Qualitati veSimulation of GeneticRegulatory Networks: Method and Application�

Hidde deJong,�

Michel Page,��� �

CelineHernandez,�

and JohannesGeiselmann�

�INRIA Rhone-Alpes,655avenuedel’Europe,Montbonnot,38334SaintIsmierCEDEX,France�

ESA,UniversitePierreMendesFrance,Grenoble�PEGM,UniversiteJosephFourier, Grenoble,France

Hidde.de-Jong,Celine.Hernandez,Michel.Page @inrialpes.fr, Johannes.Geiselmann@ujf-grenoble.fr

Abstract

Computermodelingandsimulationareindispensablefor un-derstandingthe functioning of an organismon a molecularlevel. We presentanimplementedmethodfor thequalitativesimulationof largeandcomplex geneticregulatorynetworks.The methodallows a broadrangeof regulatory interactionsbetweengenesto berepresentedandhasbeenappliedto theanalysisof a realnetwork of biological interest,thenetworkcontrolling the inititation of sporulationin the bacteriumB.subtilis.

Intr oductionIt is now commonlyacceptedin biology thatmostinterest-ing propertiesof anorganismemergefrom the interactionsamongits genes,proteins,metabolites,andothermolecules.This implies that, in orderto understandthe functioningofan organism,the networks of interactionsinvolved in generegulation,metabolism,signaltransduction,andothercellu-lar andintercellularprocessesneedto beelucidated.

A geneticregulatorynetwork consistsof a set of genesand their mutual regulatory interactions. The interactionsarisefrom thefactthatgenescodefor proteinsthatmaycon-trol theexpressionof othergenes,for instanceby activatingor inhibiting DNA transcription(Lewin 1999). The studyof geneticregulatorynetworks hasreceived a major impe-tusfrom therecentdevelopmentof experimentaltechniquespermittingthe spatiotemporalexpressionlevelsof genestoberapidly measuredin a massively parallelway (Brown &Botstein1999).However, in additionto experimentaltools,computertools for the modeling and simulation of generegulation processeswill be indispensable(de Jong2000;McAdams& Arkin 1998;Smolen,Baxter, & Byrne2000).As mostgeneticregulatorysystemsof interestinvolvemanygenesconnectedthroughinterlockingpositive andnegativefeedbackloops,anintuitiveunderstandingof theirdynamicsis hardto obtain.

Currently, only a few regulatory networks are well-understoodon themolecularlevel. In addition,quantitativeinformationonkineticparametersandmolecularconcentra-tionsareseldomavailable.Thishasstimulatedaninterestin�

A shorterversionof this paperappearsin the Proceedingsofthe 17th InternationalJoint Conferenceon Artificial Intelligence,IJCAI-01,Seattle,Washington,USA, 4-10August,2001.

modelingandsimulationtechniquesdevelopedwithin qual-itative reasoning(QR) (Heidtke & Schulze-Kremer1998;Trelease,Henderson,& Park 1999). A majorproblemwiththeseapproaches,basedonwell-known methodslikeQSIM(Kuipers1994)andQPT(Forbus1984),is their lack of up-scalability. Following approachesin mathematicalbiology,de JongandPage(2000)have proposeda qualitative sim-ulationmethodcapableof handlinglargeandcomplex net-works.

The aim of this paperis to generalizethe latter methodandto demonstrateits applicability to real networks of bi-ological interest. The generalizationof the methodallowsa broaderrangeof regulatoryinteractionsbetweengenestobeexpressed.This enablesmorecomplex systemsto bean-alyzed,suchas the network of interactionscontrolling theinititation of sporulationin the bacteriumBacillus subtilis.We have simulatedthe sporulationnetwork usinga modelconstructedfrompublishedreportsof experiments.Thesim-ulationsrevealthatanadditionalinteraction,proposedin theliteraturebeforebut not yet experimentallyidentified,maybeinvolved.

In thenext section,we will discusstheclassof equationsbeingusedto modelgeneticregulatorynetworks. Thethirdsectiondescribesthe qualitative simulationalgorithm, fo-cusingon therepresentationof thequalitativestateof a reg-ulatorysystemandthedeterminationof statetransitionsbythe simulationalgorithm. The subsequentsectionspresenttheresultsof theanalysisof thesporulationnetwork aswellasadiscussionof themethodin thecontext of relatedwork.

Modeling geneticregulatory networksApproximationsof regulatory interactions

In orderto modelageneticregulatorynetwork, wefirst haveto describetheregulatoryinteractionsin anempiricallyvalidandmathematicallyrigorousway. Considera DNA-bindingproteinencodedby gene� , activatingtheexpressionof atar-getgene . Therateof transcriptionof asa functionof theconcentration��� of theregulatoryproteinfollowsasigmoidcurve (Yagil & Yagil 1971). Below a thresholdconcentra-tion � ����� the geneis hardly expressedat all, whereasabovethis thresholdits expressionrapidlysaturates.

Sigmoidcurvesarealsofound in the caseof morecom-plex regulatory mechanisms.Considerthe proteinsJ and

K that form a dimer repressingthe transcriptionof gene (Fig. 1(b)). Analysisof a kinetic modelof this regulatorymechanismrevealsthat the rateof expressionof dependsin a sigmoidalfashionon thetotal concentrations� � and ���of J andK, respectively. That is, both J andK needto beavailableabovetheir thresholdconcentrationsfor to bere-pressed.

