arkin et al. - stochastic kinetic analysis of developmental pathway bifurcation in phage...

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Copyright1998 by the Genetics Society of America

Stochastic Kinetic Analysis of Developmental Pathway Bifurcation inPhage-Infected Escheri chi a coliCells

Adam Arkin,*,1John Ross and Harley H. McAdams*

*Department of Developmental Biology and Department of Chemistry, Stanford University, Stanford, California 94305

Manuscript received March 5, 1998Accepted for publication April 30, 1998

AB STRAC T

Fluctuations in rates of gene expression can produce highly erratic time patterns of protein production

in individual cellsa nd wide diversity in instan taneo us protein concent ration s across cell population s. When

two independentlyprod uced regulatory proteins acting at low cellular concentrations competitivelycontrol

a switch point in a pathway, stochastic variation s in their concentratio ns can produce prob abilistic pathway

selection, so that an initially homogeneous cell population partitions into distinct phenotypic subpopula-

tions. Many pathogenic orga nisms, for example, use this mechanism to rando mly switch surface featuresto evade host r esponses. This coupling between molecular-level fluctuation s and macro scopic phenotype

selection is analyzed using the phage lysis-lysogen y d ecision circuit as a mod el system. The fraction of

infected cells selecting the lysogenic pathway at different ph age:cell ra tios, predicted using a molecular-

level stocha stic kinetic model of the g enetic regulator y circuit, isco nsistent with experimenta l observation s.

The kinetic model of the decision circuit uses the stochastic formulation of chemical kinetics, stochasticmechanisms of gene expression, a nd a statistical-thermodynamic mod el of promoter regulation. Co nven-

tiona l determ inistic kinetics cann ot be used to predict statisticsof regulatory systemsth at produce proba bilis-

tic outcomes. Rather, a stochastic kinetic analysis must be used to predict statistics of regulatory outcomes

for such stochastically regulated systems.

I N McAdamsa nd Ar kin (1997), we analyzed proper- case occurs when two independently produced regula-tory proteins competitively control a developmentalties of a representative single bacterial geneticallycoupled link, that is, a configuration where one pro- switch. The independent, stochastic temporal patterns

of production of each regulatory protein can vary widelymoter controls a gene whose protein product regulates

another promoter. In that analysis, an integrated molec- from cell to cell. In this case, the path choice from the

compet itively regulat ed switch would no t be deter minis-ular-level mo del o f th e mecha nisms controlling gene

transcription and translation was developed, a nd the tic. Rather, the choice would be random with the proba-

bilities of alternative cho ices dependent on the stochas-expected time pattern of protein production from the

controlled gene was investigated using the stochastic tic properties of the gene expression mechanisms and

the design of the switch circuit. As a result an initiallyformulation of chemical kinetics (Gil l espie 1977,

1992b). The results suggested that the stocha stic fluctu- hom ogeneous cell population would partition into sub-

populations following different pathways. The pheno-ations of the reaction ra tes of gene expression reactions

can produce a highly erratic time pattern of protein types on each path could be radically different. In manypathogenic organisms random variation of surface fea-production in each individual cell and a wide diversity

of protein concentrations across a cell population at tures assists in evasion of host defenses or otherwise

enhan ces virulence (Put t ea nd Goosen 1992;Rober t -any instant of time ( McAdams and Ar kin 1997). (Sto-

chastic is used here in the technical sense of arising so n 1992; Finl ay a nd Fal kow 1997;St r auss a nd Fal -

ko w 1997). We suggest in this article that one sourcefrom a random process.)

When the protein involved is a regulatory protein, of the randomness expressed in the phenotype varia-tions can be the random thermal fluctuations in thethese fluctuations in concentration from cell to cellreaction rates of the chemical reactions comprising thecause dispersion in the time to complete regulatedregulatory circuit.events in different cells, for example, different times to

To examine this phenomenon , we ana lyze herein thecomplete regulatory cascades. A particularly interestingeffect of fluctuations in gene expression ra tes and othermolecular-level fluctuations on lysis or lysogeny pathway

Correspondin g aut hor: H arley McAdams, Department o f Develop- selection statistics by phage -infected Escheri chia colimental Biology, Stanford University School of Medicine, Stanford,

cells. This path selection is made by the lysis-lysogenyCA 94305. E-mail: [email protected] circuit wherein a well-characterized competi-1 Present address:Division of P hysical Biosciences, Calvin Labs 144,

Lawrence Berkeley National La borato ry, Berkeley, C A 94720. tive regulatory mechanism is central to the regulatory

Genetics149: 1633 1648 (August 1998)


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1635Stochastic Kinetic Ana lysis

Figu r e 1.The phage lysis-lysogeny decision cir-cuit. (a) B old horizont alli ne s in d ic at e st re tc he sof double-stranded DNA.Arrows over genes indicatedirection of transcription.Dashed boxes enclose oper-ator sites that comprise apromoter con trol complex.

The three operator sites,OR13, of the lamb da switchimplement concentration-dependent logic control-ling promoters PRM and PR.Cro and CI dimers bind tothe three siteswith differentaffinities and in opposite or-der to control the activationlevel of the PRM a nd PRpro-moters(Pt ashn e 1992; Sh eaand Acker s 1985) . The fiveboxes R1R5 contain non-genet ic p rot ein reactionsubsystems. In R1, R2, and

R5, deg indicates degra-dat ion. When p rot ein Nis available, transcribingRNAPs can be antitermina-ted at the NUTR and NUTLsites; termination sites TR1and TL1 are inoperative forantiterm inated RNAPs. TheCI d imer acts as either a re-pressor or activator of pro-moterPRM, depending on itsconcentration. See text fordiscussion of the proteaseslabeled asP1 and P2 in R3an dR4.(b) decision circuit

DNA organization. Phage-encoded genetic elementsof the decision circuit arelocated in a 5000 nucleotideregion of the phage DNA.Genes are sep arated ont o

leftward and rightward transcribed strands as indicated b y the arrows. Rightward extensions of the antiterminated PRtranscripttranscribe the O and P genes essential for phage genome replication and the Q gene that controls transcription of later geneson the lytic pathway. Leftward extension of the antiterminated PL transcript transcribes xi sa nd int genes essential for phagechromosome integration and excision into and out of the host chromosome. Locations of four termination sites are indicatedby TR12 a nd TL12.

creates the circuits bistability, a nd (ii) the Hfl proteolytic in that cell after infection. Immediately after infection, thereare no C I or Cro molecules in the cell so the regulatory circuitsystem, which integrates environmental signals into the cir-

cuits behavior. is in the state labeled S in the lower left corners of Figure2, a and b. At that point, PR is fully activated; PRMhas only aThe core of the bistable switch is the complex biochemistry

of t he PR a nd PRM promoters operator regions, which share low basal activation, and promoter PL is also activated. Tran-scription an d translation of the phag e DNA is accomplishedthree overlapping operator sites (Figure 1a, OR1, OR2, OR3),

where Cro and CI dimers bind competitively and in sequence, by the host cells machinery.Cro a nd N proteins are produced from tran scripts initiatedbut in opposite order ( Maur er et al. 1980; Meyer et al. 1980;

