Sex-ratio distortion driven by migration loads

Transcription

1 Theoretical Poulation Biology 7 (007) Sex-ratio distortion driven by migration loads Xin-Sheng Hu, Francis C. Yeh, Fangliang He Deartment of Renewable Resources, 75 General Services Building, University of Alberta, Edmonton, AB, Canada T6G H Received 0 Aril 007 Available online 5 August 007 Abstract The significance of migration load in driving the evolution of reciient oulations has long been documented in oulation genetics, but its effects have not been linked to the formation of biased sex ratios in natural oulations. In this study, we develo a single-locus model to demonstrate how the migration load can shae the rimary and secondary sex ratios in dioecious lants where sexual dimorhism is determined by the sex chromosomes (the XX XY or similar systems). Our results show that migration load can generate an array of sex ratios (from the female- to male-biased rimary/secondary sex ratios), deending on the selection systems at the gametohyte and sorohyte stages and on the sex ratio in the migrating seeds. Ovule abortion and the urging of maladative genes from the immigrating ollen at the gametohyte stage can alter the rimary sex ratio and indirectly alter the secondary sex ratio. The resence of maladative sex-linked genes from the migrating ollen and seeds of males facilitates the outcome of the female-biased secondary sex ratios, while the resence of maladative sex-linked genes from the migrating seeds of females can lead to the male-biased secondary sex ratios. The detrimental effects of the Y-chromosome from the migrating ollen and seeds can enhance the formation of female-biased rimary and secondary sex ratios. These theoretical redictions highlight an alternative aroach to the existing sex-ratio theories for interreting the formation of biased sex ratios in the oulations that are subject to the imacts of maladative genes from immigrants. r 007 Elsevier Inc. All rights reserved. Keywords: Primary sex ratio; Secondary sex ratio; Migration load; Sexual mortality; Selection; Dioecy. Introduction Sex ratio can be categorized as the rimary and secondary sex ratios. The rimary sex ratio refers to the relative freuencies of female to male gametes that are combined to roduce zygotes at the gametohyte stage. The secondary sex ratio refers to the ratio of the number of one sex to the number of its oosite sex in a oulation or the ercentage of males or females in a oulation at the sorohyte stage. Fisher s ioneer work redicted that the secondary sex ratio is exected to aroach a unity when arental exenditures on male and female offsring are eual from the inverse freuency-deendent selection (Fisher, 930). However, the ersistence of biased secondary sex ratios in natural oulations suggests that the biological mechanisms other than the inverse freuency-deendent selection might exist. Numerical Corresonding author. Fax: address: xin-sheng.hu@ualberta.ca (X.-S. Hu). theories have been roosed to infer the formation of an array of biased secondary sex ratios (Shaw and Mohler, 953; Hamilton, 967; Charnov, 98; Hardy, 00). In relation to satially structured oulations, a biased secondary sex ratio can be roduced by the mechanisms of local mate cometition (LMC; Hamilton, 967), local resource cometition (LRC; Clark, 978), and grou selection (Wilson and Colwell, 98). These existing theories can be classified into the genetic and non-genetic models (e.g., sex allocation theory), and mixture (Geber, 999). Nevertheless, relationshis between the rimary/secondary sex ratio and the gene flow among oulations adated to different environments have not been examined (Antolin, 993). The reduction in oulation fitness owing to the maladative genes from immigrants is termed as migration load (Wright, 977, ). The magnitude of migration load is related to the immigration rate and the strength of selection against maladative genes (Wright, 977; Hu and Li, 003). It is seculated that migration load /$ - see front matter r 007 Elsevier Inc. All rights reserved. doi:0.06/j.tb

2 548 ARTICLE IN PRESS X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) can freuently occur in the natural oulations that are subject to the inuts of immigrants from their surrounding oulations. Occurrence of the differential mortality of males and females cannot be excluded when maladative sex-linked genes from the immigrants exist, resulting in the biased rimary/secondary sex ratios. Furthermore, the sexratio variation among oulations can be brought about when the selection strengths for the maladative sex-linked genes from the immigrants change in sace, or when the immigration rates of sex-linked genes from a common source change in sace. The satial differential in migration loads between the sexes can lead to biased oulation sex ratios, which is distinct from the attern of the satial segregation of sexes (SSS) within oulations (a finer scale) exlained with different hyotheses (Bierzychudek and Eckhart, 988; Eley, 00; Nebel, 005). The imortance of migration load in driving the evolution of a reciient oulation has long been documented (Wright, 969, 977; Barton and Gale, 993,. 30) and its significance has been exlicitly demonstrated in shaing a secies range (Kirkatrick and Barton, 997; Hu and He, 006). Burt (995) used an indirect method, i.e. measuring the additive variance of fitness and using Fisher s fundamental theorem of natural selection, to estimate the immigration loads of seeds and ollen disersal. In Iomosis aggregate, seeds and ollen from alien oulations reduce fitness by the amounts of 0.00 and er generation, resectively (Burt, 995). The order of migration load from seeds or ollen is about 0.08 % er generation, comared with the mutation load of 0. % er generation in the secies examined by Burt (995). Early theories for lant secies did not link the sex ratio to migration loads although the effects of migrating ollen and seeds on altering the sex ratios have been demonstrated (Bulmer and Taylor, 980; De Jong et al., 00). Charnov (98) examined the euilibrium sex ratio in a