Constitutive versus Responsive Gene Expression Strategies ...

We find that the optimal constitutive expression level depends on how the costs and benefits increase with the expression level: in one case ... BrowseSubjectAreas ? ClickthroughthePLOStaxonomytofindarticlesinyourfield. FormoreinformationaboutPLOSSubjectAreas,click here. Article Authors Metrics Comments MediaCoverage ReaderComments Figures Figures AbstractMicrobesrespondtochangingenvironmentsbyadjustinggeneexpressionlevelstothedemandforthecorrespondingproteins.Adjustingproteinlevelsisslow,consequentlycellsmayreachtheoptimalproteinlevelonlybyatimewhenthedemandchangedagain.Itisthereforenotaprioriclearwhetherexpression“ondemand”isalwaystheoptimalstrategy.Indeed,manygenesareconstitutivelyexpressedatintermediatelevels,whichrepresentsapermanentcostbutprovidesanimmediatebenefitwhentheproteinisneeded.Whicharetheconditionsthatselectforaresponsiveoraconstitutiveexpressionstrategy,whatdeterminestheoptimalconstitutiveexpressionlevelinachangingenvironment,andhowisthefitnessofthetwostrategiesaffectedbygeneexpressionnoise?Basedonanestablishedmodelofthelac-andgal-operonexpressiondynamics,westudythefitnessofaconstitutiveandaresponsiveexpressionstrategyintime-varyingenvironments.Wefindthattheoptimalconstitutiveexpressionleveldiffersfromtheaveragedemandforthegeneproductandfromtheaverageoptimalexpressionlevel;dependingontheshapeofthegrowthratefunction,theoptimalexpressionleveleitherprovidesintermediatefitnessinallenvironments,ormaximizesfitnessinonlyoneofthem.Wefindthatconstitutiveexpressioncanprovidehigherfitnessthanresponsiveexpressionevenwhenregulatorymachinerycomesatnocost,andwedeterminetheminimalresponseratenecessaryfor“expressionondemand”toconferabenefit.Environmentalandinter-cellularnoisefavortheresponsivestrategywhilereducingfitnessoftheconstitutiveone.Ourresultsshowtheinterplaybetweenthedemand-frequencyforageneproduct,thegeneticresponserate,andthefitness,andaddressimportantquestionsontheevolutionofgeneregulation.Someofourpredictionsagreewithrecentyeasthighthroughputdata,forothersweproposetheexperimentsthatareneededtoverifythem. Citation:GeiselN(2011)ConstitutiveversusResponsiveGeneExpressionStrategiesforGrowthinChangingEnvironments.PLoSONE6(11): e27033. https://doi.org/10.1371/journal.pone.0027033Editor:VladimirBrezina,MountSinaiSchoolofMedicine,UnitedStatesofAmericaReceived:June20,2011;Accepted:October9,2011;Published:November30,2011Copyright:©2011NicoGeisel.Thisisanopen-accessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalauthorandsourcearecredited.Funding:ThisworkwassupportedbytheSpanishMinistryofScienceandInnovationgrantno.FIS2008–04386,andtheSpanishMinistryofEducationgrantno.FPU2007–00975.Thefundershadnoroleinstudydesign,datacollectionandanalysis,decisiontopublish,orpreparationofthemanuscript.Competinginterests:Theauthorhasdeclaredthatnocompetinginterestsexist. IntroductionInnaturalenvironmentscellsarefrequentlyfacingvariableconditions,towhichtheymustadaptinordertomaximizegrowthandsurvival.Commonenvironmentalparameterssubjecttofluctuationsarethekindsofnutrientthatareavailable,thetemperature,thesaltcontentofthesurroundings,andtheconcentrationoftoxinsandantibiotics. Understandingmicrobialbehaviorsinchangingenvironmentsprovidesinsightsintotheevolutioninnaturalhabitatswherethephysiologicdemandsareconstantlychanging[1]–[3].Manipulationofthesestrategiescanberelevantinindustrialprocessing,e.g.fermentation[4],antibiotictherapies[5]andbiotechnologicalprocessoptimization. Prokaryotesandeukaryotescopewithenvironmentalchangesbyswitchingbetweendifferentgene-expressionstates(phenotypes)[2],[3],[6]–[9],typicallyaccompaniedbymetabolicandmorphologicchanges[7],[10],[11].Aparticularphenotypeprovidesagrowthorsurvivaladvantageinoneenvironmentalcondition,butismaladaptedinotherenvironments.Themostprominentexamplesarethevegetativeandpersistentstatesofbacterialpopulations[2],[3],[12]–[14].Inthevegetativestatecellscanrapidlyproliferatebutarehighlyvulnerabletoantibioticstress.Inthepersistentstate,ontheotherhand,theycansurviveantibioticexposurebutcannotdivide.Similarsituationsariseforpili-expressionandatthelevelofmetabolicsystems:productionoflacZisenergeticallycostlyandreducesE.coli'sgrowthrateintheabsenceoflactose[15]–[18].Whenlactoseistheonlyenergysource,inturn,productionoflacZenhancesgrowth[16],[19],[20]. Howmicrobialpopulationsmaximizetheirtime-averagedgrowthrateinachangingenvironmenthasbeeninvestigatedexperimentallyandtheoreticallyalongtwomajorlines[2],[6],[21]–[26].Intheresponsiveswitchingstrategyallcellsswitchintotheadaptedstateuponanenvironmentalchange.Withstochasticswitchingapopulationfollowsabet-hedgingstrategybecausecellsalsotransitrandomlyintomaladaptedstates.Therebythepopulationmaintainsasmallmaladaptedsubpopulationwhichmaybewell-adaptedandreadyforgrowthafterafutureenvironmentalchange.Previousstudieswerebasedontheassumptionthatcellularphenotypetransitionsoccurstochasticallyatagivenrate(alsointheresponsivecase).Thereforeswitchingismodeledasaninstantaneouseventwhich,however,occursafterarandomdelay[2],[3],[6],[21],[23]–[25].Accordingly,cellsexistonlyintwostates(fit,unfit)butneverinthetransientstatesofadaptation,betweentheunfitandthefitphenotype.Animplicitassumptionisthatthetimeintervalsbetweenswitchingeventsareverylarge,i.e.,transitionsoccuronlyonceinmanygenerations[23]. Mostphenotypictransitions,however,areresponsiveandtakeseveralhours,inparticulariflargescalemetabolicandmorphologicchangesareinvolved[5],[10],[11].Theyproceedthroughasequenceofintermediatestateswherethefitstateisupregulatedwhiletheunfitphenotypeisdownregulated[5],[17],[27]–[29].Whenthetimescaleofphenotypicswitching(adaptation)iscomparabletotheenvironmentaldurationsthestatesofintermediateadaptationbecomerelevantforthetotalfitnessandshouldthereforebetakenintoaccount-unlikeatwophenotype(fit,unfit)scenario.Undertheseconsiderationsitappearsthatathirdstrategytocopewithenvironmentalfluctuationsisapassive“intermediate”one,wherecellsconstitutivelyexpressanintermediatephenotypeinallenvironments.Indeed,thisstrategyappearstobewidelyusedsincemanyprocaryoticandeucaryoticgenesareconstitutivelyexpressedalthoughthedemandforexpressionvariesintime.Giventhatregulatedgeneexpressionisadaptivebydefinition,itisnotaprioriclearwhyconstitutiveexpressioncanprovideanadvantage.Whatthendetermineswhetherageneshouldbeunderregulatedorconstitutiveexpression? Thefocusofthisarticleistounderstandhowenvironmentalfactorsdeterminetheoptimalconstitutiveexpressionlevelsthatmaximizesnetgrowthinachangingenvironment,andtounderstandwhyandunderwhichconditionsconstitutiveexpressionconfersagrowthadvantagecomparedtoregulated,responsiveexpression. Toanswerthesequestionsweproposeamodelthatbuildsonpreviouslyestablisheddescriptionsofthelac-andgal-operonexpressiondynamics[17],[22],[30],andcomparethetime-averagedgrowthratesofbothstrategiesinatwo-stateenvironment,takingaccountofenvironmentalandinter-cellularnoise. Wefindthattheoptimalconstitutiveexpressionleveldependsonhowthecostsandbenefitsincreasewiththeexpressionlevel:inonecasegrowthismaximizedbeconstitutivelyexpressingthegeneatanintermediatelevelandintheothercasethegeneiseitherfullyexpressedorfullyrepressed.Surprisingly,theoptimumconstitutiveexpressionlevelinachangingenvironmentisalwaysdifferentfromthetime-averageddemandforthegeneproduct.Wefindthataresponsivestrategycanhavelowerfitnessthanaconstitutivestrategyevenwhenthecostforsensingandregulatorymachineryisneglected,andwedeterminetheminimaladaptationratenecessaryforaresponsetoconferabenefitoverconstitutiveexpression.Environmentalandinter-cellularnoisefavortheresponsivestrategy,whereastheydecreasethefitnessoftheconstitutivestrategy.Ouranalysisillustratestheinterplaybetweendemand-frequencyforageneproduct,maladaptationcost,andthetimescaleofageneticresponse,anditraisesimportantquestionsontheevolutionofgeneexpressionstrategies. MethodsWeproposeamodelbasedontheexpressiondynamicsofmetabolicoperonsasdescribedin[16],[17],[22],[31].Wedenotetheexpressionstateofacellby,wherethefullyinducedstateisoptimal(maximizingthegrowthrate)intheenvironmentwhereastherepressedstatedenotesaphenotypethatisoptimalinenvironmentseeFigure1A[16],[22].Uponanenvironmentalchangeapopulationadaptsbyresponsivelyswitchingeitherintothe‘on’orthe‘off’state(curvedarrows).Formanysystemsthesetransitionsfollowanexponentialrelaxation[17],[22],[30]–[33].Withtheadaptedstatesbeingandandarelaxationratethisismodeledby(1) (2) Download: PPTPowerPointslidePNGlargerimageTIFForiginalimageFigure1.Modelforcellulargrowthandadaptationoftheexpressionstatex(t)inatwo-stateenvironment.(A)Inenvironment(bottom)theexpressionstate(‘off’)allowsforproliferationatthehighestrate.Uponanenvironmentalchange(top)thestateismaladaptedandthepopulationgrowsatareducedrate,whereisthecostofmaladaptation.Intheadaptationphase(,curvedarrows)cellssuppresstheunfitphenotypeandcontinuouslyupregulatethefitone.Thisincreasestheirgrowthrateuntiltheyarefullyadaptedtothenewenvironment(for:).Wealsoconsideraconstitutive-passivestrategywherecellsmaintainaconstantstatethroughoutalltimesinbothenvironments.(B)Growthrateasafunctionoftheexpressionstateinenvironment(red)andinenvironment(blue).Dotsshowtheexperimentallymeasured[16]benefitofE.coliexpressingthelac-operonatafractionoftheoptimallevel()atlactose.Wegeneralizethiscostfunctiontoaccountforconvex(,fulllines)orconcave(dashedlines)dependence.(C)Adaptationdynamics(top)andgrowthrates(bottom)inanenvironmentalcycle.Thefullblacklinecorrespondstoapopulationwhichrespondstentimesfasterthantheenvironmentalfrequency()andwhichthereforetrackstheenvironmentalchangeoccurringat,eventuallyreachingtheadaptedstates.Thegraydashedlinecorrespondstoaslowly-adaptingpopulation()whichneverreachestheadaptedstatesandinsteadoscillatesaroundanintermediateexpressionlevel.Theconstitutive-passivepopulation(dashed-greenline,)hasahighgrowthrateinenvironmentduring,butasmalloneinduring. https://doi.org/10.1371/journal.pone.0027033.g001Herereferstothetimesincethelastenvironmentalchangeandistheexpressionstatewithwhichthepopulationentersintoanewenvironment.Thismodelaccuratelyreproducestheamplitudeandphaseshiftresponseofthegal-operontoexternalglucosedriving(overagalactosebackground)withdifferentfrequencies,asmeasuredin[34](seeFigure2). Download: PPTPowerPointslidePNGlargerimageTIFForiginalimageFigure2.Amplitude-responseandphaseshiftofthemodelcomparedtotheYeastYPH499gal-operon.Wedefinethephaseshiftinourmodelastwicethetimerequiredtoreachthehalf-maximumexpressionlevelof().ThemodelaccordingtoEq.1andEq.2mimicsthegalactoseutilizationnetworkresponseoverabroadfrequencyrange(datapointsasmeasuredin[34]).Thedeviationoftheexperimentalphaseshiftfromthepredictedphaseshiftathighfrequenciesindicatesthattheresponsedoesnotexactlyfollowanexponentialrelaxation.Indeed,thefeedbackarchitectureofthegal-networkmaygiverisetoshortdelayswhichbecomenoticeableathighcyclefrequencies(phaseshifts),whichwedonottakeintoaccountinourmodel. https://doi.org/10.1371/journal.pone.0027033.g002Cellsintheoptimalstategrowatamaximalrate,whereassuboptimalstatesconferinferiorgrowthrates[2,316,17,19,22].ThedotsinFigure1BshowthegrowthbenefitofE.coliundertheassumptionthatthelacoperonisinducedatafractionoftheoptimalinductionlevelinaconstantlactoseenvironment()(datapointsasmeasuredin[16]).Asageneralizationweassumethatthereductionofthegrowthratewhennotintheoptimalstateisproportionaltocost-constantsor(dependingontheenvironment)andthatitdependsmonotonouslyontheexpressionstatewithexponents(asrecentlysuggestedin[35]):(3) (4) Here,orrespectively,isthedeviationfromtheoptimalphenotypeinagivenenvironment.Theparametersallowforconvexorconcavedependenceofthegrowthrateontheexpressionlevel[16],e.g.,forthebenefit(cost)ofproducingametabolicenzymeinthepresence(absence)ofitssubstrate.Figure1Billustratestheserelationshipsforenvironment(inblue)andenvironment(inred)with(dashedlines)and(fulllines).Incontrasttopreviousstudies[6],[33]wemaketheimportantbutplausibleassumptionthatthecostforsensingandsignalingmachineryisnegligible.Wethusfocusonlyonthedynamicalaspectsoftheresponse. Apassivepopulationconstitutivelyexpressesthesamephenotypethroughoutallenvironments.Equations3and4thenapplywith. Figure1C(top)showstheadaptationdynamicsofthephenotype(toppanel)andofthegrowthrate(bottompanel)accordingtoEq.1toEq.4.Theenvironmentchangesfromtoatwhereisthetotaldurationoftheenvironmentalcycleand. E.coliandotherprocaryotesarebelievedtobeoptimizedforfastgrowth.Wethereforetakethetime-averagedgrowthrateasameasureoffitnessinthechangingenvironment[6],[22]–[25].Withoutlossofgeneralityweassumethatanenvironmentalcyclestartswithconditionlastingforatime,andendswithenvironmentofduration().Thetime-averagedgrowthratesandoftheconstitutiveandresponsivepopulationsareobtainedbyintegratingEq.3andEq.4overthedurationofafullcycle:(5)(6) ThesecondtermsintheparenthesesofEq.5aretheintegratedcostsduringtheadaptationphasetowardsthefitstate,anddecreasewiththeresponserate. Itisinstructivetofirstconsiderperiodicenvironmentalcyclesandwechosethecycledurationasthereferencetimescale,withand.Intheperiodiccasetheup-and-downregulationdynamicsofwilleventuallybecomeperiodicwiththephenotypicstatesattheendof(beginningof),andattheendofgivenby(7)(8) ThesecorrespondtothefixedpointswhenpropagatingtheexpressionstateaccordingtoEq.1andEq.2overonecycle.Fromnowonwewillassumethatmaladaptationandgrowthratefunctionaresymmetricinbothenvironments(and)andsetthemaximalgrowthrate. Results Optimalconstitutiveexpressionlevelsinatime-varyingenvironment Asameasureoffitnesswedeterminethetime-averagedgrowthrateoftheconstitutivestrategy(seeEq.6)whichisshowninFigure3(colorcoded)inaperiodicenvironment.Theconstitutivephenotypeisshownonthex-axisandthefractionofenvironment(thedemandforexpression)isshownonthey-axis.PanelAshowsthefitnessforandpanelBfor.Themaladaptationcostis,thusinsignificantlymaladaptedstatesthepopulationhasanegativegrowthrate.Thewhitelinesdelineatetheregimesinwhichthenet-growthrateispositive.Thedashedcurvesshowtheoptimalconstitutivephenotypesthatmaximizethetime-averagedgrowthrate. Download: PPTPowerPointslidePNGlargerimageTIFForiginalimageFigure3.Time-averagedgrowthratesofconstitutivepopulationsinperiodicenvironments.Thefitnessisshownasafunctionoftheconstitutiveexpressionlevelandofthefractionofenvironment(theenvironmentwhichrequiresexpression).Regimesofpositivenet-growtharedelineatedbythewhiteline(maladaptationcost).Leftandrightpanelsshowthetime-averagedgrowthrateforaconvex()andaconcave()growthratefunction.Theoptimalconstitutiveexpressionlevelisindicatedbythedashedline.Foraconvexorlineardependence()anall-or-nothingstrategywithmaximalgrowthinoneenvironmentandnogrowthintheotherisoptimal.Instrikingcontrast,however,foraconcavedependenceanintermediatestrategywithsuboptimalgrowthinbothenvironmentsisbest.Inbothcasestheoptimalconstitutivelevelisdifferentfromtheaverageoptimum,andfromtheaveragedemandforexpression(i.e.,thediagonal).Notethattheconstitutivestrategycanonlyprovidegrowthwhenitisclosetoitsoptimumandwhentheenvironmentissufficientlyconstant(or).Insymmetricenvironments()nopositivenet-growthispossible,henceregulationbecomesimperativeinthisregime. https://doi.org/10.1371/journal.pone.0027033.g003Interestingly,whenthegrowthrateisalinearorconvexfunction(),theoptimalconstitutivestrategyisanall-or-nothingstrategy.Inthiscasenet-growthinachangingenvironmentismaximizedbymaximizinggrowthintheprevailingenvironmentwhilegrowthisminimalintheother,cf.Figure3A.Incontrast,theoptimalstrategyisanintermediateone,withintermediatefitnessinbothenvironments(cf.Figure3B),onlywhenthegrowthrateisaconcavefunction(,asforthebenefitoflac-expression).Ingeneral,andcontrarytowhatonemighthaveexpected,theoptimalconstitutivephenotypeinatime-varyingenvironmentdoesnotcorrespondtothetime-averageddemandforthisphenotypenortotheaverageoptimum,i.e.,aphenotypehassignificantlyinferiornetfitnesscomparedto. Whennoneoftheenvironmentsprevails(),theconstitutivestrategycannotprovidegrowth.Apassivestrategyisthereforenotanoptionathighmaladaptationcostsandwhenbothenvironmentsareequallyfrequent,makingresponsiveexpressionregulationanimperativeinthisregime.Wementioninadditionthatforthecurveofoptimalexpressionisstretchedtowardshigher(lower)expression,whereasitbecomeshighlynonlinearandstep-likefor. Constitutiveexpressioncanprovidehigherfitnessthanregulatedexpression