a comparative analysis of synthetic genetic...

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A comparative analysis of syntheticgenetic oscillators

Oliver Purcell1,*, Nigel J. Savery3, Claire S. Grierson4

and Mario di Bernardo2,5

1Bristol Centre for Complexity Sciences, Department of Engineering Mathematics, and2Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, UK

3Department of Biochemistry, University of Bristol, Bristol BS8 1TD, UK4School of Biological Sciences, University of Bristol, Bristol BS8 1UG, UK

5Department of Systems and Computer Science, University of Naples Federico II,Via Claudio 21, 80125, Napoli, Italy

Synthetic biology is a rapidly expanding discipline at the interface between engineering andbiology. Much research in this area has focused on gene regulatory networks that function asbiological switches and oscillators. Here we review the state of the art in the design and con-struction of oscillators, comparing the features of each of the main networks published todate, the models used for in silico design and validation and, where available, relevant exper-imental data. Trends are apparent in the ways that network topology constrains oscillatorcharacteristics and dynamics. Also, noise and time delay within the network can both haveconstructive and destructive roles in generating oscillations, and stochastic coherence is com-monplace. This review can be used to inform future work to design and implement new typesof synthetic oscillators or to incorporate existing oscillators into new designs.

Keywords: synthetic biology; gene regulatory networks; oscillators

1. INTRODUCTION

The aim of synthetic biology is to design and synthesizebiological networks or devices that perform a desiredfunction in a predictable manner (Endy 2005;Andrianantoandro et al. 2006; Serrano 2007; Haseloff &Ajioka 2009). Achieving this goal requires a combi-nation of in silico and in vivo analysis, and combinesapproaches from the fields of engineering, mathematicsand biology. Much attention has focused on the syn-thesis of gene regulatory networks. Since the earlypapers (Becskei & Serrano 2000; Elowitz & Leibler2000), two paradigmatic types of networks have beenof interest to the scientific community: switches andoscillators (Tyson et al. 2008).

Over the past few years, numerous designs have beenproposed for both types of networks in prokaryotic andeukaryotic cells. Because of the highly interdisciplinarynature of synthetic biology, results are published in aplethora of different journals ranging from dynamicalsystems and mathematics journals to computationalbiology and mainstream biological journals. Moreover,while some papers focus on the design of an oscillatorof interest, others derive its model and analyse its

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arch 2010ne 2010 1

features, while yet other journals report the in silicoor in vivo validation. A subset of the current networkshave been discussed in existing reviews (e.g. Hastyet al. 2001b; Novak & Tyson 2008). However, thesehave focused on different aspects of the oscillators andwith various degrees of detail. They also differ in theirstructure and approach. Hence, when studying the pro-blem of synthesizing a new network, it is difficult to finda coherent description of the various existing oscillatorsin a unified framework.

The aim of this paper is to address this gap in the lit-erature by expounding the features of the mainsynthetic oscillators proposed so far in a unified, coher-ent framework. In so doing, we review and compare thecharacteristics of the various oscillating networks avail-able in the literature, highlighting advantages anddisadvantages. Starting from the simplest Goodwinoscillator (Goodwin 1963), we will consider repressila-tors (Fraser & Tiwari 1974; Elowitz & Leibler 2000),several types of activator–repressor networks and thevery recent oscillators constructed in mammalian cells(Tigges et al. 2009, 2010). The aim is to present areview of the features of each network, the modelsused for their in silico design and validation and themain in vivo data available in the literature. The com-parative analysis of these existing oscillators can beused to inform the modelling, design and in vivo

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2 Review. Synthetic genetic oscillators O. Purcell et al.


implementation of future synthetic oscillators, but alsoas a guideline to synthetic biologists and engineerswishing to use existing models and designs.

For brevity and clarity, this review focuses solely onthe analysis of synthetic oscillators. However, the studyof natural genetic oscillators is itself a large and estab-lished field, with results that are relevant to the studyof synthetic oscillators in their wider context (e.g.Lewis 2003; Pomerening et al. 2005).

The rest of the paper is organized in the followingsections, each devoted to a different class of oscillators,giving details on the network, the mathematical analy-sis presented in the literature and the results of thein silico and, if available, in vivo implementation ofthe design:

— Goodwin oscillator (§2);— Repressilators (§3);— Amplified negative feedback oscillators (§4), focusing

on networks obtained by using either transcriptionalrepression, repression by sequestration or repressionby proteolysis;

— Fussenegger oscillators (§5);— Smolen oscillator (§6);— Variable link oscillators (§7); and— Metabolator (§8).

2. GOODWIN OSCILLATOR

The Goodwin oscillator, conceived over 40 years ago(Goodwin 1963), was the first synthetic genetic oscil-lator to be studied. It is also the simplest oscillator,comprising a single gene that represses itself(figure 1a). Early theoretical work was extensive, butnoted for its conflicting results on the existence of oscil-lations. Relevant references to this early work can befound within Banks & Mahaffy (1978a) and Smith(1987), while discussion of the early work can befound in Tyson (1975), Banks & Mahaffy (1978a) andSmith (1987).

More recently, different types of models have beenproposed in the literature including ordinary differentialequations (ODEs; Muller et al. 2006; electronic sup-plementary material, §S2.1), delay differentialequations (DDEs; Smith 1987 (using a model fromBanks & Mahaffy (1978b)); Bratsun et al. 2005; Strickeret al. 2008; electronic supplementary material, §S2.3and S1) and discrete stochastic simulations using theGillespie algorithm (Bratsun et al. 2005; Gillespie2007; Stricker et al. 2008).

1This was one of the first demonstrations of a constructive role fornoise in GRN dynamics.

2.1. Mathematical analysis

Both ODE (Muller et al. 2006) and DDE (Bratsun et al.2005) models suggested that oscillations in the Good-win network arise through a Hopf bifurcation, ifrepression is modelled by a nonlinear Hill functionwith a sufficiently high cooperativity coefficient(Smith 1987; Bratsun et al. 2005). The presence of atime delay in the negative feedback loop was alsosuggested to have a constructive role. In particular,the region in parameter space where oscillations are


detected can be expanded by increasing the value ofthe time delay within the feedback loop (Strickeret al. 2008). Moreover, as shown in Mallet-Paret &Smith (1990), oscillations can be proved to be associatedwith a unique limit cycle and hence to be globally asymp-totically stable.

2.2. In silico experiments

The presence of oscillations was confirmed via in silicoexperiments using a number of different techniques.Early discrete, Boolean-like simulations (Fraser &Tiwari 1974; Tiwari & Beckman 1974) suggested onlydamped oscillations were possible under biologicallyrealistic parameters. However, oscillations for biologi-cally plausible parameters could be obtained usingstochastic simulations with parameters drawnrandomly1 from either an exponential (Tiwari &Beckman 1974) or normal distribution (Fraser &Tiwari 1974). Recent studies support this early finding,with increased parameter ranges associated with oscil-latory behaviour being observed under stochasticsimulations (Lewis 2003; Bratsun et al. 2005; Strickeret al. 2008).

In one of these examples, a biologically detailedsimulation allied to in vivo work (Stricker et al. 2008),it is shown that while oscillations are predicted todecay with a deterministic model, they can persistunder Gillespie simulations (figure 1b). However, thiswas only observed under certain conditions, suggestingthat the constructive role for noise may be parameterdependent. These simulations also suggested that arobust period appears to be a feature of the Goodwinoscillator, an observation also made elsewhere (Lewis2003). Period was of the order of between 10 and 20min in simulations and was relatively insensitive to inhi-bition of repression (Stricker et al. 2008). Further inaccordance with these results, a separate study foundthat for certain parameter values, period is resistantto temperature, displaying temperature compensation(Ruoff & Rensing 1996), a common feature of naturaloscillators (Pittendrigh & Caldarola 1973).

Early simulations also found that a time delaybetween transcription and translation, characteristicof a eukaryotic system, could improve the regularityof oscillations (Tiwari & Beckman 1974). However,recent Gillespie simulations incorporating delay conver-sely suggested that increasing delay within the feedbackloop reduces regularity (Bratsun et al. 2005). Togetherwith the mathematical results demonstrating the roleof time delay in the negative feedback in improvingoscillatory capacity, studies suggest a trade-off betweenan increased oscillatory capacity and noisy oscillationsmay exist with regard to time delay.

2.3. In vivo implementation

Given the controversial nature of some of the findings ofthe theoretical work, it was important that an in vivoconstruction was performed. Construction of theGoodwin oscillator used the PLlacO-1 promoter (Lutz &


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Figure 1. (a) Goodwin oscillator topology. It comprises a single gene that represses itself. Throughout the text, the conventionused is that a solid line indicates direct transcriptional control, while dotted lines indicate an alternative or indirect regulatorymechanism. (b) Gillespie simulation of a Goodwin oscillator model. (c) In vivo implementation. The PLlacO-1 promoter (Lutz &Bujard 1997; also employed in the construction of Elowitz’s repressilator, see §3.3), active in the absence of LacI, was used tocontrol LacI expression. (d) In vivo time series of GFP fluorescence. Figure (b) and (d) are adapted from Stricker et al. (2008).

Review. Synthetic genetic oscillators O. Purcell et al. 3


Bujard 1997), repressed by LacI, but capable of drivingtranscription at a sufficiently high level when unre-pressed. By using this promoter to control LacIexpression (Stricker et al. 2008; figure 1c), a negativefeedback loop was formed. The same promoter also con-trolled expression of the fluorescent reporter proteinyemGFP (monomeric yeast-enhanced green fluorescentprotein), allowing observation of dynamics. LacI andyemGFP contained sequences termed ssrA ‘tags’,which are recognized by Escherichia coli proteases,and are used to increase the degradation rates of LacIand yemGFP to achieve faster dynamics. Each genewas constructed on a different plasmid. The systemwas introduced into E. coli deficient in the lac operon,minimizing host genome interference.

Consistent with simulations (Stricker et al. 2008),oscillations were relatively irregular, often failing toreturn to a zero level (figure 1d). Period was of theorder of 30 min, slightly longer than correspondingsimulations. However, in agreement with simulations,the period of the oscillations was largely unaffected byIPTG (an inducer molecule that binds to LacI and inhi-bits repression), varying less than 5 per cent over therange examined (Stricker et al. 2008). The percentageof oscillatory cells was not reported, meaning a properassessment of robustness is not possible.

