Abstract
The sexratio trait is the production of femalebiased progenies due to Xlinked meiotic drive in males of several Drosophila species. The driving X chromosome (called SR) is not fixed due to at least two stabilizing factors: natural selection (favoring ST, the nondriving standard X) and drive suppression by either Ylinked or autosomal genes. The evolution of autosomal suppression is explained by Fisher's principle, a mechanism of natural selection that leads to equal proportion of males and females in a sexually reproducing population. In fact, sexratio expression is partially suppressed by autosomal genes in at least three Drosophila species. The population genetics of this system is not completely understood. In this article we develop a mathematical model for the evolution of autosomal suppressors of SR (sup alleles) and show that: (i) an autosomal suppressor cannot invade when SR is very deleterious in males (c < ⅓, where c is the fitness of SR/Y males); (ii) “SR/ST, sup/+” polymorphisms occur when SR is partially deleterious (∼0.3 < c < 1); while (iii) SR neutrality (c = 1) results in sup fixation and thus in total abolishment of drive. So, surprisingly, as long as there is any selection against SR/Y males, neutral autosomal suppressors will not be fixed. In that case, when a polymorphic equilibrium exists, the average female proportion in SR/Y males' progeny is given approximately by
MENDEL'S first law states that heterozygotes produce equal proportions of the two gamete types. This equality results from the meiotic segregation of gene pairs during gamete formation. Yet several genetic elements have been found to violate Mendelian transmission by actively biasing segregation in their favor. The beststudied example of segregation distortion was first recorded by Gershenson (1928) and later named meiotic drive by Sandler and Novitski (1957).
The sexratio trait known in 12 Drosophila species is a case of meiotic drive in the sex chromosomes. Males carrying certain X chromosomes, called SR, produce femalebiased progenies due to the degeneration of Ybearing sperm. The effect of drive in sexual proportion has important evolutionary consequences. The driving X(SR) has a transmission advantage over nondriving X (ST, for standard) so one can expect SR fixation followed by population extinction due to the lack of males (Gershenson 1928; Hamilton 1967; reviewed in Carvalho and Vaz 1999; Jaenike 2001). However, SR frequency in natural populations is usually low and stable (Dobzhansky 1958). In Drosophila mediopunctata, for example, SR frequency remained between 13 and 20% for 10 years (A. B. Carvalho, M.D. Vibranovski and S. C. Vaz, unpublished data). At least two factors seem to be responsible for the stabilization of SR/ST polymorphisms in natural populations: natural selection and drive suppression by modifier genes.
Fitness measurements have been made mainly in D. pseudoobscura. The main findings from these experimental studies are that SR/Y males have lower fertility and/or viability than ST/Y males and that SR/SR female homozygosis is highly deleterious (Wallace 1948; Curtsinger and Feldman 1980; Beckenbach 1996). There are also indications of SR/ST female overdominance (Gebhardt and Anderson 1993). Edwards (1961) and Curtsinger and Feldman (1980) carried out mathematical studies with sexratio models showing that the stabilization of X polymorphism under meiotic drive is possible under a wide range of fitness values. Thus, experimental and theoretical investigations support the idea that SR drive is counterbalanced by SR deleterious effects on individual fitness, resulting in SR/ST polymorphism.
Another stabilizing mechanism may be provided by autosomal or Ylinked drive suppressors. Suppressors are genes that restore the Mendelian transmission by neutralizing the effect of genes responsible for meiotic drive (Stalker 1961; Hamilton 1967; Thomson and Feldman 1975). The spread of Ylinked suppressors of sexratio in SRbearing populations can be explained by meiotic drive theory: any Ylinked gene that increases the transmission rate of the Y chromosome (as does a sexratio suppressor) is directly favored. Therefore, Ylinked suppressors are expected to run to fixation unless they are deleterious (Clark 1987; Carvalhoet al. 1997). Autosomal suppressors of sexratio are expected to evolve in response to SR because of a notably simple mechanism known as Fisher's (1930) principle (reviewed in Bull and Charnov 1988). Fisher's argument can be put as follows. In any sexually reproducing population, half of the genes come from each sex, regardless of the population sexual proportion. If the genetic system generates excess of one sex (as does the sexratio trait), the rare sex will be effectively more fertile as a result of a greater per capita contribution to the next generation. So, the rare sex has a selective advantage. If sexual proportion is a hereditary trait, then alleles directing the progeny sexual proportion to the rare sex (the males, in the case of sexratio) are expected to invade the population. These alleles should spread until the equilibrium of equal number of males and females is reached. This mechanism of natural selection is the most accepted explanation for the commonness of the 1:1 sexual proportion in nature (Bull and Charnov 1988). When parental expenditure is different between sexes, Fisher (1930) suggested that the sex ratio evolves to a value such that expenditure is equalized between male and female offspring. A clear theoretical demonstration of Fisher's principle under this circumstance was provided by Uyenoyama and Bengtsson (1979). A clear experimental demonstration of Fisher's principle was carried out by Carvalho et al. (1998) in a study with D. mediopunctata. They founded populations fixed for SR and thus with female excess. The proportion of males rose from 16 to 32% in 49 generations due to the accumulation of sexratio autosomal suppressors. This work demonstrated that sexual proportion actually responds to natural selection as postulated by Fisher (see also Conover and Van Voorhees 1990; Basolo 1994).
