One- and Two-Locus Population Models With Differential Viability Between Sexes: Parallels Between Haploid Parental Selection and Genomic Imprinting

Abstract

A model of genomic imprinting with complete inactivation of the imprinted allele is shown to be formally equivalent to the haploid model of parental selection. When single-locus dynamics are considered, an internal equilibrium is possible only if selection acts in the opposite directions in males and females. I study a two-locus version of the latter model, in which maternal and paternal effects are attributed to the single alleles at two different loci. A necessary condition for the allele frequency equilibria to remain on the linkage equilibrium surface is the multiplicative interaction between maternal and paternal fitness parameters. In this case the equilibrium dynamics are independent at both loci and results from the single-locus model apply. When fitness parameters are additive, analytic treatment was not possible but numerical simulations revealed that stable polymorphism characterized by association between loci is possible only in several special cases in which maternal and paternal fitness contributions are precisely balanced. As in the single-locus case, antagonistic selection in males and females is a necessary condition for the maintenance of polymorphism. I also show that the above two-locus results of the parental selection model are very sensitive to the inclusion of weak directional selection on the individual's own genotypes.

PARENTAL genetic effects refer to the influence of the mother's and father's genotypes on the phenotypes of their offspring, not attributable just to the transfer of genes. Examples have been documented across a wide range of areas of organism biology; see, for example, Table 1 in Wade (1998) and Tables 1 and 2 in Rasanen and Kruuk (2007). Parental selection is a more formal concept used in theoretical modeling and concerns situations where the fitness of the offspring depends, besides other factors, on the genotypes of its parent(s) (generalizing from Kirkpatrick and Lande 1989).

TABLE 1.

Frequencies of genotypes and fitness parameterizations in model 1

	Gametes/haploids	Frequency before selection	Fitness
Zygote					Male	Female
(A)A	A	pfpm	1 − α	1 − δ
(A)a	1/2 A 1/2 a	pf(1 − pm)	1	1
(a)A	1/2 a 1/2 A	(1 − pf)pm	1 − α	1 − δ
(a)a	A	(1 − pf)(1 − pm)	1	1

Parentheses in the first column indicate maternal genotype (parental selection model) or inactivation of the maternally derived allele (imprinting model). Whether selection occurs at the diploid (first column) or subsequent haploid (second column) stage does not change the resulting allele frequencies.

TABLE 2.

Offspring genotypic proportions from different mating types, sorted among four phenotypic groups/combinations of maternal and paternal effects: model 2

		Offspring genotypes/phenotypes
Parental genotypes		Paternal (φ = 1)				Joint (φ = 4)
Male	Female	AB	Ab	aB	Ab	AB	Ab	aB	ab
AB	AB					1
	Ab
	aB
	ab	(1−r)/2	r/2	r/2	(1−r)/2
Ab	AB
	Ab		1
	aB					r/2	(1−r)/2	(1−r)/2	r/2
	ab
		Offspring genotypes/phenotypes
Parental genotypes		Maternal (φ = 2)				None (φ = 3)
Male	Female	AB	Ab	aB	Ab	AB	Ab	aB	ab
aB	AB
	Ab					r/2	(1 − r)/2	(1 − r)/2	r/2
	aB			1
	ab
ab	AB	(1 − r)/2			(1 − r)/2
	Ab
	aB
	ab								1

Another well-known parent-of-origin phenomenon is genomic imprinting. Here, the level of expression of one of the alleles depends on which parent it is inherited from. Often it is difficult to tell apart the phenotypic patterns due to parental effects and genomic imprinting, and thus a problem arises in the process of identifying the candidate genes for such effects (Hager et al. 2008). Analytic methods (Weinberg et al. 1998; Santure and Spencer 2006; Hager et al. 2008) have been developed to quantify subtle differences between the two. In this article, I point out that a simple mathematical model, first suggested for genomic imprinting at a diploid locus, can be interpreted, without any formal changes, to describe parental selection on haploids.

