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RESEARCHOpenAccessMostofthelong-termgeneticgainfromoptimum-contributionselectioncanberealisedwithrestrictionsimposedduringoptimisationMarkHenryon1,2*,TageOstersen1,BirgitteAsk1,AndersCSrensen3andPeerBerg4AbstractBackground:Wetestedthehypothesisthatoptimum-contributionselection(OCS)withrestrictionsimposedduringoptimisationrealisesmostofthelong-termgeneticgainrealisedbyOCSwithoutrestrictions.
Methods:Weusedstochasticsimulationtoestimatelong-termratesofgeneticgainrealisedbybreedingschemesthatappliedOCSwithoutandwithrestrictionsimposedduringoptimisation,wherelong-termreferstogenerations23to25(approximately).
Sixrestrictionswereimposed.
Fiveoftheseremovedsolutionsfromthesolutionspace.
Thesixthremovedrecordsofselectiondecisionsmadeatearlierselectiontimes.
Wealsosimulatedaconventionalbreedingschemewithtruncationselectionasareferencepoint.
Generationsoverlapped,selectionwasforasingletrait,andthetraitwasobservedforallselectioncandidatespriortoselection.
Results:OCSwithrestrictionsrealised67to99%oftheadditionalgainrealisedbyOCSwithoutrestrictions,whereadditionalgainwasthedifferenceinthelong-termratesofgeneticgainrealisedbyOCSwithoutrestrictionsandourreferencepointwithtruncationselection.
Theonlyexceptionswerethoserestrictionsthatremovedallsolutionsneartheoptimumsolutionfromthesolutionspaceandtherestrictionthatremovedrecordsofselectiondecisionsmadeatearlierselectiontimes.
Imposingtheserestrictionsrealisedonly12to46%oftheadditionalgain.
Conclusions:Mostofthelong-termgeneticgainrealisedbyOCSwithoutrestrictionscanberealisedbyOCSwithrestrictionsimposedduringoptimisation,providedtherestrictionsdonotremoveallsolutionsneartheoptimumfromthesolutionspaceanddonotremoverecordsofearlierselectiondecisions.
InbreedingschemeswhereOCScannotbeappliedoptimallybecauseofbiologicalandlogisticalrestrictions,OCSwithrestrictionsprovidesausefulalternative.
Notonlydoesitrealisemostofthelong-termgeneticgain,OCSwithrestrictionsenablesOCStobetailoredtoindividualbreedingschemes.
BackgroundOptimum-contributionselection(OCS)maximisesthegeneticmeritofacohortofanimalswhileconstrainingtheaveragerelationshipofthecurrentgeneration[1-5].
OCSdoesthisbyoptimisingthegeneticcontribution(i.
e.
,numberofmatings)ofeachselectioncandidatetothecohort,conditionalonpredictedbreedingvaluesandadditive-geneticrelationships.
ThebenefitofOCS,besidesreducingtherisksofinbreeding,geneticdrift,andundesirablechangesingenefrequencies,isthatitcanmaximiselong-termgeneticgain,whichisthegoalofmostbreedingschemes[2,6].
Maximisinglong-termgeneticgainisrealisedbystrikingabalancebetweenshort-termratesofgeneticgainandinbreeding.
Thispromotesshort-termgeneticgainatratesofinbreedingthatdonotsubstantiallyerodeadditive-geneticvari-ation[7].
Despitethebenefit,OCSisnotwidelyusedinpracticalbreedingschemes.
Asfarasweareaware,ithasonlybeenappliedbyafewactorsinprogressivesectorsofthebreedingindustry[8].
Amajorreasonisthattheoptimumnumberofmatings,asdefinedbyOCS,cannotbeallocatedtoalloftheselectedanimalsbecauseofbiologicalandlogisticalrestrictions.
Inpig-breedingschemes,forexample,itmayonlybepossible*Correspondence:mahe@seges.
dk1Seges,DanishPigResearchCentre,Axeltorv3,1609CopenhagenV,Denmark2SchoolofAnimalBiology,UniversityofWesternAustralia,35StirlingHighway,Crawley,WA6009,AustraliaFulllistofauthorinformationisavailableattheendofthearticleGeneticsSelectionEvolution2015Henryonetal.
;licenseeBioMedCentral.
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Henryonetal.
GeneticsSelectionEvolution(2015)47:21DOI10.
1186/s12711-015-0107-7toallocateoptimumnumbersofmatingstoselectedsiresbecauseitisdifficulttoforeseewhichfemaleswillbeavailableforreproductionateachselectiontime.
Likewise,manypigbreederswillonlyusesireswiththehighestbreedingvaluesforbreeding.
OnewaytomakeOCSmorepracticalistoimposerestrictionsonOCSdur-ingoptimisation.
NotonlywouldthismakeOCSdecisionsreadilyapplicabletoindividualbreedingschemes,itmayevenrealisemost,ifnotall,ofthelong-termgeneticgainrealisedbyOCSwithoutrestrictionsfortworeasons.
First,inmostOCSanalyses,therearemanywaystoallocatenumbersofmatingstotheselectioncandidates(i.
e.
,manypossiblesolutionsinthesolutionspace)[8,9].
Itislikelythatmanysolutionslieat,ornear,theoptimumsolution.
ImposingrestrictionsonOCSmerelyremovessomeofthesolutionsfromthesolutionspace,sosolutionscanstillexistneartheoptimum.
Second,OCSisabletocorrectforselectiondecisionsmadeatearlierselectiontimesbytakingintoaccountthatsomeselectioncandidatesandancestralanimalshavealreadygeneratedoffspring[3,5].
Basedonthislineofreasoning,wehypothesisedthatOCSwithrestrictionsimposedduringoptimisationwillrealisemostofthelong-termgeneticgainrealisedbyOCSwithoutrestrictions.
Wetestedthishypothesisbystochas-ticsimulation.
Wesimulatedbreedingschemeswithrestrictionsthatwereinspiredby,butnotunique,topigbreeding.
MethodsProcedureWeusedstochasticsimulationtoestimatelong-termratesofgeneticgainrealisedbyOCSwithoutandwithrestrictionsimposedduringoptimisation,wherelong-termreferstogenerations23to25(approximately).
Wedidthisbysimulatingbreedingschemesthatlooselyre-sembledthoseusedforpigs.
Generationsoverlapped,se-lectionwasforasingletrait,andthetraitwasobservedforallselectioncandidatespriortoselection.
Wealsosimulatedaconventionalbreedingschemewithtrunca-tionselectionasareferencepoint.
Inallschemes,bestlinearunbiasedprediction(BLUP)breedingvalueswereusedasindicatorsofgeneticmerit.
ThebreedingschemethatappliedOCSwithoutrestrictionswasunrestrictedOCS.
UnrestrictedOCS.
Atotalof300matingswereallo-catedamongapproximately2250maleand2250femaleselectioncandidatesbyOCSattimettogenerateanewcohortofanimals,wherethetimeintervalttot+1rep-resentsafemalereproductivecycle.
Maleswerecandi-datesforselectionatages3to5(i.
e.
,bornattimest-3tot-5).
Therewasnoupperlimitforthenumberofmat-ingsthatwereallocatedtoeachmale;maleswereallo-cated0,1,2…or300matingsateachtime.
Femaleswerecandidatesatages4to6(i.
e.
,bornattimest-4tot-6).
Eachfemalewasallocated0or1matingateachtimeand300femaleswereallocatedamatingateachtime.
OCSwascarriedoutbymaximisingthegeneticmeritofthenewcohortwhileapplyingapenaltytotheaveragerelationshipofthecurrentgeneration,whichincludedthenewcohort.
Ninepenaltieswereapplied:5,10,20,50,100,200,500,1000,and5000.
Theaveragerelationshipwascalculatedusinganadditive-relationshipmatrixthatincludedmaleandfemaleselectioncandidates,immatureoffspringthatweretooyoungtobecandidates,andallancestorstracedbackfromtheseanimals.
The300sireanddammatingswerepairedrandomly.
Eachpairing(dam)producedfiveoffspring,resultingin300full-sibfamiliesand1500offspring.
Offspringwereassignedasmalesandfemaleswithaprobabilityof0.
5.
Allanimalswerephenotypedforthetraitunderselec-tionatage1(i.
e.
,bornattimet-1).
ThebreedingschemesthatappliedOCSwithrestric-tionsweresimulatedbyimposingrestrictionsonthebreedingscheme,unrestrictedOCS.
