GENETIC VARIATION AND THE EVOLUTION OF EPIGENETIC REGULATION

Authors


Abstract

Epigenetic variation has been observed in a range of organisms, leading to questions of the adaptive significance of this variation. In this study, we present a model to explore the ecological and genetic conditions that select for epigenetic regulation. We find that the rate of temporal environmental change is a key factor controlling the features of this evolution. When the environment fluctuates rapidly between states with different phenotypic optima, epigenetic regulation may evolve but we expect to observe low transgenerational inheritance of epigenetic states, whereas when this fluctuation occurs over longer time scales, regulation may evolve to generate epigenetic states that are inherited faithfully for many generations. In all cases, the underlying genetic variation at the epigenetically regulated locus is a crucial factor determining the range of conditions that allow for evolution of epigenetic mechanisms.

Recent observations of epigenetic variation hint that many populations have a nongenetic supply of heritable variation that may facilitate rapid phenotypic adaptation. The transgenerational inheritance of epigenetic variation has been observed in plants (Bender and Fink 1995; Johannes et al. 2009; Reinders et al. 2009; Verhoeven et al. 2010), animals (e.g., in the nematode Caenorhabditis elegans; Greer et al. 2011, and mammals; Daxinger and Whitelaw 2012), yeast (Acar et al. 2005; 2008; Levy et al. 2012), bacteria (Lim and van Oudenaarden 2007; Maamar et al. 2007; Beaumont et al. 2009), and viruses (Stumpf et al. 2002). However, it is unclear what ecological and genetic conditions should select for the initial evolution of epigenetic regulation.

An important evolutionary consideration concerns whether the epigenetic variation is rooted in genetic variation. Johannes et al. (2009) demonstrate an elegant approach to exploring meiotically inherited epigenetic variation that is independent of genetic variation. By establishing a population from two parental lines that differed genetically at the DDM1 gene, the authors could explore the influence of inherited epigenetic states on phenotypic heritability. DDM1 is involved with the maintenance of DNA methylation, and mutants lacking a functional allele have significantly lower DNA methylation across the genome. Even after backcrossing that generated progeny with a functional DDM1 allele, epigenetic variation among individuals persisted. They called this a population of epigenetic Recombinant Inbred Lines (epiRILs). By performing selection experiments for a variety of phenotypes, Johannes et al. (2009) were able to estimate their heritabilities, finding them to be surprisingly high despite the presumably very low levels of genetic variation between individuals. This raises the possibility that adaptation could occur via selection on epigenetic variation.

Although there is little research that shows simultaneous epigenetic inheritance and adaptive epigenetic effects, several studies provide clues as to how such a system may operate. Herrera et al. (2012) studied the growth of single genotype lines of the nectar-living yeast Metschnikowia reukaufii in a variety of sugar compositions and concentrations. They found consistent methylation differences among the yeast in different conditions, and when yeast DNA methylation was inhibited by the chemical 5-azacytidine, lines were less able to proliferate in any media containing sugar. In this system, DNA methylation appeared to be the mechanism underlying the adaptive phenotypic plasticity. However, the research did not ascertain whether these epigenetic changes could be transmitted across meiotic generations. Epigenetic differences have also been found among wild violets (Viola cazorlensis) that experience different levels of herbivory (Herrera and Bazaga 2011), and between prickly and nonprickly leaves of the holly Ilex aquifolium (Herrera and Bazaga 2013). Simulating herbivory and other stresses in a laboratory experiment on clonal dandelions, Verhoeven et al. (2010) found that the epigenetic changes induced by these stresses could last for many generations.

Considering the possibility that epigenetic variation could provide a source of adaptive phenotypic variation, we must ask why there is a need for epigenetic variation. Why is genetic variation alone an insufficient source of inherited variation? In what follows we specify the type of epigenetic variation in which we are interested. Some observed epigenetic variation may stem from genetic variation; we do not investigate this here because in such a case the epigenetic states are simply phenotypes caused by a particular genotype. Instead, we consider variation to be epigenetic when there is considerable within-genotype variation in epigenetic state. Even if genetic variation influences epigenetic processes, the epigenetic states may evolve without complete dependence on genetic state.

