Rank-based statistical methodologies for quantitative trait locus mapping

This article addresses the identification of genetic loci (QTL and elsewhere) that influence nonnormal quantitative traits with focus on experimental crosses. QTL mapping is typically based on the assumption that the traits follow normal distributions, which may not be true in practice. Model-free tests have been proposed. However, nonparametric estimation of genetic effects has not been studied. We propose an estimation procedure based on the linear rank test statistics. The properties of the new procedure are compared with those of traditional likelihood-based interval mapping and regression interval mapping via simulations and a real data example. The results indicate that the nonparametric method is a competitive alternative to the existing parametric methodologies.     

Nonparametric trend statistic incorporating dispersion differences in sib pair linkage for quantitative traits

The Haseman-Elston (HE) regression method and its extensions are widely used in genetic studies for detecting linkage to quantitative trait loci (QTL) using sib pairs. The principle underlying the simple HE regression method is that the similarity in phenotypes between two siblings increases as they share an increasing number of alleles identical by descent (IBD) from their parents at a particular marker locus. In such a procedure, similarity was identified with the locations, that is, means of groups of sib pairs sharing 0, 1, and 2 alleles IBD. A more powerful, rank-based nonparametric test to detect increasing similarity in sib pairs is presented by combining univariate trend statistics not only of locations, but also of dispersions of the squared phenotypic differences of two siblings for three groups. This trend test does not rely on distributional assumptions, and is applicable to the skewed or leptokurtic phenotypic distributions, in addition to normal or near normal phenotypic distributions. The performances of nonparametric trend statistics, including nonparametric regression slope, are compared with the HE regression methods as genetic linkage strategies.     

Nonparametric functional mapping of quantitative trait loci

Summary .  Functional mapping is a useful tool for mapping quantitative trait loci (QTL) that control dynamic traits. It incorporates mathematical aspects of biological processes into the mixture model-based likelihood setting for QTL mapping, thus increasing the power of QTL detection and the precision of parameter estimation. However, in many situations there is no obvious functional form and, in such cases, this strategy will not be optimal. Here we propose to use nonparametric function estimation, typically implemented with B-splines, to estimate the underlying functional form of phenotypic trajectories, and then construct a nonparametric test to find evidence of existing QTL. Using the representation of a nonparametric regression as a mixed model, the final test statistic is a likelihood ratio test. We consider two types of genetic maps: dense maps and general maps, and the power of nonparametric functional mapping is investigated through simulation studies and demonstrated by examples.     

Statistical Genetics of an Annual Plant, Impatiens Capensis. I. Genetic Basis of Quantitative Variation

Analysis of quantitative genetics in natural populations has been hindered by computational and methodological problems in statistical analysis. We developed and validated a jackknife procedure to test for existence of broad sense heritabilities and dominance or maternal effects influencing quantitative characters in Impatiens capensis. Early life cycle characters showed evidence of dominance and/or maternal effects, while later characters exhibited predominantly environmental variation. Monte Carlo simulations demonstrate that these jackknife tests of variance components are extremely robust to heterogeneous error variances. Statistical methods from human genetics provide evidence for either a major locus influencing germination date, or genes that affect phenotypic variability per se. We urge explicit consideration of statistical behavior of estimation and testing procedures for proper biological interpretation of statistical results.     

FiSSE: A simple nonparametric test for the effects of a binary character on lineage diversification rates

It is widely assumed that phenotypic traits can influence rates of speciation and extinction, and several statistical approaches have been used to test for correlations between character states and lineage diversification. Recent work suggests that model‐based tests of state‐dependent speciation and extinction are sensitive to model inadequacy and phylogenetic pseudoreplication. We describe a simple nonparametric statistical test (“FiSSE”) to assess the effects of a binary character on lineage diversification rates. The method involves computing a test statistic that compares the distributions of branch lengths for lineages with and without a character state of interest. The value of the test statistic is compared to a null distribution generated by simulating character histories on the observed phylogeny. Our tests show that FiSSE can reliably infer trait‐dependent speciation on phylogenies of several hundred tips. The method has low power to detect trait‐dependent extinction but can infer state‐dependent differences in speciation even when net diversification rates are constant. We assemble a range of macroevolutionary scenarios that are problematic for likelihood‐based methods, and we find that FiSSE does not show similarly elevated false positive rates. We suggest that nonparametric statistical approaches, such as FiSSE, provide an important complement to formal process‐based models for trait‐dependent diversification.     

