The Evolution of Genomic Imprinting via Variance Minimization: An Evolutionary Genetic Model

Corresponding author: Hamish G. Spencer, Department of Zoology, University of Otago, P.O. Box 56, Dunedin, New Zealand., h.spencer{at}otago.ac.nz (E-mail)

	ABSTRACT

TOP ABSTRACT MODEL FORMULATION STABILITY ANALYSIS DISCUSSION APPENDIX LITERATURE CITED

A small number of mammalian loci exhibit genomic imprinting, in which only one copy of a gene is expressed while the other is silenced. At some such loci, the maternally inherited allele is inactivated; others show paternal inactivation. Several hypotheses have been put forward to explain how this genetic system could have evolved in the face of the selective advantages of diploidy. In this study, we examine the variance-minimization hypothesis, which proposes that imprinting arose through selection for reduced variation in levels of gene expression. We present an evolutionary genetic model incorporating both this selection pressure and deleterious mutations to elucidate the conditions under which imprinting could evolve. Our analysis implies that additional mechanisms such as genetic drift are required for imprinting to evolve from an initial nonimprinting state. Other predictions of this hypothesis do not appear to fit the available data as well as predictions for two alternative hypotheses, genetic conflict and the ovarian time bomb. On the basis of this evidence, we conclude that the variance-minimization hypothesis appears less adequate to explain the evolution of genomic imprinting.

A number of mammalian genes have different expression patterns depending on whether they are inherited maternally or paternally, a phenomenon known as genomic imprinting. The molecular mechanisms underlying imprinting are the subject of intense research efforts, but no simple story is emerging. It is clear that methylation of cytosine bases is involved (JAENISCH 1997 ), but this methylation need not be at the imprinted locus itself (JONES et al. 1998 ; REIK and WALTER 1998 ; THORVALDSEN et al. 2002 ). Moreover, other epigenetic marks are also implicated (DAVIS et al. 2000 ).

Because imprinting reduces or eliminates the ability to mask deleterious mutations, it would seem to be selectively disadvantageous, yet this pattern of expression appears to have evolved independently at multiple loci (BARTOLOMEI and TILGHMAN 1997 ). Several verbal hypotheses have been proposed to resolve this paradox and to explain some notable features of genomic imprinting, such as the fact that fetal growth-enhancing genes tend to be maternally inactivated, while fetal growth-inhibiting genes are more often paternally inactivated (HAIG and WESTOBY 1989 ; MOORE and HAIG 1991 ; HAIG 1992 ). Of these hypotheses, the genetic-conflict hypothesis (GCH; HAIG and WESTOBY 1989 ; HAIG 1992 ) is the most widely discussed. According to the GCH, a father's genetic interest under polygamy is best served by inactivating growth-inhibiting genes in his offspring, thus maximizing their survivorship even at the expense of any (half-) sibs. In contrast, a mother's genetic interest is enhanced by inactivating growth-enhancing genes, allowing her to maximize the total survivorship of all her offspring by maternally regulating their growth in utero. An alternative explanation, the ovarian time-bomb hypothesis (OTBH; HAIG 1994 ; MANN and VARMUZA 1994 ; MOORE 1994 ; SOLTER 1994 ; VARMUZA and MANN 1994 ), has also received significant attention. Under the OTBH, selection pressure to minimize females' mortality due to development of unfertilized eggs inside the ovary leads to maternal inactivation of growth-enhancing genes that might stimulate such development; stabilizing selection on overall fetal levels of growth factor then elicits the opposite expression pattern in males (IWASA 1998 ). Other plausible hypotheses also exist but remain by contrast relatively unexplored.

