Regulatory Dynamics of Synthetic Gene Networks with...

doi:10.1016/j.jmb.2006.03.064 J. Mol. Biol. (2006) 359, 1107–1124

Regulatory Dynamics of Synthetic Gene Networkswith Positive Feedback

Yusuke T. Maeda* and Masaki Sano

Department of Physics,Graduate School of ScienceTheUniversity of Tokyo, 7-3-1Hongo, Bunkyo-ku, Tokyo113-0033, Japan

0022-2836/$ - see front matter q 2006 E

Abbreviations used: PFS, positiveNFS, no feedback system; IPTG, isogalactosidase; wt, wild-type; GFP, gprotein.E-mail address of the correspond

[email protected]

Biological processes are governed by complex networks ranging from generegulation to signal transduction. Positive feedback is a key element in suchnetworks. The regulation enables cells to adopt multiple internalexpression states in response to a single external input signal. However,past works lacked a dynamical aspect of this system.

To address the dynamical property of the positive feedback system, weemploy synthetic gene circuits in Escherichia coli to measure the rise-time ofboth the no-feedback system and the positive feedback system. We showthat the kinetics of gene expression is slowed down if the gene regulatorysystem includes positive feedback. We also report that the transition ofgene switching behaviors from the hysteretic one to the graded one occurs.A mathematical model based on the chemical reactions shows that theresponse delay is an inherited property of the positive feedback system.Furthermore, with the aid of the phase diagram, we demonstrate thedecline of the feedback activation causes the transition of switchingbehaviors. Our findings provide a further understanding of a positivefeedback system in a living cell from a dynamical point of view.

q 2006 Elsevier Ltd. All rights reserved.

Keywords: systems biology; synthetic biology; positive feedback; dynamics;hysteresis
*Corresponding author
Introduction

One of the central focuses in post-genomicresearch is to understand how biological pheno-mena arise from the interactions between genes andproteins. These interactions are incorporated intocomplex regulatory networks. Recently, the poten-tial reduction of these systems to smaller functionalmotifs has been proposed. Tomake progress towardunderstanding how sets of regulatory motifsinteract in a large-scale network, several studieshave characterized the properties of elementalregulatory systems, e.g. feed-forward loops1,2 andfeedback systems.3–5 It is of interest to study theproperty of the regulatory structure in the biologicalnetwork.

One of the ubiquitous regulatory systems is afeedback system. This system uses its output as aregulatory input to perform a number of functions.

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feedback system;propyl thio b-D-reen fluorescent

ing author:

One type of feedback is negative feedback, in whichtranscription factors negatively regulate their owntranscription. A single-step negative feedbackreduces cell-to-cell fluctuations in the steady stateof transcription factors,3 alters the spectra of geneexpression noise6 and speeds up the dynamics ofgene expression.7 A double negative feedbacksystem in which each of two regulatory factorsnegatively regulates the synthesis of the othertoggles the gene switch and exhibits hysteresis inthe steady state.8,9 These properties play a centralrole on diverse processes, e.g. differentiation.10

Another important feedback system is the posi-tive feedback system (PFS). Theoretical and experi-mental works have found that cells with the PFSexhibit a bistable response, switching betweendiscrete stable steady states without being able torest in intermediate states.4,11,12 This implies that asingle-gene switch can be constructed as analternative to double negative feedbacks. Such aPFS is included in the bacteriophage lambda lytic-lysogeny switch,13 nutrients utilization systems,14–16 several signaling cascades17–19 and cell cycleregulatory circuits.20,21

However, bistability is not a sole property of thePFS. Savageau has predicted that positive feedback

d.

1108 Dynamics of Gene Networks with Positive Feedback

slows down the kinetics of protein synthesis.22

Although past works have not focused on adynamical aspect of positive feedback, it is import-ant to know how cells reach to the steady state.

Here, we explore properties of the dynamicalbehavior of gene expression by focusing on the rise-time; which is defined as the time required for thegene product to reach half of its steady-stateconcentration. We synthesize three positive feed-back systems in Escherichia coli by using theelements from the lambda phage lysogenicsystem.23 Employing a relatively simple geneticcircuit out of well-characterized elements that donot comparatively interfere with intrinsic com-ponents avoids the impairing cell viability. Compar-ing difference between the no-feedback system(NFS) and the PFS, we show that the kinetics ofthe PFS is slower than that of the NFS. We also showthat two of three PFSs exhibit the hysteretic geneexpression but one of them does the graded one.Next, we propose the mathematical model toanswer the following questions. (I) What mechan-ism generates the response delay? (II) Whatmechanisms convert the hysteretic gene expressionto the graded one? (III) Is there an inevitablerelation between the response delay and hysteresis?The analysis of a mathematical model together withexperimental data reveals that the response delay ofthe PFS results from the accumulation of activatorproteins. Furthermore, with the aid of the phasediagram, we find that the rate of the feedbackactivation transits the switching behaviors. Finally,we discuss the biological function of response delayand hysteresis. These results may help in furtherunderstanding of biological phenomena involvingthe PFS especially from a dynamical viewpoint.

Results

Design of the synthetic genetic networks

To explore a dynamical property of the positivefeedback system (PFS), we employed the artificialgenetic network system. The PFS was isolated fromthe promoter PRM and the cI gene of bacteriophagelambda. These genetic elements provide positivefeedback as follows. The PRM promoter transcribescI in a lysogen.13,24 This promoter consists of threetandem operational sites, OR1, OR2 and OR3. TheseOR sites also play a role as the another promoter:the lytic promoter PR, from which RNA polymerasetranscribes cro and several early lytic genes. Twokinds of repressor dimers, CI and Cro can bind tothese OR sites. The binding affinities for CI of thesesites are such that at increased concentrations CIinitially binds to OR1, then OR2 and finally OR3.The binding affinities for Cro are such that it willfirst bind to OR3 and then to OR1 and OR2. It is theoccupancy of these sites that determines which oneof the two promoters that are active. CI monomersdimerize and then CI dimer binds tightly to OR1.The binding of CI dimer to OR1 prevents the

binding of the RNA polymerase to the PR promoterand thus causes a further repression of transcriptionof cro from the PR promoter. When bound to theOR2 site, the CI protein can interact with thes-factor in the RNA polymerase holoenzyme andincrease the rate of transcription from PRM. Hence,CI positively regulates its own synthesis and itsfeedback activation will ensure that sufficientlyhigh levels of CI protein are presented to maintainthe phage in the lysogenic state. When theconcentration of CI proteins is high, the CI dimerbinds to the low-affinity OR3 site and represses thetranscription of its own gene. The OR3 occupationensures that the rate of transcription of the cI gene isnever high to respond to endogenous signalsproduced by the host cell.

