Abstract
At a small number of loci in eutherian mammals, only one of the two copies of a gene is expressed; the other is silenced. Such loci are said to be “imprinted,” with some having the maternally inherited allele inactivated and others showing paternal inactivation. Several hypotheses have been proposed to explain how such a genetic system could evolve in the face of the selective advantages of diploidy. In this study, we examine the “ovarian time bomb” hypothesis, which proposes that imprinting arose through selection for reduced risk of ovarian trophoblastic disease in females. We present three evolutionary genetic models that incorporate both this selection pressure and the effect of deleterious mutations to elucidate the conditions under which imprinting could evolve. Our findings suggest that the ovarian time bomb hypothesis can explain why some growthenhancing genes active in early embryogenesis [e.g., mouse insulinlike growth factor 2 (Igf2)] have evolved to be maternally rather than paternally inactive and why the opposite imprinting status has evolved at some growthinhibiting loci [e.g., mouse insulinlike growth factor 2 receptor (Igf2r)].
THE unequal expression in mammals of some maternally and paternally derived genes known as genomic imprinting reduces (or even eliminates) the masking benefits of diploidy. (For a review of the benefits of diploidy over haploidy see Perrotet al. 1991; Otto and Goldstein 1992.) Consequently, genomic imprinting confers an apparent selective disadvantage on any imprinted individual and yet several mammalian loci appear to have evolved from a nonimprinting state to become imprinted (Bartolomei and Tilghman 1997). Several hypotheses have been proposed to explain this paradox (reviewed in Haig and Trivers 1995; Hurst 1997; see also Spenceret al. 1999; Spencer 2000). One of the earliest suggestions notes that there are no parthenogenetic mammals and that even attempts to create such lines in the laboratory have not succeeded (Kaufman 1983). The apparent restriction of imprinting among vertebrates to mammals, therefore, has led to the hypothesis that imprinting, by requiring genetic input from both parents, evolved to prevent parthenogenesis (Solter 1988).
A number of criticisms of this hypothesis can be made (Solter 1988; Haig and Trivers 1995; Hurst 1997). Clearly, only maternal inactivation of an allele could have the necessary feature of killing a parthenogenetic embryo. But by destroying some fraction of its carriers’ progeny, such an allele would actually decrease its frequency relative to a nonimprinting allele that permitted asexual reproduction (Haig and Trivers 1995; Hurst 1997). Thus, selection at the level of the individual would oppose the evolution of imprinting to prevent parthenogenesis. By suggesting that parthenogenetic lines are evolutionary dead ends that should be selectively eliminated, the parthenogenesisprevention hypothesis apparently requires grouplevel selection, which would be easily subverted by individuallevel selection. As a result, the parthenogenesisprevention hypothesis has generally been considered insufficient to account for observed patterns of imprinting (Haig and Trivers 1995; Hurst 1997).
Nevertheless, one version of this suggestion, the ovarian time bomb hypothesis (OTBH; Varmuza and Mann 1994), explicitly envisages an individuallevel cost of parthenogenesis, namely ovarian trophoblastic disease. This cancerlike condition could arise as a consequence of an unfertilized egg spontaneously developing in the ovaries. Inactivating or downregulating the maternal copy of a growthenhancing gene in a mother’s offspring means that a paternal genetic contribution is essential for successful embryogenesis. Hence, this form of imprinting in a mother can prevent such rogue reproduction and confers on her a selective advantage over those genotypes that are not imprinted.
In this article, we develop three evolutionary genetic models of how imprinting could evolve under the assumptions of the OTBH. These models allow us to predict when an imprintable allele can invade a population originally fixed for a nonimprintable ancestral allele, when the nonimprinting allele can invade a population fixed for imprinting, and when the two alleles can coexist. The first model describes the case of maternal inactivation; the others deal with paternal inactivation.
