Statistics of natural populations. III. Sequential sampling plans for the estimation of gene frequencies

In previous work several models have been developed for genetic surveys of natural populations. Parents of unknown genotype are collected from a natural population, polymorphic at a single genetic locus. From each of these N cryptic parents a number of offspring are identified for their genotype. Our problem is to select an efficient offspring sampling plan for estimating the frequency of an allele in the cryptic adult population based on the N family profiles of juvenile genotypes. A criterion called the information per unit cost of observation is introduced to evaluate sequential sampling plans, in which the number of offspring per family examined is random. Some simple, practical schemes for stopping the sampling of offspring from a collected parent are introduced; one example is stopping when: (i) the offspring are definitive about the parental genotype(s) for the first time; (ii) a fixed number of one genotype only is seen; or (iii) a fixed maximum feasible number of offspring have been genotyped. This sampling scheme is recommended. For each sampling scheme, the best linear unbiased estimator and the sequential maximum likelihood estimator of the allele frequency are characterized. From the moments of these estimators, it is then possible to tabulate efficient sequential sampling plans, which are better (in the sense of information per unit cost), just as simple, and less costly than corresponding fixed sampling plans in use.     

Using maximum likelihood to estimate population size from temporal changes in allele frequencies.

We develop a maximum-likelihood framework for using temporal changes in allele frequencies to estimate the number of breeding individuals in a population. We use simulations to compare the performance of this estimator to an F-statistic estimator of variance effective population size. The maximum-likelihood estimator had a lower variance and smaller bias. Taking advantage of the likelihood framework, we extend the model to include exponential growth and show that temporal allele frequency data from three or more sampling events can be used to test for population growth.     

A correction for allele frequency estimates derived from isofemale lines

Isofemale lines are commonly used inDrosophila and other genera for the purpose of assaying genetic variation. Isofemale lines can be kept in the laboratory for many generations before genetic work is carried out, and permit the confirmation of newly discovered alleles. A problem not realized by many workers is that the commonly used estimate of allele frequency from these lines is biased. This estimation bias occurs at all times after the first laboratory generation, regardless of whether single individuals or pooled samples are used in each well of an electrophoretic gel. This bias can potentially affect the estimation of population genetic parameters, and in the case of rare allele analysis it can cause gross overestimates of gene flow. This paper provides a correction for allele frequency estimates derived from isofemale lines for any time after the lines are established in the laboratory. When pooled samples are used, this estimator performs better than the standard estimator at all times after the first generation. The estimator is also insensitive to multiple inseminations. After the lines have drifted oneN _e generations, multiple inseminations actually make the new estimator perform better than it does in singly inseminated females. Simulations show that estimates made using either estimator after the lines have drifted to fixation have a much greater error associated with their use than do those estimates made earlier in time using the correction. In general it is better to use corrected estimates of gene frequency soon after lines are established than to use uncorrected estimates made after the first laboratory generation. This work was supported by an NSERC fellowship to A.D.L.     

Electrotypes and formal genetics of red cell glutathione peroxidase (GPX1) in the Djuka of Surinam.

Samples of venous blood from 239 male and 476 female adults including 41 pairs of parents and 123 of their children belonging to a Surinam population called the Djuka or Bush Negroes of West African origin were screened for electrophoretic variants of red cell glutathione peroxidase (GPX1) in Cellogel. The results confirmed an earlier hypothesis that at least a part of the GPX1 variation mainly, if not exclusively, observed in the Africans and people of African origin living elsewhere, is determined by two codominant alleles (called GPX1*1 and GPX1*2), at an autosomal locus. The frequency of GPX1*2 allele in the Djuka was estimated to be .054. A rare variant provisionally designed as GPX1 Djuka (thought to be a heterozygote due to a third allele called GPX1*3 and the GPX1*1) was found in two apparently unrelated individuals. Catalytically, the product of GPX1*2 appears to be about twice more active than that of GPX1*1. For heuristic purposes, it was proposed and discussed that GPX1*2 is a South-Saharan African allele and is amenable for natural selection.     