AnoverviewofthesebreedingschemesisinTable1.
Eachbreedingschemewassimulatedattheninepenalties.
Theschemeswithrestrictionscanbelooselygroupedintothreecategories.
Inthefirstcategory,restrictionswereimposedonfemalecandidates.
Theseschemesaretruncateddamsanddamsunknown.
Inthesec-ondcategory,restrictionswereimposedonbothmaleandfemalecandidates.
Theseschemesareone-chanceOCSofsires,pre-selectionofsires,andsiremultiples.
Inthethirdcategory,whichonlyincludestheschemeoffspringunknown,immatureoffspringthatweretooyoungtobeselectioncandidateswerenotknownbeforeOCS.
Thefirstandsecondcat-egoriesarerestrictionsthatremovesolutionsfromthesolutionspace.
Offspringunknown,inthethirdcategory,removesrecordsofselectiondecisionsmadeatearliertimes.
Wealsosimulatedabreedingscheme,multiplerestrictions,inwhichseveraloftherestrictionsthatremovesolutionsfromthesolutionspacewereimposedsimultaneously.
Thefollowingsectionsprovideadescriptionofthebreedingschemeswithrestrictions.
Truncateddams.
Truncateddamsisasforunre-strictedOCSwiththerestrictionthatOCSwasonlyappliedtomalecandidates.
Three-hundreddamsweretruncationselectedbasedonbreedingvalueateachtimeandeachselecteddamwasallocatedonemating.
SireswereselectedbyOCSconditionalonthetruncation-selecteddams.
TheselecteddamswereusedtoestimatethegeneticmeritofthenewcohortandHenryonetal.
GeneticsSelectionEvolution(2015)47:21Page2of14theywereincludedintheadditive-relationshipmatrix.
Theadditive-relationshipmatrixincludedmalecandi-dates,truncation-selecteddams,immatureoffspring,andallancestralanimalstracedbackfromtheseanimals.
Damsunknown.
Damsunknownisasfortruncateddamswiththeaddedrestrictionthatthetruncation-selecteddamswerenotknownbeforecarryingoutOCSofsires.
Thetruncation-selecteddamswerenotusedtoestimatethegeneticmeritofthenewcohortandtheywereexcludedfromtheadditive-relationshipmatrix.
Thisbreedingschemeresemblespracticalbreedingschemeswhereitisdifficulttoforeseewhichfemaleswillbeavailableforreproductionateachselec-tiontime.
One-chanceOCSofsires.
One-chanceOCSofsiresisasfortruncateddamswiththeaddedrestrictionthatmaleswereonlycandidatesforOCSatage3(i.
e.
,bornattimet-3).
OldermaleswerenotcandidatesforOCS.
Thisbreedingschemeresemblespracticalbreedingschemeswheremalesareonlycandidatesforselectionduringtheinitialstagesoftheirrepro-ductivelives.
Pre-selectionofsires.
Pre-selectionofsiresisasfortruncateddamswiththeaddedrestrictionthat0.
5,1,5,10,and25%ofthemalecandidateswerepre-selectedbytruncationselectionbasedonbreedingvaluebeforeOCS.
Onlymalesthatwerepre-selectedwerecandidatesforOCS.
Thisbreedingschemeresem-blespracticalbreedingschemeswhereanimalbreederswillonlyusesireswiththehighestbreedingvaluesforbreedingand/oritisexpensivetomaintainbreedinganimals.
Siremultiples.
Siremultiplesisasfortruncateddamswiththeaddedrestrictionthatnumbersofmatingswereallocatedtosiresinmultiplesof5,10,20,50,and100.
Whenthemultiplewas5,forexample,sirescouldonlybeallocated0,5,10,…300matings.
Thisbreedingschemeresemblespracticalbreedingschemeswherethenucleuspopulationismaintainedinmultipleherdsandthenumberofmatingsallocatedtoeachselectedsirearedistributedequallyacrossherds.
Offspringunknown.
Offspringunknownisasfortrun-cateddamswiththeaddedrestrictionthatimmatureoffspring,tooyoungtobeselectioncandidates,werenotknownbeforecarryingoutOCSofsires.
Theoffspringwereexcludedfromtheadditive-relationshipmatrix.
Theadditive-relationshipmatrixonlyincludedmalecan-didates,truncation-selecteddams,andancestralanimalstracedbackfromtheseanimals.
Thisrestrictionremovesrecordsofselectiondecisionsmadebeforetimet.
Thebreedingschemeresemblespracticalbreedingschemeswhereyounganimalsarefirstrecordedindatabaseslaterinlife.
Table1Breedingschemesapplyingoptimum-contributionselection(OCS)withrestrictionsRestrictionsBreedingschemeTruncateddamsDamsunknownOne-chancemalesPre-selectionmalesSiremultiplesOffspringunknownTruncateddamsDamsunknownOne-chanceOCSofsiresPre-selectionofsires25%25Pre-selectionofsires10%10Pre-selectionofsires5%5Pre-selectionofsires1%1Pre-selectionofsires0.
5%0.
5Siremultiples55Siremultiples1010Siremultiples2020Siremultiples5050Siremultiples100100OffspringunknownMultiplerestrictions510Therestrictionsaretruncationselectionofdams(Truncateddams),truncation-selecteddamsnotknownbeforecarryingoutOCSofsires(Damsunknown),OCSlimitedtoyoungmales(One-chancemales),pre-selectionofmalesbytruncationselectionbeforeOCS(Pre-selectionmales),numbersofmatingsallocatedtosiresbyOCSinmultiples(Siremultiples),andimmatureoffspringnotknownbeforeOCS(Offspringunknown).
Filledcircles()andnumbersindicatethatrestrictionwasimposed,wherenumbersindicatetheproportion(%)ofmalesthatwerepre-selectedandthemultipleusedtoallocatenumbersofmatingstosires.
Henryonetal.
GeneticsSelectionEvolution(2015)47:21Page3of14Multiplerestrictions.
Severaloftherestrictionsthatremovesolutionsfromthesolutionspacewereimposedsimultaneously:Truncateddams:OCSwasonlyappliedtomalecandidatesand300damsweretruncationselectedateachtime.
Damsunknown:Truncation-selecteddamswerenotknownbeforecarryingoutOCSofsires.
One-chanceOCSofsires:MaleswereonlycandidatesforOCSatage3.
Pre-selectionofsires5%:5%ofthemalecandidateswerepre-selectedbeforeOCS.
Siremultiples10:Numbersofmatingswereallocatedtosiresinmultiplesof10.
Truncationselection10.
Truncationselection10isaconventionalbreedingschemewith10siresand300damstruncationselectedbasedonbreedingvalueateachtime.
Eachsirewasrandomlymatedwith30damsandeachdam(mating)producedfiveoffspringthatresultedin300full-sibfamiliesand1500offspring.
Breedingschemeswererunfor100times(t=1,…100,approximately25generations).
Inthefirst20times,trun-cationselectionwascarriedoutbyapplyingtruncationselection10.
ThisestablishedaselectedpopulationbeforeOCSwasappliedattimes21to100.
Wesimulated100replicatesforeachbreedingscheme.
ThebreedingschemesthatappliedOCSwerecomparedatpenaltiesthatmaximisedlong-termratesofgeneticgainforthetraitunderselection.
TraitThetraitunderselectionwasassumedtobenormally-distributedandgeneticallycontrolledbytheinfinitesimalmodelofadditive-geneticeffects.
Ithadaheritabilityof0.
20andadditive-geneticvarianceof1.
0.
SamplingBreedingschemeswereinitiatedbysamplinganunrelatedbasepopulationof20siresand600dams.
Thephenotypeofthetraitfortheithbaseanimal,pi,wascalculatedaspi=ai+ei,whereaiisthebaseanimal'strueadditive-geneticvalueandeiisitsresidualenvironmentalvalue.
Thetrueadditive-geneticvaluewassampledfromaieN0;σ2a1andtheenvironmentalvaluefromeieN0;σ2e4.
Phenotypesofanimalsinsubsequentgenerationswerecalculatedasdescribedforthebasepopulationwiththeexceptionthatthetruebreedingvaluesoftheithanimal,ai,wassampledasaieN1=2asiadi;1=21Fip;whereasiandadiareadditive-geneticvaluesofthesire,si,andthedam,diofanimali,andFiistheaverageinbreedingcoefficientofsianddi.