To synthesize the PFS, we positioned the PRMpromoter upstream of the cI-gfp(aav) fusion gene(Figure 1(a)). Although the PRM promoter isintrinsically the weak promoter, the binding of CI-GFP(AAV) dimer to OR1, OR2 enhances thepromoter activity.25 This molecular mechanismcan make CI dimers positively regulate its ownsynthesis. We call such a regulation the feedbackactivation. Then, PRM transcribes cI-gfp(aav) by CI-GFP(AAV) dependent manner. For the tuning of thetranscription initiation, the PRM promoter wasmodified to include the lactose operator lacOdownstream of the promoter.26 In this study, weadopted the O1 type of lacO. A chromosomallyexpressing lactose repressor protein LacI, binds tothe lactose operator and inhibits gene expressionfrom the upstream promoter because LacI excludesRNA polymerase from DNA strand. The chemicalinducer IPTG inactivates LacI. By varying the IPTGconcentration, one can tune the transcription of thedownstream gene by removing the LacI. Therefore,the PRM promoter activity is fully repressed in theabsence of IPTG. Hereinafter we call this regulatorysystem the PRM wild-type positive feedback system(PRM wt-PFS).

As a control, we constructed the Ptrc-2::gfp(aav) asa no-feedback system (NFS) (Figure 1(b)). No-feedback means the absence of feedback regulationby activator proteins while other molecular pro-cesses, a maximal promoter activity and proteindegradation, are identical with those of the PFS. Thetranscription from Ptrc-2 can be controlled by IPTGbecause this promoter includes the lacO operator,the same as PRM-lacO.

In addition, we constructed the mutated PRMpromoter: PRML1L1R3 and PRMR3R2R3 (Figure 1(c)and (d)). Further, we integrated lacO downstream ofthese promoters as same as PRM-lacO. The formerpromoter PRML1L1R3, which was constructed byreplacing each of OR1 and OR2 with the OL1 sitecontained three adjacent sites OL1(upstream),OL1(downstream), OR3. The OL1 operator is oneof the operator sites in the PL promoter that can bebound by the CI protein. The binding affinity of CIto OL1 is larger than that of each of OR1 and OR2.27

According to the in vitro studies, on both OR andOL, we assume that CI dimers initially bind to

Figure 1.Design of the gene expression systems. (a) The PRMwild-type positive feedback system. The promoter regioncontains three operator sites, known as OR1–OR3. The cI-gfp(AAV) gene encodes the activator protein. CI-GFP(AAV)dimerizes and binds to one of the three binding sites, OR1, OR2 (activation) or OR3 (repression). We fused the lactoseoperator, lacO, downstream of the PRM promoter. A chromosomally encoded lactose repressor, LacI, binds to lacO andrepresses the transcription of the cI-gfp(AAV) gene. The inducer IPTG binds and inactivates LacI, causing the expressionof CI-GFP(AAV). (b) The no-feedback system. This system has a Ptrc-2 promoter instead of PRM-lacO. Cells expressingGFP can be induced by IPTG, just as in the positive feedback system except for the lacking cI region. (c) Variants of thePRM promoter. The l-phage PRM promoter contains three neighboring binding sites: OL1, OR1 and OR2. They allowcooperative activation. (d) The OL1, OR1, OR2, OR3 sequence.

Dynamics of Gene Networks with Positive Feedback 1109

whether the upstream OL1 or the downstream onealternatively, then the remaining OL1 and finallyOR3 as the CI concentration increases. We call thePFS with PRML1L1R3-lacO as the PRML1L1R3positive feedback system (PRML1L1R3-PFS). Thelatter promoter PRMR3R2R3 was constructed bysubstituting OR1 with OR3. The order of a CI dimerbinding to these sites are such that dimers firstlybind to OR2, then whether the upstream OR3 oralternative downstream one, and finally anotherOR3.28 We also call the PFS with PRMR3R2R3-lacOas the PRMR3R2R3 positive feedback system (PRM-R3R2R3-PFS).

The dynamics of positive feedback: positivefeedback shows response-delay

To study the response kinetics of the PFS, thetime-evolution of gene expression from culturesbearing the reporter plasmid was measured by theautomated multiwell fluorimeter.7 Green fluores-cent protein (GFP) fluorescence and optical absor-bance A600 were measured at about 5 min intervals.Cultures were maintained within the mid-exponen-tial growth phase during the measurements.

Gene expression was induced by adding IPTG.The raw time dependence of the fluorescenceintensity was smoothened with a window size offour activities. We define the rise-time t, as thatrequired for a gene-expressed product to reach halfof its steady-state concentration, x(t)Zxmax/2. Theobserved GFP fluorescence reflects the total CI-GFP(AAV) fluorescence including CI-GFP(AAV)monomers, free dimers and DNA–dimercomplexes.The GFP fluorescence from cells with the PRM wt-

PFS increases slower than that with the NFS(Figure 2(a)). After the short time lag, the fluor-escence of the NFS is rapidly expressed and then thepromoter activity saturated to the constant(Figure 2(a), blue). The rise-time of the NFS is50 min (tNFSZ50 min). The kinetics differ signifi-cantly from that of the PRM wt-PFS. In the PFS, theincrement in the promoter activity of the PRM wt-PFS is small for the low CI-GFP(AAV) range(Figure 2(a), red). However, at a later time (after50 min) when the activator concentration is in anintermediate range, a small increase in the activatorconcentration results in a large change of theproduction rate. The rise-time of the PRMwt-PFS is

Figure 2. (a) Experimental dynamical behaviors of the synthetic genetic networks. Fluorescence per one cell,normalized by its maximal value, is plotted versus time. The rise-time is the time taken to reach half of the maximalproduct concentration. This is the time at which the curves intersect the horizontal line at relative fluorescenceZ0.5.Blue, the NFS (IPTGZ1 mM); black, the NFS (IPTGZ0.005 mM); red, the PRMwt-PFS; green, the PRML1L1R3-PFS;orange, the PRMR3R2R3-PFS. Maximal fluorescence of the NFS (IPTGZ0.005 mM) is comparable in intensity to that ofthe PRMwt-PFS. The error bars are standard deviation at rise-time. (b) and (c) Day-to-day reproducibility of the high-resolution measurement. Unaveraged GFP fluorescence (b) and absorbance (c) (background subtracted) for repeatedexperiments performed on different days. The mean relative errors for the absorbance and fluorescence measurementsare 10%, respectively.


