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Accumulation of Biomass and Mineral Elements with Calendar Time by Corn: Application of the Expanded Growth Model.
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Title: Accumulation of Biomass and Mineral Elements with Calendar Time by Corn: Application of the Expanded Growth Model.
Series Title: Overman AR, Scholtz RV III (2011) Accumulation of Biomass and Mineral Elements with Calendar Time by Corn: Application of the Expanded Growth Model. PLoS ONE 6(12): e28515. doi:10.1371/journal.pone.0028515
Physical Description: Journal Article
Creator: Scholtz, Richard
Overman, Allen R.
Publisher: PLoS One
Place of Publication: San Francisco, CA
Publication Date: 2011
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Abstract: The expanded growth model is developed to describe accumulation of plant biomass (Mg/ha) and mineral elements (kg/ha) in with calendar time (wk). Accumulation of plant biomass with calendar time occurs as a result of photosynthesis for green land-based plants. A corresponding accumulation of mineral elements such as nitrogen, phosphorus, and potassium occurs from the soil through plant roots. In this analysis, the expanded growth model is tested against high quality, published data on corn (Zea mays L.) growth. Data from a field study in South Carolina was used to evaluate the application of the model, where the planting time of April 2 in the field study maximized the capture of solar energy for biomass production. The growth model predicts a simple linear relationship between biomass yield and the growth quantifier, which is confirmed with the data. The growth quantifier incorporates the unit processes of distribution of solar energy which drives biomass accumulation by photosynthesis, partitioning of biomass between light-gathering and structural components of the plants, and an aging function. A hyperbolic relationship between plant nutrient uptake and biomass yield is assumed, and is confirmed for the mineral elements nitrogen (N), phosphorus (P), and potassium (K). It is concluded that the rate limiting process in the system is biomass accumulation by photosynthesis and that nutrient accumulation occurs in virtual equilibrium with biomass accumulation.
Acquisition: Collected for University of Florida's Institutional Repository by the UFIR Self-Submittal tool. Submitted by Richard Scholtz.
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AccumulationofBiomassandMineralElementswith CalendarTimebyCorn:ApplicationoftheExpanded GrowthModelAllenR.Overman,RichardV.ScholtzIII *Agricultural&BiologicalEngineeringDepartment,UniversityofFlorida,Gainesville,Florida,UnitedStatesofAmericaAbstractTheexpandedgrowthmodelisdevelopedtodescribeaccumulationofplantbiomass(Mgha2 1)andmineralelements (kgha2 1)inwithcalendartime(wk).Accumulationofplantbiomasswithcalendartimeoccursasaresultofphotosynthesis forgreenland-basedplants.Acorrespondingaccumulationofmineralelementssuchasnitrogen,phosphorus,and potassiumoccursfromthesoilthroughplantroots.Inthisanalysis,theexpandedgrowthmodelistestedagainsthigh quality,publisheddataoncorn(ZeamaysL.)growth.DatafromafieldstudyinSouthCarolinawasusedtoevaluatethe applicationofthemodel,wheretheplantingtimeofApril2inthefieldstudymaximizedthecaptureofsolarenergyfor biomassproduction.Thegrowthmodelpredictsasimplelinearrelationshipbetweenbiomassyieldandthegrowth quantifier,whichisconfirmedwiththedata.Thegrowthquantifierincorporatestheunitprocessesofdistributionofsolar energywhichdrivesbiomassaccumulationbyphotosynthesis,partitioningofbiomassbetweenlight-gatheringand