Editor:DavidUmulis,PurdueUniversity,UNITEDSTATES
Received:November9,2018;Accepted:April19,2019;Published:June6,2019
DataAvailability:AllrelevantdataarewithinthemanuscriptanditsSupportingInformationfiles.
Funding:ThisworkwassupportedinpartbyaResearchFellowshipforYoungScientists(No.18J12309toTY)andbyaGrant-in-AidforScientificResearchonInnovativeAreas(No.16H01248toHF)fromtheJapanSocietyforthePromotionofScience.Thefundershadnoroleinstudydesign,datacollectionandanalysis,decisiontopublish,orpreparationofthemanuscript.
Competinginterests:Theauthorshavedeclaredthatnocompetinginterestsexist.
Inthisstudy,were-examinedtheoriginalDCmodelsexhaustivelyundervariousparameterconditions,totestwhethertheycanproduceorixatephyllotaxis.WethenexpandedDCmodelsbyintroducingprimordialage-dependentchangesintheinhibitorypower.Ourresultsindicatethatalateandslowincreaseintheinhibitorypoweriscriticalfortheestablishmentoforixatephyllotaxisandimplythatchangingtheinhibitorypowerisgenerallyanimportantcomponentofthemechanismofphyllotacticpatterning.
TerminalwinterbudsofO.japonicathathadbeencollectedinJulyfromnineplantsgrowingattheKoishikawaBotanicalGardens,GraduateSchoolofScience,TheUniversityofTokyowereusedformorphologicalanalyses.
Forscanningelectronmicroscopy(SEM),thedehydratedsampleswereinfiltratedoncewitha1:1v/vmixtureofethanolandisoamylacetateandtwicewithisoamylacetate.Subsequently,thesampleswerecriticalpointdried,sputtercoatedwithgold–palladium,andobservedusingSEM(HitachiS-3400N).
TheshootapexisconsideredasaplaneinDC1(A)andasaconeinDC2(B).
Consideringassumptions5and7,thedistancefromthecenteroftheshootapextothemthprimordium(rm)isexpressedwiththeinitialradialvelocityV0as:(2)
Thepoint(R0cosθ,R0sinθ)whereI(θ)issmallestischosenforthepositionofanewprimordium.NotethatηandGaretheonlyrelevantparametersthatinfluencethebehaviorofI(θ)inDC1.
Becauseofassumption6,thedistancefromthecenteroftheshootapextothemthprimordiumontheconicalsurface(rm)isexpressedwiththetimeafteritsemergenceTmandtheinitialradialvelocityV0as:(6)Byusingtm≡TmV0/R0,astandardizedageofthemthprimordiumdefinedastheproductofTmandtherelativeSAMgrowthrateV0/R0,rmismoresimplyexpressedas:(7)
TheDC2modelischaracterizedbythreeparameters:α,,and.Theseparametersrepresentthesteepnessofthedeclineoftheinhibitoryeffectaroundthethreshold,theflatnessoftheshootapex,andtheratiooftheinhibitionrangetotheSAMsize,respectively.
ComputersimulationsusingDC2andDC2-derivedmodelswereinitiatedbyplacingasingleprimordiumortwoprimordiaatacentralangleof120°ontheSAMperiphery.Intheformerinitialcondition,thesecondprimordiumarisesatacertaintimeorimmediatelyafterthefirstprimordium,independenceonparametersettings,attheoppositeposition,andinsomecases,moreprimordiaareimmediatelyinsertedatmiddlepositions.Thuscomputersimulationswiththisconditionsubstantiallycoversituationsstartingwith1×2xprimordia(x=0,1,2)evenlydistributedontheSAMperiphery.Similarly,simulationswiththelatterconditionsubstantiallycoversituationsstartingwith3×2xprimordia(x=0,1,2).Wealsotestedsimulationswithanotherinitialcondition,inwhichtwoprimordiawereplacedatoppositepositionswithacentralangleof180°,buttheyreturnedcompletelysameresultsassimulationsinitiatedbyplacingasingleprimordiumdidandarethereforeomitted.
Computersimulationswereperformedwithanangleresolutionof0.1°.DC2andDC2-derivedmodelsweresimulatedwithatimestepofΔtm=0.001.
(A)Transversesection.OpointstothesummitoftheSAM,andleafprimordiaaredesignatedasP1,P2,P3,etc.,withP1beingtheyoungestvisibleprimordium.BlacklinesrepresentorthostichiesdrawnbyjoiningthegravitycentersofleafprimoridaandO.Thefourorthostichylinescanberoughlyapproximatedbytwoorthogonallines(palegraybroadlines).(B)Longitudinalsection.I1indicatestheincipientprimordium.(C)Scanningelectronmicroscopicimage.(D)Divergenceanglesmeasuredusingthetransversesections.Divergenceanglescloseto180°showoppositepositioningofthesuccessiveprimordia(blue),whileanglesnear90°or270°showadjacentpositioning(yellow).(E)ThenaturallogsofplastochronratiosOP2/OP1andOP3/OP2areplottedbasedonwhetherthetwoprimordiaarelocatedinanadjacentoroppositeposition.In(D)and(E),pointslinkedbyalinerepresentdatafromthesamesample,andredpointsindicatedataobtainedfromthesectionof(A).