Similarlyasnetproliferationrequiresthatthepassivepopulationissufficientlywelladapted,theresponsivepopulationcanonlyachieveapositivenetgrowrateathighmaladaptationcostsiftheresponserateliesaboveathreshold,cf.thewhitelineinFigure4A().Whentheenvironmentspendsequalamountsoftimeinasin()thepopulationspendssignificantamountsoftimetransitingbetweenphenotypesratherthanintheadaptedphenotypes,whichreducesthetime-averagedgrowthrate.Inparticular,whentheresponserateistoosmallthepopulationneverreachestheadaptedstate,butinsteadlow-passfilterstheenvironmentalchangeandslowlyoscillatesaroundaphenotype,whichcorrespondstothetime-averageddemand,seealsoFigure1C(dashedgrayline)and[34].Whentheenvironmentaldurationsareasymmetric(or)thepopulationremainspartiallyadaptedtothepredominantenvironmentintheenvironmentofshortduration.Thepopulationtherebyhasalowergrowthrateinthesporadicenvironment,butachievesahigheraveragegrowthrate. Download: PPTPowerPointslidePNGlargerimageTIFForiginalimageFigure4.Fitnessofaresponsivepopulation(A)andstrategyphasediagram(B).(A)Theresponsivepopulationhasnegativegrowthwhenitsresponserateistoosmall;thegrowth-thresholdisindicatedbythewhiteline.Thenetmaladaptationcostislargestat(whenbothenvironmentshaveequaldurations)becauseinthisregimethepopulationspendsmostofthetimetransitingbetweenadaptedstatesratherthanbeingadapted.(B)showstheregimesofoptimalstrategy(constitutiveorresponsive)asafunctionofthedemandforexpression(environment)andresponserate.Theregimeinwhicharesponsivestrategywithrateconfershigherfitnessthanaconstitutivestrategyisindicatedinwhite,andforastochasticenvironmentinlightgrayandwhite.Whenenvironmentsareasymmetricaslowresponsivepopulationlagsbehindtheenvironmentandcannotreachanadaptedstateinanyofthetwoconditions.Thereforeithaslowerfitnessthantheconstitutivestrategywhichprovidesimmediatealthoughintermediategrowthinbothenvironments.Thephaseboundariesareindependentofthemaladaptationcost. https://doi.org/10.1371/journal.pone.0027033.g004Geneexpressionlevelscanbeadjustedtotheiroptimumbyafewpointmutationsandwithinafewhundredgenerations[16].Wethereforemaketheplausibleassumptionthattheconstitutivepopulationisoptimallyadaptedtoanenvironmentalcycle,i.e..Sincetheresponsivestrategyfollowsenvironmentalchangesandapproachestheoptimumstateinagivenenvironment,aresponseshouldalwaysconfersuperiorgrowththanconstitutiveexpression.Figure4Bcomparesthetime-averagedgrowthrateoftheconstitutivestrategywitharesponsivestrategyofadaptationrate( = 1).Thereexistthreeregimesindicatingwhetheraconstitutiveoraresponsivestrategyconfersfastergrowth.ThewhiteareainFigure4Benclosestheregimeinwhichtheresponsivepopulationhasahighertime-averagedgrowthrate.Whentheresponseratehighlyexceedstheenvironmentalrateofchangethepopulationfollowstheenvironmentquasi-instantaneouslyandisquasi-alwaysadapted. Remarkably,however,as,thetime-averagedgrowthrateoftheresponsivepopulationbecomessmallerthantheoneoftheconstitutivelyexpressingpopulation(indicatedbythegrayshadedareas).Inparticularinasymmetricenvironments()theconstitutivepopulationcanachievesuperiorgrowthevenwhentheresponserateistentimeslargerthantheenvironmentalfrequency.Consequently,respondingtoenvironmentalchangesprovidesabenefitonlyiftheresponserateliesaboveathreshold.Interestinglythissuggeststhatafastresponsecannotevolvefromconstitutiveexpressionviaaslowresponsebecausefitnessalongthispathwouldhavelowerthanconstitutivefitness. Theslowergrowthoftheresponsivepopulationisaconsequenceofthelow-passfilteringwhichoccurswhentheadaptationtimeislongerthanthedurationoftheshortenvironment.Asexplainedabove,thephenotypicstateslowlyoscillatesaround(theaveragedemand)whichissuboptimalcomparedtotheconstitutivelevel.Inanasymmetricenvironmentthesporadicconditiondrivestheresponsivepopulationawayfromthestatewhichisadaptedintheprevailingcondition.Theresponsivepopulationthereforecannotreachtheadaptedstateinanyofthetwoenvironments.Theconstitutivepopulation,ontheotherhand,benefitsfromhavingintermediategrowthwithoutadelayinbothenvironments(at),ormaximalgrowthintheprevailingenvironment(at). Importantly,thephaseboundariesareindependentofthemaladaptationcost.Withoutgoingintodetails,wepointout,thatwhenislargethereexisttworegimesinwhichthepassivepopulationhasapositivenet-growthrate,whereastheresponsiveonehasanegativenetgrowthrate.Whenthemaladaptationcostsaredifferentinthetwoenvironments,thephaseboundariesbecomeasymmetricandareshiftedalongandthemaximalgrowthbenefitatagivenresponseratedecreasescomparedtothesymmetriccase,renderingconstitutiveexpressionevenmorefavorable.Foraconvexdependenceonthephenotype()thephaseboundaryisshiftedtolargerresponserates(becausethegrowthraterelaxesslowerthantheexpressionstate),whereasitmovestosmallerresponseratesforaconcavedependence(,becauserelaxesfasterthantheexpressionstate). Insummary,aconstitutivestrategycanconfersignificantlybettergrowththanresponsiveexpressionwhentheenvironmentsareasymmetricintheirmaladaptationcostsordurations.Wepointoutthatthisisamereconsequenceofthefiniteadaptationtimesandnotofa“cost-of-regulation”. Fastergrowthinrandomenvironments