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Systems mapping: how to improve the genetic mapping of complex traits through design principles of biological systems
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Title: Systems mapping: how to improve the genetic mapping of complex traits through design principles of biological systems
Series Title: BMC Systems Biology
Physical Description: Book
Language: English
Creator: Wu, Rongling
Cao, Jiguo
Huang, Zhongwen
Wang, Zhong
Gai, Junyi
Vallejos, Eduardo
Publication Date: 2011
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Abstract: Background: Every phenotypic trait can be viewed as a “system” in which a group of interconnected components function synergistically to yield a unified whole. Once a system’s components and their interactions have been delineated according to biological principles, we can manipulate and engineer functionally relevant components to produce a desirable system phenotype. Results: We describe a conceptual framework for mapping quantitative trait loci (QTLs) that control complex traits by treating trait formation as a dynamic system. This framework, called systems mapping, incorporates a system of differential equations that quantifies how alterations of different components lead to the global change of trait development and function through genes, and provides a quantitative and testable platform for assessing the interplay between gene action and development. We applied systems mapping to analyze biomass growth data in a mapping population of soybeans and identified specific loci that are responsible for the dynamics of biomass partitioning to leaves, stem, and roots. Conclusions: We show that systems mapping implemented by design principles of biological systems is quite versatile for deciphering the genetic machineries for size-shape, structural-functional, sink-source and pleiotropic relationships underlying plant physiology and development. Systems mapping should enable geneticists to shed light on the genetic complexity of any biological system in plants and other organisms and predict its physiological and pathological states.
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Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution.
Resource Identifier: doi - 10.1186/1752-0509-5-84
System ID: AA00012411:00001

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METHODOLOGYARTICLE OpenAccessSystemsmapping:howtoimprovethegenetic mappingofcomplextraitsthroughdesign principlesofbiologicalsystemsRonglingWu1*,JiguoCao2,ZhongwenHuang3,ZhongWang4,JunyiGai5*andEduardoVallejos6AbstractBackground: Everyphenotypictraitcanbeviewedasa system inwhichagroupofinterconnectedcomponents functionsynergisticallytoyieldaunifiedwhole.Onceasystem scomponentsandtheirinteractionshavebeen delineatedaccordingtobiologicalprinciples,wecanmanipulateandengineerfunctionallyrelevantcomponentsto produceadesirablesystemphenotype. Results: Wedescribeaconceptualframeworkformappingquantitativetraitloci(QTLs)thatcontrolcomplextraits bytreatingtraitformationasadynamicsystem.Thisframework,calledsystemsmapping,incorporatesasystemof differentialequationsthatquantifieshowalterationsofdifferentcomponentsleadtotheglobalchangeoftrait developmentandfunctionthroughgenes,andprovidesaquantitativeandtestableplatformforassessingthe interplaybetweengeneactionanddevelopment.Weappliedsystemsmappingtoanalyzebiomassgrowthdatain amappingpopulationofsoybeansandidentifiedspecificlocithatareresponsibleforthedynamicsofbiomass partitioningtoleaves,stem,androots. Conclusions: Weshowthatsystemsmappingimplementedbydesignprinciplesofbiologicalsystemsisquite versatilefordecipheringthegeneticmachineriesforsize-shape,structural-functional,sink-sourceandpleiotropic relationshipsunderlyingplantphysiologyanddevelopment.Systemsmappingshouldenablegeneticiststoshed lightonthegeneticcomplexityofanybiologicalsysteminplantsandotherorganismsandpredictits physiologicalandpathologicalstates.BackgroundPredictingthephenotypefromthegenotypeofcomplex organismsisoneofthemostimportantandchallenging questionswefaceinmodernbiologyandmedicine[1]. Geneticmapping,dissectingaphenotypictraittoits underlyingquantitativetr aitloci(QTLs)throughthe useofmolecularmarkers,hasprovenpowerfulfor establishinggenoty pe-phenotyperelationshipsandpredictingphenotypesofindividualorganismsbasedon theirQTLgenotypesresponsibleforthetrait[2].The successofthispredictiondependsonhowwellwecan maptheunderlyingQTLsandcharacterizecomplex interactionsoftheseQTLswitheachotherandwith environmentalfactors.Power fulstatisticalmodelshave beendevelopedinthepasttwodecadestodetectQTLs andstudytheirbiologicalfunctioninadiversearrayof phenotypictraits[3-9].Worldwide,asubstantialeffort hasbeenmaderesultinginthecollectionofalarge amountofdataaimedattheidentificationofQTLs [10-15].Unfortunately,despitehundredsofthousandsof QTLsdetectedinadiversityoforganisms,onlyafewof themhavebeenisolatedbypositionalcloning(see [16-18]),leavingitunsolvedhowtoconstructagenotype-phenotyperelationshipmapusinggeneticmapping. Themostlikelyreasonforthisresultmayarisefroma possibilitythattheQTLsdetectedbystringentstatistical testsarenotbiologicallyrelevant.Existingstrategiesfor QTLmappingwerebuiltontestingforadirect *Correspondence:rwu@hes.hmc.psu.edu;sri@njau.edu.cn Contributedequally1CenterforComputationalBiology,NationalEngineeringLaboratoryforTree Breeding,KeyLaboratoryofGeneticsandBreedinginForestTreesand OrnamentalPlants,BeijingForestryUniversity,Beijing100083,China5NationalCenterforSoybeanImprovement,NationalKeyLaboratoryfor CropGeneticsandGermplasmEnhancement,SoybeanResearchInstitute, NanjingAgriculturalUniversity,Nanjing210095,China FulllistofauthorinformationisavailableattheendofthearticleWu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 2011Wuetal;licenseeBioMedCentralLtd.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommons AttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproductionin anymedium,providedtheoriginalworkisproperlycited.

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associationbetweengenotypeandend-pointphenotype. Althoughsuchstrategiesaresimpleandhavebeen widelyaccepted,theyneglectthebiologicalprocesses involvedintraitdevelopment[14].Toattempttofill thisgap,astatisticalmode l,calledfunctionalmapping [19-21],hasbeendevelopedtostudytheinterplay betweengeneticsandthedevelopmentalprocessofa phenotypictraitbyintegratingmathematicalmodelsand computationalalgorithms.Ifatraitisunderstoodasa system thatiscomposedofmanyunderlyingbiological components[22-24],weshouldbeinabetterposition tocomprehendtheprocessandbehavioroftraitformationbasedoninteractiverelationshipsamongdifferent components.ThroughmappingandusingthoseQTLs thatgoverndesignprinciplesofabiologicalsystem,a newtraitthatisabletomaximizeresource-useefficiencycanbegeneratedandengineered. Asoneimportantstrategyforplantstorespondto variationintheavailabilityofresourcesintheirenvironment,biomassallocationhasbeenextensivelyusedto studytherelationshipbetweenstructureandfunctionin modernecology[25-28].Theconceptofbiomassallocationhasnowbeenincreasinglyintegratedwithplant managementandbreeding,aimedtodirectamaximum amountofbiomasstothetargetofharvest(leaves,stem, roots,orfruits)[29-31].Ifthewhole-plantbiomassis consideredasatargettrait,weneedtounderstandhow differentorgansofaplantcoordinateandinteractto optimizethecaptureofnutrients,light,water,andcarbondioxideinamannerthatmaximizesplantgrowth ratethroughaspecificdevelopmentalprogrambecause plantbiomassgrowthisnots implytheadditionofbiomasstovariousorgans.Manytheoriesandmodelshave beenproposedtopredictthepatternofbiomasspartitioninginaresponsetochangingenvironment.Chen andReynolds[27]usedcoordinationtheorytomodel thedynamicallocationofcarbontodifferentorgans duringgrowthinrelationtocarbonandwater/nitrogen supplybyagroupofdifferentialequations.Compared totheconventionaloptimizationmodelinthecontext ofmaximizingtherelativegrowthrateofaplant,the coordinationmodeldoesnotrequireanunrealistic capacitytheplantpossessesforknowingbeforehandthe environmentalconditionsitwillexperienceduringthe growthperiod.Here,weintegratethecoordinationand optimizationmodeltostudythepatternofbiomasspartitioningbyincorporatingth eallometricscalingtheory intoasystemofdifferentialequations. InaseriesofallometricstudiesbyWestetal.[32-34], apowerrelationshipthatuniversallyexistsbetween partsandthewholecanbeexplainedbytwofundamentaldesignprinciplesinbiophysicsandbiochemistry;i.e., allorganismstendtomaximizetheirmetaboliccapacity byincreasingsurfaceareasforenergyandmaterial productionaswellasinternalefficiencythroughreducingdistancesandthetimetotransportwater,nutrients,andcarbon.Theintegrationofthisoptimization theoryexpressedintermsofallometricscalingwiththe coordinationtheoryleadstoatripledgroupofordinary differentialequations(ODEs)tospecifythecoordination oftheleaf,stem,androotbiomassforaplant: dML dt = LWL LMLdMS dt = SWSdMR dt = RWR RMR (1) where ML, MS,and MRarethebiomassesoftheleaves ( L ),thestem(S ),andtheroots( R ),respectively,with whole-plantbiomass W = ML+ MS+ MR; a and b are theconstantandexponentpowerofanorganbiomass scalingaswhole-plantbiomass[32,33];and g istherate ofeliminatingageingleavesandroots.Thecomplex interactionsbetweendifferentpartsofaplantthat underliedesignprinciplesofplantbiomassgrowthcan bemodeledandstudiedbyestimatingandtestingthe ODEparameters( aL, bL, lL, aS, bS, aR, bR, lR).For example,plantsareequippedwithacapacitytooptimize theirfitnessunderlownutrientavailabilitybyshifting thepartitioningofcarbohydratestoprocessesassociated withnutrientuptakeatacostofcarbonacquisition [29].Theseparameterscanbeusedtoquantifyandpredictsuchregulationbetweendifferentplantpartsin responsetoenvironmentalanddevelopmentalchanges. Inthisarticle,weputforwardaconceptualframework toincorporatethedesignprinciplesoftraitformation anddevelopmentintoastatisticalframeworkforQTL mapping.Complementaryt oourpreviousfunctional mapping[19-21],wenamethisnewmappingframework systemsmapping inlightofitssystemsdissectionand modelingofphenotypicfor mation.AgroupofODEs like(1)orothertypesofdifferentialequationsisusedto quantifythephenotypicsystem.Muchworkinsolving ODEshasfocusedonthesimulationandanalysisofthe behaviorofstatevariablesforadynamicsystem,butthe estimationofODEparametersthatdefinethesystem basedonthemeasurementofstatevariablesatmultiple timepointsisrelativelyanewarea.Yet,intherecent years,manystatisticianshavemadegreatattemptsto developstatisticalapproachesforestimatingODEparametersbymodelingthestructureofmeasurementerrors [35-42].WeimplementedRamsayetal. s[41]penalized splinemethodforestimatingconstantdynamicparametersinourgeneticmapping.TheproblemforsystemsmappingwithODEmodelsisdifferentfromthose consideredincurrentliterat ure.First,systemsmappingWu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page2of11

