TRANSITIONS FROM REPRODUCTIVE SYSTEMS GOVERNED BY TWO SELF-INCOMPATIBLE LOCI TO ONE IN FUNGI

Authors


Abstract

Self-incompatibility (SI), a reproductive system broadly present in plants, chordates, fungi, and protists, might be controlled by one or several multiallelic loci. How a transition in the number of SI loci can occur and the consequences of such events for the population's genetics and dynamics have not been studied theoretically. Here, we provide analytical descriptions of two transition mechanisms: linkage of the two SI loci (scenario 1) and the loss of function of one incompatibility gene within a mating type of a population with two SI loci (scenario 2). We show that invasion of populations by the new mating type form depends on whether the fitness of the new type is lowered, and on the allelic diversity of the SI loci and the recombination between the two SI loci in the starting population. Moreover, under scenario 1, it also depends on the frequency of the SI alleles that became linked. We demonstrate that, following invasion, complete transitions in the reproductive system occurs under scenario 2 and is predicted only for small populations under scenario 1. Interestingly, such events are associated with a drastic reduction in mating type number.

Recognition systems involved in discriminating between self and nonself, such as mating types in fungi (Billiard et al. 2011) or protists (Phadke and Zufall 2009), and self-incompatibility (SI) in plants (Lawrence 2000) are important mechanisms in many species to prevent self-fertilization and promote outcrossing (Charlesworth and Charlesworth 1979; Lande and Schemske 1985). The genes that control these systems are present at one or several loci in the genome and several such systems show extraordinary high levels of genetic diversity. Although SI in flowering plants can be controlled by a single multiallelic locus (S-locus) (Takayama and Isogai 2005), it is controlled by two multiallelic and independent loci in grasses (S and Z, see review in Yang et al. 2008) and invertebrate chordates (Harada et al. 2008). In fungi, there are either one or two incompatibility loci (Casselton et al. 1998; Casselton 2002; Billiard et al. 2011).

The maintenance of genetic diversity at self-incompatible loci is easily explained by negative frequency-dependent selection, whereby individuals with a rare SI haplotype can mate with more partners than individuals with a common SI haplotype. As a consequence, rare or novel SI haplotypes quickly increase in frequency, and are therefore less likely than neutral alleles to be lost by genetic drift (Wright 1939). The advantage to a new allele decreases as its frequency in the population increases. In populations where two loci are involved in the recognition mechanism, recombination and the related linkage disequilibrium also affect the dynamics and evolution of incompatibility haplotypes as they determine the distribution of alleles in the population and in consequence, the advantage of an incompatibility haplotype in relationship to others in the population.

A major unsolved puzzle is the evolution of the number of incompatible loci. Evolutionary transitions in reproductive systems generally involve changes to novel forms that spread and replace ancestral ones (Barrett 2008). The emergence of new incompatibility haplotypes (Gervais et al. 2011) and the dynamics of a diploid population with a multiallelic locus have been thoroughly explored (Asmussen and Basnayake 1990; Altenberg 1991; Gavrilets and Hastings 1995; Yi et al. 1999; Asmussen et al. 2004; Trotter and Spencer 2007, 2009). Iwanaga and Sasaki (2004) provided a pioneering study of the dynamic of reproductive systems where SI and hierarchical cytoplasmic inheritance are controlled by two multiallelic loci in haploid systems. However, the transition from a reproductive system governed by two loci, to a single locus one has not been modelled. Moreover, the consequences of such transitions have no theoretical basis, although such transitions might have occurred at different levels on the tree of life.

For the fungal phylum Basidiomycota, two evolutionary scenarios have been proposed to explain transitions from two self-incompatible loci to a single locus system (Fraser et al. 2007). Examples are known within this phylum, where, in closely related species, reproduction is controlled by either one (a reproductive system called unifactorial or bipolar) or two (a reproductive system called bifactorial or tetrapolar) independently segregating multiallelic mating-type (MAT) loci (Fig. 1; Hibbett and Donoghue 2001; Casselton and Kües 2007; Coelho et al. 2010). In unifactorial populations, successful fusion between two mating types occurs only if they have different incompatibility alleles at their unique MAT locus, whereas in bifactorial populations, this event occurs only if they have different incompatibility alleles at both MAT loci. The first scenario of transition from a bifactorial to a unifactorial reproductive system assumes a translocation of the chromosomal segment containing one MAT locus to the close vicinity of the other. This has occurred in human pathogens such as Cryptococcus neoformans (Lengeler et al. 2002) and Malassezia globosa (Xu et al. 2007), and in plant pathogen Ustilago hordei (Lee et al. 1999). The second scenario hypothesizes the establishment of a loss of the incompatibility function mutation in one of the MAT loci, the B locus as in the mushrooms Coprinellus disseminatus (James et al. 2006) and Pholiota nameko (Aimi et al. 2005; Yi et al. 2009).

Figure 1.