PredictionBLUP-breedingvalueswerepredictedbyfittingananimalmodeltothephenotypes.
Themodelwas:yXbZae;whereyisavectorofphenotypesobservedforselectioncandidatesandancestralanimals,bisavectoroffixedbirth-timeeffects,aisavectorofrandomanimaleffects,eisavectorofresidualerrors,andXandZareinci-dencematrices.
The(co)variancestructureusedtopredictthebreed-ingvalueswas:aeeN00!
;Aσ2a00Iσ2e!
;whereAistheadditive-relationshipmatrixandIisanidentitymatrix.
Thevariances,σ2aandσ2e;werethesameasthoseusedtosampleanimals.
Optimum-contributionselectionOCSallocatedmatingstoselectioncandidatesattimetbymaximisingthequadraticfunction,Ut,withrespecttoc:Utcc0^aωL2cPv0AcPv1wherecisanvectorofgeneticcontributionstothenewcohortandthenumberofmatingsallocatedtoeachcan-didateisalinearfunctionofthesecontributions,isanvectorofBLUP-breedingvalues,ωisthepenaltyappliedtotheaveragerelationshipofthecurrentgeneration,Listhegenerationinterval,visakvectoroflifetime-breedingprofilesorexpectedrelativecontributionstofutureage-classes,Pisanxkmatrixofcontributionstoeachage-classofanimalsinthecurrentgeneration,Aisanxnmatrixofadditive-geneticrelationships,nisthetotalnumberofanimalsinthepopulationpedigree,whichincludesselectioncandidates,immatureoffspring,andallancestorstracedbackfromtheseanimals,andkisthenumberofsex-ageclassesinthecurrentgener-ation.
Usingthesedefinitions,itfollowsthatc'istheaveragebreedingvalueofthenewcohort,(c+Pv)/Lisavectorofcontributionstothecurrentgeneration,Pv/Lisavectorofcontributionsmadetothecurrentgener-ationbeforetimet,and((c+Pv)'A(c+Pv))/L2istheaveragerelationshipofthecurrentgeneration.
OurmethodofcarryingoutOCSissimilartothatofWrayandGoddard[1],whoalsoappliedapenaltytoaverageHenryonetal.
GeneticsSelectionEvolution(2015)47:21Page4of14relationships,butitdiffersfromthatofMeuwissen[2],whoconstrainedratesofinbreedingtopre-definedlevels.
Intheremainderofthissection,wedescribetheparameters,c,v,P,andL,andthemethodsthatweusedtoimposethere-strictionsduringoptimisation.
Geneticcontributions.
Thevectorofgeneticcontribu-tions,c,wassolvedwithlinearconstraintsimposed.
Letelementsofc,cmpandcfq,bethecontributionsofthepthmaleandqthfemale,thencmp0;1600;2600;…;300600;cfq0;1600;Xnmp1cmp0:5;andXnfq1cfq0:5withcmp=0andcfq=0formalesandfemalesthatwerenotcandidatesforselection,andnmandnfarethetotalnum-bersofmalesandfemalesinthepopulationpedigree(n=nm+nf).
Thecontributionsallocatedtothepthmaleandqthfemaleweretransformedtonumbersofmatingsby600·cmpand600·cfq.
Lifetime-breedingprofiles.
Thelifetime-breedingpro-fileofanimalsatagelistheproportionofmatingopportunitiesduringtheirlifetimethatwasexpectedtoberealisedbeyondagel[3,5].
Males,whichwerecandidatesforselectionatages3to5,hadkm=5ageswithlifetime-breedingprofiles.
Thesewereatages0to4.
Femaleswerecandidatesatages4to6.
Theyhadkf=6ageswithlifetime-breedingprofilesatages0to5.
Oldermalesandfemalesdidnothavelifetime-breedingprofilesbecausemalesandfemaleswerenolongercandidatesbeyondages5and6.
Thevectoroflifetime-breedingprofileswasdefinedasv1=2vmvf!
;wherevmandvfarekm=5andkf=6(k=km+kf=11)vectorsoflifetime-breedingprofilesformalesatages0to4andfemalesatages0to5,suchthat,vmlbpm0lbpm1lbpm2lbpm3lbpm4266664377775;vflbpf0lbpf1lbpf2lbpf3lbpf4lbpf52666666437777775;andlbpmjandlbpfjarethelifetime-breedingprofilesofmalesandfemalesatagej.
Elementslbpmjandlbpfjwerecalculatedaslbpmj1Xj1u0εmuandlbpfj1Xj1u0εfu;whereεmuandεfuaretheproportionsofoffspringfrommalesandfemalesatageu.
Thefirstthreeelementsofvmandthefirstfourofvfwere1becausemalesatages0to2andfemalesatages0to3wereimmatureoffspringthatweretooyoungtobeselectioncandidates.
Alloftheirmatingopportunitieswererealisedbeyondages2and3.
Geneticcontributionstoanimalsinthecurrentgeneration.
ThematrixPwasdefinedasPPm00Pf!
;wherePmandPfarenmxkm=nmx5andnfxkf=nfx6matrices.
Pmcontainsgeneticcontributionsofthenmmalestothekmage-classeswithlifetime-breedingprofiles[3,5].
Pfcontainscontributionsofthenffemalestothekfage-classes.
MatricesPmandPfhavethefollowingform:Pm0nml0nml0nml0nml0nml0nm90nm90nm90nm9cnm9;40nm80nm80nm8cnm8;3cnm8;40nm70nm7cnm7;2cnm7;3cnm7;40nm6cnm6;1cnm6;2cnm6;30nm60nm5cnm5;1cnm5;20nm50nm50nm4cnm4;10nm40nm40nm40nm30nm30nm30nm30nm30nm20nm20nm20nm20nm20nm10nm10nm10nm10nm1266666666666666664377777777777777775andPf0nfl0nfl0nfl0nfl0nfl0nfl0nf110nf110nf110nf110nf11cnf11;50nf100nf100nf100nf10cnf10;4cnf10;50nf90nf90nf9cnf9;3cnf9;4cnf9;50nf80nf8cnf8;2cnf8;3cnf8;40nf80nf7cnf7;1cnf7;2cnf7;30nf70nf70nf6cnf6;1cnf6;20nf60nf60nf60nf5cnf5;10nf50nf50nf50nf50nf40nf40nf40nf40nf40nf40nf30nf30nf30nf30nf30nf30nf20nf20nf20nf20nf20nf20nf10nf10nf10nf10nf10nf126666666666666666666643777777777777777777775:ThefirstcolumnofPmandPfrepresentsthegeneticcontributionsofmalesandfemalestothenewcohortandis,therefore,0.
ThesecondandsubsequentcolumnsofPmrepresentthecontributionsofmalestoanimalsatages1to4.
ThesecondandsubsequentcolumnsofPfrepresentthecontributionsoffemalestoanimalsatages1to5.
0nmland0nflarenmlandnflvectorsof0s,wherenmlandnflarethenumbersofmalesandfemalesatagel.
cnml;jandcnfl;jarenmlandnflvectorsofcontributions,whereeachelementofcnml;jandcnfl;jisthecontributionofthepthmale(p=1,…nml)orqthfemale(q=1,…nfl)atageltoanimalsatagej.
Generationinterval.
Generationinterval,L,representstheaverageageofanimalswhentheiroffspringwereborn(i.
e.
,numberoftimesforagenerationtoreplicateitself).
Itwascalculatedasthesumoftheelementsofv,Henryonetal.
GeneticsSelectionEvolution(2015)47:21Page5of14suchthat,LXkj1vj1=2Xkmj1vmjXkfj1vfj:ImposingrestrictionsonUt(c).
Restrictionswereim-posedduringoptimisationofUt(c)byfixingelementsofcandP.
Elementsofcwerefixedwhenimposingrestrictionsthatremovedsolutionsfromthesolutionspace.
ElementsofPwerefixedfortherestrictionthatremovedrecordsofselectiondecisionsmadeatearliertimes.
Truncateddamsanddamsunknown.
Inthebreedingschemetruncateddams,wheresireswereselectedbyOCSand300damsweretruncationselected,elementsofcassociatedwithtruncation-selecteddamswerefixedto1/600(i.
e.
,eachtruncation-selecteddamcontributedonemating).
Elementsofcweresetto0forallotherfemales.