MODEL FORMULATION AND ANALYSIS
Model 1: Maternal inactivation and deleterious mutation: This model adapts the analysis of Spencer et al. (1998) of the genetic conflict hypothesis to the ovarian time bomb hypothesis. Consider an autosomal growthfactor locus originally fixed for an unimprintable allele A. Now suppose a maternally imprintable allele a is introduced. Under the OTBH, this new allele would enjoy a selective advantage because, when present in unfertilized eggs, it prevents those eggs from spontaneously developing and thereby reduces that female’s risk of ovarian trophoblast disease. We can therefore write the respective fitnesses of AA and aa females as 1  s and 1, respectively (s ≤ 1). Heterozygote females would enjoy an intermediate fitness of 1  hs; in the absence of segregation distortion, the imprintable allele should be passed to onehalf of a heterozygote female’s eggs, so we would expect h = ½. Males, of course, experience no cancer risk from expressing the growth factor in their gametes, so all of their fitnesses would be unity.
Under this simple model, the imprintable allele a always confers a selective advantage over the nonimprintable allele and should therefore become fixed in the population. This analysis, however, fails to consider the costs of haploidy, particularly the loss of masking of recessive deleterious mutants (Perrotet al. 1991; Otto and Goldstein 1992). Consider a recessive mutant allele a^{*} that arises from both A and a at rate μ. This mutation will be completely masked in both Aa^{*} and a^{*}A individuals, where we write the paternally derived allele first. It will also be masked in aa^{*} individuals. However, a^{*}(a) individuals, where the parentheses denote inactivation of the maternally derived allele, will suffer a reduced fitness of 1  t (t ≤ 1) because they lack a functional copy of the gene. Note that this selection pressure, unlike that imposed by ovarian cancer, applies equally to both sexes. Assuming that a^{*} acts like a null allele, this mutant will also confer a reduced risk of cancer, since it fails to produce sufficient growth factor to initiate development when present in an unfertilized egg. We can then construct Table 1 to list the nine distinct genotypes and their relative fitnesses in both females and males.
Let us define f_{1}, f_{2}, and f_{3} as the respective frequencies of the A, a, and a^{*} alleles in females and m_{1}, m_{2}, and m_{3} as the corresponding frequencies in males (thus, Σf_{i} = Σm_{i} = 1). We then derive the following recursions for allele frequencies in the following generation,
This system of equations has multiple equilibria (i.e., values of f_{1}, f_{2}, f_{3}, m_{1}, m_{2}, and m_{3} that satisfy Equations 1a, 1b, 1c, 1d, 1e, 1f, 1g, 1h with the primes removed from the lefthand sides). However, several of these equilibria are associated with allele frequencies that are complex, less than zero, or greater than unity. The only four biologically feasible solutions are: (i) the fixation of a^{*} (i.e., f_{1} = f_{2} = m_{1} = m_{2} = 0, f_{3} = m_{3} = 1), which we denote equilibrium M1; (ii) a mutationselection balance between a and a^{*}, denoted M2 and given by f_{2} = m_{2} = 1  2μ/t(1 +μ) and f_{3} = m_{3} = 2μ/t(1 +μ); (iii) a mutationselection balance between A and a^{*}, denoted M3 (see appendix a for allele frequencies); and (iv) a mutationselection balance among all three alleles, denoted M4 (see appendix a for allele frequencies). Figure 1 plots these equilibria and the system’s evolutionary trajectories for a representative set of parameter values.