One such theory, the variance-minimization hypothesis (VMH), proposes that imprinting may have evolved as a means of regulating the level of gene expression (SOLTER 1988 ; HURST 1997 ). If a gene's mean level of expression is determined primarily by demand for the gene product, individuals may be able to reduce the variance in expression level by inactivating one copy of the gene while retaining a mean level close to the optimum value. Alternatively, stochastically expressed genes with low probability of transcription might undergo selection to decrease the risk of both alleles being silenced; genomic imprinting is one mechanism that could accomplish this goal (OHLSSON et al. 2001 ). Clearly, the VMH would predict that imprinting should occur most frequently in genes where variation in expression level has the most severe fitness consequences. Limited support for this prediction comes from studies demonstrating that mice heterozygous for knockout mutants at the nonimprinted genes Igf1 and Igf1r are phenotypically indistinguishable from wild-type sibs (LIU et al. 1993 ), while biallelic expression in humans of the IGF2 gene, which is ordinarily maternally inactivated, leads to overgrowth (MORISON et al. 1996 ) and predisposition to Wilms' tumor (OGAWA et al. 1993 ). Nevertheless, the apparent polymorphic imprinting in humans of the IGF2R (XU et al. 1993 ) and WT1 genes (JINNO et al. 1994 ) demonstrates that even large changes in the level of an imprinted gene's expression need not have deleterious phenotypic effects. Further objections have been raised to the variance-minimization hypothesis on the grounds that the verbal model fails to explain why not all genes are imprinted (HURST 1997 ) and why imprinting has been found only in certain taxonomic groups (HAIG and TRIVERS 1995 ).

In this article, we develop an evolutionary genetic model of the VMH. This model enables us to predict when an imprintable allele can invade a population initially fixed for an unimprintable allele, when the nonimprinting allele can invade a population initially fixed for imprinting, and when (if ever) the two alleles can coexist. We then compare the models with analogous models of the OTBH and identify areas in which the two hypotheses predict different evolutionary outcomes. Finally, we identify areas of disagreement in predictions of the VMH, the OTBH, and the GCH and discuss the support afforded to each by existing and possible future results.

	MODEL FORMULATION

TOP ABSTRACT MODEL FORMULATION STABILITY ANALYSIS DISCUSSION APPENDIX LITERATURE CITED

Consider an autosomal locus with three alleles: A, always expressed regardless of which parent it is inherited from; a, expressed when inherited from one parent but inactivated when inherited from the other; and a*, a nonfunctional mutant allele. Let p, q, and r be these alleles' respective frequencies. Both the A and a alleles mutate at some frequency to the a* allele; however, the mutation rates in females and males need not be identical. Several studies have found evidence for unequal mutation rates (HALDANE 1947 ; DRYA et al. 1989 ; CHARLESWORTH 1993 ), which could conceivably provide a mechanism for the sexual asymmetry characteristic of genomic imprinting. We therefore define µ_I as the mutation rate in gametes of the sex that inactivates the a allele when transmitting it (females under maternal inactivation, males under paternal inactivation) and µ_E as the mutation rate in gametes of the nonimprinting sex. This parameterization allows us to consider equal mutation rates (µ = µ_I = µ_E) as a special case of the general model. We ignore the possibility of back mutations (i.e., mutations from a* to either A or a) because a* will be rare at all biologically interesting equilibria (and so the products of such a process will be extremely rare indeed).

We next consider the effects of selection in the offspring. Let s be the fitness cost of the increased variance in the level of gene expression in biallelic individuals as posited by the VMH; the fitnesses of imprinting vs. nonimprinting individuals are then 1 and 1 - s, respectively. If neither allele is expressed, variance is of course zero but the lack of a functional allele may have other fitness consequences, so we assign these individuals a fitness of 1 - t. Thus, t represents the fitness cost of having no functional allele at the locus. The reasoning underlying the VMH suggests that s, t > 0; however, the model's general formulation allows us to consider cases in which selection acts to maximize variance (s < 0) and/or to favor biallelic silencing (t < 0). We require, however, that s, t < 1. We can now use Table 1 to derive the following recursions for the allele frequencies (see the Appendix),

(1a)

where

(1b)

is the mean fitness, obtained by summing the right-hand sides of Equation 1a. Although these equations were derived assuming maternal inactivation, repeating the calculations for paternal inactivation yields exactly the same set of equations. All results for the VMH reported in this article therefore apply regardless of whether the a allele is inactivated maternally or paternally.

To find the system's equilibria, we solve Equation 1a Equation 1b with p = p', q = q', and r = r', obtaining the five solutions shown in Table 2. It is important to note that although five distinct equilibria exist, individual equilibria are biologically feasible (i.e., p, q, r all between 0 and 1 inclusive) only for parameter values satisfying the feasibility criteria shown in the table. Furthermore, these allele frequencies and feasibility criteria were calculated under the assumption µ_I µ_E. The special case of equal mutation rates in the two sexes (µ = µ_I = µ_E) produces zero denominators in the feasibility criteria for equilibria 4 and 5; these terms must therefore be rederived, yielding the solutions shown in Table 3. (All derivations are shown in the Appendix.) Fig 1 and Fig 2 plot these equilibria and the system's evolutionary trajectories for two sets of parameter values.