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isconstructedwithinamixture-basedframework becauseQTLgenotypesthatdefinetheDEmodelsare missing.Second,systemsmappingincorporatesgenotypicdatawhicharecategoricalorbinary.Thesetwo characteristicsdeterminethehighcomplexityofourstatisticalmodelandcomputationalalgorithmusedforsystemsmapping.ResultsQTLdetectionWedevelopanewmodelforQTLmappingbytreating traitformationasadynamicsystemandfurtherincorporatingthedesignprincip lesofthebiologicalsystem intoastatisticalmappingframework(see Methods ). Thenewmodel,namedsystemsmappinginlightofits systemsfeatureofphenotypicdescription,wasusedto mapQTLsforbiomasspartitioninginamappingpopulationofsoybeanscomposedof184recombinantinbred lines(RIL)derivedfromtwocultivars,KefengNo.1and Nannong1138-2.ForanRILpopulation,therearetwo homozygousgenotypes,onecomposedoftheKefeng No.1allelesandtheothercomposedoftheNannong 1138-2alleles.Figure1illustratesthegrowthtrajectories ofleaf,stemandrootbiomassforindividualRILs.By usingthesystemofODEs(1)tofitgrowthtrajectories ofleaf,stemandrootbiomassovertime,weobtaineda meancurveforeachtrait.Itcanbeseenthatgrowth trajectoriescanbewellmodeledbythreeinterconnectingODEs(1)derivedfromcoordinationtheory[27]and allometricscaling[32-34].Themodel-fittedcurvesof leaf(Figure1A)androotbiomasstrajectories(Figure 1C)delineatereasonablythedecayofleafandrootbiomassatalatestageofdevelopmentduetosenescence. Asexpected,stembiomassgrowthdoesnotexperience suchadecay(Figure1B)althoughgrowthatthelate stagetendstobestationary. Byscanningthegeneticlinkagemapcomposedof950 molecularmarkerslocatedin25linkagegroups,we detectedtwosignificantQTLs,onenamedas biomass1 thatresidesbetweenmarkersGMKF167aand GMKF167bandthesecondas biomass2 thatresides betweenmarkerssat-274andBE801128(Additionalfile 1,FigureS1).Usingthemaximumlikelihoodestimates ofthecurveparametersinODE(1)whosestandard errorswereobtainedbyth eparameterbootstrap[43] (Table1),wedrewthegrowthtrajectoriesofleaf,stem androotbiomassfortwodifferentgenotypesateach QTL(Figure2).ThegeneticeffectsoftheQTLsdisplayeddifferenttemporalpatternsforthreeorgans.The QTLswereexpressedmorerapidlywithtimeforthe stemthanfortheleavesandroots.At biomass1 ,the allelesfromparentNannong1138-2increaseleafand stembiomassgrowth(Figure2Aand2B),whereasthe allelesfromparentKefengNo.1increaserootbiomass growth(Figure2C).Thiscouldbeinterpretedasthe Nannong1138-2allelefavoringcarbonallocationtothe shootsattheexpenseoftherootsbuttheKefengNo1 allelefavoringcarbonallocationtotherootsatthe expenseoftheshoots.Likewise,the biomass2 alleles fromNannong1138-2favorcarbonallocationtothe 2 4 6 8 0 5 10 15 20 BiomassTime A 2 4 6 8 Time B 2 4 6 8 Time C Figure1 Growthtrajectoriesofleaf(A),stem(B)androotbiomass( C)measuredatmultipletimepointsinagrowingseasonof soybeans .Eachgreylinepresentsthegrowthtrajectoryofoneof184RILs,whereasblacklinesarethemeangrowthtrajectoriesofallRILs fittedusingasystemofODEs(1). Wu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page3of11

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leaves(Figure2D)andthosefromKefengNo.1favor carbonallocationtotheroots,buttheallelesatthis QTLinheritedfromparentKefengNo.1favorcarbon allocationtothestem,whichisdifferentfromthebehaviorofQTL biomass1 .Notethatleafandrootbiomass growthtendtodecayatthelatestageforalmostall RILs.ButthegenotypesattheQTLsdetecteddonot reflectthistrend(Figure2),althoughtheydisplaymuch reducedratesofgrowthatthelatestage.Weexplained thistobeduetosomeotherQTLsthathavenotbeen detectedwiththecurrentlinkagemap. Thefunctionalrelationshipsamongleaf,stemandroot biomassweredeterminedb ytheQTLsdetected(Figure3).For biomass1 ,twogenotypesarenotonlydifferentinwhole-plantbiomasstrajectory,butalso displaypronounceddiscrepanciesinbiomassgrowth trajectoriesofindividualorgans(Figure3Aand3B). ThismeansthatthisQTLaffectsthedynamicsofboth plantsizeandbiomasspartitioning.Thegenotype composedoftheKefengNo.1alleleshasasmaller slopeofbiomassgrowth,leadingtosmallerwholeplantbiomassatlatestagesofdevelopment,thanthat composedoftheNannong1138-2alleles,buttheformerhaslargerrootbiomassovertheentireperiodof growthattheexpenseoftheshootsthanthelatter. For biomass2 ,twogenotypesaresimilarintotalplant sizeduringgrowth,buttheyhaveamarkeddistinction inbiomasspartitioning(Figure3Cand3D).Itappears thatthisQTLaffectsplantgrowthtrajectoriesthrough alteringbiomasspartitioningratherthantotalamount ofbiomass.AtthisQTL,thegenotypewiththeKefeng No.1alleleshasadominantmainstemandheavy roots,whereasthegenotypewiththeNannong1138-2 allelescarriesdenseleaves. Figure4showsthedynamicpatternofbiomasspartitioningtodifferentorgans.Ingeneral,thestemreceives increasingallocationwithtime,whereasthepartitioning totheleavesandrootsdecreaseswithtime.BothQTLs detected, biomass1 and biomass2,controlthedegreeof suchtime-dependentincreaseordecrease.Forexample, at biomass1 ,theKefengNo.1genotypealwaysexhibits alargerdegreeofincreasingbiomasspartitioningtothe stembutalargerdegreeofdecreasingbiomasspartitioningtotheleavesandrootsthantheNannong11382genotype(Figure4Avs.4B).QTL biomass2 hasa similarpatternofbiomasspartitioningforthestemand leaves,althoughitdisplaysastrongereffectthandoes QTL biomass1 .AtQTL biomass2 ,thereisalarger degreeofdecreasingbiomasspartitioningtotheroots fortheNannong1138-2genotypethantheKefeng No.1genotype(Figure4Cvs.4D).SimulationByanalyzingarealdatasetforsoybeanmapping,systemsmappingproducestheidentificationoftwosignificantQTLsthatcontrolthedynamicformationof whole-plantbiomassthroug hdevelopmentalregulation ofdifferentorgans,stem,le aves,androots.Tovalidate thenewmodel,weperformedsimulationstudiesby mimickingtheeffectsofQTL biomass2 detectedfrom theexampleofQTLmappinginsoybeans.Thesimulatedmappingpopulationcontainsthesamegenotype datafor184RILs.Thephenotypicvaluesofthreetraits, thestem,leafandrootbiomass,assumedtoobeythe systemofODE(1),weresimul atedatsixdifferenttime pointsbysummingtime-dependentgenotypicvaluesat biomass2 calculatedwithcurveparametersinTable1 andresidualerrors.Specifically,thephenotypicvaluesof Table1Themaximumlikelihoodpointestimates(PEs)ofODEparametersandstandarderrors(SEs)oftheestimates fortheQTLsdetectedQTLModelGenotypeEstimate aLbLgLaSbSaRbRgR1M 1 QQ PE 2.090.160.430.930.071.520.662.91 SE 0.021e-33e-30.012e-40.016e-30.02 qq PE 2.530.110.360.920.041.570.543.90 SE 0.014e-42e-30.011e-40.015e-30.01 M0 PE 2.300.130.390.920.051.560.603.37 SE 0.072e-30.010.024e-40.050.020.08 2M 1 QQ PE 1.890.140.441.040.071.110.561.85 SE 0.041e-30.010.011e-40.015e-30.01 qq PE 2.550.100.310.980.041.110.512.18 SE 0.041e-30.010.014e-40.015e-30.02 M0 PE 2.250.120.371.030.051.120.552.06 SE 0.083e-30.020.026e-40.020.010.05ModelM1assumestwodifferentgenotypesataQTL(undertheH1),whereasModelM0assumesasinglegenotype(undertheH0). Note:QTL1isonebetweenmarkersGMKF167aandGMKF167bonlinkagegroup12.QTL2isonebetweenmarkerssat-274andBE801128onlinkagegroup24. Theallelesofgenotype QQ arederivedfromKefengNo.1andthoseofgenotype qq derivedfromNannong1138-2.Wu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page4of11

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the k thtraitweresimulatedby addingwhitenoisewith variances 2 k totheODEcurvesforthe j thQTLgenotypewiththeprobability wj | i,i.e.,theconditionalprobabilityofthe i thRILthatcarriesthe j thQTLgenotype, giventhetwomarkersgenotypesofthisRIL.Thevalues ofnoisevariance 2 k weresetastheestimatesfromthe realdata,whichare 2 1=2.42 2 2=1.72 and 2 3=0.14 fortheleaf,stem,androotbiomass,respectively.Meanwhile,byassum ingamodestheritability 2 4 6 8 0 5 10 15 20 B iomass A: Leaf 2 4 6 8 B: Stem 2 4 6 8 C: Root 2 4 6 8 0 5 10 15 20 B iomassTim e D: Leaf 2 4 6 8 Tim e E: Stem 2 4 6 8 Tim e F: Root Nannong 1138-2 Nannong 1138-2 Kefeng No. 1 Nannong 1138-2 Kefeng No. 1 Nannong 1138-2 Nannong 1138-2 Nannong 1138-2 Kefeng No. 1 Kefeng No. 1 Kefeng No. 1 Kefeng No. 1 Figure2 Growthtrajectoriesofleaf(A,D),stem(B,E)androotbiomass(C,F)fortwodifferentgenotypes(presentedbysolidand brokenblackcurves)ataQTLdetectedonlinkagegroup12(upperpanel)and24(lowerpanel),respectively .TwogenotypesataQTL arethehomozygotefortheallelesinheritedfromKefengNo.1(solid)andthehomozygotefortheallelefromNannong1138-2(broken).Curves ingreyaregrowthtrajectoriesof184RILs. Wu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page5of11

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(0.05)foreachtraitatamiddlestageofgrowth,werescaled 2 k valueswhichwereusedtosimulateanew dataset. Systemsmapping,implementedwiththeparameter cascadingmethod,estimatesQTLgenotype-specific curveparametersintheODE(1)fromthesimulated data.Thesimulationwasrepeated100timestocalculate themeans,biases,standarddeviations,androotmean squareerrors,withresultstabulatedinTable2.Itwas foundthatthemodelcanprovidereasonablyaccurate andpreciseestimatesofQTLgenotype-specificODE parameterswithamodestsamplesize( n =184).The biasesoftheestimatesarenegligible,comparedwiththe scaleofthestandarddeviations.Giventhatthis simulateddataisamimicryoftherealsoybeandata,the resultssuggestthattheexperimentaldesignusedfor soybeanmappingisscientificallysoundandcanprovide convincingQTLdetection.Thisactuallyisnotsurprisingbecausephenotypinghaslowmeasurementerrors. Inanalyzingasimulateddatasetforthetraits assumedtohaveamodestheritability(0.05),theestimatesoftheODEparametersarereasonablyaccurate andprecise,indicatingthepowerofsystemsmappingto detectsmallQTLsinvolvedintraitformation.Weperformedanadditionalsimu lationtoinvestigatethe powerandfalsepositiveratesofthemodelbychanging levelsofnoises.Ingeneral,thepowerofthemodelis high,reaching0.80evenwhentheheritabilityofgrowth 2 4 6 8 0 5 10 15 20 25 B iomassA Ke f eng N o.1 All e l es 2 4 6 8 B Nannong 1138-2 Alleles Whole Leaf Stem Root Whole Leaf Stem Root 2 4 6 8 0 5 10 15 20 25 B iomassTim e C 2 4 6 8 Tim e D Whole Leaf Stem Root Whole Leaf Stem Root Figure3 Growthtrajectoriesofwhole-plant(red),leaf(green),stem(blue)androotbiomass(black)fortwodifferentgenotypesata QTLdetectedonlinkagegroup12(upperpanel)and24(lowerpanel) .TwogenotypesataQTLarethehomozygoteforthealleles inheritedfromKefengNo.1(A,C)andthehomozygoteforallelesinheritedfromNannong1138-2(B,D). Wu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page6of11

