Second-order approximations for selection coefficients at polygenic loci

I determine the second-order approximation for the phenotypic distribution of an arbitrary number of quantitative traits, ignoring the effects of epistasis and linkage disequilibrium, conditioned on the presence of a specified genotype at one underlying locus of small effect. Using this approximation, I determine formulae for the effects of selection at a single locus with random mating under either Gaussian stabilizing selection, or correlated selection with truncation selection for one character. These formulae apply for arbitrary phenotypic distributions, yet even with multivariate Gaussian distributions of phenotypic effects the formula for correlated selection includes a correction to the standard formula in Falconer (1989). 1 demonstrate that this approximation has an error that is third order in the allelic or genotypic effects, independent of the form of the phenotypic distribution. I show also that the approximation of analogous form for the phenotypic distribution conditioned on the presence of a specified allele at a single locus is also correct to second order. Both approximations allow for dominance and are consistent in the sense that computing marginal fitnesses from approximations based on genotypic deviations and those based on average allelic effect yield the same answers.Supported by PHS Grant ROI GM 32130     

PHYSIOLOGICAL AND BEHAVIORAL ADAPTATION TO VARYING ENVIRONMENTS: A MATHEMATICAL MODEL

We develop a mathematical model to explore the evolution of habitat selection and physiological adaptation in a heterogeneous environment. The model assumes the following conditions: 1) a panmictic population of infinite size; 2) prereproductive individuals mobile enough to move between patches; 3) alleles at one locus code for absence or presence of adaptation to detrimental patches; 4) alleles at a second locus code for absence or presence of behavior(s) that cause avoidance of the detrimental patches; 5) additive effects of alleles controlling physiology and behavior; 6) frequency-independent fitness. Results of the model indicate that nontrivial, polymorphic equilibria do not exist. The pattern of genotypic fitnesses and the initial allelic frequencies can influence whether the population adapts by physiological or behavioral mechanisms, or by both. Linkage between the two loci can alter the outcome of evolution, given specified genotypic fitness values and initial allelic frequencies.     

Moments,cumulants, and polygenic dynamics

A new approach for describing the evolution of polygenic traits subject to selection and mutation is presented. Differential equations for the change of cumulants of the allelic frequency distribution at a particular locus and for the cumulants of the distributions of genotypic and phenotypic values are derived. The derivation is based on the assumptions of random mating, no sex differences, absence of random drift, additive gene action, linkage equilibrium, and Hardy-Weinberg proportions. Cumulants are a set of parameters that, like moments, describe the shape of a probability density. Compared with moments, however, they have properties that make them a much more convenient tool for investigating polygenic traits. Applications to directional and stabilizing selection are given.     

The diffusion model for migration and selection in a plant population

 The diffusion approximation is derived for migration and selection at a multiallelic locus in a partially selfing plant population subdivided into a lattice of colonies. Generations are discrete and nonoverlapping; both pollen and seeds disperse. In the diffusion limit, the genotypic frequencies at each point are those determined at equilibrium by the local rate of selfing and allelic frequencies. If the drift and diffusion coefficients are taken as the appropriate linear combination of the corresponding coefficients for pollen and seeds, then the migration terms in the partial differential equation for the allelic frequencies have the standard form for a monoecious animal population. The selection term describes selection on the local genotypic frequencies. The boundary conditions and the unidimensional transition conditions for a geographical barrier and for coincident discontinuities in the carrying capacity and migration rate have the standard form. In the diallelic case, reparametrization renders the entire theory of clines and of the wave of advance of favorable alleles directly applicable to plant populations. Received 30 August 1995; received in revised form 23 February 1996     

A model of assortative mating with partial dominance.

