537Short Report

IntroductionBiological oscillations are observed in all organisms at widelyvarying temporal and spatial scales. Oscillations play an importantrole in maintaining homeostasis and delivering encoded information.Many cellular processes have circadian rhythms, whereas oscillatorswith time constants from milliseconds to minutes with profoundbiological consequences include the heart beat and neuraloscillations, the glycolytic system of muscle or yeast cells, andwidespread calcium ion, inositol 1,4,5-trisphosphate and calmodulinoscillations (Goldbeter, 1996; Kearns et al., 2006; Matsu-ura et al.,2006).

Many biological oscillators show synchronized oscillations, notbecause of the presence of an invariant central or controlling clock,but because the independent oscillating systems (normallyincluding an internal negative feedback; Fig. 1A) are coupled.Examples where biological data show the presence of pairs ofcoupled oscillators are given in Table 1. For most of the cases inTable 1, the coupling between biological oscillators is madethrough communication via signaling messengers such ashormones and neurotransmitters (Fig. 1A). For instance,Dictyostelium cellular oscillators are coupled by cyclic adenosinemonophosphate (cAMP), and neuronal oscillators are coupled byneurotransmitters such as glutamate (for excitatory coupling) andg-aminobutyric acid (GABA) (for inhibitory coupling). We found

that synchronization between two homogeneous biologicaloscillators is most widely induced by a positive feedback loopthat couples the local oscillators. There are two kinds of a positivefeedback loop, depending on the regulation type of the interactionconstituting the feedback: a double-positive feedback loop (PP)in which both links activate (positive) interactions (Fig. 1B), anda double-negative feedback loop (NN) in which both links inhibit(negative) interactions (Fig. 1C). Therefore, coupling of oscillatorstends to involve two negative feedback loops and one positivefeedback loop (which can have either PP or NN couplingstructure). There have been studies on the dynamics of coupledfeedback loops (Hasty et al., 2001; Brandman et al., 2005; Lockeet al., 2006; Matsu-ura et al., 2006; Kim, D. et al., 2007; Kim, J.-R. et al., 2008; Tsai et al., 2008; Shin et al., 2009) and on thesynchronization of biological oscillators (Bier et al., 2000;Takamatsu et al., 2000; McMillen et al., 2002; Gonze et al., 2005;Fukuda et al., 2007; Li and Wang, 2007; Yu et al., 2007; Menget al., 2008; Morelli et al., 2009), but these structures and theirconsequences have not been modeled in a general context. Kearnsand colleagues (Kearns et al., 2006) show that the coupling oftwo positive oscillatory signals in an antiphase relationship canproduce stable activity in response to stimulation, withcomputational simulations suggesting that the relative strength oftwo feedback mechanisms and their temporal relationship to each

A design principle underlying the synchronization ofoscillations in cellular systemsJeong-Rae Kim1, Dongkwan Shin1, Sung Hoon Jung1,2, Pat Heslop-Harrison3 and Kwang-Hyun Cho1,*1Department of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Republic of Korea2Department of Information and Communication Engineering, Hansung University, Seoul 136-792, Republic of Korea3Department of Biology, University of Leicester, Leicester LE1 7RH, UK*Author for correspondence (ckh@kaist.ac.kr)

Accepted 19 November 2009Journal of Cell Science 123, 537-543 © 2010. Published by The Company of Biologists Ltddoi:10.1242/jcs.060061

SummaryBiological oscillations are found ubiquitously in cells and are widely variable, with periods varying from milliseconds to months, andscales involving subcellular components to large groups of organisms. Interestingly, independent oscillators from different cells oftenshow synchronization that is not the consequence of an external regulator. What is the underlying design principle of such synchronizedoscillations, and can modeling show that the complex consequences arise from simple molecular or other interactions between oscillators?When biological oscillators are coupled with each other, we found that synchronization is induced when they are connected togetherthrough a positive feedback loop. Increasing the coupling strength of two independent oscillators shows a threshold beyond whichsynchronization occurs within a few cycles, and a second threshold where oscillation stops. The positive feedback loop can be composedof either double-positive (PP) or double-negative (NN) interactions between a node of each of the two oscillating networks. The differentcoupling structures have contrasting characteristics. In particular, PP coupling is advantageous with respect to stability of period andamplitude, when local oscillators are coupled with a short time delay, whereas NN coupling is advantageous for a long time delay. Inaddition, PP coupling results in more robust synchronized oscillations with respect to amplitude excursions but not period, with appliednoise disturbances compared to NN coupling. However, PP coupling can induce a large fluctuation in the amplitude and period of theresulting synchronized oscillation depending on the coupling strength, whereas NN coupling ensures almost constant amplitude andperiod irrespective of the coupling strength. Intriguingly, we have also observed that artificial evolution of random digital oscillatorcircuits also follows this design principle. We conclude that a different coupling strategy might have been selected according to differentevolutionary requirements.

