A general definition of the heritable variation that determines the potential of a population to respond to selection

Genetic selection is a major force shaping life on earth. In classical genetic theory, response to selection is the product of the strength of selection and the additive genetic variance in a trait. The additive genetic variance reflects a population's intrinsic potential to respond to selection. The ordinary additive genetic variance, however, ignores the social organization of life. With social interactions among individuals, individual trait values may depend on genes in others, a phenomenon known as indirect genetic effects. Models accounting for indirect genetic effects, however, lack a general definition of heritable variation. Here I propose a general definition of the heritable variation that determines the potential of a population to respond to selection. This generalizes the concept of heritable variance to any inheritance model and level of organization. The result shows that heritable variance determining potential response to selection is the variance among individuals in the heritable quantity that determines the population mean trait value, rather than the usual additive genetic component of phenotypic variance. It follows, therefore, that heritable variance may exceed phenotypic variance among individuals, which is impossible in classical theory. This work also provides a measure of the utilization of heritable variation for response to selection and integrates two well-known models of maternal genetic effects. The result shows that relatedness between the focal individual and the individuals affecting its fitness is a key determinant of the utilization of heritable variance for response to selection.     

Pleiotropic Models of Polygenic Variation, Stabilizing Selection, and Epistasis

We show that in polymorphic populations many polygenic traits pleiotropically related to fitness are expected to be under apparent ``stabilizing selection' independently of the real selection acting on the population. This occurs, for example, if the genetic system is at a stable polymorphic equilibrium determined by selection and the nonadditive contributions of the loci to the trait value either are absent, or are random and independent of those to fitness. Stabilizing selection is also observed if the polygenic system is at an equilibrium determined by a balance between selection and mutation (or migration) when both additive and nonadditive contributions of the loci to the trait value are random and independent of those to fitness. We also compare different viability models that can maintain genetic variability at many loci with respect to their ability to account for the strong stabilizing selection on an additive trait. Let V(m) be the genetic variance supplied by mutation (or migration) each generation, V(g) be the genotypic variance maintained in the population, and n be the number of the loci influencing fitness. We demonstrate that in mutation (migration)-selection balance models the strength of apparent stabilizing selection is order V(m)/V(g). In the overdominant model and in the symmetric viability model the strength of apparent stabilizing selection is approximately 1/(2n) that of total selection on the whole phenotype. We show that a selection system that involves pairwise additive by additive epistasis in maintaining variability can lead to a lower genetic load and genetic variance in fitness (approximately 1/(2n) times) than an equivalent selection system that involves overdominance. We show that, in the epistatic model, the apparent stabilizing selection on an additive trait can be as strong as the total selection on the whole phenotype.     

COMPARING STRENGTHS OF DIRECTIONAL SELECTION: HOW STRONG IS STRONG?

The fundamental equation in evolutionary quantitative genetics, the Lande equation, describes the response to directional selection as a product of the additive genetic variance and the selection gradient of trait value on relative fitness. Comparisons of both genetic variances and selection gradients across traits or populations require standardization, as both are scale dependent. The Lande equation can be standardized in two ways. Standardizing by the variance of the selected trait yields the response in units of standard deviation as the product of the heritability and the variance-standardized selection gradient. This standardization conflates selection and variation because the phenotypic variance is a function of the genetic variance. Alternatively, one can standardize the Lande equation using the trait mean, yielding the proportional response to selection as the product of the squared coefficient of additive genetic variance and the mean-standardized selection gradient. Mean-standardized selection gradients are particularly useful for summarizing the strength of selection because the mean-standardized gradient for fitness itself is one, a convenient benchmark for strong selection. We review published estimates of directional selection in natural populations using mean-standardized selection gradients. Only 38 published studies provided all the necessary information for calculation of mean-standardized gradients. The median absolute value of multivariate mean-standardized gradients shows that selection is on average 54% as strong as selection on fitness. Correcting for the upward bias introduced by taking absolute values lowers the median to 31%, still very strong selection. Such large estimates clearly cannot be representative of selection on all traits. Some possible sources of overestimation of the strength of selection include confounding environmental and genotypic effects on fitness, the use of fitness components as proxies for fitness, and biases in publication or choice of traits to study.     