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curvesislow(0.05).Basically,aQTLcanbefully detectedwhentheheritabilityis0.10orlarger.Inany case,thefalsepositiveratesarenotbeyond0.10,mostly beinglessthan0.05.DiscussionMappingthegeneticarchitectureofcomplextraitshas beenasubjectofinterestinboththeoreticalandempiricalaspectsofmodernbiology[3-15].Original approachesforgeneticmappingarebasedonsinglepointvariationinaphenotypictrait,neglectingthe dynamicchangeofthetraitduringdevelopment.To capturethedynamicpatternofgeneticcontrol,anew statisticalmodelcalledfunctionalmappinghasbeen developedbyincorporatingthemathematicalaspectsof traitdevelopment[19-21].De spitesignificantimprovementoverconventionalstaticmapping,functionalmappinghasstillamajorlimitationincharacterizing developmentalpathwaysthatcauseafinalphenotype andunravelingtheunderlyinggeneticmechanismsfor traitformationandprogression. Inthisarticle,wehaveforthefirsttimeputforwarda newapproach-systemsmappingbytreatingaphenotypic traitasadynamicsystemandincorporatingthedesign principlesofabiologicalsystemintoastatisticalframeworkforgeneticmapping.Variouscomponentsthat 2 4 6 8 0 20 40 60 B iomass P ercentageA Ke f eng N o.1 All e l es 2 4 6 8 B N annong 1138-2 All e l es 2 4 6 8 0 20 40 60 B iomass P ercentageTim e C 2 4 6 8 Tim e D Leaf Stem Root Leaf Stem Root Leaf Stem Root Leaf Stem Root Figure4 Time-dependentpercentagesofleaf(green),stem(blue)androotbiomass(thinblack)fortwodifferentgenotypesataQTL detectedonlinkagegroup12(upperpanel)and24(lowerpanel).TwogenotypesataQTLarethehomozygotefortheallelesinherited fromKefengNo.1(A,D)andthehomozygoteforallelesfromNannong1138-2(B,C). Wu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page7of11

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constitutethesystemthroughdevelopmentalregulation arestudiedandconnectedbyasystemofbiologically meaningfuldifferentialequations(DE).Thus,thegenetic mappingofacomplexphenotypebecomeanissueof testingandestimatinggenotype-specificcurveparametersatspecificQTLsthatdefinetheemergingpropertiesanddynamicbehavioroftheDEsystem.Systems mappingidentifiesQTLsthatcontroldevelopmental interactionsoftraits,thetemporalpatternofQTLs expressionduringdevelopment,aswellasthegenetic determinantsthatcontroldevelopmentalswitches(on/ off). SystemsmappingwasappliedtomapQTLsfor dynamictrajectoriesofbiomassfromdifferentorgans, thestem,leaves,androots,thatinteractandcoordinate todeterminewhole-plantbiomassgrowth,inanexperimentalcrossofsoybeansbetweenKefengNo.1and Nannong1138-2.Ingeneral,thestemreceivesaproportionallylargeramountofbiomasswithdevelopment, accompanyingaproportionaldecreaseofbiomasstothe leavesandroots.SpecificQTLs, biomass1 and biomass2, thatcontrolthisallometricchangewithdevelopment havebeendetectedfromsystemsmapping.Thealleles atthetwoQTLsinheritedfromKefengNo.1tendto amplifythiscontrastindevelopment-dependentbiomass partitioning,ascomparedtothosefromNannong11382.OneofthetwoQTLs, biomass2 ,wasfoundinasimilargenomicregionidentifiedbytraditionalfunctional mapping[44].Thisconsistencydoesnotonlysimply verifyoursystemsmapping,butalsogainsnewinsight intobiologicalfunctionsofthedetectedQTLs.For example,thetwoQTLsdetected, biomass1 and biomass2 ,triggergeneticeffectsontheinteractionsand coordinationofdifferentorganswhichcausethe dynamicvariationofbiomassgrowth. ThroughvarioustestsforODEparametersindividuallyorinacombination,oursystemsmappingcan revealthegeneticcontrolmechanismsforseveral mechanisticallymeaningfulrelationships.Theyare(1) size-shaperelationship -isabigplantduetoabigstem withsparseleavesorasmallstemwithdenseleaves?(2) structural-functionalrelationship -inaspecificenvironmentdoesaplanttendtoallocatemorecarbontoits leavesforCO2uptakeorrootsforwaterandnutrient uptake?(3) cause-effectrelationship -aremoreroots duetomoreleavesordomoreleavesproducemore roots?and(4) pleiotropicrelationship -differenttraits withasimilarfunctiontendtointegrateintomodularity [45].HowdothesameQTLspleiotropicallycontrolthis modularity?Abetterunderstandingoftheserelationshipshelpstogainmoreinsightsintothemechanistic responseofplantsizeandshapetodevelopmentaland environmentalsignalsand,also,provideguidanceto selectanideotypeofcropcultivarswithoptimalshape andstructuresuitedtoaparticularenvironment[46]. Themodeldescribedinthisarticleisasimpleframeworkforsystemsmapping.Itcanbeusedasastart pointtoexpandtheconceptofsystemsmappingto tacklemorecomplicatedbiologicalproblems.Aphenotypecanbedissectedtoanynumberofcomponentsat anyleveloforganization,molecule,cell,tissue,orwhole organism,dependingonthei nterestofresearchersand dataavailability.Withmoreknowledgeaboutphenotype formationanddevelopment,morecomponentscanbe involvedinasystemthatisspecifiedbyhigh-dimension differentialequations.Soph isticatedmathematicaltechniquesareneededtoobtainstablesolutionsofthese equations.Inaddition,byintegratingitwithgenomewideassociationstudies,systemsmappingwillnotonly provideaclearviewofhowdifferentcomponentsinteractandcoordinatetoformaphenotype,butalsowillbe capableofillustratingacomprehensivepictureofthe geneticarchitectureofcomplexphenotypes.Thereis alsoagoodreasontointegratesystemsmappingwith networkbiologytoexplorehow omics information contributetotheregulatorymechanismsofphenotype formation[47].Inanycase,systemsmappingwillopen anewavenueforunderstandingthegeneticarchitecture ofcomplexphenotypesfromaperspectiveofmechanisticpathwaysinsidetheirformation.ConclusionsThepasttwodecadeshaveseenaphenomenalincrease inthenumberoftoolsforthegeneticmappingofcomplextraits.Althoughgeneticmappingcontinuestobe aninterestingareaingeneticresearchowingtothesuccessofmolecularandsequencingtechnologiesingeneratingafloodofdata,aconceptualbreakthroughinthis arearemainselusive.Inthisarticle,wepresentabottom-topmodelformappingandstudyingthegenetic architectureofcomplextraits.Differentfromexisting Table2Meansofmaximumlikelihoodestimatesofcurve parametersfromtheODEsystem(1)andtheirbiases, standarddeviations(STD)andsquarerootmeansquare errors(RMSE)from100simulationreplicatesGenotypeEstimate aLbLgLaSbSaRbRgRQQ TRUE2.550.100.310.980.041.110.512.18 MEAN2.550.100.310.980.041.110.512.18 BIAS*1032.10-0.070.720.270.013.49-0.88-2.89 STD*1023.750.080.600.940.011.290.491.11 RMSE*1023.750.080.610.940.011.340.501.15 qq TRUE1.890.140.441.040.071.110.561.85 MEAN1.890.140.441.040.071.110.561.85 BIAS*103-4.230.110.55-0.61-0.051.620.92-2.02 STD*1023.600.140.920.990.041.380.451.66 RMSE*1023.620.140.920.990.041.390.461.67 Wu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page8of11

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mappingmodels,weuseasystemsapproachtoidentify specificgenesorquantitativetraitlocithatgovernthe developmentalinteractionsofvariouscomponentscomprisingthephenotype.Themapofdevelopmentalinteractionsamongdifferentcomponentsisconstructedbya systemofdifferentialequations.Thus,byestimatingand testingmathematicalparametersthatspecifythesystem, weareabletopredictoralterthephysiologicalstatusof aphenotypebasedontheunderlyinggeneticcontrol mechanisms.Wehavetestedandvalidatedourmodel byanalyzingarealdatasetforgeneticmappingofbiomassgrowthinsoybeans.ThedetectionofQTLsbythe newmodelprovidesbiologica llymeaningfulinterpretationsofQTLeffectsontraitformationanddynamics. Thenewmodelcanbereadilyusedtostudythegenetic basisofphenotypesinanyotherorganism.MethodsMappingPopulationOurmodelderivationisbasedonamappingpopulation comprisingof n recombinantinbredlines(RILs), initiatedwithtwoinbredlines.Bycontinuousselfingor inbreeding,RILsaftertheF7generationareconsidered homozygousbecausethefixationatanylocusisgiven by f =1-0.57-1 1.Inpracticalterms,allplantsfroma singleRILaregeneticallyidentical,andcanbeusedfor replicatedexperimentsunderdifferentenvironments.In addition,eachRILrepresentsauniquecombinationof allelesfromtheparentalgenotypeswheretherearetwo homozygousgenotypesateachmarkerlocus,eachcorrespondingtoaparentalallele.Themappingpopulation isgenotypedatmolecularmarkerstoconstructalinkage mapcoveringtheentiregenome.Therecombination fractionbetweentwomarkersisconvertedtothe geneticdistanceincentiMorgan(cM)throughamap function,suchastheHaldaneorKosambimapfunction. ThemapconstructedisusedtolocateQTLsthatcontrolaquantitativetraitofinterest. Weobtainedasampleof184RILsderivedfromtwo cultivars,KefengNo.1andNannong1138-2,formappingagronomictraits.TheseRILsweregenotypedfor 950molecularmarkerslocatingin25linkagegroups [48,49].Theplantsweregrowninasimplelatticedesign withtworeplicatesinaplotatJiangpuSoybeanExperimentStation,NanjingAgriculturalUniversity,China. Tenplantsinthesecondrowofaplotwererandomly selectedformeasuringleaf,stemandrootbiomassat eachtimeinthewholegrowingseason.After20daysof seedlingemergence,dryweightsseparatelyforthe leaves,stemandrootsweremeasuredonceevery5to 10daysuntilmostplantsstoppedgrowth.Atotalof6 to8measurementsweretakenforeachoftheRILsstudied.Greateffortsweremadetocontrolmeasurement errorsforsuchalarge-scalefieldtrial.Phenotypingprecisionwasestimatedtobeabove95%. Unlikeatraditionalmappin gproject,ourgoalisto mapQTLsthatcontrolthedynamicprocessofhowdifferentorgans,thestem,leaves,androots,interactand coordinatetodeterminewhole-plantbiomass.Theinteractionsandcoordinationofdifferentorgansforaplant areunderstoodusingdesignprinciplesdescribedbythe ODEsystem(1).LikelihoodLet yki= yki( ti 1) , yki timiT denotethevectorof phenotypicvaluesfortrait k ( k =1forleafbiomass( L ), 2forstembiomass( S ),and3forrootbiomass( R ))measuredonprogeny i attimepoints ti 1, timiT .Note thatthenumberoftimepointsmeasuredmaybeprogeny-specific,expressedas miforprogeny i .Assuming thatmultipleQTLs(segregatingwith J genotypes),each bracketedbytwoflankingmarkers M ,affectsthese threetraits,weconstructa mixturemodel-basedlikelihoodas L( z M )=ni =1 Jj =1[ j | ifj( zi; j, )] (2) where y =( y1, y2, y3)isajointvectorofphenotypic valuesforthethreetraits,with zi=( y1 i, y2 i, y3 i)presentingthe z -vectorforprogeny i ; j | iistheconditional probabilityofQTLgenotype j ( j =1,..., J)giventhemarkergenotypeofprogeny i ,whichcanbeexpressedasa functionoftherecombinationfractionsbetweenthe QTLandmarkers[50],and fj( zi; j, )isanMVNof leaf,stemandrootbiomassforprogeny i whichcarries QTLgenotype j ,withmeanvectors j=( 1 j, 2 j, 3 j), j =1, ... J specifiedby j,andcovariancematrixspecifiedby Ifasystemofdifferentialequations(1)isusedtojointly modelQTLgenotype-specificmeansvectorsforthe threetraits,thenwehave j=( aLj, bLj, ljL, aSj, bSj, aRj, bRj, lRj)forgenotype j .EstimationUnlikeatraditionalmixturemodelforQTLmapping, wewillmodelthegenotypicvaluesofeachQTLgenotypeinlikelihood(2)characterizedbyagroupofnonlinearODEs.Whileananalyticalsolutionisnot available,wewillimplemen tnumericalapproachesto solvetheseODEs.Let kj( t )denotethegenotypicvalue ofthe k thtraitattime t foraQTLgenotype j .Thus, thedynamicsystemofthetraitsandtheirinteractions, regulatedbyQTLgenotypes,canbemodeledbyaWu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page9of11