Althoughperiodicenvironmentsarecommoninnature,moregenerallytheenvironmentaldurationsarerandom.Thepassivestrategy,havingaconstantexpressionlevel,onlyexperiencestheaveragedurationsandthereforeisnotaffectedbytherandomness(environmentaldurationsenterlinearlyintothetimeaveragedgrowthrate).Fortheresponsivepopulation,however,itisnotclearwhetherandhowrandomnesswillaffectitslong-termgrowth. Hereweassumethattheindividualdurations(,)ofenvironmentsandarerandomanduncorrelated,drawnfromexponentialdistributionswithparametersand. Figure5Ashowsthephenotypedynamics(toppanel)andthegrowthrate(bottompanel)fortheresponsivepopulationasafunctionoftime,accordingtoEq.1to4().Inlongerthanaverageconditionsthepopulationhastimetofullyadaptandgrowintheoptimalphenotypeatamaximalrate.Duringtheshortenvironmentalconditions,inturn,thephenotypehasnotenoughtimetosignificantlyadaptandremains“close”tothepreviouslyadaptedphenotype.Uponthenextenvironmentalchangethepopulationcanquicklyreturntothefitstate.Onaveragethepopulationtherebyspendslesstimeinmaladaptedstatesthanifeveryenvironmentalchangehadafixedaverageduration.Thissuggeststhatthenetgrowthrateshouldbelargerinarandomcomparedtoaperiodicenvironment,aconditionwhichhadpreviouslybeenobservedinamodelofstochasticswitching[23]. Download: PPTPowerPointslidePNGlargerimageTIFForiginalimageFigure5.Growthdynamicsinrandomenvironments.(A)Adaptationdynamicsinarandomenvironment(top)andthecorrespondingmomentarygrowthrate(bottom).Duringshortsporadicenvironmentalchangesthephenotyperemainsclosetothepreviouslyfitstate,andtherebyremainsadaptedforthesucceedingenvironment.Asaconsequenceofthefiniteadaptationtimethepopulationlow-passfiltersenvironmentalchangesandonaveragespendslesstimeinmaladaptedstatescomparedtoaperiodicenvironment.Thetime-averagedgrowthrateinafluctuatingenvironmentsignificantlyexceedsthetime-averagedgrowthrateinaperiodicenvironment.Theirratiodefinesthebenefitin(B).Thiseffectbecomesmostrelevantwhentheenvironmentonaveragespendsequalamountsoftimeinbothstates,andwhentheresponserateiscomparabletotherateofenvironmentalchange. https://doi.org/10.1371/journal.pone.0027033.g005Sinceindividualenvironmentaldurationsarerandom,thepopulationdoesnotperiodicallycyclealongthesamephenotypetrajectories.Tocalculatethelong-termgrowth-ratewethereforeevaluateitfromalargenumberofcyclesofrandom-durations.Toensurethatthedistributionofenvironmentaldurationsissampledwithsufficientaccuracywechoosethenumberofcycles.Thetimeaveragedgrowthrateisthenobtainedbyintegratingthegrowthrateovereachcycle(providingaccordingtoEq.5)andweightingitinthesumwiththefractionaldurationofthetotaltime.(9) Foralargenumberofcyclesthetime-averagedgrowthratesettlesatanasymptoticvalue. Figure5Bshowstheratioofthelong-termgrowthrateinarandomenvironment(accordingtoEq.9)tothelong-termgrowthrateinaperiodicenvironment(obtainedaccordingtoEq.5),wherethemeandurationsinthestochasticandperiodiccaseareidentical.Forandtheenvironmentisalmostconstant,hencethereishardlyanyfitnessdifferenceinthisregime.Similarly,iftheresponserateismuchlargerthantheenvironmentalfrequency,thedifferenceissmallbecausethepopulationrapidlyadaptstoevenshortenvironmentalfluctuations.Forandwithresponseratescomparabletotheenvironmentalduration,however,growthinarandomenvironmentissignificantlyfasterthanintheperiodiccase,reminiscentofa(stochastically)resonantphenomenon.Thenetgrowthratedifferencebetweenperiodicandrandomenvironmentsdependsonthecostofmaladaptation:the(on-average)shortertimesinmaladaptedstatesresultinafasternet-growthinthestochasticenvironmentcomparedtotheperiodicone.Hence,thegreaterthecostofmaladaptation,thegreateristhegrowth-rateadvantageinarandomenvironment(here).Thisresultillustratestheimportanceofstudyingmicrobialbehaviorsinanaturalsetting. ThelightgrayareatogetherwiththewhiteareainFigure4Bindicatetheregimeinwhicharesponseisfavoredoveranoptimalconstitutivestrategyinarandomenvironment.Environmentalnoisesignificantlyincreasestheresponsiveregime.Inarandomenvironmenttheconstitutivestrategythereforeappearsasagoodstrategyonlywhenenvironmentalchangesaresporadicandwhenresponsiveregulationisveryslow. Extrinsicnoisebenefitstheresponsivestrategybutreducesfitnessoftheconstitutivestrategy Expressionofmostgenesinunicellularorganismsisstochastic.Asaresult,geneticallyidenticalcellscanshowdifferentproteinexpressionlevels[36]–[40],adoptdifferentstatesinthesameenvironment[5],[7],[13],[22],[41],andrespondtostimuliwithdifferentresponsetimes[42],[43].Differentgenesshowdifferentnoiselevels,andratherthansuppressingnoise[44]somecis-regulatoryelementsseemtopromoteexpressionnoise[45]–[47].Itthereforeisanintriguingquestionwhetherandunderwhichconditionsinter-cellularvariabilitycanprovideabenefitorwhethernoise,asaninevitablesideeffectoflowcopynumbersignaling,alwaysreducesfitness. Weassumethatatthebeginningofanenvironmentalcyclethepopulationisheterogeneousaroundthestateofitscorrespondinghomogeneouspopulation.Twokindsofpopulationheterogeneitycanbedistinguished:inthefirstcaseindividualsubpopulationsofaresponsivepopulationcanhavedifferentresponsetimes(formathematicalconveniencewerefertoresponsetimesratherthanresponserates)[42],[43].Inthesecondscenariodifferentsubpopulationsareindifferentstates.Withdenotingeitheror,thetime-averagedgrowthrateis(10) wheretheintegralisthetotalpopulationsizebythetime(forafullcycle).Hereisthedistributionof,andisthechangeinthesizeofasubpopulation,cf.Eq.5and6. Toensurethatallresponsetimesarepositiveweassumeagamma-distributionfor.Figure6Ashowstherelativefrequenciesofstatesinthepopulationasafunctionoftime.Thedashedlineisforahomogeneouspopulation(,,,and).Initiallyandbytheendofthefirstenvironment(i.e.,)allofthepopulationisinthesamestate,respectively.Duetotheheterogeneousresponse,however,fastrespondingsubpopulationscanquicklyreachtheadaptedstateandproliferateatahighrate,whereasslowlyrespondingsubpopulationsarelaggingbehindthehomogeneousone.Aheterogeneousresponsethereforeresultsintransientheterogeneityofthestatesduringtheadaptationperiod.Theexpectedbenefitofaheterogeneousresponseistwofold:first,fastrespondingsubpopulationsrapidlyadaptanddrivepopulationgrowthatthebeginningofthenewenvironment.Second,slowlyrespondingsubpopulationsremainclosetothepreviouslyfitstate(in)andcanquicklyresumegrowthiftheenvironmentchangesagain()duringtheadaptationperiod,i.e.forsmall. Download: PPTPowerPointslidePNGlargerimageTIFForiginalimageFigure6.Populationdynamicswithheterogeneousresponserates(A)andbenefitcomparedtoahomogeneouspopulation(B).