A nonexhaustive cladogram illustrating the distribution of the reproductive systems through Basidiomycota phylum. Names of species with unifactorial reproductive system are represented in boxes, white and gray boxes are, respectively, biallelic and multiallelic unifactorial species. The other species are bifactorial species. The number of mating-type alleles is indicated when available. The Ascomycota phylum have been taken as outgroup. More generally, in the subphylum Agaricomycotina, which encompasses the mushrooms, 65% of the species are bifactorial with two multiallelic loci and 35% are unifactorial. In the latter, 35% some originate from linkage of the two loci (inline image) and are biallelic, other have lost their incompatibility function (AB*) and are multiallelic or are homothallic (Raper 1966). In the subphylum Ustilaginomycotina, containing plant pathogenic species, species with bifactorial reproductive systems are closely related to biallelic unifactorial species (Bakkeren et al. 2008). In the subphylum Pucciniomycotina, containing the rust fungi, most species present a biallelic unifactorial system and some species have a multiallelic intermediary reproductive system, inline image, between the unifactorial and the bifactorial state (Coelho et al. 2010). Data are derived from Hibbett and Donoghue (2001), James (2007), Wang et al. (2009), and Kües et al. (2011).

Another interesting feature of the fungal phylum Basidiomycota is that MAT loci can display a high or low level of allele diversity (Fig. 1; Hibbett and Donoghue 2001), leading to reproductive systems where the number of MAT genotypes varies between two and thousands. If allelic diversity of MAT loci is maintained by negative frequency-dependent selection, the origin of systems with few mating types is puzzling.

Evolutionary transitions may involve intermediate forms that might come at a cost (Maynard and Szathmary 1995). In the fungal phylum Basidiomycota, intermediate forms of mating types between unifactorial and bifactorial have been described and have reduced fitness (Fraser et al. 2004; Hsueh et al. 2008; Coelho et al. 2010). Additionally, conflicts are expected to occur, for example, when mating types also control organelle transmission during reproduction (Hurst and Hamilton 1992; Hurst 1995, 1996). Other functional conflicts are expected to occur as mating types may also play a key role in triggering developmental switches along their haploid–diploid life cycle (Perrin 2012). Such costs might make some evolutionary transitions less likely than others.

The objects of the present work are as follows. First, we analytically study the population genetics of a reproductive system governed by one SI locus and of two unlinked SI loci. Second, we model and evaluate the two main scenarios for transition from a bifactorial to a unifactorial system. We derive analytically the conditions of invasion by a new mating type under the both scenarios hypothesized above, taking into account the number of alleles at MAT loci, the recombination rates between them and a potential fitness reduction of a new mating type. We determine the conditions for complete transition of a reproductive system from two loci to a single incompatibility locus. Finally, we characterize the genetic implications of such a transition, particularly for the allelic diversity of self-incompatible loci and on the number of the mating types. The results of this study provide novel insights into the dynamic of reproductive systems with SI, on how transitions in the number of SI loci can occur, and how they affect the genetic diversity and the number of SI haplotypes maintained in a population.

Self-Incompatible Reproductive Systems

We consider a haploid population in which gametes fusion is successful only between individuals carrying different MAT alleles (a condition called heterothallism) at a single MAT locus (a population with a unifactorial system) or at two MAT loci (a population with a bifactorial system). In both systems, the fitness of a genotype is determined by the proportion of compatible genotypes present in the population (rare genotype advantage induced by SI following Wright 1939). The two reproductive systems can be described analytically as follows (as also described in Iwanaga and Sasaki 2004).

UNIFACTORIAL SYSTEM

Assume a population with n alleles, Ai, at the MAT locus present at a frequency,pi. In these populations, gamete fusion occurs only between two haploids carrying different alleles, Ai and Aj, with ij. The fitness,  inline image, of genotype Ai can be defined as the proportion of compatible mates in the population or inline image. The change in one generation in the genotype Ai frequency pi is then described by

image(1)

where

image(2)

A stable equilibrium is always reached, independent of the initial frequencies, when the frequencies of all n MAT alleles in the population are at the same frequency

image(3)

BIFACTORIAL SYSTEM

In bifactorial heterothallic populations, two individuals can successfully fuse only if they carry different alleles at both MAT loci. The life cycle includes three main steps: gamete fusion, recombination, and gametogenesis.

Denoting the incompatible alleles at the A locus by Ai and at the B locus by Bj, each haploid genotype AiBj is present in the population at frequency pij and each incompatible allele Ai and Bi at frequency inline image and inline image, respectively. The change in the frequency pij of the genotype AiBj after one generation depends on its fitness, inline image, characterized by the proportion of compatible genotypes in the population, and on the recombination rate, r, between the MAT loci. The change in the frequency pij of a haploid genotype AiBj, is described by the following recursion equations (developed in Appendix A)

image(4)

with the linkage disequilibrium between the MAT loci

image(5)

and where

image(6)

and

image(7)

Equation 4 has stable and unstable equilibria depending on the recombination rate, r, and the number, n, of A alleles and, m, B alleles (Supporting Information). When inline image a stable equilibrium exists with isoplethy (all genotypes have the same frequency)

image(8)

where inline image and inline image being, respectively, the number of A and B alleles at SI loci at equilibrium.