Key words: Biological oscillations, Coupled oscillators, Design principle, Double-negative interactions, Double-positive interactions, Positivefeedback, SynchronizationJo

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other might account for cell-type-specific dynamic regulation. Tounravel the most general design principles of synchronizedoscillations, we focused on the positive feedback that couples twolocal oscillators. In general, positive feedback can amplify signals(Hasty et al., 2000), reduce a response speed, induce hysteresis(Becskei et al., 2001; Ferrell, 2002; Isaacs et al., 2003; Kim, J.-R. et al., 2008) and realize a toggle switch (Gardner et al., 2000;Hasty et al., 2000; Tyson et al., 2003; Kobayashi et al., 2004),but its role in coupling local independent oscillators and inducingsynchronized oscillations has not been investigated yet.

In this paper, we explore interesting and important features ofthe PP and NN feedback that couples two oscillators and enablessynchronized oscillation. Through mathematical modeling andextensive computational simulations in which the strength offeedbacks coupling the two oscillators (coupling strength) and otherparameters (including delay in feedback) were varied, we revealthat the two types of coupling structures have their owncharacteristic features in synchronization. In particular, we aimedto analyze the differences of the two coupling types with respectto the time delay between two oscillators, to the period andamplitude of the resulting synchronized oscillations, and to noiserobustness. Specifically, we have shown that PP coupling is moreadvantageous when local oscillators are connected with a shortcommunication time delay, whereas NN coupling is advantageousfor a long communication time delay. We also found that this designprinciple holds for various examples of synchronized oscillations(Table 1). We have also investigated whether artificial evolution ofrandom coupling of digital oscillators follows such a designprinciple for synchronized oscillations and, intriguingly, find thatthis is indeed the case.

Results and DiscussionThe coupling strength of two local oscillators determinessynchronization as well as synchronization timeLet us consider the case where two isolated oscillators oscillateindependently with a 170° phase difference and they are then coupledthrough a positive feedback loop as shown in Fig. 1B,C. For a weakcoupling strength (f), the two oscillators are not synchronized (leftarea of Fig. 1D and left panel of 1E). If not synchronized, they showa phase portrait moving around a limit cycle (red curve in Fig. 1E)located in the diagonal direction of the phase plane (f=0). On theother hand, synchronized oscillators show a limit cycle aligned inthe direction of X1=X2 (yellow curve in Fig. 1E). As coupling strengthpasses a threshold of approximately 1.068, synchronization is inducedand the time taken for synchronization rapidly decreases as couplingstrength increases to 2 (Fig. 1D). If the coupling strength becomesmuch larger (f>3.3), then the limit cycle of the yellow curve changesinto a stable steady state point (see supplementary material Movie1) implying disappearance of the oscillation (yellow region in Fig.1D). A larger coupling strength makes the positive feedbackpredominant and thereby suppresses the oscillatory behavior of twonegative feedback oscillators. Together, these imply that a certainrange of coupling strength (orange region in Fig. 1D) is required forsynchronized oscillation and a specific coupling strength is relatedto the synchronization time (see supplementary material Figs S1-S4for further details).