View larger version (13K): In this window In a new window Download PPT slide   Figure 1. Example of evolutionary trajectories for the variance-minimization model with s = 4 x 10-6, t = 5 x 10-5, µ = µI = µE = 10-6. The frequency of each allele is depicted as distance from the opposite side of the triangle. For these parameter values, equilibria V2 and V4 are locally stable; V1 and V3 are locally unstable. As a result, the unimprintable allele A and the imprintable allele a can each exist in stable mutation-selection balance with the a* allele but both cannot persist in a single population.

View larger version (12K): In this window In a new window Download PPT slide   Figure 2. Example of evolutionary trajectories for the variance-minimization model with s = -4 x 10-6, t = 5 x 10-7, µ = µI = µE = 10-6. For these parameter values, equilibria V1 and V4 are locally stable; V5 is locally unstable. The nonfunctional allele a* can therefore stably exist either by itself or in mutation-selection balance with the unimprintable allele A, but the imprintable allele a cannot persist in the population.

	STABILITY ANALYSIS

TOP ABSTRACT MODEL FORMULATION STABILITY ANALYSIS DISCUSSION APPENDIX LITERATURE CITED

Near equilibrium V1 (fixation of a*), we linearize the system given by Equation A1a Equation A1b Equation A1c Equation A1d Equation A1e Equation A1f Equation A1g Equation A1h Equation A1i Equation A1j Equation A1k Equation A1l Equation A1m Equation A1n Equation A1o (see the Appendix) and solve for the eigenvalue governing the increase of the A allele when it is rare. This eigenvalue is less than one, and hence equilibrium V1 is stable to invasion by A, if

(2a)

Using the same procedure, we find that equilibrium V1 is stable to invasion by a if

(2b)

Thus, everything else being equal, A is more likely than a to invade.

At equilibrium V2 (a/a* polymorphism, A absent), the reduced two-allele system is stable when inequality (2b) is reversed; this condition is the same as that under which equilibrium V2 is biologically feasible, so the reduced system is always stable when feasible. This same equilibrium is stable to invasion by A if

(2c)

and

(2d)

Stability and feasibility of equilibrium V3 (three-allele polymorphism) are mutually incompatible. Therefore, equilibrium V3 is never stable.

At equilibrium V4 (A/a* polymorphism, a absent), the reduced two-allele system is always internally stable when feasible. This equilibrium is also always stable to invasion by a.

At equilibrium V5 (A/a* polymorphism, a absent), the reduced two-allele system is never internally stable; however, the equilibrium is always stable to invasion by a.

The feasibility and stability domains of all five equilibria for the special case µ = µ_I = µ_E are shown in Fig 3. For realistic mutation rates, this graph is visually indistinguishable from that for the general case, implying that the VMH's partition of parameter space into feasibility and stability domains is qualitatively similar regardless of any sex-biased mutation rate. Nevertheless, we must still consider the possibility that unequal mutation rates could shift these domains, causing an allele with constant selection parameters s and t to occupy a different domain (and hence potentially experience a different evolutionary fate).

View larger version (12K): In this window In a new window Download PPT slide   Figure 3. Existence and stability of equilibria in s-t phase space for a variance-minimization model with equal mutation rates in the two sexes. For the purpose of illustration, this figure assumes µ = 10-6. (The case of unequal mutation rates is visually indistinguishable unless the rate is unreasonably high, on the order of 10-2). The equilibria listed are those taking on biologically feasible values within that region of phase space; locally stable equilibria are italicized.

To address this question, we examined the system's behavior when the feasibility criteria for equilibrium V3 (three-allele polymorphism) presented in Table 2 are satisfied. Fig 3 shows that this is the only instance in which equilibrium V2 (a/a* polymorphism) is stable and hence in which genomic imprinting can be maintained. Note that V4 (A/a* polymorphism) is also stable under these conditions, so the population's initial allele frequencies will determine whether imprinting evolves. We then focused on two statistics affecting the probability that genomic imprinting will evolve: ₃, the frequency of the a allele at unstable internal equilibrium V3 (given in Table 2), and _2A, the eigenvalue for invasion of equilibrium V2 by the A allele (given by Equation A3g in the Appendix). To determine the effect of unequal mutation rates on these statistics, we estimate the change in each statistic when µ_I and µ_E are interchanged,

(2e)

where 0 < m < n < 1.

