The University of Chicago
Antagonistic Pleiotropy, Reversal of Dominance, and Genetic Polymorphism
Author(s): James W. Curtsinger, Philip M. Service and Timothy Prout
Source: The American Naturalist, Vol. 144, No. 2 (Aug., 1994), pp. 210-228
Published by: The University of Chicago Press for The American Society of Naturalists
Stable URL: http://www.jstor.org/stable/2463157
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Vol. 144,No. 2
The American
Naturalist
August1994
ANTAGONISTIC PLEIOTROPY, REVERSAL OF DOMINANCE, AND
GENETIC POLYMORPHISM
JAMESW. CURTSINGER,*PHILIP M. SERVICE,t AND TIMOTHYPROUTt
andBehavior,University
ofMinnesota,
SaintPaul,Minnesota
ofEcology,Evolution,
*Department
ArizonaUniversity,
Arizona86011;
Flagstaff,
ofBiologicalSciences,Northern
55108;tDepartment
95616
ofCalifornia,
Davis,California
tCenterforPopulation
Biology,University
SubmittedJanuary29, 1993; Revised September23, 1993; Accepted September30, 1993
loci withantagonistic
forpolymorphism
at pleiotropic
effects
on fitness
Abstract.-Conditions
andmultiplicativity
areinvestigated,
undertheassumptions
ofadditivity
offitness
components
arerather
We showthattheconditions
forstablepolymorphism
restrictive,
especomponents.
tothedominance
arealso verysensitive
parameters;
ciallywithweakselection.Theconditions
A reviewof
in particular,
reversalof dominanceis oftenrequiredforstablepolymorphism.
of dominancesuggeststhatdominancereversalis notlikelyto be
biochemicalmechanisms
at twoantagonistic
and pleiotropic
formaintaining
common.The conditions
geneticvariation
forstablepolymorthanfortheone-locuscase. Whenconditions
lociareevenmorerestrictive
are satisfied,
substantial
dominance
variancein one or both
pleiotropy
phismby antagonistic
pleiotropy
fitness
is expectedbutis seldomobservedinexperiments.
Antagonistic
components
whichusuallytendstoreduce
onthefitness
separately,
selection
components
impliesstabilizing
infitness
trade-offs
maybe comcomponents
geneticvariance.We concludethat,eventhough
roleinexplaining
thepersistence
ofgenetic
probably
playsa limited
mon,antagonistic
pleiotropy
variation
in fitness
components.
inevolutionary
genetThe maintenance
is a centralproblem
ofgeneticdiversity
might
ics. Rose (1982,1985)has popularizedtheidea thatmanypolymorphisms
betweenfitness
be maintained
whichentailstrade-offs
pleiotropy,
byantagonistic
underthecontrolofloci withpleiotropic
geneaction.
components
fromthe
in fitnesscomponents
comes primarily
The evidencefortrade-offs
of correlated
observation
responsesto selection.In femaleDrosophilamelanoand longevity
and also bebetweenearlyfecundity
gaster,thereis antagonism
and latefecundity
tweenearlyfecundity
1981a,1981b;
(Rose and Charlesworth
andFowler1992forconflicting
etal. 1984;Rose 1984;butsee Partridge
Luckinbill
betweenearlycompetitive
thereis antagonism
results).In maleD. melanogaster,
and late matingsuccess on the
matingsuccess on theone handand longevity
in D. melanogasteris
other(Service 1993). Furtherevidenceforantagonism
beprovidedby Hiraizumi(1961),who observednegativegeneticcorrelations
inchromosomes
thatconferhighfitness.
rateandfecundity
tweendevelopmental
Simmonset al. (1980) foundthat,in D. melanogaster
recentlysampledfrom
in differwerenotreflected
effects
on viability
chromosomal
nature,differential
inotherfitness
components.
effects
suggesting
compensatory
encesinnetfitness,
Am. Nat. 1994. Vol. 144, pp. 210-228.
? 1994 by The Universityof Chicago. 0003-0147/94/4402-0002$02.00.
All rightsreserved.
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ANTAGONISTIC
PLEIOTROPY
211
The Drosophiladataare thusbroadlyconsistent
withtheantagonistic
pleiotropy
modelin thesensethatthereare documented
trade-offs
betweenfitnesscomponents.
Even if antagonistic
gene effectsare common,however,thisby no means
impliesthatantagonistic
pleiotropy
generates
polymorphisms.
Antagonism
is not
sufficient
forthemaintenance
ofpolymorphism:
positiveviability
selectioncannotprevent
theelimination
ofa dominant
allelethatinducessterility,
forinstance.
Nor is antagonism
necessaryforpolymorphism;
simple(single-component)
overdominance,frequencydependence,and nichevariationcan explainpolymorphismwithoutinvoking
specialpatterns
ofpleiotropy.
Even thoughantagonistic
pleiotropy
is neithernecessarynor sufficient,
it is a commonly
held view that
antagonism
is intimately
connectedwithmaintaining
variationin populations.
This stemsin partfroma statement
by Falconer(1981,p. 300) suggesting
that
allelesat pleiotropic
locithathavepositiveeffects
on twocharacters
underselectionare expectedto increaseto fixation,
whileallelesthathave negativeeffects
on bothcharacters
areexpectedtobe eliminated.
The onlyallelesleftsegregating
are expectedto be thosethathavepositiveeffects
on one character
andnegative
effects
on theother.
Here we studythedegreeto whichstablepolymorphisms
mightarisein large
withconstant
andantagonism
offitness
populations
selectionregimes,
pleiotropy,
ofthepacomponents.
