Selection in a subdivided population with local extinction and recolonization

In a subdivided population, local extinction and subsequent recolonization affect the fate of alleles. Of particular interest is the interaction of this force with natural selection. The effect of selection can be weakened by this additional source of stochastic change in allele frequency. The behavior of a selected allele in such a population is shown to be equivalent to that of an allele with a different selection coefficient in an unstructured population with a different size. This equivalence allows use of established results for panmictic populations to predict such quantities as fixation probabilities and mean times to fixation. The magnitude of the quantity N(e)s(e), which determines fixation probability, is decreased by extinction and recolonization. Thus deleterious alleles are more likely to fix, and advantageous alleles less likely to do so, in the presence of extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

Ages,extinction times,and first passage probabilities for a multiallele diffusion model with irreversible mutation

We derive some new results for diffusion models in population genetics of the “infinite-alleles” type. Assuming present allelic frequencies known, we find expected values for times which may be interpreted either as the age of the kth oldest allele or the time to extinction of the “kth-to-last” allele to be lost. We also find some conditional expectations and probabilities related to the order of extinction or creation of the alleles. Computations illustrate the dependence of the expected times on the mutation rate.     

The Exact Test for Cytonuclear Disequilibria

We extend the analysis of the statistical properties of cytonuclear disequilibria in two major ways. First, we develop the asymptotic sampling theory for the nonrandom associations between the alleles at a haploid cytoplasmic locus and the alleles and genotypes at a diploid nuclear locus, when there are an arbitrary number of alleles at each marker. This includes the derivation of the maximum likelihood estimators and their sampling variances for each disequilibrium measure, together with simple tests of the null hypothesis of no disequilibrium. In addition to these new asymptotic tests, we provide the first implementation of Fisher's exact test for the genotypic cytonuclear disequilibria and some approximations of the exact test. We also outline an exact test for allelic cytonuclear disequilibria in multiallelic systems. An exact test should be used for data sets when either the marginal frequencies are extreme or the sample size is small. The utility of this new sampling theory is illustrated through applications to recent nuclear-mtDNA and nuclear-cpDNA data sets. The results also apply to population surveys of nuclear loci in conjunction with markers in cytoplasmically inherited microorganisms.     

Extinction and fixation times with dominance and inbreeding

The effect of partial inbreeding on extinction and fixation times of a selected allele with partial dominance is studied using a diffusion model. Asymptotic approximations are obtained for large populations and the accuracy of the approximations was found to increase with inbreeding level. They show that inbreeding reduces extinction and fixation times compared to random mating at least by a factor 1+F, where F is Wright’s fixation index. The reduction of extinction and fixation times due to inbreeding is stronger for strong selection and if alleles are either highly recessive or highly dominant. This bears implications for the effect of inbreeding on the signature of selective sweeps. These findings extend previous results obtained for random mating populations and help clarifying previous simulation and numerical results on the effect of inbreeding on the dynamics of selected alleles.     

Selection, subdivision and extinction and recolonization

In a subdivided population, the interaction between natural selection and stochastic change in allele frequency is affected by the occurrence of local extinction and subsequent recolonization. The relative importance of selection can be diminished by this additional source of stochastic change in allele frequency. Results are presented for subdivided populations with extinction and recolonization where there is more than one founding allele after extinction, where these may tend to come from the same source deme, where the number of founding alleles is variable or the founders make unequal contributions, and where there is dominance for fitness or local frequency dependence. The behavior of a selected allele in a subdivided population is in all these situations approximately the same as that of an allele with different selection parameters in an unstructured population with a different size. The magnitude of the quantity N(e)s(e), which determines fixation probability in the case of genic selection, is always decreased by extinction and recolonization, so that deleterious alleles are more likely to fix and advantageous alleles less likely to do so. The importance of dominance or frequency dependence is also altered by extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

The genetic architecture of adaptation under migration-selection balance

Many ecologically important traits have a complex genetic basis, with the potential for mutations at many different genes to shape the phenotype. Even so, studies of local adaptation in heterogeneous environments sometimes find that just a few quantitative trait loci (QTL) of large effect can explain a large percentage of observed differences between phenotypically divergent populations. As high levels of gene flow can swamp divergence at weakly selected alleles, migration-selection-drift balance may play an important role in shaping the genetic architecture of local adaptation. Here, we use analytical approximations and individual-based simulations to explore how genetic architecture evolves when two populations connected by migration experience stabilizing selection toward different optima. In contrast to the exponential distribution of allele effect sizes expected under adaptation without migration (Orr 1998), we find that adaptation with migration tends to result in concentrated genetic architectures with fewer, larger, and more tightly linked divergent alleles. Even if many small alleles contribute to adaptation at the outset, they tend to be replaced by a few large alleles under prolonged bouts of stabilizing selection with migration. All else being equal, we also find that stronger selection can maintain linked clusters of locally adapted alleles over much greater map distances than weaker selection. The common empirical finding of QTL of large effect is shown to be expected with migration in a heterogeneous landscape, and these QTL may often be composed of several tightly linked alleles of smaller effect.     

