Equation1isusedtodeterminetheflowrateforsetupA.AsimilarequationisusedinsetupBforevaluationoftheflowratek02,withk02replacingk0ineq1andk01heldconstant.SetupCrequiresanadditionalequationforthepositivefeedbackoftheflowratek01:
τ
?k01?t)g([I-])-kkmax,1
01)-k1+(I/[I-n01
(2)
T
])
wherethemaximumflowratesmaydifferforthetwopumps,buttheremainingparametersnandITareequalinthetwoflows.ExperimentalResults
Inordertounderstandbetterthebehaviorofthissystem,webeganourinvestigationbyexploringtherolesofparameters?t,n,andτineqs1and2.Asthetimestep?tapproacheszero,thedifferenceequationsbecomeordinarydifferentialequations(ODEs).OurexperimentalmeasurementsandtheresultsofsimulationsrevealthatthereisnoobservabledifferenceindynamicbehaviorbetweenthedifferenceequationsandtheODEsif?tissmallerthan2s.Therefore,intheexperimentsdescribedbelowwechoose?t)1s.
Theparameternaffectstheslopeofthesigmoidalfunctionineqs1and2.Forn.1thesigmoidalfunctionfcanbeapproximatedbyastepfunction:f)kmaxfor[I-]<IT,f)0for[I-]>IT,andf)kmax/2for[I-])IT.Thelargerthevalueofn,themorestronglythesystemisforcedtofinddynamicswithanaverageiodideconcentrationclosetothetargetvalue.Forn>2,thecontrolfeedbackandthebehaviorofthesystemareonlyslightlyaffectedbyincreasingn.Wechoosen)5inourexperimentsandsimulations,andweconfirmbysimulationsthatthereisnosignificantchangeindynamicswhentheparameternisincreasedtenfold.
Thetimeconstantτaffectstherateofiodide-dependentmodificationoftheflowrate.Inanalogywithneuronmodelsimulations,11-13weassumethatτislargerelativetothechemicalrelaxationtimes.Thischoicecorrespondstoaslowadaptationtochangesintheextracellularenvironment.Tospeedupourexperiments,wesetτ)1000s.Wehaveverifiedbothbyexperimentsandsimulationsthatthedynamicsremainsqualitativelysimilarforotherτ(τ)500,2000,4000,10000s).
SetupA.Figure2showsseveralexamplesofoscillatorybehaviorobservedinsetupA.Thetoppartofeachfiguredisplaysthevariableflowrate,andthebottompartdisplaysthemeasureddifferencebetweenthepotentialsoftheISEandthecalomelreferenceelectrode.Higheriodideconcentrationcorrespondstomorenegativepotentialdifference.InitialtransientperiodsareexcludedfromFigure2,andeachpatternisdisplayedforadurationof1h,duringwhichallparametersarefixed.Figure2a-dshowsthedynamicalbehaviorforfourdifferenttargetconcentrationsIT.A10-foldchangeinthetargetconcentrationdoesnotresultinsignificantlydifferentbehaviorwhenITischangedfrom1×10-6to1×10-7M(Figure2a,b).Burstingbehaviorremainssimilar,preservingthreeoscillationsperburstforbothtargetconcentrations.However,the
quiescent
5150J.Phys.Chem.A,Vol.101,No.28,1997Figure2.DynamicalbehaviorinsetupAexperiments:iodide(potentialdifferencebetweenelectrodes,bottompart)andflowrate(toppart)timeseries.Maximumflowratekmax)0.05s-1andtargetiodideconcentrationIT)(a)1×10-6M,(b)1×10-7M,(c)2×10-8M,(d)5×10-9M.TargetIT)1×10-6Mandkmax)(e)0.1s-1(f)0.025s-1
.
periodbecomesshorteratsmallerIT.Asthetargetconcentrationisfurtherdecreased,thenumberofoscillationsperburstfirstincreases(Figure2c)untilperiod-oneoscillations,analogoustotonicfiringinaneuron,emerge(Figure2d).WhenITissetbelowtheminimumvalueoftheuncontrollediodideconcentra-tionoscillations(notshown),thenegativefeedbackholdstheflowrateatarelativelylowvalueatwhichonlythelowiodidesteadystateissustained.Torestoretheoscillations,thetargetiodideconcentrationhastobeincreasedtoatleastIT)1×10-7M.ThishysteresisindicatesthecoexistenceofsustainedoscillationsandthestableLIsteadystate.ThereisnobistabilityforlargevaluesofIT;whenITisincreasedabove1×10-6Mthenumberofoscillationsperburstdecreasesuntillow-frequencyperiod-oneoscillationsemerge.FurtherincreasesofITyieldonlythehigh-iodidesteadystate,withaflowratethatapproacheskmax.
AsimilarscenarioofdynamicalregimesisobservedforfixedparameterITandvariablekmaxasshowninFigure2a,e,f.ForIT)1×10-6Mandkmax)0.1s-1,burstingwithtwooscillationsperburstrepresentsastationarypattern(Figure2e).Askmaxisdecreased,thenumberofoscillationsperburstincreasesmuchasitdoeswhenITisdecreased,andeventuallyhigh-frequencyperiod-oneoscillationsemergeforsmallerkmax(Figure2f).Weobtainonlythelowiodidesteadystateforkmax<0.002s-1.Ontheotherhand,whenkmaxisincreasedabove0.1s-1,theburstingisreplacedbylowfrequencyperiod-oneoscillations.Theseoscillationsremainstableevenforkmax.1.