satially structured oulation from the oulation genetics ersective. However, all these theories are mainly related to Clark s LRC concet (978), which is difficult to verify in lants because the movement of males and females are distinct from those in the animals. The urose of this study is to formulate an alternative theory to address the differential sexual mortality owing to the immigration of maladative genes. An earlier study examined several roerties of the sexual differential in the migration load under a variety of conditions where males and females were treated as two searate suboulations connected by mating (Hu, 006). The analytical relationshis between migration load and sex ratio were not derived, but the revious results are alicable to the case of sex-linked modifier genes with indirect effects on the sex ratio (Feldman and Otto, 989). Here, we continue to study the relationshis between migration load and sex ratio in dioecious lants and necessarily extend the earlier theory to the case of sexlinked genes with direct effects on the sex ratio (Hu, 006). The roosed theory is alicable to the dioecious lants where sexual dimorhism is determined by the sex chromosomes (the XX XY or similar systems), such as in Silene latifolia (Grant et al., 994), Cannabis sativa L. (Moliterni et al., 004), and other dioecious lants (Ainsworth, 000; Moliterni et al., 004). A dioecious oulation consists of lants with staminate flowers and istillate flowers roduced on searate lants. The evolution of dioecious lants generated from the cosexuality hases remains dynamic through the rocesses of sterile mutation or/and resources reallocation in order to avoid inbreeding deression or to achieve a maximum reroductive gain (Sakai and Weller, 999; Webb, 999). Although the resent study focuses on the simlest but the most studied XX XY system at the molecular level (Ainsworth, 000), the same modeling rocedure can be alied to other comlicate systems to study the relationshis between migration load and sex ratio. Analogous to revious studies (Hu and Li, 003; Hu and He, 006; Hu, 006), selection at the gametohyte and sorohyte stages is taken into account since the exression of some sex-related genes cannot be excluded at each develoing hase (Haldane, 93; Tanksley et al., 98; Charlesworth and Charlesworth, 99; Mulcahy et al., 996). Migration load can be ascribed to either the migrating ollen or the migrating seeds, or both. This can occur at the gametohyte stage when the maladative genes from migrating ollen are resent, resulting in a biased rimary sex ratio. It can also arise at the sorohyte stage when the maladative genes from migrating seeds are resent, resulting in a biased secondary sex ratio. As demonstrated in Hu (006), the two stages of migration loads are interrelated because the gametohyte and sorohyte stages are biologically connected in the life cycle of a dioecious lant. Therefore, sexual differential in the migration load at each stage may generate an array of rimary and secondary sex ratios (from the male- to female-biased sex ratios). In the following sections, we first develo a single-locus model that is alicable to the general case under strong or weak selection. The analytical exressions are then derived for the relationshis between the migration load and the rimary/secondary sex ratio under weak selection at the gametohyte and sorohyte stages. Inferences on the relationshis between the migration load and the rimary/ secondary sex ratio in natural oulations are drawn with numerical examles.. The model.. Assumtions Consider a natural oulation of a dioecious lant secies, with constant rates of immigrating ollen and seeds er generation from source oulations. This condition may naturally occur in the mainland-island structure of oulations where the island oulations mainly receive immigrants from the mainland or in a

3 X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) Adults Gametes Gametes Gametes, * *,, y,,...,,... **, **,... Fertilization continuously distributed secies where the marginal oulations receive migrants from central oulations. The model is distinct from revious models that considered the effects of seed and ollen disersal distances on the sex ratio (De Jong et al., 00). Theoretical deduction follows the seuence of events in the lant life cycle: adult generation t, gamete (ollen and ovules) formation, ollen flow, gametohyte selection, mating, seed formation, seed flow, sorohyte selection, and adult generation t+ (Fig. ). Mutation rate is assumed to be very small and hence its effect is not included. Poulation size is assumed to be large so that the effect of genetic drift is negligible. The allele freuencies in ovules before random combinations with ollen or in ollen before the occurrence of immigration are assumed to be the same as those in the receding adult generation (Wright Fisher s model). Only the dioecious lants with the XX XY system are examined although the same theoretical rocess can be alied to dioecious lants with the sexual dimorhism being controlled by autosomal genes or other tyes of sex-linked genes, such as the ZZ ZW system in Fragaria elateria (Grant, 999). The effects of Y-chromosome on the fitness are included but its olymorhism on the locus oosite to the locus on the X-chromosome is not considered here... General case Pollen flow Seed flow Seeds Seeds Adults *, *,... **, **,... Selection Selection,,...;,,... Fig.. A scheme shows the life cycle of a dioecious lant. and are the freuencies of alleles A and a in the females, resectively. and are the freuencies of genotyes AA and Aa in the females, resectively.,, and y are the freuencies of genotyes A, a, and the Y- chromosome in the males, resectively. Different numbers of suerscrit * are used for the variables after different occurrences in the life cycle. Selection is considered after ollen flow at the gametohyte stage and seed flow at the sorohyte stage so that effects of migration load can be examined in the same generation. Suose there are N individuals in a dioecious oulation, with N xx females and N xy