Constraints on the evolution of function‐valued traits: a study of growth in Tribolium castaneum

Growth trajectories often impact individual fitness. They are continuous by nature and so are amenable to analysis using a function‐valued (FV) trait framework to reveal their underlying genetic architecture. Previous studies have found high levels of standing additive genetic (co)variance for growth trajectories despite the expectation that growth should be responding to frequent strong directional selection. In this study, the FV framework is used to estimate the additive genetic covariance function for growth trajectories in larval Tribolium castaneum to address questions about standing additive genetic (co)variance and possible evolutionary constraints on growth and to predict responses to four plausible selection regimes. Results show that additive genetic (co)variance is high at the early ages, but decreases towards later ages in the larval period. A selection gradient function of the same size and in the same direction of the first eigenfunction of the G‐function should give the maximal response. However, evolutionary constraints may be acting to keep this maximal response from being realized, through either conflicting effects on survivability and fecundity of larger body size, few evolutionary directions having sufficient additive variance for a response, genetic trade‐offs with other traits or physiological regulatory mechanisms. More light may be shed on these constraints through the development of more sophisticated statistical approaches and implementation of additional empirical studies to explicitly test for specific types of constraints.     

Speeding up microevolution: the effects of increasing temperature on selection and genetic variance in a wild bird population

The amount of genetic variance underlying a phenotypic trait and the strength of selection acting on that trait are two key parameters that determine any evolutionary response to selection. Despite substantial evidence that, in natural populations, both parameters may vary across environmental conditions, very little is known about the extent to which they may covary in response to environmental heterogeneity. Here we show that, in a wild population of great tits (Parus major), the strength of the directional selection gradients on timing of breeding increased with increasing spring temperatures, and that genotype-by-environment interactions also predicted an increase in additive genetic variance, and heritability, of timing of breeding with increasing spring temperature. Consequently, we therefore tested for an association between the annual selection gradients and levels of additive genetic variance expressed each year; this association was positive, but non-significant. However, there was a significant positive association between the annual selection differentials and the corresponding heritability. Such associations could potentially speed up the rate of micro-evolution and offer a largely ignored mechanism by which natural populations may adapt to environmental changes.     

Testing for microevolution in body size in three blue tit populations

Quantifying the genetic variation and selection acting on phenotypes is a prerequisite for understanding microevolutionary processes. Surprisingly, long-term comparisons across conspecific populations exposed to different environments are still lacking, hampering evolutionary studies of population differentiation in natural conditions. Here, we present analyses of additive genetic variation and selection using two body-size traits in three blue tit (Parus caeruleus) populations from distinct habitats. Chick tarsus length and body mass at fledging showed substantial levels of genetic variation in the three populations. Estimated heritabilities of body mass increased with habitat quality. The poorer habitats showed weak positive selection on tarsus length, and strong positive selection on body mass, but there was no significant selection on either trait in the good habitat. However, there was no evidence of any microevolutionary response to selection in any population during the study periods. Potential explanations for this absence of a response to selection are discussed, including the effects of spatial heterogeneity associated with gene flow between habitats.     

Population bottlenecks increase additive genetic variance but do not break a selection limit in rain forest Drosophila

According to neutral quantitative genetic theory, population bottlenecks are expected to decrease standing levels of additive genetic variance of quantitative traits. However, some empirical and theoretical results suggest that, if nonadditive genetic effects influence the trait, bottlenecks may actually increase additive genetic variance. This has been an important issue in conservation genetics where it has been suggested that small population size might actually experience an increase rather than a decrease in the rate of adaptation. Here we test if bottlenecks can break a selection limit for desiccation resistance in the rain forest-restricted fly Drosophila bunnanda. After one generation of single-pair mating, additive genetic variance for desiccation resistance increased to a significant level, on average higher than for the control lines. Line crosses revealed that both dominance and epistatic effects were responsible for the divergence in desiccation resistance between the original control and a bottlenecked line exhibiting increased additive genetic variance for desiccation resistance. However, when bottlenecked lines were selected for increased desiccation resistance, there was only a small shift in resistance, much less than predicted by the released additive genetic variance. The small selection response in the bottlenecked lines was no greater than that observed in the control lines. Thus bottlenecks might produce a statistically detectable change in additive genetic variance but this change has no impact on the response to selection.     

