Suppression of Sex-Ratio Meiotic Drive and the Maintenance of Y-Chromosome Polymorphism in
Drosophila
Author(s): John Jaenike
Source: Evolution, Vol. 53, No. 1 (Feb., 1999), pp. 164-174
Published by: Society for the Study of Evolution
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Evolution, 53(1), 1999, pp. 164-174
SUPPRESSION OF SEX-RATIO MEIOTIC DRIVE AND THE MAINTENANCE OF
Y-CHROMOSOME POLYMORPHISM IN DROSOPHILA
JOHN JAENIKE
Departmentof Biology, University
of Rochester,Rochester,New York14627
Abstract.-Like several otherspecies of Drosophila,D. quinariais polymorphic
forX-chromosomemeioticdrive;
matingsinvolvingmales thatcarrya "sex-ratio"X chromosome(XSR) resultin the productionof stronglyfemalebiased offspring
sex ratios(Jaenike1996). A surveyof isofemalelinesofD. quinariafromseveralpopulationsreveals
thatthereis geneticvariationforpartialsuppressionof thismeioticdrive.Crossingexperiments
show thatthereis
Y-linked,and probablyautosomal,variationforsuppressionof drive.Y-linkedsuppressorsof X-chromosomedrive
have nowbeendescribedin severalspeciesofDiptera.I developa simplemodelforthemaintenance
ofY-chromosome
in species polymorphic
polymorphism
forX-linkedmeioticdrive.One interesting
featureof this model is that,if
thereis a stableY-chromosome
thentheequilibriumfrequencyof thestandardand sex-ratioX chropolymorphism,
X chromosomes
mosomesis determined
solelyby Y-chromosome
parameters,
notby thefitnesseffectsofthedifferent
on theircarriers.This model suggeststhatY-chromosome
polymorphism
maybe easier to maintainthanpreviously
andI hypothesize
thatkaryotypic
variationin Y chromosomes
will be foundto be associatedwithsuppression
thought,
of sex-ratiomeioticdrivein otherspecies of Drosophila.
Key words.-Drosophila quinaria,geneticarmsrace, populationgenetics,segregationdistortion,
selfishgenes, Xchromosomepolymorphism,
Y-chromosome
polymorphism.
Received March 13, 1998. AcceptedSeptember10, 1998.
Adaptation to the externalenvironmentis most effectively
achieved if the differentalleles at a locus compete on a level
playing field, specifically, if their representationin future
generations is determinedsolely by theireffectson the survival and fertilityof their carriers. However, some genes,
termed "selfish genetic elements," can also spread through
a variety of nonadaptive mechanisms, even if they cause a
decrease in the fitnessof theircarriers (Werren et al. 1988).
One notable class of selfish genes are those that cause segregation distortion,which are passed to more than half of the
functional gametes of a heterozygous individual (Lyttle
1991).
Given that an allele can gain a fitnessadvantage through
such segregation distortion,it is remarkable that most loci
exhibit Mendelian segregation. To explain the general fairness of meiosis, Crow (1991) reviews theoreticalstudies that
show that alleles enhancing segregation distortion will be
favored only if they are linked to the drivinglocus, whereas
those that depress the level of distortionwill be favored if
they are linked to the targetof the drive locus. Most importantly,unlinked modifieralleles will be favored only if they
bringabout a reductionin the drive. Thus, only a small fraction of the genome is selected to increase drive, whereas the
majorityis selected to decrease it.
Nevertheless,the "parliament of genes," as Leigh (1977)
puts it, occasionally fails to suppress segregation distortion.
Most notably, it has recentlybecome apparent that X-chromosome segregationdistortion(or sex-ratio meiotic drive) is
much more common than previously believed. The "sexratio" phenomenon is characterized by the production of
stronglyfemale-biased progeny in matings involving males
thatcarry a particulartype of X chromosome (denoted XSR).
Such drive has now been discovered in six species groups of
Drosophila flies obscurea, melanica, tripunctata,testacea,
quinaria, and melanogaster; see Cazemajor et al. 1997), as
well as various otherinsects (e.g., stalk-eyedflies,Presgraves
et al. 1997). I suspect thatthe known distributionof sex-ratio
meiotic drive is more a functionof what species have been
studied in laboratory breeding experiments than its actual
distributionacross species in the wild. Lyttle (1991) and
Hurst and Pomiankowski (1991) have argued that X-chromosome drive may be more prevalent than autosomal drive.
The existence of X-chromosome meiotic drive in a species
indicates that there must be strong intragenomic conflict,
which could result in the evolution of Y-linked and/or autosomal suppressorsof drive. Because "sex-ratio" males sire
few if any sons, the Y chromosome is obviously affectedby
this meiotic drive. As a result, Y-linked suppressors of sexratio drive will be selectively favored.Autosomal suppressors
will be favored for two reasons. First, sex-ratio drive will
result in the production of an excess of females in the population (Bryant et al. 1982), thus favoring any autosomal
allele thatleads to the productionof more sons (Fisher 1930).
Second, if the frequency of XSR is at a stable equilibrium,
thentheremustalmost certainlybe a balance between meiotic
drive and natural selection acting at the individual level to
hold the driving alleles in check. Therefore,carriersof XSR
will have lower fitness,on average, than individuals carrying
the standard, nondriving X chromosome (Edwards 1961;
Curtsingerand Feldman 1980). Autosomal alleles in males
with no drive suppression are associated with XSR in all of
the resultingoffspring,whereas the association between XSR
and autosomal loci is reduced in individuals carryingalleles
thatsuppress drive. Thus, autosomal alleles thatcan suppress
drive will benefitby being associated with higher fitnessindividuals, on average, than alleles thatdo not suppress drive.
For many decades, studies of the population genetics of
sex-ratio meiotic drive focused on Drosophila pseudoobscura. Yet, despite extensive searches, no Y-linked nor autosomal suppressors of drive have been discovered (Beckenbach et al. 1982). However, recent studies of other species
harboring sex-ratio polymorphisms suggest that D. pseudoobscura may be exceptional in lacking suppressors of
drive. Autosomal and/or Y-linked suppressors of sex-ratio
164
(? 1999 The Society forthe Study of Evolution. All rightsreserved.
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SUPPRESSION OF SEX-RATIO MEIOTIC DRIVE
meiotic drive have been reportedin several species of insects,
including Drosophila paramelanica (Stalker 1961), D. nzediopunctata (Carvalho and Klaczko 1993, 1994; Carvalho et
al. 1997), D. simulans (Mercot et al. 1995; Atlan et al. 1997;
Cazemajor et al. 1997), D. subobscura (Hauschteck-Jungen
1990), and the stalk-eyed fly Cyrtodiopsis dalnmanni(Presgraves et al. 1997; Wilkinson et al. 1998).