Little epistasis for anxiety-related measures in the DeFries strains of laboratory mice

Recent advances in methodologies for testing epistatic interactions, combined with several successes in demonstrating genetic interaction effects in animal and human genetics, have rekindled interest in the role of epistatic influences on complex traits. It has even been suggested that the unacknowledged presence of epistasis vitiates the genetic dissection of human and animal behavior. Here we report a genome-wide interaction analysis of 1636 F₂ mice to show that epistasis is of minimal importance in an animal model of anxiety. By using a sufficiently large sample of F₂ animals, we provide evidence that interaction effects between any two loci contribute less than 5% to the total phenotypic variance in multiple tests of anxiety. We conclude that interactions between loci do not necessarily vitiate the genetic analysis of behavior in at least one animal model of anxiety.     

Second-order moments of segregating sites under variable population size

The identification of genomic regions that have been exposed to positive selection is a major challenge in population genetics. Since selective sweeps are expected to occur during environmental changes or when populations are colonizing a new habitat, statistical tests constructed on the assumption of constant population size are biased by the co-occurrence of population size changes and selection. To delimit this problem and gain better insights into demographic factors, theoretical results regarding the second-order moments of segregating sites, such as the variance of segregating sites, have been derived. Driven by emerging genomewide surveys, which allow the estimation of demographic parameters, a generalized version of Tajima's D has been derived that takes into account a previously estimated demographic scenario to test single loci for traces of selection against the null hypothesis of neutral evolution under variable population size.     

Molecular population genetics and the search for adaptive evolution in plants

The first papers on plant molecular population genetics were published approximately 10 years ago. Since that time, well over 50 additional studies of plant nucleotide polymorphism have been published, and many of these studies focused on detecting the signature of balancing or positive selection at a locus. In this review, we discuss some of the theoretical and statistical issues surrounding the detection of selection, with focus on plant populations, and we also summarize the empirical plant molecular population genetics literature. At face value, the literature suggests that a history of balancing or positive selection in plant genes is rampant. In two well-studied taxa (maize and Arabidopsis) over 20% of studied genes have been interpreted as containing the signature of selection. We argue that this is probably an overstatement of the prevalence of natural selection in plant genomes, for two reasons. First, demographic effects are difficult to incorporate and have generally not been well integrated into the plant population genetics literature. Second, the genes studied to date are not a random sample, so selected genes may be overrepresented. The next generation of studies in plant molecular population genetics requires additional sampling of local populations, explicit comparisons among loci, and improved theoretical methods to control for demography. Eventually, candidate loci should be confirmed by explicit consideration of phenotypic effects.     

Improving the power of sib pair quantitative trait loci detection by phenotype winsorization

OBJECTIVES: In sib pair studies, quantitative trait loci (QTL) identification may be adversely affected by non-normality in the phenotypic distribution, particularly when subjects falling in the tails of the distribution bias the trait mean or variance. We evaluated the robustness and power of reducing the influence of subjects with extreme phenotypic values by Winsorizing non-normal distributions in three versions of Haseman-Elston regression-based methods of QTL linkage analysis. METHODS: Data were simulated for normal and non-normal distributions. Phenotypic values that correspond to cutoff points at the omega and 1 - omega percentiles of the distribution were identified, and phenotypic values falling outside the boundaries of the omega and 1 - omega cutoff points were replaced by the omega and 1 - omega values, respectively. One million replications were performed for the three tests of linkage for Winsorized and non-Winsorized data. RESULTS: Winsorization reduced conservatism in the tails of the empirical type I error rate for the vast majority of the tests of linkage, increased the power of QTL detection in non-normal data and created a slight negative bias in symmetrical phenotypic distributions. CONCLUSIONS: Winsorizing can improve the power of QTL detection with certain non-normal distributions but can also introduce bias into the estimate of the QTL effect.     