+

genei

proteinJ

(a)

-

proteincomplex J� Kgenei

proteinJ

proteinK

(b)

Figure1: (a) Activation of a target gene by a regulatoryproteinJ. (b) Inhibition of by aproteincomplex J� K.

In thecaseof steepsigmoids,thecombinedeffect of reg-ulatoryproteinsongeneexpressioncanbeapproximatedbymeansof Booleanfunctions (Kauffman 1993; Thomas&d’Ari 1990). Here we will rewrite a Booleanfunction intermsof stepfunctionsandarithmeticsumsandmultiplica-tions, following the procedureof Plahteet al. (1998). Fortheactivatorproteinin Fig. 1(a),we thusobtaina regulationfunction ����� ���! #"%$'& ��� �)( � �'� , where$'& � � is astepfunction(Fig. 5) and "*�+� a rateparameter. Similarly, theeffect ofa regulatoryprotein repressinggene canbe describedby�,�-� ���. /"0$21 �-� ��( � �3� , with $21 �-� ��( � �'�4 6587#$'& �-� �)( � �3� .Thedimerrepressorexamplein Fig. 1(b) leadsto thefunc-tion �:9;�-�<� ( � � �= >" 9�� 5?7@$3& ����� ( �A� �<$3& ��� � ( � � �B� .

Althoughtheabove discussionhasfocusedon therepre-sentationof interactionsregulatingthesynthesisof proteins,it alsoappliesto the degradationof proteins. Sigmoidre-lations are observed in the latter caseas well, so that thelogicalapproximationsarevalid. In orderto formally distin-guishproteindegradationratesfrom proteinsynthesisrates,we will denotetheformerby C�9 insteadof " 9 .StateequationsThe dynamicsof geneticregulatorynetworks canbe mod-eledby asystemof differentialequationssuggestedbyMestlet al. (1995),extendingearlierproposalsby Glass& Kauff-man(1973)andThomasetal. (1990).

D��9 >E 9F�HG �I7KJ 9B��G � ��9 ( ��9=L �<(!5NM MPO=( (1)

where G RQ � � (AS:SAST( �VU�WYX is a vectorof cellularproteincon-centrations.Thestateequations(1) definetherateof changeof the concentration�V9 asthe differenceof the rateof syn-thesisE 9B��G � andtherateof degradation7ZJ 9B��G � ��9 of thepro-tein. Exogenousvariablescanbedefinedby setting

D�V9 #� .Thefunction E�9=[]\ U ^�_a` \ ^b_ is definedas

E�9 ��G �= +cdfe)g � 9 d ��G �h�i�<( (2)

where �39 d � � is a regulationfunctionand j a possiblyemptysetof indicesof regulationfunctions. The function J 9F� � is

definedanalogouslyto (2), exceptthat for reasonsthatwillbecomeclearbelow, we demandthat J 9;��G � is strictly posi-tive. Notice that for the above definitionsof E29 � � and Jk9 � � ,the stateequations(1) are piecewise-linear. Equationsofthis form, and their logical abstractions,have beenwell-studiedin mathematicalbiology(e.g.,(Lewis & Glass1991;Mestl, Plahte,& Omholt 1995; Snoussi1989; Thomas&d’Ari 1990)).

Eqs.(1) and(2) generalizeupontheformalismemployedin (de Jong& Page2000) in two respects.First, the reg-ulation functions �:9 d � � may be the mathematicalequivalentof any Booleanfunction,whereasin theearlierpaperit wasrestrictedto logical functionscomposedof Booleanprod-ucts.Second,theregulationof proteindegradationcannowbemodeled,whereasbeforethedegradationratewassettoJk9 ��G �= C 9 . Theseextensionsallow thestructureof complexregulatorynetworksto beformalizedin aconvenientway.

In Fig. 3 the stateequationscorrespondingto two of thegenesin the sporulationnetwork of Fig. 2 areshown. Thedifferentialequationin (a)statesthatspo0Eis transcribedata rate "VlBm from a npo -promoterwhenits repressorAbrB isbelow its thresholdconcentration�2qsrut (i.e., $21 �-��qAr ( �2qsrut �= 5 ). In addition, for transcriptionto commence,the sigmafactor npo encodedby sigA needsto be availableat a con-centrationabovethethreshold� l q t (i.e., $'& ��� l q ( � l q t �= v5 ).Spo0Edegradesat a rateproportionalto its own concentra-tion.

w�x y-z|{k}Y~� w'x y����)}Y~}

w'x y-z��)}Y~� � }��:�3}

w�x y���{k}Y~} w�x y���~}

� � �:� �w x y�z�~�

(a)

�

w'x y����)}Y~}w'x y���{k}Y~}

�

w�x y � ~}(b)

Figure 4: (a) Two-dimensionalphasespacedivided intoregulatory domainsby the thresholdplanes. The shadedregulatorydomaindefinedby ��� �st 1 �;�� � � � � ��� �st �� and� � ��� 1 �B�� � � � � � � ��� �� hasa target equilibrium in an ad-jacentregulatorydomain. (b) The samephasespacewithswitchingzonesaroundthethresholdplanes.