120 min (twtPFSZ120 min), which corresponds toapproximately one and one third of a cell cycle. Toevaluate the response delay, we define tKtNFS asthe delay time. The delay time of the PRMwt-PFS is70 min, which is approximately 0.8 times of a cellcycle. However, comparing the two systems at1 mM IPTG, the maximal fluorescence of the NFS isten times as large as that of the PRMwt-PFS. In orderto exclude the possibility that the response delayarises from the difference of the promoter activity,we measured the time-evolution of the NFS at0.005 mM IPTG. The maximal GFP expression ofthe NFS at 0.005 mM IPTG is approximatelycomparable to that of the PRMwt-PFS at 1 mMIPTG. We find that the dynamics do not depend onthe maximal level of protein (Figure 2(a), black andblue). This result suggests that the response delay isindependent of the maximal saturation level of the

gene product. Then, we note that the fusionconstruct of the PRMwt-PFS is identical with thebi-cistronic one at the dynamics level (see Sup-plementary Data, Figure S1). We conclude that theresponse delay is an intrinsic property of the PFS.

For the further characterization of the dynamicalproperty of the positive feedback system, weexamine whether the rise-time is increased ordecreased by genetic mutations. As seen inFigure 2(a) (green), the rise-time of the PRM-L1L1R3-PFS at 1 mM IPTG is 90 min (tL1L1R3-PFSZ90 min), which corresponds to one cell cycle. Thedelay time is 40 min, which is approximately onehalf of a cell cycle. The PRML1L1R3-PFS exhibits thesteeper, and its maximal fluorescence is 1.1-fold ofthe PRMwt-PFS.

We constructed another mutated PFS: the PRM-R3R2R3-PFS whose binding affinity of the CI dimer


to the uppermost-stream operator site becomeslower in vitro.28 In contrast to the others, thePRMR3R2R3-PFS at 1 mM IPTG shows the slowestkinetics. Its rise-time is 250 min (tR3R2R3-PFSZ250 min), which corresponds to 2.8 times a cellcycle. The delay time is 200 min, which is 2.2 times acell cycle. Further, its maximal fluorescence is0.7-fold of the PRMwt-PFS.

These results suggest that (i) the rise-time of thePFS is larger than that of the NFS and (ii) the rise-time is influenced with the genetic mutation.

Positive feedback switches gene expressionin a hysteretic or a graded manner

Another important characteristic of a PFS is itsown history dependent behavior, so-called hyster-esis. In other words, it is easier to maintain thesystem in its on state than to toggle it on and off. Thesystem requires a larger signal to switch from low tohigh (low/high) comparing with from high to low(high/low); low means the state with the loweractivator concentration, while high is the state withthe higher activator concentration. However, cellswith the PFS do not always exhibit hysteresis. Cellsalso switch gene expression in a gradedmanner. If aPFS expresses a gene gradually, the expression levelmoves reversibly between low and high. Todetermine whether the PRM PFSs exhibit thehysteretic gene expression or the graded one, weperformed the following experiment. The overnightcultures were grown in a fresh minimal mediumwith either the presence or absence of 1 mM IPTG. Itis crucial to use cells with well-defined initial states,either high (at 1 mM IPTG) or low (at 0 mM IPTG),because the response of a hysteretic system isexpected to depend on its history. Cells weremaintained in a mid-exponential growth phase.GFP fluorescence was measured after six to sevengenerations. We determined whether they werehysteretic or not if the difference of CI-GFP(AAV)fluorescence between high/low and low/highwas larger or smaller than the error values,respectively.

We find that cells with the PRMwt-PFS expressedin a critically different manner between increasing(low/high) and decreasing (high/low) IPTG(Figure 3(a)). Cells that have been initially in thehigh CI-GFP(AAV) state remain in that state at IPTGconcentrations O0.0025 mM. On the other hand,cells in the low state switch to the high state whentreated with IPTG at 0.05 mM or higher. Thehysteretic response is observed at IPTG concen-trations of between 0.0025 mM and 0.05 mM. On thecontrary, cells with the NFS exhibits not such ahysteresis but the graded response (Figure 3(b)).

Next, to determine whether cells with the variantPFS exhibit hysteresis or not, both the PRML1L1R3-PFS and PRMR3R2R3-PFS were subjected to thesame procedure. In PRML1L1R3-PFS, the hystereticgene expression is observed at IPTG concentrationsof between 0.005 mM and 0.02 mM (Figure 3(c)).However, the profile of PRMR3R2R3-PFS is graded

and reversible without showing any signs sugges-tive of hysteresis (Figure 3(d)). These resultsconfirm that cells with the PRM PFSs producewhether the hysteretic or the graded response.We mention that E. coli has an inner PFS during

lactose utilization. In this study, both the syntheticPFS and NFS can ignore the inner system. It isbecause glucose is included in the minimal mediumand inhibits LacYprotein.12,29 Furthermore, glucosedecreases the intracellular concentration of cAMP.Then, the transcriptional activator complex CRP–cAMP cannot induce the lac operon. Therefore, wecan control gene expression without the concomi-tant maintenance. Thus, we consider that thepositive feedback system could be controlled bythe single parameter; the concentration of IPTG, I.As shown, both the response delay and hysteresis

are the major characteristics of the syntheticpositive feedback systems. However, the followingquestions arise. (I) What mechanism generates theresponse delay? (II) What mechanism converts theresponse from hysteretic to the graded one? (III) Isthere any inevitable relation between the responsedelay and hysteresis? To answer these questions, wepresent mathematical analyses based on thedynamics of gene expression in the followingsections.

Mathematical analysis: the model and dynamics

First, to answer the question (I), we analyze themathematical model describing the processesdepicted in Figure 1. The model is the variant ofthe Collins model, which is derived from anapplication of the slow chemical reactions.12 Themain difference is that we ignore the influence ofcell growth on chemical reactions. In Supplemen-tary Data, we provide a detailed description of thederivation of the dynamical model. Althoughstatistical mechanics models have been alreadyproposed,30,31 the present model is not meant tobe a detailed account of the full biochemistry of thesystem, rather a simple model that captures theessential behavior.For the NFS with the constant production rate,

one assumes that:

dx

dtZaðIÞKgx; (1)

where x is the mean GFP concentration per one cell.The degradation term is defined as Kgx. Theproduction term a(I) is the function of the IPTGconcentrations, I. By simple mathematics, we obtainthe rise-time of the NFS as tZln 2/g. The effectivedegradation rate of protein is dependent on bothproteolysis and cell division. However, the rise-timeof the NFS (50 min) is larger than the expected valuefrom the simple mathematics, 36 min (see Supple-mentary Data). The difference might be caused from(i) the initial lag-time in the gene expression, e.g.GFP maturation and transcription, or (ii) the over-simplification of mathematics. In this study, because

Figure 3. Hysteretic response or graded one in the positive feedback system. The mean GFP expression of (a) thePRMwt-PFS, (b) the NFS, (c) the PRML1L1R3-PFS and (d) the PRMR3R2R3-PFS versus the IPTG concentration of threeindependent experiments. Red, low/high (going up); blue, high/low (coming down). The green region shows therange of IPTG concentrations over which the gene expression occurs hysteretically. Cells in the high or low state weresuspended and washed in a medium lacking IPTG, when diluted 1/100 into a medium containing the indicated IPTGconcentration. Six to seven generations were grown, after which GFP fluorescence was measured.


our main focus is the dynamic property of the PFS,we do not try to pursue this inconsistency. In themeantime, we define the effective half-life of GFP as50 min. It leads to gZ0.014 (1/min).

