Introduction

Most chromosomes follow ‘fair’ Mendelian segregation, resulting in each of the homologues being present in (approximately) 50% of the gametes. However, some genetic elements are recovered in more than half of the functional gametes of heterozygous individuals showing so-called segregation distortion or meiotic drive. Segregation distortion occurs in a number of taxa, ranging from fungi to plants and animals (for reviews see, for example, Jaenike, 2001; Burt and Trivers, 2006) with marked similarity between taxa (Taylor and Ingvarsson, 2003).

Segregation distortion is advantageous at the gene level, as distorter alleles have a transmission advantage and their frequency in the population will increase. Many distorters in nature show almost complete distortion when unsuppressed (the distorter allele is present in more than 90% of functional gametes). However, considerable variation exists between populations and different distorters, and an effective distortion can range from just above 0.5 to almost 1 (for example, see Sturtevant and Dobzhansky, 1936; Hickey and Craig, 1966; Gileva, 1987; Carvalho de et al., 1989; Jaenike, 1996; Van Boven and Weissing, 1998; Jaenike, 1999; Montchamp-Moreau et al., 2001; Atlan et al., 2003).

The presence of a driving chromosome is usually not neutral with respect to individual fitness, both in heterozygous and homozygous condition (for example, see Wallace, 1948; Curtsinger and Feldman, 1980; Jaenike, 1996; Taylor and Ingvarsson, 2003; Atlan et al., 2004). In some well-studied cases, homozygosity for a distorter allele causes sterility in males, or even lethality in males and females (for example, the t-complex of the house mouse and the segregation distorter of Drosophila melanogaster; see Lyttle, 1991; Burt and Trivers, 2006). Therefore, there will often be strong selection for suppressors of segregation distortion (for example, see Hurst et al., 1996). Such suppressors have been indeed found in most of the species harbouring segregation distorters (for example, see Jaenike, 2001; Burt and Trivers, 2006).

When segregation distorters are located on sex chromosomes, they not only have an effect on individual fitness but also lead to biased sex ratios in the population (Jaenike, 2001). Biased sex ratios induce selection for increased production of the rarer sex. Unbiased sex ratios can either be restored through suppressors of drive or through a change in the mechanism of sex determination (Bull and Charnov, 1977; Cosmides and Tooby, 1981; Werren and Beukeboom, 1998; Burt and Trivers, 2006). Segregation distortion has been proposed as the driving force behind a change in the sex-determining mechanism of the wood lemming, Myopus schisticolor (Bengtsson, 1977), the mole, Talpa occidentalis (McVean and Hurst, 1996), the creeping vole, Microtus oregoni (Charlesworth and Dempsey, 2001), the sciarid fly, Sciara coprophila (Haig, 1993b), the housefly, Musca domestica (Clark, 1999) and scale insects, Neococcoidea (Haig, 1993a).

To our knowledge, there are only very few mathematical models investigating the effect of sex chromosome segregation distortion on the evolution of sex determination. Present models are tailored to a particular species, and aim to explain the observed sex-determining mechanism in that species (Bengtsson, 1977; Jayakar, 1987; McVean and Hurst, 1996; Charlesworth and Dempsey, 2001). Therefore, there is currently still little understanding of the conditions under which sex chromosome segregation distortion can lead to changes in sex-determining mechanisms. We aim to fill this gap by means of a more generic model in which we analyse the conditions for the spread of a new sex-determining gene in a system with a distorting (driving) sex chromosome.

The best documented and probably most common case is segregation distortion occurring during the production of male gametes, wherein it is manifested by the dysfunction of sperm (or pollen) lacking the driving element, or, more precisely, loss of gametes carrying the sensitive allele (for example, see Lyttle, 1993; Taylor and Ingvarsson, 2003). Sex chromosomal distortion is most common in systems with male heterogamety, and usually, the X chromosome drives against the Y chromosome. Presumably, Y drive is less common, because it easily leads to population extinction (Hamilton, 1967; Taylor and Ingvarsson, 2003). Therefore, we chose to focus on the case in which the initial sex-determining mechanism is an XY system and in which segregation distortion occurs only in males.

