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vol. 165, no. 5
the american naturalist

may 2005
Maintenance of Sex-Linked Deleterious Alleles by Selfing and
Group Selection in Metapopulations of the Phytopathogenic
Fungus Microbotryum violaceum
Aurélien Tellier,*,† Lorys M. M. A. Villaréal,*,‡ and Tatiana Giraud§
Ecologie, Systématique et Evolution, Unité Mixte de Recherche
8079, Centre National de la Recherche Scientifique-Université Paul
Sabatier, Bâtiment 360, Université Paris-Sud, F-91405 Orsay,
France
Submitted July 2, 2004; Accepted December 22, 2004;
Electronically published March 14, 2005
Online enhancements: appendix, figures.
abstract: Microbotryum violaceum is a fungus that causes the sterilizing anther smut disease in many Caryophyllaceae. Its diploid teliospores are heterozygous at the mating-type locus, normally producing equal proportions of haploid sporidia of the two mating types.
However, natural populations contain high frequencies of individuals
producing sporidia of only one mating type. This mating-type ratio
bias is caused by the presence of deleterious alleles at haploid phase
(“haplo-lethals”) linked to the mating-type locus. These haplo-lethals
can be transmitted if there is conjugation among the products of
meiosis (intratetrad selfing). Haplo-lethals still suffer from selective
disadvantages, through reducing the infection probability of strains
that carry them, and thus cannot persist in a panmictic population.
We develop a realistic model of a metapopulation of M. violaceum
on its host Silene latifolia. Simulations show that if intratetrad selfing
rate is high, haplo-lethals can be maintained under a metapopulation
structure because of founder effects and selection at the population
level. Populations founded only by strains carrying haplo-lethals experience a lower extinction rate precisely because of their lower infection ability; they spread more slowly and sterilize fewer plants,
thereby allowing their host population to grow more rapidly and
therefore to be less prone to extinction.
* A. Tellier and L. M. M. A. Villaréal contributed equally to the work in this
article and should be considered as sharing the position of first author.
†
E-mail: [email protected].
‡
E-mail: [email protected].
§
Corresponding author. Address for correspondence: Ecologie, Systématique
et Evolution, Bâtiment 362, Université Paris-Sud, F-91405 Orsay Cedex,
France; e-mail: [email protected].
Am. Nat. 2005. Vol. 165, pp. 577–589. 䉷 2005 by The University of Chicago.
0003-0147/2005/16505-40512$15.00. All rights reserved.
Keywords: population-level selection, parasites, virulence, disease
transmission, protected polymorphism, transmission rate.
Metapopulation dynamics, with populations connected by migration and experiencing regular extinctionrecolonization events, can yield patterns dramatically different from those expected in large panmictic populations
at equilibrium (Olivieri et al. 1990). This is because of
stochastic founder effects, demographic disequilibrium,
different levels of selection, and heterogeneity of selection
pressures among populations. Models using a metapopulation structure have been particularly useful for explaining natural patterns of polymorphism maintenance in sex
ratios and in genes involved in host-parasite relationships.
Gynodioecy in plants for instance (i.e., the coexistence
of hermaphroditic and female individuals), caused by nucleocytoplasmic polymorphism at loci involved in gender
determination, and the high frequencies of females observed in some natural populations of gynodioecious species are more easily explained in metapopulations
(McCauley and Taylor 1997; Couvet et al. 1998) than in
large panmictic populations (Charlesworth and Ganders
1979; Couvet et al. 1990; Olivieri et al. 1990). Group selection phenomena are involved in the maintenance of
such nucleocytoplasmic polymorphism in subdivided populations; founder effects create genetic variance between
populations, and local population growth is faster when
local frequency of females is high (Couvet et al. 1998).
Heterogeneity of selection pressures among populations
can also play a role in the evolution of gynodioecy and
female frequencies in a metapopulation because the fitness
of females and hermaphrodites is a function of local sex
ratio, and sex ratio varies among populations because of
founder effects (McCauley and Taylor 1997).
Metapopulation dynamics may also yield patterns of sex
ratio distorters opposite to those expected in panmictic
populations (Van Bowen and Weissing 1999). Meiotic
drive should confer a strong transmission advantage to sex
ratio distorters, resulting in their fixation. However, the
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The American Naturalist
known sex ratio distorters (e.g., those located on X chromosomes in Drosophila species) remain polymorphic in
nature (Hartl et al. 1967). Negative pleiotropic effects
(Beckenbach 1978) and drive suppressors (Jaenike 2001)
have been proposed to explain why these sex ratio distorters on X chromosomes are not fixed in Drosophila
species. Population structure may also facilitate the persistence of polymorphism because of selection at the population level; distorter alleles rapidly spread within populations, but the productivity of populations with high
distorter frequencies can be significantly reduced. An efficient distorter will therefore be underrepresented in the
migrant pool and may furthermore increase the probability of population extinction (Van Bowen and Weissing
1999).
Metapopulation structure has also been proposed to
greatly accelerate the accumulation of mildly deleterious
mutations (Higgins and Lynch 2001; Glemin et al. 2003),
not because of selection at different levels, but merely because stochasticity renders them nearly invisible to natural
selection. In this case, however, polymorphism cannot be
maintained because mildly deleterious mutations are fixed
through stochastic effects.