Next,weexaminedwhethermodificationofDC1couldenableittoproduceorixatephyllotaxis.InanattempttomodifyDC1,wefocusedontheinhibitorypowerofeachleafprimordiumagainstnewprimordiumformation—whichisassumedtobeconstantinDCmodelsbutmaypossiblychangeduringleafdevelopment—andexpandedDC1byintroducingage-dependent,sigmoidalchangesintheinhibitorypower.InthisexpandedversionofDC1(EDC1),theinhibitoryfieldstrengthI(θ)wasredefinedasthesummationoftheproductsoftheage-dependentchangeintheinhibitorypowerandthedistance-dependentdeclineofitseffect:(9)
Fisdefinedas:(10)whereparametersaandbareconstantsthatrepresenttherateandtimingoftheage-dependentchangesintheinhibitorypower,respectively.Underthisequation,inanage-dependentmanner,theinhibitorypowerincreasesata>0anddecreasesata<0.Inthepresentstudy,ηwasfixedat2forEDC1.
PriortocomputersimulationanalysiswithEDC1,wesearchedforparametersofEDC1thatcanfittherequirementsofnormalorixatephyllotaxis.Whenthenormalpatternoforixatephyllotaxisisstablymaintained,arectangularcoordinatesystemwiththeoriginatthecenteroftheshootapexcanbesetsuchthatallprimordialieonthecoordinateaxes,andeveryfourthprimordiumislocatedonthesameaxisinthesamedirection,i.e.,thepositionofanyprimordium(mthprimordium)canbeexpressedas(rmcosθm4i,rmsinθm4i)forintegersi.Underthiscondition,weconsideredwhetheranewprimordium(nthprimordium)isproducedattheposition(R0cosθn4i,R0sinθn4i),tokeepthenormalorixatephyllotacticpattern.InEDC1,asinDC1,newprimordiumformationat(R0cosθn4i,R0sinθn4i)impliesthattheinhibitoryfieldstrengthI(θ)onthecircleMhasaminimumatθn4i.Forthisreason,wefirstattemptedtosolvethefollowingequation:(11)
SimilartotheapproachusedforDC1,weexpandedDC2byintroducingprimordialage-dependentchangesintheinhibitorypower.InthisexpandedversionofDC2(EDC2),theinhibitoryfieldstrengthI(θ)wasredefinedasthesummationoftheproductsoftheage-dependentchangeintheinhibitorypowerandthedistance-dependentdecreaseofitseffect:(12)whereFisafunctionexpressingatemporalchangeintheinhibitorypower,definedas:(13)
(A)ContourmapofthenaturallogoftheinhibitoryfieldstrengthIwithintheshootapicalregionthatgeneratedorixatephyllotaxisinthecomputersimulationusingEDC2.Avalueof0impliesthattheinhibitoryfieldstrengthisequaltothethresholdforprimordiumformation.(B)RelationshipbetweenplastochronsanddivergenceanglesinorixatepatternsgeneratedincomputersimulationsusingEDC2.Forapairofsuccessiveprimordia,LmandLm+1,astandardizedplastochronwascalculatedastm+1tm=ln(rm/rm+1).Orixatepatternswereplottedbasedontheirtwostandardizedplastochrons:oneforthepairofoppositeprimordiawithadivergenceangleofapproximately180°,andtheotherforthepairofadjacentprimordiawithadivergenceangleofapproximately±90°.
(A)GradualincreaseoftheinhibitorypowerwitharelativelysmallsizeofSAM.(B)SuddendecreaseoftheinhibitorypowerwitharelativelylargesizeofSAM.EDC1canestablishorixatephyllotaxisundereitheroftheseconditionswhileEDC2canonlyundertheformercondition.
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(A)Twodifferentsituationsofthearrangementofthefourprecedingprimordia,Ln4,Ln3,Ln2,andLn1,relativetotheincipientprimordiumLninnormalorixatephyllotaxis.(B)Theblueandredcurvesshownumericalsolutionsofinsituations1and2,respectively.TheirintersectionpointsareexpectedtogivetheparameterconditionsofEDC1thatarerequiredforstabilizingthenormalorixatephyllotaxis.(C)InhibitoryfieldstrengthontheperipheryofSAMinsituation1(blue)andsituation2(red)attheparametersettingsdeterminedassolutionsofthatarecommontobothofthesesituations.Graphsweredrawnwithθn4ias0°.
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ComputersimulationswithEDC1wereperformedundertheparameterconditionofG=0.5,a=10,andb=5,5.1,or5.2.ChangesinthedivergenceanglefromL70~L71toL99~L100areshownfortheresultantpatterns.Asmall-anglespiralwasobtainedatb=5.1(bluecircle),whilefive-cycleandsix-cyclealternatepatternswereproducedatb=5andatb=5.2,respectively.
ComputersimulationsusingEDC2wereperformedunder101settingsofB(0≤B≤1)atΓ=1or3,A=4or10,andα=1.Divergenceanglesandplastochrontimesdeterminedfromthelastnineleafprimordia(L92toL100)areshownforthepatternsobtainedwithvariousBsettings,whichrepresentcharacteristicsofphyllotacticpatternsasinfluencedbythetimingoftheincreaseoftheinhibitorypower.
ComputersimulationsusingEDC2wereperformedunder101settingsofΓ(1≤Γ≤3)atα=1or4,A=4or10,andA×B=3.Divergenceanglesandplastochrontimesdeterminedfromthelastnineleafprimordia(L92toL100)areshownforthepatternsobtainedwithvariousΓsettings,whichrepresentcharacteristicsofphyllotacticpatternsasinfluencedbytheratiooftheinhibitionrangetotheSAMsize.
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WethankMasashikoFurutani(FujianAgriculturalandForestryUniversity)andKoichiFujimoto(OsakaUniversity)forvaluablediscussions.
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