Our approachconsistsin partofnumerical
investigation
rameterspace of therelevantmodelsand in identifying
and quantifying
therebeen used to
gionsthatpreservegeneticvariation.This methodhas previously
investigatethe feasibility
of multiallelic
polymorphisms
withoverdominance
insex-linked
andautosomal
(Lewontinet al. 1978),therelativelevelsofvariation
ofpolysystems(Curtsinger
1980;Pamiloand Crozier1981),and thefeasibility
in haploidand diploidsystemsunderfrequency-dependent
selection
morphism
resultsfor
(Brooksand Curtsinger
1994).We also deriveanalyticand numerical
two-locusandquantitative
geneticmodels.An experimentally
testableprediction
regarding
themagnitude
of dominancevarianceproducedby antagonistic
pleioand show
ofdominance
tropyis described.We considerbiochemical
mechanisms
thatthe modesof gene actionthatare mostlikelyto generatepolymorphisms
or
requirea formof dominancerelationships
forwhichthereis littletheoretical
in the broader
empiricalsupport.Finally,we considerantagonistic
pleiotropy
contextof stabilizing
overdominance.
selectionand single-gene
GENETIC
MODELS
involvesone dialleliclocuswith
The simplestmodelofantagonistic
pleiotropy
as shownat thetopoftable1. The
pleiotropic
effects
on twofitness
components,
withdomipopulationis assumedto be monoeciousand withoutage structure,
and less thanor equal to
nanceparameters
(h,, h2) assumedto be nonnegative
unityand selectionparameters
positiveandless than
(f,v) assumedto be strictly
Thatis, for
or equal to unity.The parameter
constraints
guaranteeantagonism.
thanthatofA2A2
A1A1 is greater
setsthefitness
ofgenotype
anyvalidparameter
II. The
I and vice versaforfitness
withrespectto fitness
component
component
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212
THE AMERICANNATURALIST
TABLE 1
Two
PARAMETERIZATIONS OF THE ANTAGONISTIC PLEIOTROPY MODEL
GENOTYPE
AIAI
Basic model:
Fitness componentI
Fitness componentII
Alternativeparameters:
Fitness componentI
Fitness componentII
1
1- f
1 + HI V
1- F
AIA2
1-
hlv
1 - h2f
1
1
A2A2
1-
1
v
1- V
1 + H2F
withinfitnesscomponents,
constraints
also precludeover-or underdominance
thoughover-and underdominance
fornetfitness
remainspossible.Fitnesscomor multiplicatively
to producetotal
ponentsare assumedto combineadditively
fitness.
It is convenient
to classifyparameter
setsintofourcategoriesaccordingto the
dominanceparameters:
1. If h, = h2 = 0.5, gene action is additive.The heterozygotehas intermediate
fitness
forbothcomponents.
2. If h, and h2 are bothless than0.5, thenthereis "beneficialreversal"of
withrespectto component
dominance.Thatis, theA1 alleleis dominant
I, and
the A2 allele is dominantwithrespectto componentII. This is the same as
recessivenessofdeleterious
alleleswithinfitness
components.
3. If h, and h2are bothgreaterthan0.5, thenthereis "deleteriousreversal"
of dominance;the inferior
allele foreach fitnesscomponentis dominantwith
respectto thatcomponent.
4. If h, > 0.5 and h2 < 0.5, thentheA2 allele is dominant
forbothfitness
ifh, < 0.5 and h2 > 0.5, thenA1 is dominant
forboth
components.
Similarly,
We referto thesecases as "paralleldominance."
components.
The MultiplicativeCase
Now we ask howthefeasibility
ofpolymorphism
dependsonf, v, h1, and h2.
If thetwofitnesscomponents
referto twostagesin thelifecycle,suchas early
and late survival,orjuvenilesurvivaland adultfertility,
thenit is reasonableto
assumethatthecomponents
combinein a multiplicative
fashionto producenet
fitness.Existenceand stability
of theuniquepolymorphic
equilibrium
requires
overdominance
fornetfitness:
(1 - h1v)(I - h2f) > 1 - f, 1 - v.
(1)
Reversaloftheinequalities
(1) producesan unstableinterior
equilibrium.
If h, = h2 = 0, then
Analysisof a fewsimplespecialcases is informative.
thereis beneficial
reversalofdominanceand theinequalities
If
(1) are satisfied.
reversalofdominanceand inequalities
(1)
h, = h2 = 1, thenthereis deleterious
are notsatisfied
If h, = h2 =
(infact,thereis an unstableinterior
equilibrium).
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213
ANTAGONISTIC PLEIOTROPY
1.00
0.75
0.25
0.25
I
I,
FIXATION
k
ION
XFIXAT
0.00
I
0.00
0.25
0.50
0.75
1.00
V
fortheaddi-equilibria
forstablepolymorphic
conditions
FIG. 1.-Phase planeillustrating
at thetopoftable1. The
defined
tivecase h1 = h2 = 0.5;f andv are selectioncoefficients
whenselectionis
is severelyrestricted
regionofthespacethatallowsstablepolymorphism
weak.
forstablepolymorphism
thentheconditions
withincomponents),
0.5 (additivity
are ratherrestrictive:
2vl(2 + v) <f < 2vl(2 - v).
(2)
Figure
For example,ifv = 0.1, then0.095< f < 0.105forstablepolymorphism.
(2), whichis restricted
1 showstheregionofthef,vplanethatsatisfies
inequalities
underall butthemostintenseselectionregimes.In practice,extremeselection
sterility
(f = 1) and the
seemsunlikely,
suchas one allelecausinghomozygous
otherhomozygous
lethality
(v = 1); whenf,v < 0.5 thespace is smaller.
(1) become
Iff = v > 0, theninequalities
I + h1h2v- h1 - h2> 0
(3)
reversaland additivecases.
whichis satisfied
forbeneficial
reversalof dominance(or, equivasuggestthatbeneficial
These observations
roleinpromoteffects)
playsan important
ofdeleterious
therecessiveness
lently,
of selection
as notedby Rose (1982,1985).Equal intensity
ingpolymorphisms,
thatone fitness
also helpsby ensuring
arisingfromthetwofitnesscomponents
theother.We now showthatthe conjectures
componentdoes notoverwhelm
to thewholeparameter
space.
based on specialcases applymoregenerally
ofrealbiologicalparameter
Withno priorknowledge
values,we assumeequal
forthe entirespace. Later we will arguethatcertainparameter
probabilities
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THE AMERICAN NATURALIST
214
1.0
0.8
,~~~~~~~.
0.6
_
0.4
~~~~~~~~~~~.
0.0
_E
0.2
...
..-. ..
.-
~~~~~~~~~~.......
.............................
- - ..
.
_
0.0
. .
.
-....- .
.
.. :..
:~~~~~~~~........
