Population Models of Genomic Imprinting. II.

Maternal and Fertility Selection

Hamish G. Spencer

Allan Wilson Centre for Molecular Ecology and Evolution

Department of Zoology, University of Otago, Dunedin, New Zealand

Timothy Dorn1 and Thomas LoFaro

Department of Mathematics and Computer Science

Gustavus Adolphus College, Saint Peter, MN 56082-1498, USA

1Current Address: Department of Mathematics, Oregon State University,

Corvallis, OR 97331-4605, USA

Genetics: Published Articles Ahead of Print, published on June 18, 2006 as 10.1534/genetics.106.057539

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Running Title: Selection with Imprinting

Keywords: genomic imprinting, maternal effects, population-genetic model, Hopf bifurcation,

cycling

Corresponding Author: Hamish G. Spencer

Allan Wilson Centre for Molecular Ecology and Evolution

Department of Zoology

University of Otago

P.O. Box 56

Dunedin

New Zealand

Phone: (+64-3) 479 7981

Fax: (+64-3) 479 7584

Email: h.spencer@otago.ac.nz

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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 2-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.

Biologists 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.

4

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 to normal embryonic development

(TONG et al. 2000). Indeed, most of our examples of mammalian maternal-effect 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 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 growth-

affecting 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 orthologue

(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

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the mouse paternally expressed gene-3 (Peg3) affect several features of fetal and post-natal

development in 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-transferase-1 (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 paper 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 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

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.

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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 importance 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 will 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 = i or j and i, j, k, l œ {1, 2}, there are 12 different fitness

parameters, as shown in Table 1. If gij is the post-selection frequencies of adults with

genotypes AiAj (with ,

1iji j

g =∑ ), then the recursion equations for these frequencies in the

following generation are

( )

( )

( )

( )

1 111 1111 11 1211 12 2111 212 2

1 112 1112 11 1212 12 2112 212 2

1 121 1221 12 2121 21 2221 222 2

1 122 1222 12 2122 21 2222 222 2

,

,

,

,

wg p w g w g w g

wg q w g w g w g

wg p w g w g w g

wg q w g w g w g

′ = + +

′ = + +

′ = + +

′ = + +

(1)

in which

( )1

11 12 212 ,

1

p g g g

q p

= + +

= − (2)

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are the respective frequencies of A1 and A2, and w , the population’s mean fitness, is the sum

of the right-hand sides of (1) so that the iterated frequencies ( ijg ′ ) 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 wijs) 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’s (1998) model (since the latter is a special case of the imprinting model with wijkl

= wijlk = wjikl = wjilk). Thus, two distinct polymorphic equilibria (i.e., values of such

that for all i and j and 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 5 distinct equilibria (not all of which are

stable, of course) may be feasible (i.e., real solutions with

ˆij ijg g=

ˆ ˆij ijg g′ = 11 12 21 22ˆ ˆ ˆ ˆ0 , , ,g g g g≤ 1≤

111 12 21 22ˆ ˆ ˆ ˆ0 , , ,g g g g≤ ≤ ) 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 pseudo-random wijkl values

drawn independently from the uniform distribution between 0 and 1) possessed more than 5

distinct, feasible equilibria. Moreover, we did not find any sets of fitnesses that afforded 3 or

more stable polymorphic equilibria in 106 randomly generated sets, each with 100 random

initial genotype frequencies (drawn using the broken stick method). These simulation results

indicate that 5 may well be the maximum number of feasible equilibria. (Up to 10 equilibria

arise from solving ij ijg g′ = for all i and j, but many of these are complex roots or unfeasible

solutions.)

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The two fixation equilibria (A1 fixed: g11 = 1, g12 = g21 = g22 = 0; A2 fixed: g11 = g12 =

g21 = 0, g22 = 1) always exist. The former is locally stable if

( )( )2 211111 1112 2121 1112 2121 1112 1221 21214 2 2w w w w w w w w> + + + + − (3)

which, in the absence of imprinting, collapses to the condition found by GAVRILETS’s (1998).

The analogous condition for local stability of the A2 fixation is

( )( )2 212222 2221 1212 2221 1212 2221 2112 12124 2 2w w w w w w w w> + + + + − . (4)

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’s

[1998] model), the only cycles known are of period 2 (SPENCER 2003). Consequently, the

mean fitness, w , need not be maximized (Fig. 1a).

