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.
Conditionally expressed genes are defined as those genes in a population not expressed by every individual or in every generation. In particular, we examine three types of conditionally expressed genes: (1) those expressed by only a fraction, ϕ, of all individuals at every generation; (2) those expressed by all individuals but only for a subset of generations, g, over an interval of T generations, where ϕ = (g/T); and (3) those expressed by a fraction of the population over a subset of generations in an interval. In each case, the expressed fraction of all gene copies, ϕ, is subject to natural selection but the unexpressed fraction, (1 – ϕ), is affected only by the forces of mutation and drift. As a result, the strength of natural selection acting on conditionally expressed genes is weaker than it is on constitutively expressed genes. Examples of conditionally expressed genes include recessives (expressed only in homozygotes), genes with sex-limited effects (expressed only in one but not in the other sex), and inducible genes (expressed only or predominantly by individuals experiencing the inducing environments). Another type of conditionally expressed gene, “caste genes” expressed only in the sterile workers of the eusocial social insects, has been modeled by Linksvayer and Wade (2009).
Sequence polymorphism within populations
Expression in a fraction of individuals, ϕ, at every generation:
First, we consider the effects of deleterious mutations at a conditionally expressed locus. Throughout, we compare two loci with identical selective and demographic parameters, where one locus is conditionally expressed while the other is constitutively expressed. We begin by considering a haploid population for ease of demonstration, although all methods can be readily extended to diploids. Let there be two alleles, A and a, where A is wild type and a is deleterious. When not expressed, the fitness of the deleterious allele at a conditionally expressed locus, wa, is equal to 1; i.e., it is neutral because it is not expressed. When the mutation is expressed, its fitness becomes wa = 1 − s, where s is the strength of selection against the deleterious mutation manifest by the decreased viability of those carrying the gene. The average fitness of a conditionally expressed deleterious allele within a population consisting of ϕ expressers and (1 − ϕ) nonexpressers is . We denote the average selection against the allele as sc = ϕs, where the subscript, c, indicates the selection coefficient associated with the conditionally expressed allele.
Assuming large population size and small s, population genetic theory (e.g., Crow and Kimura 1970, pp. 58–62) has shown that, at mutation–selection balance, a constitutive deleterious mutant allele expressed in every individual in every generation will have an equilibrium frequency of , where u is the per-locus rate of mutation to the deleterious allele. In contrast, if conditionally expressed, a deleterious allele would have an equilibrium frequency of . Hence, the relative equilibrium allele frequency of a conditionally expressed gene to a constitutively expressed paralog is(1)Polymorphism at mutation–selection balance is , which is also approximately (1/ϕ) if we ignore terms of order (μ/s)2 (see below). By a similar derivation, it is easily shown that Equation 1 also holds for diploids with arbitrary dominance.
Expression by all individuals but only in a fraction, ϕ, of all generations:
Let ϕ be the fraction of generations, g, over an interval of T generations when the conditional gene is expressed (ϕ = g/T). With discrete generations, the average fitness of such an allele, wG, equals its geometric mean fitness (Crow and Kimura 1970; Frank and Slatkin 1990), which can also be seen from the multiplicative gene frequency recursion. Thus, we find that . This is equivalent to an average, per-generation selection coefficient, sc, equal to 1 − (1 − s)ϕ.
Following the same logic as above, we find that the relative equilibrium allele frequency of a conditionally expressed gene to a constitutively expressed paralog is(2)Numerical investigation shows that Rp is much more sensitive to ϕ than to s. That is, varying s by many orders of magnitude (1 × 10−11 < s < 1 × 10−1) results in a change in Rp of only 5% for any of a wide range of ϕ values (Figure 1A). Analytically, we can assume that s is small, so that (1 − s)ϕ ∼ 1 − sϕ. With this assumption, we see that sc ∼ s ϕ, as above, allowing us to reduce Equation 2 to(3)Our result conforms to the intuitive expectation that a conditional trait expressed half the time over an interval of T generations (ϕ = 0.50), or in half of the individuals all of the time, will accumulate twice as much mutational variation (Rp ∼ 1/0.50 = 2) as a constitutively expressed trait under approximately the same strength of selection and mutation pressure (Figure 1A).
Expression in some individuals, some of the time:
We can combine the two cases above into a single expression if we distinguish the two separate contributions to ϕ, as ϕI (the fraction of individuals within a generation) and ϕG (the fraction of generations). In this combined case, the average, per-generation selection coefficient, sc, equals 1 − (1 − s ϕI)ϕG. When s is assumed to be small, .
