Role of Autoregulation and Relative Synthesis of Operon Partners in Alternative Sigma Factor ...

25 Pages·2016·2.41 MB·English

RESEARCHARTICLE
Role of Autoregulation and Relative Synthesis
of Operon Partners in Alternative Sigma
Factor Networks
JatinNarula☯,AbhinavTiwari☯,OlegA.Igoshin*
DepartmentofBioengineering,RiceUniversity,Houston,Texas,UnitedStatesofAmerica
☯Theseauthorscontributedequallytothiswork.
*[email protected]
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Abstract
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Despitethecentralroleofalternativesigmafactorsinbacterialstressresponseandviru-
lencetheirregulationremainsincompletelyunderstood.Hereweinvestigateoneofthe
best-studiedexamplesofalternativesigmafactors:theσBnetworkthatcontrolsthegeneral
stressresponseofBacillussubtilistouncoverwidelyrelevantgeneraldesignprinciplesthat
OPENACCESS
describethestructure-functionrelationshipofalternativesigmafactorregulatorynetworks.
Citation:NarulaJ,TiwariA,IgoshinOA(2016) WeshowthattherelativestoichiometryofthesynthesisratesofσB,itsanti-sigmafactor
RoleofAutoregulationandRelativeSynthesisof
RsbWandtheanti-anti-sigmafactorRsbVplaysacriticalroleinshapingthenetworkbehav-
OperonPartnersinAlternativeSigmaFactor
Networks.PLoSComputBiol12(12):e1005267. iorbyforcingtheσBnetworktofunctionasanultrasensitivenegativefeedbackloop.Wefur-
doi:10.1371/journal.pcbi.1005267 therdemonstratehowthisnegativefeedbackregulationinsulatesalternativesigmafactor
Editor:StefanKlumpp,MaxPlanckInstituteof activityfromcompetitionwiththehousekeepingsigmafactorforRNApolymeraseand
ColloidsandInterfaces,GERMANY allowsmultiplestresssigmafactorstofunctionsimultaneouslywithlittlecompetitive
Received:August17,2016 interference.
Accepted:November23,2016
Published:December15,2016
Copyright:©2016Narulaetal.Thisisanopen AuthorSummary
accessarticledistributedunderthetermsofthe
CreativeCommonsAttributionLicense,which Understandingtheregulationofbacterialstressresponseholdsthekeytotacklingthe
permitsunrestricteduse,distribution,and problemsofemergingresistancetoanti-bacteria’sandantibiotics.Tothisend,herewe
reproductioninanymedium,providedtheoriginal studyoneofthelongestservingmodelsystemsofbacterialstressresponse:theσBpathway
authorandsourcearecredited.
ofBacillussubtilis.ThesigmafactorσBcontrolsthegeneralstressresponseofBacillussub-
DataAvailabilityStatement:Allrelevantdataare tilistoavarietyofstressconditionsincludingstarvation,antibioticsandharmfulenviron-
withinthepaperanditsSupportingInformation mentalperturbations.Recentstudieshavedemonstratedthatanincreaseinstresstriggers
files.
pulsatileactivationofσB.Usingmathematicalmodelingweidentifythecorestructural
Funding:TheresearchwassupportedbyNational designfeatureofthenetworkthatareresponsibleforitspulsatileresponse.Wefurther
InstituteofGeneralMedicalSciences(https://www. demonstratehowthesamecoredesignfeaturesarecommontoavarietyofstressresponse
nigms.nih.gov,grant#096189)andNational
pathways.Asaresultofthesefeatures,cellscanrespondtomultiplesimultaneousstresses
ScienceFoundation,DivisionofMolecularand
withoutinterferenceorcompetitionbetweenthedifferentpathways.
CellularBiosciences(https://www.nsf.gov/div/
index.jsp?div=MCB,grant#MCB-1616755)toOAI.
Thefundershadnoroleinstudydesign,data
collectionandanalysis,decisiontopublish,or
preparationofthemanuscript.
PLOSComputationalBiology|DOI:10.1371/journal.pcbi.1005267 December15,2016 1/25
DesignandFunctionalPropertiesofthePartner-Switchingσ-factorNetworks
CompetingInterests:Theauthorshavedeclared Introduction
thatnocompetinginterestsexist.
Bacteriasurviveinstressfulenvironmentalconditionsbyinducingdramaticchangesintheir
geneexpressionpatterns[1,2].Foravarietyofstresses,theseglobalchangesingeneexpression
arebroughtaboutbytheactivationofalternativeσ-factorsthatbindtheRNApolymerasecore
enzymeanddirectittowardstheappropriatestressresponseregulons[3].Consequently,to
ensurethattheseσ-factorsareonlyactiveunderspecificenvironmentalconditions,bacteria
haveevolvedregulatorysystemstocontroltheirproduction,activityandavailability[3,4].
Theseregulatorynetworkscanbehighlycomplexbutfrequentlysharefeaturessuchasanti-σ-
factors,partnerswitchingmechanismsandproteolyticactivation[4].Thecomplexityofthese
networkshasimpededaclearmechanisticunderstandingoftheresultingdynamicalproper-
ties.Inthisstudy,wefocusononeofthebeststudiedexamplesofalternativeσ-factors,the
generalstress-responseregulatingσBinBacillussubtilis[5]tounderstandhowthestructureof
theσ-factorregulatorynetworksisrelatedtotheirfunctionalresponse.