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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 of its own phenotype. These

effects thus act multiplicatively, and so

ijkl ij klw m v= (5)

for i =1, 2, 3. Equations (1) thus become

( )

( )

( )

( )

1 111 11 11 11 12 12 21 212 2

1 112 12 11 11 12 12 21 212 2

1 121 21 12 12 21 21 22 222 2

1 122 22 12 12 21 21 22 222 2

,

,

,

.

wg v p m g m g m g

wg v q m g m g m g

wg v p m g m g m g

wg v q m g m g m g

′ = + +

′ = + +

′ = + +

′ = + +

(6)

Normalization means that just 6 parameters – 3 ms and 3 vs – 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.

Moreover, since

( )12 12 11 11

12 21 11 21 11

11 22 21 12 21 11

1

g v qg v p

v v g g gv v g v v g

′ ′=

′ ′ ′− −=

′ ′+

(7)

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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

1 1

11 11 12 12 21 212 2

11 11 12 12 21 21 22 22f

m g m g m gp

m g m g m g m g+ +

=+ + +

(8)

and pm = p, we have

1 1

11 11 12 12 21 212 2

11 11 12 12 21 21 22 22

f m f m f mf

f m f m f m f

m v p p m v p q m v q pp

m v p p m v p q m v q p m v q qm

+ +′ =

+ + + (9)

and

1 1

11 12 212 2

11 12 21 22

f m f m f mm

f m f m f m f m

v p p v p q v q pp

v p p v p q v q p v q q+ +

′ =+ + +

, (10)

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 4 distinct fitnesses: wijkl = αik (i, k œ {1, 2}). As

before, this allows Equations (1) to be simplified and, indeed, they may be reduced to just two

independent recursions:

11

( ) ( )1 1

11 11 212 21 1

11 11 12 21 22 222 2

1 12 2

f m f m f mf

f m f m f m

m f m f m f m

p p p q q pp

p p p q q p q q

p p p p q q p

f m

α α αα α α α α α

+ +′ =

+ + + + +

′ = + +

(11)

where

( ) ( )1 1

11 11 11 12 21 212 21 1

11 11 11 12 12 21 22 21 22 222 2

1 111 12 212 2

f

m

g g gp

g g g

p g g g

gα α α

α α α α α α+ +

=+ + + + +

= + +

(12)

The natural interpretation of pf and pm is a little different from the multiplicative case,

however. This time, pf is the post-selection frequency of A1; pm is the pre-selection frequency.

This system affords just 3 equilibria: two trivial fixations and one potential

polymorphism given by the pseudo-Hardy-Weinberg form 211ˆ ˆ=g p , and

, in which

12 21ˆ ˆ ˆ= =g g pq

222ˆ ˆ=g q

22 11 21

11 12 21 22

2ˆ ˆ ˆf mp p pα α α

α α α α− −

= = =− − +

(13)

and . The fixation of Aˆ 1= −q p 1 is locally stable provided ( )111 12 222α α α> + ; that of A2 if

(122 21 112 )α α α> + . 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 Fig 4, for an example). No cases of cycles of length greater than 2 were found in 107

numerical examples with the 4 αij 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 mother and her offspring. Maternal effects mean that this

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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 = βjl, Equations (1) reduce to

( )

( )

( )

( )

1 111 11 11 21 12 11 212 2

1 112 12 11 22 12 12 212 2

1 121 21 12 11 21 21 222 2

1 122 22 12 12 21 22 222 2

,

,

,

,

wg p g g g

wg q g g g

wg p g g g

wg q g g g

β β β

β β β

β β β

β β β

′ = + +

′ = + +

′ = + +

′ = + +

(14)

Note that the different βjl values appear a different number of times in Equations (14) than the

corresponding αiks 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 4 βjl

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 = γij and Equations (1) reduce to Equations

(6) with vij = 1 and mij = γij for all i and j. Thus, this system behaves as if no selection acts on

males and viability selection with fitnesses γij acts on females. Like the case of complete

paternal inactivation, this system affords just 3 equilibria: two trivial fixations and one

potential pseudo-Hardy-Weinberg polymorphism, in which

13

( )

22 12 21

11 12 21 22

2ˆ2

pγ γ γ

γ γ γ γ− −

=− − +

. (15)

The stability of the equilibria is simple: fixation of A1 is locally stable provided

(111 12 212 )γ γ γ> + , fixation of A2 if ( )1

22 12 212γ γ γ> + and the polymorphism is stable if both

these inequalities are reversed – amounting to mean heterozygote advantage – and this

condition simultaneously guarantees feasibility.

