Additive Variation Maintained under Stabilizing Selection: A Two-Locus Model of Pleiotropy for Two Quantitative Characters

A model with two diallelic loci controlling two additive quantitative characters is suggested. One of the loci has a similar effect on both characters, whereas the second locus has an antagonistic effect on two characters. Both characters experience direct stabilizing selection. The model yields a stable polymorphic state, with both characters maintaining genetic variation. The genetic correlation between the characters at the equilibrium is zero, in spite of the pleiotropic effects of the loci controlling them.     

Deleterious Mutations, Apparent Stabilizing Selection and the Maintenance of Quantitative Variation

Apparent stabilizing selection on a quantitative trait that is not causally connected to fitness can result from the pleiotropic effects of unconditionally deleterious mutations, because as N. Barton noted, "...individuals with extreme values of the trait will tend to carry more deleterious alleles...." We use a simple model to investigate the dependence of this apparent selection on the genomic deleterious mutation rate, U; the equilibrium distribution of K, the number of deleterious mutations per genome; and the parameters describing directional selection against deleterious mutations. Unlike previous analyses, we allow for epistatic selection against deleterious alleles. For various selection functions and realistic parameter values, the distribution of K, the distribution of breeding values for a pleiotropically affected trait, and the apparent stabilizing selection function are all nearly Gaussian. The additive genetic variance for the quantitative trait is kQa2, where k is the average number of deleterious mutations per genome, Q is the proportion of deleterious mutations that affect the trait, and a2 is the variance of pleiotropic effects for individual mutations that do affect the trait. In contrast, when the trait is measured in units of its additive standard deviation, the apparent fitness function is essentially independent of Q and a2; and beta, the intensity of selection, measured as the ratio of additive genetic variance to the "variance" of the fitness curve, is very close to s = U/k, the selection coefficient against individual deleterious mutations at equilibrium. Therefore, this model predicts appreciable apparent stabilizing selection if s exceeds about 0.03, which is consistent with various data. However, the model also predicts that beta must equal Vm/VG, the ratio of new additive variance for the trait introduced each generation by mutation to the standing additive variance. Most, although not all, estimates of this ratio imply apparent stabilizing selection weaker than generally observed. A qualitative argument suggests that even when direct selection is responsible for most of the selection observed on a character, it may be essentially irrelevant to the maintenance of variation for the character by mutation-selection balance. Simple experiments can indicate the fraction of observed stabilizing selection attributable to the pleiotropic effects of deleterious mutations.     

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.     

Geographical Variation in a Quantitative Character

A model for the evolution of the local averages of a quantitative character under migration, selection, and random genetic drift in a subdivided population is formulated and investigated. Generations are discrete and nonoverlapping; the monoecious, diploid population mates at random in each deme. All three evolutionary forces are weak, but the migration pattern and the local population numbers are otherwise arbitrary. The character is determined by purely additive gene action and a stochastically independent environment; its distribution is Gaussian with a constant variance; and it is under Gaussian stabilizing selection with the same parameters in every deme. Linkage disequilibrium is neglected. Most of the results concern the covariances of the local averages. For a finite number of demes, explicit formulas are derived for (i) the asymptotic rate and pattern of convergence to equilibrium, (ii) the variance of a suitably weighted average of the local averages, and (iii) the equilibrium covariances when selection and random drift are much weaker than migration. Essentially complete analyses of equilibrium and convergence are presented for random outbreeding and site homing, the Levene and island models, the circular habitat and the unbounded linear stepping-stone model in the diffusion approximation, and the exact unbounded stepping-stone model in one and two dimensions.     

The Quantitative Genetic Consequences of Pleiotropy under Stabilizing and Directional Selection

The independence of two phenotypic characters affected by both pleiotropic and nonpleiotropic mutations is investigated using a generalization of M. Slatkin's stepwise mutation model of 1987. The model is used to determine whether predictions of either the multivariate normal model introduced in 1980 by R. Lande or the house-of-cards model introduced in 1985 by M. Turelli can be regarded as typical of models that are intermediate between them. We found that, under stabilizing selection, the variance of one character at equilibrium may depend on the strength of stabilizing selection on the other character (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the types of mutations that can occur. Similarly, under directional selection, the genetic covariance between two characters may increase substantially (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the kinds of mutations that are assumed to occur. Hence, even for the simple model we consider, neither the house-of-cards nor the multivariate normal model can be used to make predictions, making it unlikely that either could be used to draw general conclusions about more complex and realistic models.     

