The effect of genetic drift on the variance/covariance components generated by multilocus additive x additive epistatic systems

The effect of population bottlenecks on the components of the genetic variance/covariance generated by n neutral independent additive x additive loci has been studied theoretically. In its simplest version, this situation can be modelled by specifying the allele frequencies and homozygous effects at each locus, and an additional factor measuring the strength of the n-th order epistatic interaction. The variance/covariance components in an infinitely large panmictic population (ancestral components) were compared with their expected values at equilibrium over replicates randomly derived from the base population, after t bottlenecks of size N (derived components). Formulae were obtained giving the derived components (and the between-line variance) as functions of the ancestral ones (alternatively, in terms of allele frequencies and effects) and the corresponding inbreeding coefficient F(t). The n-th order derived component of the genetic variance/covariance is continuously eroded by inbreeding, but the remaining components may increase initially until a critical F(t) value is attained, which is inversely related to the order of the pertinent component, and subsequently decline to zero. These changes can be assigned to the between-line variances/covariances of gene substitution and epistatic effects induced by drift. Numerical examples indicate that: (1) the derived additive variance/covariance component will generally exceed its ancestral value unless epistasis is weak; (2) the derived epistatic variance/covariance components will generally exceed their ancestral values unless allele frequencies are extreme; (3) for systems showing equal ancestral additive and total non-additive variance/covariance components, those including a smaller number of epistatic loci may generate a larger excess in additive variance/covariance after bottlenecks than others involving a larger number of loci, provided that F(t) is low. Our results indicate that it is unlikely that the rate of evolution may be significantly accelerated after population bottlenecks, in spite of occasional increments of the derived additive variance over its ancestral value.     

Epistasis and the temporal change in the additive variance-covariance matrix induced by drift

The effect of population bottlenecks on the components of the genetic covariance generated by two neutral independent epistatic loci has been studied theoretically (additive, covA; dominance, covD; additive-by-additive, covAA; additive-by-dominance, covAD; and dominance-by-dominance, covDD). The additive-by-additive model and a more general model covering all possible types of marginal gene action at the single-locus level (additive/dominance epistatic model) were considered. The covariance components in an infinitely large panmictic population (ancestral components) were compared with their expected values at equilibrium over replicates randomly derived from the base population, after t consecutive bottlenecks of equal size N (derived components). Formulae were obtained in terms of the allele frequencies and effects at each locus, the corresponding epistatic effects and the inbreeding coefficient Ft. These expressions show that the contribution of nonadditive loci to the derived additive covariance (covAt) does not linearly decrease with inbreeding, as in the pure additive case, and may initially increase or even change sign in specific situations. Numerical examples were also analyzed, restricted for simplicity to the case of all covariance components being positive. For additive-by-additive epistasis, the condition covAt > covA only holds for high frequencies of the allele decreasing the metric traits at each locus (negative allele) if epistasis is weak, or for intermediate allele frequencies if it is strong. For the additive/dominance epistatic model, however, covAt > covA applies for low frequencies of the negative alleles at one or both loci and mild epistasis, but this result can be progressively extended to intermediate frequencies as epistasis becomes stronger. Without epistasis the same qualitative results were found, indicating that marginal dominance induced by epistasis can be considered as the primary cause of an increase of the additive covariance after bottlenecks. For all models, the magnitude of the ratio covAt/covA was inversely related to N and t.     

EPISTASIS AS A SOURCE OF INCREASED ADDITIVE GENETIC VARIANCE AT POPULATION BOTTLENECKS

The role of epistasis in evolution and speciation has remained controversial. We use a new parameterization of physiological epistasis to examine the effects of epistasis on levels of additive genetic variance during a population bottleneck. We found that all forms of epistasis increase average additive genetic variance in finite populations derived from initial populations with intermediate allele frequencies. Average additive variance continues to increase over many generations, especially at larger population sizes (N = 32 to 64). Additive-by-additive epistasis is the most potent source of additive genetic variance in this situation, whereas dominance-by-dominance epistasis contributes smaller amounts of additive genetic variance. With additive-by-dominance epistasis, additive genetic variance decreases at a relatively high rate immediately after a population bottleneck, rebounding to higher levels after several generations. Empirical examples of epistasis for murine adult body weight based on measured genotypes are provided illustrating the varying effects of epistasis on additive genetic variance during population bottlenecks.     