TheσB-mediatedresponseistriggeredbydiverseenergyandenvironmentalstresssignals
andactivatesexpressionofabroadarrayofgenesneededforcellsurvivalintheseconditions
[5].ActivityofσBistightlyregulatedbyapartner-switchingnetwork(Fig1Aand1B)compris-
ingσB,itsantagonistanti-σ-factorRsbW,andanti-anti-σ-factorRsbV.Intheabsenceofstress,
RsbWdimer(RsbW )bindstoσBandpreventsitsassociationwithRNApolymerasethereby
2
keepingtheσBregulonOFF.UndertheseconditionsmostofRsbViskeptinthephosphory-
latedform(RsbV~P)bythekinaseactivityofRsbW .RsbV~PhasalowaffinityforRsbW
2 2
andcannotinteractwithiteffectively[6].However,inthepresenceofstress,RsbV~Pis
dephosphorylatedbyoneorbothofthededicatedphosphatasecomplexes:RsbQPforenergy
stressandRsbTUforenvironmentalstress[7–10].DephosphorylatedRsbVattackstheσB-
RsbW complextoinduceσBrelease,therebyturningtheσBregulonON[11].Notably,the
2
genesencodingσBanditsregulatorsliewithinaσB-controlledoperon[12],therebyresulting
inpositiveandnegativefeedbackloops.
Recently,itwasshownthatunderenergystressσBisactivatedinastochasticseriesoftran-
sientpulsesandincreasingstressresultedinhigherpulsefrequencies[13].Ithasalsobeen
shownthatincreaseinenvironmentalstressorsuchasethanolleadstoasingleσBpulsewith
anamplitudethatissensitivetotherateofstressorincrease[14].Whileitisclearthatthepul-
satileactivationofσBisrootedinthecomplexarchitectureofitsregulatorynetwork(Fig1A
and1B)itsmechanismisnotfullyunderstood.PreviousmathematicalmodelsoftheσBnet-
workeitherdidnotproducethepulsatileresponse[15]ormadesimplificationstothenetwork
[13]thataresomewhatinconsistentwithexperimentallyobserveddetails.Asaresult,it
remainsunclearwhichdesignfeaturesoftheσBnetworkenableitsfunctionalproperties.
ToaddresstheseissueswedevelopadetailedmathematicalmodeloftheσBnetworkand
examineitsdynamicstounderstandthemechanisticprinciplesunderlyingthepulsatile
response.Bydecouplingthepost-translationalandtranscriptionalcomponentsofthenetwork
weshowthatanultrasensitivenegativefeedbackbetweenthetwoisthebasisforσBpulsing.
MoreoverwefindthattherelativesynthesisratesofσBanditsoperonpartnersRsbWand
RsbV,playsacriticalroleindeterminingthenatureoftheσBresponse.Wealsouseour
model,togetherwithpreviouslypublishedexperimentaldatafrom[13,14],toexplainhowthe
σBnetworkisabletoencodetherateofstressincreaseandthesizeofstochasticburstsofstress
phosphataseintotheamplitudesofσBpulses.
Wefurtherdevelopthismodeltoinvestigatehowthenetworkfunctionsinthecontextof
otherσ-factors.Asinmanyotherbacteria,σBisoneofthemanyσ-factorsthatcomplexwith
RNA-polymerasecorethatispresentinlimitedamounts[3,16].Therefore,wheninduced
thesealternativeσ-factorscompetewithoneanotherandthehousekeepingσ-factorσAfor
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DesignandFunctionalPropertiesofthePartner-Switchingσ-factorNetworks
Fig1.σBgeneralstressresponsenetwork.A.NetworkdiagramoftheσBgeneralstressresponse.The
networkhastwomodules:atranscriptionalmodulethatinputsthefreeσBlevelandoutputsthetotal
concentrationsofoperonproteinsRsbV(RsbV ),RsbW(RsbW )andσB(B );andapost-translational
T T T
modulethatusesRsbV ,RsbW andB andthestressphosphataselevelsasinputstooutputthelevelof
T T T
freeσB.Inthepost-translationalmodule,energyandenvironmentalstressesactivatethestress-sensing
phosphatasesRsbQP(QP)andRsbTU(TU)whichdephosphorylateRsbVwhichinturnactivatesσBby
releasingitfromtheσB-RsbW complex.NoteonlythemonomericformsofRsbWandRsbVhavebeen
2
shownforsimplicity.B.SimplifiedviewoftheσBnetwork.TheσBnetworkworksasafeedbackloopwherein
freelevelcontrolsRsbV ,RsbW andB levelsviaoperontranscriptionandthethreeoperoncomponents
T T T
togetherwiththestressleveldeterminethefreeσBlevel.Thefeedbackloopsigniscanbeeitherpositiveor
negativedependingonwhetherincreaseinoperoncomponentlevelsimpactsfreeσBleveleitherpositivelyor
negatively.C-E.DynamicsoffreeσBinresponsetoastep-increaseinphosphataseconcentrationfordifferent
combinationsoftherelativesynthesisratesofσBoperonpartners(λ =RsbW /B ,λ =RsbV /B ).