In addition to the agreement with simulations, anin vivo implementation has also validated the mathe-matical results on the requirement for sufficientnonlinearity in repression to obtain oscillations. Theimplementation here used LacI, which is a tetramer. How-ever, an earlier implementation used TetR (Becskei &Serrano 2000), which is dimeric. Repression by TetRwill therefore be described by a lower Hill coefficient


than LacI, and subsequently there will be less nonlinearityin repression. In agreement with the mathematical results,the implementation using TetR did not oscillate, butinstead exhibited highly stable dynamics.

2.4. Discussion

The insensitivity of period to IPTG and (theoretically)temperature (Ruoff & Rensing 1996) are potentiallyuseful features of the Goodwin oscillator. However,oscillations observed thus far have been quite irregular,which may limit its utility.

We now move to discussing multi-gene oscillators,starting with one of the most widely known geneticoscillators presented in the literature: the repressilator.

3. REPRESSILATORS

A repressilator can be thought of as an extension of theGoodwin oscillator. It is defined as a regulatory networkof one or more genes, with each gene repressing its suc-cessor in the cycle (Muller et al. 2006). The term wasfirst used to describe a cycle of three genes (Elowitz &Leibler 2000; figure 2a). A one-gene repressilator isthe previously discussed Goodwin oscillator (§2).

Repressilators were first studied over 30 years ago as alogical extension to the Goodwin oscillator, using thesame discrete, Boolean-like simulations (Fraser & Tiwari1974). Continuous models were subsequently studiedusing DDEs (Smith 1987; electronic supplementarymaterial, §S2.3), and more recently ODEs (Elowitz &Leibler 2000; El-Samad et al. 2005; Muller et al. 2006;figure 2c; electronic supplementary material, §S2.1 and2.2), other DDE models (Wang et al. 2005a) (electronic


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Figure 2. (a) Three-gene repressilator topology. Each gene represses its successor in the cycle. (b) In vivo implementation of athree-gene repressilator. LacI represses tetR through PLlacO-1, tetR represses l cI through PLtetO-1 and cI represses lacI throughl PR, completing the cycle. All genes contain an ssrA sequence tag to promote rapid degradation. (c) ODE model equations forthe in vivo implementation. mi and pi represent mRNA and protein concentrations, respectively, i and j paired order-wise, result-ing in six ODEs. a0 and a þ a0 are the number of protein copies produced per cell with saturating repressor levels (promoter‘leakyness’) and without repressor, respectively. b is the ratio of the protein decay rate to the mRNA decay rate and n theHill coefficient. (d) Time series obtained by in silico simulation of ODEs for a particular set of parameters (see Elowitz & Leibler(2000) for details). (e) In vivo time series of GFP fluorescence. Figure (d) and (e) are adapted from Elowitz & Leibler (2000).



supplementary material, §S2.4), discrete stochastic simu-lations using the Gillespie algorithm (Yoda et al. 2007;Rajala et al. 2010) and hybrid stochastic schemes(Tuttle et al. 2005) have been used. The repressilatortopology is also observed in other fields. In electronics, acycle of an odd number of NOT gates is termed a ‘ringoscillator’, and in neuroscience, cyclic networks ofneurons are referred to as ‘neural ring networks’(Pasemann 1995).


The mathematical results on the Goodwin oscillator arean invocation of results developed for repressilators ingeneral. As with the Goodwin oscillator, oscillationsarise through a Hopf bifurcation (Muller et al. 2006),results on DDEs also suggesting a common requirementof sufficiently nonlinear repression (Smith 1987).Results proving oscillations as a global attractor of thesystem are also applicable, while a complementarytheory developed for DDEs, not yet applied to theGoodwin oscillator (Mallet-Paret & Sell 1996), demon-strates that, at least for three genes, time delay can havea constructive effect, creating oscillations (Wang et al.2005a).

Repressilators comprising an even number of geneswere initially found to exhibit multi-stability (Smith1987; Muller et al. 2006). More recently, it has been


shown that repressilators with six or more genes actu-ally possess a quasi-stable periodic solution (all butone Floquet multipliers less than 1), generated througha Hopf bifurcation (Strelkowa & Barahona 2010). Thestability of the periodic solution increases for increasinggene number (Strelkowa & Barahona 2010), the magni-tude of the multiplier corresponding to the unstabledirection decreasing. This finding potentially expandsthe oscillatory capacity of the repressilator class.Indeed, only the two- and four-gene repressilatorshave not been shown to oscillate, the two-gene versiondemonstrated in vivo to function as a bi-stable switch(Gardner et al. 2000).

Bifurcation analysis of a three-gene repressilatorODE model allied to in vivo work (Elowitz & Leibler2000; figure 2c) showed oscillations were favoured bystrong promoters (high maximal transcriptional rate),efficient ribosome-binding sites (high average numberof proteins per transcript), low levels of transcriptionwhen fully repressed and cooperative repression charac-teristics. It was also important that genes in the systemall had comparable mRNA and protein decay rates.

The studies of Muller et al. (2006) and Elowitz &Leibler (2000) considered symmetrical networks posses-sing identical parameter sets for each gene. In vivoimplementation of such a balanced network is likelyto be impossible. Non-symmetric networks can alsodisplay oscillations, although the analysis of these has




not been extensive (El-Samad et al. 2005; Strelkowa &Barahona 2010).


Early discrete simulations (Fraser & Tiwari 1974)demonstrated oscillations in three-gene repressilatorsover a small parameter range, requiring a high rep-ression efficiency, in agreement with mathematicalresults. Simulations of ODEs by Elowitz (Elowitz &Leibler 2000) also exhibited oscillations, with aperiod of the order of 150 min, using parameterssuggested by their analytical work (figure 2d).While mathematical analysis had demonstrated theability for time delay to generate oscillations, simu-lations showed that it could also be used to tuneoscillation characteristics. Increasing the total timedelay in the system increased the amplitude andperiod of oscillations linearly (Wang et al. 2005a).Recent work also suggests the rate and duration oftranscriptional pausing can affect the period (Rajalaet al. 2010).

Deterministic simulations also confirmed the exist-ence of quasi-stable oscillations in even gene-numberrepressilators (Strelkowa & Barahona 2010). Althoughoscillations eventually decay, it was observed that thelifetime of oscillations improves for higher gene num-bers, as predicted by the mathematics. It was alsosuggested that this ‘quasi-stability’ can be exploitedto switch oscillations on and off in a controlled way(Strelkowa & Barahona 2010).

The constructive role for noise observed in theGoodwin oscillator (Tiwari & Beckman 1974) was alsoconfirmed for repressilators. The inclusion of stochasti-city into the initial discrete simulations, this time bydrawing parameters randomly from a normal distri-bution, significantly decreased the dependence onrepression efficiency for oscillations (Fraser & Tiwari1974). Furthermore, while the quasi-stability eventuallystopped oscillations in deterministic simulations of evengene-number networks, the stochasticity in Gillespiesimulations allowed oscillations to continue indefinitely(Strelkowa & Barahona 2010). However, as observedfor the Goodwin oscillator, a constructive role for noisecan depend on system parameters. In the adiabaticregime (a significantly higher DNA-protein binding raterelative to the rate of change in protein number) of amodel presented in Yoda et al. (2007), oscillationsoccurred in Gillespie simulations but not in a correspond-ing ODE representation, suggesting a constructive rolefor noise. Yet, the opposite was observed in a stronglynon-adiabatic regime, where noise destroyed the correctordering of oscillations, or phase coherence, observedfor the ODEs. A destructive role for noise was also pre-sent in Gillespie simulations by Elowitz (Elowitz &Leibler 2000), in which oscillations exhibited significantamplitude variability. The property of stochastic coher-ence, a maximum in the regularity of the oscillations(measure on the distance between peaks) as a functionof the system size (average number of components;Hilborn & Erwin 2008), has been documented (Yodaet al. 2007). This has not yet been observed in theGoodwin oscillator specifically, in this case, it is the


dependence of phase coherence on noise, phase coherencedepending nonlinearly on noise strength (Yoda et al.2007).

Finally, the sensitivity to the level of parametersymmetry observed mathematically persists under sto-chastic simulations; oscillations are suppressed abovesome level of asymmetry (Tuttle et al. 2005).


Based on the suggestions of their mathematical analysis(see §3.1), Elowitz & Leibler (2000) constructed a three-gene repressilator in vivo (figure 2b) using LacI from E.coli, TetR from the Tn10 transposon and cI from the l

phage (Elowitz & Leibler 2000). In their network, LacIrepresses transcription of tetR, TetR represses tran-scription of cI and cI represses transcription of lacI,completing the cycle of repression. Owing to the math-ematical results, strong, tightly repressible hybridpromoters were employed to control tetR and cI, com-bining the l PL promoter with LacI and TetRoperators, respectively (Lutz & Bujard 1997; Elowitz& Leibler 2000). The l PR promoter, which containscI operators, was used to control lacI. To increaseprotein decay rates closer to that of the mRNA, anothersuggestion of the mathematical work, ssrA tags wereinserted at the 30 end of each gene, targeting the repres-silator proteins for rapid destruction (Elowitz & Leibler2000). The repressilator was constructed on a low copy-number plasmid and introduced into E. coli deficientin the lac operon, minimizing interference from thehost genome on dynamics. Additionally, a high copy-number plasmid containing a reporter gene thatencoded an intermediate stability green fluorescentprotein (GFP) and that was repressed by TetR wasintroduced, allowing monitoring of network dynamicsby fluorescence.

At least 40 per cent of cells exhibited oscillations,with a period of 160+ 40 (mean+ s.d.) min, of theorder of deterministic simulations, and a significantvariation in amplitude, as predicted by stochastic simu-lations (Elowitz & Leibler 2000). While protein levels insimulations returned to zero between oscillations, fluor-escence by GFP increased over time (figure 2e). This isprobably due to the higher stability of the GFP relativeto the rapidly degraded repressilator proteins. However,it was not explicitly accounted for in modelling, andhence not predicted. Despite the variability in ampli-tude, oscillations in progeny were correlated for asignificant amount of time after cell division, demon-strating that network state is passed to progeny(Elowitz & Leibler 2000). The timing of oscillationswas not correlated with cell division, suggesting repressi-lator dynamics were decoupled from the cell-divisioncycle (Elowitz & Leibler 2000). However, the entry ofE. coli into stationary phase arrested oscillations, indi-cating that dynamics are coupled to the globalregulation and effects of cell growth and division(Elowitz & Leibler 2000). Observation time for oscil-lations was limited by this entry into stationary phaseto approximately 10 h, allowing three to four oscillationsto occur (Elowitz & Leibler 2000). A potential source ofinterference may be the increased concentration of the




s38 transcription initiation factor that occurs duringstationary phase. This would compete with s70 (theinitiation factor responsible for specifying RNA polymer-ase binding to the repressilator promoters) for availableRNA polymerase and may significantly affect dynamics.Designing repressilators that function with both types ofs factor may allow oscillations to continue into station-ary phase.