Theoretical studies of epigenetics, as well as studies oriented toward stochastic or sensed phenotypic switching, reveal some scenarios in which epigenetic regulation may evolve. Models of epigenetic inheritance have shown that parent–offspring correlation in phenotype depends heavily on the rates of epigenetic switching (epimutation) between generations (Bonduriansky and Day 2009; Slatkin 2009; Tal et al. 2010; Danchin and Wagner 2010; Danchin et al. 2011; Furrow et al. 2011). Because these models generally did not incorporate natural selective forces, the adaptive consequences of this inheritance are not clear. Geoghegan and Spencer (2012) explored the long-term evolutionary dynamics of epigenetic modifications that influenced fitness, finding that there were potentially more stable equilibria for epigenetic variation than for analogous models of genetic inheritance. Extending that work, these authors also showed that spatial environmental heterogeneity could lead to high levels of epigenetic variation at equilibrium, as long as several different environmental conditions were all relatively prevalent (Geoghegan and Spencer 2013). Modeling both genetic and epigenetic variation, Day and Bonduriansky (2011) demonstrated that epigenetic variation can influence the evolution of genetic variation and vice versa. However, the population in their model evolved in a static environment, and the epigenetic variation was not regulatory in its influence on fitness. Models of stochastic and sensed phenotypic switching have also explored the adaptive benefits of phenotypic inheritance of intermediate fidelity, finding that stochastic switching rates should evolve to be similar to the rate of change in the environment (Ishii et al. 1989; Lachmann and Jablonka 1996; Salathe et al. 2009). Stochastic switching models use mutation rates that are the same from any state to any other and do not depend on the environment. For the model of epigenetic inheritance described later, we relax these restrictions and allow mutation rates to vary by epigenetic state and environment. Incorporating environmentally sensitive mutation rates, Kussell and Leibler (2005) demonstrate that sensing should evolve when environmental cues reliably indicate upcoming selective pressures, although they do not explore the effect of genetic variation on these dynamics.

To understand how both genetic and environmental conditions influence the de novo evolution of an epigenetic regulatory mechanism, we introduce a model of the interaction of genetic and epigenetic variation in a temporally varying environment. Studying the evolution of epigenetic inheritance without incorporating genetic variation misses a crucial question: when is genetic variation sufficient or insufficient to buffer a population against environmental variability? By examining the conditions under which epigenetic regulation evolves, we can elucidate the costs and benefits of epigenetic regulation and inheritance relative to genetic variation. Unlike previous models of epigenetic inheritance, both genetic and epigenetic variations are assumed to influence evolutionary fitness, and rates of epigenetic mutation can depend on the environmental state. In this model, a genetic locus influences phenotype, and a modifier locus codes for a regulator of epigenetic states at the phenotypic locus. Starting with a population that may have genetic variation, but has no epigenetic variation, we introduce a modifier allele coding for epigenetic regulation, parameterized by the rates of epimutation between epigenetic states in each possible temporal environmental state. In this framework, we investigate the conditions that select for evolutionary invasion of the modifier. The environmental state can vary between generations, with different states having different selective optima and producing different epimutation rates. The environmental states could represent levels of herbivory or pathogen risk (Verhoeven et al. 2010; Herrera and Bazaga 2011), or any other stress that can influence rates of epimutation.

We find that both the number of generations between environmental changes and the initial allele frequencies at the phenotypic locus influence the conditions for evolution of epigenetic regulation. Except when recombination rates between the phenotypic locus and the modifier locus are low, the initial frequencies of active and inactive alleles at the phenotypic locus strongly affect whether an epigenetic regulator invades. This result is not surprising but has not been previously presented in the theoretical literature. The expected number of generations (period) before an environmental shift also affects this evolution, with short period lengths selecting for regulation that induces epigenetic states to switch often, and long periods selecting for faithful transmission of epigenetic states, but only when epimutation is directed toward the optimal state and is environmentally induced. This result differs from conclusions drawn from models of stochastic switching. When there is any cost to epigenetic regulation, genetic modifiers that produce undirected epigenetic mutation are unlikely to evolve, and epimutation toward the currently optimal state is a requirement for the evolutionary invasion of novel epigenetic mechanisms.

