Modeling Multiallelic Selection Using a Moran Model

We present a Moran-model approach to modeling general multiallelic selection in a finite population and show how it may be used to develop theoretical models of biological systems of balancing selection such as plant gametophytic self-incompatibility loci. We propose new expressions for the stationary distribution of allele frequencies under selection and use them to show that the continuous-time Markov chain describing allele frequency change with exchangeable selection and Moran-model reproduction is reversible. We then use the reversibility property to derive the expected allele frequency spectrum in a finite population for several general models of multiallelic selection. Using simulations, we show that our approach is valid over a broader range of parameters than previous analyses of balancing selection based on diffusion approximations to the Wright–Fisher model of reproduction. Our results can be applied to any model of multiallelic selection in which fitness is solely a function of allele frequency.NATURAL selection has long been a topic of interest in population genetics, yet the stochastic theory of genes under selection remains underdeveloped compared to the theory of neutral genes. Due to the interplay of stochastic and deterministic forces, models of selection present analytical challenges beyond those of neutral models, although a great deal of progress has been made with models that use diffusion approximations to a Wright–Fisher model of reproduction. Diffusion approximations with selection are, however, sometimes difficult to employ and always require assumptions about population parameters for tractability. These limitations suggest that there may be value in developing new methods of solving the problem of selection in a finite population, and here we do so using a Moran model of reproduction in place of the familiar Wright–Fisher model. Our approach has two major advantages over previous models: general applicability to a wide variety of selection models and accuracy over a broad range of parameter values. In this work, we propose new expressions for the full stationary distributions of allele frequencies under multiallelic selection, as well as expressions for average allele frequency distributions.We restrict our attention to exchangeable models of selection, meaning that relabeling the alleles will not change selective outcomes and thus that selection will be a function of allele frequency rather than allele identity. Many models of selection can be transformed into frequency-dependent forms (Denniston and Crow 1990), and some common models of selection have the desired property of exchangeability. For example, symmetric overdominant selection, in which heterozygotes have a selective advantage over homozygotes but the specific genotype of homozygote or heterozygote has no further selective effect, can be expressed as frequency-dependent selection on individual (exchangeable) alleles, although the direct selection is actually on diploid genotypes. Many other proposed models of multiallelic balancing selection, in which substantial variation is maintained by selection, can be viewed in this way. Such models have been of particular interest because of the potential application to highly multiallelic systems found in nature, such as self-incompatibility (SI) loci in plants and the major histocompatibility complex (MHC) loci in vertebrates, and the desire to analyze these systems is a motivation for the present work. We now review some of the population genetic theory related to these systems.Early in the history of population genetics, Wright (1939) presented a somewhat controversial stochastic model of gametophytic self-incompatibility (GSI) genes, sparking much further theoretical and empirical work. An analytic theory of multiallelic symmetric overdominance was developed along similar lines to this early model (Kimura and Crow 1964; Takahata 1990) and has been used as an approximation to the unknown mode of selection in the MHC (Takahata et al. 1992). Drawing insights from these first two applications, other biological systems where balancing selection was posited, including sex determination in honeybees (Yokoyama and Nei 1979), fungal mating systems (May et al. 1999), and heterokaryon incompatibility in fungi (Muirhead et al. 2002), have also been modeled successfully using closely related approaches. Progress has been made in using these models to address genealogical (Takahata 1990; Vekemans and Slatkin 1994) and demographic (Muirhead 2001) questions, as well as extending the models into more complex modes of selection (Uyenoyama 2003) and reproduction (Vallejo-Marin and Uyenoyama 2008).Models of genetic variation under balancing selection have traditionally been focused on specific systems, such that extensions require entirely new analyses, and have also included a number of simplifying assumptions in the interest of mathematical tractability. For example, the symmetric overdominance model has been strongly criticized as an unrealistic approximation of MHC evolution (Paterson et al. 1998; Hedrick 2002; Penn et al. 2002; Ilmonen et al. 2007; Stoffels and Spencer 2008), and yet it has proved difficult to make finite-population models of any of the more realistic frequency dependence schemes using the same approaches. A constraint on further progress is the fact that the standard model of stochastic population genetics, the Wright–Fisher model, is in fact quite difficult to analyze.The Wright–Fisher model of reproduction employs nonoverlapping generations, so that for a diploid population of size N, all 2N allele copies are chosen simultaneously when forming a new generation of individuals. While it is straightforward to describe this reproduction scheme mathematically as a discrete-time Markov chain, that chain unfortunately appears intractable even in simple cases (Ewens 2004). Traditionally, then, diffusion approximations have been used to obtain quantities of interest, such as the equilibrium expected number of alleles, allele frequency spectra, and fixation probabilities and times. Diffusion approximations are derived in the limit , but are applicable to problems of finite N, provided that the strengths of other forces such as mutation and selection can be assumed to be weak, of O(N⁻¹) (Ewens 2004). Watterson (1977) derived such a diffusion approximation for multiallelic symmetric overdominance using these assumptions. More recently, as interest in population genetics has turned to problems of inference, Grote and Speed (2002) considered sampling probabilities under the diffusion approximation for symmetric overdominance, while Donnelly et al. (2001) and Stephens and Donnelly (2003) proposed computational methods for some asymmetric models.Although strong selection can be modeled using diffusion approximations by making the product of the population size and the selection coefficient (Ns) large, the assumption of weak selection is not in fact appropriate for the canonical biological systems of balancing selection. Specifically, selection coefficients are defined by the differences in fitness (the expected number of offspring) among individuals in the population at a given time. These differences may be large in systems such as GSI, where the fitness of a very common allele may be very small while the fitness of other alleles may be greater than one.In an attempt to deal with the extremely strong selection of gametophytic self-incompatibility, Wright''s (1939) original model focused attention on the dynamics of a single representative allele. He collapsed the influence of all other alleles into a single summary statistic: the homozygosity, F, which is a function of the frequencies of all alleles, and which Wright (1939) assumed to be constant. The analysis is essentially that of a two-allele system, using a one-dimensional diffusion analysis. This approach, while shown by simulation to be very effective in the appropriate parameter range (Ewens and Ewens 1966), received substantial criticism on mathematical grounds (Fisher 1958; Moran 1962; Ewens 1964b). Ewens (1964b), in particular, objected to the use of diffusion theory for GSI, pointing out that strong frequency-dependent selection violates the diffusion requirement that both the mean and the variance of the change in allele frequencies be small and of O(N⁻¹). Ewens (1964a) then applied Wright''s basic one-dimensional diffusion approach to modeling symmetric overdominance, but assumed that selection was weak and of O(N⁻¹) to stay within the strict limits of the diffusion approximation.Kimura and Crow (1964) and Wright (1966), on the other hand, presented alternative one-dimensional diffusion approximations to symmetric overdominance, closer in spirit to Wright''s original model of GSI, that did not make the weak-selection assumption. Watterson (1977) was concerned about both the inconsistencies of the approximations used in these models and the treatment of F as a constant rather than as a random variable dependent upon allele frequencies. Using his own multiallelic diffusion approximation for symmetric overdominance (Watterson 1977), he derived an alternative (small-Ns) approximation to the frequency of a single representative allele. We consider this approximation, as well as the best-known one-dimensional symmetric overdominance diffusion, the strong-selection approximation of Kimura and Crow (1964), in comparison with our alternative approach to deriving allele frequency spectra under general multiallelic selection with exchangeable alleles.To avoid the approximations required to employ Wright–Fisher/diffusion-based methods, we turn to an alternative model of reproduction in a finite population: the overlapping-generations model of Moran (1962). In the Moran model, a single allele copy dies and another reproduces in each time step, rather than all 2N allele copies simultaneously being replaced by offspring each generation. As in the Wright–Fisher model, this reproduction scheme is represented mathematically by a Markov chain. Unlike the Wright–Fisher model, however, the Moran model can sometimes yield tractable, exact solutions to the underlying Markov chain, without the need to resort to diffusion approximations. We exploit this trait to develop a new stochastic theory of multiallelic selection with minimal dependence on assumptions about population parameter values. Our method has the additional benefit of being flexible: it can accommodate any exchangeable model of multiallelic selection and either of two general models of parent-independent mutation, the infinite-alleles and k-allele models of mutation. Our Moran-model predictions agree well with the results of Wright–Fisher simulations.     

Reproductive Value and Fluctuating Selection in an Age-Structured Population

Fluctuations in age structure caused by environmental stochasticity create autocorrelation and transient fluctuations in both population size and allele frequency, which complicate demographic and evolutionary analyses. Following a suggestion of Fisher, we show that weighting individuals of different age by their reproductive value serves as a filter, removing temporal autocorrelation in population demography and evolution due to stochastic age structure. Assuming weak selection, random mating, and a stationary distribution of environments with no autocorrelation, we derive a diffusion approximation for evolution of the reproductive value weighted allele frequency. The expected evolution obeys an adaptive topography defined by the long-run growth rate of the population. The expected fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population. Simulations of the age-structured model verify the accuracy of the diffusion approximation. We develop statistical methods for measuring the expected selection on the reproductive value weighted allele frequency in a fluctuating age-structured population.THE evolutionary dynamics of age-structured populations were formalized by Charlesworth (1980, 1994) and Lande (1982) on the basis of earlier ideas of Fisher (1930, 1958), Medawar (1946, 1952), and Hamilton (1966), showing that the strength of selection on genes affecting the vital rates of survival or fecundity depends on their age of action (reviewed by de Jong 1994; Charlesworth 2000). Fisher defined the reproductive value of individuals of a given age as their expected contribution to future population growth, determined by the age-specific vital rates. This has the property that in a constant environment the total reproductive value in a population always increases at a constant rate. The total population size, however, undergoes transient fluctuations as the stable age distribution is approached, and the total population size only asymptotically approaches a constant growth rate (Caswell 2001).Environmental stochasticity creates continual fluctuations in age structure, producing temporal autocorrelation in population size and in allele frequencies, which seriously complicate demographic and evolutionary analyses. Fisher (1930, 1958, p. 35) suggested for analysis of genetic evolution that individuals should be weighted by their reproductive value to compensate for deviations from the stable age distribution. Here we apply this suggestion to study weak fluctuating selection in an age-structured population in a stochastic environment.One of the central conceptual paradigms of evolutionary biology was described by Wright (1932). His adaptive topography represents a population as a point on a surface of population mean fitness as a function of allele frequencies. Assuming weak selection, random mating, and loose linkage (implying approximate Hardy–Weinberg equilibrium within loci and linkage eqilibrium among loci), natural selection in a constant environment causes the population to evolve uphill of the mean fitness surface (Wright 1937, 1945, 1969; Arnold et al. 2001; Gavrilets 2004). Evolution by natural selection thus tends to increase the mean fitness of a population in a constant environment.Lande (2007, 2008) generalized Wright''s adaptive topography to a stochastic environment, allowing density-dependent population growth but assuming density-independent selection, showing that the expected evolution maximizes the long-run growth rate of the population at low density, . Here r is population growth rate at low density in the average environment and is the environmental variance in population growth rate among years, which are standard parameters in stochastic demography (Cohen 1977, 1979; Tuljapurkar 1982; Caswell 2001; Lande et al. 2003). In this model of stochastic evolution the adaptive topography describing the expected evolution is derived by expressing r and as functions of allele frequencies with parameters being the mean Malthusian fitnesses of the genotypes and their temporal variances and covariances. These results are based on diffusion approximations for the coupled stochastic processes of population size and allele frequencies in a fluctuating environment.Diffusion approximations are remarkably accurate for many problems in evolution and ecology (Crow and Kimura 1970; Lande et al. 2003). Because a diffusion process is subject to white noise with no temporal autocorrelation, the approximation is most accurate if the noise in the underlying biological process is approximately uncorrelated among years. Despite temporal autocorrelation in total population size produced by age-structure fluctuations, the stochastic demography of age-structured populations over timescales of a generation or more can nevertheless be accurately approximated by a diffusion process (Tuljapurkar 1982; Lande and Orzack 1988; Engen et al. 2005a, 2007). The success of the diffusion approximation for total population size occurs because the noise in the total reproductive value is nearly white, with no temporal autocorrelation to first order, and the log of total population size fluctuates around the log of reproductive value with a return time to equilibrium on the order of a few generations (Engen et al. 2007). Hence the diffusion approximation is well suited to describe the stochastic dynamics of total reproductive value as well as total population size.This article extends Lande''s (2008) model of fluctuating selection without age structure by deriving a diffusion approximation for the evolution of an age-structured population in a stochastic environment. Assuming weak selection at all ages, random mating, and a stationary distribution of environments with no temporal autocorrelation, we show that the main results of the model remain valid, provided that the model parameters are expressed in terms of means, variances, and covariances of age-specific vital rates and that allele frequencies are defined by weighting individuals of different age by their reproductive value, as suggested by Fisher (1930, 1958). We perform simulations to verify the accuracy of the diffusion approximation and outline statistical methods for estimating the expected selection acting on the reproductive value weighted allele frequency.     

Genomic Consequences of Background Effects on scalloped Mutant Expressivity in the Wing of Drosophila melanogaster