Best linear unbiased allele-frequency estimation in complex pedigrees

Many types of genetic analyses depend on estimates of allele frequencies. We consider the problem of allele-frequency estimation based on data from related individuals. The motivation for this work is data collected on the Hutterites, an isolated founder population, so we focus particularly on the case in which the relationships among the sampled individuals are specified by a large, complex pedigree for which maximum likelihood estimation is impractical. For this case, we propose to use the best linear unbiased estimator (BLUE) of allele frequency. We derive this estimator, which is equivalent to the quasi-likelihood estimator for this problem, and we describe an efficient algorithm for computing the estimate and its variance. We show that our estimator has certain desirable small-sample properties in common with the maximum likelihood estimator (MLE) for this problem. We treat both the case when parental origin of each allele is known and when it is unknown. The results are extended to prediction of allele frequency in some set of individuals S based on genotype data collected on a set of individuals R. We compare the mean-squared error of the BLUE, the commonly used naive estimator (sample frequency) and the MLE when the latter is feasible to calculate. The results indicate that although the MLE performs the best of the three, the BLUE is close in performance to the MLE and is substantially easier to calculate, making it particularly useful for large complex pedigrees in which MLE calculation is impractical or infeasible. We apply our method to allele-frequency estimation in a Hutterite data set.     

Estimates of natural selection due to protein tertiary structure inform the ancestry of biallelic loci

We consider the inference of which of two alleles is ancestral when the alleles have a single nonsynonymous difference and when natural selection acts via protein tertiary structure. Whereas the probability that an allele is ancestral under neutrality is equal to its frequency, under selection this probability depends on allele frequency and on the magnitude and direction of selection pressure. Although allele frequencies can be well estimated from intraspecific data, small fitness differences have a large evolutionary impact but can be difficult to estimate with only intraspecific data. Methods for predicting aspects of phenotype from genotype can supplement intraspecific sequence data. Recently developed statistical techniques can assess effects of phenotypes, such as protein tertiary structure on molecular evolution. While these techniques were initially designed for comparing protein-coding genes from different species, the resulting interspecific inferences can be assigned population genetic interpretations to assess the effect of selection pressure, and we use them here along with intraspecific allele frequency data to estimate the probability that an allele is ancestral. We focus on 140 nonsynonymous single nucleotide polymorphisms of humans that are in proteins with known tertiary structures. We find that our technique for employing protein tertiary structure information yields some biologically plausible results but that it does not substantially improve the inference of ancestral human allele types.     

The frequency of multiple paternity suggests that sperm competition is common in house mice (Mus domesticus)

Sexual selection is an important force driving the evolution of morphological and genetic traits. To determine the importance of male-male, postcopulatory sexual selection in natural populations of house mice, we estimated the frequency of multiple paternity, defined as the frequency with which a pregnant female carried a litter fertilized by more than one male. By genotyping eight microsatellite markers from 1095 mice, we found evidence of multiple paternity from 33 of 143. Evidence for multiple paternity was especially strong for 29 of these litters. Multiple paternity was significantly more common in higher-density vs. lower-density populations. Any estimate of multiple paternity will be an underestimate of the frequency of multiple mating, defined as the frequency with which a female mates with more than a single male during a single oestrus cycle. We used computer simulations to estimate the frequency of multiple mating, incorporating observed reductions in heterozygosity and levels of allele sharing among mother and father. These simulations indicated that multiple mating is common, occurring in at least 20% of all oestrus cycles. The exact estimate depends on the competitive skew among males, a parameter for which we currently have no data from natural populations. This study suggests that sperm competition is an important aspect of postcopulatory sexual selection in house mice.     

Decompositions of Price’s formula in an inhomogeneous population structure

The central tool for the study of allele frequency change due to selection is the remarkably simple but powerful formula of Price [Nature 227 (1970) 520] . Here, I provide what might be called a structural analysis of this formula. The formula essentially accumulates the average allele frequency change over many instances of a fitness‐determining interaction, but there are different ways of organizing this average and these lead to quite different computational algorithms. I present three of these: an analysis by population state, an analysis by recipient and an analysis by actor. A comparison of these can lead to a heightened understanding of the different factors behind selective allele frequency change. In particular, I pay attention to the effects of structural inhomogeneity on reproductive value (RV) and emphasize that Price’s formula measures RV‐weighted allele frequency change. I examine in detail a simple example as a crucial way of cementing the different theoretical pathways. My aim was to produce a simple transparent presentation and therefore I work with a simple population structure and have omitted a number of technical details that are found elsewhere.     

