Evolution of Autosomal Suppression of the Sex-Ratio Trait in Drosophila

Corresponding author: Suzana Casaccia Vaz, Instituto de Biologia, Universidade Federal do Rio de Janeiro, Caixa Postal 68011, CEP 21944-970, Rio de Janeiro, Brazil., suzana{at}biologia.ufrj.br (E-mail)

Communicating editor: M. W. FELDMAN

	ABSTRACT

TOP ABSTRACT THE MODEL NUMERICAL SIMULATIONS EQUILIBRIUM FREQUENCIES STABILITY OF EQUILIBRIA DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

The sex-ratio trait is the production of female-biased progenies due to X-linked 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 Y-linked 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, sex-ratio 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 < ¹/₃, 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 (ac + 1 - a + )/4ac, where a is the fitness of SR/ST females.

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 best-studied example of segregation distortion was first recorded by GERSHENSON 1928 and later named meiotic drive by SANDLER and NOVITSKI 1957 .

The sex-ratio trait known in 12 Drosophila species is a case of meiotic drive in the sex chromosomes. Males carrying certain X chromosomes, called SR, produce female-biased progenies due to the degeneration of Y-bearing 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 sex-ratio 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 Y-linked 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 Y-linked suppressors of sex-ratio in SR-bearing populations can be explained by meiotic drive theory: any Y-linked gene that increases the transmission rate of the Y chromosome (as does a sex-ratio suppressor) is directly favored. Therefore, Y-linked suppressors are expected to run to fixation unless they are deleterious (CLARK 1987 ; CARVALHO et al. 1997 ). Autosomal suppressors of sex-ratio 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 sex-ratio 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 sex-ratio) 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 sex-ratio 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 SR-bearing 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 suppressor-free 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 (CAZEMAJOR et 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 Y-linked or autosomal sex-ratio suppressor was ever found in this species despite directed search (POLICANSKY and DEMPSEY 1978 ; BECKENBACH et 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 SR-bearing 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 sex-ratio 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 (, defined as the average female proportion in SR/Y males progeny) is given by (ac + 1 - a + )/4ac (where a and c are the fitness of SR/ST females and SR/Y males, respectively) and, since drive is a known parameter from natural populations, estimates for fitness combinations can be made from the above formula. Second, as long as there is selection against SR/Y males (¹/₃ < c < 1), neutral autosomal suppressors always remain polymorphic; this result contrasts with the dynamics of Y-linked suppressors, expected to run to fixation unless they are deleterious. These conclusions are relevant for the understanding of naturally occurring sex-ratio polymorphisms in Drosophila.

	THE MODEL

TOP ABSTRACT THE MODEL NUMERICAL SIMULATIONS EQUILIBRIUM FREQUENCIES STABILITY OF EQUILIBRIA DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

The model we describe below represents a typical sex-ratio 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 egg-to-adult viability component (sex-ratio 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. Sex-ratio expression in SR/Y males depends on an autosomal locus that affects the sexual proportion only: sup denotes the suppressor allele and "+" is the wild-type 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 X-bearing 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 (CARVALHO et 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

TOP ABSTRACT THE MODEL NUMERICAL SIMULATIONS EQUILIBRIUM FREQUENCIES STABILITY OF EQUILIBRIA DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

Numerical simulations covering a biologically meaningful set of the a x 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 x c scanning for two different values of b are shown in Fig 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).

View larger version (42K): In this window In a new window Download PPT slide   Figure 1. Numerical simulations with a model for sex-ratio autosomal suppression. The parameters a, b, and c are the fitnesses of ST/SR, SR/SR, and SR/Y genotypes, respectively. (a) b = 1/2. (b) b = 1. SR/ST polymorphisms occur for a and c values in the shaded space. An autosomal suppressor does not invade the population in the region denoted by +, remains polymorphic in the sup/+ region, and is fixed in the sup region. Region 1 is SR fixation (with fixed sup) and region 2 is ST fixation (with sup/+ neutral polymorphism).

Fig 2 shows the relation between , 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 is a linear function of sup frequency (see Equation A11).