Thr esholdand equilibrium inequalitiesIn general,a proteinencodedby a genewill be involvedindifferent interactionsat different thresholdconcentrations.Although exact numericalvalueswill not usuallybe avail-able,wecanorderthe � 9 thresholdconcentrationsof gene ,whichgivesthethresholdinequalities

� � � � �;�9 � SAS:S � � ���A� �9 ���.�A� 9 S (3)

phosphorelay

-

-

-

+

-

+

-

+

--

+

+

-

-

-

-

Legenda

SinR

codingregion

promoter

gene

protein

Spo0A

Spo0A� P

Spo0A� P

abrB

(spo0H)sigH

[2,3,6,7,23]

[1,2,4,5]

[13]

[15,16]

[10]

[15]

[10]

kinA

SinR

sinI

hpr (scoR)

spo0A

�3� �3�

KinA

sinR

spoIIA...

�3�

� ��3�

AbrB

� �

�3�[3,11]

Spo0E

AbrB

Hpr

SinI

spo0E

SinR/SinI

[3]

[20,21]

[3,7]

�3�� �

�3��3���3�

� �

[8]

[12] [13]

[3,11]

[3,11]

[9]

[12]

[9]

[14]

[17]

[18] [19]

[22]

[10]

Figure2: Geneticregulatorynetwork underlyingtheinititation of sporulationin B. subtilis. For every gene,thecodingregionandthepromotersareshown. Promotersaredistinguishedby thespecific n factordirectingDNA transcription.Theregulatoryactionof aproteintendingto activate(inhibit) expressionis indicatedby a ‘+’ (‘-’). As anotationalconvention,namesof genesareprintedin italic andnamesof proteinsstartwith acapital.Thenumbersin thefigurearereferencesto theliteraturedescribingthe structureof genesandthe regulatory interactions:[1] Jaackset al. (1989),J. Bacteriol., 171(8):4121;[2] Predichet al.(1992),J.Bacteriol., 174(9):2771;[3] FuyitaandSadaie(1998),J.Biochem., 124:98;[4] LeDeauxet al. (1995),J.Bacteriol.,177(3):861;[5] Jianget al. (2000),Mol. Microbiol., 38(3):535;[6] Chibazakuraet al. (1991),J.Bacteriol., 173(8):2625;[7]Strauchet al. (1992),Biochimie, 74:619;[8] Mandic-Mulic et al. (1995),J. Bacteriol., 177(16):4619;[9] Bai andMandic-Mulic (1993),GenesDev., 7:139;[10] Strauchet al. (1989),EMBO J., 8(5):1615;[11] Strauch(1993),In: Sonensheinet al.,BacillusSubtilisandotherGram-Positive Bacteria, ASM Press,757; [12] Gauret al. (1988),J. Bacteriol., 170(3):1046;[13]StrauchandHoch(1993),Mol. Microbiol., 7(3):337;[14] Kallio et al. (1991),J.Biol. Chem., 266(20):13411;[15] Weir et al.(1991),J.Bacteriol., 173(2):521;[16] Healyetal. (1991),Mol. Microbiol., 5(2):477;[17] Trachetal. (1991),Res.Microbiol.,142:815;[18] Wu et al. (1991),Gene, 101(1):113;[19] Mandic-Mulic et al. (1992),J.Bacteriol., 174(11):3561;[20] Burbulyset al. (1991),Cell, 64:545;[21] Hoch (1993),Annu. Rev. Microbiol., 47:441;[22] Perego andHoch (1988),J. Bacteriol.,170(6):2560.[23] Yamashitaetal. (1989),J.Gen.Microbiol., 135:1335.

Stateequationfor Spo0E:��<�-�I�h�� H¡V¢'£¥¤Y��¦u§©¨uªs«©¬ t�­ ¢s®¯¤Y� « ¨uª « t ­b°²±  H¡;�< �¡A¨Thresholdinequalities:³]´ ªA H¡ t ´ ªA H¡ � ´ ª: �¡¶µ ´@·N¸©¹  H¡Equilibrium inequalities:

³0´ �  H¡�º ±  H¡ ´ ª  H¡ t (a)

Stateequationfor AbrB:���«�¬I�Z�)«�¬�¢3£I¤Y��«�¬»¨Fª�«�¬ � ­ ¢s®=¤Y� « ¨Fª « t ­0¼¤¶½ ° ¢s®¯¤Y�<  « ¨FªA  « t ­ ¢s®¯¤Y�<¾;«2¨FªA¾;« t�­ ¢'£¥¤Y�< H¡s¨FªA H¡¶µ ­u­�°¿± «�¬)�,«�¬Thresholdinequalities:³]´ ªs«©¬ t ´ ªs«©¬ � ´À·Á¸�¹ «�¬Equilibrium inequalities: ª�«�¬ � ´ ��«©¬ º ± «�¬ ´@·Á¸�¹ «�¬ (b)

Figure3: Stateequations,thresholdinequalities,andequilibriuminequalitiesfor thegenes(a) spo0Eand(b) abrB in Fig. 2.Thesubscriptsin theequationsreferto thesporulationgeneskinA ( Â�à ), spo0E( $:Ä ), spo0A( $ à ), sigA ( à ), sigH ( Å ), andabrB( Ã,Æ ).