Joint effect of selection, linkage and partial inbreeding on additive genetic variance in an infinite population

Under the inifinitesimal model of gene effects, selection reduces the additive genetic variance by inducing negative linkage disequilibrium among selected genes. If the selected genes are linked, the decay of linkage disequilibrium is delayed, and the reduction of additive genetic variance is enhanced. Inbreeding in an infinite population also alters the additive genetic variance through the generation of positive association among genes within a locus. In the present study, the joint effect of selection, linkage and partial inbreeding (partial selfing or partial full-sib mating) on the additive genetic variance was modeled. The recurrence relations of the additive genetic variance between successive generations and the prediction equation of the asymptotic additive genetic variance were derived. Numerical computation showed that although partially inbred populations initially maintain larger genetic variances, the accumulated effect of selection overrides the effect of inbreeding. Stochastic simulation was carried out to check the precision of prediction, showing that the obtained equations give a satisfactory prediction during initial generations. However, the predicted values always overestimate the simulated values, especially in later generations. Based on these results, possible extensions and perspectives of the assumed model were discussed.     

The effect of epistasis on the excess of the additive and nonadditive variances after population bottlenecks

The effect of population bottlenecks on the components of the genetic variance generated by two neutral independent epistatic loci has been studied theoretically (VA, additive; VD, dominant; VAA, additive x additive; VAD, additive x dominant; VDD; dominant x dominant components of variance). Nonoverdominance and overdominance models were considered, covering all possible types of marginal gene action at the single locus level. The variance components in an infinitely large panmictic population (ancestral components) were compared with their expected values at equilibrium, after t consecutive bottlenecks of equal size N (derived components). Formulae were obtained in terms of allele frequencies and effects at each locus and the corresponding epistatic value. An excess of VA after bottlenecks can be assigned to two sources: (1) the spatiotemporal changes in the marginal average effects of gene substitution alpha(i), which are equal to zero only for additive gene action within and between loci; and (2) the covariance between alpha2(i) and the heterozygosity at the loci involved, which is generated by dominance, with or without epistasis. Numerical examples were analyzed, indicating that an increase in VA after bottlenecks will only occur if its ancestral value is minimal or very small. For the nonoverdominance model with weak reinforcing epistasis, that increase has been detected only for extreme frequencies of the negative allele at one or both loci. With strong epistasis, however, this result can be extended to a broad range of intermediate frequencies. With no epistasis, the same qualitative results were found, indicating that dominance can be considered as the primary cause of an increase in VA following bottlenecks. In parallel, the derived total nonadditive variance exceeded its ancestral value (V(NA) = V(D) + V(AA) + V(AD) + V(DD)) for a range of combinations of allele frequencies covering those for an excess of VA and for very large frequencies of the negative allele at both loci. For the overdominance model, an increase in V(A) and V(NA) was respectively observed for equilibrium (intermediate) frequencies at one or both loci or for extreme frequencies at both loci. For all models, the magnitude of the change of V(A) and V(NA) was inversely related to N and t. At low levels of inbreeding, the between-line variance was not affected by the type of gene action. For the models considered, the results indicate that it is unlikely that the rate of evolution may be accelerated after population bottlenecks, in spite of occasional increments of the derived V(A) over its ancestral value.     

The effect of a population bottleneck on the evolution of genetic variance/covariance structure

It is well known that standard population genetic theory predicts decreased additive genetic variance (V(a) ) following a population bottleneck and that theoretical models including interallelic and intergenic interactions indicate such loss may be avoided. However, few empirical data from multicellular model systems are available, especially regarding variance/covariance (V/CV) relationships. Here, we compare the V/CV structure of seventeen traits related to body size and composition between control (60 mating pairs/generation) and bottlenecked (2 mating pairs/generation; average F = 0.39) strains of mice. Although results for individual traits vary considerably, multivariate analysis indicates that V(a) in the bottlenecked populations is greater than expected. Traits with patterns and amounts of epistasis predictive of enhanced V(a) also show the largest deviations from additive expectations. Finally, the correlation structure of weekly weights is not significantly different between control and experimental lines but correlations between necropsy traits do differ, especially those involving the heart, kidney and tail length.     