Meyer and P t ashn e 1980; P t as hn e 1992). Figu re 2, a and at p romot ers PRa ndPLand both proteins begin to accumulateimmediately after infection. Initially terminators TR1 a nd TL1b, shows contour maps of the activation level of PR a nd PRM,

respectively, as a function of CI and Cro dimer concentration. partially block RNA polymerase (RNAP) transcription: about50%at TR1(Fr iedman a nd Got t esman 1983) a nd 80%a tTL1The activation levels are calculated using the model and pa-

rameters in Shea and Acker s (1985). ( Dr aho sandSzyba l ski1981). Ho wever, asth e concen trationof N increases, the N protein ( with other molecules from theThe lysis or lysogeny outcome in each cell is determined

by the specific temporal pattern of CI and Cro accumulation host cell) acts to antiterminate RNAP at NUT sites upstream


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probability of PRE activation. So the probability of lysogeny is TABLE 1reduced in well-fed cells. Dependence of lysogeny on MOI

Parametersfor promoters PRE and PLarises because the concentration of the host-encoded Hfl sys-tem is independent of MOI, while the number of cII an d cIII

Stategenes is proportional to MOI. Thus, cells with higher MOIG kochave a higher prob abilityof achieving CII concentration neces-

No. O 1 O 2 (kcal mol1) (sec1)sary to activate PRE and kickstart CI production.

If a cell reaches a state where (i) the Cro feedback loopPromoter PREis established, (ii) PRM an d PL are repressed, and (iii) CII 1 0.0 0.0

concentration is low, there is a high probability that the cell

2 RNAP 9.9 0.00004will continue on the lytic path. On the other hand , if a cell 3 CII 9.7 0.0reaches a state where ( i) the CI feedba ck loop is established,4 CII RNAP 21.5 0.015and (ii) PR and PL are repressed, there is a high probability

Promoter PLtha t the cell will continue to lysogeny (McAdamsa nd Sha pir o1 0.0 0.01995).2 Cro2 10.9 0.03 Cro2 12.1 0.04 CI 2 11.7 0.0

SOU R CE DATA AND G E NE TI C CI R CU I T MODE L5 CI2 10.1 0.06 RNAP 12.5 0.011Kourilskys measurementsof lysogeny versus API: We use7 Cro2 Cr o2 22.9 0.0the experimental assays of percent lysogeny versus API in8 Cro2 CI 2 20.9 0.0Kou r il sky(1973) to compa re with predictions of the stochas-9 CI 2 Cr o2 22.8 0.0tic kinetic model. In Kourilskys experiments, lambda phages10 CI2 CI 2 23.7 0.0were added at various API to exponentially growing E. coli

cultures for an incubatio n time that ensured near 100%phag e

Promoter PRE parameters are estimated from H oyt et al.absorption. Kourilskys measurements included O a n d P (1982), and Shih an d Gussin (1983, 1984). The value ofstrains incapable of phage chromosome replication. Since thekOC, the reaction for closed- to open-complex formation, isphage chromosome count does not increase, the postinfectionestimated from Gil ad i et al. (1990). Binding free energies ofdistribution of the O or P phage particlesamong the targetCI 2, Cro2, and RNAP to promoter PL operators are assumedcells can be computed using the Poisson infection statisticsto be the same as for operators OR2and OR3of promoter PRmodel described below. Selection of the O a n d P mutantsin Shea an d Acker s (1985). PRa nd PRMparameters are fromfor modeling eliminates the need to model chromosome repli-Shea and Acker s (1985).cation.

In Kourilskys plots of log API versus the log of the percentcells lysogenized, th e shape o f th e rate of lysogenization versusAPI curves was similar for starved a nd unstarved cells, but the algorithm described by Gil l espie (1977). The Gillespie algo-starved curves were systematically shifted to a 50 100 times rithm produ ces a stochastic realization of the tempora l beha v-higher lysogenization rate with little effect on the qualitative ior of the system by calculating the probabilistic outcome ofdependence of lysogenization rate on the infection ratio [Fig- each discrete chemical event and the resulting chan ges in theure 2 in Kou r il sky (1973)]. The starved cell results with number of each molecular species. In the application of the

50 100 times higher ra tes of lysogeny are used for compa rison Gillespie algorith m to simulation of bacteria l regulation , eachsince the num ber of simulation runs necessary to estimate the simulation run provides a representative case of the sequencefraction, f, of lysogeny varies as 1/f. and timing of events and the regulator y outcome in an individ-

Stochastic kinetic model: The stochastic kinetic mo del used ual cell starting from specified condition s. Multiple runs withhere to analyze operation of the lysis-lysoge ny decision cir- the same initial conditions (e.g., the same MOI) are used tocuit includes the geneticm echanisms and the coupled protein estimate the proba bility tha t cells will enter lysogeny for thesedimerization and degradation reactions shown in Figure 1a. cond itions. [About 4(1 p)/f2ePsamples are required to esti-Genetic mechanisms are modeled using explicit , though mate the probability, P, of a binary random event with 95%approximate, reaction m odels of each submechanism an d ex- con fiden ce where fe is the desired ma ximum fractional errorplicitly including features such as termina tion sites. Thus, pro- in P( Fel l er 1968). In the present case, Pis the probabilitymoter o perator sites are mod eled using the statistical-thermo - of lysogeny in each cell.] Computations were performed ondynamic approach described by Shea and Acker s (1985). SGI workstations and parallela rray supercomputers( 200 node(The stochastic version of the Shea and Ackers model of pro- Cray T3D machine at Eglin AFB, and 400 node SP2 machinemoter kinetics is produced by calculating the instantaneous at Maui High Performance Computer Center). Additionalprobability of each distinct tran scriptionally active state of a details on the software are available on the website: www.promoter using the partition function, and then using this lbl.gov/aparkin/Lambd aMod.html/. Criteria d escribed be-probability in calculating the reaction probabilities for the low were used to categorize the outcome of each individualtranscript initiation reactions.) Transcript elongation is mod- run as a lysogenic outcome or not.eled as a sequence of individual nucleotide steps. Translation To compare the percent lysogenization predicted by thecontrol is modeled as described by McAdams an d Ar kin simulation at various infection levels to experimenta l observa-(1997). Assumptions and reaction models used for elements t ions obtained by Kou r il sky ( 1973), the Po isson-weightedof the system are described below and listed in Tables 1 and average of the probabilityof lysogenyat each MOI iscomputed2. The reaction models are represented as a set of coupled to estimate the expected percent lysogeny as a function ofsto ch astic kin etic eq ua tion s. AP I. Th e th eo retica l P oisso n pro ba bility th at a given cell will

Analytical solution to such systems of stochastic reaction be infected with exactly MOI M phage when API A isequations is only practical for simple reaction systems. How-ever, numerical solutions can be computed for complex sys- P(M,A)

A M

A! eA, (1)

tems of coupled stochastic reactions using the Monte Carlo


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1638 A. Arkin, J. Ross and H . H . McAdams