structuralcomponentsoftheplants,andanagingfunction.Ahyperbolicrelationshipbetweenplantnutrientuptakeand biomassyieldisassumed,andisconfirmedforthemineralelementsnitrogen(N),phosphorus(P),andpotassium(K).Itis concludedthattheratelimitingprocessinthesystemisbiomassaccumulationbyphotosynthesisandthatnutrient accumulationoccursinvirtualequilibriumwithbiomassaccumulation.Citation: OvermanAR,ScholtzRVIII(2011)AccumulationofBiomassandMineralElementswithCalendarTimebyCorn:ApplicationoftheExpandedGrowth Model.PLoSONE6(12):e28515.doi:10.1371/journal.pone.0028515 Editor: RandallP.Niedz,UnitedStatesDepartmentofAgriculture,UnitedStatesofAmerica Received July25,2011; Accepted November9,2011; Published December14,2011 Copyright: 2011Overman,Scholtz.Thisisanopen-accessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense,whichpermits unrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalauthorandsourcearecredited. Funding: ThisanalysiswasfundedbytheFloridaAgriculturalExperimentStation.Thefundershadnoroleinstudydesign,datacollectionandanalysis,decis ion topublish,orpreparationofthemanuscript. CompetingInterests: Theauthorshavedeclaredthatnocompetinginterestsexist. *E-mail:rscholtz@ufl.eduIntroductionInarecentpublicationtheauthorsdiscussedamodelofyield responseofcorntoplantpopulationandabsorptionofsolar energywithintheplantcanopy[1].Datafromthreefieldstudies formedtheempiricalfoundationforthemathematicalmodel.The simpleexponentialmodelcontainedtwoparameters:onefor upperlimitonyieldathighplantpopulationandanexponential responsecoefficient.Themodeldescribedthedataverywelland exhibitedsimilaritiesamongthethreestudies.Inatextbookthe authorshavediscussedvariousaspectsofcropgrowthandyield [2],includingamathematicalmodelofcropgrowthwithcalendar time.Theexpandedgrowthmodelincorporatesthethreebasic processesofanenergydrivingfunction,partitioningofbiomass betweenlight-gatheringandstructuralcomponentsoftheplants, andanagingfunction.Thismodelisusedinthepresentanalysis. Asimplifiedtheoryofbiomassproductionbyphotosynthesishas beenpublishedbytheauthors[3].Thetheoryincorporatesbasic principlesfrommathematicsandphysicsandusesdatafromthe literatureforawarm-seasonperennialgrassasanempiricalbase. Thestrategyfollowstheprocedureof emergence asdescribedby RobertLaughlin[4]whichmeansthatdevelopmentofthetheory isguidedbymeasurementandobservation.Thisapproach examinesbehaviorofalargeassemblageofmatter,incontrast totheclassicalreductionistapproachwhichbreaksasystemintoits smallestpartsandthendescribesinteractionsamongtheparts. DatafromafieldstudyatFlorence,SC,USAareusedto evaluateapplicationofthemodel[5]forcorn( Zeamays L.).Many fieldstudieshavebeenconductedonthegrowthofcorn,notedas examplesinreferences[6–11].Keymeasurementsofaccumulationofplantbiomassaswellasthemineralelementsnitrogen, phosphorus,andpotassiumwereestablishedinthe1948studyby Sayre[6].EffectofappliedNandPfertilizationonbiomass accumulationwasevaluatedbyBar-YosefandKafkafi[7]. InteractionsofplantpopulationandappliedNonbiomass accumulationweremeasuredbyRhoadsandStanley[8]. Dependenceofyieldonmeasuredevapotranspirationforthree differentsoilswasreportedbyTolkandHowell[9].An exponentialrelationshipwasclearlydemonstrated.Thepresent articleisnotintendedasageneralliteraturereviewofeither mathematicalmodelsorfieldstudiesoncropgrowth.Itdescribes conceptsandproceduresfortheexpandedgrowthmodelonhigh qualitydatafromafieldstudywithgrowthofcorn.MethodsThefirststepistodefinerelevantquantities(variablesandmodel parameters): t iscalendartime(referencedtoJan.1),wk; Y isbiomass yield(drymatter),Mgha2 1; Nuisplantnutrientuptake(N,P,orK), kgha2 1; Nc= Nu/ Y isplantnutrientconcentration(N,P,orK), gkg2 1.Acommonreferencetimeisusedtofacilitatecomparison amongvariousstudies.Thesecondstepistoutilizeamathematical PLoSONE|www.plosone.org1December2011|Volume6|Issue12|e28515