We consider three scenarios: a driving X chromosome (scenario 1); a driving Y chromosome (scenario 2); and a driving autosome with a male-determining factor (scenario 3). Segregation distortion associated with all such chromosomes has been found in natural populations of various species (Clark, 1999; Jaenike, 2001; Burt and Trivers, 2006). The presence of driving chromosomes leads to female-biased (scenario 1) or male-biased (scenarios 2 and 3) sex ratios, presumably promoting the spread of new masculinizing or feminizing factors, respectively. Throughout, we assume that a distorting allele has a detrimental effect on fitness in homozygous condition, either by reducing male fertility or by reducing viability in males and/or females.

By means of a dynamic model, we address the following questions. Under which circumstances does sex chromosomal segregation distortion lead to changes in the mechanism of sex determination? More specifically, how is the invasion prospect of a new sex-determining factor affected by the strength of distortion and the fitness effect of a distorter present in the population? When invasion is possible, will a new factor spread to fixation, leading to a switch to a different sex-determining mechanism? How is the frequency of the segregation distorter affected by the invasion of a new sex-determining factor?

The model

We model the evolutionary dynamics of the sex-determining system with a set of recurrence equations. We assume an infinite diploid population with random mating and non-overlapping generations. We analyse three scenarios in an initial XY system: (1) a driving X chromosome, (2) a driving Y chromosome and (3) a driving autosomal male-determining factor. First, we will present our general model assumptions and then introduce modifications specific to the three scenarios.

Genotypes and sex determination

As the number of ways in which sex could be determined is virtually unlimited, we decided to base our model on a relatively general mode of sex determination. We consider a sex-determination system consisting of three independent gene loci (on three different chromosomes). In the absence of segregation distortion, each locus has two basic alleles, but additional alleles can be present in specific models (see below). The first locus corresponds to the standard XY system of sex determination with two basic alleles: X and Y, in which Y corresponds to a dominant male-determining factor. As X drive leads to female-biased sex ratios, one might expect selection in favour of male-determining factors. Therefore, we also consider a second locus, which harbours a dominant male-determining M allele and a standard m allele. We assume that this locus is autosomal (not linked to the X or Y chromosome). Such autosomal male-determining factors have been found in a number of species (Martin et al., 1980; Traut and Willhoeft, 1990; Dübendorfer et al., 2002), and are a likely step in the evolution of sex-determining mechanisms (Bull, 1983). Y chromosomal drive leads to male-biased sex ratios and should favour feminizing genes. Therefore, we also consider a third locus, which has a female-determining F allele and a standard f allele. There are many possibilities of how female and male-determining factors interact during sexual development. We consider the simplest case, in which F is dominant over M and Y, meaning that the presence of F always leads to female development, even if both Y and M are present in homozygous state. We assume that the lack of an X or Y chromosome does not have negative effects on fitness (see Discussion). If F is absent, but Y and/or M are present, an individual becomes a male, otherwise (neither Y nor M is present) it becomes a female. All possible male and female genotypes are summarized in Table 1. This sex-determining system resembles that of the housefly (Dübendorfer et al., 2002; Kozielska et al., 2006), but we believe that the model is more generally applicable.

Table 1 All possible genotypes considered in the model (without distinguishing between driving and non-driving alleles)
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Drive

Segregation is random in females and in males that do not possess a driving chromosome or are homozygous for it. In males heterozygous for a driving chromosome and sensitive chromosomes (also see below), k denotes the frequency of the driving allele in gametes and, hence, indicates the strength of distortion.

Fitness

As described in the Introduction, the presence of distorter allele is often detrimental for fitness. Our invasion analysis (Appendix B) is based on quite general assumptions on the fitness effects of a driving allele in homozygous condition: homozygous males have fertility u (0⩽u⩽1), and the viability of homozygous males and females is given by νm and νf, respectively (0⩽νm, νf⩽1). For the numerical analysis, which allows us to study the full dynamics of the system, we considered three, more specific, fitness schemes: (a) no cost of drive; (b) males homozygous for the driving allele (that is XdXd, YdYd or MdMd, depending on the scenario) are sterile; and (c) homozygosity for the driving allele is lethal, both in males and females.

