Mathematical modelling of the agr operon in Staphylococcus aureus

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Mathematical modelling of the agr operon in
Staphylococcus aureus
Jabbari, S.; King, J.R.; Koerber, A.J.; Williams, P.
DOI:
10.1007/s00285-009-0291-6
Document Version
Peer reviewed version
Citation for published version (Harvard):
Jabbari, S, King, JR, Koerber, AJ & Williams, P 2010, 'Mathematical modelling of the agr operon in
Staphylococcus aureus', Journal of Mathematical Biology, vol. 61, no. 1, pp. 17-54.
https://doi.org/10.1007/s00285-009-0291-6
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JournalofMathematicalBiologymanuscriptNo.
(willbeinsertedbytheeditor)
Mathematical modelling of the agr operon in Staphylococcus
aureus
SaraJabbari JohnR.King AdrianJ.
· ·
Koerber PaulWilliams
·
Received:date/Accepted:date
Thefinalpublicationisavailableathttp://link.springer.com/article/10.1007/s00285-
009-0291-6
Abstract Staphylococcusaureusisapathogenicbacteriumthatutilisesquorumsens-
ing(QS),acell-to-cellsignallingmechanism,toenhanceitsabilitytocausedisease.
QSallowsthebacteriatomonitortheirsurroundingsandthesizeoftheirpopulation,
andS.aureusmakesuseofthistoregulatetheproductionofvirulencefactors.Here
we describe a mathematical model of this QS system and perform a detailed time-
dependentasymptoticanalysisinordertoclarifytherolesofthedistinctinteractions
thatmakeuptheQSprocess,demonstratingwhichreactionsdominatethebehaviour
ofthesystematvarioustimepoints.Wecouplethisanalysiswithnumericalsimula-
tionsandarethusabletogaininsightintohowalargepopulationofS.aureusshifts
fromarelativelyharmlessstatetoahighlyvirulentone,focussingontheneedforthe
threedistinctphaseswhichformthefeedbackloopofthisparticularQSsystem.
Keywords Staphylococcus aureus Quorum sensing Mathematical modelling
· · ·
Asymptoticanalysis Twocomponentsystems
·
MathematicsSubjectClassification(2000) 34E05 92B05 37N25
· ·
SaraJabbari(B),JohnR.King
DivisionofTheoreticalMechanics,SchoolofMathematicalSciences,UniversityofNottingham,Notting-
ham,NG72RD,UK
E-mail:[email protected]
AdrianJ.Koerber
SchoolofMathematicalSciences,UniversityofAdelaide,SA5005,Australia
PaulWilliams
CentreforBiomolecularSciences,UniversityofNottingham,Nottingham,NG72RD,UK
2 S.Jabbarietal.
1 Introduction
1.1 Staphylococcusaureus
Staphylococcusaureusisanopportunisticandinvasivepathogencapableofpromot-
ing disease in almost any tissue of the human body [24]. While it forms part of the
natural flora in a large number of people without causing any harm, S. aureus can
cause a broad spectrum of infections, ranging from minor ailments such as super-
ficial infections, boils and subcutaneous abscesses, to much more serious ones, for
example pneumonia, endocarditis (inflammation of the heart valves), osteomyelitis
(inflammationofboneandbonemarrow),sepsis(infectionofthebloodstream),sep-
ticarthritisandtoxicshocksyndrome[5,17,24,37].
One increasingly important factor motivating the study of this pathogenic bac-
teriumis itsability todevelop resistance toantibiotics previously used successfully
inthetreatmentofstaphylococcalinfections.Methicillin-resistantS.aureus(MRSA)
strainsaremulti-antibioticresistantandarealeadingcauseofhospital-acquiredin-
fections.Theyhavebecomeincreasinglydifficulttotreat,especiallyascertainstrains
havealsodevelopedresistancetovancomycin,oneofthelasttherapeuticresortsfor
fighting S. aureus infection [13,29]. In contrast to hospital-acquired MRSA strains,
community-acquired(CA-MRSA)strainsaremuchmorevirulentandcancausedis-
ease in healthy individuals [8]. Consequently S. aureus infections are the cause of
muchmorbidityandmortalityandarebecomingextremelydifficulttotreatusingcon-
ventionalantibiotictherapy.Newtargetsforthedevelopmentofanti-staphylococcal
agentsareurgentlyrequired.Inthiscontext,attenuatingthepathogenicity(theability
ofoneorganismtocausediseaseinanother)ofS.aureusbyinhibitingtheabilityof
thebacteriumtoproducevirulencefactors(i.e.productsthatspecificallyallowbacte-
riatocausedisease)offerssuchatarget.Inparticular,S.aureusemploysacell-to-cell
communicationsystemtermed‘quorumsensing’(QS)tocontrolvirulencegeneex-
pression.ByinhibitingQS,itshouldbepossibletopreventinfection.