Applying this procedure to ₃, we find that

(2f)

(See the Appendix for details.) Because unstable internal equilibrium V3 lies between stable equilibria V2 and V4, ₃ corresponds approximately to the minimum frequency of a needed for the system to move from the stability domain of V4 (A/a* polymorphism) to that of V2 (a/a* polymorphism). As shown in Fig 4, ₃ is a rapidly decreasing function of s, so relatively small fluctuations in allele frequencies may suffice to permit the evolution of imprinting when the selective advantage of such imprinting is large. Inequality (2f) also implies that imprinting can evolve over a broader range of allele frequencies when µ_I > µ_E than when µ_I < µ_E and that this difference is almost exactly proportional to the arithmetic difference between the two mutation rates (for small values of µ_I and µ_E).

View larger version (9K): In this window In a new window Download PPT slide   Figure 4. Plot of 3, the frequency of a at equilibrium V3, as a function of s on a log-log scale. For the purpose of illustration, this figure assumes t = 0.3 and µI = µE = 10-6. As the benefit of imprinting s increases, this allele frequency becomes very small, making it easier for genetic drift or other processes to push the system past the selective barrier that would otherwise prevent the evolution of imprinting.

Similarly, analysis of the eigenvalue _2A yields

(2g)

Small values of _2A mean that a population within the stability domain of equilibrium V2 will converge to that equilibrium more rapidly, reducing the chance that stochastic perturbations will drive the system outside the equilibrium's stability domain. Inequality (2g) therefore indicates that unequal mutation rates will affect the system's susceptibility to perturbation and that this effect is again proportional to µ_I - µ_E. For plausible mutation rates, however, this difference will be very small (approximately on the order of 10^-6); it therefore appears unlikely that unequal mutation rates in females and males will significantly affect the evolution of genomic imprinting under the variance-minimization hypothesis.

	DISCUSSION

TOP ABSTRACT MODEL FORMULATION STABILITY ANALYSIS DISCUSSION APPENDIX LITERATURE CITED

The preceding analysis demonstrates that the variance-minimization hypothesis predicts four possible evolutionary scenarios; which of these unfolds will depend on the values of the evolutionary parameters and the initial allele frequencies. When the selective cost t of lacking a functional allele is small or negative, the null mutant allele a* will go to fixation and remain there. If s, the selective benefit of minimizing variance in gene expression, is also negative, fixation of a* and mutation-selection balance between a* and the unimprintable allele A are both stable outcomes. (We note that neither of these equilibria is probably of any real interest, since t is not likely to be small or negative given the postulate of the variance-minimization hypothesis that the expression level of the gene is critical.) For slightly larger values of t, the system will always evolve (or retain) this mutation-selection balance of the A and a* alleles. Finally, when both s and t are large relative to the mutation rate, either the unimprintable allele A or the imprintable allele a can be stably maintained in mutation-selection balance with a*. If, as seems plausible, we assume a large disadvantage for individuals lacking a functional allele (t >> µ_I, µ_E), then variance minimization need exert only a small selective pressure (s > (µ_I + µ_E)/(1 - µ_E)) to maintain imprinting once it has evolved; in Fig 3, this criterion corresponds to choosing selection parameters from the region above the hyperbola given by (2d), bearing in mind that t 1.

It is worth observing that this mathematical model of the variance-minimization hypothesis also applies to some other hypotheses that may seem conceptually unrelated to the VMH. The results described in this article follow from applying fundamental principles of population genetics to the set of genotypic frequencies and fitnesses presented in Table 1. But this table will be identical for any hypothesis that combines an intrinsic individual advantage of imprinting with an opposing disadvantage due to loss of masking. All such hypotheses are therefore formally equivalent to the VMH. Other hypotheses for the evolution of genomic imprinting, however, may present unique features distinguishing them from this VMH-related class. Under our previously published evolutionary genetic models of the ovarian time-bomb hypothesis (WEISSTEIN et al. 2002 ), for example, the selective advantage of imprinting accrues only to females, whereas both sexes benefit equally from imprinting under the VMH. Furthermore, the OTBH posits that the selective effect of imprinting depends on the expression levels in a female's unfertilized eggs rather than on her own individual expression level (cf. Table 1 in this article with Table 1 in WEISSTEIN et al. 2002 ).