PP and NN couplings have contrasting propertiesdepending on time delay in feedback between oscillatorsConsidering the dynamics of coupled oscillators, the couplingstrength and the communication time delay between two oscillators

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Table 1. Various examples of synchronized oscillations and the corresponding coupled oscillators

Coupling type secnerefeR erutcurts krowteN yawhtap noitcennoC metsys lacigoloiB

Zebrafish segmentation clock Notch r DeltaC r Notch (Mara et al., 2007)

Dictyostelium cAMP oscillations ACA r E_cAMP r CAR1 r ACA ACA r E_cAMP r CAR1 r I_cAMP

(Kim, J. et al., 2007)

Pulsatile secretion of GnRH GnRH r Common pool of GnRH (Khadra and Li, 2006)

Circadian oscillators Clock gene r Average neurotransmitter (VIP) (Gonze et al., 2005)

Synchronization of LIP and MT neurons

LIP neurons } MT neurons (Saalmann et al., 2007)

PP

Corticospinal coherence Motor cortex neurons } spinal cord neurons (Schoffelen et al., 2005)

Sensory pyramidal neurons Pyramidal neurons r bipolar cells pyramidal neurons

(Doiron et al., 2003)

Cortical feedback control in the visual thalamus

LGN r PGN LGN (Bal et al., 2000)

Synchronized oscillations in interneuron network

Pyramidal neuron r inhibitory interneuron pyramidal neuron

(Whittington et al., 1995; Bartos et al., 2002)

Ovulation cycle regulation Ovulation oscillator r pheromones ovulation cycle

(Stern and McClintock, 1998; Brennan and Zufall, 2006)

NN

Insulin secretion Calcium r insulin glucose r PFK r calcium (Bertram et al., 2004; Pedersen et al., 2005)

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are the two main parameters that determine the dynamics such assynchronization, and the period and amplitude of synchronizedoscillations. We have considered a parameter region forsynchronization of two oscillators having a phase difference of 170°(Fig. 2A for PP and Fig. 2D for NN). Fig. 2A shows that coupledoscillators of type PP can be synchronized when the time delay(which corresponds to t in the mathematical model in Materialsand Methods) between the oscillators is short compared to theirperiods of uncoupled oscillators. By contrast, coupled oscillatorsof type NN can be synchronized when the time delay between theoscillators is relatively long (Fig. 2D). In Fig. 2B, when twouncoupled oscillators with a phase difference of 170° are coupledby PP with a time delay of zero, two oscillators can be synchronized.This is because if the time delay is zero, the two oscillators coupledby PP simultaneously enhance each other as soon as they arecoupled. If the time delay is, however, long (here, 0.5 cycles), thetwo oscillators cannot be synchronized (Fig. 2C). To explain this,let us consider two oscillators coupled at two nodes X1 and X2 asshown in Fig. 1A. Since the time delay between X1 and X2 is 0.5cycles, the value of X1 at time 0 is positively influenced by thevalue of the other node X2 at time –0.5 cycles. Note that thedecreasing pattern of X1 around time 0 is similar to that of X2 around–0.5 cycles. Since X1 is positively regulated by X2, the phase of X1

will not so much change by the coupling. The same also holds forthe phase of X2. This means that the phase difference between X1

and X2 will still be kept around 180°, implying no synchronization.

Hence, if the time delay is long, two oscillators cannot besynchronized. The synchronization characteristics of NN in Fig.2E,F can also be explained similarly.

Even homogeneous biological oscillators can exhibit differentoscillatory patterns, such as different periods or amplitudes as wellas noise. To address this issue, for each time delay, we haverandomly perturbed all the parameter values of the coupledoscillators by 10% and examined the phase difference of twooscillators. The insets in Fig. 2A,D show the average phasedifference obtained from 100 repetitions of the above procedure foreach time delay. Since a smaller phase difference implies bettersynchronization, we find that coupled oscillators of type PP are wellsynchronized for a short time delay (see the inset in Fig. 2A) butcoupled oscillators of type NN are well synchronized for a longtime delay (see the inset in Fig. 2D). Taken together, we concludethat oscillators show increased synchronization behavior with PPcoupling when they have a short communication time delay,whereas NN is better for a longer communication time delay (Fig.2G).