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modelwhichrelatesbiomassaccumulationtocalendartime.Forthis purposeweadopttheexpandedgrowthmodeldiscussedinSection 3.5ofOvermanandScholtz[2],whichcanbewrittenas Y ~ AQ 1 where A isayieldfactor,Mgha2 1;and Q isadimensionless growth quantifier ,definedbyQ ~ 1 { kxi erf x { erf xi { k p p exp { x2 { exp { x2 i : exp 2 p s cxi 2 inwhichthedimensionlesstimevariable x isdefinedby x ~ t { m 2 p s z 2 p s c 2 3 withtheparametersinEq.(3)definedas m timetothemeanofthe solarenergydistribution(referencedtoJan.1forthenorthern hemisphere),wk; 2 p s thetimespreadofthesolarenergydistribution, wk; k thepartitioncoefficientbetw eenlight-gatheringandstructural componentsoftheplants,and c anagingcoefficientfortheplant species,wk2 1.Itfollowsthat xicorrespondstothetimeofinitiationof significantplantgrowth ti,wk.Theseparametersarediscussedinmore detailinthenextsectiononapplic ationtothecornstudyatFlorence, SC,USA.The‘errorfunction’,erf x ,inEq.(2)isdefinedby erf x ~ 2 p p x 0exp { u2 du 4 with u asthevariableofintegration fortheGaussiandistribution exp { u2 .Valuesoftheerf x canbeobtainedfromahandbookof mathematicalfunctions( seeTable7.1 [12]). Examinationofdataofcouplingbetweenplantnutrientaccumulation Nuandplantbiomass Y leadstothehyperbolicphaserelation Nu~ NumY Kyz Y 5 with Numaspotentialmaximumplantnutrientaccumulationathigh Y and Kyisthevalueof Y atwhich Nu= Num/2.Thissubjectisexploredin moredetailinthenextsectionofapplicationtodatafromFlorence, SC,USA.ResultsDataforthisanalysisareadaptedfromafieldstudybyD.L. Karlenandassociates[5]with‘Pioneer3382’cornonNorfolk loamyfinesand(fine-loamy,siliceous,thermicTypicPaleudult)at theUSDA-ARSCoastalPlainsSoil,Water,andPlantResearch CenteratFlorence,SC,USA.Datafor1982andplantpopulation densityof7plantsm2 2areusedhere.PlantingdatewasApril2 ( t =15.0wk).FertilizerapplicationwasN-P-K=268-36224kgha2 1.DataaregiveninTable1foreachsamplingtime forcalendartime t ,biomassyield Y ,andplantnutrientuptakeand plantnutrientconcentrationfornitrogen,phosphorus,and potassium.Analysisofdatafromvariousstudieshasledto parameterestimates: m ~ 26 : 0wk, 2 p s ~ 8 : 00wk, k ~ 5, c ~ 0 : 20wk{ 1.Nowbyvaryingthetimeofinitiation, xi,itcanbe shownthatmaximumutilizationofsolarenergyisobtainedfor xi=0.Thischoiceoftheparametersleadsto x ~ t { m 2 p s z 2 p s c 2 ~ t { 26 : 0 8 : 00 z (8 : 00)(0 : 20) 2 ~ t { 19 : 6 8 : 00 6 ItfollowsfromEq.(6)that xi=0correspondsto ti=19.6wk. Apparentlyatimeintervalof4.6weeksisrequiredforgermination ofseedsanddevelopmentofcornplantstoreachoptimumcapture ofsolarenergyandsignificantplantgrowth.Notethattheeffectof theagingfunctionistoshiftreferencetimefrom26.0wkto 19.6wk.Thegrowthquantifierequationbecomes Q ~ 1 { kxi erf x { erf xi { k p p exp { x2 { exp { x2 i : exp 2 p s cxi ~ erf x { 0 { 2 : 821exp { x2 { 1 7 ValuesadaptedfromtheexperimentarelistedinTable2.Linear regressionofbiomassyieldonthegrowthquantifierleadsto ^ Y Y ~ 0 : 290 z 7 : 274 Qr ~ 0 : 9973 8 Table1. Accumulationofbiomassandmineralelementsby cornatFlorence,SC,USA.TYNuNcPuPcKuKcwkMgha2 1kgha2 1gkg2 1kgha2 1gkg2 1kgha2 1gkg2 115.0planting(April2) 19.60.406133212.51844.2 21.01.836032.863.311864.5 22.04.2111026.1133.121952.1 24.010.217116.8222.229829.2 26.818.819810.5341.626712.5 29.522.52079.2341.529313.0 doi:10.1371/journal.pone.0028515.t001 Table2. Correlationofbiomassaccumulation( Y )withthe growthquantifier( Q )forcornatFlorence,SC,USA.t xerfxexp( 2 x2)Q YwkMgha2 119.600100.406 21.00.1750.19990.96980.2851.83 22.00.3000.32860.91390.5714.21 24.00.5500.56330.73901.30010.2 26.80.9000.79690.44492.36318.8 28.81.1500.89610.26652.96521.3 29.51.23750.91980.21623.13122.5  103.821----doi:10.1371/journal.pone.0028515.t002 ApplicationoftheExpandedGrowthModel PLoSONE|www.plosone.org2December2011|Volume6|Issue12|e28515