Here, the relative equilibrium allele frequency of a conditionally expressed gene to a constitutively expressed paralog is(4)For example, the standing polymorphism of genes with male-limited expression in facultatively sexual species, such as Caenorhabditis elegans (Chasnov and Chow 2002, Siedel et al. 2008) and the pea aphid, Acrythosiphon pisum (Brisson and Nuzhdin 2008), might well be represented by Equation 4. Chasnov and Chow (2002) discuss the poor mating efficiency of males in the hermaphroditic species, C. elegans, relative to dioecious species. They provide theoretical support for the hypothesis that deleterious mutations have accumulated in those genes with male-limited expression and this is responsible for poor male-mating efficiency. They argue further that even a very low frequency of functional males “can support male-specific genes against mutational degeneration” even in a species like C. elegans that is 99.9% hermaphroditic. Our Equation 4 shows that the relative degree of mutational degradation of genes with sex-limited or environment-limited expression can be quantified and equals a simple function of ϕ. For example, imagine that males appear only once every five generations (ϕG = 0.20) and, when males do appear, they represent only 5% of the population (ϕI = 0.05). Thus, relative to a constitutively expressed gene that is expressed in each individual, male and female, at every generation, we would expect a gene with male-limited expression in C. elegans to be at least 100 times more polymorphic at equilibrium (1/ϕIϕG = [1/.20][1/0.05] = 100). No adaptive explanation for high diversity in terms of frequency-dependent sexual selection or sex-ratio selection is necessary in this case. Or, differently put, polymorphism alone is not convincing evidence of frequency-dependent sexual selection; only polymorphism levels that exceed (1/ϕIϕG) (i.e., 100 times normal in this example) are evidence for such selection. In fact, it may well be the case that species of Caenorhabditis with infrequently expressed male genes are at risk of losing male function to mutation and relaxed selection as Chasnov and Chow (2002) suggested. Expression (4) may be useful in identifying a quantitative threshold of gene expression sufficient to engender the loss of males.
In all the cases above, the generations of neutrality experienced by conditionally expressed genes necessarily slow the removal of deleterious mutations by natural selection. Computer simulations show that periodic neutrality leads to equilibrium allele frequencies predicted by our analytical approximation (i.e., Equations 1 and 3), supporting our analytical results (Figure 1B; see methods). Significantly, Equation 2 is equivalent to the results obtained investigating maternal-effect genes, using an arithmetic mean selection coefficient of (s/2), as is appropriate for a trait expressed in half of the individuals within a single generation (i.e., the females) (Wade 1998; Demuth and Wade 2007). Thus, Equation 2 brings conditional gene expression between individuals within a single generation and conditional gene expression between generations into a single, general framework. However, an explicitly spatial model of selection on a conditionally expressed trait is required to validate this result.
Migration in a spatially heterogeneous environment:
When the cause of conditional expression is spatial heterogeneity in the distribution of trait-inducing cues or of selection pressures across the environment, a spatially explicit model is required to accurately specify the dynamics of natural selection. Two types (or “grains”) of spatial heterogeneity exist, fine grained and coarse grained (Levins 1968). In a fine-grained environment, individuals experience multiple environments throughout their lifetime, while in a coarse-grained environment an individual completes its entire life cycle in a single habitat (Levins 1968). Levins (1968) showed that selection in a fine-grained environment is governed by the arithmetic mean selection coefficient over habitat types, while in a coarse-grained environment selection is governed by the geometric mean selection coefficient. Alternatively, Nagylaki (1980) showed that, in a (coarse-grained) spatially heterogeneous environment with migration between habitat patches, the outcome of natural selection is determined by the arithmetic mean selection coefficient over individuals in the population as long as migration between habitats is sufficiently high (e.g., the “high migration limit”).
As we showed above, the geometric and arithmetic mean fitnesses are approximately equivalent for conditionally expressed traits, meaning that we do not necessarily need to make explicit assumptions about the pattern of environmental heterogeneity. The question, then, becomes at what value for migration is the high-migration limit achieved for conditionally expressed genes?
To answer this, we consider for simplicity a haploid population divided into two subpopulations, S and N, representing selective and nonselective habitats, respectively. The subpopulations are composed of different numbers of individuals, NS and NN, such that ϕ is defined as the fraction of the total population occupying the selective habitat, ϕ = NS/(NS + NN). The life cycle of an individual consists of migration → selection → reproduction/mutation, with selection occurring only in habitat S. We assume irreversible, recurrent mutation from the wild-type to the deleterious allele in accordance with the infinite sites model (Watterson 1975). For simplicity, we also ignore the effects of drift.
We assume that migration is conservative, meaning that the number of migrants sent out from each subpopulation in each generation is equal, e.g., when organisms assort in space according to the ideal free distribution (Fretwell 1972; Rice 2004). In the current model, this would be the case if the subpopulation experiencing selection is occupying a marginal habitat. Letting m1 equal the fraction of subpopulation S composed of immigrants from subpopulation N in each generation (e.g., the “backward migration rate”), we have, NSm1 = NNm2. Defining ψ = NN/NS gives m1 = ψm2. The parameter ψ is related to ϕ by ϕ = 1/(1+ ψ) or, equivalently, ψ = (1/ϕ) − 1. Noting that the mean fitness in the selective habitat is w̄S = 1 − sqS, at migration–mutation–selection equilibrium, the frequency of deleterious mutant alleles in the selective and neutral subpopulations, respectively, is exactly(5a)(5b)Ignoring terms of second order or higher in s and u, substituting ψm2 for m1, and using the relation ϕ = 1/(1 + ψ), we find that(6a)Assuming that m2qS ≫ u, Equation 5b gives,(6b)Thus, with conservative migration between selective and nonselective habitats, the equilibrium frequency of deleterious mutants in the total population is approximately equal to the value we find for conditionally expressed genes in our unstructured models.