The high degree of dependence of most pathogens on
their hosts for long-term survival ensures that the size,
structure, and distribution of host populations are primary
determinants of the dynamics of pathogen populations in
natural ecosystems. Heterogeneity of host spatial distribution thus often results in parasite populations functioning as a metapopulation, which has dramatic consequences on the dynamics of gene frequencies of parasites
(Burdon and Jarosz 1992; Burdon 1993; Thrall and Jarosz
1994a, 1994b; Thrall et al. 1995). Metapopulation structure
is indeed known to affect patterns of local adaptation
(Lively 1999; Dybdahl and Storfer 2003 and references
therein), coexistence of hosts and pathogens (Thrall and
Burdon 1997; O’Keefe and Antonovics 2002), evolutionarily stable strategy (ESS) values for virulence and transmission rate (Boots and Sasaki 1999; O’Keefe and Antonovics 2002), and polymorphism maintenance of genes
involved in host-parasite relationships (May and Nowak
1994; Dybdahl and Storfer 2003; Thrall and Burdon 2003).
The main mechanism responsible for protected polymorphism in genes involved in host-parasite interaction is
frequency-dependent selection caused by the arms race
between hosts and parasites and the existence of costs of
infectivity and resistance (e.g., Antonovics and Thrall 1994;
Thrall and Burdon 2003). Population-level selection probably also often plays a role but has been evidenced only
in a few cases, such as the maintenance of polymorphism
in parasite virulence when superinfections occur (May and
Nowak 1994; Frank 1996) or ESS values of parasite virulence and transmission rate (Haraguchi and Sasaki 2000).
Because of the dramatic impacts of metapopulation dynamics on sex ratio evolution, on maintenance of polymorphism in genes involved in host-parasite relationships,
and on accumulation of deleterious alleles, we employ a
metapopulation approach in exploring the conditions of
maintenance of sex-linked deleterious alleles in the fungus
Microbotryum violaceum parasitizing the plant Silene latifolia. The S. latifolia–M. violaceum pathosystem has already been successfully explored with metapopulation
modeling and experiments to explain the maintenance of
polymorphism in quantitative resistant traits of the host
plants (Thrall et al. 1993a, 1995; Thrall and Jarosz 1994a,
1994b). Population genetics analyses (Delmotte et al. 1999;
Giraud 2004) have also supported the representation of
this plant-pathogen system as a spatially subdivided population model.
The anther-smut fungus M. violaceum (formerly Ustilago violacea) is a basidiomycete obligate parasite of many
Caryophyllaceae (Thrall et al. 1993b). In an infected plant,
the fungus produces teliospores in the place of pollen (in
dioecious plant species, infected females develop sporebearing anthers, and ovaries are aborted; Day and Garber
1988). The disease sterilizes but does not kill its host plant.
Teliospores are the dispersing form of M. violaceum; they
are transported from diseased to healthy plants by insects
that usually serve as pollinators for the host plant (Roche
et al. 1995; Shykoff and Bucheli 1995). Once deposited on
a host plant, diploid teliospores undergo meiosis and give
rise to four haploid cells (two of mating type A1 and two
of mating type A2), forming a three-celled promycelium
attached to the wall of the teliospore containing the fourth
cell. Each of these cells buds off yeast-like sporidia locally
on the plant surface. New infectious dicaryons are rapidly
produced by conjugation of two cells (sporidia or promycelial cells) of opposite mating type (Day and Garber
1988; Hood and Antonovics 1998). Microbotryum violaceum can perform either outcrossing or selfing, and selfing
can occur either between products of a single meiosis (intratetrad conjugation) or between cells from different meiotic tetrads of the same fungal individual (intertetrad conjugation). Dicaryotic hyphae then enter the host tissue and
grow endophytically. The plant becomes systemically infected the next year and produces only diseased flowers.
Recovery from disease is rare (Alexander and Antonovics
1988).
Several studies (Garber et al. 1978; Kaltz and Shykoff
1997; Oudemans et al. 1998; Garber and Ruddat 2000;
Hood and Antonovics 2000; Thomas et al. 2003) have
shown that some M. violaceum strains produce teliospores
all giving rise to sporidia of only one mating type (complete individual bias toward A1 or A2). This mating-type
ratio distortion has been explained by the presence of
highly deleterious alleles exposed at the haploid phase,
Sex-Linked Deleterious Alleles in Metapopulations
“haplo-lethals,” linked to one mating type (Hood and Antonovics 2000). These strains with mating-type ratio distortion (hereafter called “biased strains”) still produce four
cells following meiosis, but sporidia carrying the haplolethal allele quickly die after a few mitotic divisions (Hood
and Antonovics 2000). This mating-type ratio distortion
is therefore very different from sex ratio distortion described in animals (e.g., Hartl et al. 1967); it does not
result from the presence of distorters in overrepresented
gametes but from the presence of lethal alleles in nonviable
gametes. Such an original mechanism of segregation distortion has evolutionary dynamics completely different
from those of the usual distorters, and its existence appears
even more puzzling.
Teliospores of a strain with a biased mating-type ratio
are still fully able to produce an infectious dicaryon by
early intrapromycelial (intratetrad) conjugation between
the two adjacent sporidia of complementary mating type
(Oudemans et al. 1998; Hood and Antonovics 2000).
Haplo-lethals would therefore have absolutely no deleterious effect if all strains, biased and unbiased, underwent
only intratetrad matings (intratetrad mating rate of 1) because each teliospore would then give rise to a single conjugating dicaryon in both biased and unbiased strains.
