A consequence of genomic imprinting is that offspring are more similar to one parent than to the other, depending on which parent's genes are inactivated in those offspring. We hypothesize that genomic imprinting may have evolved at some loci because of selection to be similar to the parent of one sex or the other. We construct and analyze an evolutionary-genetic model of a two-locus two-deme system, in which one locus codes for a character under local selection and the second locus is a potential cis-acting modifier of imprinting. A proportion of males only migrate between demes every generation, and prebreeding males are less fit, on average, than females. We examine the conditions in which an imprinting modifier allele can invade a population fixed for a nonimprinting modifier allele and vice versa. We find that the conditions under which the imprinting modifier invades are biologically restrictive (high migration rates and high values of recombination between the two loci) and thus this hypothesis is unlikely to explain the evolution of imprinting. Our modeling also shows that, as with several other hypotheses, polymorphism of imprinting status may evolve under certain circumstances, a feature not predicted by verbal accounts.

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GENOMIC imprinting—the differential expression of mammalian genes depending on the sex of the parent from which they are inherited—is the subject of a significant research effort. As a consequence, some 83 transcriptional units have been identified as being imprinted in mice and/or humans, as well as several other mammals (MORISON et al. 2005), although a recent estimate predicts that up to 600 mouse genes may be imprinted (LUEDI et al. 2005). Because imprinting alters the usual biallelic genic expression to a monoallelic state in at least some tissues for some period of development, an imprinted locus is effectively hemizygous for that period. This pseudohemizygosity removes the masking effect of diploidy and exposes the organism to the consequences of various mutations that might otherwise be hidden by a functionally normal allele. Hence, the fact that imprinting has evolved at so many mammalian loci implies that imprinting must confer some selective advantage (SPENCER 2000).

HURST and MCVEAN (1998) and SPENCER (2000), for example, have reviewed a number of hypotheses about why imprinting may be selectively favored, but new ideas are constantly being proposed. Mathematical modeling of some of these hypotheses has greatly helped our understanding of their applicability, predictions, and plausibility. For instance, WEISSTEIN and SPENCER (2003) showed that the "variance minimization" hypothesis, which holds that imprinting arose to facilitate tighter control of gene expression at critical loci, was an unlikely evolutionary explanation. Partly as a result of such work, MORISON et al. (2005) considered just four hypotheses to be plausible: genetic conflict, ovarian time bomb, X-linked sex-specific selection, and sexually antagonistic selection.

In this article, we mathematically model a fifth hypothesis, adumbrated by SPENCER et al. (2004), that proposes that imprinting will evolve in situations in which selection favors offspring being similar to the parent of one sex. This "chip-off-the-old-block" hypothesis points out that a direct consequence of imprinting is that offspring will phenotypically resemble one parent more than the other, independent of the offspring's sex (SPENCER 2002; SANTURE and SPENCER 2006). Thus, if such resemblance is selectively favored, one possible evolutionary outcome is imprinting.

A potential scenario for the generation of selection for parental resemblance arises because of the male-biased differential migration rate in most mammals (see GREENWOOD 1980 and DOBSON 1982 for reviews). In a heterogeneous habitat, mothers are likely to be better adapted to the local environment than fathers, because the latter are more likely to have been subject to different selection pressures elsewhere. Thus, at least early in life, before any migration, offspring of both sexes would be selected to resemble their mothers. Thus the chip-off-the-old-block hypothesis would predict that almost any locus exhibiting local adaptation, especially in juveniles (e.g., coat color in rodents: HOEKSTRA et al. 2006), could be imprinted. This prediction contrasts with those of the genetic conflict and ovarian time bomb hypotheses that focus on genes active during fetal development and, in the case of genetic conflict, soon after birth.

Alternatively, in species that are highly mobile, fathers may be more fit than mothers because they have been exposed to a wider range of habitats. In this case, both male and female offspring will do better if they resemble their fathers.