Equations 5a and 5b can be used to determine the conditions under which the analytical approximation leading to Equations 6a and 6b hold. This is done by choosing an arbitrary accuracy threshold, say 10%, and then setting the ratios (Equation 5a/Equation 6a and Equation 5b/Equation 6b) equal to 0.9 or 1.1 (depending on whether the approximate result under- or overestimates the exact result, respectively) and solving for m1. The result is a curve that dissects the parameter space into a region where the approximation is valid to within our chosen threshold and a region where it is not. These curves are independent of the selection coefficient because s cancels when dividing Equation 5a by Equation 6a. Figure 2A plots the range of parameter values under which Equation 6a holds to within 10% of the exact result of Equation 5a for u = 10−4, 10−5, 10−6. The approximation is valid over a wider parameter range when the mutation rate is low, and in general, the approximation of Equation 6a is valid even when migration rates are quite small (Figure 2A). While the accuracy of Equation 6a is independent of the selection coefficient, this is not true for Equation 6b. Approximation (6b), however, is less sensitive to mutation rate and more sensitive to the value of the selection coefficient. Figure 2B shows how the range of parameter values under which Equation 6b holds is greater for smaller selection coefficients. Numerical investigation shows that Equation 6b will hold to within η% as long as m1 > s/η, although the exact value depends on ϕ (see Figure 2B).
Figure 2C shows the range of parameter values under which both Equations 6a and 6b hold to within 10% of the exact result of Equations 5a and 5b for s = 0.01 and u = 10−4. This figure is divided into three regions: in the region below the lower curve both approximations fail; between the curves, Equation 6a holds but 6b does not; and above the upper curve both hold to within 10% of the exact results. The parameter range where Equation 6a holds is much more extensive than that where Equation 6b holds (Figure 2C), suggesting that in nature there will often be substantial differences in allele frequencies measured between habitat types. Thus, the outcome of genetic analysis of samples from natural populations will be sensitive to the sample locations across a selectively variable environment.
It can be seen from Figures 2, A–C, that, as ϕ gets smaller, the above approximation requires higher migration rates to be valid. In addition, Figures 2A and 2C show that the parameter range for Equation 6a to hold is smaller for higher mutation rates. With realistic per-gene rates of deleterious mutation (u = 10−6), the approximation holds for a large range of ϕ values even when migration is quite limited (clearly seen in Figure 2B). Significantly, Kawecki (1994) derived Equation 6a by a different route, but under the assumption of complete random migration in every generation (m = 0.5); our results validate Equation 6a for a less restrictive parameter range with conservative migration.
Stochastic simulations testing the validity of our analytical results
To test the validity of the approximations leading to Equations 6a and 6b, we conducted extensive individual-based stochastic simulations of a population subdivided into selective and neutral subpopulations with migration between them (Figure 2B; see methods). Sample outputs of individual simulation runs are given in Figure 2D. Figure 2D plots the outcome of individual runs for , and with parameter values u = 0.001, s = 0.01, m1 = 0.1, and NT = 50,000.
Simulation results are given in Table 1, and plotted in Figure 2C (see below). Three simulations were run for each set of parameter values, and the mean and standard error of these runs are reported in Table 1. We note that the standard deviations of allele frequency are larger for lower ϕ values (and that the fluctuations of allele frequencies in Figure 2D are greater for lower values of ϕ). This is a consequence of the smaller census size of the selected subpopulation when ϕ is small, causing drift with respect to natural selection to become stronger. We chose parameter values that specifically test the validity of the curves in Figure 2C: (1) open circles are parameter values (m1 and ϕ) where Equations 6a and 6b did not hold; (2) solid triangles indicate where Equation 6a but not Equation 6b holds; and, (3) solid squares indicate where both hold. The simulation results match the analytical predictions, giving robust support for our approximations.