When inline image, the solutions of equation 4 depend on the number of alleles at the MAT A locus (n) and the number of alleles at the MAT B locus (m). When the number of alleles at the MAT A and B alleles is the same (n=m), and when inline image, Iwanaga and Sasaki (2004) found two other equilibria (Appendix B). When the number of A alleles differs from the number of B alleles (inline image), multiple solutions can be found and chaotic behavior is observed. The behaviors of the solutions found for equation 4 are summarized in the bifurcation diagrams in Figure 2, which also shows the changes in MAT genotype frequencies. Solutions depend on the initial frequency of MAT genotypes in the population and on the linkage disequilibrium between the incompatibility loci. With a small recombination rate, the A and B alleles in the population are expected to be nonrandomly associated (in linkage disequilibrium). Therefore, when A and B alleles have a high and positive linkage disequilibrium, they are more likely to co-occur in the same mating type and are rarely found in different mating types. Mating types carrying the rarest SI alleles and the rarest association of SI alleles in the population are therefore compatible with several other mating types, and are thus favored, and will increase in frequency (Wright 1939). Under these conditions, groups of matching mating types with strong linkage disequilibrium can be favored. This process is well illustrated first in Figure 2 where we can observe that the frequency of some mating types with rare SI allele strongly increase, although others are maintained at a low frequency and, second, in Figure S1, where we can observe that linkage disequilibrium is maintained when recombination rate is small inline image. Thus, the frequency, the distribution, and the association of the A and B alleles drive the dynamics of MAT frequencies in the population.

Figure 2.

Dynamics of mating-type (MAT) genotype frequencies in a bifactorial population. The first column represents the bifurcation diagrams for equation 4. The bifurcation diagram shows the possible long-term values of genotype frequencies as a function of the recombination rate r (the bifurcation parameter). The second and third columns illustrate the change of MAT frequencies as a function of time (number of generations) when inline image and inline image, respectively, starting from random initial frequencies (each line represents a different genotype). Each row in the panel represents the dynamics of MAT genotypes frequencies considering different numbers of MAT A alleles (n) and MAT B alleles (m): upper row n= 2, m= 2 (r= 0.248 and 0.251), middle row n= 5; m= 7 (r= 0.0236 and 0.0789) and lower row n= 2; m= 100 (r= 0.001 and 0.1).

Transitions From two loci to single locus in SI systems

To examine the two transition scenarios from bifactorial to unifactorial system, we first derive the analytical expressions for the MAT frequencies when a new mating type with a unifactorial reproductive strategy arises in a bifactorial population, assuming either a chromosome rearrangement causing linkage between the two MAT loci (scenario 1) or loss of function at one MAT locus (scenario 2). We study the conditions for invasion of a bifactorial population by a mating type with a unifactorial reproductive strategy. Invasion is characterized in relation to the allele numbers at both MAT loci, n and m, the recombination rate, r, of the bifactorial population, and the potential fitness reduction (zygote mortality), α, associated with the new unifactorial mating type. Finally, we estimate the equilibrium MAT frequencies in the resulting population. In a later section, we investigate how analytical predictions are affected by genetic drift in finite populations.

DETERMINISTIC INVASION OF MATING TYPES WITH ONE SI LOCI IN A BIFACTORIAL POPULATION

First scenario of transition: a translocation causing the two MAT loci to become linked

We assume that the rearrangement causing linkage of the two MAT loci brings together either: (1) a common A allele and a common B allele, (2) a novel A allele that has recently arisen by mutation, and a common B allele, or (3) new mutant alleles at both the A and B loci. After fusion between a linked and a recombinant (unlinked) mating type, we assume that fitness is reduced by a fixed factor, α, that varies between 0 and 0.5. This factor is intended to take account of potential cost (fitness reduction in zygote) due to conflicts in cytoplasmic organelle inheritance (Hurst and Hamilton 1992; Hurst 1995, 1996) and/or of the formation of intermediate reproductive system, such as tripolar or the pseudounifactorial reproductive system (described by Fraser et al. 2004; Hsueh et al. 2008; Coelho et al. 2010). For each of the three situations considered ((1)–(3)), we present the recursion equations that describe the evolution and the conditions for invasion and coexistence.

The dynamics and evolution of a new linked MAT genotype, AB-linked, differ and modify the dynamic of the MAT genotypes present in a bifactorial population (as described in Appendix B). The change in frequency of the initial resident MAT alleles in the population, inline image (inline image denotes inline imagein the presence of the mating type inline image) and the new linked mating type inline image composed by i and j alleles that are either common or novel in the population are described as follow (Appendix B)

image(9)

with

image(10)

The fitness of recombinant and nonrecombinant MAT genotypes are

image(11)

and the linkage equilibrium for each possible MAT genotype present in the population is described by

image(12)

Finally, the mean fitness, inline image, in the population is

image(13)

In the domain where inline image, the new linked genotype is either lost in the population (the frequency of this linked mating type is zero, inline image) or it is maintained in the population (inline image). From equations 9, when inline image, the frequency of the MAT genotypes still present in the population after invasion (those that are compatible with the AB-linked genotype, that is, which do not have MAT alleles Ai and/or Bj) can be determined (Appendix C) and are, for each case as follows:

image(14)

Stability of these equilibria is analyzed in Supporting Information.