The OTBH and VMH models are thus not mathematically equivalent. Nevertheless, they do employ the same parameterization: in both cases, s represents the fitness cost of biallelic expression (attributable under the OTBH to elevated risk of ovarian cancer) and t the cost of lacking a functional allele. We can therefore easily compare the results of the two models. Fig 5 and Fig 6 depict the regions of parameter space in which the models predict stable fixation of each of the three alleles. We observe that, although the models' predictions match in some regions, there are many parameter values for which they predict different results. For example, the VMH predicts that a maternally inactivated allele with s = 7.5 x 10^-6, t = 2.5 x 10^-6 cannot persist within a population, while the OTBH predicts that such an allele will go almost to fixation, eventually reaching mutation-selection balance with a*. Empirical observation of a common allele with these parameters would therefore constitute evidence favoring the OTBH over the VMH if the system could be assumed to have achieved equilibrium. In general, the VMH permits simultaneous stability of two equilibria (bistability) over a broader range of parameter values than does the OTBH; this trend is particularly pronounced for the large values of t that seem most plausible in real biological systems. In the absence of information regarding the population's initial allele frequencies, a prediction of bistability is equally consistent with observation of either stable equilibrium state, so the extensive bistability predicted under the VMH may make the hypothesis relatively difficult to falsify. For lower values of t in maternally inactivated systems, however, the two hypotheses yield distinct and unambiguous predictions (such as the numerical example given above).

View larger version (17K): In this window In a new window Download PPT slide   Figure 5. Stability of allele-fixation equilibria in s-t phase space under the variance-minimization and ovarian time-bomb hypotheses for maternal inactivation. For the purpose of illustration, this figure assumes µI = µE = 10-6. Where more than one allele is listed, fixation of each allele is a stable outcome. Dashed lines separate different outcomes under the OTBH; dotted lines separate different outcomes under the VMH.

View larger version (15K): In this window In a new window Download PPT slide   Figure 6. Stability of allele-fixation equilibria in s-t phase space under the variance-minimization and ovarian time-bomb hypotheses for paternal inactivation. Parameters and notation are as in Fig 5.

Beyond these parameter-specific differences, evolutionary genetic models of the three hypotheses discussed in this article also yield disparate predictions for the taxonomic distribution, possible evolutionary fate, and parental directionality of genomic imprinting, as summarized in Table 4. Under our model of the VMH, for example, imprinting does not depend on reproductive features specific to mammals and could therefore plausibly occur in other groups as well. The OTBH, on the other hand, explicitly assumes a risk of ovarian cancer and thus cannot easily explain the presence of imprinting in Drosophila (LLOYD et al. 1999 ), while the GCH, which invokes between-sib competition for maternal resources, could apply to any taxonomic group with sufficient maternal investment (HAIG and WESTOBY 1989 ; MOORE and HAIG 1991 ). The GCH also predicts that multiple paternity either contributes to (SPENCER et al. 1998 ) or is essential for (HAIG 1992 , HAIG 1999 ) the evolution of genomic imprinting, while neither of the other models shares this feature. These discordant predictions underscore the importance of determining the distribution of genomic imprinting across and within taxa, as such data could provide crucial evidence for or against each of these hypotheses.

Models of the three hypotheses also predict different sets of possible scenarios for the evolution of imprintable alleles. Under the GCH, for example, imprintable and unimprintable alleles can stably coexist within the same population, a result not found under the VMH or the OTBH. The apparent polymorphic imprinting status of the IGF2R (XU et al. 1993 ) and WT1 genes (JINNO et al. 1994 ) is therefore better explained by the GCH than by the other hypotheses discussed here. Conversely, bistable systems in which biallelic expression and imprinting are both stable outcomes for a given set of parameter values are possible under the latter two hypotheses but not under the GCH. This prediction may prove difficult to test; however, if two genes with similar selective parameters but opposite imprinting status were found, this observation might indicate support for the VMH and/or the OTBH.