Stability analysis: Near equilibrium M1 (fixation of a^{*}), we linearize the system (1) and solve for the eigenvalue that governs increase of A when it is rare. This eigenvalue is less than unity, and equilibrium M1 is stable to invasion by A, if
For the two remaining equilibria, we explicitly assume h = ½ for the sake of algebraic tractability. Numerical iteration of recursions (1) indicates that equilibrium M3 (A/a^{*} polymorphism, a absent) exists only when equilibrium M1 is unstable to invasion by A [i.e., t > [s +μ(4  s)]/4, the reverse of 2a]. In fact, M3 is the root of the fourthdegree polynomial given by (A27) in appendix a. It is a matter of algebra to see that to order μ, this polynomial is positive at f_{1} = 1 and, if t > [s + μ(4  s)]/4, it is negative at f_{1} = 0. Hence, there is at least one mutationselection balance equilibrium M3. For reasonable μ values, we have found no other solutions for an A/a^{*} mutationselection equilibrium. Further numerical investigation suggests that M3 is stable to invasion by allele a when
Inequality (2e) is also relevant for the existence of M4, the internal equilibrium, which is given by a cubic equation that factors to give the value of f_{1} reported in (A33) of appendix a. After some algebra, the allele frequencies at this equilibrium are seen to be biologically feasible if both (2c) with h = ½ and (2e) hold. We have not been able to derive analytical conditions for the local stability of M4. However, the fact that its existence entails the local stability of M2 and M3 suggests that it should be unstable when it exists. Indeed, there are no values of s < 1 for which inequalities (2b), (2c), and (2e) hold simultaneously. For numerical verification of M4’s instability, we set μ= 10^{6}, allowed both s and t to vary from 10^{6} to 1 in logarithmic increments of 10^{1/4}, and at each such point (set of parameters) determined the equilibrium’s stability. For all points at which equilibrium M4 was feasible, it was also unstable; for nearly all points at which the equilibrium was unfeasible, it was stable (the exceptions being several cases with t < 2μ). There were no instances when the equilibrium was simultaneously stable and biologically feasible. We also performed simulations for ∼50 specific combinations of s, t, and μ, in which we started the system near equilibrium M4 and then iterated equations (1a, 1b, 1c, 1d, 1e, 1f, 1g, 1h) over 10^{4} generations. The equilibrium was unstable in all such trials. These combined findings strongly suggest that equilibrium M4 is never stable.
These results are summarized in Figure 2, which plots equilibrium stability over st phase space (assuming h = ½ and μ= 10^{6}). From this figure, we see that equilibrium M1 (fixation of a^{*}) is stable only for very small values of t, at which selection against mutant alleles is too weak to counteract the pressure of recurrent irreversible mutations. Likewise, only equilibrium M3 (A/a^{*} polymorphism) is stable for very small values of s, at which selection for A alleles to mask mutant a^{*} alleles overcomes the opposing selection for decreased cancer risk. For these values of s and t, imprinting will neither increase when rare nor be maintained if already present. We contrast this result with the wide range of s and t values at which only equilibrium M2 is stable. In these regions, the decreased cancer risk associated with the a allele more than compensates for the inability to mask a^{*} alleles, so imprinting should invade and be maintained.
Figure 2 also demonstrates that, for small values of s and slightly larger values of t, equilibria M2 and M3 can be simultaneously stable. In this region of phase space, an imprintable allele will be maintained if present but cannot invade. For some sets of selection coefficients, therefore, maternal inactivation and full expression can both be stable evolutionary outcomes. (In a finite population, of course, genetic drift may enable the system to switch from one stable equilibrium to another.) This feature does not appear in verbal statements of the OTBH or in the quantitativegenetic model of Iwasa (1998); it is a novel prediction of our mathematical model. In contrast, the model finds no stable equilibria at which imprinting can be polymorphic (i.e., with both A and a present), contrary to the predictions of a model for the geneticconflict hypothesis (Spenceret al. 1998).
Model 2: Paternal inactivation and deleterious mutation: We can modify the previous model to make the inactivation of the a allele paternal. Verbal arguments based on the OTBH (Varmuza and Mann 1994) imply that imprinting should not evolve in this circumstance. Using Table 2, where the first allele written now represents the maternally derived allele, we can derive the following iterations:
Once again, this system appears to have four biologically feasible equilibria: (i) the fixation of a^{*} (i.e., f_{1} = f_{2} = m_{1} = m_{2} = 0, f_{3} = m_{3} = 1), denoted P1; (ii) a mutationselection balance between a and a^{*}, denoted P2 (see appendix b for allele frequencies); (iii) a mutationselection balance between A and a^{*}, denoted P3 (see appendix b for allele frequencies); and (iv) a second mutationselection balance between a and a^{*}, denoted P4 (see appendix b for allele frequencies).
