How might genotype frequencies of alleles for body size change under directional selection?

Journal List
Heredity [Edinb]
v.118[1]; 2017 Jan
PMC5176114

Heredity [Edinb]. 2017 Jan; 118[1]: 96–109.

Abstract

Much of quantitative genetics is based on the ‘infinitesimal model', under which selection has a negligible effect on the genetic variance. This is typically justified by assuming a very large number of loci with additive effects. However, it applies even when genes interact, provided that the number of loci is large enough that selection on each of them is weak relative to random drift. In the long term, directional selection will change allele frequencies, but even then, the effects of epistasis on the ultimate change in trait mean due to selection may be modest. Stabilising selection can maintain many traits close to their optima, even when the underlying alleles are weakly selected. However, the number of traits that can be optimised is apparently limited to ~4Ne by the ‘drift load', and this is hard to reconcile with the apparent complexity of many organisms. Just as for the mutation load, this limit can be evaded by a particular form of negative epistasis. A more robust limit is set by the variance in reproductive success. This suggests that selection accumulates information most efficiently in the infinitesimal regime, when selection on individual alleles is weak, and comparable with random drift. A review of evidence on selection strength suggests that although most variance in fitness may be because of alleles with large Nes, substantial amounts of adaptation may be because of alleles in the infinitesimal regime, in which epistasis has modest effects.

Introduction

The relation between an organism's DNA sequence and its fitness is extremely complex, being mediated by gene expression, physiology, development and behaviour, all in interaction with the environment. Population and quantitative genetics use simple and abstract models to explain the evolutionary consequences of this relationship—a bold undertaking. Many have questioned whether this approach can account for the complexities of gene interaction [that is, of epistasis], and have suggested that properly incorporating epistasis will radically change our ability to determine the causes of quantitative variation, and our understanding of evolution [Carlborg and Haley, 2004; Carter et al., 2005; Huang et al., 2012; Hansen, 2013; Nelson et al., 2013].

In fact, classical quantitative and population genetics do allow for an arbitrary relation between genotype and phenotype, and for evolution across a ‘rugged fitness landscape'. Phenotypic traits [including fitness] depend on interactions among sets of alleles, as well as on the marginal effects of individual alleles. Remarkably, the variance associated with sets of one, two or more genes can be estimated from correlations of the trait between relatives, without any need to know the detailed genetic basis of trait variation [Fisher, 1918; Lynch and Walsh, 1998]. Within this framework, epistasis has two distinct roles. First, it generates nonrandom associations among alleles [that is, linkage disequilibria]. However, in a sexual population these are broken up by recombination, and hence have no long-term consequence. More important, epistasis makes the marginal [that is, additive] effects of alleles depend on the current genetic background. Thus, even though the immediate response of allele frequencies to selection is due to the additive component of genetic variance, these additive effects may change over time. Indeed, an amino acid that is benign in one species may be lethal when in the genetic background of even a closely related species [Kondrashov et al., 2002].

Fisher and Wright developed methods that can describe arbitrary epistasis: the analysis of variance [Fisher, 1918], leading to the ‘Fundamental Theorem of Natural Selection' [Fisher, 1930], and selection gradients on the ‘adaptive landscape' [Wright, 1931]. In both models, the response of allele frequencies to selection is primarily because of the additive effects of individual alleles: nonadditive variance does not contribute directly to long-term evolution. However, Fisher and Wright held very different views on the evolutionary significance of epistasis [Provine, 1988]. Wright [1931] argued that gene interactions lead to multiple ‘adaptive peaks', and that progressive evolution is limited by the difficulty of crossing between these. In contrast, Fisher [1930] held that because environments fluctuate, and because evolution occurs in a space of extremely high dimension, there can be a continuing response to selection without the need ever to cross a fitness valley in opposition to selection.

In the following, I bring together theoretical results that show that the evolution of complex traits can be described by an ‘infinitesimal model' that is not sensitive to the detailed way in which genes interact. Epistasis has surprisingly little effect on the response to either directional or stabilising selection, even when substantial fractions of the genetic variance are because of gene interactions, and the underlying fitness landscape is rugged. This leads to robust limits to the number of traits that can be kept close to an intermediate optimum, and suggests that selection is most efficient in the infinitesimal regime, when it is comparable with the strength of random drift on individual alleles.

The infinitesimal model

Practical quantitative genetics depends on the infinitesimal model, under which the components of genetic variance remain approximately constant despite selection. Defined at the individual level, in its simplest form this model states that two parents produce offspring whose breeding values are normally distributed around the mean breeding value of the parents, with variance independent of these parental values. This definition extends to the whole pedigree, such that the distribution of descendants is multivariate normal, with a covariance that is independent of the ancestral values. This implies that selection of specific individuals as parents only affects the offspring means and not their covariance [Lange, 1978; Bulmer, 1980].