1.2 Geneexpressionandquorumsensing
In both eukaryotic and prokaryotic cells, most genes are not expressed (i.e. tran-
scribed into mRNA and then translated into proteins) constitutively but are tightly
regulated.Forexample,geneswhichcodefortheproductionofvirulencefactorsare
expressedonlyunderparticularenvironmentalconditionssuchasthoseencountered
within host tissues. The transcription of individual genes is switched ‘on’ or ‘off’
bygeneregulatoryproteinswhichbindtospecificDNAsequences(namelythepro-
moter/operator region(s) found upstream of the gene) and either activate or inhibit
transcription.
S.aureusemploysQStoregulatetheexpressionofspecificgenes.QSdependson
the synthesis of small molecules (often referred to as pheromones or autoinducers)
thatarebothproducedanddetectedbythebacterialcells.Asthebacterialpopulation
densityincreases,sodoestheconcentrationofQSsignalmolecules(indeed,iftheQS
systemhaspositivefeedbackthesynthesisofthesemoleculesshouldalsoincrease).
MathematicalmodellingoftheagroperoninStaphylococcusaureus 3
This is often viewed as being associated with a critical threshold concentration of
bacterial cells being reached, thus activating a target sensor or response regulator,
sofacilitatingtheexpressionofQS-dependentgenes.However,theinterpretationof
such behaviour for our model will be somewhat different given that the population
sizeisassumedtobealreadyatorabovethisthresholdlevel.Instead,wewillseethat
the process can be understood as the density of the QS signal molecule becoming
sufficientlyhigh,ratherthanthepopulationitself.Thiscouldbepertinentwhen,for
example,apopulationofcellsentersanewenvironmentsothattheconcentrationof
theQSsignalmoleculeisinfluencedbydiffusivitywithinthatenvironment,inaddi-
tiontothesizeofthepopulation.Thisismorecloselylinkedtothemorerecentview
thatQScaninfactbeamechanismforbacteriatorecognisetheirsurroundingsrather
thansimplydetecttheirpopulationsize[30](andcouldthusinsteadbereferredtoas
‘diffusion sensing’), i.e. the concentration of signal molecule will depend upon the
mediuminwhichthebacteriaresideasmuchasthenumberofbacteriaproducingthe
molecule (for a mathematical discussion of this see [25]). A more appropriate term
forthiscell-signallingsystemisperhaps‘efficiencysensing’[12]whichencapsulates
bothoftheseideas.
It is evident therefore that QS can be used for multiple purposes. For example,
S.aureusemploysQStoregulatetheproductionofvirulencefactors[17,37].Inthe
contextofinfection,ithasbeensuggestedthat,onceinsidethebody,S.aureuscoordi-
natesthedeploymentofvirulencefactorswithbacterialcellpopulationsizethrough
QS so as to delay alerting the immune system until there are sufficient numbers of
bacteria present which are capable of overwhelming the host defences [32], corre-
spondingtotheconventionalinterpretationofQS.Ontheotherhand,S.aureususes
QSinendosomeescapewherebyasinglebacteriumcanbecomeinternalisedwithin
a host cell. QS signal molecule accumulation will occur, causing production of the
virulence factors allowing for endosome escape, so that here the signalling system
is really being used for ‘diffusion sensing’. Other uses of QS by diverse bacterial
speciesincludeDNAuptakeorexchange(transformationandconjugation),biolumi-
nescence,secondarymetabolite(e.g.antibiotic)production,swimmingandswarming
motilityandbiofilm(accumulationofbacteriaonasurfacewherethemicro-organism
isenmeshedina‘slime’matrix)development[3,4].