Stability analysis: Using the same procedure as in the maternal inactivation case, we find that equilibrium P1 (fixation of a^{*}) is stable to invasion by A if
In contrast to model 1, numerical iterations of system (3) show that the paternal inactivation model has a fairly simple outcome (Figure 3). We note that equilibrium P1 (fixation of a^{*}) is stable over a much broader range of parameter values than is M1; even for large values of t, sufficiently high cancer risk s can lead to the fixation of the deleterious mutant. For most plausible values, however (i.e., where t is much greater than s), the only stable equilibrium is P3 (A/a^{*} polymorphism). For h = ½, equilibrium P2 is always unstable and equilibrium P4 requires biologically unfeasible allele frequencies; consequently, the a allele is always lost. According to this model, therefore, growthenhancing genes should not evolve paternal inactivation.
We can also use model 2 to analyze the behavior of growthinhibiting genes. At such a locus, the unimprintable allele A would now enjoy the selective advantage because it reduces both the chance of an egg spontaneously developing and the probability of expressing a deleterious mutant allele. We can therefore describe this system by simply requiring that s < 0 (rather than s > 0 as at growthpromoting loci). Modifying Tables 1 and 2 appropriately, we see that the A allele now has higher fitness than the a allele under both the maternal and the paternal inactivation models, so the imprintable allele should always be selectively eliminated. Therefore, our simple model of the OTBH leaves us with no adaptive explanation for observed instances of paternal inactivation (e.g., Bartolomei and Tilghman 1997).
Model 3: Paternal inactivation, deleterious mutation, and stabilizing selection: Alternative formulations of the OTBH, more complex than the model described above, have incorporated possible mechanisms for the evolution of paternal inactivation. The “innocent bystander” hypothesis suggests that paternal inactivation results from imprinting machinery aimed at specific physical features of trophoblastspecific genes but also present in some other genes, both paternal and maternal (Varmuza and Mann 1994). It therefore explains paternal imprinting at a molecular but not at an evolutionary level: Until the putative physical targets of the imprinting machinery are identified, the hypothesis remains descriptive rather than predictive. Iwasa (1998) provided an alternative explanation, proposing that the risk of an unfertilized egg spontaneously developing can be reduced by increasing expression of the maternal growthinhibiting alleles (as well as decreasing expression of her growthpromoting alleles as Varmuza and Mann originally suggested). Stabilizing selection for constant overall levels of growth inhibition in the zygote would then select for the contrary pattern in paternal alleles, i.e., inactivation of inhibitor alleles. To mimic the effect of such stabilizing selection, we begin with the paternal inactivation model of a growthinhibiting gene, described above. In this analysis, we replace the a allele with a new allele a^{†}, which is underexpressed when inherited paternally and overexpressed when inherited maternally. Each a^{†} allele present in a female therefore inhibits spontaneous development in onehalf of her unfertilized eggs, increasing her fitness by an amount r/2; each unimprintable A allele she carries likewise decreases the risk of such development in onehalf of her eggs, decreasing her fitness by an amount s/2. (We assume here that h = ½; given that this is a growthinhibiting locus, we also expect r > 0 > s.) Using Table 3, where the first allele written again represents the maternally derived allele, we can derive the following iterations,
This system has three biologically feasible equilibria: (i) the fixation of a^{*} (i.e., f_{1} = f_{2} = m_{1} = m_{2} = 0, f_{3} = m_{3} = 1), denoted S1; (ii) a mutationselection balance between a^{†}and a^{*}, denoted S2 (see appendix c for allele frequencies); and (iii) a mutationselection balance between A and a^{*}, denoted S3 (allele frequencies same as P3; see appendix c).
Stability analysis: Again using the same procedure as in the maternal inactivation case, we find that equilibrium S1 (fixation of a^{*}) is stable to invasion by A if
Now, when both S2 and S3 are feasible, one must be stable and the other unstable (since S1 is unstable and no other equilibria exist). We performed 40 simulations using random parameter values 1 < s < 0 < r, t < 1; in each case, S2 was stable when r + s > 0 and unstable when r + s < 0. We tested this pattern by performing 20 additional simulations with r =s + 0.01 and 20 with r =s  0.01: The former group of simulations all converged to equilibrium S2 and the latter to S3. We conclude that S2 is stable and S3 unstable for r + s > 0 and that the reverse holds true for r + s < 0. (When r + s = 0, the “overexpression” of maternally inherited a^{†} alleles is equal to the normal expression of A alleles; this case is therefore equivalent to paternal inactivation without stabilizing selection, which we have already examined under model 2.)