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 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 (1965) and leads to the recursion equations for the genotype

frequencies in the following generation of

( ) ( )( )

( )

2 1 11111 11 1112 1211 11 12 1121 2111 11 212 2

11 11 2 21 1 11221 2112 12 21 1212 12 2121 214 4 4

21 1 11112 11 12 1121 11 21 1122 11 22 1212 122 2 4

12 12 1 11221 2112 12 21 1222 14 2

f g f f g g f f g gwg v

f f g g f g f g

f g g f g g f g g f gwg v

f f g g f g

⎛ ⎞+ + + +′ = ⎜ ⎟⎜ ⎟+ + + +⎝ ⎠

+ + +′ =

+ + +

( )

( )

21 12 22 2121 21 2122 21 224 2

21 1 1 11211 11 12 1212 12 1221 2112 12 21 2111 11 212 4 4 2

21 21 21 1 12121 21 2211 11 22 2212 12 22 2221 21 224 2 2

21 11212 12 1221 21124 4

22 22

g f g f g g

f g g f g f f g g f g gwg v

f g f g g f g g f g g

f g f fwg v

⎛ ⎞⎜ ⎟⎜ ⎟+ +⎝ ⎠

⎛ ⎞+ + + +′ = ⎜ ⎟⎜ ⎟+ + + +⎝ ⎠

+ +′ =

( )( )

112 21 1222 2212 12 222

2 21 12121 21 2122 2221 21 22 2222 224 2

.g g f f g g

f g f f g g f g

⎛ ⎞+ +⎜ ⎟⎜ ⎟+ + + +⎝ ⎠

(16)

At first glance, BODMER’s (1965) model apparently requires nine fertility parameters,

corresponding to the 3 × 3 = 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

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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, a

A1A1 × A1A2 cross, with fertility f1112, produces one half A1A2 offspring, but a A1A2 × A1A1

cross, with fertility f1211, gives none. Hence, the parameter f1112 appears in the recursion for

the frequency of A1A2, , but f12g ′ 1211 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

( ) ( )222 2222 12 1222 21 2221 12 1222 21 2221 12 21 2122 2212 1222 22214 4v f v f v f v f v f v v f f f f> + + + + − (17)

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 non-imprinting case;

DOEBELI and DE JONG, 1998) but it must be quite rare: we found no cycling in 107

simulations.

DISCUSSION

The models above demonstrate that when selection acts on loci that engender maternal

genetic effects and that are subject to genomic imprinting, novel genotype-frequency

dynamics may arise. These behaviors include oscillations between two distinct polymorphic

values, as well as longer-period cycling lasting many generations due to Hopf bifurcations

(also known as Andronov-Hopf bifurcations). This last phenomenon is particularly

15

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 multi-locus

mutation with selfing (YANG and KONDRASHOV 2003).

The way in which the cycling between 2 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., Fig 3, 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 (Fig. 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 Fig. 1 and Fig. 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.

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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. 1996, 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 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 quantitative-genetic 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. Financial support

for this work was provided by the Marsden Fund of the Royal Society of New Zealand

contract U00-315 (H.G.S.). Additional thanks are due to Gustavus Adolphus College and the

Allan Wilson Centre at Massey University for funding and hosting the sabbatical for TL that

facilitated this collaboration.

17

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20

APPENDIX: HOPF BIFURCATIONS IN 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 ),(1 nn xFx λ=+ with λ∈ , , and F three times

differentiable. For a Hopf bifurcation to occur there must exist an equilibrium point x

nx∈

0 and a

parameter value 0λ such that the linearization of ),( xF λ with respect to x at ),( 00 xλ has a

pair of complex conjugate eigenvalues a bi± with a and b non-zero and (There

is one additional restriction that if a = -1/2 then

.122 =+ ba

.2/3±≠b ) Moreover, one must check that

as the parameter λ is varied monotonically about 0λ , the quantity goes from a value

less than 1 to a value greater than 1 or vice versa.

22 ba +

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 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 4 eigenvalues of the linearization at this critical point.

We note for completeness that there is one other sequence plotted in Figure A1. This

represents the real, non-zero eigenvalue of the linearization. The fourth eigenvalue equals

zero for all parameter values due to the normalization of equations (1).

Figure A1 shows the result of this process. The upper sequence plots the moduli of the

complex conjugate pair of eigenvalues as a function of w1111. The horizontal line at y = 1

represents the threshold that must be crossed for a Hopf bifurcation to occur. Note that this

sequence crosses this line at approximately w1111 = 0.02 and again at w1111 = 0.33. Further

refinements of this program yielded more accurate bifurcation values of w1111 = 0.0254 and

w1111 = 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 = 1 at both w1111 = 0.0254 and w1111 = 0.3395 verifies the final

condition stated above.

21

3. Plot each of the moduli against the corresponding parameter value.

c. Compute the modulus 22 ba +

for each of these eigenvalues.