3.4. Discussion

Although conceived after the Goodwin oscillator, thethree-gene repressilator was the first synthetic geneticoscillator to be successfully implemented in vivo(Elowitz & Leibler 2000). However, less than half ofobserved cells exhibited oscillations, suggesting itlacks robustness. The role of noise is not yet well under-stood, and may be partly responsible. Furthermore,although the repressilator can function with someparameter asymmetry, significant asymmetry couldcontribute to the fact that the majority of cells didnot oscillate. Interference from native cellular systemsmay also play a role, with the oscillations decaying toan equilibrium value. The limited studies on time delaysuggest the repressilator may function in eukaryotes,although there are indications that temporal differencesin how DNA is duplicated in eukaryotes may affect therepressilator dynamics (Chen et al. 2004).

Regarding improvements, it has been observed thatthe addition of a positive feedback loop to any of therepressilator genes expands the region in parameterspace over which the core repressilator oscillates, enhan-cing robustness (Tsai et al. 2008). This addition alsoallows the repressilator to exhibit a far greater rangeof frequencies for a given amplitude (Tsai et al. 2008),increasing its potential utility as a component of morecomplex synthetic networks (Tsai et al. 2008). Finally,the controllable oscillatory capacity of even-gene repres-silators may significantly expand the utility of thetopology. More generally, this highlights the potentialfor transient dynamics and in particular quasi-stability,used elsewhere in flight and fluid control (Strelkowa &Barahona 2010), in gene network function.

The presence of a positive self-feedback loop is afeature of the class of oscillators presented in thenext section, the amplified negative feedback oscillators.

4. AMPLIFIED NEGATIVE FEEDBACKOSCILLATORS

The Goodwin oscillator and the repressilators areformed from solely repressive links. The logical nextstep in oscillator design was to explore oscillators alsoincorporating activating links between genes. Theamplified negative feedback oscillator comprising twogenes is possibly the simplest type of amplified negativefeedback oscillator. Here, one gene promotes (amplifies)its own transcription via a positive self-feedback loopand also activates transcription of the other gene. Atthe same time, the second gene represses transcriptionof the first gene, forming a negative feedback loop(figure 3a). Three different versions of this topologyhave been considered; repression by transcriptional


control, repression through sequestration by dimeriza-tion and repression by proteolysis. In what follows, wewill describe the features of each of these three mainrealizations of the network.

4.1. Case I: repression by transcriptional control

ODE models and Gillespie-based stochastic simulationsof these networks have been presented in the literature(Atkinson et al. 2003; Guantes & Poyatos 2006; Conradet al. 2008 (uses model from Guantes & Poyatos (2006));figure 3c; electronic supplementary material, §S3.1).

4.1.1. Mathematical analysis. The analysis of themodels proposed in the literature suggests two alterna-tive bifurcation mechanisms to explain the presence ofoscillations in the network. In Guantes & Poyatos(2006), the occurrence of oscillations is explained interms of a saddle-node bifurcation on an invariantcircle (SNIC), while under different parameter valuesConrad et al. (2008) show oscillations can arise fromeither a SNIC or a subcritical Hopf bifurcation.Furthermore, in Atkinson et al. (2003), a Hopf bifur-cation is also proposed as causing the onset ofoscillatory behaviour. The discrepancies are presumablydue to both model and parameter differences. However,in all cases, the bifurcation can occur because of signifi-cantly faster activator than repressor dynamics, eitherthrough faster activator degradation and translationrates (Guantes & Poyatos 2006; Conrad et al. 2008),or activator mRNA degradation rates (Atkinson et al.2003). This has the effect of causing the activator con-centration to reach a significant amplitude before therepressive effect gets too great (Guantes & Poyatos2006), encouraging sustained oscillations. The Hopfbifurcation in Atkinson et al. (2003) can also arisethrough changes in activator and repressorcooperativity.

4.1.2. In silico experiments. The simulations reported inGuantes & Poyatos (2006) suggest that oscillationsemerging from a SNIC do so at almost zero frequency,as the trajectories are still influenced by the slow trajec-tories of the region where the stable equilibrium waspresent before the bifurcation. This slow region cor-responds to the saturated part of the repressor-responsecurve, where repressor concentration is little affected bysmall activator changes. This translates into oscillationswith an extended region of high activator and repressorconcentrations and therefore large periods (Guantes &Poyatos 2006). Immediately after the bifurcation, afurther increase in the ratio of the activator to repressordegradation rates decreases the period rapidly, whichquickly plateaus. Repression is via a homodimer, requiringa significant concentration to achieve strong repression,permitting relatively large amplitudes (Guantes &Poyatos 2006). Finally, while the Hopf bifurcations gener-ate sinusoidal oscillations, the SNICs produce ‘relaxation’oscillations, which rise sharply but decay slowly.

Undamped oscillations owing to the Hopf bifurcationin Atkinson et al. (2003) have been observed, althoughfor biologically realistic parameters only damped oscil-lations were found (figure 3d). A period of 20 h was


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Figure 3. (a) Amplified negative feedback topology, with repression by transcriptional control. Gene A activates its own tran-scription and that of gene B, while B represses transcription from A. (b) In vivo implementation. Details provided in the maintext. (c) ODE model equations for the in vivo implementation. xi represent concentrations of network components (normalizedwith respect to their steady-state values), odd and even numbered variables describe mRNA and protein concentrations, respect-ively. The model comprises six ODEs, two describing the reporter LacZ. fi are tri-phasic transcriptional rates (see electronicsupplementary material, §S3.2), while bi are constants describing numerous rates. (d) In vivo (squares) and simulation (solidlines) time series for the three-cycle implementation. b-Galactosidase is encoded by the lacZ reporter encoded as part of thelacYZA operon on the E. coli chromosome, but under the control of the oscillator. (e) In vivo and simulation time series forthe four-cycle implementation. Figure (d) and (e) are adapted from Atkinson et al. (2003).



observed for a cell doubling time of 2 h, a considerablylonger period than the three -gene repressilator. Periodand amplitude were shown to depend on cell doublingtime, and while halving the doubling time halvedthe period, it approximately doubled the amplitude(Atkinson et al. 2003). The suggestion from mathemat-ical work that increasing activator cooperativity on itsown transcription could shift dynamics towards anundamped regime was implemented by increasing acti-vator gene copy number. Higher gene copy numberleads to higher activator concentration, allowing theactivator to get further along its sigmoidal activationcurve, better realizing the cooperative dynamics mani-fested in the rapidly changing ‘switch’ section of thecurve (Atkinson et al. 2003). Although undamped oscil-lations were still not obtained, the number of dampedoscillations could be improved from three to four(figure 3e).

Gillespie simulations confirmed the presence of oscil-lations from the SNIC bifurcation in Guantes &Poyatos (2006), demonstrating weak stochastic coher-ence when the parameters are such that the networkdynamics are close to the bifurcation point. By modifyingtranscription and translation rates such that the averagenumber of mRNA molecules could be varied while keep-ing average protein levels unchanged, mRNA number


could be seen to contribute significantly to noise andits effect on the regularity of oscillations. Specifically,the effect of the average mRNA number on regularitywas comparable to the effect of the average number ofall other molecules combined (Guantes & Poyatos 2006).

4.1.3. In vivo implementation. Only the Hopf-drivenoscillator in Atkinson et al. (2003) has beenimplemented in vivo. The network consists of two mod-ules, one encoding the activator, NRI, and the secondencoding the repressor LacI. The activating module ofthe oscillator was constructed by fusing glnG, encodingNRI, to a control region based on a glnA promoter(glnAP2; figure 3b). The promoter was regulated bytwo upstream adjacent high-affinity binding sites forphosphorylated NRI (NRIp), which formed an ‘enhan-cer’, and two ‘perfect’ LacI operators, one downstreamof the promoter, and one immediately upstream of theenhancer (Atkinson et al. 2003). The design intendedto make use of DNA looping; during activation, NRIpinteracts with promoter-bound RNA polymerase via aDNA loop, while during repression, LacI bound to thetwo operators also forms a loop, ensuring stable repres-sion. The two loops act antagonistically, as formation ofthe loops is mutually exclusive (Atkinson et al. 2003).




The repressor module was constructed by fusing lacI toa control region comprising the glnK promoter and anupstream enhancer formed of adjacent high-affinityand low-affinity NRIp binding sites (Atkinson et al.2003). Consequently, the glnK promoter is only fullyactivated at high NRIp levels (Atkinson et al. 2003).The modules were incorporated into the E. coli chromo-some as a single copy within regions flanked bytranscription termination sequences, ensuring tran-scriptional isolation (Atkinson et al. 2003). E. coliwere mutated such that the only LacI source was therepressor module, and mutant NRI phosphataseensured NRI phosphorylation under all conditions(Atkinson et al. 2003). LacI dynamics were assayed ona population level with b-galactosidase encoded bylacZ present in the lacYZA operon (under the controlof the oscillator) on the E. coli chromosome, and on asingle-cell level using CFP fused to an LacI repressiblepromoter present as a single chromosomal copy(Atkinson et al. 2003).

In vivo dynamics of b-galactosidase closely matchedmodelling predictions both qualitatively and quantitat-ively. Initially, three damped oscillations were observed(figure 3d) as predicted by simulation. This could beincreased to four (figure 3e) by moving the activatormodule next to the replication origin where copy numbersexist approximately fourfold higher than at the replica-tion terminus (Atkinson et al. 2003), again, aprediction of simulations (Atkinson et al. 2003). Further-more, amplitude and period were comparable betweensimulation and experiment, experiments also matchingpredicted effects of doubling time, controlled usinggrowth media, on period and amplitude (Atkinsonet al. 2003). Single-cell observations agreed with thoseat the population level, demonstrating damping is notdue to individual cells oscillating out of phase (Atkinsonet al. 2003). The percentage of oscillating cells was notreported, so it is not possible to assess robustness.

4.1.4. Discussion. Although only damped oscillationshave been obtained in vivo, the significant agreementbetween simulations and experiment shows relativelysimple models can capture both qualitative and quanti-tative behaviour. However, this quantitative agreementis only relevant at the population level and furtheranalysis is required to ascertain the applicability ofthe model at a single-cell level. Such an analysiswould also help define the role of noise, so far suggestedby Guantes & Poyatos (2006) to be of little importance.