males. Let l be the rimary sex ratio generated by the current adults before the occurrence of migration and natural selection. Let l be the secondary sex ratio in the current generation and then is set as the ratio N xx /N xy. It can be seen that l ¼ l under the assumtion of Wright Fisher s model. Consider a diallelic sex-linked locus with alleles A and A. In the case of multile alleles, consider one allele as A and the remaining alleles as A. There are three genotyes A A, A A,andA A in the females, with freuencies being,, and, resectively. There are two genotyes A Y and A Y in the males, with freuencies being y and y, resectively. The sum of all genotyic freuencies euals unity, i.e. þ þ þ y þ y ¼. The allele freuencies are for allele A residing in the females, for allele A residing in the females, for allele A residing in the males, for allele A residing in the males, and y for the freuency of the counterart locus on the Y-chromosome. These allele freuencies can be calculated from genotyic freuencies, i.e. ¼ þ =, ¼ þ =, ¼ y =, ¼ y =, and y ¼ y = þ y =. The secondary sex ratio can be exressed in terms of the genotyic freuencies l ¼ þ þ y þ y. () When there is an eual sum of all genotyic freuencies in the males and females ( þ þ ¼ y þ y ), an unbiased sex ratio exists (l ¼ ). E. () can also be rewritten in terms of allele freuencies, i.e. l ¼ð þ Þ= ð þ þ y Þ. From this relationshi we can obtain þ þ y ¼ðþl Þ and þ ¼ l ð þ l Þ. Furthermore, we can show that an eual freuency of the X- and Y-chromosomes is resent in the males, i.e. y ¼ þ ¼ =ð þ l Þ. Suose that a roortion of ollen, denoted by m P,is relaced with the immigrating ollen grains at the gametohyte stage er generation. Let Q y and Q y be the freuencies of genotyes A Y and A Y in the source oulation. From the assumtions, let Q, Q,andQ y be the freuencies of alleles A, A, and the locus (only one allele) on the Y-chromosome, resectively. It can be seen that Q ¼ Q y =, Q ¼ Q y =, and Q y ¼ Q y = þ Q y = ¼ Q þ Q. Since only ollen grains can migrate but ovules cannot, the sum of all allele freuencies in the migrating ollen is less than, i.e. Q + Q + Q y o. Thus, it is necessary to re-scale these freuencies by letting Q 0 ¼ Q =ðq þ Q þ Q y Þ, Q 0 ¼ Q =ðq þ Q þ Q y Þ, and Q 0 y ¼ Q y=ðq þ Q þ Q y Þ so that Q 0 þ Q0 þ Q0 y ¼. After ollen flow, the allele freuencies in ollen are derived as m P ¼ð m PÞ þ Q 0 þ l, (a) m P ¼ð m PÞ þ Q 0 þ l, (b) m P y ¼ð m PÞ y þ Q 0 y þ l. (c)

4 550 ARTICLE IN PRESS X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) The exressions Q 0 y ¼ Q0 þ Q0 and y ¼ þ remain valid. The allele freuencies in ovules after ollen flow, denoted by and for alleles A and A, resectively, are the same as in the receding adults, i.e. ¼ and ¼. There is no change in both the rimary and the secondary sex ratios at this stage (l ¼ l ) since immigration does not alter the relative freuencies of males and females. Now consider selection at the gametohyte stage. Let the fitness be and su for alleles A and A in the ovules, resectively, in which allele A is assumed to be maladative; sv and sv for A, A, resectively, in the ollen, and sv y for the locus on the Y-chromosome. When v ¼ 0, the fitness of A in the ollen is the same as its fitness in the ovules. When sv y 40, the locus on the Y-chromosome has negative effects on the ollen fitness. This may occur when deleterious mutation accumulates due to the absence of recombination between the X- and Y-chromosomes (Fllatov et al., 000). With these settings, the average fitness at the gametohyte stage is w ¼ l xx l xy where l xx and l xy are the migration loads from ovules and ollen, resectively, and they are l xx ¼ s u and l xy ¼ sð v þ v þ y v yþ. After selection at the gametohyte stage, the gamete freuencies are 0 y 0 ð su Þ ¼ ð sv Þ C w, (3) B ð sv Þ C A ð sv y Þ y where þ þ þ þ y ¼. The rimary sex ratio denoted by l can be exressed as l ¼ð þ Þ=ð þ þ y Þ. From E. (3), we can obtain l ¼ þ l xx þ þ y l ¼ l ð l xx ð þ l ÞÞ. (4) xy l xy ð þ l Þ Note that the relationshis þ þ y ¼ðþl Þ and þ ¼ l ð þ l Þ are alied in deriving E. (4). The rimary sex ratio l is changed after selection and not the same as the secondary sex ratio l in the receding adults. E. (4) exlicitly indicates that a bias of the rimary sex ratio from the secondary sex ratio is related to selection at the gametohyte stage, such as the occurrence of differential ollen cometitive abilities or differential ovule abortions. The genotyic freuencies after combinations between ollen and ovules can be readily obtained. Let,, and be the freuencies of female genotyes A A, A A, and A A in seeds, resectively; y and y be the freuencies of male genotyes A Y and A Y in seeds, resectively. These genotyic freuencies can be exressed as ¼, ¼ þ, ¼, y ¼ y, and y ¼ y. The sum of all these genotyic freuencies is less than, i.e. þ þ þ y þ yo. This arises because only the combinations of ollen with ovules are included and the other combinations such as ollen with ollen and ovules with ovules are not biologically meaningful. These freuencies are rescaled so that their sum euals unity. Let T ¼ þ þ þ y þ y ¼ ð þ Þð þ þ y Þ. The re-scaled genotyic freuencies are 0 ¼ =T, 0 ¼ =T, 0 ¼ =T, 0 y ¼ y =T, and 0 y ¼ y =T. Let m S be the immigration rate of seeds er generation. Thus, the freuencies of these genotyes after seed immigration can be written as ij ¼ð m S Þ 0 ij þ m SP ij ði; j ¼ ; ; ijþ, (5a) iy ¼ð m SÞ 0 iy þ m SQ iy ði ¼ ; Þ, (5b) where P, P, P, Q y, and Q y are the freuencies of genotyes A A, A A, A A, A Y, and A Y in the migrating seeds (P +P +P +Q y +Q y ¼ ), resectively. The secondary sex ratio can be altered after seed immigration because the secondary sex ratios in the migrating seeds may be uneual to that in the focal oulation. The effects of immigrating seeds may be different from the effects of immigrating ollen. Now consider selection at the sorohyte stage. Let the genotyic fitness be for A A, hsu for A A, su for A A, sv y for A Y, and sv y for A Y. The arameter h is the degree of dominance in heterozygotes. When h euals, the dominance