While there has been much progress in understanding the evolution of genomic imprinting (Hunter 2007), including advances in modeling (Spencer 2000, 2008), the population genetics theory of parental effects received less attention. Existing major-locus effect models of parental selection are single-locus, two-allele, and mostly concern uniparental (maternal) selection (Wright 1969; Spencer 2003; Gavrilets and Rice 2006; Santure and Spencer 2006), with only one specific case where the fitness effects of both parents interact studied by Gavrilets and Rice (2006). No attempt to extend this theory into multilocus systems has yet been made. Considering a two-locus model with both parents playing a role in selection on the offspring is called for by the observation that many maternal and paternal effects aim at the different traits or different life stages of their progeny. Among birds, for example, body condition soon after hatching is largely determined by the mother, while paternally transmitted sexual display traits develop much later in life (Price 1998). Such effects are therefore unlikely to be regulated within a single locus. Sometimes the effects are on the same trait, but still attributed to different loci: expression of gene A^vy that causes the “agouti” phenotype (yellow fur coat and obesity) in mice is enhanced by maternal epigenetic modification (Rakyan et al. 2003), while paternal mutations at the other locus, MommeD4, contribute to a reverse phenotypic pattern in the offspring (Rakyan et al. 2003). The epigenetic state of the murine Axin^Fu allele is both maternally and paternally inherited (Rakyan et al. 2003).

Focusing selection on haploids reduces the number of genotypes that need to be taken into account, while preserving the main properties of the multilocus system. Genes with haploid expression and a potential of parental effects can be found in two major taxonomic kingdoms. A notable candidate is Spam1 in mice, which is expressed during spermogenesis and encodes a factor that enables sperm to penetrate the egg cumulus (Zheng et al. 2001). This gene remains a target for effectively haploid selection, because its product is not shared via cytoplasm bridges between developing spermatides. Mutations at Spam1 alter performance of the male gametes that carry it and might indirectly, perhaps by altering the timing of fertilization, affect the fitness of the zygote. The highest estimated number of mouse genes expressed in the male gametes is currently 2375 (Joseph and Kirkpatrick 2004), and one might expect some of them to have similar paternal effects. Plants go through a profound haploid stage in their life cycles, and genes involved at this stage have an inevitable effect on the fitness of the future generations. In angiosperms, seed development is known to be controlled by both maternal (Chaudhury and Berger 2001; Yadegari and Drews 2004) and paternal (Nowack et al. 2006) effect genes, expressed, respectively, in female and male gametophytes.

Under haploid selection, there can be no overdominance, and thus polymorphism is much more difficult to maintain than in diploid selection models (summarized in Feldman 1971). Nevertheless, differential or antagonistic selection between sexes can lead to a new class of stable internal equilibria in the diploid systems (Owen 1953; Bodmer 1965; Mandel 1971; Kidwell et al. 1977; Reed 2007), and I make use of this property in the haploid models developed below. In the experiment by Chippindale and colleagues (Chippindale et al. 2001), ∼75% of the total fitness variation in the adult stage of Drosophila melanogaster was negatively correlated between males and females, which suggests that a substantial portion of the fruit fly expressed genome is under sexually antagonistic selection. I assume that the effect of either parent on the fitness of the individual depends on the sex of the latter, which in respect to modeling is equivalent to the assumption of differential viability between the sexes in the progeny of the same parent(s). Biological systems that satisfy the latter assumptions can be found among colonial green algae: many members of the order Volvocales are haploid except for the short zygotic stage, and during sexual reproduction, they are also dioecious and anisogametic. I return to this example in the discussion. The possibility that genes expressed in animal gametes may be under antagonistic selection between sexes has been discussed (Bernasconi et al. 2004). For example, a (hypothetical) mutation increasing the ATP production in mitochondria would be beneficial in sperm, because of the increased mobility of the latter, but neutral or detrimental in the egg, due to a higher level of oxidative damage to DNA (Zeh and Zeh 2007).

My main purpose was to derive conditions for existence and stability of the internal equilibria of the model(s). I begin with a simple one-locus case, which can be analyzed explicitly, and show how these one-locus results can be extended to the case of two recombining loci with multiplicative fitness. Then, I assume an additive relation between the maternal and paternal effect parameters and study the special cases where parental effects are symmetric.

ONE LOCUS

Model 0—haploid viability selection:

I first examine a standard haploid single-locus diallelic model with viability (gametic) selection (Bürger 2000). Constant selection pressure, random mating, unlimited population size, and discrete generations are assumed. When there are no differences between sexes, the fitter of the two gametes, A or a, eventually gets fixed and no polymorphic equilibrium is possible. Let us examine a special case where viabilities of the gamete A differ between males and females, and the viabilities of the gamete a = 1 regardless of sex. The frequency of A in the sex i (i = m for males and i = f for females), after meiosis and selection, is , where and are the preselection frequencies of A in males and females, respectively, and is the fitness of A in sex i. Let and , for males and females, respectively. A single polymorphic equilibrium is given by

(1)

The equilibrium (1) exists, and is stable, if and only if

(2)

Reversing left and right parts of the inequality (2) gives the conditions for stability of fixations at p_m,f = 0, p_m,f = 1, respectively. [Analyses of stability of all gene frequency equilibria presented in this article were done by examining the Jacobian matrix of partial derivatives at the equilibrium point, unless stated otherwise. Details are available as supporting information, File S1 (Mathematica 5.2 notebook).]

