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“Coarse”Notes
PopulationGenetics
Evolutionaryquantitativegeneticsandone-locuspopulationgenetics
READING:Nielsen&Slatkinpp.215–230
•Most“interesting”evolutionaryproblemsinvolvequestionsaboutphenotypicmeans
•Goal:determinehowselectioncausesevolutionarychangeinthemeanofaquantitative
character.
•Aside:
Quantitativevs.QualitativeTraits
•Backtoourmainstory....selectiononatraitsuchasbodysize
–bodysizeofindividualisdenotedz
–meanbodysizeinapopulation:𝑧
–Supposeselectionaltersallelefrequenciesatlocithataffectz
–Considertheeffectonbodysizeofonespecificlocus,i
–Ingeneral,todeterminehowmuch𝑧changes,oneneedstodetermine:
1)howselectionchanges𝑝 # atalllociaffecting𝑧
2)howchangesin𝑝 # combinetochange𝑧.
–We'llfocusontheeffectsofoneofthelocithataffects •Thephenotypeofanindividualdependson:
1)itsgenotypeatlocusi
2)itsgenotypeatotherlocithataffectthetrait
3)itsenvironmentalexperience
VII-1
“Coarse”Notes
PopulationGenetics
-lump2&3intoasinglefactor,c
•Canthinkofanindividual'sphenotypeas𝑧 = 𝑎&' + 𝑐
•Theaveragephenotypecansimilarlybebrokendownintotheaveragecontributionof
locusiandtheaverageoverotherlociandenvironmentaleffects:𝑧 = 𝑎 + 𝑐
•Considertheeffectofachangein𝑝 # on𝑧(holdingallotherlociconstant):
𝑑𝑧
∆𝑧 # = 𝑧 𝑝 # + ∆𝑝 # − 𝑧 𝑝 # ≈ ∆𝑝 # 𝑑𝑝
/0 2
/ 78 9
(dropi)
=
𝑝(1 − 𝑝)
/1 3
/1
(terminbracketsisjustouroldfriendtheadaptivetopography)
(usingthechainrule) =
-Summarizing:∆𝑧
#
/0 2
/1 3
≈
𝑝(1 − 𝑝)
2 /0 3
3 /1
/ 78 9 /0
/0
/1
𝑝(1 − 𝑝)
/ 78 9
/0
(Thisformemphasizeshowmeanfitnessdependsonmeanphenotype)
/0
•Whatis ?
/1
𝑑𝑧
𝑑
𝑑 3
=
𝑎 +𝑐 =
𝑝 𝑎22 + 2𝑝𝑞𝑎23 + 𝑞3 𝑎33 + 𝑐
𝑑𝑝
𝑑𝑝
𝑑𝑝
= 2𝑝𝑎22 + 2𝑞𝑎23 − 2𝑝𝑎23 + 2𝑞𝑎33 + 0
= 2(𝑝𝑎22 + 𝑞𝑎23 − 𝑝𝑎23 + 𝑞𝑎33 )
Puttingthesepiecestogether:
1
∆𝑧 # = ×𝑝𝑞×4(𝑝𝑎22 + 𝑞𝑎23 − 𝑝𝑎23 + 𝑞𝑎33 )3 2
VII-2
“Coarse”Notes
PopulationGenetics
or
∆𝑧
where
𝐺
and
#
#
= 𝐺 # 𝛽
= 2𝑝𝑞 𝑝𝑎22 + 𝑞𝑎23 − 𝑝𝑎23 + 𝑞𝑎33
𝛽=
𝑑 ln 𝑤
𝑑𝑧
3
•Whatis𝛽?
-calledtheselectiongradient
-measuresthestrengthof“directional”selectionactingonatraitz
•Whatis𝐺 # ?
-Itisacentrallyimportantquantitycalledtheadditivegeneticvariancecontributed
bylocusi
-Why“additive”?
•DominanceVariance
-Considertotalvariationinzduetovariationatlocusi:
var(z(i))=var(ajk+c)=var(aji)sinceassumecisfixed.
3
3
So,var(z(i))=var(aji)=𝐸 𝑎&'
− 𝐸 𝑎&' (assumingHWequilibrium)
3
3
3
= 𝑝3 𝑎22
+ 2𝑝𝑞𝑎23
+ 𝑞3 𝑎33
− 𝑝3 𝑎22 + 2𝑝𝑞𝑎23 + 𝑞3 𝑎33 3 =
(afteralotoftediousalgebra)
𝑎22 + 𝑎33 3
= 𝐺 # + 2𝑝𝑞 𝑎23 −
2
=𝐺 # +𝐷 # where 𝐷
#
= 2𝑝𝑞 𝑎23 −
FGG HFII
3
3
-D(i)iscalledthedominancevariancecontributedbylocusi
-Why“dominance”?
•RelationshipbetweenadditiveanddominanceVariance
VII-3
“Coarse”Notes
PopulationGenetics
-Case1:nodominance
-Case2:symmetricover-dominance
•Whataboutotherlocithatcontributetovariationinthetrait?
-Generallydifficult(recallcomplicationsof2-locuspopulationgenetics)to
understand
-Assuminglinkageequilibriumamongalllocicontributingtovariationinthetrait:
'
∆𝑧 = ∆𝑧
(2)
+ ∆𝑧
Ifwedefine𝐺 =
∆𝑧 = 𝐺𝛽
(3)
+ ⋯ + ∆𝑧
'
#
#K2 𝐺
(')
= 𝐺
,then
2
+𝐺
3
+ ⋯+ 𝐺
'
𝛽=
𝐺
#
𝛽
#K2
-Thisisaversionofthebreeder’sequation,whichisthecentralequationof
quantitativegenetics
-Noteitssimilaritytotheequationofselectionatonelocus
-Amoretraditionalformofthebreeder’sequationisR=h2S
•R=∆𝑧isthe“responsetoselection”
•S=the“selectiondifferential”=differencebetweenthemeansofthebreeders
andthepopulation
•h2isthe“heritability”=G/PwherePisthetotal(“phenotypic”)variabilityinz
duetoallgeneticandenvironmentalcauses
•Thisimpliesthat𝑆 = 𝑃𝛽
•P=G+D+Ewhere𝐷 = '#K2 𝐷 # tothetotal“dominancevariance”andEis
thevariationdueto“environmental”effects(includinggeneticinteractions
besidesdominance,suchasepistatis)
•Important:onlyGdeterminestheresponsetoselectionsinceadditiveeffects
aretheonly“predictable”componentsofgeneticvariation.
-Notethatadditive,dominancevarianceschangeasallelefrequencies
change.
VII-4