TABLE 2

Parametersfor transcriptionand translation reactions

Reaction/event Parameter References and comments

Transcription reactions

k22 30 nt sec1 Selected as an average rate. Measured elongationRNAPDNAn

k22RNAPDNAn1

rates vary widely, depending on DNA templateand cell state (Got t a

et al. 1991; Kennel l and

Riezman 1977; Kor nber g and Baker 1992;Vogel and Jensen 1994)

k23 5 nt sec1RNAPDNANut(L,R)

k23RNAPDNANut(L,R)1

k24 0.145 (msec)1 Selected to produce termination and antitermina-RNAPDNANut(L,R) N

k24

k25RN APNDNANut(L,R)1

k25 0.1 sec1 tion consistent with Li et al. (1992) an d Wh al en

et al. (1988)

k26 30 nt sec1RN APNDNANut(L,R)

k26RN APNDNANut(L,R)1

k27 15 nt sec1 Selected to yield 50% termination at N 0 n mRNAPDNATR1

k27RNAPDNATR11

(Dambl y-Ch au dier e et al. 1983; Fr iedman andGot t esman 1983)

k28 15 sec1RNAPDNATR1

k28RNAP D NATR1

k29 30 nt sec1 Assumption that a ntitermina ted RNAP passes termi-RN APNDNATR1

k29 RN APNDNATR11nator freely

k31 5 nt sec1RNAPDNATL1

k31RNAPDNATL11 Selected to yield 80% termina tion at N 0 n m

k32 25 sec1RNAPDNATL1

k32RNAP D NATL1 Selected to yield 80% termina tion at N 0 n m

k33 30 nt sec1 Assumption: antiterminated RNAP passes termina-RN APNDNATLI

k33RN APNDNATL11

tor freelyTranslation reactions

k34 0.002 (msec)1 (Kennel l and Riezman 1977; Sor ensen and Ped-Ribosome R NARBS

k34Ribo some RNARBS

er sen 1991)

k35 100 nt sec1 (Adhyaa nd Got t esman 1982;Kennel l an d Riez-Ribosome R NAn

k35Ribo some RNAn1

man 1977; Sor ensen and Peder sen 1991)

k36RN ase 0.2 sec1 Adjusted to get an average of 10 proteins per tran-RNase R NARBS k36 RNase

scriptAverage number of proteins per transcript

(all transcripts) 10 (Kepes 1963; Yar ch uket al. 1992)

whereP(M,A) is the probability of a cell having MO I M, at growing cells roughly of fset t he effects in slower grow ingAPI A ( El l is a nd Del br uck1939) . Th e e xp ec te d f ra ct io n c el ls.of lysogens at a given API, Flysogens, is th en 2. Th e vo lum e o f th e cell g ro ws a ppro xim ately lin ea rly f ro m

1 1015 to 2 1015 liters. (The maximum differenceFlysogens(A)

M

P(M,A).F(M), (2) in volume between linear and exponential cell growthmodels is 6% with negligible effect on simulation re-

where F(M) is the estimated probability of lysogeny for cells sults.)with various MOIs as estimated using the stochastic kinetic 3. Ho st housekeeping molecules relevan t to pha ge gene ex-model. pression an d phag e protein degra da tion are constitutively

expressed and regulated at constant concentration, whichis the same in all cells. This implies, for example, tha t

ModelingAssumptions all enzymes required for metabolic pathways, etc. , areexpressed at levels consistent with a healthy ba cteriumand that cytoplasmic concentrations of proteases, RNAP,1. Dispersion in cell genera tion times can be neglected. Thusribosomes, and metabolic substrates are maintained d ur-all runs used a cell cycle time of 35 min, consistent withing the early postinfection period when the lysis-lysogenythe cell cycle time reported by Kour il sky ( 1973). E. colidecision is being resolved. Both RNA polymerase a ndcell generation times are observed to be approximatelyribosomes are present in the cell in relatively large num-normal distributed with standa rd deviation of a bout 22%bers, however, the freepolymerase and ribosome concen-o f t h e m e a n (Pl ank and Har vey 1979). The neglecttrations are thought to be a fraction of the total and toof dispersion in cell generation times is equivalent to

assu ming t hat any growt h rat e-relat ed effects in fast er be bu ffered by exchange with u nit s t hat are engaged in


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other reactions. U nder these cond itions, fluctuations inpolymerase and ribosome con centrations would be rela-tively small.

4. Regulatory effects of host proteins such as integrationhost factor and RNase III on phage gene expressionare a ssumed to be equivalent for all cells and constantover t ime. For example, the effect of integration hostfactor on PLactivity ( Gil ad i et al. 1990) is a ssumed to beincluded in the kinetic parameters of the promoter andto be independent of phage MOI.

5. Effects such as macro molecular crowding or two-step bind -ing to DNA or RNA that might affect reaction kineticsare assumed to be subsumed into the kinetic parametercharacterizing th e reaction.

6. Intermediate reactions (as with sigma-factorso r other sub-elements) in assembly of the RNAP and ribosome com-plexes are not rate limiting. Instead, we assume either (i)that the host cell maintains an effective concentration oftranscriptionally an d translationally available concentra-tions of these molecular complexes that are in rapid ex-change with their binding sites on the DNA or RNA, or(ii) that the component subunits are in rapid eq uilibriumwith functionally active assemblies (Shea an d Acker s1985; P t ashn e and Gann 1997). The rate-limiting stepin tra nscript initiation is a ssumed to be the closed- to

open-complex isomerization reaction (McCl ur e 1980).7. Phage gene expression is stochastic, consistent with the

mechanisms described by McAdams and Ar kin (1997).8. An average of 10 proteins are prod uced per tran script for

allgenes( Kepes 1963;Sh ea a ndAcker s1985). Tran scriptdegrada tion rates and ribosome binding rates are chosento produce that average yield.

9. In Kourilskys experiment the E. colicellswere unsynchro-nized, hence they were presumably infected at randomtimes in the cell cycle ( Kou r il sky 1973). We assume allinfections occur early enough in the cell cycle so that cellgrowth only affects operation of the decision logic bydilution effects on concentrations of phag e-encoded mol-ecules. The initial rates of phage protein prod uction fromPR- a n d PL-initiated tra nscripts in ea ch ho st cell are ind e-

pendent of cell volume. H owever, for the same rate ofprotein production, the consequent rate of change inphage protein concentrationis cell size d ependent so thattiming of subsequent events could be slowed somewhatfor larger cell size at infection time. Most cells that arefated to become lysogens are comm itted by 10 to 15 min Figu r e 3.The solid lines in (a) show the time course of(causing, for example, the cessation of Cro 2 production th e averageintracellular Cro 2 and CI2 concentrations at MOI

6. The shaded region indicates the 1 range as estimatedshown in Figure 3c). Most infections early in the cellby determining the 16th and 84th percentile points in thecycle are thus resolved before the next cell division. Forpopulation at each time. (b, c) show the same d ata, but forinfections occurring late in the cycle, if commitment hasthe two subpopulations with different phen otypic fates. Thenot occurred before division, the phage chromosomesconcentration profiles of th e two regulatory dimers in eachand proteins at division are randomly shared between thesubpopulation a re similar for th e first few minutes, but divergedaughter cells when the cell divides. Then the phageinto a substantially different time pattern after about 7 min.infection continues, but with lower MOI. For the O orCommon experimental method s for assaying time evolutionP pha ge mutants, the average postdivision MOI ish alved,of protein concentrations would yield data equivalent to thesince phag e chromo some replication is not possible. Ha lv-average value curves in (a), masking the differences in theing the MOI reduces the probability of lysogeny in thediverging subpopulations.da ughter cells. This suggests tha t n eglecting cell division

leads to some degree of overestimation of the probabilityof lysogens in the simulation.