 The frequency of a new linked mating type, once a bifactorial population is established, depends on the recombination rate, r, and the fitness reduction associated with the linked MAT genotype, α (Fig. 3). With no fitness reduction (α= 0), the new linked mating type invades in all situations considered ((1)–(3): linkage composed by common or novel alleles, Fig. 3). However, the final frequency of the new MAT genotype in the bifactorial population depends on the initial frequency of the MAT alleles that become linked and on the number of MAT alleles in the initial population (equation 14, Fig. 3). Trajectories of MAT frequencies after the introduction of a new linked mating type, AB-linked, are shown in Figures 4 (r = 0.5) and S2 (r = 0.001). The new genotype, AB-linked composed by MAT alleles, Ai, Bj (Figs. 4, S2, plain line) invades the population and eliminates all the recombinant mating types carrying Ai, Bj (Figs. 4, S2, dashed line). Once established, AB-linked segregates, at high frequency, with all compatible MAT (i.e., all those in the population which do not carry of Ai and/or Bj MAT alleles, Figs. 4, S2, gray line). From these results, it follows that when Ai and/orBj are alleles new to the population, no MAT genotypes initially present are eliminated. The equilibrium frequency of the linked mating type in the population is lower than when the MAT alleles in the linked genotype were initially common (Figs. 4, S2). Recombinant mating types that are maintained in the population (the ones that are compatible with the novel linked mating type) are all maintained at the same frequencies: inline image, where n* et m* are the number of A and B alleles that differ from Ai and Bj, respectively.

Figure 3.

Theoretical parameter domains for invasion of a new mating-type genotype as a function of recombination rate, r, and fitness reduction, α, for the two scenarios: (1) linkage between the two MAT loci brings together either: a common SI A allele and a common SI B allele (the first column, inline image), a novel SI A allele and a common SI B allele (the second column, inline image) or two novel SI alleles at A and B locus (the third column, inline image) and (2) a new mating type that has lost its self-incompatible function at the B locus (the fourth column, AB*). Each row in the panel represents a different population with a different number of MAT A alleles (n) and MAT B alleles (m) in the invaded bifactorial population: upper row n= 2, m= 2; middle row n= 5, m= 7; lower row n= 2, m= 100. The dashed line represents the threshold value inline image and the dotted line represents the threshold value of α that determines invasion (Supplementary Information Table 1). Values are obtained with a numerical analysis of equation 9.

Figure 4.

Trajectories of the changes in frequency of mating-type (MAT) genotypes when a new AB-linked MAT genotype arises in a bifactorial population. Linkage between the two MAT loci brings together either: a common SI A allele and a common SI B allele (the first column, inline image), a novel SI A allele and a common SI B allele (the second column, inline image) or two novel SI alleles at A and B locus (the third column, inline image). Results are presented for the domain where the recombination rate, r, exceed the threshold inline image (r= 0.5) and there is no fitness reduction (α= 0). Each row in the panel represents a different population with a different number of MAT A alleles (n) and MAT B alleles (m) in the invaded bifactorial population: upper row n= 2, m= 2; middle row n= 5, m= 7; lower row n= 2, m= 100. The black line represents the linked mating type, the gray line represents all mating types compatible with the linked mating type, and the dashed line represents all mating types incompatible with the linked mating type. Values are obtained with a numerical analysis of equation 9.

When the new linked mating type is assumed to suffer no fitness reduction, recombination suppression gives an advantage to this new mating type over the recombining mating type. This arises because the new mating type in each generation will produce more compatible mating types. Indeed, half of the progeny of a linked mating type is compatible with the parental linked mating type and progeny of a linked mating type are compatible with each other (fusion between a AB-linked mating type, composed by Ai and Bj MAT allele, and a recombinant mating type AkBl produces 50% of AiBj - linked genotype and 50% of AkBl genotype). In contrast, genotypes produced by the same recombinant mating type AiBj will produce only a proportion r/2 of compatible mating types (fusion between theAiBj genotype and AkBl genotype produce the following proportion of genotypes (1 −r)/2  AiBj, (1 −r)/2  AkBl, r/2  AiBl, and r/2  AkBj as the two MAT loci are recombining).

The strength of the advantage of recombination suppression depends thus also on linkage disequilibrium in the population. For example, when the linkage disequilibrium between MAT allele Ai and Bj is strong and negative (the alleles Ai and Bj are mostly present in different mating types), recombination can significantly increase the frequency of the AiBj mating type in the population. In contrast, when the linkage disequilibrium between MAT allele Ai and Bj is strong and positive (the alleles Ai and Bj are mostly associated to the same mating types), recombination can then favor groups of matching genotypes, and linkage disequilibrium is maintained (Figs. S3, S4). Nevertheless, when the recombination rate decreases and drops to zero, all mating types have the same fitness and they coexist.