4.2. Case II: repression by sequestration

An alternative implementation of the amplified nega-tive feedback oscillator is based on repressionoccurring through sequestration of proteins from thefirst gene, by dimerization with proteins from thesecond gene (figure 4a). This implementation was firstconsidered in Barkai & Leibler (2000). ODEs havebeen used to study this network (Vilar et al. 2002;Hilborn & Erwin 2008; figure 4b; see electronic sup-plementary material, §S4.1, and §S4.2), SDEs(Hilborn & Erwin 2008; electronic supplementarymaterial, §S4.3) and also Gillespie simulations


(Barkai & Leibler 2000; Vilar et al. 2002; Steuer et al.2003; Hilborn & Erwin 2008). The network has neverbeen implemented in vivo.

4.2.1. Mathematical analysis. The existence of oscil-lations can be proved using the Poincare–Bendixontheorem on a simplified model (figure 4b) describingonly the evolution of the repressor and the activator–repressor complex. Such a model has been observedin silico to capture qualitatively the main features ofa more complete model (reported in the electronic sup-plementary material, §S4.1; Vilar et al. 2002).Oscillatory behaviour was detected over a broad rangeof parameter values, suggesting robustness, but requiresan intermediate repressor degradation rate (Vilar et al.2002), a potential point of fragility.

4.2.2. In silico experiments. In addition to their impor-tance in determining the existence of oscillations,deterministic and stochastic simulations highlightedthe significant effect of the repressor degradation rateon the period of the oscillations; the lower the rate,the longer the period (Vilar et al. 2002). The amplitudeof the oscillations was found to be sensitive to transcrip-tion and translation rates (Barkai & Leibler 2000).Period is predicted to be of the order of 20 h (Vilaret al. 2002), comparable to the Hopf-driven transcrip-tionally controlled network in Atkinson et al. (2003).

In addition to oscillations existing (figure 4c) over abroad parameter range, simulations suggested oscil-lations could exist even at very low average mRNAlevels (Barkai & Leibler 2000; Vilar et al. 2002), oscil-lations persisting even when the number of mRNAmolecules mainly alternates between zero and one(Vilar et al. 2002). This is in contrast to the sensitivityseen with transcriptional repression, and is a conse-quence of the fact that average protein levels appearto be the important factor in permitting oscillations,which can be maintained at low mRNA levels byincreasing the translation rate (Vilar et al. 2002). A con-structive role for noise is also apparent, with oscillationsoccurring over a wider range of parameter values in sto-chastic simulations than in deterministic simulations(Vilar et al. 2002; Hilborn & Erwin 2008). Furthermore,stochastic coherence is also a feature of the network,with maximum oscillation regularity being observedfor an intermediate system size in Gillespie simulations(Steuer et al. 2003; Hilborn & Erwin 2008).

However, the details of the stochastic coherence aredependent on where noise exists. Under an SDEmodel the regularity of oscillations exhibits a verydifferent dependence on noise subject to whether noiseis added to activator or repressor concentrations(Hilborn & Erwin 2008). This dependence can probablybe attributed to the difference in dynamics of the com-ponents (Hilborn & Erwin 2008). Strikingly, the abilityto observe stochastic coherence at all in these simu-lations is dependent on the model used. UnderGillespie simulations of a reduced model (electronicsupplementary material, §S4.2), using steady-stateassumptions on fast variables, stochastic coherence isnot observed, even though deterministic simulations of


A B

(a)

time (h)

num

ber

of m

olec

ules

(R

, C)

(c)

where,

dRdt

=bR

dMR

aRqR+aR¢g R A (R)

qR + g R A (R)−gC A (R)R + dAC − dRR

dCdt

= g C A(R)R−dAC

(b)

(a ¢Ar (R) − Kd) +12

12 a ¢Ar (R) − Kd)2 + 4aAr(R)Kd

r(R) =

A(R) =

bA

dMA(g CR + dA)

Kd =qA

gA

0

500

1000

1500

2000

50 100 150 200

Figure 4. (a) Amplified negative feedback topology, with repression by dimerization. This topology is identical to that of §4.1.However, rather than repression occurring at the level of transcription, repression occurs through sequestration of the proteinencoded by the first gene, by dimerization with the protein encoded by the second gene. Solid lines represent direct transcriptionalcontrol, while the dotted line represents the repression by dimerization. (b) ODE model equations describing this network. R and Care the number of repressor molecules and activator–repressor complexes, respectively, A(R) is the quasi-equilibrium value of thenumber of activator molecules and MA and MR are the number of activator and repressor mRNAs, respectively. ai and ai

0 are thebasal and activated rates of transcription, bi the translation rates, di the degradation rates, gi and ui the rates of binding and unbind-ing, respectively, of A to other components. (c) In silico simulation of the ODEs. Solid lines and dashed lines represent R and C,respectively. Figure (c) adapted from Vilar et al. (2002). Copyright # National Academy of Sciences, USA.



the full and reduced model are very similar (Hilborn &Erwin 2008). The model reduction has thereforeobscured the stochastic coherence.

4.2.3. Discussion. The ability to function over a relativelybroad parameter range confers a level of robustness toparameter fluctuations, suggesting an in vivo implemen-tation may be successful. Period is comparable torepression by transcriptional control. However, in con-trast, low mRNA levels appear to be less consequential,and noise plays a more significant role, expanding theoscillatory parameter region, and generating stochasticcoherence. Finally, it is worth emphasizing that low-dimensional approximate models can indeed yield infor-mation on whether a network can oscillate, but mightfail to capture more subtle features of the system ofinterest such as stochastic coherence.

4.3. Case III: repression by proteolysis

A third implementation of the amplified negative feed-back topology where repression is obtained by


degradation of the protein from the first gene by a pro-tease encoded by the second gene was studied byGuantes & Poyatos (2006) and Conrad et al. (2008)(figure 5a). The network has been studied usingODEs (Guantes & Poyatos 2006; figure 5b; Conradet al. 2008; uses the same model as Guantes & Poyatos(2006)) and Gillespie simulations (Guantes & Poyatos2006), but never implemented in vivo.

4.3.1. Mathematical analysis. Oscillations in Guantes &Poyatos (2006) were found to arise through a subcriticalHopf bifurcation, requiring the activator to have signifi-cantly faster dynamics than the repressor, achieved byconsiderably faster activator degradation and trans-lation rates. This is a requirement in common withrepression by transcriptional control. The existence ofa subcritical Hopf bifurcation was confirmed byConrad et al. (2008) who also demonstrated, as withthe transcriptionally repressed oscillator, that oscil-lations could additionally arise through a SNIC.Oscillations exist for a significant range of values of the


A

(a)

(b)

(c)

B

dimensionless time 1

0

10

20

30

40

3 5 7

dim

ensi

onle

ss c

once

ntra

tion

dxdt

= D xx1 + rx2

1 + x2

1 + x2

− x − s¢xy

dydt

= Dxy1 + rx2

− y

s¢xy =dAR

dA K1K

Figure 5. (a) Amplified negative feedback topology, withrepression by proteolysis. This topology is identical to thatof §4.1 and 4.2. However, repression is instead by degradationof the protein encoded by the first gene by a protease encodedby the second gene. Solid lines represent direct transcriptionalcontrol, while the dotted line represents the repression by pro-teolysis. (b) ODE model equations describing this network. xand y are dimensionless concentrations of A and B, respect-ively. All other parameters are identical to those defined forelectronic supplementary material, §S3.1. Repression by pro-tease action is captured by the term s0xy. (c) In silicosimulation of the ODEs. The blue solid line and red dashedline correspond to x and y, respectively. Figure (c) adaptedfrom Guantes & Poyatos (2006).



ratio between the activator and repressor degradationrates (Guantes & Poyatos 2006).

4.3.2. In silico experiments. Under the Hopf bifurcation,oscillations appear possessing a finite frequency, whichincreases as activator dynamics become increasinglyfaster than repressor dynamics. Frequency reaches aplateau approximately fourfold higher than oscillationsgenerated by the transcriptionally repressed SNIC-driven oscillations (§4.1.2). In addition to existing fora significant variation of the activator to repressordegradation rate, oscillations are observed for a largerparameter range than for the SNIC-driven oscillations(Guantes & Poyatos 2006). This suggests a certainlevel of robustness to parameter variations. The subcri-tical nature of the bifurcation can also permit dampedoscillations for a brief window in the degradation ratio(Guantes & Poyatos 2006). Repression in the transcrip-tionally repressed SNIC-driven oscillator occurs with


homodimers, whereas repression here occurs via mono-meric proteases. As such, the required level ofrepression is reached much sooner, resulting in signifi-cantly lower activator amplitudes (figure 5c; Guantes& Poyatos 2006).

Stochastic coherence is observed under Gillespiesimulations, and is notably stronger than in thetranscriptionally repressed SNIC oscillator (Guantes &Poyatos 2006). However, as for that oscillator, themRNA number contributes significantly to noiseand its effect on oscillation regularity (Guantes &Poyatos 2006).

4.3.3. Discussion. The subtle change from nonlineartranscriptional repression via homodimers, to linearrepression via monomeric proteases, can produce anoscillator with higher frequency, lower amplitude oscil-lations and stronger stochastic coherence. This is aclear demonstration of how relatively small changes inthe repression mechanism can actually have largeeffects on dynamics, and therefore function.

4.4. Remarks

The amplified negative feedback oscillators discussed sofar do not demonstrate sustained oscillations in vivo, ordemonstrably robust dynamics. However, they are valu-able for their ability to inform on the relationshipbetween topology and dynamics. Three of the four net-works described here require the same condition forbifurcation, significantly faster activator than repressordynamics. However, despite this commonality, theydisplay a wide range of bifurcation mechanisms. Theperiod of the oscillations is comparable between theHopf-driven transcriptionally repressed oscillator inAtkinson et al. (2003) and the dimerization repressedoscillator. However, the period of the oscillations exhib-ited by the proteolytically repressed oscillator driven bythe subcritical Hopf is instead approximately four timesshorter than the SNIC-driven transcriptionallycontrolled oscillator.

5. FUSSENEGGER OSCILLATORS

The requirement of relatively faster activator thanrepressor dynamics for oscillations is present in threeout of the four oscillator topologies based on amplifiednegative feedback discussed so far. This has the sameeffect as that of adding a delay in the negative feedbackloop. Both allow the activator concentration to reach asignificant level before repression becomes too great,encouraging sustained oscillations. It is thereforeexpected that increasing the delay in the negative feed-back loop may increase the likelihood of observingundamped oscillations in vivo, as opposed to thedamped oscillations obtained in vivo for the amplifiednegative feedback oscillators reviewed so far.

