Population Models of Genomic Imprinting. II

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q ¼ 1 А p. р2Ю are the respective frequencies of A1 and A2, and ¯w, the population's ..... pseudo-Hardy–Weinberg form g11 ¼ p2, g12 ¼ g21 ¼ pq, and g22 ¼ ...
Copyright Ó 2006 by the Genetics Society of America DOI: 10.1534/genetics.106.057539

Population Models of Genomic Imprinting. II. Maternal and Fertility Selection Hamish G. Spencer,*,1 Timothy Dorn†,2 and Thomas LoFaro† *Allan Wilson Centre for Molecular Ecology and Evolution, Department of Zoology, University of Otago, Dunedin, 9054 New Zealand and † Department of Mathematics and Computer Science, Gustavus Adolphus College, Saint Peter, Minnesota 56082-1498 Manuscript received February 22, 2006 Accepted for publication June 4, 2006 ABSTRACT Under several hypotheses for the evolutionary origin of imprinting, genes with maternal and reproductive effects are more likely to be imprinted. We thus investigate the effect of genomic imprinting in single-locus diallelic models of maternal and fertility selection. First, the model proposed by Gavrilets for maternal selection is expanded to include the effects of genomic imprinting. This augmented model exhibits novel behavior for a single-locus model: long-period cycling between a pair of Hopf bifurcations, as well as two-cycling between conjoined pitchfork bifurcations. We also examine several special cases: complete inactivation of one allele and when the maternal and viability selection parameters are independent. Second, we extend the standard model of fertility selection to include the effects of imprinting. Imprinting destroys the ‘‘sex-symmetry’’ property of the standard model, dramatically increasing the number of degrees of freedom of the selection parameter set. Cycling in all these models is rare in parameter space.

B

IOLOGISTS have long recognized the fundamental importance of the maternal environment to the developing organism (see Wade 1998 for an overview). Although we have as yet no comparable data from mammals, it is clear from DNA microarray studies on Drosophila melanogaster that a significant proportion of the maternal genome is expressed in offspring. For example, Arbeitman et al. (2002) found that 1212 different RNA transcripts present in the first hours of development were maternally deposited during oogenesis. Crucially, this maternal genetic effect was compounded by standard genetic expression since all but 27 of these same genes were subsequently transcribed from the embryo’s own copies. The list of mammalian maternal-effect genes is small, although growing rapidly. Perhaps the best-known example is the mouse chromosome 7 gene Mater, which has no known phenotypic effect on its bearers, except that females homozygous for a null mutant are sterile because the maternally derived protein is necessary for normal embryonic development (Tong et al. 2000). Indeed, most of our examples of mammalian maternaleffect genes affect the early development of the offspring of homozygous females: the offspring of Zar1 null females, for example, usually die at the two-cell

1 Corresponding author: Allan Wilson Centre for Molecular Ecology and Evolution, Department of Zoology, University of Otago, 340 Great King St., P.O. Box 56, Dunedin, 9054 New Zealand. E-mail: [email protected] 2 Present address: Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605.

Genetics 173: 2391–2398 (August 2006)

stage (Wu et al. 2003). The mouse gene stella (also called PGC7) exhibits a maternal effect, but the paternal contribution is also relevant: when mated to stella-deficient males, stella-deficient females have no live pups, but when mated to wild-type males they produce about a third of the usual number of offspring (Payer et al. 2003). Genes that are active early in mammalian development are also likely targets for genomic imprinting, according to a number of explanations for the evolutionary origin of imprinting (Spencer 2000). For instance, the genetic-conflict hypothesis argues that growthaffecting genes active during fetal development may be agents of genetic conflict and thus become imprinted (Haig 1992). The ovarian time-bomb hypothesis (Varmuza and Mann 1994; see also Iwasa 1998) posits that genes essential for the initiation of embryogenesis will be imprinted. Thus mammalian genes with a strong maternal effect may also be those more likely to be subject to imprinting. Indeed, both the murine paternally expressed gene-1 (Peg1) and its human ortholog (PEG1) are imprinted (Kaneko-Ishino et al. 1995; Kobayashi et al. 1997) and the former is known to have strong direct and maternal effects (Lefebvre et al. 1998). Mice paternally inheriting a mutation of Peg1 (also known as mesoderm-specific transcript, Mest), exhibited growth retardation and reduced survival, and adult females were unable to successfully raise pups, irrespective of the pups’ own genotype (Lefebvre et al. 1998). Similarly, mutations in the mouse paternally expressed gene-3 (Peg3) affect several features of fetal and postnatal development in

