An explicit formula for frequency changes in genetic algebras

Summary A general situation in population genetics is considered with any hereditary system described by a genetic algebra. It is assumed that there is random mating, no selection, and infinite population size. A formula is derived for the distribution of genetic types in the general jth generation given the distribution in the initial generation. Special attention is paid to the case of one locus with polyploidy and chromosome segregation. The calculations are carried out as far as possible in the situation with triploid gametes.Work supported by the Alexander von Humboldt Foundation.     

The distribution of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">st</Emphasis></Subscript> and other genetic statistics for a class of population structure models

We examine genetic statistics used in the study of structured populations. In a 1999 paper, Wakeley observed that the coalescent process associated with the finite island model can be decomposed into a scattering phase and a collecting phase. This decomposition becomes exact in the large population limit with the coalescent at the end of the scattering phase converging to the Ewens sampling formula and the coalescent during the collecting phase converging to the Kingman coalescent. In this paper we introduce a class of limiting models, which we refer to as G/KC models, that generalize Wakeley’s decomposition. G in G/KC represents a completely general limit for the scattering phase, while KC represents a Kingman coalescent limit for the collecting phase. We show that both the island and two-dimensional stepping stone models converge to G/KC models in the large population limit. We then derive the distribution of the statistic F _st for all G/KC models under a large sample limit for the cases of strong or weak mutation, thereby deriving the large population, large sample limiting distribution of F _st for the island and two-dimensional stepping stone models as a special case of a general formula. Our methods allow us to take the large population and large sample limits simultaneously. In the context of large population, large sample limits, we show that the variance of F _st in the presence of weak mutation collapses as O(\frac1logd){O(\frac{1}{\log d})} where d is the number of demes sampled. Further, we show that this O(\frac1logd){O(\frac{1}{\log d})} is caused by a heavy tail in the distribution of F _st . Our analysis of F _st can be extended to an entire class of genetic statistics, and we use our approach to examine homozygosity measures. Our analysis uses coalescent based methods.     

The Determination of Sample Sizes in the Comparison of Two Multinomial Proportions from Ordered Categories

We consider sample size determination for ordered categorical data when the alternative assumption is the proportional odds model. In this paper the sample size formula proposed by Whitehead (Statistics in Medicine, 12 , 2257–2271, 1993) is compared with the methods based on exact and asymptotic linear rank tests with Wilcoxon and trend scores. We show that Whitehead's formula, which is based on a normal approximation, works well when the sample size is moderate to large but recommend the exact method with Wilcoxon scores for small sample sizes. The consequences of misspecification in models are also investigated.     

Evaluation of Noether's Method of Sample Size Determination for the Wilcoxon-Mann-Whitney Test

NOETHER (1987) proposed a method of sample size determination for the Wilcoxon-Mann-Whitney test. To obtain a sample size formula, he restricted himself to alternatives that differ only slightly from the null hypothesis, so that the unknown variance o² of the Mann-Whitney statistic can be approximated by the known variance under the null hypothesis which depends only on n. This fact is frequently forgotten in statistical practice. In this paper, we compare Noether's large sample solution against an alternative approach based on upper bounds of σ² which is valid for any alternatives. This comparison shows that Noether's approximation is sufficiently reliable with small and large deviations from the null hypothesis.     

Variance of Neutral Genetic Variances within and between Populations for a Quantitative Character

The variances of genetic variances within and between finite populations were systematically studied using a general multiple allele model with mutation in terms of identity by descent measures. We partitioned the genetic variances into components corresponding to genetic variances and covariances within and between loci. We also analyzed the sampling variance. Both transient and equilibrium results were derived exactly and the results can be used in diverse applications. For the genetic variance within populations, sigma 2 omega, the coefficient of variation can be very well approximated as [formula: see text] for a normal distribution of allelic effects, ignoring recurrent mutation in the absence of linkage, where m is the number of loci, N is the effective population size, theta 1(0) is the initial identity by descent measure of two genes within populations and t is the generation number. The first term is due to genic variance, the second due to linkage disequilibrium, and third due to sampling. In the short term, the variation is predominantly due to linkage disequilibrium and sampling; but in the long term it can be largely due to genic variance. At equilibrium with mutation [formula: see text] where u is the mutation rate. The genetic variance between populations is a parameter. Variance arises only among sample estimates due to finite sampling of populations and individuals. The coefficient of variation for sample gentic variance between populations, sigma 2b, can be generally approximated as [formula: see text] when the number of loci is large where S is the number of sampling populations.     