The promoters used in the experiment areregulated by the lac operator system. According tothe simple model for the lac regulated promoter,32

the production term is defined as:

aðIÞZamaxsI C I

2

KI C I2; (2)

where sI is the basal rate of synthesis under the fullrepression, and KI is the Michaelis constant of anti-repression. This function represents the promoteractivity, which is regulated by IPTG. amax is themaximal rate of production from promoter.

The simulated result agrees well with exper-iments in the case with the NFS (Figure 4(a), blue).

A PFS is one in which the production ratedepends on the intracellular concentration of thegene product. The following model is derived froma straightforward application of the chemicalkinetics describing the processes depicted inFigure 1(a). The time evolution of the number ofCI-GFP(AAV) monomers is:

dx

dtZ

1

hðxÞ ðf ðx; IÞKgxÞ; (3)

where x is the mean concentration of the CI-GFP(AAV) monomer per one cell. The productionterm f(x,I) represents the effect of the transcriptionand translation, whereas h(x) arises from the vastseparation of time scales set by the transcriptionand protein dimerization rates. This model isclosely similar to the Collins model.12 Their


functional forms are as follows:

f ðx; IÞZaðIÞ 1CK1K2x2CbK1K2K3x

4

1CK1K2x2CK1K2K3x

4CK1K2K3K4x6

(4)

hðxÞZ1C4K1xC4K1K2xO0ðxÞC16K1K2K3x3O0ðxÞC36K1K2K3K4x

5O0ðxÞ; ð5Þ

O0ðxÞZm

1CK1K2x2CK1K2K3x

4CK1K2K3K4x6;

(6)

Figure 4 (legend

XZxC2K1x2C2K1K2x

2O0ðxÞC4K1K2K3x4O0ðxÞC6K1K2K3K4x

6O0ðxÞ; ð7Þwhere X is the total concentration of CI-GFP(AAV)per one cell, and O0(x) represents the promoterconcentration with no CI dimer. The form of theproduction term f(x,I) dictates the equilibriumnumber of CI monomers. The even polynomials inx occur due to the dimerization and the subsequentbinding to the operator sites. We ignore cooperativenon-specific DNA binding by octamerizing CIproteins because the differences between non-specific and specific binding affinities for CIrepressor are large.33 K1 represents the dimerizationaffinity of CI monomers. Ki (iZ2–4) represents the

on page 1116)

Figure 4 (legend on page 1116)


binding affinity of the activator to operator sites. bdenotes the ratio of the transcriptional activity ofthe two CI dimers binding state to the no activatorbinding state. Ki/K2 (iZ3,4) denotes the relativeaffinities for dimer binding to OR1 versus that ofbinding to OR2 or OR3. The fact that bO1 on the x4

term is present because transcription is enhancedwhen the two operator sites OR1 and OR2 areoccupied. The x6 term represents the occupation ofall three operator sites and disappears from thenumerator, because the OR3 occupation by the CIdimer inhibits polymerase binding and shuts offtranscription. Because only CI can activate tran-scription, the promoter activity increases with theCI concentration. Utilizing equation (2), we can

rewrite the production function as:

f ðx; IÞZamaxsICI

2

KICI2

!1CK1K2x

2CbK1K2K3x4

1CK1K2x2CK1K2K3x

4CK1K2K3K4x6;

(8)

where amax is the maximal rate of synthesis.Then, we performed the best-fit random para-

meterization (see Materials and Methods) with thereference to the molecular biological research. Thetheoretical kinetics of each of the systems repro-duces the experiment well (Figure 4(a)). The

Figure 4 (legend next page)


parameters appearing in equations are listed inTable 1.

For the operator region, we obtain K3/K2z5, K4/K2z0.03, bz18 in PRMwt-PFS. In the PRML1L1R3-PFS, the expected increase of both K2 and K3 areobserved. In the PRML1L1R3-PFS, the ratios of theCI binding affinities are K3/K2z4 and K4/K2z0.02.Further, bz21 is larger than that of the PRMwt-PFS.In the PRMR3R2R3-PFS, parameters b and K4 aresmaller than those of the PRMwt-PFS (bz3.5, K4/K2z0.005), while the absolute K2 and K3 values arethe same as those of the PRMwt-PFS. To sum up, wefind that a higher rise-time system has higher b andKi (iZ2–4), whereas a lower rise-time systemhas lower ones. Of course, the rise-time dependson various parameters. However, we especially

investigate the dependence of a rise-time on b andKi for a representative example.Figure 4(b) shows the rise-time as a function of b

and K3 for the PRMwt-PFS. We find that the PFS witha larger b value has a smaller rise time. It can beinterpreted as follows. If b is increased, the feedbackactivation enhances. This enhancement leads to anincrease in the number of the transcriptionalproducts for a given value of x. Hence, many ofthe products will provide the fast CI accumulation,and in turn, CI will induce further transcriptionagain. As a result, the large feedback activationdecreases the rise-time. Next, we consider the K3dependency. The increase of K3 reduces rise-timeunder the condition that gene expression isfully induced (XmaxO30,000). If K3 increases, the

Figure 4. Simulated dynamical behaviors of the positive feedback systems. (a) Simulated gene expression dynamics.Shown is normalized total protein concentration over time. Experiments (points), and the mathematical model using thequantified parameters (continuous line). Blue, the NFS; red, the PRMwt-PFS; green, the PRML1L1R3-PFS; orange, thePRMR3R2R3-PFS. (b)–(g) The map of rise-time andmaximum level of total CI concentration for the PRMwt-PFS. Shown in(b), (d) and (f) are the contour-like maps of rise-time as a function of (b, K3), (b, K2) and (b, K4), respectively. (c), (e) and (g)Maximal GFP concentration as a function of (b, K3), (b, K2) and (b, K4), respectively.


formation of the CI–operator complex becomesfrequent. The frequent CI binding increases thechance of transcription per unit time. So, thefrequent transcription yields the faster CI accumu-lation. In turn, the fast accumulation will induce therapid activation of PRM promoter. As a result, thekinetics of the gene expression become faster, andthe rise-time becomes smaller.