PP and NN induce different features of the resultingsynchronized oscillationsWe have examined the period and amplitude of the resultingoscillations produced by the coupled oscillators for a wide rangeof parameter values (Fig. 3A,B). For PP, the coupled oscillator showsno oscillation when the coupling strength is larger than a certain

Fig. 1. An example of the coupling structure of twobiological oscillators. (A) The coupling is made throughsignaling messengers such as hormones, neurotransmitter,light, or sound. Coupling occurs at all scales fromsubcellular to inter-ecosystem, and might involve identicalor non-identical oscillators, and physical or chemicalcouplers. (B) Two oscillators coupled by double-positiveinteractions (PP) with coupling strength f. (C) Twooscillators coupled by double-negative interactions (NN).(D) The synchronization time after two oscillators with aphase difference of 170° are coupled. (E) The phaseportraits in the (X1, X2)-plane (upper figures) and temporaloscillation patterns (lower figures) of the coupledoscillators for various values of f (left to right) 0, 1.07, 1.5and 2.5. The red curve denotes the limit cycle of theoscillator with f=0 (no coupling) and the yellow curvedenotes the limit cycle of synchronized oscillators (f>1.068;for f=1.07, shown, synchronization is reached after some300 cycles). As f increases, the phase portrait of the coupledoscillators moves from around the red curve towards theyellow curve (see supplementary material Movie 1). Violetlines denote X1; green lines denote X2 before (dotted lines)and after (solid lines) coupling.

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threshold (f>3.3), regardless of the time delay (Fig. 3A, left) (zeroamplitude in yellow region of Fig. 3A,B means no oscillation). Inthis case, the variance of the period of the resulting synchronizedoscillations is very large as the coupling strength changes (the leftinset in Fig. 3A). We note that the period of unsynchronizedoscillations does not change much with respect to the variation ofeither coupling strength or time delay. Compared to PP, NN showsdifferent features of the period (Fig. 3A, right). For NN, the rangeof coupling strength for synchronized oscillation changes along withthe time delay. In addition, the period of oscillations produced bythe coupled oscillator does not change much with either thecoupling strength or the time delay; the period of synchronizedoscillations changes only a little as the coupling strength increases(the right inset in Fig. 3A). In this case, the range of the couplingstrength for synchronized oscillation is also wider compared to thatof PP coupling (compare two insets in Fig. 3A). The characteristicsof amplitude are similar to those of period (Fig. 3B). For PP (Fig.3B, left), the variance of the amplitude of the resulting synchronizedoscillations is very large as the coupling strength changes (the leftinset in Fig. 3B). In addition, the amplitude of synchronizedoscillation increases rapidly and then oscillation switches off as thecoupling strength increases. Compared to PP, NN also shows

different features in terms of amplitude (Fig. 3B, right). As thecoupling strength increases, the amplitude decreases very slowly.Hence, coupled oscillators of type NN produce synchronizedoscillations with almost constant period and amplitude, even if thecoupling strength changes. In other words, NN can induce morestable oscillation patterns with respect to parameter changes,especially for the coupling strength.

In addition to the period and amplitude, we also investigated thenoise robustness (Fig. 3C). Following application of noise to theamplitude and period of one of the pair of coupled oscillators, wefound that the amplitude of type PP remained close to theundisturbed range, whereas the amplitude of type NN showedexcursions well outside the range. Type PP lost cycle period andsynchronization more than type NN, although type PP regainedsynchronization more rapidly than NN coupling after the noisystimulus disappeared.

We can conclude that the consequences of coupling oscillatorswith PP or NN show contrasting features with respect to stabilityof period and amplitude (favored by type NN), toggling of theoscillation phenomenon with changing coupling strength (favoredby PP), robustness to noise (favored by PP for amplitude but notperiod), and effects of delay in feedback. To investigate a design

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Fig. 2. The parameter regions ofcoupling strength and communicationtime delay between two oscillators forsynchronized oscillations with respect toeach coupling type. Insets show theaverage of the phase differences of twooscillators obtained from 100 perturbationsof the parameters. (A) PP coupling type.(B,C) Temporal oscillation patterns of thetwo oscillators with coupling strength f=3with communication time delay of t=0cycles (B) and t=0.5 cycles (C). (D) NNcoupling type. (E,F) Temporal oscillationpatterns with communication time delay oft=0 cycles (E) and t=0.5 cycles (F). (G)Two oscillators with a short communicationtime delay can be synchronized by PPcoupling, whereas those with a longcommunication time delay can besynchronized by NN coupling. Violet linesdenote X1; green lines denote X2 before(dotted lines) and after (solid lines)coupling.

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principle in synchronized oscillators, we tested the effect of selectionon the evolution of a number of digital oscillators that result insynchronized oscillations by coupling, and compared the resultingcoupled structure obtained from such artificial evolution with thoseidentified in Table 1 involving a range of situations.