Sex-ratio meiotic drive has been found in two species of
the Drosophila quinaria group, D. quinaria and D. recens
(Jaenike 1996). Mitochondrial sequence data suggest that
these two species diverged about 1.5 million years ago (Spicer and Jaenike 1996). If the sex-ratio condition was present
in the common ancestor of these two species, this would
indicate thattherehas been considerable time forsuppressors
of drive to evolve.
In this paper, I reportevidence forY-linked and autosomal
suppressorsof sex-ratiomeiotic drive in Drosophila quinaria.
A survey of isofemale strainsindicates the existence of both
suppressor and nonsuppressor types within populations.
Based on observations of Y-chromosome polymorphismin
D. quinaria and other species harboring sex-ratio meiotic
drive, I explore theoreticallythe conditions underwhich such
Y-chromosome polymorphismscan be maintained in natural
populations and the effects of such Y-chromosome polymorphismson the dynamics of the X-chromosome typeswith
which they interact.
MATERIALS
AND METHODS
Strains
The " sex-ratio" strainof Drosophila quinaria used in these
experiments was derived from a single wild-caught female
collected near Rochester, New York, in 1994. This female
produced only female offspringwhen placed in culture, indicating that it had mated with a sex-ratio male in nature.
The "sex ratio" X chromosome (XSR) was isolated and has
been maintained in the laboratoryby crossing males froma
standard (non-"sex-ratio") strainto XSRXSRfemales, yielding XSRY sons. These XSRY males are crossed to XSRXSR
females to yield XSRXSRdaughters.In this "sex-ratio" strain
there is no selection in favor of Y-linked or autosomal suppressors of sex-ratio meiotic drive, because only the daughters of XSRY by XSRXSR crosses are retained. Because XSR
is rare in natural populations of D. quinaria (PSR = 0.03 in
Rochester; Jaenike 1996), it is likely thatthisisofemale strain
carried only a single XSR chromosome.
Other strainsof D. quinaria were collected by sweep netting over cucumber baits set out in skunk cabbage patches
in the summer of 1995 at Rochester, New York; Deer Isle,
Maine; and Lineville, Pennsylvania. Flies were kept as isofemale strains in the laboratory on Instant Drosophila Medium (Carolina Biological Supply, Burlington,NC) plus cucumber.Sixteen isofemale strainswere established fromNew
York populations, 17 fromMaine, and 28 fromPennsylvania.
Strain Variationfor Suppression of Sex-Ratio Meiotic Drive
The initial screen for suppressors of sex-ratio drive involved crossing one male fromeach isofemale strainto one
XSRXSRfemale. Such crosses would yield F1 males thatcarry
165
the XSR chromosome fromthe laboratory "sex-ratio" strain,
a Y chromosome from the tested isofemale strain, and autosomes from both strains. Three of these XSRY sons from
each strain were then crossed individually to virginXSRXST
females (where XST is the standard,nondrivingX), and the
numbers of sons and daughters among the offspringwere
determined.This survey of strainsindicated whetherstrains
varied in the degree to which theysuppressed sex-ratiodrive
and whetherthe average degree of suppression varied among
populations in differentregions (states).
Genetic Variationfor Suppression
To determine whetherXSR males siring substantial numbers of male offspringcarried Y-linked or autosomal suppressors of drive, I chose several isofemale lines in which
meiotic drive was incomplete. For each strain,10 single-pair
crosses were made between males of the tested strain and
XSRXSR females fromthe laboratory "sex-ratio" strain.For
each of these 10 crosses, three of the resultingF1 males, all
of which carry the XSR chromosome, were crossed individually to virginXSTXST females to assay the level of sex-ratio
drive. Significant differences among strains in the F2 offspring sex ratios would be due to Y-linked or autosomal
suppressors of drive,ratherthan maternaleffects,because all
of the tested XSRY males are the sons of stock XSRXSR females. The use of 10 independentcrosses per strainensures
that any environmentaleffects on sex-ratio drive would be
also be randomly associated with the differentisofemale
strains.
Once it had been established thatisofemale strainsdiffered
in the degree to which they could suppress sex-ratio meiotic
drive, I triedto determinewhetherthis suppression is due to
Y-linked or autosomal factors. In April 1996, six isofemale
strains that had shown evidence of drive suppression were
crossed reciprocally to a laboratorystrainof D. quinaria that
carries only the nondriving,standard XST chromosome. The
resulting F1 males would be identical, on average, with respect to theircomplementof autosomes, but theywould differ
in the source of their X and Y chromosomes. Specifically,
the Y chromosomes of these F1 males would be derived from
the strain of the male used in the parental cross. These F1
males obtained fromthe reciprocal crosses were then mated
to XSRXSRfemales to yield F2 XSR-carryingsons thatdiffered
in the source of theirY chromosome,but which were similar,
on average, with respect to autosomal loci. The F2 males
were then crossed to XSTXST females and the offspringsex
ratios were determined.Any consistent differencein the F3
sex ratios between males obtained from the two reciprocal
crosses for a given tested strainwould indicate the existence
of Y-linked suppressors of sex-ratio meiotic drive.
In November 1996, fourof these six isofemale strainswere
tested furtherto look for autosomal as well as Y-linked suppressors of drive. As before, each strain was crossed reciprocally to a laboratory strainof D. quinaria thatcarries only
the nondrivingXST chromosome. Because the only consistent
differencebetween the reciprocal crosses is the source of the
Y chromosome in the F2 males used in the test of drive
suppressors, a significantY-source effectis indicative of Ylinked suppressors of drive.
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166
TABLE
JOHN JAENIKE
1. Analysisof varianceof offspring
sex ratiosamongall
18
isofemale strains in offspringsex ratio of XSR males (PROC GLM;
SAS Institute1994).
Source
df
16
(0
C
SS
MS
F (denominator)
P
2 0.815 0.407 0.23 (strain[state]) 0.80
State
Strain(state) 32 56.64 1.77 4.11 (error)
0.0001
Error
58 24.99
14
12
10
L
8
E6
Z 4
Each reciprocal parental cross was a single-pair mating
and was replicated threetimes foreach straintested.Variation
among the F2 males in theiroffspringsex ratios among replicate crosses (but withina specificcross directionfora given
strain) is referredto as cross (Y source) in the analysis of
variance. Significantcross variation would reflectthe existence of genetic variation in drive suppression eitherwithin
the isofemale strainor the stock strainof D. quinaria.
From each replicate single-pair cross (within cross direction), I obtained three F1 males, which were mated individually to XSRXSR females to yield F2 males carryingthe XSR
chromosome. Variation among the F1 males obtained froma
single replicate cross in the offspringsex ratios of theirF2
sons is referredto as subcross (cross) variationin the analysis
of variance. The F2 sons produced were then mated to virgin
XSTXST females and the offspringsex ratios scored. Significant subcross (within cross) variation would indicate the
existence of autosomal variation in drive suppression either
withinthe isofemale strainor the stock strainof D. quinaria.
