The Journal of Cell Biology

JCB

http://www.jcb.org/cgi/doi/10.1083/jcb.200301125 The Journal ofCellBiology  applied tomanysystems,rangingfromafewisolated computationally. Thisstandardmodelingapproachhasbeen of ordinarydifferentialequations,whichcanbesolved methods ofchemicalreactionkinetics,onecanobtainaset with geneticormetabolicnetworks.Then,followingstandard heterogeneity inherenttocellularorganizationwhendealing (Eldar etal.,2002),itiscustomarytoneglectallthespatial tain systems,e.g.,earlydevelopmentof cer- of cellularprocessesaretakenintoaccountinmodeling in anidealmacroscopicreactor.Althoughsomespatialaspects can beapproximatedbyanetworkofbiochemicalreactions assumption thatinteractionsbetweenmolecularcomponents or incomputersimulations. parts, eitherasanalyticalsolutionsofmathematicalequations enough toreproducethebehavioroforganism,orits of themolecularcomponentsmaybeunderstoodwell that agrowingnumberofbiologistsbelievetheinteractions this data(EndyandBrent,2001;Kitano,2002).Itseems of fastcomputerscapable,atleastinprinciple,toprocess generating vastamountsofdata,andtothegeneralaccessibility largely duetothedevelopmentofnewexperimentalmethods there isarenewalofinterestinmodelingbiologicalsystems, tative informationhaveprecludedasimilarsuccess.Currently, complexity oflivingsystemsandthelackreliablequanti- however, thesituationhasbeendifferent.Theenormous in disciplinessuchasengineeringandphysics.Inbiology, Modeling hashadalongtradition,andremarkablesuccess, We usethe 2 1 eling; regulation José M.G. Vilar, *Abbreviation usedinthispaper:TMG, thiomethyl- 7642. Fax:(212)327-7640.E-mail: [email protected] 1230 YorkAvenue,Box34,New York,NY10021.Tel.:(212)327- Address correspondencetoJoséM.G. Vilar,TheRockefellerUniversity, system. which seemstocapture many experimental aspectsofthe cellular, andthatofcellpopulation)intoasinglemodel, We integrate three different levels ofdescription(molecular, limitations ofmodelingthedynamics ofcellularnetworks. system toillustrate thecurrent state,applicability, and Key words: geneexpression;induction; a casestudy Modeling network dynamics: the

Department ofMolecularDepartment Biology, Princeton University, Princeton, NJ08544 The Rockefeller University, New York, NY10021

The Rockefeller University Press, 0021-9525 Mini-Review Modeling ofcellularprocessesistypicallybaseduponthe

lac

operon in

1

ClinC.Guet, C

, a ˘

Volume 161,Number 3,May 12, 2003471–476 Escherichia coli lac operon;computationalmod- 1,2 /2003/05/471/6 $8.00 Drosophila melanogaster andStanislas Leibler asaprototype - D -galactoside. 1 example ofthe ology. tant in ways toovercomethem,willbecomeincreasinglyimpor- governing thereactions.Understandingtheselimitations,and the natureofnonlinearitiesandvaluesparameters without numerous,andoftenarbitrary,assumptionsabout vivo processesareknownthatitisverydifficulttoproceed 2001). Ontheotherhand,sofewdetailsaboutactualin the molecularcomponentscannotbeneglected(Kuthan, are present(Ellis,2001),andinwhichthediscretenatureof which phenomenalikemolecularcrowdingorchanneling highly heterogeneousandcompartmentalizedstructure,in On theonehand,cellisnotawell-stirredreactor.It spread usemightindicate,suchmodelinghasmanylimitations. components toentirecells.Incontrastwhatthis ments demonstratedtwointerestingfeaturesofthe experiments ofNovickandWeiner(1957).Theseexperi- (1996). Here,weconcentrateourattentionontheelegant we referthereadertolivelyaccountbyMüller-Hill genetic systemhasbeendescribedinmanyplaces;forinstance, genes requiredfortheuptake andcatabolismoflactose.A The The regulatory network.First,theinductionof modeled. different levelsatwhichbiologicalnetworksneedtobe reactions. Thisexamplewillalsoallowustoexplainthe standard approachformodelingofnetworksbiochemical features cannotbequantitativelyunderstoodusingthe (Fig. 1b).Wewillarguebelowthateventhesetwosimple simplest examplesofphenotypic,orepigenetic,inheritance mitted throughmanygenerations;thisprovidedoneofthe state ofasinglecell(inducedoruninduced)couldbetrans- ments ofNovickandWeiner(1957)alsoshowedthatthe of thesetwotypescells(Fig.1a).Second,theexperi- in thecellpopulationareaconsequenceofcoexistence duced). Intermediatelevelsofenzymeproductionobserved be viewedaseitherswitchedon(induced)orshutoff(unin- duction oflactose-degradingenzymesinasinglecellcould was revealedasanall-or-nonephenomenon;i.e.,thepro- We willillustratethemainissuesofmodelingusing lac lac order tofullyintegratemodelingintoexperimentalbi- operonconsistsofaregulatory domainandthree operon lac lac operonin operon, Escherichia coli . Thisclassical lac operon wide-