As expected by theory, autosomal suppressors have been found in some SRbearing Drosophila populations. In D. mediopunctata there are at least four suppressor genes in different chromosomes (Carvalho and Klaczko 1993). Female proportion averages 95.1% in a suppressorfree strain and 51.7% in a strain full of suppressors, while in a hybrid strain the average is 72.3% (n = 6, 5, and 7 SR/Y males, respectively; Carvalho and Klaczko 1993, Table 1). Hence, there seems to be no dominance in expression, although the experimental design would not detect fully recessive suppressors. In D. simulans suppression seems to be partially recessive in the two main chromosomes (Cazemajoret al. 1997). Autosomal suppression also seems to be present in D. quinaria (Jaenike 1999) and D. paramelanica (Stalker 1961).
D. pseudoobscura is an interesting exception. No Ylinked or autosomal sexratio suppressor was ever found in this species despite directed search (Policansky and Dempsey 1978; Beckenbachet al. 1982). Wu (1983) investigated this fact with a mathematical model for the evolution of autosomal suppressors. He showed that a neutral suppressor (i.e., that suppresses meiotic drive but has no fitness effect) is not expected to invade a SRbearing population if the fitness of SR/Y males is < ∼0.3 in relation to ST/Y males (in that case the stabilization of a “SR/ST” polymorphism requires female overdominance). So, according to this model a very low viability and/or fertility of SR/Y males can explain the absence of suppressors in D. pseudoobscura. It remains to be shown what happens when fitness configurations allow the initial spread of these suppressors. Will they remain polymorphic as suggested by Varandas et al. (1997, Figure 5) or will they run to fixation?
In this article we develop and study a theoretical model for the evolution of sexratio autosomal suppressors. Numerical simulations show three possible outcomes for a neutral suppressor in a population with SR/ST polymorphism: (i) noninvasion, (ii) polymorphism, and (iii) fixation. Through mathematical analysis we define the stability conditions for the two trivial equilibria (noninvasion and fixation) whereas the polymorphic equilibrium was studied mainly with simulations. Two results can be outlined. First, meiotic drive in a polymorphic equilibrium (t̂, defined as the average female proportion in SR/Y males progeny) is given by
THE MODEL
The model we describe below represents a typical sexratio system with natural selection on males and females and meiotic drive restricted to X sperm excess in SR/Y males. It follows the usual assumptions of population genetics modeling: random mating, large population size, nonoverlapping generations, and constant selection coefficients. Fitness is given by the eggtoadult viability component (sexratio models including fecundity selection produce the same general results as viability models; Curtsinger and Feldman 1980). In accordance with Edwards' (1961) notation, a, b, and c refer to the fitness of ST/SR, SR/SR, and SR/Y genotypes, respectively, relative to the fitness of ST/Y and ST/ST, which are set to 1. Sexratio expression in SR/Y males depends on an autosomal locus that affects the sexual proportion only: sup denotes the suppressor allele and “+” is the wildtype nonsuppressor allele. We assumed absence of dominance in suppression, which is somewhat simpler to study and seems to be the case in D. mediopunctata (Carvalho and Klaczko 1993). Males with the +/+ genotype produce 100% of Xbearing sperm, +/sup males produce 75%, and totally suppressed sup/sup males produce 50% (see Table 1; numerical simulations assuming other dominance relations produced essentially the same results). The sup allele is not expressed in females or ST/Y males. Autosomal suppression in D. mediopunctata and D. simulans is known to be polygenic; however, a monogenic model simplifies the problem considerably. Besides, monogenic and polygenic models on the evolution of sexual proportion (Nur 1974; Bulmer and Bull 1982) predict the same evolutionary rate and the same sexual proportion in the equilibrium (Carvalhoet al. 1998, pp. 729–730).
Let the frequency of SR chromosomes be given by p while the frequency of ST chromosomes is 1 – p. The frequency of sup is r and that of the nonsuppressor allele (+) is 1 – r. The p and r variables are listed in Table 2.
The complete system consists of eight recurrence equations (for p_{e}, p_{s}, p_{m,} r_{e1}, r_{e2}, r_{s1}, r_{s2}, and r_{sY}) deduced in appendix a.