Inthebreedingschemedamsunknown,wherethetruncation-selecteddamswerenotknownbeforecarryingoutOCSofsires,allelementsofcassociatedwithfemaleswerefixedto0withtheexceptionofanunrelated'dummy'dam,whichwasaddedtothedatasetwithacontributionof0.
5.
One-chanceOCSofsires,pre-selectionofsires,andsiremultiples.
Restrictionswereimposedonfemalesbyfix-ingelementsofcto1/600fortruncation-selecteddamsand0forallotherfemales.
Inthebreedingschemeone-chanceOCSofsires,restrictionswereimposedonmalesbyfixingelementsofcto0forallmalesthatwerenotatage3.
Inthebreedingschemepre-selectionofsires,onlymalesthatwerepre-selectedwerecandidatesforOCS.
Elementsofcassociatedwithallothermaleswerefixedto0.
Inthebreedingschemesiremultiples,elementsofcassociatedwithmalecandidateswererestrictedto0,x/600,2x/600,…0.
5,wherexisthesiremultiple(x=2,5,10,20,50,and100).
Offspringunknown.
Contributionstoimmatureoff-springwereremovedbyfixingcolumns2and3ofPmandcolumns2–4ofPfto0.
DataanalysesForeachbreedingscheme,weplottedthelong-termrateofgeneticgainrealisedatthepenaltythatmaximisedlong-termrateofgeneticgainforthetraitunderselec-tion.
Long-termrateswereassessedastheproportionofadditionalgainrealised,whereadditionalgainwasthedifferenceinthelong-termratesofgeneticgainrealisedbyunrestrictedOCSatpenalty50andourreferencepoint,truncationselection10.
Preliminaryanalysisshowedthatthelong-termrateofgeneticgainrealisedbyunrestrictedOCSwasmaximisedatpenalty50.
Ourreasoningforchoosingtruncationselection10asareferencepointisoutlinedintheAppendix.
TheAppendixalsohighlightsthatchoosingareferencepointwassubjectivesinceitisdifficulttofindconventionalbreedingschemeswithtruncationselectionthatmatchOCS.
WealsopresentfindingsthatprovideinsightintothemechanismsthatunderlieOCSwithrestrictions:Penaltiesthatmaximisedlong-termratesofgeneticgain.
Short-termratesofgeneticgainandinbreeding.
Plotofshort-termratesofgeneticgainagainstshort-termratesofinbreedingrealisedatthepenaltiesthatmaximisedlong-termratesofgeneticgain.
Weoverlaidthisplotwiththeshort-termresponsefrontierforunrestrictedOCS,wheretheshort-termresponsefrontierisshort-termrateofgeneticgainrealisedateachpenaltyplottedasafunctionofshort-termrateofinbreeding.
Short-termgenerationintervals.
Short-termnumbersofsireswithallocatedmatings.
Shortandlong-termresponsefrontiersforunrestrictedOCS.
Shortandlong-termratesrefertoanimalsbornattimest=26to35(approximatelygenerations6to8)andt=91to100(approximatelygenerations23to25).
RatesofgeneticgainwerecalculatedasthelinearregressionofStont,whereStistheaveragetrue-breedingvalueofanimalsbornattimet.
Ratesofinbreedingwerecalcu-latedas1-exp(β),whereβisthelinearregressionofln(1-Ft)ontandFtistheaveragelevelofinbreedingforanimalsbornattimet.
Theratesofgeneticgainandinbreedingwerescaledbysettingto100thelong-termratesofgeneticgainandinbreedingrealisedbyunre-strictedOCSatpenalty50.
Preliminaryanalysisshowedthatrateofgeneticgainat100wasequivalentto0.
215genetic-standarddeviationspertimeandapproximately0.
9genetic-standarddeviationspergeneration.
Rateofinbreedingat100wasequivalentto0.
0020pertimeandapproximately0.
008pergeneration.
Allresultsarepre-sentedasmeans(±s.
d.
)ofthe100simulationreplicates.
RatesofgeneticgainandinbreedingarepresentedasfunctionsoftimebecauseOCSmaximisesratesofgen-eticgainineachcohort.
Weacknowledgetheargumentsforpresentingratesofinbreedingpergeneration,namelythatgeneticvariationerodesandmutationsac-cumulatepergeneration.
However,preliminaryanalysesshowedthatgenerationintervaldidnotdiffermarkedlybetweenschemes.
Therelativeratesofinbreedingbe-tweenschemeswere,therefore,similarwhenpresentedpertimeorpergeneration.
SoftwareTheschemesweresimulatedusingtheprogram,ADAM[10].
Eachschemereplicatewasinitiatedwitharandomseed.
BLUP-breedingvalueswerepredictedusingtheHenryonetal.
GeneticsSelectionEvolution(2015)47:21Page6of14program,DMU6[11].
OCSwascarriedoutusingthepro-gram,EVA[12].
EVAmaximisedthequadraticfunction(Equation(1))usinganevolutionaryalgorithm[9,13].
ResultsLong-termratesofgeneticgainOCSwithrestrictionsrealisedmostoftheadditionallong-termgeneticgainthatwasrealisedbyOCSwithoutrestric-tions(Figure1).
OCSwithoutrestrictionsrealised18%additionalgain,whereadditionalgainwasthedifferenceinthelong-termratesofgeneticgainrealisedbyunrestrictedOCSatpenalty50andourreferencepoint,truncationse-lection10.
Inallbutafewexceptions,OCSwithrestric-tionsatpenaltiesthatmaximisedlong-termratesofgeneticgainrealised67to99%ofthisadditionalgain.
Evenmultiplerestrictions,whereseveraloftherestrictionsthatremovesolutionsfromthesolutionspacewereimposedsimultaneously,realised89%oftheadditionalgain.
Theexceptionswerepre-selectionofsires0.
5%,siremultiples50and100,andoffspringunknown.
Imposingtheserestric-tionsrealisedonly12to46%oftheadditionalgain.
ThefollowingsectionspresentfindingsthatprovideinsightintothemechanismsthatunderlieOCSwithrestrictions.
Penaltiesthatmaximisedlong-termratesofgeneticgainThepenaltiesthatmaximisedlong-termratesofgeneticgainineachbreedingschemewithOCSwere50and100(Table2).
Theexceptionswerepre-selectionofsires0.
5and1.
0%andmultiplerestrictions.
Inpre-selectionofsires0.
5and1.
0%,thepenaltiesthatmaximisedlong-termratesofgeneticgainwere5000and200.
Itwas200inmultiplerestrictions.
Short-termratesofgeneticgainandinbreedingOCSwithrestrictionsrealisedshort-termratesofgen-eticgainandinbreedingthatdifferedfromtheshort-termratesrealisedbyOCSwithoutrestrictions(Table2).
Atthepenaltiesthatmaximisedlong-termgeneticgain,OCSwithrestrictionsrealised94to102%oftheshort-termratesofgeneticgainand76to235%oftheshort-termratesofinbreedingrealisedbyunrestrictedOCS.
Notonlydidtheseratesdiffer,theyformedadistinctpatternwhenweplottedtheshort-termratesofgeneticgainagainstshort-termratesofinbreedingforeachre-strictionandoverlaidthisplotwiththeshort-termre-sponsefrontierforunrestrictedOCS(Figure2).
Inthisplot,theshort-termratesofgeneticgainandinbreedingrealisedbyOCSwithmostofourrestrictionsalignedthemselvesalongtheresponsefrontierandwerecentredroundtheshort-termratesrealisedbyunrestrictedOCSatpenalty50,thebreedingschemethatmaximisedlong-termratesofgeneticgain.
Theexceptionswerepre-selectionofsires0.
5%,siremultiples50and100,andoffspringunknown,namelythesamerestrictionsthatfailedtorealisemostofthelong-termgeneticgain.
TheLong-termrateofgeneticgain80859095100UnrestrictedOCSDamsunknownOnechanceMultipleTruncateddamsPre-selectionofsires2550.
5101Siremultiples1005201050TSOffspringunknownFigure1Maximumlong-termratesofgeneticgainrealisedbyoptimum-contributionselection(OCS)withoutandwithrestrictions.
BreedingschemeapplyingOCSwithoutrestrictionsisUnrestrictedOCS(dark-shadedbar).
SchemesapplyingOCSwithrestrictionsareTruncateddams,Damsunknown,One-chance(representingbreedingscheme,one-chanceOCSofsires),Pre-selectionofsires25,10,5,1,and0.