Hood and Antonovics (2000), however, have shown that
if intrapromycelial conjugations are frequent in vivo, even
for unbiased strains, many intertretrad conjugations are
still observable. With an intratetrad mating rate lower than
1, biased strains are expected to suffer from disadvantages
that become stronger as the intratetrad mating rate decreases. For a given number of teliospores, unbiased strains
will indeed produce many more conjugating dicaryons
than biased strains; all teliospores not performing intratetrad mating will give no dicaryon in biased strains (when
alone on a plant), whereas they will produce many conjugating dicaryons in unbiased strains that produce viable
sporidia of both mating types. Kaltz and Shykoff (1999)
and Roche et al. (1995) have shown that infection probability increases with the number of conjugating dicaryons.
Infection probability is therefore expected to be lower for
biased strains than for nonbiased strains, and this has been
checked experimentally using artificial inoculations (T. Giraud and J. A. Shykoff, unpublished data). Given this disadvantage of haplo-lethal alleles, the high occurrence of
strains with a biased mating-type ratio in some natural
metapopulations of M. violaceum parasitizing S. latifolia
is surprising: 52% on 252 tested strains in Virginia (Oudemans et al. 1998), 14% on 29 tested strains in Switzerland
(Kaltz and Shykoff 1997), and 9% on 587 tested strains
in Essonne, France (Thomas et al. 2003). In addition, very
different percentages of biased strains were found in populations collected from the various host species, for example, 30%–100% on Dianthus carthusianorum in Swit-
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zerland against 9% on S. latifolia in France (Thomas et
al. 2003).
Antonovics et al. (1998), using a model of an infinite
panmictic population of M. violaceum, showed that
mating-type-linked haplo-lethal alleles could be maintained only if they exhibited a diploid advantage if the
intratetrad mating rate was at all lower than 1. However,
to date, no such advantage has been demonstrated. Furthermore, Antonovics et al. (1998) simulated a single gamete pool in the population, whereas distance-dependent
intrapopulation dispersal (Giraud 2004) and heterozygote
deficits (Delmotte et al. 1999; Giraud 2004) in M. violaceum rather suggest that gametes are strongly separated
on the different plants in a population. The selective disadvantage of haplo-lethals is therefore more likely to result
from a reduction of infection ability of strains that carry
them than from competition with other gametes. In fact,
infection probability of biased strains is similarly lower
than that of unbiased strains, whether in single inoculations or in competition with an unbiased strain (T. Giraud
and J. A. Shykoff, unpublished data). In this study, using
a realistic metapopulation model of the system S. latifolia–
M. violaceum, we examine under which conditions and
through which selective forces the maintenance of polymorphism for the mating-type ratio bias in M. violaceum
is possible. In particular, we explore whether this polymorphism can be maintained despite reduction in infection ability of biased strains in the absence of compensatory diploid advantage of these biased strains and with
intratetrad mating rates lower than 1.
The Model
We developed a spatially explicit simulation model of sets
of local populations (patches) incorporating both withinpopulation dynamics and among-population dispersal.
This model was built by modifying and combining previous models, some of them for exploring the conditions
of host and pathogen coexistence using metapopulation
dynamics (Thrall and Jarosz 1994a; Thrall et al. 1995) and
another for simulating the evolution of haplo-lethals but
without metapopulation structure or separation of gametes on the different plants within the population (Antonovics et al. 1998). Each diseased plant was considered
to be infected by a single strain, which is usually the case
in natural populations (Baird and Garber 1979; Day 1980;
T. Giraud, unpublished data; but see Hood 2003). We also
assumed that the resistant or sensitive state of a plant was
independent of the biased or unbiased state of a fungal
strain (M. E. Hood, personal communication). The model
was parameterized using data from previous field studies
(Thrall and Jarosz 1994a, 1994b; Thrall et al. 1995) because
our aim was to investigate the conditions of haplo-lethal
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The American Naturalist
maintenance in conditions as close as possible to natural
conditions. Because some parameters may, however, be
variable in natural conditions, we also investigated how
changing parameter values influenced the results of the
simulations.
The model assumed deterministic within-population
dynamics, whereas dispersal and extinction/recolonization
steps were stochastic. Each generation was thus composed
of three steps. First, the within-patch dynamics of disease
transmission were simulated through the deterministic
equations (1)–(6) described below. Second, a percentage
of seeds and teliospores of each occupied site was allowed
to disperse stochastically to other occupied sites or to colonize empty sites. Third, to insure population turnover,
some patches went extinct stochastically before the completion of each generation. Model structure and design of
the simulations are detailed below, and the variables used
in the model are listed in table 1.
Within-Patch Dynamics
Sexually transmitted diseases, including M. violaceum disease on S. latifolia populations with pollinators for vectors
(Thrall et al. 1993a), are considered to follow a frequencydependent transmission (Getz and Pickering 1983; May
and Anderson 1990). Reproduction by healthy hosts was
assumed to occur early during the growing season and to
be followed by infection and possible overwinter host
death. This assumption is based on the long time lag between the receipt of spores and the appearance of newly
infected flowers (Alexander 1990). The simplifying assumption of no host recovery was made. Our model sim-
ulated disease transmission in which vectors were likely to
carry fungal teliospores from the two previously visited
hosts, if diseased. Competition for infection then occurred
between the two types of teliospores, biased and unbiased,
if both were present.