Sandhoff disease heterozygote detection: a component of population screening for Tay-Sachs disease carriers. II. Sandhoff disease gene frequencies in American Jewish and non-Jewish populations.

Carrier frequencies for the allele(s) causing Sandhoff disease have been estimated for the U.S. Jewish and non-Jewish populations. The estimates have been made directly, with data from 22,043 Jewish and 32,342 non-Jewish individuals measured for total serum hexosaminidase activity and the heat-labile fraction. These values have been shown to identify potential carriers of the Sandhoff allele(s) with 95% sensitivity. Subsequent leukocyte assays of total hexosaminidase activity and the heat-labile fraction in those identified in serum tests have been shown to provide a much finer discrimination between those who carry the allele(s) and those who do not. Results from such assays were used to generate these carrier frequency estimates. Carrier frequency estimates have also been made indirectly from Sandhoff disease incidence data collected during the period 1979-84. These estimates are in agreement with data for the Jewish population under analysis, but in the non-Jewish population the estimate derived from data on screened individuals is greater than the estimate derived from incidence figures. The possible causes for such a difference are discussed. In a study of non-Jewish individuals each of whose grandparents derives from a single country of origin, the distribution of countries among Sandhoff disease carriers differs significantly from that in the non-Jewish sample under analysis, indicating possible ethnic groups with increased or decreased carrier frequencies. These analyses suggest an increased Sandhoff disease carrier frequency among Mexican and Central-American populations and a decreased carrier frequency among non-Jewish German populations.     

A human dimorphism resulting from loss of an Alu.

The molecular phylogeny of Alu and other repeated sequences in the human genome provides clues to events during primate evolution. A subclass of human Alu's has been previously identified as dimorphic insertions within members of the medium reiteration frequency (mer) class of repeats, reflecting the complicated sequence of insertion and radiation events leading to the current human genome structure. One dimorphic Alu is located within a previously unidentified mer family member, in the first intron of the human T4 (CD4) gene. The insertion (Alu+ allele) has a frequency of approximately 70% in Europeans and Africans and is homozygous in 20 Asian samples. Polymerase chain reaction amplification, direct DNA sequencing, and Southern analysis using oligonucleotide probes revealed that the Alu- allele was derived from the Alu+ allele by loss of part of the inserted sequence. Comparison with a tightly linked marker within the human genome and studies of baboon DNA samples revealed that the original insertion was a relatively early event in primate evolution, but that the Alu sequence loss leading to the dimorphism has occurred much more recently. Loss of Alu insertions therefore represents one mechanism for the generation of human Alu dimorphisms.     

Excision of one of two defective P elements as the cause of alternate mutational events (sn+ and sne) of the singed bristle allele snw in Drosophila melanogaster

The X-linked singed locus is concerned with the bristle phenotype and female sterility, and is known as a hot spot of P element insertion. A moderate allele of singed, singed-weak (snw) (Engels, 1979; 1984) is inserted with P elements. It is used as an index of P element activity, since it mutates at a high frequency to either a more extreme allele, singed-extreme (sne), or to a phenotype that is equivalent to the wild type (sn+) when an autonomous P element exists. We show here that snw is inserted with two defective P elements in reverse orientation, and the two alternate mutational events (sn+ and sne) are caused by the excision of one or the other of the P elements present in the singed gene. It is interesting that sn+ and sne are inserted with a single P element in the same position, but show very different phenotypes. The insertional sites of P elements in the singed locus possibly contain an unidentified repetitive sequence, which is repeated dozens of times per haploid genome of the wild-type strain Canton-S.     