10. The target cells are infected effectively simultan eously so 12. A cell becomes committed to lysogeny if there is (i) athat no temporal infection effects or phage infection- sufficient time-integra ted concen tration of CII to activatedependent immunity occurs. PRE, and (ii) [CI2] [Cro2] at the en d of the 35-min cell

11. The cell is assumed to b e a homogeneous, well-stirred cycle. Activation of PRE was defined as an average activationmedium so the concept of protein concentration isvalid level of one open-complex per 2 min over a contiguousand spatial effects are averaged out. [E. colisignaling p ro- 4-min p eriod. This level of CI I p rodu ct ion wou ld alsoteins have been shown to diffuse distances comparable to activate the other CII-dependent promoters, Panti-Q a ndthe cell dimensions in much less than a second ( Ishih ar a PI, that function in execution of the lysogenic pathway

(McAdams a ndSha pir o 1995). CI 2 concentration greateret al. 1983).]


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than that of Cro 2 at the end of the cell cycle isan additional regions,neglecting the differencesbetween transcription ratesfor different nucleotides. (Transcription through terminationindication that activation of PRE occurred early enough

and was productive enough to lock on the CI feedback and antitermination sites is described below.)Termination:At termination sites, transcribing RNAPs slowloop.

down ( i.e., the average interstep time parameter is larger) andthere is a probability of transcript termination at the site.

Reaction Models When the RNAP is antiterm inated upstream of the terminato rsite, the termination site is then modeled as the normalThe following paragraphs describe the models used for theDNA described above.mecha nisms of th e lysis-lysogeny decision circuit. Reactions

Antitermination: The reactions to assemble the antitermi-and parameters are listed in Tables 13. Parameters in thenated form of RNAP at NUTL a n d N U TR sites (Figure 1a)kinetic model are derived from the sources cited in Tablesdepend on N protein concentration. The antitermination13. Considerations underlying selection o f the CII and CIIIreaction complex also involves Rho and at least four add itionalproteolysisreaction models and rate parametersa re describedhost factors: NusA, NusB, NusG, and S10 ( Wh al en et al.1988;below.Mason and Gr eenbl at t 1991; Li et al. 1992; DeVit o andPhagegeneexpression: The genetic mechanisms associatedDa s 1994). These factors a re assumed to be constitutivelywith tran script initiation and translation con trol produce theexpressed and present in the necessary concen tration s so thatlargest component of the stochastic effects that lead to diver-the concentration of N is limiting. In vitro studies suggestgent phenotypes in the infection system. The following ge-that in the presence of the proper h ost factors, the fractionalnetic reactions are modeled: operator/promoter binding,readthrough of a downstream terminator increases in propor-transcript initiation, tran scription, initiation of translation,tion to the concentration of N un til there is full antitermina-translation, and initiation of mRNA degradation. The tran-tion at N concentrations between 50 and 100 nm (Wh al en etscription mo del includesmecha nisms for th e two RNAP termi-al. 1988; Li et al. 1992; DeVit o and Da s 1994). Under thesenation sites, TR1 an d TL1, and antitermination at the NUTRconditionsthe antitermination processat theNUTsite ismod-and NUTL sites (Figure 1a).eled as a single-step reaction assumed to be a pseudo-first-Operator/ promoter bin din g and contr ol of transcri pt i ni ti ation:

order reaction with the ra te chosen such that an titerminationShea and Acke r s (1985) describe a statistical mechanical/ of RNAP is nea r 100% for N concentrations above about 75thermodynamic approach to modeling the PR/PRMpromotern m. This parameter ch oice fits the experimen tal depen den cecomplex. We use the same approach to mod el PL repressionof fractional readthrough of a downstream terminator as ab y C I 2 and Cro2 andPREactivation by CII using the parametersfunction of N concentration ( Wh al en et al. 1988; Li et al.in Table 1. The instantaneous probability of each distinct1992; DeVit o and Da s 1994).occupancy state of a promoter is assumed to be determined

Translation: Translation control is modeled as d escribedby the partition function d efined in a ccord with the Shea/by McAdams and Ar kin (1997), based on the mechanismAckers formula tion, an d we use the proba bility in the stochas-described byYar c h uket al. (1992). In that model, a competi-tic formulation of kinetics as d efined by Gil l espie (1977,tion between ribosome and RNase E binding determines the1992b). A key assumption is that effector molecule bindingaverage number of proteins produced per transcript. In thereactions at a promoter occur much faster than the rate ofstochastic kinetic model of the circuit,th is ribosome-RNase Etranscript initiation at the promoter. With this rapid equilib-competitive binding reaction istreated a s a stochastic chemicalriuma ssumption, the binding state ofth e promoter ismod eledreaction. The temporary occlusion of the ribosome bindingby rando mlychoo sing the promoter state at each instant usingsite after a successful ribosome binding event is modeled.the probabilities given by the partition function ( Shea an dMotion of a translating ribosome on a transcript is modeled

Acker s 1985). If the promo ter state selected is one from which similarly to the model of motion of a transcribing RNAP onRNAP can in itiate transcription, then tha t tran script initiationDNA described above. If one ribosome by chance overtakesreaction is included in the list of possible reactions for theanother in the mod el, the progression o f the former is haltednext Monte Carlo calculation in accord with the Gillespieuntil the latter moves ahead. The average ribosome step timealgorithm. Whenever the Monte Carlo calculation determinesis selected to be shorter tha n th e RNAP step-time para meter,that a transcript is initiated from on e of the promoters, a newproducing ribosome queuing as is observed ( Kennel l an dtranscribing RNAP is initiated on the corresponding DNARiezman1977;Yar c h uket al.1992). As with the transcriptiontranscription object. The promo ter activation function s in Fig-model, the statistics of the interstep times are assumed to beure 2 show the resulting average rates of transcript initiationdescribed by the exponential probability function.as a function of CI2 and Cro 2 concentrations. Occlusion of

Phageprotein dimerizationand degradation: The principalthe promoter site by the footprint of a recently launchedreactions involving phage-encoded proteins in the decisionRNAP is included in the model.circuit are iden tified in t he boxes labeled R1 to R5 in FigureTranscription: A transcript elongation model estimates the1a. Reactions in R1 include degrada tion and d imerization of CI;time delays between tran script initiation and arrival at the endR2 includes dimerization and degradation of Cro; R3 and R4of each coding region on the operon. Thisd elay, plusthe delayinclude competitive degrad ation of CII and CIII by the two ho stuntil an effective level of signaling molecules is accumulated,cell proteases (see below); and R5 includes degradation of N.determines the timing of regulatory molecule concentrations