This is precisely the idea behind the oscillatorspresented in Tigges et al. (2009, 2010) that we shallrefer to, for the sake of brevity, as the Fusseneggeroscillators. The Fussenegger oscillators are the only oscil-lators to have been implemented in a eukaryotic system.This is an important step towards using eukaryotic cells




as hosts for more complex networks and for exploitingthe powerful but often subtle regulatory mechanismsthey possess. The original Fussenegger oscillator(Tigges et al. 2009) comprises two genes, with bothsense and antisense transcription occurring from one ofthem. The sense transcript is translated, the resultingprotein feeding back to itself by promoting transcription,and also activating the second gene. In a first for asynthetic genetic network, this second gene activatesantisense transcription from the first gene, the transcriptnot translated, instead hybridizing with the sense tran-script, repressing sense protein production at translation.This completes a negative feedback loop (figure 6a).The Fussenegger oscillator is therefore an amplified nega-tive feedback oscillator (figure 6b). However, importantly,in comparison to the previous negative feedback oscil-lators, it contains an additional step in the negativefeedback loop that ‘delays’ the repressive effect.

The biological implementation of the Fusseneggeroscillator was modelled using ODEs (electronic sup-plementary material, §S5) and Gillespie simulations(modified t-leap). Models considered the potentialinteractions of RNA polymerases transcribing senseand antisense strands simultaneously, a significant con-sequence of the sense–antisense repression mechanism.The final model was obtained after rounds of improve-ment using extensive parameter estimation fromin vivo data as detailed in Tigges et al. (2009), butso far no mathematical analysis of its dynamics orbifurcations have been presented.


Simulations demonstrated the occurrence of oscillationsrobust to variations in mRNA and protein degradationrates, but highly sensitive to absolute and relative genedosage (Tigges et al. 2009; figure 6e). Gene dosage alsoaffected period and amplitude nonlinearly, exposing apotential method of tuning the oscillator. Further con-trol was available by inhibiting activation by the senseprotein, allowing oscillations to be switched on or off.Oscillations were insensitive to inhibition of antisensetranscription.

Stochastic simulations showed significant variability atlow gene levels, while at high gene levels, low-amplitudeoscillations not present in ODE simulations wereobserved, suggesting a constructive role for noise undercertain conditions.


The sense–antisense construct used for the in vivoimplementation comprised the gene encoding thetetracycline-dependent transactivator protein (tTA),controlled in the sense direction by the tTA activatedpromoter PhCMV*-1, forming the positive feedbackloop. tTA also promoted production of pristinamycin-dependent transactivator protein (PIT) throughPhCMV*-1. PIT promoted production of antisense tTARNA through a PPIR promoter (figure 6c). Rapidlydegraded GFP controlled by tTA through PhCMV*-1

reported on dynamics. Interactions between tTA andPhCMV*-1 could be inhibited by tetracycline, while the


PIT–PPIR interaction could be inhibited by pristinamy-cin. The tTA, PIT and gfp genes were each placed ondifferent plasmids, and introduced into Chinesehamster ovary (CHO-K1) cells (Tigges et al. 2009).

Undamped oscillations were observed with periods of170+ 71 (mean+ s.d.) min (Tigges et al. 2009;figure 6f ), confirming the effectiveness of consideringa delay in the negative feedback. However, the factthat the implementation is eukaryotic may also play ayet undetermined role. Significant cell–cell variabilitywas also observed, as predicted by stochastic simu-lations. The sensitive and nonlinear relationshipbetween period and amplitude, and relative genedosage, also matched predictions by simulations,demonstrating that network output can be predictablytuned. Further in accordance with simulations, inhi-bition by tetracycline of transcriptional activation bythe sense protein could abolish oscillations. However,in contrast to simulations, inhibition of antisense tran-scription by pristinamycin also had a marked effect,fluorescence continually growing rather than oscillating.

A ‘low-frequency’ variant of the Fussenegger oscil-lator has recently been constructed (Tigges et al.2010) using direct interference by siRNAs in place ofthe indirect repression by production of an antisensemRNA. Although a step in repression has beenremoved, the time delay for the repressive process isstill adequate to support sustained oscillations. Theoscillations possess a period of approximately 26 h,the longest of any synthetic oscillator. In contrast tothe tunability of the original Fussenegger oscillator,and in agreement with theoretical predictions, theperiod is insensitive to relative plasmid dosages.However, the oscillator is not robust, at most onlyapproximately 18 per cent of cells exhibiting oscillations.

5.3. Discussion

In contrast to the damped oscillations displayed in theprevious amplified negative feedback in vivo implemen-tation (Atkinson et al. 2003), the Fussenegger oscillatorand the recent low-frequency variant produce sus-tained, undamped oscillations, probably owing to theadditional time delay in the negative feedback loop.They are the only eukaryotic synthetic oscillators todate, and the first to incorporate sense–antisense andsiRNA-mediated regulation, potential mechanisms avail-able for exploitation by synthetic biologists. The resultsprovide an insight into the non-intuitive relationshipbetween gene dosage and dynamics, and suggest thatthis can be used to fine-tune network output. In bothimplementations, a full mathematical study would helplink the underlying mechanisms, together with the impli-cations for time delay of a eukaryotic implementation, tothe behaviour observed. This would provide a valuableand experimentally verified first insight into thedynamics of eukaryotic oscillators.

6. SMOLEN OSCILLATOR

None of the preceding oscillators are demonstrablyrobust. In particular, the percentage of oscillating cellsis either low, or not reported. However, robustness is


PPIR

(a) (b)

(e) ( f )

tTA

PhCMV*-1

PIT

PhCMV*-1

(d ) see electronic supplementary material, §S5 for model equations

A CB A C

(c)

20 40 600

0.1

0.2

0.3

0.4

0.5

0

0.2

0.6

0.8

1.0

1.2

1.4

0.4

–0.212000 1000800600400200

time (min)

fluo

resc

ence

(ar

b. u

nits

)

time (h)

prot

ein

conc

entr

atio

n (µ

M)

Figure 6. (a) Fussenegger oscillator topology. A promotes its own transcription, while also activating gene C. C promotes tran-scription of B (antisense A), which, through hybridization, represses A at the level of translation. B is represented by a square tosignify that it is an RNA not a protein. Solid lines represent direct transcriptional control, while the dotted line represents therepression by sense–antisense hybridization. (b) The Fussenegger oscillator topology in (a) is an example of an amplified negativefeedback topology; the combination of the link from C to B and B to A effectively forms one repressive link from C to A. (c)In vivo implementation. Details provided in the main text. (d) ODE model equations for the in vivo implementation aregiven in electronic supplementary material, §S5. (e) In silico simulation of ODEs for (1:1:1) plasmid ratios and no antibiotics.The black and blue lines are tTA and PIT, respectively, while the dashed and solid green lines are unfolded and active GFP,respectively. These are the results of an initial qualitative model, intended to demonstrate the capacity for oscillations. Thetimescale does not agree with the final model or the in vivo results. ( f ) Single-cell fluorescence trajectory, equimolar plasmidratios (100 ng each), no antibiotics. Figure (e) and (f) are adapted from Tigges et al. (2009).



vital if synthetic oscillators are to become componentsof larger synthetic systems, or interface with and regu-late natural systems (Lu et al. 2009).

The Smolen oscillator2 (Smolen et al. 1998) com-prises two genes (figure 7a). The first gene (gene A)promotes its own transcription and that of the othergene, while the second gene (gene B) represses itsown transcription and that of the first gene. The self-repression loop acting on the second gene is the extralink that differentiates the topology of this oscillatorfrom the amplified negative feedback oscillators (§4).Different types of models have been proposed to charac-terize the dynamics of the Smolen oscillator. Inparticular, ODE models can be found in Smolen et al.(1998) (electronic supplementary material, §S6.2),Hasty et al. (2002) (electronic supplementary material,§S6.1) and Stricker et al. (2008) (electronic supplemen-tary material, §S6.3), this latter model being far moredetailed and higher dimensional than the previoustwo. DDE models were also proposed in Smolen et al.(1998, 1999) (electronic supplementary material,§S6.4); Smolen et al. (1999) also considering a spatial

2This is sometimes referred to as the Hasty oscillator.


component. Finally, SDE-based models were discussedin Wang et al. (2005b; electronic supplementarymaterial, §S6.5) and Gillespie simulations were usedto study the oscillator dynamics (Wang et al. 2005b;Stricker et al. 2008).


Oscillations were first demonstrated mathematicallyusing a simple model and shown to arise from either asupercritical or subcritical Hopf bifurcation (Hastyet al. 2002). It was shown that such a bifurcation ismore likely to occur when the activator degradationrate is two or three times faster than that of the repres-sor. A similar requirement was also suggested foramplified negative feedback oscillators and might bedue to the similarity between the topologies. The exist-ence of a subcritical Hopf bifurcation is dependent onthe degree to which the activator promotes the tran-scription of the activator and repressor genes uponbinding to a specific site. Its occurrence yields narrowregions in parameter space where the coexistence ofoscillations and a stable equilibrium state is detected(see Hasty et al. (2002) for further details).


see electronic supplementary material, §S6.3 for model equations

1.0

0.5

0 30time (min) time (min)

fluo

resc

ence

(ar

b. u

nits

)

norm

aliz

ed c

once

ntra

tion

60 600

4

8

12

Plac/ara-1Plac/ara-1lacIaraC

120 180

(c)

(a) (b)

(e)(d )

BA

Figure 7. (a) Smolen oscillator topology. Gene A activates its own transcription and that of gene B, while gene B represses its owntranscription and that of gene A. (b) In vivo implementation. Details provided in the main text. (c) ODE model equations for thein vivo implementation are given in electronic supplementary material, §S6.3. (d) In silico simulation of ODEs for 0.7% arabinoseand 2 mM IPTG. There red and green lines are LacI tetramers and AraC dimers, respectively. The black line is LacI mRNA. (e)Single-cell fluorescence trajectories, with 0.7% arabinose and 2 mM IPTG. Figure (d) and (e) are adapted from Stricker et al.(2008).