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systemofODE(1), d kj( t ) dt = gk( j( t ), j), k =1, ,3, (3) where j( t )=( 1 j( t ),..., 3 j( t ))T,andjisavectorof ODEparametersassociatedwithQTLgenotype j .ForJ possiblegenotypesinthemappingpopulation,wehave = T 1,..., T JT .Thequestionnowishowto estimatefromnoisymeasurements.Thefunctionalmean kj( t )mayberepresentedasalinearcombinationof basisfunctions: kj( t )=Rr =1ckjrkjr( t )= cT kjkj( t ) (4) where jkj( t ) = ( jkj 1( t ),..., jkjR( t ))Tisavectorofbasis functionswith R ordersand ckj=( ckj 1,...,ckjR)Tisavectorofbasiscoefficients.Define c = ckj3; J k =1; j =1b asa length( R J )vectorofbasiscoefficients.Thecubic B-splinesareoftenchosenasbasisfunctions,sinceany B-splinebasisfunctionisonlypositiveoverashortsubintervalandzeroelsewhere.Thisiscalledthe compact support property,andisessentialforefficientcomputation.TheflexibilityoftheB-splinebasisfunctions dependonthenumberandlocationofknotswechoose. Itisaninfinite-dimensionoptimizationproblemto choosetheoptimalnumberofknotsandtheirlocations. Apopularapproachtoavoidthisdilemmaischoosinga saturatednumberofknotsandusingaroughnesspenaltytocontrolthesmoothnessofthefittedcurveand avoidover-fitting[40]. Weestimatethebasiscoefficient c andODEparameter basedonatwo-nestedlevelofoptimization.In theinnerlevelofoptimization, c isestimatedbyoptimizingacriterion U ( c | ),givenanyvalueof .Therefore,theestimate c maybeviewedasafunctionof whichisdenotedas c ( ) .Sincenoanalyticformula for c isavailable, c ( ) isanimplicitfunction.Inthe outerlevelofoptimization, isestimatedbyoptimizing acriterion H c ( ) .Theparameter c isremovedin theparameterspaceintheouterlevelbytreatingitas animplicitfunctionof .Although c ( ) doesnot haveananalyticformula,theouterlevelofoptimization onlyrequirestocalculatethederivative d c / d ,which canbeobtainedbyusingtheimplicitfunctiontheorem. Theaboveoptimizationprocedureiscalledtheparametercascadingmethod.Notethatwhenthetwocriteria U ( c | )and H c ( ) arethesame,the parametercascadingmethodisequivalenttotheprofilingmethod.AdditionalmaterialAdditionalfile1:FigureS1 .Theprofilesofthelog-likelihoodratios(LR) betweenthefullmodel(thereisaQTL)andreducedmodel(thereisno QTL)forsoybeanheightgrowthtrajectoriesthroughoutthesoybean genomecomposedof25linkagegroups. Acknowledgements ThisworkissupportedbytheChangjiangScholarsAward, One-thousand Person Award,NSF/IOS-0923975,adiscoverygrantoftheNaturalSciences andEngineeringResearchCouncilofCanada(NSERC)(J.Cao),theNational KeyBasicResearchProgramofChina(2009CB1184 2010CB1259 2011CB1093),TheNationalHighTechnologyR&DProgramofChina (2009AA1011),andtheMOE111Project(B08025).Partofthisworkwas carriedwhenRWandJCwereinvitedResearchFellowsattheStatisticaland AppliedMathematicalSciencesInstitute(SAMSI),sponsoredbyDuke University,UniversityofNorthCarolinaatChapelHill,andNorthCarolina StateUniversity.ThanksareduetoProf.ShouyiChenandProf.DeyueYufor kindpermissiontousethejointlydevelopedNJRIKYgeneticlinkagemap. Authordetails1CenterforComputationalBiology,NationalEngineeringLaboratoryforTree Breeding,KeyLaboratoryofGeneticsandBreedinginForestTreesand OrnamentalPlants,BeijingForestryUniversity,Beijing100083,China.2DepartmentofStatistics&ActuarialScience,SimonFraserUniversity, Burnaby,B.C.CanadaV5A1S6.3DepartmentofAgronomy,HenanInstituteof ScienceandTechnology,Xinxiang453003,China.4CenterforStatistical Genetics,PennsylvaniaStateUniversity,Hershey,PA17033,USA.5National CenterforSoybeanImprovement,NationalKeyLaboratoryforCropGenetics andGermplasmEnhancement,SoybeanResearchInstitute,Nanjing AgriculturalUniversity,Nanjing210095,China.6DepartmentofHorticultural Sciences,UniversityofFlorida,Gainesville,FL32611,USA. Authors contributions RWconceivedoftheideaofsystemsmapping,coordinatedthewhole study,andwrotethemanuscript.JCderivedthemodel,computedthereal data,runsimulationstudies,andparticipatedinthewritingoftheMethods section.ZHconductedthesoybeanexperimentandcollectedphenotypic data.ZWpackedthemodelintoacomputerpackageSysMap.JGdirected theexperimentaldesignanddatacollectionofsoybeans.EVoversawthe projectandhighlightedthebiologicalrelevanceacomputationalmodel mustpossess.Allauthorsreadandapprovedthefinalmanuscript. Received:7February2011Accepted:27May2011 Published:27May2011 References1.LeeSH,vanderWerfJHJ,HayesBJ,GoddardME,VisscherPM: Predicting unobservedphenotypesforcomplextraitsfromwhole-genomeSNP data. PLoSGenet 2008, 4(10) :e1000231.. 2.LynchM,WalshB: GeneticsandAnalysisofQuantitativeTraits Sinauer Associates,Sunderland,MA;1998. 3.LanderES,BotsteinD: MappingMendelianfactorsunderlying quantitativetraitsusingRFLPlinkagemaps. Genetics 1989, 121 :185-199. 4.ZengZB: Precisionmappingofquantitativetraitloci. Genetics 1994, 136 :1457-1468. 5.XuS,AtchleyWR: Arandommodelapproachtointervalmappingof quantitativegenes. Genetics 1995, 141 :1189-1197. 6.WuRL,MaCX,CasellaG: Jointlinkageandlinkagedisequilibrium mappingofquantitativetraitlociinnaturalpopulations. Genetics 2002, 160 :779-792. 7.YiN,XuS: Bayesianlassoforquantitativetraitlocimapping. Genetics 2008, 179 :1045-1055. 8.ZouF,NieL,WrightFA,SenPK: ArobustQTLmappingprocedure. JStat PlannInfer 2009, 139 :978-989. 9.EhrenreichIM,TorabiN,JiaY,KentJ,MartisS,ShapiroJA,GreshamD, CaudyAA,KruglyakL: Dissectionofgeneticallycomplextraitswith extremelylargepoolsofyeastsegregants. Nature 2010, 464 :1039-1042.Wu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page10of11

PAGE 11

10.SteinmetzLM,SinhaH,RichardsDR,SpiegelmanJI,OefnerPJ,McCuskerJH, DavisRW: Dissectingthecomplexarchitectureofaquantitativetrait locusinyeast. Nature 2002, 416 :326-330. 11.SingerJB,HillAE,BurrageLC,OlszensKR,SongJ,JusticeM,O BrienWE, ContiDV,WitteJS,LanderES,NadeauJH: Geneticdissectionofcomplex traitswithchromosomesubstitutionstrainsofmice. Science 2004, 304 :445-448. 12.AltshulerD,DalyMJ,LanderES: Geneticmappinginhumandisease. Science 2008, 322 :881-888. 13.ShaoH,BurrageLC,SinasacDS,HillAE,ErnestSR,O BrienW,CourtlandHW, JepsenKJ,KirbyA,KulbokasEJ,DalyMJ,BromanKW,LanderES,NadeauJH: Geneticarchitectureofcomplextraits:Largephenotypiceffectsand pervasiveepistasis. ProcNatlAcadSciUSA 2008, 105 :19910-19914. 14.MackayTFC,StoneEA,AyrolesJF: Thegeneticsofquantitativetraits: challengesandprospects. NatRevGenet 2009, 10 :565-577. 15.BalasubramanianS,SchwartzC,SinghA,WarthmannN,KimMC,MaloofJN, LoudetO,TrainerGT,DabiT,BorevitzJO,ChoryJ,WeigelD: QTLmapping innew Arabidopsisthaliana advancedintercross-recombinantinbred lines. PLoSONE 2009, 4(2) :e4318. 16.FraryA,NesbittTC,GrandilloS,KnaapE,CongB,LiuJ,MellerJ,ElberR, AlpertKB,TanksleySD: fw2.2 :aquantitativetraitlocuskeytothe evolutionoftomatofruitsize. Science 2000, 289 :85-88. 17.LiC,ZhouA,SangT: Ricedomesticationbyreducingshattering. Science 2006, 311 :1936-1939. 18.HuangX,QianQ,LiuZ,SunH,HeS,LuoD,XiaG,ChuC,LiJ,FuX: NaturalvariationattheDEP1locusenhancesgrainyieldinrice. Nat Genet 2009, 41 :494-497. 19.MaCX,CasellaG,WuRL: Functionalmappingofquantitativetraitloci underlyingthecharacterprocess:atheoreticalframework. Genetics 2002, 161 :1751-1762. 20.WuRL,LinM: Functionalmapping-howtomapandstudythegenetic architectureofdynamiccomplextraits. NatRevGenet 2006, 7 :229-237. 21.LiY,WuRL: Functionalmappingofgrowthanddevelopment. BiolRev 2010, 85 :207-216. 22.KitanoH: Systemsbiology:abriefoverview. Science 2002, 295 :1662-1664. 23.JansenRC: Studyingcomplexbiologicalsystemsusingmultifactorial perturbation. NatRevGenet 2003, 4 :145-151. 24.CseteME,DoyleJC: Reverseengineeringofbiologicalcomplexity. Science 2002, 295 :1664-1669. 25.CannellMGR,DewarRC: Carbonallocationintrees:areviewofconcepts formodelling. AdEcolRes 1994, 25 :59-104. 26.LuoY,FieldCB,MooneyHA: Predictingresponsesofphotosynthesisand rootfractiontoelevatedCO2:Interactionsamongcarbon,nitrogen,and growth. PlantCellEnviron 1994, 17 :1195-1204. 27.ChenJ,ReynoldsJ: Acoordinationmodelofcarbonallocationinrelation towatersupply. AnnBot 1997, 80 :45-55. 28.WeinerJ: Allocation,plasticityandallometryinplants. PerspectPlantEcol EvolSyst 2004, 6 :207-215. 29.HermansC,HammondJP,WhitePJ,VerbruggenN: Howdoplants respondtonutrientshortagebybiomassallocation? TrendsPlantSci 2006, 11 :610-617. 30.MarcelisLFM,HeuvelinkE: Conceptsofmodellingcarbonallocation amongplantorgans. In Functional-StructuralPlantModellinginCrop Production. Editedby:VosJ,MarcelisLFM,deVisserPHB,StruikPCand EversJB.Springer.PrintedintheNetherlands;2007:103-111. 31.GenardM,DauzatJ,FranckN,LescourretF,MoitrierN,VaastP, VercambreG: Carbonallocationinfruittrees:Fromtheorytomodelling. Trees-StructureandFunction 2008, 22 :269-282. 32.WestGB,BrownJH,EnquistBJ: Ageneralmodelfortheoriginof allometricscalinglawsinbiology. Science 1997, 276 :122-126. 33.WestGB,BrownJH,EnquistBJ: Thefourthdimensionoflife:Fractal geometryandallometricscalingoforganisms. Science 1999, 284 :1677-1679. 34.WestGB,BrownJH,EnquistBJ: Ageneralmodelforontogeneticgrowth. Nature 2001, 413 :628-631. 35.PutterH,HeisterkampSH,LangeJMA,DeWolfF: ABayesianapproachto parameterestimationinHIVdynamicalmodels. StatMed 2002, 21 :2199-2214. 36.HuangY,WuH: ABayesianapproachforestimatingantiviralefficacyin HIVdynamicmodels. JApplStat 2006, 33 :155-174. 37.HuangJZ,LiuN,PourahmadiM,LiuL: Covarianceselectionand estimationviapenalizednormallikelihood. Biometrika 2006, 93 :85-98. 38.LiL,BrownMB,LeeKH,GuptaS: Estimationandinferenceforasplineenhancedpopulationpharmacokineticmodel. Biometrics 2002, 58 :601-611. 39.RamsayJO: Principaldifferentialanalysis:Datareductionbydifferential operators. JRoyStatSocSerB 1996, 58 :495-508. 40.RamsayJO,SilvermanBW: FunctionalDataAnalysis. 2edition.Springer,New York;2005. 41.RamsayJO,HookerG,CampbellD,CaoJG: Parameterestimationfor differentialequations:ageneralizedsmoothingapproach(with discussion). JRoyStatSocSerB 2007, 69 :741-796. 42.LiangH,WuHL: Parameterestimationfordifferentialequationmodels usingaframe-workofmeasurementerrorinregressionmodel. JAm StatAssoc 2008, 103 :1570-1583. 43.EfronB,TibshiraniRJ: AnIntroductiontotheBootstrap Chapman&Hall; 1993. 44.LiQ,HuangZ,XuM,WangC,GaiJ,HuangY,PangX,WuRL: Functional mappingofgenotype-environmentinteractionsforsoybeangrowthby asemiparametricapproach. PlantMethods 2010, 6 :13. 45.McCarthyMC,EnquistBJ: Consistencybetweenanallometricapproach andoptimalpartitioningtheoryinglobalpatternsofplantbiomass allocation. FunctEcol 2007, 21 :713-720. 46.JiaoY,WangY,XueD,WangJ,YanM,LiuG,DongG,ZengD,LuZ,ZhuX, QianQ,LiJ: RegulationofOsSPL14byOsmiR156definesidealplant architectureinrice. NatGenet 2010, 42 :541-544. 47.WuS,YapJS,LiY,LiQ,FuGF,LiJH,DasK,BergA,ZengYR,WuRL: Networkmodelsfordissectingplantdevelopmentbyfunctional mapping. CurrBioinform 2009, 4 :183-187. 48.LiH,HuangZ,GaiJ,WuS,ZengY,WuRL: Aconceptualframeworkfor mappingquantitativetraitlociregulatingontogeneticallometry. PLoS ONE 2007, 2(11) :e1245. 49.ZhangW-K,WangY-J,LuoG-Z,ZhangJ-S,HeC-Y,WuXL,GaiJY,ChenSY: QTLmappingoftenagronomictraitsonthesoybean( Glycine maxL. Merr.)geneticmapandtheirassociationwithESTmarkers. TheorAppl Genet 2004, 108 :1131-1139. 50.WuRL,MaCX,CasellaG: StatisticalGeneticsofQuantitativeTraits:Linkage, Maps,andQTL Springer-Verlag,NewYork;2007.doi:10.1186/1752-0509-5-84 Citethisarticleas: Wu etal .: Systemsmapping:howtoimprovethe geneticmappingofcomplextraitsthroughdesignprinciplesof biologicalsystems. BMCSystemsBiology 2011 5 :84. Submit your next manuscript to BioMed Central and take full advantage of: Convenient online submission Thorough peer review No space constraints or color gure charges Immediate publication on acceptance Inclusion in PubMed, CAS, Scopus and Google Scholar Research which is freely available for redistribution Submit your manuscript at www.biomedcentral.com/submit Wu etal BMCSystemsBiology 2011, 5 :84 http://www.biomedcentral.com/1752-0509/5/84 Page11of11