It has been already noted that the maximal levelsof fluorescence are different among the PFSs.Figure 4(c) presents the total CI-GFP(AAV) concen-tration as a function of b and K3. As shown inFigure 4(c), the maximal activator concentrationsignificantly depends on b but not K3. This resultcan be interpreted as follows. If x is near themaximal value, we can approximate the productionterm as f ðx; IÞyaðIÞb because both K3[K4 andK1K2K3x

4[1 are satisfied near the steady state.From simple mathematics, we obtain a(I)b/g as theapproximate maximal total CI concentration. These

Table 1. The variables and the parameters used in the model

Nomenclature W

amax The maximal promoter activityK1 (!10

K4) The dimerization constant 1K2 [!10

K6) CI dimer-O0 affinity 1K3 (!10

K6) CI dimer-O1 affinity 6K4 (!10

K8) CI dimer-O2 affinity 4b The rate of feedback activationg The degradation ratem The plasmid copy numbersI (!10

K4) The leakiness of lac regulationKI (!10

K4) The repression factor of lacregulation

results suggest that the change of b causes theobserved changes of the maximal fluorescence inthe variant PRM PFSs. In the PRML1L1R3-PFS, bz21is larger than that of the PRMwt-PFS. Whereas in thePRMR3R2R3-PFS, bz3.5 is substantially smallerthan that of the PRMwt-PFS. The same results holdfor the rise-time as a function of b and K2(Figure 4(d) and (e)). However, the rise-time isdecreased as K4 becomes larger (Figure 4(f)). This isbecause K4 represents repression by the OR3occupation, so the increase of K4 leads to the strongnegative feedback. Gene expression with thestronger negative feedback is faster than that withthe weaker one because the stronger one shuts off itsown production to reach the required steady-stateconcentration7 (Figure 4(g)). As a result, the rise-time becomes smaller.

It is plausible that the other parameters also affectthe rise-time. For example, the model suggests thatthe increase of the degradation rate of protein g

s

ild-type L1L1R3 R3R2R3

68G4.7 68G4.7 74G2.0.0G0.3 1.0G0.4 1.0G0.2.2G0.4 1.8G0.7 1.1G0.1.7G2.2 7.1G2.8 6.7G0.5.1G2.0 4.1G0.8 0.55G0.05718G3.7 21G3.6 3.5G0.20.014 0.014 0.01410 10 100.25 0.25 0.256.8 6.8 6.8


shifts the rise-time toward the smaller value (datanot shown).

Although g is considered as the constant value, gcan also affect the speed of gene expression (datanot shown). We also need to note the difference ofthe time-evolution at the later time betweenexperiment and model. This discrepancy wouldbe caused by the normalization. The maximal valueof gene expression is defined as the CI-GFP(AAV) atthe end of this phase, 600 min. However, as thesystem does not reach the steady state at that time,the experimental maximal values may be under-estimated.

Reconstruction of switching behaviors fromexperimental dynamics

Second, we address the question (II). Thisexploration is important to understand the hystere-tic gene switch.

In this model, the feedback activation and proteinmultimerization yield non-linearities. These non-linearities lead to a multistable regime in the steadystate. The transition from graded (one stable fixedpoint) to hysteresis (two stable fixed pointsseparated by one unstable fixed point) occurswhen equation (3) admits precisely three solutions:this signifies the onset of a saddle-node bifurcation.The behavior of the hysteretic gene network can bedescribed by using equation (3). To analyze thegeometric structure of the model, the nullclines ofequation (3) are determined as follows.34 First, dx/dtZ0 is set to 0, from which follows:

gxZaðIÞ 1CK1K2x2 CbK1K2K3x

4

1CK1K2x2 CK1K2K3x

4 CK1K2K3K4x6:

(9)

equation (9) has fixed points. These fixed points canbe found graphically by plotting the left and theright-hand sides of equation (9) and by seeking forthe intersections. Figure 5(a) shows that there areone or three intersections, depending on the valueof a. Other parameters in the model equations arefixed at the values of the PRMwt-PFS. Assume a isan intermediate value am for which there are threeintersection x1, x2, and x3 (x1!x2!x3) (Figure 5(a),red). As a is decreased from am, the two intersec-tions x2 and x3 approach each other and eventuallycoalesce in a saddle-node bifurcation when the lineintersects the curve tangentially. If a is decreasedfurther, there is one intersection, x1, and thereforeno additional solutions apart from xZx1(Figure 5(a), green). On the other hand, as a isincreased from am, the two intersections x1 and x2approach each other and eventually disappear. Thegene expression develops to the sole intersectionxZx3 (Figure 5(a), blue). We also find the decreaseof f(x,I) around larger values of x. This decline isresulted from the x6 term. The repression by theOR3 occupation becomes significant as x increases.

In the region with the three intersections, the topand bottom intersections are stable, so that protein

concentrations near these values will remain. At themiddle point xZx2, the activator degradation andproduction are equal. However, if x decreasesslightly, the resulting positive feedback expressionis weaker and x drops to x1. If x increases slightly,the activator production increases dramatically andx increases to the new fixed point x3. The middlepoint is unstable so that tiny fluctuations will drivethe protein concentration toward one of the stablestates. Hence, within this region, the system willhave two experimentally accessible states and thusis multistable. In summary, the fate of the system isdetermined by the initial state x0. xZx3 is achievedfor x0Ox2. xZx1 is achieved for x0!x2.On the other hand, if equation (9) has only one

intersection over 0!a!70 in which corresponds to0 mM![IPTG]!1 mM, the model equation pro-duces a graded switch. The sole stable fixed pointdetermines the final state of the system.Figure 5(a) also reveals an important feature of

the system. Hysteresis exclusively occurs if thereare one unstable and two stable fixed points. Thisresult is only the case if the curve intersects theline three times. The reason to have three intersec-tions comes from the inflected shape of a curve froma low to a high state. The shape of this curve isdetermined by the weight of binding activities, x2,x4, x6 and the parameters b and Ki, as described bythe term f(x,I). We define that hysteresis occurs ifcells exhibit bistable gene expression. The modelequations predict that hysteresis occurs between0.01 mM and 0.018 mM IPTG at the experimentalparameter condition (Figure 5(b)). Furthermore, themodel equations in which the parameters are fixedat the value of the PRML1L1R3-PFS also showhysteresis (Figure 5(c)). The parameter regionwhere the hysteretic response appears is includedin such a region observed in the experiment.However, the model in which the parameters arefixed at the value of the PRMR3R2R3-PFS switchesgene expression in the graded manner the as sameas in the experimental result (Figure 5(d)). This isbecause the model has only a mono-stable fixedpoint over a (Figure 5(e)). Therefore, the switchingbehavior is graded, and its gene expression alwaysdevelops toward a mono-stable point. Thesesimulated results qualitatively agree with theobservation on the experiments. However, theparameter region where hysteresis appears isnarrower than the experimental results. We willdiscuss the expansion of the experimental hystereticregion in the Discussion.