W T T V T T
doi:10.1371/journal.pcbi.1005267.g001
RNApolymerase.Weuseourmodeltoinvestigatehowthedesignofthisnetworkenablesitto
functioneveninthepresenceofcompetitionfromσAwhichhasasignificantlyhigheraffinity
forRNApolymerase[17].Lastly,weinvestigatehowmultiplealternativeσ-factorscompete
whencellsareexposedtomultiplestressessimultaneously.Usingourmodelweidentifydesign
featuresthatareubiquitousinstressσ-factorregulationandcriticaltobacterialsurvivalunder
diversetypesofstresses.
Results
BiochemicallyaccuratemodelofσBpulsing
Inarecentstudy,Lockeetal.[13]demonstratedthatastep-increaseinenergystressresultsin
pulsatileactivationofσB.Thestudyalsoproposedaminimalmathematicalmodelofthenet-
workwhichreproducedpulsinginσB.However,thismodelincludedseveralassumptions
inconsistentwithexperimentallyobserveddetails:(i)Phosphorylationanddephosphorylation
reactionswereassumedtofollowMichaelis-Mentenkineticsdespitethefactthatkinase
(RsbW)andphosphataseconcentrationsareknowntobecomparabletosubstrate(RsbV)
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DesignandFunctionalPropertiesofthePartner-Switchingσ-factorNetworks
concentrations[18]sotheapproximationbreaksdown[19],(ii)σBandRsbVarerepresented
asasinglelumpedvariableratherthanseparatespeciesand,(iii)partner-switching,andthe
formationanddissociationofvariousRsbW complexeswerenotincludedexplicitly.Though
2
thisminimalmodelproducespulsesresemblingtheirexperimentalobservations,itdoesnot
depictabiochemicallyaccuratepictureoftheσBnetwork.Consequentlyitcannotbeusedto
uncoverthedesignfeaturesthatenableσBpulsing.
TounderstandtheσBnetworkresponsewebuiltonourearlierstudy[15]todevelopa
detailedmathematicalmodelthatexplicitlyincludesallknownmolecularinteractionsinthe
network.Notethatwemadeonesignificantchangetothemodeldiscussedin[15].Themodel
in[15]assumedthatthesynthesisratesforσBanditsoperonpartners(RsbWandRsbV)fol-
lowthestoichiometryoftheirbindingratios(i.e.RsbW /B =2andRsbW /RsbV =1;where
T T T T
B ,RsbW andRsbV representtotalσB,RsbWandRsbVconcentrationsrespectively).How-
T T T
everexperimentalmeasurementshaveshownthatσB,RsbWandRsbVareproducedinnon-
stoichiometricratios[18].Theexactmechanismunderlyingthesenon-stoichiometricratiosis
currentlyonlyincompletelyunderstood.However,analysisoftheopen-readingframesinthe
operonshowedthatrsbVandrsbWmaybetranslationallycoupledduetooverlappingtermi-
nationandinitiationcodons[20]whichmayensurethattheyareexpressedinsimilar
amounts.ThesameanalysisalsoshowedthatthersbWandsigBreadingframesoverlapped
andthatthisoverlapwasprecededbyaregionofdyadsymmetrywhichmayformastem-loop
structure[20].ThesefeaturesmayinterferewithsigBtranslationandleadtolowerexpression
ofσBthanitsbindingpartnersRsbVandRsbW.Toaccountforthesefeatures,incontrastto
ourearlierstudy,weassumedσB,RsbWandRsbVcanbeproducedinnon-stoichiometric
ratiosandstudiedhowchangesinrelativesynthesisratesofσBoperonpartnersaffectthe
responseoftheσBnetworktostep-increasesinenergystressphosphataselevels.Wenotethat
RsbX,anegativeregulatorofRsbTUphosphatase[21],isnotincludedinourmodel.RsbXwas
excludedforsimplicitysinceitisnotessentialforthepulsatileresponseoftheσBnetwork[14].
SimulationsofthisdetailedmodelshowedthatdifferentcombinationsofRsbW:σB(λ )
W
andRsbV:σB(λ )relativesynthesisratesleadtoqualitativelydifferentdynamicalresponsesof
V
theσBnetwork.Foroperonpartnersynthesisratiossimilartothoseestimatedin[18],(i.e.
RsbW >2B andRsbW (cid:25)RsbV )ourmodelrespondedtoastep-upincreaseofthephos-
T T T T
phatasewithapulsatileσBresponse(Fig1C)thatresembledtheexperimentallyobserved
behavior[13].Incontrast,whenRsbW:σBandRsbV:σBrelativesynthesisratesfollowthestoi-
chiometryoftheirbindingratiospulsingisnotobservedandtheσBactivitymonotonically
increasesovertime(Fig1D).PulsingalsodisappearswhenRsbWsynthesisishighenoughto
neutralizebothitsbindingpartners(Fig1E).