It is instructive to compare the feasibility of equilibria in models 2 and 3 (Figures 3 and 4, respectively). Because model 3 includes a parameter r not present in model 2, the former plot has one more dimension than the latter. Moreover, this extra parameter effectively decouples the system’s equilibria: The feasibility of S2 depends on r but not on s, and the feasibility of S3 depends on s but not on r, whereas the feasibilities of P2 and P3 from model 2 both depend on s. As a result of this decoupling, equilibrium S2 (mutationselection balance of the imprintable allele under stabilizing selection) is stable over a wide range of parameter space, although P2 (the corresponding equilibrium without stabilizing selection), as we observed, is never stable. This analysis therefore elucidates the range of parameter values for which Iwasa’s (1998) modified version of the OTBH holds.
DISCUSSION
The results of our models validate the most basic prediction of the ovarian time bomb hypothesis—that the risk of ovarian cancer in females can lead to the evolution of maternal inactivation. The feasibility of such an evolutionary event will obviously depend on the parameter values present in a particular system. Specifically, imprinting of growthenhancing genes can evolve most easily when the cancer risk, s, exceeds a specific threshold, calculated from (2e). Once imprinting has evolved, however, it can be maintained even if s decreases significantly below that threshold. Moreover, our results imply that polymorphism in imprinting status will not evolve under the selection regime envisaged by the OTBH, a finding in contrast to that of a similar model for the geneticconflict hypothesis (Spenceret al. 1998). Our models also confirm the verbal hypothesis in predicting that the original form of the OTBH should not lead to paternal imprinting of growthenhancing genes.
The models also suggest possible answers to some objections that have been raised against the OTBH. One such objection states that mammalian ovarian teratomas are too rare to apply the selective pressure needed to fix an imprintable allele (Solter 1994). Our first model implies that if this pressure falls below a threshold (s = 4μ/(1 +μ)), selection should indeed be too weak to maintain imprinting. For realistic mutation rates, however, this threshold is very low (e.g., if μ= 10^{6}, then s ≈ 4 × 10^{6}). We therefore conclude that the OTBH can explain the maintenance of imprinting even when the risk of ovarian cancer is very low, although that risk must be higher for imprinting to evolve in the first place.
A second objection that has been raised to the OTBH suggests that a single maternally imprinted growthfactor locus would avert the risk of ovarian trophoblast disease and that the existence of multiple such alleles represents evidence against the hypothesis (Haig 1994; Haig and Trivers 1995). This argument is valid if imprinting at one locus can effectively eliminate all possibility of unfertilized eggs spontaneously developing; however, it seems feasible that the effect could be less definitive (Iwasa 1998). We can generalize our model by assuming the existence of N loci with imprintable alleles, each of which reduces the fitness of a homozygous unimprinted female by amount s_{i}. As each imprintable allele becomes fixed, the selective pressure imposed by the remaining cancer risk is reduced, until the risk becomes too low for further imprinting alleles to invade. Given the ease with which selection at a single locus can exceed the invasion threshold given by inequality (2e), our first model suggests that imprinting could easily evolve at multiple loci via the OTBH mechanism. Thus, the OTBH can be consistent with the presence of multiple imprinting alleles (see also Iwasa 1998).
A third objection notes that while the original version of the ovarian time bomb hypothesis might elucidate a mechanism for maternal imprinting, it cannot explain the inactivation of paternal alleles (Haig 1994; Hurst 1997), except by positing that such alleles might be “innocent bystanders” (Varmuza and Mann 1994). Indeed, our second model demonstrates that neither growthenhancing nor growthinhibiting genes should evolve paternal inactivation under the original form of the OTBH. Our third model, however, confirmed the validity of Iwasa’s (1998) suggestion that paternal inactivation of growthinhibiting genes could evolve from stabilizing selection augmenting the standard OTBH. This revised version of the OTBH is therefore a viable explanation for the evolution of both maternal and paternal inactivation of growthaffecting loci in mammals.