The rate of the feedback activation transitsswitching modes

Although our mathematical model reproducesthe experimental results, it remains unknown whatthe condition is necessary for observing hysteresis,and what molecular mechanism critically deter-mines the transition of switchingmodes.We answerthis problem by using the phase diagram. Thephase diagram is useful to know the transition

0

100

200

300

400

0 5 10 4 15 20 × 103

20 × 103

degradation

0.06mM IPTG

0.005mM IPTG0.015mM IPTG

x (CI monomer) [a.u]

f(x,

I), γ

x [a

.u]

Nor

mal

ized

tot

al C

I con

cent

ratio

n pe

r ce

ll

IPTG [mM]

StableUnstable

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1

IPTG [mM]

0.001 0.01 0.1 1

IPTG [mM]

0.001 0.01 0.1 10

0.2

0.4

0.6

0.8

1

Nor

mal

ized

tot

al C

I con

cent

ratio

n pe

r ce

ll

StableUnstable Stable

Nor

mal

ized

tot

al C

I con

cent

ratio

n pe

r ce

ll

0

50

100

150

200

0 5 10 15

x (CI monomer) [a.u]

degradation0.005mM IPTG0.015mM IPTG0.035mM IPTG0.06mM IPTG

f(x,

I),

γ x

[a.u

]

(a) (b)

(c) (d)

(e)

Figure 5. Hysteretic and graded responses reconstructed from the experimental dynamics. Graphical plot of the leftand right-hand side of equation (9). For a given a value (which represents the promoter activity and corresponds to theIPTG concentration), the fate of the system is determined by x0. For x0Ox2, the final state is xZx3. For x0!x2, the resultingfinal state is xZx1 (x1!x2!x3). (a) Nullclines of the model of the PRMwt-PFS. Green line, for small a (corresponding tolow IPTG concentration and representing low promoter activity), the stable fixed point is only one point (green, filledcircle xZx1). Red line, for intermediate a (corresponding to intermediate IPTG concentration and representingintermediate promoter activity am), there are three fixed points; filled circle xZx1, open circle xZx2 and filled circle xZx3. The two points are stable (x1, x3), but one is unstable (x2). Blue line, for very large a (increasing IPTG concentration),the two fixed points x1 and x2 converge and disappear. The x3 (blue, Filled circle) is the final state of the system. (b)–(d)Hysteresis in gene expression of positive feedback systems. The mean GFP expression of (b) the model of the PRMwt-PFS



points of switching manners as a function of theparameter.14

Figure 6(a) shows the phase diagram for theproperties of gene expression of the model in whichthe other parameters are fixed at the value of thePRMwt-PFS as a function of a and K3. This Figureshows that gene expression of this model alwaystakes place hysterically. On the contrary, the a-K3phase diagram of the model in which the otherparameters have the value of the PRMR3R2R3-PFSalways shows the graded response (data notshown). As well as K3, the change of both K2 andK4 do not transit the switching behavior (Figure 6(b)and (c)). These results suggest that the affinitybetween the operator and the activator does notcause the transition of switching behavior.

Next, we examine whether b changes the switch-ing behaviors or not. The a-b phase diagram of themodel of PRMwt-PFS shows that the cells exhibithysteresis around bz18 (Figure 6(d), dotted line).While b is decreased from the critical point bwtc z6:2,the cell also switches gene in a graded fashion, withthe expression levels of cells moving continuouslybetween low and high values (Figure 6(d)). Assimilar, the a-b phase diagram of the model of thePRMR3R2R3-PFS shows that the switching behaviormoves from the hysteretic manner to the graded oneon reaching a critical value of b, bR3R2R3c Z3:6(Figure 6(e)). The b value of the PRMR3R2R3-PFS isbelow the critical point bR3R2R3c (Figure 6(e), dottedline). These results suggest that the experimentallyobserved change is a shift over the critical point of aa–b phase diagram.

Finally, we consider the reason why the PRM-R3R2R3-PFS has the lower b value as observed inthe experiment. We propose the following mechan-ism as a probable one. Because the highest affinityoperator is OR2 in the PRMR3R2R3-PFS, a first CIdimer binds to OR2. But whether a second CI dimerbinds to the upstream OR3 or the downstream onewith an equal probability. If a second CI dimerbinds to the upstream OR3, it functions as thetranscriptional activator. On the contrary, if asecond CI dimer binds to the downstream OR3, itfunctions as the transcriptional repressor becausethe binding competition with RNA polymeraseoccurs. Such transcriptional repression decreasesthe effective feedback activation. Such a decrease ofthe feedback activation causes the transition from ahysteretic to a graded response.

Discussion

This study has demonstrated experimentally andtheoretically that the PRM PFS slows down thedynamics of the genetic networks. It is remarkablethat the mathematical model predicts that the two

(c) themodel of the PRML1L1R3-PFS and (d) themodel of the Phigh (going up); blue, high/low (coming down). The mathepositive feedback systems. (e) Nullclines of the model ofoccurred at one point. It means the PRMR3R2R3-PFS does no

mechanisms change the rise-time. One is the rate ofthe feedback activation and the other is the affinityof the operator–CI dimer interaction. For example,the former decreases the rise-time by the enhance-ment of the transcription but the latter increases therise-time by reducing the frequency of the operator–CI complex formation. In addition, we find thatcells with the PFS exhibit both the hysteretic and thegraded response. The transition of switchingbehaviors is controlled by the rate of the feedbackactivation. In this section, we discuss about (1) thebiological function of the response delay, (2) therelation between response delay and hysteresis and(3) the origin of the differences between theexperimental results and the theoretical calcu-lations.

Biological function of response delay

The important question is whether a responsedelay induced by the effect of positive feedback isused to perform a certain function in a transcrip-tional regulatory network. We take the lytic-lysogenic decision of the bacteriophage lambda asa representative example. The CI and Cro regula-tors define the lysogenic and lytic states, respect-ively. After an infection, host bacterium almost goesinto a lytic state by Cro. This differentiation is madethat is dependent upon environmental signals andthe number of infecting phages. But if the lysogenicstate is chosen, the stable lysogenic prophage ismaintained by the PRM-cI PFS. Cro is produced fromthe PR promoter without regulation, whereas CI issynthesized from the PRM promoter with positivefeedback. According to our results, it is expectedthat the initial rise of Cro production is typicallyfaster than that of CI if the degradation rates of tworegulators are identical. Moreover, the rise-time ofthe PRMwt-PFS is about 120 min. It is larger than thetime required for a decision of cell fates, which is20–30 min.35 The establishment of the lysogenicstate is much later than the period of judgment.Thus, the response delay allows cells to carefullydecide their own fate and to turn the lytic stateexclusively.A similar mechanism has been proposed in the

flagella synthesis system.36 The flagella synthesissystem also has the positive feedback systemcomposed of the master regulator fliA. This PFSacts to prolong the expression of FliA after signaldeactivation, and thus to protect flagella productionfrom transient loss of input signal.

Relation between response delay and hysteresis

The second important question is the question(III); what kind of inevitable relation exists or not

RMR3R2R3-PFS versus the IPTG concentration. Red, low/matical model reproduces the switching behaviors of thethe PRMR3R2R3-PFS. Independent of a, the intersectiont have multi-stability.