Pulsingoriginatesfromemergentnegativefeedbackinthenetwork
Tounderstandwhythepulsatileresponseisonlyobservedforcertainoperonpartnersynthesis
rates,weinvestigatedourmathematicalmodelbydecouplingthenetwork’stranscriptional
andpost-translationalresponses(asshowninFig1A).ByvaryingtheσBoperontranscription
rate,whilekeepingtherelativesynthesisratesofRsbW:σB(λ )andRsbV:σB(λ )fixed,we
W V
wereabletocalculatethepost-translationalresponse(Fig2A,bluecurve)oftheσBnetwork:
[σB]=F ([B ],[P ]).ThisfunctiondescribeshowthefreeσBconcentrationvariesasafunc-
p T T
tionofB (totalconcentrationofσB)andP (totalphosphataseconcentration).Notethat
T T
althoughwereferonlytoB forbrevity,RsbW andRsbV arealwaysassumedtoincreasein
T T T
proportiontotheB forthispost-translationalresponse.Thispost-translationalfunctionis
T
analogoustoaninvitroassaywhereinvariouscombinationsoftotalσB(BT−andproportional
amountsofRsbW andRsbV )andtotalphosphatase(P )aremixedtogetherandthenthe
T T T
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DesignandFunctionalPropertiesofthePartner-Switchingσ-factorNetworks
Fig2.NegativefeedbackdrivesthepulsatileresponseoftheσBnetwork.A.Decoupledpost-
translational(bluecurve)andtranscriptional(blackcurve)responsesoftheσBnetworkforλ =RsbW /B =
W T T
4,λ =RsbV /B =4.5.σBandB representtheconcentrationsoffreeandtotalσB.Redcirclemarksthe
V T T T
steadystatesofthefullsystem.NotethatRsbW andRsbV areassumedtoalwaysincreaseinproportionto
T T
theB forbothpost-translationalandtranscriptionalresponses.B.Sensitivityofthepost-translational
T
response(LG )tochangesintotalσBconcentration(operonproduction).C.RepresentationoftheσB
P
pulsatiletrajectoryintheσB-B phaseplane(greencurve).Blueandcyancurvesarethedecoupledpost-
T
translationalresponsesathighandlowphosphataseconcentrations.Blackcurveisthetranscriptional
response.D.(λ ,λ )relativesynthesisparameterspaceisdividedintoregionswithpositive(RegionI),
W V
negative(RegionII)andzero(RegionIII)post-translationalsensitivitythatrespectivelycorrespondtoan
effectivepositive,negativeandnofeedbackintheσBnetwork.Redandblacklinesrepresenttheanalytically
calculatedregionboundariesλ =2+λ andλ =2(1+λ k /k).
W V W V deg k
doi:10.1371/journal.pcbi.1005267.g002
resultingfreeσBconcentrationismeasured.Inparallel,wecalculatedthetranscriptional
response(Fig2A,blackcurve)[B ]=F ([σB])whichanalogoustoatranscriptionalreporter
T T
constructinvivo,describeshowchangesinthefreeσBconcentrationaffecttotalσBconcentra-
tions(andRsbW andRsbV concentrationswhicharealwaysproportionaltoB ).Inthis
T T T
analysisframework,thesteadystateofthecompleteclosedloopnetworkcanbedetermined
bysimultaneouslysolvingthepost-translationalandtranscriptionalequations,[σB]=F ([B ],
P T
[P ])and[B ]=F ([σB])ateachphosphataseconcentrationP .Graphingbothfunctionspro-
T T T T
videdthesteady-statesolutionastheirintersectionpoint(Fig2A,redcircle).
Thisdecouplingapproximationallowsustoquantifythesignandstrengthoffeedbackin
thefullmodel.TheeffectivesignofthefeedbackintheσBnetworkisgivenbythesignofthe
productofthesensitivitiesoftworesponsefunctions,i.e.sign((@F /@[σB])(cid:1)(@F /@[B ])).
T P T
Sinceσ-factorsfunctionasactivatorsoftranscription,F ([σB])isamonotonicallyincreasing
T
functionofσB(i.e.@F /@[σB]>0).Consequently,thesignofthefeedbackintheσBnetworkis
T
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DesignandFunctionalPropertiesofthePartner-Switchingσ-factorNetworks
givenbythesignofthesensitivityofthepost-translationalresponsetoRsbB (i.e.@F /@[B ]).
T P T
Inotherwords,ifincreaseintheoperonproductionleadstoanincreaseinfreeσBthenthe
feedbackispositive,whereasifincreaseintheoperonproductionleadstoadecreaseinfreeσB
thenthefeedbackisnegative.OurresultsshowthatfortheparameterschoseninFig1CF isa
P
non-monotonicfunctionofB (Fig2A,bluecurve).AtlowRsbB ,freeσBincreasesasafunc-
T T
tionofB becauseRsbWissequesteredintheW V complex.HoweverathigherB ,thekinase
T 2 2 T
fluxdominatesthephosphatasefluxresultinginanincreasedRsbV~Pandthefreeingof
RsbW fromRsbV.FreedRsbW sequestersσBintheW σBcomplex.Furthermore,inthetotal
2 2 2
σBconcentrationrangewhere@F /@[B ]<0inFig2B,thepost-translationalresponseisquite
P T
steep(Fig2A),i.e.smallchangesinB leadtosignificantdecreasesinfreeσB.Thisultrasensitiv-
T
itycanbequantifiedbycalculatingtheslopeinlogarithmicspace,i.e.