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SupplementaryMaterial Aconceptualframeworkforgeneticdis-sectionofcomplextraitsthroughde-signprinciplesofbiologicalsystemsRonglingWu 1 ,JiguoCao 2 ,ZhongwenHuang 3 ,ZhongWang 4 ,JunyiGai 5 andC.Eduardo Vallejos 6 1 CenterforComputationalBiology,NationalEngineeringLa boratoryforTreeBreeding,Key LaboratoryofGeneticsandBreedinginForestTreesandOrna mentalPlants,BeijingForestry University,Beijing100083,China2 DepartmentofStatistics&ActuarialScience,SimonFraser University,Burnaby,B.C. CanadaV5A1S63 DepartmentofAgronomy,HenanInstituteofScienceandTech nology,Xinxiang453003, China4 CenterforStatisticalGenetics,PennsylvaniaStateUnive rsity,Hershey,PA17033,USA 5 NationalCenterforSoybeanImprovement,NationalKeyLabo ratoryforCropGenetics andGermplasmEnhancement,SoybeanResearchInstitute,Na njingAgriculturalUniversity, Nanjing210095,China6 DepartmentofHorticulturalSciences,UniversityofFlori da,Gainesville,FL32611,USA 1

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Additionalle,FigureS1 250 500 750 1000 12 345 LR 250 500 750 1000 678910111213 LR 250 500 750 1000 14151617181920 LR 0 250 500 750 1000 21222324 25 LRTest Position (cM) 10 cM Theprolesofthelog-likelihoodratios(LR)betweenthefu llmodel(thereisaQTL)and reducedmodel(thereisnoQTL)forsoybeanheightgrowthtra jectoriesthroughoutthe soybeangenomecomposedof25linkagegroups.Thegenomicpo sitioncorrespondingtothe peakofthecurveisthemaximum-likelihoodestimateoftheQ TLlocation.Tickmarkson thex-axisrepresentthepositionsofmicrosatellitemarke rsoneachchromosome(bar,10cM). Thecriticalthresholdsforacclaimingthegenome-wideexi stenceofaQTLareobtainedfrom permutationtests.The95thpercentile(indicatedathoriz ontallines)ofthedistributionof themaximumLRvaluesobtainedfrom200permutationtestsis usedasanempiricalcritical valuetodeclaregenome-wideexistenceofaQTLatthe5%sign icancelevel. 2