The chip-off-the-old-block hypothesis is related to both the X-linked sex-specific selection hypothesis (IWASA and POMIANKOWSKI 1999) and the sexually antagonistic selection hypothesis (DAY and BONDURIANSKI 2004). All three ideas invoke selection pressures on offspring, rather than various forms of selection on mothers, fathers, and offspring as under genetic conflict (HAIG 1992) and the ovarian time bomb (VARMUZA and MANN 1994; IWASA 1998). Chip off the old block differs, however, in that it allows identical selection pressures on both male and female offspring, although differential selection is not prohibited. In contrast, under both the X-linked sex-specific selection and the sexually antagonistic selection hypotheses, differential selection pressure on males and females is the motor for the evolution of imprinting.

We construct a two-locus, two-allele modifier model, akin to those used to investigate the evolution of recombination (NEI 1967; FELDMAN et al. 1980) and migration (KARLIN and MCGREGOR 1974). We assume that variation at the selected locus is maintained by differential selection in two demes connected by migration. In deme X, the AA genotype is selectively favored, with Aa genotypes having a fitness 1 – s and aa genotypes 1 – 2s; in deme Y these fitnesses are reversed. Thus, there is no dominance in either deme, but alternative alleles are favored. To keep the model as tractable as possible, we assume that only males migrate at a rate m. Later we consider the effects of breaking down the model symmetry in several ways.

Whether or not alleles at the selected locus are imprinted depends on the alleles at a second, modifier locus. We assume that the modifier is a cis-acting, gametic modifier (SPENCER and WILLIAMS 1997) that inactivates the paternal allele: sperm with the imprinting modifier allele, e, inactivate the allele at the A locus in the resulting offspring; sperm with the Mendelizing modifier allele, E, do not. Maternal inactivation can be modeled similarly. Individuals with an inactivated allele are assumed to have the same fitness as homozygotes for the active allele. Let the four possible haplotypes, AE, Ae, aE, and ae, be numbered 1–4.

We assume that selection occurs after zygote formation but before migration. Recombination between the two loci is assumed to occur in both males and females at a rate r. Mating is at random within each deme and we ignore the effects of genetic drift. As in keeping with such two-locus models, we need to keep track of haplotype frequencies, rather than simply allele frequencies, and the two-deme structure requires that we follow the haplotype frequencies in each deme. Nevertheless, we do not need to track male and female frequencies separately, because these are regenerated afresh each generation from the postselection, premigration haplotype frequencies. Let x_i and y_i denote these frequencies in deme X and deme Y, respectively, where i denotes the haplotype number. Then the postmigrational frequencies, denoted by the addition of subscripts f and m for females and males, respectively, are given by

(1)

The required recursions, in which the prime indicates the respective frequencies in the following generation, are thus given by

(2)

(3)

(4)

(5)

where

(6)

and is the sum of the right-hand sides of Equations 2–5. Similarly, in deme Y

(7)

(8)

(9)

(10)

where

(11)

and is the sum of the right-hand sides of Equations 7–10. The symmetry between the selection for A in deme X and for a in deme Y is reflected in these equations, in that Equations 7–11 are simply Equations 2–6 with replaced by , etc.

We have not been able to solve the above model to find all equilibria and characterize their local stability. Nevertheless, it is not necessary to do so to understand the possible origination of imprinting. The evolution of imprinting corresponds to the replacement of the E allele by e. Hence, we are interested in the local stability of the fixation of the E allele to possible invasion by e and vice versa. At the former of these fixations, the haplotypes involving the e allele all have frequency zero: x₂ = x₄ = y₂ = y₄ = 0. We first solve the reduced system of four equations {, ; i = 1, 3} with the other variables identically zero, to find the nontrivial migration–selection balance, and then examine the eigenvalues of the reduced system in x₂, x₄, y₂, and y₄ linearized around this equilibrium. If the leading eigenvalue of this reduced system is less than unity, the equilibrium is locally stable and the e allele cannot invade. This method was used to investigate the evolution of dominance by FELDMAN and KARLIN (1971) and the parallels between dominance and imprinting by SPENCER and WILLIAMS (1997).