More complex dynamical behaviour was detected inmore recent work. For example, using a higher dimen-sional, detailed model allied to experiments (§6.3),it was shown that two limit cycles can existsimultaneously (Stricker et al. 2008). Specifically, along-period limit cycle at low arabinose (an enhancerof activating links) levels can coexist with a limitcycle characterized by a smaller period with a verysmall basin of attraction. As the arabinose levelincreases, this latter limit cycle coalesces onto a stableequilibrium via a Hopf bifurcation, resulting in the pres-ence of damped oscillations whose basin of attractionexpands for increasing arabinose concentration(Stricker et al. 2008).


All the ODE models studied in the literature predictedthe existence of oscillations with periods of approxi-mately 40 min (Smolen et al. 1998; Stricker et al.2008; figure 7d). Such a period is far shorter than theperiod of oscillations detected in the Hopf-driven tran-scriptionally controlled amplified negative feedbackoscillator in Atkinson et al. (2003) or the Fusseneggeroscillators. Relative amplitudes of the activator andthe repressor differ between models (Smolen et al.1998; Stricker et al. 2008), although Gillespie simu-lations (Stricker et al. 2008) agree with the earlierODE model in Smolen et al. (1998), suggesting activa-tors achieve a significantly higher concentration. Thisis a characteristic of the Smolen oscillator shared withthe SNIC-driven transcriptionally controlled amplifiednegative feedback oscillator.

The effects of time delays have also been consideredin the literature. Namely, it was shown that theinclusion of delays in the models of greater than 2 h,accounting for transcription factor diffusion, eliminatedoscillations observed in Smolen et al. (1998), but


created a new stable (delay-induced) limit cycle witha period of the same order as the delay (Smolen et al.1999). Also, when spatial diffusion models for mRNAwere considered, oscillations were made to disappearby a narrow region of low diffusivity representing anuclear membrane (Smolen et al. 1999). Conversely,effectively increasing delay by decreasing reactionrates in the detailed model increased the region in par-ameter space associated to oscillatory behaviour andhence increased robustness to parameter variations(Stricker et al. 2008). It is clear that the effects ofdelay are not yet fully understood.

A certain degree of tunability of the oscillationperiod was observed. In particular, IPTG that inhibitsrepressive links or arabinose that enhances activatinglinks had differing effects on period in the detailedODE model (Stricker et al. 2008). At low concen-trations, period rapidly increases for increasing IPTGconcentration to a peak, after which it decreases asIPTG concentration increases further, while periodincreases monotonically to a plateau for increasingarabinose (Stricker et al. 2008). Using a parameterrelating E. coli cell cycle time to temperature in placeof the Arrhenius constant, period was observed todecrease almost linearly for increasing temperature(Stricker et al. 2008). In general, oscillations appear tobe tunable over a wide range of conditions (Strickeret al. 2008), and could possibly be tuned over a stillgreater range by using one or more of these tuningparameters in conjunction.

The simultaneous existence of a long-period limitcycle and a stable equilibrium with damped oscillations(§6.1) manifests itself as a bimodal distribution ofperiod lengths in Gillespie simulations (Stricker et al.2008). The majority of oscillations correspond to thelong-period limit cycle, far fewer small-amplitude oscil-lations occurring by noise-induced incursions into thebasin of attraction of the stable equilibria (Stricker




et al. 2008). As the basin of attraction expands, thenumber of small-amplitude oscillations increases(Stricker et al. 2008). This is a novel behaviour, directlyattributable to noise.

Surprisingly, oscillations have higher regularity whenthe Goodwin oscillator subnetwork itself (B with itsself-repressing feedback only) generates weak, low-amplitude oscillations with the repressor not reachingzero (Mather et al. 2009), rather than more regularoscillations that reach zero (Stricker et al. 2008). Thisfinding is undoubtedly a consequence of the nonlinear-ity of genetic networks, and highlights the potentialpitfalls of taking a modular approach to gene networkdesign.

Stochastic coherence has so far only been observed inthe simple model (Wang et al. 2005b). By positioningthe Chemical Langevin Equation (CLE) and Gillespierepresentations of the model (Hasty et al. 2002) closeto the subcritical Hopf bifurcation, a peak in thesignal-to-noise ratio for oscillations occurs for anintermediate system size (Wang et al. 2005b).


In vivo construction (figure 7b) based on the detailedmodel used the hybrid promoter Plac/ara-1 (Lutz &Bujard 1997) that contained two adjacent AraC bind-ing sites upstream of the promoter (at a position fromwhich bound AraC will activate transcription) andthree LacI operators (two upstream of the promoterand one downstream; Stricker et al. 2008). Plac/ara-1 isactivated by AraC in the presence of arabinose andrepressed by LacI. Repression can be inhibited byIPTG (Stricker et al. 2008). The araC and lacI geneswere each placed independently under the control ofPlac/ara-1 promoters. yemGFP, also under the controlof Plac/ara-1, reported on dynamics. All proteins con-tained ssrA tags, promoting rapid degradation. Theactivator and repressor genes were placed on separateplasmids and transformed into E. coli deficient inaraC and lacI, minimizing host genome interference(Stricker et al. 2008).

Over 99 per cent of cells displayed oscillations, with aperiod of approximately 40 min (Stricker et al. 2008;figure 7e), in accordance with simulations, and a cleardemonstration of robustness. Oscillatory state wastransmitted to progeny and synchrony lost after a fewperiods (Stricker et al. 2008). In agreement with theor-etical results, oscillations were highly tunable,oscillating over a wide range of IPTG and arabinoseconcentrations, temperatures and media sources, allow-ing period tuning between 13–58 min (Stricker et al.2008). Modelling also closely captured the effects of ara-binose and IPTG on period, and accurately predictedthe relationship between temperature and period, from258C to 378C. However, individual cells showed anunexpected gradual increase in period, possibly owingto the dynamics of the reporter used. As with thethree-gene repressilator, this was not accounted for inthe model. Although cell doubling time in minimalmedia and Luria Broth differed by approximately 1 h,period lengths were comparable, demonstrating thatdynamics are de-coupled from the cell cycle (Stricker


et al. 2008). The bi-modality of period lengths predictedby Gillespie simulations has not yet been observed(Stricker et al. 2008), this novel behaviour remainingtheoretical for the time being.

6.4. Discussion

The Smolen oscillator is robust and highly tunable, ableto display periods over a fivefold range, down to a rapid13 min. These are both characteristics observed in natu-ral oscillators; robustness provides reliability, whiletunability provides utility. This is encouraging for thefuture construction of synthetic oscillators. The rapid-ity of the oscillations appears to be due to theaddition of a negative feedback loop to what wouldotherwise be an amplified negative feedback topology.It probably also contributes towards robustness andtunability. Predicted bi-modality is a novel and poten-tially desirable attribute that also appears to betunable. If realized in vivo, this may expand thefuture applications of the oscillator. Studies on timedelays are inconclusive, but the construction ofthe Fussenegger oscillators suggests a eukaryoticimplementation may be successful.

7. VARIABLE LINK OSCILLATORS

A gene regulatory network is generally abstracted as atopology comprising a number of nodes connected byactivating or repressing links. However, not all networkscan be described in this way. For instance, the PRM pro-moter of the l phage is controlled by cI at threeoperators: OR1, OR2 and OR3. Binding affinities aresuch that binding typically proceeds sequentially,OR1 before OR2, and OR2 before OR3 (Hasty et al.2001a). Transcription is enhanced by cI binding toOR2, but repressed by binding to OR3; therefore, atlow to medium cI concentrations, OR2 will be boundand transcription enhanced, while at high concen-trations OR3 will also be bound, repressingtranscription (Hasty et al. 2001a). The link formed bythe regulation of this promoter is therefore variable.

This promoter has been used in an amplified nega-tive feedback-like topology (figure 8a; Hasty et al.2001a). The first gene regulates itself and a secondgene, through the variable promoter, while repressionby the second gene is via a protease acting on theproduct of the first gene. To study the network insilico, an ODE-based model was proposed in Hastyet al. (2001a) (figure 8c).

The implementation relies on the fact that a singlegene regulating itself through the variable promoteracts as a bi-stable switch (Hasty et al. 2001a), display-ing hysteresis with respect to its degradation rate. Theremainder of the topology exists to effectively movethe degradation rate back and forward, pushing thesystem round the hysteric loop, generating oscillations.Oscillations are favoured with smaller protein degra-dation rates and lower protease activity (Hasty et al.2001a). Gene copy number also plays a role (Hastyet al. 2001a).

Oscillations were found to exhibit periods of theorder of tens of minutes via in silico simulations of


A

(a)

(c) (d )

(b)

B rcsAcIl PRM l PRM

dxdt

= mx

(1 + x2 + as1x4)

(1 + x2 + as1x4)

1 + x2 + s1x4 + s1s 2x6

1 + x2 + s1x4 + s1s 2x6

−(g xx + g xyxy)

dydt

= my −g yy

time (min)0

10

20

30

20 40 60

CI

conc

entr

atio

n (n

M)

Figure 8. (a) Variable link oscillator topology. At low concentrations of protein A, gene A promotes its own transcription andthat of gene B, while at high A concentrations, A represses its own transcription and that of gene B. B is a protease and repressesA by degradation. The variable links are denoted by a ball. Solid lines represent direct transcriptional control, while the dottedline represents the repression by proteolysis. (b) Possible in vivo implementation using the PRM promoter of the l phage. Detailsprovided in main text. (c) ODE model equations. x and y are dimensionless quantities of cI and RcsA, respectively, t is dimension-less time (see Hasty et al. (2001a) for details) and mi are the respective plasmid copy numbers. si represents the relative affinitiesof cI dimer binding to OR1 relative to OR2 (s1) and OR3 (s3) binding. a is the degree to which transcription is enhanced bydimer occupation of OR2, while gx and gy capture degradation of cI and RcsA, respectively, and gxy represents the rate of cIdegradation by RcsA. (d) In silico simulation of ODEs demonstrates oscillations. Figure (d) adapted from Hasty et al. (2001a).



the topology (figure 8d). This is comparable to theSmolen oscillator and significantly more rapid thaneither the Hopf-driven amplified negative feedbackoscillator in Atkinson et al. (2003) or the Fusseneggeroscillators. In addition to their importance in determin-ing the existence of oscillations, degradation rates andgene copy number were also found to affect the period(Hasty et al. 2001a).

Although no in vivo implementation has yet beenundertaken, it has been proposed that it might be pos-sible to use the repressor cI as gene A, and the proteaseRcsA as gene B (Hasty et al. 2001a), both controlled bythe PRM promoter (figure 8b).