Figure 6. Phase diagram of switching behaviors. The phase diagram of hysteretic and graded responses. It is possibleto move from an uninduced state to an induced state either hysteretically (gray region) or in a graded manner (whiteregion). (a) A a-K3 phase diagram for the model of the PRMwt-PFS: the vertical line indicates a, which corresponds to theIPTG concentrations and also represents the promoter activity. The horizontal line indicates K3, which represents theaffinity of OR2-CI dimer. The dotted line represents the experimental PRMwt-PFS. The region of hysteresis is robust to K3.(b) A a-K2 phase diagram for the model of the PRMwt-PFS. (c) A a-K4 phase diagram for the model of the PRMwt-PFS. Theregion of hysteresis is also robust to both K2 and K4. (d) and (e) a-b phase diagrams; (d) for the model of the PRMwt-PFS;and (e) for the model of the PRMR3R2R3-PFS. The dotted lines in (d) and (e) represent the experimental PRMwt-PFS andthe experimental PRMR3R2R3-PFS, respectively. The horizontal line indicates b, which represents the rate of feedbackactivation. The region of hysteresis grows smaller as b is decreased, eventually reaching a critical point (C) at the rate offeedback activation bwtc Z6:2 and b

R3R2R3c Z3:6, respectively. A graded gene switch is predicted to occur beyond this

cusp.



between the response delay and hysteresis? In fact,there is no inevitable relation between the responsedelay and hysteresis. Cooperativity is not necessaryfor the PFS to generate the response delay. Even ifthere is no cooperativity in the PFS (Hill coefficientnZ1), the response delay arises (Appendix of Kalir,et al.36). On the other hand, the emergence ofbistability (hysteresis) needs cooperativity (Hillcoefficient nR2). Therefore, the delayed geneexpression does not always guarantee the hystereticgene expression.

For another example, the system with thecoherent feed-forward loop has been known togenerate the sign-sensitive delay.37 However, such asystem does not exhibit hysteresis because thissystem only has one stable fixed point. So, theresponse delay is not also sufficient for hysteresis.

On the other hand, we examine whether thesystem with the double positive feedback loops,one responds fast and another slower, exhibitsthe irreversible response the same as hysteresis.However, the significant response delay has notalways been observed in such a system.38 Thus,hysteresis is not also sufficient for the responsedelay.

From the viewpoint of dynamics, the responsetime is related to the stability of the equilibrium. Ifa gene product is initially in an equilibrium state,external or internal perturbations to the steadystate will be performed by an exponential decayor rise back to equilibrium. The stability of thefixed point is given by the time constant in theexponential response; high stability corresponds toa fast return to equilibrium, whereas low stabilitycorrelates with a slow return. With regard tostability, the central relation is that stability isweakened with activation.39 So, the responsedelay is related to the stability of the fixed points,which are weakened by the positive feedbacksystem. However, we need further experimentalevidence to prove this consideration at a singlecell level.

Comparison between experiments and models

Models based on the experimental dynamicshave correctly explained the overall switchingbehaviors. However, the quantitative differenceshave been observed between the theoretical calcu-lations and the experimental results. The hysteresisobtained by the experiment is wider than thetheoretical prediction. What mechanisms do causethese differences? We propose possible mechanismsby the mathematical analyses.

The experimental hysteresis is observed between0.0025 mM and 0.05 mM IPTG. However, the modelshows the hysteresis at the narrower range near the0.02 mM IPTG. Concerning the lower IPTG concen-tration region, we suppose the long-distanceinteraction between the LacI-lacO and the CI-OR2as a probable mechanism. The long-distanceinteraction means the spatial interaction of CIdimers with the LacI complex. If the formation of

the operator–CI dimer complex inhibits the LacIbinding to lacO, the relative promoter activitybecomes higher at a certain IPTG concentrationdue to the lower LacI repression. As a result, cellsmaintain the high state at a lower IPTG concen-tration. Thereafter, the region which hysteresisoccurs will expand.On the other hand, concerning the higher IPTG

concentration region, the observed differences canbe reconstructed by several mechanisms: (1) thehigher affinity of CI dimer-OR2 binding; (2) non-linear protein degradation.40 As K3 is increased,the hysteretic region moves toward the higherIPTG concentration region. However, the discre-pancy of dynamics between experiment andmodel grows larger because the response delayis increased. A more likely mechanism is non-linear protein degradation. If we employ themodified model that involves the non-lineardegradation derived from proteolysis of dimerproteins at higher concentrations, instead of thelinear degradation, the theoretical hysteretic regionexpands between 0.02 mM and 0.05 mM IPTG(data not shown). Moreover, the time-evolutionagrees with experiments. However, we have notobserved the non-linear dependence of the pro-teolysis of ssrA-tagged proteins yet. The observedexpansion of the hysteretic region warrants furtherexploration.The present experimental and theoretical study

adds the dynamical aspect of the positive feedbacksystem to previously studied motifs such as thefeed-forward loop that can act as a sign-sensitivedelay element,37 negative feedback.7 It would beimportant to characterize and study additionalmotifs and their complexes at the dynamicalviewpoints in order to understand basic networkelements and their functions.

Materials and Methods

Culture and measurements

Single colonies grown on agar plates were inoculatedin 1.5 ml LB medium with 50 mg/ml ampicillin andgrown overnight at 37 8C with shaking at 200 rpm. Thecultures were diluted (1:100(v/v), yielding A600Z0.005)into fresh minimal medium (solution 1: 2 g of(NH4)2SO4, 6 g of Na2HPO4, 3 g of KH2PO4, 3 g ofNaCl, 0.011 g of Na2SO4; and solution 2: 0.2 g of MgCl2,0.01 g of CaCl2, 0.005 g of FeCl3$7H2OC2.5 mg thiamineper liter) containing 1 mM glucose as the main carbonsource, 0.05%(w/v) Casamino acids as the mainnitrogen source, 0.2%(v/v) glycerol and 50 mg/mlampicillin41) in a flat-bottom 96-well plate, at a finalvolume of 200 ml per well and covered them with 100 mlof mineral oil (Sigma) to avoid evaporation. Cells weregrown in a Arvo FX (Perkin Elmer) multiwell fluori-meter at 37 8C and shaken, until A600Z0.1. Subsequentlythe dilution 1:10 (v/v) into a fresh medium was done atthe same temperature, with various concentrations ofIPTG to induce the promoters. After an additional 1.5 hincubation, cultures were assayed with an automatically


repeated protocol of shaking (2 mm double-orbit,normal speed, 30 s), absorbance (A) measurements(600 nm filter, 0.1 s, absorbance through approximately0.5 cm of fluid), fluorescence readings (ex. 485 nm andem. 535 nm filters, 0.5 s; CW lamp energy 10,000 units),and a delay (120 s). The interval between the repeatedmeasurements was 5 min. The day-to-day reproduci-bility error of the fluorescence is less than10%(Figure 2(b) and (c)).