This is because the Y chromosome is identical in all males
descended froma given parental cross. Finally, variation in
the offspringsex ratio among the differentF2 sons within a
single subcross could indicate segregation of the strain and
stock alleles at autosomal suppressor loci in the F1 males
produced in the parental crosses. Because the original matings involved a single pair of flies, variation in the Y chromosome could not contributeto subcross (within cross) variation. (See Appendix 1 for crossing scheme.)
All statistical analyses of sex ratio utilized PROC GLM
(SAS Institute1994), withtheproportionof males siredtransformed to arcsinV/(numberof males + %/8)/(total
number of
offspring+ ?/4)(Sokal and Rohlf 1995). Each progeny sex
ratio was weighted by the total number of offspring(SAS
Institute1982).
Fertilityof Sons of XSRMales and Source of Their
X Chromosome
From the crosses outlined in the previous section, numerous sons of XSR males were obtained. Thirtyof these were
crossed individually to single virgin XSTXST females to determinewhetherthey could sire viable progeny.
The second set of crosses described in the previous section
used female XSRXSR flies that carried a recessive eye-color
mutation (brick) that exhibits complete linkage to the factor(s) causing sex-ratio meiotic drive. Thus, the tests of offspring sex ratio involved crosses between XSRbricky males
and XSTXST females. The eye color in any sons produced in
these crosses will indicate the source of theirX chromosome:
a son withwild-typeeye color inheritedan X fromits mother,
while one with brick-colored eyes got its X fromits father.
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Offspring
sex ratio
Variationamongisofemalestrainsof Drosophilaquinaria
in offspring
sex ratios of XSR males. Each male carriedan XSR
chromosomefromthesame stock"sex-ratio"strainand a Y chromosomefroma wild-typeisofemalestrain.The autosomeswere
50% fromthe laboratorystock "sex-ratio" strainand 50% from
thetestedisofemalestrain.
FIG. 1.
The wild-type and brick sons of XSR males were dissected
at one week of age and scored for sperm motilityto gauge
theirpotential fertility.
RESULTS
Strain Variationfor Suppression of Sex-Ratio Meiotic Drive
The "sex-ratio" males obtained by crossing males of the
differentisofemale strainsto XSRXSRfemales exhibited substantialvariationin theiroffspringsex ratios. The differences
among strains within regions were highly significant(P <
0.0001; Table 1). For the majorityof strains,fewerthan 10%
of the offspringof these XSR males were male. However, in
nine of 61 strains,XSR males sired > 20% male offspring,
and in one strainthey sired approximatelyequal numbersof
sons and daughters (Fig. 1).
The mean fractionof sons was similarin strainsoriginating
fromthe differentareas: 0.15 forPennsylvania, 0.1 1 forNew
York, and 0.14 for Maine. These differencesare not significant (P = 0.39; Table 1). It is also worth noting that the
frequencies of XSR are similar in these three areas (3% in
Rochester, and 6% in both Deer Isle, ME, and Linesville,
PA; Jaenike 1996). Therefore,thereappears to be local, rather
than regional variation for suppression of sex-ratio meiotic
drive.
Genetic Variationfor Suppression
A subset of strainsthathad exhibited apparentsuppression
of drive was used to determineif there are consistent interstraindifferences.The strainstestedincluded one fromPennsylvania, two from New York, and three from Maine. Examining all of the strains simultaneously showed a highly
significantstrain effect,as well as significanteffectof replicate cross within strains (Table 2). Recall that three sons
from each of the 10 replicate crosses within strains were
tested, allowing a test of the effectof replicate cross within
strain.The significantstraindifferencesconfirmthatthereis
genetic variation among them in the degree of drive suppression. The significantcross-within-straineffectsuggests
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167
SUPPRESSION OF SEX-RATIO MEIOTIC DRIVE
TABLE 2.
Analysis of variance of offspringsex ratios among six isofemale strains in offspringsex ratio of XSR males (PROC GLM;
SAS Institute1994).
Source
df
Strain
Cross (strain)
113.09
70
81.01
37
Error
MS
F (denominator)
P
22.62
11.11 (cross[strain])
0.0001
SS
5
75.35
2.04
that individual strains may be genetically variable for drive
suppression. Using data fromonly those males siring 20 or
more offspring,the mean fractionof sons sired by XSR males
ranged from0.10 ? 0.016 (n = 16 progeny sex ratios of XSR
males) to 0.33 ? 0.03 (n = 17) among the six strainstested.
Reciprocal crosses between selected isofemale strainsand
the stock cultureof XST-bearingD. quinaria yielded F1 males
that were then mated to XSRXSR females. The resulting F2
males, all of which were XSR, were similar, on average, in
theircomplement of autosomes, but differedin the source of
theirY chromosomes. Four isofemale strains were tested in
two separate experiments,whereas two were testedonly once.
Table 3 shows the mean fractionof sons among the offspring
of the tested XSR males. The table also includes the probabilityassociated with a testof a Y-chromosome effectin each
experiment,obtained from an analysis of variance for each
strain,with offspringsex ratio examined as a functionof the
source of the Y chromosome (PROC GLM, SAS Institute
1994).
For strains that were tested in both experiments,Fisher's
combined probabilityis presented (Sokal and Rohlf 1995).
For two of the strains that were tested twice (LM29 from
New York and DPT16 from Maine), XSR males with a Y
obtained fromthese strainssired significantlymore male offspring than genetically equivalent males with a stock Y in
one experiment,but not in the other,although in the nonsignificantcases, the males with the strain Y sired more sons
than did those with the stock Y. The combined probabilities
were significantfor both of these strains (Table 3). Of the
two strainsthatwere testedonce, DPT-28 fromMaine showed
a significanteffectof the Y.
Another way to look at these data is to examine the differencein sex ratio as a functionof the Y chromosome,using
each independenttest of a strain as a single observation. A
Wilcoxon signed ranks test indicates that,in the aggregate,
1.76 (error)
0.021
XSR flies bearing a Y from the tested strains (which were
selected for testingbecause of suppression of meiotic drive
found in previous experiments)sired significantlymore sons
than did the XSR males carryinga Y fromthe stock culture
(P = 0.009, two-tailed test). These results indicate that at
least some of the suppression of sex-ratio drive is due to the
Y chromosome, although the effect is not statistically apparent(and may notbe present)in several of thetestedstrains.
Variation among replicate crosses withina particularcross
direction for a given straincould be due to eitherautosomal
or Y-chromosome variation within eitherthe tested strainor
the XST stock to which is was crossed. However, significant
variation among subcrosses (with replicate crosses) cannot
be due to the Y, because all of the F2 males tested were
descended froma single male and must,therefore,carryidentical Y chromosomes. Thus, significantvariation among subcrosses within a cross is most likely due to autosomal variation for drive suppression. Examples of subcross (within
cross) variation are presentedin Table 4. The males on which
these data are based were obtained from crosses between
males of strain LM29 and stock females, so that all of the
males within a cross carry identical Y chromosomes from
strain LM29. The differences among subcrosses (within
crosses) are significantfor two of the threestrainsforwhich
adequate replication allowed a test of this effect (Table 5).
