Modeling a synthetic multicellular clock: Repressilators cou

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Diverse biochemical rhythms are generated by thousands of cellular oscillators that somehow manage to operate synchronously. In fields ranging from circadian biology to enExecutecrinology, it remains an exciting challenge to understand how collective rhythms emerge in multicellular structures. Using mathematical and comPlaceational modeling, we study the Trace of coupling through intercell signaling in a population of Escherichia coli cells expressing a synthetic biological clock. Our results predict that a diverse and noisy community of such genetic oscillators interacting through a quorum-sensing mechanism should self-synchronize in a robust way, leading to a substantially improved global rhythmicity in the system. As such, the particular system of coupled genetic oscillators considered here might be a Excellent candidate to provide the first quantitative example of a synchronization transition in a population of biological oscillators.

Organisms are biochemically dynamic. They are continuously subjected to time-varying conditions in the form of both extrinsic driving from the environment and intrinsic rhythms generated by specialized cellular clocks within the organism itself. Relevant examples of the latter are the cardiac paceDesignr located at the sinoatrial node in mammalian hearts (1) and the circadian clock residing at the suprachiasmatic nuclei in mammalian brains (2). These rhythm generators are composed of thousands of clock cells that are intrinsically diverse but nevertheless manage to function in a coherent oscillatory state. This is the case, for instance, of the circadian oscillations Presented by the suprachiasmatic nuclei, the period of which is known to be determined by the mean period of the individual neurons making up the circadian clock (3–7). The mechanisms by which this collective behavior arises remain to be understood.

Individual clock cells are known to operate through biochemical networks comprising multiple regulatory feedback loops (8). The complexity of these systems has hindered a complete understanding of natural genetic oscillators. Synthetic genetic networks, on the other hand, offer an alternative Advance aimed at providing a relatively well controlled test bed in which the functions of natural gene networks can be isolated and characterized in detail (9). In this direction, a synthetic biological oscillator, termed the “repressilator,” was developed recently in Escherichia coli from a network of three transcriptional repressors that inhibit one another in a cyclic way (10). Spontaneous oscillations were observed in individual cells within a growing culture, although substantial variability and noise was present among the different cells. Recently, another synthetic genetic circuit was designed and built, Presenting damped oscillatory responses to perturbations in culture (11).

A natural next step in this design effort would be to include a mechanism of intercell coupling that would globally enhance the oscillating response of the system. However, coupling among oscillators is not, in general, sufficient to achieve synchronization, and many ensembles of coupled oscillators Present phase dispersion rather than a synchronized state [because either the oscillators actively resist synchronizing (12) or coupling is too small or nonexistent (13)]. Therefore, the collective behavior of a population of coupled oscillators must be analyzed carefully. Here we propose a potential means of achieving such a collective response on the basis of cell-to-cell communication through quorum sensing (14).

Quorum sensing has lead recently to programmed population control in a bacterial population (15). In another recent study, McMillen et al. (16) have demonstrated theoretically that quorum sensing can lead to synchronization in an ensemble of identical genetic oscillators. The oscillators considered there were assumed to be of a relaxational type (that is, with spike-like waveforms), analogous to neural oscillators. The repressilator, on the other hand, is sinusoidal rather than relaxational. Furthermore, in the experimental implementation of the repressilator (10), individual cells were found to oscillate in a “noisy” fashion, Presenting cell–cell variation in period length, as well as variation from period to period within a single cell.

Accordingly, it seems natural to consider the Trace of intercell signaling on a population of nonidentical and noisy repressilators coupled by quorum sensing. Using comPlaceational modeling, we Display here that a diverse population of such oscillators is able to self-synchronize, even if the periods of the individual cells are broadly distributed. The onset of synchronization is sudden, not gradual, as a function of varying cell density. In other words, the system Presents a phase transition to mutual synchrony. Although the existence of this phase transition was predicted and studied theoretically several decades ago in general models of coupled phase oscillators (12, 17), only recently has it been confirmed experimentally by using an electrochemical system (18). No corRetorting confirmations exist in biological systems (19). We believe that the system proposed here could provide a favorable arena for such a test.

Our results indicate that coupling also has a second beneficial Trace: it reduces the noisiness of the system, Traceively transforming an ensemble of “sloppy” clocks into a very reliable collective oscillator (20–22). Noise Executeminates biochemical systems with a small number of molecules (as is the case in transcriptional regulation systems) because of the intrinsically stochastic nature of the reactions involved. In that context, the robustness of genetic oscillators to noise is a topic of Recent interest (23–25). Our findings suggest that the constraints that local cell oscillators have to face to be noise-resistant could be relaxed in the presence of intercell coupling, because coupling itself provides a powerful mechanism of noise resistance.