We have evaluated the extent to which SNPs identified by genomewide surveys as showing unusually high levels of population differentiation in humans have experienced recent positive selection, starting from a set of 32 nonsynonymous SNPs in 27 genes highlighted by the HapMap1 project. These SNPs were genotyped again in the HapMap samples and in the Human Genome Diversity Project–Centre d''Etude du Polymorphisme Humain (HGDP–CEPH) panel of 52 populations representing worldwide diversity; extended haplotype homozygosity was investigated around all of them, and full resequence data were examined for 9 genes (5 from public sources and 4 from new data sets). For 7 of the genes, genotyping errors were responsible for an artifactual signal of high population differentiation and for 2, the population differentiation did not exceed our significance threshold. For the 18 genes with confirmed high population differentiation, 3 showed evidence of positive selection as measured by unusually extended haplotypes within a population, and 7 more did in between-population analyses. The 9 genes with resequence data included 7 with high population differentiation, and 5 showed evidence of positive selection on the haplotype carrying the nonsynonymous SNP from skewed allele frequency spectra; in addition, 2 showed evidence of positive selection on unrelated haplotypes. Thus, in humans, high population differentiation is (apart from technical artifacts) an effective way of enriching for recently selected genes, but is not an infallible pointer to recent positive selection supported by other lines of evidence.IN the last 50,000–100,000 years (KY), humans have expanded from being a rare species confined to parts of Africa and the Levant to their current numbers of >6 billion with a worldwide distribution (Jobling et al. 2004). Paleontological and archaeological evidence suggests that key aspects of modern human behavior developed ∼100–50 KYA in Africa (Henshilwood et al. 2002) and behaviorally modern humans then expanded out of Africa ∼60–40 KYA (Mellars 2006). The physical and biological environments encountered outside Africa would have been very different from those inside and included climatic deterioration reaching a glacial maximum ∼20 KYA and subsequent amelioration that permitted the development of agricultural and pastoral lifestyles in multiple independent centers after ∼10 KYA. Neolithic lifestyles would have led to further changes including higher population densities, close contact with animals, and novel foods, in turn leading to new diseases (Jobling et al. 2004). It is likely that genetic adaptations accompanied many of these events.Adaptation, or positive natural selection, leaves an imprint on the pattern of genetic variation found in a population near the site of selection. This pattern can be identified by comparing the DNA variants in multiple individuals from the same and different populations and searching for signals such as unusually extended haplotypes (extended haplotype homozygosity, EHH) (Voight et al. 2006; Sabeti et al. 2007; Tang et al. 2007), high levels of population differentiation (International Hapmap Consortium 2005; Barreiro et al. 2008; Myles et al. 2008), or skewed allele frequency spectra (Carlson et al. 2005). These signals become detectable at different times after the start of selection and are all transient, being gradually eroded by both molecular processes such as mutation, recombination, or further selection and population processes such as migration or demographic fluctuations, with the survival order extended haplotypes < population differentiation < allele frequency spectra (Sabeti et al. 2006). The absolute timescales of survival are not well understood, but extended haplotype tests typically detect selection within the last 10 KY (Sabeti et al. 2006) while unusual allele frequency spectra may detect much older selection. For example, it has been suggested that the signal associated with the FOXP2 gene (Enard et al. 2002) may predate the modern human–Neanderthal split ∼300–400 KYA (Krause et al. 2007), although such an interpretation has been questioned (Coop et al. 2008). However, despite significant uncertainties and limitations, population-genetic analyses are well placed to provide insights into many of the important events within the timescale of recent human evolution.In principle, it should be possible to survey the genome for sites of selection and then interpret this catalog in the light of archaeological, climatic, and other records. Progress toward such a goal has, however, been limited: many factors can confound the detection of selection and only genotype data from previously ascertained SNPs, rather than full resequence data, have thus far been available throughout the whole genome. In practice, the strategy used has therefore been to search the genome for signals that can be detected in available genotype data, such as extended haplotypes or population differentiation, and evaluate the significance of the regions identified by comparing them with empirical distributions of the same statistic, models that incorporate information about the demography, or biological expectations (McVean and Spencer 2006). However, it remains unclear how effective this strategy is: What false positive and false negative rates are associated with its applications? Further evaluation is desirable.The International HapMap Project has carried out the highest-resolution study so far of genetic variation in a set of human populations. In an article published in 2005, genotypes of >1 million SNPs were reported from 270 individuals with ancestry from Africa (Yoruba in Ibadan, Nigeria: YRI), Europe (Utah residents with ancestry from northern and western Europe: CEU), China (Han Chinese in Beijing, China: CHB), and Japan (Japanese in Tokyo, Japan: JPT) (International HapMap Consortium 2005). This article highlighted 32 SNPs from 27 genes that showed particular evolutionary interest because of a combination of two factors: they were nonsynonymous, that is, they changed an amino acid within a protein-coding gene and thus were likely to alter biological function, and they also exhibited a high level of population differentiation equal to or exceeding that of rs2814778, a SNP that is associated with strong biological evidence for population-specific selection. This SNP underlies the FY*0 (Duffy blood group negative) phenotype; FY*0 homozygotes do not express the Duffy blood group antigen on red blood cells and are consequently highly resistant to infection by the malarial parasite, Plasmodium vivax. The *0 allele is nearly fixed in Africa and rare outside, and it is widely believed that this is due to selection for resistance to vivax malaria.However, a number of studies have emphasized that large differences in allele frequency between populations can arise without positive selection: for example, a highly differentiated SNP in the Neuregulin I gene was not accompanied by unusual patterns in adjacent SNPs (Gardner et al. 2007), and large frequency differences can be quite common in empirical data sets, particularly in comparisons between Africa or America and the rest of the world, where population bottlenecks and “allele surfing” may have occurred during the exit from and entrance to these continents, respectively (Hofer et al. 2009). We wished to measure the extent to which the high population differentiation observed at the 27 HapMap genes might have resulted from positive selection and the extent to which it reflected other origins such as demographic factors, chance, or errors. We therefore retyped the same SNPs in the HapMap samples and in a large additional set of human populations and applied alternative tests for selection, either based on long-range haplotypes or based on full resequence data. For the latter, sequence data for 5 of the genes were available from public sources, and four new data sets were generated for this project. We found that, while genotyping errors led to some artifactual high differentiation signals, population differentiation was a useful but by no means infallible guide to recent selection detected by other methods.     

The Genealogical Consequences of Fecundity Variance Polymorphism

The genealogical consequences of within-generation fecundity variance polymorphism are studied using coalescent processes structured by genetic backgrounds. I show that these processes have three distinctive features. The first is that the coalescent rates within backgrounds are not jointly proportional to the infinitesimal variance, but instead depend only on the frequencies and traits of genotypes containing each allele. Second, the coalescent processes at unlinked loci are correlated with the genealogy at the selected locus; i.e., fecundity variance polymorphism has a genomewide impact on genealogies. Third, in diploid models, there are infinitely many combinations of fecundity distributions that have the same diffusion approximation but distinct coalescent processes; i.e., in this class of models, ancestral processes and allele frequency dynamics are not in one-to-one correspondence. Similar properties are expected to hold in models that allow for heritable variation in other traits that affect the coalescent effective population size, such as sex ratio or fecundity and survival schedules.THE population genetics of within-generation fecundity variance has been studied from two perspectives. Beginning with Wright (1938), several authors have investigated the relationship between the effective size of a panmictic population with seasonal reproduction and the variance of the number of offspring born to each adult within a season (Crow and Denniston 1988; Nunney 1993, 1996; Waples 2002; Hedrick 2005; Engen et al. 2007). Although the precise form of this relationship depends on other biological factors such as the mating system and the manner in which population regulation operates, each of these studies shows that the effective population size is a decreasing function of fecundity variance. Furthermore, provided that the variance and the coalescent effective population sizes coincide (Ewens 1982; Nordborg and Krone 2002; Sjodin et al. 2005), these results imply that both the rate at which neutral allele frequencies fluctuate from generation to generation and the rate at which lineages coalesce will be positively correlated with within-generation fecundity variance. For example, it has been suggested that the shallow genealogies that have been documented in many marine organisms are a consequence of the high variance of reproductive success in the recruitment sweepstakes operating in these species (Hedgecock 1994; Árnason 2004; Eldon and Wakeley 2006).These results hold in models in which all individuals have the same within-generation (or within-season) fecundity variance. However, the evolutionary genetics of populations that are polymorphic for alleles that influence demographic traits have also been investigated. The first results of this kind were derived by Gillespie (1974, 1975, 1977), who used diffusion theory to show that natural selection can act directly on within-generation fecundity variance in a haploid population with nonoverlapping generations. By studying a simple model of a population composed of two genotypes, say A₁ and A₂, Gillespie (1974) showed that the fluctuations in the frequency of allele A₁ can be approximated by a diffusion process with the following drift and variance coefficients,where p is the frequency of A₁, N is the number of adults, and 1 + μ_i and are the mean and the variance, respectively, of the number of offspring produced by an individual of type A_i. Most discussions of this class of models have focused on the fitness consequences of differences in fecundity variance, which are summarized by the drift coefficient, m(p), of the diffusion approximation. There are two main conclusions. The first is that because m(p) is an increasing function of the difference − , selection can favor alleles that reduce within-generation fecundity variance even if these have lower mean fecundity. Such variance–mean trade-offs can be interpreted as a kind of bet hedging and could explain the evolution of certain risk-spreading traits such as insect oviposition onto multiple host plants (Root and Kareiva 1986) or multiple mating by females (Sarhan and Kokko 2007). On the other hand, because the strength of selection on fecundity variance is inversely proportional to population size, selection for mean–variance trade-offs will usually be dominated by changes in mean fecundity. For this reason, it has been suggested that within-generation bet hedging will be favored only in very small populations (Seger and Brockman 1987; Hopper et al. 2003), although recent theoretical studies have shown that bet hedging can evolve under less restrictive conditions in subdivided populations (Shpak 2005; Lehmann and Balloux 2007; Shpak and Proulx 2007).Less consideration has been given to the diffusion coefficient, v(p), which differs from the familiar quadratic term, p(1 − p), of the Wright–Fisher diffusion. Because the variance effective population size of a monomorphic population depends on the fecundity variance, it is not surprising that v(p) has an additional dependence on the frequency of A₁ whenever the two alleles have different offspring variances. However, as noted by Gillespie (1974), the relationship between allele frequency fluctuations and the allelic composition of the population is counterintuitive. For example, when p is close to 1, so that the population is composed mainly of A₁-type individuals, the rate of allele frequency fluctuations is dominated by the variance of the A₂ genotype. In particular, if we define the variance effective population size by the expression Np(1 − p)/v(p) (Ewens 1982), then not only is this quantity frequency dependent, but also it depends on the life history traits of the missing genotype whenever the population is fixed for one of the two alleles. In contrast, the coalescent effective population size of a monomorphic population depends only on the offspring distribution of the fixed allele. The discrepancy between these two quantities raises the following question: namely, How does fecundity variance polymorphism affect the statistical properties of the genealogy of a random sample of individuals?The answer to this question is of interest for several reasons. First, although the effects of selection on genealogies have received considerable attention (Przeworski et al. 1999; Williamson and Orive 2002; Barton and Etheridge 2004), little is known about the genealogical consequences of variation in traits that alter the coalescent rate. Extrapolating from models in which the effective population size varies under the control of external factors, we might expect the coalescent process in a model with fecundity variance polymorphism to be a stochastic time change of Kingman''s coalescent. However, the results derived in the next section show that this intuition is usually wrong. The second motivation is more practical. Even if changes in fecundity variance are usually controlled by selection on other traits, the existence of interspecific differences in fecundity variance suggests that there must be periods when populations are polymorphic for alleles that alter the fecundity variance. In these instances, it might be possible to use sequence data to identify the loci responsible for these changes, but to do so will require the development of methods that exploit patterns that are unique to models in which the effective population size depends on the genetic composition of the population. For example, whereas the effects of genetic hitchhiking are usually restricted to linked sites (Maynard Smith and Haigh 1974; Kim and Stephan 2002; Przeworski 2002; Przeworski et al. 2005), we will see later that selective sweeps by mutations that affect fecundity variance would have a genomewide impact on polymorphism.Kingman (1982a,b) showed that the genealogy of a sample of individuals from a panmictic, neutrally evolving population of constant size can be described by a simple stochastic process known as the coalescent (or Kingman''s coalescent). One of the most important properties of Kingman''s coalescent is that it is a Markov process, a fact that is heavily exploited in mathematical analyses and that also allows for efficient simulations of genealogies. Unfortunately, this property generally does not hold in populations composed of nonexchangeable individuals. For example, if there are selective differences between individuals, then although the genealogy of a sample of individuals can still be regarded as a stochastic process, selective interactions between individuals cause this process to also depend on the history of nonancestral lineages. The key to overcoming this difficulty is to embed the genealogical process in a larger process that does satisfy the Markov property. This can be done in two ways. One approach is to embed the coalescent tree within a graphical process called the ancestral selection graph (Krone and Neuhauser 1997; Neuhauser and Krone 1997; Donnelly and Kurtz 1999) in which lineages can either branch, giving rise to pairs of potential ancestors, or coalesce. The intuition behind this construction is that the effects of selection on the genealogy can be accounted for by keeping track of a pool of potential ancestors that includes lineages that have failed to persist due to being outcompeted by individuals of higher fitness. Because the branching rates are linear in the number of lineages, while the coalescence rates are quadratic, this process is certain to reach an ultimate ancestor in finite time. The process can be stopped at this time, and both the ancestral and the genotypic status of individual branches can be resolved by assigning random mutations to the graph and then traversing it from the root to the leaves.The second approach is due to Kaplan et al. (1988), who showed that the genealogical history of a sample of genes under selection can be represented by a structured coalescent process. Here we think of the population as being subdivided into several demes, or genetic backgrounds, consisting of individuals that share the same genotype at the selected locus. Because individuals with the same genotype are exchangeable, the rate of coalescence within a background depends only on the size of the background and the number of ancestral lineages sharing that genotype. In addition, mutations at the selected site will move lineages between backgrounds. To obtain a Markov process, we need to keep track of two kinds of information: (i) the types of the ancestral lineages and (ii) the frequencies of the alleles segregating at the selected locus. Fortunately, because one-dimensional diffusion processes are reversible with respect to their stationary distributions (i.e., the detailed balance conditions are satisfied), the ancestral process of allele frequencies at a locus segregating two alleles has the same law as the forward process. Subsequently, Hudson and Kaplan (1988) showed that the genealogy at a linked neutral locus can be described by a structured coalescent defined in terms of the genetic backgrounds at the selected locus; in this case, recombination between the selected and neutral loci can also move lineages between backgrounds.The objective of this article is to extend the structured coalescent to population genetic models in which within-generation fecundity variance is genotype dependent. (The genealogical consequences of polymorphism affecting between-generation fecundity variance will be described in a separate article.) In these models, exchangeability is violated not only by selective differences between individuals, but also by differences in life history traits that affect coalescent rates and allele frequency fluctuations. Nonetheless, because lineages are exchangeable within backgrounds, the coalescence and substitution rates can still be calculated conditional on the types of the lineages and the genetic composition of the population. In the next two sections, I derive structured coalescent processes that describe the genealogy at a neutral marker locus that is linked to a second locus (the “selected locus”) that affects fecundity variance. This is first done for a haploid model and then extended to a diploid model in which there may be both sex- and genotype-specific differences in fecundity variance. Results for both models are summarized in

Estimating the Parameters of Selection on Nonsynonymous Mutations in Drosophila pseudoobscura and D. miranda