This result indicates that autosomal loci can also contribute
to suppression of sex-ratio meiotic drive.
Finally, apparent segregation of autosomal factors can be
seen among the tested XSR males that all carry identical Y
chromosomes. This is shown for one of the subcrosses involving strainLM29, in which the tested males carry the Y
chromosome from this strain (Fig. 2). The data shown in
Figure 2 suggest that the degree to which a Y chromosome
is capable of suppressing sex-ratio meiotic drive depends on
interactions with autosomal loci. For all crosses involving
Effect of Y chromosome on offspringsex ratio of XSR males. In these crosses, the autosomal backgrounds were similar in
males from the same strain in each experiment.
TABLE 3.
Strain (origin)
DPT16 (ME)
DPT28 (ME)
LM15 (NY)
LM29 (NY)
PY40 (PA)
WP42 (ME)
Date of
experiment
April 96
Nov. 96
April 96
April 96
April 96
Nov. 96
April 96
Nov. 96
April 96
Nov. 96
* Combined probability: x2 =
-2E1n(P)
Fraction male offspring
Difference
Difference________Combi____ed_
Stock Y
Strain Y
strainY - stock Y
0.236
0.070
0.110
0.049
0.285
0.135
0.117
0.090
0.079
0.077
0.062
0.051
0.016
0.041
0.097
0.071
0.086
0.050
0.058
0.088
0.174
0.019
0.094
0.008
0.188
0.064
0.031
0.04
0.021
-0.011
(Sokal and Rohlf 1995).
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Combined
Pr
0.006
0.23
0.034
0.88
0.002
0.068
0.49
0.25
0.063
0.54
Probabilityt
0.012
0.002
0.4
0.15
168
JOHN JAENIKE
4. Variationin offspring
sex ratiosamongsubcrossesinvolving strainLM29. All males withina given subcrosscarried
identicalY chromosomes,so thatvariationin offspring
sex ratio
is due to autosomalor nongeneticfactors.Only males siringmore
than20 offspring
are included.
5-
TABLE
Cross
Subcross
Total no.
offspring
3
1
2
1
2
142
309
234
307
4
Fractionsons
(mean + SE)
No. males
tested
0.13 ? 0.05
0.28 + 0.07
0.04 + 0.01
0.22
2 0.02
4
5
5
6
V) 4
a)
- 3
0
-oG)2
E
z
0
0
0.05
0.1
0.15
0.2
0.25
0.3
strainLM29 (Fig. 3), it is apparent thatXSR males carrying
sex ratio
Offspring
the LM29 Y chromosome exhibit considerable variabilityin
the degree to which sex-ratiomeiotic is suppressed, although FIG. 2. Offspring
sex ratiosin strainLM29 subcrossA4. All males
on average these males suppress drive more than otherwise carriedan XSRchromosomefromthe stock"sex-ratio" strainand
genetically similar males carryingthe stock Y chromosome. thesame Y chromosomefromisofemalestrainLM29. Anygenetic
variationin offspringsex ratio is due to variationin autosomal
suppressors.
Fertilityof Sons of SR Males and Source of Their
X Chromosome
degree to which sex-ratio meiotic drive is expressed in XSR
Of the 30 sons of XSR males thatwere crossed individually
males. If one assumes that XSR males siring more than 10%
to single virginfemales, 21 produced viable offspring,which
male offspringcarryeitherY-linked or autosomal suppressors
indicates thatmost of these males are fertile.Crosses between
of sex-ratiomeiotic drive,thenabout one-halfof all isofemale
XSTXST females that were homozygous for wild-type eye
strainscarrysuch suppressors(see Fig. 1). Reciprocal crosses
color and XSR males thatcarried the recessive X-linked eyebetween several isofemale strains and a stock laboratory
color mutation brick yielded a total of 9878 offspring,instrain show that the Y chromosome can contributeto supcluding 847 males with wild-typeeye color and seven males
pression of meioticdrive (Table 3; Fig. 3). Variation among
withbrickeye color. The numbersof sons of XSR males with
flies carryingidentical Y chromosomes indicates that automotile versus nonmotile sperm as a functionof eye color are
somal loci can also contributeto drive suppression (Table 5;
presentedin Table 6. The sterilityof all brick males, which
Figs. 2, 3). If the strengthof suppression depends on a Yobtained theirX chromosome fromtheirXSR father,indicates
autosome interaction,then changes in allele frequencies at
thattheywere probably XSRO. The frequencyof such males
autosomal loci withinstrainscould account for the betweenamong all of the offspringin these crosses indicates thatthe
experimentdifferencesin the magnitude of the Y effectin
rate of X-chromosome nondisjunction in females of D. quistrainsLM29 and DPT16 (Table 3), as the second experiment
naria is about 0.1%. These data show that less than 1% of
was conducted six monthsafterthe first.Alternatively,these
the sons of XSR males resultfromfemale nondisjunction.The
strains may have initially been polymorphic for Y-chromotests of male sperm motilityand male offspringproduction
some types, with selection reducing the frequency of the
demonstratethatmost sons of XSR males are fertileand obtain
suppressor Y chromosomes during laboratory maintenance.
an X chromosome from their mother and a Y chromosome
Because these strains were monomorphic for XST chromofromtheirfather.
somes, selection against suppressor Y chromosomes would
not be unexpected (see below).
DISCUSSION
It should be noted that I used only one XSR chromosome
These results show that isofemale strains of Drosophila fromnaturalpopulations of D. quinaria in the testsof meiotic
quinaria exhibit substantial and significantvariation in the drive suppression and thereforehave no informationabout
5. Analysisof varianceof offspring
sex ratiosamongsubcrossesinvolvingvariousisofemalestrains(PROC GLM; SAS Institute
1994). Significant
subcrossvariationindicatestheexistenceof autosomalsuppressors
of sex-ratiomeioticdrive,because all maleswithin
a givensubcrosscarriedidenticalY chromosomes.The Y sourceindicatestheeffectof Y-linkedsuppressorsof meioticdrive.