The OTBH has also been criticized for its inability to explain imprinting in organisms other than mammals, for example, in insects and plants (Haig 1994; Moore 1994). Clearly, the reproductive systems of these groups rule out the possibility of ovarian cancer and hence any selective advantage of imprinting under the assumptions of this hypothesis. Mann and Varmuza (1994) suggest that imprinting in different phyla may have arisen for different reasons. We can only concur that, regardless of the OTBH’s ability to explain the evolution of mammalian imprinting, alternative hypotheses may be needed to account for the same phenomenon in other taxonomic groups.
Using evolutionarygenetic models to evaluate verbal hypotheses for the evolution of imprinting has been controversial. Haig (1999), for instance, claimed that such models are inappropriate because they do not apply to longterm evolutionary change. In his view, only gametheoretic models should be used to investigate hypotheses for the evolutionary origin of imprinting. There are two difficulties with Haig’s (1999) analysis, however, the first about the relationship between the two classes of models and the second concerning his description of the sort of evolutionary genetic model used above. This article is not the place for a full comparison of the applicability domains of gametheoretic and evolutionarygenetic models, but it is sufficient here to point out that the relationship between them is more complex than Haig (1999) implies and that gametheoretic models can potentially be misleading. In the context of imprinting, the models of Spencer et al. (1998; see also Spencer 2000) constructed genotypic fitnesses and tracked the dynamics of genotype frequencies. It is well known (see, e.g., CavalliSforza and Feldman 1978) that these dynamics will often differ from those believed to follow from the use of inclusivefitness arguments such as those of Haig (1992, 1997). The latter described a game “in which the players are alleles at a locus, strategies are alleles’ patterns of expression...” (Haig 1999, p. 1229). As these arguments reduce to initial increase properties of alleles, they cannot be expected to reveal the more complex dynamics of differential genotypic fitnesses. The same restrictions apply to the analyses of Mochizuki et al. (1996) and Iwasa and Pomiankowski (2001), whose models of quantitative genetic determination of imprinting also ignore the dynamic complexity that results from full genotypic analysis when fitnesses are determined by genotypes. Thus arguments phrased in terms of an “allele’s strategy” have difficulty coping with evolution that occurs subsequent to the attainment of Spencer et al.’s polymorphic equilibrium. This problem is common to a number of geneticconflict models where the treatment involves more complex genetics, such as two linked loci (Eshel and Feldman 1984; Eshelet al. 1998).
The second of Haig’s (1999) criticisms concerned Spencer et al.’s (1998) model of the geneticconflict hypothesis, which considered two alleles, an unimprintable A and an imprintable a. As in the model above, Spencer et al. (1998) were interested in the conditions under which A was displaced by a and vice versa and when the two alleles could coexist. Haig (1999) alleged that this model assumed that a was always completely inactivated when passed on by the imprinting sex and hence only qualitative differences in levels of expression were studied. But none of the algebra used this assumption; indeed Spencer et al. (1998) explicitly pointed out that such a restriction was unnecessary. We point out that a similar generalization applies to the model above: Although we motivate our modeling by describing a as being inactivated when imprinted and a^{*} as a null mutant, the algebra requires neither of these properties to be assumed. (It is crucial, however, that a and a^{*} have similar effects when passed on by the imprinting sex; this assumption could, in principle, be relaxed, at the expense of some sordid algebraic complications.) In the framework of allelic strategies, Haig (1999) is correct in pointing out that by treating the level of imprinting as continuous, he is concerned with longterm evolution in the sense of Eshel and Feldman (1984, 2001; Eshel 1996). However, if the shortterm evolutionary scale studied by Spencer et al. (1998) permits polymorphism or prevents allelic fixation (as they showed was indeed possible), then longterm arguments in which evolution steps between such states of fixation may be misleading or even wrong.