Genetic evaluation methods for populations with dominance and inbreeding

Summary The effect of inbreeding on mean and genetic covariance matrix for a quantitative trait in a population with additive and dominance effects is shown. This genetic covariance matrix is a function of five relationship matrices and five genetic parameters describing the population. Elements of the relationship matrices are functions of Gillois (1964) identity coefficients for the four genes at a locus in two individuals. The equivalence of the path coefficient method (Jacquard 1966) and the tabular method (Smith and Mäki-Tanila 1990) to compute the covariance matrix of additive and dominance effects in a population with inbreeding is shown. The tabular method is modified to compute relationship matrices rather than the covariance matrix, which is trait dependent. Finally, approximate and exact Best Linear Unbiased Predictions (BLUP) of additive and dominance effects are compared using simulated data with inbreeding but no directional selection. The trait simulated was affected by 64 unlinked biallelic loci with equal effect and complete dominance. Simulated average inbreeding levels ranged from zero in generation one to 0.35 in generation five. The approximate method only accounted for the effect of inbreeding on mean and additive genetic covariance matrix, whereas the exact accounted for all of the changes in mean and genetic covariance matrix due to inbreeding. Approximate BLUP, which is computable for large populations where exact BLUP is not feasible, yielded unbiased predictions of additive and dominance effects in each generation with only slightly reduced accuracies relative to exact BLUP.     

Effects on phenotypic variability of directional selection arising through genetic differences in residual variability

In standard models of quantitative traits, genotypes are assumed to differ in mean but not variance of the trait. Here we consider directional selection for a quantitative trait for which genotypes also confer differences in variability, viewed either as differences in residual phenotypic variance when individual loci are concerned or as differences in environmental variability when the whole genome is considered. At an individual locus with additive effects, the selective value of the increasing allele is given by ia/sigma + 1/2 ixb/sigma2, where i is the selection intensity, x is the standardized truncation point, sigma2 is the phenotypic variance, and a/sigma and b/sigma2 are the standardized differences in mean and variance respectively between genotypes at the locus. Assuming additive effects on mean and variance across loci, the response to selection on phenotype in mean is isigma2(Am)/sigma + 1/2 ixcov(Amv)/sigma2 and in variance is icov(Amv)/sigma + 1/2 ixsigma2(Av)/sigma2, where sigma2(Am) is the (usual) additive genetic variance of effects of genes on the mean, sigma2(Av) is the corresponding additive genetic variance of their effects on the variance, and cov(Amv) is the additive genetic covariance of their effects. Changes in variance also have to be corrected for any changes due to gene frequency change and for the Bulmer effect, and relevant formulae are given. It is shown that effects on variance are likely to be greatest when selection is intense and when selection is on individual phenotype or within family deviation rather than on family mean performance. The evidence for and implications of such variability in variance are discussed.     

Why breeding time has not responded to selection for earlier breeding in a songbird population

A crucial assumption underlying the breeders' equation is that selection acts directly on the trait of interest, and not on an unmeasured environmental factor which affects both fitness and the trait. Such an environmentally induced covariance between a trait and fitness has been repeatedly proposed as an explanation for the lack of response to selection on avian breeding time. We tested this hypothesis using a long-term dataset from a Dutch great tit (Parus major) population. Although there was strong selection for earlier breeding in this population, egg-laying dates have changed only marginally over the last decades. Using a so-called animal model, we quantified the additive genetic variance in breeding time and predicted breeding values for females. Subsequently, we compared selection at the phenotypic and genetic levels for two fitness components, fecundity and adult survival. We found no evidence for an environmentally caused covariance between breeding time and fitness or counteracting selection on the different fitness components. Consequently, breeding time should respond to selection but the expected response to selection was too small to be detected.     