When fitness reduction is assumed (α > 0), invasion of the new linked MAT genotype depends on the novelty of the linked MAT alleles, on the number of MAT alleles present in the invaded population, and on the threshold value inline image.

When r is above the threshold value, inline image, and a fitness reduction is assumed, α > 0, invasion occurs only when the AB-linked genotype carries at least one new (rare) MAT allele (Fig. 3). The frequency at which the AB-linked genotype is maintained in the population decreases as the fitness reduction α increases. When, for example, α= 0.5, the advantage to the AB-linked genotype due to the recombination suppression (through its production of 50% of compatible offspring) is cancelled out by the fitness reduction (that reduces by 50% the number of compatible offspring). This demonstrate that the dynamics are driven by two processes: the advantage (for the recombinant mating type) of generating progeny of many different MAT genotypes versus the advantage (for the mutant AB-linked mating type) of not recombining and thus producing offspring that are compatible with one another. This is illustrated in Figure S5 (α= 0.5, r = 0.5), where the AB-linked genotype (black line) invades the bifactorial population only when it carries at least one novel MAT allele and the number of MAT allele is low.

When r is below the threshold value, inline image, the frequency at which the AB-linked genotype is maintained in the population increases with the recombination rate and decreases with the fitness reduction, α, (Fig. 3). In this restricted domain, as inline image tends to zero as n and m increases, the behavior of the system is also affected by the chaotic dynamics of the mating type presented in previous section (Fig. 2). Frequency of mating types after invasion depends on the initial frequencies of the genotypes and on linkage disequilibrium.

In all cases investigated, invasion by an AB-linked genotype leads to the coexistence of the bifactorial and the unifactorial reproductive systems in the population. Nevertheless, invasion causes elimination from the population of all MAT genotypes incompatible with the AB-linked genotype, thus reducing the number of MAT genotypes in the population. At equilibrium, the AB-linked genotype is present at a much higher frequency than the other MAT genotypes, which all segregate at low frequency (inline image, equation 14) in the population (Figs. 4, S2, S5, S6). In consequence, if the number of MAT alleles is high, the frequency of the recombinant mating type can be very low (Figs. 4, S6). As some mating types are present at very low frequencies, genetic drift is also expected to induce reduction of MAT genotypes. This aspect will be investigated considering finite population size with a stochastic model in “Transitions in the number of self-incompatible loci in finite populations”.

Second scenario of transition: loss of self-incompatible function at the B locus

The second scenario postulates that a new self-compatible MAT allele arises and replaces one of the incompatibility loci in a bifactorial population. We consider the case where the new genotype, inline image, is self-compatible at the B MAT locus, B*, and we assume again a fitness reduction in zygote, α.

The dynamics and evolution of a new MAT genotype, AB*, differ and modify the dynamic of the MAT genotypes present in a bifactorial population. Two types of mating types are produced, one type is composed by two self-incompatible MAT alleles, inline image, (inline imagedenotes inline image in the presence of the mating type, AiB*, at frequency inline image) and the other type is composed by one self-incompatible allele and the other the self-compatible allele inline image. We derive the change of MAT genotypes frequencies in the population from the dynamics of MAT genotypes through the life cycle (Appendix D)

image(15)

where, respectively, inline image and inline image correspond to the fitness and the frequency of the MAT genotype that carries the self-compatible MAT allele. The fitnesses in the population are

image(16)

and the linkage disequilibrium becomes

image(17)

Finally, the mean fitness of the population becomes

image(18)

As shown in Figure 3, this new MAT genotype always invades the population when its fitness is not reduced (α= 0). When α > 0 and inline image, invasion is successful only when inline image, with m, the number of B alleles (Supporting Information). Once the new mating type invades the bifactorial population, it replaces all MAT genotypes that carry a self-incompatible B allele (see illustrative trajectories in Fig. S7). Thus, the number of MAT genotypes is reduced to n, the number of A alleles, which are all at frequency inline image. Complete transition from a bifactorial to a unifactorial reproductive system is achieved.

TRANSITIONS IN THE NUMBER OF SI LOCI IN FINITE POPULATIONS

We use stochastic simulations to explore the effect of genetic drift on the conditions for transition in a self-incompatible reproductive system under the two scenarios already studied analytically.

We consider a bifactorial populations of size N in which SI is controlled by two biallelic MAT loci (n= 2 and m= 2). The four resulting genotypes (A1B1, A1B2, A2B1 and A2B2) are assumed to be at equal frequency and the recombination rate, r, is assumed to be 0.5. At the beginning of the simulations, we introduce a new unifactorial MAT genotype, following scenarios 1 and 2, in the population at a frequency of 1/N. A fitness reduction, α= 0.5, is assumed for the AB-linked genotype. We simulated genetic drift by multinomial sampling of N genotypes from the distribution of genotype frequencies in the population. We followed the new MAT genotype until is lost by genetic drift, or maintained for long term, for populations of sizes N= 30, 50, 100, or 1000 and record the frequencies of the different outcomes in 1000 simulations for each set of parameters, for up to 10,000 generations.