5%,Siremultiples5,10,20,50,and100,Offspringunknown,andMultiple(multiplerestrictions).
Theseschemesarepresentedaslight-shadedbars,exceptthosethatfailedtorealisemostofthelong-termgeneticgainrealisedbyUnrestrictedOCS(unshadedbars).
Long-termrateofgeneticgainrealisedbyaconventionalschemewithtruncationselection(TS)ispresentedasareferencepoint(dottedline).
Long-termreferstogenerations23to25(approximately).
Theratesaremeansof100simulationreplicates.
Themeanshadstandarddeviationsrangingfrom5.
8to12.
0.
Therateswerescaledbysettingto100themaximumlong-termrateofgeneticgainrealisedbyOCSwithoutrestrictions.
Rateofgeneticgainat100isequivalentto0.
215genetic-standarddeviationspertimeandapproximately0.
9genetic-standarddeviationspergeneration.
Henryonetal.
GeneticsSelectionEvolution(2015)47:21Page7of14short-termratesrealisedbyOCSwiththeserestrictionsdeviatedfurthestfromtheresponsefrontierandtheshort-termratesrealisedbyunrestrictedOCSatpenalty50.
Short-termgenerationintervalsShort-termgenerationintervalsatthepenaltiesthatmaximisedlong-termratesofgeneticgainrangedfrom4.
0to4.
1times(Table2).
Theexceptionswereone-chanceOCSofsires,pre-selectionofsires0.
5and1.
0%,whichhadslightlyshortergenerationintervals(3.
8to3.
9).
Damsunknownandoffspringunknownhadslightlylongergenerationintervals(4.
2and4.
3).
Short-termnumbersofsireswithallocatedmatingsMatingswereallocatedtoapproximately13siresduringtheshort-termtimeperiodofunrestrictedOCSatpenalty50(Table2).
Similarnumbersofsireswereallocatedmat-ingsintruncateddams,pre-selectionofsires25%,siremultiples5,andmultiplerestrictionsatthepenaltiesthatmaximisedlong-termgeneticgain(11.
9to13.
6sirespertime).
Moresireswereallocatedmatingsindamsun-knownandone-chanceOCSofsires(23.
4and17.
6).
Fewerwereallocatedmatingsinoffspringunknown(9.
7).
Inpre-selectionofsires,thenumberofsiresallocatedmatingsincreasedfrom12.
7to20.
0astheproportionofpre-selectedsiresfellfrom25%to10%.
Itthendecreasedto7.
3astheproportionfellto0.
5%.
Thenumberofallocatedmatingsinsiremultiplesdecreasedfrom11.
9to3.
0asthemultiplewasincreasedfrom5to100.
Shortandlong-termresponsefrontiersLong-termratesofgeneticgainweremaximisedatpenaltiesthatpromotedshort-termratesofgeneticgainwithoutsubstantiallyerodingadditive-geneticvariation.
Thisisillustratedbyourshortandlong-termresponsefrontiersforunrestrictedOCS(Figure3).
InunrestrictedOCS,wherelong-termrateofgeneticgainwasmaxi-misedatpenalty50,adecreaseinpenaltyfrom50to5reducedlong-termrateofgeneticgainto62%.
Thiswasbecauseadecreaseinpenaltyfrom50to5realisednomorethan105%oftheshort-termgeneticgainrealisedatpenalty50,butitincreasedtheshort-termrateofinbreedingby564%.
Ontheotherhand,increasingthepenaltyfrom50to5000reducedlong-termgeneticgaintoonly34%.
Thiswasbecauseanincreaseinpenaltyfrom50to5000realisedonly16%oftheshort-termTable2Outputparametersthatprovideinsightintothemechanismsunderlyingoptimum-contributionselection(OCS)withrestrictionsBreedingschemePenaltyΔGshortΔFshortLshortnSiresshortUnrestrictedOCS50109±9.
7114±27.
04.
1±0.
0413.
3±1.
19Truncateddams50109±9.
6132±45.
74.
1±0.
0412.
8±1.
10Damsunknown100102±7.
687±30.
24.
2±0.
0423.
4±1.
72OnechanceOCSofsires50106±8.
795±22.
23.
8±0.
0117.
6±1.
25Pre-selectionofsires25%50108±10.
2137±51.
04.
1±0.
0412.
7±1.
08Pre-selectionofsires10%100103±8.
088±26.
94.
1±0.
0320.
0±2.
03Pre-selectionofsires5%100107±10.
5105±31.
14.
0±0.
0317.
9±2.
07Pre-selectionofsires1%200111±10.
7192±60.
83.
9±0.
0311.
7±1.
69Pre-selectionofsires0.
5%5000110±10.
6268±85.
23.
9±0.
047.
3±0.
70Siremultiples550108±10.
2139±50.
64.
1±0.
0511.
9±1.
04Siremultiples1050108±9.
7137±43.
34.
1±0.
0510.
8±0.
77Siremultiples2050107±12.
1134±52.
04.
1±0.
058.
9±0.
58Siremultiples5050104±11.
1148±51.
44.
0±0.
065.
5±0.
21Siremultiples1005095±12.
2172±55.
94.
0±0.
073.
0±0.
00Offspringunknown50106±15.
1189±76.
14.
3±0.
059.
7±1.
11Multiplerestrictions50108±8.
5102±24.
43.
8±0.
0313.
6±0.
80Truncationselection10112±10.
0413±161.
14.
1±0.
0310BreedingschemeapplyingOCSwithoutrestrictionsisUnrestrictedOCS.
AllotherschemesappliedOCSwithrestrictions,exceptTruncationselection10,whichisaconventionalschemewithtruncationselection.
Theoutputparametersarepenaltyonaveragerelationshipthatmaximisedlong-termrateofgeneticgain,short-termratesofgeneticgainandinbreeding(ΔGshort,ΔFshort),short-termgenerationinterval(Lshort),andshort-termnumberofsireswithallocatedmatingsperselectiontime(nSiresshort),whereshortandlong-termrefertogenerations6to8andgenerations23to25(approximately).
Theratesofgeneticgainandinbreeding,generationintervals,andnumbersofsiresarepresentedasmeans±s.
d.
of100simulationreplicates.
Ratesofgeneticgainandinbreedingwerescaledbysettingto100thelong-termratesofgeneticgainandinbreedingrealisedbyUnrestrictedOCSatpenalty50.
Rateofgeneticgainat100isequivalentto0.
215genetic-standarddeviationspertimeandapproximately0.
9genetic-standarddeviationspergeneration.
Rateofinbreedingat100isequivalentto0.
0020pertimeandapproximately0.
008pergenerationontheobservedscale.
Henryonetal.
GeneticsSelectionEvolution(2015)47:21Page8of14geneticgainrealisedatpenalty50,althoughitdecreasedshort-termrateofinbreedingto8%.
Thereweretwofurtherobservationsfromthefron-tiers.
First,shortandlong-termratesofinbreedingweremoresensitivetochangesinpenaltythanshortandlong-termratesofgeneticgainatpenaltiessmallerthan100.
Atpenaltieslargerthan100,ratesofgeneticgainweremoresensitivethanratesofinbreeding.
Second,short-termratesofgeneticgainwerehigherthanlong-termratesatpenaltiessmallerthan200.
Forpenaltieslargerthan200,long-termratesofgeneticgainwerehigherthanshort-termrates.
DiscussionOurfindingssupportedourhypothesisthatOCSwithrestrictionsimposedduringoptimisationrealisesmostofthelong-termgeneticgainrealisedbyOCSwithoutre-strictions.
Realising67to99%oftheadditionalgainwithmanyofourrestrictionsdemonstratesthatOCSisaro-bustselectionmethod.
Thisrobustnesswasevenevidentwithmultiplerestrictions,whereseveralrestrictionsthatremovesolutionsfromthesolutionspacewereimposedsimultaneously.
InbreedingschemesforwhichOCScannotbeappliedoptimallybecauseofbiologicalandlogisticalrestrictions,OCSwithrestrictionsprovidesausefulalternative.
Notonlydoesitrealisemostofthelong-termgeneticgain,OCSwithrestrictionsenablesOCStobetailoredtoindividualbreedingschemes,wheretheoptimumnumberofmatings,giventhere-strictions,canbereadilyallocatedtoavailableselectioncandidates.