Theflowratek0variessignificantlyduringaburstperiod.AroughlythreefoldchangeoccursintheflowrateduringburstinginallexperimentsinsetupA.High-frequencyperiod-oneoscillationsareassociatedwithsmalleramplitudeoftheflowratevariation(Figure2d).Theamplitudeincreaseswith
Dolniketal.
Figure3.DynamicalbehaviorinsetupBexperiments:iodideandflowratetimeseries.Flowratek01)0.005s-1,targetconcentrationIT)1×10-6M,andmaximumflowratekmax)(a)0.05and(b)0.01667s-1.k01)0.005s-1,kmax)0.01667s-1,andIT)(c)1×10-7M,(d)5×10-8M.IT)1×10-6M,kmax)0.01667s-1,andk01)(e)0.0025s-1,(f)0.008s-1.
increasingperiodofiodideoscillations(seeFigure2d,f),aresultthatisexplainedintheDiscussion.
SetupB.ThescenarioofdynamicregimesfoundinsetupBresemblesthatinsetupA.Figure3showsseveralexamples.Figure3adisplaysthebehaviorwiththesamecontrolparametersasusedinFigure2a,butwiththefixedflowratek01)0.005s-1.Low-frequencyperiod-oneoscillationsaretheoutcomeofcontrolinsetupB,incontrasttotheburstingoscillationsinsetupA.Theperiod-oneoscillationsinsetupBarepreservedforawiderangeofIT,whichindicatesthatforasimilarsetofparametersthecontrolofdynamicalbehaviorinsetupBismoredirect(andstronger)thaninsetupA.Weweakentheeffectofthecontrolmechanismbyreducingtheparameterkmaxbytwo-thirdsandsetkmax)0.01667s-1.ThischoicereflectsthatinsetupBtheflowrateofonlyoneoutofthreeinputspeciesisdynamicallyvaried.Figure3bdisplaystheburstingoscillationsforkmax)0.01667s-1andtheotherparametersasinFigure3a.ChangingITfrom1×10-6to1×10-7Mhasarelativelysmalleffectontheburstingbehavior(Figure3b,c).Asthetargetconcentrationisfurtherdecreased,thenumberofoscillationsperburstcontinuestoincreaseuntilthequiescentperioddisappearsandhigh-frequencyperiod-oneoscillationsemerge(Figure3d).WealsofindherethecoexistenceofsustainedoscillationsandthestableLIsteadystateforsmallvaluesofIT(IT<1×10-7),butthereisnobistabilityforlargevaluesofIT.
Similardynamicalregimesareobservedwhenwevarythetime-independentflowratek01,asshowninFigure3b,e,f.ForIT)1×10-6Mandk01)0.0025s-1weobserveburstingwithtwooscillationsperburst(Figure3e).Ask01becomeslarger,thenumberofoscillationsperburstincreasesasitdoeswhenwedecreaseIT(Figure3b,f),andeventuallyhigh-frequencyperiod-oneoscillationsemergeforsmallerk01>0.01s-1.Furtherincreasesink01yieldtheLIsteadystate.
Theamplitudeofoscillationoftheflowratek02issmallerinsetupBthaninsetupA.Theamplitudedecreaseswithdecreasingkmax,andthetime-averagedvalue?k02?decreaseswithdecreasingk01.Theaveragefrequencyofoscillations
(spiking)
OscillatoryChemicalReaction
duringaburstandthefrequencyofburstingarehigherinsetupBthaninsetupA.
SetupC.WeemployheretwoindependentfeedbackssnegativefeedbackfortheflowoftheKIsolution(eq1)andpositivefeedbackfortheflowoftheacidicsolutionofClO2(eq2).Figure4ashowsthetransientandstationarybe-haviorinthissystemwithcontrolparameterskmax,1)kmax,2)0.00833s-1andIT)1×10-6M.Thetransientdynamicsdependsontheinitialvaluesofk01andk02.InFigure4a,thesearealmostequalinthebeginning.Becauseinitially[I-]>IT,theflowofKIdecreasesandtheflowofClO2increases.Duringtheperiodofhigh[I-]theratioof[ClO2]o/[I-]oincreases,leadingtothedepletionof[I-]inthesystemandtotheonsetofoscillations.
Duringtheoscillatoryperiodtheflowratek02oscillateswithanincreasingenvelope,whilethechangeintheflowratek01hastheoppositesign.Theresultingchangeintheratio[ClO2]o/[I-]osteersthedynamicsonceagaintowardtheHIstateandformsasingleburst.Afterthetransientperiodofapproximately30minthechangein[ClO2]o/[I-]oisnegligible,andtheCDIreactionremainsinasimpleperiod-oneoscillatorymodeasifunderconstantflowinaCSTR.Thenegativeandpositivefeedbackschangetheindividualflowratesinsuchawaythatthetotalflowrateintothesystemfollowsasmoothtrendwithoutwaveringthroughlocalminimaandmaxima.
Wedonotobserveburstingforequalvaluesofkmax,1andkmax,2whentheflowratek01<0.01s-1.However,burstingarisesasaresultofunequalvaluesofkmax,1andkmax,2asshowninFigure4b,c.Figure4bdisplaysburstingbehaviorwithoutthetransientperiodforIT)1×10-5and1×10-6M.AswasthecaseinsetupsAandB,a10-foldchangeinITdoesnotproduceasignificantchangeintheburstingbehavior.
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