View larger version (19K): In this window In a new window Download PPT slide   Figure 2. Numerical simulations with a model for sex-ratio autosomal suppression. The points represent 1000 random fitness combinations that result in SR/ST, sup/+ polymorphism. The parameters a, b, and c are the fitnesses of ST/SR, SR/SR, and SR/Y genotypes, respectively. is the equilibrium value of the drive parameter t ( = 1 - 1/2m1, see Equation A11). (a) as a function of a; (b) as a function of b; (c) as a function of c.

It is clear from Fig 2 that c is the parameter with the greatest effect on the value of . 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 = ¹/₂; see Fig 1 and Fig 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 varying from 0.5 to 1, is that when SR frequency is high, sup frequency is also high (not shown).

	EQUILIBRIUM FREQUENCIES

TOP ABSTRACT THE MODEL NUMERICAL SIMULATIONS EQUILIBRIUM FREQUENCIES STABILITY OF EQUILIBRIA DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

Numerical simulations indicate that it is possible to maintain a polymorphism for a neutral autosomal suppressor in a SR-bearing 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 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 = 0.78, the average female proportion in the progenies of SR/Y males from a natural population (VARANDAS et 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 , and (see Table 2 for variables definitions). The system of equations (Equation A4, and Equation A6) has two trivial solutions ( = 0 and = 1) and a third one,

(1)

where and

Equation 1 agree with Edwards' results, where corresponds to a fixed-drive 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 (; see Fig 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 five-equation system (setting r' = r = for all five recurrence equations—A9, A10, A12, A13, and A14—and substituting p with for the four p variables given in Equation 1, where = 1 - ¹/₄(_e1 + _sY); see Table 2 for the variables listing). A straightforward solution was not possible so we solved the problem by reducing the system step-by-step with the help of Maple computer software (not shown). The solutions we found for are ¹/₂, 1, and

(2)

Given _m1 = 2 - 2 (from Equation A11), the suppressor equilibrium frequency in SR/Y males is

Fig 3 compares the algebraic value of (_alg, given by the formula in Equation 2) to the true value of (suggested by _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 but provides an excellent approximation since _sim and _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 _sim had a perfect match with those predicted by _alg (not shown). Thus, we can safely affirm that the expression (ac + 1 - a + )/4ac is a very good estimate of for any value of b between 0 and 1. Note that this interval (0 < b < 1), implying selection against SR/SR females, is the biologically meaningful range for this parameter (WALLACE 1948 ; CURTSINGER and FELDMAN 1980 ; BECKENBACH 1996 ).

View larger version (18K): In this window In a new window Download PPT slide   Figure 3. Comparison between simulated and algebraically estimated meiotic drives under autosomal suppression (sim and alg, respectively). sim was obtained by iterating the recurrence Equation A13, and Equation A14 until an equilibrium was attained, with 1000 random values for the parameters a (between 0 and 3), b (0–1), and c (0–1) that resulted in SR/ST, sup/+ polymorphism. alg is the value of given by the formula (ac + 1 - a + )/4ac (see Equation 2), with the same set of a and c values used in the simulations. Note that we also carried out simulations with b = 0 (not shown) and in this case sim matches perfectly with alg, confirming that the small discrepancy between them in the figure is due solely to the assumption b = 0, used to obtain the formula for alg.

	STABILITY OF EQUILIBRIA

TOP ABSTRACT THE MODEL NUMERICAL SIMULATIONS EQUILIBRIUM FREQUENCIES STABILITY OF EQUILIBRIA DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

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 Fig 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:

(3)

(4)

Note that if bct > ¹/₂ the determining condition is (4). If bct < ¹/₂ the determining condition is (3) and in this case the polymorphism stability does not depend on b.

Suppressor noninvasion:
The equilibrium corresponding to a population bearing X polymorphism with no sex-ratio 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 = ¹/₂ (see Fig 1). Besides, since c has a very low value in this equilibrium (c < 0.3) and since = 1 and b is between 0 and 1, we know that bc < ¹/₂. 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 Perron-Frobenius theorem (ORTEGA 1987 ) for nonnegative matrices, which allows one to set the eigenvalue equal to 1 ( = 1) to find the stability boundaries. By setting = 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)].