Effect of conditional expression on the distribution of allele frequencies under drift, mutation, and selection:
We now introduce random genetic drift into our deterministic analysis and extend the model to diploidy to explicitly account for dominance. With drift the equilibrium state of the population is a probability distribution of allele frequencies, rather than a single equilibrium allele frequency as above. The distribution of allele frequencies under mutation, drift, and natural selection is given by the diffusion approximation (Kimura 1962)(7)
The terms M(x) and V(x) in Equation 7 represent the mean and variance in allele frequency change per generation, respectively. Expressions for M(x) and V(x) can be obtained using standard population genetic formulas (e.g., Crow and Kimura 1970). M(x) incorporates directional causes of allele frequency change such as selection and mutation, while V(x) incorporates nondirectional sources such as drift or temporal variation in selection intensity (Kimura 1962; Ewens 1979). Following Kimura (1962), we define(8a)(8b)where x is mutant allele frequency, u1 and u2 are mutation rates from and to the deleterious allele, respectively, h is degree of dominance (for h = 0 the mutation is recessive, h = 1 dominant), Vs is the variance in the selection coefficient s, and Ne is the effective population size. The first term on the righthand side of Equation 8b accounts for variance in allele frequency change owing to variance in selection intensity, while the second term accounts for the effects of random drift. For conditionally expressed genes,(9)Even for relatively strong selection (e.g., 4Nes > 1), s2 will be on the order of , and the product ϕ(1− ϕ) ≪ 1 for many values of ϕ. Given this, the first term of Equation 8b is negligible and is thus omitted throughout for simplicity without appreciable loss in precision. Substituting Equation 8 into Equation 7, setting h = ½ (e.g., the mutation has additive effects on fitness), and evaluating x at q, we find,(10)where C is a normalizing constant that ensures that the area under the distribution equals one.
Equation 10 suggests that conditional expression increases the strength of genetic drift at a locus (Figure 3). When Ne is small (Figure 4A), conditional expression flattens the allele frequency distribution consistent with increased neutrality so that there is greater probability that deleterious alleles can reach high frequency or even fix in the population owing to conditional expression. With large Ne (Figure 4B), the distribution shifts to the right with decreasing ϕ and is centered about the deterministic expectation, independently corroborating the deterministic results (Equation 6).
DNA sequence polymorphism under conditional expression:
We need to extend the theory to use sequence data from natural populations to test its predictions. Sequence data can be used to measure the nucleotide diversity parameter, π (Nei and Li 1979; Tajima 1983), a commonly used measure of nucleotide polymorphism. π equals , where qi is the frequency of an allele at site i, and the sum is taken over all polymorphic sites. For rare and deleterious mutations, qi is small, and ignoring terms of order gives the standard approximate result, . With the approximation sc ∼ sϕ and the standard assumptions of large population size and independence among sites, we can substitute the equilibrium allele frequency under mutation–selection balance (i.e., Equation 6) for qi to find the expected replacement site diversity, πAC, for a conditionally expressed trait at equilibrium: πAC ∼ (1/ϕ)∑2u/s. Under the same assumptions, the expression for a constitutively expressed gene is . Taking the ratio of diversities, Rh, we again obtain Equation 2:(11)However, Equation 11 will not hold for the ratio of polymorphism when there is drift, when some nonsynonymous sites evolve neutrally, or when ϕ becomes very small. The latter case occurs because, as ϕ becomes small, q → 1, which clearly violates the assumption of small q. Although a full treatment of the effects of ϕ on nucleotide diversity in a finite population is beyond the scope of this article, we can derive the following general results.
The expected replacement (i.e., nonsynonymous) site nucleotide diversity at a locus under selection and drift is given by(12)where cn is the fraction of neutral sites, θ = 4Neu (in a diploid population), and HP is the mean equilibrium diversity at nonneutral sites (Loewe et al. 2006). Neutral sites are those sites where − 1/(4Ne) < s < 1/(4Ne) (Ohta and Kimura 1971). In the case of a conditionally expressed trait, this inequality becomes − 1/(4Neϕ) < s < 1/(4Neϕ). Since ϕ is a fraction between 0 and 1, it is clear that a conditionally expressed gene will always experience random genetic drift (i.e., approximate neutrality) more strongly than a constitutively expressed gene. Put differently, conditional expression increases the neutral range of mutations by reducing Ne to ϕNe. The increased strength of drift with lower values of ϕ is seen in stochastic simulations, manifesting as an increase in the variance of mean allele frequency across replicated simulations (Table 1).
If we define Ψ(s) as the distribution of fitness effects (DFE) of new mutations (Eyre-Walker and Keightley 2007), then the fraction of neutral sites for a conditionally expressed trait is given by the equation(13)In the limit as ϕ → 0, 1/(4Neϕ)→ ∞, and −1/(4Neϕ) → −∞, giving cn = 1 in the limit by the definition of a probability distribution. Thus, as ϕ becomes small, the nonsynonymous site polymorphism at a locus approaches the expectation for complete neutrality, πA = θ = 4Neu. As expected, conditional expression increases the neutral range of mutations, and in the limit, genes that are never expressed should become pseudogenized by mutation.