In a given population at time t ⫹ 1, the numbers of
healthy plants (X t⫹1), of plants infected by a biased strain
(Yt⫹1), and of plants infected by an unbiased strain
(Z t⫹1) can be written as follows (summary of the notations
can be found in table 1):
{
[
(
X t⫹1 p X t # b ⫹ (1 ⫺ d) 1 ⫺ Yt b
)
fa
Yt
Zt
⫹ fa 2 2 ⫹ D B a 2 2
Nt
Nt
Nt
(
⫺ Z tb
)]}
a
Zt
Y
⫹ a 2 2 ⫹ D NB a 2 2 ,
Nt
Nt
Nt
(
Yt
Yt 2
X t ⫹ bfa 2 2 X t
Nt
Nt
⫹ D B ba 2
Yt Z t
Xt ,
Nt2
Yt⫹1 p (1 ⫺ d) # Yt ⫹ bfa
(
Z t⫹1 p (1 ⫺ d) # Z t ⫹ ba
(1)
)
(2)
)
Zt
Z t2
Z tYt
X t ⫹ ba 2 2 X t ⫹ D NB ba 2 2 X t ,
Nt
Nt
Nt
(3)
where d is the intrinsic death rate (the mean death rates
were set identical for healthy and diseased host, d p 0.1;
Thrall et al. 1995; Shykoff and Kaltz 1998), a is the probability that a pollinator visiting a healthy flower leaves
spores of M. violaceum (a p 0.25), b is the host birth rate,
and Nt is the total host population at time t (total number
of diseased and healthy plants). The b denotes the infection
probability of unbiased strains when alone on a plant
Table 1: List of the variables used in the model
Notation
Definition
Xt
Yt
Zt
Nt
b
d
a
l
g
f
b
DNB
DB
k, v
m
g
Kext
N0
Number of healthy plants in a population at generation t
Number of plants infected by a biased strain in a population at generation t
Number of plants infected by an unbiased strain in a population at generation t
Total number of plants in a population (healthy ⫹ diseased plants)
Host birth rate
Host death rate
Probability for a pollinator visiting a healthy flower to leave spores of Microbotryum violaceum
Maximum reproductive rate of the host
Constant determining the strength of the density dependence
Intratetrad selfing rate
Infection probability of an unbiased strain alone on a plant
Proportion of competition events won by unbiased strains
Proportion of competition events won by biased strains
Parameters of the Weibull dispersal function (shape of the curve)
Proportion of dispersed seeds from a given occupied patch to other patches
Probability of establishment of dispersing seeds
Extinction coefficient
Initial number of plants in each population
Sex-Linked Deleterious Alleles in Metapopulations
(b p 0.4; Thrall and Jarosz 1994b). The DNB is the proportion of competition events won by unbiased strains,
and DB the proportion of competition events won by biased strains.
Thrall et al. (1993a) showed that a regulation mechanism of the host population independent from the disease
dynamics was required for stable coexistence of hosts and
parasites in the metapopulation. We therefore assumed a
density-dependent per capita reproductive rate of S. latifolia (Thrall and Jarosz 1994b), and the host birth rate b
was written as
bp
l
,
gNt ⫹ 1
(4)
where l is the maximum reproductive rate of the host
(l p 2; Thrall and Jarosz 1994a) and g is a constant that
determines the strength of the density dependence (g p
0.4; Thrall and Jarosz 1994a).
In equations (2) and (3), we considered three types of
newly infected plants in each patch. First, there were plants
that received spores from only one diseased plant (second
term in brackets). Second, there were plants that received
spores from two diseased plants, but all spores were of the
same type (biased or unbiased) of spores (third term in
brackets). Third, there were plants that received spores
from two diseased plants, one with biased spores and the
other with unbiased spores (fourth term in brackets).
Because infection probability should be linked to the
quantity of infectious dicaryons produced by conjugation
(Roche et al. 1995; Kaltz and Shykoff 1999) and because
only sporidia involved in intratetrad conjugations produce
dicaryons in biased strains, the basic infection probability
of biased strains was multiplied by the intratetrad selfing
rate f to give the effective infection probability (f # b in
eq. [2]). Unbiased strains were also considered to have an
intratetrad selfing rate of f, but their sporidia not involved
in intratetrad matings can still produce dicaryons via intertetrad conjugation, in contrast to those of biased strains.
The intratetrad selfing rate f therefore does not influence
the infection probability of unbiased strains in our model.
The disadvantage of biased strains caused by their haplolethals depends in our model on the quantity of sporidia
not involved in intratetrad matings (1 ⫺ f) and consists
in a reduction of infection probability.
Some plants simultaneously received spores from both
types of strains, biased and unbiased. To determine which
type of spores won if infection was successful, we assumed
that the plants received a panmictic population of biased
and unbiased strains, as in the unique population simulated by Antonovics et al. (1998). In their model, they
considered a unique pool of gametes in a panmictic population and did not simulate the infection dynamics. The
581
disadvantage of haplo-lethals thus resulted only from competition with other gametes. In our model, gametes are
separated on the different plants in the population. When
a single biased strain is deposited on a plant, the selective
disadvantage of the haplo-lethal results only from the
lower infection probability of the strain. When two strains
are deposited on a plant, competition among gametes occurs. In the cases where one biased strain and one unbiased
strain are deposited on the same plant, we applied the
formula from the model of Antonovics et al. (1998), with
no pleiotropic advantage associated to biased strain during
the dicaryotic phase (s p 0). Following their model, the
proportion at time t ⫹ 1 of unbiased strains (Ut⫹1) in a
panmictic population of M. violaceum composed of biased
and unbiased strains can be written as
Ut⫹1 p Ut
f ⫹ [2(1 ⫺ f)/(Ut ⫹ 1)]
,
f ⫹ [2U(1
⫺ f)/(Ut ⫹ 1)]
t
(5)
with Ut being the proportion of unbiased strains at time
t and f being the fraction of biased strains that undergoes
intratetrad selfing. We assumed, in cases of competition
between the two types of strains, that teliospores from
biased and unbiased strains were deposited in equal proportions; that is, Ut p 0.5. The proportion of competition
events won by unbiased strains DNB was therefore computed as Ut⫹1 in equation (5), with Ut p 0.5. Hence
DB p
4⫺f
.