We present the results of surveys of diversity in sets of >40 X-linked and autosomal loci in samples from natural populations of Drosophila miranda and D. pseudoobscura, together with their sequence divergence from D. affinis. Mean silent site diversity in D. miranda is approximately one-quarter of that in D. pseudoobscura; mean X-linked silent diversity is about three-quarters of that for the autosomes in both species. Estimates of the distribution of selection coefficients against heterozygous, deleterious nonsynonymous mutations from two different methods suggest a wide distribution, with coefficients of variation greater than one, and with the average segregating amino acid mutation being subject to only very weak selection. Only a small fraction of new amino acid mutations behave as effectively neutral, however. A large fraction of amino acid differences between D. pseudoobscura and D. affinis appear to have been fixed by positive natural selection, using three different methods of estimation; estimates between D. miranda and D. affinis are more equivocal. Sources of bias in the estimates, especially those arising from selection on synonymous mutations and from the choice of genes, are discussed and corrections for these applied. Overall, the results show that both purifying selection and positive selection on nonsynonymous mutations are pervasive.SURVEYS of DNA sequence diversity and divergence are shedding light on a number of questions in evolutionary genetics (for recent reviews, see Akey 2009; Sella et al. 2009). Two of the most important questions of this kind concern the distribution of selection coefficients against deleterious mutations affecting protein sequences and the proportion of amino acid sequence differences between related species that have been fixed by positive selection. Several different methods have been proposed for studying each of these questions, using different features of data on polymorphism and divergence at nonsynonymous and silent sites.For example, the parameters of the distribution of selection coefficients against deleterious amino acid mutations have been estimated by contrasting the numbers of nonsynonymous and silent within-species polymorphisms and fixed differences between species (Sawyer and Hartl 1992; Bustamante et al. 2002; Piganeau and Eyre-Walker 2003; Sawyer et al. 2007); by fitting the frequency spectra of nonsynonymous and silent variants to models of selection, mutation, and drift (Akashi 1999; Eyre-Walker et al. 2006; Keightley and Eyre-Walker 2007; Kryukov et al. 2007; Boyko et al. 2008; Eyre-Walker and Keightley 2009); or by comparing levels of nonsynonymous and silent diversities between species with different population sizes (Loewe and Charlesworth 2006; Loewe et al. 2006). The results of these different approaches generally agree in suggesting that there is a wide distribution of selection coefficients against nonsynonymous mutations and that the mean selection coefficient against heterozygous carriers of such mutations is very small. The results imply that a typical individual from a human population carries several hundred weakly deleterious mutations (Eyre-Walker et al. 2006; Kryukov et al. 2007; Boyko et al. 2008); for a typical Drosophila population, with its much higher level of variability, the number is probably an order of magnitude greater (Loewe et al. 2006; Keightley and Eyre-Walker 2007).The presence of this large load of slightly deleterious mutations in human and natural populations, most of which are held at low frequencies by natural selection, has many implications. From the point of view of understanding human genetic disease, it means that we have to face the likelihood that susceptibility to a disease can be influenced by variants at many loci, each with small effects (Kryukov et al. 2007). The pervasive presence of deleterious mutations throughout the genome contributes to inbreeding depression (Charlesworth and Willis 2009) and may mean that the effective population size is reduced by background selection effects, even in regions of the genome with normal levels of genetic recombination (Loewe and Charlesworth 2007). Their presence may contribute so strongly to Hill–Robertson effects (Hill and Robertson 1966; Felsenstein 1974) that they cause severely reduced levels of diversity and adaptation in low-recombination regions of the genome (Charlesworth et al. 2010) and create a selective advantage to maintaining nonzero levels of recombination (Keightley and Otto 2006; Charlesworth et al. 2010). In addition, having an estimate of the distribution of selection coefficients against deleterious nonsynonymous mutations allows their contribution to between-species divergence to be predicted, providing a way of estimating the fraction of fixed nonsynonymous differences caused by positive selection (Loewe et al. 2006; Boyko et al. 2008; Eyre-Walker and Keightley 2009).It is thus important to collect data that shed light on the properties of selection against nonsynonymous mutations in a wide range of systems and also to compare the results from different methods of estimation, since they are subject to different sources of difficulty and biases. In a previous study, we proposed the use of a comparison between two related species with different effective population sizes for this purpose (Loewe and Charlesworth 2006; Loewe et al. 2006), using Drosophila miranda and D. pseudoobscura as material. These are well suited for this type of study, as they are closely related, live together in similar habitats, and yet have very different levels of silent nucleotide diversity, indicating different effective population sizes (N_e). This study was hampered by our inability to compare the same set of loci across the two species and by the small number of loci that could be used. We here present the results of a much larger study of DNA variation at X-linked and autosomal loci for these two species, using D. affinis as a basis for estimating divergence. We compare the results, applying the method of Loewe et al. (2006) with that of Eyre-Walker and Keightley (2009) for estimating the distribution of deleterious selection coefficients and with McDonald–Kreitman test-based methods for estimating the proportion of nonsynonymous differences fixed by positive selection. While broadly confirming the conclusions from earlier studies, we note some possible sources of bias and describe methods for minimizing their effects.     

Estimating Selection Intensity on Synonymous Codon Usage in a Nonequilibrium Population

Codon usage bias is the nonrandom use of synonymous codons for the same amino acid. Most population genetic models of codon usage evolution assume that the population is at mutation–selection–drift equilibrium. Natural populations, however, frequently deviate from equilibrium, often because of recent demographic changes. Here, we construct a matrix model that includes the effects of a recent change in population size on estimates of selection on preferred vs. unpreferred codons. Our results suggest that patterns of synonymous polymorphisms affecting codon usage can be quite erratic after such a change; statistical methods that fail to take demographic effects into account can then give incorrect estimates of important parameters. We propose a new method that can accurately estimate both demographic and codon usage parameters. The method also provides a simple way of testing for the effects of covariates such as gene length and level of gene expression on the intensity of selection, which we apply to a large Drosophila melanogaster polymorphism data set. Our analyses of twofold degenerate codons reveal that (i) selection acts in favor of preferred codons, (ii) there is mutational bias in favor of unpreferred codons, (iii) shorter genes and genes with higher expression levels are under stronger selection, and (iv) there is little evidence for a recent change in population size in the Zimbabwe population of D. melanogaster.CODONS specifying the same amino acid are called synonymous codons. These are often used nonrandomly, with some codons appearing more frequently than others. This biased usage of synonymous codons has been found in many organisms such as Drosophila, yeast, and bacteria (Ikemura 1985; Duret and Mouchiroud 1999; Hershberg and Petrov 2008). Conventionally, synonymous codons for a given amino acid are divided into two classes: preferred and unpreferred codons (Ikemura 1985; Akashi 1994; Duret and Mouchiroud 1999). Several observations indicate that codon usage is affected by natural selection. First, in species with codon usage bias, preferred codons generally correspond to the most abundant tRNA species (Ikemura 1981). Second, highly expressed genes usually have higher codon usage bias than genes with low expression (Sharp and Li 1986; Duret and Mouchiroud 1999; Hey and Kliman 2002). Third, the synonymous substitution rate of a gene has been shown to be negatively correlated with its degree of codon usage bias (Sharp and Li 1986; Bierne and Eyre-Walker 2006). The most commonly cited explanations of the apparent fitness differences between preferred and unpreferred codons are selection for translation efficiency, translational accuracy, and mRNA stability (Ikemura 1985; Eyre-Walker and Bulmer 1993; Akashi 1994; Drummond et al. 2005). Recently, it has been proposed that exon splicing also affects codon usage bias (Warnecke and Hurst 2007).From a population genetics perspective, the extent of codon usage bias is ultimately a product of the joint effects of mutation, selection, genetic drift, recombination, and demographic history. The Li–Bulmer model of drift, selection, and reversible mutation between preferred and unpreferred codons at a site is the most widely used model (Li 1987; Bulmer 1991; McVean and Charlesworth 1999). Applications of this model generally assume that the population is at mutation–selection–drift equilibrium. However, empirical studies have suggested that changes in the strengths of various driving forces may not be unusual. For example, in Drosophila melanogaster, there is evidence that the population size (Li and Stephan 2006; Thornton and Andolfatto 2006; Keightley and Eyre-Walker 2007; Stephan and Li 2007), recombinational landscape (Takano-Shimizu 1999), and mutational process (Takano-Shimizu 2001; Kern and Begun 2005) may have changed significantly over the species'' evolutionary history.Such changes cause departures from equilibrium. Theoretical models show that it takes a very long time, proportional to the reciprocal of the mutation rate, for the population to approach a new equilibrium state (Tachida 2000; Comeron and Kreitman 2002). Before reaching equilibrium, the population often shows counterintuitive patterns of evolution (Eyre-Walker 1997; Takano-Shimizu 1999, 2001; Comeron and Kreitman 2002; Comeron and Guthrie 2005; Charlesworth and Eyre-Walker 2007). Despite these theoretical results, details of the patterns of polymorphism and substitution rates following a recent change in population size, and their effects on estimates of strength of selection, have not been determined.The above findings point to the importance of incorporating nonequilibrium factors into the study of codon usage bias. To this end, we extend the Li–Bulmer model to allow population size to vary over time, by representing the evolutionary process by a transition matrix. By analyzing this matrix model, we show that a recent change in population size can result in erratic patterns of codon usage and that methods failing to take into account these demographic effects can give false estimates of the intensity of selection.To solve these problems, we propose a new method, which does not require polarizing ancestral vs. derived states using outgroup data (cf. Cutter and Charlesworth 2006), but requires only knowledge of preferred vs. unpreferred states defined by patterns of codon usage. We use information on both polymorphic and fixed sites, which enables both mutational bias and the strength of selection to be estimated, in contrast to previous methods that use information on polymorphisms alone. Simulations indicate that this method can accurately estimate both demographic and codon usage parameters and can distinguish between selection and demography. We use the new method to analyze a large D. melanogaster polymorphism data set (Shapiro et al. 2007) and find evidence for natural selection on synonymous codons. We use our approach to show that genes with shorter coding sequences and higher levels of expression are under significantly stronger selection than longer genes with lower expression.     

The Genetic Signature of Conditional Expression

Conditionally expressed genes have the property that every individual in a population carries and transmits the gene, but only a fraction, φ, expresses the gene and exposes it to natural selection. We show that a consequence of this pattern of inheritance and expression is a weakening of the strength of natural selection, allowing deleterious mutations to accumulate within and between species and inhibiting the spread of beneficial mutations. We extend previous theory to show that conditional expression in space and time have approximately equivalent effects on relaxing the strength of selection and that the effect holds in a spatially heterogeneous environment even with low migration rates among patches. We support our analytical approximations with computer simulations and delineate the parameter range under which the approximations fail. We model the effects of conditional expression on sequence polymorphism at mutation–selection–drift equilibrium, allowing for neutral sites, and show that sequence variation within and between species is inflated by conditional expression, with the effect being strongest in populations with large effective size. As φ decreases, more sites are recruited into neutrality, leading to pseudogenization and increased drift load. Mutation accumulation diminishes the degree of adaptation of conditionally expressed genes to rare environments, and the mutational cost of phenotypic plasticity, which we quantify as the plasticity load, is greater for more rarely expressed genes. Our theory connects gene-level relative polymorphism and divergence with the spatial and temporal frequency of environments inducing gene expression. Our theory suggests that null hypotheses for levels of standing genetic variation and sequence divergence must be corrected to account for the frequency of expression of the genes under study.IN genetically and ecologically subdivided populations, some individuals will experience a local environment very different from others, making it difficult to evolve a single adaptation adequate for all local conditions. Phenotypic plasticity allows organisms to respond adaptively to spatially and temporally varying environments by developing alternative phenotypes that enhance fitness under local conditions (Scheiner 1993; Via et al. 1995). Examples of alternative phenotypes, i.e., polyphenisms, include the defensive morphologies in Daphnia and algae induced by the presence of predators (e.g., Lively 1986; DeWitt 1998; Harvell 1998; Hazel et al. 2004); the winged and wingless morphs of bean beetles responding to resource variation (e.g., Abouheif and Wray 2002; Roff and Gelinas 2003; Lommen et al. 2005); and bacterial genes involved in traits such as quorum sensing, antibiotic production, biofilm formation, and virulence (Fuqua et al. 1996). The developmental basis of such alternative phenotypes often lies in the inducible expression of some genes in some individuals by environmental variables. That is, all individuals carry and transmit the conditionally expressed genes but only a fraction of individuals, φ, express them when environmental conditions are appropriate.The genes underlying plastic traits should experience relaxed selection due to conditional expression. Wade and co-workers have shown that genes hidden from natural selection in a fraction of individuals in the population by X-linked (Whitlock and Wade 1995; Linksvayer and Wade 2009) or sex-limited expression (Wade 1998; Demuth and Wade 2007) experience relaxed selective constraint. In Drosophila spp., sequence data for genes with maternally limited expression quantitatively support the theoretical predictions both for within-species polymorphism (Barker et al. 2005; Cruickshank and Wade 2008) and for between-species divergence (Barker Et Al 2005; Demuth and Wade 2007; Cruickshank and Wade 2008). Furthermore, male-specific genes in the facultatively sexual pea aphid have been shown to have elevated levels of sequence variation due to relaxed selection (Brisson and Nuzhdin 2008). Genes with spatially restricted expression in a heterogeneous environment should likewise experience relaxed selection. Adaptation to the most common environment in an ecologically subdivided population (Rosenzweig 1987; Holt and Gaines 1992; Holt 1996) allows deleterious mutations to accumulate in traits expressed in rare environments (Kawecki 1994; Whitlock 1996).Here we extend these results by quantifying the consequences of relaxed selection on conditionally expressed genes. Specifically, we show that, with weak selection, spatial and temporal fluctuations in selection intensity generate approximately equivalent effects on mean trait fitness, even with low rates of migration between habitats, resulting in a great simplification of analytical results. Our analytical approximations are supported with deterministic and stochastic simulations, and we note the conditions under which the approximations fail. We then derive general expressions for (1) the expected level of sequence polymorphism within populations under mutation, migration, drift, and purifying selection with conditional gene expression; (2) the rate of sequence divergence among populations, for dominant and recessive mutations; and (3) the reduction in mean population fitness due to accumulation of deleterious mutations at conditionally expressed loci. We find that the rate of accumulation of deleterious mutations for conditionally expressed genes is accelerated and the probability of fixation of beneficial mutations is reduced, causing a reduction in the fitness of conditional traits and an inflation in sequence variation within and between species. Our results suggest that evolutionary null hypotheses must be adjusted to account for the frequency of expression of genes under study, such that signatures of elevated within- or between-species sequence variation are not necessarily evidence of the action of diversifying natural selection. Furthermore, if conditional expression is due to spatial heterogeneity, we show that the level of genetic variation in a sample will often depend on whether or not genotypes were sampled from the selective habitat, the neutral habitat, or both. In the discussion we address the scope and limitations of our theory, as well as its implications for the maintenance of genetic variation, adaptive divergence between species, constraints on phenotypic plasticity, and evolutionary inference from sequence data.     