In conclusion, simulations seem to suggest that theuse of variable promoters in an amplified negative feed-back topology will increase the frequency of oscillations,to a level comparable with that exhibited by the Smolenoscillator. A pressing open problem is the lack of an invivo implementation of variable link oscillators. Thisis required to assess to what extent the in silico predic-tions can be matched in vivo, and to ascertain whethersuch an implementation can also match the robustnessproperties of the Smolen oscillator. The promoter struc-ture discussed has also been used in another, morecomplex transcriptional level controlled oscillatorreported in Wang et al. (2007).

8. METABOLATOR

The metabolator (Fung et al. 2005) is the first oscillatorto be reported in the literature that incorporates metab-olites as a core component. Conceptually, it comprisestwo genes. One gene produces an enzyme that convertsone metabolic pool (M2) to another (M1); its transcrip-tion being activated by M2. At the same time, the othergene produces an enzyme that converts M1 to M2; its


transcription being repressed by M2. Additionally,there is an influx into M1, and an efflux from M2

(figure 9a). To facilitate comparison with other oscil-lators, the topology of the metabolator can berepresented schematically as a gene regulatory network(figure 9b). Here, two genes, A and B, activate eachother and are self-repressed by increasing/decreasingthe metabolic pool M2. A particular implementationof this concept (figure 9c) was modelled with ODEs(electronic supplementary material, §S7) and CLEs,which explicitly represent a GFP reporter (Fung et al.2005). All following discussion refers to this model.

8.1. Mathematical analysis and in silicoexperiments

Mathematical analysis showed oscillations arise througha Hopf bifurcation, dependent on a number of factors. Itwas observed that a high-enough influx rate is essentialfor the presence of oscillations which can be instead sup-pressed by high M2 concentrations. Oscillations werealso found to be sensitive to relative gene copy numbers.Simulations predicted the oscillation period to beapproximately 40 min and confirmed the requirementof an adequate in flux rate (figure 9f ), while CLE simu-lations, as per many of the other oscillators, showed theaddition of noise created significant amplitudevariation.


The E. coli acetate pathway was exploited for in vivoimplementation. Acetyl coenzyme A (acetyl-CoA) wasused as M1, and acetyl phosphate was used as M2.Acetyl-CoA is converted to acetyl phosphate by phos-phate acetyltransferase (Pta), which corresponds togene A in the network (figure 9c,d). Acetyl phosphate


ptaPLlacO-1

acsglnAp2

lacIglnAp2

implementation

C A B

A B

M1

M2

influx

efflux

A B

concept

see electronic supplementary material, §S7 for modelequations

time (min)0

1000

2000

3000

(d )

(e)

(c)(a)

(b)

( f ) ( g)

0

10–4

10–2

100

250 500

acetyl-CoA

AcP

60 120 180 240

fluo

resc

ence

(ar

b. u

nits

)

dimensionless time

met

abol

ite c

once

ntra

tions

(dim

ensi

onle

ss u

nits

)

Figure 9. (a) Metabolite-centric conceptual view of the metabolator. Gene A produces an enzyme that converts one metabolicpool (M1) to another (M2), its transcription repressed by M2, while gene B produces an enzyme that converts M2 to M1, its tran-scription activated by M2. In addition, there is an influx into M1 and an efflux from M2. (b) Schematic GRN diagram. Gene Arepresses itself (as it increases the level of M2, which represses it), while promoting gene B (it is activated by M2). Gene B alsoeffectively represses itself (as it decreases the level of M2, which activates it), while effectively activating gene A (as M2 repressesgene A). (c) Implementation topology. Three genes were used in the implementation. The negative feedback link from gene A toitself via gene C, is equivalent to gene A repressing itself in (b). Solid lines represent direct transcriptional control, while thedotted lines represent indirect repression. (d) In vivo implementation. Details are provided in the main text. (e) ODE modelequations for the in vivo implementation are given in electronic supplementary material, §S7. ( f ) In silico simulation ofODEs with a relatively high glycolytic flux. (g) Single-cell fluorescence trajectories. Figures (a), ( f ) and (g) are adapted fromFung et al. (2005).



represses transcription indirectly: it results in the phos-phorylation of Nitrogen Regulator I (NRI), whichstimulates production of LacI by activating a glnAp2promoter. LacI represses pta expression through promo-ter PLlacO-1 (Lutz & Bujard 1997). Acetyl-CoAsynthetase (AcS—synthesises acetyl-CoA) served as


gene B in the network (figure 9c,d). Its expressionwas also under the control of a glnAp2 promoter, andwas thus upregulated by acetyl phosphate via NRI.Acetyl-CoA is a metabolic product of sugars, fattyacids and amino acids. Sugars present in growth mediaserved as the influx to the network. Acetyl phosphate




is converted to acetate by acetate kinase (AcK), andafter protonation is permeable across the cell membrane,serving as the efflux. The lacI gene was incorporated intothe E. coli chromosome in single copy, with pta and acsgenes placed on a single plasmid. A reporter geneencoding GFP, under the control of LacI, was placedon a separate plasmid. All proteins contained ssrAtags, ensuring rapid degradation. E. coli containednon-functioning pta and lacI mutants, ensuring mini-mal host genome interference, and a glnL knockout,allowing phosphorylation of NRI by acetyl phosphate.

Oscillations were observed in approximately 60 percent of cells, with a period of 45+ 10 min (mean+s.d.; figure 9g), of the order of in silico predictions,and lasted at least 4 h. As predicted by stochastic simu-lations, amplitudes varied significantly. Cell divisionsappeared uncorrelated with oscillations, suggestingdynamics are de-coupled from the cell cycle.

In agreement with the mathematical analysis, influxrates from media containing glucose, fructose and man-nose were high enough to support oscillations, while alower rate from glycerol was not. Furthermore, additionof high concentrations of acetate, which maintains ahigh acetyl phosphate (M2) concentration throughAcK, suppressed oscillations. Oscillations can thereforebe easily controlled externally through metabolites.Natural accumulation of acetate also suppressed oscil-lations, suggesting removal could support longer termoscillations (Fung et al. 2005).

8.3. Discussion

Theoretical models of the metabolator account for invivo observations, and dynamics can be predictablycontrolled by external metabolite sources. Sensitivityof oscillations to gene copy number, which will varytemporally, may account for the relatively low per-centage of oscillating cells. Although a GRNrepresentation has been proposed here, the metabolatornaturally lends itself to a metabolite-centric represen-tation (i.e. figure 9a; Fung et al. 2005). This raises thequestion of whether GRNs are always the most usefulway of representing biological networks containinggenes.

9. CONCLUSIONS

Synthetic genetic oscillators, both constructed in vivoand studied theoretically, have been reviewed andpresented, where possible, in a coherent unified frame-work. A path has been taken from the earlytheoretical work of Goodwin, to the recent constructionof the robust Smolen oscillator, and the more sophisti-cated Fussenegger oscillators. A summary of oscillatorfeatures can be found in tables 1 and 2, collating oscil-lators using solely transcriptional regulation and atleast one non-transcriptional regulation mechanism,respectively.

Although outside the scope of this review, there is anincreasing amount of investigation into coupled geneticoscillators. The focus has been on coupled repressilators(Garcia-Ojalvo et al. 2004; Ullner et al. 2007; Zhou et al.2008), although coupling of other networks has also


been studied (McMillen et al. 2002; Li et al. 2006).The analysis has almost exclusively been theoretical,but coupling has recently been used in vivo to generatea population of cells displaying synchronizedoscillations (Danino et al. 2010).

Comparisons between oscillators are hindered by thefact that, despite the drive towards a standardizedcharacterization of synthetic biological componentsand devices (The BioBricks Foundation, http://bbf.openwetware.org/), there currently exists no character-ization standard for published synthetic oscillators.Reported characteristics vary significantly betweenoscillators in both type and detail, for instance, the per-centage of oscillating cells, a vital measure ofrobustness, is only reported in approximately half ofthe in vivo implementations of the oscillators describedin this paper.

In vivo implementations have almost exclusivelybeen in prokaryotes, the eukaryotic Fussenegger oscil-lators being a recent exception. The prokaryoticsystems exploited thus far are generally easy to manip-ulate and their mechanisms of gene regulation aregenerally simpler than those of eukaryotes. This relativesimplicity means models are more likely to capture thesystems essential features and hence be more predictive.

Oscillatory periods vary between 13 min and 26 h(Smolen and the low-frequency Fussenegger oscillator,respectively), a significant range. The reason for thisvariation is not yet clear, and period can be tunedthrough a number of different means in different oscil-lators. For instance, period is tuned by temperature,and small-molecule concentrations affecting transcrip-tion regulation, in the Smolen oscillator, and byrelative plasmid dosage in the original Fusseneggeroscillator. Noise is likely to also play a role in determin-ing period. Amplitude can be tuned through similarmeans. In the original Fussenegger oscillator, this isagain through relative plasmid dosage. In terms ofreported oscillating cell percentages, only the Smolenoscillator can be termed robust.

Simulations suggest a dual role for intrinsic noise indynamics. On the one hand, noise accounts for amp-litude variations, but on the other, it enlargesoscillatory parameter regions, enhances access toregions of phase space and generates stochastic coher-ence. These constructive effects of noise will probablybe exploited in future oscillators through copy numbercontrol. However, the ability to observe these effectsin silico, in particular stochastic coherence, dependson the parameters and representation used. Stochasticcoherence is abolished in the non-adiabatic regime ofthe repressilator, and by quasi-equilibrium assumptionsin the dimerization-controlled amplified negative feed-back oscillator. It should be noted that as of yet,there appears to have been no in vivo validation ofstochastic coherence within genetic oscillators.

This review found that modelling has been signifi-cantly predictive and often in agreement with in vivoobservations, but model parameters were often iden-tified by heuristic procedures. Generally, simplermodels have been used to determine the network poten-tial for oscillations, with more detailed modelssubsequently being developed to obtain better

http://bbf.openwetware.org/




Tab

le1.

Asu

mm

ary

ofth

eos

cilla

tors

usin

gso

ley

tran

scri

ptio

nalco

ntro

l.P

erio

dis

reco

rded

asm

ean+

s.d.

;n.

a.,no

tap

plic

able

.

nam

e

gene

num

ber/

tota

llin

knu

mbe

rlin

kty

pebi

furc

atio

nm

echa

nism

nois

eef

fect

s

invi

voim

plem

enta

tion

(osc

illat

ing

cell

perc

enta

ge)

osci

llati

onch

arac

teri

stic

sro

bust

aspe

cts

tuna

ble

aspe

cts

key

refe

renc

e(s)

repr

essi

lato

rN

/NN

nega

tive

Hop

fsi

gnifi

cant

ampl

itud

eva

riat

ion.