d½sB(cid:138)=½sB(cid:138) dlog½sB(cid:138)
LG ¼ ¼
P d½B (cid:138)=½B (cid:138) dlog½B (cid:138)
T T T
ThisdimensionlessquantitycharacterizestheratioofrelativechangesinσBandB atsteady
T
state(Fig2B).ThesignofLG definestheeffectivesignofthefeedbackloopandifthemagni-
P
tudeof|LG |>1definesanultrasensitiveresponse.FortheσBnetwork,intheregionaround
P
thesteadystateLG <−1indicatingthattheσBnetworkoperatesinanultrasensitivenegative
P
feedbackregime.Twotypesofpost-translationalreactionsthatareknowntoproduceultrasen-
sitivityplayarolehere(S1AandS1BFig):(1)Zero-orderultrasensitivityduetocompetition
betweenRsbWkinaseandRsbQP/RsbTUphosphatasesforRsbVand(2)moleculartitration
duetosequestrationofσBbyRsbW.Notablyaroundthesteadystate,whereasboththefraction
ofunphosphorylatedRsbVandthefractionoffreeσBdecreaseultrasensitivelyasafunctionof
increaseinoperonexpression(proportionaltoB )thelatterisfarmoresensitive(S1CFig).
T
ThisindicatesthatmoleculartitrationbetweenσBanditsbindingpartnersmaycontribute
moretotheultrasensitivityofthepost-translationalresponsethanthezero-ordercompetition
betweenRsbWandstressphosphatases.Irrespectiveoftheirrelativecontributionshowever,
ourresultsshowthatbothmechanismscombinetoensurethatnearthesteadystatetheσBnet-
workoperatesinanultrasensitivenegativefeedbackregime.
Notably,negativefeedbackisoneofthefewnetworkmotifscapableofproducingadaption-
likepulsatileresponses[22].Moreover,ultrasensitivityofthefeedbackensureshomeostatic
behavior—makingthesteadystaterobusttovariationsofparameters[22].Thisexplainswhy
inFig1Castep-increaseinthephosphataseconcentrationinourmodelleadstoaσBpulsefol-
lowedbyreturntonearlythesamesteadystate.PlottingthetrajectoryoftheσBpulse(green
curve,Fig2C)onthe([σB],[B ])planeandoverthepost-translationalandtranscriptional
T
responses(Fig2C)illustratesthemechanismdrivingthispulsatileresponse.Startingattheini-
tialsteadystate(redcircle),anincreaseinphosphataseshiftstheultrasensitivepost-transla-
tionalresponse(cyantobluecurve)sothatfreeσBisrapidlyreleasedfromtheRsbW -σB
2
complexwhereastotalσBlevelsremainrelativelyunchanged.TheincreaseinσBoperontran-
scriptioneventuallycausesaccumulationoftotalσBandtheanti-σ-factorRsbW.Thisinturn
forcestheσBleveltodecrease,followingthepost-translationalresponsecurve,tothenew
steadystate(graycircle)whichhasverylittlefreeσBtherebycompletingtheσBpulse.
Thesameanalysiscanbeappliedfordifferentvaluesofrelativesynthesisrates,i.e.those
thatcorrespondtoFig1Dand1E.AsshowninS2Figtheseparametervaluesdonotproduce
anultrasensitivenon-monotonicpost-translationalresponse.Consequentlytheydonotlead
totheemergenceofoverallnegativefeedbackexplainingtheirnon-pulsingdynamics.To
determineifthepresenceorabsenceofnegativefeedbackmoregenerallyexplainsthedifferent
dynamicalresponsesinFig1C–1E,wesampleddifferentcombinationsofrelativesynthesis
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DesignandFunctionalPropertiesofthePartner-Switchingσ-factorNetworks
rates([RsbW ]/[B ]=λ and[RsbV ]/[B ]=λ )andcalculatedthepost-translationalsensi-
T T W T T V
tivities.Ourcalculationsshowedthatbasedonthesignofpost-translationalsensitivity(LG )
P
therelativesynthesisparameterspacecanbedividedintothreeregions(Fig2D).For(λ ,λ )
W V
combinationsinRegionIthesensitivityisalwayspositive.Increaseinλ leadsthesysteminto
W
anultrasensitivenegativeregime(LG <0and|LG |(cid:29)1)inRegionII.Afurtherincreasein
P P
λ oradecreaseinλ transitionsthesystemintoanon-responsive(LG *0)stateinRegion
W V P
III.Dynamicsimulationsforsampled(λ ,λ )combinationsconfirmthatpulsatileresponses
W V
tostep-upinphosphataseconcentrationarerestrictedtoRegionIIwheretheeffectivefeed-
backisnegative(S2Fig).