Evolutionary dynamics

We assume random mating and discrete non-overlapping generations. Under these conditions, the dynamics of the system can be easily expressed by a set of recursion equations for the frequency of each genotype in each generation, depending on the frequency of all genotypes in the previous generation (see Appendix A). Iteration of these equations allows studying the dynamics of the system.

In our numerical iterations, we started with the standard XY system (females: XX; mm; ff and males: XY; mm; ff), and at generation 0, introduced a distorter allele (step 1) at a frequency of 0.001. We let the system evolve and when the equilibrium was reached, a new sex-determining factor was introduced (step 2) at a frequency of 0.001. Again, frequencies of different genotypes were calculated at every generation until a new equilibrium appeared to be reached. All simulation results were confirmed by a mathematical invasion analysis (see Appendix B for details).

The three scenarios for sex chromosomal distortion were implemented as follows:

Scenario 1: driving X chromosome

In this version of the model, three alleles segregate at the XY locus: standard Y, standard X and driving X. The latter will be denoted Xd, and it is assumed to drive only against Y. Therefore, the frequency of Xd gametes produced by XdY males is equal to the drive strength k and the frequency of Y gametes equal 1−k. For all other male genotypes, segregation of alleles is Mendelian (k=1/2). We start with the standard XY system, and in step 1 (see above), introduce a driving Xd chromosome. This causes female-biased sex ratios, and in step 2, an M allele is introduced to regain equal sex ratios.

Scenario 2: driving Y chromosome

In this version, three alleles segregate at the XY locus: standard X, standard Y and driving Y. The later will be denoted Yd and it is assumed to drive only against X. Therefore, XYd males produce X gametes in frequency 1−k and Yd gametes in frequency k. For all other male genotypes, segregation of alleles is Mendelian (k=1/2). We start with the standard XY system, and in step 1, introduce a driving Yd chromosome. This causes male-biased sex ratios, and in step 2, an F allele is introduced.

Scenario 3: driving M

In this scenario, segregation occurs at an autosomal locus, in which M drives against m. The driving alleles will be denoted as Md. The frequency of Md in the gametes of Mdm males is equal to k, and the frequency of m gametes equals 1−k. For all other male genotypes, segregation of alleles is Mendelian (k=1/2). We start with the standard XY system, and in step 1, introduce a driving Md allele. This causes male-biased sex ratios, and in step 2, an F allele is introduced.

For each scenario, we run numerical iterations of the recursion equations (Appendix A) to investigate the effects of distortion strength k and various fitness schemes on the dynamics of the system. The results of the numerical analysis are fully in line with the (much more general) mathematical invasion analysis given in the Appendix B.

Results

Scenario 1: driving X chromosome

For any value of k>½, the driving X chromosome (Xd) invades the population, leading to a female-biased sex ratio. When M is introduced, it increases the production of males, and it is selected for, as long as selection against biased sex ratios is not overcome by selection against sterile genotypes. Once M invades, it causes the loss of Y from the population and restores the equal sex ratio. Therefore, in most cases, there is a switch from an XY system to an XX system, in which sex is determined by M. Males are again the heterogametic sex, therefore, a new system can be seen as male heterogamety for M (Table 2). Interestingly, even though the presence of the driving Xd chromosome is often responsible for the switch from XY to an autosomal system, at equilibrium, the Xd chromosome may be no longer present in the population (see below).

Table 2 Overview of changes in the sex-determining system induced by a driving X chromosome (Xd)
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No cost of drive

Owing to its transmission advantage and no fitness costs, Xd fixates in the population, removing the standard X chromosome independently of the value of k. This leads to a female-biased sex ratio with the proportion of females equal to the drive strength k. This sex ratio bias facilitates the invasion of M, which very quickly spreads replacing Y as a sex-determining factor, leading to male heterogamety for M (Figure 1a). Xd reaches fixation in both males and females. Although driving chromosomes are still present in the population, the sex ratio is unbiased, as the chromosomes sensitive to drive are absent.