Model

We model a haploid population with variation at two genetic loci. The first locus affects the phenotype, and has two possible alleles: G and g. The second locus is a modifier with two alleles, m and M, where M allows epigenetic modification at the first locus. The modifier allele M could represent any of several genetic features: a novel CpG site, a transferase gene involved in the maintenance of DNA methylation or histone modifications, or even an RNA-coding gene. The environment varies temporally with two different states, A and B. We analyze cases in which the environment switches periodically or with a waiting time that has a geometric distribution. The different environmental states could represent climatic differences such as temperature or rainfall, micro-environmental differences such as the presence or absence of chemical stressors, or an array of other conditions.

To explore the invasion of allele M, we consider six haploid geno-epigenotypes. Individuals with the m allele can either be of type math formula or math formula. Epigenetic regulation of the math formula locus occurs only in individuals that carry the M allele. The math formula locus can be in either of the epigenetic states z or w. There are four possible types with the allele M: math formula, math formula, math formula, and math formula. To specify the effect on the phenotype, we suppose that one epigenetic state, z, is active, and the other, w, is inactive or less active. The active state may correspond to reduced CpG methylation at a promoter of a gene (Deaton and Bird 2011) or trimethylation of the lysine residue K4 on histone H3 (Muramoto et al. 2010). Similarly, we consider G to be an active allele, whereas g is an inactive allele. The active state G is favored in environment A, and the inactive state g is favored in environment B. The active and inactive states may correspond to different phenotypes, such as different proportions of prickly and nonprickly leaves among holly plants (Herrera and Bazaga 2013). Prickly leaves may be beneficial in environments with high density of herbivores, while having mostly nonprickly leaves may result in higher fitness when there is minimal risk of herbivory. The fitnesses and activity states of each type are summarized in Table 1. math formula is completely inactive and is identical to g, and math formula is identical to G. We impose a cost c on allele M, as we expect that the epigenetic regulatory machinery would require some cellular resources. Our model does not assume that the chemical changes that comprise the epigenetic state are physically located at the math formula locus; they may also reside on DNA or histones near the M allele or as small segments of RNA. They are written as subscripts of the phenotypic alleles to represent the interacting effects that produce the fitness of each type.

Table 1. Fitness regime
  Fitness 
TypeFrequencyEnv. AEnv. BState
mGx1math formulamath formulaActive
math formulax2math formulamath formulaInactive
math formulax3math formulamath formulaActive
math formulax4math formulamath formulaInactive
math formulax5math formulamath formulaInactive
math formulax6math formulamath formulaInactive

We also incorporate mutation of the epigenetic states, called epimutation. Unlike what is typically assumed for genetic mutation, the epimutation rates can depend on the environment, with different epimutation rates in environments A and B.

ONE GENERATION

Populations first undergo selection, followed by reproduction and epigenetic transmission to offspring. Incorporating selection first, the postselection frequencies math formula can be expressed in terms of the initial frequencies as

display math

where E indicates the current environmental state (math formula), as in Table 1.

Because it is unclear exactly how recombination should disrupt epigenetic marks, we propose a simple model that requires only one parameter: the rate of recombination between the modifier locus and the phenotypic locus. When a recombination event places a G or g allele with an epigenetic mark onto a background with the m allele, the epigenetic mark is lost. The m allele does not code for regulatory machinery that influences epigenetic variation at the math formula locus, so a G or g allele cannot retain any epigenetic state in this event. However, when recombination places an unmarked G or g allele onto a background with the M allele, the phenotypic allele (G or g) takes on the epigenetic mark of the allele that was formerly on that background. In this way, recombination disrupts associations between the modifier locus (math formula) and the phenotypic locus (math formula), but does not influence the associations between the math formula alleles and the epigenetic marks z and w. This model of recombination is meant to represent situations in which the modifier allele M controls the epigenetic inheritance. For example, allele M could enable the production of RNA that mediates the inheritance of the epigenetic state (as has been observed in mice; Rassoulzadegan et al. 2006), or could represent the presence of new CpG sites in the promoter of a gene. In either of these cases, allele M may determine the epigenetic state of the new G or g allele after recombination. An alternate interpretation of this system of recombination is that the epigenetic state is attached to the M allele, but even in this case, our model assumes that fitness is influenced only by the interaction of epigenetic state and the math formula locus.