. -,
..... ..
.
,
1.0-0
Ol 0.25
a0 0.00
0.4
0.6
08
1.0
hi
equilibrium,
FIG. 2.-Proportionofparameter
sets(P) thatgenerate
a stablepolymorphic
model.Lower-left
as a function
h1and h2,forthemultiplicative
parameters
of dominance
upperleftandlowerright,paralleldominance;
reversalofdominance;
quadrant,beneficial
of the surfaceis by distancereversalof dominance.Smoothing
upperright,deleterious
1988).
leastsquares(Wilkinson
weighted
thefullparameter
we investigate
valuesaremorelikelytooccur.Forthemoment,
space by testingcondition(1) in a latticeof pointsspaced by 4% in the four
sections.To studyefin two-dimensional
dimensions,
whichcan be illustrated
of all cases thatproducea
fectsof dominance,we computeP, the proportion
witha givenh, and h2,forpointson the h1,h2
stablepolymorphic
equilibrium
we computeP forpointson
of selectioncoefficients,
plane.To studytheeffects
thef,v plane.
is shownin figure2. Stable
The dependenceof P on dominanceparameters
incases inwhichthereis beneficial
reversalof
are mostfrequent
polymorphisms
incases inwhichthereis delenonexistent
dominance
andarevirtually
parameters
withparalleldomiteriousreversal.It is possibleto have stablepolymorphisms
as h1or h2increases.
restrictive
becomeincreasingly
nance,buttheconditions
3. ThepossibilThe dependenceofP on selectioncoefficients
is showninfigure
wheref = v andtendto increasewith
itiesforstablepolymorphism
are greatest
selection.Inequality
offand v reducesP. Strongselectionandequality
stronger
of the intensity
of selectionbetweenthe two componentsof fitnesspromote
in the multiplicative
model, as suggestedby the preliminary
polymorphism
analysis.
In summary,
fitness
setssampledfromthetotalparameter
30% ofantagonistic
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215
ANTAGONISTIC PLEIOTROPY
1.0
1~~~~~~~~~~~~~~~.0
--......_..
.......
0.8
....
.. .. ..........
0.6
i:l: .:::.......
:.:::.:-.
i-.............. ::::-:::....
::::::
.
!::
. . ....I.il!
...,,. ., ,. ... . .;.-;. . 1. ...........
.
.
.
.
..
..
,.
,.
..
.
,.
i.
......
,...........
,.
, ,'
,
::::-n7
u_.-..
.,,,.,,,X . . . . . . . . . . .:.:.::.. ....:::
. . . . . . . . . . . . ..........
_f -.
* ,,,,........
. -.;.
. . . . . . . . . . .02
~~~~~~~~~~~~~~~~~~~~~..
.I.................
..........
rL. . .
~~~~~~~~~~~~~~~.
. ......
.U
.... . . -:..
....
.-,
.,
;,.,,,
.................
.,
0.4.......
0.2
. .
-
. +. . .
iF
. .
.......
. ..
. .
0.0
0.0
;..
.
.........
.....
. .
0.2
. .. ..
...
...
...
...
.......
...
............*
*
a
1.00
0.75
0.50
o 0.2
5
.....
0 0.00
0o4
0.6
O0
1.0
f
a stablepolymorphic
equilibrium,
FIG. 3.-Proportionofparameter
sets(P) thatgenerate
ofthe
model.Smoothing
ofselectioncoefficients
f andv,forthemultiplicative
as a function
surfacewas doneas in fig.2.
If we examineonlythe regionin which
space generatedstablepolymorphism.
quadrantof fig.2), we get 67%. With
beneficial
reversaloccurs(thelower-left
andlower-right
quadrantsoffig.2), 25% ofparamparalleldominance(upper-left
quadrant
withdeleterious
reversal(upper-right
etersetsgeneratepolymorphism;
is reducedto 25% withweak selection.
offig.2), only2%. The overallfigure
TheAdditiveCase
selectionin successivetime
The additivecase appliesto instancesoffertility
plantin year1 andyear2, or Rose's
periods,forexample,seed setina perennial
wouldthenbe expectedto be relatedto the
Net fitness
earlyand latefecundity.
thecondiUnderadditivity,
components.
sumratherthantheproductoffitness
are
equilibrium
ofthepolymorphic
tionsforexistenceand stability
(1 - hlv) + (1 - h2f)>2
-f,2 - v,
(4)
oftheinterior
equilibrium.
instability
withreversaloftheinequalities
producing
Ifh1 = h2 = 0, thenthe
ofa fewspecialcases is againinformative.
Examination
of
existsand is stable.IfhI = h2 = 1, thenthereis no possibility
polymorphism
Iff = v, thenthe condition(4) becomesh, + h2 < 1,
stablepolymorphism.
reversalcase andalso forsomeparalleldomiforanybeneficial
whichis satisfied
nance cases. As in the multiplicative
case, beneficialreversaland equalityof
selectioncoefficients
promotepolymorphism.
The P contoursfortheadditivemodelare ofthesamegeneralshapeas forthe
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216
THE AMERICANNATURALIST
multiplicative
modelbutare relatively
depressed.The overallP is 25%. We concludethatbothmodelstendto maintain
polymorphisms
whenthereis beneficial
reversalof dominance,strongselection,and approximate
equalityof selection
of all possibleantagonistic,
coefficients.
For bothmodelsa minority
pleiotropic
fitnesssets producestablepolymorphism;
onlyin the specialcase of beneficial
reversalofdominanceis polymorphism
verylikely.
Two Loci
We nowconsidera specialcase oftwodiallelicloci,forthepurposeofshowing
forkeepingbothloci polymorthatthereare moreconstraints
on theparameters
phicthanthereare forkeepingone locuspolymorphic.