We used these equations in the numerical simulations and stability analysis described in the next sections. Our aim is to answer if it is possible to maintain SR/ST, sup/+ polymorphism and, in this case, verify the fitness conditions (a, b, and c parameter values, see Table 1) in which it happens.
NUMERICAL SIMULATIONS
Numerical simulations covering a biologically meaningful set of the a × c parametric space were carried out. Each value of c between 0 and 1.5 with a 0.01 interval was tested with each value of a between 0 and 3 with the same interval. Initial allele frequencies set to either 0.01 or 0.99 converged to the same equilibrium point (the system was considered to be in equilibrium when all allele frequencies varied <10^{–5} in one generation). The results of the a × c scanning for two different values of b are shown in Figure 1. When there is SR/ ST polymorphism, there are three possible fates for the autosomal suppressor depending on SR fitness values: sup does not invade (r = 0; SR/ST, + equilibrium), sup invades but is not fixed (r between 0 and 1; SR/ST, sup/+ polymorphism), and sup invades and is fixed (r = 1; SR/ST, sup equilibrium). Some important observations can be made: (i) sup does not invade when c is very low (< ∼0.3), as shown by Wu (1983); (ii) when sup invades it is not fixed when there is any selection against SR/Y males (c < 1); and (iii) SR/ST, sup/+ polymorphisms occur when a > 1 only, i.e., when there is overdominance. These results suggest a role for selection against SR/Y males and female overdominance in species that are polymorphic for X and autosomal alleles (e.g., D. mediopunctata and D. simulans).
Figure 2 shows the relation between t̂, the equilibrium value of the drive parameter t, and each of the selection coefficients: a, b, and c, the three variables of our model. Note that t̂ is a linear function of sup frequency (see Equation A11).
It is clear from Figure 2 that c is the parameter with the greatest effect on the value of t̂. Biologically, it means that suppressor frequency in the population and thus drive intensity in SR/Y males are basically determined by the degree of selection against these males. As selection becomes less intense (high c values) sup frequency rises up to the point where no selection (c = 1) results in a totally suppressed drive (fixed sup and t̂ = ½; see Figures 1 and 2c). It should be noted that the male proportion in the equilibrium (Mz, see Equation A5), is always close to 0.5 in the cases of SR/ST, sup/+ polymorphism (it varied from ∼0.46 to 0.50 in the numerical simulations). The explanation for this small variation of Mz, in spite of t̂ varying from 0.5 to 1, is that when SR frequency is high, sup frequency is also high (not shown).
EQUILIBRIUM FREQUENCIES
Numerical simulations indicate that it is possible to maintain a polymorphism for a neutral autosomal suppressor in a SRbearing population. The suppressor equilibrium frequency (and the intensity of drive) is a function of selection coefficients, where c has the strongest effect. But what function is it? A formula for t̂ would be very useful because drive is easy to measure in natural populations. Take D. mediopunctata as an example: it would be interesting to predict fitness configurations that result in t̂ = 0.78, the average female proportion in the progenies of SR/Y males from a natural population (Varandaset al. 1997).
SR equilibrium frequency: The equilibrium frequency of SR, as a function of constant selection coefficients and meiotic drive in the absence of suppression, was first obtained by Edwards (1961; see also stability of equilibria).
The equilibrium frequencies of SR can be obtained in our model by equating
Equations 1 agree with Edwards' results, where t̂ corresponds to a fixeddrive parameter. This parameter is not constant in our model but dependent on suppressor frequency (see Equation A11).
Suppressor equilibrium frequency: Numerical simulations indicate that the value of b (when between 0 and 1) has practically no influence on the equilibrium value of t (t̂_{;} see Figure 2b). This result suggested that we could simplify the algebraic solution assuming b = 0. A direct approach to obtain the equilibrium frequencies would be to solve the fiveequation system (setting r′= r = r̂ for all five recurrence equations—A9, A10, A12, A13, and A14—and substituting p with p̂ for the four p variables given in Equations 1, where t̂ = 1 – ¼(r̂_{e1} + r̂_{sY}); see Table 2 for the variables listing). A straightforward solution was not possible so we solved the problem by reducing the system stepbystep with the help of Maple computer software (not shown). The solutions we found for t̂ are ½, 1, and
Given r̂_{m1} = 2 – 2t̂ (from Equation A11), the suppressor equilibrium frequency in SR/Y males is
Figure 3 compares the algebraic value of t̂ (t̂_{alg}, given by the formula in Equation 2) to the true value of t̂ (suggested by t̂_{sim}, obtained from 1000 computer simulations with b varying from 0 to 1). The estimate given by Equation 2, which used the simplification b = 0, slightly overestimates the true value of t̂ but provides an excellent approximation since t̂_{sim} and t̂_{alg} are highly correlated (r = 0.998; p ≪ 10^{–3}). The accuracy of our algebraic solution was confirmed by simulations with b = 0 where the values of t̂_{sim} had a perfect match with those predicted by t̂_{alg} (not shown). Thus, we can safely affirm that the expression
STABILITY OF EQUILIBRIA
In this section we apply a stability analysis to outline the conditions for the two trivial equilibria (+ and sup). Next, we deduce the conditions for the polymorphic equilibrium (sup/+) with the help of numerical simulations. In other words, we find the mathematical functions for the boundaries shown in Figure 1. The SR/ST polymorphism: Edwards' (1961) theoretical studies showed that the ratio between SR and ST equilibrium frequencies in adult females in the case of polymorphism is equal to [a(2ct + 1) – 2]/[a(2ct + 1) – 4bct] and that stable SR/ST polymorphisms occur when both numerator and denominator of the expression are greater than zero:
Suppressor noninvasion: The equilibrium corresponding to a population bearing X polymorphism with no sexratio suppression (i.e., full drive expression) is referred to as SR/ST, +. A natural example could be D. pseudoobscura.