Evolution of additive and nonadditive genetic variance in development time along a cline in Drosophila serrata

Abstract. Latitudinal clines provide natural systems that may allow the effect of natural selection on the genetic variance to be determined. Ten clinal populations of Drosophila serrata collected from the eastern coast of Australia were used to examine clinal patterns in the trait mean and genetic variance of the life-history trait egg-to-adult development time. Development time significantly lengthened from tropical areas to temperate areas. The additive genetic variance for development time in each population was not associated with latitude but was associated with the population mean development time. Additive genetic variance tended to be larger in populations with more extreme development times and appeared to be consistent with allele frequency change. In contrast, the nonadditive genetic variance was not associated with the population mean but was associated with latitude. Levels of nonadditive genetic variance were greatest in the region of the cline where the gradient in the change in mean was greatest, consistent with Barton's (1999) conjecture that the generation of linkage disequilibrium may become an important component of the genetic variance in systems with a spatially varying optimum.     

The change in quantitative genetic variation with inbreeding

Inbreeding is known to reduce heterozygosity of neutral genetic markers, but its impact on quantitative genetic variation is debated. Theory predicts a linear decline in additive genetic variance (V(A)) with increasing inbreeding coefficient (F) when loci underlying the trait act additively, but a nonlinear hump-shaped relationship when dominance and epistasis are important. Predictions for heritability (h2) are similar, although the exact shape depends on the value of h2 in the absence of inbreeding. We located 22 published studies in which the level of genetic variation in experimentally inbred populations (measured by V(A) or h2) was compared with that in outbred control populations. For life-history traits, the data strongly supported a nonlinear change in genetic variation with increasing F. V(A) and h2 were, respectively, 244% and 50% higher at F = 0.4 than in outbred populations, and dominance plus epistatic variance together exceeded additive variance by a factor of four. For nonfitness traits the decline was linear and estimates of nonadditive variance were small. These results confirm that population bottlenecks frequently increase V(A) in some traits, and imply that life-history traits are underlain by substantial dominance or epistasis. However, the importance of drift-induced genetic variation in conservation or evolutionary biology is questionable, in part because inbreeding depression usually accompanies inbreeding.     

Partitioning of population genetic variance under multiplicative-epistatic gene action

Determining the way in which different QTLs interact (epistasis) in their effects on the phenotype is crucial to many areas in population genetics and evolutionary biology. For example, in the founder event, a separated population readapts to a new environment through the release of cryptic gene-gene interactions. In hybrid zones, hybrid speciation must be subjected to natural selection for epistasis resulting from genomic recombinations between different species. However, there is a severe shortage of relevant methodologies to estimate epistatic genetic effects and variances. A statistical model has recently been proposed to estimate the number of QTLs, their genetic effects and allelic frequencies in segregating populations. This model is based on multiplicative gene action and derived from a two-level intra- and interspecific mating design. In this paper, we formulate a statistical procedure for partitioning the genetic variance into additive, dominant and various kinds of epistatic components in an intra- or mixed intra- and interspecific hybrid population. The procedure can be used to study the genetic architecture of fragmented populations and hybrid zones, thus allowing for a better recognition of the role of epistasis in evolution and hybrid speciation. A real example for two Populus species, P. tremuloides and P. tremula, is provided to illustrate the procedure. In this example, we found that considerable new genetic variation is formed through genomic recombination between two aspen species. Received: 1 May 1999 / Accepted: 27 July 1999     

Order-preserving principles underlying genotype-phenotype maps ensure high additive proportions of genetic variance