Itwasonlywhentherestrictionsbecametoostrict,wherebyallsolutionsneartheoptimumsolutionwereremovedfromthesolutionspaceorrecordsofselectiondecisionsmadeatearlierselectiontimeswereremoved,thatwefailedtorealisemostoftheadditionalgain.
So,providedtherestrictionsarenottoostrict,mostofthelong-termgeneticgainrealisedbyOCSwithoutrestrictionscanberealisedbyOCSwithrestric-tionsimposedduringoptimisation.
OCSwithmanyofourrestrictionsrealisedmostofthelong-termgeneticgainfortworeasons.
First,solutionsneartheoptimumstillexistedinthesolutionspacesaftertherestrictionshadbeenimposed.
Byapply-ingdifferentpenaltiestotheaveragerelationship,wegaveOCStheopportunitytosearchthereducedsolu-tionspacestofindalternatesolutions.
Second,OCSwasabletocorrectforearlierselectiondecisionsbytakingintoaccountthatsomeanimalshadalreadycontributed90100110120Short-termrateofgeneticgain0100200300400500Short-termrateofinbreeding205020010010Figure2Short-termratesofgeneticgainandinbreedingrealisedbyoptimum-contributionselection(OCS)withrestrictions.
Short-termratesrealisedbybreedingschemesapplyingOCSwithrestrictionsareatpenaltiesonaveragerelationshipthatmaximisedlong-termratesofgeneticgain,whereshortandlong-termrefertogenerations6to8andgenerations23to25(approximately).
Theratesarerepresentedbyshadedsquares,exceptforschemesthatfailedtorealisemostofthelong-termgeneticgainrealisedbyOCSwithoutrestrictions,namelypre-selectionofsires0.
5%,siremultiples50and100,andoffspringunknown(emptycircles).
Theplotisoverlaidwiththeshort-termresponsefrontierforOCSwithoutrestrictions(linewithfilledcircles).
Theshort-termresponsefrontierisshort-termrateofgeneticgainrealisedatfivepenaltiesplottedasafunctionofshort-termrateofinbreeding.
Thepenaltiesare10,20,50,100,and200.
Thelong-termrateofgeneticgainrealisedbyOCSwithoutrestrictionswasmaximisedatpenalty50.
Theratesaremeansof100simulationreplicates.
Therateswerescaledbysettingto100thelong-termratesofgeneticgainandinbreedingrealisedbyOCSwithoutrestrictionsatpenalty50.
Rateofgeneticgainat100isequivalentto0.
215genetic-standarddeviationspertimeandapproximately0.
9genetic-standarddeviationspergeneration.
Rateofinbreedingat100isequivalentto0.
0020pertimeandapproximately0.
008pergenerationontheobservedscale.
Henryonetal.
GeneticsSelectionEvolution(2015)47:21Page9of14tothecurrentgeneration.
Whenweapplieddifferentpenaltiestofindalternatesolutions,wealsoshiftedthebalancebetweentheshort-termratesofgeneticgainandinbreeding.
Thiswasillustratedbyourplotoftheshort-termratesofgeneticgainagainsttheshort-termratesofinbreedingrealisedbyeachrestriction.
Theserateslaidalongtheshort-termresponsefrontierforOCSwithoutrestrictionsandcentredroundtheratesrealisedbytheoptimumsolution.
Thisimpliesthat,tocopewithre-strictions,weneedtogiveOCStheopportunitytoshiftthebalancebetweenshort-termratesofgeneticgainandinbreeding.
Italsounderlinesthatshort-termratesofgeneticgainandinbreedingaremerelyintermediarypa-rameters,a'meanstoanend',giventhatlong-termrateofgeneticgainisthegoalofmostbreedingschemes.
Therefore,OCSwithrestrictionsshouldalwaysrealisemostofthelong-termgeneticgainprovidedsolutionsneartheoptimumstillexistinthesolutionspace,OCSisabletocorrectforselectiondecisionsmadeatearlierselectiontimes,andOCSisgiventheopportunitytoshiftthebalancebetweenshort-termratesofgeneticgainandinbreedinginsearchofalternatesolutions.
ItwasforthesereasonsthatOCSwasabletocopewiththefirstofourrestrictions,truncateddamsanddamsunknown,whererestrictionswereimposedonfe-malecandidates.
TruncateddamsremovedpotentialsolutionsfromthesolutionspacebyfixingelementsofcinEquation(1)thatwereassociatedwithtruncation-selecteddams.
Elementsforallotherfemalesweresetto0.
Indamsunknown,allelementsofcassociatedwithfemaleswerefixedto0.
Thereweretwostrikingfeaturesoftheserestrictions.
First,OCSonlyneededtobeap-pliedtomalestorealisemostofthelong-termgeneticgainrealisedbyOCSwithoutrestrictions.
Thiswasbecausetheintensityofselectionandthevariationingeneticcontributionswerehigherforsiresthandams.
Selectionofsireshadagreaterimpactonratesofgeneticgainandinbreeding.
Second,theidentityofthetruncation-selecteddamswasnotneededbyOCStorealisemostofthegeneticgain.
Theimpactoftheselectedsiresonthegeneticmeritofthenewcohortdidnotdependonwhichdamsweretruncation-selected,whiletherewassufficientinformationfromearlierselec-tiondecisionsforOCStoestimatetheaveragerelation-shipofthecurrentgenerationviaPinEquation(1).
Pcontainedgeneticcontributionstothecurrentgener-ation,includingcontributionsfromtheparentsofthetruncation-selecteddamsanddamsthathadalreadygeneratedoffspring.
Thus,OCScancopewithrestric-tionsimposedonfemalecandidatesinbreedingschemeswheretheselectionintensityformalesishigherthanforfemales.
ThecapacityforOCStocopewithrestrictionswasfur-therevidentwhenrestrictionswerealsoimposedonmalecandidates.
SolutionsneartheoptimumstillexistedwhenelementsofcassociatedwithbothmalesandfemaleswereShort-termLong-term5102050500100050002001000100200300500700Rateofinbreeding40060004080120Rateofgeneticgain1006020Figure3Shortandlong-termresponsefrontiersrealisedbyoptimum-contributionselection(OCS)withoutrestrictions.
Theshortandlong-termresponsefrontiersareshortandlong-termratesofgeneticgainrealisedatninepenaltiesonaveragerelationshipplottedasafunctionofshortandlong-termratesofinbreeding,whereshortandlong-termrefertogenerations6to8andgenerations23to25(approx.
).
Thepenaltiesare5(),10(),20(Δ),50(Χ),100(),200(),500(),1000(),and5000().
Therateswerescaledbysettingto100thelong-termratesofgeneticgainandinbreedingrealisedatpenalty50.
Rateofgeneticgainat100isequivalentto0.
215genetic-standarddeviationspertimeandapproximately0.
9genetic-standarddeviationspergeneration.
Rateofinbreedingat100isequivalentto0.
0020pertimeandapproximately0.
008pergenerationontheobservedscale.
Henryonetal.
GeneticsSelectionEvolution(2015)47:21Page10of14fixedbyone-chanceOCSofsires,pre-selectionofsires25,10,5,and1%,andsiremultiples5,10,and20.
Withone-chanceOCSofsires,onlyyoungmaleswereallocatedmat-ings.
Elementsofcassociatedwitholdermaleswerefixedto0.
OCScopedbecauseyoungmalestendedtohavehigherbreedingvaluesthanoldermales.
Therewasalsoenoughgeneticdiversitytoconstrainaveragerelationshipdespitethefactthatfewermaleswereavailableforselec-tion,youngmaleswerefromfewerfamilies,andtheavail-ablemalestendedtoberelated.
Solutionsstillexistedwithpre-selectionofsires25,10,5,and1%despitethefactthatonlymalesthatwerepre-selectedbasedonbreedingvaluewereallocatedmatingsbyOCS.
Elementsofcassociatedwithallothermaleswerefixedto0.
OCSdidthisbyover-comingseveralchallengesposedbypre-selection:feweravailablemalesforOCS,increasedbetween-familyvari-ationformalecandidates,andincreasedaveragerelation-shipofthemalecandidates.
Pre-selectedmalesalsotendedtoberelatedtothetruncation-selecteddams,be-causetruncation-selecteddamsalsorankedhighestforbreedingvalue.
Siremultiples5,10,and20realisedmostofthelong-termgainalthoughincreasesinthemultiplereducedthemaximumnumberofsiresthatcouldhavebeenallocatedmatings.