Metabolic control analysis as a mechanism that conserves genetic variance during advanced cycle breeding

The recycling of elite inbreds (i.e., advanced cycle breeding) has led to significant genetic gains but also to a narrow gene pool in plant breeding programs. Sustained yield improvements in many crops have suggested that genetic variance is not depleted at a rate predicted by an additive genetic model. Unlike the additive model in classical quantitative genetic theory, metabolic control analysis relates the variation in a biochemical process with the genetic variation in a quantitative trait. Our objective was to determine whether metabolic control analysis is a mechanism that slows the decrease in genetic variance during advanced cycle breeding. Three cycles of advanced cycle breeding were simulated with 10, 50, or 100 quantitative trait loci (QTL) controlling a trait. In metabolic control analysis, these QTL coded for enzymes involved in a linear metabolic pathway that converted a substrate into a product. In the absence of selection, both the additive model and the metabolic control analysis model led to about a 50% reduction in genetic variance from cycle to cycle. With selection, the additive model led to a 50–58% reduction in genetic variance, but the metabolic control analysis model generally led to only a 12–54% reduction. We suggest selection in a metabolic control analysis model as a mechanism that slows the decrease in genetic variance during advanced cycle breeding. This conservation of genetic variance would allow breeders to achieve genetic gains for a longer period than expected under the additive model.Communicated by H.C. Becker     

Very low levels of direct additive genetic variance in fitness and fitness components in a red squirrel population

A trait must genetically correlate with fitness in order to evolve in response to natural selection, but theory suggests that strong directional selection should erode additive genetic variance in fitness and limit future evolutionary potential. Balancing selection has been proposed as a mechanism that could maintain genetic variance if fitness components trade off with one another and has been invoked to account for empirical observations of higher levels of additive genetic variance in fitness components than would be expected from mutation–selection balance. Here, we used a long‐term study of an individually marked population of North American red squirrels (Tamiasciurus hudsonicus) to look for evidence of (1) additive genetic variance in lifetime reproductive success and (2) fitness trade‐offs between fitness components, such as male and female fitness or fitness in high‐ and low‐resource environments. “Animal model” analyses of a multigenerational pedigree revealed modest maternal effects on fitness, but very low levels of additive genetic variance in lifetime reproductive success overall as well as fitness measures within each sex and environment. It therefore appears that there are very low levels of direct genetic variance in fitness and fitness components in red squirrels to facilitate contemporary adaptation in this population.     

RESPONSE TO SELECTION IN PARTIALLY SELF-FERTILIZING POPULATIONS. I. SELECTION ON A SINGLE TRAIT

Self-fertilization is a common form of reproduction in plants and it has important implications for quantitative trait evolution. Here, I present a model of selection on quantitative traits that can accommodate any level of self-fertilization. The “structured linear model” (SLM) predicts the evolution of the mean phenotype as a function of three distinct quantities: the mean additive genetic value, the directional dominance, and the mean inbreeding coefficient. Stochastic simulations of truncation selection demonstrate the accuracy of the SLM in predicting changes in the mean and variance of a quantitative trait over the full range of selfing rates. They also illustrate how complex interactions between selection and mating system determine the population distribution of inbreeding coefficients and also the amount of linkage disequilibrium. Changes in the genetic variance due to linkage disequilibria, which are commonly referred to as the “Bulmer effect,” are greatly magnified by selfing. This complicates the relationship between selfing rate and response to selection. Like the random mating theory, the parameters of the SLM can be estimated from phenotypic data.     

Secondary theorem of natural selection in biocultural populations.