effect is absent in heterozygote A A comared with the fitness of homozygote A A. When h is eual to zero, a comlete dominance effect is resent. When v y and v y are uneual, the locus on the Y-chromosome has different effects on the fitness under different backgrounds of the alleles on the X-chromosome. With these settings, the average oulation fitness is w ¼ L xx L xy where L xx ¼ sðh þ Þ u, the migration load in the females, and L xy ¼ sð y v y þ y v yþ, the migration load in the males. After selection at the sorohyte stage, the genotyic freuencies are 0 0 ð shu Þ ¼ ð su Þ C w. (6) B y A ð sv y Þ y A y ð sv y Þ y The secondary sex ratio at the next generation, denoted by l, is given by l ¼ þ þ y þ y or þ þ þ y where the allele freuencies are ¼ þ ð y þ y, (7) þ =, ¼ =, ¼ y =, ¼ y =, and y ¼ Þ=. The above modeling rocess, which can be numerically calculated, resents a general situation that can be alied to the case of either strong or weak

5 X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) selection. In the next section, we assess the analytical relationshis between the migration loads and the rimary/ secondary sex ratios under weak selection..3. Weak selection Under weak selection, all terms containing the second or higher orders of the selection coefficients (s ) are small and negligible. The immigration rates of seeds and ollen are assumed to be small as well so that the balance between the effects of migration and selection can be attained. The terms containing the second or higher orders of the immigration rate (m P, m S,orm Sm P, or higher orders) or the roducts of the migration rates and selection coefficients (sm P or sm S ) are neglected. Under the assumtions mentioned in the above, the migration load at the gametohyte stage is aroximated by l xy ¼ sð v þ v þ y v y Þ, (8) which indicates that the migration load is of the order similar to the selection coefficients. The inverse of the average fitness at the gametohyte stage is aroximated by ¼ þ l xx þ l xy. (9) w From E. (3), the sum of all allele freuencies in the females and males is resectively given by þ ¼ð þ Þð þ l xx þ l xy Þ l xx ¼ ðl l xx þ l l xy Þ, ð0aþ þ l þ þ y ¼ð þ þ y Þð þ l xx þ l xy Þ l xy ¼ ð þ l xx l l xy Þ. ð0bþ þ l Thus, the rimary sex ratio can be rewritten as l ¼ l l xx þ l l xy. () þ l xx l l xy E. () gives a clear relationshi between the rimary and the secondary sex ratios. It can be shown that a biased rimary sex ratio is resent (l a) when l að þ l xx Þ= ð þ l xy Þ. This exression also reveals that the rimary sex ratio is not biased from the secondary sex ratio when selection is absent at the gametohyte stage, irresective of the differential mortality between the sexes at the sorohyte stage. The following euations are derived from E. (3): þ ¼ ð þ l Þ ð þ l xx ðþl Þl xy þ sv y Þ, (a) y ¼ ð þ l Þ ð þ l xx þ l xy sv y Þ, (b) which indicates the exression of y a þ after selection at the gametohyte stage. The sum of allele freuencies at the locus on the X-chromosome is not the same as that on the Y-chromosome. The relative freuencies of the X- and Y-chromosomes in the male gametes are changed owing to selection at the gametohyte stage. The inverse value of T used for scaling the genotyic freuencies after random combinations between ollen and ovules is given by T ¼ ð þ l Þ ð l Þl xx þ l l l xy. (3) Similarly, the inverse average fitness at the sorohyte stage is algebraically simlified as ¼ þ L xx þ L xy. (4) w The five genotyic freuencies after selection at the sorohyte stage are aroximated by ¼ ¼ ¼ y y þ L xx þ L xy T þ L xx þ L xy shu T þ L xx þ L xy su T ¼ y þ L xx þ L xy sv y T ¼ y þ L xx þ L xy sv y T, (5a) ð þ Þ, (5b), (5c) y, y. (5d) (5e) Although the analytical exressions for allele freuencies are difficult to obtain, the relationshis between migration load and rimary/secondary sex ratio can be derived. Let l m be the secondary sex ratio in the migrants that is assumed to be constant in the source oulations. Thus, P þ P þ P ¼ l m ð þ l m Þ and Q y þ Q y ¼ðþl m Þ. From Es. (7) and (5), the exression for the secondary sex ratio is l ¼ c 0 L xx T þ m S l m ð þ l m Þ c L xy T þ m S ð þ l m Þ, (6) where c 0 ¼ð m S þ L xx þ L xy Þ and c ¼ y þ c 0. þ þ þ y Substituting Es. () and (3) into (6) yields the recurrent euation for the secondary sex ratio, i.e. l ¼ að þ l mþþm S l m bð þ l m Þþm S, (7)

6 55 ARTICLE IN PRESS X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) where a ¼ ð m S þ L xx þ L xy Þð þ l xx ðþl Þl xy þ sv y Þ ð l l xy þ l xx Þ L xxð þ l Þ ð l Þl xx þ l xy, l l and b ¼ ð m S þ L xx þ L xy Þð þ l xx þ l xy sv y Þ ð l l xy þ l xx Þ L xyð þ l Þ ð l Þl xx þ l xy. l l At steady state, the euilibrium euation can be obtained from E. (7) by setting l ¼ l. With the assumtions mentioned at the beginning of this section, the following cubic euation is obtained: L xy l 3 al þ bl L xx ¼ 0, (8) where a ¼ m S þ 3L xx L xy þ l xx þ l xy sv y þ m S, þ l m and b ¼ m S 3L xx þ 3L xy þ l xx l xy þ sv y þ m Sl m. þ l m The solution to E. (8) can be numerically calculated with Mathematica (Wolfram, 99) when the migration loads at steady state are available. Although the migration rate of ollen is not in E. (8), its effects on the secondary sex ratio is realized by changing the allele freuencies and shaing the migration loads at both the gametohyte and sorohyte stages. Substitution of the steady state l into E. () gives the rimary sex ratio l at steady state. Next, we examine secific cases involving different combinations of sex and selection and comare the estimates with the exact values calculated from the general model at steady state..3.. One-sex and one-stage selection When selection takes lace only in the ollen grains, the reduction in oulation fitness comes from the maladative genes in immigrating ollen (l xx ¼ L x ¼ L xx ¼ 0 but l xy a0). This can naturally occur in ollen cometition for fertilizing ovules (Winsor et al., 000) or in the genes exressing during ollen-tube growth. From Es. () and (8), the exressions for the rimary and secondary sex ratios at steady state can be simlified (Table ). It can be seen that l 4l when l xy a0. This can also be viewed from the exact rimary sex ratio in E. (4) l ¼ l ð l xy ð þ l ÞÞ 4l. The exact secondary sex ratio (the timedeendent exression; E. (7)) is l ¼ w ð m S Þþm S =ð þ l m Þ ð w Þð m S Þþm S l m =ð þ l m Þ, (9) where ð sv Þ w ¼ þð sv Þ ð sv Þ þð sv Þ þð sv yþ. y In the resence of an unbiased sex ratio in the migrants (l m ¼ ), l 4 when sv y ¼ (comletely detrimental); l ¼ when sv y ¼ sv ¼ sv a0; and l o when sv y ¼ 0 (neutral effects of Y-chromosome). Numerical results show that the redictions with the aroximate euation (Table ) are slightly higher in comarison with the exact values estimated from the general model at steady state (Section.). Note that in estimating the redicted results with the aroximate euations (Table ), the steady-state migration loads are calculated from the general model. The same atterns can be observed with the change in migration rates of ollen and seeds (e.g., Fig. a and b). When the detrimental effect from the Y-chromosome is absent (sv y ¼ 0), a male-biased secondary sex ratio is enhanced, otherwise a female-biased secondary sex ratio is enhanced (Fig. a and b). When selection is only resent in the males at the sorohyte stage, migration load exists in the males (l xx ¼ l xy ¼ L xx ¼ 0 but L xy a0). This case can naturally occur when the heterogametic sex has a higher inviability than the homogametic sex, although few reorts are recorded in lants but not in animals (e.g., Haldane s rule; Haldane, 9). Under this condition, the rimary sex ratio is eual to the secondary sex ratio, i.e. l ¼ l at steady state. The exression for the secondary sex ratio at steady state can be simlified from E. (8) (Table ). It is exected that a high inviability in the males can result in a female-biased secondary sex ratio. This can also be inferred from the general model (Section.), i.e. l ¼ ð m S Þ= þ m S =ð þ l m Þ, (0) ð m S Þ= þ m S l m =ð þ l m Þ w where w ¼ sð m SÞ v y þ þ sm S ðv y Q y þ v y Q y Þ. þ v y þ When l m ¼ and sv y a0 or/and sv y a0, then l 4. Numerical results show that the female-biased rimary or secondary sex ratios are enhanced with an increase in the migration rate of ollen or seeds (Fig. 3). The exact rimary/secondary sex ratios are slightly smaller than the redicted values. When selection is only resent in the ovules, the reduction in fitness comes from ovule abortion (l xy ¼ L xy ¼ L xx ¼ 0 but l xx a0). The immigrating ollen and seeds in the current generation contribute maladative alleles to ovules in the next generation. Ovule abortion often occurs, as imlied from the observations of the low ratios of seeds to ovules in many lants (Burd, 994; Melser and Klinkhamer, 00). The exact rimary sex ratio at steady state is l ¼ l ð þ l Þl xx ol, which is aroximated

7 X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) Table Sex ratios and migration loads under a variety of selection schemes in the resence of an unbiased sex ratio in migrants Selection Secondary sex ratio (l ) Primary sex ratio (l ) Migration load Pollen l ¼ m S l xy þ sv y þ m S l m ð þ l m Þ m S þ l xy sv y þ m S ð þ l m Þ l ¼ l ð þ l xyþ l l xy 4l l xy 6¼0 o (Y-chromosome not detrimental); 4, otherwise. l 4l Ovules Male (sorohyte) l ¼ m S þ l xx þ m S l m ð þ l m Þ m S þ l xx þ m S ð þ l m Þ ¼ l ¼ l l xx þ l xx ol l xx 6¼0 l ¼ a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 8bL xy, 4L xy l X0; a ¼ m S L xy þ m S ; þ l m b ¼ m S þ 3L xy þ m Sl m þ l m 4 l ¼ l L xy 6¼0 Female (sorohyte) l ¼ m S 3L xx þ m S l m ð þ l m Þ m S þ 3L xx þ m S ð þ l m Þ 4 l ¼ l L xx 6¼0 Pollen+male (sorohyte) l ¼ a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 8bL xy l ¼ ð þ l xyþl 4L xy ð l l xy Þ 4l l X0; a ¼ m S L xy þ l xy sv y þ m S ; þ l m b ¼ m S þ 3L xy l xy þ sv y þ m Sl m þ l m 4 l xy 6¼0, L xy 6¼0 Ovules+female (sorohye) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ b b 8aL xx a l X0; a ¼ m S þ 3L xx þ l xx þ m S ; þ l m b ¼ m S 3L xx þ l xx þ m Sl m þ l m o l ¼ l l xx þ l xx ol l xx 6¼0, L xx 6¼0 Pollen+ovules l ¼ m S þ l xx þ l xy sv y þ m S ð þ l m Þ l 4 þ l xy l xx l xx 6¼0, L xy 6¼0 m S þ l xx l xy þ sv y þ m S l m ð þ l m Þ þ l xx l xy 4 (Y-chromosome more detrimental than sex-linked genes); o, otherwise. l o þ l xy l xx þ l xx l xy Male+female (sorohyte) L xy l 3 al þ bl L xx ¼ 0 l ¼ l L xx 6¼0, L xy 6¼0 a ¼ m S þ 3L xx L xy þ m S ; þ l m b ¼ m S 3L xx þ 3L xy þ m Sl m þ l m 4 (male s selection stronger than female s); o, otherwise. l ¼ l Two-sex+two-stage Various sex ratios Various sex ratios All migration loads exist

8 554 ARTICLE IN PRESS X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) Sex ratio Secondary Primary Primary or secondary sex ratio Migration rate of ollen Migration rate of ollen 0. Sex ratio Fig. 3. Effects of the immigrating ollen when selection occurs only in the males at the sorohyte stage. Parameters are the migration rate of seeds fixed at m S ¼ 0.0, and the selection coefficient of allele A at the sorohyte stage sv y ¼ 0:0 and sv y ¼ 0:04, and all other selection coefficients ¼ 0. The freuencies of genotyes in the immigrating seeds are P ¼ 0:5, P ¼ 0:5, P ¼ 0:, Q y ¼ 0:3, and Q y ¼ 0:. The genotyic freuencies in the immigrating ollen are Q y ¼ 0:3 and Q y ¼ 0:..0 Secondary Primary Migration rate of seeds Fig.. Effects of the immigrating ollen and seeds on the rimary and secondary sex ratios: (a) ollen disersal and (b) seed disersal. In (a), the migration rate of seeds is fixed at m S ¼ 0.0, and the other arameters are the selection coefficient of allele A at the gametohyte stage (sv ) ¼ 0.04 in the ollen and all other selection coefficients ¼ 0. In (b), the immigration rate of ollen is fixed at m P ¼ 0.0, and the other arameters are the selection coefficient of allele A at the gametohyte stage (sv ) ¼ 0.04 in the ollen, the selection coefficient for the Y-chromosome s y ¼ 0:04, and all other selection coefficients ¼ 0. In each Figure, the freuencies of genotyes in the immigrating seeds are P ¼ 0:5, P ¼ 0:5, P ¼ 0:, Q y ¼ 0:3, and Q y ¼ 0:. The genotyic freuencies in the immigrating ollen