Analysis of multilocus zygotic associations

While nonrandom associations between zygotes at different loci (zygotic associations) frequently occur in Hardy-Weinberg disequilibrium populations, statistical analysis of such associations has received little attention. In this article, we describe the joint distributions of zygotes at multiple loci, which are completely characterized by heterozygosities at individual loci and various multilocus zygotic associations. These zygotic associations are defined in the same fashion as the usual multilocus linkage (gametic) disequilibria on the basis of gametic and allelic frequencies. The estimation and test procedures are described with details being given for three loci. The sampling properties of the estimates are examined through Monte Carlo simulation. The estimates of three-locus associations are not free of bias due to the presence of two-locus associations and vice versa. The power of detecting the zygotic associations is small unless different loci are strongly associated and/or sample sizes are large (>100). The analysis of zygotic associations not only offers an effective means of packaging numerous genic disequilibria required for a complete characterization of multilocus structure, but also provides opportunities for making inference about evolutionary and demographic processes through a comparative assessment of zygotic association vs. gametic disequilibrium for the same set of loci in nonequilibrium populations.     

Quantitative variability and multilocus polymorphism under epistatic selection

We study multilocus polymorphism under selection, using a class of fitness functions that account for additive, dominant, and pairwise additive-by-additive epistatic interactions. The dynamic equations are derived in terms of allele frequencies and disequilibria, using the notions of marginal systems and marginal fitnesses, without any approximations. Stationary values of allele frequencies and pairwise disequilibria under weak selection are calculated by regular perturbation techniques. We derive conditions for existence and stability of the multilocus polymorphic states. Using these results, we then analyze a number of models describing stabilizing selection on additive characters, with some other factors, and determine the conditions under which genetic quantitative variability is maintained.     

Multilocus Population Genetics with Weak Epistasis. II. Equilibrium Properties of Multilocus Models: What Is the Unit of Selection?

Using perturbation techniques, I study the equilibrium of deterministic discrete time multilocus models with weak epistasis. The most important results are on the relationship between epistasis and disequilibrium. Disequilibrium involving a particular set of loci reflects only epistasis simultaneously involving those loci. Moreover, all the disequilibria of all orders vary approximately as the inverse of the probability of at least one recombination event among the loci involved. Finally, higher order disequilibria among loci will be lower than lower order ones, even if the level of epistasis is the same at all orders. In this sense, the unit of selection is small. However, given the larger number of higher order disequilibria, these higher order disequilibria may play an important role in the computation of gametic frequencies from allelic frequencies in models with a large number of loci. Finally, I show that epistasis between blocks of loci will be averages of epistatic effects, not additions of epistatic effects. Thus, failure to find significant epistasis on a chromosomal basis does not rule out the importance of epistatic effects.     

Dynamics of correlated genetic systems. II. Simulation studies of chromosomal segments under selection

The dynamics of chromosomal segments under selection are investigated by comparing experimental data to simulations of simple models of selection. The simulations assume 93 loci distributed evenly along an entire chromosome. The two issues addressed in this paper concern rates of decay of linkage disequilibria for chromosomes under selection and rates of gene frequency change after perturbation of gametic frequencies to states near the edge of the gametic frequency simplex. The findings are: (1) If reasonable values of inbreeding depression are assumed, linkage disequilibria decays to zero but at a rate nearly twice that expected from neutral theory. Experimental results also show accelerated decay rates. The acceleration of decay seems to be a simple consequence of the increased heterozygosity produced by selection. It is, therefore, argued that massive linkage disequilibria, of the kind found by Franklin and Lewontin (1970) in their simulations, are unlikely to characterize the genetic structure of natural populations of random mating organisms. (2) It is possible to distinguish between two time-honored models of multilocus selection, known as the symmetric overdominant and classical models, on the basis of gene frequency change near the edge of the gametic frequency simplex assuming linkage disequilibria is intense. (3) Examination of experimental data from perturbation experiments shows that neither of these elementary models adequately account for observed rates of gene frequency change, although the symmetric overdominant model does provide the better fit. Instead the experimental data suggest a markedly nonuniform distribution of selective effects along the chromosome. The data also suggest that these selective effects combine in markedly nonadditive ways in determining joint fitness.     