(A)showsthestatedensityasafunctionoftime.Duringtheadaptationphaseapopulationwithheterogeneousresponseratesshowstransientheterogeneityinthestates.Thisresultsinatwofoldbenefit;i)fastrespondingsubpopulationsrapidlyadaptanddrivethegrowthofthewholepopulation,whereasii)forenvironmentsofshortduration(small)slowlyadaptingsubpopulationsremainclosetothestatethatwillbefitwhenoccursnexttime.Thiscausesaslightasymmetryofthebenefitdiagram(B)atlargeresponseratesandsmallvs.large.Thebenefitofheterogeneity,definedastheratioofheterogeneousandhomogeneouspopulationgrowthrates,ishighestwhentheresponserateiscomparabletotheenvironmentalrateofchange. https://doi.org/10.1371/journal.pone.0027033.g006Asameasureofbenefit,Figure6Bshowstheratioofthetime-averagedgrowthratesforaheterogeneousandahomogeneouspopulationinenvironmentsofdifferentdemand,startingwithenvironment.Herecorrespondstotheinverseoftheaverageresponse-time().Forconsistencewekeepthecoefficientofvariationconstantforall().Figure6Bshowsthatresponsetimevariabilityconsistentlyincreasesthepopulationfitness,inparticularwhentheaverageresponsetimeiscomparabletothecycleduration:clearly,whenthenmostcellsrespondmuchfasterthantheenvironmentchanges,hencemostcellsarequasi-instantaneouslyadaptedtoanewenvironmenttherebyrenderingtheeffectofvariabilitysmall.Ontheotherhand,ifcellsrespondmuchslowerthantherateofchangeoftheenvironment,thentheirstateisquasi-constantduringacycle,alsodecreasingtheeffectofvariability.Theslightasymmetryofthebenefitatsmallvs.large,isduetotheaforementionedeffectofslowlyrespondingsubpopulationswhentheenvironmentrapidlyreturnstoitspreviousstate(atsmall).Hence,atshortenvironmentaldurationsslower-than-averagerespondingsubpopulationsprovideabenefit,whereasatlong-lastingenvironmentalconditions,thebenefitoffastrespondingsubpopulationsoutweighsthecostoftheslowlyrespondingones. Forasingleenvironmentalcondition,e.g.,andsmallvariability,thisbenefitcanbeunderstoodstraightforwardly.Assumingthatthedurationofoneenvironmentalconditionislongenoughforallsubpopulationstoreachtheadaptedstatewemaywriteforthepopulationsizeattime(11) asfollowsfromEq.5.UsingEq.11fortheintegralinEq.10andanormaldistributionofresponsetimeswithintegrationlimitsfromto(applicableforsmall),weobtainfortheheterogeneouspopulationsizeattime(12)(13) wherewealsousedthewellknowngaussianintegral.Equation13showsthatresponsetimevariabilityalwaysprovidesabenefitafteranenvironmentalchangecomparedtoahomogeneouspopulationwhichhasthesameaverageresponsetime.ThispropertycanbeattributedtotheconvexdependenceofthepopulationsizeontheresponsetimeasfollowsfromJensen'sInequality.Inparticularweseethatthebenefitincreaseswiththemaladaptationcost,withthesteadystategrowthrate,andwiththevariability.Thisisaplausibleresultwhenconsideringthatthebenefitofrapidlyadaptingcellsincreasesthefasterthesecellscandivideoncetheyhaveadapted,andthatthenumberofrapidlyadaptingcellsincreaseswith. Inthesecondkindofheterogeneity,cellshaveidenticalresponserates,butnoisedrivesthemintodifferentstates.Whentheresponseissufficientlyfast(andintheabsenceofmultistability)itisreasonabletoassumethatmostcellssettleintheoptimumstate(i.e.,in)whereasafewcellsleakintothenearbystates().Wethereforeassumeanexponentialdistributionofstatesatthebeginningofacycle(whentheenvironmentswitchesfromto).BycarryingouttheintegrationinEq.10weobtainthetime-averagedgrowthrateofaheterogeneouspopulation(asourstatespaceislimitedtotheinterval,weonlyconsiderdistributionswithaprobability).InFigure7Awecompareittothenetgrowthrateofahomogeneouspopulationwhereallcellsstartin,asafunctionofthemaladaptationcostandthevariability().Thebenefitofstateheterogeneityincreaseswiththevariabilityandwiththemaladaptationcost,clearlybecausecellsthatareslightlypre-adaptedtothenewenvironmentprovideahigherbenefitwhenthemaladaptationcostislarge.Ontheotherhandwefoundthatforstate-variabilityrepresentsadisadvantagebecausethebenefitofcellsthatarepre-adaptedtoconditionisoutweighedbyacostwhichthesecellsrepresentiftheenvironmentrapidlychangesbacktocondition(notshown).Henceexpressionnoiseappearstoprovideabenefitonlyifdifferentenvironmentalconditionshavesimilardurations,butnotwhenoneenvironmentstronglyprevails.ForasingleenvironmentalconditionandwiththeintegralinEq.10canagainbecarriedoutanalytically,yielding(14) Download: PPTPowerPointslidePNGlargerimageTIFForiginalimageFigure7.Benefitandcostofstateheterogeneity.Benefitsandcostsaremeasuredbytheratioofheterogeneousandhomogeneouspopulationgrowthratesoveronecycle,foraresponsivepopulationin(A)andforaconstitutivepopulationin(B).Fortheresponsivestrategythebenefitofheterogeneityincreaseswiththemaladaptationcostandwiththevariability.Thefitnessofaconstitutivepopulation(B)whichiswelladaptedtoenvironmentalcycleswhereprevails()isreducedbyvariability.Onlywhenthepopulationissignificantlymaladaptedheterogeneityprovidesabenefit.Notethatbenefitvaluesin(B)areclippedat