471 lac

on December 30, 2005 2005 30, December on www.jcb.org from Downloaded The Journal of Cell Biology 472 three genes.Inductionofthe tor andpreventtheRNApolymerasefromtranscribing regulatory protein,theLacIrepressor,canbindtoopera- uninduced. maintenance concentration,theyandtheirprogenywillremain uninduced cellsatlowinducerconcentrationaretransferredtothe tration, theyandtheirprogenywillremaininduced.Similarly,when inducer concentrationaretransferredtothemaintenanceconcen- (b) Maintenanceconcentrationeffect.Wheninducedcellsathigh here byfullellipses.Emptyellipsescorrespondtouninducedcells. increase isproportionaltothenumberofinducedcells,represented content ofthepopulationincreasescontinuouslyintime.This For lowinducerconcentrations,theenzyme( Figure 1. lecular aspects of generegulation. affects thesystem,andhowinduction dependsonthemo- process, howtheintrinsicrandomness ofmolecularevents quantitative approachestounderstand thedynamicsofthis switch fromtheuninducedto theinducedstate.Oneneeds why thecellsremaininagiven state,orwhatmakesthecells presence oftwophenotypes,but itdoesnotactuallyexplain high andtheproductionofpermeasesremainshigh. number ofpermeasesishigh,theinducerconcentration is production ofpermeasesremainslow.Incontrast,ifthe low, theinducerconcentrationinsidecellislowand ducer intothecell.Inthisway,ifnumberofpermeases is encoded byoneofthetranscribedgenes,whichbringsin- cell. Theinductionprocessisthushelpedbythepermease repressor dependsontheinducerconcentrationinside at agivenrate.Theprobabilityfortheinducertobind sor cannotbindtotheoperatorandtranscriptionproceeds ducer moleculebindstotherepressor.Asaresult,repres- This heuristicargumentisusefulforunderstandingthe The JournalofCellBiology Different inductionstates.