Numerical simulations suggested that there is no difference between the boundaries of the SR/ST, + equilibrium in the cases where b = 1 and b = ½ (see Figure 1). Besides, since c has a very low value in this equilibrium (c < ∼0.3) and since t̂ = 1 and b is between 0 and 1, we know that bct̂ < ½. Consequently, the stability condition of SR/ST polymorphisms is given by (3), which does not depend on b. Then, to simplify the problem, we could assume b = 0 in the analysis detailed in appendix b. In short, the analysis consisted in applying the PerronFrobenius theorem (Ortega 1987) for nonnegative matrices, which allows one to set the eigenvalue equal to 1 (λ= 1) to find the stability boundaries. By setting λ= 1 in the characteristic equation of the SR/ST, + Jacobian matrix we find the following solutions: c = 0, a = 2/(2c + 1), and a = (c + 1)/[2c(2c + 1)].
Figure 1 indicates the boundaries of SR/ST, + equilibria according to numerical simulations. In fact, the curves limiting this equilibrium are the two nontrivial solutions obtained with λ= 1 (see Figure 4). Thus, the SR/ST, + equilibrium is predicted when
Regarding the condition in (5), note that a > 2/(2c + 1) is the SR/ST polymorphism stability condition when bct < ½ [see (3) for t = 1]. In fact, bct < ½ always holds for SR/ST, + equilibria since here t = 1, b < 1, and c is very low (< ∼0.3). In short, SR/ST, + equilibria depend on two basic conditions: stability of the SR/ST polymorphism [in (5)] and stability of the + allele fixation [in (6)].
Wu's (1983) studies showed that the noninvasion of a suppressor allele requires strong selection against SR/Y males and SR/ST female overdominance (c < ∼0.3 and a > 1). Our findings agree with and extend those previous results. The above analysis allows the formal deduction of Wu's conditions, as follows. In accordance with (5) and (6) (and knowing that a and c are positive) we have 2/(2c + 1) < (c + 1)/[2c(2c + 1)] ⇒ c < ⅓. Therefore, the upper limit of c is c_{max} = ⅓. And, since a > 2/(2c + 1), the lower limit of a can be calculated: a_{min} = 2/(2c_{max} + 1) = 1.2.
Suppressor fixation: The SR/ST, sup equilibrium corresponds to a SR/ST population with a totally suppressed SR (t̂ = ½). Carvalho and Vaz (1999) suggest that Ylinked suppressors are in fact fixed in some populations and, therefore, SR remains undetectable (no sexratio phenotype). It is possible that the same happens with autosomal suppressors. As we can see in Figure 1, suppressor fixation occurs when c ≥ 1 (when c = 1 sup frequency reaches 100% very slowly).
The analysis for this equilibrium also consisted in setting the eigenvalue equal to 1 (λ= 1) as allowed by PerronFrobenius theorem for allpositive matrices (appendix b). In addition to four nonrelevant solutions there are three from which we find the stability boundary conditions:
The solutions obtained in Equations 8 and 9 represent the SR/ST polymorphism stability boundaries, which can be demonstrated as follows. In this equilibrium sexratio is totally suppressed so t̂ = ½. We know that if bct > ½ (i.e., bc > 1), the condition determining the SR/ST polymorphism is given by (4). It can be simplified to a > 2bc/(c + 1) for t = ½. If bct < ½ (i.e., bc < 1) stability is determined by (3) that (given t = ½) simplifies to a > 2/(c + 1).
We assumed bc < 1, which seems compatible with biological values for b. The equations limiting the SR/ST, sup parametric space are (7) and (9) (Figure 5). Therefore, the SR/ST, sup equilibrium is stable provided that c > 1 and a > 2/(c + 1).