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offspring, as well as various aspects of maternal care in mothers (Curley et al. 2004). Moreover, mutations at loci involving imprinting— such as those involved in the maintenance of DNA methylation, crucial to the monoallelic expression that defines imprinting—will almost always exhibit a maternal effect if the usual imprinting pattern is disrupted. For instance, heterozygous offspring of female mice homozygous for a deletion in the DNA methyl-transferase1 (Dnmt1) locus showed biallelic expression at the H19, which is normally expressed maternally, and died during gestation (Howell et al. 2001). There is, consequently, a natural link between loci subject to imprinting and those exhibiting maternal effects. The population genetic effects of such an association, however, have yet to be fully explored. In this article we make a start by examining the mathematical properties of a simple model of maternal selection at a locus subject to imprinting. We do so by incorporating the effect of imprinting into the two-allele single-locus model propounded by Gavrilets (1998; see also Spencer 2003), which describes the population-genetic consequences of fitness differences among both the maternal and the zygote’s own gene products. Genetic conflict may also be manifested in fertility selection, where the relative number of offspring is a property of the maternal and paternal phenotypes. It thus makes sense to investigate the consequences of incorporating imprinting into the standard model of fertility selection at a single diallelic locus (Bodmer 1965). Although imprinting is often depicted as the inactivation of either the paternal or the maternal copy of a gene, there is usually significant variation among different tissues. Murine insulin-like growth factor 2 (Igf2), for example, is maternally inactive in most tissues during embryogenesis, but has standard biallelic (Mendelian) expression in two structures associated with the central nervous system, the choroid plexus and leptomeninges (DeChiara et al. 1991; Pedone et al. 1994). Isoform 2 of human PEG1/MEST is not imprinted in most organs, but exhibits substantial variation among individuals in the level of imprinting in the placenta (McMinn et al. 2006). Thus, in its most general form, imprinting is the differential expression of paternally and maternally inherited genes. One important consequence for population genetics is that reciprocal heterozygotes at an imprinted locus will have different mean phenotypes that are not necessarily the same as either homozygote (Spencer 2002).

MODELS OF MATERNAL SELECTION

General maternal selection model: Consider a single locus with two autosomal alleles, A1 and A2, in a randomly mating, dioecious population, in which the

effects of mutation and genetic drift are negligible. Imprinting means that we must distinguish between maternally and paternally derived alleles: by AiAj we mean a genotype with a maternally derived Ai allele and a paternal Aj. Suppose wijkl is the fitness of individuals of genotype AkAl with genotype AiAj mothers. Since k ¼ i or j and i, j, k, l 2 {1, 2}, there are 12 different fitness parameters, as shown in Table 1. If gij is the postselection P frequencies of adults with genotypes AiAj (with i;j gij ¼ 1), then the recursion equations for these frequencies in the following generation are   wg  911 ¼ p w1111 g11 1 12w1211 g12 1 12w2111 g21 ;   wg  912 ¼ q w1112 g11 1 12w1212 g12 1 12w2112 g21 ;   wg  921 ¼ p 12w1221 g12 1 12w2121 g21 1 w2221 g22 ;   ð1Þ wg  922 ¼ q 12w1222 g12 1 12w2122 g21 1 w2222 g22 ; in which p ¼ g11 1 12ðg12 1 g21 Þ; q ¼1p

ð2Þ

 the are the respective frequencies of A1 and A2, and w, population’s mean fitness, is the sum of the right-hand sides of (1) so that the iterated frequencies (g 9ij ) also add to one. To our knowledge, Equations 1 are not formally equivalent to those previously used to describe any other population genetic system. Note that the normalization of Equations 1 means that the relative rather than absolute sizes of the fitness parameters (the wij’s) determine the dynamical behavior and so just 11 parameters are needed to describe the system. The model exhibits a number of interesting properties, including those possessed by Gavrilets’ (1998) model (since the latter is a special case of the imprinting model with wijkl ¼ wijlk ¼ wjikl ¼ wjilk). Thus, two distinct polymorphic equilibria (i.e., values of gij ¼ g^ij such that g^ 9ij ¼ g^ij for all i and j and 0 # g^11 ; g^12 ; g^21 ; g^22 # 1 with each inequality strict for at least one value) may be locally stable, or an internal equilibrium and one or both fixations may simultaneously be so. This result implies that at least five distinct equilibria (not all of which are stable, of course) may be feasible (i.e., real solutions with 0 # g^11 ; g^12 ; g^21 ; g^22 # 1) for certain parameter values, and, indeed, we give an example in Table 2. We have not been able to analytically solve this model to find all possible equilibria, but not one of 105 randomly generated fitness sets (i.e., sets of 12 pseudorandom wijkl values drawn independently from the uniform distribution between 0 and 1) possessed .5 distinct, feasible equilibria. Moreover, we did not find any sets of fitnesses that afforded $3 stable polymorphic equilibria in 106 randomly generated sets, each with 100 random

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TABLE 1 Mating table for general model, showing frequencies and fitnesses Mother Father

A1A1 2 g11

A1A1

A1A2

A1A1 w1111

A1A2

1 2g11 g12 1 2g11 g12

A1A1 w1111 A1A2 w1112

A2A1

1 2g11 g21 1 2g11 g21

A1A1 w1111 A1A2 w1112

A2A2

g11 g22 A1A2 w1112

1 2g12 g11 1 2g12 g11 1 4g12 g12 1 4g12 g12 1 4g12 g12 1 4g12 g12 1 4g12 g21 1 4g12 g21 1 4g12 g21 1 4g12 g21 1 2g12 g22 1 2g12 g22