Model 1—parental selection/imprinting:

In this model, the fitness of the haploid individual depends on the genotype of its haploid parent, but not on its own genotype. An allele A is assumed to have an effect on the offspring fitness, when it is found in only one of the parents (assumed to be a father in the following), whereas an a allele has no effect. This model is mathematically equivalent to the model of maternal imprinting with complete inactivation of the imprinted allele (Pearce and Spencer 1992; Anderson and Spencer 1999). Since in the latter model individuals are functionally haploid at the imprinted locus, their viabilities depend on the genotype of the paternal gamete, which in terms of the parental selection model is equivalent to a haploid father (Table 1). When sex is irrelevant to selection, both models have the same dynamics as model 0 (as pointed out by Pearce and Spencer 1992), with selection acting on paternal gametes only. When individual fitness depends on sex, the imprinting model is not formally equivalent to any other two-sex viability or fertility model, but it remains equivalent to the parental selection model. Its dynamics have been examined by Anderson and Spencer (1999). I repeat their results here, using the interpretation of parental selection. Let the paternal frequencies of the A and a alleles be and , respectively, and let and be the maternal frequencies of the same alleles. As before, when equations are the same in both sexes, a subscript i is used in the gene frequency and fitness notations to indicate sex (i = m for males and i = f for females). Let the fitnesses of the progeny of a father of genotype A be w_i, and since a was postulated to have no parental effect, it is reasonable to assume that the fitness of the offspring from this paternal genotype is 1. At the next generation,

(3)

If written down for and , and with appropriate change in the fitness notations, Equation 3 is identical to (1) in Anderson and Spencer (1999). Let and . Two fixation and one internal equilibria are possible, the latter taking the form

(4a)

(4b)

As Anderson and Spencer (1999) have shown, conditions for existence and stability of the equilibrium (4) are identical to those derived by Bodmer (1965) for his model of differential fertility/viability between sexes (also see Gavrilets and Rice 2006, p. 3033) and by Cooper (1976) for his model of the X-inactivation system in marsupials. Comparison with the results from the previous section reveals that they are also exactly the same as conditions (2) in model 0 of haploid viability selection. As is shown later on, there are cases where the same or very similar sets of inequalities define existence and stability of the polymorphic equilibria in the two-locus model.

TWO LOCI

Model 2—maternal and paternal selection:

This model is a two-locus extension of model 1. As before, the allele A at the locus A/a is assumed to have a paternal effect on the offspring fitness. An allele B with the frequency u at the second locus, B/b, is assumed to alter the offspring fitness only when it is found in the mother, i.e., it has a maternal effect, while an allele b with the frequency has no effect. The offspring fitness is therefore dependent on the genotypes of both parents; I say that those resulting from the mating between A father and B mother experience a joint parental effect, while those from the union of a father and b mother have received “none” effect. The offspring phenotypes are thus divided into four groups (paternal, maternal, none, and joint parental effects), and the subscript , ranging from 1 to 4, respectively, added to the genotypic frequency notation, denotes the frequency of this genotype within the corresponding phenotypic group. Proportions of the mating types resulting in different genotypes/phenotypes are given in Table 2; the frequency of a particular type after meiosis is obtained by summing the products of the corresponding male and female parental frequencies, multiplied by the factors from respective table entries. The recursion

(5)

can be drawn, where is the frequency of a genotype x, i.e., AB, Ab, aB, or ab, within the phenotypic group (ranging from 1 to 4), before selection. is the frequency of x in sex i, after sex-specific selection. is the fitness of the phenotype in sex i, and is the mean fitness of this sex. can be conveniently expressed in terms of allele frequencies, by noting that the frequencies of offspring phenotypes are simply the outer products of the allele frequencies at the paternal effect locus in males and the maternal effect locus in females. Following selection, we have

(6)

where is the corresponding (with subscript ranging from first to fourth) element of before selection and is the postselection frequency of the phenotype in sex i. The mean fitness, , is therefore

(7)

To adequately link the allele and genotype frequencies, two measures of disequilibria between loci, D_m and D_f, among males and females, respectively, need to be included (Table 3). Note from Tables 1 and 2 that genotype frequencies now depend on the recombination rate between loci, r.

TABLE 3.

