative value of the normalized linkage disequilibrium. The analysis assumes zero linkage disequilibrium for all allelic combinations initially. The same basic results continue to apply if the selection involves a new mutant, the occurrence of which creates linkage disequilibrium for some haplotypes. The disequilibrium pattern predicted under selection is robust with respect to the influence of migration and random genetic drift. This method is applicable to population data having linked polymorphic loci including that determined from protein or DNA sequencing.     

Modifiers of mutation rate: a general reduction principle

A deterministic two-locus population genetic model with random mating is studied. The first locus, with two alleles, is subject to mutation and arbitrary viability selection. The second locus, with an arbitrary number of alleles, controls the mutation at the first locus. A class of viability-analogous Hardy-Weinberg equilibria is analyzed in which the selected gene and the modifier locus are in linkage equilibrium. It is shown that at these equilibria a reduction principle for the success of new mutation-modifying alleles is valid. A new allele at the modifier locus succeeds if its marginal average mutation rate is less than the mean mutation rate of the resident modifier allele evaluated at the equilibrium. Internal stability properties of these equilibria are also described.     

Polygenic Mutation in Drosophila Melanogaster: Genetic Interactions between Selection Lines and Candidate Quantitative Trait Loci

We have investigated genetic interactions between spontaneous mutations affecting abdominal and sternopleural bristle number that have accumulated in 12 long-term selection lines derived from an inbred strain, and mutations at 14 candidate bristle number quantitative trait loci. The quantitative test for complementation was to cross the selection lines to an inbred wild-type strain (the control cross) and to a derivative of the control strain into which the mutant allele at the candidate locus to be tested was substituted (the tester strain). Genetic interactions between spontaneous mutations affecting bristle number and the candidate locus mutations were common, and in several cases the interaction effects were different in males and females. Analyses of variance of the (tester - control) differences among and within groups of replicate lines selected in the same direction for the same trait showed significant group effects for several candidate loci. Genetically, the interactions could be caused by allelism of, and/or epistasis between, spontaneous mutations in the selection lines and the candidate locus mutations. It is possible that much of the response to selection was from new mutations at candidate bristle number quantitative trait loci, and that for some of these loci, mutation rates were high.     

The Sampling Distribution of Disease-Associated Alleles

A theory is developed that provides the sampling distribution of low frequency alleles at a single locus under the assumption that each allele is the result of a unique mutation. The numbers of copies of each allele is assumed to follow a linear birth-death process with sampling. If the population is of constant size, standard results from theory of birth-death processes show that the distribution of numbers of copies of each allele is logarithmic and that the joint distribution of numbers of copies of k alleles found in a sample of size n follows the Ewens sampling distribution. If the population from which the sample was obtained was increasing in size, if there are different selective classes of alleles, or if there are differences in penetrance among alleles, the Ewens distribution no longer applies. Likelihood functions for a given set of observations are obtained under different alternative hypotheses. These results are applied to published data from the BRCA1 locus (associated with early onset breast cancer) and the factor VIII locus (associated with hemophilia A) in humans. In both cases, the sampling distribution of alleles allows rejection of the null hypothesis, but relatively small deviations from the null model can account for the data. In particular, roughly the same population growth rate appears consistent with both data sets.     

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.     

Conditional ancestral times and related properties of an extant mutant at a locus A given the number of segregating sites observed at a linked locus B

 It is shown how the mean ancestral times at one locus are affected in a two- locus model with recombination when information is given regarding the number of segregating sites at another locus. For samples of n genes, recursive equations are derived that describe precisely the evolution of the time-depth of such a linked genealogy. Exact numerical solutions and Markov chain Monte Carlo simulations are discussed and compared. The dependence of some properties of a singleton mutation on waiting times between events in the two-locus genealogy is quantified and illustrates the effect of recombination on these properties. The following cases are presented: (1) the distribution of the number of mutant genes in a sample arising from a singleton mutation; (2) the probability that an allele observed in a genes of a sample of size n is the ancestral type (the oldest); (3) the expectation and variance of the age of a mutant having b copies in a sample of n genes. Received: 1 September 2000 / Revised version: 1 October 2001 / Published online: 8 May 2002     

Invasion fitness,inclusive fitness,and reproductive numbers in heterogeneous populations

How should fitness be measured to determine which phenotype or “strategy” is uninvadable when evolution occurs in a group‐structured population subject to local demographic and environmental heterogeneity? Several fitness measures, such as basic reproductive number, lifetime dispersal success of a local lineage, or inclusive fitness have been proposed to address this question, but the relationships between them and their generality remains unclear. Here, we ascertain uninvadability (all mutant strategies always go extinct) in terms of the asymptotic per capita number of mutant copies produced by a mutant lineage arising as a single copy in a resident population (“invasion fitness”). We show that from invasion fitness uninvadability is equivalently characterized by at least three conceptually distinct fitness measures: (i) lineage fitness, giving the average individual fitness of a randomly sampled mutant lineage member; (ii) inclusive fitness, giving a reproductive value weighted average of the direct fitness costs and relatedness weighted indirect fitness benefits accruing to a randomly sampled mutant lineage member; and (iii) basic reproductive number (and variations thereof) giving lifetime success of a lineage in a single group, and which is an invasion fitness proxy. Our analysis connects approaches that have been deemed different, generalizes the exact version of inclusive fitness to class‐structured populations, and provides a biological interpretation of natural selection on a mutant allele under arbitrary strength of selection.     