Figure 1
figure 1

Dynamics of the sex-determining system in the presence of a driving Xd chromosome. This is a representative example with strength of drive k=0.8. Introduction of new alleles is indicated by arrows. The spread of Xd leads to biased sex ratios. When the system appeared to reach equilibrium, an autosomal masculinizing factor M was introduced, leading to a switch in sex-determining mechanism to male heterogamety for M and restoration of equal sex ratios. (a) No cost of drive. Standard X and Y are lost from the population. (b) Sterility of XdXd males. A driving Xd leads to the loss of standard X. In generation, 200 M and X are introduced simultaneously, Xd and Y disappear from the population. Introducing M on its own has no effect (see Results). (c) Lethality of XdXd genotypes. Initially polymorphism for Xd and X is maintained. Introduction of M leads to the removal of Xd and Y.

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Sterility of XdXd males

The Xd invades the population and replaces the standard X, as above, as males are always XdY and, therefore, do not have a fertility disadvantage. However, now the introduction of M leads to the production of sterile XdXd males and M cannot invade the population. Only if the standard X is reintroduced into the population together with M can they both invade, as some males produced by M possess the standard X and are fertile. These males have an advantage in strongly female-biased populations, and both M and standard X increase in frequency. An increase in the frequency of M leads to the production of sterile XdXd males and selection against Xd. Xd is eventually lost from the population and replaced by standard X (Figure 1b). Y is also removed from the population, and a male heterogametic system for M establishes. It should be noted that, although the driving X chromosome induced changes in the sex-determining mechanism, it is no longer present in the population.

This happens only for strong drive k>¾, as this will lead to the population sex ratio being very female biased and selection against the biased sex ratio is strong enough to overcome selection against infertile males. Therefore, even though the presence of M initially leads to the production of many sterile XdXd males, if X is present in the population, this cost is compensated for by the increased production of males thanks to M. For weaker drive, selection against biased sex ratio is not strong enough to overcome selection against sterile genotypes and M cannot invade (Table 2).

Lethality of XdXd genotypes

The Xd invades the population, but it cannot fixate, as homozygous genotypes are lethal. Therefore, the sex ratio bias induced by Xd is less pronounced than that for the scenarios presented above (Figure 1c). The frequency of Xd is equal to (that is, equals ½ for k=½, and ¼ for k=1). A female-biased sex ratio facilitates the invasion of the masculinizing factor M, which in turn leads to a decrease in the frequency of Y and increased homozygosity for Xd; also in males. As the standard X is present in the population at high frequency, the decrease in fitness of M-bearing males through the production of rare lethal XdXd offspring is outweighed by the advantage from an increased production of males. Eventually Xd is removed from the population because of its detrimental effect in the homozygous state, which is not counterbalanced by a transmission advantage, as XdY males are rare. A male heterogametic system for M gets established (Table 2).

Scenario 2: driving Y chromosome

Independent of the strength of drive k and the fitness of homozygous YdYd individuals, Yd always spreads in the population replacing standard Y, as females never harbour Yd and, therefore, males can never be homozygous for Yd. As all males are XYd, the sex ratio is biased with a proportion of males equal to the drive strength, k. Whether F invades depends on the fitness costs of homozygosity for Yd (Table 3). However, similar to scenario 1, a change in the sex-determining mechanism may lead to the complete removal of the driving chromosome from the population.

Table 3 Overview of changes in the sex-determining system induced by a driving Y chromosome (Yd)
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No cost of drive

The feminizing F allele always spreads in the population, leading to fixation of Yd and a switch to a female heterogametic sex-determining system. Although Yd is present in the population, the sex ratio equals 1:1, as the X chromosome sensitive to drive is lost.

Sterility of YdYd males

Selection against biased sex ratios facilitates the spread of F. It is present only in females, hence it does not have direct negative fitness effects. However, F females produce some sterile YdYd males, which prevent F from fixation. Polymorphism for X and Yd, and F and f is maintained, but the sex ratio in the population equals 0.5 (Figure 2). The frequencies of F and Yd increase with the strength of drive, k. At this state, reintroduction of the standard Y in the population will lead to its spread, as it does not cause a fitness cost in males. The segregation advantage of Yd is low, as owing to the low frequency of X, the frequency of YdX males is low. Therefore, the segregation advantage is outweighed by selection against sterile YdYd males. The frequency of Yd decreases, and standard Y replaces Yd and spreads to fixation. X is removed, F is present in all females and a female heterogametic system establishes (Figure 2).

