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

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, Wade (1998) and ​and22 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

Open in a separate windowParentheses 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

Open in a separate windowAnother 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.