Fig 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 = 1 (see Fig 4). Thus, the SR/ST, + equilibrium is predicted when

(5)

and

(6)

View larger version (21K): In this window In a new window Download PPT slide   Figure 4. Stability analysis of a model for sex-ratio autosomal suppression: suppressor noninvasion (SR/ST, + equilibrium). The parameters a and c are the fitnesses of ST/SR and SR/Y genotypes, respectively. The SR/ST, + equilibrium is stable when a > 2/(2c + 1) (solid line) and a < (c + 1)/[2c(2c + 1)] (dotted line).

Regarding the condition in (5), note that a > 2/(2c + 1) is the SR/ST polymorphism stability condition when bct < ¹/₂ [see (3) for t = 1]. In fact, bct < ¹/₂ 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 < ¹/₃. Therefore, the upper limit of c is c_max = ¹/₃. 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 ( = ¹/₂). CARVALHO and VAZ 1999 suggest that Y-linked suppressors are in fact fixed in some populations and, therefore, SR remains undetectable (no sex-ratio phenotype). It is possible that the same happens with autosomal suppressors. As we can see in Fig 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 ( = 1) as allowed by Perron-Frobenius theorem for all-positive matrices (Appendix B). In addition to four nonrelevant solutions there are three from which we find the stability boundary conditions:

(7)

(8)

(9)

The solutions obtained in Equation 8 and Equation 9 represent the SR/ST polymorphism stability boundaries, which can be demonstrated as follows. In this equilibrium sex-ratio is totally suppressed so = ¹/₂. We know that if bct > ¹/₂ (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 = ¹/₂ . If bct < ¹/₂ (i.e., bc < 1) stability is determined by (3) that (given t = ¹/₂) 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) (Fig 5). Therefore, the SR/ST, sup equilibrium is stable provided that c > 1 and a > 2/(c + 1).

View larger version (40K): In this window In a new window Download PPT slide   Figure 5. Stability analysis of a model for sex-ratio autosomal suppression: suppressor fixation (SR/ST, sup equilibrium) for bc < 1. The parameters a, b, and c are the fitnesses of ST/SR, SR/SR, and SR/Y genotypes, respectively. The SR/ST, sup equilibrium is stable when c > 1 (dashed line) and a > 2/(c + 1) (solid line).

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 sex-ratio autosomal suppressors. We can observe from Fig 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 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 could be set to 0 or 1) and from our simulation results (Fig 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 < ¹/₂ (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 in Equation 2 and solve a > 2/(2c + 1) for a, we obtain the surprisingly simple expression: a > 4/(c + 3).

Fig 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.

View larger version (50K): In this window In a new window Download PPT slide   Figure 6. Stability analysis of a model for sex-ratio autosomal suppression: polymorphism (SR/ST, sup/+ equilibrium). The parameters a and c are the fitnesses of ST/SR and SR/Y genotypes, respectively. The SR/ST, sup/+ equilibrium is stable when c < 1 (dashed line), a > (c + 1)/[2c(2c + 1)] (dotted line), and a > 4/(c + 3) (solid line).

	DISCUSSION

TOP ABSTRACT THE MODEL NUMERICAL SIMULATIONS EQUILIBRIUM FREQUENCIES STABILITY OF EQUILIBRIA DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

Autosomal suppressors of sex-ratio 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 (ac + 1 - a + )/4ac.