Theparameter�4�:� 9 denotesa maximumconcentrationfortheproteinÇ denotedby .

For thesporulationgeneabrBtwo thresholdsaredefined,� qsr t and � qsr¶� . AbrB hastwo thresholdconcentrations:� qsr tand �2qsr � . The first thresholdcorrespondsto the repressionof spo0E, sigH, andotherearlysporulationgenesby AbrB.The secondthresholdcorrespondsto the autoregulationofabrBduringvegetativegrowth,whenAbrB levelsareattheirhighest.This motivatestheorderingof theAbrB thresholdsin Fig. 3(b): � qsr t � � qsr¶� .

The O.7È5 -dimensionalthresholdhyperplanes� 9¥ ��� � � �9 ,5ÉM Â)9 M � 9 divide thephasespacebox into regions,calledregulatorydomains(Mestl,Plahte,& Omholt1995)(Fig.4).Within eachregulatory domain, the step function expres-sionsin (2) canbe evaluated,which reducesE�9 � � and Jk9 � �to sumsof rateconstants.That is, E 9F� � simplifies to someÊ 98Ë�Ì@90Í E 9B��G �.Î¥ÏÐM G MRÑ>Ò Gh , and J 9B� � to someÓ 9 ËÕÔ 9 Í Jk9 ��G �]Î�ÏÖM G M�Ñ�Ò Gh . ThesetsÌ 9 and Ô 9collect the differentsynthesisanddegradationratesof theproteinin differentdomainsof thephasespace.

It can be easily shown that all trajectoriesin a regu-latory domain tend towards a single, stable steadystateG /×ZØ2Ù , the target equilibrium, lying at the intersectionof the hyperplanes� 94 Ê 9uØ Ó 9 (Glass& Kauffman 1973;Mestl,Plahte,& Omholt1995;Thomas& d’Ari 1990).Thetarget equilibrium level Ê 9 Ø Ó 9 of the proteinconcentration� 9 givesan indicationof the strengthof geneexpressionintheregulatorydomain.

As in the caseof thresholdparameters,exact numericalvaluesfor the rate constantswill not usually be available.However, it is possibleorderthepossibletargetequilibriumlevelsof � 9 in differentregionsof the phasespacewith re-spectto the thresholdconcentrations.1 The resultingequi-librium inequalitiesdefinethe strengthof geneexpressionin a regulatorydomainin a qualitative way, on thescaleoforderedthresholdconcentrations.More precisely, for everyÊ 9 ËPÌ 9 , Ó 9 ËPÔ 9 , we specifysome Ú 9 , 5¿M Ú 9 � � 9 , suchthat

� � d � �9 � Ê 9 Ø Ó 9 � � � d � & �B�9 ( (4)

with specialcases� � Ê 9 Ø Ó 9 � � � �B�9 and � ��� � �9 � Ê 9 Ø Ó 9 ��.�A� 9 .Inspectionof the state equationsof AbrB shows thatÌ@qsr �Û(»"¥Ü�Ý and Ôaqsr C Ü�Ý . The equilibrium level" qsr Ø C qAr is placedabove the highestAbrB threshold,since

otherwisethe concentrationof AbrB would never be ableto reachor maintaina level at which negative autoregula-tion takesplace. This leadsto the equilibrium inequalities� qsr¶� � " qsr Ø C qsr �>�4�:� qAr .

Thethresholdandequilibriuminequalitiesdeterminethesign of

D�V9 in a regulatorydomain,andhencethe local dy-namicsof the system. In fact, given a regulatory domaindefinedin dimension by � � � � �9 � �V9 � � � � � & �;�9 , it canbe

1The equilibrium inequalitieshave also beencalled nullclineinequalities(deJong& Page2000),becausetheequilibriumlevelscorrespondto nullcline hyperplanesin thephasespace.

shown that,if Ê 9¶Ø Ó 9 � ��� � � �9 , thenD� 9 � � everywhereinside

the regulatorydomain. Similarly, if Ê 9 Ø Ó 9 � � � � � & �;�9 , thenD�V9 �Ð� . On theotherhand,if � � � � �9 � Ê 9 Ø Ó 9 � � � � � & �;�9 , thenthesignof

D��9 in theregulatorydomainis notunique,writtenasD�V9NÞ � . In particular,

D�V9 � � on onesideof the hyper-plane� 9¥ Ê 9HØ Ó 9 , D� 9=�i� on theothersideof theplane,andD�V9 ß� insidetheplane.Noticethat in a regulatorydomaineither

D��9 � � , D��9 �P� , orD�V9àÞ � .

Discontinuities in stateequationsThe questionmust be raisedwhat happenson the thresh-old hyperplanes�V9 � � � � �9 separatingthe regulatory do-mains,wherethe stepfunctions,andhencethe stateequa-tions,arenot defined.Severalsolutionsarepossiblefor thisnon-trivial problem.In thispaperwe follow theapproachofPlahteetal. (1994)andreplacethediscontinuousstepfunc-tions by continuousrampfunctions(Fig. 5). The solutionof the PLDEswith stepfunctionsis thendefinedto be thesolutionof the DEs with rampfunctionsconsideredin thelimit á ` � .