CI and Cro dimeri zation and degradati on: Degradation of CIthat control regulatory networks. The movement of transcrib-and Cro is modeled as occurring predominantly by proteolysising RNAP along the DNAis modeled as a sequence of indepen-of the monomeric form, a common degradation mode forden t one-nucleotide reaction steps. Each such reaction is as-multimericproteins( Shea a ndAcker s1985;Got t esman a ndsumed to be unidirectional, that is, RNAP movement isMaur izi 1992). The dimerization reactions are assumed toassumed to be strongly forward -biased. It is assumed tha t therebe characterized by fast forward and reverse reaction ra tesis a single ra te-determ ining rea ction fo r each RNAP step andwhose ra tio, the dissociation constant, is 20 n m ( Shea an dthat each forward step has constant probability of occurringAcker s 1985). The dynamics of decay of multimeric proteinper unit time, leading to an exponential distribution of in-populations is not exponential in gen eral. Rather, the dynam-terstep times. The exponential ischa racterized with an averageics and specifically, the mea sured lifetimes, depend stronglystep-time param eter. The same a verage step time was used

at each nucleotide position and for coding and n oncoding on the forward and reverse rates of the multimerization reac-


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TABLE 3

Parametersfor housekeeping and nongenetic reactions

Reaction/event Parameter References and comments

Housekeeping reactionsAvailable RNAP RNAP 30 nm Mc Cl u r e (1980, 1983)Available ribosomes Ribosomes 500 n mCell volume ( t) ( 1 k0* t) 10

15 liters k0 4.76 1018 l it ers To dou ble init ial cell volu me of 1015 liters

sec1

in 35 minNongenetic reactionsa

k1 0.0007 sec1 Selected to yield a Cl/Cl2 life time of approxi-CI

k1( )

mat ely 40 min (Reinit z and Vaisnys1990) in the co ncentra tion ran ge between20 and 100 n m

k2 0.05 m1 sec1 Bur z et al. ( 1994); Shea and Acker s (1985)2CI

k2

k3CI 2

k3 0.5 sec1

k4 0.0025 sec1 Selected to match Cro/Cro 2 lifetime of ap-Cr o

k4( )

proximately 30min ( Reinit z an dVaisnys1990) in the co ncentra tion ran ge between20 and 100 n m

k5 0.05 m1 sec1 Reinit z an d Vaisnys (1990); Sauer (1979)2Cro k

5k6

Cr o2

k6 0.5 sec1

k7 0.00231 sec1 Got t esman an d Got t esman (1981)N

k7( )

P1 concentration b P1 35 nm Adjusted to match the % lysogeny vs. APIdat a (Kou r il sky 1973)

k8 0.01 m1 sec1 Selected to ma tch CII half-life inGot t esmanCI I P1

k8

k9P1CII

and Got t esman (1981)

k9 0.01 sec1P1CII

k10P1

k10 0.002 sec1

k11 0.01m1 sec1 Selected to match C III protection of CII deg-

CIII P1 k11

k12P1CIII

radation (H oyt et al.1982;Rat t r ay et al.

1984) and CIII half-lifeKor nit zer et al.(1991a,b)

k12 0.001 sec1

P1CIII k13

P1k13 0.0001 sec

1

P2 concentration P2 140 n m

k14 0.00025 m1 sec1 Selected to ma tch CII half-life inGot t esmanCI I P2

k14

k15P2CII

and Got t esman (1981)

k15 0.065 sec1P2CII

k16P2

k16 0.6 sec1

k17 0.01m1 sec1 Selected to match CIII protection of CII fromCIII P2

k17

k18P2CIII

degradation (H oyt et al. 1982; Rat t r ayet al. 1984) and CIII half-life (Kor nit zerk18 0.01 sec1P2CIII

k19P2

et al. 1991a,b)k19 0.001 sec

1

a The ( ) notation indicates degradation.b The pa rameters on this an d following lines for the CII/CIII proteases (here labeledP1 and P2) are those corresponding to

the Full curve in Figure 6a. The Hfl-related para meters below a re a djusted to match half-lives of th eir targ eted pro teins andto match the percent lysogeny v s. API data in Kou r il sky (1973) as described in the text.


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tions and on the initial concentration of each subspecies. Lon protease, and Lon is also thought to be responsible fordegrading t he H fl p roteins (Got t esman an d Got t esmanReaction parameters for the stochastic kinetic model here

were selected to ma tch experimental measurements of life- 1981). Reported data on N degrada tion fit a first-order decaycurve (Got t esman an d Got t esman 1981); thus N probablytimes.

Degradation of CII and CIII: Two membrane-bound protein does not saturate the protease. Accordingly, N degradation ismod eled as a first-ord er reaction.complexes, H flA ( Cheng et al. 1988) and HflB ( Banu et t et

al. 1986;Her man et al. 1995;Kih ar a et al.1997), act to degra de Cell growth: The linear cell growth assumption was imple-mented as a constant probab ility of a dding a small fixed vol-CII. HflB is thought to degrade CIII ( Her man et al. 1995).

Initial studies identified HflA as a non essential membrane- ume increment each instant of time. Each run was started atan initial cell volume of 1 1015 liter and continued untilboun d G TP-utilizing p rotease ( Cheng et al.1988; Zor icka nd

Ech ol s1991; Nobl e et al. 1993). However, later experiments the volume doubled to 2 1015 liter over 35 min o f simulatedcell time.report that HflA is probably a regulator of HflB activity and

that HflB is the major protease responsible for CII degrada-tion. Additional evidence for proteolytic activity of HflB isprovided by in vitroexperiments with purified pr otein (Her - RESU LTSman et al. 1995; Kiha r a et al. 1997; Shot l and et al. 1997).

Time course of pathway selection: Figures 4 and 5The Hfl proteolytic system has other host protein targets inadd ition to CII ( Cheng a ndE chol s1987;Her man et al.1993; show the temporal trajectory of the concentration ofHer man et al. 1995). Both HflA and HflB respond to host key protein molecules in one lytic and one lysogenicenvironmenta l signa ls. There is some eviden ce tha t HflA activ- case selected from runs at MOI 6. (Figure 2c is basedity is directly or in directly affected by the catab olite-activating

on the same two cases.) The two cases show the ran-protein/cAMP system, which has been shown to reduce pro-dom ness in the intracellular regulatory protein concen-teolytic activity in respon se to car bon -source starvation (H oyt

et al. 1982; Banuet t and H er skowit z 1987). CIII protects tration trajectories and the differences in the trajector-CII from proteolysis (H oyt et al. 1982; Rat t r ay et al. 1984; ies for the divergent developmental paths possible in

Banu et t et al. 1986) even in the absence of H flA and HflB two initially identical cells. O f th e ph age-encoded pro-activity, which implies the existence of yet ano ther proteo lyticteins shown in Figures 4 and 5, Cro 2 and CII are ex-pathwayfor CII degradation (Kih ar a et al.1997). In summ ary,pressed earliest in both the lytic and lysogenic cases.although phenomenology of the proteolysis of CII and CIIICro 2 appeared within 1 min of infection (Figure 5b)is relatively well chara cterized, the exact mechanisms whereby

HflA, HflB, and perha ps anot her uniden tified pro tein con trol and CII appeared within 2 min (Figure 4a). Proteindegradat ion of CI I and p rot ect ion of CI I by CI I I are not expression in the two cases began to diverge after aboutknown.