The post-recombination frequencies math formula are functions of the post-selection frequencies:

display math

After recombination, all of the possible epigenetic mutational processes are incorporated into a single epimutation step, ultimately producing the adult individuals that start the next generation. We recognize that epimutation may stem from the combination of stochastic effects, induced effects, and incomplete resetting, so these epimutation rates should be thought of as statistical parameters, not as parameters that directly reflect the dynamics of some process. Rates of mutation between different epigenetic states may depend on the temporal environment, so our epimutation rates from z to w (called math formula) and from w to z (called math formula) must also be indexed by the temporal environmental state (represented here by math formula). Very different epimutation rates in the different environments represent epigenetic regulation that is highly environmentally induced, as may be the case with DNA methylation in dandelions in high stress versus low stress environments (Verhoeven et al. 2010).

Incorporating epimutation, the frequencies math formula at the start of the next generation can be written as

display math

Results

The rates of temporal environmental change and epimutation combine to determine whether a modifier will invade. For an overview, we focus on the case where the cost of epigenetic regulation is low relative to selection against the maladaptive expression state (in Table 1, math formula). We will refer to the rates of epimutation directed toward the currently optimal epigenetic state (math formula and math formula) as the adaptive switching rates (ASRs). Note that although we call them adaptive, these rates could produce a switch to a suboptimal state if the temporal environment were to switch at exactly the same time. We expect the ASRs to play a key role in determining the adaptive benefit of epigenetic regulation, so we first examine the case where both ASRs are equal (math formula). Conversely, we will call the rates of switching away from the currently optimal state (math formula and math formula) maladaptive switching rates (MSRs). To determine whether the modifier allele M successfully invades, we introduce M into the population at frequency 0.05, and iterate the recursions presented in the section MODEL for 10,000 generations. If the final frequency of the M allele is strictly greater than 0.05 after this period, we declare that the modifier has successfully invaded.

Figure 1 shows the conditions that allow for invasion of the epigenetic modifier allele M into a population lacking epigenetic regulation at the math formula locus. The environment switches periodically, with period given at the top of each column of panels. Each row of panels corresponds to a different ASR indicated on the right edge of the figure. Within each panel, the black squares indicate successful invasion of the modifier allele, with the axes indicating the MSRs of that modifier. The initial frequency of the active allele G is 0.95, and the recombination rate has no influence on these results (here math formula).

Figure 1.

Invasion of a genetic modifier of epigenetic regulation in a periodic environment. The colors in each panel indicate whether the modifier invaded. Within a panel, the axes are the maladaptive switching rates. The numbers heading the columns of panels indicate different periods between environmental switches. The numbers bordering the right side of each row of panels indicate the adaptive switching rates (math formula). The other parameters here are math formula, math formula, and math formula.

When the environment changes every generation, a modifier invades when it produces high MSRs (Fig. 1, first column). As the ASRs increase, the range of MSRs that allow invasion becomes more and more restrictive. This result is intuitive; what we refer to as maladaptive switching actually allows parents to produce offspring with the opposite phenotype. Because the environment switches every generation, the optimal genotype would produce opposite phenotypes each generation. However, if the environment switches only after four or more generations, higher ASRs broaden the conditions that allow invasion of the modifier (Fig. 1, fourth and fifth columns). The higher rates of directed switching allow a lineage to switch to the optimal state after the environment changes, even if most individuals are maladapted during the first generation after an environmental change.

For period 2 or 3, the results are less intuitive. If the environment switches every two generations, evolution of an epigenetic modifier occurs when the ASRs are low and MSRs are intermediate, or when the ASRs are high and exactly one of the MSRs is high (Fig. 1, second column). Results for period 3 are similar to those for higher periods, although we note invasion is also possible when all epimutation rates are 1. This is because continual phenotypic switching allows some individuals to attain the optimal phenotypic state in two out of every three generations. This odd-period effect has been noticed before in models of phenotypic switching (Liberman et al. 2011), although it seems to disappear for higher periods in our model.