Tounderstandtheboundariesbetweenthethreeregionsandhowthelevelofthephospha-
taseaffectsthenetwork,wedevelopedasimplifiedanalyticalmodelthatisbasedontheobser-
vationthatRsbWandRsbVbindstronglytoeachother[18](seeS1Textfordetails).This
approximationallowedustodeterminetheboundariesinFig2D(blackandredlines)and
resultedinaclearbiologicalinterpretationofthethreeregions.InRegionItheamountof
RsbW,irrespectiveofphosphataselevel,isinsufficienttobindallofitspartnersandconse-
quentlysomefractionofσBalwaysremainsfreeorunboundtoRsbW.IncontrastinRegion
II,theamountofphosphatasedetermineshowmuchRsbVisinitsinactivephosphorylated
formRsbV~PandthereforewhethertheamountofRsbWissufficienttobindallofitspart-
nersdependsonthelevelsofRsbV~P.Asaresult,forthisregion,theratioofkinaseandphos-
phatase(P )fluxesdeterminesthepost-translationalresponse.Lastly,RegionIIIisthe
T
oppositeofRegionIinthattheamountofRsbWismorethansufficienttobindallofitspart-
ners,evenwhenallRsbVisunphosphorylated.Asaresult,irrespectiveofphosphataselevels,
verylittleσBisfreeanditslevelisnearlyinsensitivetochangesintotalσB.Thusnegativefeed-
backandconsequentlypulsingareonlypossibleinRegionIIwherechangesinphosphatase
canshiftthebalancebetweentheprevalentpartnercomplexes.
Theroleofnegativefeedbackinproducingapulsatileresponsealsoexplainswhypulsing
doesnotoccurinstrainswhereσBoperonistranscribedconstitutively[13].Inthiscase,theσB
networklacksthenegativefeedbacknecessarytoproduceapulsatileresponse.Astep-increase
inphosphatasestillleadstoanincreaseinfreeσBduetothechangeinthepost-translational
response;however,thisnotfollowedbyanincreaseintotalσBlevels(S2CFig).Consequently,
anincreaseinphosphataseresultsinamonotonicincreaseinfreeσBratherthanapulse
(S2FFig).
Theonlyactualmeasurementsofλ andλ weremadebyDelumeauetal.[18]usinga
W V
quantitativewesternblotassay.Interestinglytheyreportthatλ =2.9,λ =1.7intheabsence
W V
ofstressandλ =2.4,λ =2.65inthepresenceofstress.Thesemeasurementssuggestthatthe
W V
ratiosmightchangedependingonwhethercellsareunderstress.Althoughthemechanism
underlyingthischangeisunclear,ourmodelpredictsthatbothmeasuredratiopairsliewithin
thenegativefeedbackregimeshowninFig2D.Accordinglyoursimulationsshowthatthenet-
workrespondstostep-increasesinphosphataselevelswithapulsatileresponseforbothpre-
stressandpost-stress(λ ,λ )values(S3Fig).However,duetoreducedultrasensitivityofthe
W V
systemfortheseparameters,concertationoffreeσBfollowingincreaseinthestress(phospha-
tase)doesnotperfectlyadapttothepre-stressvalue(S3Fig).Inanattempttomatchthenear-
perfectadaptationreportedinRefs.[13,14]we’vechosentodofurtheranalysiswithλ =4
W
andλ =4.5.
V
Notably,oursimulationsalsoshowedthatitisnotessentialforthephosphataselevelto
remainfixedafterastress-inducedstep-increase.Infact,wefoundthatadilutionmediated
declineinphosphataselevelpost-step-increasehaslittleimpactonthepulseamplitude(S4
Fig).Thisobservationcanbeexplainedbytherelativelyrapiddynamicsofthepost-transla-
tionalresponseascomparedtothegradualnatureofdilutionandsuggeststhatpulsatile
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DesignandFunctionalPropertiesofthePartner-Switchingσ-factorNetworks
dynamicsarerelevantevenforexperimentalconditionswherephosphataselevelsdonot
remainfixedinstressfulconditions[14,23].
Furtherourdecouplingmethodalsoshedslightonanotherexperimentalobservationby
Lockeetal.[13]:thedependenceofσBpulseamplitudeonthephosphataselevel.Specifically,
wefoundthatσBpulseamplitudeisathreshold-linearfunctionofthephosphataseconcentra-
tion(S5Fig).Ourdecouplingmethodshowsthatthisthreshold-linearbehaviorarisesbecause
theσBnetworkonlyoperatesinanegativefeedbackregimeforphosphataseconcentrations
higherthanathreshold.Belowthephosphatasethreshold,thepost-translationalresponse
[σB]=F ([B ],[P ])*0andisinsensitivetoRsbB (S5BandS5CFig).Thus,thefullsystem
P T T T
lacksthenegativefeedbackandasaresultσBdoesnotpulse.Usingouranalyticalapproxima-
tionwefoundthatthisphosphatasethresholdisproportionaltothebasallevelofRsbWkinase
synthesisrateandtheratioofthekinaseandphosphatasecatalyticrateconstants(S5Dand
S5EFig).IncreaseinthebasalσBoperonexpressionrateincreasesthephosphatasethreshold.
Further,anincreaseintherelativesynthesisrateofRsbW(λ =[RsbW ]/[B ])makesthe
W T T
phosphatasethresholdmoresensitivetotheσBoperonexpressionrate,whereasadecreasein
ratioofthekinaseandphosphatasecatalyticrateconstantsmakesitlesssensitive(S5Dand
S5EFig).Thisshowsthatthephosphatasethresholdrepresentstheconcentrationatwhichthe
phosphataseisabletomatchthebasalkinaseflux.