1.3 Theagroperon
InS.aureus,QSisperformedbytheagroperon(see,forexample,[10]or[26])which
consists of two transcription units (termed agrBDCA and RNAIII respectively) that
are driven by regulatory proteins which bind to promoters termed P2 and P3, per-
mitting RNA polymerase to transcribe the DNA into mRNA, prior to translation of
thisagrmRNAintoproteins.TheP2transcriptconsistsoffourgeneswhicharetran-
scribedandtranslatedtogivefourproteins(AgrB,AgrD,AgrCandAgrA),seeFigure
1.AgrBisatransmembraneproteinwhichprocessestheAgrDproteintogeneratea
QSsignalmoleculewhich,inthecaseofthestaphylococci,isasmallmodifiedpep-
tidecalledanAIP(autoinducingpeptide).TheAIPissecretedintotheexternalenvi-
ronmentwhereitisdetectedbyareceptorprotein(AgrC)presentonthebacterialcell
4 S.Jabbarietal.
AIP
Outsidethecell
AgrB AgrC Cellmembrane
)
AgrD
Insidethecell
mRNA AgrA
Virulence
agrBDCA P2 P3 RNAIII Infection
factors
Fig.1 Aschematicrepresentationoftheagrfeedbackloop.Thearrowswithafilledheadillustratethe
positivefeedbackloop.Thisloopisunusualinthateverycytoplasmic(aswellastheextracellular)com-
ponentoftheloopisup-regulated,ratherthansimplythesignal,whichisinprincipleallthatisrequired.
ThedottedboxenclosestheelementsoftheTCS.InFigure2wedescribeagenericTCS.
Signal Outsidethecell
Receptor Cellmembrane
)
Insidethecell
Response
regulator
Fig.2 AschematicrepresentationofagenericTCS.Thereceptorproteincandetectthepresenceofa
specificsignalandactivatetheresponseregulatorwhichlieswithinthecell.FortheagroperonAIPisthe
signal,AgrCthereceptorandAgrAtheresponseregulator.
surface.AIPbindingtoAgrCinducesaphosphorylation/dephosphorylationcascade
whichresultsintheactivationofAgrA,aDNA-bindingproteinwhichinteractswith
boththeP2andP3promoters.AgrAandAgrCare,respectively,theresponseregula-
torandsensorkinaseofatwo-componentsystem(TCS).Thistypeofsignalrecogni-
tionmechanismiscommonthroughoutthebacterialkingdom,seeFigure2.Although
the precise details of the AgrC and AgrA phosphorylation/dephosphorylation reac-
tionsoccurringfollowingAIPbindingtoAgrCarenotknown,inmostclassicalTCSs
thedetectionofthecognatesignalbythesensorkinaseresultsinautophosphorylation
followed by transfer of the phosphate to the response regulator protein. The phos-
phorylatedresponseregulatorusuallyhasahigheraffinityfortheDNAbindingsite
thantheunphosphorylatedformandisthereforeabletoalterregulationofthetarget
gene(s) [35]. This also seems to be the case for AgrA: see [19], where the purified
phosphorylated AgrA protein is shown to bind to both the P2 and the P3 promoter
withahigheraffinitythantheunphosphorylatedform.Inthispaperwethereforeas-
sumethatAgrAandAgrCformaclassicalTCSbutin[15]weexplorethepossibility
thatalessconventionalphosphorylationcascademayhold,wherebyeitherAgrAor
MathematicalmodellingoftheagroperoninStaphylococcusaureus 5
AgrC can be constitutively phosphorylated (it is suggested in [26] that this may be
possible).
On binding to the P2 promoter, AgrA upregulates the transcription of the agr
mRNA which will be translated into each of the four Agr proteins. Thus the agr
systemissubjecttopositivefeedbackinthatAIPsaregeneratedwhichinturndrive
the production of further AIP synthesis. This is usual in QS systems as it allows a
celltoswitchquicklybetweentwostates.Itisnoteworthy,however,thatalltheother
components of the feedback system are also subject to upregulation and our model
allowsustoshedsomelightontowhythismaybethecase.