|

Volume 161,Number3,2003

lac

operonoccurswhenthein- (a)All-or-nonephenomenon. -galactosidase) can measuretherateofproduction function directlyfromtheexperimentaldata.Indeed,one at thepresentstageofknowledgewouldbetoextractthis lecular interactions.Therefore,amorereasonableapproach tion, onewouldneeddetailedinformationaboutmanymo- tain theoreticallyevenaroughapproximationofthisfunc- function oftheinducerconcentrationinsidecell.Toob- seems relevantistheproductionofpermeasesexpressedasa nately, notallthedetailsareneeded.Atthislevel,what filled outwithassumptionsbasedonparsimony.Fortu- cesses areunknown.Thelackofinformationistypically quantitative aspectsoftheinvivodynamicsthesepro- the message,proteinfolding,andsoforth.Almostall tion oftranscription,productionmRNA,translation tor, bindingoftheRNApolymerasetopromoter,initia- pressor conformation,bindingoftherepressortoopera- the bindingofinducertorepressor,changesinre- estimate formodelingthemolecularlevelofwild-typecells. mRNA. Theresultsobtainedinthiswaycouldbeusedasan dase, asbothareproducedfromthesamepolycistronic Despite itsapparentsimplicity,the Levels oforganizationandmodeling population levels. proximation, proportionaltotheproductionof mation isthattheproductionofpermeasesis,toagoodap- import orexportmechanisms.Theotherkeypieceofinfor- toplasm isreached.Thisreliesontheabsenceofnonspecific the sameonceequilibriumbetweenmediumandcy- case, externalandinternalinducerconcentrationsareboth tant strainslackingthepermease(Herzenberg,1958).Inthis question ofhow manyarefunctionalhas notbeenad- grate inthemembrane(Ito and Akiyama,1991),yetthe however, showedthatthemajority ofthepermeasesinte- membrane andbecomefunctional. Recentexperiments, only afewpercentofthe permeases integrateintothe Novick andWeiner(1959)inferred fromexperimentsthat eventually gotothemembrane andbringmoreinducer. at thecellularlevel.Someofpermeasesproducedwill cules totheDNA-bindingproteins.Inaddition, of differentproteinstoDNA,andbindingsmallmole- translation, proteinassembly,degradation,binding clude, amongmanyothercellularprocesses,transcription, regulation. Inprinciple,itsdetailedmodelingshouldin- plays muchofthecomplexityandsubtletyinherenttogene separation ofthe corporated intoanother.Fig.2illustratesschematicallythe relevant levels,theirinteractions,andthewayonelevelisin- study. Thefirststepofmodelingis,therefore,toidentifythe are notgoingtoberelevantfortheparticularprocessunder available. Moreover,mostofthemoleculardetailscell tional informationaboutcellularprocessesthatisnotreadily level allthewayuptocellpopulationrequiresaddi- growth. Therefore,extrapolatingdirectlyfromthemolecular a gratuitousinducerisused,inductionwillslowdowncell turn alsoaffectsthecellpopulationbehavior.Forinstance,if tion changesthegrowthrateofindividualcells,whichin eron systemisnotisolatedfromtherestofcell.Induc- Molecular level. Cellular level. The coreoftheall-or-noneprocessresides lac The molecularlevelexplicitlyincludes systemintomolecular,cellular,and lac -galactosidase inmu- operonsystemdis- -galactosi- lac

op-

on December 30, 2005 2005 30, December on www.jcb.org from Downloaded The Journal of Cell Biology from asinglepromoterP1.Thegene the uptakeoflactoseinsidecell.Therole or catalyzetheconversionoflactoseintoallolactose,actualinducer.Theproduct niques thatarenowavailable(Thompsonetal.,2002). state ofthepermeaseswouldbeextremelyusefulusingtech- single-cell studiesontheconcentrationandfunctional point ofourmodel.Inviewtheall-or-nonephenomenon, constant probabilityrate.Webelievethistobetheweakest serted intothemembraneandbecomefunctionalwitha tion formodelingisthattheproducedpermeasesarein- nisms ofinsertionintothemembrane.Thesimplestassump- which includesmanyopenquestions,suchasthemecha- al., 2001),itsinvivofunctioningisstillachallengingissue, dressed. Despiteintensestudiesonthepermease(Kabacket Figure 2. come uninduced. duced celltobecomeinduced andforaninducedcelltobe- cellular levelbycomputing the probabilityforanunin- ing rates.Theseratescanbe obtainedbymodelingatthe lation levelbyconsideringthe induced–uninducedswitch- experiments. Thecellularlevel isintegratedintothepopu- rates forinducedanduninduced cellsareknownfromthe dard two-speciespopulationdynamicsmodel.Thegrowth (Koch, 1983).Atthislevel,itseemsadequatetouseastan- connected withthenumberofpermeasesinmembrane slows downthegrowthrate.Thisslowingseemstobe periments, thesituationisjustopposite:induction ducers, liketheoneusedinNovickandWeiner(1957)ex- source, inductionallowscellstogrow.Forgratuitousin- growth rateofthecells.Whenlactoseissolecarbon grow faster,bothtypesofcellscouldcoexist;ifnot,theentirepopulationwilleventuallybeinduced. lac concentration, thenonfunctionalpermeases,andfunctionalpermeases.(c)Populationlevel.Coexistenceoftwotypesnet metabolized bythecell.Inthiscase,itispossibletostudydynamicsofinductionconsideringasvariablesonlyi such asTMG,alsousethelactosepermeasetoentercell;butincontrastlactose,theybindthemselvesrepressor Additionally, theCAP–cAMPcomplexmustbindtoactivatorsite,A,forsignificanttranscription.(b)Cellularlevel.Some the repressortooneofauxiliaryoperatorsites O2andO3.Theinducerinactivates therepressorbybindingtoitandchan of therepressortomainoperatorsite O1preventstranscription.Repressionisgreatlyenhancedbytheadditionalsimultan required forlactosemetabolism(Wangetal.,2002).The Population level. operonisapopulationeffect.Uninducedcells(emptycircles)havesomeprobabilitytobecomeinduced(fullcircles).Ifunin Three levelsofdescription. Induction ofthe lacZ (a)Molecularlevel.Thethreegenes encodesforthe lac