A model of assortative mating incorporating partial dominance is proposed for a single locus with two alleles. It is derived by starting from an arbitrary genotypic distribution and finding symmetric and non-selective mating frequencies which duplicate this distribution. Numerical values are imputed to genotypes, the homozygotes having numerically equal values, opposite in sign, and the heterozygote having a value determined by the gene and heterozygote frequencies. The model is specified in a canonical form which reveals the correlation between mates based on genotypic values, and relates the correlation to the fixation index. It permits negative as well as positive values of the fixation index. It is shown that this general model includes several particular cases, in equilibrium phase, occurring in the literature.     

Genotypic correlations and QTL correspondence between line per se and testcross performance in sugar beet (Beta vulgaris L.) for the three agronomic traits beet yield,potassium content,and sodium content

The genotypic correlation between line per se and testcross performance is an important quantitative genetic parameter in the design of hybrid breeding programs. The main goal of this survey was to study the association of line per se and testcross performance at the phenotypic and molecular levels by applying multiple-line cross quantitative trait locus (QTL) mapping. We used experimental data from line per se and testcross performance of three segregating sugar beet (Beta vulgaris L.) populations. The segregating progenies were genotyped with 481 single nucleotide polymorphism and 40 simple sequence repeat markers and evaluated in field trials for beet yield as well as potassium and sodium content. We observed a decrease in the genotypic correlations between testcross and line per se performance with increasing complexity of the analyzed trait. This picture was also reflected at a molecular level by the presence of overlapping QTLs. A more detailed analysis of the forces causing low genotypic correlation between line per se and testcross performance could not rule out a possible relevance of epistasis and suggested the presence of masking dominance effects.     

Inconsistencies in standard approximations for selection coefficients at loci affecting a polygenic character

Inconsistencies exist in the standard expansions used to approximate selection coefficients for alleles at a locus underlying a quantitative character. Allelic (marginal) fitnesses obtained from expansions based on average excesses differ from allelic fitnesses obtained from expansions based on genotypic values. Similarly, the mean population fitness based on summing over either allelic or genotypic fitnesses usually differs mean population fitness obtained by averaging over the unrestricted phenotypic distribution. A consistent value of requires no variation in genotypic values. If, as suggested by Nagylaki (1984), expansions are corrected for the decrease in phenotypic variance resulting from conditioning on the presence of a particular allele or genotype, inconsistencies still exist. Unless W(z)[V _z p(z) + zp(z) + p(z)] dz = 0, where p(z) is the phenotypic probability density function, V _z the phenotypic variance, W( _z ) the fitness of phenotypic value z, the primes denote differentiation with respect to z, allelic fitnesses based on average effects differ from allelic fitnesses based on genotypic values. This condition must also be satisfied in order for either expansion to give a consistent , as first shown by Nagylaki. For arbitrary W(z), this is satisfied if and only if phenotypes are normally distributed.     

Identifying quantitative trait locus by genetic background interactions in association studies

Association studies are designed to identify main effects of alleles across a potentially wide range of genetic backgrounds. To control for spurious associations, effects of the genetic background itself are often incorporated into the linear model, either in the form of subpopulation effects in the case of structure or in the form of genetic relationship matrices in the case of complex pedigrees. In this context epistatic interactions between loci can be captured as an interaction effect between the associated locus and the genetic background. In this study I developed genetic and statistical models to tie the locus by genetic background interaction idea back to more standard concepts of epistasis when genetic background is modeled using an additive relationship matrix. I also simulated epistatic interactions in four-generation randomly mating pedigrees and evaluated the ability of the statistical models to identify when a biallelic associated locus was epistatic to other loci. Under additive-by-additive epistasis, when interaction effects of the associated locus were quite large (explaining 20% of the phenotypic variance), epistasis was detected in 79% of pedigrees containing 320 individuals. The epistatic model also predicted the genotypic value of progeny better than a standard additive model in 78% of simulations. When interaction effects were smaller (although still fairly large, explaining 5% of the phenotypic variance), epistasis was detected in only 9% of pedigrees containing 320 individuals and the epistatic and additive models were equally effective at predicting the genotypic values of progeny. Epistasis was detected with the same power whether the overall epistatic effect was the result of a single pairwise interaction or the sum of nine pairwise interactions, each generating one ninth of the epistatic variance. The power to detect epistasis was highest (94%) at low QTL minor allele frequency, fell to a minimum (60%) at minor allele frequency of about 0.2, and then plateaued at about 80% as alleles reached intermediate frequencies. The power to detect epistasis declined when the linkage disequilibrium between the DNA marker and the functional polymorphism was not complete.     