The nontrivial E fixation occurs when

(12)

from which all other equilibrium values may be calculated (since the haplotype frequencies in each deme add to one). Whether or not the leading eigenvalue, _E, for the reduced system in x₂, x₄, y₂, and y₄ linearized around this equilibrium is greater than or smaller than one depends on the values of m, s, and r, as shown in Figure 1. For small values of r and m, _E < 1 and the imprinting modifier, e, cannot invade. Smaller values of m mean that there is less mismatch of offspring with their mothers, since fewer males from the other deme become fathers. Hence, smaller values of m give imprinting less advantage. Nevertheless, whether or not _E is > 1 is relatively insensitive to s, although e is slightly favored by larger values of s. Tight linkage between the selected and modifier loci prevents the invasion of e unless m is fairly large, presumably because this does not allow sufficient recombinant types to give e an advantage in both demes.

View larger version (41K): In this window In a new window Download PPT slide   FIGURE 1.— The division of parameter space. For values of r above the bottom arrowed surface the e allele can invade a population fixed for E. For large values of m, all biologically realistic values of r (r 0) allow e to invade. Below the top arrowed surface the E allele can invade the e fixation. Hence, above the top surface, imprinting is stably fixed, below the bottom surface Mendelian expression is stably fixed, and both alleles can be found in the population for parameter values between the two surfaces.

 
The nontrivial e fixation occurs when

(13)

and the effect of m, s, and r on the leading eigenvalue, _e, for the reduced system in x₁, x₃, y₁, and y₃ linearized around this equilibrium is shown in Figure 1. Again, the system is relatively insensitive to s. For small values of m, E will invade unless r is large; larger values of m mean that e will remain fixed for tighter linkage. Between the surfaces shown in Figure 1, both E and e will be present in the population, and imprinting status may be different in different individuals.

Our model of the chip-off-the-old-block hypothesis shows that imprinting may evolve in response to selection pressures to resemble a parent. In our model, this form of selection arises from migrational load, the conflict between local adaptation and immigration, which is represented by opposite selection pressures at a single locus in two demes connected by symmetrical two-way male-only migration. The evolution of imprinting occurs by the invasion and fixation of an imprinting modifier allele at a second locus. Nevertheless, the model implies some restrictions on the way imprinting may evolve. Most importantly, the imprinting modifier cannot invade if linkage between the modifier locus is tight (recombination rate, r < 0.15) and migration between the demes is low to moderate (migration rate, m < 0.2). This restriction is serious: most known imprinting mechanisms involve genes close by (MORISON et al. 2005) and a migration rate of 20% is rather high.

The conditions for the imprinting modifier to fix are even more stringent, suggesting that imprinting status may be polymorphic. This possibility also occurs under the genetic conflict hypothesis at both autosomal (SPENCER et al. 1998) and X-linked loci (SPENCER et al. 2004) and under the sexually antagonistic selection hypothesis (DAY and BONDURIANSKI 2004). Interestingly, there is empirical evidence for polymorphism in imprinting status at at least three human loci, the Wilm's tumor suppressor gene, WT1 (JINNO et al. 1994), the serotonin-2A (5-HT_2A) receptor gene, HTR2A (BUNZEL et al. 1998), and the insulin-like growth factor receptor 2 gene, IGF2R (XU et al. 1993; MONK et al. 2006).

Nevertheless, the high levels of recombination and migration required for the evolution of imprinting suggest that the chip-off-the-old-block hypothesis is unlikely to explain the evolution of imprinting. Indeed, the difficulty of evolving imprinting via this hypothesis could provide an explanation of why imprinting is not more widespread, either across the genome or phylogenetically. Our model suggests that imprinting will probably not evolve in the verbally plausible scenario of the chip-off-the-old-block hypothesis, which would appear to apply to almost any loci that allow local adaptation, as well as to most groups of sexually reproducing plants or animals.