Uniquely, our models of the VMH suggest that such bistable cases are the only instances in which imprintable alleles can be stably fixed; in other words, imprinting is never globally stable. Under the variance minimization hypothesis, therefore, imprinting can evolve from initial biallelic expression only if other mechanisms such as genetic drift perturb the system from one allele's stability domain to the other's. Similarly, once imprinting has evolved within a population, it can be lost by introduction of an unimprintable allele (via invasion or mutation) followed by genetic drift across the stability boundary. Both the GCH and the OTBH, by contrast, predict that fixation of imprinting can be globally stable for certain parameter values. This finding suggests that fewer loci should be imprinted under the VMH than under these other hypotheses, so the relatively small number of imprinted genes appears more consistent with the VMH than with either the GCH or the OTBH (contra HURST 1997 ).

Finally, our mathematical models of the variance-minimization hypothesis do not explain why most growth-enhancing genes are maternally inactivated while most growth-inhibiting genes are paternally inactivated. The models do indicate that if mutation rates differ between males and females, it is easier to evolve inactivation of the allele inherited from the more mutation-prone sex. However, this effect is extremely weak: the additional range of allele frequencies over which imprinting can evolve is on the order of the mutation rate. Moreover, this finding does not elucidate why growth-enhancing and growth-inhibiting genes should be imprinted in opposite directions. The failure of the VMH to explain this pattern, which both the GCH and the OTBH successfully predict, thus reinforces earlier objections to the verbal statement of this hypothesis on the same grounds (HURST 1997 ). As a result, if further investigations support the apparent correlation between growth-enhancing genes and maternal inactivation, this finding could be interpreted as evidence that the variance-minimization hypothesis cannot adequately explain the evolution of genomic imprinting.

	ACKNOWLEDGMENTS

We thank A. Cree for instructive discussions of reproductive physiology as well as A. Santure and two anonymous referees for comments on the manuscript. This work was funded by the Marsden Fund of the Royal Society of New Zealand contract UOO916.

Manuscript received August 14, 2002; Accepted for publication April 16, 2003.

	APPENDIX

TOP ABSTRACT MODEL FORMULATION STABILITY ANALYSIS DISCUSSION APPENDIX LITERATURE CITED

Equilibrium allele frequencies:
Recall that the A allele is always expressed, the a allele is expressed when inherited from one parent but inactivated when inherited from the other, and the a* allele is a nonfunctional mutant arising from both A and a; these alleles' respective frequencies are p, q, and r. Let us consider the case of maternal inactivation: in this case, µ_I represents the mutation rate in females and µ_E the mutation rate in males. For each of the nine possible genotypes listed in Table 1, we now multiply the frequency of the A allele in that genotype by the genotype's postmutational frequency by the genotype's fitness. We then sum over all genotypes to calculate the recursion for the frequency of the A allele in the next generation:

(A1a)

Repeating this process for the a and a* alleles and simplifying the resulting equations yields Equation 1a Equation 1b. The case of paternal inactivation yields identical recursions; all subsequent results will therefore apply to both the maternal inactivation and the paternal inactivation cases.

Substituting (1b) into (1a) and recalling that q = 1 - p - r, we obtain

(A1b)

and

(A1c)

At equilibrium, p' = p and r' = r. Making these substitutions and solving (A1b) for p, we find

(A1d)

or

(A1e)

or

(A1f)

Case A:
Substituting (A1d) into (A1c) and solving for r, we obtain

(A1g)

or

(A1h)

Solution (A1g) corresponds to equilibrium V1; solution (A1h), to equilibrium V2.

Case B:
Substituting (A1e) into (A1c) and solving for r, we obtain

(A1i)

or

(A1j)

where

(A1k)

Solution (A1i) corresponds to equilibrium V3, the plus root of (A1j) corresponds to equilibrium V4, and the minus root of (A1j) corresponds to equilibrium V5. For each of these equilibria, once r is known, we can use Equation A1e to find p and then calculate q as 1 - p - r.

Case C:
Solving (A1f) for r yields

(A1l)

Therefore,

(A1m)

We then plug this result back into (A1l) to obtain

(A1n)

Finally, we can substitute (A1m) and (A1n) into Equation A1b to obtain

(A1o)

However, these allele frequencies are identical to those obtained by substituting Equation A1m into equilibria V4 and V5. Therefore, this case does not yield any novel equilibria.

Feasibility criteria:
At equilibrium V2 (a/a* polymorphism), p = 0. Therefore, q = 1 - r, so the equilibrium is feasible if

(A2a)

and

(A2b)

Solving (A2a) yields

(A2c)

Solving (A2b) given that (A2c) is true yields

(A2d)

By definition, t < 1, so (A2c) is always true whenever inequality (A2d) is satisfied. Therefore, equilibrium V2 is feasible if (A2d) holds.