Unlike the deterministic case, the upper limit on polymorphism set by the neutral expectation (πA = θ) prevents the ratio of polymorphisms of conditional to constitutive genes (Rh) from increasing without bound as ϕ becomes small. We now derive an equation for Rh that takes this into account. We consider the case where there are two classes of nonsynonymous sites: neutral and selected (this is the most extreme case, because it results in the highest variance in s among sites, and thus places an upper bound on the possible effect of Vs). As selection is relaxed by conditional expression, previously selected sites will become neutral (Equation 13) and will have a level of polymorphism equal to the neutral expectation, θ. We define πs as the polymorphism at sites under selection due to mutation–selection balance, cn1 and cn2 are the proportion of neutral nonsynonymous sites at conditional and constitutive loci, respectively, and ρ = θ/πs. With the assumption of two discrete types of nonsynonymous sites, . Applying these assumptions and definitions to Equation 12, taking the ratio of conditional πA to constitutive πA, and rearranging gives the relative polymorphism of a conditional gene,(14)When cn1 = 1 (e.g., all nonsyonymous sites at the conditional locus evolve neutrally), Equation 11 reduces to Rh = θ/πA as expected (where πA is given by Equation 12).
The effect of ϕ on Equation 14 depends on the DFE (through the parameters cn1 and cn2) and the relative magnitude of πs and θ. To investigate Equation 14, we define the DFE as a Gamma-distribution with shape parameter α = 0.25, which accords with empirical results from a number of species (Eyre-Walker et al. 2006; Loewe et al. 2006; Eyre-Walker and Keightley 2007). Figure 4A shows that the fraction of neutral nonsynonymous sites at a conditionally expressed locus (cn1) depends on Ne, as expected (see Equation 10), and that it is sensitive to ϕ only when ϕ is small. Figure 4B shows that the deterministic value of Rh (Equation 11) is a good approximation of Equation 14 when Ne is large, but that it consistently overestimates Rh for smaller Ne. The degree of overestimation depends on Ne because with small Ne drift is stronger at selected sites causing πA to approach θ, such that Rh → 1 in the limit when drift overwhelms selection at all nonsynonymous sites. For realistic population sizes (Ne > 103), however, Equation 11 is a reasonable approximation for the effect of conditional expression on sequence polymorphism.
Sequence divergence between populations or species
We extend our results for polymorphism within species to sequence divergence among species, which is determined by the fixation process. The probability of fixation of an allele in a population experiencing selection and drift is given by the diffusion approximation (Kimura 1962)(15)where(16)The terms M(x) and V(x) are given above by Equation 8. As above, we assume that Vs is negligible because it is typically many orders of magnitude smaller than drift variance.
We combine Equations 8a, 8b, 15, and 16 and ignore terms of order q2 because q = ½N for a new mutation (Rice 2004). Making this approximation and setting q = ½N, we find that the probability of fixation of a new conditionally expressed mutation is approximately(17)
Equation 17 was also found by Whitlock (1996). However, this is an approximate result that fails for completely recessive mutations (where h = 0). We obtain exact results by numerical integration of Equation 15 to determine the effect of dominance on divergence of conditionally expressed loci. We are interested in the relative probability of fixation for a conditionally expressed mutation, which is the ratio of fixation probabilities of conditional to constitutive mutations. This ratio, Rf, is plotted against ϕ for dominant (h = 1), additive (h = ½), and recessive (h = 0) mutations for constant values of Ne and s (see Figure 5A). Figure 5 demonstrates that degree of dominance has little effect on the relative fixation probability when ϕ > 0.5. However, when ϕ is small, dominant mutations are disproportionately affected by conditional expression because deleterious recessive mutations are always hidden from selection when rare, whereas dominant mutations are not. Conditional expression, then, provides a selective reprieve for dominant mutants that they normally do not receive, making them more sensitive to infrequent expression.
Figure 5B plots the ratio of fixation probabilities against the strength of selection (Nes) for the case of dominant mutations (h = 1) and shows that conditional expression greatly increases the probability of fixation of deleterious mutations. On the other hand, the spread of beneficial mutations (s > 0) in conditionally expressed traits is slowed (see also Whitlock 1996). To show this latter result analytically, we assume that Ne = N as per Haldane (1927) and take the limit of Equation 14 as Ne → ∞. We find the probability of fixation of a dominant, beneficial mutation (s > 0) is (for noninfinitesimal ϕ)(18)analogous to Haldane's (1927) classic result that the probability of fixation is twice the selective benefit. Taking the ratio of Equation 18 to the classic result of Haldane (1927), who assumed constant s, we find(19)That is, conditionally expressed genes suffer a reduction in probability of fixation directly proportional to ϕ, the frequency of gene expression (Figure 5B), slowing the rate of adaptive divergence between species.
The combined effect of increased probability of fixation of deleterious mutations and a decreased probability of fixation of beneficial mutations should be a net increase in overall sequence divergence between species at conditionally expressed loci, since the former is believed to be orders of magnitude greater than the latter. The logic is as follows: the relative probability of fixation of slightly deleterious alleles increases exponentially with decreasing ϕ (Figure 5B). However, the probability of fixation of beneficial mutations decreases linearly with the frequency of trait expression, ϕ. The distribution of fitness effects of new mutations is highly skewed toward deleterious mutations (Eyre-Walker and Keightley 2007), so that the relaxation of selective constraint vis à vis purifying selection outweighs its effect on adaptive substitution, increasing the rate of sequence divergence among species.