To facilitate
comparison
withtheresultsofRose (1982),we employthealternative
parameterization
shown
at the bottomof table 1. Parameters
HI and H2 describedominancerelations,
withHi = 0 forcompletedominanceofthefavoredallele,Hi = 1 foradditivity,
and Hi > 1 fordominanceof thedeleterious
allele. Selectioncoefficients
are F
and V. The Hi are greaterthanor equal to zero,whileF and V are greaterthan
zeroandless thanorequaltounity.Constraints
ensurethatfitnesses
arepositive,
withinfitnesscomponents.
selectionoperates,and thereis no overdominance
We assumethespecialcase in whichtheloci haveequal effects
thatare additive
andinwhichnetfitness
is a multiplicative
acrossloci foreach fitness
component
function
oftheindividual
fitness
components.
forstablepolymorphism
conditions
For one locus,thenecessaryand sufficient
are
(1-F)(1
+ HI V) < 1 > (1-V)(1
+ H2F),
(5)
whichrequires
H2F
< V<
F
(6)
F and V in equation(6).
Similarrestrictions
on F are obtainedby exchanging
the conditionson F and V fortwo-locus
Now our objectiveis to determine
are moreor less restrictive
and to see whether
thoseconditions
polymorphism
thantheone-locuscase. Sincewe are interested
in maintenance
ofvariation,
we
The necessaryconditionsfor
considerconditionsforprotectedpolymorphism.
withtwolociare(a) instability
ofthefourcornerequilibprotected
polymorphism
of singlegametictypes,and (b) instability
of
ria, whichcorrespondto fixation
to fixation
at one locus and polymorthefouredgeequilibria,
whichcorrespond
phismat theotherlocus. We showbelowthatthenecessarycornerconditions
thanthenecessaryandsufficient
condiforthetwo-locuscase aremorerestrictive
makesit unnecessary
to examinethe
tionsfortheone-locuscase. This strategy
fortheedgeequilibria,
whichare difficult.
conditions
stability
The fullmodelis specifiedin the Appendix.For instability
of the corners,
the double homozygote
musthave lowerfitnessthanthe two adjacentsingle
are
It is shownin theAppendixthatthecornerconditions
heterozygotes.
H2FH2F
~~~<v <
F
F(7)
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ANTAGONISTIC
PLEIOTROPY
217
Comparing
equations(6) and(7), itis readilyseenthattheone-locusupperbound
is largerthanthetwo-locusupperboundandalso thattheone-locuslowerbound
is lowerthanthetwo-locuslowerbound.It followsthatthenecessaryconditions
forprotected
polymorphism
in thetwo-locusmodelare morerestrictive
thanthe
necessaryand sufficient
conditions
fortheone-locusmodel.
A specialcase illustrates
theincreasedconstraints
forthetwo-locusmodelas
comparedwiththe one-locus model. Withno dominance(H1 = H2 = 1), a one-
locus polymorphism
can be maintained
(eq. [6] and fig.1). However,equation
(7) is neversatisfied
whenthereis no dominance,
andthereis no possibility
ofa
two-locuspolymorphism.
theconditions
Presumably
forinstability
oftheedgeequilibria
wouldconstrain
in thetwo-locuscase even further.
polymorphism
Similarconclusionsapplyto
thecase in whichnetfitness
is an additiveratherthanmultiplicative
function
of
thefitnesscomponents.
These resultssuggestthatformultilocus
cases theconstraintson the parameters
forkeepingall loci polymorphic
increasewiththe
numberofloci.
Many Loci
In this sectionwe turnto the classic componentsof varianceapproachto
in whichphenotypic
quantitative
characters,
is presumedto arisefrom
variation
at manyloci. This approachis morecloselyrelatedto experimental
segregation
because fitnesscomponentsare typicallyinherited
as polygenic
observations,
characters.
Since dominancereversalplays an important
role in maintaining
polymorphismsin the one-locusantagonistic
pleiotropy
model,one mightexpectthat
suchpolymorphisms
addedup overmanyloci in a polygenicsystemwouldproduce largeamountsof dominancevariance.However,Rose (1982) has argued
thatthe dominancevariancearisingfromantagonistic
need not be
pleiotropy
large.In thissectionwe willreconstruct
thatargument,
pointoutitslimitations,
and derivea different
and testableprediction
themagnitude
of domiregarding
nancevariancein polymorphisms
involving
antagonistic
pleiotropy.
We retainthealternative
shownat thebottomof table 1 to
parameterization
facilitate
withtheresultsofRose (1982).Scalingtherelativefitnesses
comparison
and computing
dominancevariance(Vd)and additivevariance(Va) by standard
quantitative
geneticmethods(Falconer1981),theratioof variancecomponents
forfitnesscomponent
I is foundto be
Vd/Va = pq(1
-
H1)212(Hlp + q)2.
(8a)
-
H2)2/2(H2q + p)2.
(8b)
Forfitness
component
II, itis
Vd/Va= pq(1
Withmanyloci contributing
to variationin thefitnesscomponents,
additively
totalgeneticcomponents
of varianceare obtainedby summing
overloci. Rose
thedominance
thevariancecomponent
ratiosby computing
(1982)investigated
varianceforvariousvaluesofH relativeto theadditivevariancein thepurely
additivecase. Thatis, the ratiosgivenin equations(8a) and (8b) wereplotted
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218
THE AMERICAN NATURALIST
(A)
1.00
0.75 -
0.75
0.50
0.50
0.25
(B)
1.00
-
h
d
0.25
a
b
0.00
0.0
C
0.5
1.0
0.00g
0.0
0.5
1.0
H
FIG. 4.-A,
Ratio of dominance variance (Vd) to additive variance (Va) for two fitness
as a function
of thedominanceparameter
components
subjectto antagonistic
pleiotropy,
(H), followingthe method of Rose (1982); a, p = q = 0.5; b, pq
=
(0.8)(0.2); c, pq
=
is compared
to Vaina different
withpurely
population
(0.9)(0.1). HereVd inone population
reversalofdominance
(H < 1) areconsidered.
additivegeneaction.Onlycases ofbeneficial
Thisanalysissuggestslittledominance
ratiocannot
variance,butthevariancecomponent
is computed
inrealpopulations.
bycomparing
Vd to
be estimated
B, As inA, exceptVd/Va
withoutreference
to an
Va in the same populationand forthe same fitnesscomponent,
ratiois identical
forthetwofitness
arbitrary
standard;
d,p = q = 0.5, variancecomponent
components;e, p = 0.2, q = 0.8, fitnesscomponentI;f, p = 0.2, q = 0.8, fitnesscomponent
II; g, p = 0.1, q = 0.9, fitnesscomponentI; h, p = 0.1, q = 0.9, fitnesscomponentII.
ratiosforfitness
I andII are
components
Whenp differs
from0.5, thevariancecomponent
different.