Multilocus Structure of Natural Populations of HORDEUM SPONTANEUM

The association of alleles among different loci was studied in natural populations of Hordeum spontaneum, the evolutionary progenitor of cultivated barley. The variance of the number of heterozygous loci in two randomly chosen gametes affords a useful measure of such association. The behavior of this statistic in several particular models is described. Generally, linkage (gametic phase) disequilibrium tends to increase the variance above the value expected under complete independence. This increase is greatest when disequilibria are such as to maximize the sum of squares of the two-locus gametic frequencies.—When data on several loci per individual are available, the observed variance may be tested for its agreement with that expected under the hypothesis of complete interlocus independence, using the sampling theory of this model. When applied to allozyme data from 26 polymorphic populations of wild barley, this test demonstrated the presence of geographically widespread multilocus organization. On average, the variance was 80% higher than expected under random association. Gametic frequencies for four esterase loci in both of these populations of wild barley and two composite crosses of cultivated barley were analyzed. Most generations of the composites showed less multilocus structure, as measured by the indices of association, than the wild populations.     

The strength of the selection barrier between populations

Genetic differences among populations exposed to selection form barriers against genetic exchange by mortality among hybrids. The strength of such a selection barrier, with which one (recipient) population reacts against immigration from another (donor) population, may be measured as the cumulative mean fitness of hybrids and their descendants relative to the fitness of the recipient population. Previous work analysed a case of weak selection with pairwise epistatic interactions by assuming small genetic distance between two populations in contact. The present study allows large genetic difference between the donor and recipient populations and considers weak multilocus selection with arbitrary epistatic interactions between two or more linked loci. An approximate analytical expression for the barrier strength is obtained as an expansion in which the strength of selection plays the role of a small parameter. It is shown that allele frequencies and gametic linkage disequilibria contribute in different ways to the strength of the selection barrier.     

Levels of multiallelic overdominance fitness, heterozygote excess and heterozygote deficiency

Concepts and results on selection balance in multiallelic systems are described. These include a multidimensional concept of heterozygote excess and heterozygote deficiency, a hierarchy of means of assessment of heterozygote advantage, comparisons and contrasts of allelic versus gametic polymorphic states, and conditions defining stable equilibria of complementary gametic sets. The concepts are illustrated in the context of viability selection and behavioral models of kin selection and for two major categories of multilocus selection regimes.     

The Evolution of Multilocus Systems under Weak Selection

The evolution of multilocus systems under weak selection is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. The genotypic fitnesses may depend on the gametic frequencies and time. The results hold for s << c(min), where s and c(min) denote the selection intensity and the smallest two-locus recombination frequency, respectively. After an evolutionarily short time of t(1) ~ (ln s)/ln(1 - c(min)) generations, all the multilocus linkage disequilibria are of the order of s [i.e., O(s) as s -> 0], and then the population evolves approximately as if it were in linkage equilibrium, the error in the gametic frequencies being O(s). Suppose the explicit time dependence (if any) of the genotypic fitnesses is O(s(2)). Then after a time t(2) ~ 2t(1), the linkage disequilibria are nearly constant, their rate of change being O(s(2)). Furthermore, with an error of O(s(2)), each linkage disequilibrium is proportional to the corresponding epistatic deviation for the interaction of additive effects on fitness. If the genotypic fitnesses change no faster than at the rate O(s(3)), then the single-generation change in the mean fitness is ΔW = W(-1)V(g) + O(s(3)), where V(g) designates the genic (or additive genetic) variance in fitness. The mean of a character with genotypic values whose single-generation change does not exceed O(s(2)) evolves at the rate ΔZ = W(-1)C(g) + O(s(2)), where C(g) represents the genic covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character). Thus, after a short time t(2), the absolute error in the fundamental and secondary theorems of natural selection is small, though the relative error may be large.     