Neutral mutation as the source of genetic variation in life history traits

The mechanism underlying the maintenance of adaptive genetic variation is a long-standing question in evolutionary genetics. There are two concepts (mutation-selection balance and balancing selection) which are based on the phenotypic differences between alleles. Mutation - selection balance and balancing selection cannot properly explain the process of gene substitution, i.e. the molecular evolution of quantitative trait loci affecting fitness. I assume that such loci have non-essential functions (small effects on fitness), and that they have the potential to evolve into new functions and acquire new adaptations. Here I show that a high amount of neutral polymorphism at these loci can exist in real populations. Consistent with this, I propose a hypothesis for the maintenance of genetic variation in life history traits which can be efficient for the fixation of alleles with very small selective advantage. The hypothesis is based on neutral polymorphism at quantitative trait loci and both neutral and adaptive gene substitutions. The model of neutral - adaptive conversion (NAC) assumes that neutral alleles are not neutral indefinitely, and that in specific and very rare situations phenotypic (relative fitness) differences between them can appear. In this paper I focus on NAC due to phenotypic plasticity of neutral alleles. The important evolutionary consequence of NAC could be the increased adaptive potential of a population. Loci responsible for adaptation should be fast evolving genes with minimally discernible phenotypic effects, and the recent discovery of genes with such characteristics implicates them as suitable candidates for loci involved in adaptation.     

Population genetics of short tandem repeat (STR) loci

To investigate the population genetics of short tandem repeat (STR) polymorphisms in human populations, we have studied the allele frequency distributions of four STR loci (HUMTH01, HUMVWA31, HUMF13A1 and HUMFES) in 16 different population surveys which can be categorised within three broadly defined ethnic groups: Caucasian, Asian (Indian subcontinent), and African (Afro-Caribbean and US black). We have observed that allele frequency distributions of populations within ethnic groups are similar; consequently, genetic distances are an order of magnitude lower than between ethnic groups. Inbreeding coefficients (F-statistics) and calculations of the number of mean heterozygous loci per individual, along with estimates of variance, did not suggest that the populations were substructured. This included a study of an immigrant Asian population known to comprise at least three different sub-groups. Finally, an indication of the discriminating power is given by calculation of likelihood ratios (LR) of each individual tested across all four loci. Approximately 70% of Caucasians give an LR of greater than 10,000; the test is even more discriminating in Afro-Caribbeans-approximately 90% of tests are greater than 10,000.Editor's commentsThe authors present data generated by the move from VNTR to STR loci for human identification. The data they present for samples within major racial groupings address some of the concerns about population substructuring discussed by Balding and Nichols in this volume.     

Quantifying the genetic component of phenotypic variation in unpedigreed wild plants: tailoring genomic scan for within-population use

The study of adaptive genetic variation in natural populations is central to evolutionary biology. Quantitative genetics methods, however, are hardly applicable to long-lived organisms, and current knowledge on adaptive genetic variation in wild plants mostly refers to annuals and short-lived perennials. Studies on long-lived species are essential to explore possible life-history correlates of genetic variation, selection, and trait heritability. In this paper, we propose a method based on molecular markers to quantify the genetic basis of individual phenotypic differences in wild plants under natural conditions. Rather than focusing on inferring individual relatedness to estimate the heritability of phenotypic traits, we directly estimate the proportion of observed phenotypic variance that is statistically accounted for by genotypic differences between individuals. This is achieved by (i) identifying loci that are correlated across individuals with the phenotypic trait of interest by means of an amplified fragment length polymorphism (AFLP)-based explorative genomic scan, and (ii) fitting multiple regression and linear random effect models to estimate the effects of genotype, environment and genotype × environment on phenotypes. We apply this method to estimate genotypic and environmental effects on cumulative maternal fecundity in a wild population of the long-lived Viola cazorlensis monitored for 20 years. Results show that between 56–63% (depending on estimation method) of phenotypic variance in fecundity is accounted for by genotypic differences in 11 AFLP loci that are significantly related to fecundity. Genotype × environment effects accounted for 38% of fecundity variance, which may help to explain the unexpectedly high levels of genetic variance for fecundity found.     