Thisanalysissuggestsmuchdominance
variance.Variancecomponent
ratiosdefinedin thiswayare estimable
in realpopulations.
and was fixedat unity
againstH, whereH variedfrom0 to 1 in thenumerator
in thedenominator
(fig.1 in Rose 1982).Cases in whichH > 1 werenotshown
becausedominanceofdeleterious
allelesseldomproduces
by Rose, presumably
inourfigure
4A, suggestthatdominance
The results,reproduced
polymorphism.
varianceis smallcomparedwithadditivevariancefora wide rangeof allelic
and modesofgeneaction.
frequencies
as a comparisonof
The approachtakenby Rose (1982) can be interpreted
two populationsthathave identicalgene frequenciesand different
degreesof
dominance,one beinga purelyadditivestandard.However,it is notclearthat
variancelies in comparing
themostmeaningful
measureofdominance
itsmagniwiththeadditivevariancein anotherpopulationthathas
tudein one population
a different
has no obviousrelationto
modeof geneaction.Such a comparison
measurablequantitiesin real populations,since purelyadditivestandardsare
nonexistent.
thecomparison
betweenVdand Vaas infigure
Furthermore,
making
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ANTAGONISTIC
PLEIOTROPY
219
SettingHi = 1 in the
betweenfitnesscomponents.
4A obscuresthe difference
denominatorsand H1 = H2 = H in the numeratorsartificiallyguarantees the
component
I andfitness
comporatiosforfitness
equalityofvariancecomponent
nentII,whichis notgeneral.
of
the magnitude
It is moreuseful,as well as moregeneral,to investigate
it withthe additive
dominancevariancein fitnesscomponentI by comparing
and likewiseforfitness
I in thesamepopulation,
varianceforfitness
component
II. Thiseliminates
standardand makes
anyreference
to an arbitrary
component
populations.
theratioestimablein experimental
are
foreach fitness
component
The Vd/Va
ratioscomputedwithinpopulations
4B. Here thevariancecomponent
ratiosare plottedas funcillustrated
in figure
4A, butH rangesfrom0 to 1 in
tionsof thedominanceparameter,
as in figure
fora within-population
andthedenominator,
as is appropriate
boththenumerator
comparison. If p = q = 0.5 (curve d), then the Vd/Varatio is identicalforthe
two fitnesscomponentsand approaches0.50 forsmallvalues of H. If allelic
ratiosdiffer
forfitness
frequencies
departfrom0.5, thenthevariancecomponent
I and II. WithsmallH, theratioforone fitness
component
is large,
components
evengreaterthanunity,whiletheotheris small.For instance,ifp = 0.2, q =
I is
ratioforfitness
component
0.8, andH is small,thenthevariancecomponent
II is large(curvef).
small(curvee) and theratioforfitness
component
and deratioson allelicfrequencies
The dependenceof variancecomponent
=
=
5
h
case
(using
in
for
the
special
shown
figure
greesofdominanceis
h, h2
theparameterization
at the top of table 1). For each value of h thereare two
It is readilyseen in figure5 thatwith
curves,one foreach fitnesscomponent.
the
smallh's (beneficial
reversalof dominance)and centralallelicfrequencies
comdominancevarianceapproacheshalfoftheadditivevarianceinbothfitness
variance
thedominance
allelicfrequencies,
ponents.For smallh's andnoncentral
and is dwarfed
component
equals or exceedstheadditivevariancein one fitness
by the additivevariancein the otherfitnesscomponent.As the h's increase
toward0.5 (additivity),
the dominancevariancebecomessmallin bothfitness
as discussed
butlargerh's do nottendto producepolymorphisms,
components,
previously.
Withparalleldominance,the variancecomponentratiosforthe two fitness
case h2 =
are similarin magnitude
and are identicalin thelimiting
components
1 - hI. The ratiois largewhenthedominant
alleleis commonand smallwhen
the dominantallele is rare.However,paralleldominanceproducesmuchless
thandoes beneficial
reversalofdominance.
polymorphism
relations
thatare mostlikelyto lead to stable
We concludethatthedominance
tendto createlargeamountsof dominancevariancein at least
polymorphisms
one fitness
forthecase h, = h2.Forthegeneralcase ofunequalh's,
component,
allelicfrequencies
spaceandcomputeequilibrium
we can studythefullparameter
and variancecomponent
ratiosforthecases thatproducestablepolymorphism.
ratios
ofselectionintensity,
Resultsare summarized
intable2. Irrespective
Vd/Va
averof
the
ratios
and
the
about
for
each
fitness
larger
component
average
50%
pleiotropy
ages almost100%. These resultsdo not meanthatthe antagonistic
withlargeamountsofdomimodelalwaysgenerates
stablepolymorphic
equilibria
nancevariance;rather,theydescribetheaveragebehaviorofthemodelovera
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2.0
1.0
h=O
h=O
h=0.2/
h=02
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
p
FIG. 5. Ratio Vd/Vaas a functionofallelic frequenciesforseveral values ofthedominance
parameter(h). For any value ofp and h thereare two functions,one foreach fitnesscomponent. With central polymorphisms(0.4 < p < 0.6), the two variance component ratios
are approximatelyequal, while noncentrality
of allelic frequencyproduces large dominance
variancein one fitnesscomponentand smalldominancevariancein theother.As h increases,
dominancevariance decreases, and the likelihoodof polymorphismalso decreases.
TABLE 2
AVERAGEEQUILIBRIUMRATIOS OF DOMINANCEVARIANCETO
ADDITIVE VARIANCEFOR THE ANTAGONISTICPLEIOTROPYMODEL
AVERAGE Vd/Va
Fullspace
0 <f,v < .5
0 <f,v < .1
0 <f,v < .05
Fitness
ComponentI
Fitness
ComponentII
.513
.557
.580
.584
.513
.557
.580
.584
Larger
Fitness
Component
.935
1.022
1.070
1.079
NOTE.-Each entryis based on testingapproximately456 x 103
parametersets. Selection coefficientsare f and v.