Reliable selfing rate estimates from imperfect population genetic data

Genotypic frequencies at codominant marker loci in population samples convey information on mating systems. A classical way to extract this information is to measure heterozygote deficiencies (FIS) and obtain the selfing rate s from FIS = s/(2 - s), assuming inbreeding equilibrium. A major drawback is that heterozygote deficiencies are often present without selfing, owing largely to technical artefacts such as null alleles or partial dominance. We show here that, in the absence of gametic disequilibrium, the multilocus structure can be used to derive estimates of s independent of FIS and free of technical biases. Their statistical power and precision are comparable to those of FIS, although they are sensitive to certain types of gametic disequilibria, a bias shared with progeny-array methods but not FIS. We analyse four real data sets spanning a range of mating systems. In two examples, we obtain s = 0 despite positive FIS, strongly suggesting that the latter are artefactual. In the remaining examples, all estimates are consistent. All the computations have been implemented in a open-access and user-friendly software called rmes (robust multilocus estimate of selfing) available at http://ftp.cefe.cnrs.fr, and can be used on any multilocus data. Being able to extract the reliable information from imperfect data, our method opens the way to make use of the ever-growing number of published population genetic studies, in addition to the more demanding progeny-array approaches, to investigate selfing rates.     

Multivariate Analysis of Gametic Disequilibrium in the Yanomama

The gametic disequilibria between all possible pairs of loci were examined for a set of eight codominant loci in each of fifty Yanomama villages, using a multivariate correlation analysis which reduces the results to a single measure of departure from multiple-locus-gametic equilibrium. Thirty-two of the fifty villages departed significantly from multiple-locus gametic equilibrium. The largest contributions to the departure from multiple-locus equilibrium were due to the disequilibria between MN and Ss and between Rh(Cc) and Rh(Ee), indicating the effects of tight linkage. After removing the effects of these obvious sources of disequilibrium, sixteen of the fifty villages still remained significantly out of equilibrium. The disequilibrium between any particular pair of loci was highly erratic from village to village, and (with the exception of the MN-Ss and Cc-Ee disequilibria) averaged out very close to zero overall, suggesting a lack of systematic forces (epistatic selection). The departure from equilibrium in any one village is in excess of that expected from random sampling alone, and is attributed primarily to the fission-fusion mode of village formation operative in the Yanomama and the fact that a single village consists of a few extended lineages. Village allele frequencies are highly correlated across loci, and most of the non-independence is accounted for by large correlations in the average allelic frequencies of different loci for related villages. It is suggested that these correlations also are due to territorial expansion and population growth. For the tribe as a whole, all but the tightly linked markers of the MNSs and Rh complexes are approximately uncorrelated, and large departures from multiple-locus Hardy-Weinberg expectation are primarily due to substantial Wahlund variance within the tribe. There is no need to postulate a role for selection in these disequilibria.     

Definition and Properties of Disequilibrium Statistics for Associations between Nuclear and Cytoplasmic Genotypes

We define and establish the interrelationships of four components of statistical association between a diploid nuclear gene and a uniparentally transmitted, haploid cytoplasmic gene: an allelic (gametic) disequilibrium (D), which measures associations between alleles at the two loci; and three genotypic disequilibria (D1, D2, D3), which measure associations between two cytotypes and the three respective nuclear backgrounds. We also consider an alternative set of measures, including D and the residual disequilibrium (d). The dynamics of these disequilibria are then examined under three conventional models of the mating system: (1) random mating; (2a) assortative mating without dominance (the "mixed-mating model"); and (2b) assortative mating with dominance ("O'Donald's model"). The trajectories of gametic disequilibria are similar to those for pairs of unlinked nuclear loci. The dynamics of genotypic disequilibria exhibit a variety of behaviors depending on the model and the initial conditions. Procedures for statistical estimation of cytonuclear disequilibria are developed and applied to several real and hypothetical data sets. Special attention is paid to the biological interpretations of various categories of allelic and genotypic disequilibria in hybrid zones. Genetic systems for which these statistics might be appropriate include nuclear genotype frequencies in conjunction with those for mitochondrial DNA, chloroplast DNA, or cytoplasmically inherited microorganisms.     