Strains and plasmids

Synthetic genetic networks were constructed usingstandard molecular cloning techniques as described inbasic cloning manuals.42 All other enzymes werepurchased from TOYOBO. All genes, promoters, andtranscription terminators were obtained by PCR amplifi-cation using Pyrobest polymerase from TaKaRa andprimers were from Invitrogen. Genes, promoters andtranscription terminators were obtained as follows: Ptrc-2promoter was synthesized using two oligonucleotideprimers by PCR; PRM promoter and lambda repressorgene, cI, were amplified by using isolated prophagegenome DNA; gfpmut3* and gfp(aav) including T0T1terminators from pJBA27 and pJBA112 (gift from S.Molin).41 The genetic networks were constructed in amiddle copy ColE1 plasmid pBR322 because of the ease ofgenetic manipulation installing networks on the plasmidthan in the chromosome.The plasmid pJMY203 has the PRMwt-PFS where

transcription from the PRM-lacO promoter drives theexpression of the cI-gfp(aav) fusion gene, PRM-lacO::rbsA-cI-gfp(aav)-T0T1 (Figure 1(a)). When we use the fusionconstruct, the number of GFP(AAV) and that of CI has aone-to-one correspondence. We inserted lacO betweenPRM and rbsA-cI-gfp(aav). Gene expression was controlledby IPTG. The background fluorescence was defined as thenatural fluorescence from the cells containing pBR322.Another negative control pJMY200, in which encodesPRM-lacO::rbsA-gfp(aav)-T0T1, was also constructed. Forconstructing pJMY200, cIwas deleted from pJMY203. Thefluorescence of pJMY200 is comparable to the backgroundfluorescence. The low expression level of cells withpJMY200 is due to the absence of CI, because PRMpromotes gene expression toward the opposite directionwithout CI.However, gene expression of PRM-locO::rbsA-gfp(aav)-

T0T1 is too low to distinguish the background fluor-escence. In order to compare the rise-time of the PFS withthat of the NFS, we adopted the Ptrc-2::rbsA-gfp(aav)-T0T1construct, in which Ptrc-2 drives the expression of gfp(aav)(Figure 1(b)) as the NFS. We constructed the NFS on theplasmid pJMY303. Ptrc-2 is also controllable with IPTG,because this promoter includes lacO. Furthermore, toevaluate the effect of the genetic mutation, we constructedthe other plasmids pJMY213 and pJMY233 by introducingseveral point mutations into PRM in pJMY203. pJMY213has the promoter, PRML1L1R3. Conversely, pJMY233 hasthe promoter, PRMR3R2R3.The plasmid was introduced into E. coli JM109 (geno-

type: e14-(McrA-) recA1 endA1 gyrA96 thi-1 hsdR17(rK-mKC) supE44 relA1 D(lac-proAB) (F 0 traD36 proABlacIqZDM15)). At a given IPTG concentration, themaximal expression level of cells with the controlpJMY303 is about tenfold of those with pJMY203. Themaximal expression levels of cells with pJMY213 andpJMY233 are 1.1-fold and 0.7-fold of those with pJMY203,respectively.

Hysteresis measurements

A single colony grown on agar plates was inoculated in1.5 ml of LB medium with ampicillin at 37 8C. Culturescontaining the PFS were initially grown at 1 mM IPTGwith shaking at 200 rpm to confirm a high monostablesteady state. The cultures were diluted 1:100(v/v) into afresh minimal medium either containing or lacking 1 mMIPTG and grown until A600Z0.1. Cells were suspendedand washed in a medium without IPTG. The dilution1:100 (v/v) into fresh medium with the prescribedconcentration of IPTG was done and the cultures weregrown for six to seven generations. Fluorescence wasmeasured in cell density A600Z0.1 using both Arvo FXand a fluorescence spectrophotometer FL-2500(HITACHI). We measured GFP fluorescence by FL-2500during logarithmic growth at A600 of 0.25. The excitationwavelength was 488 nm, and the emission detection was514 nm. Cell density was also measured by a spectropho-tometer V-5390 (JASCO) at a wavelength of 600 nm.

Mathematical models and parameterization

We provide a detailed description of the derivation ofthe equations in supporting information. In experiments,we obtained the maximal fluorescence of the total CI-GFP(AAV) from hysteresis measurement at 1 mM IPTGbecause the gene expression did not saturate during thetime-course measurement. In numerical calculations, wedefined the maximal total CI concentration Xmax as X(tZ1000 min). We normalized the kinetics of gene expressionby the maximal fluorescence.For the parameter estimation, we performed by

random “best-fit” parameter search on a subset of theparameters (but fixed gZ0.014). In choosing parameters,we imposed on the parameters as the conditions K3OK2,K4!K2, bO1. This step involved simulating the time-evolution of gene expression. The quality of the model indescribing the data is given by the mean error for eachPFS:

Ej Z1

T

XT

iZ1

jXexperimentij KXtheoryij jX

experimentij

; (10)

where Xexperimentij (or X

theoryij ) denotes the experimental (or

theoretical) total CI-GFP(AAV) fluorescence per cell attime tZti and the PFS j, respectively. We set E

ThresholdZ0.02 for the threshold of each E-value. We found out theparameter sets whose E-values are below the threshold.The mathematical calculation was performed withMatlab 7.0.4.

Acknowledgements

We thank K. Saeki, M. Inaki, E. Takasu H. Iwasakiand T. Kando for their kind experimental advice;and also U. Alon, H.R. Ueda, S. Kuroda, Q. Ouyang,T. Shibata, K. Fujimoto, M. Matsuo, A. Awazu andY. Murayama for useful discussions. Y.T.M. thanksthe Research Fellowships for Young Scientistsfrom the Japan Society for the Promotion of Science17-12029.


Supplementary Data

Supplementary data associated with this articlecan be found, in the online version, at doi:10.1016/j.jmb.2006.03.064

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Edited by R. Ebright

(Received 29 March 2006; accepted 29 March 2006)Available online 27 April 2006

ith Positive FeedbackIntroductionResultsDesign of the synthetic genetic networksThe dynamics of positive feedback: positive feedback shows response-delayPositive feedback switches gene expression in a hysteretic or a graded mannerMathematical analysis: the model and dynamicsReconstruction of switching behaviors from experimental dynamicsThe rate of the feedback activation transits switching modes

DiscussionBiological function of response delayRelation between response delay and hysteresisComparison between experiments and models

Materials and MethodsCulture and measurementsStrains and plasmidsHysteresis measurementsMathematical models and parameterization

AcknowledgementsSupplementary DataReferences

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