A1A1 w1211 A2A1 A1A1 A1A2 A2A1 A2A2

w1221 w1211 w1212 w1221 w1222

A1A1 A1A2 A2A1 A2A2

w1211 w1212 w1221 w1222

A1A2 w1212 A2A2 w1222

initial genotype frequencies (drawn using the brokenstick method). These simulation results indicate that 5 may well be the maximum number of feasible equilibria. (Up to 10 equilibria arise from solving g 9ij ¼ gij for all i and j, but many of these are complex roots or unfeasible solutions.) The two fixation equilibria (A1 fixed, g11 ¼ 1, g12 ¼ g21 ¼ g22 ¼ 0; A2 fixed, g11 ¼ g12 ¼ g21 ¼ 0, g22 ¼ 1) always exist. The former is locally stable if  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 w1111 . 14 w1112 1 w2121 1 w1112 1 w2121 1 2w1112 ð2w1221  w2121 Þ ;

ð3Þ TABLE 2 Example of a parameter set affording five distinct equilibria for the general maternal selection model i, j wijkl k, l

1, 1, 2, 2,

1 2 1 2

1, 1

1, 2

2, 1

0.191 0.134

0.278 0.599 0.667 0.734

0.483 0.083 0.765 0.194

2, 2

0.062 0.209

Equilibria g^11 0 0.072 0.125 0.319 1

g^12

g^21

g^22

lmax

0 0.214 0.208 0.138 0

0 0.148 0.214 0.374 0

1 0.566 0.453 0.169 0

1.456 0.960 1.031 0.888 2.315

A2A1

Locally stable? No Yes No Yes No

Open cells in the top part correspond to impossible types: no A2A1 offspring have A1A1 mothers, for example. lmax is the leading eigenvalue of the Jacobian of the system of Equations 1, linearized around the equilibrium.

1 2g21 g11 1 2g21 g11 1 4g21 g12 1 4g21 g12 1 4g21 g12 1 4g21 g12 1 4g21 g21 1 4g21 g21 1 4g21 g21 1 4g21 g21 1 2g21 g22 1 2g21 g22

A2A2

A1A1 w2111 A2A1 A1A1 A1A2 A2A1 A2A2

w2121 w2111 w2112 w2121 w2122

A1A1 A1A2 A2A1 A2A2

w2111 w2112 w2121 w2122

A1A2 w2112 A2A2 w2122

g22 g11 A2A1 w2221 1 2g22 g12 1 2g22 g12

A2A1 w2221 A2A2 w2222

1 2g22 g21 1 2g22 g21

A2A1 w2221 A2A2 w2222

2 g22

A2A2 w2222

which, in the absence of imprinting, collapses to the condition found by Gavrilets (1998). The analogous condition for local stability of the A2 fixation is  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 w2222 . 14 w2221 1 w1212 1 w2221 1 w1212 1 2w2221 ð2w2112  w1212 Þ :

ð4Þ These inequalities reveal the importance of the asymmetry of imprinting in even simple matters, in this case the stability of monomorphisms. The greater a certain imprinting effect—the difference between fitnesses of heterozygotes inheriting the rare allele from the two sorts of heterozygous mothers (e.g., for the fixation of A1, those heterozygotes inheriting A2 from their mothers and having respective fitnesses w1221 and w2121)—the less likely (other things being equal) that monomorphism is locally stable. It is interesting that it is the maternal difference between otherwise identical heterozygous offspring that is crucial, rather than the difference between the offspring themselves. One novel behavior is the potential for long-period cycling of genotype frequencies, an example of which is shown in Figures 1 and 2. Further analysis (see appendix) revealed that this dynamical behavior was due to a pair of supercritical Hopf bifurcations. In a supercritical Hopf bifurcation, a stable equilibrium point becomes unstable and is encircled by an attracting, invariant closed curve. A numerical investigation of this example revealed that the first Hopf bifurcation occurs at w1111  0.0254 and the second at w1111  0.3395. The asymptotic dynamics on this closed curve can be either periodic or aperiodic and both types of behaviors are exhibited in this case. With standard biallelic expression (i.e., Gavrilets’ 1998 model), the only cycles known are of period 2 (Spencer 2003). Consequently, the mean fitness, w,  need not be maximized (Figure 1a).

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H. G. Spencer, T. Dorn and T. LoFaro Figure 1.—An example of long-period genotypefrequency cycling with w1211 ¼ 0.75, w1221 ¼ 0.02, w2121 ¼ 0.96, w2111 ¼ 0.07, w2221 ¼ 0.01, w1112 ¼ 0.04, w1212 ¼ 0.72, w1222 ¼ 0.13, w2122 ¼ 0.31, w2112 ¼ 0.00, and w2222 ¼ 0.24. (a) Mean fitness and genotype frequencies g11 (solid line) and g12 (dotted line) over time with w1111 ¼ 0.25. (b) Bifurcation diagram for g11 as w1111 is varied. Note the gradual appearance and disappearance of cycling as w1111 is increased.