Genotype frequencies in parents expressed through allele frequencies: model 2

	AB	Ab	aB	Ab
Males (Fm)	pmum + Dm	pmvm − Dm	qmum − Dm	qmvm + Dm
Females (Ff)	pfuf + Df	pfvf − Df	qfuf − Df	qfvf + Df

It is easy to see that model 2 applies to both parental selection and genomic imprinting just as model 1 does. The former model might be interpreted such that locus A/a is maternally inactivated, while locus B/b is silenced paternally. Fitness parameters therefore correspond to individual alleles, and selection acts on the genotypically diploid, but functionally haploid individuals. To avoid confusion, the results of model 2 are presented using the terms of parental selection, but the fact that they can have dual biological interpretation must be kept in mind.

Polymorphism is maintained at linkage equilibrium (D = 0) under multiplicative fitness:

Let us first examine a simplified case where loci are unlinked (r = ) and the population is at linkage equilibrium (). It is easily shown (appendix) that in the next generation, linkage equilibrium is maintained (), if and only if

(8)

That is, in the absence of the physical linkage between loci, the population remains at linkage equilibrium, if, and only if, for both male and female offspring, the fitness product of the individuals provided with only paternal and only maternal effect is equal to the fitness product of the individuals provided with neither effect and with the joint effect of both parents. Note that for r ≠ , can be nonzero for a range of allele frequencies and fitness parameters (Equation 2 in the appendix). Equation 8 is analogous to those defining multiplicative nonepistasis in the standard two-locus viability selection models (Karlin 1975). Let us now assume, as in model 1, that the fitness of individuals not experiencing any (none) parental effect(s) is . The modified condition (8), , sets limits of a special multiplicative fitness case in the parental selection model. Now relax the initial assumptions of r = and , and substitute with in the general recursions (5). The resulting dynamic system has a single polymorphic equilibrium at , where the allele frequencies at maternal and paternal effect loci are independent of each other. That is, (i) multiplicative fitness ensures that any association between loci is eventually eliminated from the population as the allele frequency equilibrium is approached, and (ii) the frequencies and of the A allele at equilibrium are determined solely by the fitness parameters and , i.e., by a paternal effect on the fitness of males and females, but not by maternal effect parameters and ; analogously, equilibrium dynamics at the B/b locus depend only on and but not on and . The internal equilibria at each locus are (independently) described by Equation 4 and conditions for their existence and stability by Equation 2, as in the single-locus model 1. Note that stability conditions of equilibria in the multiplicative case are also independent of the recombination rate r (see the appendix).

Additive fitness—general case:

An alternative to the assumption that parental selection acts multiplicatively is to assume that maternal and paternal effects are added when provided to the same individual. The resulting fitness scheme is different, however, from the additive nonepistasis sensu Karlin (1975), because I make no assumption about the parental effects of the alternative alleles a and b, but simply leave the phenotypes not experiencing any (none) effect with the fitness = 1. Fitnesses of the parentally induced phenotypes are hereby divided into two additive components, one of which is the constant = 1, and the second is the actual contribution from parental effect(s). This approach is reflected in the following fitness matrix:

Table 4.

	Effect on fitness
Phenotype	Paternal (w1)	Maternal (w2)	None (w3)	Joint (w4)
Males (wm)	1 − α	1 − β	1	1 − α − β
Females (wf)	1 − δ	1 − γ	1	1 − δ − γ

Unlike the multiplicative fitness case in the previous section, the additive fitness case cannot be simply reduced to a one-locus system. It is useful therefore to visualize the number of possible equilibria in the genotype frequency tetrahedron (Figure 1). The population at any time is described by two points, indicating genotype frequencies among males and females, respectively, inside the tetrahedron. The space inside the tetrahedron indicates possible two-locus polymorphisms, the corner vertices represent fixations at both loci, and the edges represent polymorphisms with one locus fixed and the other polymorphic. The equilibria at the edges have close parallels with the single-locus model 1. For example, the equilibrium at the AB–aB edge has the form

(9a)

(9b)

And the equilibrium at the Ab–ab edge is obtained from (9) by substituting β, γ with 0. The resulting equations are identical to (4) that describes internal equilibrium in the single-locus case (model 1).

Figure 1.—

Geometrical representation of genotype frequencies in a haploid two-locus system. The population at any time is described by two points in a tetrahedron, one each for males and females (shown exaggeratedly different). The corners of the tetrahedron are fixation equilibria, and the edges marked by the shaded and solid lines are the equilibria fixed at A/a and B/b loci, respectively, and polymorphic for the other locus. The space within the tetrahedron shows all possible two-locus polymorphisms in the population.