The Genetical Response to Natural Selection by Varied Environments. IV. Gametic Disequilibrium in Spatially Varied Environments

Gametic disequilibria between allozyme loci were related to spatial variation of the environment in caged populations of Drosophila melanogaster . Two experiments, one with flies collected at "Chateau Tahbilk," South Australia, and the other with flies from "Groningen," The Netherlands, were sampled at generations 16 and 32. Spatial variation of the environment was stimulated using three food media. Eight polymorphic allozyme loci were used to estimate gametic disequilibria from digenic combinations of allotypes. All populations were duplicated within an environment and maintained at about 2500 adults. Standardized gametic disequilibria were compared by a weighted least squares analysis of the z-transformed statistical correlation of allele frequencies. Gametic disequilibria were strongly dependent upon food niche and food-niche interactions. The effects also varied with sampling time and were similar in duplicate populations. Gametic disequilibria were most often detected in the "Groningen"-derived populations and their strength was not strongly associated with recombination fraction. Many of the disequilibria concerned unlinked loci. The strength of selection was probably considerable and populations were evolving genetic architectures which reflected niche selection by the different foods without marked genetic isolation between foods; gene frequencies did not vary between niches within a population cage.     

Inference from Clines Stabilized by Frequency-Dependent Selection

Frequency-dependent selection against rare forms can maintain clines. For weak selection, s, in simple linear models of frequency-dependence, single locus clines are stabilized with a maximum slope of between square root of s/square root of 8 sigma and square root of s/square root of 12 delta, where sigma is the dispersal distance. These clines are similar to those maintained by heterozygote disadvantage. Using computer simulations, the weak-selection analytical results are extended to higher selection pressures with up to three unlinked genes. Graphs are used to display the effect of selection, migration, dominance, and number of loci on cline widths, speeds of cline movements, two-way gametic correlations ("linkage disequilibria"), and heterozygote deficits. The effects of changing the order of reproduction, migration, and selection, are also briefly explored. Epistasis can also maintain tension zones. We show that epistatic selection is similar in its effects to frequency-dependent selection, except that the disequilibria produced in the zone will be higher for a given level of selection. If selection consists of a mixture of frequency-dependence and epistasis, as is likely in nature, the error made in estimating selection is usually less than twofold. From the graphs, selection and migration can be estimated using knowledge of the dominance and number of genes, of gene frequencies and of gametic correlations from a hybrid zone.     

Evolution of Multilocus Genetic Structure in an Experimental Barley Population

Data from 311 selfed families isolated from four generations (F8, F13, F23, F45) of an experimental barley population were analyzed to determine patterns of change in character expression for seven quantitative traits, and in single-locus allelic frequencies, and multilocus genetic structure, for 16 Mendelian loci that code for discretely recognizable variants. The analyses showed that large changes in single-locus allelic frequencies and major reorganizations in multilocus genetic structure occurred in each of the generation-to-generation transitions examined. Although associations among a few traits persisted over generations, dynamic dissociations and reassociations occurred among several traits in each generation-transition period. Overall, the restructuring that occurred was characterized by gradual decreases in the number of clusters of associated traits and increases in the number of traits within each cluster. The observed changes in single-locus frequencies and in multilocus genetic structure were attributed to interplay among various evolutionary factors among which natural selection acting in a temporally heterogeneous environment was the guiding force.     

Multilocus and Multitrait Measures of Differentiation for Gene Markers and Phenotypic Traits

Multilocus measures of differentiation taking into account gametic disequilibrium are developed. Even if coupling and repulsion heterozygotes cannot be separated at the multilocus level, a method is given to calculate a composite measure of differentiation (CF(st)) at the zygotic level, which accounts for allelic associations combining both gametic and nongametic effects. Mean and maximum differentiations may be relevant when multilocus measures are computed. Maximum differentiation is the highest eigenvalue of the F(st) matrix, whereas mean differentiation corresponds to the mean value of all eigenvalues of the F(st) matrix. Gametic disequilibrium has a stronger effect on maximum differentiation than on mean differentiation and takes into account the anisotropy that may exist between within- and between-population components of disequilibria. Multilocus mean and maximum differentiation are calculated for a set of 81 Quercus petraea (sessile oak) populations assessed with eight allozyme loci and two phenotypic traits (bud burst and height growth). The results indicate that maximum differentiation increases as more loci (traits) are considered whereas mean differentiation remains constant or decreases. Phenotypic traits exhibit higher population differentiation than allozymes. The applications and uses of mean and maximum differentiations are further discussed.     