The bifurcation diagrams of some of the examples of 2-cycles are also worthy of note: in no cases did we find further bifurcations, giving 4-cycles. Indeed, as shown in Figure 3, 2-cycling appeared and then disappeared as one parameter was varied. This behavior has the appearance of two conjoined pitchfork bifurcations, one reversed, with their prongs aligned. So far as we know, this is also a novel finding in any population genetics model. A number of special cases deserve further analysis. Multiplicative maternal selection model: This special case further assumes that the selective pressures of the maternal effects and ordinary viability selection are independent, as, for example, when selection occurs at two separate stages in the life cycle of each individual, the first as the result of its mother’s phenotype and the second as the result of its own phenotype. These effects thus act multiplicatively, and so

Normalization means that just six parameters—three m’s and three v’s—now specify the dynamical and equilibrial behavior of the system. Again, up to two distinct polymorphic equilibria may be locally stable, but no cases of cycling were found.

Figure 2.—Long-period cycles of g11 and g12 as w1111 is incremented in steps on 0.025. Other parameters are as in Figure 1.

Figure 3.—Bifurcation diagram for g11 as w1211 is varied, with w1111 ¼ 0.01, w1221 ¼ 0.20, w2121 ¼ 0.26, w2111 ¼ 0.24, w2221 ¼ 0.02, w1112 ¼ 0.97, w1212 ¼ 0.01, w1222 ¼ 0.70, w2122 ¼ 0.29, w2112 ¼ 0.87, and w2222 ¼ 0.08. Note the appearance and disappearance of 2-cycles. The dotted line indicates the unstable polymorphic equilibrium.

wijkl ¼ mij vkl

ð5Þ

for i ¼ 1, 2, 3. Equations 1 thus become   wg  911 ¼ v11 p m11 g11 1 12m12 g12 1 12m21 g21 ;   wg  912 ¼ v12 q m11 g11 1 12m12 g12 1 12m21 g21 ;   wg  921 ¼ v21 p 12m12 g12 1 12m21 g21 1 m22 g22 ;   wg  922 ¼ v22 q 12m12 g12 1 12m21 g21 1 m22 g22 :

ð6Þ

Population Models of Genomic Imprinting, II

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Figure 4.—Cycling of genotype frequencies under complete paternal inactivation, with a11 ¼ 0.04, a21 ¼ 0.62, and a22 ¼ 0.01. (a) Mean fitness and genotype frequencies g11 (solid line) and g12 (dotted line) over time with a12 ¼ 0.54. (b) Bifurcation diagram for g11 as a12 is varied. The dotted line indicates the unstable polymorphic equilibrium.

Moreover, since g 912 ¼ v12 qg 911 =v11 p ¼

p9f ¼

v12 v21 ð1  g 911  g 921 Þg 911 v11 v22 g 921 1 v12 v21 g 911

ð7Þ

only two of Equations 6 are truly independent. In fact, they can be rewritten in terms of the frequencies, pf and qf (pm and qm) of maternally (paternally) derived A1 and A2 alleles in zygotes after the maternal-effect selection has acted, in the same way as in the absence of imprinting (Spencer 2003). Since pf ¼

m11 g11 1 12m12 g12 1 12m21 g21 m11 g11 1 m12 g12 1 m21 g21 1 m22 g22

ð8Þ

and pm ¼ p, we have p9f ¼

m11 v11 pf pm 1 12m12 v12 pf qm 1 12m21 v21 qf pm m11 v11 pf pm 1 m12 v12 pf qm 1 m21 v21 qf pm 1 m22 v22 qf qm

ð9Þ

p9m ¼ pf pm 1 12pf qm 1 12qf pm ;

v11 pf pm 1 12v12 pf qm 1 12v21 qf pm ; v11 pf pm 1 v12 pf qm 1 v21 qf pm 1 v22 qf qm

ð10Þ

which are the recursions for different selection pressures on males and females at a single diallelic locus (Pearce and Spencer 1992). The fitness of AiAj males is vij and that of females, mijvij, so (as in the absence of imprinting; Spencer 2003) the maternal component of selection effectively acts on females only. In contrast to this result, the formal equivalence between the multiplicative case of maternal selection and fertility selection in the absence of imprinting (Gavrilets 1998) does not extend to our model with imprinting (see below). Complete paternal inactivation: For many imprinted loci (e.g., Igf2-r in rodents), the paternal copy of a gene is effectively silenced. If this silencing occurs to genes in both the mother and offspring, then there are just four distinct fitnesses: wijkl ¼ aik (i, k 2 {1, 2}). As before, this allows Equations 1 to be simplified and, indeed, they may be reduced to just two independent recursions:

ð11Þ

where pf ¼ pm ¼

a11 g11 1 12a11 g12 1 12a21 g21 1 a11 g11 1 2ða11 1 a12 Þg12 1 12ða21 1 a22 Þg21 g11 1 12g12 1 12g21 :