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ANTAGONISTIC
PLEIOTROPY
221
thatdiffers
rangeofparameter
values.We havethusderiveda testableprediction
components
oftenplaysa rolein
frompreviousanalyses:ifantagonism
offitness
varianceforfitness
components
maintaining
polymorphisms,
thenthedominance
should,on theaverage,be abouthalfas largeas theadditivegeneticvariancefor
thosesamefitness
components.
charactersdo not show large
Withfew exceptions,Drosophilaquantitative
amountsof dominancevariancerelativeto theiradditivevariance(Tachidaand
viability
are on theorder
Cockerham1988).Estimatesof Vd/IVa
foregg-to-adult
of 0.10 (Mukaiet al. 1974;Mukaiand Nagano 1983;Tachidaet al. 1983).For a
dozenDrosophilaquantitative
characters
reviewedbyRoffandMousseau(1987),
11 had negligible
amountsof dominancevariance.A similarconclusionapplies
to 33 traitsin insectspecies otherthanDrosophilaand to one bird species
(Mousseauand Roff1987,p. 183).Kearseyand Kojima(1967)reportedsubstantialdominanceeffects
on egghatchability,
butbecausetheyisolatedwholechrooflinkedrecessivedeleterimosomestheirestimates
probablyincludetheeffects
ous alleles. Rose and Charlesworth
(1981a) reportedabout 10 timesas much
in an outbredpopulation
dominancevarianceas additivevarianceforlongevity
ofDrosophilamelanogaster,
error
butthestudywas small,and theunreported
associatedwiththevariancecomponent
estimateswas probablylarge.Hutchininlinesselectedforlongevity
sonandRose (1991)estimated
variancecomponents
that were derived fromthe same base populationstudied by Rose and
in
Charlesworth
(1981a) butfoundnegligible
dominanceeffect.Adultlongevity
buteven
pleiotropy,
Drosophilais oftenmentioned
inthecontextofantagonistic
forthis"best" case thepublisheddominanceestimatesare notconsistentand
be more
thelargerstudydocuments
littledominance.Dominancevariancemight
we concurwithRose
difficult
to estimatethanadditivevariance.Nevertheless,
ofdominance
et al. (1987),whonotedthatthereis "littleevidenceforthepattern
forabundant
requiredifantagonistic
pleiotropy
is to be used as an explanation
geneticvariationin fitness
components"(p. 97).
PLEIOTROPY
AND DOMINANCE
A principalresultof ouranalysisis thatstablepolymorphisms
are unlikelyin
the absence of beneficialdominancereversal.Thus,the powerof antagonistic
reversalof
dependson whether
pleiotropy
to accountforgeneticpolymorphisms
traitsis likely.This issue is largelyunexplored.
dominancebetweenpleiotropic
of
dominanceandtheimplications
Herewe considerthemechanisms
underlying
in quantiinterested
thosemechanisms
fordominance
reversal.We areprimarily
domiinformation
concerning
tative,polygenic
traits,butthereis littleempirical
fromgenes
relyon extrapolation
nancerelationsat polygenic
loci. We therefore
metabolic
thatrelatemultienzyme
of majoreffectand on theoretical
constructs
characters.
pathwaysto quantitative
Mechanisms of Dominance
Fisher(1928a,1928b)proposedthatdominancewas theresultofevolutionby
could accountforthe
naturalselectionon modifier
loci. The putativemodifiers
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222
THE AMERICANNATURALIST
of dominanceof wild-type
alleles over most
commonly
observedphenomenon
new mutations.
Wright
(1929,1934)criticizedFisher'shypothesis,
arguingthat
modifiers
wouldbe too weakto influence
modselectionon dominance
generally
ifierfrequencies
(see also Ewens 1965;Sved andMayo 1970;FeldmanandKarlin
withevidencethatmanygeneproducts
areenzymes,Wright
1971).Starting
(1934)
proposedthatdominancearisesas a consequenceof reactionkineticsand the
UnderWright's
existenceofa "factorof safety"in wild-type
homozygotes.
hypothesis,halvingtheenzymeconcentration
(as wouldoccurin a wild-type/null
wouldresultin less (perhapsmuchless) thana 50% reductionin
heterozygote)
allele.
flux,thatis, dominanceofthewild-type
Modernanalysissupportstheidea thatdominanceis a resultof thereaction
kineticsof multistep,
enzyme-catalyzed
metabolicpathways(Kacser and Burns
or activityat
1981and references
therein).A changein enzymeconcentration
to have a largeeffecton theoutputofthe
anyone stepin thesystemis unlikely
is to be exsystem,unlessenzymeactivityat thatstepis verylow. Additivity
a conclupectedforallelesthatproduceenzymeswithsmallactivity
differences,
sion also reachedby Wright(1934). Kacser and Burns(1981) also notedthat,
of enzymeactivitiesat all thesteps
because thefluxof a pathwayis a function
in the pathwayand because pathwaysare linkedto one anotherthrough
their
universalepistasisas wellas pleiotropy.
inputsand outputs,thereis essentially
ofanyparticThatis, therearetheoretically
manylocithatcouldactas modifiers
ular reactionstep. Thus, evolutionary
changesin the dominancerelationsare
clearlypossible,butitis notnecessaryto invoketheevolutionofspecialmodifier
ofdominance
ofwild-type
alleles.A recent
locito explainthegeneralobservation
in a haploidalga are typically
studyby Orr (1991) has shownthatmutations
recessivein artificial
diploids.This arguesagainstthe evolutionof dominance
modifiers
because selectionwouldhave had to occurin theheterozygous,
that
is, diploid,state.
Thereis, however,some empiricalsupportforthe existenceof dominance
thedominance
oftwoallelescontrolling
modifiers.