Parallel evolution and inheritance of quantitative traits

Parallel phenotypic evolution, the independent evolution of the same trait in closely related lineages, is interesting because it tells us about the contribution of natural selection to phenotypic evolution. Haldane and others have proposed that parallel evolution also results from a second process, the similarly biased production of genetic variation in close relatives, an idea that has received few tests. We suggest that influence of shared genetic biases should be detectable by the disproportionate use of the same genes in independent instances of parallel phenotypic evolution. We show how progress in testing this prediction can be made through simple tests of parallel inheritance of genetic differences: similar additive, dominance, and epistasis components in analysis of line means and similar effective numbers of loci. We demonstrate parallel inheritance in two traits, lateral plate number and body shape, in two lineages of threespine stickleback that have adapted independently to freshwater streams on opposite sides of the Pacific Ocean. Notably, reduction of plate number in freshwater involves a substitution at the same major locus in both lineages. Our results represent only a first step in the study of the genetics of parallel phenotypic evolution in sticklebacks. Nevertheless, we have shown how such studies can be employed to test the genetic hypothesis of parallel evolution and how study of parallel evolution might yield insights into the roles of both selection and genetic constraint in phenotypic evolution.     

Population structure, demographic history, and evolutionary patterns of a green-fruited tomato, Solanum peruvianum (Solanaceae), revealed by spatial genetics analyses

? Premise of the study: Wild relatives of crop species have long been viewed as an important genetic resource for crop improvement, but basic information about the population biology of these species is often lacking. This study investigated the population structure, demographic history, and evolutionary patterns of a green-fruited relative of the cultivated tomato, Solanum peruvianum. ? Methods: We investigated spatial genetics of S. peruvianum and screened for loci potentially under natural selection by integrating amplified fragment length polymorphism (AFLP) genotypes, phenotypic data, geography, and geographic information system (GIS)-derived climate data of 19 natural populations. ? Key results: Solanum peruvianum had a moderate degree of population differentiation, likely reflecting partial geographic isolation between species. Populations had a distribution pattern consistent with north-to-south "stepping-stone" dispersal with significant isolation by distance (IBD), similar to other tomato species. Several AFLP loci showed evidence of selection and associated with climate variables. However, phenotypic traits generally did not correlate with climate variables. ? Conclusions: Geographic features of the coastal Andes is likely an important factor that determines the migration pattern and population structure of S. peruvianum, but climatic factors do not appear to be critical for its phenotypic evolution, perhaps due to a high degree of phenotypic plasticity. Spatial genetics of wild relatives of crop species is a powerful approach to understand their evolutionary patterns and to accelerate the discovery of their potential for crop improvements.     

Landscape genetic analyses reveal cryptic population structure and putative selection gradients in a large-scale estuarine environment

Disentangling the relative contributions of selective and neutral processes underlying phenotypic and genetic variation under natural, environmental conditions remains a central challenge in evolutionary ecology. However, much of the variation that could be informative in this area of research is likely to be cryptic in nature; thus, the identification of wild populations suitable for study may be problematic. We use a landscape genetics approach to identify such populations of three-spined stickleback inhabiting the Saint Lawrence River estuary. We sampled 1865 adult fish over multiple years. Individuals were genotyped for nine microsatellite loci, and georeferenced multilocus data were used to infer population groupings, as well as locations of genetic discontinuities, under a Bayesian model framework ( geneland ). We modelled environmental data using nonparametric multiple regression to explain genetic differentiation as a function of spatio-ecological effects. Additionally, we used genotype data to estimate dispersal and gene flow to parameterize a simple model predicting adaptive vs. plastic divergence between demes. We demonstrate a bipartite division of the genetic landscape into freshwater and maritime zones, independent of geographical distance. Moreover, we show that the greatest proportion of genetic variation (31.5%) is explained by environmental differences. However, the potential for either adaptive or plastic divergence between demes is highly dependent upon the strength of migration and selection. Consequently, we highlight the utility of landscape genetics as a tool for hypothesis generation and experimental design, to identify focal populations and putative selection gradients, in order to distinguish between phenotypic plasticity and local adaptation.     