AllmaleswerecandidatesforOCS,butasthemultipleincreased,fewersiresand,inturn,fewerancestorscouldhavemadegeneticcontribu-tionstoeachgeneration.
So,notonlycanOCScopewithrestrictionsimposedonfemalecandidates,itcanalsohandlerestrictionsimposedonmales.
AlthoughOCScopedwithmanyofourrestrictions,therewasalimitwheretherestrictionsbecametoostrictandwefailedtorealisemostofthelong-termgeneticgain.
Pre-selectionofsires0.
5%andsiremultiples50and100failedtorealisemostofthegainbecausetheyremovedallsolutionsneartheoptimumfromthesolu-tionspace.
Offspringunknownremovedrecordsofselectiondecisionsmadeatearliertimes.
Theserestrictionsimpactedonshort-termratesofgeneticgain,buttheirgreatestimpactwasthroughincreasedratesofinbreeding,whicherodedadditive-geneticvariation.
Pre-selectionofsires0.
5%increasedratesofinbreedingbecauseonlyabout11maleswereavailableforOCSandthesemalestendedtoberelated.
Noallocationofmatingstothesemalescouldhaverealisedlowratesofinbreeding.
Thiswassupportedbythefactthatthepenaltythatmaximisedlong-termgeneticgainforpre-selectionofsires0.
5%was5000,thelargestpenaltythatweappliedtotheaveragerelationship.
Insiremultiples50and100,OCScouldonlyallocatematingstoamaximumofsixandthreesires.
Withsofewsiresandancestorsmakinggeneticcontributionstoeachgeneration,anaccumulationofinbreedingwasunavoidable.
Inoffspringunknown,contributionstoyoung,immatureoffspringwereremovedbyfixingcolumnsofPto0.
ThisledOCStounderestimatetheaveragerelationshipofthecurrentgeneration.
OCSwasmorelikelytoallocatematingstosiresthathadalreadycontrib-utedtothecurrentgeneration,whichincreasedratesofinbreeding.
Theseexplanationswerereinforcedbyourplotofshort-termratesofgeneticgainagainstshort-termratesofinbreeding.
TheratesrealisedbyOCSwithpre-selectionofsires0.
5%,siremultiples50and100,andoffspringunknowndeviatedfromtheshort-termresponsefrontierforOCSwithoutrestrictionsandfromtheratesrealisedbytheoptimumsolution.
Thus,althoughOCSisarobustselectionsystem,itdoeshavelimits.
Itissensitivetore-strictionsthatremoveallsolutionsneartheoptimumfromthesolutionspaceorremoverecordsofearlierselectiondecisions.
Long-termratesofgeneticgainweremaximisedatpenal-tiesthatstruckanappropriatebalancebetweenshort-termratesofgeneticgainandinbreeding.
Thiswasillustratedbyourshortandlong-termresponsefrontiersforunrestrictedOCS.
Themainfeatureofthesefrontierswasthatthepen-altythatmaximisedlong-termratesofgeneticgainrealisedalmostasmuchshort-termgeneticgainasOCSatsmallerpenalties.
Sacrificingsmallamountsofshort-termgainledtovastlyreducedratesofinbreeding.
ThishighlightsthatOCSwiththeappropriatepenaltypromotesshort-termgeneticgainwhilemaintainingadditive-geneticvariationandthepotentialtorealiselong-termgain.
Therefore,itmakesgoodsensetouseOCSwithpenaltiesthatstrikeanappropriatebalancebetweenshort-termratesofgeneticgainandinbreeding.
Notonlydoesitmakegoodsensetousepenaltiesthatstrikeanappropriatebalance,thepenaltiesshouldalsorealiseacceptableratesofinbreeding.
Acceptableratescanbedifficulttodefine,giventhatthenatureofgeneticvariation,thegenerationofnewvariationthroughmuta-tion,andtheimpactsofinbreedingonfitnessarepoorlyunderstood.
Currentknowledgesuggeststhatratesbe-tween0.
005and0.
01pergenerationareacceptable[14].
Thisequatestoratesbetween60and125pertime(ap-proximately)afterwescaledourratesofinbreedingbysettingto100thelong-termrateofinbreedingrealisedbyunrestrictedOCSatpenalty50.
AcceptingBijma's[14]rateshastwoimplications.
First,itprovidesanimalbreederswithanadditionalcriterionbywhichtodefinepenaltiesforOCS.
Second,Bijma's[14]ratesindicatethatourfindingsarerobusttothetimehorizonthatweassumedforoursimulationsandthattheygeneralisebeyondthishorizon.
So,strikinganappropriatebalancebetweenshort-termratesofgeneticgainandinbreeding,andrealisingratesofinbreedingthatfallwithinaccept-ablelevels,providetwoworthwhilecriteriabywhichtodefinepenaltiesforOCS.
Afurtherfeatureoftheresponsefrontierswasthatsmallpenaltiesresultedinshort-termratesofgeneticgainthatwerehigherthanlong-termrates,whilelargeHenryonetal.
GeneticsSelectionEvolution(2015)47:21Page11of14penaltiesresultedinlong-termratesthatwerehigherthanshort-termrates.
Therearetwointerrelatedexpla-nationsforthis.
First,ateachpenalty,variationinbreed-ingvaluesandvariationinadditive-geneticrelationshipsdecreasedatdifferentratesovertime,wherebreedingvaluesandadditive-geneticrelationshipsarerepresentedbyandAinEquation(1).
Second,therelativecon-straintofeachpenaltyonaveragerelationshipchangedastheratioofthetwovariationschangedovertime.
Decreasingthevariationinbreedingvaluesfasterthanthevariationinadditive-geneticrelationshipsincreasedtherelativeconstraintonaveragerelationship,whiledecreasingthevariationinadditive-geneticrelationshipsfasterthanthevariationinbreedingvaluesreducedtheconstraint.
Thesetwoexplanationsmeanthat,atsmallpenalties,whereemphasiswasongeneticgain,OCSde-creasedthevariationinbreedingvaluesfasterovertimethanitdecreasedthevariationinadditive-geneticrelation-ships.
Notonlydidthislimitthepotentialtorealisegen-eticgain,itincreasedtherelativeconstraintonaveragerelationship.
Athighpenalties,wheretheemphasiswasonconstrainingaveragerelationship,variationinadditive-geneticrelationshipsdecreasedfasterthanvariationinbreedingvalues.
Thisretainedthepotentialtorealisegen-eticgainwhilereducingtherelativeconstraintonaveragerelationship.
Thisreasoningleadsustospeculatethat,inpractice,penaltiesthatmaximiseratesoflong-termgen-eticgainarethosethatmaintainaconstantrelativecon-straintonaveragerelationshipovertime.
Achievingthiswillinvolvedynamicpenalties,derivedateachselectiontimeasfunctionsofthevariationsinbreedingvaluesandadditive-geneticrelationships,giventhatwedonotknowtheimpactofindividualpenaltiesonlong-termratesofgeneticgain.
Therefore,thechoiceofpenaltyplaysanimportantroleinmanagingvariationinbreedingvaluesandadditive-geneticrelationships.
Itmayevenbepossibletoincreaselong-termgeneticgainbyapplyingdynamicpenaltiesovertime.
Althoughtherestrictionsthatweimposedwerein-spiredbypigbreeding,OCSshouldalsocopewiththeseandotherrestrictionsinmostbreedingschemesfortworeasons.
First,theprinciplethatunderliesOCSisthesameforallbreedingschemes,irrespectiveofthespe-cies,numberortypeoftraits,phenotypingstrategy,andgeneticandphenotypic(co)variances.
ThisisbecauseOCSusessummarystatisticsasinput.
Geneticmeritisanaggregatebreedingvalue,whileaveragerelationshipisderivedfromamatrixofgeneticrelationships.
Second,themechanismsbywhichOCScopedwitheachrestric-tionapplyacrossschemes.
Whilethemechanismsarethesame,whatissuretodifferbetweenschemesisthepointatwhichtherestrictionsbecometoostrict.
Forexample,OCScouldbecomesensitivetodamsunknowninschemeswheretheselectionintensityandvariationingeneticcontributionsoffemalesareincreased.
OCSwillprobablyalsobecomesensitivetoone-chanceOCSofsireswhentheintervalbetweenselectiontimesisshortenedandtherearefewermalecandidatesavailableforselectionateachtime.