Stabilizing and Disruptive Selection on a Mutant Character in Drosophila. IV. Selection on Sensitivity to Temperature

Individual selection on temperature sensitivity was applied to the relative length of the 4th vein of the mutant ci^D-G in four selection lines according to different selection schemes indicated in Figure 1. These selection systems include, besides the selection on temperature, disruptive selection. In all lines the disruptive component causes an increase of the phenotypic variance, but there are large differences in its composition. The selection changed temperature sensitivity in the expected direction in all lines. In one line (C-D^-) however, the change in temperature sensitivity and the increase in variance were only small because the disruptive component and the canalizing component of the selection scheme restricted each others' effect. The results are discussed in relation to earlier results obtained by disruptive selection at one temperature.     

Survival of the Curviest: Noise-Driven Selection for Synergistic Epistasis

A major goal of human genetics is to elucidate the genetic architecture of human disease, with the goal of fueling improvements in diagnosis and the understanding of disease pathogenesis. The degree to which epistasis, or non-additive effects of risk alleles at different loci, accounts for common disease traits is hotly debated, in part because the conditions under which epistasis evolves are not well understood. Using both theory and evolutionary simulation, we show that the occurrence of common diseases (i.e. unfit phenotypes with frequencies on the order of 1%) can, under the right circumstances, be expected to be driven primarily by synergistic epistatic interactions. Conditions that are necessary, collectively, for this outcome include a strongly non-linear phenotypic landscape, strong (but not too strong) selection against the disease phenotype, and “noise” in the genotype-phenotype map that is both environmental (extrinsic, time-correlated) and developmental (intrinsic, uncorrelated) and, in both cases, neither too little nor too great. These results suggest ways in which geneticists might identify, a priori, those disease traits for which an “epistatic explanation” should be sought, and in the process better focus ongoing searches for risk alleles.     

Maintenance of Genetic Variability under Strong Stabilizing Selection: A Two-Locus Model

We study a two locus model with additive contributions to the phenotype to explore the relationship between stabilizing selection and recombination. We show that if the double heterozygote has the optimum phenotype and the contributions of the loci to the trait are different, then any symmetric stabilizing selection fitness function can maintain genetic variability provided selection is sufficiently strong relative to linkage. We present results of a detailed analysis of the quadratic fitness function which show that selection need not be extremely strong relative to recombination for the polymorphic equilibria to be stable. At these polymorphic equilibria the mean value of the trait, in general, is not equal to the optimum phenotype, there exists a large level of negative linkage disequilibrium which ``hides' additive genetic variance, and different equilibria can be stable simultaneously. We analyze dependence of different characteristics of these equilibria on the location of optimum phenotype, on the difference in allelic effect, and on the strength of selection relative to recombination. Our overall result that stabilizing selection does not necessarily eliminate genetic variability is compatible with some experimental results where the lines subject to strong stabilizing selection did not have significant reductions in genetic variability.     

A Model Selection Approach for the Identification of Quantitative Trait Loci in Experimental Crosses, Allowing Epistasis

The identification of quantitative trait loci (QTL) and their interactions is a crucial step toward the discovery of genes responsible for variation in experimental crosses. The problem is best viewed as one of model selection, and the most important aspect of the problem is the comparison of models of different sizes. We present a penalized likelihood approach, with penalties on QTL and pairwise interactions chosen to control false positive rates. This extends the work of Broman and Speed to allow for pairwise interactions among QTL. A conservative version of our penalized LOD score provides strict control over the rate of extraneous QTL and interactions; a more liberal criterion is more lenient on interactions but seeks to maintain control over the rate of inclusion of false loci. The key advance is that one needs only to specify a target false positive rate rather than a prior on the number of QTL and interactions. We illustrate the use of our model selection criteria as exploratory tools; simulation studies demonstrate reasonable power to detect QTL. Our liberal criterion is comparable in power to two Bayesian approaches.     

Permutation Tests for Multiple Loci Affecting a Quantitative Character

The problem of detecting minor quantitative trait loci (QTL) responsible for genetic variation not explained by major QTL is of importance in the complete dissection of quantitative characters. Two extensions of the permutation-based method for estimating empirical threshold values are presented. These methods, the conditional empirical threshold (CET) and the residual empirical threshold (RET), yield critical values that can be used to construct tests for the presence of minor QTL effects while accounting for effects of known major QTL. The CET provides a completely nonparametric test through conditioning on markers linked to major QTL. It allows for general nonadditive interactions among QTL, but its practical application is restricted to regions of the genome that are unlinked to the major QTL. The RET assumes a structural model for the effect of major QTL, and a threshold is constructed using residuals from this structural model. The search space for minor QTL is unrestricted, and RET-based tests may be more powerful than the CET-based test when the structural model is approximately true.     