Associative overdominance: Evaluating the effects of inbreeding and linkage disequilibrium

Expressions are obtained for the expected phenotypic values of homozygous and heterozygous genotypes for a neutral marker locus linked to a locus segregating for a recessive deleterious gene. The phenotypic values are functions of the allele frequencies at the marker locus, the inbreeding coefficient and the degree of association of the deleterious gene with the marker alleles. The analysis is extended to more than two alleles at the marker locus. Either linkage disequilibrium or inbreeding alone can produce an apparent superiority of heterozygotes for the marker locus (unless specified otherwise, the terms ‘homozygote’ and ‘heterozygote’ will refer to the marker locus). The effect of linkage disequilibrium on the difference between the heterozygote and homozygote values can be positive (associative overdominance) or negative (associative underdominance), depending on the frequencies of the marker alleles and the degree of their association with the deleterious gene. Inbreeding has always a positive effect. In general, the expected value of a homozygote is a positive function of its allele frequency. When the various homozygous genotypes are combined into one class and the various heterozygous genotypes into another, the phenotypic difference of the two classes is a function of the evenness of the allelic frequency distribution. Inbreeding is a more likely explanation of associative overdominance if the frequency of the deleterious gene is low, but its effect on the character high. Conversely, linkage disequilibrium is more likely if the frequency is high and the effect low. The degrees of association between marker alleles and the deleterious gene can, in principle, be estimated from the observed phenotypic scores and used to calculate expected multi-locus genotype scores. This could provide the basis for statistical tests of the associative overdominance hypothesis as an explanation of observed correlations between multi-locus heterozygosity and phenotypic traits.     

The effect of positive assortative mating at one locus on a second linked locus

Summary A model for positive assortative mating based on genotype for one locus is employed to investigate the effect of this mating system on the genotypic structure of a second linked locus as well as on the joint genotypic structure of these two loci. It is shown that the second locus does not attain a precise positive assortative mating structure, but yet it shares a property that is characteristic of positive assortative mating, namely an increase in the frequency of homozygotes over that typically found in panmictic structures. Given any arbitrary genotypic structure for the parental population, the resulting offspring generation possesses a structure at the second locus that does not depend on the recombination frequency, while the joint structure of course does. In case assortative mating as well as linkage are not complete, there exists a unique joint equilibrium state for the two loci, which is characterized by complete stochastic independence between the two loci as well as by Hardy-Weinberg proportions at the second locus. For the second locus alone, Hardy-Weinberg equilibrium is realized if and only if gametic linkage equilibrium and an additionally specified condition are realized.     

Studies of Esterase-6 in DROSOPHILA MELANOGASTER. II. the Genetics and Frequency Distributions of Naturally Occurring Variants Studied by Electrophoretic and Heat Stability Criteria

Measurements of the electrophoretic mobility and thermostability of esterase-6 allozymes have been used to determine the amount of allelic variation at the esterase-6 locus in Drosophila melanogaster. We studied 398 homozygous lines obtained from four natural populations. Use of a spectrophotometric assay for esterase-6 activity has allowed precise quantitation of heat-stability variants. Using these methods, eight putative alleles were detected within the two most common electrophoretic classes. Analyses of F1 and F2 progeny show that the behavior of stability variants is consistent with the hypothesis that this variation is due to allelic variation at the Est-6 locus. Analyses of the gene-frequency distributions within and between populations show (1) that observed allele-frequency distributions do not deviate significantly from those expected for neutral variants, and (2) that there is little evidence for an increase in apparent divergence of the different populations at the genotypic or phenotypic levels when the additional variation detected is considered. These findings suggest that gene-frequency analysis alone is unlikely to resolve the question of the selective significance of allozyme variation.     

