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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 578 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- 579 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 580 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. 582 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. 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