The plasticity load and the mutational cost of complexity
The mutational load is the decrease in mean population fitness due to recurrent deleterious mutation (Muller 1950). In a haploid population, with a deleterious allele, in frequency q, the mean fitness of a population is w̄ = 1 − qs. Substituting the allele frequency at mutation–selection balance, q̂ (Equation 6), for q gives, w̄ = 1 − u. Thus, the mutational load is equal to the deleterious mutation rate (Muller 1950).
For a conditionally expressed locus, the mean population fitness is . Thus, conditional expression does not affect the mutational load over the total population. However, if we consider the mutational load only in generations or habitats where the gene is expressed, then s is not diminished by averaging over nonexpressive conditions and we have , where the asterisk denotes marginal fitness (e.g., the fitness only in the selective environment). Defining the load, Lc, as , we have(20)We refer to this as the plasticity load. For diploids, the same methods lead to Lc = 2u/ϕ. For haploids, the plasticity load is maximal (i.e., Lc = 1) when ϕ = u, which corresponds to complete mutational decay. Equation 20 demonstrates the connection between the accumulation of deleterious mutations under conditional expression (e.g., Equation 6) and the resulting decline in adaptedness of the population to the conditional environment. Kawecki (1994) and Whitlock (1996) derived the equivalent of Equation 20 for a spatially subdivided population with random migration between habitat patches. Our methods extend these results to any conditionally expressed locus (conditional in time and space). Furthermore, because the ϕ approximation is accurate in the selective environment over a much wider range of parameter values than it is over the total population (see Figures 2, A and B), our results for the plasticity load hold much more generally than suggested by Kawecki (1994) and Whitlock (1996).
We can extend these results to account for the load experienced when more than a single locus contributes to fitness. Consider multiple nucleotide sites contributing multiplicatively to total fitness such that W = Π(1 − qs). Taking the logarithm of both sides, using the approximation log(1 + x) ∼ x for small x, and taking the exponent gives W = e−nu and L = 1 – e−nu, where n is the total haploid number of sites at the locus under consideration and u is the average mutation rate over these sites (for diploids, W = e−2nu; e.g., Gillespie 2004). The mutational load for a conditionally expressed trait when expressed is(21)
A population expressing a trait conditionally will have a lower mean fitness than a population of specialists that express a trait unconditionally by a factor of(22)Thus, specialists will always be better adapted to the conditional environment than nonspecialists, even when antagonistic pleiotropy is not a factor, as long as specialism involves suites of genes that are conditionally expressed (Kawecki 1994; Whitlock 1996). Since Equations 21 and 22 depend on the product nu, the cost to plasticity relies heavily on the number of loci coding for the plastic phenotype. More elaborate plastic phenotypes, then, are more evolutionarily constrained by mutation than simple plastic phenotypes as demonstrated in Figure 6 by the different curves corresponding to different values of nu.
Temporal variation simulation:
To simulate temporal fluctuations in gene expression, we carried out a deterministic simulation in Mathematica (Wolfram), using the standard haploid allele frequency recursion, q = q(1 − s)/W̄ + u(1 − q) (Crow and Kimura 1970), where the mutation rate, u, was set at 0.001. Periodicity of selection was simulated by switching the value of the selection coefficient, s, between 0 and 0.01 at a frequency determined by ϕ. For instance, for ϕ = ½, s alternated between 0 and 0.01 every other generation. For ϕ = 1/10, s = 0 for every generation except generations that were multiples of 10, when s would become equal to 0.01. The results of the simulation are plotted in Figure 1B for multiple values of ϕ. Simulations were run for 10,000 generations, with data points sampled and plotted every 100 generations.
Spatial variation simulation:
To simulate spatial variation in gene expression, we carried out a stochastic simulation in Mathematica. A population of NT individuals was divided into a selective and a neutral subpopulation, each population composing an array of individuals denoted by 0 for wild type and 1 for mutant, where NS = ϕNT individuals composed the selective subpopulation and NN = (1 − ϕ)NT the neutral subpopulation. Individuals were sampled at random with replacement within each subpopulation in accordance with the Wright–Fisher model, and wild-type individuals were mutated with probability u before being placed into the next generation. All randomly chosen individuals in the neutral population were placed into the next generation, whereas mutant individuals in the selective habitat were retained with probability (1 − s). Generations were discrete such that a “generation” consisted of sampling, selection (in the selective habitat) and mutation until NS and NN individuals occupied the next generation in the selective and neutral habitats, respectively. To simulate migration, at the end of each generation, mNS individuals from each subpopulation were chosen at random and exchanged between subpopulations, where m is the migration rate. This constitutes “conservative migration,” as was assumed in our analytical results. All simulations began with zero mutants, except for the simulation for plotted in Figure 2C, which was initiated with q = 0.5, in order that it reached equilibrium within 10,000 generations. Each simulation was run for 10,000 generations. The parameter values for Figure 2C were u = 0.001, s = 0.01, m = 0.1, and NT = 50,000.