operonchangesthe

LacA

isnotyetfullyunderstoodbecauseitsproduct,agalactosideacetyltransferase,

lac

-galactosidase, whichcaneitherbreakdownlactoseinto

repressorisencodedby

side thecell( Here, ical equationsforthesevariables: tion processbywritingdownthephenomenologicaldynam- tion enterstheequationsthrough thespecificformof of theirrespectivearguments. Themolecularleveldescrip- permease ( plained above, itcanbeobtainedfromexperiments. Itbe- function oftheinternalinducer concentration.Asex- f a lac ables arerelevantforthedescriptionoffunctioning The precedingdiscussionseemstoindicatethatthreevari- The model explicitly inthemodelisconcentrationof (

2

Z 1

, and

lacZ , Now, wearereadytomodelthedynamicsofinduc- ), whichisthequantitymeasuredinexperiments.

system.Thesearetheconcentrationsofnonfunctional

a

2

, and

,

I lacY

ex

f

lacI 3

istheexternalinducerconcentration;

.

Y , and ---- dI dt lacY a f , whichisimmediatelyupstreamoftheoperon.Binding ), offunctionalpermease( - 1 3 I ( areconstants;and ). Anothervariablethatweneedtoincorporate I = isthe ) istheproductionrateof permeases asa lacA [] f 2 ----- dZ arecotranscribedasapolycistronicmessage dt () ------dY ----- dY dt I dt - -galactoside permease, which is in charge of -galactoside permease,whichisinchargeof ex Modeling networkdynamics| - f = = – = g f f

3 b 1 f () () 1 1 I I Ya () I – – Y f – 1 f a , + 2 a 1 - Y f Y D 3 2 b Y f , and Z -galactose and , , 2 f . ), andofinducerin- I ex ging itsconformation. gratuitous inducers, – f 3 a nternal inducer arefunctions eous bindingof 3 -galactosidase Vilaretal. I , andarenot works inthe duced cells g D , -glucose b 1 , 473 b f