The failure of invasion in certain cases demonstrates that selection must be weak for stochastic switching, where mutation rates do not depend on the current environmental state. Our model represents stochastic switching when the ASRs and MSRs all equal the same value. In Figures 1 or 2, the stochastic switching case occurs in each panel at the point along the diagonal (where the MSRs are equal) that is equal to the ASR specified by the panel row. Theoretical studies of stochastic switching have shown that switching rates should evolve to approximately the inverse of the length of time between environmental changes (Ishii et al. 1989; Lachmann and Jablonka 1996; Salathe et al. 2009), but some empirical research has suggested that the strength of selection for these mechanisms may be weak (Philippi 1993). For example, if the environmental period were 2, we would expect to see invasion when the MSRs and ASRs are all equal to 1/2 (Fig. 1, second column, sixth row, center of panel). Instead, the relatively small cost math formula of epigenetic regulation inhibits the evolution of switching. This indicates that the indirect strength of selection on switching rates cannot counter a selective disadvantage of even 1%. We do still see many cases in which the modifier does invade, demonstrating that selection for environmentally induced epimutation may be considerably stronger than selection for spontaneous, environment-independent epimutation. Although we do not present a particular metric to represent the degree of environmental induction, it is clear that greater differences between mutation rates in different environments correspond to more environmental influence. For example, if the environmental period is 4 we see invasion when the ASRs are high and the MSRs are low (Fig. 1, fourth column, lower rows).

Figure 2.

Invasion of a genetic modifier of epigenetic regulation in a geometrically fluctuating environment. The colors in each panel indicate the fraction of 10 simulations in which the modifier successfully invaded. Within a panel, the axes are the maladaptive switching rates. The numbers heading the columns of panels indicate different expected periods between environmental switches (the expected period is the inverse of the probability of the environment switching during any particular generation). The rows of panels are as in Figure 1. The other parameters here are math formula, math formula, and math formula.

When the environment switches after a geometrically distributed waiting time instead of a fixed period, the conditions for invasion are slightly different, although no less broad. Within each panel of Figure 2, the horizontal and vertical axes are the MSRs. Each column of panels corresponds to a different expected waiting (the inverse of the per-generation environmental switching probability), and each row corresponds to different ASRs between 0 and 1. Because the environmental state is a random variable, the dynamics are no longer deterministic. The shade of each square within a panel indicates the probability that the modifier allele M invades, obtained by averaging the results of 10 runs of simulated evolution. Figure 2 demonstrates that M no longer invades when the expected period is 2. For higher periods, M is actually able to invade over a broader range of ASRs, although the threshold MSRs allowing invasion seem to be lower than in the fixed period case (e.g., compare the rightmost columns of Figs. 1 and 2).

In the previous cases, the allele that modifies epigenetic regulation was introduced into a population dominated by the active phenotypic-influencing allele, G. When the initial frequency of G is high, recombination has no influence on the invasion dynamics. However, epigenetic regulation only matters on a G background, so the relative benefit and cost of the M allele may change depending on the initial frequencies. Figure 3 shows the interaction between recombination rates and initial G frequency for the case of fixed period 4. The ASRs are 0.6 and the MSRs are 0.1. When there is no recombination, we have invasion for any initial allele frequencies. However, increasing recombination leads to a sharp spike in the threshold G frequency that allows invasion. This pattern is similar for many other cases, although the specific thresholds both of G frequency and recombination vary.

Figure 3.

Invasion with respect to recombination and initial frequencies at the regulated locus. The colors in each panel indicate whether the modifier will invade. The horizontal axis indicates recombination rate, whereas the vertical axis shows the initial frequency of the active allele G. The environment switches every four generations. The other parameters here are math formula, math formula, adaptive switching rates equal to 0.6, maladaptive switching rates equal to 0.1.

In the event of a successful invasion and fixation, higher recombination rates reduce the rate at which the modifier M increases in frequency. This effect appears to be mediated by a slower increase in the frequency of allele G. Figure 4 shows the change in frequency of the modifier allele M and the regulated allele G for different recombination rates. The lines starting from 0.05 at 0 generations are the frequencies of the M allele. The G allele is initially at a frequency of 0.7, and the opaque lines represent the change in G frequency. The M allele begins to increase in frequency only after the G allele is nearly fixed. For higher recombination rates, G, and subsequently M, may take more than twice as long to approach fixation. Again the ASRs are 0.6, the MSRs are 0.1, and the period is 4. Some lines appear thicker than others because the frequencies fluctuate over the eight generations that comprise one cycle through both environments.

Figure 4.