TABLE
Strain
LM29
PY40
DPT16
Source
Y source
Cross(Y source)
Subcross(cross)
Y source
Cross(Y source)
Subcross(cross)
Y source
Cross(Y source)
Subcross(cross)
df
SS
1
6
9
1
1
6
1
2
3
11.722
13.927
22.912
4.620
0.733
9.732
0.443
0.226
1.293
MS
11.722
2.321
2.083
4.620
0.733
1.622
0.443
0.113
0.431
F (denominator)
P
5.05 (cross[Y source])
1.11 (subcross[cross])
2.78 (withinsubcross)
6.30 (cross[Y source])
0.45 (subcross[cross])
3.21 (withinsubcross)
3.92 (cross[Y source])
0.26 (subcross[cross])
1.22 (withinsubcross)
0.068
0.20
0.004
0.25
>0.5
0.0088
0.23
>0.5
0.31
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169
SUPPRESSION OF SEX-RATIO MEIOTIC DRIVE
of sons of XSR males. Fisher'sexact test(two6. Fertility
tailed),P = 0.005.
30
TABLE
25
a)
I)n 20
Eye color of sons
2
4-
oL_
a)
E
z
O QLM29
* stock
15
15
brick
wild-type
SoUrce of
X chromosome
Motile sperm
Nonmotilesperm
father
mother
0
44
5
20
1l
males. They then used numerical methods to
explore the parameter space for conditions under which Y
polymorphismscould be maintained, varying the fitnessof
the XSR chromosome and the ySLIppressorchromosome. The
principal conclusion fromtheirmodel is thatY-chromosome
polymorphismsmay be maintained under a broader range of
conditions than Clark's (1987) model suggested.
Clark's model involved picking selection coefficientsrandomly. However, in natural populations of Drosophila harboring XSR chromosomes, selection coefficientsare already
restrictedto a subset that allows an X-linked polymorphism.
Thus, the question I ask is somewhat differentfromClark's:
given an existingX-linked polymorphisminvolving sex-ratio
meiotic drive, under what conditions will a Y-chromosome
polymorphism be maintained? The following model reinforces the conclusion of Carvalho et al. (1997) that maintenance of a Y-linked polymorphismmay be less constrained
than previously thought.
As an evolutionary startingpoint, this model assumes that
a population harborsa stable polymorphismfordriving(XSR)
and nondriving(XST) X chromosomes and thatit is initially
monomorphic for a single type of Y chromosome that is
completely susceptible to sex-ratio meiotic drive. Edwards
(1961; see also Curtsingerand Feldman 1981) gives the conditions required for stable maintenance of an XSR/XSTpolymorphismin termsof the fitnessesof the differentgenotypes
and the strengthof the drive parameter(t), which is defined
as the fractionof daughters in the offspringof XSR males.
Following Edwards' notation,let the various genotypeshave
the following relative fitnesses: XSTXST = 1, XSTXSR = a,
XSRXSR = b, XSTy = 1, and XSRY = c.
Under what conditions will a new Y-chromosome type (denoted Y') invade this population? Assume that the Y' chromosome is less susceptible to drive (t < 1) and thatY' males
have lower fitnessthan males carryingthe ancestral Y chromosome. If Y' males did not sufferreduced fitness,then the
Y' chromosome would quickly spread to fixation;similarly,
if Y' males sufferedlower fitness,but were equally susceptible to sex-ratio meiotic drive, then the Y' would be lost
fromthe population. In accordance withwhat is known about
sex-ratio meiotic drive in species lacking suppressors (e.g.,
D. pseuidoobscutraand D. neotestacea), let the driveparameter
(t) equal 1 for XSRY males. For notational simplicity,let t'
denote the intensityof drive in XSRY' males. The fitnessof
Y'-carrying males relative to Y-carryingmales is denoted as
d. Finally, assume that male fitnessis a multiplicativefunction of the effectsof the X and Y chromosomes (e.g., WxSR y,
=cd).
The new Y' chromosome will invade a population if its
transmissionrate is greaterthanthatof the prevailing Y chromosome. Because XSR males bearing the ancestral Y chroXSRYsL'ppressor
5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
sex ratio
Offspring
FIG. 3. Effectof Y chromosomeon offspring
sex ratio.All males
carriedan XSRchromosome,
50% oftheirautosomesfromthestock
"sex-ratio"strain,and 50% of theirautosomesfromstrainLM29.
Males differed
inthesourceoftheirY chromosome.
Notethatmales
was obtainedfromstrainLM29 sireda greatwhoseY chromosome
of sonsthanmalescarrying
thelaboratory
stockY chroer fraction
mosome.
XSR chromosome variation in resistance to Y-linked or autosomal suppression of drive. Finally, I found that the vast
majority of sons of XSR males inherittheir X chromosome
fromtheirmotherand thatmost of these sons are capable of
siring viable progeny. Because the sons of XSR males are
generally fertilein D. quinaria, unlike the case in D. pseltdoobscura, where the sons of XSR males are XO sterile (Henehan and Cobbs 1983), genes thatsuppress sex-ratiomeiotic
drive can be selectively favored in D. quinaria.
A Model of Y-ChromosornePolymorphismAssociated With
Suppression of Sex-Ratio Meiotic Drive
Wu (1983) has considered the conditions under which an
autosomal suppressor of sex-ratio meiotic drive can invade
a population. Because Y-linked suppressors of meiotic drive
have now been reportedin several species of Drosophila and
in stalk-eyed flies, and because the Y chromosome is the
directtargetof the drive locus, I consider here the conditions
under which a Y-linked suppressor of drive can be favored
by selection and the effects of this on an X-chromosome
polymorphisminvolving meiotic drive.
Clark (1987) has modeled the effectsof viability selection
and meiotic drive (operating in both sexes) on X- and Ychromosome polymorphisms.Parametervalues were chosen
randomly in his numerical simulations, and stable Y-chromosome polymorphismswere rarelyobserved. This theoretical result is in contrast to the observed Y-linked polymorphisms for suppression of sex-ratio meiotic drive observed
in several species of flies, including D. paramelanica, D.
affinis,D. siniulans,D. mediopunctata,D. quinaria, and Cyrtodiopsis dalmanni.
Carvalho et al. (1997) considered a more specific, but biologically realistic, model of meiotic drive and viability selection. They assumed thatX-chromosome drive occurs only
in males, as in Drosophila. To simplify the analysis, they
assumed thatthe fitnessof XSR males equaled thatof XSRXSR
females and that drive was completely suppressed in
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170
JOHN JAENIKE
1/2
0.8
d/2
Ea
E
0C/
E
.l
0.6
20E
Cr
CZ,
a)
Li. 0.-Psr(male)
0
PSRIY fixed PSRIY polymorphicPSRIY fixed
Py'
1
0.2 -
FrequencyofXSR
FIG.4. Y-chromosome
transmission
of thefreratesas a function
quencyof theXSR chromosome.In thisexample,thefitnessof Y'0
1000
200
400
600
800
carryingmales is 0.7 relativeto thatof Y-carrying
males,and the
driveparameter
(t) fortheXSRY' males is only0.75, comparedto
Generation
1.0 forXSRYmales.The dashedlines indicatea set of equilibrium
ofthesex ratioXSR chromosome
thatwillyielda stable FIG. 5. Numericalsimulationof invasion of a Y' (suppressor)
frequencies
Note thatwhen eitherY-chromo- chromosomeand its effecton the equilibriumfrequencyof XSR.