1 a22 g22

ð12Þ

The natural interpretation of pf and pm is a little different from the multiplicative case, however. This time, pf is the postselection frequency of A1; pm is the preselection frequency. This system affords just three equilibria: two trivial fixations and one potential polymorphism given by the pseudo-Hardy–Weinberg form g^11 ¼ p^ 2 , g^12 ¼ g^21 ¼ p^^ q, and g^22 ¼ q^2 , in which p^ ¼ p^f ¼ p^m ¼

and p9m ¼

a11 pf pm 1 12a11 pf qm 1 12a21 qf pm a11 pf pm 1 12ða11 1 a12 Þpf qm 1 12ða21 1 a22 Þqf pm 1 a22 qf qm

2a22  a11  a21 a11  a12  a21 1 a22

ð13Þ

and q^ ¼ 1  p^. The fixation of A1 is locally stable provided that a11 . 12ða12 1 a22 Þ and that of A2 if a22 . 12ða21 1 a11 Þ. These fixations can, of course, be simultaneously locally stable. If both these inequalities are satisfied, the polymorphic equilibrium is feasible but unstable; if both are reversed the polymorphic equilibrium is feasible and may be locally stable. If the polymorphic equilibrium remains unstable, genotype frequencies oscillate between two values (see Figure 4, for an example). No cases of cycles of length .2 were found in 107 numerical examples with the four aij values independently sampled from the uniform distribution between zero and one. Complete maternal inactivation: We now investigate the case in which the maternal allele is completely silenced in both the mother and her offspring. Maternal effects mean that this counterpart to the previous model is not its formal equivalent, in contrast to most pairs of population genetic models of maternal and paternal inactivation (see, e.g., Pearce and Spencer 1992). Supposing that wijkl ¼ bjl, Equations 1 reduce to

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H. G. Spencer, T. Dorn and T. LoFaro



 wg  911 ¼ p b11 g11 1 12b21 g12 1 12b11 g21 ;   wg  912 ¼ q b12 g11 1 12b22 g12 1 12b12 g21 ;   wg  921 ¼ p 12b21 g12 1 12b11 g21 1 b21 g22 ;   wg  922 ¼ q 12b22 g12 1 12b12 g21 1 b22 g22 :

(1965) and leads to the recursion equations for the genotype frequencies in the following generation of wg  911 ¼ v11

wg  912 ¼ v12

ð14Þ wg  921 ¼ v21

Note that the different bjl values appear a different number of times in Equations 14 than the corresponding aik’s in this form of the equations for complete paternal inactivation and, consequently, the system cannot be rewritten in terms of two variables. Nevertheless, as in the complete paternal inactivation case, for various fitnesses the two fixations and a polymorphism are possible equilibria. In contrast, however, oscillations in genotype frequency do not appear to occur: none were found in 107 numerical cases, each with four bjl values independently drawn from U[0, 1]. Nor, in spite of this apparent simplicity of the system, have we been able to find an analytical expression for the polymorphic equilibrium, and we cannot show that it is unique. Nevertheless, some 107 random numerical examples, each with 100 random initial genotype frequencies, failed to reveal any cases with more than one locally stable polymorphism. Maternal selection only: This case assumes that selective differences are due only to the mother’s genes. This assumption means that wijkl ¼ gij and Equations 1 reduce to Equations 6 with vij ¼ 1 and mij ¼ gij for all i and j. Thus, this system behaves as if no selection acts on males and viability selection with fitnesses gij acts on females. Like the case of complete paternal inactivation, this system affords just three equilibria: two trivial fixations and one potential pseudo-Hardy–Weinberg polymorphism, in which p^ ¼

2g22  g12  g21 : 2ðg11  g12  g21 1 g22 Þ

ð15Þ

The stability of the equilibria is simple: fixation of A1 is locally stable provided that g11 . 12ðg12 1 g21 Þ, fixation of A2 is stable if g22 . 12ðg12 1 g21 Þ, and the polymorphism is stable if both these inequalities are reversed— amounting to mean heterozygote advantage—and this condition simultaneously guarantees feasibility.

wg  922 ¼ v22

2 f1111 g11 1 12ð f1112 1 f1211 Þg11 g12 1 12ð f1121 1 f2111 Þg11 g21

!

2 2 1 14ð f1221 1 f2112 Þg12 g21 1 14 f1212 g12 1 14 f2121 g21 1 2 f1112 g11 g12

!