2f ⫹ 4
(6)
The proportion of competition events won by biased
strains is then D B p 1 ⫺ D NB.
Among-Patch Dynamics
The metapopulation was composed of a one-dimensional
array of 200 patches with two extreme absorbing boundaries to mimic linear roadside metapopulations of S. latifolia, with defined size beyond which propagules are lost
from the system (Thrall et al. 1995). Field data from Thrall
et al. (1995) indicate a high rate of population turnover,
with substantial colonization and extinction for the host.
The plant colonization rate, equal to the population extinction rate, was estimated over several years to be 20%
per year (Thrall et al. 1995). Parameter values detailed
below resulted in an extinction/colonization mean rate for
plant populations of 24% per generation.
A fraction, m p 0.05, of the plants in a given patch
produces seeds that disperse to the four neighboring
patches, according to a Weibull probability distribution.
This fat-tailed function shows low resolution at very low
dispersal distances (e.g., within population; Clark et al.
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The American Naturalist
1999) but was fitted to describe the pattern of seed dispersal over long distances (Thrall and Jarosz 1994a). The
probability of landing in a patch that is i units from the
source patch is given by
k
k
P(i) p k i k⫺1e⫺(i/v) ,
v
population status, that is, healthy or diseased hosts (Thrall
and Jarosz 1994a; Thrall et al. 1995). The extinction probability associated with a given occupied patch was therefore
P(Nt) p
(7)
where the parameters k and v, respectively, control the
scale and the shape of the dispersal curve (Martz and
Waller 1982; Thrall and Jarosz 1994a). In our model, we
set these parameters at k p 0.9 and v p 0.7, that is, at
higher values than Thrall and Jarosz (1994a). This was to
balance the dispersal restriction to the four neighboring
patches and thus obtain stable coexistence of plants and
fungi. The probability of establishment of dispersing seeds
was fixed at g p 0.8 (Thrall and Jarosz 1994a), whereas
seeds that remained within their patch were subjected to
density-dependent regulation (eq. [4]).
Metapopulations of M. violaceum follow an island
model dynamic rather than a stepping-stone model because there is no isolation by distance between populations
(Delmotte et al. 1999; Giraud 2004). In our model, a fraction, 0.125, of the diseased plants in a given patch send
teliospores randomly toward any of the other occupied
patches, where they infect the plants with a probability of
b or b # f, respectively, for unbiased and biased strains.
Thus, biased strains also have a disadvantage in colonizing
new patches. For unbiased strains, the fraction of dispersing spores is then 0.125 # b p 0.05, as in Thrall and
Jarosz (1994a). These parameters result in similar dispersal
rates of M. violaceum and S. latifolia, whereas in natural
populations, the fungus disperses less than does the plant
(Delmotte et al. 1999). However, relative dispersal abilities
are important only when plant-pathogen coevolution is
studied, for instance, for local adaptation (Kaltz et al.
1999). The pattern of plant dispersal should not influence
the conditions of maintenance of haplo-lethals here because the resistance of plant genotypes and the virulence
of fungi genotypes were not taken into account. As a control, we performed simulations using a stepping-stone
model of pathogen dispersal, and similar results indeed
emerged (data not shown). However, because existence of
the fungus requires the presence of the plant, very low
rates of plant migration added to highly disturbed habitat
(high extinction rates) should have a large impact on the
persistence of this plant-pathogen system and may result
in the extinction of the fungus or even in the crash of the
entire metapopulation.
After among-patch dispersal, a probability of extinction
was calculated for each occupied patch. Previous analysis
of host extinction rate in natural populations showed that
there was a large effect of population size but no effect of
1
,
1 ⫹ Nt # K ext
(8)
where Nt is the total number of plants in a patch and Kext
is the extinction coefficient. Because our model assumed
a greater dispersal of M. violaceum than did the model by
Thrall et al. (1995), to balance the above approximations
of probability of host dispersal, a greater extinction probability was used to stabilize the metapopulation dynamics
(K ext p 0.9 in our model, compared with K ext p 1.133 in
Thrall et al. 1995).
Initialization
To start each simulation, 40 out of the 200 possible patches
were randomly filled with N0 p 50 plants each. Twenty
percent of these initial populations were then randomly
chosen to be diseased at a prevalence of 0.25 (Thrall and
Jarosz 1994a). Each initial fungal population had a proportion M0 of unbiased strains and a proportion 1 ⫺
M 0 of biased strains. Each simulation was allowed to run
for 300 generations, by which time spatial and temporal
patterns generally became stable. The metapopulation dynamics were considered to be stable when values of the
number of occupied patches, the population’s extinction/
colonization rates, and biased strains frequencies were virtually unchanged for 100 generations.
Modeling and Statistical Analyses
Simulations were performed using the Scilab software (version 2.7; INRIA, France). The code is available on request.
Depending on the studied parameters, 15–40 repetitions
were run. These numbers of replicates appeared sufficient,
given the low variances in results among replicates (see
next section). SAS software (SAS 2001) was used to test
the significance of mean differences by applying several
simple comparison tests of means (Student t-test) or multiple comparison tests (Bonferroni and Tukey HSD under
GLM procedure; Zar 1984). In cases of inequality of variances, nonparametric tests were used to confirm the results of the above tests (Zar 1984) because transformations
of data (arcsine or log) failed to stabilize the variances.