Y-chromosome
polymorphism.
some typeis fixedin thepopulation,the otherhas a highertrans- Fitnessesparameterswere set as follows:a(WST SR females) = 0.9,
missionrateand can therefore
invade.
b(WSR SR females) = 0.6, C(WSR males) = 0.8, d(Wy,males) = 0.95, ty
= 1, and ty = 0.65. These parameters
yieldthepredicted(Edwards
of XSR of 0.39 whenonlytheY chro1961) equilibriumfrequency
mosome produce only daughters,the transmissionrate of the mosome is present.The Y' chromosomewas introducedat 1%
in generation100. The predictedequilibriumofXSR with
frequency
Y chromosome averaged across the population is
a Y/Y' polymorphism
is 0.0699, whichwas reachedin under1000
generations.
(1)
WY = 0.5(1 - PSR),
where PSR is the frequency of the XSR chromosome in the
population. Transmission of the Y' chromosome is reduced
by a factord, because of the reduced fitnessof Y' males, but
it is increased as a result of reduced susceptibilityto sexratio melotic drive. Thus, the transmission rate of the Y'
chromosome is
Wy = d[0.5(1
- PSR) + (1
-
t')PSR].
(2)
As shown in Figure 4, the transmissionrates of the two Ychromosome types depend on the frequencyof the XSR chromosome in the population.
If both Y-chromosome types are present at equilibrium,
they will experience equal transmissionrates, in accordance
with haploid selection models (Hartl and Clark 1989), that
is,
0.5(1
- PSR) = d[0.5(1
- PSR) + (1
-
t')PSR].
(3)
This indicates that if there is a Y-chromosome polymorphism involving suppressor and nonsuppressor Y chromosomes, thefrequencyof the XSR chromosome will be adjusted
to an equilibrium of:
=S
1-- d
1
1 + d(l -2t')(
(4)
In other words, the XSR chromosome is broughtto a new
equilibrium frequencythatdepends solely on parametersaffectingthe relative transmissionrates of the two types of Y
chromosome,specifically,the degree to which sex-ratiomeiotic drive is expressed in XSRY' males (t') and the relative
fitnessof these males (d). The most interestingresult is that
if there is a Y/Y' polymorphism,the new equilibrium frequency of XSR is independentof the fitnesseffectsit has on
its carriers. In essence, the change in PSR is brought about
by changes in the mean level of t,the meiotic driveparameter,
whose magnitudedepends on an interactionbetween the driving XSR chromosome and the target Y chromosomes. This
result has been confirmedwith numerical simulations,using
a modification of the model presented in Edwards (1961).
The basis forthe numerical results are presentedin Appendix
2 and one example is shown in Figure 5. It should be noted
that these simulations do incorporate the effects of "sex
ratio" on female fitness(see Appendix 2).
As the frequency of the Y' chromosome increases, the
mean level of the drive parameter (t) in the population will
decrease. As shown in Appendix 3, d(PSR)/dtis positive for
all parametersets that lead to a stable "sex-ratio" polymorphism. Therefore, the spread of Y' will cause a decline in
the equilibrium frequencyof XSR.
Under what conditions will a stable Y-chromosome polymorphism be maintained? A protected polymorphism requires that each Y-chromosome type experience a greater
transmissionrate than the other when the other is near fixation. Because the relative transmissionrates of the two Ychromosome types depend on the frequencyof the XSR chromosome, the conditions for a stable Y-chromosome polymorphism are
PSRIY'
fixed <
PSRIY polymorphic<
PSRIY fixed
As shown above, PSRIY polymorphic = (1 - d)/[1 + d(l
2t')]. Edwards (1961) presents equilibrium gene ratios (r =
PSR/PST) for a model of sex-ratio meiotic drive with a single
type of Y chromosome. If we assume thatthe typical Y chromosome is completely susceptible to the drive (t = 1, as in
Drosophila), then the equilibrium gene ratio (r = PSR/PST) in
males is
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SUPPRESSION OF SEX-RATIO MEIOTIC DRIVE
_
A
\
r
.Zg....
\/EREE'i'.,'.,..'.'~~~~~~~
~~~~~~~~~..'--,...
Drive
.7
fX
FitnessofY' (d)
FIG. 6. Minimumvalue of SR requiied forinvasionof a Y' (suppressor)chromosomeas functionof thedriveparameter(to.) and
fitness(d) of the Y' chromosome.It is assumedthatthe Y chromosomehas a relativefitnessof one and thatits transmission
is
completelysuppressedbyan XSR chromosome
(ty 1). The shaded
area shows the parameterspace in whicha Y' chromosomecan
invadeif theequilibriumfrequency
ol XSR is 0.3 in theabsenceof
suppressor
s.
_.c[a(1 + 2c).-
rYfixed -a(1
+2c)-4bc
2]
(5)
Similarly, if the Y' chromosome is fixed, then
rYfxe
r~ie=a(1
c[a(1 + 2ct') - 2]
+ 2ct') -4bct''
(6)
where c in this case is the fitnessof XSRY' relative to XSTY'
males. Thus, the condition required for maintenance of a Ychromosome polymorphismcan be put as
ray fixed <
ypoyorphic
<
Y fixed'
or
c[a(1 +2ct') -2] <
1- d
a(1 + 2ct') - 4bct'
2d(1 - t')
+ 2c) - 2]
a(1 + 2c) - 4bc
<c[a(1
(7
If, in the absence of Y', the equilibrium frequencyof XSR
is greater than (1 - d)I(1 + d[1 - 2t']), then the Y' can
invade the population, because under such conditions Wyr>
Wy. If the equilibrium frequencyof XSR is below this critical
value, then Y' cannot invade. A plot of the value of PSR
necessary forinvasion of the suppressorY' chromosometype
as a functionof d and t' is shown in Figure 6. If PSR exceeds
this value, then the Y'-chromosome type can invade. In Drosophilaspecies lacking Y-linked or autosomal suppressorsof
drive (e.g., D. pseudoobscura and D. neotestacea), the equilibrium frequency of XSR is often 20-30% in natural populations (Dobzhansky and Epling 1944; James and Jaenike
1990). Assuming a 30% frequencyof XSR, it can be seen that
a fair fractionof the d-t' parameterspace allows invasion of
suppressor Y' chromosomes.
The model presented above (eq. 4) indicates that if one
knows the equilibrium frequencyof XSR in a population and
the intensityof meiotic drive against the suppressorY' chro-
171
mosome (t'), then one can predict the fitnessof Y'-bearing
individuals (d). Conversely, if one knows d and t', it should
be possible to predict the equilibrium frequency of XSR in
populations that are stably polymorphicfor the two Y-chromosome types.