https://doi.org/10.1371/journal.pone.0027033.g007Asweareconsideringanexponentialdistributionofstates,wehave.Hence,forahomogeneouspopulation()thisexpressiondirectlyexplainstheincreasingbenefitatincreasingmaladaptationcostsandvariability,afteranenvironmentalchange.Notethatthebenefitdecreaseswithincreasingresponserates,becausethepopulationwillbenefitmorefromcellsthatareslightlypre-adaptedwhentheresponseisslowthanwhentheresponseisfast.Aslowlyrespondingpopulationmightthereforeincreaseitsfitnessbyincreasinggeneexpressionnoise.Wementionthattheresultsremainqualitativelysimilarwhenweassumeasymmetricdistributionaroundaninitialstate. Figure7Bshowstheratioofthetime-averagedgrowthratesofaconstitutive-heterogeneoustoaconstitutive-homogeneouspopulationasobtainedfromEq.10.Weassumedthattheconstitutivepopulationisoptimizedforgrowthinenvironmentalcycleswhereprevails,andhasanexponentialdistributionoverneighboringstates.Forthecasethatthehomogeneouspopulationisreasonablywelladapted(i.e.for),heterogeneityrepresentsasignificantcostbecauseasmallerfractionofthepopulationresidesintheoptimumstate(thisissimilartoaresponsivepopulationinanenvironmentwhereoneconditionstronglyprevails).Onlyifthepopulationissufficientlymaladapted(),diversificationcanincreasefitnessduetothepresenceofasmallwell-adaptedsubpopulation. Geneexpressionlevelscanevolvetoanoptimumwithinafewhundredgenerations[16].Theaboveresultsthereforeindicatethattheexpressionofaconstitutivegenewillbeselectedagainstnoise[39],[48]–[50].Ontheotherhand,wefindthataresponsivestrategycanbenefitboth,fromheterogeneousstatesandfromheterogeneousresponsetimes,inparticularwhenmaladaptationcostsarehighandwhenbothenvironmentaldurationsarecomparabletothepopulation-averagedresponsetime. DiscussionItisageneralbeliefthatrespondingtoanenvironmentalchangeisbetterthannotresponding.Itisnotaprioriclear,however,whetherrespondingisindeedthebeststrategyinarapidlychangingenvironment.Infact,manygenesarenotresponsivelyregulatedbutexpressedconstitutivelydespiteavaryingdemandforthegeneproduct.Inthisarticleweexplainedwhichconditionsselectforaconstitutiveoraresponsivegeneexpressionstrategyinatime-varyingenvironment,takingaccountofenvironmentalandinter-cellularnoise. Witharesponsivestrategyapopulationcanswitchbetweentwoadaptedphenotypes,whereeachoneconfersmaximalgrowthinoneenvironmentwhileminimizinggrowthintheother,cf.Figure1.Afteranenvironmentalchangetheresponsivepopulationismaladaptedandrequirestimeforthetransitionintotheadaptedstate,eventuallyreachingitbyatimewhentheenvironmentchangesonceagain.Withaconstitutive-passivestrategyapopulationcanevolutionarilytuneitsphenotypetoanoptimalintermediatelevel,whichononehandallowssuboptimal(intermediate)growthatalltimesinbothenvironments[16],andwhichontheotherhandbypassestheadaptationlag.Asafunctionofthemaladaptationcost,thetimescalesofenvironmentalchanges,andofthegeneticresponsewehavestudiedwhichistheoptimalconstitutiveexpression-levelandunderwhichconditionsitconfersfastergrowththanaresponsivelyregulatedexpression. Wefoundthattheoptimalconstitutivelevelinachangingenvironmentisdifferentfromtheaverageoptimalexpressionlevel(Figure3):whenthegrowthrateisaconvexfunctionoftheexpressionlevel,theoptimummaximizesgrowthinoneenvironmentwhileprovidingminimalgrowthintheother.Whenthegrowthrateisaconcavefunction,theoptimalconstitutivelevelisanintermediateone,providingintermediategrowthinbothenvironments.Interestingly,whetherconvexorconcave,theoptimumlevelisgenerallydifferentfromtheaveragedemandforthegeneproduct. Atlargemaladaptationcoststheconstitutivestrategyconfersnet-growthonlywhenoneoftheenvironmentalconditionsprevails,otherwiseafastrespondingstrategybecomesimperativetoachievenetgrowth.Aresponsivepopulationthatcannotrespondsufficientlyfast,however,lagsbehindtheenvironment:itsexpressionstateslowlyoscillatesaroundtheaverageddemandforthegeneproductanddoesnotreachtheoptimuminanyofthetwoenvironments(cf.Figure1C).Undertheseconditionsconstitutiveoptimalexpressionprovidesalargertime-averagedgrowthratethanresponsivelyregulatedexpression,cf.Figure4. Theresponsivevs.constitutiveregimesareseparatedbyafirstorderphase-transition.Thisindicatesthatafastgeneticresponsecannotevolvestartingfromconstitutiveexpressionviaaslowresponse,becauseitwouldhavetogothrougharegimeoflowerfitness.Thisconditionmaygiverisetoevolutionaryhysteresisasrecentlysuggestedfortheevolutionofstochasticswitching[24].Aninterestingquestionthatarises,ishowresponsivegeneexpressioncanevolvefromconstitutiveexpression. Previousstudies[6],[33]hadfoundthatconstitutiveexpressioncanonlybebetterthanresponsiveexpressionifthecostforsensingandregulatorymachineryishigh.Inotherwords,whenregulationcomes“forfree”thesestudiespredictthatregulationwillalwaysbeselectedfor.Instrikingcontrast,weneglectedthecostofregulationandstillfindthatconstitutiveexpressioncanbebetterthanadaptiveexpression.Thisresultisamereconsequenceofexplicitlytakingintoaccountthe(slow)adaptationdynamicsandtheintermediatestates,whichwereneglectedinpreviousstudies. Wefindthataresponsivestrategyhassignificantlylargergrowthratesinrandomenvironmentscomparedtoperiodicenvironmentswhenthetimescalesofthegeneticresponseandenvironmentalchangearecomparable,seeFigure5B.Asimilareffectwaspreviouslyobservedinamodelofstochasticswitching[23],thereforeitwouldbeveryinterestingtoverifythispredictionexperimentally.Furthermorewefindthatinachangingenvironmentaresponsivestrategycanbenefitfrominter-cellularnoise,inparticularwhenenvironmentaldurationsarecomparabletothepopulationaveragedresponsetime,whereasthefitnessoftheconstitutivestrategyisimpaired,cf.Figure6and7. Thus,ourmainconclusionsare:i)thataconstitutivegene-expressionstrategyisbetterthanaresponsivestrategywhentheenvironmentsareasymmetricorwhenaresponseisnotsufficientlyfastandii)thatconstitutivelyexpressedstatesareselectedagainstnoisewhereasgenesthatrespondtoenvironmentalchangesmaybenefitfromnoise. Recentanalysisofyeasthigh-throughputdataindeedconfirmthisresult[39],[48]–[50]:geneswhichareconstitutivelyexpressedandunderanalmostconstantdemand(commonlyreferredtoashousekeepinggenes)havebelow-averagelevelsofgeneexpressionnoise,theproteasomehavingtheleast[39].Thisisinagreementwithothertheoreticalworksongeneexpressioninaconstantenvironment[17],[35],[35,51].Tightlyregulatedgenes,whichrespondtoenvironmentalperturbations,however,showsystematicallyhigherlevelsofgeneexpressionnoise.Thisisparticularlystrikingforstressresistancegenesandfortheproductsofmetabolicsystemsintherepressedstate[39],[42],[46],[49]–[52].Inprinciplethenoiselevelscanbetunedbythecell[36];thereforeitwouldbeinterestingtoexperimentallyverifyacorrelationbetweengeneexpressionnoise,generesponsetime,andthefrequencyofdemandforageneproduct.Morespecificallythismaybeachievedinalaboratoryevolutionexperimentwhereanoisygenethatrespondstoenvironmentalperturbationsisputunderconstantdemand.Accordingtoouranalysisevolutionwillthenselectagainstgeneexpressionnoise. Complementingpreviousworksonphenotypicswitching[6],[23]–[25]ourresultsallowtodividetheenvironmentalparameterspaceintothreeregimesofoptimalgrowthstrategies:a)Whenpopulationscanrapidlyswitchbetweenadaptedstatestheresponsiveswitchingstrategyisthebest.b)Whenadaptationisslowandenvironmentsaresymmetric,stochasticswitchingispreferredoverresponsiveswitching[23]–[25].c)Whenaresponseisslowandtheenvironmentisasymmetric,constitutiveexpressionisbetterthanresponsiveswitching(thisstudy),whereasstochasticswitchingcanbeworse[24],[25]. Ourpredictionsonoptimalityofconstitutiveexpressioncanbeverifiedexperimentallyasfollows.Inaconstantlactoseenvironmentwithsaturatinginducerconcentrations,theexpressionlevelofthelac-operonwasshowntoadapttoanoptimumwithinafewhundredgenerations[16].Usingasimilarprotocoltheconstitutivemutants,or,canbeevolvedinachangingenvironmentwherethedemandforLacproteinsoscillatesintime.Ourmethodpredictsthe(optimal)expressionlevelstowhichaconstitutivelyexpressingstrainwillevolveasafunctionoftheexpressiondemand. Recentlydevelopedpromotersallowforagradedinductionofvarioussugarsystems[53],[54].Byvaryingtheinducerconcentrationthesepromoterscanbeusedtomeasurethegrowth-ratedependenceontheexpressionlevelofdifferentgenes(i.e.,thecost-benefitrelationship)andtodeterminetheoptimalconstitutiveexpressionlevelsatdifferentdemandsforthegeneproduct.Itwouldalsobeveryinterestingtousethesepromoterstocharacterizealargesetofcost-benefitfunctions:doallfunctionsfallintoacertainclass?Aretherethreshold-likecost-benefitfunctions?Classifyingandunderstandingtheshapeofthesefunctionsmayprovideprofoundinsightsintocellularexpressionregulation. Usingmicrofluidicdevices[55]thetime-averagedgrowthratesinarapidlychangingenvironmentcanbemeasured[22,34,allowingcomparisontoourconstitutivevs.responsivestrategydiagrams,seeFigure4B.ForE.coli,usingtheexperimentallydeterminedcost-benefitdataofthelac-operon(notexpressingLacZwhenlactoseisavailable:,expressingLacZwhenlactoseisunavailable,,at[16])ouranalysispredictsthatconstitutivelac-expressionwillhaveagrowthrateadvantagewhentheexpressiondemandliesaboveatanenvironmentalcycleduration(whereistheresponserateofthelac-operon).Whenthecycledurationislonger,e.g.,thentheresponsivestrategyhassufficienttimetofullyadaptandtheintermediate-constitutivestrategywillonlyhavehigherfitnesswhenthedemandforlac-expressionalsoincreasesabove.Togetherthisindicatesthatthelac-operonwasoptimizedforrarelactoseavailabilitywithlongcycledurations,consistentwithpreviousworks[16],[17].Indeed,inanaturalsettingE.colifindslactoseonlyduringhourswhiletraversingtheprimarymammalianintestine[1],whereaslactoseisunavailableinmostotherhabitats(colon,soil,water).Assumingacycleduration(hencethedemand),ouranalysisconsistentlypredictsthatregulated-inductionofLac-proteinsconfershigherfitnessthanoptimalconstitutiveexpression. Finally,acomparisonofregulatorystrategiesacrossdifferentspeciesthatevolvedindifferenthabitatswouldprovidefurtherinsightintotheinterplayofenvironmentaldemandfrequencyandtherequirementsfortheregulationofgenes[1].Specifically,constitutivegeneexpressionlevelsandgeneinductionpatternsmaydiffersignificantlybetweenthewildtypeS.Cerevisiaepopulations,andpopulationswhichwereusedovermanygenerationsinindustrialfermenters,e.g.,breweries. Inthispaperwehavestudiedoptimalgeneexpressionstrategiesinarapidlychangingenvironment.Weanalyzedtheinterplaybetweenthetimescalesofgeneticresponseandthedemandforaphenotype,themaladaptationcosts,andthefitness.Someofourpredictionsagreewithexperimentalobservations,andwesuggesttheexperimentsneededtoverifyothers.Webelievethiswillstimulatefurtherexperimentalworkand–inlinewithourpredictions–deepenourunderstandingofmicrobialgeneexpressionstrategies. Acknowledgments TheauthorwouldliketothankMiguelRubi,JavierBucetaandMarcWeberforhelpfuldiscussionsandreadingthemanuscript. 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