5 m in. Both the lytic- and lysogeny-fated cases experi-The half-life of unprotected CII h as been observed to be

enced a nearly equal burst of CII production at thisanywhere from 5 min ( H oyt et al. 1982) to less than 30 sectime (Figure 4a), however, in the lysogeny-fated case,depending on conditions (Rat t r ay et al. 1984). The short

half-life of CII and the relatively low concentrations of CII there was a simultaneous burst of CIII production (Fig-protein and HflA/B suggest that binding of CII to the Hfl ure 4b). So lysogeny resulted in this case because, byproteins is tight an d fast. Two alternative mechanisms have

chance, the bursts of CII a nd CIII were both large andbeen hypothesized for the protection of CII by CIII (Cheng

simultaneous so that CII degradation was slowed and it

et al.1988): (i) competitive binding of CII a nd CIII to the H fl

survived long enough to activate PRE and kickstart CI(and perhaps another) proteolytic complex, or (ii) direct CIIIbind ing to CI I to form a pr oteolysis-resistant complex. Avail- production. Figure 4c shows that the CII/CIII proteasesable experimental data does not differentiate between the two were strongly inhibited by th e bursts of CI II pro ductionalternatives. In order to select among candidate models, we in th e lysogenic case. CI2 concentration (Figure 5a) ininvestigated alternative reaction mechanisms seeking a reac-

the lysogenic case began to grow at about 12 min justtion system that meets four con straints: (a) yields a 2-min ha lf-after CII concentration peaked. The growing CI 2 con-life for CII in the range of initial concentrations spanningcentration repressed PR and stopped Cro production.50 100 nm( Got t esman and Got t esman 1981; Cheng et al.

1988), (b ) prod uces an a pproximately 6-min half-life for CIII As a result, the Cro 2concentration declined in the lyso-in t he absence of CI I ( Kor nit zer et al. 1991a), (c) yields geny-fated case after 12 min (Figure 5b). In con trast,CIII protection of CII consistent with H oyt et al. (1982) an d

in th e lytic-fated case no CIII production occurred soRat t r ayet al. (1984), an d (d ) functions in the overall simula-

the unprotected C II rapidly degraded and did not acti-tion model to produce the simulated percent lysogeny as avatePRE enough to start the CI expression feedback loop.function o f API con sistent withKour il sky (1973). Con straint

(d) proved the most restrictive. The best satisfaction of the Without expression of CI, Cro2 production continuedconstraints (a) through (d) was obtained using a proteolytic (Figure 5b) and lysis ensued.system in which CI I an d CIII are competitive substrates for two

Figure 2c shows the intracellular CI and Cro dimerindependent proteases. The resulting reaction mechanism is

concentrat ion trajectory for the lysogenic-fated case atshown in Figure 1a an d rate para meters used are in Table 3.MOI 6 (identical data as Figure 5) superimposed onOne protease ( called P1) is more specific than the other but

saturates at very low concentrations of CII; the other (called th e PR and PRM promoter activation contours. The CI 2P2) is only slightly less specific, d oes no t saturate, a nd has a repressor concentration began to autoregulate its ownmuch h igher ma ximal a ctivity. These proteases correspond to concentration 20 min after infection; thereafter, theHflB and the putative second protea se identified b y Kih ar a

CI2concentration remained constant and PRMactivationet al. ( 1997).slowly increased as Cro 2 concentration was diluted inDegradati on of N:The half-life of the antiterm ination -contro l-

ling protein N is approximately 5 min. Degrad ation is by the the growing cell (Figure 2c, arrow). The concentrat ion


11/16


Figu r e5.Time evolution of Cro and CI dimer concentra-tions for the same two simulation runs at MOI 6 as Figure 4.For the lysogenic case (bold), the high CII concentration after6 min (Figure 4a) leads to the accumulation of CI 2 (a) andcessation of Cro production (b). Dilution and degradationcauses Cro 2concentration to decline thereafter. For the lyticcase, in contrast, the initial burst of CII isn ot sustained ( Figure4a) so that PREis not significantly activated and CI prod uctionis negligible (a) . Cro 2growth begins immediately after infec-tion (b ) and , in lytic cases, continues building until it repressesboth PL a nd PRM thus ending the possibility of lysogeny.

Figu r e 4.Time evolution of CII and CIII concentration

for two casesa t MOI 6illustrating a lytic and a lysogen ic (bold) cells with correspond ing initial conditions,e.g., a partic-outcome. The pattern of protein concentration growth is dis-ular MOI . Figure 3a shows the estimated statistical distri-tinctive and different for every simulation run. Lysogeny re-bution of the CI 2 an d Cro 2 concentration trajectoriesquires early, higher CII concentration as in (a) so that pro-

moter PRE is activated, and protein CI and its dimer begin to for the subset of cells at MOI 6. (For all plots in Figureaccumulate to turn on PRMand repress Cro production. The 3, the bold lines are the average concentration of thehigh CIII concentration in the lysogenic case in (b) protected

indicated species an d the lighter lines a re the 1CII from d egradation. The percent proteolytic activity in (c)

ran ge.) The lysis- an d lysogen y-fated subsets shown incalculated by ((k10( [P1] [ P1 CII])/(total P1)) ( k16Figure 3, b and c, each experience a different pattern([P2] [P2 CII])/(total P2)))/(k10 k16) indicates the

percentage of the total protein activity available for the degra- of Cro 2an d CI 2concentra tion growth statistics, distinctdation of CII. (Figure 5 shows additional protein concentra- from each other and from the combined statistics. Fig-tions from the same simulation runs.)

ure 2d shows the same a verage C ro 2an d CI2concentra-

tion trajectories for the lysogenic-fated and lytic-fated

cases at MOI 6 superimposed on the PR and PRM pro-trajectory for the lytic case in Figure 5 is not shown inmoter activation contours.Figure 2c to avoid confusing the figure. H owever, t he

Lysogenic fraction: Kinetic model estimates com-oval on Figure 2c indicatesth e region where the concen-pared to experiment: The experimental lysogeny frac-trations stabilized at 12 min after infection.tion data shown in Figure 6b for starved O () an dEstimatedstatisticsof concentrationtrajectories: TheP () mutants are from Figure 2 of Kou r il sky( 1973).Monte Carlo solution to the stocha stic kinetic equation sFor the high er API values in Figure 6b, P oisson statisticsproduces a data base of representative time-dependentof infection (Equa tion 1) predicts that some cells wouldsamples of the concentration trajectories as the infec-ha ve q uite high in fection levels. At some infect ion level,tion progresses for each molecular species in the reac-the phage processes must become disruptive to the hosttion system. Analysis of this database provides estimates

of the statistical parameters of the infection progress in cell processes so that the assumptions underlying the


12/16


kinetic model become invalid. To addressthis possibility range of difference between the resulting estimates of

the fra ction of lysogens. The hatched area in Figure 6awe solved for two sets of CII/C III prot eolysis para meters:

indicates the correspond ing ra nge of d ifferences in thefirst, for the best ma tch to the experimental O and

probab ility of lysogeny vs. MOI resulting from the differ-P data for all available API (Figure 6a, labeled Full,

ent proteolytic model para meters. Both curves in Figuresymbol:), an d, second, for the best match co nsidering