The theoreticalresultspresentedhere indicate thatthe stable maintenance of a Y-chromosome polymorphismmay be
considerably more likely than suggested by Clark's (1987)
more general model. A major reason is that the presence of
a stable polymorphisminvolving a driving X chromosome
restrictsthe parameterspace to the region in which stable Ychromosome polymorphismsare possible (Clark 1987). Thus,
Y-chromosome polymorphisms,when theydo occur, may often be associated with variation in suppression of X-chromosome meiotic drive. It is possible that cytologically distinguishableY-chromosome types vary in theirsusceptibility
to such drive. For instance, in the D. affinissubgroup, karyotypic Y-chromosome polymorphismis known in D. affinis
and in the eastern semispecies of D. athabasca, which harbor
drivingXSR chromosomes,but notin D. algonquin, D. tolteca,
or the western-northern
semispecies of D. athabasca, which
are not polymorphic for sex-ratio meiotic drive (Miller and
Stone 1962; Miller and Roy 1964). It would be interesting
to determinewhetherthe differentY-chromosome types differ is their susceptibility to meiotic drive in these species.
Y-chromosome polymorphisms are also known in D. pseudoobscura and D. persirnilis(Dobzhansky and Epling 1944),
but as yet thereis no evidence thatthese are associated with
variation in susceptibilityto X-chromosome meiotic drive in
these species (Beckenbach et al. 1982).
With respect to the autosomal segregrationdistorter(SD)
polymorphismin Drosophila melanogaster,Crow (1991) has
argued that suppressors of SD have not evolved because the
driving chromosomes are so rare in nature. Why then have
such suppressorsevolved in D. quinaria, in which frequencies
of XSR chromosomes in natural populations are also low,
ranging from only 3% to 6% (Jaenike 1996). The model
presented above suggests that the low frequency of XSR in
this species is due, at least in part,to the presence of Y-linked
suppressorsof meiotic drive. Thus, the presentlow frequency
of a driving element does not necessarily reflectthe original
level of selection in favor of suppressors.
The above equilibrium model indicates the sort of rapid
gene frequencychanges to be expected over 10s to 100s of
generations,given thata species is polymorphicforX-linked
drive and Y-linked suppression of drive. Drosophila quinaria
is well suited for experimental field studies of these genetic
interactions,because it occurs in discrete semi-isolated populations inhabiting skunk cabbage patches. Molecular evidence reveals modest, but significant,genetic differentiation
among these populations (Shoemaker and Jaenike 1997). One
can predict that an experimental manipulations of XSR (or
Y') frequencies will bring about rapid changes in the frequency of Y' (or XSR) chromosomes.
The equilibrium model presented in this paper does not
account for why some species of insects harboringsex-ratio
X chromosomes are polymorphic for Y-linked or autosomal
suppressors, whereas others are not. I suspect thatthe interactions between a driving X chromosome and these suppressors amount to an endless genetic arms race, ratherthan
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172
JOHN JAENIKE
a single long-termequilibrium. Although suppressors may
evolve rapidly, as demonstrated experimentally by Lyttle
(1979), the absence of suppressors in D. pseudoobscuaraindicates thatrapid evolution of suppressorsis not guaranteed.
Because the sex-ratio polymorphismin D. pseudoobsciura
is quite ancient,dating back 700,000 years or more (Babcock
and Anderson 1996), suppressors may have existed in the
past. Perhaps the sex-ratio complex of genes (Wu and Beckenbach 1983) has evolved insensitivityto the suppressors.In
supportof this possibility,D. mediopunctatahas been found
to be polymorphic for XSR chromosomes that vary in the
degree to which meiotic drive can be suppressed (Carvalho
et al. 1997). In contrast,although D. sirnulansharbors XSR
chromosomes, meiotic drive is not expressed in the normal
genetic background in natural populations (Cazemajor et al.
1997), which indicates thatthe suppressorsof drive are presently ascendant in this species. Similarly, the X-linked Stellate gene in D. nmelanogastermay have formerlybeen involved in X-chromosome meiotic drive, with such drive now
being suppressed by the Y-linked crystal (= Suippressorof
Stellate) gene (Hurst 1992, 1996; but see Robbins et al. 1996).
Thus, although X- and Y-chromosome polymorphisms can
be stable, given certain parameters, there will be ongoing
selection on the X chromosome, the Y chromosome, and the
autosomes to destabilize the polymorphism. An arms race
between the X and the Y (or autosomes) might proceed as
follows throughevolutionary time: (1) No X-linked meiotic
drive (many species of Drosophila); (2) :X-chromosomepolymorphism(XSR/XST),all Y chromosomes susceptible to drive
(e.g., D. neotestacea); (3) X-chromosome polymorphism
(XSR/XST), polymorphismforY-linked and/orautosomal suppressors of drive (e.g., D. quinaria, althoughvariationamong
XSR chromosomes in resistance to suppression has not yet
been examined); (4) Polymorphism for X-linked resistance
to the suppressors (XSR suppressible XSR resistant,XST), polymorphism for Y-linked and/orautosomal suppressors (e.g.,
D. mediopunctata,D. paramelanica, D. affinis);(5a) X-chromosome polymorphism(loss of XSR suppressible, leaving XSR
resistant and XST), no apparent polymorphismfor suppressors
(e.g., D. pseudoobscura); and (5b) No apparent X-chromosome polymorphism (XST and XSR suppressible ), fixation of
suppressors (e.g., D. sisnulans).
As the final stage in the process, there could be accumulation of mutationsdeleterious to driving ability in XSR 5L1
pressible alleles in species where suppressors are fixed stage
(5b), because drive is no longer expressed. The loss of potential X-linked drive could then lead to mutational deterioration of the suppressors, thus bringing a species back to
stage 1 and leaving only a molecular trace of the former
meiotic drive interaction.
A phylogenetic analysis of sex-ratio meiotic drive and its
suppression within a group of species, such as the obsciura
group of Drosophila, could be used to inferthe sequence of
such changes, unless they occur so rapidly that any phylogenetic signal is rapidly lost. That such changes could occur
rapidly lies behind the idea that rapid coevolution of meiotically driven sex-linked genes and their suppressors are
responsible forreproductiveisolation between incipient species (Frank 1991; Hurst and Pomiankowski 1991). However,
Coyne et al. (1991) have pointed out that normal sex ratios
are typically produced in crosses involving fertile hybrid
males, implyingan absence of sex-ratiomeiotic drive in these
hybrids.The absence of drive in hybridmales is difficultto
reconcile with the meiotic drive theory of speciation, but it
does not rule out the possibility of a genetic arms race within
species between driving X chromosomes and Y-linked or
autosomal suppressors of drive. Perhaps meiotic drive requires a specific targetsequence thatoccurs only in the species in which the drive has evolved. To test the notion that
drive is species-specific, which is essentially the opposite of
the Frank (1991) and Hurst and Pomiankowski (1991) hypothesis, one could determinewhetheran XSR chromosome,
which exhibits drive against a nonspecific Y chromosome,
can also drive against a heterospecificY chromosome in hybrid males.