are Q y ¼ 0:3 and Q y ¼ 0:. by E. (). The simlified exression for the secondary sex ratio at steady state is listed in Table. The secondary sex ratio is determined by the sex ratio in migrants (l m ). When l m ¼, the secondary sex ratio is unbiased (l ¼ ) but the rimary sex ratio is male-biased (Fig. 4). The load from the maladative genes in the ovules (l xx a0) affects the rimary sex ratio but does not affect the secondary sex ratio. This is because the secondary sex ratio is determined by the relative freuencies of the X- and Y- chromosomes in the male gametes that are fused with ovules to roduce offsing. Although the allele freuencies in the ovules are changed after selection, the secondary sex ratio can be obtained by substituting w ¼ / into E. (9), i.e. l ¼ Sex ratio Primary Secondary Migration rate of ollen Fig. 4. Effects of the immigrating ollen when selection is only in the females at the gametohyte stage, with the migration rate of seeds fixed at m S ¼ 0.0, the selection coefficient of allele A at the gametohyte stage su ¼ 0:04 and all other selection coefficients ¼ 0. The freuencies of genotyes in the immigrating seeds are P ¼ 0:5, P ¼ 0:5, P ¼ 0:, Q y ¼ 0:3, and Q y ¼ 0:. The genotyic freuencies in the immigrating ollen are Q y ¼ 0:3 and Q y ¼ 0:. in the resence of unbiased sex ratio in the immigrants. The exact values calculated from the general model are the same as the redicted values based on the aroximate exression in a range of l m. When selection is resent only in the females at the sorohyte stage, the reduction in oulation fitness is due to the maladative genes in females (l xy ¼ l xx ¼ L xy ¼ 0 but L xx a0). The immigrating seeds can directly affect the secondary sex ratio but the immigrating ollen can

9 X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) Primary or secondary sex ratio h=0.0 h=0.5 Sex ratio Primary Secondary Migration rate of ollen Migration rate of ollen Fig. 5. Effects of the immigrating ollen when selection exists at the sorohyte stage, with the migration rate of seeds fixed at m S ¼ 0.0, the selection coefficient of allele A at the sorohyte stage su ¼ 0:04 and all other selection coefficients ¼ 0. The freuencies of genotyes in the immigrating seeds are P ¼ 0:5, P ¼ 0:5, P ¼ 0:, Q y ¼ 0:3, and Q y ¼ 0:. The genotyic freuencies in the immigrating ollen are Q y ¼ 0:3 and Q y ¼ 0:. Fig. 6. Effects of the immigrating ollen when selection exists in the ollen and the males at the sorohyte stage, with the migration rate of seeds fixed at m S ¼ 0.0, the selection coefficients sv ¼ 0:0, sv y ¼ 003, sv y ¼ 0:03, sv y ¼ 0:04, and all other selection coefficients ¼ 0. The freuencies of genotyes in the migrating seeds are P ¼ 0:5, P ¼ 0:5, P ¼ 0:, Q y ¼ 0:3, and Q y ¼ 0:. The genotyic freuencies in the immigrating ollen are Q y ¼ 0:3 and Q y ¼ 0:. indirectly influence the secondary sex ratio through the union with the ovules. This can naturally occur for the females carrying maladative genes to retain a lower fertility. The rimary sex ratio is eual to the secondary sex ratios at steady state (l ¼ l ). It can be seen from the exression of the secondary sex ratio at steady state (Table ) that the secondary sex ratio is male-biased (l o) in the resence of migration loads in the females at the sorohyte stage. The exression from the general model is l ¼ ð m SÞ= þ m S =ð þ l m Þ w 3, () ð m S Þ= þ m S l m =ð þ l m Þ where w 3 ¼ su ðh þ Þ. If su a0 and l m ¼, then l o. When the degree of dominance decreases, the migration load increases and the male-biased rimary or secondary sex ratios are enhanced (Fig. 5). The secondary sex ratios redicted from aroximate euation are slightly lower than the exact values..3.. One-sex and two-stage selection Some genes can be exressed at either the gametohyte stage or the sorohyte stage, or both (Charlesworth and Charlesworth, 99). Many structural genes exress in the sorohytic ortion of angioserm life cycle and also exerience selection in the ollen (Tanksley et al., 98; Mulcahy et al., 996). When selection is resent only in the ollen and the males at the sorohyte stage, migration load is resent in the males but absent in the females (l xx ¼ L xx ¼ 0 but l xy a0 and L xy a0). The rimary sex ratio at steady state is greater than the secondary sex ratio, i.e. l 4l (Table ). The urging of maladative gametes in the ollen increases the relative freuency of the female gametes and augments a female-biased rimary sex ratio. Several factors can influence the secondary sex ratio. With an increase in the migration rates of ollen or seeds, the migration loads at both the gametohyte and the sorohyte stages increase, and so are the relative migration loads of l xy =L xy (results not shown here). The secondary sex ratio slightly decreases but the rimary sex ratio slightly increases with the migration rates of seeds and ollen (e.g., Fig. 6). A female-biased secondary sex ratio is brought about in the resence of selection in the males and an unbiased sex ratio in the migrants (l m ¼ ). When selection is resent only in the ovules and the females at the sorohyte stage, migration load is resent in the females but absent in the males (l xy ¼ L xy ¼ 0 but l xx a0 and L xx a0). The rimary sex ratio at steady state is smaller than the secondary sex ratio, i.e. l ol (Table ). The removal of ovules increases the relative freuency of male gametes and leads to the trend of a male-biased rimary sex ratio. Numerical results show that a male-biased secondary sex ratio is brought about when there is selection in the males and an unbiased sex ratio in the migrants (l m ¼ ). With an increase in the migration rates of seeds and ollen, the migration loads at both the gametohyte and the sorohyte stages increase but the secondary sex ratio decreases. A negative relationshi between l xx and L xx is seen with the change in the degree of dominance. The secondary sex ratio decreases when the degree of dominance decreases. The sex ratios redicted from the aroximate euation (Table ) are slightly lower than the exact values at steady state (results not shown here).