Enl

arge

dos

cilla

tory

para

met

erra

nge.

Stoc

hast

ic(p

hase

)co

here

nce.

3ge

ne.

prok

aryo

tic

(app

rox

40%

)

peri

od:

160+

40m

in.

Sign

ifica

ntam

plit

ude

vari

atio

n

dyna

mic

sde

coup

led

from

cell

cycl

e

dyna

mic

sco

uple

dto

glob

alre

gula

tion

ofce

llgr

owth

Elo

wit

z&

Lei

bler

(200

0),

Mul

ler

etal

.(2

006)

and

Stre

lkow

a&

Bar

ahon

a(2

010)

Goo

dwin

1/1

1ne

gati

veH

opf

enla

rged

osci

llato

rypa

ram

eter

regi

onpr

okar

yoti

c(n

.a.)

peri

od:�

30m

ins.

Irre

gula

r.R

arel

yre

ach

zero

peri

odre

sist

ant

toIP

TG

conc

.

n.a.

Stri

cker

etal

.(2

008)

and

Mul

ler

etal

.(2

006)

Am

p.N

eg.

F.B

ack.

Tra

nsc.

2/3

2po

siti

ve,

1ne

gati

veH

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sub-

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for

SNIC

wea

kst

ocha

stic

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renc

efo

rSN

ICdr

iven

osci

llato

r

prok

aryo

tic

(n.a

.)os

cilla

tion

sda

mpe

d.P

erio

d:10

/20

h

n.a.

peri

odan

dam

plit

ude

via

cell

doub

ling

Atk

inso

net

al.

(200

3)an

dG

uant

es&

Poy

atos

(200

6)Sm

olen

2/4

2po

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ve,

2ne

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vesu

per/

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f,un

defin

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odel

bi-m

odal

osci

llati

ons

(in

vivo

impl

emen

tati

on).

Stoc

hast

icco

here

nce.

prok

aryo

tic

(.99

%)

peri

od:1

3–58

min

deco

uple

dfr

omdo

ublin

gti

me/

cell

cycl

e

peri

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IPT

G,

arab

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008)

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ifica

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c(6

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ifica

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on/o

ffby

influ

xan

def

flux

rate

s

Fun

get

al.

(200

5)





Tab

le2.

Asu

mm

ary

ofth

eos

cilla

tors

usin

ga

leas

ton

ere

gula

tory

mec

hani

smth

atdo

esno

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tat

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leve

lof

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scri

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n.P

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dis

reco

rded

asm

ean+

s.d.

;n.

a.,no

tap

plic

able

.

nam

e

gene

num

ber/

tota

llin

knu

mbe

rlin

kty

pebi

furc

atio

nm

echa

nism

nois

eef

fect

s

invi

voim

plem

enta

tion

(osc

illat

ing

cell

perc

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ge)

osci

llati

onch

arac

teri

stic

sro

bust

aspe

cts

tuna

ble

aspe

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key

refe

renc

e(s)

Fus

sene

gger

2/4

3po

siti

ve,

1ne

gati

ven.

a.si

gnifi

cant

vari

abili

tyat

low

gene

dosa

ge,

addi

tion

alos

cilla

tion

sat

high

dosa

ge

euka

ryot

ic(n

.a.)

peri

od:1

70+

71m

in.S

igni

fican

tce

ll–ce

llva

riab

ility

n.a.

peri

odan

dam

plit

ude

via

gene

dosa

ge

Tig

ges

etal

.(2

009)

low

-fre

quen

cyFus

sene

gger

2/3

2po

siti

ve,

1ne

gati

ven.

a.n.

a.eu

kary

otic

(�18

%)

peri

od:2

6+

8.5

h.Lon

gest

peri

odof

any

synt

heti

cos

cilla

tor

peri

odin

sens

itiv

eto

rela

tive

plas

mid

dosa

ge

n.a.

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ges

etal

.(2

010)

Am

p.N

eg.

F.B

ack.

Dim

.

2/3

2po

siti

ve,

1ne

gati

velo

ssof

equi

libri

umst

abili

ty

enla

rged

osci

llato

rypa

ram

eter

regi

on.

Stoc

hast

icco

here

nce

no(-

)pe

riod

:�20

hos

cilla

tion

spo

ssib

leat

very

low

aver

age

mR

NA

leve

ls

peri

odby

degr

adat

ion

rate

.A

mpl

itud

eby

tran

scri

ptio

n/tr

ansl

atio

nra

te

Vila

ret

al.

(200

2)

Am

p.N

eg.

F.B

ack.

Pro

t.

2/3

2po

siti

ve,

1ne

gati

vesu

b-H

opfor

SNIC

stoc

hast

icco

here

nce

no(-

)hi

gher

freq

uenc

yan

dlo

wer

ampl

itud

eth

antr

ansc

.re

pres

sed

SNIC

.D

ampe

dos

cilla

tion

spo

ssib

le

larg

eros

cilla

tory

para

met

erre

gion

than

wit

hSN

ICdr

iven

tran

sc.

cont

rolle

dos

cilla

tor

peri

odby

rela

tive

degr

adat

ion

rate

s

Gua

ntes

&P

oyat

os(2

006)

and

Con

rad

etal

.(2

008)

vari

able

link

2/3

2va

riab

le,

1ne

gati

velo

ssof

equi

libri

umst

abili

ty

n.a.

no(-

)pe

riod

:ten

sof

min

utes

n.a.

peri

odvi

ade

grad

atio

nra

tes

and

gene

dosa

ge

Has

tyet

al.

(200

1a)







qualitative and quantitative predictions of the networkbehaviour. Detailed models can also uncover more com-plex behaviour as, for instance, the coexistence ofmultiple limit cycles observed in the Smolen oscillator.However, where more complex behaviour has not yetbeen observed in vivo, it is important to check thatsuch behaviours are not just an artefact of the increasedmodel detail and the subsequent higher dimensionality,which might serve to distract from the main dynamicalbehaviour.

Surprisingly, not all models explicitly take intoaccount the dynamics of the reporter, usually a fluor-escent protein, as the means of assessing dynamics.For example, the models proposed for both the repres-silator and the Smolen oscillator do not, and bothshow unexpected effects that might be explained bythe reporter characteristics, specifically, the continuallygrowing fluorescence of the repressilator, and the gradu-ally increasing period of the Smolen oscillator.

Both mathematical analysis and in silico simulationsconfirmed that the most common source of oscillationsis the presence of a Hopf bifurcation in the model,although other bifurcation scenarios, such as a saddle-node of limit cycles or homoclinic bifurcation, are alsopotential mechanisms. Despite the importance of thesemechanisms, their implications are only marginally dis-cussed in the literature. This is rather surprising asthey can have dramatic effects on the dynamics exhibitedby the oscillators. Good examples are the transcription-ally and proteolytically controlled amplified negativefeedback oscillators where in both cases oscillations canbe generated either through a SNIC or a subcriticalHopf bifurcation (Guantes & Poyatos 2006; Conradet al. 2008). In the transcriptionally repressed oscillator,the SNIC leads to oscillations with arbitrarily low fre-quencies and large periods, while in the proteolyticallyrepressed oscillator, the subcritical Hopf bifurcationyields higher frequencies and damped oscillations. Differ-ences are further apparent when responses to externalstimuli are considered. In both oscillators, the SNICcauses the respective oscillator to act as an integratorof signals, while the subcritical Hopf bifurcationcauses it to act as a frequency-sensitive resonator, bothbehaviours directly attributable to the underlying bifur-cation mechanism (Guantes & Poyatos 2006; Conradet al. 2008).

There is an interesting relationship between topologyand dynamics. Although bifurcation mechanisms variedsignificantly across the amplified negative feedbackimplementations, the bifurcation requirement of a signifi-cant difference in the speed of activator and repressordynamics was well conserved. This requirement evenextended through to the Smolen oscillator, presumablyowing to the similarity between the two topologies. Pre-vious studies demonstrated correlation between topologyand the frequency–amplitude relationship of oscillators(Tsai et al. 2008). Together with the findings here, itappears that correlations between topology anddynamics exist within certain levels of classification,and over certain ranges of topologies.

Time delay has both constructive and destructiveeffects. In the Goodwin oscillator, it increases the oscil-latory parameter space, but also reduces regularity,


while under one representation of the Smolen oscillator,it probably increases robustness, yet under anotheroscillations are abolished when it is too great, or in amanner representing a nuclear membrane. However,increased time delay is the likely reason why theFussenegger oscillators display undamped oscillationswhile the earlier amplified negative feedback implemen-tation does not.

Even in light of the apparent robustness of theSmolen oscillator, robustness is still a significant issuefor synthetic oscillators. Although current oscillatorsgenerally appear decoupled from the cell cycle, net-works exist in cellular environments that are not fullycharacterized or understood. Interference from thehost cell is likely, possibly accounting for the low per-centages of oscillating cells reported. The halting ofoscillations in the three-gene repressilator by entryinto the stationary phase is an indication of such inter-ference (Elowitz & Leibler 2000). With the exception ofthe Fussenegger oscillators, all in vivo implementationsused mutation or removal of particular host genes tomitigate against interference. However, as networksbecome larger, the number of potentially interferinghost components will increase. Unknown componentswill form interactions and known components willform unknown interactions. Both of these are morelikely in lesser characterized and larger eukaryotic sys-tems into which networks will increasingly be placed.These issues also extend to purely synthetic com-ponents. A measure of how likely network function isto be compromised if subjected to reasonable levels ofhost cell interactions is required. In this direction, twoempirical rules have been developed for the repressilator(Goh et al. 2008). Generally, non-specific interactionsforming coherent couplings (couplings preserving regu-latory signs between nodes) are more likely tomaintain oscillations than those forming incoherentcouplings, and non-specific interactions having higherregulatory homogeny (a greater fraction of their inter-actions with the network positive or negative) are alsomore likely to preserve oscillations (Goh et al. 2008).Effects of the cell cycle, through changes in volume,component levels at division (Chen et al. 2004; Yodaet al. 2007) and levels of host cell components requiredfor network function e.g. RNA polymerase and ribo-somes (Tuttle et al. 2005), are also starting to beconsidered. Together, these should aid the design andimplementation of new types of reliable, robust, tunablebiological oscillators, and their incorporation into newsynthetic genetic regulatory network designs.

We acknowledge funding from the UK EPSRC via the BristolCentre for Complexity Sciences. We thank MatthewR. Bennett for providing model details.

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a comparative analysis of synthetic genetic...

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