Allelic genealogies in sporophytic self-incompatibility systems in plants.

Expectations for the time scale and structure of allelic genealogies in finite populations are formed under three models of sporophytic self-incompatibility. The models differ in the dominance interactions among the alleles that determine the self-incompatibility phenotype: In the SSIcod model, alleles act codominantly in both pollen and style, in the SSIdom model, alleles form a dominance hierarchy, and in SSIdomcod, alleles are codominant in the style and show a dominance hierarchy in the pollen. Coalescence times of alleles rarely differ more than threefold from those under gametophytic self-incompatibility, and transspecific polymorphism is therefore expected to be equally common. The previously reported directional turnover process of alleles in the SSIdomcod model results in coalescence times lower and substitution rates higher than those in the other models. The SSIdom model assumes strong asymmetries in allelic action, and the most recessive extant allele is likely to be the most recent common ancestor. Despite these asymmetries, the expected shape of the allele genealogies does not deviate markedly from the shape of a neutral gene genealogy. The application of the results to sequence surveys of alleles, including interspecific comparisons, is discussed.     

The number of lines of descent and fixation probabilities of alleles in the pure genetic drift process: analytical approximations

Analysis of diversity between populations diverged from an evolutionarily small number of generations cannot be done under the assumption that allele frequencies reflect an equilibrium between genetic drift and mutations. An alternative to analysis through coalescence theory is proposed in this situation by developing analytical approximations. Using a Poisson approximation of its distribution, the expected number of genes from one generation whose copies make up the population after a number of generations characterized by fixation index F can be shown to be approximated by 2/F-1, irrespective of population size, and probabilities of fixation of alleles over a finite period of time can be also approximated. These expressions, which were checked numerically, should make it possible to calculate approximate likelihoods for allele frequency distributions promoted by drift.     

Properties of a neutral allele model with intragenic recombination

An infinite-site neutral allele model with crossing-over possible at any of an infinite number of sites is studied. A formula for the variance of the number of segregating sites in a sample of gametes is obtained. An approximate expression for the expected homozygosity is also derived. Simulation results are presented to indicate the accuracy of the approximations. The results concerning the number of segregating sites and the expected homozygosity indicate that a two-locus model and the infinite-site model behave similarly for 4Nu less than or equal to 2 and r less than or equal to 5u, where N is the population size, u is the neutral mutation rate, and r is the recombination rate. Simulations of a two-locus model and a four-locus model were also carried out to determine the effect of intragenic recombination on the homozygosity test of Watterson (Genetics 85, 789-814; 88, 405-417) and on the number of unique alleles in a sample. The results indicate that for 4Nu less than or equal to 2 and r less than or equal to 10u, the effect of recombination is quite small.     

Fixation of advantageous alleles in partially self-fertilizing populations. The effect of different selection modes

The expected fixation probability of an advantageous allele was examined in a partially self-fertilizing hermaphroditic plant species using the diffusion approximation. The selective advantage of the advantageous allele was assumed to be increased viability, increased fecundity, or an increase in male fitness. The mode of selection, as well as the selfing rate, the population size, and the dominance of the advantageous allele, affect the fixation probability of the allele. In general it was found that increases in selfing rate decrease the fixation probability under male sexual selection, increase fixation probability under fecundity selection, and increase when recessive and decrease when dominant under viability selection. In some cases the highest fixation probability of advantageous alleles under fecundity or under male sexual selection occurred at an intermediary selfing rate. The expected mean fixation times of the advantageous allele were also examined using the diffusion approximation.     

Power calculations for the transmission/disequilibrium and affected sib pair tests using elementary probability methods

The transmission/disequilibrium test (TDT) and the affected sib pair test (ASP) both test for the association of a marker allele with some conditions. Here, we present methods for calculating the probability of detecting the association (power) for a study examining a fixed number of families for suitability for the study and for calculating the number of such families to be examined. Both calculations use a genetic model for the association. The model considered posits a bi-allelic marker locus that is linked to a bi-allelic disease locus with a possibly nonzero recombination fraction between the loci. The penetrance of the disease is an increasing function of the number of disease alleles. The TDT tests whether the transmission by a heterozygous parent of a particular allele at a marker locus to an affected offspring occurs with probability greater than 0.5. The ASP tests whether transmission of the same allele to two affected sibs occurs with probability greater than 0.5. In either case, evidence that the probability is greater than 0.5 is evidence for association between the marker and the disease. Study inclusion criteria (IC) can greatly affect the necessary sample size of a TDT or ASP study. IC considered by us include a randomly selected parent at least one parent or both parents required to be heterozygous. It also allows a specified minimum number of affected offspring to be required (TDT only). We use elementary probability calculations rather than complex mathematical manipulations or asymptotic methods (large sample size approximations) to compute power and requisite sample size for a proposed study. The advantages of these methods are simplicity and generality.     