10 556 ARTICLE IN PRESS X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) Secondary Primary selection intensity is greater in the males than in the females, or a male-biased sex ratio is generated when selection intensity is greater in the females than in the males (results not shown here). Sex ratio Migration rate of seeds Fig. 7. Effects of the immigrating seeds when selection is resent at the gametohyte stage, with the migration rate of ollen m P ¼ 0:0, the selection coefficients su ¼ 0:0, sv ¼ 0:03, sv y ¼ 0:04, and all other selection coefficients ¼ 0. The freuencies of genotyes in the immigrating seeds are P ¼ 0:5, P ¼ 0:5, P ¼ 0:, Q y ¼ 0:3, and Q y ¼ 0:. The genotyic freuencies in the immigrating ollen are Q y ¼ 0:3, and Q y ¼ 0: Two-sex and one-stage selection When selection is resent only in the ovules and ollen, the migration load is absent at the sorohyte stage but resent at the gametohyte stage (L xx ¼ L xy ¼ 0 but l xx a0 and l xy a0). This likely occurs in natural oulations for the genes that are exressed only in the haloid sage. The rimary sex ratio aroximated by E. () can be male- or female-biased, deending uon the relative migration loads of l xx and l xy. In the aroximate euation for the secondary sex ratio at steady state (Table ), the difference between l xy and sv y at steady state is given by sv y l xy ¼ sð þ Þv y þ s ðv y v Þþs ðv y v Þ. When the locus on the Y-chromosome is detrimental, i.e. l xy osv y, the female-biased rimary and secondary sex ratios are roduced owing to a larger load in the ollen (e.g., Fig. 7), which otherwise may roduce a male-biased sex ratio. However, both the rimary and secondary sex ratios decrease with an increase in the migration rates of ollen and seeds. In the resence of selection only at the sorohyte stage, the migration load is absent at the gametohyte stage but resent at the sorohyte stage (l xx ¼ l xy ¼ 0 but L xx a0 and L xy a0). Differential seed germination and seedling mortality between the males and females in dioecious lants have been documented (Eley, 00). The malebiased secondary ratios are artly due to the different ostreroductive growth in many dioecious angioserms (Lloyd, 973). The rimary sex ratio is eual to the secondary sex ratio (l ¼ l ) since the relative freuencies of male and female gametes are not altered. The secondary sex ratio can be estimated from E. (8) with a ¼ m S þ 3L xx L xy þ m S =ð þ l m Þ and b ¼ m S 3L xx þ 3L xy þ m S l m =ð þ l m Þ. The results show that a femalebiased rimary or secondary sex ratio is generated when.3.4. Two-sex and two-stage selection In natural oulations, some sex-linked genes can exress in both the gametohyte and sorohyte stages (Tanksley et al., 98; Winsor et al., 000). The focus here is to study the joint effects of multile migration loads on the sex ratio on the basis of the general model described in Section.. Only the steady state results are resented here although the transient results are estimated to obtain the steady-state results. The effects of seeds and ollen flow remain the same as those in revious examles on shaing the sex ratio and migration loads (e.g., Fig. 8). The secondary sex ratios redicated from E. (8) are slightly Migration load Sex ratio lxx Lxx lxy Lxy Migration rate of seeds Secondary Primary Migration rate of seeds Fig. 8. Effects of the immigrating seeds when selection is resent at the gametohyte and sorohyte stages: (a) the migration loads and (b) the rimary and secondary sex ratios. The immigration rate of seeds is changed with the immigration rate of ollen fixed at m P ¼ 0:0. Other arameters are the selection coefficients su ¼ 0:0, su ¼ 0:0, sv y ¼ 0:03, sv y ¼ 0:04, sv y ¼ 0:03, and sv ¼ 0:0. The freuencies of genotyes in the immigrating seeds are P ¼ 0:5,P ¼ 0:5, P ¼ 0:, Q y ¼ 0:3, and Q y ¼ 0:. The genotyic freuencies in the immigrating ollen are Q y ¼ 0:3, and Q y ¼ 0:

11 X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) larger than the exact values at steady state. An array of rimary and secondary sex ratios can occur under different conditions. 3. Discussion This study has demonstrated how the maladative genes from immigrating seeds and ollen influence the rimary and secondary sex ratios in a dioecious lant. The change in the rimary and secondary sex ratios can be substantial even when the immigration rates are small. The migration load at the gametohyte stage and/or the sorohyte stage increases with an increasing immigration rate of seeds and ollen, which in turn roduces an array of rimary and secondary sex ratios. The resence of maladative sexlinked genes in the males can enhance the outcome of a female-biased secondary sex ratio, while the resence of maladative sex-linked genes in the females or ollen can lead to a male-biased secondary sex ratio. The detrimental effect from the Y-chromosome can generate a femalebiased rimary or secondary sex ratio. These results highlight the significance of an alternative aroach to the existing sex-ratio theories in interreting the formation of biased sex ratios in a artially isolated oulation. It is significant to understand that maladative genes can be raidly eliminated when the focal oulation is comletely isolated from the other oulations excet under stabilizing selection or when other evolutionary forces counteract the directional selection. Maintenance of maladative genes has been examined in theories for oulation genetics (Wright, 969) and a secies s range (Kirkatrick and Barton, 997; Hu and He, 006). Maladative genes should be freuent in lant oulations due to the influences from immigrating ollen and seeds in comarison with the effects of deleterious mutation (Mulcahy et al., 996). In our numerical examles under weak selection (Figs. 8), the migration loads at a single locus are about 0. %. The cumulative migration load could be substantial when there are multile maladative loci from immigrating seeds and ollen. The migration load-driven mechanism could be imortant for interreting biased sex ratios in lant secies with different sex systems, although the resent study only addresses the lant secies with sexuals determined by the sexual chromosomes. The selection coefficient at the gametohyte stage refers to the relative intensity of selection before the fusion of ollen with ovules, not at a secific time. The selection coefficient at the sorohyte stage refers to the selection intensity from zygote formation to the matured adults. The variation in viability from seed germination to seedling and to adults at the srorohyte stage is not searately considered (Eley, 00). Note that the resence of migration load is deendent on the difference between the freuencies of maladative genes in the migrants and reciient oulation. This result is related to that addressed by Wright (969, ) for the joint effects of immigration rate and selection excet that the immigration here refers to both seed and ollen disersal. One caveat is that we have not considered the effects of emigration. In the model of multile oulations such as a finite island model, the exchange of migrants among oulations can reduce the variation of gene freuencies in the migrants and reciient oulations. The conseuence is that the migration load is reduced and the sex ratio is altered. In comarison with revious non-genetics-based theories of dioecious lants (Bulmer and Taylor, 980; Charnov, 98; De Jong et al., 00), our genetics-based theory accentuates a different biological mechanism. The existing sex-ratio theories for lants are concetually related to the model of Clark s LRC concet that is originally develoed for animal secies. Sib mating and LRC are viewed as two imortant comonents in the early sex-ratio theories for dioecious lants (De Jong et al., 00) and the atterns of seeds and ollen disersal can roduce biased secondary sex ratios. The resent study does not address the effects of relative distances