For the mean and standard deviation of allele frequency reported in the text, each simulation was run three times, and the allele frequency from each run was computed as the mean allele frequency of the last 5000 generations.
Our results show that spatial or temporal fluctuations in the induction of gene expression generate relaxed selective constraint, increasing the level of standing polymorphism within a species and the rate of nonadaptive divergence between species. Specifically, relative to genes expressed in all individuals all the time, conditional expression increases polymorphism by a factor of approximately (1/ϕ), where the approximation is closer for populations of large effective size. This factor is the inverse of ϕ, the fraction of gene-expressing individuals. When gene expression is both spatially (ϕI) and temporally (ϕΓ) conditional, the equilibrium polymorphism at a locus is proportional to the product of the inverses, (1/ϕIϕG). We show that the assumption of random mixing between selective and neutral habitats every generation is not necessary and that using the mean selection coefficient (sϕ) for a conditionally expressed locus in a spatially variable population is valid even for small rates of migration (Figures 2, A–C). In general, our theory connects gene-level relative polymorphism and divergence with the spatial and temporal frequency of environments inducing gene expression.
Conditionally expressed genes suffer increased mutational and drift loads due to the increased maintenance and fixation of slightly deleterious mutations. As suggested previously (Kawecki 1994, 1997; Whitlock 1996; Kawecki et al. 1997), this poses a serious constraint on the elaboration and maintenance of phenotypically plastic traits. In contrast to other hypotheses for constraints on phenotypic plasticity such as antagonistic pleiotropy and energetic costs (reviewed in DeWitt et al. 1998), mutation accumulation due to relaxed selection is a necessary consequence of conditional expression experienced by any conditionally expressed gene. Our theory provides quantitative predictions for how such effects can be observed in gene sequence data. Furthermore, Equation 22 demonstrates that the constraint is greater not only for more infrequently expressed genes, but for traits encoded by a larger suite of genes. If complexity is measured by the number and diversity of distinct morphs encoded by a single genotype, then mutation accumulation due to relaxed selection imposes a potentially severe cost to complexity, limiting the number and elaborateness of distinct morphs in polyphenic species.
Relaxed selection has been proposed as a cause of the observed stability of species' niches over long periods of geologic time (“niche conservatism,” Holt and Gaines 1992; Holt 1996). Our theory extends these previous results by demonstrating how parameter values affect the predicted plasticity load. According to Figure 2A, the plasticity load will exist even with relatively low rates of migration between selective and nonselective habitats. The level of migration necessary to eliminate the load may be low enough to facilitate speciation as suggested by Kawecki (1997).
Importantly, the relaxed selection described here is not experienced by genes expressed conditionally within a single individual's lifetime. In many cases, genes are expressed with varying intensity across tissues or cell types or at different life stages in an individual's lifetime. If there is a relationship between the intensity of gene expression and the intensity of selection (i.e., the magnitude of s), then it might well be possible to extend our theory. If the relationship between expression level and selection was given by the linear regression coefficient, β, then two genes with different expression levels could be compared for standing polymorphism and divergence using our theory by substituting β for ϕ. Recent empirical work has shown that levels of gene expression within an individual are correlated with rates of divergence (Lemos et al. 2005) suggesting that such relationships may exist and permit tests of our theory. Similarly, if the tissue specificity of gene expression could be related to strength of selection, predictions regarding the relative rate and diversity of evolution of one gene to another could be made. This is an active area of current research and it is too early to determine whether such relationships will be discovered and whether, if discovered in one species, they hold in another.
Applications of the Theory and Tests of its Predictions:
Our theory is supported by previous studies, which serve as a guide for future testing. One way of testing the theory is to apply the methods used in testing the predictions of maternal effects theory (Wade 1998; Linksvayer and Wade 2009), which is a special case of the current model with ϕ = ½. Theory predicts that, all else being equal, maternal effect genes should experience relaxed selection relative to genes expressed in both sexes. The specific predictions are: (1) standing variation for maternal effect genes should be twice that of genes expressed in both sexes, and (2) maternal effect genes should be differentiating more rapidly across taxa. Barker et al. (2005) tested these two predictions in two species of fruit flies, using the tandem duplicate gene pair bicoid (bcd, maternal expression) and zerknult (zen, zygotic expression in both sexes). For the ratios of (bcd/zen) polymorphism, they found it to be 1.08 for silent sites and 2.18 for nonsynonymous sites within Drosophila melanogaster, and 0.89 and 2.68 for silent and nonsynonymous sites, respectively, in Drosophila simulans. Both ratios provide quantitative support for the theory predictions. In a broader test of the theory, Cruickshank and Wade (2008; Figure 3, A and B) compared sequence diversity within D. simulans and sequence divergence among several species for 42 genes, critical to early embryo development (9 maternal and 33 zygotic). For five maternal-zygotic homologs, like bcd and zen, they observed a nonsynonymous site diversity to average 0.55 ± 0.22 SE for the maternal genes and 0.28 ± 0.15 SE for the zygotic genes for a ratio of 1.96, very nearly identical to the theoretically predicted ratio of 2.0. Across species, Cruickshank and Wade (2008, Figure 2) found that “…the average level of between-species sequence divergence ranges from two to four times greater for maternal than for zygotic genes.” These empirical findings also quantitatively support the theoretical predictions.