Detecting Selection in Population Trees: The Lewontin and Krakauer Test Extended

Detecting genetic signatures of selection is of great interest for many research issues. Common approaches to separate selective from neutral processes focus on the variance of F_ST across loci, as does the original Lewontin and Krakauer (LK) test. Modern developments aim to minimize the false positive rate and to increase the power, by accounting for complex demographic structures. Another stimulating goal is to develop straightforward parametric and computationally tractable tests to deal with massive SNP data sets. Here, we propose an extension of the original LK statistic (T_LK), named T_F–LK, that uses a phylogenetic estimation of the population''s kinship () matrix, thus accounting for historical branching and heterogeneity of genetic drift. Using forward simulations of single-nucleotide polymorphisms (SNPs) data under neutrality and selection, we confirm the relative robustness of the LK statistic (T_LK) to complex demographic history but we show that T_F–LK is more powerful in most cases. This new statistic outperforms also a multinomial-Dirichlet-based model [estimation with Markov chain Monte Carlo (MCMC)], when historical branching occurs. Overall, T_F–LK detects 15–35% more selected SNPs than T_LK for low type I errors (P < 0.001). Also, simulations show that T_LK and T_F–LK follow a chi-square distribution provided the ancestral allele frequencies are not too extreme, suggesting the possible use of the chi-square distribution for evaluating significance. The empirical distribution of T_F–LK can be derived using simulations conditioned on the estimated matrix. We apply this new test to pig breeds SNP data and pinpoint outliers using T_F–LK, otherwise undetected using the less powerful T_LK statistic. This new test represents one solution for compromise between advanced SNP genetic data acquisition and outlier analyses.THE development of methods aiming at detecting molecular signatures of selection is one of the major concerns of modern population genetics. Broadly, such methods can be classified into four groups: methods focusing on (i) the interspecific comparison of gene substitution patterns, (ii) the frequency spectrum and models of selective sweeps, (iii) linkage disequilibrium (LD) and haplotype structure, and (iv) patterns of genetic differentiation among populations (for a review see Nielsen 2005). Tests based on the comparison of polymorphism and divergence at the species level inform on mostly ancient selective processes. Population-based approaches, however, are designed to pinpoint modern processes of local adaptation and speciation occurring among populations within a species. Such approaches also become crucial in the fields of agronomical and biomedical sciences, for instance, to pinpoint possible interesting (QTL) regions and disease susceptibility genes. Especially, human, livestock, and cultivated plants genetics may benefit from such methods while whole-genome single-nucleotide polymorphisms (SNPs) genotyping technologies are becoming routinely available (e.g., Barreiro et al. 2008; Flori et al. 2009).In the population genomic era (Luikart et al. 2003), identifying genes under selection or neutral markers influenced by nearby selected genes is a task in itself for quantifying the role of selection in the evolutionary history of species. Conversely, the accurate inference of demographic parameters such as effective population sizes, migration rates, and divergence times between populations relies on the use of neutral marker data sets. One approach of detecting loci under selection (outliers) with population genetic data is based on the genetic differentiation between loci influenced only by neutral processes (genetic drift, mutation, migration) and loci influenced by selection.Lewontin and Krakauer''s (LK) test for the heterogeneity of the inbreeding coefficient (F) across loci was the first to be developed with regard to this concept (Lewontin and Krakauer 1973). The LK test was immediately subject to criticisms (Nei and Maruyama 1975; Lewontin and Krakauer 1975; Robertson, 1975a,b; Tsakas and Krimbas 1976; Nei and Chakravarti 1977; Nei et al. 1977). Indeed, its assumptions are likely to be violated due to loci with high mutation rate, variation of F due to unequal effective population size (N_e) among demes, and correlation of allele frequencies among demes due to historical branching. The robustness of the LK test to the effects of demography was tested through coalescent simulations by Beaumont and Nichols (1996). They tested the influence of different models of population structure on the joint distribution of F_ST (i.e., the inbreeding coefficient F) and heterozygosity (H_e). The F_ST distribution under an infinite-island model is inflated for low H_e values under both the infinite-allele model (IAM) and the stepwise mutation model (SMM) (Beaumont and Nichols 1996). This tendency becomes, however, more marked when strong differences in effective size N_e and gene flow among demes occur, that is, when allele frequencies are correlated among local demes. This suggests an excess of false significant loci when one assumes an infinite-island model as a null hypothesis, while correlations of gene frequencies substantially occur. However, the F_ST distribution shows robustness properties for high H_e values (typical from microsatellite markers). Therefore, Beaumont and Nichols (1996) suggested the possibility of detecting outliers by using the distribution of neutral F_ST conditionally on H_e under the infinite-island model of symmetric migration, with mutation.The problem of accounting for correlations of allele frequencies among subpopulations was discussed by Robertson (1975a), who showed how these correlations inflated the variance of the LK test. Different approaches were taken to cope with the problem. It was, for instance, proposed to restrict the analysis to pairwise comparisons (Tsakas and Krimbas 1976; Vitalis et al. 2001). However, as pointed out by Beaumont (2005), reducing the number of populations to be compared to many pairwise comparisons raises the problem of nonindependence in multiple testing and may reduce the power to detect outliers. Another way was to assume that subpopulation allele frequencies are correlated through a common migrant gene pool, that is, the ancestral population in a star-like population divergence. In this case, subpopulations evolve with an unequal number of migrants coming from the migrant pool and/or to different amounts of genetic drift. This demographic scenario can be explicitly modeled using the multinomial-Dirichlet likelihood approach (Balding 2003). This multinomial-Dirichlet likelihood (or Beta-binomial for biallelic markers such as SNPs) was implemented by Beaumont and Balding (2004) and subsequently by Foll and Gaggiotti (2008), Gautier et al. (2009), Guo et al. (2009), and Riebler et al. (2010), in a Bayesian hierarchical model in which the F_ST is decomposed into two components: a locus-specific (α) effect and a population-specific (β) effect. This Bayesian statistical model together with prior assumptions on α and β was implemented in a Markov chain Monte Carlo (MCMC) algorithm. A substantial improvement made by Foll and Gaggiotti (2008) was to use a reverse-jumping (RJ)-MCMC to simultaneously estimate the posterior distribution of a model with selection (with α and β) and of a model without selection (with β only). More recently, Excoffier et al. (2009) addressed the issue of accounting for “heterogeneous affinities between sampled populations”—in other words, accounting for migrant genes that do not necessarily originate from the same pool—by using a hierarchically structured population model. They showed by simulations that the false positive rate is lower under a hierarchically structured population model than under a simple island model, for the IAM and the SMM applicable to microsatellite markers and for a SNP mutation model. Excoffier et al.(2009) thus proposed to extend the Beaumont and Nichols (1996) method to a hierarchically structured population model.Nowadays, a computational challenge is to analyze data sets with increasing numbers of markers and populations, under complex demographic histories, in a reasonable amount of time. This is especially the case in agronomical and biomedical sciences with the increasingly used biallelic SNP markers. A question arises as to whether F_ST-based methods would be sufficiently powerful to detect outliers with SNP markers. Indeed, for low H_e values, the inflation of the F_ST distribution under the infinite-island model accentuates dramatically when assuming a mutation model typical for SNPs (simulations of Eveno et al. 2008). Excoffier et al. (2009) corroborated these results and also indicated that the F_ST distribution is generally broader under a model of hierarchically structured populations when using SNP markers. In addition, as the authors pinpoint, although the hierarchical island model is more conservative than the island model, an excess of false positives can be obtained “if the underlying genetic structure is more complex …, for instance in case of complex demographic histories, involving population splits, range expansion, bottleneck or admixture events” (Excoffier et al. 2009, p. 12). The Bayesian hierarchical models developed by Beaumont and Balding (2004) and Foll and Gaggiotti (2008) effectively account for strong effective size and migration rate variation among subpopulations, but they still impose a star-like demographic model in which the current populations share a common migrant pool and are not supposed to have undergone historical branching. More practically, MCMC-based methods might suffer from a computational time requirement when analyzing large marker data sets such as SNP chips data sets. Therefore, the development of simple parametric tests potentially dealing with a summary of the population tree, including historical branching as well as population size variation, remains an alternative solution to achieve a good compromise between advanced genetic data acquisition and outlier analyses.In this article, we describe an extension of the original parametric LK test for biallelic markers that deals with complex population trees through a statistic that takes into account the kinship (or coancestry) matrix between populations, under pure drift with no migration. The statistics of the classical test (T_LK) and its extension (T_F–LK) are expected to follow a chi-square distribution with (n – 1) d.f., where n is the number of populations studied. Through forward simulations of neutral SNPs data under increasingly complex demographic histories, we obtained the empirical distribution of both statistics and showed that they follow a chi-square distribution provided the ancestral allele frequencies are not too extreme. These results also emphasize the robustness of these statistics to variation in demographic histories. Forward simulations of the same demographic models but including selection in one population allowed us to evaluate the power of both statistics to detect selection. We show that the extension of the LK test is more powerful at detecting outliers than the classical LK test for complex demographic histories. A comparison with one of the MCMC methods for multinomial-Dirichlet models (Foll and Gaggiotti 2008) also revealed substantial additional power. We apply this new statistical test to a data set of SNP markers in known genes of the pig genome, taking advantage of the availability of microsatellite markers for the estimation of the kinship matrix. This new parametric test can help to screen large marker data sets and large numbers of populations for outliers in a reasonable amount of time, although we recommend to simulate the empirical distribution of the T_F–LK statistics conditionally on the estimated kinship matrix.     

The Total Branch Length of Sample Genealogies in Populations of Variable Size

We consider neutral evolution of a large population subject to changes in its population size. For a population with a time-variable carrying capacity we study the distribution of the total branch lengths of its sample genealogies. Within the coalescent approximation we have obtained a general expression—Equation 20—for the moments of this distribution with a given arbitrary dependence of the population size on time. We investigate how the frequency of population-size variations alters the total branch length.MODELS for gene genealogies of biological populations often assume a constant, time-independent population size N. This is the case for the Wright–Fisher model (Fisher 1930; Wright 1931), for the Moran model (Moran 1958), and for their representation in terms of the coalescent (Kingman 1982). In real biological populations, by contrast, the population size changes over time. Such fluctuations may be due to catastrophic events (bottlenecks) and subsequent population expansions or just reflect the randomness in the factors determining the population dynamics. Many authors have argued that genetic variation in a population subject to size fluctuations may nevertheless be described by the Wright–Fisher model, if one replaces the constant population size in this model by an effective population size of the form(1)where N_l stands for the population size in generation l. The harmonic average in Equation 1 is argued to capture the significant effect of catastrophic events on patterns of genetic variation in a population: if, for example, a population went through a recent bottleneck, a large fraction of individuals in a given sample would originate from few parents. This in turn would lead to significantly reduced genetic variation, parameterized by a small value of N_eff. (See, e.g., Ewens 1982 for a review of different measures of the effective population size and Sjödin et al. 2005 and Wakeley and Sargsyan 2009 for recent developments of this concept.)The concept of an effective population size has been frequently used in the literature, implicitly assuming that the distribution of neutral mutations in a large population of fluctuating size is identical to the distribution in a Wright–Fisher model with the corresponding constant effective population size given by Equation 1. However, recently it was shown that this is true only under certain circumstances (Kaj and Krone 2003; Nordborg and Krone 2003; Jagers and Sagitov 2004). It is argued by Sjödin et al. (2005) that the concept of an effective population size is appropriate when the timescale of fluctuations of N_l is either much smaller or much larger than the typical time between coalescent events in the sample genealogy. In these limits it can be proved that the distribution of the sample genealogies is exactly given by that of the coalescent with a constant, effective population size.More importantly, it follows from these results that, in populations with variable size, the coalescent with a constant effective population size is not always a valid approximation for the sample genealogies. Deviations between the predictions of the standard coalescent model and empirical data are frequently observed, and there are a number of different statistical tests quantifying the corresponding discrepancies (see, for example, Tajima 1989, Fu and Li 1993, and Zeng et al. 2006). The analysis of such deviations is of crucial importance in understanding, for example, human genetic history (Garrigan and Hammer 2006). But while there is a substantial amount of work numerically quantifying deviations, often in terms of a single number, little is known about their qualitative origins and their effect upon summary statistics in the population in question.The question is thus to understand the effect of population-size fluctuations on the patterns of genetic variation, in particular for the case where the scale of the population-size fluctuations is comparable to the time between coalescent events in the ancestral tree. As is well known, many empirical measures of genetic variation can be computed from the total branch length of the sample genealogy (the expected number of single-nucleotide polymorphisms, for example, is proportional to the average total branch length).The aim of this article is to analyze the distribution of the scaled total branch length T_n for a sample genealogy in a population of fluctuating size, as illustrated in Figure 1. For the genealogy of n ≥ 2 lineages sampled at the present time, the expression ⌊NT_n⌋ gives the total branch length in terms of generations. Here ⌊Nt⌋ is the largest integer ≤Nt, and the scaling factor N is a suitable measure of the number of genes in the population and serves as a counterpart of the constant generation size of the standard Wright–Fisher model.Open in a separate windowFigure 1.—The effect of population-size oscillations on the genealogy of a sample of size n = 17 (schematic). Left, genealogy described by Kingman''s coalescent for a large population of constant size, illustrated by the light blue rectangle; right, sinusoidally varying population size. Coalescence is accelerated in regions of small population sizes and vice versa. This significantly alters the tree and gives rise to changes in the distribution of the number of mutations and of the population homozygosity.A motivating example is given in Figure 2, which shows numerically computed distributions ρ(T_n) of the total branch lengths T_n for a particular population model with a time-dependent carrying capacity. The model is described briefly in the Figure 2 legend and in detail in a model for a population with time-dependent carrying capacity. As Figure 2 shows, the distributions depend in a complex manner on the form of the size changes. We observe that when the frequency of the population-size fluctuations is very small (Figure 2a), the distribution is well described by the standard coalescent result(2)(Hein et al. 2005). When the frequency is very large (Figure 2e), Equation 2 also applies, but with a different time scaling reflecting an effective population size: t on the right-hand side (rhs) in Equation 2 is replaced by t/c with c = N/N_eff. Apart from these special limits, however, the form of the distributions appears to depend in a complicated manner upon the frequency of the population-size variation. The observed behavior is caused by the fact that coalescence proceeds faster for smaller population sizes and more slowly for larger population sizes, as illustrated in Figure 1. But the question is how to quantitatively account for the changes shown in Figure 2.Open in a separate windowFigure 2.—Numerically computed distributions of the scaled total branch lengths T_n in genealogies of samples of size n = 10. The model employed in the simulations is outlined in a model for a population with time-dependent carrying capacity. It describes a population subject to a time-varying carrying capacity, K_l = K₀(1 + ɛ sin(2πνl)). The frequency of the time changes is determined by ν, and l = 1, 2, 3, … labels discrete generations forward in time. The parameter N = K₀ describes the typical population size, which is taken here to be equal to the time-averaged carrying capacity. a–e show for populations with increasingly rapidly oscillating carrying capacity. The dashed red line in a shows that in the limit of low frequencies the standard coalescent result, Equation 2, is obtained. The dashed red line in e shows that also in the limit of large frequencies the standard coalescent result is obtained, but now with an effective population size. The dashed red line in d is a two-parameter distribution, Equation 41, derived in comparison between numerical simulations and coalescent predictions. Further numerical and analytical results on the frequency dependence of the moments of these distributions are shown in Figure 4. Parameter values used: K₀ = 10,000, ɛ = 0.9, and r = 1 (see a model for a population with time-dependent carrying capacity for the exact meaning of the intrinsic growth rate r) and (a) νN = 0.001, (b) νN = 0.1, (c) νN = 0.316, (d) νN = 1, and (e) νN = 100.We show in this article that the results of the simulations displayed in Figure 2 are explained by a general expression—Equation 20—for the moments of the distributions shown in Figure 2. Our general result is obtained within the coalescent approximation valid in the limit of large population size. But we find that in most cases, the coalescent approximation works very well down to small population sizes (a few hundred individuals). Our result enables us to understand and quantitatively describe how the distributions shown in Figure 2 depend upon the frequency of the population-size oscillations. It makes possible to determine, for example, how the variance, skewness, and the kurtosis of these distributions depend upon the frequency of demographic fluctuations. This in turn allows us to compute the population homozygosity and to characterize genetic variation in populations with size fluctuations.The remainder of this article is organized as follows. The next section summarizes our analytical results for the moments of the total branch length. Following that, we describe the model employed in the computer simulations. Then, corresponding numerical results are compared to the analytical predictions. And finally, we summarize how population-size fluctuations influence the distribution of total branch lengths and conclude with an outlook.     

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.     