of seed and ollen disersal, but emhasizes the relative rates of immigrating seeds and ollen in a focal oulation. This is fundamentally a different model. Maladative genes from immigrating seeds and ollen are imortant comonents for roducing a biased rimary/secondary sex ratio although a small roortion of sib mating occurs due to the assumtion of a random combination between ollen and ovules. The LRC model is actually related to the two-stage of selection intensities. A larger selection coefficient is euivalent to a smaller gain of natural resources for the maladative gene carriers. In comarison with the exlanations of biased secondary sex ratios from the ersective of life history traits (Delh, 999), their distinction from the resent theory is evident. The effects from immigrating seeds and ollen are not taken into account in the early exlanations although lant oulations are freuently affected by migrating seeds and ollen that may carry maladative genes. Maladative genes are maintained by the joint effects of immigration and natural selection. One joint exlanation between the resent and the life history traits-associated theories is that the difference in first reroduction (the biased rimary sex ratio) can bring about a biased secondary sex ratio, which can be exlicitly inferred from E. (). This study assumes that mutation rate is very small and negligible comared with the immigration rates of seed and ollen. This is lausible for the oulations with a short history because lants are more influenced by migrants than by deleterious mutants. In the oulations with a long history, however, mutation load is likely an imortant comonent in reducing oulation fitness (Charlesworth and Charlesworth, 99). The mutation load at a single locus is about m where m is the mutation rate; the mutation load over the whole genome (without eistasis) is about exð UÞ where U is the sum of total deleterious mutation over the whole genome (Wright, 977). If U ranges from 0. to.0 (Lynch and Walsh, 998), the

12 558 ARTICLE IN PRESS X.-S. Hu et al. / Theoretical Poulation Biology 7 (007) mutation load can be substantial, at %. Under such a condition, the effects of migration and mutation loads would likely coexist. The concets resented here is for individual genes where the interaction among multile maladative genes is excluded. As imlied from Hu (006), the rimary and secondary sex ratios may be altered by linkage diseuilibria since immigration can generate linkage diseuilibrium. Another kind of interaction among multile genes is the ossible effects of sex ratio modifiers that may restore the rimary and/or secondary sex ratios to euality (Feldman and Otto, 989). Under this situation, maintenance of unbiased sex ratios becomes more comlicated and the theory that addresses the joint effects of migration load and sex-ratio modifiers is needed. The resent study only addresses the case where sexual dimorhisms are determined by the sexual chromosomes (Ainsworth, 000; references therein). In lants where sexual is controlled by the autosomal genes, sex determination becomes comlicated and multile loci are often involved (Grant, 999). In such cases, the relationshi between migration load and sex ratio should be assessed. Numerical alications can be obtained from the theoretical redications concerning the relationshis between migration loads and sex ratios. One is to exlain the formation of a range of secondary sex ratios in a artially isolated oulation. In dioecious lants, about 9% of the surveyed secies have eual numbers of males and females and 57% have male-biased secondary sex ratios (Lloyd, 973; Delh, 999). A few dioecious lants exhibit femalebiased secondary sex ratios, such as the secies Rumex nivalis (Stehlik and Barrett, 005). Several biological mechanisms were roosed to interret the underlying rocesses (Lloyd, 973; Delh, 999; Stehlik and Barrett, 005). An eventual secondary sex ratio can be modified by life history occurrences that cause differential sexual mortality. In reference to the concet resented in this study, these occurrences may occur in the stage of the gametohyte or sorohyte, or in both. With the maladative genes from immigrating ollen, the gametohytic selection through differential growth in the ollen tube or ovule abortion can distort the secondary sex ratio. The rimary sex ratio (rior to fertilization) is an imortant reason to cause the secondary sex-ratio distortion (Stehlik and Barrett, 005), as demonstrated in the resent theory. Evidence suorting the secondary sexratio distortion from ostzygote is widely recorded in the literature. The reduction in oulation fitness was reorted owing to the immigrating seeds and ollen, such as in Imomosis aggregate and Imatiens caensis (Burt, 995). Our results also exlicitly show that the differential sexual mortality from the migrating seeds can be an imortant reason for a artially isolated oulation. The second alication is to interret the homogeneity or heterogeneity in the rimary/ secondary sex ratio among oulations that are maintained by the mechanism of gene migration-natural selection. Both the rimary and secondary sex ratios are the function of immigration rates of seeds and ollen and the selection coefficients. The ecological factors that cause the differences in migration rates of seeds and ollen among local oulations can result in the variation of oulation sex ratios in sace, such as the shifting of insect ollination to wind ollination (Sakai and Weller, 999). When there is heterogeneity among oulations in urging the maladative genes from the immigrating ollen, differential migration loads at the gametohyte stage exist and the satial variation in the rimary/ secondary sex ratio evolves. Our results imly that differences in the viability of microgametohytes and megagametohytes among natural oulations can confirm our redictions although emirical evidence is rare at the oulation level. Evidence is also infreuently recorded to show the satial variation of oulation sex ratios that is attributable to the differences in removing maladative genes from the immigrating seeds, contrast to the case of SSS occurring within oulations (Bierzychudek and Eckhart, 988; Iglesias and Bell, 989; Korelainen, 99; Eley, 00; references therein). Thus, emirical studies that examine the satial migration loads in the males and females would be of articular significance for insights into the satial attern of sex ratio at the oulation level. The third alication of our concet is to interret the detrimental effects of the Y-chromosome on changing the rimary and secondary sex ratios. Since the Y-chromosome does not recombine with its counterart X-chromosome, the detrimental mutations often accumulate and this in turn increases male mortality (Fllatov et al., 000). Furthermore, Y-chromosome may have detrimental effects on the X chromosome-linked genes (Charlesworth, 00), which may take lace in the males at the sorohyte stage. The resent study rovides a choice that is comliant to studying different sets of occurrences in setting different selection coefficients relevant to the Y-chromosome at each stage of the life cycle. Our results indicate that the Y-chromosome can affect the viability of ollen or the viability of males at the sorohyte stage, or both. In each case, the resence of detrimental Y-chromosome from immigrating seeds and ollen can enhance the formation of female-biased rimary and secondary sex ratios. Acknowledgments We areciate Marcus W. Feldman and two anonymous reviewers for useful suggestions and constructive comments that substantially imroved the earlier version of this aer. References Ainsworth, C., 000. Boys and girls come out to lay: the molecular biology of dioecious lants. Ann. Bot. 86,. Antolin, M.F., 993. Genetics of biased sex ratios in subdivided oulations: models, assumtions, and evidence. Oxford Surveys in Evol. Biol. 9, 39 8.