âYãäå

æ-ç:è�é ã�ê â ãYë

é ã

ìäå é ãâYã

í çAè�é ã�ê â ã�ê ì ë

Figure 5: Step function $'& �-��� ( �:� � and ramp functionÚ & �-� �)( � �k( á � . Ú & approaches$'& as á ` � .The useof ramp functionsdivides the phasespaceinto

regulatory domainsseparatedby switching zones, regionsin which one or more ��9 have a value in the á -intervalQ ��� � � �9 7 á Økî�( ��� � � �9 ï á Ø2î W arounda threshold��� � � �9 . An ex-amplephasespacewith switchingzonesis shown in Fig. 4.Insidetheregulatorydomains,theDEswith rampfunctionsareequivalentto thePLDEswith stepfunctions.Outsidetheregulatory domains,in the switching zones,the DEs withrampfunctionsmay be nonlinearfunctionsof the concen-tration variables. The switching zonesseparatingthe reg-ulatory domainsvanishandapproach(intersectionsof) thethresholdhyperplanesas á ` � .

Qualitati vesimulation of geneticregulatorynetworks

The goal of qualitative simulation is to exploit the quali-tative constraintson parametervaluesin order to predictthe qualitative dynamicsof a regulatory network. Morespecifically, we would like to know which regulatory do-mainscan be reachedby somesolution trajectorystartingin the initial regulatorydomain,for parametervaluescon-sistentwith thespecifiedthresholdandequilbriuminequal-ities. A sequenceof regulatory domainsthus generatedgivesan indicationof the evolution of the functionalstateof thesystem,astransitionsbetweenregulatorydomainsre-flect changesin the synthesisanddegradationratesof pro-

teins(Lewis & Glass1991;Mestl, Plahte,& Omholt1995;Thomasð & d’Ari 1990).

Qualitati ve values,states,and behaviors

The analysis of PLDEs of the form (1) motivates theintroduction of the qualitative value of a state variableand its derivative, as well as definitionsof the qualitativestateand qualitative behavior of the system. Let ñT��ò �Õ ñT��G ( G _ (©óô(©õö(B÷Z( ò � bethesolutionof PLDEswith stepfunc-tions describinga regulatory network on the time-intervalQ ò _ ( ò;ø Q for given parametervaluesand initial conditionsGZ�-ò _ �à G _ .Def. 1 (Qualitati vevalue) Supposethat at some ò � ò _it holds that ñb��ò � lies in a regulatory domain, such that� � � � �9 �úù 9B��ò � � � � � � & �B�9 for every ( 5RM MûO and5iM  9 � � 9 ). The qualitative value ü'ý�� ù 9;( ò � of ù 9 ��ò � is

given by the inequalities� � � � �9 � �V9 � � � � � & �B�9 , while thequalitative value

Dü'ý�� ù 9»( ò � ofDù 9 �-ò � is givenby oneof thein-

equalitiesD�V9 � � , D�V9 �#� , D�V9hÞ � , dependingon thesignofDù 9 �-ò � in theregulatorydomain.

The definition can be straightforwardly generalizedto thecaseof regulatory domainsboundedby �V9 þ� or �V9 �.�A� 9 . Noticethat thequalitative valueis not definedin theswitchingzonesaroundthethresholdhyperplanes.

Def. 2 (Qualitati vestate) Thequalitative stateü $ ��ñ ( ò � forñ at ò is givenby thevectorsü'ýV�-ñ ( ò � andDü'ý���ñ ( ò � of qualita-

tivevalues.

A qualitative stateassociatedwith a regulatory domaincanbe interpretedasrepresentinga functionalstateof theregulatorysystem. Eachproteinconcentrationhasa valuelying betweentwo consecutivethresholds,andis eithertend-ing towardsoneof thethresholdvalues,or evolving towardsa valuebetweenthethresholds.

Def. 3 (Qualitati vebehavior) Thequalitative behaviorü2Æ'�-ñ � of ñ is given by the sequenceof qualitative statesü $ ��ñ ( ò � on Q ò _ ( ò;ø Q .A qualitative behavior definesa successionof qualitative

statesof the regulatorysystem. It is not difficult to showthatevery solution ñ of (1) canbe abstractedinto a uniquequalitativebehavior.

Qualitati ve simulation algorithm

In terms of the above definitions, the qualitative simula-tion procedurecanbeformulatedasfollows(Kuipers1994).Given initial qualitative values ÿ�� _ , describingthe initialproteinconcentrationsG , thesimulationalgorithmcomputestheinitial qualitative stateü $ _ , andthendeterminesall pos-sibletransitionsfrom ü $ _ to successorqualitativestates.Thegenerationof successorstatesis repeatedin arecursiveman-neruntil all qualitativestatesreachablefrom theinitial qual-itativestatehavebeenfound.

Thepossibletransitionsfrom aqualitativestatearedeter-minedby thefollowing definition.