Genotypic diversity: estimation and prediction in samples

We show that a commonly used statistic of genotypic diversity can be used to reflect one form of deviation from panmixia, viz. clonal reproduction, by comparing observed and predicted sample statistics. The characteristics of the statistic, in particular its relationship with population genotypic diversity, are formalised and a method of predicting the genotypic diversity of a sample drawn from a panmictic population using allelic frequencies and sample size is developed. The sensitivity of some possible tests of significance of the deviation from panmictic expectations is examined using computer simulations. Goodness-of-fit tests are robust but produce an unacceptably high level of type II error. With means and variances calculated either from Monte Carlo simulations or from distributional and series approximations, t-tests perform better than goodness-of-fit tests. Under simulation, both forms of t-test exhibit acceptable rates of type I error. Rates of type II are usually large when allele frequencies are severely skewed although the latter test performs the better in those conditions.     

Association analysis of some drought related characters in hexaploid spring wheat (Triticum aestivum L.).

The association among yield components and their direct and indirect influence on the grain yield of wheat were investigated. 24 breeding lines were tested in a randomized complete block experiment design with three replications. According to the results the phenotypic correlation among the traits and their path coefficient were estimated. Positive significant correlation coefficients were obtained for association between survival rate treatment I, III, leaf venation, stomatal frequency, osmotic pressure, flag leaf area and number of tillers per plant with grain yield per plant at both phenotypic and genotypic levels. Negatively significant correlation between hygrophilic colloids and epidermal cell size with grain yield per plant was obtained at phenotypic and genotypic levels. Path coefficient was also computed to estimate the contribution of character to the yield. Path coefficient analysis revealed that flag leaf area, root/shoot ratio and survival rate II had the highest positive direct effects on grain yield, while hygrophilic colloids and osmotic pressure had negative direct effect on grain yield. These results thus obtained suggested that flag leaf area is an important component of yield and hence needs special attention in selection strategies.     

Marked genetic polymorphism of the swine steroid 21-hydroxylase gene, and its location between the SLA class I and class II regions

Restriction fragment length polymorphism (RFLP) analysis of the swine 21-hydroxylase (CYP21) region was conducted on 31 unrelated SLA class I typed pigs, mainly Large Whites, including 15 haplotypes. Ten haplotypes were from SLA genotypic homozygotes and five were from SLA class I phenotypic homozygotes. DNA digestion with Hin dIII, TaqI and PstI, and hybridization to a 4.5-kb swine CYP21 genomic probe yielded respectively two, four and three RFLP patterns. Six patterns were identified with combined RFLP. In addition, analysis of the CYP21 region in families comprising several SLA recombinants demonstrated that the CYP21 gene lies in the DNA segment between the SLA class I and class II regions. These overall results reinforce our previous conclusion about the existence in the pig of a single 21-hydroxylase gene. The characterization of at least six CYP21 allelic patterns provides a new tool for studying the associations between the SLA region and zootechnical traits.     

Novel bivariate moment-closure approximations

Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.     