The agr-P3 transcript gives rise to an un-translated regulatory RNA termed
RNAIIIandalsotoaproteintoxin,d -haemolysin.RNAIIIistheintracellulareffec-
toroftheagr systemwhichactsbyupregulatingtranscriptionofmanyextracellular
proteingeneswhiledownregulatingthecellwallcolonisationfactorgenes.Thusthe
cellcanusetheAIPconcentrationtocoordinatevirulencegeneexpressionwithcell
density. This means that itshould be possible to prevent S. aureus infections by in-
hibiting AIP-dependent QS by blocking either AIP synthesis or AIP action or by
destroyingtheAIPasitaccumulatesextracellularly(see[3,4,28]).Interestingly,the
studyofagrsystemsindifferentS.aureusstrainshasalreadyhighlightedthepoten-
tialforQSinhibitionthroughblockade ofAgrCactivation.S.aureusstrainscanbe
divided into four groups (I to IV) depending on the structure of the AIP produced;
group I strains are activated by the group I AIP but are inhibited by the AIPs made
byS.aureusstrainsbelongingtogroupsII,IIIorIV[22].Indeed,[22]demonstrates
that, in mice, S. aureus skin abscess infections caused by a strain producing AIP-I
can be prevented by coadministering a group II AIP. Since the AIPs are amenable
tolaboratory synthesis itispossible todesign inhibitors based on the AIPstructure
whichbindtoAgrCandcaninhibitallS.aureusagrgroups[23].Suchpeptidesneed
tobeabletobindtotheAIP-bindingsiteonAgrCwithoutactivatingthekinaseand
soblockbindingofthenative(andhenceactivating)AIP.Thiswouldensurethatthe
bacterialcellremainsintheQSdown-regulatedstate.
1.4 MathematicalmodelsofQS
Most QS models produced so far have been mathematical models of the slightly
simpler Gram-negative lux-system (here the signal molecule is produced within the
cellandisfreelydiffusibleacrossthecellmembrane,sothatonlytwoelementsare
required to complete the system: a signal synthase and a signal receptor) or its ho-
mologues,seeforexample[2,6,7,16,34].Toourknowledge,onlytwofocusonQS
in S. aureus (which is Gram-positive): Koerber et al. [20] model endosome escape,
asdescribedin 1.2,atthecellularlevel,whileGustafssonetal.[11]investigatethe
§
TCS of the agr operon (which is taken to be of a classical form) and the influence
of the regulatory protein SarA on the basal transcription of the operon. The latter
is more akin to the current paper as the modelling is performed at the sub-cellular
level.Howeverin[11]theTCSistreatedinisolationinthesensethattheAIPcon-
centration is taken to be a parameter of the model, so that the full feedback circuit
containedwithintheQSloopisnotconsidered;bycontrast,weincludethecomplete
6 S.Jabbarietal.
circuit shown in Figure 1, calculating the AIP level as part of the solution. By in-
corporatingthefullcircuit,weaimtoshedlightinparticularontheimplicationsof
theagr operonleadingtomultiplenetworkelementsbeingup-regulated.Alongside
numericalsimulationswewillperformanasymptoticanalysisofthetime-dependent
modelinordertocharacteriseitsbehaviour.Thisanalysiswillbothdemonstratehow
a population of S. aureus shifts from a relatively harmless state to a highly virulent
oneandprovidesimplermodelswhichcanbeusedtoextendthisstudy.Forinhibitor
therapytohaveafutureincombattingstaphylococcalinfections,itiscrucialthatwe
gainafullunderstandingofhowtheagroperonworksandthismodeliswellsuited
tostudyingtheeffectsofinhibitors;thisgeneralisationwillbeaddressedelsewhere.
2 Formulation
2.1 Dimensionalmodel
Wefollow[14]inusingamodellingapproachsimilartothatemployedbyDockery
andKeener[6]intheirmodelofvirulence-relatedQSinP.aeruginosa,wherebywe
shallformulateasystemofordinarydifferentialequationsrepresentingtheintracel-
lularcomponentsofthefullagr operon.AkeyvariableinourmodelisP(t),which
representstheproportionofcellswithaboundagrpromoter.Roughlyspeaking,this
willbeequivalenttotheproportionofagrup-regulatedcellsandwewillhenceforth
refertoPinthisway,makingtheproportionofdown-regulatedcells1 P(t).
−
Inordertobuildthemodelweadoptthefollowingassumptions.
– The bacteria are in a well-mixed environment, so spatial dependencies can be
ignored,andthepopulationsizeisconstant(i.e.birthratematchesdeathand/or
removal rate) and large enough to make a continuum model appropriate. These
are conditions which are most similar to those of a chemostat, thus making it
plausiblethat,toanextent,ourresultscouldbetestedexperimentallyforpossible
validationandcalculationofparametervalues.
– agrmRNAisproducedatsomebasalrateinadown-regulatedpopulationandits
averagerateoftranscriptionincreaseslinearlywiththeaveragelevelofupregu-
lationinthepopulation,P(t).