����� and���� � ������ and

�������ª ¾ � �� ´ ��� ´ ª ¾ � ®� �� ,������ ³ ª ¾ � ®� ��

´ ��� ´ ª ¾ � ®�� �� ,������ ³ª ¾ � �� ´ ��� ´ ª ¾ � ®� �� ,

������ ³ ª ¾ � ®� ��´ ��� ´ ª ¾ � ®�� �� ,

������ ³ª ¾ � �� ´ � � ´ ª ¾ � ®� �� ,�� � ´Ö³ ª ¾ � £� ��

´ � � ´ ª� ¾ � �� ,�� � ´Ö³ª ¾ � �� ´ � � ´ ª ¾ � ®� �� ,

�� � ´Ö³ ª ¾ � £� ��´ � � ´ ª� ¾ � �� ,

�� � � ³Figure 6: Continuity constraintsfor the qualitative valuesü'ý29 ( Dü'ý 9 and ü'ý�X9 ( Dü'ý X9 of two qualitative statesü $ and ü $ X de-fined on adjacentregulatorydomains.Valid for 5 �  9 �� 9 7Ð5 , theconstraintscanbeeasilygeneralizedto thecaseof qualitativevalues�4M � 9 � ��� �;�9 and ���|�s� �9 � � 9!M �.�A� 9 .Def. 4 (Statetransition) Let ü $ and ü $ X be two qualitativestatesassociatedwith adjacentregulatorydomains.A transi-tion from ü $ to ü $ X is possible,if for every , 5]M M O , suchthat ü'ý29�� ü'ý,X9 , the qualitative values ü'ý29 ( Dü'ý 9 and ü'ý�X9 ( Dü'ý X9satisfythecontinuityconstraintsin Fig. 6.

Fig. 7(a) illustratestheapplicationof therule. Intuitivelyformulated,the rule saysthat a transitionfrom onequali-tative stateto anotheris possible,if a trajectorymay crosstheswitchingzoneseparatingtheregulatorydomainsof thequalitativestates.

A simulationalgorithmbasedonDef.4 is describedin (deJong& Page2000). The qualitative statesand transitionsgeneratedby the algorithm form a statetransition graph.Thegraphmaycontaincyclesandstateswithoutsuccessors,which aretogetherreferredto asattractors. Sincethenum-ber of possiblequalitative statesis finite, every pathin thestatetransitiongraphwill reachan attractorat somepoint.Eachpathrunningfrom theinitial qualitativestateü $ _ to anattractorformsa possiblequalitativebehavior of theregula-tory system.w'x y z �)}Y~�

w'x y z {k}Y~�w x y-z�~�

w'x y � �)}Y~}w'x y � ~}w'x y � {k}Y~}(a)

w'x y � �)}Y~}w'x y � {k}Y~} w'x y � ~}(b)

Figure7: (a) Phasespacewith derivative vectors �G in theregulatorydomains( ` ) and the statetransitions( � ) per-mitted by Def. 4. (b) Solutiontrajectoryescapingthroughswitchingzones.

Propertiesof simulation algorithmGiven a modelandinitial qualitative values ÿ�� _ , what canbe saidaboutthe correctnessof the behaviors producedbyqualitative simulation?We definea set � of possiblesolu-tionsof (1) on Q ò _ ( ò;ø Q , suchthatfor every ñ Ë�� thenumer-ical valuesof ó , õ , and ÷ satisfy (3)-(4), and ü'ý���ñ ( ò _ �4

ÿ�� _ . Theultimateaim of qualitative simulationis to deter-mine� the set ��� of qualitative behaviors, suchthat (1) forevery ñ Ë�� thereis a Æ%Ë ��� , suchthat ü2Æ'�-ñ �à Æ (sound-ness), and(2) for every Æ8Ë!��� thereis a ñÕË"� , suchthatü2Æ'�-ñ �= Æ (completeness).

Unfortunately, the simulationalgorithmbasedon Def. 4is not sound.A trajectorymayentera switchingzonefroma regulatorydomain,andthenescapethroughotherswitch-ing zonesto enteranother, possiblynon-adjacentregulatorydomain(Fig. 7(b)). As a consequence,thesimulationalgo-rithm may overlook qualitative statetransitions.We foundthat the practicalconsequencesof suchomissionsare lim-ited,sincequalitativestatesnot directly reachableby a tran-sition areoftenindirectly reachableby a sequenceof transi-tions.

Completenessof the simulation algorithm has neitherbeenproven nor disproven, but seemsdifficult to guaran-teegiventhebehavioral complexity that canbeattainedbymodelsof theform (1) (Lewis & Glass1991).

GeneticNetwork AnalyzerThesimulationmethodhasbeenimplementedin Java1.2,ina programcalledGNA (GeneticNetwork Analyzer).2 Theprogramreadsandparsesinput files specifyingthe modelof thesystem(stateequations,thresholdandequilibriumin-equalities)andtheinitial state.Fromthis informationit pro-ducesa statetransitiongraph.Extensionsof thesimulationalgorithmallow all qualitative statesandtheir transitionstobe generated,aswell as the completionandsimulationofmodelswith unspecifiedthresholdandequilibriuminequal-ities.

GNA is accessiblethrough a graphical user-interface,which allows the network of interactionsbetweengenestobe displayed,aswell asthe statetransitiongraphresultingfrom the simulation. In addition, the usercananalyzetheattractorswith their basinsof attraction,andfocuson qual-itative behaviors to studythe temporalevolution of proteinconcentrationsin moredetail(Fig. 8).