QUANTITATIVE GENETICS AND POPULATION DYNAMICS

This study examines the dynamics of a competition and a host-parasite model in which the interactions are determined by quantitative characters. Both models are extensions of one-dimensional difference equations that can exhibit complicated dynamics. Compared to these basic models, the phenotypic variability given by the quantitative characters reduces the size of the density fluctuations in asexual populations. With sexual reproduction, which is described by modeling the genetics of the quantitative character explicitly with many haploid loci that determine the character additively, this reduction in fitness variance is magnified. Moreover, quantitative genetics can induce simple dynamics. For example, the sexual population can have a two-cycle when the asexual system is chaotic. This paper discusses the consequences for the evolution of sex. The higher mean growth rate implied by the lower fitness variance in sexual populations is an advantage that can overcome a twofold intrinsic growth rate of asexuals. The advantage is bigger when the asexual population contains only a subset of the phenotypes present in the sexual population, which conforms with the tangled bank theory for the evolution of sex and shows that tangled bank effects also occur in host-parasite systems. The results suggest that explicitly describing the genetics of a quantitative character leads to more flexible models than the usual assumption of normal character distributions, and therefore to a better understanding of the character's impact on population dynamics.     

Wald Tests of Location for Symmetric Nonnormal Data

For nonnormal data we suggest a test of location based on a broader family of distributions than normality. Such a test will in a sense fall between the standard parametric and non parametric tests. We see that the Wald tests based on this family of distributions have some advantages over the score tests and that they perform well in comparison to standard parametric and nonparametric tests in a variety of situations. We also consider when and how to apply such tests in practice.     

Mathematical models in genetics

In this study, we present some of the basic ideas of population genetics. The founders of population genetics are R.A. Fisher, S. Wright, and J. B.S. Haldane. They, not only developed almost all the basic theory associated with genetics, but they also initiated multiple experiments in support of their theories. One of the first significant insights, which are a result of the Hardy–Weinberg law, is Mendelian inheritance preserves genetic variation on which the natural selection acts. We will limit to simple models formulated in terms of differential equations. Some of those differential equations are nonlinear and thus emphasize issues such as the stability of the fixed points and time scales on which those equations operate. First, we consider the classic case when selection acts on diploid locus at which wу can get arbitrary number of alleles. Then, we consider summaries that include recombination and selection at multiple loci. Also, we discuss the evolution of quantitative traits. In this case, the theory is formulated in respect of directly measurable quantities. Special cases of this theory have been successfully used for many decades in plants and animals breeding.     

Comparisons of likelihood and machine learning methods of individual classification

Classification methods used in machine learning (e.g., artificial neural networks, decision trees, and k-nearest neighbor clustering) are rarely used with population genetic data. We compare different nonparametric machine learning techniques with parametric likelihood estimations commonly employed in population genetics for purposes of assigning individuals to their population of origin ("assignment tests"). Classifier accuracy was compared across simulated data sets representing different levels of population differentiation (low and high F(ST)), number of loci surveyed (5 and 10), and allelic diversity (average of three or eight alleles per locus). Empirical data for the lake trout (Salvelinus namaycush) exhibiting levels of population differentiation comparable to those used in simulations were examined to further evaluate and compare classification methods. Classification error rates associated with artificial neural networks and likelihood estimators were lower for simulated data sets compared to k-nearest neighbor and decision tree classifiers over the entire range of parameters considered. Artificial neural networks only marginally outperformed the likelihood method for simulated data (0-2.8% lower error rates). The relative performance of each machine learning classifier improved relative likelihood estimators for empirical data sets, suggesting an ability to "learn" and utilize properties of empirical genotypic arrays intrinsic to each population. Likelihood-based estimation methods provide a more accessible option for reliable assignment of individuals to the population of origin due to the intricacies in development and evaluation of artificial neural networks.