– SinceeachmoleculeofthismRNAcontainsalltheinformationrequiredforthe
translation of all four Agr proteins, the same numbers of each of these are pro-
duced, i.e. we assume that on each pass a ribosome translates the entire strand
of mRNA. We also assume there to be a plentiful supply of ribosomes within
thecellssincetheyarerequiredfortranslationofallproteins,notonlythosein-
volvedinQS.Wedonotthereforeneedtotracktheconcentrationofribosomes
and can take the rates of translation of each of the proteins to be the same, and
proportionaltotheconcentrationofagrmRNA.
– The levels of proteins and mRNA inside the cells are limited by natural degra-
dationandthroughdilution,proteinshavingarelativelylowrateofdegradation.
Bacterial cells grow until they undergo binary fission to produce two daughter
cellsofequalsize.Eachdaughterreceivesacopyofthechromosomeandsuffi-
cientnumbersofallthedifferentchemicalsrequiredforsurvival.Weassumethat
MathematicalmodellingoftheagroperoninStaphylococcusaureus 7
Fig.3 Bacterialcellsreproducethroughbinaryfission,producingtwodaughtercells.Thecontentsofthe
parentcellaredividedbetweenthedaughtercells.Heredeathratematchesbirthrate,asinourmodel,so
theoriginalcontentsofthecells(showedschematicallyasfilledcircles)arereducedthroughdilution.
all the contents of the parent cell are divided equally between the two daughter
cells. Through this dilution process the number of proteins in each cell would,
intheabsenceofproteinproduction,bereducedastimegoeson(seeFigure3).
We assume that the contents of a dead cell are degraded and have no effect on
the remaining cells. The dilution rate, r, can be calculated as ln2/t wheret is
d d
thetimeittakesforoneS.aureusgenerationtoundergobinaryfission.Wedefine
d =l +r(seeTable1)forallintracellularconcentrationsX.
X X
– HousekeepingphosphatasesareabletodephosphorylateAgrAataratem .
– After membrane-anchored AgrD has been (post-translationally) modified into
AIP (by AgrB) we can ignore what remains of the AgrD protein as it has no
further effect on the QS loop and will eventually be lost through dilution or
metabolism.
– Receptor-boundAIPcanunbindspontaneously,atrateg .
– WhenanAIPbindstoareceptor,i.e.toAgrC,thelatterautophosphorylates.To
simplifythesystemwewillassumethatthisprocessissufficientlyfastthatitin
effecthappensassoonastheAIPbindstothereceptor.Whenthereceptortrans-
fersitsphosphatetotheAgrAprotein(ataratef )itisfreetoautophosphorylate
again, and the phosphorylated AgrA is able to bind to the promoter site of the
DNAandincreasemRNAproduction.
– ThereisaplentifulsupplyofSarAproteinssothat,unlikein[11],wedonottake
intoconsiderationtheirspecificeffectuponAgrAbindingtothepromotersites.
The resulting equations are, for conciseness, shown only in Figure 4; see Tables 1
and 2 for definitions of the parameters and variables. Notice that taking the equa-
tiongoverningP(t)tobequasi-steadywouldgiveusMichaelis-Mentenkineticsfor
mRNA transcription, as is often done in modelling of this kind. However, we shall
see that P(t) is among the slower evolving variables and we accordingly retain the
fulldynamicbalance.
We take the initial conditions of the system to be the steady state which would
arise were no AIP produced (k=0), i.e. a totally down-regulated state, so QS has
no effect on the cells and all protein levels are controlled by translation, transcrip-
tion,degradationanddilution.Thiscouldbereproducibleinachemostatexperiment
through manipulation of AIP synthesis, i.e. by bringing the agr operon under the
control of an inducible promoter. As a reflection of bacteria in nature it might be
interpretedasalargepopulationofcellsinwhichtheagrsystemissuppressedbyan-
8 S.Jabbarietal.
Table1 Definitionsoftheparameters.