Numerical results:

Analytical treatment of the two-locus model with additive fitness is difficult due to the number of independent parameters involved. I therefore start with numerical analysis of the general additive case, while the analytic results for several special cases are presented in the next section. A series of 10⁵ iterative simulations was run with parameters , and sampled uniformly from the interval [1, −2], ensuring that and do not exceed 1. Fifty sets of the initial frequencies of genotypes, with neither genotype fixed, were drawn by the broken-stick method for each parameter set, with recombination rate r varying uniformly from 0 to . A series of iterations was considered to reach its equilibrium point when the computed change in the genotype frequencies was <10⁻³ in 1000 generations.

Two-locus polymorphism requires symmetry in fitness parameters:

Among the numerically obtained equilibria, the two-locus polymorphisms were rare: only 21 of 10⁵ parameter combinations were able to maintain intermediate frequencies at both loci. Furthermore, all these combinations were characterized by a certain symmetry of parameter values and fell into the following categories: (i) antagonistic selection between phenotypes, i.e., and ; (ii) equal contributions from both parents, i.e., and ; and (iii) antagonistic selection between sexes, i.e., and . Here, the approximation sign means that the difference between the corresponding parameter values used in the simulations was <10⁻³. Further analysis of cases i–iii above is presented in the next section.

Edge equilibria (one-locus polymorphism):

Whether the edge equilibrium (4) or (9) was the outcome of the simulation appeared to be determined by a few simple inequalities. First, in all runs that resulted in the equilibrium point with one locus fixed and the other retaining polymorphism, the sex-specific fitness parameters at the polymorphic locus were of opposite sign; that is, equilibrium with polymorphism at the A/a locus and fixation at the B/b locus required that or ; likewise, or was required for polymorphism at the B/b locus and fixation at the A/a locus. Another strictly necessary, though by no means complete, condition for the equilibrium to be found at the Ab–ab (polymorphism at the A/a locus and fixation for the b allele) and AB–Ab (polymorphism at the B/b locus and fixation for the A allele) edges of the genotype frequency tetrahedron was

(10a)

That is, when paternally induced phenotypes were under weaker selection than maternally induced ones, the frequency of the A allele in the edge equilibrium was always higher than the frequency of the B allele. Likewise, fixation at the AB–aB (polymorphism at the A/a locus and fixation for the B allele) and aB–ab (polymorphism at the B/b locus and fixation for the a allele) edges of the genotype frequency tetrahedron was possible only if

(10b)

That is, a maternal effect weaker relative to the paternal one always resulted in the higher frequency of the B allele at the edge equilibrium.

Analytical results—special cases:

The following simplifications (i–iv) of the additive fitness case allowed the analysis of existence and stability of fixation and polymorphic equilibria. Specifically, cases i–iii correspond to the three combinations of parameter values for which the numerical simulations suggested occurrence of the two-locus polymorphism.

i. Antagonistic selection between phenotypes:

Let and in the general additive case. That is, among the affected phenotypes, the increase (decrease) in fitness due to the paternal effect is equal to the fitness loss (gain) due to the maternal effect. Analysis shows that neither fixations for the AB or ab genotype nor edge equilibria analogous to (4) and (9) are stable. Conditions for the stability of fixations for the Ab and aB genotypes are in fact the same as those for the corresponding fixations for A and B alleles in the one-locus case (Figure 2). That is,

(11a)

(11b)

One can see that inequalities (11a) and (11b) are symmetrical relative to the line and inequality (11b) is identical to the reversed right-hand part of (2) in the haploid viability model 0 and in model 1 of single-locus parental selection/genomic imprinting [see inequality (7) in Anderson and Spencer 1999; compare also Figure 2 here with Figure 1 in their article]. Recall that in the single-locus model 1, instability of fixations means stability of the internal equilibrium. Since it can be shown that the edge equilibria are always unstable in the case above (see File S1), it is worth checking whether a two-locus polymorphism is stable in the interval of . Repeating the numerical results from the previous section, with the parameter values forced to satisfy the above inequality, demonstrates that the two-locus polymorphism is always stable, though no analytical treatment is possible. In the following cases (ii and iii), conditions for stability of the two-locus polymorphism are confirmed numerically in a similar manner.