Unbiased estimation of gene diversity in samples containing related individuals: exact variance and arbitrary ploidy

Gene diversity, a commonly used measure of genetic variation, evaluates the proportion of heterozygous individuals expected at a locus in a population, under the assumption of Hardy-Weinberg equilibrium. When using the standard estimator of gene diversity, the inclusion of related or inbred individuals in a sample produces a downward bias. Here, we extend a recently developed estimator shown to be unbiased in a diploid autosomal sample that includes known related or inbred individuals to the general case of arbitrary ploidy. We derive an exact formula for the variance of the new estimator, H, and present an approximation to facilitate evaluation of the variance when each individual is related to at most one other individual in a sample. When examining samples from the human X chromosome, which represent a mixture of haploid and diploid individuals, we find that H performs favorably compared to the standard estimator, both in theoretical computations of mean squared error and in data analysis. We thus propose that H is a useful tool in characterizing gene diversity in samples of arbitrary ploidy that contain related or inbred individuals.     

The temperature‐size rule in Daphnia magna across different genetic lines and ontogenetic stages: Multiple patterns and mechanisms

Ectotherms tend to grow faster, but reach a smaller size when reared under warmer conditions. This temperature‐size rule (TSR) is a widespread phenomenon. Despite the generality of this pattern, no general explanation has been found. We therefore tested the relative importance of two proposed mechanisms for the TSR: (1) a stronger increase in development rate relative to growth rate at higher temperatures, which would cause a smaller size at maturity, and (2) resource limitation placing stronger constraints on growth in large individuals at higher temperatures, which would cause problems with attaining a large size in warm conditions. We raised Daphnia magna at eight temperatures to assess their size at maturity, asymptotic size, and size of their offspring. We used three clonal lines that differed in asymptotic size and growth rate. A resource allocation model was developed and fitted to our empirical data to explore the effect of both mechanisms for the TSR. The genetic lines of D. magna showed different temperature dependence of growth and development rates resulting in different responses for size at maturity. Also, at warm temperatures, growth was constrained in large, but not in small individuals. The resource allocation model could fit these empirical data well. Based on our empirical results and model explorations, the TSR of D. magna at maturity is best explained by a stronger increase in development rate relative to growth rate at high temperature, and the TSR at asymptotic size is best explained by a size‐dependent and temperature‐dependent constraint on growth, although resource limitation could also affect size at maturity. In conclusion, the TSR can take different forms for offspring size, size at maturity, and asymptotic size and each form can arise from its own mechanism, which could be an essential step toward finding a solution to this century‐old puzzle.     

Minimum Sample Size Ensuring Validity of Classical Confidence Intervals for Means of Skewed and Platykurtic Distributions

COCHRAN (1953) and BARTCH (1957) gave formulae for the magnitude of the sample size (n) ensuring the validity of the limiting normal distribution of the sample mean x(ⁿ) obtained from a non-normal distribution with marked asymmetry and kurtosis. These formulae have been checked empirically in this paper using (a) simulated data with given asymmetry and kurtosis and (b) real data gathered from a coronary heart disease study. We find that our results are in general agreement with Bartch's formula. However, in a number of cases, the asymptotic normal distribution is attained for smaller sample size than that required by Bartch's formula.     

Robust covariate-adjusted log-rank statistics and corresponding sample size formula for recurrent events data

Summary .   Recurrent events data are frequently encountered in clinical trials. This article develops robust covariate-adjusted log-rank statistics applied to recurrent events data with arbitrary numbers of events under independent censoring and the corresponding sample size formula. The proposed log-rank tests are robust with respect to different data-generating processes and are adjusted for predictive covariates. It reduces to the Kong and Slud (1997, Biometrika 84, 847–862) setting in the case of a single event. The sample size formula is derived based on the asymptotic normality of the covariate-adjusted log-rank statistics under certain local alternatives and a working model for baseline covariates in the recurrent event data context. When the effect size is small and the baseline covariates do not contain significant information about event times, it reduces to the same form as that of Schoenfeld (1983, Biometrics 39, 499–503) for cases of a single event or independent event times within a subject. We carry out simulations to study the control of type I error and the comparison of powers between several methods in finite samples. The proposed sample size formula is illustrated using data from an rhDNase study.     