Application: sporulation in B. subtilisThemethodandits implementationhavebeenusedto studytheregulatorynetwork underlyingtheinitiationof thesporu-lation processin the Gram-positive soil bacteriumBacil-lus subtilis (Grossman1995; Hoch 1993). Under condi-tions of nutrient deprivation, B. subtilis can decidenot todivide andform a dormant,environmentally-resistantsporeinstead.The decisionto eitherdivide or sporulateis madeby a complex regulatorynetwork integratingvariousenvi-ronmental,cell-cyle, andmetabolicsignals.

A schematicrepresentationof the coreof this network,displaying key genesand their regulatory interactions,isshown in Fig. 2. The centralcomponentof the network isa phosphorylationpathway, a phosphorelay, which trans-fers phosphatesfrom the KinA kinaseto the Spo0Aregu-lator. Aboveacertainthreshold,thephosphorylatedform ofSpo0A(Spo0A# P) activatesvariousgenesthatcommit thebacteriumto sporulation.An exampleis thespoIIA operon,

2GNA is availablefrom theauthorsuponrequest.

which encodesthetranscriptionfactor n�$ , essentialfor thedevelopmentof theforespore.Theflux of phosphateacrossthephosphorelay, andhencetheconcentrationof Spo0A# P,is controlledby variousexternalsignalsinfluencingtheac-tivity of pathwaycomponents.In addition,theflux of phos-phateis regulatedby Spo0A# P itself, througha numberofdirectandindirectfeedbackloopsinvolvingabrB, sinI, sinR,andothergenes.

Using the extensive literatureon B. subtilis sporulation,and the Subtilist databaseat the Institut Pasteur(Moszer,Glaser, & Danchin1995),we have formulatedstateequa-tions andappropriateparameterinequalitiesfor every genein thenetwork. In total, themathematicalmodelconsistsof10 stateequations,10 thresholdinequalities,and30 equi-librium inequalities.In the rarecasesthat the literaturedidnotunambiguouslydeterminetheparameterinequalities,wehave systematicallyexplored the alternatives and selectedthosethatpermit theobservedbehavior of thebacteriumtobereproduced.

Thebehavior of B. subtilishasbeensimulatedfrom a va-riety of initial states,reflectingdifferentphysiologicalcon-ditions. For example,fig. 8(a)shows thesimulationresultsfor aninitial statereflectinga perturbationof thevegetativegrowth conditions,whentheproteinkinaseKinA autophos-phorylatesin responseto anexternalsignalindicatingastateof nutritional deprivation. Under theseconditions,a statetransitiongraphwith two attractorsis produced,correspond-ing to statesin whichthebacteriumcontinuesto divide(V3)or initiatessporeformation(V15 andV18). Bothstatesmaybereached,dependingon theexactvaluesof theparameterssatisfyingthethresholdandequilibriuminequalities.

In order to obtain result consistentwith experimentaldata,we foundthat thetargetequilibriumconcentrationsofSpo0Ehaveto beplacedbelow thelowestthresholdconcen-trations( � � "�lFmAØ C lBm � � lFm t ). That is, we needto assumethat spo0Eexpressionlevels are quite weak. When otherequilibriuminequalitiesarechosen,thesimulationspredictthat sporulationcannotbe initiated underappropriatecon-ditions, contrary to what is observed (Fig. 8(b)). In fact,Spo0Emediatesthenegativeautoregulationof Spo0A# Patlow concentrationsof the latter, andthuspreventsa criticalconcentrationof Spo0A# Pto accumulate.

Theabovechoiceof parameterconstraintsis troublesome,becauseit impliesthatSpo0Ecannotexertany influenceonthe decisionto sporulate,since its concentrationwill notreachthethresholdlevelsabovewhichit canblockthephos-phateflux throughthephosphorelay. Thesimulationresultsthussuggestthatthenetwork in Fig.2, basedon interactionsreportedin the literature,may be incomplete. As a rem-edy, we could postulatethat an unknown signal decreasestheactivity of Spo0Eat theonsetof sporulation.Molecularstudiesof the interactionof Spo0Ewith componentsof thephosporelaysuggestthe existenceof sucha cellular factorwhich remainsasof yet unidentified(Ohlsen,Grimsley, &Hoch1994).

DiscussionWe have presentedan implementedmethodfor thequalita-tivesimulationof geneticregulatorysystemsthatcanhandle

(a)

³]´ �  H¡©º ±  H¡ ´ ª  H¡ t23 statesattractors:sporulationanddivisionªA H¡ t ´ �, H¡ º ±  �¡ ´ ªA H¡ �44 statesattractors:divisionª  H¡ � ´ �  H¡�º ±  �¡ ´ ª  H¡Hµ65 statesattractors:divisionªA H¡¶µ ´ �, H¡ º ±  �¡ ´À·Á¸�¹  H¡1155statesattractors:division

(b)

Figure8: (a) GNA outputfor the equilibrium inequalities� � "VlFm:Ø C lFm � � lBm t . The left window shows the statetransitiongraphwith a qualitative behavior runningfrom the initial qualitative stateV1 to the attractorstateV15. In the right windowthetemporalevolution of thequalitativevalueof Hpr andKinA for this behavior canbefollowed. (b) Summaryof simulationresultsfor aninitial stateexpectedto inducesporulation,while varyingtheequilibriuminequalitiesfor Spo0E.Thesimulationstakebetween0.5and3 secondsto completeon a SunUltra 10 workstation.