First scenario: linkage of the two MAT loci

As in the analytical model, we first investigate the situations where recombination is suppressed between A and B incompatible locus with the same three situations with respect to allele frequencies at the MAT locus, when (1) A and B alleles are common; (2) the A allele is new and the B allele is common; or (3) A and B alleles are new. Simulation results confirm our analytical predictions. Although AB-linked mating type is quickly lost from the population when both the alleles A and B are common, this genotype invades and is maintained when one of the alleles is rare (when the A allele is rare and the B allele is common, case (2)). The genotype is maintained in a proportion of runs ranging from 0.062 to 0.021 with population sizes ranging from N= 30 to N= 1000 (Fig. 5A). When the alleles A and B are new, case (3), the genotype is even more likely to be maintained (0.62–0.79 of the runs, Fig. 5B). Once maintained, the AB-linked MAT genotype segregates with a single one of the compatible genotypes present in the initial population (in our example, either A2B2 or A1B2 for case (2) and any initial genotypes for the case (3)). All other genotypes are lost from the population. The transition from a bifactorial to a unifactorial system is achieved. In small populations, transition can be rapid (less than 50 generations for case (2) and for case (3), from about 280 generations in a population of N= 30 to N= 4700 generation for N= 50). Thus, the populations resulting from the transition from a bifactorial to a unifactorial system have drastic reductions in the number of mating types. However, a complete transition in the reproductive system was observed only in small populations with our parameter values. In larger populations, the results are (as expected) similar to the deterministic ones and the number of MAT genotype in the population increases, that is, no transition in the reproductive system occurred but both the unifactorial and bifactorial systems coexist (Fig. 5B).

Figure 5.

Probability of invasion and time to invasion of a new mating-type (MAT) genotype in a bifactorial population (n= 2, m= 2) as a function of the population size, N. Three forms of new MAT genotypes are considered (A and B): a linked MAT genotype carrying a novel SI A allele and a common SI B allele (inline image) or carrying two novel SI alleles at A and B locus (inline image) and (C and D) a new MAT genotype that has lost its self-incompatible function at the B locus associated with either a common or novel SI allele at the A locus (A1B*and An+1B*, respectively). After invasion, the new MAT genotype is maintained with one of the initial genotype (either genotype 1 in dark or genotype 2 in middle gray) or is maintained with all initial genotypes (light gray). (A) The new linked genotype, inline image, is maintained either with A1B2, (genotype 1) or A2B2 (genotype 2). (B) The new linked genotype, inline image is maintained with either A1B1 or A1B2 (genotype 1) or A2B1 or A2B2 (genotype 2). (C) The new MAT genotype A1B* is maintained with A2B* (genotype 1). (D) The new MAT genotype An+1B* is maintained with either with A1B* (genotype 1) or A2B* (genotype 2). Results are over 1000 simulations after 10,000 generations.

Second scenario: loss of SI capacity at the B locus

For all population sizes, there is a very high probability that a new MAT genotype in which SI is lost at the MAT B locus invades the population (P= 0.76–0.79 if the A allele is common, A1B* or A2B* genotypes, Fig. 5C; P= 0.95–0.98 if the A allele is novel, A3B*, Fig. 5D), replacing all self-incompatible B alleles by the self-compatible B* allele. The resulting population has a unifactorial-like reproductive system with a number of MAT genotypes that equals to the number of A alleles. The frequency of the A allele carried by the individual in which B3* arose slightly affects the speed of the transition. When MAT genotype with a common MATA alleles loses its functional B allele, the transition to the unifactorial system occurs within just a few generations (24–122 generations for population sizes of N = 30 to N= 1000 A1B*; Fig. 5C). When the MAT A allele is rare (A3B*), this transition is completed within 42–480 generations for population sizes from N = 30 to N = 100 (Fig. 5D). Simulation results confirm thus analytical results; under this scenario, transition from a bifactorial to a unifactorial system is easily achieved.

Discussion

SCENARIOS OF TRANSITION FROM A BIFACTORIAL TO A UNIFACTORIAL SYSTEM

Our analytical results show that both scenarios proposed for the transition from a bifactorial to a unifactorial reproductive system are likely to occur. However, conditions for complete transition differ as well as the genetic implications of such transition (genetic diversity of MAT alleles and the number of mating types maintained) of the resulting populations (Table 1).

Table 1.  Theoretical parameter domains for invasion, number, and genotypes of mating types maintained after invasion and frequency at equilibrium of a new mating-type genotype after invasion for the two scenarios. Scenario 1: linkage between the two MAT loci brings together either a common SI A allele and a common SI B allele (inline image), a novel SI A allele and a common SI B allele (inline image), or two novel SI alleles at A and B locus (inline image). Scenario 2: a new mating type that has lost its self-incompatible function at the B locus (AB*). Results are presented for the domain where the recombination rate, r, exceed the threshold inline image. n and m are the initial number of SI alleles at both the A and B loci, respectively, in the bifactorial population and n* and m* are the number of A and B alleles at SI loci at equilibrium, respectively, after invasion and α is the fitness reduction associated to the new unifactorial mating type.
 Scenario 1Scenario 2
inline image inline image inline image A 1 B*
 