K‐Sample Test and Sample Size Calculation for Comparing Slopes in Data with Repeated Measurements

Sample size calculations based on two‐sample comparisons of slopes in repeated measurements have been reported by many investigators. In contrast, the literature has paid relatively little attention to the design and analysis of K‐sample trials in repeated measurements studies where K is 3 or greater. Jung and Ahn (2003) derived a closed sample size formula for two‐sample comparisons of slopes by taking into account the impact of missing data. We extend their method to compare K‐sample slopes in repeated measurement studies using the generalized estimating equation (GEE) approach based on independent working correlation structure. We investigate the performance of the sample size formula since the sample size formula is based on asymptotic theory. The proposed sample size formula is illustrated using a clinical trial example. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effect of recurrent mutation on the frequency spectrum of a segregating site and the age of an allele

The sample frequency spectrum of a segregating site is the probability distribution of a sample of alleles from a genetic locus, conditional on observing the sample to be polymorphic. This distribution is widely used in population genetic inferences, including statistical tests of neutrality in which a skew in the observed frequency spectrum across independent sites is taken as a signature of departure from neutral evolution. Theoretical aspects of the frequency spectrum have been well studied and several interesting results are available, but they are usually under the assumption that a site has undergone at most one mutation event in the history of the sample. Here, we extend previous theoretical results by allowing for at most two mutation events per site, under a general finite allele model in which the mutation rate is independent of current allelic state but the transition matrix is otherwise completely arbitrary. Our results apply to both nested and nonnested mutations. Only the former has been addressed previously, whereas here we show it is the latter that is more likely to be observed except for very small sample sizes. Further, for any mutation transition matrix, we obtain the joint sample frequency spectrum of the two mutant alleles at a triallelic site, and derive a closed-form formula for the expected age of the younger of the two mutations given their frequencies in the population. Several large-scale resequencing projects for various species are presently under way and the resulting data will include some triallelic polymorphisms. The theoretical results described in this paper should prove useful in population genomic analyses of such data.     

Genetic diversity patterns in Curcuma reflect differences in genome size

The relationships between genome size and the systematic and evolutionary patterns in vascular plants are equivocal, although a close relationship between genome size and evolutionary patterns has been previously reported. However, several studies have also revealed the dynamic nature of genome size evolution and its considerable ‘ups’ and ‘downs’. Thus, in this study, the phylogenetic relationships among three previously revealed genome size groups and among species of the highly polyploid genus Curcuma were evaluated using AFLP. Our results suggest two main lineages within Indian Curcuma reflecting evolution of genome size. The first one includes hexaploids and higher polyploids of the previously recognized genome size group I, and the second one includes mainly hexaploids of genome size groups II and III. Within genome size group I, relationships among species seem to be influenced by reticulate evolution and higher polyploids are likely to be of allopolyploid origin. Reproductive systems in Indian Curcuma vary considerably among ploidy levels and these differences considerably affect morphological and genetic variation. In general, clonally reproducing species are expected to exhibit low genotypic diversity, but, at the same time, species of allopolyploid origin are expected to maintain higher levels of heterozygosity compared with their progenitors. We investigated intra‐populational genetic variability in Curcuma spp. to evaluate whether mode of reproduction or ploidy represent the main factor influencing the degree of genetic diversity. We found that hexaploid species exhibited significantly higher genetic diversity than higher polyploids (9x, 15x). Our results suggest that this genetic diversity pattern is largely influenced by the mode of reproduction, as higher polyploids reproduce exclusively vegetatively, whereas hexaploids reproduce mainly sexually. © 2011 The Linnean Society of London, Botanical Journal of the Linnean Society, 2011, 165 , 388–401.     

A Complementary Method for Production of Tetraploid Crassostrea gigas Using Crosses Between Diploids and Tetraploids with Cytochalasin B Treatments

We present a new method to produce tetraploid Crassostrea gigas by cytochalasin B inhibition of polar body 2 expulsion in diploid females crossed with tetraploid males. This offers a means of direct introgression of genetic characters from selected diploid to tetraploid lines, avoiding a triploid step. Offspring larval ploidy shifted over time and depended on size, with tetraploids more frequent among the smaller larvae and triploids among the large. Viable tetraploids were found at 4 and 6 months, indicating the technique was successful. The possibility that gynogenesis occurred was tested by microsatellite analysis to confirm the presence of paternally inherited alleles. These were present in all animals of the 2n × 4n + CB (female first) cross. However, a 4n × 2n + CB cross produced triploids, including some gynogens. Our method illustrates for the first time that diploid C. gigas eggs, if selected for large size, can give viable tetraploid offspring.     