Results
We first analyzed the evolution of haplo-lethals in a single
population of our model, that is, using only the withinpatch dynamics of disease transmission simulated through
Sex-Linked Deleterious Alleles in Metapopulations
the deterministic equations (1)–(6), considering that the
number of healthy plants was not limiting. Biased strains
always disappeared from the population unless the intratetrad selfing rate was exactly 1 (fig. 1). This is consistent
with the results of the nonspatial model by Antonovics et
al. (1998), although their model considered a single gamete
pool in the population, whereas ours takes into account
the separation of gametes on the different plants within
the population and the different infection abilities of biased and unbiased strains. Corresponding to one of the
model assumptions, when both biased and unbiased
strains undergo only intratetrad selfing, biased strains incur no disadvantage. In our model with f p 1, equation
(2) indeed reduces to equation (3). When f ! 1, biased
strains suffer from a disadvantage in infection ability and
from gamete competition when the two types of spores
are deposited on the same plant, and they therefore decrease in frequency until extinction.
We then explored the conditions of maintenance of
haplo-lethals using the metapopulation model. With an
intratetrad selfing rate of 1, biased strains could also persist
in the metapopulation for the same reason as in the single
population. In the results of the simulations with f p 1,
the mean percentage of biased strains did not differ from
0.5 at generation 300 (fig. 2; Student t-test, df p 24,
P p .74). When biased strains had a disadvantage in infection probability (f ! 1), the maintenance of haplolethals without any pleiotropic advantage was still possible
under metapopulation dynamics when the intratetrad conjugation rate was greater than 0.7 (fig. 2). For values of
the intratetrad conjugation rate below or equal to 0.7, the
percentages of biased strains decreased to values not significantly different from 0 (Bonferroni test, 1% at generation 300, df p 13; fig. 2). In contrast, for values of the
intratetrad conjugation rate above 0.7, the percentages of
biased strains reached stable values significantly different
from 0 (Bonferroni test, 1% at generation 300, df p 13;
fig. 2). The stability of the ratio between biased and nonbiased strains after generation 200 is indicated by the plateau and the small standard errors in fig. 2 and by a few
simulations run over 500 generations (not shown). These
equilibrium values of the percentage of biased strains in
the metapopulation increased with the intratetrad conjugation rate f (fig. 2).
Biased strains could be maintained in the metapopulation even when they incurred slight disadvantages (1 1
f 1 0.7) because of founder effects and selection at the
population level. Indeed, recently colonized populations
(small populations, younger than 20 generations) represented a large majority (∼70%) of the total number of
fungal populations and exhibited mostly extreme frequencies of biased or unbiased strains (figs. 3, 4 in the
online edition of the American Naturalist). This shows that
583
Figure 1: Change in the mean percentage of biased strains in a single
population for different values of the intratetrad selfing rate f (M0 p
0.5).
drift rather than selection was responsible for the evolution
of biased strains in young populations; recurrent population extinctions allowed recolonization of healthy populations by only biased strains because of founder effects.
Older populations, with larger sizes, exhibited an unbiased
strain distribution skewed toward 1 (figs. 3, 4). Selection
for unbiased strains, rather than drift, was thus responsible
for the evolution of biased strains frequencies within populations. Biased strains have indeed a lower infection probability (f # b) than unbiased strains (b). This is true
within populations but also for colonization of new populations. The colonization ability of biased strains is therefore lower than that of unbiased strains. They were indeed
underrepresented in the migrant pool (not shown).
However, populations with only biased strains experience a lower rate of extinction than populations with only
unbiased strains, the difference being greatest between
generations 20 and 70 for lower intratetrad selfing rates
and for larger populations (fig. 5). In fact, the proportion
of extinct populations was greater in our simulations
among the populations with only unbiased strains than
among those with only biased strains (fig. 6). This was
precisely because of the lower infection ability of biased
strains; they spread more slowly than unbiased strains
within their populations, sterilizing fewer host plants and
thereby allowing their host population to reproduce more
and to grow more rapidly. Because the extinction probability depends on the number of plants in a population
(eq. [8]), this leads to a lower extinction rate of populations with biased strains than of populations with unbiased
strains. Populations with both biased and unbiased strains
have intermediate extinction rates (fig. 6). The balance
between extinction rates and infection abilities thus leads
to a stable equilibrium of the percentages of biased and
584
The American Naturalist
Figure 2: Change in the mean percentage of biased strains in the metapopulation for different values of the intratetrad selfing rate f (M0 p 0.5;
means and SE over populations and 40 repetitions). Groups with different letters have significantly different means (Bonferroni test, 1% at generation
300, df p 13).
unbiased strains in the metapopulation. Biased strains are
selected against within populations and for colonization,
but founder effects allow many populations to have only
biased strains, or high percentages of them, and these populations live longer. Founder effects and selection at the
population level are thus responsible of the maintenance
of a stable polymorphism of individual mating-type ratio
in the metapopulation. For intratetrad conjugation rates
lower than 0.7, the extinction rate of populations with
biased strains was still lower than that of populations with
unbiased strains, but this advantage was not sufficient to
balance the stronger disadvantage in infection ability of
biased strains.
The maintenance of biased strains was also dependent
on the initial frequency of unbiased strains introduced in
the metapopulation (M0). In the absence of mutation (as
assumed in our model), the biased (respectively unbiased)
strains could not appear when starting with an initial proportion of biased strains of M 0 p 0 (respectively, M 0 p
1). For extreme values of M0 (M 0 ! 0.2 or M 0 1 0.8), genetic drift led to rapid loss of the rare strain type. For
intermediate values of the frequency of unbiased strains
initially introduced in the metapopulation, the precise values of M0 did not influence the long-term values of the
proportion of biased strain (fig. 7 in the online edition of
the American Naturalist).