A haplotype-based algorithm for multilocus linkage disequilibrium mapping of quantitative trait loci with epistasis

For tightly linked loci, cosegregation may lead to nonrandom associations between alleles in a population. Because of its evolutionary relationship with linkage, this phenomenon is called linkage disequilibrium. Today, linkage disequilibrium-based mapping has become a major focus of recent genome research into mapping complex traits. In this article, we present a new statistical method for mapping quantitative trait loci (QTL) of additive, dominant, and epistatic effects in equilibrium natural populations. Our method is based on haplotype analysis of multilocus linkage disequilibrium and exhibits two significant advantages over current disequilibrium mapping methods. First, we have derived closed-form solutions for estimating the marker-QTL haplotype frequencies within the maximum-likelihood framework implemented by the EM algorithm. The allele frequencies of putative QTL and their linkage disequilibria with the markers are estimated by solving a system of regular equations. This procedure has significantly improved the computational efficiency and the precision of parameter estimation. Second, our method can detect marker-QTL disequilibria of different orders and QTL epistatic interactions of various kinds on the basis of a multilocus analysis. This can not only enhance the precision of parameter estimation, but also make it possible to perform whole-genome association studies. We carried out extensive simulation studies to examine the robustness and statistical performance of our method. The application of the new method was validated using a case study from humans, in which we successfully detected significant QTL affecting human body heights. Finally, we discuss the implications of our method for genome projects and its extension to a broader circumstance. The computer program for the method proposed in this article is available at the webpage http://www.ifasstat.ufl.edu/genome/~LD.     

Testing for concordant equilibria between population samples

A substantial body of theory has been developed to assess the effect of evolutionary forces on the distribution of genotypes, both single and multilocus, within populations. One area where the potential for application of this theory has not been fully appreciated concerns the extent to which population samples differ. Within populations, the divergence of genotype or haplotype frequencies from that expected under Hardy-Weinberg (HW) or linkage equilibrium can be measured as disequilibria coefficients. To assess population samples for concordant equilibria, an analytical framework for comparing disequilibria coefficients between populations is necessary. Here we present log-linear models to evaluate such hypotheses. These models have broad utility ranging from conventional population genetics to genetic epidemiology. We demonstrate the use of these log-linear models (1) as a test for genetic association with disease and (2) as a test for different levels of linkage disequilibria between human populations.     

Multilocus structure in Pinus contorta Dougl.

We studied isozyme variation at 21 loci in 66 populations from three subspecies of Pinus contorta Dougl.; 35 in spp. latifolia, 20 in spp contorta and 11 in spp. murrayana. The objectives were to assess gametic disequilibria and multilocus structure. There was considerable differentiation of allele frequencies at 19 polymorphic loci across the 66 populations and within the subspecies. Allele frequencies at many loci correlated with geographic variables. Genetic variability varied considerably among populations within subspecies but the subspecies means were similar. The mean number of polymorphic loci and the mean heterozygosity over 19 polymorphic loci were, respectively, 13 and 0.194 in latifolia, 12 and 0.196 in murrayana, and 12 and 0.180 in contorta. The mean heterozygosity correlated with longitude and altitude across the 66 populations and with latitude in latifolia. Gametic disequilibria were evident in 40 populations; 29 in latifolia, eight in murrayana and three in contorta. Gametic disequilibria correlated with latitude across the 66 populations and with longitude in latifolia. The single-locus F _ST averaged 0.0339 in latifolia, 0.0567 in murrayana, and 0.0764 in contorta. The multilocus F _STM was 0.1227 in latifolia, 0.2926 in murrayana, and 0.3328 in contorta. Multilocus Wahlund and founder effects, migration patterns, and natural selection, probably played significant roles in generating and maintaining the multilocus genetic structure in P. contorta in general and the subspecies latifolia in particular.