1 2

, ,

on December 30, 2005 2005 30, December on www.jcb.org from Downloaded The Journal of Cell Biology between simulationsandexperimentalresultscomefromtheearlystagesofinduction. and Weiner(1957).Thedashedlineisthesameasblackbutshiftedtoleft0.33generations.Itillustratesthat experimental valuesobtainedbyNovick same asinpanelc.Reddotsrepresent population level.Theblacklineisthe ments. Inductionfor500 guishable. (d)Simulationversusexperi- the cellaverageandpopulation 474 that thereareinducerconcentrations, on theirargument. out ofthecellandareassumedtodependhyperbolically count fortheinducertransportbypermeaseinand solutions forsuchvaluesof terms, thishappensbecausetheequationshavetwostable uninduced, iftheywereuninduced.Inmathematical cells remaininduced,iftheywerepreviouslyor for highconcentrations.Thefunctions constants) andincreasesmonotonicallyuntilitsaturates trations ( haves likeaquadraticpolynomialforlowinducerconcen- for cellsat500 scale. (c)Sameasinpanelabutnow TMG) attime0.Notethesemilogarithmic lations forsinglecellsat7 content obtainedfromcomputersimu- tative timecoursesof blue) colorlinescorrespondtorepresen- behavior. Thethin(yellow,green,and cell, cellaverage,andpopulation Figure 3. just bychance, acellbecomesinduced.The classicalap- into accountstochastic eventstoexplainwhy, atsomepoint, dard approachis,however,not enough.Oneneedstotake more intricatekinetics(Chung andStephanopoulos,1996). showed thattheobservedbehavior isalsocompatiblewith provide anysubstantialadditional insight.Theybasically the standardbiochemicalreactionkineticsapproach,didnot In fact,subsequent,muchmorecomplexmodels,basedon observed intheexperiments(CohnandHoribata,1959a,b). even simplerandexplainedtosomeextentthemainfeatures one, proposedalreadybyNovickandWeiner(1957),was tenance concentration. can apparentlyexplaintheexistenceofso-calledmain- hibits “hysteresis.”Thus,thestandardmodelingapproach the maintenanceconcentration(5 uninduced (bottom)cellstransferredto content obtainedforinduced(top)and tative (b) Maintenanceconcentration.Represen- grow slowerthanuninducedones. in thesimulationsthatinducedcells population results,wehaveconsidered dashed blacklineisthepopulation is theaverageover2,000cells.The model. Thethickcontinuousblackline from thestochasticbehaviorof from noninducedtoinducedstatesresult observed differencesinswitchingtimes -galactosidase contentareindistin- -galactosidase content.Toobtainthe With onlythesefourequationsonecanexplainthefact To fullyunderstandtheall-or-none phenomena,thestan- There aremanyvariationsofthissimplemodel.Thefirst The JournalofCellBiology time coursesofthe Modeling results. f 1 [ I ] M TMG.Inthiscase,

c 1 -galactosidase

-galactosidase M TMGatthe M TMG.The

(a)Single-

| c 2

I Volume 161,Number3,2003

I ex andthesystemtherebyex- M

c 3 I 2 , with f I 2 ex ( , forwhichthe I ex c ) and 1 , c 2 , and f 3 ( I ) ac- c 3 lespie, 1977). governed bysuchstochasticequationsonacomputer(Gil- ulate thedynamicalbehavioroffourrandomvariables tions intonumbersofmoleculespercell.Then,onecansim- dation, etc.)intoprobabilitytransitionratesandconcentra- done bytransformingthedifferentrates(production,degra- a stochasticcounterpartofthepreviousequations.Thisis to theinducedstate.Fortunately,itispossiblewritedown proach isunabletoexplaintheswitchfromuninduced unable tofully explaintheexperiments.In the simulations, cesses (Raoetal.,2002). fort tounderstandtheroleof stochasticityincellularpro- recently, however,hastherebeen arenewedwidespreadef- lished inthelate1950s(Montroll andShuler,1958).Only in the1940s(Delbrück,1940) andwasalreadywellestab- chastic approach.Thistypeofapproachstartedtobeapplied tion kineticscannotbeusedandhastoreplacedbyasto- thus completelydifferent.Asaconsequence,classicalreac- ior ofthesinglecellandbehavioraverageare average exhibitsasmoothbehavior.Inthiscase,thebehav- yellow, green,andbluelinesinFig.3a).Incontrast,thecell chemical reactionsandstronglyvariesfromcellto(e.g., happens isaresultoftheintrinsicstochasticnaturebio- duced totheinducedstate.Thetimeatwhichthistransition the single-celllevel,thereisafastswitchfromnonin- for cellsplacedundersuboptimalinductionconditions.At tosidase contentobtainedfromsuchcomputersimulations Fig. 3ashowsrepresentativetimecoursesofthe One shouldstress thateventhestochasticapproach isstill the maindifferences -galac-