In short, the SR/ST, sup equilibrium depends on two basic conditions: the stability of the SR/ST polymorphism [a > 2/(c + 1), for bc < 1] and the stability of the sup allele fixation (c > 1).
Polymorphism: This equilibrium may represent D. mediopunctata, D. simulans, and other species known to be polymorphic for sexratio autosomal suppressors. We can observe from Figure 1 that the double polymorphism occurs when there is overdominance (a > 1) and selection against SR/Y males (c between ∼0.3 and 1).
The Jacobian elements for the SR/ST, sup/+ equilibrium are functions of suppressor equilibrium frequencies (the r̂ variables) and these happen to be quite extended polynomials in a and c (not shown). Therefore, we could not solve the characteristic equation and perform a formal stability analysis for this equilibrium. However, the boundaries for a preserved polymorphism can be inferred from our previous analysis on sup noninvasion and sup fixation (where r̂ could be set to 0 or 1) and from our simulation results (Figure 1). The SR/ST, sup/+ equilibrium is found between + and sup trivial equilibria. The first boundary of the polymorphism is that of the suppressor invasion: a > (c + 1)/[2c(2c + 1)] [obtained from the noninvasion condition in (6) with the simplification b = 0]. The second boundary (c < 1) is obtained from the suppressor fixation condition [in (7)]. The third and last boundary should be the stability condition of SR/ST polymorphisms. In fact, it can be obtained as follows. First, we verified by simulations that SR/ST, sup/+ polymorphisms occur in the space where bct < ½ (when b ranges between 0 and 1; not shown). Therefore, the stability condition for the X polymorphism is given by (3): a > 2/(2ct + 1). If we substitute t for the formula we found for t̂ in Equation 2 and solve a > 2/(2ct̂ + 1) for a, we obtain the surprisingly simple expression: a > 4/(c + 3).
Figure 6 summarizes the results for the SR/ST, sup/+ equilibrium. Note that conditions c < 1 and a > 4/(c + 3) imply a > 1, i.e., SR/ST female overdominance. Table 3 outlines the analysis results for all equilibria.
DISCUSSION
Autosomal suppressors of sexratio were first investigated in theory by Wu (1983) who demonstrated that they are not expected to spread under some fitness configurations. He aimed to explain the absence of suppression in D. pseudoobscura. In this work we developed a different model to study the evolution of these suppressors in Drosophila. We showed that an invading suppressor either remains polymorphic or runs to fixation. Essentially, a preserved polymorphism occurs when SR is deleterious in males (c < 1) and suppressor fixation occurs when SR is neutral or positively selected (c ≥ 1). Our main conclusions are (i) a polymorphism for suppression can be preserved even if the suppressor allele is neutral in fitness; (ii) the conditions for this preserved polymorphism (SR/ST, sup/+ equilibrium) are a > (c + 1)/[2c(2c + 1)], a > 4/(c + 3), and c < 1, where a and c are the ST/SR female and SR/Y male selection coefficients, respectively; and (iii) the meiotic drive in the equilibrium (i.e., the average female proportion in SR/Y males progeny) is given by
Experimental vs. theoretical data: Three species bear sufficient data to weigh against our theoretical results: D. mediopunctata, D. simulans, and D. pseudoobscura. Such comparison is based on the assumption that our model is valid for them, in particular that autosomal suppressors are neutral (see Limitations of the model). The first two species harbor SR/ST, sup/+ polymorphisms (Carvalho and Klaczko 1993; Cazemajoret al. 1997), while D. pseudoobscura lacks suppression (Policansky and Dempsey 1978; Beckenbachet al. 1982). D. mediopunctata SR/Y males sire progenies with 78% of females on average (t ∼ 0.78; Varandaset al. 1997). Figure 7 presents fitness combinations from simulations resulting in t̂ values compatible with this species (dotted region). The polymorphism in this case occurs when ∼0.2 < c < ∼0.5 and a > ∼1.2. Regarding D. simulans, SRbearing populations differ in SR frequency but drive expression is usually highly suppressed. The hatched region in Figure 7 presents fitness combinations that explain t̂ values compatible with this species (0.55–0.60; Atlanet al. 1997). Two natural examples could be the population of Nairobi, Kenya, where p̂_{m} ∼ 15% and t ∼ 0.58 and the population of St. Martin where p̂_{m} ∼ 22% and t ∼ 0.57 (Atlanet al. 1997). According to our model, the SR/ST, sup/+ polymorphism for such populations requires overdominance (a > 1) and c between ∼0.4 and ∼0.8. The example of D. simulans illustrates that even when c < 1 we might be dealing with undetectable sexratio due to high suppressor frequency. In fact, despite more than 70 years of research with this species, only recently Merçot et al. (1995) crossed distant populations revealing a high frequency of masked SR, almost totally neutralized by populationspecific Ylinked and autosomal suppressors. Heterospecific crosses with D. sechellia and D. mauritiana also suggest cryptic sexratio in D. simulans (Dermitzakiset al. 2000; Taoet al. 2001). If this phenomenon is common, known sexratio populations of Drosophila could be just a biased sample of what actually exists in nature: “known” populations (where c ≪ 1) plus “hidden” populations (where c is close to 1). A similar observation was made by Carvalho and Vaz (1999).