Novel moment closure approximations in stochastic epidemics

Moment closure approximations are used to provide analytic approximations to non-linear stochastic population models. They often provide insights into model behaviour and help validate simulation results. However, existing closure schemes typically fail in situations where the population distribution is highly skewed or extinctions occur. In this study we address these problems by introducing novel second-and third-order moment closure approximations which we apply to the stochastic SI and SIS epidemic models. In the case of the SI model, which has a highly skewed distribution of infection, we develop a second-order approximation based on the beta-binomial distribution. In addition, a closure approximation based on mixture distribution is developed in order to capture the behaviour of the stochastic SIS model around the threshold between persistence and extinction. This mixture approximation comprises a probability distribution designed to capture the quasi-equilibrium probabilities of the system and a probability mass at 0 which represents the probability of extinction. Two third-order versions of this mixture approximation are considered in which the log-normal and the beta-binomial are used to model the quasi-equilibrium distribution. Comparison with simulation results shows: (1) the beta-binomial approximation is flexible in shape and matches the skewness predicted by simulation as shown by the stochastic SI model and (2) mixture approximations are able to predict transient and extinction behaviour as shown by the stochastic SIS model, in marked contrast with existing approaches. We also apply our mixture approximation to approximate a likehood function and carry out point and interval parameter estimation.     

Transformation of QTL genotypic effects to allelic effects

The genotypic and allelic effect models are equivalent in terms of QTL detection in a simple additive model, but the QTL allelic model has the advantage of providing direct information for marker-assisted selection. However, the allelic matrix is four times as large as the genotypic IBD matrix, causing computational problems, especially in genome scans examining multiple positions. Transformation from genotypic to allelic effects, after estimating the genotypic effects with a smaller IBD matrix, can solve this problem. Although the validity of transformation from genotypic to allelic effects has been disputed, this work proves that transformation can successfully yield unique allelic effects when genotypic and allelic IBD matrixes exist.     

Definition and Estimation of Higher-Order Gene Fixation Indices

Fixation indices summarize the associations between genes that arise from the joint effects of inbreeding and selection. In this paper, fixation indices are derived for pairs, triplets and quadruplets of genes at a single multiallelic locus. The fixation indices are obtained by dividing cumulants by constants; the cumulants describe the statistical distribution of alleles and the constants are functions of gene frequency. The use of cumulants instead of moments is necessary only for four-gene indices, when the fourth cumulant is used. A second type of four-gene index is also required, and this index is based upon the covariation of second-order cumulants. At multiallelic loci, a large number of indices is possible. If alleles are selectively neutral, the number of indices is reduced and the relationship between gene identity and gene cumulants is shown.--Two-gene indices can always be estimated from genotypic frequency data at a single polymorphic locus. Three-gene indices are also estimable except when allele frequency equals one-half. Four-gene indices are not estimable unless selection is assumed to have an equal effect upon each allele (such as under selective neutrality) and the locus contains at least three alleles of unequal frequency. For diallelic or selected loci, an alternative four-gene fixation index is proposed. This index incorporates both types of four-gene associations but cannot be related to gene identity.     

Multiple Mating But Not Recombination Causes Quantitative Increase in Offspring Genetic Diversity for Varying Genetic Architectures

Explaining the evolution of sex and recombination is particularly intriguing for some species of eusocial insects because they display exceptionally high mating frequencies and genomic recombination rates. Explanations for both phenomena are based on the notion that both increase colony genetic diversity, with demonstrated benefits for colony disease resistance and division of labor. However, the relative contributions of mating number and recombination rate to colony genetic diversity have never been simultaneously assessed. Our study simulates colonies, assuming different mating numbers, recombination rates, and genetic architectures, to assess their worker genotypic diversity. The number of loci has a strong negative effect on genotypic diversity when the allelic effects are inversely scaled to locus number. In contrast, dominance, epistasis, lethal effects, or limiting the allelic diversity at each locus does not significantly affect the model outcomes. Mating number increases colony genotypic variance and lowers variation among colonies with quickly diminishing returns. Genomic recombination rate does not affect intra- and inter-colonial genotypic variance, regardless of mating frequency and genetic architecture. Recombination slightly increases the genotypic range of colonies and more strongly the number of workers with unique allele combinations across all loci. Overall, our study contradicts the argument that the exceptionally high recombination rates cause a quantitative increase in offspring genotypic diversity across one generation. Alternative explanations for the evolution of high recombination rates in social insects are therefore needed. Short-term benefits are central to most explanations of the evolution of multiple mating and high recombination rates in social insects but our results also apply to other species.