www.jcb.org from Downloaded on December 30, 2005 2005 30, December on

The Journal of Cell Biology can providevaluable informationabout the mechanistic The typeofmodels andexperimentsthatwe havediscussed Evolutionary andphysiological levels sults. Therearesomedifferences:therisein proach wouldbesufficienttodescribethegivensystem. method totellwhetherornotthestandardmodelingap- main problem,however,isthattherenogeneralapriori hassle encounteredbeyondthestandardapproach.The here tousestandardkineticequationsandavoidmostofthe as canbeseeninFig.3c.Therefore,itshouldpossible cells, andthepopulationaverageallgiveverysimilarresults, case, thesingle-cellpicture,averageoverindependent lac count allthreelevelsofdescriptionisnotneededwhenthe lar conditions.Forinstance,anapproachtakingintoac- pends notonlyonthegivensystembutalsoparticu- whether ornotweshouldconsideralloftheseeffectsde- greatly increasesthecomplexityofmodeling.Ingeneral, come them.Consideringstochasticandpopulationeffects tions ofthestandardmodelingapproachandhowtoover- ence ofthemaintenanceconcentration. stochastic modelseemstobethuscompatiblewiththeexist- such switchingeventswouldbetooraretoobserved.The one statetoanother;forrealisticvaluesofprobabilityrates formed for1,000cells,werecordednosingleswitchingfrom tenance concentration.Indeed,inthesimulationsweper- were previouslyinducedoruninducedattheexpectedmain- other? Fig.3bshowsthesingle-cellbehaviorforcellsthat do notswitchatanappreciableratefromonestatetoan- Is therearangeofinducerconcentrationsforwhichthecells there reallyexistsamaintenanceconcentrationinthemodel. induced tothestateforcesusreconsiderwhether by thedashedlineinFig.3a. are thesimulationsinagreementwithexperiments,asshown population. Onlywhenthisistakenproperlyintoaccount rate. Therefore,wehavetoconsiderthedynamicsofcell that theinducedanduninducedcellsgrowatadifferent uninduced cells.Asexplainedbefore,thereasonforthisis which isanindicationofthecoexistenceinducedand concentrations seemsnottosaturateatthemaximumvalue, the productionof all thecellseventuallybecomeinduced.Inexperiments, is asmalldropin In addition,comingbacktoFig.3b,onecanseethatthere activity isfasterintheexperimentsthansimulations. with theexperimentalobservations. The ling, 1999)doesnotnecessarilyleadtobetteragreement of moremoleculardetailsintoamodel(CarrierandKeas- a morefundamentalaspectofthe model whetherthesedifferencesareamatterofdetailsor in NovickandWeiner,1957).Onecannotinferfromthe Fig. 3 tions. Thisdropisnotpresentintheexperiments(see transferred fromhightomaintenanceinducerconcentra- plest networks. clearly illustratesthecomplexity ofmodelingeventhesim- In Fig.3dwecompareexperimentalandsimulationre- So far,wehavepointedoutjustafewofthemanylimita- The factthatfluctuationsmakecellsswitchfromtheun- operonisinducedathighinducerconcentrations.Inthis -galactosidase forsuboptimalinducer -galactosidase contentwhencellsare lac system.Theaddition lac operonexample -galactosidase structure ofthe started tobeinvestigated. al., 1999a,b)!Thephysiologicalroleofthesepumpshasjust that surprisinglycanpumplactoseoutsideofthecell(Liuet have uncoveredanovelsetofsugareffluxpumpsin duction processitself.Forinstance,recentgeneticstudies In addition,therecanbeotherfactorsthataffectthein- puters. of quantitativebiologiststhan onthesheerpowerofcom- era,” willstillrelymoreongoodintuitionandskills genomic ductive modelingofbiological systems,eveninthe“post- lation, orevolutionary. molecular, cellular,physiological,population,intrapopu- ori anyofthemanycomplementarylevelsdescription: through evolution.Itisimportantnottodisregardapri- time. Theyformpartofaunitythathasbeenshaped eters intheequations. are changed.Thisallowstheuseofeffective(fitting)param- are keptunderconstantconditionsandonlyafewvariables in matchingtheexperimentalresultsisthatexperiments overlooked, aretakenintoaccount. fluctuations andpopulationeffects,whichareusually experiments ofNovickandWeiner(1957),providedthat equation modelisabletoexplainthemainresultsof than technical.Inthecasewehavediscussed,afour- remain useless.Theproblemisthusmoreconceptual less therelevantelementsareidentified,modelwill one caneasilyendupwithhugesetsofequations,butun- usually help.Ifmoremoleculardetailsareconsidered, equations toincludemoredetailsofinteractionsdoesnot variables, adequateapproximations,etc.Addingmore we havedescribed.Oneneedsfirsttoidentifytherelevant plied “automatically”eveninacaseassimpletheone Some ideasthatwewouldliketoemphasizeareasfollows. ples toshowtheintricacyofmodelingbiologicalnetworks. have triedtouseoneofthesimplestandbest-studiedexam- many publishedreviewsadvertisingthisaspect.Rather,we all thepotentialthatmodelingpossesses;therearenow modeling ofcellularprocesses.Wehavenottriedtoexpose illustrate thecurrentstate,applicability,andlimitationsof The exampleofthe Discussion Accepted: 17March 2003 Revised: 17March 2003 Submitted: 28January2003 ers, suchasIPTGorthiomethyl- metabolic productoflactose,ratherthangratuitousinduc- ditions (Savageau,1983),andtheinducerisallolactose,a place inthemammaliandigestivetractunderanaerobiccon- the caseof tion ithasacquiredthroughevolution(Savageau,1977).In then isitpossibletorelatethenetworkstructurefunc- needs toconsiderthemintheirnaturalenvironment.Only functioning andunderlyinglogicsofcellularnetworks,one In ouropinion,becauseoftheseandsimilarreasons,pro- Third, networksareisolatedneitherinspacenor Second, oneofthemainreasonsforsuccessmodels First, standardizedmodelingmethodscannotbeap- lac lac operonof operon.But,toreallyunderstandthe lac Modeling networkdynamics| operonswitchhasbeenusedhereto E. coli , inductionusuallytakes - D -galactoside (TMG).* Vilaretal. E. coli