In spite of direct search efforts, no suppression has ever been found in natural populations of D. pseudoobscura. A possible explanation is that suppressors are not expected to invade when there is strong selection against SR/Y males, i.e., a very low value of c (Wu 1983). An alternative though unlikely explanation is that suppression has not yet arisen by mutation in that species. Here we showed that the stability conditions for suppressor noninvasion are overdominance—a is always >1.2—and strong selection against SR/Y males—c is always <⅓ (Figure 7, crosshatched area). These results confirm and extend those obtained by Wu (1983).
What holds sup in check? Our model indicates that as long as there is any selection against SR/Y males an autosomal suppressor (even with no deleterious effect) will not run to fixation. This result contrasts with Ylinked suppressors: in the presence of SR chromosomes a neutral suppressor allele will always run to fixation. For this reason naturally occurring polymorphisms for Ylinked suppression can be explained only by a deleterious effect of the suppressor allele (Carvalhoet al. 1997). If a neutral autosomal suppressor (sup) is not fixed then there is at least some female bias; this means that Fisher's principle should be favoring sup. Thus, what holds sup in check? In the meiosis of SR/Y males, autosomal suppression decreases the proportion of SR gametes, increasing the proportion of Y gametes. Therefore sup is associated with Y gametes while + is associated with SR gametes. Because of this linkage disequilibrium, sup and + frequencies are different not only between sexes but also between ST/ST, ST/SR, SR/SR, ST/Y, and SR/Y individuals (which explains why eight recurrence equations were required to follow SR and sup frequencies!). Since sup and + are associated with different genotypes with different fitnesses (a, b, and c parameters), they are indirectly selected. This indirect selection most likely holds sup in check. We have done some preliminary calculations on the marginal fitness of sup and + alleles, which indicate that the + alleles are associated with bestfit genotypes (ST/SR females, for example). A complete investigation of this issue is beyond the scope of this article and should be considered elsewhere.
Limitations of the model: We have focused our investigation on the case of neutral suppressors, and it will be interesting to explore the consequences of selection. A suggestion of selection against autosomal suppressors appeared in Carvalho et al. (1998). They followed the sexual proportion in experimental populations of D. mediopunctata fixed for SR and the frequency of males rose from 16 to 32% in 49 generations due to the accumulation of sexratio autosomal suppressors. However, this rate of change was slower than that expected by Fisherian selection (Carvalhoet al. 1998, p. 726). A possible explanation for this difference is that autosomal suppressors are slightly deleterious. If suppression does have a cost then suppressor equilibrium frequency may be quite different from what a neutral model predicts. We carried out numerical simulations assuming a 1% fitness loss in all males with the +/sup genotype and a 2% loss in all males with the sup/sup genotype. The essence of our previous findings remains: suppressors will not invade when selection against SR is strong and will remain polymorphic when SR is moderately deleterious (c ≥ ∼0.6; Figure 8). However, there are some significant changes: ST/SR female overdominance is no longer obligatory for the stability of SR/ST, sup/+ polymorphisms and suppressor equilibrium frequency is drastically decreased even by weak selection (for example, sup does not run to fixation when SR is not deleterious, i.e., c ≥ 1). It should be noted that a 1% selection is very hard to detect experimentally.
Another limitation of our model is the existence of Ylinked suppressors of sexratio in natural populations (Carvalhoet al. 1997; Jaenike 1999; MontchampMoreauet al. 2001). Since Ylinked suppressors are directly favored by meiotic drive, their evolution is expected to be faster than that caused by Fisher's principle. In fact, the frequency of a Ylinked suppressor, even being deleterious, will rapidly run to equilibrium in simulations (∼1000 generations; not shown). In our simulations, an autosomal suppressor might take ∼2500 generations to attain the equilibrium. In that sense, autosomal suppressors might be less important than Ylinked ones. A model including both types of suppression may be useful, if it does not call for too many arbitrary assumptions.
Suppression and the stability of SR/ST polymorphisms: At least two factors have a role in the stabilization of SR/ ST polymorphisms: natural selection and suppression. Both effects can be measured by the conditions determined by Edwards (1961) for the stability of the X polymorphism [see (3) and (4)]. Selection is given by the a, b, and c parameters while suppression affects t, the drive parameter. Figure 9 shows the effect of autosomal suppression on the stability of SR/ST polymorphisms.