Figure 2.—

Stability of equilibria in the fitness parameter space under the assumption of antagonistic selection between phenotypes: special case i of the additive fitness scheme. Fixations for Ab and aB genotypes are stable above the and below the lines, respectively. The space in between the lines represents the stability range of the two-locus polymorphism. The dashed line divides the stability ranges of the internal (Equation 4) and fixation (p = 0) equilibria in the single-locus model 1 (see text).

ii. Equality of maternal and paternal effects:

Let and . Also let the fitnesses of phenotypes affected by both parents be and , for males and females, respectively. The resulting fitness scheme is not a variant of the general additive case, except for t = 2. The outcome of the local stability analysis depends on whether parameter k is larger or <1: if t > 1, fixations for Ab and aB genotypes are always unstable, while conditions for stability of fixations for AB and ab genotypes take the form

(12a)

(12b)

and the two-locus polymorphism, as seen from the numerical explorations, appears to be stable when both inequalities are reversed:

(12c)

If t < 1, conditions (12b) and (12c) hold true, but for the fixation of the AB genotype to be stable, inequality (12a) must be reversed. The fixation for AB therefore shares stability range with either two-locus polymorphic equilibrium or the fixation for the ab genotype. The corner equilibria Ab and aB, which were unstable for t > 1, can both be simultaneously stable if t < 1 and if condition (12a) is satisfied (Figure 3). As numerical tests show, which of the corner equilibria is reached appears to be determined by the initial frequency ratio of A and B alleles: genotype Ab becomes fixed if , and, respectively, aB is fixed when in the initial population. Note that this condition holds for whatever small values of p and u and is independent of α and δ. For example, a maternal effect allele can reach fixation from small initial frequency, provided this frequency is still higher than that of the allele with paternal effect, even if it benefits sons at the expense of daughters.

Figure 3.—

Stability of equilibria in the fitness parameter space under the assumption of equality of maternal and paternal effects: special case ii. The solid line is , and the other two represent for t = 2 (thick dashed line) and t = (thin dashed line). At the point only a two-locus polymorphism is stable for t = 2 but fixation for the AB genotype and the two-locus polymorphism are simultaneously stable for t = .

iii. Antagonistic selection between sexes:

Let and . That is, selective advantage in males (females) brought about by the paternal (maternal) effect is precisely balanced by the fitness loss due to the same effect in the other sex. In this case, fixation is stable at neither one nor both loci, but numerical results from the previous section suggest that a two-locus polymorphism is globally stable.

iv. No difference in fitness between sexes:

Let ; i.e., there is no difference in fitness between males and females. I have already mentioned that, without selection being sex-specific, the single-locus model 1 reduces to the haploid viability model, which has only trivial equilibria. Similar equilibrium dynamics are observed at the D = 0 surface in the two-locus model 2 with additive fitnesses. That is, global stability of the fixations of A and/or B alleles is ensured by the corresponding fitness parameter ( and/or ) being negative, and fixations for a and/or b are globally stable otherwise. Away from the linkage equilibrium (D ≠ 0), a pair of feasible polymorphic equilibria exists, in which the frequency of one allele is conditioned by the other, the recombination rate r depends on the allele frequency, and D is equal among males and females,

(13a)

and

(13b)

and

(13c)

where .

It can be seen that if u = 1 or 0 in Equation 13a, p can also take only values of 1 and 0, which implies that (13) cannot be reduced to the edge equilibrium. Although equilibria (13) can be feasible (the full conditions of existence and feasibility can be found in File S1), one eigenvalue of the system's Jacobian at the corresponding points is always greater than unity, and thus they are never stable.

Effect of selection on the individual's own genotype:

It is not unlikely for alleles with parental effects to alter the individual's own fitness (Arbeitman et al. 2002; Spencer 2003). To see how sensitive the equilibria in the parental selection model are to the weak viability selection acting on the individual's own genotype, two new parameters were included in the general model. Let G₁ and G₂ be the fitnesses of the genotypes containing A and B alleles, correspondingly, while the fitnesses of a and b alleles are set to 1. Multiplicative nonepistasis is assumed between these viability parameters (Karlin 1975), with no difference between sexes. While it was not my purpose to analyze the complete dynamics of the resulting complex model, I could show that only a weak selection on the offspring genotypes can significantly shift the model's equilibria from one type to another. In particular, for a parental selection parameter set that would otherwise result in the corner or the edge equilibrium, the two-locus polymorphic equilibria become possible for a certain range of individual viabilities (Figure 4).

Figure 4.—

Examples of shifts of the model's equilibria by selection on the individual's own genotype. The dashed and solid lines are the frequencies p and u of the A and B alleles, respectively, averaged between sexes, at equilibrium. Additive parental selection parameters are α = 0.14, β = −0.1, δ = −0.12, γ = 0.11. The viability fitness G₂ of the genotypes containing the B allele is varied, as the viability G₁ of the A genotypes is kept constant and shown at the top of each plot. (A) Change from fixation AB to the Ab–ab edge equilibrium; (B) change from AB–Ab to AB–aB edge equilibrium; (C) change from AB–Ab edge to ab fixation equilibrium.