In quantitative genetics, the degree of resemblance between parents and offspring is described in terms of the additive variance (V(A)) relative to genetic (V(G)) and phenotypic (V(P)) variance. For populations with extreme allele frequencies, high V(A)/V(G) can be explained without considering properties of the genotype-phenotype (GP) map. We show that randomly generated GP maps in populations with intermediate allele frequencies generate far lower V(A)/V(G) values than empirically observed. The main reason is that order-breaking behaviour is ubiquitous in random GP maps. Rearrangement of genotypic values to introduce order-preservation for one or more loci causes a dramatic increase in V(A)/V(G). This suggests the existence of order-preserving design principles in the regulatory machinery underlying GP maps. We illustrate this feature by showing how the ubiquitously observed monotonicity of dose-response relationships gives much higher V(A)/V(G) values than a unimodal dose-response relationship in simple gene network models.     

Persistence of changes in the genetic covariance matrix after a bottleneck

Abstract.— Genetic variance, phenotypic variance, and the genetic covariance matrix ( G ) can change as a result of genetic drift. These changes will persist over time to some extent and will continue if population size remains relatively small. Nine populations founded by a single pair of Drosophila melanogaster were measured for a series of six morphological characteristics for a large number of parent-offspring families at both the third generation after the bottlenecks and after 20 generations. From these data, the phenotypic variance, additive genetic variance, and G were estimated for each line at each generation. Phenotypic and genetic variances were highly correlated over time, so that the measurements made at the third generation were predictive of the state of the population 17 generations later. Genetic covariances were also somewhat stable over time; however, the G matrices of some lines changed significantly over the intervening generations. This change did not return the populations toward their original state before the population bottlenecks. We conclude that the genetic covariance matrix can change as a result of mild genetic drift over a short span of time.     

Dissecting total genetic variance into additive and dominance components of purebred and crossbred pig traits

The partition of the total genetic variance into its additive and non-additive components can differ from trait to trait, and between purebred and crossbred populations. A quantification of these genetic variance components will determine the extent to which it would be of interest to account for dominance in genomic evaluations or to establish mate allocation strategies along different populations and traits. This study aims at assessing the contribution of the additive and dominance genomic variances to the phenotype expression of several purebred Piétrain and crossbred (Piétrain × Large White) pig performances. A total of 636 purebred and 720 crossbred male piglets were phenotyped for 22 traits that can be classified into six groups of traits: growth rate and feed efficiency, carcass composition, meat quality, behaviour, boar taint and puberty. Additive and dominance variances estimated in univariate genotypic models, including additive and dominance genotypic effects, and a genomic inbreeding covariate allowed to retrieve the additive and dominance single nucleotide polymorphism variances for purebred and crossbred performances. These estimated variances were used, together with the allelic frequencies of the parental populations, to obtain additive and dominance variances in terms of genetic breeding values and dominance deviations. Estimates of the Piétrain and Large White allelic contributions to the crossbred variance were of about the same magnitude in all the traits. Estimates of additive genetic variances were similar regardless of the inclusion of dominance. Some traits showed relevant amount of dominance genetic variance with respect to phenotypic variance in both populations (i.e. growth rate 8%, feed conversion ratio 9% to 12%, backfat thickness 14% to 12%, purebreds-crossbreds). Other traits showed higher amount in crossbreds (i.e. ham cut 8% to 13%, loin 7% to 16%, pH semimembranosus 13% to 18%, pH longissimus dorsi 9% to 14%, androstenone 5% to 13% and estradiol 6% to 11%, purebreds-crossbreds). It was not encountered a clear common pattern of dominance expression between groups of analysed traits and between populations. These estimates give initial hints regarding which traits could benefit from accounting for dominance for example to improve genomic estimated breeding value accuracy in genetic evaluations or to boost the total genetic value of progeny by means of assortative mating.     

The genetic rescue of two bottlenecked South Island robin populations using translocations of inbred donors

Populations forced through bottlenecks typically lose genetic variation and exhibit inbreeding depression. ‘Genetic rescue’ techniques that introduce individuals from outbred populations can be highly effective in reversing the deleterious effects of inbreeding, but have limited application for the majority of endangered species, which survive only in a few bottlenecked populations. We tested the effectiveness of using highly inbred populations as donors to rescue two isolated and bottlenecked populations of the South Island robin (Petroica australis). Reciprocal translocations significantly increased heterozygosity and allelic diversity. Increased genetic diversity was accompanied by increased juvenile survival and recruitment, sperm quality, and immunocompetence of hybrid individuals (crosses between the two populations) compared with inbred control individuals (crosses within each population). Our results confirm that the implementation of ‘genetic rescue’ using bottlenecked populations as donors provides a way of preserving endangered species and restoring their viability when outbred donor populations no longer exist.     