2 1 12 f1121 g11 g21 1 f1122 g11 g22 1 14 f1212 g12

2 1 14ð f1221 1 f2112 Þg12 g21 1 12 f1222 g12 g22 1 14 f2121 g21 1 12 f2122 g21 g22 ! 2 1 1 1 1 2 f1211 g11 g12 1 4 f1212 g12 1 4ð f1221 1 f2112 Þg12 g21 1 2 f2111 g11 g21 2 1 14 f2121 g21 1 f2211 g11 g22 1 12 f2212 g12 g22 1 12 f2221 g21 g22 2 1 4 f1212 g12

1 14ð f1221 1 f2112 Þg12 g21 1 12ð f1222 1 f2212 Þg12 g22

2 2 1 14 f2121 g21 1 12ð f2122 1 f2221 Þg21 g22 1 f2222 g22

! :

ð16Þ

At first glance, Bodmer’s (1965) model apparently requires nine fertility parameters, corresponding to the 3 3 3 ¼ 9 possible matings between the three different genotypes. But as Feldman et al. (1983) pointed out, reciprocal matings between unlike parental genotypes produce the same proportions of offspring genotypes. Consequently, the fertility parameters corresponding to these three crosses always appear together in the recursions, and their average, rather than their individual values, is what matters. This ‘‘sex-symmetry’’ property means there are just six independent parameters in Bodmer’s model, since the iterations are unchanged if we set each of these three pairs of fertilities to the same value as their arithmetic means. Fertility selection with imprinting—Equations 16— does not display this property, however, because imprinting destroys the symmetry of reciprocal matings. For instance, an A1A1 3 A1A2 cross, with fertility f1112, produces one-half A1A2 offspring, but an A1A2 3 A1A1 cross, with fertility f1211, gives none. Hence, the parameter f1112 appears in the recursion for the frequency of A1A2, g 912 , but f1211 does not, and so both values matter. The local stability condition for the fixations again reveals the importance of imprinting. For instance, the condition for local stability of the fixation of A2 is 4v22 f2222 . v12 f1222 1 v21 f2221 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðv12 f1222 1 v21 f2221 Þ2 1 4v12 v21 ðf2122 f2212  f1222 f2221 Þ:

ð17Þ In the absence of imprinting, this inequality collapses to simple heterozygote disadvantage. Not surprisingly, the large number of parameters of the above model allows some unusual dynamical behavior. Cycling is possible (since it is in the nonimprinting case; Doebeli and de Jong 1998) but it must be quite rare: we found no cycling in 107 simulations.

MODELS OF FERTILITY SELECTION

General fertility selection model: Using the same conventions as for the models of maternal selection, let us suppose that fijkl is the fertility of the cross of AiAj females with AkAl males and that AmAn individuals have viability vmn. This parameterization of 16 fertility and 4 viability parameters is simply the imprinting version of the model of fertility selection proposed by Bodmer

DISCUSSION

The models above demonstrate that when selection acts on loci that engender maternal genetic effects and that are subject to genomic imprinting, novel genotypefrequency dynamics may arise. These behaviors include oscillations between two distinct polymorphic values, as well as longer-period cycling lasting many generations

Population Models of Genomic Imprinting, II

due to Hopf bifurcations (also known as Andronov– Hopf bifurcations). This last phenomenon is particularly interesting because, as one fitness parameter is continuously altered (as in a standard bifurcation diagram), these cycles appear at once (at the Hopf bifurcation), rather than as the culmination of a sequence of bifurcations. Moreover, these cycles disappear at a second Hopf bifurcation, as the fitness parameter is further changed. We know of no other single-locus population-genetic models exhibiting such behavior, although single Hopf bifurcations do arise in models of (i) constant viability selection and recombination for two loci each with two alleles (Hastings 1981; Akin 1982, 1983) and (ii) constant selection and multilocus mutation with selfing (Yang and Kondrashov 2003). The way in which the cycling between two polymorphic values occurs (for certain fitness values) in some of the models is also noteworthy. In all cases examined, a single locally stable polymorphic equilibrium becomes unstable at a pitchfork bifurcation, as one parameter is varied (e.g., Figures 3 and 4). The population then oscillates between two polymorphic values. In the general model of maternal effects with viability selection, however, further changes in the parameter sometimes led to these two values and the unstable equilibrium coalescing into a locally stable equilibrium again (Figure 3). Nevertheless, although these mathematical properties are interesting, examination of the various models with randomly assigned fitnesses showed that cycling of all types was rare in parameter space. Moreover, the examples in Figures 1 and 3 exhibit strong interactions between maternal and offspring genotypes that generate large differences in fitnesses between, say, the same offspring genotypes with different maternal genotypes. Whether or not evolution disproportionately favors such parameters is a separate question, of course (Spencer and Marks 1988; Marks and Spencer 1991), but the biological importance of cycling in these models is certainly open to question. Nevertheless, any random allele frequency changes due to genetic drift are unlikely to prevent cycling (unless they lead to the fixation of one allele) because in all cases these cycles are attracting. Two of the special cases—complete paternal inactivation and maternal selection only—possess another interesting feature: the ‘‘pseudo-Hardy–Weinberg’’ form of the sole polymorphic equilibrium even though the recursions cannot be reduced to those in allele frequencies. Previous models concerned with the evolution of imprinting from standard expression (Spencer et al. 1998, 2004) have found a similar result, but these latter models all involved both imprintable and unimprintable alleles. Pearce and Spencer (1992) investigated the effect of imprinting on standard models of viability selection. They found that in all cases—except for that of different selection pressures on males and females—the models were formally equivalent to models without imprinting (but with suitably adjusted viabilities). In contrast, none