DISCUSSION

A model of genomic imprinting, first proposed by Pearce and Spencer (1992), with complete inactivation of the imprinted allele, is mathematically equivalent to the model of parental selection on haploids. Similarity of its equilibrium behavior to the earlier models of sex-specific selection has been noted before (Anderson and Spencer 1999). Only when selection acts in the opposite directions in respect to males and females is a unique internal equilibrium possible, with the same stability conditions as in Bodmer's (1965) model of differential viability/fertility between sexes (Anderson and Spencer 1999). Very similar inequalities determine stability of polymorphism in a diploid model of combined maternal selection on males and direct selection on females, with the respective parameter space reduced by dominance (Gavrilets and Rice 2006). In this article, I have extended Pearce and Spencer's original model into a two-locus system and shown that it has the same equilibrium dynamics as a pair of one-locus models if and only if the fitness parameters at maternal and paternal effect (or paternally and maternally imprinted) loci are multiplicative. I have obtained results that are quite different from those of a well-known diploid two-locus viability selection theory (Karlin 1975; Bürger 2000). In the latter, henceforth termed the viability selection model (VSM), internal equilibria on the D = 0 surface exist in both multiplicative and additive fitness cases, whereas polymorphic equilibria involving associations between loci (D ≠ 0) are possible only in the multiplicative case and when the recombination rate is low. In contrast, the additive fitness case in the parental selection/imprinting model (PSM) does not allow for a stable two-locus polymorphism at the D = 0 surface, but is the only case where equilibria at D ≠ 0 were detected numerically. Away from equilibrium, in the multiplicative case of the PSM, association between loci can be generated by selection, provided that there is linkage between loci (r ≠ , see the appendix); whereas in the multiplicative case of the VSM, linkage equilibrium, once reached, is preserved for any r, even when the allele frequencies are not in equilibrium (Bürger 2000). Conditions for existence and stability of equilibria are also not analogous: in the VSM, polymorphism is ensured by heterozygote superiority at each locus, while in the PSM, it is antagonistic selection between sexes that allows the polymorphic equilibria (on or away from the D = 0 surface) to exist.

Results of the analysis of the PSM generally support intuition. For the parental effect/nonimprinted allele to reach fixation in the multiplicative case, its average fitness effect must exceed 1, and it is the fitter of maternal and paternal effect alleles that becomes fixed or is maintained at the higher frequency in the additive case. Only in the special additive case ii, with the fitness parameter t < 1, are the results somewhat counterintuitive, admitting two simultaneously stable equilibria. Analogy of the latter condition with the dominance relation in diploid systems is apparent: previous studies on the sex-specific selection show that simultaneous existence/stability of two or more equilibria requires fitness of a heterozygote to be different from 1; in particular, there must be overdominance in one (Mérat 1969) or both sexes (Owen 1953; Bodmer 1965; Kidwell et al. 1977).

Although I have examined the fitness effects conveyed only in one direction, i.e., from a parent to its offspring, it is likely that the fitness of the former is also affected by the advance of the corresponding parental effect allele. With both maternal and paternal effects taking place, a sexual conflict might be expected to arise, particularly when sons benefit from their father's contribution but daughters do not, and vice versa. For example, when males of the dung beetle Onthophagus taurus help females to provide food to the offspring, their sons inherit a specific paternal phenotype characterized by large body size and the presence of horns. Males with such phenotypes are generally more successful in securing mating, thus increasing the chances of their patrilinear genes spreading in the population (Hunter 2007). The above theoretical study suggests that in a two-locus system, a conflict between parents is relevant only if maternal and paternal fitness effects are not multiplicative, since only with multiplicative fitnesses are the equilibria at either locus independent. Under the assumption of additive fitness, however, a hypothetical parental conflict may not be easily resolved. In fact, the only stable equilibria in which maternal and paternal effect alleles can both be found at intermediate frequencies require exact balance between the respective fitness parameters and therefore are likely to be very rare. Additivity of the fitness effects is expected when a common resource, such as food, is simultaneously provided by both parents. The above interpretation is not much different when viewed in terms of genomic imprinting (to which the present results equally apply), only here the effect of a parent is mediated through conventional inheritance. The role of sexual conflict in the evolution of genomic imprinting has been widely discussed before (Moore and Haig 1991; Chapman et al. 2003; Day and Bonduriansky 2004).