At equilibrium V3 (three-allele polymorphism), feasibility requires that

(A2e)

(A2f)

(A2g)

(A2h)

(A2i)

(A2j)

We consider four cases in turn.

Case A: s < 0 and s + t > 0:
In this case, (A2g) implies that

(A2k)

The left-hand side of this inequality is positive, so the right-hand side must also be positive, implying t < 0. But this contradicts our assumption that s + t > 0. Therefore, feasibility is not possible in this case.

Case B: s < 0 and s + t < 0:
In this case, (A2g) implies that

(A2l)

which is just the reverse of (A2k). The left-hand side of (A2l) is negative, so the right-hand side must also be negative, implying t > 0. However, solving (A2j) under the assumptions of case B yields t < 0. Thus, feasibility is also not possible in this case.

Case C: s > 0 and s + t < 0:
As in case B, (A2j) yields t < 0. Solving (A2i), we obtain µ_Et > s(1 - µ_E). But the left-hand side is negative and the right-hand side positive, so this inequality cannot hold. Therefore, feasibility is again not possible.

Case D: s > 0 and s + t > 0:
Solving (A2j), we obtain

(A2m)

Solving (A2g) yields inequality (A2l). But 0 < s < 1 and 0 < t < 1, so

(A2n)

and inequality (A2l) is always true.

Solving (A2f), we obtain

(A2o)

Since s > 0, the right-hand side of (A2o) is positive. If

(A2p)

(A2o) will always be true. If instead (A2p) is violated, then it can be proven that

(A2q)

Since t < 1, this again implies that (A2o) is satisfied.

Solving (A2e) yields

(A2r)

and

(A2s)

Solving (A2h), we obtain

(A2t)

and

(A2u)

Finally, solving (A2i) yields

(A2v)

We then combine feasibility criteria (A2m), (A2r), (A2s), (A2t), (A2u), and (A2v), obtaining

(A2w)

Therefore, equilibrium V3 is feasible if inequalities (A2w) are satisfied.

At equilibrium V4 (A/a* polymorphism), q = 0 and p = 1 - r. Therefore, feasibility requires that

(A2x)

and

(A2y)

Solving (A2y), we obtain

(A2z)

We then combine (A2x) with (A2z) and simplify to obtain the feasibility conditions for equilibrium V4,

(A2a)

where

(A2b)

In the special case µ = µ_I = µ_E, one of the feasibility conditions given by (A2aa) becomes undefined. We must therefore replace µ_I and µ_E with µ in both (A2x) and (A2z) before combining the two inequalities to obtain the special case's feasibility conditions for equilibrium V4:

(A2c)

At equilibrium V5 (A/a* polymorphism), we again have p = 1 - r, so the equilibrium is feasible if both (A2x) and

(A2d)

are satisfied. If s + t > 0, (A2dd) implies that

(A2e)

But, given that t must be less than one, (A2ee) is never true, so feasibility is possible only if s + t < 0. We then combine (A2x) with (A2dd) and simplify to obtain the feasibility conditions for equilibrium V5:

(A2f)

As before, if µ = µ_I = µ_E, we must replace µ_I and µ_E with µ before solving for equilibrium V5's feasibility conditions in the special case of equal mutation rates:

(A2g)

Stability criteria:
At equilibrium V1 (fixation of a*), the eigenvalue governing the increase of A when rare is given by

(A3a)

We then derive the stability condition by solving < 1, which yields

(A3b)

At this same equilibrium, the associated eigenvalue for a is

(A3c)

The equilibrium is therefore stable to invasion by a if

(A3d)

At equilibrium V2 (a/a* polymorphism, A absent), the eigenvalue of the two-allele system is given by

(A3e)

Solving < 1, we find that the corresponding stability condition is

(A3f)

At this same equilibrium, the eigenvalue governing the increase of the A allele when rare is given by

(A3g)

The stability conditions are then

(A3h)

At equilibrium V3 (three-allele polymorphism), the two eigenvalues are given by

(A3i)

where

(A3j)

Stability requires -1 < ₁ < 1 and -1 < ₂ < 1. Solving ₂ < 1 yields

(A3k)

We have already shown that equilibrium V3 can be feasible only if s + t > 0. Furthermore, we know from (A2w) that

(A3l)

and from (A2w) that

(A3m)

The last term of (A3k) is therefore positive, so the inequality reduces to

(A3n)

However, we know from the equilibrium's feasibility conditions that s + t - 2st > s + t > 0, so inequality (A3n) cannot hold. Therefore, ₂ > 1, so equilibrium V3 is never stable.