Mother

Father A1A1 A1A2 A2A1 A2A2

A1A1 A211g 1A1 w1111

112 112 g g A1A1 w1211

112 112 g g A2A1 w1221

121 112 g g A1A1 w2111

121 112 g g A2A1 w2121

22 11g g A2A1 w2221

A1A2 111 122 g g A1A1 w1111

111 122 g g A1A2 w1112

112 124 g g A1A1 w1211

112 124 g g A1A2 w1212

112 124 g g A2A1 w1221

112 124 g g A2A2 w1222

121 124 g g A1A1 w2111

121 124 g g A1A2 w2112

121 124 g g A2A1 w2121

121 124 g g A2A2 w2122

122 122 g g A2A1 w2221

122 122 g g A2A2 w2222

A2A1 111 212 g g A1A1 w1111

111 212 g g A1A2 w1112

112 214 g g A1A1 w1211

112 214 g g A1A2 w1212

112 214 g g A2A1 w1221

112 214 g g A2A2 w1222

121 214 g g A1A1 w2111

121 214 g g A1A2 w2112

121 214 g g A2A1 w2121

121 214 g g A2A2 w2122

122 212 g g A2A1 w2221

122 212 g g A2A2 w2222

A2A2 A11 22g g 1A2 w11121

12 222 g g A1A2 w1212

112 222 g g A2A2 w1222

121 222 g g A1A2 w2112

121 222 g g A2A2 w2122

A222g 2A2 w2222

Table 1 – Mating table for general model, showing frequencies and fitnesses

Table 2 – Example of a Parameter Set Affording 5 Distinct Equilibria for General Maternal

Selection Model.

wijkl i, j

1,1 1,2 2,1 2,2

1, 1 0.191 0.278 0.483

1, 2 0.134 0.599 0.083

2, 1 0.667 0.765 0.062 k, l

2, 2 0.734 0.194 0.209

Empty cells correspond to impossible types: no A2A1 offspring have A1A1 mothers, for

example.

Equilibria:

11g 12g 21g 22g maxλ Locally

Stable?

0 0 0 1 1.456 No

0.072 0.214 0.148 0.566 0.960 Yes

0.125 0.208 0.214 0.453 1.031 No

0.319 0.138 0.374 0.169 0.888 Yes

1 0 0 0 2.315 No

maxλ is the leading eigenvalue of the Jacobian of the system of equations (1), linearized

around the equilibrium.

24

Figure Legends

Figure 1 – An example of long-period genotype-frequency cycling with w1211 = 0.75, w1221 =

0.02, w2121 = 0.96, w2111 = 0.07, w2221 = 0.01, w1112 = 0.04, w1212 = 0.72, w1222 = 0.13, w2122 =

0.31, w2112 = 0.00 and w2222 = 0.24. (a) Mean fitness and genotype frequencies g11 (solid line)

and g12 (dotted line) over time with w1111 = 0.25. (b) Bifurcation diagram for g11 as w1111 is

varied. Note the gradual appearance and disappearance of cycling as w1111 is increased.

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

parameters as in Fig. 1.

Figure 3 – Bifurcation diagram for g11 as w1211 is varied, with w1111 = 0.01, w1221 = 0.20, w2121

= 0.26, w2111 = 0.24, w2221 = 0.02, w1112 = 0.97, w1212 = 0.01, w1222 = 0.70, w2122 = 0.29, w2112 =

0.87 and w2222 = 0.08. Note the appearance and disappearance of 2-cycles. The dotted line

indicates the unstable polymorphic equilibrium.

Figure 4 – Cycling of genotype frequencies under complete paternal inactivation, with α11 =

0.04, α21 = 0.62 and α22 = 0.01. (a) Mean fitness and genotype frequencies g11 (solid line) and

g12 (dotted line) over time with α12 = 0.54. (b) Bifurcation diagram for g11 as α12 is varied.

The dotted line indicates the unstable polymorphic equilibrium.

Figure A1 – Modulus of complex eigenvalues as a function of w1111 for the numerical

example of Fig.1.

0.0

0.1

0.2

0.3

0.4

0.5

0.00.2

0.40.6

0.81.0

0.00.10.20.3

w1111

g11g12

w1211

0.0 0.1 0.2 0.3 0.4 0.5

g11

0.00

0.05

0.10

0.15

Fig. 3

Generation

0 20 40 60

g11 or g12

0.0

0.4

0.8

0.1

0.2

w

a

Fig. 4

α12

0.0 0.2 0.4 0.6 0.8 1.0

g11

0.0

0.5

1.0

b