Selection for Genetic Variation Inducing Pro-Inflammatory Responses under Adverse Environmental Conditions in a Ghanaian Population

Background

Chronic inflammation is involved in the pathogenesis of chronic age-associated, degenerative diseases. Pro-inflammatory host responses that are deleterious later in life may originate from evolutionary selection for genetic variation mediating resistance to infectious diseases under adverse environmental conditions.

Methodology/Principal Findings

In the Upper-East region of Ghana where infection has remained the leading cause of death, we studied the effect on survival of genetic variations at the IL10 gene locus that have been associated with chronic diseases. Here we show that an IL10 haplotype that associated with a pro-inflammatory innate immune response, characterised by low IL-10 (p = 0.028) and high TNF-α levels (p = 1.39×10⁻³), was enriched among Ghanaian elders (p = 2.46×10⁻⁶). Furthermore, in an environment where the source of drinking water (wells/rivers vs. boreholes) influences mortality risks (HR 1.28, 95% CI [1.09–1.50]), we observed that carriers of the pro-inflammatory haplotype have a survival advantage when drinking from wells/rivers but a disadvantage when drinking from boreholes (p_interaction = 0.013). Resequencing the IL10 gene region did not uncover any additional common variants in the pro-inflammatory haplotype to those SNPs that were initially genotyped.

Conclusions/Significance

Altogether, these data lend strong arguments for the selection of pro-inflammatory host responses to overcome fatal infection and promote survival in adverse environments.     

Drosophila Alcohol Dehydrogenase Polymorphism and Carbon-13 Fluxes: Opportunities for Epistasis and Natural Selection

The influence of genetic variations in Drosophila alcohol dehydrogenase (ADH) on steady-state metabolic fluxes was studied by means of (13)C NMR spectroscopy. Four pathways were found to be operative during 8 hr of ethanol degradation in third instar larvae of Drosophila. Seven strains differed by 18-25% in the ratio between two major pathway fluxes, i.e., into glutamate-glutamine-proline vs. lactate-alanine-trehalose. In general, Adh genotypes with higher ADH activity exhibit a twofold difference in relative carbon flux from malate into lactate and alanine vs. α,α-trehalose compared to low ADH activity genotypes. Trehalose was degraded by the pentose-phosphate shunt. The pentose-phosphate shunt and malic enzyme could supply NADPH necessary for lipid synthesis from ethanol. Lactate and/or proline synthesis may maintain the NADH/NAD(+) balance during ethanol degradation. After 24 hr the flux into trehalose is increased, while the flux into lipids declines in Adh(F) larvae. In Adh(S) larvae the flux into lipids remains high. This co-ordinated nature of metabolism and the genotype-dependent differences in metabolic fluxes may form the basis for various epistatic interactions and ultimately for variations in organismal fitness.     

Modifications in Estimating the Number of Genes for a Quantitative Character

In estimating the minimum number of genes contributing to a quantitative character, it is suggested that the squared difference between the means of the two parents be corrected for experimental variance and that the genetic variance stemming from differences in gene frequencies of the parents be estimated by least squares utilizing information on all entries.     

A Simulation Study of Truncation Selection for a Quantitative Trait Opposed by Natural Selection

A quantitative character controlled at one locus with two alleles was submitted to artificial (mass) selection and to three modes of opposing natural selection (directional selection, overdominance and underdominance) in a large random-mating population. The selection response and the limits of the selective process were studied by deterministic simulation. The lifetime of the process was generally between 20 and 100 generations and did not appear to depend on the mode of natural selection. However, depending on the values of the parameters (initial gene frequency, selection intensity, ratio of the effect of the gene to the environmental standard deviation, fitness values) the following outcomes of selection were observed: fixation of the allele favored by artificial selection, stable nontrivial equilibrium, unstable equilibrium and loss of the allele favored by artificial selection. Finally, the results of the simulation were compared to the results of selection experiments.     

Selection for Chaperone-Like Mediated Genetic Robustness at Low Mutation Rate: Impact of Drift,Epistasis and Complexity