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of the models derived above have any formal equivalence to known models without imprinting. Moreover, the properties of some models without imprinting are destroyed by imprinting: the sex-symmetry property of fertility selection (Feldman et al. 1983), for example, does not hold in the presence of imprinting. This work thus adds to the growing literature that shows how standard biallelic Mendelian expression permits several simplifications in population-genetic and quantitativegenetic models (Spencer 2002). Thus, although the number of imprinted genes is small (Morison et al. 2005), their very existence illuminates our understanding of population-genetic and other processes. We thank Mike Paulin for discussions about the models and Ken Miller for assistance with the figures. Two anonymous reviewers also provided helpful suggestions. Additional thanks are due to Gustavus Adolphus College and the Allan Wilson Centre at Massey University for funding and hosting the sabbatical for T.L. that facilitated this collaboration. Financial support for this work was provided by the Marsden Fund of the Royal Society of New Zealand contract U00-315 (H.G.S.).

LITERATURE CITED Akin, E., 1982 Cycling in simple genetic systems. J. Math. Biol. 13: 305–324. Akin, E., 1983 Hopf bifurcation in the two locus genetic model. Memoirs of the American Mathematical Society No. 284. American Mathematical Society, Providence. Arbeitman, M. N., E. E. M. Furlong, F. Imam, E. Johnson, B. H. Null et al., 2002 Gene expression during the life cycle of Drosophila melanogaster. Science 297: 2270–2275. Bodmer, W. F., 1965 Differential fertility in population genetics models. Genetics 51: 411–424. Curley, J. P., S. Barton, A. Surani and E. B. Keverne, 2004 Coadaptation in mother and infant regulated by a paternally expressed imprinted gene. Proc. R. Soc. Lond. Ser. B 271: 1303–1309. DeChiara, T. M., E. J. Robertson and A. Efstratiadis, 1991 Parental imprinting of the mouse insulin-like growth factor II gene. Cell 64: 849–859. Doebeli, M., and G. de Jong, 1998 A simple genetic model with non-equilibrium dynamics. J. Math. Biol. 36: 550–556. Dorn, T., 2005 Hopf bifurcations in a maternal genomic imprinting model. Senior Mathematics Honors Thesis, Gustavus Adolphus College, Saint Peter, MN. Feldman, M. W., F. B. Christiansen and U. Liberman, 1983 On some models of fertility selection. Genetics 105: 1003–1010. Gavrilets, S., 1998 One-locus two-allele models with maternal (parental) selection. Genetics 149: 1147–1152. Haig, D., 1992 Genomic imprinting and the theory of parentoffspring conflict. Semin. Dev. Biol. 3: 153–160. Hastings, A., 1981 Stable cycling in discrete-time genetic models. Proc. Natl. Acad. Sci. USA 78: 7224–7225. Howell, C. Y., T. H Bestor, F. Ding, K. E. Latham, C. Mertineit et al., 2001 Genomic imprinting disrupted by a maternal effect mutation in the Dnmt1 gene. Cell 104: 829–838. Iwasa, Y., 1998 The conflict theory of genomic imprinting: How much can be explained? Curr. Top. Dev. Biol. 40: 255–293. Kaneko-Ishino, T., Y. Kuroiwa, N. Miyoshi, T. Kohda, R. Suzuki et al., 1995 Peg1/Mest imprinted gene on chromosome 6 identified by cDNA subtraction hybridization. Nat. Genet. 11: 52–59. Kobayashi, S., T. Kohda, N. Miyoshi, Y. Kuroiwa, K. Aisaka et al., 1997 Human PEG1/MEST, an imprinted gene on chromosome 7. Hum. Mol. Genet. 6: 781–786. Lefebvre, L., S. Viville, S. C. Barton, F. Ishino, E. B. Keverne et al., 1998 Abnormal maternal behaviour and growth retardation associated with loss of the imprinted gene Mest. Nat. Genet. 20: 163–169. Marks, R. W., and H. G. Spencer, 1991 The maintenance of singlelocus polymorphism. II. The evolution of fitnesses and allele frequencies. Am. Nat. 138: 1354–1371.