The Theory and Applications of Measuring Broad-Range and Chromosome-Wide Recombination Rate from Allele Frequency Decay around a Selected Locus

Recombination is the exchange of genetic material between homologous chromosomes via physical crossovers. High-throughput sequencing approaches detect crossovers genome wide to produce recombination rate maps but are difficult to scale as they require large numbers of recombinants individually sequenced. We present a simple and scalable pooled-sequencing approach to experimentally infer near chromosome-wide recombination rates by taking advantage of non-Mendelian allele frequency generated from a fitness differential at a locus under selection. As more crossovers decouple the selected locus from distal loci, the distorted allele frequency attenuates distally toward Mendelian and can be used to estimate the genetic distance. Here, we use marker selection to generate distorted allele frequency and theoretically derive the mathematical relationships between allele frequency attenuation, genetic distance, and recombination rate in marker-selected pools. We implemented nonlinear curve-fitting methods that robustly estimate the allele frequency decay from batch sequencing of pooled individuals and derive chromosome-wide genetic distance and recombination rates. Empirically, we show that marker-selected pools closely recapitulate genetic distances inferred from scoring recombinants. Using this method, we generated novel recombination rate maps of three wild-derived strains of Drosophila melanogaster, which strongly correlate with previous measurements. Moreover, we show that this approach can be extended to estimate chromosome-wide crossover interference with reciprocal marker selection and discuss how it can be applied in the absence of visible markers. Altogether, we find that our method is a simple and cost-effective approach to generate chromosome-wide recombination rate maps requiring only one or two libraries.     

Genomic control for association studies under various genetic models

Case-control studies are commonly used to study whether a candidate allele and a disease are associated. However, spurious association can arise due to population substructure or cryptic relatedness, which cause the variance of the trend test to increase. Devlin and Roeder derived the appropriate variance inflation factor (VIF) for the trend test and proposed a novel genomic control (GC) approach to estimate VIF and adjust the test statistic. Their results were derived assuming an additive genetic model and the corresponding VIF is independent of the candidate allele frequency. We determine the appropriate VIFs for recessive and dominant models. Unlike the additive test, the VIFs for the optimal tests for these two models depend on the candidate allele frequency. Simulation results show that, when the null loci used to estimate the VIF have allele frequencies similar to that of the candidate gene, the GC tests derived for recessive and dominant models remain optimal. When the underlying genetic model is unknown or the null loci and candidate gene have quite different allele frequencies, the GC tests derived for the recessive or dominant models cannot be used while the GC test derived for the additive model can be.     

Detection of a cryptic paracentric inversion within band 11p13 in familial aniridia by fluorescence in situ hybridization

We report the first familial case of dominantly inherited aniridia with a cryptic inversion within band 11p13. High-resolution chromosome analysis gave a suspicion of a tiny constitutional aberration around band 11p13 and fluorescence in situ hybridization using 11p cosmids successfully confirmed that the aniridia patients of this family have an inversion within band 11p13. The distal breakpoint of the inversion is telomeric to a candidate aniridia gene (AN2) and suggests that more genes might be involved in the etiology of aniridia. In situ hybridization is a powerful tool to detect cryptic rearrangements in sporadic or familial patients with aniridia. This family indicated the importance of careful observation of the 11p13 region of aniridia patients, even if the aniridia was autosomal dominantly inherited.     

A nonlinear method for estimating nucleotide polymorphism from restriction maps

Restriction mapping is used to estimate nucleotide sequence polymorphism when the regions to be studied are too long or too numerous to be sequenced. Restriction mapping is less costly than DNA sequencing, but it does not allow direct measurement of underlying nucleotide polymorphism. It is therefore useful to be able to estimate underlying nucleotide polymorphism from observations of polymorphism in restriction maps, as this offers some of the resolution afforded by DNA sequencing at a reduced cost. Previous estimators of underlying nucleotide polymorphism have assumed that each restriction-enzyme- binding site contains, at most, a single polymorphic nucleotide position (the low-polymorphism-frequency assumption), and this assumption has placed an upper limit on the level of polymorphism that can be resolved by these estimators. The present study documents an estimator which allows relaxation of this assumption. The new estimator more accurately estimates underlying nucleotide polymorphism when the polymorphism level is high enough to falsify the low-polymorphism- frequency assumption. The new estimator therefore yields good results for data sets that are too divergent for analysis by present methods.      

Estimation of the frequency of hexosaminidase a variant alleles in the American Jewish population.