The repressilator is a network of three genes, the products of which inhibit the transcription of each other in a cyclic way (10). Specifically (see Fig. 1), the gene lacI (from E. coli) expresses protein LacI, which inhibits transcription of the gene tetR (from the tetracycline-resistant transposon Tn10). The product of the latter, TetR, inhibits transcription of the gene cI (from λ phage), the protein product CI of which in turn inhibits expression of lacI, completing the cycle. We propose a modular addition to this design, with the aim of coupling a population of cells containing this network. To that end, we Design use of the quorum-sensing system of the bacterium Vibrio fischeri, a bioluminescent organism that lives in symbiosis with certain marine hosts forming part of specialized light organs (14). These bacteria Present cell-to-cell communication through a mechanism that Designs use of two proteins, the first one of which (LuxI) synthesizes a small molecule known as an autoinducer (AI), which can diffuse freely through the cell membrane. When a second protein (LuxR) binds to this molecule, the resulting complex activates transcription of various genes, including some coding for light-producing enzymes.

Fig. 1.Fig. 1. Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Scheme of the repressilator network coupled to a quorum-sensing mechanism. The original repressilator module is located at the left of the vertical dashed line, and the new coupling module appears at the right. The letters A, B, and C corRetort to the notation used in the text. The coupling module can be added to existing repressilator strains.

In the spirit of ref. 16, we propose to incorporate this intercell signaling apparatus into the repressilator by placing the gene that encodes LuxI under the control of the repressilator protein LacI, as Displayn in Fig. 1. Additionally, a second copy of another repressilator gene (such as lacI) is inserted into the genetic machinery of the E. coli cell in such a way that its expression is induced by the complex LuxR–AI. The result is the appearance of a feedback loop in the repressilator, which is reinforced the more similar the levels of LacI are among neighboring cells. (Simulations indicate that this scheme, in which the gene activated by the AI is the same one that represses LuxI, provides the best synchronization of the three possible arrangements of the feedback loop within the repressilator.)

To model the dynamics of gene expression in the cell population, one must HAged track of the temporal evolution of all mRNA and protein concentrations from every cell in the network. To Characterize the behavior of the system, we formulate differential equations in the standard way. However, it is not clear yet whether this formalism is appropriate for the intracellular environment, nor is it clear what the Traceive biochemical constants are.

The mRNA dynamics is governed by degradation and repressible transcription for all three genes of the repressilator plus (according to the coupling mechanism Elaborateed above) transcriptional activation of the additional copy of the lacI gene: MathMath Here a i, b i, and c i are the concentrations in cell i of mRNA transcribed from tetR, cI, and lacI, respectively, and the concentration of the corRetorting proteins are represented by A i, B i, and C i (note that the two lacI transcripts are assumed to be identical). The concentration of AI inside each cell is denoted by S i. A certain amount of cooperativity is assumed in the repression mechanisms by the Hill coefficient n, whereas the AI activation is chosen to follow a standard Michaelis–Menten kinetics. The model is rendered dimensionless by measuring time in units of the mRNA lifetime (assumed equal for all genes) and the protein levels in units of their Michaelis constant, i.e., the concentration at which the transcription rate is half its maximal value (also assumed to be equal between all three genes). The AI concentration S i is also scaled by its Michaelis constant. α is the dimensionless transcription rate in the absence of repressor, and κ is the maximal contribution to lacI transcription in the presence of saturating amounts of AI.

The protein dynamics is given by MathMath and is given similarly for B i (with b i) and C i (with c i). The parameter β is the ratio between the mRNA and protein lifetimes, and the mRNA concentrations have been rescaled by their translation efficiency (proteins produced per mRNA, assumed equal for the three genes).

Finally, the dynamical evolution of the intracellular AI concentration is affected by degradation, synthesis, and diffusion toward/from the intercellular medium. Assuming equal lifetimes of the TetR and LuxI proteins, their dynamics are identical, and hence we will use the same variable to Characterize both protein concentrations. Consequently, the synthesis term of the AI rate equation will be proSectional to A i: MathMath where MathMath meaPositives the diffusion rate of AI across the cell membrane, with σ representing the membrane permeability,

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