Polymorphic Genes of Major Effect: Consequences for Variation,Selection and Evolution in Arabidopsis thaliana

The importance of genes of major effect for evolutionary trajectories within and among natural populations has long been the subject of intense debate. For example, if allelic variation at a major-effect locus fundamentally alters the structure of quantitative trait variation, then fixation of a single locus can have rapid and profound effects on the rate or direction of subsequent evolutionary change. Using an Arabidopsis thaliana RIL mapping population, we compare G-matrix structure between lines possessing different alleles at ERECTA, a locus known to affect ecologically relevant variation in plant architecture. We find that the allele present at ERECTA significantly alters G-matrix structure—in particular the genetic correlations between branch number and flowering time traits—and may also modulate the strength of natural selection on these traits. Despite these differences, however, when we extend our analysis to determine how evolution might differ depending on the ERECTA allele, we find that predicted responses to selection are similar. To compare responses to selection between allele classes, we developed a resampling strategy that incorporates uncertainty in estimates of selection that can also be used for statistical comparisons of G matrices.THE structure of the genetic variation that underlies phenotypic traits has important consequences for understanding the evolution of quantitative traits (Fisher 1930; Lande 1979; Bulmer 1980; Kimura 1983; Orr 1998; Agrawal et al. 2001). Despite the infinitesimal model''s allure and theoretical tractability (see Orr and Coyne 1992; Orr 1998, 2005a,b for reviews of its influence), evidence has accumulated from several sources (artificial selection experiments, experimental evolution, and QTL mapping) to suggest that genes of major effect often contribute to quantitative traits. Thus, the frequency and role of genes of major effect in evolutionary quantitative genetics have been a subject of intense debate and investigation for close to 80 years (Fisher 1930; Kimura 1983; Orr 1998, 2005a,b). Beyond the conceptual implications, the prevalence of major-effect loci also affects our ability to determine the genetic basis of adaptations and species differences (e.g., Bradshaw et al. 1995, 1998).Although the existence of genes of major effect is no longer in doubt, we still lack basic empirical data on how segregating variation at such genes affects key components of evolutionary process (but see Carrière and Roff 1995). In other words, How does polymorphism at genes of major effect alter patterns of genetic variation and covariation, natural selection, and the likely response to selection? The lack of data stems, in part, from the methods used to detect genes of major effect: experimental evolution (e.g., Bull et al. 1997; Zeyl 2005) and QTL analysis (see Erickson et al. 2004 for a review) often detect such genes retrospectively after they have become fixed in experimental populations or the species pairs used to generate the mapping population. The consequences of polymorphism at these genes on patterns of variation, covariation, selection, and the response to selection—which can be transient (Agrawal et al. 2001)—are thus often unobserved.A partial exception to the absence of data on the effects of major genes comes from artificial selection experiments, in which a substantial evolutionary response to selection in the phenotype after a plateau is often interpreted as evidence for the fixation of a major-effect locus (Frankham et al. 1968; Yoo 1980a,b; Frankham 1980; Shrimpton and Robertson 1988a,b; Caballero et al. 1991; Keightley 1998; see Mackay 1990 and Hill and Caballero 1992 for reviews). However, many of these experiments report only data on the selected phenotype (e.g., bristle number) or, alternatively, the selected phenotype and some measure of fitness (e.g., Frankham et al. 1968, Yoo 1980b; Caballero et al. 1991; Mackay et al. 1994; Fry et al. 1995; Nuzhdin et al. 1995; Zur Lage et al. 1997), making it difficult to infer how a mutation will affect variation, covariation, selection, and evolutionary responses for a suite of traits that might affect fitness themselves. One approach is to document how variation at individual genes of major effect affects the genetic variance–covariance matrix (“G matrix”; Lande 1979), which represents the additive genetic variance and covariance between traits.Although direct evidence for variation at major-effect genes altering patterns of genetic variation, covariation, and selection is rare, there is abundant evidence for the genetic mechanisms that could produce these dynamics. A gene of major effect could have these consequences due to any of at least three genetic mechanisms: (1) pleiotropy, where a gene of major effect influences several traits, including potentially fitness, simultaneously, (2) physical linkage or linkage disequilibrium (LD), in which a gene of major effect is either physically linked or in LD with other genes that influence other traits under selection, and (3) epistasis, in which the allele present at a major-effect gene alters the phenotypic effect of other loci and potentially phenotypes under selection. Evidence for these three evolutionary genetic mechanisms leading to changes in suites of traits comes from a variety of sources, including mutation accumulation experiments (Clark et al. 1995; Fernandez and Lopez-Fanjul 1996), mutation induction experiments (Keightley and Ohnishi 1998), artificial selection experiments (Long et al. 1995), and transposable element insertions (Rollmann et al. 2006). For pleiotropy in particular, major-effect genes that have consequences on several phenotypic traits are well known from the domestication and livestock breeding literature [e.g., myostatin mutations in Belgian blue cattle and whippets (Arthur 1995; Grobet et al. 1997; Mosher et al. 2007), halothane genes in pigs (Christian and Rothschild 1991; Fujii et al. 1991), and Booroola and Inverdale genes in sheep (Amer et al. 1999; Visscher et al. 2000)]. While these data suggest that variation at major-effect genes could—and probably does—influence variation, covariation, and selection on quantitative traits, data on the magnitude of these consequences remain lacking.Recombinant inbred line (RIL) populations are a promising tool for investigating the influence of major-effect loci. During advancement of the lines from F₂''s to RILs, alternate alleles at major-effect genes (and most of the rest of the genome) will be made homozygous, simplifying comparisons among genotypic classes. Because of the high homozygosity, individuals within RILs are nearly genetically identical, facilitating phenotyping of many genotypes under a range of environments. In addition, because of recombination, alternative alleles are randomized across genetic backgrounds—facilitating robust comparisons between sets of lines differing at a major-effect locus.Here we investigate how polymorphism at an artificially induced mutation, the erecta locus in Arabidopsis thaliana, affects the magnitude of these important evolutionary genetic parameters under ecologically realistic field conditions. We use the Landsberg erecta (Ler) × Columbia (Col) RIL population of A. thaliana to examine how variation at a gene of major effect influences genetic variation, covariation, and selection on quantitative traits in a field setting. The Ler × Col RIL population is particularly suitable, because it segregates for an artificially induced mutation at the erecta locus, which has been shown to influence a wide variety of plant traits. The Ler × Col population thus allows a powerful test of the effects of segregating variation at a gene—chosen a priori—with numerous pleiotropic effects. The ERECTA gene is a leucine-rich receptor-like kinase (LRR-RLK) (Torii et al. 1996) and has been shown to affect plant growth rates (El-Lithy et al. 2004), stomatal patterning and transpiration efficiency (Masle et al. 2005; Shpak et al. 2005), bacterial pathogen resistance (Godiard et al. 2003), inflorescence and floral organ size and shape (Douglas et al. 2002; Shpak et al. 2003, 2004), and leaf polarity (Xu et al. 2003; Qi et al. 2004).Specifically, we sought to answer the following questions: (1) Is variation at erecta significantly associated with changes to the G matrix? (2) Is variation at erecta associated with changes in natural selection on genetically variable traits? And (3) is variation at erecta associated with significantly different projected evolutionary responses to selection?     

Directionality of Epistasis in a Murine Intercross Population

Directional epistasis describes a situation in which epistasis consistently increases or decreases the effect of allele substitutions, thereby affecting the amount of additive genetic variance available for selection in a given direction. This study applies a recent parameterization of directionality of epistasis to empirical data. Data stems from a QTL mapping study on an intercross between inbred mouse (Mus musculus) strains LG/J and SM/J, originally selected for large and small body mass, respectively. Results show a negative average directionality of epistasis for body-composition traits, predicting a reduction in additive allelic effects and in the response to selection for increased size. Focusing on average modification of additive effect of single loci, we find a more complex picture, whereby the effects of some loci are enhanced consistently across backgrounds, while effects of other loci are decreased, potentially contributing to either enhancement or reduction of allelic effects when selection acts at single loci. We demonstrate and discuss how the interpretation of the overall measurement of directionality depends on the complexity of the genotype–phenotype map. The measure of directionality changes with the power of scale in a predictable way; however, its expected effect with respect to the modification of additive genetic effects remains constant.EPISTASIS is present when the effect of a genetic substitution depends on the genotypes at other loci. At the population level, this means that average allelic effects change as allele frequencies at other loci change, and thus that gene effects can evolve. The evolutionary significance of epistasis has been recognized mainly in relation to the allele-frequency changes that are caused by genetic drift (e.g., Goodnight 1987, 1988, 1995; Cheverud and Routman 1996; Barton and Turelli 2004; De Brito et al. 2005; Turelli and Barton 2006), whereas the epistatic effects under directional selection have been treated only recently (e.g., Carter et al. 2005; Weinreich et al. 2005; Carlborg et al. 2006; Hansen et al. 2006; Yukilevich et al. 2008). Technically, the effects of epistatic interaction can be considered as having two aspects: the architecture itself (i.e., the existence of a nonadditive component, the so-called functional aspect, see below), and the effect of allele frequency on genetic variance (i.e., the statistical aspect). An additional consideration is crucial for the response to selection of a given trait, namely that the response is generated by the joint action of many epistatic interactions. Each of the interactions can either enhance or diminish the additive genetic effect in any specific phenotypic dimension. Their composite effect depends on the pattern, i.e., whether the effects accumulate or cancel each other out (Hansen and Wagner 2001a,b; Carter et al. 2005; Hansen et al. 2006). In the following we provide a brief general account of epistasis and then focus on the effect of its composite pattern, the empirical assessment of which is the goal of this study.The traditional population-genetic approach to selection response initially emphasized additive genetic variance and treated any variance unexplained by the additive effects, including variance due to interactions within or between loci, as residual variance (Fisher 1918). Later this model was extended to account for epistasis (Cockerham 1954, Kempthorne 1954). The interaction component of this residual variance is dependent on population allele frequencies at the interacting loci. Starting with nonadditive effects within a single locus (dominance), Falconer (1960) described the effect of allele frequencies on the statistical measure of average allelic effect. Cheverud and Routman (1995) explored the analogous effect at the two-locus level, leading to distinction between allele-frequency-dependent statistical epistasis (contributing to epistatic variance) on population level, and allele-frequency-independent physiological (or functional) epistasis on the individual level, which contributes to all the genetic variance components, i.e.,, additive, dominance, and epistatic. Thus while physiological epistasis describes the genetic architecture of a given phenotype defining the potential for epistatic effects, statistical epistasis describes the realization of individual-level epistatic effects in terms of allele-frequency-dependent genetic variance components. Several authors have worked out tools to estimate gene interaction effects at the individual level independently of the allele frequencies to distinguish between the physiological and statistical epistasis (Cheverud and Routman 1995; Wagner et al. 1998; Hansen and Wagner 2001a; Barton and Turelli 2004; Yang 2004; Zeng et al. 2005; Wang and Zeng 2006). These methods have been recently generalized in a common framework for measuring epistasis and translating between the population and individual levels (NOIA: Alvarez-Castro and Carlborg 2007; Alvarez-Castro et al. 2008; see Le Rouzic 2008 for R software package). An understanding of the pattern of epistatic architecture enables prediction of its effects at any given set of allele frequencies.The original physiological epistasis has often referred to isolated pairwise interactions. The genetic basis of a complex trait is affected by more than two interacting loci; thus the trait''s genetic variance is affected by the combination of these interaction effects. Effects at the individual loci can add up, or cancel out. For example, when the alleles at background loci become fixed (e.g., due to a bottleneck), previously background-dependent genetic effects at loci A and B can become additive effects of the same or of the opposite sign. Depending on the sign and size of their effects, the two allele substitutions at loci A and B therefore add up to increase, decrease, or have no overall effect on additive genetic variance. Hansen and Wagner (2001a) introduced the notion of directionality of epistasis to emphasize the importance of the pattern of epistatic architecture for the system''s evolvability. Directionality measures the consistency of epistatic effects on additive variance for a specific locus and trait across the genome, given a defined reference genotype. It describes whether epistasis tends to enhance or diminish the additive effects of interacting loci on a trait in a specified phenotypic direction. The population-dynamic studies analyzing effects of the epistatic pattern show that the directionality of epistasis can be a major determinant of evolution on time scales beyond a few generations (Hermisson et al. 2003; Carter et al. 2005; Hansen et al. 2006; Yukilevich et al. 2008; Alvarez-Castro et al. 2009; Fierst and Hansen 2010). Averaged across interactions, positive directional epistasis increases additive variance and the response to positive selection relative to that predicted by the additive genetic effects alone, while negative directional epistasis tends to decrease the response to selection in that same phenotypic direction. An absence of epistatic directionality occurs when positive and negative directional epistatic effects cancel out on average or, trivially, from the absence of epistasis. Thus directionality describes the local, population-specific curvature of the genotype–phenotype map (Figure 1).Open in a separate windowFigure 1.—Directionality of epistasis describes the local curvature of the genotype–phenotype map. The genotypic values of the trait of interest in the two parental inbred populations and in an intercross are plotted on y-axis. Given a defined direction on a trait axis and a reference point for measurement of genetic effects, the directionality () describes whether the epistasis increases or decreases the effect of allelic substitutions relative to the value predicted by additive effects alone. The extrapolation of the curvature beyond the local effects requires the knowledge of higher-order epistatic effects; however, the local effects can be calculated for different reference points and different phenotypic directions (see Hansen and Wagner 2001a,b).Aspects of epistatic directionality have been addressed previously with respect to the effect on fitness (reviewed in Phillips et al. 2000), with focus on the synergistic epistasis for deleterious effect enhancement (e.g., Kondrashov 1988; Charlesworth 1990; Hansen and Wagner 2001b). However, the empirical study of overall directionality of epistasis, and its measurement in morphological or physiological traits, is still lacking (Hansen 2006). Implicit indications from empirical studies are ambiguous (see examples in Carter et al. 2005). Here, we present an assessment of directionality of two-way epistasis between QTL in an intercross population of laboratory mice, paying special attention to scale effects.     

Parallel Adaptation: One or Many Waves of Advance of an Advantageous Allele?