Invasion condition α≃ 0 inline image inline image inline image
Number of mating types inline image inline image nm # 1 n
Mating-type genotypes inline image inline image inline image inline image
Frequency of the new mating type inline image inline image inline image inline image

First scenario of transition: a translocation event links the two MAT loci

Under the first scenario, invasion of a unifactorial mating type is driven by its potential fitness reduction and its novelty, but also by the linkage disequilibrium and by the number of MAT alleles at the two MAT loci in the invaded bifactorial population. Recombination suppression gives a fitness advantage to the new mating type over the bifactorial recombinant mating types. AB-linked mating type produces 50% of compatible progeny, whereas other bifactorial recombinant resident mating types produce only (1 −r)/2 of compatible progeny. When the AB-linked mating type arising in the bifactorial population suffers from fitness reduction, α, its invasion depends on the novelty of the two MAT alleles caught within the AB-linked locus as well as on the number of incompatible alleles present in the bifactorial population. Conditions for invasion, in the domain where recombination rate is medium to high (inline image), are established (Supporting Information) and presented in Table 1. In the situation where the AB-linked mating type brings together a common and a novel MAT allele, it invades only when inline image (n being the number of alleles at the same locus already present in the bifactorial population). When the AB-linked mating type brings together two novel MAT alleles, it invades only when inline image, (n and m being the number of alleles at each locus already present in the bifactorial population). The resulting population has a reduced number of mating types (Table 1) in which both reproductive systems coexist.

The stochastic simulation model confirms results predicted by determinist analytical models: invasions of bifactorial population by the linked unifactorial mating type, coexistence of both reproductive systems, and reduction in the number of mating types. However, after invasion, in small-sized populations, we also observe a complete transition from the bifactorial to unifactorial reproductive system. Indeed, after invasion, the linked mating type is maintained at a much higher frequency than other mating types (Table 1), the mating type that are not compatible with the linked mating type are eliminated by drift, allowing for a complete transition in the reproductive system. Thus, when transitions occur, they are associated with a drastic reduction in the number of mating types and allelic diversity.

It is remarkable that all known species with a unifactorial heterothallism that originate from a linkage of the two MAT loci—U. hordei, M. globosa, and C. neoformans (reviewed by Bakkeren and Kronstad 1994; Hsueh and Heitman 2008)—as well as their bifactorial phylogenetically close relatives are pathogenic species. They might therefore become isolated or strongly structured into small demes (Litvintseva et al. 2011) and thus meet the condition of small population sizes.

Other mechanisms are expected to play a role in the transition. Empirical studies suggest, when events corresponding to scenario 2 occur, nonMAT genes within the nonrecombining region of the MAT locus might increase the selective advantage of this new MAT allele. For example, in C. neoformans, genes encoding components of the pheromone response pathway that regulate mating, fruiting, or virulence are within the AB-linked locus (Lengeler et al. 2002).

Second scenario of transition: loss of self-incompatible capacity at the B locus

Under the second scenario, transition from bifactorial to unifactoral reproductive system occurs under a large range of conditions. When the recombination rate is medium to high (inline image), invasion and complete transition is achieved when inline image, that is, when the potential fitness reduction, α, of the novel mating type is small in relation to the allelic diversity at the locus were SI is lost in the bifactorial population (with m the number of alleles present in the bifactorial population at the locus were SI is lost). The resulting population is a unifactorial one, in which the number of mating types is reduced to the number of MAT alleles at the remaining self-incompatible locus (n). After an invasion and fixation of the new self-compatible B allele within the population, we observe a purge of all self-incompatible alleles at the B locus (m). These results are confirmed by the stochastic simulation model.

Species such as C. disseminatus are thought to have evolved via this scenario (James et al. 2006). As in our model, they present a high allelic diversity at their single incompatibility MAT locus and a single allele at the locus that is hypothesized to have lost its incompatibility.

GENETIC IMPLICATIONS OF TRANSITION

Interestingly, we show that a complete transition from two loci to a single locus in SI systems is associated with a reduction in the number of mating types in the population. In consequence, we found a mechanism that reduces the number of mating types in the population and that can come as an advantage of being unifactorial. As mating types are expected to control potential inheritance conflicts and developmental switches along the sexual cycle (Hurst and Hamilton 1992; Hurst 1995; Perrin 2012), the reduction in the number of mating types also reduces the intragenomic and functional conflicts or the costs related to the regulation of the inheritance of cytoplasmic genetic elements (Billiard et al. 2011; Perrin 2012). Having a reduced number of mating types might be advantageous and thus be maintained. Indeed, having numerous mating types implies the activation of various functions associated to each mating type (Perrin 2012), for example, functionality and activation of new transcription factors or pheromone receptors in relation to existing ones. The benefit of a reduction in costs and conflicts within the life cycle of a unifactorial species might thus be stronger than the benefit of maintaining genetic variability and having a larger proportion of compatible mate, especially in haploid lineage where inbreeding avoidance is questioned (Billiard et al. 2011; Perrin 2012).