6a show negligible lysogenization at MOI 3 a n d aon ly API values 6 (Figure 6a, labeled Lower, symbol:rapid increase in lysogeny for MOI 3. Correspond ing). Points in Figure 6a ( and ) reflect the estimat edpoints in Figure 6b yield the solid lines bounding theprobability of lysogeny in individual infected cells vs.

hatch ed region. These estimates of th e fraction of lyso-MOI from solution of the stochastic kinetic model equa -gens in an infected cell populatio n versus API are ca lcu-tions for these two different cho ices of H fl param eters.lated as the P oisson-weighted sum of points for differentThe vertically hatched area in Figure 6b indicates theMOIs in Figure 6a for correspond ing cases using Eq ua-

tion 2. The match with experiment is good at th e critical

low MOI values, but falls above observed values at high

MOI. We attribute th e overestimation o f the percentage

lysogeny at high API in Figure 6b predominantly to

disruption o f host cell processesa t high infect ion levels.

The thr ee curveslabeled Poisson -nshowthe hypothet-ical fraction of lysogens vs. API expected for a threshold

model, where all cells with MOI n are assumed to

become lysogens. The solution of the stochastic kinetic

model exhibits rapid onset of lysogeny for MOI 2(Figure 6a), representing an approximation to a th resh-

old process in the d ecision circuit prod uced by the rein-

forcing effects of production from multiple promoters

and earlier antitermination as MOI increases. At low

APIs, both the experimental points and the stochastic

model predictions for the O a n d P mutants lie be-

tween the idealized threshold model predictions for

thresholds at MOI 3 and MOI 4.Digital mutants: Add itional tests of the kinetic mod el

by predictions of other experimental observations are

needed; however, we a re unaware of ad ditional, inde-

pendent measurements for similar strains and condi-

Figu r e 6.Comparison of results from stochastic kineticmodel and experimental results from Kou r il sky (1973) forfraction of lysogens produced v s. MOI. (a) Simulation resultsfor fraction of cells prod ucing lysogen s (, ). Curve labeledFull results from choice of proteolysis parameters to matchthe full experimental data set in (b); curve labeled Lowerresults from proteolysis parameters chosen fo r best matchto experimental points at lower API values in ( b). Verticallyhatched area in (b) indicates the range of difference betweenthe resulting estimates of th e fra ction of lysogens. Results withseveral digital mutants are shown in (a): O T, terminationsites removed (x); ON, hence no-antitermination ();OCoop, noncooperative binding of CI dimers atOR13().(b) Solid linesb ounding the hatched region are the predictedfraction of lysogens for the Full and Lower cases in (a) calcu-lated by weighting the results shown in ( a) by the theoreticalPoisson statistical distribution of the number of phage percell at each API. Experimental points for the fraction of lyso-gens for O () and P () strains. Experimental points arefrom Figure 2 in Kou r il sky (1973) for cells starved beforeinfection. The three curves labeled Poisson-n in (b) show the hypothetical fraction of lysogens vs. API expected for athreshold model where all cells with MOI n are assumedto become lysogens. (c) Curves labeled ON/50 and OT/50 are predictions for digital mutants. See text for explanation.


13/16


tions. Accordingly, we include in Figure 6c testable pre- gene expression. The random developmental path

choice between the lysogenic or lytic path in individualdictions of rates of lysogeny for several digital mutan ts

based on changes in the stochastic kinetic model re- cellswas shown to result from the inevitable fluctuations

in the temporal pattern of protein concentration growthflecting several mutant cases. The curve labeled ON

in Figure 6c reflects our prediction of the percent lysog- caused b y the molecular-level therma l fluctuations in

rates of rate-determining reactions within gene expres-eny for a digital mutant with the function of the Nprotein disabled in the kinetic model and all other pa- sion mechanisms. The resulting differences in concen-

tration between the regulatory proteins controlling therameters as for the curve labeled Full. (We use the

O

nota tion to ind icate replication deficient, i.e., ei- bistable switching elements of the decision circuit ledto different path selections in different cells. The esti-ther O or P, mutants.) This is the prediction of per-

cent lysogeny from the kinetic mod el for a starved ON mated variation with API of the fraction of a phage

-infected cell population that become lysogenic wasmutant; the curve labeled N/50 is the correspond ing

prediction for an unstarved ON mutant. [The un- shown to be consistent with experimental observations.

This analysis indicates how molecular level thermalstarved estimate is derived by dividing the starved

estimate by 50, consistent with the observation by Kou- fluctuations can be exploited by the regulatory circuit

designs of developmental switches to produce differentr il sky (1973) tha t star ved cells systematically exhibited

increased lysogeny by 50 100.] The predicted level phenotypic outcomes. Such regulatory mechanisms will

produce diverse phenotypes even in clonal cell popula-of lysogeny is reduced for the ON case because thetranscribing polymerases are never antiterminated and tions maintained in the most homogeneous laboratory

environments. In such systems environm ental signalsproduction o f CII a nd C III is always lower than when N

production leading to RNAP antitermination is possible. can act on the parameters of the regulatory circuit to

bias the probabilities of path choice under differentThe cur ve in Figure 6a labeled OCoop shows thepredicted probability of lysogeny for a digital mutant conditions.

Even in nondifferentiated cell populations, the con-where CI binding to the PR PRM operator sites is made

noncooperative in the kinetic model, reducing the effec- centration of regulatory proteins within individual cells

will vary widely from the average concentration mea-tiveness of positive auto regulation o fPRM. The pr edicted

experimental fraction of lysogens (not shown) is close sured by laboratory procedures. In situations where

there are divergent phenotype subpopulations as ana-to the curves for the ON cases in Figure 6c. The

dashed line labeled OT in Figure 6c is the estimate lyzed herein, the concentration trajectories of each sub-

population can differ radically from each other andfor another starved digital mutant with the TR and TLtermination sites disabled (the line labeled OT/50 is from the average measured for the full population (Fig-

ure 3). For these situations, experimental methodsfor unstarved mutants). The reduced slope of lines for

the OT mutant in Figure 6c is due to th e predicted (such as various cell sorting techniques) that profile the

distribution of individual cell parameters are necessary.increase in the estimated probability of lysogeny for cells

with MOI 1 and 2 for this mutant as shown in Figure 6a. In any case, the function of regulatory circuits that deter-mine cell fates is determined by protein concentration s

within each cell, and th e temporal pattern of these intra-D I S C U S S I O N

cellular concentra tion trajectories is completely differ-

ent from any averaged measurement (compare FigureStochastic gene expression and competitive geneticregulatorymechanisms: In preceding sections a stochas- 3 with Figures 4 and 5).