Robustness of the Results
Our model was run using the range of parameter values
available from previous field studies (Thrall and Jarosz
1994a, 1994b; Thrall et al. 1995) because our aim was to
be as realistic as possible. Some parameters, however, may
be variable in natural conditions, so we investigated how
changing parameter values affected the results. The parameter values that allow the coexistence of plants and
pathogens produce similar results, in particular regarding
the intratetrad selfing rate required for haplo-lethal maintenance (see appendix in the online edition of the American Naturalist). The results obtained with the limited
range of parameters extensively studied above therefore
seem to be robust. The threshold value of 0.7 for the
intratetrad selfing rate should not, however, be taken too
strictly because it still could vary a little, depending on
parameter values and some model assumptions such as
the relationship between the disadvantage in infection ability by biased strains and the intratetrad selfing rate.
Discussion
Metapopulation Effect on the Maintenance
of Haplo-Lethals
Many species exist as metapopulations, in balance between
local population extinction and recolonization (Thrall et
al. 1993a; Delmotte et al. 1999; Kaltz and Shykoff 1999),
processes that have been shown to affect the distribution
of genetic diversity within populations and in the metapopulation as a whole because of stochastic founder effects,
demographic disequilibrium, different levels of selection,
and heterogeneity of selection pressures among populations (Olivieri et al. 1990). Peculiarities of metapopulation
dynamics often lead to different evolution of polymor-
Sex-Linked Deleterious Alleles in Metapopulations
Figure 5: Ratio of the probability of extinction of populations with only
biased strains over that of populations with only unbiased strains, derived
from the deterministic equations of within-population dynamics, for different values of the intratetrad selfing rate f and of the initial number
of plants N0.
phism than is expected in a same finite-sized panmictic
population.
The simulation results of the model presented in this
study are in accordance with this idea. They show that
mating-type-linked deleterious alleles that cause a reduction in infection ability can persist in a polymorphic state
even in the absence of compensatory advantage, with their
maintenance being caused only by a peculiar selfing mechanism and a metapopulation structure. Under metapop-
585
ulation dynamics and without any advantage linked to
haplo-lethal alleles, we indeed observed the stable maintenance of biased strains when the intratetrad selfing rate
was higher than ∼0.7. In contrast, in a single panmictic
population, biased strains always became extinct when
they had no pleiotropic advantage, as soon as the intratetrad selfing rate was even slightly lower than 1 (Antonovics et al. 1998; this study). The maintenance of haplolethals in our study was permitted by high intratetrad
selfing rates that reduced the disadvantage of strains carrying these mating-type-linked deleterious alleles and by
metapopulation dynamics, with founder effects and selection at the population level. The recurrent population extinctions allowed recolonization of healthy populations by
only biased strains because of founder effects, and these
populations with only biased strains experienced lower
extinction rates than populations with unbiased strains.
The balance between extinction rates and infection abilities
of biased and unbiased strains thus leads to a stable equilibrium of the percentages of haplo-lethals in the metapopulations. Biased strains are selected against within populations and for colonization, but their populations live
longer. This mechanism is consistent with patterns in natural populations, where large plant populations showed
high percentages of biased strains, whereas small populations exhibited more extreme proportions of biased
strains (Thomas et al. 2003).
Metapopulation structure, with subdivided populations
and environmental stochasticity, had already been proposed to greatly accelerate the accumulation of mildly deleterious mutations (Higgins and Lynch 2001; Glemin et
al. 2003). However, in these cases, polymorphism was not
Figure 6: Percentage of extinction in the different types of populations—with only unbiased strains, with both biased and unbiased strains, and
with only biased strains (M0 p 0.5, f p 0.8; mean over 20 repetitions).
586
The American Naturalist
maintained; mildly deleterious mutations were fixed because stochasticity rendered them nearly invisible to natural selection. Our study rather provides another example
of maintenance of polymorphism through group selection
in metapopulations, as in the case of gynodioecy (Couvet
et al. 1998), sex ratio distorters (Van Bowen and Weissing
1999), and virulence in superinfections (May and Nowak
1994). Our results can be generalized to the evolution of
transmission rates in parasites; individuals with different
transmission rates can coexist in a metapopulation if those
that have a lower transmission rate and hence lower fitness
within populations also induce a lower extinction rate of
the populations where they are prominent, for instance,
by allowing their host population to reproduce more or
by keeping their hosts alive longer. Our study thus provides
one of the very few examples of population-level selection
in host-parasite evolution, through a mechanism allowing
protected polymorphism in parasite transmission rate
without any associated cost or host variability in resistance.
Intratetrad Selfing Rate
Using a realistic range of parameter values, this model
showed that it is crucial to estimate the intratetrad selfing
rate parameter if we are to explain the maintenance of
haplo-lethals. If the intratetrad selfing rate is 1, biased
strains have no real disadvantage because haplo-lethals are
then never in a haploid stage, and unbiased strains never
have the opportunity to express the ability of their sporidia
to replicate. Considering that the metapopulation is stable
over a long-term period (a in nature has a value ∼0.25,
and metapopulations indeed appear stable), if the intratetrad selfing rate is lower than 1 but still high (higher
than ∼0.7), there is no need to look for pleiotropic advantages of haplo-lethals because metapopulation dynamics allows them to be maintained despite their deleterious
effect. Conversely, if the intratetrad selfing rate is low (approximately lower than 0.7), compensatory advantages
linked to haplo-lethals are expected.