Associate Editor: A. Agrawal

ACKNOWLEDGMENTS

The authors thank J. Pannell, N. Perrin, L. Lehman, T. Giraud, D. Charlesworth, and A. Hirzel for comments or discussions on the manuscripts. We are also strongly grateful to A. Agrawal and two anonymous reviewers for their comments that considerably improved our manuscript. The study was supported by the Swiss National Science Foundation grants #PZ00P3_121702, #PZ00P3_139421/1, and #31003A-130065 to SV and PMPDP3_129027 to HNH.

Appendices

Appendix A

The analytical description and analysis of the dynamics of the bifactorial reproductive system presented hereafter is analogue to the work of Iwanaga and Sasaki (2004) on the hierarchical cytoplasmic inheritance in the plasmodial slime mold. In consequences, we will use similar notation and follow their approach when applicable. The life cycle is composed by the three main steps: gamete fusion, recombination, and gametogenesis. First, two random haploid genotypes (AiBk and AjBl) meet; if the two gametes carry different mating type alleles at the two loci (ij and kl), they fuse and form a zygote inline image:

image((A1))

Fusion is followed by recombination between the MAT loci, the frequency of the zygote inline image after recombination is

image((A2))

Finally, gametogenesis gives the new haplotypes from the zygote

image((A3))

Replacing A1 and A2 into A3 gives

image((A4))

The nonrecombinant contribution is then

image

And the recombinant contribution is

image

Finally, we obtain the expected genotype frequency after the life cycle

image((A5))

Where the term inline image represents the fitness of pijunder frequency-dependent dynamics, (pijxiyj) is the linkage disequilibrium and where inline image is the average fitness of the haplotypes in the population.

Appendix B

In the presence of a new AB-linked genotype, inline image, we can derive the evolution of the frequency of the genotypes in the population after the life cycle (gamete fusion, recombination, and gametogenesis, as in Appendix A). The frequency of the new linked mating type at generation t depends only on its frequency at generation t– 1 and on its fitness (the proportion of compatible mate in the population), weighted by its survival (1 −α). Whereas, formation of AkBl recombinant mating types in the presence of a new linked mating type, AB-linked comes from three processes: (1) from the nonrecombination portion (1 – r) of a zygote inline image composed by two recombinant mating-types genotypes, AkBl and AoBp, (2) from the recombination portion, r, of a zygote inline image composed by two recombinant mating-types genotypes, AkBp and AoBl, and finally (3) from the portion of the surviving zygote inline image composed by one recombinant mating-types genotypes, AkBl, and an AB-linked genotype. Thus, when two random haploid genotypes meet; if the two gametes carry different mating-type alleles at the two loci, they fuse and form a zygote, two type of zygote can be formed.

image
image((B1))

The frequencies of the zygotes after recombination and accounting for fitness reduction are

image((B2))

We have then

image((B3))

After simplification, the system leads to equations 9–13.

Appendix C

Here, we compute the frequency at which the mating types are maintained after an invasion event.

Assuming that the population reaches an equilibrium, where all inline imageare at the same frequency, and AiBj is at frequency p*, we have

image((C1))

And at equilibrium, we have

image((C2))

which has two equilibrium values, one is trivial where the frequency of the AB-linked mating type is zero, inline image, and one when it is maintained when,inline image. In the latter case, equation C2 leads to

image((C3))

With equations C1, C3, 11, and 13, we obtain the frequency of the new linked mating-type genotype at equilibrium after invasion

image((C4))

And thus the other mating-type genotypes reach the following frequency

image((C5))

Appendix D

In the presence of a new mating type carrying a self-compatible allele in one of the MAT loci, and assuming a fitness reduction in zygote, α, we can rewrite the evolution of the frequency of the mating types in the population after the life cycle (gamete fusion, recombination, and gametogenesis, as in Appendix A). Two types of mating types are produced, one type is composed by two self-incompatible MAT alleles, inline image, and the other type is composed by one self-incompatible allele and the other the self-compatible allele inline image. inline imagecan be produced through four processes: (1) from the nonrecombination portion (1 – r) of a zygote inline image, (2) from the recombinant fraction, r, of the zygote composed by two SI MAT alleles, inline image, (3) from the nonrecombinant fraction that survived fusion between a mating-type genotype composed by one self-incompatible allele and the other mating-type genotype composed by two self-incompatible alleles inline image, and finally (4) from the recombinant fraction that survived fusion between two mating-type genotypes composed each by one self-incompatible allele inline image. inline image is produced through three processes: (1) from the nonrecombination portion (1 – r) of a zygote inline image, (2) from a recombination contribution of a zygote inline image, and (3) from a zygote inline image. Thus, when two random haploid genotypes meet, depending on their mating-type alleles at the two loci, they can fuse and form a zygote, as follow

image
image
image((D1))

The frequencies of the zygotes after recombination and accounting for fitness reduction are

image((D2))

with

image((D3))

Thus, we have

image((D4))

After simplification, the system becomes

image((D5))

With the corresponding fitness for each mating type

image((D6))

And the linkage disequilibrium for each possible genotype present in the population

image((D7))

Finally, the mean fitness of the population becomes

image((D8))

Ancillary