There appear to be several alleles of the hexosaminidase A (HEX A) gene that lead to different clinical syndromes. In addition to the infantile-onset Tay-Sachs disease (TSD), there is a juvenile-onset and an adult-onset form, which are also characterized by low HEX A levels. There are also apparently healthy adults with low HEX A activity. Based primarily on data from population screening for TSD carrier status, we estimate the allele frequency of the combined variant alleles for which data are available to be about 4.5 x 10(-4) and the frequency of adults showing zero HEX A levels (when tested using artificial substrate) to be about 1:67,000. The implications for population screening and prenatal diagnosis are discussed.     

The Yank of Dobzhansky’s Bequest

Dobzhansky studied mechanisms of balancing selection using systems of inversions in Drosophila and he soon found that changes in inversion frequencies along generations in experimental populations conformed to the expectation for a simple model of heterosis. However, other more complex modes of selection, like rare male advantage, were later found to affect the maintenance of inversion polymorphisms. Here we show that a more realistic (and complex) model than heterosis—integrating all known fitness component estimates obtained in independent experiments for the ST/CH system of inversions in Drosophila pseudoobscura—not only conforms to but actually also predicts the inversion frequencies. This concludes this line of work and points to other selection mechanisms than heterosis that were also considered by Dobzhansky—frequency- and sex-dependent selection—as potential mechanisms of balancing selection responsible for the maintenance of the inversion polymorphisms in Drosophila.     

Weighted normality-based estimator in correcting correlation coefficient estimation between incomplete nutrient measurements

Consider the problem of estimating the correlation between two nutrient measurements, such as the percent energy from fat obtained from a food frequency questionnaire (FFQ) and that from repeated food records or 24-hour recalls. Under a classical additive model for repeated food records, it is known that there is an attenuation effect on the correlation estimation if the sample average of repeated food records for each subject is used to estimate the underlying long-term average. This paper considers the case in which the selection probability of a subject for participation in the calibration study, in which repeated food records are measured, depends on the corresponding FFQ value, and the repeated longitudinal measurement errors have an autoregressive structure. This paper investigates a normality-based estimator and compares it with a simple method of moments. Both methods are consistent if the first two moments of nutrient measurements exist. Furthermore, joint estimating equations are applied to estimate the correlation coefficient and related nuisance parameters simultaneously. This approach provides a simple sandwich formula for the covariance estimation of the estimator. Finite sample performance is examined via a simulation study, and the proposed weighted normality-based estimator performs well under various distributional assumptions. The methods are applied to real data from a dietary assessment study.     

The effect of an inversion system and the time interval between matings on postcopulatory sexual selection in the seaweed fly, Coelopa frigida

The seaweed fly, Coelopa frigida, exhibits LMSP. A large chromosomal inversion system affects many traits including egg-to-adult viability via heterosis. Consequently, there is also considerable potential for cryptic female mate choice to operate on the basis of sperm karyotype. Here, we investigated the effect of time interval and chromosomal inversion karyotype on postcopulatory sexual selection. Homokaryotypic females were mated with a male of the same and a male of the opposite homokaryotype. The order of the matings was varied so cryptic female mate choice could operate either in concert or antagonistically with LMSP. LMSP was found when there was a 24 h time interval between matings, irrespective of the order in which the males were mated. However, when the males were mated in quick succession the order of mating was important. When LMSP and cryptic female mate choice work in concert a high level of LMSP was found. However, when the male of opposite homokaryotype mated first, then first male sperm precedence was observed. This suggests that polyandrous females might be able to bias paternity but only when matings occur in quick succession. Consequently, population density is likely to affect the operation of postcopulatory sexual selection.     

Allele age and a test for selection on rare alleles

An approximate expression for the probability distribution of the age of a neutral allele as a function of its frequency is derived for a population undergoing arbitrary changes in population size. A simple maximum-likelihood estimator of allele age based on frequency is also obtained. The distribution of allele age, combined with a model predicting the extent of intra-allelic variability generated by mutation and recombination, leads to a statistical test of whether a rare allele has experienced natural selection. The test is based on finding whether there is too little or too much intra-allelic variability to be consistent with the observed frequency. The test is applied to the locus, BRCA1, associated with early-onset breast cancer in humans and shows that two common disease-associated alleles (5382insC and 185delAG) appear to have been subject to natural selection.