Circumstantial evidence that intratetrad selfing rates are
high in Microbotryum violaceum come from analyses of
karyotypes and of regions of heterozygosity (Hood and
Antonovics 2004). To precisely determine the proportion
of intratetrad conjugations, markers linked to a centromere other than the one to which the mating-type locus
is linked are needed. For heterozygous individuals at such
a marker, intratetrad matings will always restore heterozygosity, whereas self intertetrad matings will yield half
heterozygous dicaryons and half homozygous ones (Hood
and Antonovics 2000, 2004). The color markers used by
Baird and Garber (1979) in their inoculation experiments
met these requirements, and we can derive from their
results an estimated intratetrad selfing rate of 90.8%
(100% ⫺ twice the percentage of homozygous genotypes
recovered from their heterozygous inoculated genotype).
However, the possible nonneutrality of the color markers
used by Baird and Garber (1979) may have biased the
results of their study. Giraud et al. (2005) performed artificial inoculations of mixtures of M. violaceum strains
with different genotypes for microsatellite markers. These
microsatellite markers are more likely to be neutral, but
their degree of linkage to centromeres is unknown. The
outcrossing rate was estimated between 14.3% and 28.3%,
depending on the identity of inoculated pair of strains.
From this, intratetrad mating rates between 43.3% and
71.4% could be calculated. The range of this estimate is
very large and different from the one derived from the
results of Baird and Garber (1979), but it still overlapped
our threshold of 0.7, suggesting that pleiotropic advantages
of strains carrying haplo-lethal alleles may not be necessary
to explain their maintenance. However, given the variability of the estimations of intratetrad mating rates available to date, additional studies, specifically designed to
obtain precise estimates of the intratetrad selfing rate in
M. violaceum and its degree of variability among strains,
may be needed. Ideally, selfing conjugations of numerous
strains should be analyzed with neutral markers such
as microsatellites linked to a centromere other than the
mating-type locus.
Initial Percentage of Unbiased Strains
The initial percentage of unbiased strains in the metapopulation (M0) was also an important parameter for the
maintenance of haplo-lethals. Intermediate values of the
initial percentage of unbiased strains had no influence on
the final value of the percentage of biased strains in the
metapopulation, but for extreme values of this parameter,
genetic drift led the rare type of strains to extinction.
Haplo-lethals should therefore vanish if they appear in a
metapopulation by a single mutation. However, haplolethals have been suggested to appear regularly because of
recurrent mutations caused by transposable element insertions (Garber and Ruddat 2000; Thomas et al. 2003).
Thus, there should be a significant probability that haplolethals sometimes will appear alone in a few populations,
multiplying until they reach a sufficiently high percentage
to then be maintained by metapopulation dynamics.
Linkage of Haplo-Lethals to the Centromere
Our model assumed complete linkage between the haplolethal locus and the mating-type locus. In M. violaceum,
there is indeed strong evidence of crossover suppression
in the vicinity of the mating-type locus but also throughout
the genome (Garber et al. 1987; Hood and Antonovics
Sex-Linked Deleterious Alleles in Metapopulations
2000). Haplo-lethal alleles in particular were most often
found completely linked to the mating-type locus (Hood
and Antonovics 2000) and sometimes linked to another
centromere (bias at the teliospore level; Oudemans et al.
1998; Thomas et al. 2003). This is compatible with the
theoretical expectation that a deleterious mutation can be
protected only if it is completely linked to the region of
heterozygosity (Leach et al. 1986). However, Antonovics
and Abrams (2004) showed that intratetrad mating allows
protection of deleterious recessive alleles even if there is
partial linkage between the mating-type locus and the
haplo-lethal locus if an advantage is associated with it in
the heterozygous state.
Differences among Host Races
The various Caryophyllaceous hosts of M. violaceum exhibit significant differences in their patterns of disease level
and percentages of biased strains (Thomas et al. 2003).
Our model may allow a comprehensive study of haplolethal frequency patterns on the different host species by
indicating which ecological traits are important to investigate and to contrast among host species. Such a study
could in turn validate our model. First, it appears important to measure the intratetrad selfing rate and its range
of variability on each species of host plant. The intratetrad
selfing rate is a key component of the maintenance of
haplo-lethals and of the expected frequencies in populations. If the various host races of M. violaceum have different intratetrad selfing rates, as suggested by differences
in karyotype variability (Hood and Antonovics 2004) and/
or have different infection rates (appendix), our model
may be able to explain their levels of biased strains. The
second factor that could be responsible for the observed
differences in haplo-lethal frequencies among host races
of M. violaceum is the dynamics of metapopulations of the
different host plants. The relationship between extinction
rate and size of the populations in particular should
strongly impact on haplo-lethal frequencies because it is
the basis of the mechanism allowing the maintenance of
biased strains. Third, the probability that pollinators transfer spores also represents a key parameter in the change
of proportions of biased strains in metapopulations (appendix). Such ecological parameters, however, are often
difficult to obtain, and significant efforts to improve the
knowledge of the ecology of these plant species may be
required in order to integrate modeling of pollinator behaviors and responses of wild plant populations to the
environment.
587
Acknowledgments
We thank A. Garnier, C. Lavigne, C. Montchamp-Moreau,
J. A. Shykoff, and A. Thomas for helpful discussions; J. A.
Shykoff, M. Veuille, R. Wyand, and anonymous reviewers
for useful comments on the manuscript; F. Austerlitz for
suggestions on seed dispersal modeling; and E. Klein and
E. Porcher for advice on statistical analyses.
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