Artificial evolution of digital oscillation circuits follows thesame design principle for synchronized oscillationsBiology and engineering share many interesting design principles(Gardner et al., 2000; Tyson et al., 2003; Guantes and Poyatos, 2006;Kim, T.-H. et al., 2007). We simulated evolution using an evolvabledigital system. We first generated a number of digital oscillationcircuits (corresponding to biological oscillators), evolved the circuitsusing a genetic algorithm (Kim, J. et al., 2008) (see supplementarymaterial Fig. S5) and examined the resulting Boolean logic betweentwo digital oscillation circuits. Experimental results from 1000simulation runs showed that about 60% of the evolved synchronizeddigital oscillators have coupled oscillator circuits of type PP (Fig.3D) and that the average time delay between oscillators is short(Fig. 3E), whereas about 5% have the coupling type NN (Fig. 3D)and the time delay is about ten times longer than that of PP (Fig.3E). These results are consistent with the analysis results of thebiological systems above. Here, the prevalence of PP is due to theevolution process that selects in its early stage a simpler structure(combination logic gradually evolves from simple to complex),

which means a relatively short communication time delay betweenoscillators, and this supports our previous analysis. Hence, we canconclude that artificial evolution of digital oscillation circuits alsofollows the same design principle as biological evolution forsynchronization of biological oscillators.

ConclusionSynchronized oscillations are important for biological functioning(Glass, 2001; Buzsaki and Draguhn, 2004; Schoffelen et al., 2005;Mara et al., 2007), and although the biochemical and geneticcomponents involved in many types of interactions have beencharacterized, the key factors and parameters involved insynchronization have not been explored in detail. Here, we showa design principle of a synchronization mechanism, the advantageof a particular coupling structure for synchronization, and theconsequences of changes in synchronization time (Fig. 1), time delay(Fig. 2), the switching and loss of synchronized oscillations (Fig.3), and a group of parameters that are relevant to diverse biologicalsystems in which synchronized oscillation of independent oscillatorshas been observed (Table 1).

In summary, PP coupling is advantageous when local oscillatorsare connected with a short time delay, whereas NN coupling isrequired for a longer time delay. In addition, PP coupling results inmore robust synchronized oscillations with respect to noisedisturbances compared to NN coupling. However, PP coupling can

Fig. 3. Synchronization properties of coupled oscillators with various coupling parameters or artificial evolution. (A) The period of the resultingsynchronized oscillations produced by the coupled oscillators for each time delay and coupling strength in coupling types PP (left) and NN (right). Insets show thechange of period as the coupling strength varies when the time delay is fixed to 0.13 (left) or 0.5 (right) cycles. (B) The amplitude of the resulting synchronizedoscillations produced by the coupled oscillators for each time delay and coupling strength in coupling types PP (left) and NN (right). Insets show the change ofamplitude as the coupling strength varies when the time delay is fixed to 0.13 (left) or 0.5 (right) cycles. (C) Response patterns when we apply random perturbation(upper panel) to one node (Y1) of the synchronized oscillators for coupling types PP (left) and NN (right) with f=3.0. (D) The numbers of different coupling types(PP, NN and PN) observed out of 1000 artificial evolutions of digital oscillation circuits for synchronized oscillations. (E) The average time delay of each couplingtype (PP and NN) calculated from the evolved digital oscillators. Error bar denotes s.d. Violet lines denote X1; green lines denote X2 before (dotted lines) and after(solid lines) coupling.