In some species, sex-limited gene expression may result in values of ϕ that differ from ϕ = ½. In parthenogenic species with facultative sexuality, male-specific genes experience a frequency of expression such that ϕ < ½. Brisson and Nuzhdin (2008) identified male-specific, female-specific, and sex-neutral genes in the facultatively sexual pea aphid, A. pisum, and measured levels of sequence divergence between species for these classes of genes. This species reproduces clonally for 10 to 20 generations before producing a sexual generation composed of males and females in the fall (Moran 1992; Brisson and Nuzhdin 2008). Consistent with the predictions of relaxed selective constraint, male-specific genes had more than a twofold increase in divergence compared to female-specific and sex-neutral genes, which had approximately the same levels of divergence, and tests of selection showed that this variation accumulated neutrally (Brisson and Nuzhdin 2008). The pattern of expression for this species gives a frequency of expression of between ϕ = 0.05 and ϕ = 0.1. While our theory shows that the level of standing polymorphism at equilibrium should be inflated 10- to 20-fold, the consequence for divergence is not as straightforward. From Equation 17 and Figure 5, a ϕ value or 0.1 will produce a 2-fold increase in divergence if the mutations accumulated experience an average selection intensity of Nes ∼ −1.3, although this requires the assumption that deleterious mutations are dominant. Further empirical work on the effect sizes of spontaneous mutations in this species, or estimation of Ne and comparison with mutational effect sizes from other species, will be necessary to determine if these data support our theory. This highlights the fact that less data are required to apply our theory to polymorphism data than to divergence data. While divergence data are sensitive to the selection coefficient and dominance, polymorphism data are not. Unfortunately, at present, polymorphism data are not as easily obtained as divergence data.
With polymorphism data, our theory can be used in multiple ways. Given a known value of ϕ, a null expectation for levels of sequence polymorphism can be obtained. Our theory shows that null expectations should always be adjusted to account for the effects of the frequency of expression. Alternatively, the parameter ϕ can be estimated from the data, or if ϕ is known, differences in selection pressures can be measured between paralogs or classes of genes. The two major barriers to such an analysis are the availability of polymorphism data and of expression data across individuals.
The results presented here for conditionally expressed genes can also be applied to social genes. Recent theory (Linksvayer and Wade 2009) hypothesizes that genes with indirect genetic effects or genes whose primary fitness effect is at the level of the social group (called “kin selection” genes) should experience relaxed selective constraint in a manner analogous to conditionally expressed genes. In the Linksvayer–Wade theory, the coefficient of relatedness, r, between the individual expressing the gene and the group experiencing its fitness effects, plays the same role as ϕ in our theory for conditionally expressed genes. The quantitative predictions developed here, then, can be applied with modification to kin selection genes.
While we focused primarily on relaxed selection as an evolutionary constraint on conditionally expressed traits, we cannot discount the possibility that much of the accumulated mutational variation at conditional loci may provide the raw material for adaptive evolutionary change under appropriate conditions. The increased level of standing genetic variation caused by relaxed selection may facilitate the rapid emergence of evolutionary novelties when formerly rare, low ϕ environments become more common. Further, the larger neutral range experienced by conditionally expressed genes may allow populations to explore previously inaccessible regions of sequence space, allowing populations to escape local optima in favor of globally higher fitness peaks. The possibility of such adaptive peak shifts (Wright 1931) is of considerable medical relevance since virulence and antibiotic resistance are both conditionally expressed traits, and since all new host species and drug therapies begin as rare, low ϕ environments. Similarly, given the broad but predictable patterns of climate change associated with global warming, it is likely that, at many latitudes, formerly rare environments will become more common. Conditionally expressed genes, then, may contribute disproportionately to the evolution of novel form and function under appropriate ecological and population genetic conditions.
We thank M. Feldman, M. Hahn, and M. Lynch for helpful discussions and Y. Brandvain, T. Cruickshank, T. Platt, and P. Zee for helpful comments on the manuscript. This work was supported by National Institutes of Health grant R01 GM65414 to M.J.W. and a National Science Foundation Integrative Graduate Education and Research Traineeship (NSF IGERT) Fellowship to J.D.V.D.
Communicating editor: M. W. Feldman
- Received September 23, 2009.
- Accepted December 1, 2009.
- Copyright © 2010 by the Genetics Society of America