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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
2
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: [email protected]
3
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
5
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.
6
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)
7
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.)
8
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).
9
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)
10
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
12
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
14
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.
16
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