The effect of ploidy level on fitness in parthenogenetic flatworms

Although polyploidy plays an important role in speciation, its impact on fitness is still debated. One problem is that its adaptive significance can only be inferred by comparing forms with different ploidy that are identical in all other traits. This situation is uncommon, presumably because ploidy types often differ in reproduction mode, genetic background or habitat. Here we compare fitness in a system of triploid and tetraploid karyotypes of the planarian flatworm Schmidtea polychroa . Both types have the same type of sperm-dependent parthenogenesis and share the same genetic background and habitat. Hence, fitness differences, if any, can be attributed to different ploidy levels only. Contrary to the general assumption of a positive correlation between fitness and ploidy level, we showed that triploids produced 58% more offspring than tetraploids. Within each ploidy type, we identified groups of highly related clones using microsatellites. Significant variation among clonal groups in body size, offspring and cocoon number and hatching time indicated a genetic basis for variance in these traits. A small model shows that despite low fitness of tetraploids, stable coexistence of triploids and tetraploids can be explained by the recurrent origin of triploids from tetraploids and vice versa.  © 2005 The Linnean Society of London, Biological Journal of the Linnean Society , 2005, 85 , 191–198.     

Ploidy tug‐of‐war: Evolutionary and genetic environments influence the rate of ploidy drive in a human fungal pathogen

Variation in baseline ploidy is seen throughout the tree of life, yet the factors that determine why one ploidy level is maintained over another remain poorly understood. Experimental evolution studies using asexual fungal microbes with manipulated ploidy levels intriguingly reveals a propensity to return to the historical baseline ploidy, a phenomenon that we term “ploidy drive.” We evolved haploid, diploid, and polyploid strains of the human fungal pathogen Candida albicans under three different nutrient limitation environments to test whether these conditions, hypothesized to select for low ploidy levels, could counteract ploidy drive. Strains generally maintained or acquired smaller genome sizes (measured as total nuclear DNA through flow cytometry) in minimal medium and under phosphorus depletion compared to in a complete medium, while mostly maintained or acquired increased genome sizes under nitrogen depletion. Improvements in fitness often ran counter to changes in genome size; in a number of scenarios lines that maintained their original genome size often increased in fitness more than lines that converged toward diploidy (the baseline ploidy of C. albicans). Combined, this work demonstrates a role for both the environment and genotype in determination of the rate of ploidy drive, and highlights questions that remain about the force(s) that cause genome size variation.     

The use of MapPop1.0 for choosing a QTL mapping sample from an advanced backcross population

QTL detection is a good way to assess the genetic basis of quantitative traits such as the plant response to its environment, but requires large mapping populations. Experimental constraints, however, may require a restriction of the population size, risking a decrease in the quality level of QTL mapping. The purpose of this paper was to test if an advanced backcross population sample chosen by MapPop 1.0 could limit the effect of size restriction and improve the QTL detection when compared to random samples. We used the genotypic and phenotypic data obtained for 280 genotypes, considered as the reference population. The “MapPop sample” of 100 genotypes was first compared to the reference population, and genetic maps, genotypic and phenotypic data and QTL results were analysed. Despite the increase in donor allele frequency in the MapPop sample, this did not lead to an increase of the genetic map length or a biased phenotypic distribution. Three QTL among the 10 QTL found in the reference population were also detected in the MapPop sample. Next, the MapPop sample results were compared to those from 500 random samples of the same size. The main conclusion was that the MapPop software avoided the selection of biased samples and the detection of false QTL and appears particularly interesting to select a sample from an unbalanced population.     