Dominance and the maintenance of polymorphism in multiallelic migration-selection models with two demes

The maintenance of genetic variation in a spatially heterogeneous environment has been one of the main research themes in theoretical population genetics. Despite considerable progress in understanding the consequences of spatially structured environments on genetic variation, many problems remain unsolved. One of them concerns the relationship between the number of demes, the degree of dominance, and the maximum number of alleles that can be maintained by selection in a subdivided population. In this work, we study the potential of maintaining genetic variation in a two-deme model with deme-independent degree of intermediate dominance, which includes absence of G×E interaction as a special case. We present a thorough numerical analysis of a two-deme three-allele model, which allows us to identify dominance and selection patterns that harbor the potential for stable triallelic equilibria. The information gained by this approach is then used to construct an example in which existence and asymptotic stability of a fully polymorphic equilibrium can be proved analytically. Noteworthy, in this example the parameter range in which three alleles can coexist is maximized for intermediate migration rates. Our results can be interpreted in a specialist-generalist context and (among others) show when two specialists can coexist with a generalist in two demes if the degree of dominance is deme independent and intermediate. The dominance relation between the generalist allele and the specialist alleles play a decisive role. We also discuss linear selection on a quantitative trait and show that G×E interaction is not necessary for the maintenance of more than two alleles in two demes.     

Approximations for expected generation number

A deterministic formula is commonly used to approximate the expected generation number of a population of growing cells. However, this can give misleading results because it does not allow for natural variation in the times that individual cells take to reproduce. Here we present more accurate approximations for both symmetric and asymmetric cell division. Based on the first two moments of the generation time distribution, these approximations are also robust. We illustrate the improved approximations using data that arise from monitoring individual yeast cells under a microscope and also demonstrate how the approximations can be used when such detailed data are not available.     

The robustness of Hamilton's rule with inbreeding and dominance: kin selection and fixation probabilities under partial sib mating

Assessing the validity of Hamilton's rule when there is both inbreeding and dominance remains difficult. In this article, we provide a general method based on the direct fitness formalism to address this question. We then apply it to the question of the evolution of altruism among diploid full sibs and among haplodiploid sisters under inbreeding resulting from partial sib mating. In both cases, we find that the allele coding for altruism always increases in frequency if a condition of the form rb>c holds, where r depends on the rate of sib mating alpha but not on the frequency of the allele, its phenotypic effects, or the dominance of these effects. In both examples, we derive expressions for the probability of fixation of an allele coding for altruism; comparing these expressions with simulation results allows us to test various approximations often made in kin selection models (weak selection, large population size, large fecundity). Increasing alpha increases the probability of fixation of recessive altruistic alleles (h<1/2), while it can increase or decrease the probability of fixation of dominant altruistic alleles (h>1/2).     

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.     

‘True’ null allele detection in microsatellite loci: a comparison of methods,assessment of difficulties and survey of possible improvements

Null alleles are alleles that for various reasons fail to amplify in a PCR assay. The presence of null alleles in microsatellite data is known to bias the genetic parameter estimates. Thus, efficient detection of null alleles is crucial, but the methods available for indirect null allele detection return inconsistent results. Here, our aim was to compare different methods for null allele detection, to explain their respective performance and to provide improvements. We applied several approaches to identify the ‘true’ null alleles based on the predictions made by five different methods, used either individually or in combination. First, we introduced simulated ‘true’ null alleles into 240 population data sets and applied the methods to measure their success in detecting the simulated null alleles. The single best‐performing method was ML‐NullFreq_frequency. Furthermore, we applied different noise reduction approaches to improve the results. For instance, by combining the results of several methods, we obtained more reliable results than using a single one. Rule‐based classification was applied to identify population properties linked to the false discovery rate. Rules obtained from the classifier described which population genetic estimates and loci characteristics were linked to the success of each method. We have shown that by simulating ‘true’ null alleles into a population data set, we may define a null allele frequency threshold, related to a desired true or false discovery rate. Moreover, using such simulated data sets, the expected null allele homozygote frequency may be estimated independently of the equilibrium state of the population.     

Between‐year variation of MHC allele frequencies in great reed warblers: selection or drift?