Rate of modifier invasion and fixation with respect to recombination rate. As recombination rate increases (here math formula), the rate of invasion of the modifier M becomes considerably slower. Also plotted is the change in frequency of the G allele over time, starting from an initial frequency of 0.7. The M allele tends to increase in frequency only after G has nearly reached fixation. For each combination of selection coefficient s (math formula) and cost c, a black square indicates invasion whereas a white square indicates that the modifier cannot invade. The environment switches every four generations. The other parameters for these simulations are math formula, math formula, adaptive switching rates equal to 0.6, maladaptive switching rates equal to 0.1.

Unlike the case of stochastic switching, a modifier that enables epigenetic regulation can pay a substantial fixed fitness cost c and still invade. Figure 5 demonstrates a linear relationship between cost and selection for the threshold between successful and unsuccessful invasion of the epigenetic modifier. In this case, where the ASRs are 0.9, the MSRs are 0.1, and the period is 5, allele M is able to increase in frequency after introduction as long as the cost is less than approximately one third of the strength of selection. The selective coefficient math formula represents the cost of being phenotypically maladapted to the current temporal environment.

Figure 5.

Invasion of modifier across a range of costs and selection. There is a linear relationship between cost and selection strength that determines the threshold for invasion of the modifier M. For each combination of selection coefficient s (math formula) and cost c, a black square indications invasion whereas a white square indicates that the modifier could not invade. The adaptive switching rates are 0.9, and the maladaptive switching rates are 0.1. The period of environmental change is 5, and there is no recombination (math formula).

Discussion

We have analyzed the evolution of epigenetic regulation in a population with genetic sources of phenotypic variation. An individual's phenotype determines its survival, with any particular phenotype being optimal in one temporal environment state and suboptimal in the other. We model epigenetic regulation by specifying rates of switching (epimutation) between epigenetic states in each environmental state; the epigenetic state of the phenotype-influencing locus math formula determines whether an allele is active or inactive. Exploring a range of initial levels of genetic variation, temporal fluctuation regimes, and recombination rates between the modifier and phenotypic loci, we find that all three of these features combine to determine the range of epimutation rates that can evolve. Recombination rate and initial variation at the math formula locus interact, with higher recombination rates restricting the range of G allele frequencies that allows an epigenetic modifier to invade. Because we model the epigenetic regulation as an influence only on the “active” genotype G at the phenotypic locus, the higher the initial frequency of G, the broader the range of epimutation parameters that allows invasion of a genetic modifier of this regulation.

For different periods of environmental change, qualitatively different epimutation parameters allow epigenetic regulation to evolve. When the environment changes every generation, invasion occurs when there are high rates of switching away from the currently optimal phenotype (Fig. 1, first column). We refer to these rates as MSRs, although they are clearly not maladaptive in this case. Here epigenetic regulation prepares individuals for the new temporal environment. As the period increases, invasion generally occurs for high rates of directed switching toward the phenotype that has the highest fitness in the current environment. We call these rates ASRs. As long as MSRs are low, individuals with the epigenetic modifier will produce a lineage that rapidly adapts to a new environment, followed by relatively stable inheritance of the epigenetic state until there is an environmental shift. Such a pattern was seen in a study of epimutation rates in dandelions (Verhoeven et al. 2010), where stresses induced epigenetic changes that were then stably inherited. Furthermore, these results hold whether the environment switches in a deterministic manner, or with some probability during each generation. If the environment changes every two generations, the conditions for invasion are more complex. Invasion occurs when the ASRs are low and the MSRs are intermediate, or when the ASRs are higher and exactly one MSR is high. In this case, epigenetic regulation is adaptive when it generates some epigenetic variation, but not when it causes an overly strong response to the environment. Regardless of the rate of environmental change, the epimutation rates that can evolve differ from those found in previous models of stochastic switching (Ishii et al. 1989; Lachmann and Jablonka 1996; Salathe et al. 2009). Previous models did not impose a cost to the epigenetic regulation, and even a small cost precludes the evolution of regulation that is not directed and environmentally sensitive. This suggests that environmentally insensitive, multigenerational epigenetic inheritance is unlikely to evolve. Instead, we expect epigenetic regulation to be responsive to environmental cues, increasing the frequency of adaptive epialleles when novel stresses occur.