These disagreements are important in determining the novelty of our findings. For example, the evolutionarygenetic models developed here and in Spencer et al. (1998) imply that the OTBH and the geneticconflict hypothesis make an important distinguishing prediction: Polymorphism in imprinting status is present only under the latter. Crucially, this prediction can be made only using evolutionarygenetic models because only they can find polymorphic equilibria. The gametheoretic models are designed to find only fixation equilibria (Haig 1999), whereas Mochizuki et al.’s (1996) and Iwasa’s (1998) quantitativegenetic models follow only the population’s mean levels of gene expression. Our finding that polymorphism is not possible under the OTBH is thus novel because it is tacitly assumed in these other models.
Our models also suggest a second distinguishing prediction between the OTBH and the geneticconflict hypothesis: The simultaneous stability of imprinting and nonimprinting for biologically plausible parameter values occurs only under the OTBH. The possibility of such a bistable system has been previously noted in a quantitativegenetic study of the geneticconflict hypothesis (Mochizukiet al. 1996), although those authors note that this occurs only for a narrow range of parameter values (which are of questionable biological relevance; Spencer 2000). Under the present model, the simultaneous equilibria exist over a large and biologically plausible region of parameter space; moreover, the model requires no assumptions about multiple paternity or stabilizing selection. In principle, such a bistable system could explain interspecific variation in imprinting patterns, although we agree with Iwasa (1998) that parametervalue differences between species (e.g., at human and mouse Igf2r) seem a more plausible explanation. Hence, given the polymorphic imprinting status of both the IGF2R gene (Xuet al. 1993) and the WT1 gene (Jinnoet al. 1994) in humans, our models appear to lend greater support to the geneticconflict hypothesis than to the ovarian time bomb hypothesis, at least for these loci.
APPENDIX A: MATERNAL INACTIVATION
To derive equilibrium allele frequencies under model 1 (maternal inactivation), we plugged (1g) and (1h) into (1a1d), substituted f_{3} = 1  f_{1}  f_{2} and m_{3} = 1  m_{1}  m_{2}, and set
Case IA: Plugging (A5) into (A1), we find that
Case IB: Plugging (A6) into (A1) and (A2), we find
Case IC: Plugging (A7) into (A4), we find that
Subcase IC1: The first condition presented in (A13) can be written as
Subcase IC1a (h < ½): The discriminant of (A16) can be rewritten as
Subcase IC1b (h > ½): Define Q = 2s(1 μ  2h + μh), c = 8s(1 μ)(1  hs)(2h  1), d = 4s(2h  1), e = 8μs^{2}(1  h)(2h  1), and f = 4μ^{2}s^{2}(1  h)^{2}. Then c, d, e, f > 0 > Q, so we can write the minus root of (A16) as
Subcase IC2: We can plug (A7) and (A12) into (A3) to obtain
Subcase IC2b: Substituting (A14) into (A21) and solving for f_{1}, we find that
Subcase IC2c: Substituting (A14) into (A22) and solving for f_{1}, we find that
APPENDIX B: PATERNAL INACTIVATION
To derive equilibrium allele frequencies under model 2 (paternal inactivation without stabilizing selection), we plugged (3g) and (3h) into (3a3d), substituted f_{3} = 1  f_{1}  f_{2} and m_{3} = 1  m_{1}  m_{2}, and set
Case IIB: Plugging (B6) into (B4) and solving for m_{2}, we obtain
Case IIC: Solving (B4) for m_{2}, we find that
Subcase IIC1: Solving (B12) for m_{1}, we obtain
For this solution to be feasible, we must have both m_{1} > 0 and m_{1} < 1. The former condition requires that the denominator of (B12) is negative, so the latter condition can be expressed as
Subcase IIC2: Substituting both (B7) and (B13) into (B3), we obtain
Subcase IIC2a: Combining the two conditions of (B16), we find that
Subcase IIC2b: Substituting (B7), (B13), and (B17) into (B1), we obtain
By numerical analysis, at most one root of this equation is biologically feasible over the range 0 < s, t < 1, producing a unique solution for m_{1} that corresponds to equilibrium P3 (A/a^{*} polymorphism). As under the maternal case, the easiest way to calculate the other allele frequencies is to repeat the above analysis under the assumption f_{2} = m_{2} = 0; this procedure gives the same coefficients as (B23). Both approaches will therefore yield the same solution for m_{1} and hence for f_{1}, m_{2}, and f_{2}, implying that f_{2} = m_{2} = 0. Moreover, we note that equations (B17), (B22), and (B23) are, respectively, identical to (A29), (A30), and (A31) from the maternal inactivation model. Therefore, allele frequencies at equilibrium P3 are identical to those at equilibrium M3.