One reason that imprinting is difficult to evolve by a chip-off-the-old-block mechanism arises because the selection pressure for imprinting is a secondary force, arising as a by-product of selection for local conditions. To see the quantitative effect, we note that the difference in the population's mean fitness between the fixations of E and e given by Equations 12 and 13 is very small. At the former,

(14)

which when m = 0.2 and s = 0.1, for example, is 0.9207. At the latter equilibrium we find

(15)

which when m = 0.2 and s = 0.1 is only marginally greater, 0.9253. Thus the effective selection pressure in favor of imprinting is in fact very slight, even in the presence of nontrivial migration rates and local selection pressures under which imprinting is predicted to evolve. We note that this argument may well apply to other hypotheses, providing another explanation for the relative rarity of imprinting.

The form of selection we model may also have alternative outcomes. For instance, reducing the migration rate will automatically mean that more fathers come from the same deme as mothers, reducing the effective selection pressure for maternal similarity. Of course, male-biased migration will affect all loci, in contrast to imprinting, which affects just the target locus. Thus, the benefits of migration (e.g., lowering of inbreeding) may outweigh the potential benefit of a decreased migration rate. Interestingly, the model implies that the selection coefficient, s, is relatively unimportant in whether or not imprinting evolves, whereas it is important in determining the level of migrational load. Thus if s is small, a reduction in the migration rate is less likely, whereas chances for the evolution of imprinting are largely unchanged.

One shortcoming of our modeling above is that it represents a special case of selection for parental similarity. Moreover, the combination of local adaptation and male-only migration means that offspring are better off resembling their mothers, and so only paternal inactivation can evolve. Maternal inactivation could arise only under the chip-off-the-old-block hypothesis if offspring were selected to resemble their fathers. One scenario in which such selection might arise would be if migrating males were subject to selection so that those who successfully migrated were more fit. In these circumstances, species with higher migration rates would be expected to show higher levels of maternal inactivation compared to species with lower levels of migration, which should exhibit more paternal inactivation. Interestingly, such selection pressure, stronger on males than on females, is similar to that modeled by DAY and BONDURIANSKI (2004) when considering the sexually antagonistic selection hypothesis. In their models, however, the affected loci were those underlying sexually selected traits, whereas in the scenario above variation at the loci affects viability. A synthesis of the two hypotheses might allow the evolution of imprinting at more loci under a wider range of conditions.

Of course, offspring can resemble their parents very well if there is no genetic variation at the relevant loci. Selection for parental similarity can result simply in the fixation of variation. To model this hypothesis, therefore, we needed to separate the processes maintaining genetic variation (in our scenario the opposite selection pressures in the two demes) from the selection to resemble a parent (arising secondarily from male-only migration). An alternative approach might be to have frequency-dependent selection or heterozygote advantage maintaining variation at the target locus, with a second set of selection pressures to resemble one parent or the other. The biological justification for this sort of scenario, however, is not immediately obvious, and it is unlikely that imprinting would evolve under heterozygote advantage since it essentially prevents heterozygous expression. Nevertheless, our finding that the conditions for the evolution of imprinting are rather restrictive may be a consequence of the selection for parental similarity being a secondary force, emerging from migrational load. This problem is a reflection of the long-standing issue in evolutionary genetics of how to maintain suitable variation in models of selection. We note that quantitative-genetic models avoid this issue, agnostically assuming that variation will be maintained; such an approach may be useful in modeling the chip-off-the-old-block hypothesis.

Much of this work was done while H.G.S. was on sabbatical leave at Cornell University. We are grateful to Scott Davidson for discussions of various models and to two anonymous reviewers for a number of helpful comments. Financial support for this work was provided by the Allan Wilson Centre for Molecular Ecology and Evolution (H.G.S.) and National Science Foundation grants DEB 9419631 and 9527592 (A.G.C).

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