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 ; CAZEMAJOR et al. 1997 ), while D. pseudoobscura lacks suppression (POLICANSKY and DEMPSEY 1978 ; BECKENBACH et al. 1982 ). D. mediopunctata SR/Y males sire progenies with 78% of females on average (t 0.78; VARANDAS et al. 1997 ). Fig 7 presents fitness combinations from simulations resulting in 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, SR-bearing populations differ in SR frequency but drive expression is usually highly suppressed. The hatched region in Fig 7 presents fitness combinations that explain values compatible with this species (0.55–0.60; ATLAN et al. 1997 ). Two natural examples could be the population of Nairobi, Kenya, where _m 15% and t 0.58 and the population of St. Martin where _m 22% and t 0.57 (ATLAN et 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 sex-ratio due to high suppressor frequency. In fact, despite more than 70 years of research with this species, only recently MERCOT et al. 1995 crossed distant populations revealing a high frequency of masked SR, almost totally neutralized by population-specific Y-linked and autosomal suppressors. Heterospecific crosses with D. sechellia and D. mauritiana also suggest cryptic sex-ratio in D. simulans (DERMITZAKIS et al. 2000 ; TAO et al. 2001 ). If this phenomenon is common, known sex-ratio 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 .

View larger version (31K): In this window In a new window Download PPT slide   Figure 7. Fitness configurations compatible with meiotic drive data for natural populations of Drosophila. The parameters a and c are the fitnesses of ST/SR and SR/Y genotypes, respectively. Simulations were carried out with the formula (Equation 2), selecting a and c values that resulted in a given range of . In D. simulans, 0.55 < < 0.60. For this range, the frequency of SR in male adults (m; see Equation 1) is between 1 and 30% in the simulations. In D. mediopunctata 0.75 < < 0.80. For this range, m is between 3 and 13%. In D. pseudoobscura, = 1 and m varied from 1 to 8%.

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 <¹/₃ (Fig 7, cross-hatched 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 Y-linked suppressors: in the presence of SR chromosomes a neutral suppressor allele will always run to fixation. For this reason naturally occurring polymorphisms for Y-linked suppression can be explained only by a deleterious effect of the suppressor allele (CARVALHO et 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 best-fit 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 sex-ratio autosomal suppressors. However, this rate of change was slower than that expected by Fisherian selection (CARVALHO et 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; Fig 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.

View larger version (77K): In this window In a new window Download PPT slide   Figure 8. Numerical simulations with a model for deleterious autosomal suppressors. This figure should be compared to Fig 1A. Recurrence equations different from those used to produce Fig 1 were developed to include selection against sup. The fitness parameters a, b (set to 1/2), and c are defined in Table 1 with the difference that the fitness of males with the sup/+ and sup/sup genotypes was multiplied by 0.99 and 0.98, respectively. SR/ST polymorphisms occur for a and c values in the shaded space. Autosomal suppressors do not invade the population in the region denoted by + and there is polymorphism in the sup/+ region. The open region represents SR or ST fixation.

Another limitation of our model is the existence of Y-linked suppressors of sex-ratio in natural populations (CARVALHO et al. 1997 ; JAENIKE 1999 ; MONTCHAMP-MOREAU et al. 2001 ). Since Y-linked 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 Y-linked 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 Y-linked 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. Fig 9 shows the effect of autosomal suppression on the stability of SR/ST polymorphisms.

View larger version (19K): In this window In a new window Download PPT slide   Figure 9. Role of autosomal suppression in the stability of SR/ST polymorphisms. The parameters a, b, and c are the fitnesses of ST/SR, SR/SR, and SR/Y genotypes, respectively. All regions above the dashed line represent a SR/ST population before sup invasion (conditions set by Equation 3 and Equation 4 for t = 1). All regions above the solid line represent a SR/ST population after sup invasion (see conditions marked * and ** in Table 3). Depending on fitness values suppression can convert SR fixation to SR/ST polymorphism (increasing the polymorphism's parametric space; region 1) or convert SR/ST polymorphism to ST fixation (reducing the polymorphism's parametric space; region 2). Note that the region denoted by SR/ST means polymorphism if suppressor is either present or absent.

Suppression reduces the value of 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 suppressor-bearing 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; VARANDAS et al. 1997 ; CARVALHO and VAZ 1999 ). The second situation (SR is eliminated; region 2 in Fig 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; Fig 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 (FUJB-UFRJ), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Sub-Reitoria de Ensino para Graduados (SR2-UFRJ).

Manuscript received May 29, 2003; Accepted for publication October 3, 2003.