At equilibrium V4 (A/a* polymorphism, a absent), the eigenvalue of the two-allele system is given by

(A3o)

where

(A3p)

and R is given by (A1k). Solving < 1 yields

(A3q)

where

(A3r)

It can be shown that (A3q) follows from the feasibility criteria (A2aa) and the requirements s, t < 1 and 0 < µ < 1. Therefore, equilibrium V4 is internally stable whenever it is feasible.

At this same equilibrium, the eigenvalue governing the increase of the a allele when rare is

(A3s)

Solving < 1 yields

(A3t)

where

(A3u)

Again, (A3t) follows from feasibility criteria (A2aa), so equilibrium V4 is always stable to invasion by a.

At equilibrium V5 (A/a* polymorphism, a absent), the eigenvalue of the two-allele system is given by

(A3v)

where all terms are as defined in (A3p). Solving < 1 yields

(A3w)

where c₁, c₂, and c₃ are given by (A3r). Using the "InequalityInstance" function in Mathematica (WOLFRAM RESEARCH 2000 ), it can be proven that (A3w) is incompatible with the equilibrium feasibility conditions given by (A2ff) and therefore that equilibrium V5 is never internally stable.

At this same equilibrium, the eigenvalue governing the increase of the a allele when rare is

(A3x)

The equilibrium can therefore be invaded by a when > 1. Solving this inequality yields

(A3y)

where B₁, B₂, and B₃ are given by (A3u). Again using the InequalityInstance function in Mathematica (WOLFRAM RESEARCH 2000 ), we find that inequalities (A3y) and (A2ff) are incompatible, so the a allele can never invade equilibrium V5.

Quantitative effect of unequal mutation rates:
Let us consider the region of parameter space in which equilibrium V3 (three-allele polymorphism) is feasible; that is, when inequalities (A2w) are satisfied. Define ₃ as the frequency of the a allele at this equilibrium:

(A4a)

Let m and n be the mutation rates in the two sexes, with 0 < m < n < 1. Therefore, when µ_I > µ_E, we have (µ_I, µ_E) = (n, m), and ₃ for this case is given by

(A4b)

Likewise, if µ_I < µ_E, then (µ_I, µ_E) = (m, n), yielding

(A4c)

The difference in the equilibrium allele frequency of a between the two cases is therefore given by

(A4d)

From (A2w), we have that

(A4e)

and

(A4f)

These inequalities imply that s, t > 0. Therefore,

(A4g)

The InequalityInstance function in Mathematica (WOLFRAM RESEARCH 2000 ) also reveals that

(A4h)

Now let _2A be the eigenvalue for invasion of equilibrium V2 by the A allele, given by Equation A3g. As for ₃, we can then define

(A4i)

when (µ_I, µ_E) = (n, m), and

(A4j)

when (µ_I, µ_E) = (m, n). The difference between the eigenvalues in the two cases is therefore

(A4k)

Differentiating this result with respect to t, we obtain

(A4l)

Because this partial derivative is negative, we can find an upper bound for _2A by setting t equal to its minimum value, given by (A4f):

(A4m)

Plugging (A4m) into (A4k) and differentiating the result with respect to s yields

(A4n)

This partial derivative is positive, so to find the upper bound for _2A we must set s equal to its maximum value, which is one. Doing so and substituting again into (A4k), we find that

(A4o)

To find the lower bound for _2A, we set t equal to its maximum value [because the partial derivative in (A4l) is negative], which is one. Plugging in, solving, and differentiating with respect to s now yields

(A4p)

This partial derivative is also negative, so to obtain the lower bound we must set s also equal to its maximum value of one. We then find that

(A4q)

The InequalityInstance function in Mathematica (WOLFRAM RESEARCH 2000 ) reveals that for realistic mutation rates (i.e., n < 10^-2)

(A4r)

Combining (A4o) and (A4r),

(A4s)

	LITERATURE CITED

TOP ABSTRACT MODEL FORMULATION STABILITY ANALYSIS DISCUSSION APPENDIX LITERATURE CITED

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