Figure 2
figure 2

Dynamics of the sex-determining system with a driving Yd chromosome for the case in which YdYd males are sterile. Introduction of new alleles is indicated by arrows. Yd (with drive strength k=0.8) replaces standard Y and leads to a strongly male-biased sex ratio. Introduction of F leads to a polymorphic system. If standard Y is reintroduced into this system, it invades the population and fixates, replacing the driving Yd and leading to a female heterogametic system.

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Lethality of YdYd genotypes

If drive strength is low, F does not invade the population, as selection against a (slightly) biased sex ratio is weaker than selection against the lethal genotypes produced when F is present. Therefore, an XY system with the driving Yd chromosome (that is, no standard Y) and a biased sex ratio is stable for k<3/5 (see Appendix B). For higher values of k, the F factor invades the population. Initially, a polymorphism on both the F and XY loci is maintained, with a population sex ratio that is only slightly male biased, similar to the scheme with male sterility. For k between 3/5 and 0.655 (see Appendix B), this polymorphic sex-determining system is stable and Y cannot reinvade the population. However, for k>0.665, the reintroduction of the standard Y also leads to an increase of F in frequency and eventually to removal of Yd and X. F reaches a frequency of ½ in females and female heterogamety with equal sex ratios is established.

When F is present in the population, but before fixation of standard Y, in some circumstances, autosomal M can invade. However, it never reaches fixation, and standard Y will fixate in the population once it is reintroduced. Therefore, female heterogamety with the frequency of standard Y equal to 1 is an equilibrium state of this system, although polymorphism for M and m can be present.

Scenario 3. driving autosomal M

Similar to the case with driving Yd, driving Md invades the population regardless of drive strength and the fitness of individuals homozygous for Md, as it is only present in males in heterozygous state. Y is removed from the population and all individuals are XX. The population sex ratio is male biased and equal to drive strength, k. Whether F invades depends on the fitness costs of homozygosity for Md (Table 4).

Table 4 Overview of changes in the sex determining system induced by a driving autosomal factor Md
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No cost of drive

For any drive strength k, selection against male-biased sex ratios facilitates the spread of the feminizing F allele. The invasion of F leads to fixation of Md, and a switch to a female heterogametic system with 1:1 sex ratio.

Sterility of MdMd males

F always invades irrespective of drive strength. However, fixation of F and Md is impossible, as this will lead to all males being sterile. Therefore, there is no full switch to female heterogamety, but polymorphism for both Md and m, and F and f is maintained in the population. The population sex ratio equals 0.5. With increasing k, the equilibrium frequency of Md and F increases. If at any point Y is reintroduced into this system, it will reinvade the population, as it assures maleness without fitness costs. Eventually, Y spreads to fixation and all females become heterozygous for F. The frequency of Md decreases, but it is not removed from the population. Therefore, at the stable equilibrium, in effect, a female heterogametic system is present, but with a polymorphism at the M locus (Figure 3).

Figure 3
figure 3

Dynamics of the sex-determining system with a driving autosomal Md factor for the case of sterility in MdMd males. Introduction of new alleles is indicated by arrows. Spread of Md (with drive strength k=0.8) leads to the removal of Y, the switch to a male heterogametic system for Md, and male-biased sex ratios. Introduction of F leads to a polymorphic system. When Y is reintroduced, it leads to a switch to a female heterogametic system, but polymorphism on the M locus is maintained.