	APPENDIX A

TOP ABSTRACT THE MODEL NUMERICAL SIMULATIONS EQUILIBRIUM FREQUENCIES STABILITY OF EQUILIBRIA DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

RECURRENCE EQUATIONS
Consider a generation cycle starting with the production of gametes. Union of gametes in G₀ results in zygotes that grow to adults of G₀. These adults produce the gametes of G₁ and so on. Let p be SR frequency in G₀ (as defined in Table 2) while p' is SR frequency in the next generation (G₁).

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) x (1 - p_s). The frequencies of SR/SR, ST/SR, and ST/ST female adults (F₁₁, F₁₂, and F₂₂) can be calculated from the respective zygotic frequencies by applying the selection coefficients (see Table 1),

(A1)

where w_F = bp_ep_s + a[(1 - p_e)p_s + p_e(1 - p_s)] + (1 - p_e)(1 - p_s).

SR frequency in G₀ female adults will be F₁₁ + ¹/₂ F₁₂:

(A2)

Since we assume no drive in females and no selection on fecundity, SR frequency in eggs from G₁ is equal to SR frequency in female adults from G₀:

(A3)

Let t be the proportion of X-bearing sperm resulting from SR/Y male meiosis (and 1 - t is the proportion of Y-bearing sperm). Since this proportion is ¹/₂ for ST/Y males, the proportion of SR among X sperm from G₁ is

(A4)

Similarly, the proportion of Y-bearing sperm in the population sperm pool, i.e., the zygotic male proportion in G₁, is

(A5)

SR frequency in male zygotes is equal to SR frequency in eggs (p_e). SR frequency in male adults from G₀ 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 (from Equation A3) to obtain SR frequency in male adults from G₁:

(A6)

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 gamete-to-zygote and zygote-to-adult 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:

and

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:

(A7)

(A8)

Now, let r_e and r_s be the frequency of the sup allele in eggs and sperm (as defined in Table 2) while r^'_e and r^'_s are these same frequencies in the next generation (G₁). The frequency of sup in SR and ST eggs from G₁ can be obtained by

and

where F₁₁, F₁₂, and F₂₂ are defined in Equation A1. These frequencies can be simplified as

(A9)

The frequencies of sup in each of the three sperm types in G₁ (SR, ST, and Y) can be calculated if we follow G₀ male meiosis. Table A11 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 X-bearing sperm resulting from SR/Y male meiosis in G₀ is t = [SR_sup] + [SR_+], where [SR_sup] = ³/₈(r_e1 + r_sY) - ¹/₄r_e1r_sY and [SR_+] = 1 - ⁵/₈(r_e1 + r_sY) + ¹/₄r_e1r_sY. Therefore, t can be simplified as 1 - ¹/₄(r_e1 + r_sY), which, given Equation A7, equals

(A11)

We can now calculate sup frequency in SR and Y sperm in G₁ (r^'_s). The frequency of sup in SR sperm equals [SR_sup]/([SR_sup] + [SR_+]):

(A12)

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 G₁ equals

(A13)

Similarly, the frequency of Y_sup haplotype in total sperm produced by ST/Y males will be [Y_sup]₂ = ¹/₂r_m2 = ¹/₄(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 = ¹/₈(r_e1 + r_sY) + ¹/₄r_e1r_sY (see Table A11). Finally, the frequency of Y_sup sperm in the population sperm pool in G₁, given by r^'_sY, is the weighted average of what came from SR/Y and ST/Y meiosis: , where Mz' is defined in Equation A5. Appropriate substitutions lead to

(A14)

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 Equation A2 and Equation A11, respectively.

	APPENDIX B

TOP ABSTRACT THE MODEL NUMERICAL SIMULATIONS EQUILIBRIUM FREQUENCIES STABILITY OF EQUILIBRIA DISCUSSION APPENDIX A APPENDIX B LITERATURE CITED

JACOBIANS AND EIGENVALUES
The general Jacobian of the system is a five-by-five matrix with the system's partial derivatives:

(B1)

Suppressor noninvasion:
Matrix J₁, the Jacobian for the SR/ST, + equilibrium, can be obtained from B1 by substituting b = 0, = 0, and p = , where