Altogethertheseresultsshowhowtheultrasensitivenegativefeedbackplaysacriticalrole
indeterminingmanypropertiesoftheσBnetworkpulsatileresponseandhowthedecoupling
methodcanfacilitatetheidentificationofessentialdesignfeaturesthatenabletheexistenceof
thisnegativefeedback.
UnderenergystressconditionsσBnetworkencodesphosphataseburst
sizeintopulseamplitudes
IntheprecedingsectionswehaveshownhowtheσBnetworkrespondstoastep-increasein
RsbQPorRsbTUphosphatasesbyproducingasinglepulseofactivity.However,Lockeetal.
[13]haveshownthatanincreaseinenergystressleadstoasustainedresponsewithaseriesof
stochasticpulsesinσBactivity.Thisstudyfurthershowedthatthissustainedpulsingresponse
isdrivenbynoisyfluctuationsinlevelofenergy-stress-sensingphosphataseRsbQP.Whilethe
meanlevelofRsbQPisregulatedtranscriptionallybyenergystress,itsconcentrationinsingle
cellscanfluctuateduetothestochasticityofgeneexpression[8,13].Todetermineifourmodel
couldexplainthisresponsetostochasticfluctuationsinRsbQP,wemodifiedittoincludefluc-
tuationsintheconcentrationofthisphosphatase.
Basedonprevioustheoretical[24,25]andexperimental[26]studiesweassumethatfluctu-
atingphosphataselevelfollowsagammadistributionwhichisdescribedbytwoparameters—
burstsize(b,averagenumberofmoleculesproducedperburst)andburstfrequency(a,num-
berofburstspercellcycle).Themeanphosphataseinthiscaseistheproductofburstsizeand
burstfrequency(hP i=ab).Thus,energystresscanincreasemeanphosphatasebychanging
T
burstsizeorburstfrequencyorboth.Inotherwords,stressconditionscanincreasephospha-
taselevelsbyeitherproducingmorephosphatasemoleculespertranscription-translation
eventorbymakingtheseeventsmorefrequent.Whiletheresultsof[13]cannotexcludeeither
mechanism,wecanuseourmodeltouncoverwhichmechanismsisdominant.
First,weperformedstochasticsimulationsinwhichmeanphosphataseconcentrationwas
variedbychangingburstsize.Thesesimulationsreproducedalltheexperimentally-observed
featuresoftheσBpulsatileresponse.Specificallyourresultsshowthatstochasticburstsinstress
phosphataselevelsleadtopulsesofσBactivity(Fig3A).Moreover,consistentwiththeexperi-
mentalobservationsof[13],ourmodelshowedthattheamplitudeofσBpulsesincreases
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DesignandFunctionalPropertiesofthePartner-Switchingσ-factorNetworks
Fig3.PulsatileresponseoftheσBnetworktostochasticphosphataseburstsduringenergystress.
ModelsimulationsforσBnetworkresponsewhereenergystressleadstoanincreaseinstress-sensing
phosphataseRsbQPburstsize(A-D)orRsbQPburstfrequency(E-H).A,E.Simulationsshowstochastic
burstsinlevelsofRsbQPleadtopulsesofσBtargetpromoteractivity.Lightanddarkgreencurvesaresample
trajectoryfromstochasticsimulationathighandlowstressrespectively.NotethatσBtargetpromoteractivity
pulseamplitudeincreasessignificantlywithincreasingstressforburstsizemodulation(A)butnotforburst
frequencymodulation(E).B,F.MeanσBpulseamplitudeincreaseslinearlyasafunctionofmean
phosphataselevelforburstsizemodulation(B)butisinsensitivetomeanphosphataselevelforburst
frequencymodulation(F).Greencirclesanderrorbarsshowmeansandstandarddeviationscalculatedfrom
stochasticsimulations.Blacklineisalinearfit.C,G.Withincreasingmeanphosphataselevel,meanσBpulse
frequencyincreasesultrasensitivelyforburstsizemodulation(C)andlinearlyforburstfrequencymodulation
(G).Greencirclesanderrorbarsshowmeansandstandarddeviationscalculatedfromstochasticsimulations.
BlackcurvesareaHill-equationfitwithn =5.6in(C)andalinearfitin(G)respectively.D,H.MeanσBtarget
Hill
expressionincreasesultrasensitivelyasafunctionofmeanphosphataselevelforbothburstsize(D)and
burstfrequency(H)modulation.GreencirclesarethemeanσBtargetexpressioncalculatedfromstochastic
simulations.BlackcurveisaHill-equationfitwithn =2in(D)andinn =1.2(H).
Hill Hill
doi:10.1371/journal.pcbi.1005267.g003
linearlywiththestressphosphataselevel(Fig3Aand3B).Finally,wefoundthatstress-medi-
atedincreasesinphosphataseconcentrationleadtoanultrasensitive(effectiveHillcoefficient
~5.6)increaseinthefrequencyofσBpulsing(Fig3C)andanultrasensitive(effectiveHillcoef-
ficient~2)increaseinthelevelofσBtargetexpression(Fig3D).