Models for detecting the effect of adaptation on population genomic diversity are often predicated on a single newly arisen mutation sweeping rapidly to fixation. However, a population can also adapt to a new environment by multiple mutations of similar phenotypic effect that arise in parallel, at the same locus or different loci. These mutations can each quickly reach intermediate frequency, preventing any single one from rapidly sweeping to fixation globally, leading to a “soft” sweep in the population. Here we study various models of parallel mutation in a continuous, geographically spread population adapting to a global selection pressure. The slow geographic spread of a selected allele due to limited dispersal can allow other selected alleles to arise and start to spread elsewhere in the species range. When these different selected alleles meet, their spread can slow dramatically and so initially form a geographic patchwork, a random tessellation, which could be mistaken for a signal of local adaptation. This spatial tessellation will dissipate over time due to mixing by migration, leaving a set of partial sweeps within the global population. We show that the spatial tessellation initially formed by mutational types is closely connected to Poisson process models of crystallization, which we extend. We find that the probability of parallel mutation and the spatial scale on which parallel mutation occurs are captured by a single compound parameter, a characteristic length, which reflects the expected distance a spreading allele travels before it encounters a different spreading allele. This characteristic length depends on the mutation rate, the dispersal parameter, the effective local density of individuals, and to a much lesser extent the strength of selection. While our knowledge of these parameters is poor, we argue that even in widely dispersing species, such parallel geographic sweeps may be surprisingly common. Thus, we predict that as more data become available, many more examples of intraspecies parallel adaptation will be uncovered.THERE are many dramatic examples of convergent evolution across distantly related species, where a phenotype independently evolves via parallel changes at orthologous genetic loci (Wood et al. 2005b; Arendt and Reznick 2008), indicating that adaptation can be strongly shaped by pleiotropic constraints (Haldane 1932; Orr 2005; Stern and Orgogozo 2008; Kopp 2009). There are also a growing number of examples of the parallel evolution of a phenotype within a species due to independent mutations at the same gene (Arendt and Reznick 2008) (which are sometimes referred to as genetically redundant). Some of the best-studied examples come from the repeated evolution of resistance to insecticides within several insect species (Ffrench Constant et al. 2000) and the resistance of malaria to antimalarial drugs (Anderson and Roper 2005; Pearce et al. 2009). Another example is the loss of pigmentation in Drosophlia santomea through at least three independent mutations at a cis-regulatory element (Jeong et al. 2008), while the evolution of pigmentation within vertebrate species provides further examples (Protas et al. 2006; Gross et al. 2009; Kingsley et al. 2009). There are also a number of examples of parallel evolution within our own species (Novembre and Di Rienzo 2009). For example, various G6PD mutations have spread in parallel in response to malaria (Tishkoff et al. 2001; Louicharoen et al. 2009), and lactase persistence has evolved independently in at least three different pastoral populations (Tishkoff et al. 2007; Enattah et al. 2008). A particularly impressive example in humans is offered by the sickle cell allele at the β-globin gene that confers malaria resistance, where multiple changes have putatively occurred at a single base pair (see Flint et al. 1998, for discussion). In each of these examples, multiple, independent mutations have led to the same or a functionally equivalent adaptive phenotype, although they differ in the degree to which the functional consequences and equivalences of the different mutations have been explored. Such repeated adaptive evolution via similar changes within a species, which we term parallel adaptation, may therefore be common. As we also address repeated evolution of a similar phenotype via changes at different genetic loci, this could more broadly be termed “convergent adaptation” (Arendt and Reznick 2008).In many of these examples the selection pressure is patchy and rates of gene flow are low, increasing the chance of parallel adaptation. However, parallel adaptation can occur even in a panmictic population. For example, adaptation may occur from multiple independent copies of the selected allele present in standing variation at mutation–selection balance within the population (Orr and Betancourt 2001; Hermisson and Pennings 2005). Even when there is no standing variation for a trait in a panmictic population, a selected allele could arise independently several times during the course of a selective sweep, if mutation is sufficiently fast relative to the spread of the selected allele. This idea was formalized by Pennings and Hermisson (2006a,b), who showed that such soft sweeps may be expected when the population scaled mutation rate (the product of the effective population size and mutation rate) toward the adaptive allele is >1. Thus, repeated mutation may be quite common for species with large populations or where the mutation target is large, e.g., knocking out of a gene. Pennings and Hermisson (2006a) showed that the number of independently arisen selected alleles in a sample has approximately a Ewens distribution, and properties of neutral variation at a closely linked site can be derived from this (Pennings and Hermisson 2006b). Such a selective sweep has been termed a soft sweep, as the population can adapt without the dramatic reduction in diversity at linked selected sites that is usually associated with a full sweep (Maynard Smith and Haigh 1974); see Pennings and Hermisson (2006a,b), Hermisson and Pfaffelhuber (2008), and Pritchard et al. (2010) for discussion and Schlenke and Begun (2005) or Jeong et al. (2008) for potential examples.Clearly, if parallel mutations can occur during adaptation in a large panmictic population, then limited dispersal should further increase the chance of parallel adaptation, as other mutations can arise and spread during the time it takes one to move across the species range. Intuitively, a low rate of dispersal and a large mutational target should increase the chance of parallel adaptation (as in Coop et al. 2009; Novembre and Di Rienzo 2009), but it is unclear exactly how other dispersal, population, and mutational parameters play into the probability of parallel adaptation. However, in the absence of a formal model, many simple questions remain: Does parallel adaptation occur only in species with strong population structure? Weak selection pressures lead to slowly spreading mutations. Is parallel adaptation more likely in this case? This leaves us unable to understand the likelihood of parallel adaptation in particular examples (such as Flint et al. 1998) and more generally its role in geographic patterns of adaptation (such as Coop et al. 2009).Here we study parallel adaptation in a homogeneous, geographically spread population. We focus on the case where a population is exposed to a novel selection regime throughout a homogeneous species range, and the population is initially entirely devoid of standing variation for the trait, assumptions that favor the fixation of only a single new allele in the population. We use simple approximations to derive theoretical results for the properties of parallel adaptation in a continuous spatial population with strong migration for a range of dispersal distributions (also called dispersal kernels, including fat-tailed examples). We are able to describe fairly completely the resulting patterns and show that they are well captured by a single compound parameter combining the rate of mutation and the speed at which the mutation spreads. For an introduction to the patterns of genetic diversity that can be expected from such geographic structure at both neutral and selected loci, see Lenormand (2002), Charlesworth et al. (2003), and Novembre and Di Rienzo (2009).We show that when population sizes are sufficiently large and dispersal distances are small compared to the species range, parallel adaptation within a species is likely to be common, and quantify this relationship. Furthermore, we describe how separately arisen mutations will—at least for some time—leave behind a spatial pattern reflecting their separate origins.The structure of this article is as follows: In methods we introduce and analyze our model of a continuous population, first in the classical context and then in a more general context that allows for accelerating waves (arising from fat-tailed dispersal distributions). In Simulations we present the results of some simulations of the continuous process, intended to assess the robustness of our results to deviations from the assumptions. In Biological parameters and the characteristic length and Applications we present and discuss the theoretical results in a few biologically reasonable contexts, providing numerical results to illustrate how the different parameters play into the probability of parallel adaptation. In the discussion we discuss consequences and extensions. Some mathematical arguments are postponed until the appendixes.

Modeling assumptions:

Here we describe the assumptions behind our model and give some background, before introducing in methods the model we analyze. First, we assume each mutation under consideration confers a selective advantage such that, upon appearing in the population, it quickly rises locally to some equilibrium frequency. Second, there is significant spatial structure; namely, migration is weak enough that the selected trait reaches an equilibrium frequency locally before spreading to the entire population. Third, the parallel mutations are distinguishable and confer the same selective benefit. Fourth, these mutations are neutral relative to each other, in the sense that in a population at equilibrium frequency (e.g., fixation) for any collection of these mutations, the dynamics of their relative proportions occur on a longer timescale than their dynamics in the original background (examples are given below). We call this last assumption allelic exclusion, since it implies that areas fixed for one adaptive allele will not be rapidly overtaken by another.Under these assumptions, a newly arisen advantageous mutation, if it is initially successful, will spread through the population in a more-or-less wavelike manner (more on this later). If another allele conferring the same advantage arises in a location the first has not yet reached, then the two waves spread toward each other and will at some point collide. What happens when they collide will generally depend on the details of their epistatic interaction or, if they occur at a single site, on their dominance interaction. However, by our assumption of allelic exclusion, the dynamics are slower than the spread of the selected alleles. This allows us to neglect the slower mixing of types and genetic drift that will happen in this phase, instead focusing on the first process by which independently arisen alleles partition the population.In Figure 1 we show a cartoon to illustrate our model, and in Figures 5 and ​and66 we show the results of a simulation (described in Simulations).Open in a separate windowFigure 1.—A cartoon representation of our model of spatial parallel mutation. In the top row, each panel represents a two-dimensional (2D) species range with time increasing from the left to the right panel. In the bottom row, a 1D species range is represented by the vertical axis and time is the horizontal axis, with more recent times closer to the right side of the page. Stars represent a new mutation arising and escaping drift. The three colors represent the area occupied by three different alleles. Note that I and II are not different views of the same process, although they are similar.Open in a separate windowFigure 5.—A space–time plot of a single run of a simulation on a linear array of 500 demes each of size 100 over 20,000 generations. The parameters were s = 0.1, m = 0.01, and μ = 4 × 10⁻⁶, and migration was nearest neighbor. Time runs down the plot; different colors label different types, and areas occupied by more than one type are colored by a mixture of the colors (local drift is strong in this simulation, so most demes have only one type). Each distinct “cone” has a unique type despite similarities in color choice. Note that types expand at roughly constant speed until encountering another type, and that mixing, while present, happens on a longer timescale. Types that appear where the advantageous type is already fixed (e.g., the orange bit between the purple and blue regions on the left) are unlikely to survive, even if they locally escape drift.Open in a separate windowFigure 6.—Six time slices of an example simulation in a two-dimensional range, showing initial establishment and expansion of types and the beginning of mixing (which happens much slower than expansion). The population was composed of a 60 × 60 grid of demes with 1000 individuals in each. Different colors correspond to different types, and white is the ancestral type; when more than one color occupies a deme, the colors are mixed, so that eventually, if all colors spread to all demes, the entire population will be gray.

Allelic exclusion:

The allelic exclusion assumption is fundamental to our approach. It will hold, for instance, if there is a single advantageous mutation, and we treat each time it arises independently as a distinct allele, identifiable by examination of linked neutral variation. It will also hold if mutations at multiple sites within a gene are genetically redundant, such as loss-of-function mutations, and no additional selective benefit is conferred by having a mutation at more than one site (though this may be an approximation, since even loss-of-function changes within the same gene may differ in their characteristics, as in Rosenblum et al. 2010).Another important consequence of allelic exclusion is that a mutation occurring in a location where the advantageous allele already exists in large numbers is unlikely to persist or achieve high frequency—indeed, if the interaction is neutral and 999 other individuals already exist in the same location with the selected trait, then a new mutation will contribute on average only 0.001 of the future population and has high probability of being lost from the population by drift. This fact allows us to ignore all new mutations that occur after any selected allele has risen in that location to a nonzero frequency. In particular, the shape of the wave front will not be important, only how its leading edge spreads. Below, for convenience we often talk about the probability or rate of local fixation, but it follows from this observation that we need require only that the allele escapes loss from the population by drift and that some intermediate equilibrium frequency is reached, as would occur in the case of overdominance.

Selection:

We also assume that the advantageous, derived alleles have a reproductive advantage of (1 + s) relative to the ancestral type. In practice, in a diploid model with dominance or epistasis, or in the presence of density dependence, we require that both the manner in which a new mutation escapes drift and the way that it subsequently spreads through the population be well approximated by the simple haploid (or additive) model. Roughly speaking, this holds if the growth and spread of the allele are driven by growth where the allele is at very low frequency (and primarily occurring in heterozygotes). This implies that the probability a new mutation escapes drift is well approximated by 2s divided by the variance in offspring number [which is quite robust to the details of spatial structure (Maruyama 1970, 1974)] and that per-capita growth is fastest when at low frequency. In the usual formulation of diploid systems (Aronson and Weinberger 1978), this is satisfied if the fitness advantage of the homozygote is no more than twice the fitness advantage of the heterozygote. In other cases, e.g., an Allee effect, the behavior can be quite different; see Stokes (1976).

Background on the wave of advance:

We model the spread of a selected allele by making use of existing work on traveling waves, a link first established independently by Fisher (1937) and by Kolmogorov, Petrovskii, and Piscunov (KPP) (Kolmogorov et al. 1937). We introduce and review the wave of advance literature here, as much of the subsequent development has occurred in fields other than population genetics. Suppose that individuals produce a random number of offspring with mean r and that offspring disperse a random distance with standard deviation σ, and let p(t, x) be the expected proportion of mutants at time t and location x. Suppose also that the selection coefficient s is small and the advantage is additive and that the population density ρ is fairly large. Both articles argued that if the dispersal distance is Gaussian, or if σ is small (so that the “long-time” dispersal distribution is Gaussian), then barring the appearance of new mutations, the time evolution of p is well described by the reaction–diffusion equation now known as the Fisher–KPP equation,(1)where d is the dimension of the species range. They furthermore showed in d = 1 that a “wave of advance” occurs as the solution to this equation and that for initial conditions where the allele is only polymorphic within a spatially bounded region, the solution moved asymptotically with speed . Kolmogorov et al. (1937) also covered the more general case in which p(x, t)(1 − p(x, t)) is replaced by F(p(x, t)) for an appropriate function F, which gives the density-dependent growth rate of the selected type, subject to certain conditions.For many other choices of dispersal distribution and growth function F the advancing front of a new type also approaches a constant wave shape that advances at constant speed through time—a “traveling wave” solution, but with a speed not given by the same formula. Then the frequency of individuals of the selected type at x  at time t  can be expressed p(x, t) = h(x − νt), where h(·) gives the shape of the wave and ν is its speed. These traveling wave solutions have been studied for the Fisher–KPP equation for a range of appropriate F (Aronson and Weinberger 1975); the speed can often be found more easily than the wave shape (Hadeler and Rothe 1975). Radially symmetric solutions also exist, in which the new type travels outward from an initial origin; the behavior of such radially spreading waves depends on initial conditions, but will asymptotically move with the same constant speed and fixed wave shape as in one dimension.Since the introduction of the Fisher–KPP equation, traveling wave solutions to reaction–diffusion equations have been studied in the ecological literature as a model of invading species (Skellam 1951; Kot et al. 1996), as well as in a range of other fields. See Aronson and Weinberger (1978) for some classical theory, general discussion, and context or Volpert et al. (1994) for a more extensive reference. Related models, using integrodifference or integrodifferential equations have been used by various authors to include various important biological factors such as age structure and fluctuating environments (Neubert and Caswell 2000; Neubert et al. 2000; Kot and Neubert 2008); see Hastings et al. (2005) or Zhao (2009) for a review. Density regulation is often discussed in these models, but important behaviors can usually be determined by a linearization, on the basis of how the new type grows when rare. Common to these models is the existence of traveling wave solutions, whose forms and speeds are often known only implicitly; most natural models of the spread of a new selected type can be translated into one of these frameworks. There is also a fruitful connection of these Fisher–KPP models to branching random walks that is beyond the scope of this article; see McKean (1975), Biggins (1979, 1995), and Kot et al. (2004). A similar model, the contact process, has also been widely studied in the probabilistic literature; see Bramson et al. (1989).The qualitative behavior of the spread of an organism or an allele in a population can depend on the organism''s dispersal kernel, defined as the probability density of the distance between mother and child''s birth locations (see Shigesada and Kawasaki 1997 or Cousens et al. 2008 for discussion). Most mathematical models of invasions assume that the dispersal kernel has tails bounded by an exponential and obtain a constant wave speed. In some species this is appropriate, while in others, rare, long-distance migration events are important (Shigesada and Kawasaki 1997). In such organisms, dispersal may be better modeled by a kernel that is not bounded by an exponential (i.e., a “fat-tailed” kernel), although there is generally insufficient evidence so far (Cousens et al. 2008, Chap. 5). Mollison (1972) showed that in a certain model, if the kernel is fat tailed, the range occupied by the expanding type will be patchy and will grow faster than linearly: the spread accelerates and eventually moves faster than any constant-speed traveling wave. Moreover, Lewis and Pacala (2000) established a link between leptokurtic kernels (kernels whose kurtosis exceeds that of the standard Gaussian) and patchy invasion dynamics. Leptokurtic but exponentially bounded kernels can lead to waves that initially accelerate but settle to a constant speed. We shall see that the important behavior of the model is not determined by the asymptotic, long-time speed of the wave, but rather by its behavior at intermediate times. Therefore, kernels that have similar short-time behavior but different long-time behavior can give rise to similar dynamics on the scale we are interested in. Consideration of other wave behaviors leads to a more general model, which we study in The general case.The models reviewed above are haploid models; traveling waves in diploid models have been much less studied. Aronson and Weinberger (1975) show that in the diploid analog to Equation 1, if the difference in selection coefficient is small, then allele frequency dynamics are approximately governed by (1). If local populations are in Hardy–Weinberg equilibrium, then more general results apply, demonstrating the existence of traveling waves (Weinberger 1982; Zhao 2009). If dispersal occurs over a distance comparable to the width of the wave, then this will no longer be the case, and while recently developed general theory (Zhao 2009) might be applied, the existence and characterization of traveling waves in other diploid models is to our knowledge an open question. However, we certainly expect the behavior to be wavelike, and since our theory takes wave behavior as an input, we have no qualms about using our model to discuss the diploid organisms in Biological parameters and the characteristic length.     