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epdcx:valueString Systems mapping: how to improve the genetic mapping of complex traits through design principles of biological systems
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Abstract
Background
Every phenotypic trait can be viewed as a "system" in which a group of interconnected components function synergistically to yield a unified whole. Once a system's components and their interactions have been delineated according to biological principles, we can manipulate and engineer functionally relevant components to produce a desirable system phenotype.
Results
We describe a conceptual framework for mapping quantitative trait loci (QTLs) that control complex traits by treating trait formation as a dynamic system. This framework, called systems mapping, incorporates a system of differential equations that quantifies how alterations of different components lead to the global change of trait development and function through genes, and provides a quantitative and testable platform for assessing the interplay between gene action and development. We applied systems mapping to analyze biomass growth data in a mapping population of soybeans and identified specific loci that are responsible for the dynamics of biomass partitioning to leaves, stem, and roots.
Conclusions
We show that systems mapping implemented by design principles of biological systems is quite versatile for deciphering the genetic machineries for size-shape, structural-functional, sink-source and pleiotropic relationships underlying plant physiology and development. Systems mapping should enable geneticists to shed light on the genetic complexity of any biological system in plants and other organisms and predict its physiological and pathological states.
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Wu, Rongling
Cao, Jiguo
Huang, Zhongwen
Wang, Zhong
Gai, Junyi
Vallejos, Eduardo
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dochead Methodology article
bibl
title
p Systems mapping: how to improve the genetic mapping of complex traits through design principles of biological systems
aug
au ca yes id A1 ce snm Wufnm Ronglinginsr iid I1 email rwu@hes.hmc.psu.edu
A2 CaoJiguoI2 jiguo_cao@sfu.ca
A3 HuangZhongwenI3 huangdou373@126.com
A4 WangZhongI4 zwang@hes.hmc.psu.edu
A5 GaiJunyiI5 sri@njau.edu.cn
A6 VallejosEduardoI6 vallejos@ufl.edu
insg
ins Center for Computational Biology, National Engineering Laboratory for Tree Breeding, Key Laboratory of Genetics and Breeding in Forest Trees and Ornamental Plants, Beijing Forestry University, Beijing 100083, China
Department of Statistics & Actuarial Science, Simon Fraser University, Burnaby, B.C. Canada V5A 1S6
Department of Agronomy, Henan Institute of Science and Technology, Xinxiang 453003, China
Center for Statistical Genetics, Pennsylvania State University, Hershey, PA 17033, USA
National Center for Soybean Improvement, National Key Laboratory for Crop Genetics and Germplasm Enhancement, Soybean Research Institute, Nanjing Agricultural University, Nanjing 210095, China
Department of Horticultural Sciences, University of Florida, Gainesville, FL 32611, USA
source BMC Systems Biology
issn 1752-0509
pubdate 2011
volume 5
issue 1
fpage 84
url http://www.biomedcentral.com/1752-0509/5/84
xrefbib pubidlist pubid idtype doi 10.1186/1752-0509-5-84pmpid 21615967
history rec date day 7month 2year 2011acc 2752011pub 2752011
cpyrt 2011collab Wu et al; licensee BioMed Central Ltd.note This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
abs
sec
st
Abstract
Background
Every phenotypic trait can be viewed as a "system" in which a group of interconnected components function synergistically to yield a unified whole. Once a system's components and their interactions have been delineated according to biological principles, we can manipulate and engineer functionally relevant components to produce a desirable system phenotype.
Results
We describe a conceptual framework for mapping quantitative trait loci (QTLs) that control complex traits by treating trait formation as a dynamic system. This framework, called systems mapping, incorporates a system of differential equations that quantifies how alterations of different components lead to the global change of trait development and function through genes, and provides a quantitative and testable platform for assessing the interplay between gene action and development. We applied systems mapping to analyze biomass growth data in a mapping population of soybeans and identified specific loci that are responsible for the dynamics of biomass partitioning to leaves, stem, and roots.
Conclusions
We show that systems mapping implemented by design principles of biological systems is quite versatile for deciphering the genetic machineries for size-shape, structural-functional, sink-source and pleiotropic relationships underlying plant physiology and development. Systems mapping should enable geneticists to shed light on the genetic complexity of any biological system in plants and other organisms and predict its physiological and pathological states.
bdy
Background
Predicting the phenotype from the genotype of complex organisms is one of the most important and challenging questions we face in modern biology and medicine abbrgrp
abbr bid B1 1
. Genetic mapping, dissecting a phenotypic trait to its underlying quantitative trait loci (QTLs) through the use of molecular markers, has proven powerful for establishing genotype-phenotype relationships and predicting phenotypes of individual organisms based on their QTL genotypes responsible for the trait
B2 2
. The success of this prediction depends on how well we can map the underlying QTLs and characterize complex interactions of these QTLs with each other and with environmental factors. Powerful statistical models have been developed in the past two decades to detect QTLs and study their biological function in a diverse array of phenotypic traits
B3 3
B4 4
B5 5
B6 6
B7 7
B8 8
B9 9
. Worldwide, a substantial effort has been made resulting in the collection of a large amount of data aimed at the identification of QTLs
B10 10
B11 11
B12 12
B13 13
B14 14
B15 15
. Unfortunately, despite hundreds of thousands of QTLs detected in a diversity of organisms, only a few of them have been isolated by positional cloning (see
B16 16
B17 17
B18 18
), leaving it unsolved how to construct a genotype-phenotype relationship map using genetic mapping.
The most likely reason for this result may arise from a possibility that the QTLs detected by stringent statistical tests are not biologically relevant. Existing strategies for QTL mapping were built on testing for a direct association between genotype and end-point phenotype. Although such strategies are simple and have been widely accepted, they neglect the biological processes involved in trait development
14
. To attempt to fill this gap, a statistical model, called functional mapping
B19 19
B20 20
B21 21
, has been developed to study the interplay between genetics and the developmental process of a phenotypic trait by integrating mathematical models and computational algorithms. If a trait is understood as a "system" that is composed of many underlying biological components
B22 22
B23 23
B24 24
, we should be in a better position to comprehend the process and behavior of trait formation based on interactive relationships among different components. Through mapping and using those QTLs that govern design principles of a biological system, a new trait that is able to maximize resource-use efficiency can be generated and engineered.
As one important strategy for plants to respond to variation in the availability of resources in their environment, biomass allocation has been extensively used to study the relationship between structure and function in modern ecology
B25 25
B26 26
B27 27
B28 28
. The concept of biomass allocation has now been increasingly integrated with plant management and breeding, aimed to direct a maximum amount of biomass to the target of harvest (leaves, stem, roots, or fruits)
B29 29
B30 30
B31 31
. If the whole-plant biomass is considered as a target trait, we need to understand how different organs of a plant coordinate and interact to optimize the capture of nutrients, light, water, and carbon dioxide in a manner that maximizes plant growth rate through a specific developmental program because plant biomass growth is not simply the addition of biomass to various organs. Many theories and models have been proposed to predict the pattern of biomass partitioning in a response to changing environment. Chen and Reynolds
27
used coordination theory to model the dynamic allocation of carbon to different organs during growth in relation to carbon and water/nitrogen supply by a group of differential equations. Compared to the conventional optimization model in the context of maximizing the relative growth rate of a plant, the coordination model does not require an unrealistic capacity the plant possesses for knowing beforehand the environmental conditions it will experience during the growth period. Here, we integrate the coordination and optimization model to study the pattern of biomass partitioning by incorporating the allometric scaling theory into a system of differential equations.
In a series of allometric studies by West et al.
B32 32
B33 33
B34 34
, a power relationship that universally exists between parts and the whole can be explained by two fundamental design principles in biophysics and biochemistry; i.e., all organisms tend to maximize their metabolic capacity by increasing surface areas for energy and material production as well as internal efficiency through reducing distances and the time to transport water, nutrients, and carbon. The integration of this optimization theory expressed in terms of allometric scaling with the coordination theory leads to a tripled group of ordinary differential equations (ODEs) to specify the coordination of the leaf, stem, and root biomass for a plant:
display-formula M1
graphic file 1752-0509-5-84-i1.gif
where it Msub L
, MS
, and MR
are the biomasses of the leaves (L), the stem (S), and the roots (R), respectively, with whole-plant biomass W = ML
+ MS
+ MR
; α and β are the constant and exponent power of an organ biomass scaling as whole-plant biomass
32
33
; and γ is the rate of eliminating ageing leaves and roots. The complex interactions between different parts of a plant that underlie design principles of plant biomass growth can be modeled and studied by estimating and testing the ODE parameters (αL
, βL
, λL
, αS
, βS
, αR
, βR
, λR
). For example, plants are equipped with a capacity to optimize their fitness under low nutrient availability by shifting the partitioning of carbohydrates to processes associated with nutrient uptake at a cost of carbon acquisition
29
. These parameters can be used to quantify and predict such regulation between different plant parts in response to environmental and developmental changes.
In this article, we put forward a conceptual framework to incorporate the design principles of trait formation and development into a statistical framework for QTL mapping. Complementary to our previous functional mapping
19
20
21
, we name this new mapping framework "systems mapping" in light of its systems dissection and modeling of phenotypic formation. A group of ODEs like (1) or other types of differential equations is used to quantify the phenotypic system. Much work in solving ODEs has focused on the simulation and analysis of the behavior of state variables for a dynamic system, but the estimation of ODE parameters that define the system based on the measurement of state variables at multiple time points is relatively a new area. Yet, in the recent years, many statisticians have made great attempts to develop statistical approaches for estimating ODE parameters by modeling the structure of measurement errors
B35 35
B36 36
B37 37
B38 38
B39 39
B40 40
B41 41
B42 42
. We implemented Ramsay et al.'s
41
penalized spline method for estimating constant dynamic parameters in our genetic mapping. The problem for systems mapping with ODE models is different from those considered in current literature. First, systems mapping is constructed within a mixture-based framework because QTL genotypes that define the DE models are missing. Second, systems mapping incorporates genotypic data which are categorical or binary. These two characteristics determine the high complexity of our statistical model and computational algorithm used for systems mapping.
Results
QTL detection
We develop a new model for QTL mapping by treating trait formation as a dynamic system and further incorporating the design principles of the biological system into a statistical mapping framework (see b Methods). The new model, named systems mapping in light of its systems feature of phenotypic description, was used to map QTLs for biomass partitioning in a mapping population of soybeans composed of 184 recombinant inbred lines (RIL) derived from two cultivars, Kefeng No. 1 and Nannong 1138-2. For an RIL population, there are two homozygous genotypes, one composed of the Kefeng No. 1 alleles and the other composed of the Nannong 1138-2 alleles. Figure figr fid F1 1 illustrates the growth trajectories of leaf, stem and root biomass for individual RILs. By using the system of ODEs (1) to fit growth trajectories of leaf, stem and root biomass over time, we obtained a mean curve for each trait. It can be seen that growth trajectories can be well modeled by three interconnecting ODEs (1) derived from coordination theory
27
and allometric scaling
32
33
34
. The model-fitted curves of leaf (Figure 1A) and root biomass trajectories (Figure 1C) delineate reasonably the decay of leaf and root biomass at a late stage of development due to senescence. As expected, stem biomass growth does not experience such a decay (Figure 1B) although growth at the late stage tends to be stationary.
fig Figure 1caption Growth trajectories of leaf (A), stem (B) and root biomass (C) measured at multiple time points in a growing season of soybeanstext
Growth trajectories of leaf (A), stem (B) and root biomass (C) measured at multiple time points in a growing season of soybeans. Each grey line presents the growth trajectory of one of 184 RILs, whereas black lines are the mean growth trajectories of all RILs fitted using a system of ODEs (1).
1752-0509-5-84-1
By scanning the genetic linkage map composed of 950 molecular markers located in 25 linkage groups, we detected two significant QTLs, one named as biomass1 that resides between markers GMKF167a and GMKF167b and the second as biomass2 that resides between markers sat-274 and BE801128 (Additional file supplr sid S1 1, Figure S1). Using the maximum likelihood estimates of the curve parameters in ODE (1) whose standard errors were obtained by the parameter bootstrap
B43 43
(Table tblr tid T1 1), we drew the growth trajectories of leaf, stem and root biomass for two different genotypes at each QTL (Figure F2 2). The genetic effects of the QTLs displayed different temporal patterns for three organs. The QTLs were expressed more rapidly with time for the stem than for the leaves and roots. At biomass1, the alleles from parent Nannong 1138-2 increase leaf and stem biomass growth (Figure 2A and 2B), whereas the alleles from parent Kefeng No. 1 increase root biomass growth (Figure 2C). This could be interpreted as the Nannong 1138-2 allele favoring carbon allocation to the shoots at the expense of the roots but the Kefeng No1 allele favoring carbon allocation to the roots at the expense of the shoots. Likewise, the biomass2 alleles from Nannong 1138-2 favor carbon allocation to the leaves (Figure 2D) and those from Kefeng No. 1 favor carbon allocation to the roots, but the alleles at this QTL inherited from parent Kefeng No. 1 favor carbon allocation to the stem, which is different from the behavior of QTL biomass1. Note that leaf and root biomass growth tend to decay at the late stage for almost all RILs. But the genotypes at the QTLs detected do not reflect this trend (Figure 2), although they display much reduced rates of growth at the late stage. We explained this to be due to some other QTLs that have not been detected with the current linkage map.