Roleof termination sitesin the circuit: The simula-tic kinetic model of the lysis-lysogeny genetic regula-tory circuit is used to estimate the dynamical behavior tion results with hypothetical digital-mutations affect-

ing termination effectiveness of th eTR1 andTL1 termina-of the circuit, including effects of random patterns of

TABLE 4

Examplesof other casesof bistable regulatorymechanisms producing stochastic phenotypic outcomes

O rganism Bistable locking mechanism Function

E. coli Pap system (Woude et al. 1996) Different ial methylation of Phase variat ion in p il i exp ression, af fect ingalternat ive L rp binding sites viru lence

E. coli Fim system (Rober t son 1992) I n ve rt ib le D NA se gm en ts P h a se va r ia t io n , t yp e I p il i, a f fe ct in g vir ule n cephage Mu ( Put t e and Goosen 1992) I n ve rt ib le D NA se gm en ts P h a se va r ia t io n in t a il l ea d in g t o d if fe re n t h o st

spec ificit ySalmonell a typhimuri umH in syst em I nve rt ib le D NA se gm en ts P h ase va ria tio n in fla ge llin a lt er s a n tig en

(Put t e an d Goosen 1992) responseM oraxell a bovis( Mar r s et al. 1988) I nver tib le D NA se gm en ts P h ase va ria tio n in p ilin a lt er s a n tig en r esp on se


14/16


tion sites demon strate tha t both remo val of these sites formulation of chemical kinetics, the kinetics of the rate-

and removal of the antitermination effects of the N limiting step(s) will dominate the overall kineticsof a series

protein have major effects on the level of lysogeny, but of cascaded chemical reactions. So, stochastic processes

in opposing directions (Figure 6a). We conjecture that affecting transcription control, posttranscriptional edit-

the function of these phage-encoded features in th e ing, or message transport may be screened by the final

phage regulatory circuit design may be to adjust the translation contro l mechanism. Anoth er possibility islevel of the phage lysogenic response to an optimum that the stochastic pattern of signal protein productionrange for phage survival. may only cause uncertainty in timing of regulatory

Otherstochasticswitchingmechanisms:Although the events, not uncertainty in outcome. Within broa d limitsspecific ana lysis reported here d eals with regulat ion of the duration of many cellular functions may be lessthe phage infection, regulatory circuits based on bista- important to proper cellular function than the properble genetic regulatory mechanisms are used in many sequencing of events. For example, cells halt at variousorgan isms to produce subpopulations of distinct pheno- checkpoints until conditions (e.g., restoration of essen-types by ran dom phenotype switching. Table 4 shows a tial nutrients, completion of precursor cellular events)small sample of well-known cases of bistable regulatory for furth er prog ress are satisfied (Ha r t wel l and Wein-mechanisms in regulatory circuits that produce stochas- er t 1989; Kaufmann and Paul es 1996; Wel l s 1996).tic phenotype outcomes. Many examples are found in In this case, the indeterminism relates to whether thepathogenic organisms. Conventional deterministic ki- cell will progresso r not progress along a developmentalnetics d oes not model statistics of regulatory systems path at any instant, rather than to a choice amon g alter-that produce probab ilistic o utcomes. A stochastic ki- nate stable pathways. So, the regulatory decision logic is:netic an alysis as used in th is paper for the decision H ALT until CO NDITIONS are met then PROC EED,

circuit can be used to predict statistics o f regulatory where CONDITIONS are sensed environmental oroutcomes fo r these stocha stically regulated systems. cellular signa ls. The result is dispersion acro ss the cellSuch stochastic kinetic an alyses may also permit im- population in the rate of progression along prescribedproved exploitation of information in the statistics of pathways rather than dispersion in outcome.phenotypic outco mes.

This work was supported by Office of NavalResearch Gra nt N00014-For bistable switching phenomena that are integral 96-1-0564. A.A. was partially supported by National Science Founda-

to the operation of the cell, some complications encoun- t ion grant CH E9109301. The work was also supported in part bya grant of computer t ime from the Department of Defense Hightered in analysis of the decision circuit would not bePerformance Computing Centers at Eglin Air Force Base and Maui.present. On e example is the uncertainty in ph age geneWe thankLucy Sh apir o for valuable comments on the manuscript.dosage and of timing of infection in the cell cycle due

to the random nature of the phage infection process.

Also, pha ge infection is disruptive o f norma l cell pro-

cesses, whereas the cells integral processes would pre- LI TERATU RE CI TEDsumably be less so.

Adh ya, S.,an d M.Got t esman ,1982 Promo ter occlusion: transcrip-Switching dynamics: We expect that many a spects of

tion through a pro moter may inhibit itsa ctivity. Cell 29: 939944.th e switch dynamics will be found in other multi- Banu et t , F., an d I . Her skowit z,1987 Identification of polypep-

tides encoded by an Escherichia colilocus ( hflA) that governs thepotent regulato ry switches. Examples include: ( i) tran-lysis-lysogenydecision of bacteriophage . J. Bacteriol. 169:4076

sient, low-level expression o f key regulat ory prot eins (a s4085.

with CII, CIII, and N), (ii) stochastic progress toward Banu et t , F., M. A. Hoyt , L. McFar l an e, H. Ech ol sa nd I . Her sko-wit z, 1986 hflB, a newEscherichia colilo cusregulating lysogenycommitment a s concentrations of the controlling pro-and the level of bacteriophage lambda CII protein. J. Mol. Biol.teins in each cell change from moment to moment and187: 213224.

the probability of each potential outcome also changes Bel for t , M., a nd D. L. Wul ff, 1974 The roles of lambda cIII geneand the Escheri chia coli catabolite gene activation system in theuntil eventually the cells fate is determined, and (iii)establishment of lysogeny by bacteriophage lambda. Proc. Natl.design featuresth at adjust timing of reactions as with theAcad. Sci. USA 71: 779782.

delayof CII and CIII concentration growth produced by Bur z, D. S., D. Becket t , N. Benso n an d G. K. Acker s, 1994 Self-termination sitesTL1a nd TR1. assembly of bacteriophage cI repressor: effects of single site

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ism: If random processes of gene expression tend toC h e n g , H . H . , an dH . E c h o l s,1987 Aclasso fEscheri chia coliproteins

make the pattern of protein production inherently er- controlled by the h flA locus. J. Mol. Biol. 196: 737740.Cheng, H. H. , P . J. Muhl r ad, M. A. Hoyt an d H. Echol s, 1988ratic, how do cells achieve the regulatory determinism

Cleavage of the CII protein of phage lambda by purified H flAnecessary for most functions? One possibility is that theprotease: control of the switch b etween lysis and lysogeny. Proc.

overall regulatory a rchitecture can suppress deleterious Natl. Acad. Sci. USA 85: 7882 7886.Dambl y-Cha ud ier e, C., M. Go t t esman , C. Debouk an d S. Adyha,effects of molecular-level reaction fluctuation s. For exam-

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DeVit o, J., an d A. Das, 1994 Con trol of transcription pro cessivity Kiha r a, A., Y. Akiyamaa nd K.It o, 1997 Ho st regulation of lysogenicdecision in bacteriophage lambda: transmembrane modulationin phage : Nus facto rs stren gthe n the termin ation -resistan t state

of RNA polymerase induced by N antiterminator. Proc. Natl. of FtsH (HflB), the cII degrading protease, by HflKC (HflA).Proc. Natl. Acad. Sci. USA 94: 5544 5549.Acad. Sci. USA91: 8660 8664.

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