Perspective: Here's to Fisher,additive genetic variance,and the fundamental theorem of natural selection

Fisher's fundamental theorem of natural selection, that the rate of change of fitness is given by the additive genetic variance of fitness, has generated much discussion since its appearance in 1930. Fisher tried to capture in the formula the change in population fitness attributable to changes of allele frequencies, when all else is not included. Lessard's formulation comes closest to Fisher's intention, as well as this can be judged. Additional terms can be added to account for other changes. The "theorem" as stated by Fisher is not exact, and therefore not a theorem, but it does encapsulate a great deal of evolutionary meaning in a simple statement. I also discuss the effectiveness of reproductive-value weighting and the theorem in integrated form. Finally, an optimum principle, analogous to least action and Hamilton's principle in physics, is discussed.     

A test of quantitative genetic theory using Drosophila- effects of inbreeding and rate of inbreeding on heritabilities and variance components

Inbreeding is expected to decrease the heritability within populations. However, results from empirical studies are inconclusive. In this study, we investigated the effects of three breeding treatments (fast and slow rate of inbreeding - inbred to the same absolute level - and a control) on heritability, phenotypic, genetic and environmental variances of sternopleural bristle number in Drosophila melanogaster. Heritability, and phenotypic, genetic and environmental variances were estimated in 10 replicate lines within each of the three treatments. Standard least squares regression models and Bayesian methods were used to analyse the data. Heritability and additive genetic variance within lines were higher in the control compared with both inbreeding treatments. Heritabilities and additive genetic variances within lines were higher in slow compared with fast inbred lines, indicating that slow inbred lines retain more evolutionary potential despite the same expected absolute level of inbreeding. The between line variance was larger with inbreeding and more than twice as large in the fast than in the slow inbred lines. The different pattern of redistribution of genetic variance within and between lines in the two inbred treatments cannot be explained invoking the standard model based on selective neutrality and additive gene action. Environmental variances were higher with inbreeding, and more so with fast inbreeding, indicating that inbreeding and the rate of inbreeding affect environmental sensitivity. The phenotypic variance decreased with inbreeding, but was not affected by the rate of inbreeding. No inbreeding depression for mean sternopleural bristle number was observed in this study. Considerable variance between lines in additive genetic variance within lines was observed, illustrating between line variation in evolutionary potential.     

Metapopulation patterns of additive and nonadditive genetic variance in the sea bass (Dicentrarchus labrax)

Describing and explaining the geographic within‐species variation in phenotypes (“phenogeography”) in the sea over a species distribution range is central to our understanding of a variety of eco‐evolutionary topics. However, phenogeographic studies that have a large potential to investigate adaptive variation are overcome by phylogeographic studies, still mainly focusing on neutral markers. How genotypic and phenotypic data could covary over large geographic scales remains poorly understood in marine species. We crossed 75 noninbred sires (five origins) and 26 dams (two origins; each side of a hybrid zone) in a factorial diallel cross in order to investigate geographic variation for early survival and sex ratio in the metapopulation of the European sea bass (Dicentrarchus labrax), a highly prized marine fish species. Full‐sib families (N = 1,950) were produced and reared in a common environment. Parentage assignment of 7,200 individuals was performed with seven microsatellite markers. Generalized linear models showed significant additive effects for both traits and pleiotropy between traits. A significant nonadditive genetic effect was detected. Different expression of traits and distinct relative performances were found for reciprocal crosses involving populations located on each side of the main hybrid zone located at the Almeria‐Oran front, illustrating asymmetric reproductive isolation. The poor fitness performance observed for the Western Mediterranean population of sea bass is discussed as it represents the main source of seed hatchery production, but also because it potentially illustrates nonadaptive introgression and maladaptation.     