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Lethality of MdMd genotypes

If the drive strength is weak (k<3/5), then the population sex ratio is only slightly male-biased and F does not invade, because the advantage of producing females by F is outweighed by the disadvantage of producing lethal, homozygous Md individuals. Therefore, for weak drive, a system with male heterogamety for Md and a biased sex ratio is stable. For higher k, F invades, leading to a system polymorphic for F and f, and Md and m, with the equilibrium frequency of F and Md increasing with k (as it does in the scheme with sterile males). The population sex ratio is slightly male biased. For k between 3/5 and 0.8 (see Appendix B), this polymorphic sex-determining system is stable and Y cannot reinvade the population. For stronger drive (k⩾0.8), if Y is reintroduced it will spread to fixation, leading to female heterogamety. Although the frequency of Md decreases, it is not removed from the population and polymorphism at the M locus is maintained (similar to the case with sterility of MdMd males).

Discussion

Sex determination is a fundamental developmental process and one might expect sex-determining mechanisms to be highly stable. Yet, sex-determining mechanisms vary greatly between different taxonomic groups and even between closely related species (Bull, 1983; Kraak and Pen, 2002; Janzen and Phillips, 2006). Although steady empirical progress is being made in unravelling the genetics of sex-determining mechanisms, there are still many unanswered questions about how new sex-determining systems evolve. It has been proposed that ‘genetic conflict is the most likely general explanation for the diversity of sex-determining mechanisms’ (Werren and Beukeboom, 1998). Recent theoretical work has largely focused on maternal–offspring conflict over sex ratio (reviewed in Uller et al., 2007).

Intragenomic conflict (between different genes within an individual), in general, and segregation distortion, in particular, is another selective force that can lead to changes in sex-determining systems (see Introduction). However, there is little theoretical understanding of the process. Most models considered thus far were tailored to one particular species and aimed at explaining the sex-determining mechanisms found in that species (Bengtsson, 1977; McVean and Hurst, 1996). The more general models make some unrealistic assumptions. For example, Jayakar (1987) assumed that the driving chromosome shows the same segregation distortion in males and females. Charlesworth and Dempsey (2001) assumed that the distorter has a complete transmission advantage (present in 100% of gametes) in females and a complete transmission disadvantage in males.

In this paper, we developed some general models to investigate the change in the mechanisms of sex determination in systems in which a driving sex chromosome is present. Although trying to be as general as possible, we had to make some assumptions concerning sex determination and segregation distortion. Although we believe that our assumptions reflect the most common patterns observed in natural systems (see the Introduction), we briefly discuss how our results might be affected if these assumptions were not met.

In our model, we consider segregation distortion only in males, the heterogametic sex. Male heterogamety is the prevailing sex-determining mechanism in species with genetic sex determination, and most of the described cases of segregation distortion affect sperm or pollen production (Jaenike, 2001; Taylor and Ingvarsson, 2003). However, meiotic drive has also been found during female meiosis (for example, in maize; Buckler et al., 1999), and sex ratio segregation distortion in XY (ZW) females has been described in wood lemmings (Bengtsson, 1977) and postulated in the butterfly Danaus chrysippus (Smith et al., 1998). For reasons of symmetry, the results of our model immediately carry over to systems with female heterogamety and meiotic drive occurring in females. For example, if females are ZW, and Z drives against W (symmetric to a driving X) and homozygosity for driving Z causes lethality, a driving Z should invade but not reach fixation, and introduction of an autosomal feminizing gene would lead to a switch to female heterogamety of an autosomal factor and removal of the driving Z from the population. This is assuming that the autosomal M in our model becomes a feminizing factor and F becomes a masculinizing factor. Our scenario of sterility of males that are homozygous for the driving allele would, by symmetry, apply to cases in which females that are homozygous for the driving allele are sterile. We do not know of such cases in nature, but this scenario seems plausible, taking into account that a driving chromosome disturbs meiosis.