Next,wecomparedtheseresultswithstochasticsimulationsinwhichburstfrequencywas
modulated(Fig3E–3H).ThesesimulationsalsoledtoanincreaseinσBpulsing(Fig3E)anda
non-linearincreaseinthelevelofσBtargetexpressionasmeanphosphataselevelwasincreased
withmorefrequentbursts(Fig3H).However,wefoundthatσBpulseamplituderemainscon-
stantforburstfrequencymodulation(Fig3Eand3F)unlikethe~5-foldincreaseforburst-size
modulation(Fig3B).Moreover,thefrequencyofσBpulsesincreaselinearlywithphosphatase
levelunlikethenon-linearincreaseobservedwithburst-size-increasesimulations(compare
Fig3Cand3G).Notablytheexperimentalobservationsreportedin[13]showthatσBpulse
amplitudedoesincrease(~3-fold)withanincreaseinenergystressthussuggestingthat
increaseinphosphataseconcentrationathighstressisprimarilytheresultofincreaseinburst
size.
Tofurtherreinforcetheroleofmeanburst-sizemodulationincontrollingtheσBpulsatile
responsewenextexaminedthecumulativehistogramsofpulseamplitudesatdifferentphos-
phataseconcentrations.Thesehistogramscarrydifferentsignaturesforburst-sizeorburst-fre-
quencyencoding.Thedistributionofpulseamplitudesisunchangedwithincreaseinburst
frequency(S6AFig)becauseσBpulseamplitudeisdeterminedbyphosphataseburstsizeand
notburstfrequency.Incontrast,ifphosphataselevelsarecontrolledbychangingmeanburst
sizethenthedistributionofpulseamplitudeschangesaccordingly.Consequently,the
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DesignandFunctionalPropertiesofthePartner-Switchingσ-factorNetworks
normalizedcumulativehistogramsofpulseamplitudesoverlapforburst-frequencyencoding
(S6AFig)butnotburst-sizeencoding(S6BFig).Applyingthistesttothedatafrom[13],we
foundthatthenormalizedcumulativepulseamplitudeshistogramsdonotoverlap(S6CFig).
TheseresultspredictthatstressaffectstheσBnetworkviaburst-sizemodulationofphospha-
taseproductionwhichisthenencodedintoσBpulseamplitudes.Whilethemolecularmecha-
nismthatintroducesenergystresstothenetworkisstillnotfullyunderstood,ourprediction
placesanimportantconstraintonit.
σBnetworkencodesrateofenvironmentalstressincreaseintopulse
amplitudes
OurmodelcanalsobeusedtostudytheresponseofσBnetworktoenvironmentalstress.
Unliketheenergystressphosphatase,theenvironmentalstressphosphataseRsbUisregulated
post-translationallybybindingofRsbT[27–29].RsbTistrappedbyitsnegativeregulators
underunstressedconditionsbutisreleaseduponstress.Consequently,theconcentrationof
RsbTUcomplexistightlycontrolledatthepost-translationallevelandisthereforeexpectedto
berelativelyinsensitivetogeneexpressionfluctuationsbutsensitivetothelevelofenviron-
mentalstress.Asaresult,step-upincreasesinenvironmentalstressagentslikeethanolproduce
rapidincreasesinRsbTUandresultinonlyasinglepulseofσBactivity[14].Howeverithas
beenshownthatforgradualincreasesinstress,σBpulseamplitudedependsontherateof
stressincrease[14].Toexplainthisresponse,wemodeledgradualstresswithrampedincrease
inRsbTUcomplexconcentration(Fig4A).Oursimulationsshowedthatthedetailedmodelof
σBnetworkisindeedabletocapturetheeffectofrateofstressincreaseonσBpulseamplitudes.
SpecificallyforafixedincreaseinRsbTUcomplex,thepulseamplitudedecreasesnon-linearly
asafunctionofthedurationofphosphataseramp(Fig4Band4E).
Fig4.RatesensitivityoftheσBpulsatileresponsetoenvironmentalstress.A.Rampedincreasesin
RsbTUcomplexconcentrationwereusedasmodelinputstosimulatedifferentratesofstressincreaseinσB
network.B.σBpulseamplitudesinthewildtypemodel(k =0.72hr-1isthedegradationrateofσBoperon
deg
proteins)resultingfromtherampedincreasesinphosphataseconcentrationshownin(A).C,D.σBpulse
amplitudesresultingfromtherampedincreaseinphosphataseconcentrationshownin(C)forvarious
degradation/dilutionrates(D).E.Non-lineardependenceσBpulseamplitudeonphosphataserampduration
forvariousdegradation/dilutionrates.CirclesandsolidcurvesrepresentsimulationresultsandHill-equation
fitsrespectively.Colorsrepresentdifferentk valuesasin(D).F.K ,thehalf-maximalconstantofthe
deg ramp
non-lineardependenceofamplitudeonrampduration,asafunctionofk .
deg
doi:10.1371/journal.pcbi.1005267.g004
PLOSComputationalBiology|DOI:10.1371/journal.pcbi.1005267 December15,2016 10/25