Population decline is accompanied by loss of genetic diversity in the Lesser Grey Shrike Lanius minor

The Lesser Grey Shrike has suffered successive declines in population size and a marked contraction of its breeding range since the early 20th century, largely because of long‐term agricultural intensification. This has resulted in a severely fragmented distribution in Western Europe, with isolated breeding nuclei in Spain, France and Italy and a more continuous distribution in Eastern Europe and Asia. Using a combination of nuclear and mitochondrial markers, we assessed the genetic structure and diversity of Lesser Grey Shrike populations from Western Europe, Central Europe and Asia. There was significant genetic differentiation among three major regional groups, one European and two Asian. Genetic diversity measures were lowest in the smallest and most marginal Spanish population. Limited genetic diversity, combined with rapid population decline, suggests the Spanish population may face extinction in the near future.     

Additive and over-dominant effects resulting from epistatic loci are the primary genetic basis of heterosis in rice

A set of 148 F9 recombinant inbred lines (RILs) was developed from the cross of an indica cultivar 93-11 and japonica cultivar DTT13,showing strong F1 heterosis.Subsequently,two backcross F1 (BCF1) populations were constructed by backcrossing these 148 RILs to two parents,93-11 and DT713.These three related populations (281BCF1 lines,148 RILs) were phenotyped for six yield-related traits in two locations.Significant inbreeding depression was detected in the population of RILS and a high level of heterosis was observed in the two BCF1 populations.A total of 42 main-effect quantitative trait loci (M-QTLs) and 109 epistatic effect QTL pairs (E-QTLs) were detected in the three related populations using the mixed model approach.By comparing the genetic effects of these QTLs detected in the RILs,BCF1 performance and mid-parental heterosis (HMp),we found that,in both BCF1 populations,the QTLs detected could be classified into two predominant types:additive and over-domlnant loci,which indicated that the additive and over-dominant effect were more important than complete or partially dominance for M-QTLs and E-QTLs.Further,we found that the E-QTLs detected collectively explained a larger portion of the total phenotypic variation than the M-QTLs in both RILs and BCF1 populations.All of these results suggest that additive and over-dominance resulting from epistatic loci might be the primary genetic basis of heterosis in rice.     

Genetic characterisation of dough rheological properties in a wheat doubled haploid population: additive genetic effects and epistatic interactions

Doubled haploid lines (n=160) from a cross between wheat cultivars Cranbrook (high dough extensibility) and Halberd (low dough extensibility) were grown at three Australian locations. The parents differ at all high- and low-molecular-weight glutenin loci. Dough rheological parameters were measured using small-scale testing procedures, and quantitative trait locus (QTL) mapping procedures were carried out using an existing well-saturated genetic linkage map for this cross. Genetic parameters were estimated using three software packages: QTLCartographer, Epistat and Genstat. Results indicated that environmental factors are a major determinant of dough extensibility across the three trial sites, whereas genotypic factors are the major determinants of dough strength. Composite interval mapping analysis across the 21 linkage groups revealed that as expected, the main additive QTLs for dough rheological properties are located at the high- and low-molecular-weight glutenin loci. A new QTL on chromosome 5A for M-extensibility (a mixograph-estimated measure of extensibility) was detected. Analysis of epistatic interactions revealed that there were significant conditional epistatic interactions related with the additive effects of glutenin loci on dough rheological properties. Therefore, the additive genetic effects of glutenins on dough rheological properties are conditional upon the genetic background of the wheat line. The molecular basis of the interactions with the glutenin loci may be via proteins that modify or alter the gluten protein matrix or variations in the expression level of the glutenin genes. Reverse-phase high performance liquid chromatography analysis of the molar number of individual glutenin subunits across the population showed that certain conditional epistases resulted in increased expression of the affected glutenin. The epistatic interactions detected in this study provide a possible explanation of the variable genetic effects of some glutenins on quality attributes in different genetic backgrounds and provide essential information for the accurate prediction of glutenin related variance in marker-assisted wheat breeding.