The major histocompatibility complex (MHC) genes are extremely polymorphic and this variation is assumed to be maintained by balancing selection. Cyclic interactions between pathogens and their hosts could generate such selection, and specific MHC alleles or heterozygosity at certain MHC loci have been shown to confer resistance against particular pathogens. Here we compare the temporal variation in allele frequencies of 23 MHC class I alleles with that of 23 neutral microsatellite markers in adult great reed warblers (a passerine bird) in nine successive cohorts. Overall, the MHC alleles showed a significantly higher variation in allele frequencies between cohorts than the microsatellite alleles, using a multi-variate genetic analysis (amova). The frequency of two specific MHC alleles, A3e (P = 0.046) and B4b (P = 0.0018), varied more between cohorts than expected from random, whereas none of the microsatellite alleles showed fluctuations exceeding the expectation from stochastic variation. These results imply that the variation in MHC allele frequencies between cohorts is not a result of demographic events, but rather an effect of selection favouring different MHC alleles in different years.     

Allele frequency of VNTR locus D1S80 observed in Hb D-Los Angeles carrires

One of our previous studies presented the allele frequencies of D1S80 VNTR locus in province Denizli including the high frequencies of allele 24 and 18. In Denizli province of Turkey, the most common abnormal variant is Hb D-Los Angeles with a frequency of 57.8?% of the total abnormal Hbs. The aim of this study is to identify the allele frequencies of D1S80 VNTR locus in Hb D-Los Angeles carriers in Denizli province of Turkey. We studied unrelated 36 Hb D-Los Angeles carriers residing in Denizli province of Turkey. The size range of the D1S80 VNTR locus PCR products was determined first by agarose gel electrophoresis and then by a capillary electrophoresis system. For all subjects, DNA sequencing was performed. Allele frequency, theta (k) value, and observed and expected heterozygosity were calculated using Arlequin Software version 3.11. The most common alleles were the 24 (32?%), 18 (18.1?%) and 29 (16.7?%) alleles, and frequencies of these alleles were 0.329, 0.186 and 0.171 respectively. Other observed alleles percentages were 33, 2?%. We did not observe alleles 6, 15, 27 and 35, but we observed alleles 20 and 33. Results were in Hardy–Weinberg linkage disequilibrium. Observed heterozygosity was 0.889, and expected heterozygosity was 0.847. Theta (k) value was 4.91 (95?% confidence interval limits). According to our results, we concluded that Hb D-Los Angeles carriers have different allele frequencies in D1S80 VNTR and also have their own D1S80 VNTR locus divergence.     

Assignment of allelic configuration in polyploids using the MAC-PR (microsatellite DNA allele counting—peak ratios) method

Polysomic inheritance frequently results in the simultaneous occurrence of several microsatellite DNA alleles on a single locus. The MAC-PR (microsatellite DNA allele counting—peak ratios) method was recently developed for the analysis of polyploid plants and makes use of the quantitative values for microsatellite allele peak areas. To date, this approach has only been used in plants with known genetic relationships. We report here the application of MAC-PR for the first time to random samples of unknown pedigrees. We analysed six microsatellite loci using a set of tetraploid ornamental rose (Rosa × hybrida L.) varieties. For each locus, all alleles were analysed in pairwise combinations in order to determine their copy number in the individual samples. This was accomplished by calculating the ratios between the peak areas for two alleles in all of the samples where these two alleles occurred together. The allele peak ratios observed were plotted in a histogram, and those histograms that produced at least two well-separated groups were selected for further analysis. Mean allelic peak ratio values for these groups were compared to the relationships expected between alleles in hypothetical configurations of the locus investigated. Using this approach, we were able to assign precise allelic configurations (the actual genotype) to almost all of the varieties analysed for five of the six loci investigated. MAC-PR also appears to be a very effective tool for detecting null alleles in polyploid species.Communicated by C. Möllers     

The Genetic Basis of Phenotypic Adaptation I: Fixation of Beneficial Mutations in the Moving Optimum Model

We study the genetic basis of adaptation in a moving optimum model, in which the optimal value for a quantitative trait increases over time at a constant rate. We first analyze a one-locus two-allele model with recurrent mutation, for which we derive accurate analytical approximations for (i) the time at which a previously deleterious allele becomes beneficial, (ii) the waiting time for a successful new mutation, and (iii) the time the mutant allele needs to reach fixation. On the basis of these results, we show that the shortest total time to fixation is for alleles with intermediate phenotypic effect. We derive an approximation for this “optimal” effect, and we show that it depends in a simple way on a composite parameter, which integrates the ecological parameters and the genetic architecture of the trait. In a second step, we use stochastic computer simulations of a multilocus model to study the order in which mutant alleles with different effects go to fixation. In agreement with the one-locus results, alleles with intermediate effect tend to become fixed earlier than those with either small or large effects. However, the effect size of the fastest mutations differs from the one predicted in the one-locus model. We show how these differences can be explained by two specific effects of multilocus genetics. Finally, we discuss our results in the light of three relevant timescales acting in the system—the environmental, mutation, and fixation timescales—which define three parameter regimes leading to qualitative differences in the adaptive substitution pattern.