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where ^ Y Y signifiesanestimatorofbiomassyield Y .Thequalityof thecorrelationisconfirmedinFigure1.Equation(8)isin agreementwithEq.(1),thesimplelinearmodel.Thenextstepisto explorethecouplingbetweenaccumulationofplantnutrientsand plantbiomass.Equation(5)canberearrangedtothelinearform Y Nu~ KyNumz 1 NumY 9 Datafornitrogen,phosphorus,andpotassiumarenowusedtotest thevalidityofEq.(9).Thevalueof Y / Nucorrespondingtoeach valueof Y iscalculatedfromTable1.Linearregressionthenleadsto Nitrogen: Y Nu~ 0 : 0251 z 0 : 00367 Yr ~ 0 : 9967 10 ^ N Nu~ 273 Y 6 : 84 z Y 11 Phosphorus: Y Pu~ 0 : 269 z 0 : 0172 Yr ~ 0 : 9965 12 ^ P Pu~ 58 : 1 Y 15 : 6 z Y 13 Figure2.Correlationofplantbiomasstonitrogenuptakeratio (A)andofplantnitrogenuptake(B)withbiomassyield. Crop dataforcornatUSDAresearchcenteratFlorence,SC,USA[5].Lineis drawnfromEq.(10).CurveisdrawnfromEq.(11). doi:10.1371/journal.pone.0028515.g002 Figure3.Correlationofplantbiomasstophosphorusuptake ratio(A)andofplantphosphorusuptake(B)withbiomass yield. CropdataforcornatUSDAresearchcenteratFlorence,SC,USA [5].LineisdrawnfromEq.(12).CurveisdrawnfromEq.(13). doi:10.1371/journal.pone.0028515.g003 Figure1.Correlationofbiomassyield(Y)withgrowth quantifier(Q). BiomassyielddataforcornatUSDAresearchcenter atFlorence,SC,USA[5].LineisdrawnfromEq.(8). doi:10.1371/journal.pone.0028515.g001 ApplicationoftheExpandedGrowthModel PLoSONE|www.plosone.org3December2011|Volume6|Issue12|e28515

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Potassium: Y Ku~ 0 : 00635 z 0 : 00316 Yr ~ 0 : 9894 14 ^ K Ku~ 317 Y 2 : 01 z Y 15 ResultsareshowngraphicallyinFigures2,3,and4fornitrogen, phosphorus,andpotassium,respectively.Thehighcorrelations confirmthevalidityofthephaserelationsforthisstudy.DiscussionThenextstepistoprovidesimulationofbiomass( Y )andplant nitrogen( Nu)withcalendartime( t ).Accumulationofthegrowth quantifier( Q )withcalendartimefollowsfromEq.(7).Couplingof biomassyieldwithgrowthquantifierfollowsfromEq.(8). CouplingofplantnitrogenwithbiomassyieldfollowsfromEq. (11).Couplingofplantnitrogenconcentrationisthendefinedby Nc= Nu/ Y .SimulationcurvesareshowninFigure5alongwith valuesfromtheexperiment.Closeagreementbetweenthe estimatedandmeasuredvaluesshouldbenoted.Thedeclinein plantnitrogenconcentrationwithcalendartimeisexplainedby theshiftfromdominanceoflight-gathering(leaf)fractionwith higherplantNinyoungplantstowarddominanceofstructural (stalk)fractionwithlowerplantNinolderplants.Incaseswhere correlationsaremuchlowerthanobtainedinthisanalysiscould signifyeitherlargescatterinthedataand/orthatthelinear relationshipisnotvalid.Thehighefficiencyofnitrogenutilization bytheplantsinthisstudymaybenoted.Potentialnitrogenuptake fromEq.(11)is273kgha2 1forappliednitrogenof268kgha2 1, foranefficiencyratioof273/268=1.02. Numerousfactorsinfluencethevalueoftheyieldfactor A Theseincludeplantpopulation,levelofappliednutrients,water availability(suchasrainfallorirrigation),andfrequencyofharvest (forperennialgrasses).Someoftheseinteractionshavebeen detailedinOvermanandScholtz[1&2]. Wecannowinterpretthemeaningofthisanalysis.Both Y and Nuareaccumulatingwithcalendartimeandthereforerepresent rateprocesses.Theratelimitingprocessinthesystemisbiomass accumulationbyphotosynthesis.Phaserelations(Eqs.