We have performed a rather general invasion analysis of new sex-determining genes for a broad range of fitness schemes. The full dynamics of the system could only be studied numerically, and hence could only be performed for a few specific cases. To this end, we focused on lethality and/or male sterility in individuals homozygous for the driving allele. However, our analytical results (Appendix B) suggest that the numerical analysis can be extrapolated to other fitness schemes as well, as the invasion prospects of a new sex-determining factor do not often depend on the precise fitness values. Examples of reduced fitness of individuals possessing driving alleles, not only in homozygous but also in heterozygous state, or even increased fitness of heterozygous females, are known from nature (for example, Wallace, 1948; Curtsinger and Feldman, 1980; Jaenike, 1996; Atlan et al., 2004; Wilkinson et al., 2006). In general, fitness values of different genotypes will influence the balance between sex ratio selection (favouring spread of a new sex-determining factor) and viability selection acting on new sex-determining factors. Strongly reduced viability may prevent the spread of a new factor. However, according to our results, once new factors invade, the equilibrium sex-determining mechanism is largely independent of the specific fitness effects. This may hold at least as long as segregation distorters have a negative effect on fitness. When distorter alleles show overdominance in females (Wallace, 1948; Curtsinger and Feldman, 1980; Wilkinson et al., 2006), the outcome of evolution may be different, as both fixation and removal of the distorter allele may not be favoured.

In addition, one could imagine that a new sex-determining gene has fitness costs on its own (for example, because of some genetic incompatibilities), which would hamper its spread (Werren and Beukeboom, 1998). Here again, results would probably depend on the balance between selection favouring spread of a new factor (sex ratio bias caused by drive) and selection against it (reduced fitness). A few simulations that we did show was that if a new sex-determining factor has only a slightly reduced fitness (a reduction by 5%), the resulting system is the same as that with no fitness costs, at least for the case with a driving Y chromosome. However, a complete switch may not occur if the M factor causes a fitness reduction. Interestingly, in case of lower fitness of F, we sometimes observed cycling behaviour. An increase in the frequency of F leads to a decrease in the frequency of Yd, which reduces sex ratio bias and selection favouring F, which in turn decreases F frequency, which allows an increase in the frequency of Yd, which re-creates a biased sex ratio and increase in the frequency of F and the cycle repeats (results now shown).

In our model, we also assumed that the absence of a Y or X chromosome does not have negative effects on fitness. This is probably not realistic in species with old, strongly differentiated sex chromosomes, in which Y usually lacks some of the genes that are present on the X chromosome, or when Y possesses genes crucial for gametogenesis. In such cases, some of our results will not hold, as a full switch to another sex-determining system requires removal of one of the original sex chromosomes (either X or Y). However, absence of one of the sex chromosomes does not necessarily lead to (drastic) reduction in fitness (Juchault and Rigaud, 1995). Undifferentiated sex chromosomes have been found in a number of animal and plant species (Charlesworth et al., 2005; Fraser and Heitman, 2005), and even if the chromosomes are morphologically differentiated, they may not possess any genes necessary for viability or fertility (Hiroyoshi, 1977). More theoretical and empirical work is needed to understand the effect of different fitness values associated with different genotypes.

The switch to a new sex-determining system is driven by selection against biased sex ratios. When there are no fitness costs associated with homozygosity for the distorter allele, even weak sex ratio selection leads to changes in sex-determining mechanism. However, often sex ratio selection has to be sufficiently strong to overcome selection against less-fit genotypes (Appendix B). Therefore, in some cases, only strong segregation distorters will lead to changes in sex-determining mechanisms. This suggests that when segregation distortion is weak, the evolution of suppressors may be the only way to decrease sex ratio bias. Interestingly, Lyttle (1981) in his model on the evolution of new sex-determining mechanism through aneuploidy, also concluded that strong drive favours change in sex-determining mechanisms and weak drive favours the accumulation of suppressors. We suspect that this is a general phenomenon. Moreover, once suppressors start accumulating in the population decreasing the effective drive, this may prevent changes in the sex-determining system, as long as suppressors are less costly than a new sex-determining mechanism.

According to our model, the invasion prospects of a new sex-determining factor may depend on whether the distorter has any effects on the fertility of males or on the viability of both sexes. Sometimes fitness of only one sex matters, that is, either viability of females or the product of viability and fertility in males (which is their lifetime reproductive success). Once the sex-determining factor does invade, however, the final state of the system (full switch to different sex-determining system) is usually independent of whether male or female fitness is affected (at least for the numerical scenarios considered by us). Therefore, our assumptions on the fertility or viability effects of segregation distorters do not seem to substantially affect our predictions. The situation may be different if the population is structured into small local mating groups or when different mating patterns occur (Van Boven and Weissing, 1999; Pen, 2006).