Ford(1940)modified
relations
inonlythreegenerations
ofartificial
wingcolorinthemothAbraxasgrossulariata
selection(but see Ewens 1965).Indirectevidenceforthe modification
of the
dominanceof melanismin Bistonbetulariawas providedby Kettlewell(1965),
who crossed melanicEuropean B. betulariabetulariawithB. betulariacognitaria
fromNorthAmerica.The F1 showeda breakdownin the degreeof melanism.
whether
thiswas due to modifiers
ofdominance
However,itwas notdetermined
andheterozygotes.
orgeneralmodifiers
ofhomozygotes
results
Also,Kettlewell's
werenotreproduced
byWest(1977).Otherexamplesofdominancemodification
on dominancerelationsare discussedby
and the effectof geneticbackground
relations
can be modFord(1955,1965,pp. 28-39). We concludethatdominance
ified.
Implicationsfor Dominance Reversal
If dominanceresultsfromthe actionof modifiers
in the sense proposedby
modifier
loci couldnot
Fisher,thenthereis no reasonto supposethatdifferent
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PLEIOTROPY
ANTAGONISTIC
223
charactersindependently
controlthedominancerelationsof severalpleiotropic
(Sheppard1953; 1975,pp. 287-288).An examplethatis oftencited(see, e.g.,
Sheppard1953,1975,p. 286; Ford 1965,p. 27) in supportofdominancereversal
effects
thathaspleiotropic
a locusinthemothEphestiakuhniella
is thatinvolving
on viability,developmenttime,and male matingability(Caspari 1950). In
betweentwo of the
theredid appearto be antagonism
Caspari'sexperiments,
heteroeffects.Unfortunately,
whenanalyzinghomozygous
fitnesscomponents
fromhomozygotes.
The evidencesuggested
zygotescould notbe distinguished
wereeithersuperiorto or equal
forviability,
and heterozygotes
overdominance
timeandmating
ability.Theevidence
fordevelopmental
tothebetterhomozygote
as
withinall threefitnesscomponents
withoverdominance
is thusas consistent
it is withdominancereversal,a pointnotedby Caspari(1950,p. 379).
Kacser and Burns's(1981) analysiswas extendedby Keightleyand Kacser
characofthedominancerelationsofpleiotropic
(1987)to explicitconsideration
pathway
ters.The latteranalysisis basedon a modelofa branchedmultienzyme
In general,thedomicharacters).
witha singleinputandtwooutputs(pleiotropic
wereidenticalundervariationin encharacters
nanceindicesofthepleiotropic
zymeactivityat any stepin thepathway(i.e., paralleldominance).Differences
reversal,wereseenundercertain
dominance
inthedominanceindices,including
(i) Theremust
are thefollowing:
The necessaryconditions
conditions.
restrictive
pools,as wouldoccurifan
be strongnonlinearity
betweenfluxesand metabolite
feedback.(ii)Theallelicvariation
oriftherewerenegative
enzymeweresaturated
(iii)The
in questionmustnotaffectenzymesinthebranchwiththenonlinearity.
(iv)Theallelic
branchesmustcompeteforthesameinputsubstrate.
twodivergent
sensitive
a stepwhichis reasonably
variation
at thelocusinquestion"mustaffect
andKacserconcludethatthese
tochangesinenzymeactivity"(p. 326).Keightley
in
invivoandthattherefore
largedifferences
willnotoftenbe satisfied
conditions
search
charactersare unlikely.Theirliterature
dominancebetweenpleiotropic
ofpleiorelations
indominance
revealedonlythreeexamplesoflargedifferences
tropiccharacters,all cases of genes withlargeeffect.We also note thatthe
carriersof humanrecessivedisorders,such as
to detectheterozygous
inability
evidencethat
effects
is further
pleiotropic
by meansofdominant
cysticfibrosis,
dominancereversalis notcommon.However,we also notethatconditioniii,
be expectedtogiveriseto antagonisthatmight
above,isjustthesortofsituation
ticpleiotropy.
We are unawareof any systematic
surveyof dominancerelationsforalleles
Because itisjust suchloci
inphenotypic
effects.
withrelatively
smalldifferences
traits(such as fitness
variationin quantitative
in controlling
thatare important
thisis an important
empiricalquestion.The availableevidence
components),
suggeststhatalleles withsmalleffectshouldbe additiveand that,if thereis
characofdominanceshouldbe similarforpleiotropic
thedirection
dominance,
andRose (1991),whofound
ters.Thisconclusionis also supported
byHutchinson
in an experimental
relatedto longevity
fora varietyof characters
netadditivity
populationof Drosophila melanogasterS
of dominancesuggeststhat
of the mechanisms
To sum up, a consideration
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224
THE AMERICAN NATURALIST
(A)
w
(B)
w~~~~
W
wl
FIG. 6.-If fitnesscomponentswI and wlI are measureddirectlyand one is foundto be a
linear decreasingfunctionof the other,say, wlI = a - bwl, and if net fitnessW = wlwll,
then W = aw, - bwi, which is a parabola givingan intermediateoptimumvalue for the
fitnesscomponentwI. The relationis symmetricalso thatthe same procedurewill give an
intermediateoptimumforfitnesscomponentwll.
willbe mostcommonforpleiotropic
orparalleldominance
additivity
lociaffecting
or paralleldominance,polymorphism
traits.Withadditivity
is unquantitative
likely.
DISCUSSION
We open our discussionby considering
thetheoryof antagonistic
pleiotropy
in thelargercontextofan old andfamiliar
problem.Supposethatone measures
fitnesscomponents
as in the studiescitedabove by Rose, Luckinbill.
directly,
thatgeneratesa
Service,and others.Furthersupposethatthereis antagonism
negativelinearrelationship
betweenthetwofitness
components.