AC KNOWLEDGMENTS
This work was supportedby grantsfromthe National Science Foundation (DEB 9206916 and DEB 9615065). I thank
I. Dombeck for technical help, D. Shoemaker for collecting
flies from Maine, S. Tonsor and S. Kalisz for providing facilities for collection of the Pennsylvania samples, R. Rothwell for permission to collect D. quinaria on his land, M.
Cranston for mathematical advice, and T. Starmer and two
anonymous reviewers fortheirvery helpfulcommentson the
manuscript.
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yotype in Drosophila affinisand related species. J. Hered. 53:
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SHOEMAKER, D. D., AND J. JAENIKE. 1997. Habitatcontinuity
the geneticstructure
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1326-1332.
strain male
X
stock female
XY A1A2
X
XX A3A4
XY A3A1, XY A3A2 etc.
F1 son
X
stock "sex-ratio" female
XY A3A1
X
XSRXSR
XSRY A1Aj, XSRY A3Aj,
AjAj
etc.
Testcross F, sons fromone subcross X stock "standard" female
XSRY A1Aj
x
XSTXSTAA
XSRY AjAj
x
XSTXSTAA
XSRY A3Ai
x
XSTXSTAA
XSRY A3AJ
X
XSTXSTAA
Determine offspringsex ratios
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174
JOHNJAENIKE
APPENDIX 2
Numericalanalysisof Y-chromosomedynamics.Fitnessesof the fivegenotypesforthe example shownin Figure 5 (WSTST= 1,
WSTSR[a] = 0.9, WSRSR[b]= 0.6, WSTY= 1. WSRY[C]= 0.8) were chosen to maintain a stable polymorphismat the sex-ratio locus, which
requiresthata(l + 2ct) > 2 and a(l + 2ct) > 4bc (Edwards 1961). Drive parametert (whichis thefractionof daughterssiredby XSR
males) is 1 forXSRY males and 0.65 in XSRY' males; d, the fitnessof Y'-bearingmales relativeto theY-bearingmales,is set at 0.95.
of
frequencyof each matingtypeby proportion
The frequencyof each genotypein nextgenerationwas obtainedby (1) multiplying
genotype;(3) summingfor
each genotypeamongtheresultingoffspring;
(2) multiplying
thisproductby the fitnessof each offspring
each genotypeacross all matingtypes;and (4) dividingthissum by totalnumberof femalesor males.
Mating type
Male genotype
and frequency
XSRY
x(1
Y')
XSRy' xY'
XSTY
XSTyf
Z(l
zy'
Female genotype
and frequency
Y')
Mating
frequency
XSTXST(u)
XSRXST(v)
(1
(1
-
Y')xu
Y')xv
XSRXSR(w)
(1
-
Y')xw
XSTXST
XSTXSR
XSRXSR
-
Offspringgenotypes
(1
-
XSRXSR
(1
-
Y')zu
(1 - Y')zlv
XSTXST
Y')zwI
Y'zu
XSRXSR
Y'zW
XSTXSR
Y'zv
Fitness
XSRXST
1
0.5
XSRY
XSRy'
XSTy
0.5
0.25
0.5
0.25
0.25
0.5
0.5
0.5
0.25
0.25
0.25
0.5
0.5
a
(Al)
For thereto be a stable polymorphism
at the "sex-ratio" locus,
boththenumerator
and denominator
of equation(Al) mustbe positive.The derivativein Pwithrespectto t,whichcan be determined
by addingand subtracting
4bctfromthenumerator
and thenrearranging,is (aot + 2f)I(a + ft)2, where(x = 4bc and 3 = 2ac dPldtwill be positiveif (aot + 23) > 0, whichcan
4bc. Therefore,
be simplifiedto b < al(2 - a) or a > 2b/(l + b). We now show
thatif thereis a stablepolymorphism,
dP/dt
mustbe positive.
= 0, thena = 2b(l + b). Substituting
First,ifdP/dt
thisexpression
fora intothenumerator
of equation(Al) yields
t
t'/2
(1 - t')/2
(1 - t')
0.25
1
XSTYI
0.5
1
APPENDIX 3
Demonstration
thatequilibriumfrequencyof XSR decreases as
thedriveparametert decreasesdue to partialsuppressionof drive. or
Insteadof PSR, considerthe equilibriumgene ratio(P = PSR/PST)
in females(Edward 1961) as:
a(l + 2ct)-2
a(l + 2ct) - 4bct'
XSRXSR
(1 - t')
(1 - t')/2 (1 - t')/2
t
Y'xu
Y'XV
Y xw
XSTXST
XSTXSR
XSTXST
c
b
(2b
-
cd
E)(1 +
1 +b
4bct< 2 -
0.5
0.25
1
d
2ct) > 4bct,
E+ b2Ect
(A3)
Combininginequalities(A2) and (A3) yields the impossibleinequality
e + 2Ect
2> 2 + E + 2Ect,
b
because b, c, t,and e are all positive.ThereforedP/dt
cannotbe less
thanzero.
> 0. This impliesthata > 2b/(l + b). This can
Finally,let dP/dt
be set up as a = (2b + E)(1 + b). Substituting
thisexpressionfor
a intothenumerator
of (Al) yields
(2b + E)(1 + 2ct) > 2,
1 +b
2b(1 + 2ct) > 2
or,
1 +b
4bct> 2-E-2Ect.
(A4)
a = 2b(l + b) intothedenominator
or 4bct> 2. Substituting
yields
thisexpressionfora intothenumerator
of (Al) yields
Substituting
2b(1 + 2ct) > 4bct,
(2b + E)(1 + 2ct) > 4bct,
1 +b
or 2 > 4bct. Because 4bct cannotsimultaneously
be greaterthan or
and less than2, dPldtcannotequal 0.
Next considerthepossibilitythatdP/dt< 0, whichmeansthata
4bct< 2 +
b
(A5)
< 2b(1 + b). This can be setup as a = (2b - E)(1 + b). Substituting
thisexpressionfora intothenumerator
of (Al) yields
Combininginequalities(A4) and (A5) yields the permissiblein(2b - E)(1 + 2ct) > 2
equality
1 +b
E + 2ECt
2+
>+2c>2-E-2Ect.
b
or
at the
4bct> 2 + e + 2Ect.
(A2) These resultsshow thatif thereis a stable polymorphism
mustbe positive,whichmeansthatthe
"sex-ratio"locus,thedP/dt
a = (2b - E)(1 + b) fora intothenumerator
of (Al)
of XSRwill increasewiththedriveparameSubstituting
equilibriumfrequency
ter t.
yields
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