Flow cytometric evidence for endopolyploidy in seedlings of some Brassica species

Flow cytometric analysis of the nuclear DNA contents of somatic tissues from seedlings of Brassica rapa L. and B. oleracea L. revealed extensive endoreduplication, resulting in tissues that contain cells with multiple ploidy levels (also called ’endopolyploidy’). Multiples of the haploid nuclear genome complement (1C) corresponding to 2C, 4C, 8C, 16C, 32C and 64C were observed in Brassica rapa, while B. oleracea exhibited a mixture of cells with five ploidy levels, 2C, 4C, 8C, 16C and 32C. The distribution of cells with the different ploidy levels was tissue-specific and characteristic of the stage of development. Multiploidy was not found in the embryos of dry seeds. Rapid endoreduplication occurred during seedling development. It is most probable that multiploidy is, if not a general feature, at least very common in Brassica species. The physiological and genetic implications of this original feature are discussed. Received: 17 March 2000 / Accepted: 17 April 2000     

Multiple aspects of condition influence a heritable sexual trait: a synthesis of the evidence for capture of genetic variance in red junglefowl

A long‐standing question in the study of sexual selection is: if an attractive male trait signals heritable aspects of quality, how does genetic variation associated with that trait persist in the face of directional selection? It has been proposed that sensitivity to condition in general could allow a sexual signal to show heritable variation even under persistent directional selection. The multitude of genes underlying components of condition present a large target for mutation, and so genetic variance in condition may not be readily exhausted by selection. Insights gained in studies of the red junglefowl are relevant to this hypothesis, and provide a model for research in other systems. The fleshy comb of male red junglefowl is among the best‐studied sexual signals. Multiple components of condition, such as health, immune function, testosterone, and social status, influence comb growth, and, in the present study, we provide the first comprehensive integration of this large body of evidence to lay out the compelling case for condition‐dependence of the junglefowl comb. Variation in comb size is heritable and, most important to this issue, is genetically correlated with heritable variation in a body condition index. Although understanding of the red junglefowl's comb is far from complete, it exceeds our understanding of many other sexual signals and provides an empirical model for the study of condition‐dependent signals in general. © 2007 The Linnean Society of London, Biological Journal of the Linnean Society, 2007, 92 , 651–660.     

Evaluation of tuber bearing Solanum species belonging to different ploidy levels for its yielding potential at low soil fertility

Virus free tubers from thirty-six clones of tuber bearing Solanum species belonging to different ploidy levels, from the world gene bank at the International Potato Center, were cultivated under field conditions at low and high fertilizer rates. Nine of the clones were chosen for their high yielding potential (advanced group) and twenty-seven for their extensiveness (native group). The tuber yields of the thirty-six clones at the two fertilizer levels were positively correlated and the advanced group yielded more than the native group, even under conditions of low soil fertility.A broad genetic variation in root size at both fertilizer levels was apparent. Although some native clones showed rather large root systems as a group, the advanced clones had a similar root dry weight to the native clones.Tuber dry weight was significantly correlated with root dry weight but the advanced clones had a higher tuber yield per unit of root dry matter produced.     

Quantifying the percent increase in minimum sample size for SNP genotyping errors in genetic model-based association studies

Kang et al. [Genet Epidemiol 2004;26:132-141] addressed the question of which genotype misclassification errors are most costly, in terms of minimum percentage increase in sample size necessary (%MSSN) to maintain constant asymptotic power and significance level, when performing case/control studies of genetic association in a genetic model-free setting. They answered the question for single nucleotide polymorphisms (SNPs) using the 2 x 3 chi2 test of independence. We address the same question here for a genetic model-based framework. The genetic model parameters considered are: disease model (dominant, recessive), genotypic relative risk, SNP (marker) and disease allele frequency, and linkage disequilibrium. %MSSN coefficients of each of the six possible error rates are determined by expanding the non-centrality parameter of the asymptotic distribution of the 2 x 3 chi2 test under a specified alternative hypothesis to approximate %MSSN using a linear Taylor series in the error rates. In this work we assume errors misclassifying one homozygote as another homozygote are 0, since these errors are thought to rarely occur in practice. Our findings are that there are settings of the genetic model parameters that lead to large total %MSSN for both dominant and recessive models. As SNP minor allele approaches 0, total %MSSN increases without bound, independent of other genetic model parameters. In general, %MSSN is a complex function of the genetic model parameters. Use of SNPs with small minor allele frequency requires careful attention to frequency of genotyping errors to insure that power specifications are met. Software to perform these calculations for study design is available, and an example of its use to study a disease is given.