suppl
Additional file 1
Figure S1. The profiles of the log-likelihood ratios (LR) between the full model (there is a QTL) and reduced model (there is no QTL) for soybean height growth trajectories throughout the soybean genome composed of 25 linkage groups.
name 1752-0509-5-84-S1.PDF
Click here for file
tbl Table 1The maximum likelihood point estimates (PEs) of ODE parameters and standard errors (SEs) of the estimates for the QTLs detected.tblbdy cols 12
r
c left
QTL
center
Model
Genotype
Estimate
αL
βL
γL
αS
βS
αR
βR
γR
cspan
hr
1
M1
QQ
PE
2.09
0.16
0.43
0.93
0.07
1.52
0.66
2.91
SE
0.02
1e-3
3e-3
0.01
2e-4
0.01
6e-3
0.02
10
qq
PE
2.53
0.11
0.36
0.92
0.04
1.57
0.54
3.90
SE
0.01
4e-4
2e-3
0.01
1e-4
0.01
5e-3
0.01
11
M0
PE
2.30
0.13
0.39
0.92
0.05
1.56
0.60
3.37
9
SE
0.07
2e-3
0.01
0.02
4e-4
0.05
0.02
0.08
2
M1
QQ
PE
1.89
0.14
0.44
1.04
0.07
1.11
0.56
1.85
SE
0.04
1e-3
0.01
0.01
1e-4
0.01
5e-3
0.01
qq
PE
2.55
0.10
0.31
0.98
0.04
1.11
0.51
2.18
SE
0.04
1e-3
0.01
0.01
4e-4
0.01
5e-3
0.02
M0
PE
2.25
0.12
0.37
1.03
0.05
1.12
0.55
2.06
SE
0.08
3e-3
0.02
0.02
6e-4
0.02
0.01
0.05
tblfn
Model M1 assumes two different genotypes at a QTL (under the H1), whereas Model M0 assumes a single genotype (under the H0).
Note: QTL 1 is one between markers GMKF167a and GMKF167b on linkage group 12. QTL 2 is one between markers sat-274 and BE801128 on linkage group 24. The alleles of genotype QQ are derived from Kefeng No. 1 and those of genotype qq derived from Nannong 1138-2.
Figure 2Growth trajectories of leaf (A, D), stem (B, E) and root biomass (C, F) for two different genotypes (presented by solid and broken black curves) at a QTL detected on linkage group 12 (upper panel) and 24 (lower panel), respectively
Growth trajectories of leaf (A, D), stem (B, E) and root biomass (C, F) for two different genotypes (presented by solid and broken black curves) at a QTL detected on linkage group 12 (upper panel) and 24 (lower panel), respectively. Two genotypes at a QTL are the homozygote for the alleles inherited from Kefeng No.1 (solid) and the homozygote for the allele from Nannong 1138-2 (broken). Curves in grey are growth trajectories of 184 RILs.
1752-0509-5-84-2
The functional relationships among leaf, stem and root biomass were determined by the QTLs detected (Figure F3 3). For biomass1, two genotypes are not only different in whole-plant biomass trajectory, but also display pronounced discrepancies in biomass growth trajectories of individual organs (Figure 3A and 3B). This means that this QTL affects the dynamics of both plant size and biomass partitioning. The genotype composed of the Kefeng No. 1 alleles has a smaller slope of biomass growth, leading to smaller whole-plant biomass at late stages of development, than that composed of the Nannong 1138-2 alleles, but the former has larger root biomass over the entire period of growth at the expense of the shoots than the latter. For biomass2, two genotypes are similar in total plant size during growth, but they have a marked distinction in biomass partitioning (Figure 3C and 3D). It appears that this QTL affects plant growth trajectories through altering biomass partitioning rather than total amount of biomass. At this QTL, the genotype with the Kefeng No. 1 alleles has a dominant main stem and heavy roots, whereas the genotype with the Nannong 1138-2 alleles carries dense leaves.
Figure 3Growth trajectories of whole-plant (red), leaf (green), stem (blue) and root biomass (black) for two different genotypes at a QTL detected on linkage group 12 (upper panel) and 24 (lower panel)
Growth trajectories of whole-plant (red), leaf (green), stem (blue) and root biomass (black) for two different genotypes at a QTL detected on linkage group 12 (upper panel) and 24 (lower panel). Two genotypes at a QTL are the homozygote for the alleles inherited from Kefeng No.1 (A, C) and the homozygote for alleles inherited from Nannong 1138-2 (B, D).
1752-0509-5-84-3
Figure F4 4 shows the dynamic pattern of biomass partitioning to different organs. In general, the stem receives increasing allocation with time, whereas the partitioning to the leaves and roots decreases with time. Both QTLs detected, biomass1 and biomass2, control the degree of such time-dependent increase or decrease. For example, at biomass1, the Kefeng No. 1 genotype always exhibits a larger degree of increasing biomass partitioning to the stem but a larger degree of decreasing biomass partitioning to the leaves and roots than the Nannong 1138-2 genotype (Figure 4A vs. 4B). QTL biomass2 has a similar pattern of biomass partitioning for the stem and leaves, although it displays a stronger effect than does QTL biomass1. At QTL biomass2, there is a larger degree of decreasing biomass partitioning to the roots for the Nannong 1138-2 genotype than the Kefeng No. 1 genotype (Figure 4C vs. 4D).
Figure 4Time-dependent percentages of leaf (green), stem (blue) and root biomass (thin black) for two different genotypes at a QTL detected on linkage group 12 (upper panel) and 24 (lower panel)
Time-dependent percentages of leaf (green), stem (blue) and root biomass (thin black) for two different genotypes at a QTL detected on linkage group 12 (upper panel) and 24 (lower panel). Two genotypes at a QTL are the homozygote for the alleles inherited from Kefeng No.1 (A, D) and the homozygote for alleles from Nannong 1138-2 (B, C).
1752-0509-5-84-4
Simulation
By analyzing a real data set for soybean mapping, systems mapping produces the identification of two significant QTLs that control the dynamic formation of whole-plant biomass through developmental regulation of different organs, stem, leaves, and roots. To validate the new model, we performed simulation studies by mimicking the effects of QTL biomass2 detected from the example of QTL mapping in soybeans. The simulated mapping population contains the same genotype data for 184 RILs. The phenotypic values of three traits, the stem, leaf and root biomass, assumed to obey the system of ODE (1), were simulated at six different time points by summing time-dependent genotypic values at biomass2 calculated with curve parameters in Table 1 and residual errors. Specifically, the phenotypic values of the kth trait were simulated by adding white noise with variances inline-formula
1752-0509-5-84-i2.gif
to the ODE curves for the jth QTL genotype with the probability w
j|i
, i.e., the conditional probability of the ith RIL that carries the jth QTL genotype, given the two markers genotypes of this RIL. The values of noise variance
were set as the estimates from the real data, which are
1752-0509-5-84-i3.gif
,
1752-0509-5-84-i4.gif
and
1752-0509-5-84-i5.gif
for the leaf, stem, and root biomass, respectively. Meanwhile, by assuming a modest heritability (0.05) for each trait at a middle stage of growth, we re-scaled
values which were used to simulate a new data set.
Systems mapping, implemented with the parameter cascading method, estimates QTL genotype-specific curve parameters in the ODE (1) from the simulated data. The simulation was repeated 100 times to calculate the means, biases, standard deviations, and root mean square errors, with results tabulated in Table T2 2. It was found that the model can provide reasonably accurate and precise estimates of QTL genotype-specific ODE parameters with a modest sample size (n = 184). The biases of the estimates are negligible, compared with the scale of the standard deviations. Given that this simulated data is a mimicry of the real soybean data, the results suggest that the experimental design used for soybean mapping is scientifically sound and can provide convincing QTL detection. This actually is not surprising because phenotyping has low measurement errors.
Table 2Means of maximum likelihood estimates of curve parameters from the ODE system (1) and their biases, standard deviations (STD) and square root mean square errors (RMSE) from 100 simulation replicates.
Genotype
Estimate
αL
βL
γL
αS
βS
αR
βR
γR
QQ
TRUE
2.55
0.10
0.31
0.98
0.04
1.11
0.51
2.18
MEAN
2.55
0.10
0.31
0.98
0.04
1.11
0.51
2.18
BIAS*10sup 3
2.10
-0.07
0.72
0.27
0.01
3.49
-0.88
-2.89
STD*102
3.75
0.08
0.60
0.94
0.01
1.29
0.49
1.11
RMSE*102
3.75
0.08
0.61
0.94
0.01
1.34
0.50
1.15
qq
TRUE
1.89
0.14
0.44
1.04
0.07
1.11
0.56
1.85
MEAN
1.89
0.14
0.44
1.04
0.07
1.11
0.56
1.85
BIAS*103
-4.23
0.11
0.55
-0.61
-0.05
1.62
0.92
-2.02
STD*102
3.60
0.14
0.92
0.99
0.04
1.38
0.45
1.66
RMSE*102
3.62
0.14
0.92
0.99
0.04
1.39
0.46
1.67
In analyzing a simulated data set for the traits assumed to have a modest heritability (0.05), the estimates of the ODE parameters are reasonably accurate and precise, indicating the power of systems mapping to detect small QTLs involved in trait formation. We performed an additional simulation to investigate the power and false positive rates of the model by changing levels of noises. In general, the power of the model is high, reaching 0.80 even when the heritability of growth curves is low (0.05). Basically, a QTL can be fully detected when the heritability is 0.10 or larger. In any case, the false positive rates are not beyond 0.10, mostly being less than 0.05.
Discussion
Mapping the genetic architecture of complex traits has been a subject of interest in both theoretical and empirical aspects of modern biology
3
4
5
6
7
8
9
10
11
12
13
14
15
. Original approaches for genetic mapping are based on single-point variation in a phenotypic trait, neglecting the dynamic change of the trait during development. To capture the dynamic pattern of genetic control, a new statistical model called functional mapping has been developed by incorporating the mathematical aspects of trait development
19
20
21
. Despite significant improvement over conventional static mapping, functional mapping has still a major limitation in characterizing developmental pathways that cause a final phenotype and unraveling the underlying genetic mechanisms for trait formation and progression.
In this article, we have for the first time put forward a new approach-systems mapping by treating a phenotypic trait as a dynamic system and incorporating the design principles of a biological system into a statistical framework for genetic mapping. Various components that constitute the system through developmental regulation are studied and connected by a system of biologically meaningful differential equations (DE). Thus, the genetic mapping of a complex phenotype become an issue of testing and estimating genotype-specific curve parameters at specific QTLs that define the emerging properties and dynamic behavior of the DE system. Systems mapping identifies QTLs that control developmental interactions of traits, the temporal pattern of QTLs expression during development, as well as the genetic determinants that control developmental switches (on/off).
Systems mapping was applied to map QTLs for dynamic trajectories of biomass from different organs, the stem, leaves, and roots, that interact and coordinate to determine whole-plant biomass growth, in an experimental cross of soybeans between Kefeng No. 1 and Nannong 1138-2. In general, the stem receives a proportionally larger amount of biomass with development, accompanying a proportional decrease of biomass to the leaves and roots. Specific QTLs, biomass1 and biomass2, that control this allometric change with development have been detected from systems mapping. The alleles at the two QTLs inherited from Kefeng No. 1 tend to amplify this contrast in development-dependent biomass partitioning, as compared to those from Nannong 1138-2. One of the two QTLs, biomass2, was found in a similar genomic region identified by traditional functional mapping
B44 44
. This consistency does not only simply verify our systems mapping, but also gains new insight into biological functions of the detected QTLs. For example, the two QTLs detected, biomass1 and biomass2, trigger genetic effects on the interactions and coordination of different organs which cause the dynamic variation of biomass growth.
Through various tests for ODE parameters individually or in a combination, our systems mapping can reveal the genetic control mechanisms for several mechanistically meaningful relationships. They are (1) size-shape relationship is a big plant due to a big stem with sparse leaves or a small stem with dense leaves? (2) structural-functional relationship in a specific environment does a plant tend to allocate more carbon to its leaves for CO2 uptake or roots for water and nutrient uptake? (3) cause-effect relationship are more roots due to more leaves or do more leaves produce more roots? and (4) pleiotropic relationship different traits with a similar function tend to integrate into modularity
B45 45
. How do the same QTLs pleiotropically control this modularity? A better understanding of these relationships helps to gain more insights into the mechanistic response of plant size and shape to developmental and environmental signals and, also, provide guidance to select an ideotype of crop cultivars with optimal shape and structure suited to a particular environment
B46 46
.
The model described in this article is a simple framework for systems mapping. It can be used as a start point to expand the concept of systems mapping to tackle more complicated biological problems. A phenotype can be dissected to any number of components at any level of organization, molecule, cell, tissue, or whole organism, depending on the interest of researchers and data availability. With more knowledge about phenotype formation and development, more components can be involved in a system that is specified by high-dimension differential equations. Sophisticated mathematical techniques are needed to obtain stable solutions of these equations. In addition, by integrating it with genome-wide association studies, systems mapping will not only provide a clear view of how different components interact and coordinate to form a phenotype, but also will be capable of illustrating a comprehensive picture of the genetic architecture of complex phenotypes. There is also a good reason to integrate systems mapping with network biology to explore how "omics" information contribute to the regulatory mechanisms of phenotype formation
B47 47
. In any case, systems mapping will open a new avenue for understanding the genetic architecture of complex phenotypes from a perspective of mechanistic pathways inside their formation.
Conclusions
The past two decades have seen a phenomenal increase in the number of tools for the genetic mapping of complex traits. Although genetic mapping continues to be an interesting area in genetic research owing to the success of molecular and sequencing technologies in generating a flood of data, a conceptual breakthrough in this area remains elusive. In this article, we present a bottom-top model for mapping and studying the genetic architecture of complex traits. Different from existing mapping models, we use a systems approach to identify specific genes or quantitative trait loci that govern the developmental interactions of various components comprising the phenotype. The map of developmental interactions among different components is constructed by a system of differential equations. Thus, by estimating and testing mathematical parameters that specify the system, we are able to predict or alter the physiological status of a phenotype based on the underlying genetic control mechanisms. We have tested and validated our model by analyzing a real data set for genetic mapping of biomass growth in soybeans. The detection of QTLs by the new model provides biologically meaningful interpretations of QTL effects on trait formation and dynamics. The new model can be readily used to study the genetic basis of phenotypes in any other organism.