This standard infinitesimal model can be justified as the limit of an additive model, when the number of loci tends to infinity [Fisher, 1918; Bulmer, 1980]. However, the model extends to allow for substantial epistasis [Barton et al., 2016; Paixao and Barton, 2016]. The key assumption is that phenotypes occupy a narrow range, relative to the range of multilocus genotypes that are possible, given the standing variation. This is consistent with the basic observation that artificial selection can shift the mean phenotype far outside its original range, within a few tens of generations, and implies that very many genotypes are consistent with any specific trait value; recombination between these different genotypes generates new variation. Thus, knowing the trait value gives little information about individual genotype, and hence hardly influences the distribution of allele frequencies. Therefore, selection on a trait that depends on very many loci hardly perturbs the variance components away from their neutral evolution.

This is illustrated by Figure 1 which includes strong epistatic interactions, so that most of the additive variance is due to epistatic coefficients; as is typical, the epistatic variance is much smaller than the additive component [lower pair of curves at right]. Selection rapidly changes the mean, by ~11 genetic s.d. over 100 generations, However, the variance components are only modestly changed from their neutral evolution: the additive component is reduced by 25%, and the nonadditive component by 31%, after 100 generations [compare dashed and solid lines at right].

The effect of selection on the mean and variance components in the presence of epistasis. Directional selection, β=0.2 [solid line] is contrasted with the neutral case [dashed line]; shaded areas indicate ±1 s.d. The left panel shows the change in mean from its initial value and the right panel shows the additive variance, VA, and the additive × additive variance, VA A [lower pair of curves]. Only the genic components of variance are shown; random linkage disequilibria make no appreciable difference on average. There are M=3000 loci, and N=100 haploid individuals. Alleles are given equal main effects but random sign

. Sparse pairwise epistasis is represented by choosing a fraction 1/M of pairwise interactions, ωι j, from a normal distribution with s.d.

. The trait is now defined as z=δ.γ+δ.ω.δT, where δ=±[1/2]. Initial allele frequencies are drawn from a U-shaped β-distribution, mean p̂ =0.2 and variance 0.2 p̂q̄. Individuals are produced by Wright–Fisher sampling from parents chosen with probability proportional to W=eβ z. For each example, three sets of allelic and epistatic effects are drawn and for each of those, three populations are evolved; this gives 9 replicates in all.

To account for epistasis, the basic infinitesimal model must be extended, such that individual trait values are represented by components due to sets of one, two or more loci. However, the distribution of these components among offspring follows rules that depend only on the components of genetic variance, and not on the values of the parents [Figure 2]. The infinitesimal model can apply even when much of the genotypic variance is due to epistatic interaction. The covariance between relatives is given by the rules of classical quantitative genetics, and just as in the additive case, these covariances depend only on the components of genetic variance in the base population. Allele frequencies may change substantially as a result of random drift: the crucial assumption is that selection on the phenotype causes only a small perturbation away from neutrality. The cumulative effects of these small perturbations change the genetic components of the trait mean significantly, but not the variance components themselves.

The mean and variance of offspring plotted against components of the parents' trait values. Top left: additive component of offspring, AO, against the mean of the parents' additive component, AP. The line represents AO=AP. Top right: the same, but for the additive × additive components. The line shows a linear regression. Bottom left: additive variance among offspring, VA,O against the mean additive components of the parents, AP. Bottom right: additive × additive variance of offspring against the mean additive × additive component of the parents. Lines in the bottom row show quadratic regressions. The example shows a nonadditive trait under selection β=0.2, with M=3000 loci and N=100 haploid individuals, as in Figure 1. At generation 20, 200 pairs of minimally related parents [F=0.165] were chosen, and 1000 offspring were generated for each pair. For each offspring, the components of trait value were calculated relative to the allele frequencies, p, in the base population. Defining genotype by X=0, 1, these components are A=ζ.[α+[ω+ωT].[p−1/2]], AA=ζ.ω.ζT, where ζ=X−p.

Though only a few examples are shown here, the infinitesimal model applies very widely. Nevertheless, it clearly does not apply to all forms of epistasis: systematically positive or negative interactions, such as might be produced by a scale change, would cause offspring distributions to be non-Gaussian, and to be centred away from the mean of the parents. The key point in this section, however, is that random epistatic interactions are consistent with an infinitesimal limit, in which the response to selection on quantitative traits can be predicted from classical quantitative genetics. The complications of epistasis are entirely absorbed into a few variance components that are hardly perturbed by selection.

The overall sign of epistasis has received much attention: systematically negative epistasis would give an advantage to sex and recombination, and would allow a higher mutation rate without leading to excessive load [Kondrashov, 1988]. However, invoking systematic epistasis raises the question of why the effects of interactions between alleles should be biased with respect to the marginal effects of these alleles on fitness. This might be a side effect of how organisms are built, which might in turn be because of past selection for [say] robustness to environmental or genetic perturbation. There is a close analogy here with the evolution of dominance. The immediate cause of dominance may be that losing function from one copy of a gene causes little fitness loss, whereas losing both is strongly deleterious. However, this raises the question of why organisms should typically have excess capacity; that redundancy may itself be because of selection for robustness against environmental fluctuations [Wright, 1929; Bourguet, 1999].