These results suggest that both existing genetic variation and the way environmental conditions change shape the evolution of epigenetic mechanisms. Epigenetic variation is only beneficial if there are multiple genetically influenced phenotypic states that are optimal under different environmental exposures. Thus, we may expect attainment of a mutation–selection balance at a genetic locus to be a precursor to the invasion of a modifier of epigenetic regulation. This balance should generate a high frequency of alleles in an active state in order for the modifier to invade, and invasion should be a two-part process, beginning with the active phenotypic allele increasing in frequency before the modifier begins to increase. These initial conditions are less important if the modifier has very low rates of recombination with the phenotype-influencing locus. It would be interesting to study the empirical (i.e., naturally occurring) distribution of recombination rates between genes regulating expression and the loci upon which they act.

To understand the results in terms of the fidelity of transgenerational epigenetic inheritance, we must consider both the rate of environmental change and the rates of epimutation. In a rapidly changing environment, adaptive epigenetic regulation might produce variation that is not at all heritable, because of the environmental sensitivity of the epimutation rates. However, environmentally induced epigenetic changes do not preclude the possibility of faithful epigenetic inheritance. When environmental changes are rare, there may be periods of many generations during which epigenetic marks are inherited stably. Only after an environmental perturbation will there be a period of instability. The influence of environmental effects makes epigenetic inheritance considerably more difficult to measure and predict than genetic inheritance.

Our haploid model focuses specifically on the adaptive role of transgenerational epigenetic inheritance and the epigenetic variation that can result. It does not address questions related to sex-specific non-Mendelian inheritance, such as the adaptive benefits of genomic imprinting or maternal effects. In its current form, the model results may also extend to a diploid population in which epimutation occurs independently on either chromosome, and fitness is determined multiplicatively by the number of modifier alleles M with fitness factor math formula and on the number of active (G, math formula) and inactive alleles (g, math formula, math formula, math formula). Extensions that explicitly track sexes and allow for sex-dependent epimutation would be of considerable interest. Considering the influence of prenatal environment on human epigenetic variation (Heijmans et al. 2008; Waterland et al. 2010), we may expect maternal and paternal environment to have different effects on epigenetic regulation and inheritance.

Other models of the relationship between the modifier locus, the epigenetic locus, and fitness may yield different results. When recombination rates between the two loci are low, the particular formulation of recombination's influence on epigenetic state is relatively unimportant. This may not be true for higher recombination rates. Alternative models of the influence of epigenetic state on fitness could also offer different insights. A model could consider each type to have a different fitness, or incorporate incomplete penetrance of epigenetic silencing. Incomplete penetrance ought to reduce the effectiveness of selection on the modifier, contracting the parameter range in which invasion can occur. Other complications may produce less predictable results, although the determination of a biologically well-motivated function that maps from type to fitness will also become more difficult.

Epigenetic regulation may serve as a mechanism for plasticity, responsive phenotypic switching, or stable inheritance of phenotypic states. By framing the model in terms of environment-dependent rates of epimutation, we are able to investigate a range of regulatory regimes from stochastic switching (when all epimutation rates are equal) to responsive switching (when rates depend on the environment). Our results demonstrate that most cases in which there is selection for epigenetic regulation involve epimutation rates that are highly sensitive to the environment. However, the consequences of epigenetic variation at evolutionary equilibrium depend strongly on whether the environment switches rapidly or slowly. When the environment fluctuates rapidly, high rates of epimutation result in many individuals being prepared for new selective pressures, and there should be high levels of epigenetic variation. If the environment changes only after several generations, then high rates of epimutation toward the optimal phenotype will not actually produce large amounts of epigenetic variation. The population will adapt rapidly after an environmental shift, then epigenetic variation will dissipate until the next shift. Thus, the importance of epigenetic variation may not be obvious from a snapshot of it in a population at a single point in time. To better understand epigenetic inheritance and its adaptive significance, epigenetic variation and environmental influences should be observed over many generations. A clearer picture of these dynamics may inform conservation, adaptive responses to climate change, and the general importance of regulatory variation in evolution.

ACKNOWLEDGMENTS

The authors thank Associate Editor K. Donohue, two anonymous reviewers, and members of the Feldman laboratory for helpful comments on the article.

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