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on December 30, 2005 2005 30, December on www.jcb.org from Downloaded The Journal of Cell Biology 476 Delbrück, M.1940.Statisticalfluctuationsinautocatalyticreactions. Chung, J.D.,andG.Stephanopoulos.1996.Onphysiologicalmultiplicity Cohn, M.,andK.Horibata.1959b. Analysis ofthedifferentiationandhet- Cohn, M.,andK.Horibata.1959a.Inhibitionbytheglucoseofinducedsyn- Carrier, T.A.,andJ.D.Keasling.1999.Investigatingautocatalyticgeneexpression References Eldar, A.,R.Dorfman,D.Weiss,H.Ashe,B.Z.Shilo,andN.Barkai.2002.Ro- Ellis, E.J.2001.Macromolecularcrowding:obviousbutunderappreciated. Gillespie, D.T.1977.Exactstochasticsimulationofcoupledchemicalreactions. Endy, D.,andR.Brent.2001.Modellingcellularbehaviour. Herzenberg, L.A.1958.Studiesontheinductionof Ito, K.,andY.Akiyama.1991.Invivoanalysisofintegrationmembrane-pro- Kaback, H.R.,M.Sahin-TothandA.B.Weinglass.2001.Thekamikazeapproach The JournalofCellBiology Chemical Physics. 1509–1521. population heterogeneityofbiologicalsystems. erogeneity withinapopulationof maintenance. thesis ofthe systems throughmechanisticmodeling. ing. bustness oftheBMPmorphogengradientin Journal ofPhysicalChemistry. 391–395. Biochem. Sci strain of teins in to membranetransport. -gal Nature. actoside synthesis. Escherichia coli Escherichia coli 419:304–308. . 26:597–604. J. Bacteriol. -galactoside-enzyme systemof 8:120–124. . . Mol. Microbiol. J. Bacteriol Nat. Rev.Mol.Cell.Biol. Biochim. Biophys. 78:601–612.

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A

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