largeandcomplex networksof genesandinteractions.Themethodis ageneralizationof themethodin (deJong& Page2000),in that it allows a largerclassof regulatoryrelation-shipsbetweengenesto bemodeled.In thefirst place,therearenorestrictionson thelogical functionsthatcanberepre-sentedby (2). This permitscomplex regulatoryinteractionsto beincludedin themodels,asillustratedby thestateequa-tion for abrBin Fig. 3(b). In thesecondplace,theregulationof proteindegradationcanbetakeninto account.Althoughthis featureof the methodhasnot beenusedin the sporu-lation example,it turnedout to be crucial in modelingthenetwork controlling the inductionof the lytic cycle follow-ing phageÊ infectionof E. coli (resultsnot shown here).

The applicability of the methodto actualregulatorynet-workshasbeendemonstratedby ananalysisof thelargeandcomplex network underlyingtheinitiation of sporulationinB. subtilis. Theanalysishasresultedin asuggestionto com-plete the model compiled from the sporulationliterature,which shows the potentialof the methodasa tool to focusfurtherexprimentation.To our knowledge,qualitative sim-ulationof geneticregulatorynetworksof thesizeandcom-plexity consideredin thispaperhasnotbeenundertakenthusfar (but see(Mendoza,Thieffry, & Alvarez-Buylla1999)).

Upscalingof thesimulationmethodisachievedbymodel-ing geneticregulatorysystemsby aclassof piecewise-lineardifferentialequationsimposingstrongconstraintson thelo-cal dynamicsof thesystem.Besidesin moleculargenetics,PLDEsof this form have beenusedin otherbiological do-mains,for instancein populationbiology (Plahte,Mestl, &Omholt1995). In orderto effectively applytheconstraints,the representationof thequalitative stateof thesystemandthe simulationalgorithm are adaptedto the mathematicalstructureof theequations.

Adaptationto a specificclassof modelsis the principal

respectin which the methodpresentedin this paperdiffersfrom well-known QR methodslike QPTandQSIM (Forbus1984;Kuipers1994).A majordifferencewith QSIM is thatthequalitativestateof a regulatorysystemis describedon ahigherlevel of abstraction.In particular, thebehavior insidea regulatory domainis abstractedinto a single qualitativestate,makinguseof thefactthatinsidea regulatorydomaineither

D�V9 � � , D��9 �#� , orD�V9öÞ � . In QSIM onewould have

to distinguishqualitative statesinsideandon the boundaryof a regulatorydomain.Thenumberof thesestatesis expo-nentially growing in O , the sizeof the regulatorynetwork,andmayleadto spuriousbehaviors.

Approximating step functions by infinitely steeprampfunctionsallows a precisedefinition of the behavior of thesystemin the thresholdplanes,and henceof the possiblesuccessorsof a qualitative state.Thecalculationof succes-sorstatesisbasedonDef.4 andaccomplishedbyaninequal-ity reasonermanipulatingthe stateequationsandthresholdandequilibrium inequalities(de Jong& Page2000). Thestatetransitionrule in Def. 4 is simpleandintuitively clear,but doesnot preserve soundnessof the algorithm,contraryto QSIM. Eventhoughthepracticallimitationsof this maybelimited, thedevelopmentof statetransitionrulesguaran-teeingsoundnessis animportanttopic for furtherresearch.

The work most closely related to the approachpre-sentedin this paperis the generalizedlogical methodofThomasand colleagues(Thomas, Thieffry, & Kaufman1995; Thomas& d’Ari 1990). Snoussihasdemonstratedthat the logical formalismcanbe seenasan abstractionofEq.(1), whereeachregulationfunctionin E 9B��G � is themath-ematicalequivalentof a simple Booleanfunction, namelya Booleanvariableor its negation,andwhere Jk9 ��G � is setequalto C,9 (1989).Thetransitionrule in thelogical methodis different,in thatit allowstransitionsthatarenotpermitted

by thecontinuityconstraintsin Fig. 6, which aremotivatedby the% definition of stepfunctionsas infinitely steeprampfunctions.SnoussiandThomasprovidecriteriafor theiden-tification of singularsteadystateslocatedin the thresholdplanes(Snoussi& Thomas1993). An interestinggeneral-izationof theirwork would beto providesimilar criteriaforthemoregeneralclassof equationsemployedin this paper.

The B. subtilis examplesuggestan approachto validatehypothesizedmodelsof geneticregulatorynetworks. Giventemporalgeneexpressionpatternsobserved under certainphysiologicalconditionsin wild-type or mutantstrainsofthebacterium,onecandevelopalgorithmsto systematicallysearchthe spaceof freely adjustableparameterinequalitiesto find constraintsfor which themodelis ableto accountforthe observations. Extensionsof this type would allow thesimulationmethodpresentedin this paperto evolve into amoregeneralmodelingtool.

Acknowledgments The authors would like to thankFrancoisRechenmann,IvaylaVatcheva,Alain Viari andtwoanonymousrefereesfor commentson a previousversionofthis paper.

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