(11),(13), and(15))implythataccumulationofthemineralelements(N,P, andK)occurinvirtualequilibriumwithbiomassaccumulation. Thisconclusionissupportedbythesimplifiedtheoryofbiomass production[3].Anexcellentdiscussionofphotosynthesishasbeen Figure4.Correlationofplantbiomasstopotassiumuptake ratio(A)andofplantpotassiumuptake(B)withbiomassyield. CropdataforcornatUSDAresearchcenteratFlorence,SC,USA[5].Line isdrawnfromEq.(14).CurveisdrawnfromEq.(15). doi:10.1371/journal.pone.0028515.g004 Figure5.Accumulationofbiomassyield(A),plantnitrogen uptake(B),andplantnitrogenconcentration(C)withcalendar time. CropdataforcornatUSDAresearchcenteratFlorence,SC,USA [5].CurvesaredrawnfromEqs.(7),(8),and(11).(A). doi:10.1371/journal.pone.0028515.g005 ApplicationoftheExpandedGrowthModel PLoSONE|www.plosone.org4December2011|Volume6|Issue12|e28515

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presentedbyMorton[13],withemphasisonwhathasbeen learnedandwhatremainsasopenquestions. Theexpandedgrowthmodelisderivedviaclassicalmethods. Bymakingsimplifyingassumptions,ananalyticalsolutioncanbe foundfromlineardifferentialequationsthatarebasedonkey fundamentalprocesses.Thus,theexpandedgrowthmodel eliminatestheneedtousecomputeralgorithmstosolvethe inherentcomplexity,andalongwiththenutrientaccumulation phaserelationshiphasbeenshowntocloselyagreewiththedata presentedbyKarlenetal.[5].Itissuggestedthatthisprocedure shouldbetestedforothercases,includingothercropspeciesand experimentalconditions,andothermineralelements(suchasCa, Mg,etc.).Somedatasuggestthattheprocedureutilizingnutrient accumulationphaserelationshipdoesapplyforCaandMg[11]. Theauthorsplantoexaminetheyieldsoflight-gatheringand structuralcomponentsandtheeffectonnutrientaccumulation usingtheexpandedgrowthmodelinafuturepublication.AuthorContributionsConceivedanddesignedtheexperiments:AO.Analyzedthedata:AORS. Contributedreagents/materials/analysistools:AORS.Wrotethepaper: AO.Reviewedandpreparedthemanuscriptforpublication,thisincludes graphsandtables:RS.References1.OvermanAR,ScholtzRV(2011)ModelofYieldResponseofCorntoPlant PopulationandAbsorptionofSolarEnergy.PLoSOne6(1):e16117. doi:10.1371/journal.pone.0016117. 2.OvermanAR,ScholtzRV(2002)MathematicalModelsofCropGrowthand Yield.NewYork:TaylorandFrancis.328p. 3.OvermanAR,ScholtzRV(2010)AMemoironASimplifiedTheoryofBiomass ProductionbyPhotosynthesis.UniversityofFlorida.GainesvilleFL.19p. 4.LaughlinRB(2005)ADifferentUniverse:ReinventingPhysicsfromtheBottom Down.CambridgeMA:BasicBooks.254p. 5.KarlenDL,SadlerEJ,CampCR(1987)Drymatter,nitrogen,phosphorus,and potassiumaccumulationratesbycornonNorfolkloamysand.AgronomyJ79: 649–656. 6.SayreJD(1948)Mineralaccumulationincorn.PlantPhysiology23:267–281. 7.Bar-YosefB,KafkafiU(1972)Ratesofgrowthandnutrientuptakeofirrigated cornasaffectedbyNandPfertilization.SoilSciSocAmerProc36:931–936. 8.RhoadsFM,StanleyRL(1979)Effectofpopulationandfertilityonnutrient uptakeandyieldcomponentsofirrigatedcorn.SoilCropSciSocFlaProc38: 78–81. 9.MuttiLSM(1984)Dynamicsofwaterandnitrogenstressesoncorngrowth, yield,andnutrientuptake.PhDDissertation.UniversityofFlorida,Gainesville, FL.128p. 10.TolkJA,HowellTA,EvettSR(1998)Evapotranspirationandyieldofcorn grownonthreehighplainssoils.AgronomyJ90:447–454. 11.OvermanAR,ScholtzRV(1999)Modelforaccumulationofdrymatterand plantnutrientsbycorn.CommunSoilSciPlantAnal30:2059–2081. 12.AbramowitzM,StegunIA(1965)HandbookofMathematicalFunctions.New York:DoverPublications.1046p. 13.MortonO(2007)EatingtheSun:HowPlantsPowerthePlanet.London: HarperCollinsPublishers.457p.ApplicationoftheExpandedGrowthModel PLoSONE|www.plosone.org5December2011|Volume6|Issue12|e28515