For multiplicativecomponents
thissituation
translates
intostabilizing
selectionwithoptimum
values forthe individualfitnesscomponents,
as shownin figure6. Put in this
context,thequestionat handbecomestheclassicone ofwhethertheoptimum
understabilizing
selectionis producedby heterozygotes
phenotype
or homozythetrait(Falconer1981,p. 309).Stabilizing
lociunderlying
gotesat thepolygenic
selectionforhomozygotes
tendsto reducegeneticvariance(Robertson1956;
Bulmer1971,1976).The question,then,is whether
selectionon fitness
stabilizing
overdominance.
arisesfrommultilocus
components
The totalnumberofconvincing
cases of single-locus
overdominance
is debatable but is probablyl6ss than10. Sickle-cellanemiacan be interpreted
as an
thatexhibitsbeneficialdominancereversal
exampleof antagonistic
pleiotropy
and overdominance
fortotalfitness.However,we doubtthatthereare many
andintensity
ofselectioncomparable
to sicklecell. It seems
geneswitha pattern
fortotalfitnessmaintains
allelicvariationat
veryunlikelythatoverdominance
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ANTAGONISTIC
PLEIOTROPY
225
suchas fitness
characters
quantitative
each ofthepossiblymanyloci underlying
components.
charquantitative
at thelociunderlying
Apartfromtheissueofoverdominance
thatcause us to questionthe role of
acters,thereare fourotherobservations
First,thetwo-locuscondipolymorphisms.
ingenerating
pleiotropy
antagonistic
thantheone-locus
to satisfy
are moredifficult
polymorphism
tionsforprotected
withmoreloci. Second, whenit
increasingconstraints
conditions,suggesting
on the
modelgenerates,
pleiotropy
theantagonistic
does producepolymorphism,
variancerelativetotheadditivevariance,which
dominance
average,considerable
Third,ifdomiobservations.
withmost(butnotall) experimental
is inconsistent
and Kacser
unlikely
nancereversalis relatively
to occurat one locus (Keightley
1987)or if it requiresthe questionableFisherianselectionprocesson different
loci foreach polygene,thenit seemsevenless likelyto occursimultamodifier
subjecttoweakselectionon
areprobably
neouslyat manyloci. Fourth,polygenes
by antagonisforpolymorphism
conditions
a per-locusbasis, buttherestrictive
withweak selection.If mostloci
tic pleiotropy
becomeeven morerestrictive
show paralleldominance,as arguedabove, thenthe likelihoodof overdominancefornetfitnesswithweak selectionis muchless thantheP = 0.25 given
earlier.
as
pleiotropy
ofantagonistic
aboutthegenerality
Whilewe have reservations
we do nothavereservations
a mechanism
formaintaining
geneticpolymorphisms,
Trade-offs
arewelldocuinfitness
components.
abouttheexistenceoftrade-offs
sexuallyselectedtraits,and charactersthat
characters,
mentedforlife-history
resources(see, e.g., Houle 1991and
are connectedvia allocationof limiting
of trade-offs
therein;Muelleret al. 1991).However,theobservation
references
meanthatthereis underlying
antagoat thephenotypic
leveldoes notnecessarily
corThe variation
mustbe geneticandthenegativephenotypic
nisticpleiotropy.
forthereto be antagonistic
relationmustbe due to a negativegeneticcorrelation
populationsthatthis
Also, it is even possiblewithsmalllaboratory
pleiotropy.
Since such information
is usually
is due to linkagedisequilibrium.
correlation
itis notpossibleto know,at thispoint,theactual
lackingin studiesoftrade-offs
pleiotropy.
prevalenceofantagonistic
that
whenitdoes occurourconclusionis thatitis veryunlikely
Nevertheless,
role in maintaining
geneticvariation
playsan important
pleiotropy
antagonistic
Othermodesofselectionshouldbe studiedandcompared
forfitness
components.
to see whetherthereare morefeasibleways that
withantagonistic
pleiotropy
We notethat,whileRose's original
theory
selectioncan maintain
polymorphism.
of antagonistic
(Rose 1982,1985)continuesto be citedextensively,
pleiotropy
to havereservations
his subsequentstudieshavecausedhimand hisco-workers
aboutthetheoryforsomeof thesamereasonsgivenin thisarticle.Rose et al.
as an inadequate
pleiotropy
(1987,p. 101) statedthat"we regardantagonistic
variation."We concurwiththis
basis forthe explanationof mostquantitative
The maintenance
of geneticvariationforfitness-related
quantitative
statement.
geproblemin evolutionary
traitshas been and continuesto be a fundamental
netics.
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226
THE AMERICAN NATURALIST
ACKNOWLEDGMENTS
This research is supported by U.S. Public Health Service fellowship F32
AGO5394 to P.M.S. and by grantsfromthe National Institutesof Health (K04
HD 00638 and PO1 AG 08761) to J.W.C. We thankM. Kirkpatrick,M. Rose,
and R. Shaw forcomments.
APPENDIX
A Two-LocusANTAGONISTICPLEIOTROPYMODEL
WithallelesAl and A2 at thefirstlocus and B, and B2 at the secondlocus, gametes
A,B,, A1B2, A2B1, and A2B2are denoted1-4, respectively.
The fitnessesof the nine
genotypesare
A1A, A1A2 A2A2
W33
BIB,
B,B2
WI,
W12
W13
W14
W34
B2B2
W22
W24
W44
whereWijis thefitness
ofthegenotype
formed
fromgametesi andj:
=
+
2H1
W11 (1
V)(1 2F);
W12= (1
-
F)(1 + HI V);
W22= [1 - V(l - HI)][1 - F(1 - H2)];
W13= (1 - F)(1 + HI V);
W14=
W23=
W24= (1
-
V)(I + H2F);
W33= [1 - V(1 - HI)][1W34 = (1
(Al)
1;
F(1-H2)];
V)(l + H2F);
-
W44= (1 + 2H2F)(1 - 2V).
The necessaryconditions
fora stablepolymorphic
in thetwo-locusmodel
equilibrium
are thatthefitness
ofthedoublehomozygote
at each cornerbe smallerthanthefitnesses
of bothof theadjacentsingleheterozygotes.
Solvingeach oftheeightinequalities
forV
resultsin thefollowing:
V
( F
VHI(I
-
(A2)
3F)'
FH2
V > 1F1H)1H
F
HI + F(1 - HI)(I - H2)
V>
H2F
(A3)
(A4)
(A5)
1 + 3H2F'
reducesthe eightinequalities
to thefourshownabove. Equations(A2) and
Symmetry
(A4) giveupperboundson V, whileequations(A3) and(A5) givelowerbounds.It is easily
shownthatupperboundequation(A4) is smaller"
thanupperboundequation(A2) andalso
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ANTAGONISTIC PLEIOTROPY
227
that lower bound equation (A3) is larger than lower bound equation (A5). Combining
equations (A3) and (A4) gives the conditionsstated in the text(eq. [7]).
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