Genetic robustness is defined as the constancy of a phenotype in the face of deleterious mutations. Overexpression of chaperones, to assist the folding of proteins carrying deleterious mutations, is so far one of the most accepted molecular mechanisms enhancing genetic robustness. Most theories on the evolution of robustness have focused on the implications of high mutation rate. Here we show that genetic drift, which is modulated by population size, organism complexity, and epistasis, can be a sufficient force to select for chaperone-mediated genetic robustness. Using an exact analytical solution, we also show that selection for costly genetic robustness leads to a paradox: the decrease of population fitness on long timescales and the long-term dependency on robustness mechanisms. We suggest that selection for genetic robustness could be universal and not restricted to high mutation rate organisms such as RNA viruses. The evolution of the endosymbiont Buchnera illustrates this selection mechanism and its paradox: the increased dependency on chaperones mediating genetic robustness. Our model explains why most chaperones might have become essential even in optimal growth conditions.MUTATIONAL (or genetic) robustness is defined as the constancy of a phenotype in the face of deleterious mutations (Sanjuan et al. 2007). Selection drives populations to adapt to their environment by the fixation of successive advantageous mutations. However, in approaching a fitness optimum—i.e., a genotype that is maximally adapted—they have to cope with an increasing proportion of deleterious mutations and, when at the optimum, they experience only neutral and deleterious mutations (Silander et al. 2007). Therefore any mechanism that would reduce the effect of deleterious mutations, i.e., increase mutational robustness, could be favored by natural selection when at, or near, an optimum of fitness. Indeed, the general observation that for a large range of organisms, mutations have little effect on fitness, suggests that selection for robustness is pervasive (Melton 1994; Winzeler et al. 1999). Three main mechanisms that are not mutually exclusive could explain how genetic robustness has arisen. First, in the “intrinsic hypothesis” (de Visser et al. 2003) robustness could simply be a by-product of some biologically relevant functions. Second, mutational robustness could be a by-product of the selection for nongenetic perturbations such as environment changes or intrinsic noise (Wagner 2005). Third, mutational robustness could be selected for because it is adaptive in itself. In the following we restrict our attention to this “adaptive hypothesis” (de Visser et al. 2003).Chaperone proteins, proteins that help other proteins to fold properly, have been shown to buffer the effect of deleterious mutations in diverse organisms (Rutherford 2003). In lineages that have accumulated deleterious mutations, the overexpression of the chaperone GroESL in Escherichia coli (Fares et al. 2002) or Salmonella typhymurium (Maisnier-Patin et al. 2005) resulted in an improved fitness. However, such robustness appears to come at a cost, as the buffering was visible only in carbon-rich media (Fares et al. 2002), and it is also known that GroESL-mediated refolding of proteins is ATP dependent. Chaperones can also buffer against environmental perturbations (such as heat shock); however, the observation that groESL evolved under positive selection and is overproduced in obligate intracellular endosymbionts (Moran 1996; Fares et al. 2004), for which environmental perturbations are assumed to be very weak, suggests that genetic robustness could be the direct target of selection.Selection for a modifier of genetic robustness, i.e., a gene modulating the effect of mutations, has been mainly studied in the context of high mutation rates, as the effect of the modifier allele affects the fitness of mutants (Wagner 2005). Under some theoretical frameworks, it has been suggested that the intensity of selection acting on a modifier of robustness would be of the order of the mutation rate (Gardner and Kalinka 2006). Therefore it has been presumed that selection for genetic robustness is relevant only in very large populations having a high mutation rate, such as RNA virus populations. In agreement with these ideas, artificial life experiments (Wilke and Adami 2001; Azevedo et al. 2006) and experimental data on viruses (Montville et al. 2005; Sanjuan et al. 2007) have shown that robustness varies between organisms and that it can be selected for under high mutation rates. It has also been shown by Krakauer and Plotkin (2002) that drift, i.e., stochastic effects due to the finite size of populations, can promote selection for robustness even when more robust alleles are costly, as suggested in the case of chaperone overexpression. However, again this effect was examined only under high mutation rates.When mutations are very rare, populations experience at the most the presence of a single mutant. In such conditions, the population fitness at equilibrium does not depend on the mutation rate but only on drift (Sella and Hirsh 2005; Tenaillon et al. 2007). Two factors modulate how drift affects fitness:

1. Epistasis, defined here as a local property of the adaptive landscape, describes how the selective effects of mutations depend on the genetic background in which they arise. Epistasis is negative (positive) if two mutations have a lower (higher) fitness when simultaneously present within a genome than expected if they did not interact. Negative epistasis increases selection against mutation-loaded individuals and therefore reduces the effect of drift on population fitness (Charlesworth 1990; Tenaillon et al. 2007).
2. Phenotypic complexity, defined as the number of independent mutable traits that contribute to fitness (Orr 2000; Tenaillon et al. 2007), also affects population fitness in finite populations: complex organisms are more sensitive to the action of drift (Hartl and Taubes 1998; Poon and Otto 2000; Tenaillon et al. 2007).

In this article, we attempt to further clarify the role of drift on the evolution of chaperone-like genetic robustness and to decouple the effect of drift from the effect of the mutation rate. We use Fisher''s geometric model of adaptation (Fisher 1930), to map phenotype to fitness under an assumption of a vanishing mutation rate and extract exact analytical solutions for the genetic properties of the population at mutation–selection–drift equilibrium (MSDE). We examine how these genetic properties change under various population sizes and epistasis parameters and in organisms ranging in phenotypic complexity.