Mutation-Selection Balance under Genomic Imprinting at an Autosomal Locus

I model the effect of genomic imprinting on the equilibrium allele frequencies at an autosomal diallelic locus subject to viability selection and mutation. The population size is assumed to be very large; male and female mutation rates may be unequal. Different models examine cases of the inactivation of one gene (with both complete and partial penetrance) and of differential expression of genes according to the parent of origin. In the simplest cases the frequency of the deleterious allele is approximately twice that of a dominant nonimprinting mutant, but considerably less than that of a recessive nonimprinting mutant. Under imprinting, selection and unequal mutation rates interact: other things being equal, male-biased mutation leads to lower mutant frequencies under maternal imprinting and higher frequencies under paternal imprinting. I also model cases where just one allele is imprintable (and the other not). These models allow us to predict the frequency of a failure to imprint in a normally imprinting system, as well as the frequency of imprinting at a standard nonimprinting locus.     

Duality,ancestral and diffusion processes in models with selection

The ancestral selection graph in population genetics was introduced by Krone and Neuhauser [Krone, S.M., Neuhauser, C., 1997. Ancestral process with selection. Theor. Popul. Biol. 51, 210–237] as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura [Kimura, M., 1955. Stochastic process and distribution of gene frequencies under natural selection. Cold Spring Harbor Symposia on Quantitative Biology 20, 33–53]. It is shown that the process of fixation of the allele in the diffusion model corresponds to convergence of the ancestral process to its stationary measure. The time to fixation of the allele conditional on fixation is studied in terms of the ancestral process.     

Population genetics of modifiers of meiotic drive. I. The solution of a special case and some general implications

This is a study of the formal population genetics of a two locus model where the alleles at one locus are subject to meiotic drive and zygotic selection and the only effect of the other locus is the modification of drive intensity. A complete analytic solution is obtained for a biologically reasonable special case. It is then argued, partially with the aid of computer analysis, that with moderate relaxation of assumptions of the special case, the conclusions derived from that case still hold. These conclusions are that if there is linkage a stable two locus polymorphism can result. There is permanent linkage disequilibrium with the loosing allele at the drive locus in coupling with the suppressor allele at the modifier locus, and the driven allele coupled with the modifier allele which enhances drive. It is suggested that this result explains how the SD system in Drosophila maintains its integrity in natural populations.     

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.     

Cloning, mapping and molecular analysis of the pyrG (orotidine-5'-phosphate decarboxylase) gene of Aspergillus nidulans

We have modified the transformation procedures of Ballance et al. [Biochem. Biophys. Res. Commun. 112 (1983) 284-289] to give increased rates of transformation in Aspergillus nidulans. With the modified procedures we have been able to complement pyrG89, a mutation in the orotidine-5'-phosphate decarboxylase gene of A. nidulans, by transformation with a library of wild-type (wt) sequences in pBR329. We have recovered, by marker rescue from one such transformant, a plasmid (pJR15) that carries an A. nidulans sequence that complements pyrG89 efficiently. In three experiments, this plasmid gave an average of 1985 stable transformants/micrograms of transforming DNA. We have analyzed ten of these genetically and by Southern hybridization. In five transformants a single copy of the transforming plasmid had integrated at the pyrG locus, in one transformant several copies of pJR15 had integrated at this locus, in one transformant several copies of the plasmid had integrated into other sites, and in three transformants, the wt allele had apparently replaced the mutant allele with no integration of pBR329 sequences. Sequence and S1 nuclease protection analysis revealed that pJR15 contains a gene that predicts an amino acid sequence with regions of strong homology to the orotidine-5'-phosphate decarboxylases of Neurospora crassa and Saccharomyces cerevisiae. We conclude that this gene is the wt pyrG allele. Finally, we have compared the 5'- and 3'-noncoding sequences and intron splice sequences to other genes of A. nidulans and have mapped the pyrG locus to a region between the fpaB and galD loci on linkage group I.     

Selection at the Esterase-2 Locus of Drosophila buzzatii? Perturbation-Reperturbation Experiments

Apparent selection affecting starch gel electrophoretic alleles at the Esterase-2 locus of Drosophila buzzatii has been detected in laboratory and natural populations. Perturbation-reperturbation of allele frequencies in replicated laboratory populations attempts to test direct selective effects at the locus versus effects of linked loci. Sequential gel electrophoresis has identified more alleles within starch classes, and three of these alleles (within the a, b and c starch alleles) were used in cage population experiments. Allele a/1.00/1.00/1.00 was set up in 10 replicate populations with allele c/1.00/1.00/1.00, and in an independent 10 replicate populations with allele b/0.99/1.01/1.00. For each set, three reperturbations were done. Replicate populations generally showed similar patterns of allele frequency change and clear directionality: effects of selection, not drift. However, four populations deviated from their replicates, indicating dissipation of linkage disequilibrium. Estimates of pre-adult viability in the F₂ of pair-wise crosses among 12 sequential gel electrophoretic alleles showed very variable modes of inheritance and relative viability fitnesses. Together with the diversity of patterns of allele frequency change in the cage populations, these results suggest a gene complex, with selection acting on an interacting set of loci which may include Esterase-2.