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induce a large fluctuation in the amplitude and period of the resultingsynchronized oscillation depending on the coupling strength,whereas NN coupling ensures almost constant amplitude andperiod irrespective of the coupling strength. The biologicalsignificance of our findings can be found from the examples shownin Table 1. For instance, oscillators coupled through PP (such assegmentation clocks and circadian clocks) might have been evolvedto robustly oscillate with respect to noisy cue signals, whereasoscillators coupled through NN (such as ovulation oscillators andinsulin secretion oscillators) might have been evolved to be robustlysynchronized for a wide range of variation in the communicationsignal between the oscillators. Moreover, our results imply thatindependent oscillators can be synchronized with only weakinteractions, which might include small intercellular fluxes betweenidentical pathways. Furthermore, with estimates showing that a highproportion of genes show some sort of diurnal rhythm (McDonaldand Rosbach, 2001; Mockler et al., 2007), our results havesignificant implications by providing a mechanism whereby a largenumber of oscillating cellular events can become synchronized whenonly weakly coupled to the environment-driven master oscillator.The abrupt transitions, which biologically correspond to robustswitching behavior, were noted with respect to the transitionbetween unsynchronized (UO), synchronized (SO) and no (NO)oscillations as coupling strength increases (Fig. 1C, Fig. 3). Thereis an interesting study on brain rhythms (Kopell et al., 2000),supporting our design principle, in which it was shown that asynchronized b-rhythm is generated by PP coupling of adjacent(thereby having a short time delay) neurons, whereas it is generatedby NN coupling of distant (thereby having a long time delay)neurons. The contrasting synchronization of oscillators betweendifferent organisms is shown with the short time delay but robustsynchronization seen in Dictyostelium synchronization via cAMPdiffusion [see Kim, J. et al. (Kim, J. et al., 2007), with PP coupling]whereas synchronization of much longer ovulation cycles inmammals involving pheromone signaling is supported by NNfeedback (Stern and McClintock, 1998; Brennan and Zufall, 2006).We have further found that artificial evolution of random couplingamong digital oscillation circuits also follows the same designprinciple we unraveled. From these results, we conclude that aparticular type of coupling might have been selected according toevolutionary requirements such as time delay of the communicationbetween oscillators and the dynamic characteristics of resultingsynchronized oscillation patterns. Recently, engineered control ofcellular function through the design and manipulation of geneticoscillators is within the reach of current technology (Hasty et al.,2000; Hasty et al., 2001). Since most biological oscillators areoriginated from genetic oscillators, our results can provide a usefulexperimental guide for synchronization study of genetic oscillators,particularly for engineering synchronized genetic oscillators.

Materials and MethodsAs most of the biochemical reactions such as gene transcriptional regulations can beapproximated by Hill-type stimulus-response curves (Yagil and Yagil, 1971; Lemmeret al., 1991; Gardner et al., 2000), biological oscillators can also be described bysuch Hill-type equations. There have been a number of experimental case studiesshowing the validity of the Hill-type modeling of various biological oscillators suchas circadian oscillators (Goldbeter, 1995; Scheper et al., 1999; Ruoff et al., 2001;Smolen et al., 2001; Forger and Peskin, 2003; Smolen et al., 2004; Gonze et al.,2005; Locke et al., 2005; Locke et al., 2006; Bernard et al., 2007; Kuczenski et al.,2007; Leise and Moin, 2007; Bagheri et al., 2008), calcium oscillators (Tang andOthmer, 1994; Friel, 1995; Li and Wang, 2007), segmentation clocks (Meinhardt andGierer, 2000; Rida et al., 2004; Yoshiura et al., 2007; Zeiser et al., 2007; Momijiand Monk, 2008) and NF-kB oscillators (Krishna et al., 2006; Ashall et al., 2009)(see supplementary material Table S1 for details). So, we have adopted such a well-

established Hill-type mathematical model in this paper to explore the general designprinciples of synchronized biological oscillations. Since the time delays betweenoscillators are important factors for synchronization, we used delayed differentialequations for our mathematical models.

In particular, we have constructed the PP model as follows:

and the NN model as follows:

where we set H=3, V1=V2=V3=V4=1, time delay t1=t2=t (0<t<15), t3=t4=t5=t6=2,coupling strength F12=F21=f (0<f<8), K31=K42=K13=K24=0.5, Kd1=Kd2=Kd3=Kd4=0.5,and Kb1=Kb2=Kb3=Kb4=0.1 for simplification. The delayed differential equations weresolved numerically by dde23 in MATLAB 7.0 (R14). Further details are availableon request.

This work was supported by the National Research Foundation ofKorea (NRF) grant funded by the Korea Ministry of Education,Science and Technology (MEST) through the BRL (Basic ResearchLaboratory) grant (2009-0086964), the Systems Biology grant(20090065567) and the 21C Frontier Microbial Genomics andApplication Center Program (Grant MG08-0205-4-0).

Supplementary material available online athttp://jcs.biologists.org/cgi/content/full/123/4/537/DC1

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