Subcase IIC2c: For the purposes of algebraic tractability, we hereafter assume that h = ½. We then separately examine the cases a = b = 0 (Subcase IIC2ci), a = 0 ≠ b (Subcase IIC2cii), and a ≠ 0 (Subcase IIC2ciii).
Subcase IIC2ci (a = b = 0): Substituting h = ½ into (B19), setting b = 0, and solving for m_{1}, we find that
We have already examined the first two solutions; by inspection, the third solution is negative and hence unfeasible.
Subcase IIC2cii (a = 0 ≠ b): Substituting h = ½ into (B19), setting a = 0, and solving for m_{1}, we derive
We can then solve (B18) and (B19) for f_{1} and plug in (B25) to obtain
Equation B27 implies that s = 4/(3 μ); this value is greater than one and hence unfeasible.
Subcase IIC2cii.a: Substituting back into (B25) and (B26), we obtain f_{1} = m_{1} = 0. This case has already been analyzed (Case IIB) and can therefore be excluded from further consideration.
Subcase IIC2cii.b: Solving Equation B29 for t, we obtain
Feasibility requires t < 1, which implies
This value of s is greater than one and hence unfeasible. This concludes our analysis of subcase IIC2cii.
Subcase IIC2ciii (a ≠ 0): From (B18),
As we saw in examining Equation B27, (B33) is unfeasible, so we can turn to the other two possibilities.
Subcase IIC2ciii.a: Substituting (B34) into (B19), we find that a = 0. But this contradicts our assumption for Subcase IIC2ciii. Therefore, this subcase can be excluded.
Subcase IIC2ciii.b: Solving (B35) for m_{1}, we find
Let us separately analyze the minus root (Subcase IIC2ciii.b1) and the plus root (Subcase IIC2ciii.b2) of this equation.
Subcase IIC2ciii.b1: Feasibility requires m_{1} > 0, which implies
Since s > 0, this would require
This value is greater than one and hence unfeasible.
Subcase IIC2ciii.b2: Again, feasibility requires m_{1} > 0, which implies (B37) or
We have shown that inequality (B37) is unfeasible. But inequality (B39) implies that
This value is less than zero and hence also unfeasible. This concludes our analysis of Subcase IIC2c and therefore of model 2 (paternal inactivation).
APPENDIX C: PATERNAL INACTIVATION WITH STABILIZING SELECTION
Our procedures for calculating equilibrium allele frequencies for model 3 closely followed those already outlined in appendix b; we therefore omit a detailed derivation. Equilibrium S1 (fixation of a^{*}) occurs at f_{1} = f_{2} = m_{1} = m_{2} = 0, f_{3} = m_{3} = 1. Allele frequencies at equilibrium S2 (a^{†}/a^{*} polymorphism) are given by
At equilibrium S3 (A/a^{*} polymorphism), allele frequencies are given by
These frequencies are identical to those for equilibrium P3 when h = ½.
Acknowledgments
Financial support for this work was provided by the Marsden Fund of the Royal Society of New Zealand contract UOO916 (A.E.W. and H.G.S.) and the U.S. National Institutes of Health grants GM 28016 and GM 28428 (M.W.F.).
Footnotes

Communicating editor: M. A. Asmussen
 Received February 26, 2002.
 Accepted May 28, 2002.
 Copyright © 2002 by the Genetics Society of America