The formal equivalence of the genomic imprinting model with that of haploid parental selection is not surprising given the similarity of the nature of these two phenomena. The possibility that imprinted genes may have evolved from those with a maternal (paternal) effect has been discussed by Chandra and Nanjundiah (1990). In the context of the model presented here, an evolutionary switch from the predominantly haploid to the predominantly diploid phase would serve as a driving force for a pair of the parental selection loci to start functioning as diploid counterparts with the opposite imprinting patterns. Genomic imprinting with respect to determination of the mating type is known for yeasts (Klar 1990), so discoveries supporting the above reasoning are not improbable among other lower eukaryotes with both haploid and diploid stages active. Note, however, that my model concerns only those cases in which the imprinted allele is completely inactivated.

It is not easy to find a combination of haploidy, parental, and sex-specific selection in one realistic biological example. At least one system, however, possesses features that resemble all the main assumptions of the model of parental selection presented above: the colonial algae Volvox carteri reproduces asexually under normal conditions but can enter the sexual stage once its environment becomes harsh. This change is triggered by a sexual pheromone, which, even at very low concentrations, interacts with the extacellular matrix surrounding the colony and, consequently, switches on the expression of a number of genes that are known to play a role both in gametogenesis and in wound healing (Amon et al. 1998). The sex-inducing pheromone itself is a glycoprotein analogous to those composing the extracellular matrix and thus appears to have a similar genetic background. If any mutations in this set of genes were to alter the penetrability of the sexual inducing signal or the timing of the switch to sexual reproduction, etc., they would inevitably affect the fitness of the offspring of the mutants. Since V. carteri is anisogamous and has a strictly genetic mechanism of sex determination (Kirk and Nishii 2001), it is not implausible that different mutations may have different fitness effects on the male and female colonies/gametes in the subsequent generations. There has been a recent rise of interest in V. carteri as a model organism for studying the evolutionary earliest molecular mechanisms of cell differentiation and sexual vs. asexual reproduction (Kirk and Kirk 2004; Armin 2006). One might reasonably expect more empirical findings relevant to the theoretical modeling of parental selection at the haploid level to be made soon.

Previous studies of parental (maternal) selection that use a major locus effect approach (Wright 1969; Wade 1998; Spencer 2003; Gavrilets and Rice 2006; Santure and Spencer 2006) are complemented by those that use the methods of quantitative genetics (Kirkpatrick and Lande 1989; Wade 1998; Hager et al. 2008). None, to my knowledge, has focused on the interaction between the effects of both parents, despite empirical evidence that both mother and father contribute to the selection pattern on their offspring (Chapman et al. 2003; Rakyan et al. 2003; Hunter 2007). It will be interesting to discover if considering maternal and paternal effects as quantitative traits would result in similar predictions about sexual conflict to those derived in the present study. Another important direction for future research is to focus on the interaction between parental and offspring genotypes, an approach used by Spencer (2003) and Gavrilets and Rice (2006). The ease with which the parental selection model's equilibria are changed by inclusion of weak directional selection on the offspring's own genotypes suggests that such interactions might have a complex nature.

APPENDIX

The fact that linkage equilibrium in model 2 can be maintained only if fitness is multiplicative can be demonstrated as follows. Assume that loci are unlinked (r = ) and the population is currently at linkage equilibrium. After meiosis and selection, the associations between loci take the form

(A1a)

(A1b)

Substituting in (A1) returns the only solution that allows polymorphism at both loci:

This is Equation 8 in the text. Let us now relax the assumption of free recombination between loci, holding that Equation 8 is true and . Equation A1 will take the form

(A2)

For r ≠ , in Equation A2 can be nonzero for a range of allele frequencies and fitness parameters. Setting r = , however, remains a sufficient condition for the maintenance of linkage equilibrium in the multiplicative case, even before the allele frequency equilibrium is reached.

A polymorphic two-locus equilibrium in the multiplicative fitness case, which represents a pair of one-locus polymorphisms similar to (4) in model 1, exists and retains its stability conditions even when the loci do not recombine freely and the initial D ≠ 0. For any and r, but letting and , the Jacobian of the system of recurrence Equations 5 is

(A3)

This matrix is independent of the recombination rate r, and since at equilibrium, the corresponding elements in the top right and bottom left of the matrix become zero. Thus the remaining elements at the bottom right and top left corners of the matrix independently determine the local stability of the A/a and the B/b locus, respectively. Further substitutions, , , and bring the top left corner of (A3) to the form identical to (4) in Anderson and Spencer (1999), which suggests that the stability conditions derived therein remain exactly the same.