Suppression reduces the value of t̂ and so can (i) avoid SR fixation, increasing the SR/ST parametric space (region 1), or (ii) eliminate SR, decreasing the SR/ST parametric space (region 2). The first situation corresponds to the idea that suppression stabilizes the polymorphism because it avoids SR fixation. Polymorphism stabilization due to suppression happens when selection against SR is weaker (b and c close to 1) and, hence, SR frequency is higher (in this case when c ≥ 1 sup is fixed and when c < 1 there is sup/+ polymorphism). It is possible that some suppressorbearing present populations have suffered the risk of extinction in the past due to a high SR frequency in the absence of suppression (D. simulans and D. mediopunctata are candidate species; Varandaset al. 1997; Carvalho and Vaz 1999). The second situation (SR is eliminated; region 2 in Figure 9) occurs when selection against SR is stronger. In this case the equilibrium resulting from sup invasion is ST fixation and a neutral +/sup polymorphism. Thus, perhaps some populations devoid of SR chromosomes were once balanced SR/ST polymorphisms but SR was eliminated by natural selection when suppressors spread and diminished drive. Note that this evolutionary scenario is a very likely outcome if SR is deleterious (b = 0.5; Figure 9). Should we question the common idea that suppression stabilizes SR/ST polymorphisms? Atlan et al. (1997) studied several D. simulans populations from America, Europe, Asia, and Africa that lacked SR but exhibited resistance (suppression) to the SR of a different population. Maybe SR chromosomes were once present but were eliminated due to suppression.
Acknowledgments
We are very grateful to A. Clark, C. Struchiner, P. Otto, A. Peixoto, M. Vibranovski, C. Codeço, and three anonymous referees for valuable suggestions on the manuscript. We also thank C. Landim, C. Tomei, J. Koiller, M. Shinobu, R. Chasse, C. Guerra, and J. Vaz for all the mathematical assistance; G. Vaz for graphical assistance; and Pennsylvania State University for computer facilities. Financial support was provided by Fundação Universitária José Bonifácio (FUJBUFRJ), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and SubReitoria de Ensino para Graduados (SR2UFRJ).
APPENDIX A: RECURRENCE EQUATIONS
Consider a generation cycle starting with the production of gametes. Union of gametes in G_{0} results in zygotes that grow to adults of G_{0}. These adults produce the gametes of G_{1} and so on. Let p be SR frequency in G_{0} (as defined in Table 2) while p′ is SR frequency in the next generation (G_{1}).
SR frequency: Assuming random mating and random union of gametes, the frequency of ST/ST female zygotes, for example, is the product of ST frequency in eggs and sperm, i.e., (1 – p_{e}) × (1 – p_{s}). The frequencies of SR/SR, ST/SR, and ST/ST female adults (F_{11}, F_{12}, and F_{22}) can be calculated from the respective zygotic frequencies by applying the selection coefficients (see Table 1),
SR frequency in G_{0} female adults will be F_{11} + ½ F_{12}:
Since we assume no drive in females and no selection on fecundity, SR frequency in eggs from G_{1} is equal to SR frequency in female adults from G_{0}:
Let t be the proportion of Xbearing sperm resulting from SR/Y male meiosis (and 1 – t is the proportion of Ybearing sperm). Since this proportion is ½ for ST/Y males, the proportion of SR among X sperm from G_{1} is
Similarly, the proportion of Ybearing sperm in the population sperm pool, i.e., the zygotic male proportion in G_{1}, is
SR frequency in male zygotes is equal to SR frequency in eggs (p_{e}). SR frequency in male adults from G_{0} can then be calculated by applying the selection coefficient c (see Table 1): p_{m} = cp_{e}/[cp_{e} + (1 – p_{e})]. It suffices to substitute
Suppressor frequency: As we assumed that autosomal suppression is selectively neutral (Table 1), the frequency of sup in SR/Y adults, for example, is equal to its frequency in SR/Y zygotes from the same generation (r_{m1}, see Table 2). The same holds true for any other genotype (ST/Y, SR/SR, ST/SR, and ST/ST). In this way, sup frequency in adults can be calculated directly from sup frequency in the gametes that originated these adults (instead of separately modeling the gametetozygote and zygotetoadult transitions). It is worth stating that this approach was essential to bring forward the analytical and algebraic solutions of the model.
The frequency of sup in SR/SR, ST/SR, and ST/ST female zygotes and adults (r_{f11}, r_{f12}, and r_{f22}, respectively) is the average between sup frequency in eggs and in X sperm:
Similarly, sup frequency in SR/Y and ST/Y male zygotes and adults (r_{m1} and r_{m2}, respectively) is the average between sup frequency in eggs and in Y sperm:
Now, let r_{e} and r_{s} be the frequency of the sup allele in eggs and sperm (as defined in Table 2) while
The frequencies of sup in each of the three sperm types in G_{1} (SR, ST, and Y) can be calculated if we follow G_{0} male meiosis. Table A1 shows the proportion of each sperm haplotype produced by every SR/Y and ST/Y male considering the autosomal genotype (see also the meiotic drive pattern defined in Table 1).