The "Secondary Theorem of Natural Selection," an extension of Fisher's fundamental theorem, states that the rate of change in the mean of an arbitrary character in response to selection is proportional to the additive genetic covariance between the character and fitness. Here I derive an expression for the change in the mean value of a trait subject to both genetic and cultural transmission. I start with the one-locus case under generalized mating and cultural transmission from parents to offspring, then proceed to the two-locus case. My results support previous work on the effects of nongenetic inheritance by showing that (i) cultural transmission introduces a timelag in the population response to selection; (ii) with cultural transmission the effects of selection persist even after selection is relaxed; and (iii) cultural transmission can either enhance or retard phenotypic evolution relative to that obtained under purely genetic transmission.     

Genetic variation maintained in multilocus models of additive quantitative traits under stabilizing selection.

Stabilizing selection for an intermediate optimum is generally considered to deplete genetic variation in quantitative traits. However, conflicting results from various types of models have been obtained. While classical analyses assuming a large number of independent additive loci with individually small effects indicated that no genetic variation is preserved under stabilizing selection, several analyses of two-locus models showed the contrary. We perform a complete analysis of a generalization of Wright's two-locus quadratic-optimum model and investigate numerically the ability of quadratic stabilizing selection to maintain genetic variation in additive quantitative traits controlled by up to five loci. A statistical approach is employed by choosing randomly 4000 parameter sets (allelic effects, recombination rates, and strength of selection) for a given number of loci. For each parameter set we iterate the recursion equations that describe the dynamics of gamete frequencies starting from 20 randomly chosen initial conditions until an equilibrium is reached, record the quantities of interest, and calculate their corresponding mean values. As the number of loci increases from two to five, the fraction of the genome expected to be polymorphic declines surprisingly rapidly, and the loci that are polymorphic increasingly are those with small effects on the trait. As a result, the genetic variance expected to be maintained under stabilizing selection decreases very rapidly with increased number of loci. The equilibrium structure expected under stabilizing selection on an additive trait differs markedly from that expected under selection with no constraints on genotypic fitness values. The expected genetic variance, the expected polymorphic fraction of the genome, as well as other quantities of interest, are only weakly dependent on the selection intensity and the level of recombination.     

Comparing Evolvability and Variability of Quantitative Traits

There are two distinct reasons for making comparisons of genetic variation for quantitative characters. The first is to compare evolvabilities, or ability to respond to selection, and the second is to make inferences about the forces that maintain genetic variability. Measures of variation that are standardized by the trait mean, such as the additive genetic coefficient of variation, are appropriate for both purposes. Variation has usually been compared as narrow sense heritabilities, but this is almost always an inappropriate comparative measure of evolvability and variability. Coefficients of variation were calculated from 842 estimates of trait means, variances and heritabilities in the literature. Traits closely related to fitness have higher additive genetic and nongenetic variability by the coefficient of variation criterion than characters under weak selection. This is the reverse of the accepted conclusion based on comparisons of heritability. The low heritability of fitness components is best explained by their high residual variation. The high additive genetic and residual variability of fitness traits might be explained by the great number of genetic and environmental events they are affected by, or by a lack of stabilizing selection to reduce their phenotypic variance. Over one-third of the quantitative genetics papers reviewed did not report trait means or variances. Researchers should always report these statistics, so that measures of variation appropriate to a variety of situations may be calculated.     

BUTTERFLY EYESPOTS: THE GENETICS AND DEVELOPMENT OF THE COLOR RINGS

The butterfly Bicyclus anynana has a series of distal eyespots on its wings. Each eyespot is composed of a white pupil, a black disc, and a gold outer ring. We applied artificial selection to the large dorsal eyespot on the forewing to produce a line with the gold ring reduced or absent (BLACK) and another line with a reduced black disc and a broad gold ring (GOLD). High heritabilities, coupled with a rapid response to selection, produced two lines of butterflies with very different phenotypes. Other eyespots showed a correlated change in the proportion of their color rings. Surgical experiments were performed on pupal wings from the different lines at the time of eyespot pattern specification. They showed that the additive genetic variance for this trait was in the response of the wing epidermis to signaling from the organizing cells at the eyespot center (the focus). This response was found to vary across different regions of the wing and also between the sexes. The particular eyespot color composition found for each sex, as well as the maintenance of the high genetic variation, are discussed with reference to the ecology of the butterfly, sexual selection, and visual selection by predators.     