Likewise,thelevelofpre-selectionwherepre-selectionofsiresbecomestoostrictwillpresumablyvarybetweenschemesthatdifferforpopulationsize,familystructure,andselectionintensity.
Thus,theunderlyingmechanismsbywhichOCScopeswithrestrictionsshouldapplyacrossbreedingschemes,butthepointatwhichtherestrictionsbecometoostrictwillprobablydiffer.
AppendixChoosingareferencepointforOCSChoosingaconventionalbreedingschemewithtrunca-tionselectionasareferencepointforthelong-termratesofgeneticgainrealisedbyOCSwassubjective.
Thereasonwasthatratesandlevelsofgeneticgainandin-breedingrealisedbyOCSandtruncationselectiondif-feredovertime.
Itwas,therefore,difficulttofindaconventionalbreedingschemewithtruncationselectionthatmatchedOCS.
Thisisillustratedbyapreliminaryanalysis,wherewetracedthegeneticgainandinbreedingrealisedby(i)abreedingschemewithunrestrictedOCSatpenalty50,and(ii)threeschemeswithtruncationselection.
Thethreeschemeswithtruncationselectionweretruncationselection10,truncationselection60,andtruncationse-lection100.
UnrestrictedOCSatpenalty50andtrunca-tionselection10weresimulatedasdescribedintheMethodssection.
Truncationselection60andtruncationselection100weresimulatedasdescribedfortruncationselection10withtheexceptionthat60and100siresweretruncationselectedateachtime.
Geneticgainwaspresentedonthegenotypicscaleandinbreedingontheobservedscale.
Tracinggeneticgainandinbreedingovertimeillus-tratedsixpoints(Figure4):Rateofgeneticgainwasnotconstantovertime.
TherateofgeneticgainrealisedbyunrestrictedOCSfellfrom0.
238genetic-standarddeviationspertimeduringtimes26to35to0.
215genetic-standarddeviationspertimeduringtimes91to100.
Theraterealisedbytruncationselection10fellfrom0.
252to0.
182.
Itfellfrom0.
221to0.
201fortruncationselection60andfrom0.
207to0.
196fortruncationselection100.
UnrestrictedOCSrealisedhigherratesofgeneticgainthantruncationselection.
TherateofgeneticgainrealisedbyunrestrictedOCSduringtimes26to35was7.
4and15.
0%higherthantruncationselection60andtruncationselection100(0.
238vsHenryonetal.
GeneticsSelectionEvolution(2015)47:21Page12of140.
221and0.
207genetic-standarddeviationspertime).
Itwas7.
0and9.
7%higherthantruncationselection60andtruncationselection100duringtimes91to100(0.
215vs.
0.
201and0.
196).
Theexceptionwastruncationselection10,whereunrestrictedOCSonlyrealisedhigherratesofgeneticgainaftertime45(approximately).
TherateofgeneticgainrealisedbyunrestrictedOCSwas5.
6%lowerthantruncationselection10duringtimes26to35(0.
238vs.
0.
252).
Itwas18.
1%higherduringtimes91to100(0.
215vs.
0.
181).
UnrestrictedOCSrealisedmoregeneticgainthantruncationselection.
ThelevelofgeneticgainrealisedbyunrestrictedOCSattime35was0.
4and4.
0%higherthantruncationselection60andtruncationselection100(8.
7vs.
8.
7and8.
4genotypicstandarddeviations).
Itwas5.
0and9.
5%higherthantruncationselection60andtruncationselection100attime100(23.
3vs.
22.
2and21.
3).
Theexceptionwastruncationselection10,whereunrestrictedOCSonlyrealisedmoregeneticgainaftertime80.
ThelevelofgeneticgainrealisedbyunrestrictedOCSwas5.
8%lowerthantruncationselection10attime35(8.
7vs.
9.
2).
Itwas2.
6%higherattime100(23.
3vs.
22.
7).
RateofinbreedingrealisedbyunrestrictedOCSwasnotconstantovertime.
Itwasapproximatelyconstantfortruncationselection.
TherateofinbreedingrealisedbyunrestrictedOCSfellfrom0.
0023pertimeduringtimes26to35to0.
0020pertimeduringtimes91to100.
Theraterealisedbytruncationselection10wasapproximately0.
0075pertimeduringtimes21to100.
Itwasapproxi-mately0.
0020and0.
0014duringtimes21to100fortruncationselection60andtruncationselection100.
UnrestrictedOCSrealisedlowerratesofinbreedingthantruncationselection10,similarratesastrun-cationselection60,andhigherratesthantruncationselection100.
TherateofinbreedingrealisedbyunrestrictedOCSwas70.
8and73.
4%lowerthantruncationselection10duringtimes26to35and91to100(0.
0023vs.
0.
0078and0.
0020vs.
0.
0075pertime).
UnrestrictedOCSrealised11.
8%moreinbreedingthantruncationselection60duringtimes26to35(0.
0023vs.
0.
0020)and3.
9%lessduringtimes91to100(0.
0020vs.
0.
0021).
Itrealised75.
4and44.
2%moreinbreedingthantruncationselection100duringtimes26to35and91to100(0.
0023vs.
0.
0013and0.
0020vs.
0.
0014).
UnrestrictedOCSrealisedlowerlevelsofinbreedingthantruncationselection.
ThelevelofinbreedingrealisedbyunrestrictedOCSattime35was46.
2,27.
1,and22.
4%lowerthantruncationselection10,truncationselection60,andtruncationselection100(0.
12vs.
0.
22,0.
17,and0.
16).
Itwas55.
6and14.
8%lowerthantruncationselection10andtruncationselection60attime100(0.
23vs.
0.
52and0.
27)andapproximatelythesameastruncationselection100attime100(0.
23vs.
0.
23).
ThesepointshighlightthatOCSandtruncationselec-tionarecontrastingselectionmethodsthatrealisediffer-entratesandlevelsofgeneticgainandinbreeding.
Withnoobviousmatch,wechosetruncationselection10asthereferencepointforoursimulations.
Ourreasoningwasthat,whenwecomparedtruncationselection10withunrestrictedOCS,wesawtheoriginalmotivationforOCS:sacrificingsmallamountsofshort-termgeneticgaincanvastlyreducerateofinbreeding[1,2].
Thispro-motesgeneticgainwhilemaintainingadditive-geneticvariationandthepotentialtocontinuerealisinggain.
Theotherschemeswithtruncationselectionalsohave020406080100Time00.
10.
20.
30.
40.
50.
6Inbreedingcoefficient0510152025Geneticgain(genetic-standarddeviation)OCSTS10TS60TS100OCSTS10TS60TS100Figure4Geneticgainandinbreedingrealisedbyoptimum-contribution(OCS)withoutrestrictionsandtruncationselectionovertime.
BreedingschemeapplyingOCSwithoutrestrictionsisrepresentedbyOCS(dottedline).
Threeschemeswithtruncationselectionaretruncationselection10,truncationselection60,andtruncationselection100(TS10,TS60,TS100,solidlines).
Henryonetal.
GeneticsSelectionEvolution(2015)47:21Page13of14meritasreferencepoints.
Forexample,truncationselec-tion60realisedsimilarratesofinbreedingasunrestrictedOCS,althoughitdidsoathigherlevelsofinbreeding.
Usingtruncationselection60andtruncationselection100asreferencepointswouldnothavechangedtheconclu-sionofourstudy.
CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
Authors'contributionsAllauthorsdesignedthestudy,interpretedthefindings,andrevisedthemanuscript.
MHranthesimulations,analysedthesimulateddata,anddraftedthemanuscript.
MH,ACS,andPBco-wroteADAM.
PBwroteEVA.
Allauthorsreadandapprovedthefinalmanuscript.
AcknowledgementsThisstudywasfundedbyGUDPCenterforInnovation,MinistryofFoodAgricultureandFisheries,andthePigResearchCentre,DanishAgricultureandFoodCouncil.
Threerefereesprovidedusefulcomments.
Authordetails1Seges,DanishPigResearchCentre,Axeltorv3,1609CopenhagenV,Denmark.
2SchoolofAnimalBiology,UniversityofWesternAustralia,35StirlingHighway,Crawley,WA6009,Australia.
3AarhusUniversity,InstituteforMolecularBiologyandGenetics,P.
O.
Box50,8830Tjele,Denmark.
4NordGen,NordicGeneticResourceCenter,P.
O.
Box115,1431s,Norway.
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