In some of our simulations, a complete switch to a new sex-determining system could only be achieved if the standard X or Y chromosome was reintroduced in the population. This is more plausible than it may appear at first sight. In our model, the standard X or Y chromosome was completely driven out of the population. However, in natural systems, a population often consists of many subpopulations with limited gene flow between them. Therefore, one can easily imagine that a driving element will initially appear only in one subpopulation and spread within it, and subsequently to other populations. However, if a new sex-determining factor appears before the whole population is fixed for a driving allele, migration of standard chromosomes from other subpopulations will lead to spread of new sex-determining factor even in subpopulations in which the driving element is fixed. Alternatively, even if the driving allele is fixated in the whole population, any mutation restoring vitality or fertility of homozygotes will be favoured and spread in the population, even if it does not have a segregation advantage.

Perhaps the most interesting result of our simulations is that the segregation distorter, whose presence initiated the change in the sex-determination system, is often subsequently lost from the population. Perhaps surprisingly, such an effect has never been found in previous models (Bengtsson, 1977; Jayakar, 1987; Haig, 1993a, 1993b; McVean and Hurst, 1996; Charlesworth and Dempsey, 2001). In these models, a change in the sex-determining system usually neutralizes drive, for example, by removing sensitive alleles, but never leads to the complete loss of the driving allele. However, our results show that the presence of a segregation distorter can only be a transient state, the traces of which are no longer visible once a new sex-determination system has become established.

Loss of the driver would make it difficult to detect the role of segregation distortion in the evolution of extant sex-determining mechanisms, especially given that changes to a new sex-determining mechanism are often very rapid (Figures 1, 2 and 3). Indication of the role of segregation distortion in the evolution of sex-determining mechanisms could be found indirectly, for example, by looking at closely related species of a species in which sex chromosome meiotic drive has been found, to check whether change in sex determination occurred and whether they are consistent with our results (for example, if X drives, invasion of a new male-determining factor is expected). Alternatively, in a species or a closely related group of species in which sex-determining genes are known to be located on different chromosomes (for example, medaka fish, Takehana et al., 2007; the fly Megasela scalaris, Traut and Willhoeft, 1990), one could investigate whether some of the populations (species) possess driving sex chromosomes. If in such populations, the sex-determining mechanism is ancestral to other populations, this could suggest that segregation distortion was responsible for the change in sex-determining mechanisms. However, this is only circumstantial evidence.

A more direct test of the predictions of our model would probably demand laboratory experiments. They could be set up using (closely related) species in which multiple sex-determining factors exist together with segregation distorters. The housefly, Musca domestica, seems to be an especially good candidate. In this species, a variety of sex-determining mechanisms coexist in natural populations, such as a standard XY system, as well as autosomal male and female-determining factors, with effects on sex determination, as described in our model (Dübendorfer et al., 2002). In addition, in some North American populations, autosomal M factors show segregation distortion of 0.75–0.9 (Clark, 1999; although driving M's are probably absent in European populations; Kozielska, 2008). Unfortunately, changes in the frequency of sex-determining factors in these North American populations have not been studied. Even though segregation distortion is probably not the main driving force for the evolution of sex determination in natural populations of the housefly (Feldmeyer et al., 2008; Kozielska et al., 2008), experiments could be set up to test predictions of our model. In addition, one could study frequencies of different factors in natural populations with segregation distortion over long periods of time. In addition, in other species with multiple sex-determining factors, segregation distortion could be induced with molecular techniques (Windbichler et al., 2008) and subsequent dynamics of sex-determining mechanisms could be followed.

In summary, our simulations show that in many cases segregation distortion creates selective pressure allowing novel sex-determining alleles to spread and induce shifts to a new sex-determining system. Interestingly, the segregation distorter, the presence of which initiated this shift, can be subsequently lost from the population. Therefore, the role of segregation distortion in the evolution of sex-determining mechanisms may be underestimated.