The frequency of Xbearing sperm resulting from SR/Y male meiosis in G_{0} is t = [SR_sup] + [SR_+], where [SR_sup] = ⅜(r_{e1} + r_{sY}) – ¼r_{e1}r_{sY} and [SR_+] = 1 – ⅝(r_{e1} + r_{sY}) + ¼r_{e1}r_{sY}. Therefore, t can be simplified as 1 – ¼(r_{e1} + r_{sY}), which, given Equation A7, equals
We can now calculate sup frequency in SR and Y sperm in G_{1} (r′_{s}). The frequency of sup in SR sperm equals [SR_sup]/([SR_sup] + [SR_+]):
As for ST/Y male meiosis the reasoning is straightforward. Because of Mendelian segregation, the frequency of sup in either ST or Y sperm equals to r_{m2}. Therefore, given Equation A8, sup frequency in ST sperm in G_{1} equals
Similarly, the frequency of Y_sup haplotype in total sperm produced by ST/Y males will be [Y_sup]_{2} = ½r_{m2} = ¼(r_{e2} + r_{sY}). And the frequency of Y_sup haplotype in sperm produced by SR/Y males can be simplified to [Y_sup]_{1} = ⅛(r_{e1} + r_{sY}) + ¼r_{e1}r_{sY} (see Table A1). Finally, the frequency of Y_sup sperm in the population sperm pool in G_{1}, given by r′_{sY}, is the weighted average of what came from SR/Y and ST/Y meiosis:
The complete system consists of eight recurrence equations (for p_{e}, p_{s}, p_{m}, r_{e1}, r_{e2}, r_{s1}, r_{s2}, and r_{sY}): (A3), (A4), (A6), (A9), (A10), (A12), (A13), and (A14), where p_{f} and t are defined in Equations A2 and A11, respectively.
APPENDIX B: JACOBIANS AND EIGENVALUES
The general Jacobian of the system is a fivebyfive matrix with the system's partial derivatives:
Suppressor noninvasion: Matrix J_{1}, the Jacobian for the SR/ST, + equilibrium, can be obtained from B1 by substituting b = 0, r̂ = 0, and p = p̂_{,} where t̂ = 1 in Equations 1,
The characteristic equation can be obtained by setting the determinant of the J_{1} –λI matrix equal to 0, where I is the fivebyfive identity matrix. The roots of this equation are the eigenvalues (λ) of J_{1}: 0 and the roots of a 4° polynomial for λ (with extensive coefficients on a and c; not shown). Now we can check if J_{1} contains only positive (or null) elements so that the PerronFrobenius theorem can be applied (Ortega 1987). We know that Q_{1} is always positive since c ≥ 0. Thus, all elements of J_{1} are positive simply when Q_{2} > 0: 2ac + a  2 > 0 ⇒ a > 2 / ( 2c + 1), which is precisely one of the stability conditions we will find for this equilibrium (see Table 3). So, J_{1} is always positive and the PerronFrobenius theorem validates the procedure λ= 1 to get the stability boundaries of the equilibrium. Given λ= 1, the 4° polynomial is reduced to c(4ac^{2} + 2ac – c – 1)(2ac + a – 2) = 0 whose solutions are c = 0, a = 2/(2c + 1) and a = (c + 1)/[2c(2c + 1)].
Suppressor fixation: The Jacobian matrix for the SR/ST, sup equilibrium, J_{2} (not shown), can be obtained from B1 by substituting r̂ = 1 and p = p̂_{,} where t̂ = ½ in Equations 1.
The eigenvalues (λ) of J_{2} are 0 and the roots of a 4° polynomial for λ (with extensive coefficients on a, b, and c; not shown). We did not find a general condition that assured positive elements for J_{2}. Therefore, we checked 1000 random simulations that resulted in SR/ST, sup equilibria. A short Maple algorithm was developed to verify each element from the 1000 SR/ST, sup matrices. J_{2} was always positive and the PerronFrobenius theorem could also be applied. By setting λ= 1 the polynomial is reduced to ac(c – 1)(c + 1)(ab + a – 2b)(ac + a – 2)(ac + a – 2bc) = 0 whose solutions are four nonrelevant ones, a = 0, c = 0, c =–1, and a = 2b/(b + 1), and three from which we will find the stability boundary conditions c = 1, a = 2bc/(c + 1), and 2/(c + 1).
Footnotes

Communicating editor: M. W. Feldman
 Received May 29, 2003.
 Accepted October 3, 2003.
 Copyright © 2004 by the Genetics Society of America