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McMinn, J., M. Wei, Y. Sadovsky, H. M. Thaker and B. Tycko, 2006 Imprinting of PEG1/MEST isoform 2 in human placenta. Placenta 27: 119–126. Morison, I. M., J. P. Ramsay and H. G. Spencer, 2005 A census of mammalian imprinting. Trends Genet. 21: 457–465. Payer, B., M. Saitou, S. C. Barton, R. Thresher, J. P. C. Dixon et al., 2003 stella is a maternal effect gene required for normal early development in mice. Curr. Biol. 13: 2110–2117. Pearce, G. P., and H. G. Spencer, 1992 Population genetic models of genomic imprinting. Genetics 130: 899–907. Pedone, P. V., M. P. Cosma, P. Ungaro, V. Colantuoni, C. B. Bruni et al., 1994 Parental imprinting of rat insulin-like growth-factor-II gene promoter is coordinately regulated. J. Biol. Chem. 269: 23970–23975. Spencer, H. G., 2000 Population genetics and evolution of genomic imprinting. Annu. Rev. Genet. 34: 457–477. Spencer, H. G., 2002 The correlation between relatives on the supposition of genomic imprinting. Genetics 161: 411–417. Spencer, H. G., 2003 Further properties of Gavrilets’ one-locus twoallele model of maternal selection. Genetics 164: 1689–1692. Spencer, H. G., and R. W. Marks, 1988 The maintenance of singlelocus polymorphism. I. Numerical studies of a viability selection model. Genetics 120: 605–613. Spencer, H. G., M. W. Feldman and A. G. Clark, 1998 Genetic conflicts, multiple paternity and the evolution of genomic imprinting. Genetics 148: 893–904. Spencer, H. G., M. W. Feldman, A. G. Clark and A. E. Weisstein, 2004 The effect of genetic conflict on genomic imprinting and modification of expression at a sex-linked locus. Genetics 166: 565–579. Tong, Z.-B., L. Gold, K. E. Pfeifer, H. Dorward, E. Lee et al., 2000 Mater, a maternal effect gene required for early embryonic development in mice. Nat. Genet. 26: 267–268. Varmuza, S., and M. Mann, 1994 Genomic imprinting—defusing the ovarian time bomb. Trends Genet. 10: 118–123. Wade, M. J., 1998 The evolutionary genetics of maternal effects, pp. 5–21 in Maternal Effects as Adaptations, edited by T. A. Mousseau and C. W. Fox. Oxford University Press, New York. Wu, X., M. M. Viveiros, J. J. Eppig, Y. Bai, S. L. Fitzpatrick et al., 2003 Zygote arrest 1 (Zar1) is a novel maternal-effect gene critical for the oocyte-to-embryo transition. Nat. Genet. 33: 187–191. Yang, H.-P., and A. S. Kondrashov, 2003 Cyclical dynamics under constant selection against mutations in haploid and diploid populations with facultative selfing. Genet. Res. 81: 1–6. Communicating editor: M. K. Uyenoyama

APPENDIX: HOPF BIFURCATIONS IN THE GENERAL MATERNAL SELECTION MODEL

The structure shown in the bifurcation diagram of Figure 1b suggests a supercritical Hopf bifurcation since the equilibrium point bifurcates not via period doubling cascade, but immediately into a region where the dynamics are more complicated. The oscillatory behavior illustrated in Figure 1a reinforces the suggestion that a Hopf bifurcation has occurred. In a supercritical Hopf bifurcation, a stable equilibrium becomes unstable and is surrounded by an attracting invariant circle as the parameter is varied. The dynamics on this circle may be periodic or aperiodic, but in either case are oscillatory in nature. The Hopf bifurcation theorem for maps describes the necessary conditions for the existence of such a bifurcation. Let xn11 ¼ F ðl; xn Þ with l 2 R, x 2 Rn , and F three times differentiable. For a Hopf bifurcation to occur an equilibrium point x0 and a parameter value l0 must exist such that the linearization of F ðl; xÞ with respect to x at ðl0 ; x0 Þ has a pair of complex conjugate eigenvalues a 6 bi with a and b nonzero and a 2 1 b 2 ¼ 1.

Figure A1.—Modulus of complex eigenvalues as a function of w1111 for the numerical example of Figure 1. 1 (Therepis ffiffiffi one additional restriction that if a ¼ 2 then b 6¼ 6 3=2.) Moreover, one must check that as the parameter l is varied monotonically about l0 , the quantity a 2 1 b 2 goes from a value ,1 to a value .1 or vice versa. The algebraic complexity of this model has so far precluded an analytical verification of the hypotheses of the Hopf bifurcation theorem. However, we have numerically verified these conditions using the software Maple 10. The program (Dorn 2005) works as follows:

1. Increment the parameter w1111 from 0 to 0.4. 2. For each parameter value perform the following: a. Numerically compute the critical point. b. Compute the four eigenvalues of the linearization at this critical point. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c. Compute the modulus a 2 1 b 2 for each of these eigenvalues. 3. Plot each of the moduli against the corresponding parameter value. Figure A1 shows the result of this process. The top sequence plots the moduli of the complex conjugate pair of eigenvalues as a function of w1111. The horizontal line at y ¼ 1 represents the threshold that must be crossed for a Hopf bifurcation to occur. Note that this sequence crosses this line at w1111 ¼ 0.02 and again at w1111 ¼ 0.33. Further refinements of this program yielded more accurate bifurcation values of w1111 ¼ 0.0254 and w1111 ¼ 0.3395. For these two approximate bifurcation values we manually verified that these eigenvalues satisfy the conditions on a and b given above. Finally, the crossing of the sequence through the line y ¼ 1 at both w1111 ¼ 0.0254 and w1111 ¼ 0.3395 verifies the final condition stated above. We note for completeness that there is one other sequence plotted in Figure A1. This represents the real, nonzero eigenvalue of the linearization. The fourth eigenvalue equals zero for all parameter values due to the normalization of Equations 1.