Age Structure,Changing Demography and Effective Population Size in Atlantic Salmon (Salmo salar)

Effective population size (N_e) is a central evolutionary concept, but its genetic estimation can be significantly complicated by age structure. Here we investigate N_e in Atlantic salmon (Salmo salar) populations that have undergone changes in demography and population dynamics, applying four different genetic estimators. For this purpose we use genetic data (14 microsatellite markers) from archived scale samples collected between 1951 and 2004. Through life table simulations we assess the genetic consequences of life history variation on N_e. Although variation in reproductive contribution by mature parr affects age structure, we find that its effect on N_e estimation may be relatively minor. A comparison of estimator models suggests that even low iteroparity may upwardly bias N_e estimates when ignored (semelparity assumed) and should thus empirically be accounted for. Our results indicate that N_e may have changed over time in relatively small populations, but otherwise remained stable. Our ability to detect changes in N_e in larger populations was, however, likely hindered by sampling limitations. An evaluation of N_e estimates in a demographic context suggests that life history diversity, density-dependent factors, and metapopulation dynamics may all affect the genetic stability of these populations.THE effective size of a population (N_e) is an evolutionary parameter that can be informative on the strength of stochastic evolutionary processes, the relevance of which relative to deterministic forces has been debated for decades (e.g., Lande 1988). Stochastic forces include environmental, demographic, and genetic components, the latter two of which are thought to be more prominent at reduced population size, with potentially detrimental consequences for average individual fitness and population persistence (Newman and Pilson 1997; Saccheri et al. 1998; Frankham 2005). The quantification of N_e in conservation programs is thus frequently advocated (e.g., Luikart and Cornuet 1998; Schwartz et al. 2007), although gene flow deserves equal consideration given its countering effects on genetic stochasticity (Frankham et al. 2003; Palstra and Ruzzante 2008).Effective population size is determined mainly by the lifetime reproductive success of individuals in a population (Wright 1938; Felsenstein 1971). Variance in reproductive success, sex ratio, and population size fluctuations can reduce N_e below census population size (Frankham 1995). Given the difficulty in directly estimating N_e through quantification of these demographic factors (reviewed by Caballero 1994), efforts have been directed at inferring N_e indirectly through measurement of its genetic consequences (see Leberg 2005, Wang 2005, and Palstra and Ruzzante 2008 for reviews). Studies employing this approach have quantified historical levels of genetic diversity and genetic threats to population persistence (e.g., Nielsen et al. 1999b; Miller and Waits 2003; Johnson et al. 2004). N_e has been extensively studied in (commercially important) fish species, due to the common availability of collections of archived samples that facilitate genetic estimation using the temporal method (e.g., Hauser et al. 2002; Shrimpton and Heath 2003; Gomez-Uchida and Banks 2006; Saillant and Gold 2006).Most models relating N_e to a population''s genetic behavior make simplifying assumptions regarding population dynamics. Chiefly among these is the assumption of discrete generations, frequently violated in practice given that most natural populations are age structured with overlapping generations. Here, theoretical predictions still apply, provided that population size and age structure are constant (Felsenstein 1971; Hill 1972). Ignored age structure can introduce bias into temporal genetic methods for the estimation of N_e, especially for samples separated by time spans that are short relative to generation interval (Jorde and Ryman 1995; Waples and Yokota 2007; Palstra and Ruzzante 2008). Moreover, estimation methods that do account for age structure (e.g., Jorde and Ryman 1995) still assume this structure to be constant. Population dynamics will, however, likely be altered as population size changes, thus making precise quantifications of the genetic consequences of acute population declines difficult (Nunney 1993; Engen et al. 2005; Waples and Yokota 2007). This problem may be particularly relevant when declines are driven by anthropogenic impacts, such as selective harvesting regimes, that can affect age structure and N_e simultaneously (Ryman et al. 1981; Allendorf et al. 2008). Demographic changes thus have broad conservation implications, as they can affect a population''s sensitivity to environmental stochasticity and years of poor recruitment (Warner and Chesson 1985; Ellner and Hairston 1994; Gaggiotti and Vetter 1999). Consequently, although there is an urgent need to elucidate the genetic consequences of population declines, relatively little is understood about the behavior of N_e when population dynamics change (but see Engen et al. 2005, 2007).Here we focus on age structure and N_e in Atlantic salmon (Salmo salar) river populations in Newfoundland and Labrador. The freshwater habitat in this part of the species'' distribution range is relatively pristine (Parrish et al. 1998), yet Atlantic salmon in this area have experienced demographic declines, associated with a commercial marine fishery, characterized by high exploitation rates (40–80% of anadromous runs; Dempson et al. 2001). A fishery moratorium was declared in 1992, with rivers displaying differential recovery patterns since then (Dempson et al. 2004b), suggesting a geographically variable impact of deterministic and stochastic factors, possibly including genetics. An evaluation of those genetic consequences thus requires accounting for potential changes in population dynamics as well as in life history. Life history in Atlantic salmon can be highly versatile (Fleming 1996; Hutchings and Jones 1998; Fleming and Reynolds 2004), as exemplified by the high variation in age-at-maturity displayed among and within populations (Hutchings and Jones 1998), partly reflecting high phenotypic plasticity (Hutchings 2004). This diversity is particularly evident in the reproductive biology of males, which can mature as parr during juvenile freshwater stages (Jones and King 1952; Fleming and Reynolds 2004) and/or at various ages as anadromous individuals, when returning to spawn in freshwater from ocean migration. Variability in life history strategies is further augmented by iteroparity, which can be viewed as a bet-hedging strategy to deal with environmental uncertainty (e.g., Orzack and Tuljapurkar 1989; Fleming and Reynolds 2004). Life history diversity and plasticity may allow salmonid fish populations to alter and optimize their life history under changing demography and population dynamics, potentially acting to stabilize N_e. Reduced variance in individual reproductive success at low breeder abundance (genetic compensation) will achieve similar effects and might be a realistic aspect of salmonid breeding systems (Ardren and Kapuscinski 2003; Fraser et al. 2007b). Little is currently known about the relationships between life history plasticity, demographic change and N_e, partly due to scarcity of the multivariate data required for these analyses.Our objective in this article is twofold. First, we use demographic data for rivers in Newfoundland to quantify how life history variation influences age structure in Atlantic salmon and hence N_e and its empirical estimation from genetic data. We find that variation in reproductive contribution by mature parr has a much smaller effect on the estimation of N_e than is often assumed. Second, we use temporal genetic data to estimate N_e and quantify the genetic consequences of demographic changes. We attempt to account for potential sources of bias, associated with (changes in) age structure and life history, by using four different analytical models to estimate N_e: a single-sample estimator using the linkage disequilibrium method (Hill 1981), the temporal model assuming discrete generations (Nei and Tajima 1981; Waples 1989), and two temporal models for species with overlapping generations (Waples 1990a,b; Jorde and Ryman 1995) that differ principally in assumptions regarding iteroparity. A comparison of results from these different estimators suggests that iteroparity may often warrant analytical consideration, even when it is presumably low. Although sometimes limited by statistical power, a quantification and comparison of temporal changes in N_e among river populations suggests a more prominent impact of demographic changes on N_e in relatively small river populations.     

Sex-Linked Inheritance in Macaque Monkeys: Implications for Effective Population Size and Dispersal to Sulawesi

Sex-specific differences in dispersal, survival, reproductive success, and natural selection differentially affect the effective population size (N_e) of genomic regions with different modes of inheritance such as sex chromosomes and mitochondrial DNA. In papionin monkeys (macaques, baboons, geladas, mandrills, drills, and mangabeys), for example, these factors are expected to reduce N_e of paternally inherited portions of the genome compared to maternally inherited portions. To explore this further, we quantified relative N_e of autosomal DNA, X and Y chromosomes, and mitochondrial DNA using molecular polymorphism and divergence information from pigtail macaque monkeys (Macaca nemestrina). Consistent with demographic expectations, we found that N_e of the Y is lower than expected from a Wright–Fisher idealized population with an equal proportion of males and females, whereas N_e of mitochondrial DNA is higher. However, N_e of 11 loci on the X chromosome was lower than expected, a finding that could be explained by pervasive hitchhiking effects on this chromosome. We evaluated the fit of these data to various models involving natural selection or sex-biased demography. Significant support was recovered for natural selection acting on the Y chromosome. A demographic model with a skewed sex ratio was more likely than one with sex-biased migration and explained the data about as well as an ideal model without sex-biased demography. We then incorporated these results into an evaluation of macaque divergence and migration on Borneo and Sulawesi islands. One X-linked locus was not monophyletic on Sulawesi, but multilocus data analyzed in a coalescent framework failed to reject a model without migration between these islands after both were colonized.THE effective size of a population (N_e) determines the relative impact of genetic drift and natural selection on mutations with mild effects on fitness (Charlesworth 2009). Differences in N_e are hypothesized to affect virtually every aspect of genome evolution, including rates of molecular evolution, abundance of introns and transposable elements, and persistence of duplicate genes, and this has important implications for the evolution of complexity via both adaptive and degenerative processes (Lynch 2007). Of relevance are not only the number of different individuals in a population, but also the number of copies of a gene within each individual. In diploid species with separate sexes, sex chromosomes and mitochondrial DNA (mtDNA) differ in copy number from autosomal DNA (aDNA): both sexes have two alleles at autosomal loci whereas in species with male heterogamy, males have one X and one Y chromosome, females have two Xs, and a female/male pair has effectively only one copy of mtDNA due to maternal inheritance. Sex-specific differences in demographic parameters such as migration, adult sex ratio, and variance in reproductive success also affect relative copy number and associated levels of neutral polymorphism at mtDNA, aDNA, the X chromosome (xDNA), and the Y chromosome (yDNA) (Hedrick 2007).The effective population size is the number of individuals in a Wright–Fisher idealized population (Fisher 1930; Wright 1931) that have the same magnitude of genetic drift as an observed population, where ideal individuals are diploid, and have discrete (nonoverlapping) generations, constant population size, and random mating. N_e can be quantified in terms of variance in allele frequency over generations (variance N_e) or variance in inbreeding over time (inbreeding N_e). If population size is constant with random mating, these approaches for quantifying N_e produce identical results (Kimura and Crow 1963; Whitlock and Barton 1997). At mutation–drift equilibrium with an equal number of males and females and a Poisson distributed number of offspring with a mean of two offspring per individual, N_e-aDNA and N_e-xDNA are expected to be four and three times as large, respectively, as N_e-yDNA and N_e-mtDNA; we refer to this as the “ideal expectation with an equal proportion of males and females.”Demography can alter relationships between N_e of different parts of the genome. For example, extreme skew in adult sex ratio can cause N_e of uniparentally inherited portions of the genome to exceed N_e of biparentally inherited portions (Figure 1A; Nunney 1993; Caballero 1994; Hoelzer 1997; Hedrick 2007). With a skewed sex ratio, the more common sex has a higher variance in reproductive success than the rare one, and this causes the overall variance in reproductive success to increase as the sex-ratio bias increases (Nunney 1993). Sex-biased dispersal such as female philopatry also alters relationships between N_e-aDNA, N_e-xDNA, N_e-yDNA, and N_e-mtDNA (Figure 1B), causing N_e of portions of the genome that disperse less to increase (Nei and Takahata 1993; Hoelzer 1997; Wang and Caballero 1999).Open in a separate windowFigure 1.—N_e of aDNA, xDNA, mtDNA, and yDNA as a function of (A) sex ratio skew and (B) the probability of female dispersal. In B, a finite island model of subdivided populations of constant size is assumed with a population size of 10,000 individuals, 10 subpopulations, and a male probability of migration equal to 0.1.At least five factors related to natural selection also can cause the relative N_e of aDNA, xDNA, yDNA, and mtDNA to depart from expectations: (1) very low or absent recombination in mtDNA and a portion of yDNA, (2) haploidy of mtDNA and yDNA, (3) hemizygosity of xDNA in males, (4) sexual selection and differences in gene content, and (5) differences in the rate and variance of mutation. “Selective sweeps” in which an advantageous mutation is fixed by natural selection, reduces N_e of linked sites (Maynard Smith and Haigh 1974) and this can affect the entire mitochondrial genome and nonrecombining portion of the Y chromosome. Nonrecombining portions of yDNA and mtDNA are also affected by stochastic loss of alleles containing the fewest deleterious mutations (“Muller''s ratchet”; Muller 1964; Felsenstein 1974), which results in a gradual decline of fitness of these chromosomes over time. N_e of nonrecombining DNA is further reduced by elimination of variation linked to substantially deleterious mutations (“background selection”; Charlesworth et al. 1993), by interference between linked polymorphisms that impedes fixation of advantageous alleles and extinction of deleterious ones (the “Hill–Robertson effect”; Hill and Robertson 1966; McVean and Charlesworth 2000), and by increased frequency of deleterious mutations linked to advantageous ones during a selective sweep (“genetic hitchhiking”; Rice 1987). Hemizygous X-linked and haploid Y-linked loci in males and mtDNA loci in both sexes are more vulnerable to recessive deleterious mutations because they are not masked by a second allele (Otto and Goldstein 1992). Hemizygosity on the X chromosome can also increase the rate of selective sweeps when advantageous mutations are recessive (Charlesworth et al. 1987). Similarly, these loci are also susceptible to recessive species incompatibilities—a factor that at least partially accounts for Haldane''s rule for hybrid sterility (Haldane 1922; Orr 1997). Sexual selection differentially influences the probability of fixation of mutations depending on mode of inheritance (Wade and Shuster 2004), especially mutations with antagonistic fitness effects between the sexes (Gibson et al. 2002). Additionally, the rate of evolution of animal mtDNA is much higher than aDNA, xDNA, and yDNA (Haag-Liautard et al. 2008) and this presumably contributes to variation in the frequency of nonneutral mutations in different parts of the genome.Differences among N_e of mtDNA, yDNA, xDNA, and aDNA are thought to be particularly pronounced in papionin monkeys (macaques, baboons, geladas, mandrills, drills, and mangabeys). These monkeys have a highly sex-biased adult demography; females form stable philopatric groups of close relatives, whereas males generally change social groups and disperse more widely (Dittus 1975). Often adult sex ratio of papionins is female biased (Dittus 1975; Melnick and Pearl 1987; O''Brien and Kinnard 1997; Okamoto and Matsumura 2001), and males have higher variance in reproductive success than females (Dittus 1975; de Ruiter et al. 1992; Keane et al. 1997; Van Noordwijk and Van Schaik 2002; Widdig et al. 2004). These sex differences predict strong population subdivision of mtDNA with little or no subdivision of aDNA, deep mtDNA coalescence times, and frequent mtDNA paraphyly among species, and discordant genealogical relationships between mtDNA and yDNA—and this has been observed in multiple studies (Melnick and Pearl 1987; Melnick 1988; Melnick and Hoelzer 1992; Melnick et al. 1993; Hoelzer et al. 1994; Evans et al. 1999, 2001, 2003; Tosi et al. 2000, 2002, 2003; Newman et al. 2004). Female philopatry and obligate male migration is a common social system in mammals (Greenwood 1980; Dobson 1982; Johnson 1986), though less so in humans (Seielstad et al. 1998), and molecular variation provides an effective tool for exploring the impact of natural selection and demography on aDNA, the sex chromosomes, and mtDNA (Nachman 1997; Bachtrog and Charlesworth 2002; Stone et al. 2002; Berlin and Ellegren 2004; Hellborg and Ellegren 2004; Wilder et al. 2004; Hammer et al. 2008).We explored the genetic effects of demography and linked selection in structuring sequence polymorphism of a papionin monkey—the macaques—at two levels. We first tested whether levels of polymorphism in aDNA, xDNA, yDNA, and mtDNA in a Bornean population of the pigtail macaque, Macaca nemestrina, match expectations under scenarios involving natural selection and also whether the data might be explained by simple demographic models with sex-specific dispersal or a biased sex ratio. We then explored demography on a larger, inter-island scale by estimating the time of divergence between macaques on Borneo and Sulawesi islands and by testing for evidence of ongoing migration between these islands.     