Methods
Mapping Population
Our model derivation is based on a mapping population comprising of n recombinant inbred lines (RILs), initiated with two inbred lines. By continuous selfing or inbreeding, RILs after the F7 generation are considered homozygous because the fixation at any locus is given by f = 1 0.57-1 ≈ 1. In practical terms, all plants from a single RIL are genetically identical, and can be used for replicated experiments under different environments. In addition, each RIL represents a unique combination of alleles from the parental genotypes where there are two homozygous genotypes at each marker locus, each corresponding to a parental allele. The mapping population is genotyped at molecular markers to construct a linkage map covering the entire genome. The recombination fraction between two markers is converted to the genetic distance in centiMorgan (cM) through a map function, such as the Haldane or Kosambi map function. The map constructed is used to locate QTLs that control a quantitative trait of interest.
We obtained a sample of 184 RILs derived from two cultivars, Kefeng No. 1 and Nannong 1138-2, for mapping agronomic traits. These RILs were genotyped for 950 molecular markers locating in 25 linkage groups
B48 48
B49 49
. The plants were grown in a simple lattice design with two replicates in a plot at Jiangpu Soybean Experiment Station, Nanjing Agricultural University, China. Ten plants in the second row of a plot were randomly selected for measuring leaf, stem and root biomass at each time in the whole growing season. After 20 days of seedling emergence, dry weights separately for the leaves, stem and roots were measured once every 5 to 10 days until most plants stopped growth. A total of 6 to 8 measurements were taken for each of the RILs studied. Great efforts were made to control measurement errors for such a large-scale field trial. Phenotyping precision was estimated to be above 95%.
Unlike a traditional mapping project, our goal is to map QTLs that control the dynamic process of how different organs, the stem, leaves, and roots, interact and coordinate to determine whole-plant biomass. The interactions and coordination of different organs for a plant are understood using design principles described by the ODE system (1).
Likelihood
Let
1752-0509-5-84-i6.gif
denote the vector of phenotypic values for trait k (k = 1 for leaf biomass (L), 2 for stem biomass (S), and 3 for root biomass (R)) measured on progeny i at time points
1752-0509-5-84-i7.gif
. Note that the number of time points measured may be progeny-specific, expressed as mi
for progeny i. Assuming that multiple QTLs (segregating with J genotypes), each bracketed by two flanking markers M, affects these three traits, we construct a mixture model-based likelihood as
M2
1752-0509-5-84-i8.gif
where y = (y
1, y
2, y
3) is a joint vector of phenotypic values for the three traits, with z
i
= (y
1i
, y
2i
, y
3i
) presenting the z-vector for progeny i; ω
j|i
is the conditional probability of QTL genotype j (j = 1,..., J) given the marker genotype of progeny i, which can be expressed as a function of the recombination fractions between the QTL and markers
B50 50
, and fj
(z
i
; Θ
j
, Ψ) is an MVN of leaf, stem and root biomass for progeny i which carries QTL genotype j, with mean vectors
1752-0509-5-84-i9.gif
specified by Θ
j
, and covariance matrix specified by Ψ. If a system of differential equations (1) is used to jointly model QTL genotype-specific means vectors for the three traits, then we have Θ
j
= (αLj
, βLj
, λ jL
, αSj
, βSj
, αRj
, βRj
, λ Rj
) for genotype j.
Estimation
Unlike a traditional mixture model for QTL mapping, we will model the genotypic values of each QTL genotype in likelihood (2) characterized by a group of nonlinear ODEs. While an analytical solution is not available, we will implement numerical approaches to solve these ODEs. Let μkj
(t) denote the genotypic value of the kth trait at time t for a QTL genotype j. Thus, the dynamic system of the traits and their interactions, regulated by QTL genotypes, can be modeled by a system of ODE (1),
M3
1752-0509-5-84-i10.gif
where
μ
j
(t) = (μ
1j
(t), ⋯, μ
3j
(t))
T
, and Θ
j
is a vector of ODE parameters associated with QTL genotype j. For J possible genotypes in the mapping population, we have
1752-0509-5-84-i11.gif
. The question now is how to Θ estimate from noisy measurements. The functional mean μkj
(t) may be represented as a linear combination of basis functions:
M4
1752-0509-5-84-i12.gif
where
ϕ
kj
(t) = (ϕ
kj1(t), ⋯, ϕkjR
(t))
T
is a vector of basis functions with R orders and c
kj
= (c
kj1, ⋯, ckjR
)
T
is a vector of basis coefficients. Define
1752-0509-5-84-i13.gif
as a length (R × 3 × J) vector of basis coefficients. The cubic B-splines are often chosen as basis functions, since any B-spline basis function is only positive over a short subinterval and zero elsewhere. This is called the compact
support property, and is essential for efficient computation. The flexibility of the B-spline basis functions depend on the number and location of knots we choose. It is an infinite-dimension optimization problem to choose the optimal number of knots and their locations. A popular approach to avoid this dilemma is choosing a saturated number of knots and using a roughness penalty to control the smoothness of the fitted curve and avoid over-fitting
40
.
We estimate the basis coefficient c and ODE parameter Θ based on a two-nested level of optimization. In the inner level of optimization, c is estimated by optimizing a criterion U(c|Θ), given any value of Θ. Therefore, the estimate
1752-0509-5-84-i14.gif
may be viewed as a function of Θ, which is denoted as
1752-0509-5-84-i15.gif
. Since no analytic formula for
is available,
is an implicit function. In the outer level of optimization, Θ is estimated by optimizing a criterion
1752-0509-5-84-i16.gif
. The parameter
is removed in the parameter space in the outer level by treating it as an implicit function of Θ. Although
does not have an analytic formula, the outer level of optimization only requires to calculate the derivative
1752-0509-5-84-i17.gif
, which can be obtained by using the implicit function theorem. The above optimization procedure is called the parameter cascading method. Note that when the two criteria U(c|Θ) and
are the same, the parameter cascading method is equivalent to the profiling method.
Authors' contributions
RW conceived of the idea of systems mapping, coordinated the whole study, and wrote the manuscript. JC derived the model, computed the real data, run simulation studies, and participated in the writing of the Methods section. ZH conducted the soybean experiment and collected phenotypic data. ZW packed the model into a computer package SysMap. JG directed the experimental design and data collection of soybeans. EV oversaw the project and highlighted the biological relevance a computational model must possess. All authors read and approved the final manuscript.
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ack
Acknowledgements
This work is supported by the Changjiang Scholars Award, "One-thousand Person" Award, NSF/IOS-0923975, a discovery grant of the Natural Sciences and Engineering Research Council of Canada (NSERC) (J. Cao), the National Key Basic Research Program of China (2009CB1184,2010CB1259,2011CB1093), The National High Technology R&D Program of China (2009AA1011), and the MOE 111 Project (B08025). Part of this work was carried when RW and JC were invited Research Fellows at the Statistical and Applied Mathematical Sciences Institute (SAMSI), sponsored by Duke University, University of North Carolina at Chapel Hill, and North Carolina State University. Thanks are due to Prof. Shouyi Chen and Prof. Deyue Yu for kind permission to use the jointly developed NJRIKY genetic linkage map.
refgrp Predicting unobserved phenotypes for complex traits from whole-genome SNP dataLeeSHvan der WerfJHJHayesBJGoddardMEVisscherPMPLoS Genet2008410e1000231.pmcid 2565502link fulltext 18949033LynchMWalshBGenetics and Analysis of Quantitative Traitspublisher Sinauer Associates, Sunderland, MA1998Mapping Mendelian factors underlying quantitative traits using RFLP linkage mapsLanderESBotsteinDGenetics1989121185lpage 19912036012563713Precision mapping of quantitative trait lociZengZBGenetics19941361457146812059248013918A random model approach to interval mapping of quantitative genesXuSAtchleyWRGenetics19951411189119712068408582623Joint linkage and linkage disequilibrium mapping of quantitative trait loci in natural populationsWuRLMaCXCasellaGGenetics2002160779792146197211861578Bayesian lasso for quantitative trait loci mappingYiNXuSGenetics20081791045105510.1534/genetics.107.085589242985818505874A robust QTL mapping procedureZouFNieLWrightFASenPKJ Stat Plann Infer200913997898910.1016/j.jspi.2008.06.009Dissection of genetically complex traits with extremely large pools of yeast segregantsEhrenreichIMTorabiNJiaYKentJMartisSShapiroJAGreshamDCaudyAAKruglyakLNature20104641039104210.1038/nature08923286235420393561Dissecting the complex architecture of a quantitative trait locus in yeastSteinmetzLMSinhaHRichardsDRSpiegelmanJIOefnerPJMcCuskerJHDavisRWNature200241632633010.1038/416326a11907579Genetic dissection of complex traits with chromosome substitution strains of miceSingerJBHillAEBurrageLCOlszensKRSongJJusticeMO'BrienWEContiDVWitteJSLanderESNadeauJHScience200430444544810.1126/science.109313915031436Genetic mapping in human diseaseAltshulerDDalyMJLanderESScience200832288188810.1126/science.1156409269495718988837Genetic architecture of complex traits: Large phenotypic effects and pervasive epistasisShaoHBurrageLCSinasacDSHillAEErnestSRO'BrienWCourtlandHWJepsenKJKirbyAKulbokasEJDalyMJBromanKWLanderESNadeauJHProc Natl Acad Sci USA2008105199101991410.1073/pnas.0810388105260496719066216The genetics of quantitative traits: challenges and prospectsMackayTFCStoneEAAyrolesJFNat Rev Genet20091056557719584810QTL mapping in new Arabidopsis thaliana advanced intercross-recombinant inbred linesBalasubramanianSSchwartzCSinghAWarthmannNKimMCMaloofJNLoudetOTrainerGTDabiTBorevitzJOChoryJWeigelDPLoS ONE200942e431810.1371/journal.pone.0004318262984319183806fw2. 2: a quantitative trait locus key to the evolution of tomato fruit sizeFraryANesbittTCGrandilloSKnaapECongBLiuJMellerJElberRAlpertKBTanksleySDScience2000289858810.1126/science.289.5476.8510884229Rice domestication by reducing shatteringLiCZhouASangTScience20063111936193910.1126/science.112360416527928Natural variation at the DEP1 locus enhances grain yield in riceHuangXQianQLiuZSunHHeSLuoDXiaGChuCLiJFuXNat Genet20094149449710.1038/ng.35219305410Functional mapping of quantitative trait loci underlying the character process: a theoretical frameworkMaCXCasellaGWuRLGenetics200216117511762146219912196415Functional mapping how to map and study the genetic architecture of dynamic complex traitsWuRLLinMNat Rev Genet2006722923716485021Functional mapping of growth and developmentLiYWuRLBiol Rev20108520721619930171Systems biology: a brief overviewKitanoHScience20022951662166410.1126/science.106949211872829Studying complex biological systems using multifactorial perturbationJansenRCNat Rev Genet2003414515112560811Reverse engineering of biological complexityCseteMEDoyleJCScience20022951664166910.1126/science.106998111872830Carbon allocation in trees: a review of concepts for modellingCannellMGRDewarRCAd Ecol Res19942559104Predicting responses of photosynthesis and root fraction to elevated CO2: Interactions among carbon, nitrogen, and growthLuoYFieldCBMooneyHAPlant Cell Environ1994171195120410.1111/j.1365-3040.1994.tb02017.xA coordination model of carbon allocation in relation to water supplyChenJReynoldsJAnn Bot199780455510.1006/anbo.1997.0406Allocation, plasticity and allometry in plantsWeinerJPerspect Plant Ecol Evol Syst2004620721510.1078/1433-8319-00083How do plants respond to nutrient shortage by biomass allocation?HermansCHammondJPWhitePJVerbruggenNTrends Plant Sci20061161061710.1016/j.tplants.2006.10.00717092760Concepts of modelling carbon allocation among plant organsMarcelisLFMHeuvelinkEFunctional-Structural Plant Modelling in Crop ProductionSpringer. Printed in the Netherlandseditor Vos J, Marcelis LFM, de Visser PHB, Struik PC and Evers JB2007103111Carbon allocation in fruit trees: From theory to modellingGenardMDauzatJFranckNLescourretFMoitrierNVaastPVercambreGTrees Structure and Function200822269282A general model for the origin of allometric scaling laws in biologyWestGBBrownJHEnquistBJScience199727612212610.1126/science.276.5309.1229082983The fourth dimension of life: Fractal geometry and allometric scaling of organismsWestGBBrownJHEnquistBJScience19992841677167910.1126/science.284.5420.167710356399A general model for ontogenetic growthWestGBBrownJHEnquistBJNature200141362863110.1038/3509807611675785A Bayesian approach to parameter estimation in HIV dynamical modelsPutterHHeisterkampSHLangeJMADe WolfFStat Med2002212199221410.1002/sim.121112210633A Bayesian approach for estimating antiviral efficacy in HIV dynamic modelsHuangYWuHJ Appl Stat20063315517410.1080/02664760500250552Covariance selection and estimation via penalized normal likelihoodHuangJZLiuNPourahmadiMLiuLBiometrika200693859810.1093/biomet/93.1.85Estimation and inference for a spline-enhanced population pharmacokinetic modelLiLBrownMBLeeKHGuptaSBiometrics20025860161110.1111/j.0006-341X.2002.00601.x12229995Principal differential analysis: Data reduction by differential operatorsRamsayJOJ Roy Stat Soc Ser B199658495508RamsayJOSilvermanBWFunctional Data AnalysisSpringer, New Yorkedition 22005Parameter estimation for differential equations: a generalized smoothing approach (with discussion)RamsayJOHookerGCampbellDCaoJGJ Roy Stat Soc Ser B20076974179610.1111/j.1467-9868.2007.00610.xParameter estimation for differential equation models using a frame-work of measurement error in regression modelLiangHWuHLJ Am Stat Assoc20081031570158310.1198/016214508000000797263193719956350EfronBTibshiraniRJAn Introduction to the BootstrapChapman & Hall1993Functional mapping of genotype-environment interactions for soybean growth by a semiparametric approachLiQHuangZXuMWangCGaiJHuangYPangXWuRLPlant Methods201061310.1186/1746-4811-6-13290357820525184Consistency between an allometric approach and optimal partitioning theory in global patterns of plant biomass allocationMcCarthyMCEnquistBJFunct Ecol20072171372010.1111/j.1365-2435.2007.01276.xRegulation of OsSPL14 by OsmiR156 defines ideal plant architecture in riceJiaoYWangYXueDWangJYanMLiuGDongGZengDLuZZhuXQianQLiJNat Genet20104254154410.1038/ng.59120495565Network models for dissecting plant development by functional mappingWuSYapJSLiYLiQFuGFLiJHDasKBergAZengYRWuRLCurr Bioinform2009418318710.2174/157489309789071093A conceptual framework for mapping quantitative trait loci regulating ontogenetic allometryLiHHuangZGaiJWuSZengYWuRLPLoS ONE2007211e124510.1371/journal.pone.0001245208075818043752QTL mapping of ten agronomic traits on the soybean (Glycine max L. Merr.) genetic map and their association with EST markersZhangW-KWangY-JLuoG-ZZhangJ-SHeC-YWuXLGaiJYChenSYTheor Appl Genet20041081131113910.1007/s00122-003-1527-215067400WuRLMaCXCasellaGStatistical Genetics of Quantitative Traits: Linkage, Maps, and QTLSpringer-Verlag, New York2007