Parameter Rateconstantfor Units
m basalproductionofmRNA moleculescells 1s 1
− −
v mRNAtranscription moleculescells 1s 1
− −
k proteintranslation s 1
−
a T,a R AgrBandAgrCtakenupintocellmembrane s−1
a S AgrDanchorstocellmembrane s−1
l X naturaldegradationofvariableX(t) s−1
r dilutionthroughcelldivision s 1
−
dX degradationanddilution(dX=l X+r) s−1
k AIPproductionfromAgrD,mediatedbyAgrB molecules 1cm3s 1
− −
b bindingofAIPtoAgrC molecules 1cm3s 1
− −
g separationofAIPfromAgrC s 1
−
f activationofAgrAbyAIP-boundAgrC molecules 1cm3s 1
− −
m dephosphorylationofAgrAbyphosphatases s 1
−
b bindingofthepromotersite molecules 1cellss 1
− −
u unbindingofthepromotersite s 1
−
N totalnumberofbacteriaperunitvolume cellscm 3
−
Table2 Definitionsofthevariables.
Variable Concentrationof Units
M mRNA moleculescm 3
−
A,B,C,D cytoplasmicAgrA,AgrB,AgrC,AgrD moleculescm−3
T,R transmembraneAgrB,AgrC moleculescm−3
S anchoredAgrD moleculescm 3
−
a freeAIP moleculescm 3
−
R AIP-boundreceptor moleculescm 3
∗ −
AP phosphorylatedAgrA moleculescm−3
P proportionofcellsthatisup-regulated -
otherofthemanygeneregulationnetworksinvolvedincellularregulation,beforethis
lattermechanismisswitchedoff,throughsomeenvironmentalormetabolicchange,
toallowagrupregulationtobegin.Usingsuchinitialconditionsallowsustomonitor
howalargepopulationofbacteriacanshiftintoanup-regulatedstateasAIPlevels
increase.Hencewetakethefollowinginitialconditions(whicharethesteadystates
oftheequationsinFigure4withk=0)
a(0)=R (0)=A (0)=P(0)=0,
∗ P
Nm Nk m Nk m
M(0)= , A(0)= , B(0)= ,
d d d d (a +d )
M M A M T B
Nk m Nk m Na k m (1)
T
C(0)= , D(0)= , T(0)= ,
d (a +d ) d (a +d ) d d (a +d )
M R C M S D M T T B
Na k m Na k m
R S
R(0)= ,S(0)= .
d d (a +d ) d d (a +d )
M R R C M S S D
MathematicalmodellingoftheagroperoninStaphylococcusaureus 9
Outsidethecell
ddat =kTS−b Ra+gR∗−laa
Cellmembrane
ddSt =a SD−dSS−kTS ddTt =a TB−dTT ddRt∗ =b Ra−(g +dR∗)R∗ ddRt =a RC−b Ra+gR∗−dRR
Intshideecell ddAtP =f AR∗−(m +dAP)AP
ddBt =k M−(a T+dB)B ddAt =k M−f AR∗+m AP−dAA
dP b
ddDt =k M−(a S+dD)D dt =NAP(1−P)−uP ddCt =k M−(a R+dC)C
dM
dt =Nm+NvP−dMM
Fig.4 AschematicrepresentationofthecompletemodelfortheagrcircuitwithaclassicalTCS.See
Tables1and2fordefinitionsofparametersandvariables.Thedimensionlessversionofthismodelis
showninFigure5.
2.2 Nondimensionalmodel
Wenondimensionalisetherelevant1 variablesusing(1),i.e.weset
d d d d (a +d ) d d (a +d )
M M A M Y X M Y Y X
M = M, A = A, X = X, Y = Y,
′ Nm ′ Nk m ′ Nk m ′ Na k m
Y
(2)
forX =B,C,DandY =T,R,Srespectively.Theremainingscalingsare
bf bNa k 2m2 f bk m b
a′= d M5d Ad R(aRR+dC)a, R∗′= d M3d AR∗, AP′= Nd MAP, t =d Mt, (3)
Palreadybeingdimensionless.Timeisthusscaledwithd ,therateofmRNAdegra-
M
dationwhiletheothersarechosentosimplifythecorrespondingequationsasmuchas
possible,i.e.theyarechosentosetthecoefficientsofbasalmRNAtranscription,AIP-
receptorbinding,AgrAactivationandphosphorylatedAgrAbindingtothepromoter
siteincertainequationstounity.Thefollowingdimensionalparametersemerge:
d
l X′ = d X for X =A,T,R,S,R∗,AP, (4)
M
a +d
a ′ = Y X for (X,Y)=(B,T),(C,R)or(D,S), (5)
X,Y d
M
1 NamelymRNAandallunphosphorylatedproteins,sincethesewouldbetheonlynon-zerovariables
inatotallydown-regulatedcell.