Bidirectional selection for female propensity to remate in the bean beetle, <Emphasis Type="Italic">Callosobruchus chinensis</Emphasis>

The evolution of female multiple mating, or polyandry, is difficult to comprehend and thus has been the subject of a large number of studies. However, there is only a little evidence for genetic variation in polyandry, although the evolution of a trait via selection requires genetic variation that enables the trait to respond to selection. We carried out artificial selection for increased and decreased female propensity to remate as a measure of polyandry to investigate whether this trait has a genetic component that can respond to selection in the adzuki bean beetle, Callosobruchus chinensis. Artificial selection produced responses in both directions and divergence between the selection lines in the female propensity to remate. Although the experimental design adopted in this study selected jointly for female receptivity to remating, which is a trait of females, and male ability to inhibit female remating—both of which are associated with female propensity to remate—the observed response to selection was attributable only to the female receptivity to remating. This study indicates that the female receptivity to remating has significant additive genetic variation and can evolve according to whether remating is advantageous or disadvantageous to females in C. chinensis.     

Predicted accuracy of and response to genomic selection for new traits in dairy cattle

Genomic selection relaxes the requirement of traditional selection tools to have phenotypic measurements on close relatives of all selection candidates. This opens up possibilities to select for traits that are difficult or expensive to measure. The objectives of this paper were to predict accuracy of and response to genomic selection for a new trait, considering that only a cow reference population of moderate size was available for the new trait, and that selection simultaneously targeted an index and this new trait. Accuracy for and response to selection were deterministically evaluated for three different breeding goals. Single trait selection for the new trait based only on a limited cow reference population of up to 10 000 cows, showed that maximum genetic responses of 0.20 and 0.28 genetic standard deviation (s.d.) per year can be achieved for traits with a heritability of 0.05 and 0.30, respectively. Adding information from the index based on a reference population of 5000 bulls, and assuming a genetic correlation of 0.5, increased genetic response for both heritability levels by up to 0.14 genetic s.d. per year. The scenario with simultaneous selection for the new trait and the index, yielded a substantially lower response for the new trait, especially when the genetic correlation with the index was negative. Despite the lower response for the index, whenever the new trait had considerable economic value, including the cow reference population considerably improved the genetic response for the new trait. For scenarios with a zero or negative genetic correlation with the index and equal economic value for the index and the new trait, a reference population of 2000 cows increased genetic response for the new trait with at least 0.10 and 0.20 genetic s.d. per year, for heritability levels of 0.05 and 0.30, respectively. We conclude that for new traits with a very small or positive genetic correlation with the index, and a high positive economic value, considerable genetic response can already be achieved based on a cow reference population with only 2000 records, even when the reliability of individual genomic breeding values is much lower than currently accepted in dairy cattle breeding programs. New traits may generally have a negative genetic correlation with the index and a small positive economic value. For such new traits, cow reference populations of at least 10 000 cows may be required to achieve acceptable levels of genetic response for the new trait and for the whole breeding goal.     

The effects of selection for gain in mice on the direct-maternal genetic correlation

Components of genetic variation for postweaning growth traits were estimated for both control and growth stocks of mice. The effect of phenotypic selection for gain, which genetically combines selection for additive direct and maternal effects, on additive genetic variance components, heritability, and additive genetic correlationsis discussed. Quantitative genetic theory predicts that simultaneous selection for two metric traits in the same direction will cause the genetic correlation between the two traits to become more negative. The results presented in this paper conflict with this theory. The direct-maternal additive genetic correlation was more negative in the control line (with 356 mice) than in the growth-selected line (with 320 mice) for the three traits analyzed (0.310 vs 0.999 for 21-day weight, 0.316 vs 1.000 for 42-day weight, and 0.506 vs 1.000 for gain from 21–42 days). Estimates were obtained by restricted maximum likelihood (REML) computed under a derivative free algorithm (DFREML).