Directional selection

This extension of the infinitesimal model immediately leads to a remarkably general expression for the effect of epistasis on the limits to directional selection on standing variation [Paixao and Barton, 2016]. [Note that here, I use directional selection to refer to an exponential relation between fitness and trait; other forms of selection—for example, truncation selection—will select on the variance as well as the mean]. Under the infinitesimal model, the additive variance, VA, decreases by a factor [1−1/[2Ne]] per generation, whereas the mean increases by βVA, where β is the selection gradient. Therefore, the total change in mean sums to 2Neβ which is just 2Ne times the change in the first generation [V0A being the initial additive variance]. Robertson [1960] showed that this result can be derived by considering the slight increase in fixation probability of favourable alleles because of selection—a derivation that makes clear that the infinitesimal model implicitly assumes selection on individual alleles, s, to be weaker than drift [that is, Nes50 generations, say; Hill, 1982], mutation makes a significant contribution, increasing additive variance by VAm per generation. Under the standard infinitesimal model, the additive variance approaches an equilibrium between mutation and random drift of 2NeVAm, and the mean will change under directional selection in proportion to this variance. In the short term, mutation generates negligible epistatic variance, unless mutations have large effect, as it introduces alleles at low frequency [Hill and Rasbash, 1986]. However, epistasis makes additive effects conditional on genotype, so that the effect of new mutations may change with the mean. In the long term, the genetic variance will evolve unpredictably, as new alleles introduced by mutation become common enough to interact with each other. Nevertheless, as mutational variance is ubiquitous [Houle et al., 1996; Lynch and Walsh, 1998], an indefinite response to directional selection is expected.

When multiple traits are selected, the mean changes in proportion to the additive genetic covariance matrix [termed the ‘G matrix'] that in turn is proportional to the mutational covariance in the infinitesimal limit. The G matrix has received much attention on the grounds that it constrains adaptation. However, artificial selection has proved successful even when deliberately applied to trait combinations that show minimal variance [see, for example, Weber et al., 1999; Hill and Kirkpatrick, 2010; Marchini et al., 2014]: as long as there is some additive variance in the direction of selection, selection can change the mean. Of course, the G matrix has very high dimension, and some directions may have zero variance [that is, there may be some zero eigenvalues]. Even then, however, the G matrix does not necessarily constrain adaptation in the long term: it inevitably changes as new mutations arise, with effects in different directions. Imagine that traits may be influenced by a very large number of sites, n, of which only a much smaller number, 1 ns. Thus, evolution is constrained by the total number of sites that could affect the traits, and not by the number segregating at any particular time. Therefore, observation of the G matrix at any one time would not inform us about constraint on long-term evolution. This is illustrated in Figure 3. The left panel shows that in any one generation, most variance is explained by 0. Rescaling, we find that ℐ/VW[a]=Ne/2.

The gain in information, ℐ, and the expected total heterozygosity,

, plotted against initial allele frequency, p0 [left and middle respectively]. The right plot shows the ratio between the information gain and the expected total fitness variance. Selection strength is 4Nes=α=0.125, 0.25,…, 8 [black…red]. In the limit α→0, the scaled ratio tends to 1/16, independent of p0. A full colour version of this figure is available at the Heredity journal online.

Appendix B

Minor errors inMustonen and Lässig [2010]

Note that there is a pervasive factor of two error in Mustonen and Lässig [2010]. Their Equation [2], which corresponds to Wright's stationary distribution [Equation 1], should include a factor of 2 in the exponent. This is not due to the assumption of haploidy versus diploidy: Wright's formula is the same in both cases, with the factor of 2 in the number of genes compensating a factor of 2 in the definitions of mean fitness. The factor of 2 in Wright's formula arises from the factor of 1/2 in the stochastic term of the diffusion equation that is missing from Mustonen and Lässig [2010, Equation [S20]].

There is also a sign error in Ω in Equation [S74].

Notes

The author declares no conflict of interest.

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Articles from Heredity are provided here courtesy of Nature Publishing Group

How does directional selection affect allele frequency?

Directional selection leads to increase over time in the frequency of a favored allele. Consider three genotypes [AA, Aa and aa] that vary in fitness such that AA individuals produce, on average, more offspring than individuals of the other genotypes.

What happens when there is a directional selection?

Directional selection can also be compared to disruptive selection, or a selection that causes an increase in both extremes of a trait spectrum. If a directional selection is applied to a population over time, the traits that are selected for will permanently increase, while the traits selected against will be lost.

How does genotype frequency affect allele frequency?

The relative genotype frequencies show the distribution of genetic variation in a population. Relative allele frequency is the percentage of all copies of a certain gene in a population that carry a specific allele. This is an accurate measurement of the amount of genetic variation in a population.

How does directional selection change a population?

In directional selection, a population's genetic variance shifts toward a new phenotype when exposed to environmental changes. Diversifying or disruptive selection increases genetic variance when natural selection selects for two or more extreme phenotypes that each have specific advantages.