Rate of Adaptation in Large Sexual Populations

Adaptation often involves the acquisition of a large number of genomic changes that arise as mutations in single individuals. In asexual populations, combinations of mutations can fix only when they arise in the same lineage, but for populations in which genetic information is exchanged, beneficial mutations can arise in different individuals and be combined later. In large populations, when the product of the population size N and the total beneficial mutation rate U_b is large, many new beneficial alleles can be segregating in the population simultaneously. We calculate the rate of adaptation, v, in several models of such sexual populations and show that v is linear in NU_b only in sufficiently small populations. In large populations, v increases much more slowly as log NU_b. The prefactor of this logarithm, however, increases as the square of the recombination rate. This acceleration of adaptation by recombination implies a strong evolutionary advantage of sex.IN asexual populations, beneficial mutations arising on different genotypes compete against each other and in large populations most of the beneficial mutations are lost because they arise on mediocre genetic backgrounds or acquire further beneficial mutations less rapidly than their peers—the combined effects of clonal interference and multiple mutations (Gerrish and Lenski 1998; Desai and Fisher 2007). Exchange of genetic material between individuals allows the combination of beneficial variants that arose in different lineages and can thereby speed up the process of adaptation (Fisher 1930; Muller 1932). Indeed, most life forms engage in some form of recombination, e.g., lateral gene transfer or competence for picking up DNA in bacteria, facultative sexual reproduction in yeast and plants, or obligate sexual reproduction in most animals. Some benefits of recombination for the rate of adaptation have recently been demonstrated experimentally in Caenorhabditis reinhardtii (Colegrave 2002), Escherichia coli (Cooper 2007), and Saccharomyces cerevisiae (Goddard et al. 2005); for a review of older experiments, see Rice (2002).Yet the benefits of sex become less obvious when one considers its disadvantageous effects: recombination can separate well-adapted combinations of alleles and sexual reproduction is more costly than asexual reproduction due to resources spent for mating and, in some cases, the necessity of males. The latter—in animals often termed the twofold cost of sex—implies that sexual populations can be unstable to the invasion of asexual variants. As a result, the pros and cons of sex have been the subject of many decades of debate in the theoretical literature (Crow and Kimura 1965; Maynard Smith 1968; Felsenstein 1974; Barton 1995a; Barton and Charlesworth 1998), and several different potentially beneficial aspects of sex have been identified, including the pruning of detrimental mutations (Peck 1994; Rice 1998) and host–parasite coevolution or otherwise changing environments (Charlesworth 1993; Ladle et al. 1993; Bürger 1999; Waxman and Peck 1999; Gandon and Otto 2007; Callahan et al. 2009). In the opposite situation of relatively static populations, it has been proposed that recombination is favored in the presence of negative epistasis (Feldman et al. 1980; Kondrashov 1984, 1988)—a situation when the combined detrimental effect of two unfavorable alleles is greater than the sum of the individual effects. While this may sometimes be a significant effect, most populations, especially microbes, are likely to be under continuing selection and the benefits of sex for speeding up adaptation are likely to dominate.The Fisher–Muller hypothesis is that sex speeds up adaptation by combining beneficial variants. Moreover, it has been demonstrated by Hill and Robertson (1966) that linkage decreases the efficacy of selection. This detrimental effect of linkage, known as the “Hill–Robertson effect,” causes selection for higher recombination rates, which has been shown by analyzing recombination modifier alleles at a locus linked to two competing segregating loci (Otto and Barton 1997; Iles et al. 2003; Barton and Otto 2005; Roze and Barton 2006; Martin et al. 2006). Hitchhiking of the allele that increases the recombination rates with the sweeping linked loci results in effective selection for increased recombination.Experiments and simulation studies suggest that the Hill–Roberston effect is more pronounced and selection for recombination modifiers is stronger in large populations with many sweeping loci (Felsenstein 1974; Colegrave 2002; Iles et al. 2003). However, the quantitative understanding of the effect of recombination in large populations is limited. Rouzine and Coffin have studied the role of recombination in the context of evolution of drug resistance in human immunodeficiency virus, finding that recombination of standing variation speeds up adaptation by producing anomalously fit individuals at the high fitness edge of the distribution (Rouzine and Coffin 2005; Gheorghiu-Svirschevski et al. 2007). The effects of epistatic interactions between polymorphisms and recombination on the dynamics of selection have recently been analyzed by Neher and Shraiman (2009). Yet none of these works consider the effects of new beneficial mutations. In the absence of new mutations (and in the absence of heterozygous advantage that can maintain polymorphisms) the fitness soon saturates as most alleles become extinct and standing variation disappears. Thus the crucial point that must be addressed is the balance between selection and recombination of existing variation and the injection of additional variation by new mutations.Here, we study the dynamics of continual evolution via new mutations, selection, and recombination using several models of recombination. Our primary models most naturally apply when periods of asexual reproduction occur between matings, so that they approximate the life style of facultatively outcrossing species such as S. cerevisiae, some plants, and C. elegans, which reproduce asexually most of the time but undergo extensive recombination when outcrossing. The models enable us to study analytically the explicit dependence of the rate of adaptation and of the dynamics of the beneficial alleles on the important parameters such as the outcrossing rate and population size. In an independent study N. H. Barton and J. Coe (personal communication) calculate the rate of adaptation for obligate sexual organisms using several different multilocus models of recombination, including the free recombination model studied here. The relation of our work to theirs, as well as to that of Cohen et al. (2005, 2006) who have also studied the effects of recombination with multiple new mutations, is commented on in the discussion.When deleterious mutations can be neglected, the rate of adaptation is the product of the rate of production of favorable mutations NU_b (N being the population size and U_b the genomewide beneficial mutation rate), the magnitude of their effect, and their fixation probability. The fixation probability is dominated by the probability that the allele becomes established, i.e., that it rises to high enough numbers in the population that it is very unlikely to die out by further stochastic fluctuations. In a homogeneous population a single beneficial mutation with selective advantage s has a probability of establishment and eventual fixation of (in discrete generation models, P_e≈2s) (Moran 1959). In a heterogeneous population, however, a novel beneficial mutation can arise on different genetic backgrounds and its establishment probability will thus vary, being greater if it arises in a well-adapted individual. But even well-adapted genotypes soon fall behind due to sweeps of other beneficial mutations and combinations. To avoid extinction, descendants of the novel mutation thus have to move to fitter genetic backgrounds via recombination in outcrossing events (Rice 2002). As a result the establishment probability decreases as the rate of average fitness gain, v, in the population increases. But the rate of average fitness gain or, equivalently, the rate of adaptation itself depends on the establishment probability. These two quantities therefore have to be determined self-consistently.In this article we analyze several models via self-consistent calculations of the fixation probability of new mutations. For a given production rate of beneficial mutations NU_b, we find that interference between mutations is of minor importance if the recombination rate r exceeds . In this regime, the rate of adaption is v ≈ NU_bs² as found for sequential mutations or in the absence of linkage. At recombination rates below , however, v grows only logarithmically with log NU_b. We find this behavior in all our models and argue that it obtains more generally. The prefactor of the log NU_b increases with the square of the recombination rate, implying a strong benefit of recombination in large populations.     

Increase in Quantitative Variation After Exposure to Environmental Stresses and/or Introduction of a Major Mutation: G × E Interaction and Epistasis or Canalization?

Why does phenotypic variation increase upon exposure of the population to environmental stresses or introduction of a major mutation? It has usually been interpreted as evidence of canalization (or robustness) of the wild-type genotype; but an alternative population genetic theory has been suggested by J. Hermisson and G. Wagner: “the release of hidden genetic variation is a generic property of models with epistasis or genotype–environment interaction.” In this note we expand their model to include a pleiotropic fitness effect and a direct effect on residual variance of mutant alleles. We show that both the genetic and environmental variances increase after the genetic or environmental change, but these increases could be very limited if there is strong pleiotropic selection. On the basis of more realistic selection models, our analysis lends further support to the genetic theory of Hermisson and Wagner as an interpretation of hidden variance.A common experimental observation in quantitative genetics is a higher phenotypic variance for quantitative traits in populations that carry a major mutation or are exposed to environmental stresses (e.g., heat shock) (Scharloo 1991; for a recent review see Gibson and Dworkin 2004). Part of the added variance must be genetic because the population responds to artificial selection. The lower variability of the wild type than that of the mutants has been interpreted as evidence for robustness or canalization (Waddington 1957): that is, under the new condition the magnitudes of gene effects across all trait loci increase relative to the original condition. The importance of canalization has been recognized for a long time and has been the subject of renewed interest recently (see de Visser et al. 2003 and Hansen 2006 for reviews).An alternative population genetic theory has been proposed by Hermisson and Wagner (2004), who suggest that the increase in genetic variance V_G after the change in environmental conditions or genetic background is a generic property of the population, with no need to introduce canalization (Waddington 1957). The theory appears simple. Under mutation–selection balance (MSB), the mutant alleles are at a selective disadvantage and there is a negative correlation between frequencies and effects of mutations: mutant alleles of small effects on the trait segregate at intermediate frequencies. After the change in genetic or environmental background, gene effects consequently change due to G × E interaction or epistasis, which reduces the negative correlation because genes that were previously of small effects and at intermediate frequencies may now have large effects. That is, the frequencies of alleles are determined by the previous MSB, while their new effects are at least partly determined by the new conditions. The genetic variance will therefore increase.Hermisson and Wagner (2004) found that the predicted increase in genetic variance can be substantial; however, the predicted increase is highly sensitive to the population size and can increase without bound with increasing population size (see their Figure 2 and Equation 16). Genetic variance would enlarge with the population size within a small population (Lynch and Hill 1986; Weber and Diggins 1990), but becomes insensitive to the population size within large populations (Falconer and Mackay 1996, Chap. 20). Hence the unbounded increase under the novel environmental condition appears to us as a downside of their theory, even though the predicted increase can be reduced if the changed environmental condition is not novel but there is previous adaptation to it (see their Figure 3).Open in a separate windowFigure 2.—Influence of the pleiotropic effect (s_p) on the increase of genetic variance Δ_G in units of the interaction parameter ξ for a “typical” situation with strength of stabilizing selection ω² = 0.1μ², mutation rate λ = 0.1 per haploid genome per generation, and population size N_e = 10⁶. The allelic pleiotropic effect on fitness and its variance effect on the trait independently follow gamma distributions with shape parameters β_s and β_v, respectively. The mean of a² across loci is E(v) = E(a²) = 10⁻⁴μ².Open in a separate windowOpen in a separate windowFigure 3.—Influence of shapes of distributions of mutational effects on (a) the variances at mutation–selection balance and (b) their increases after the genetic or environmental change. The squares represent the genetic variance and its increase and the triangles the environmental variance and its increase. The mutation rate is λ= 0.1 per haploid genome per generation, the population size is N_e = 10⁹, and the strength of real stabilizing selection is ω² = 0.1μ². Allelic effects on trait value (a), fitness (s), and residual variance (b) are assumed to be independently distributed such that v = a² follows a gamma () distribution with mean 10⁻⁴μ², s follows gamma (β_s) with mean s_p = 0.05, and b follows gamma (β_b) with mean 10⁻⁴μ².The basic model that Hermisson and Wagner (2004) employed is that the quantitative trait is under real stabilizing selection and mutant alleles have effects on the focal trait only by changing its so-called locus genetic variance. At the mutation–real stabilizing selection balance, some mutants can segregate at intermediate frequencies because of their small effects and therefore weak selection; and there are more such mutants the more strongly leptokurtic is the distribution of effects at individual loci. The unbounded increase of Hermisson and Wagner (2004) results from such a gene-frequency distribution; but it has been shown (see Barton and Turelli 1989; Falconer and Mackay 1996; Lynch and Walsh 1998) that solely stabilizing selection, whether modeled with a Gaussian (Kimura 1965) or a house of-cards approximation (Turelli 1984) or even the generalized form of Hermisson and Wagner (2004) (i.e., their Equation 14), cannot provide a satisfactory explanation for the high levels of genetic variance observed in natural populations under realistic values of mutation and selection parameters.A common observation is that one trait is controlled by many genes and one gene can influence many traits; i.e., pleiotropy is ubiquitous (Barton and Turelli 1989; Barton and Keightley 2002; Mackay 2004; Ostrowski et al. 2005). Recent detailed studies suggest that pleiotropy calculated as the number of phenotypic traits affected varies considerably among quantitative trait loci (QTL) (Cooper et al. 2007; Albert et al. 2008; Kenney-Hunt et al. 2008; Wagner et al. 2008). Such pleiotropic effects must influence the magnitude of the variance. Though some genes have little effect on the focal trait, they almost certainly affect other traits and therefore are not neutral. The inclusion of pleiotropic effects on fitness strengthens the overall selection on mutant alleles and, assuming such pleiotropic effects are mainly deleterious, maintains them at low frequencies. The genetic variance for a trait is therefore likely to be maintained at lower levels than that under only real stabilizing selection on the trait alone (Tanaka 1996). Although the gene-frequency distribution is much more extreme under this joint model, the relevant rate of mutation is genomewide and hence is much larger than that where mutation affects only the focal trait as is assumed in the real stabilizing selection model (Turelli 1984; Falconer and Mackay 1996). Taking into account empirical knowledge of mutation parameters, a combination of both pleiotropic and real stabilizing selection appears to be a plausible mechanism for the maintenance of quantitative genetic variance (Zhang et al. 2004). If pleiotropic selection is much stronger than real stabilizing selection, the association between frequency and effect of mutant alleles is weaker than that for a real stabilizing selection model. Further, if overall selection is stronger than recurrent mutation, the frequency distribution of mutant alleles will be extreme. Under those situations, the increase of genetic variance after the genetic or environmental change will be kept at lower levels than that of Hermisson and Wagner (2004), and hence the unbounded increase could be avoided.Further, Hermisson and Wagner (2004) assume that the environmental variance is not under genetic control (i.e., the variance of phenotypic value given genotypic value is the same for all genotypes) and therefore is not subject to change. This assumption conflicts with the increasingly accumulating empirical data that indicate otherwise (Zhang and Hill 2005; Mulder et al. 2007 for reviews). Direct experimental evidence is available that mutation can directly affect environmental variance, V_E (Whitlock and Fowler 1999; Mackay and Lyman 2005), and Baer (2008) provides what is perhaps the first clear demonstration that mutations increase environmental variances, on the basis of data for body size and productivity of Caenorhabditis elegans, and finds that the magnitudes of the increases are of the same order as those in the genetic variance.As real stabilizing selection on phenotype favors genotypes possessing low V_E (Gavrilets and Hastings 1994; Zhang and Hill 2005), a mutant that contributes little to V_E is more favored by stabilizing selection than one that contributes a lot. With all else being the same, mutants with small effect on V_E thus segregate at relatively high frequencies at MSB. That is, there is a negative correlation between the effect on V_E and the frequency of mutant genes. After the genetic or environmental change, some mutants that were previously of small effects on V_E have large effects due to G × E interaction or epistasis while their frequencies remain roughly the same as in the previous MSB. This certainly increases environmental variance.In this note, we first assume that mutant alleles can affect only the mean value of a focal quantitative trait and otherwise affect fitness through their pleiotropic effects (Zhang et al. 2004) and try to answer the following questions: How will the conclusion of Hermisson and Wagner (2004) be affected by taking into account the pleiotropic effect of mutants? Can the “unbounded increase” be avoided? We then further assume that mutant alleles can also directly affect the environmental variance of the focal trait (Zhang and Hill 2008) and investigate how both V_G and V_E change following the genetic or environmental change in the population.