Robust dynamical pattern formation from a multifunctional minimal genetic circuit
© Rodrigo et al; licensee BioMed Central Ltd. 2010
Received: 17 November 2009
Accepted: 22 April 2010
Published: 22 April 2010
A practical problem during the analysis of natural networks is their complexity, thus the use of synthetic circuits would allow to unveil the natural mechanisms of operation. Autocatalytic gene regulatory networks play an important role in shaping the development of multicellular organisms, whereas oscillatory circuits are used to control gene expression under variable environments such as the light-dark cycle.
We propose a new mechanism to generate developmental patterns and oscillations using a minimal number of genes. For this, we design a synthetic gene circuit with an antagonistic self-regulation to study the spatio-temporal control of protein expression. Here, we show that our minimal system can behave as a biological clock or memory, and it exhibites an inherent robustness due to a quorum sensing mechanism. We analyze this property by accounting for molecular noise in an heterogeneous population. We also show how the period of the oscillations is tunable by environmental signals, and we study the bifurcations of the system by constructing different phase diagrams.
As this minimal circuit is based on a single transcriptional unit, it provides a new mechanism based on post-translational interactions to generate targeted spatio-temporal behavior.
Synthetic Biology aims to engineer genetic networks with defined dynamics . For this, it usually relies on the use of design principles derived from the analysis of natural genetic networks. Those networks are large and complex systems with many unknown interactions that can dramatically affect the system dynamics. Then, for a complete understanding of the mechanisms underlying gene networks it is valuable the engineering of synthetic circuits that have a minimal complexity. In addition, such small circuits would allow the modular design of complex hierarchical structures with targeted spatial and temporal behaviors. However, even the design of small circuits with existing genetic components is very challenging due to the lack of enough parameters to fine-tune the system. In fact, the use of properly characterized genetic components favors an accurate prediction of the dynamics of an in vivo implemented circuit [2–5]. The extreme case being the design of a genetic network composed of a single transcriptional unit showing a specified spatio-temporal dynamics. As all the protein concentrations shall be coupled, it is very difficult to have a non-trivial dynamics unless the time scales of protein interactions and of cell-to-cell communication are conveniently coupled.
In higher organisms, development results from the coordinated action of thousands of genes at any moment during the cell cycle. However, small regulatory circuits control the execution of genetic programs by triggering cell differentiation according to spatial patterns . These patterns result from gradients of signaling molecules, which diffuse in the medium and are sensed at each moment by the cell circuitry. Quantitative models based on reaction-diffusion equations have been successfully applied to understand the principles of organism's development [7–9]. Furthermore, synthetic patterns have been previously engineered in bacteria  and flies . However, genetic systems with defined spatial and temporal behavior have not been artificially constructed yet. In such a synthetic system, the fate of every cell within the population could be controlled, for instance, by oscillators working in a specific manner in response to spatial location or by the state of an internal memory. It is of particular interest to apply the same design principles underlying naturally occurring molecular clocks, where rythmicity is mainly based on negative feedback loops , to the in vivo engineering of synthetic oscillatory circuits [13, 14].
Results and Discussion
The system, a single transcriptional unit, consists in a combinatorial promoter, lactose-luciferase, which controls the expression of two TFs LacI and LuxR, and the enzyme LuxI (see Methods for further details). Being all the concentrations of protein species proportional, it would make a priori especially difficult our targeted dynamics. Fortunately, we can still have a rich dynamics at single cell owed to the suitable design of molecular interactions (multimerization and binding events). Furthermore, this model is coupled to a population model, where cell-to-cell communication introduces the spatial dimension. Usually the models including the spatial dimension require the use of several genes with uncoupled dynamics. Here we will show that a dynamical pattern behavior can be generated by using genes expressing the same concentration of proteins (up to a proportionality factor).
Kinetic parameters used in the spatiotemporal transcription model.
α = 125 nM/min
θ Y = 10 nM
Λ = 75 nM
θ X = 5 nM
δ = 300 nM/min
θ I = 15 μ M
L0 = 4.1
T = 5.5 min
K1 = 100 nM
K2 = 10 nM
K3 = 55 nM
K4 = 300 nM
κ = 10 nM/min
η = 2000 min-1
δ A = 0.002 min-1
λ A = 1 min-1
N m = 105 cells/nL
D = 10-3 mm2/min
Interestingly, our circuit does not need further genetic manipulation to change the behavior regime. Environmental signals control the dynamics leading a fully tunable circuit, since by varying the concentration of IPTG and AHL we can change the dynamical regime. According to our model, this minimal genetic unit can display complex dynamics. The integration in a single circuit of the ability of oscillating and having memory may have important applications in Synthetic Biology.
Synchronization of oscillations
Secondly, we have introduced an intrinsic white noise to account for the stochasticity raised from the small number of molecules. It is important to notice that this noise is always present in a discrete system of molecules, although at high number of molecules it is usually neglected. We have not accounted for external sources of noise. Power spectral density (PSD) analysis has been performed showing that QS enhances regular oscillations, since in that case there is one frequency with a PSD significantly higher (Figs. 5c, d). The global effect of AHL together with the different temporal scales in the system, first producing the activator then the repressor with a delay, enables to sustain in time the oscillatory behavior of the whole population.
The in vivo implementation of such a circuit and its characterization could be the matter of further work, beyond the scope of this study. Experimentally, a fluorescent protein such as GFP could be inserted into the operon as reporter. Time-lapse microscopy could be used to visualize fluorescence in solid-phase in a scale of various cm in space and tens of generations in time , which would be sufficient to observe the dynamical patterns. Even though, this could also be tested readily with the use of microfluidic devices. In that way, the experimental work would help to validate or refine the mathematical model.
In this manuscript, we have shown for the first time the design of a single trancriptional unit with a post-translational dynamics that couples spatial and temporal scales to generate dynamical spatial structures. We have followed a model-based design approach to obtain a minimal genetic unit with multiple functionalities and displaying certain homeostasis to both environmental and mutational perturbations, since external fluctuations or variations in kinetic parameters are compensated due to QS. We have relied on nonlinear dynamics and stochastic modeling to analyze our system under biological noise. We have provided a new mechanism able to switch from oscillatory to bistable regimes using an external inducer and to produce complex spatio-temporal patterns by using a single transcriptional unit. In addition, this sort of small functional modules could be hierarchically assembled to generate more complex systems . That these simple units are able to generate such a complex behavior provides new avenues to understand natural genetic circuits by designing synthetic minimal systems. The bottleneck in the construction of a synthetic network consists in the number of independent transcriptional units, as transcription carries the largest source of intrinsic and extrinsic noises . Although a system consisting in a single operon would be very appealing, it may have serious problems due to its minimalism unless the system is properly designed. In the past, most authors have used at least two operons to construct systems with oscillatory behavior or producing a patterning. Although for the former, it was already known several examples of oscillatory circuits consisting in a single operon. That our minimal circuit can display such a rich behavior highlights the fact that rational design techniques take advantage of engineering principles for constructing genetic circuits with specified functions.
Dynamics at the single cell level
A synthetic lactose-luciferase (lac-lux) promoter controls the transcription of the lac repressor LacI (X), the activator LuxR (Y) and the enzyme LuxI (Z). Here, we consider the bacterium Escherichia coli as cellular chassis, and all proteins are assumed to be ssrA-tagged for enzymatic degradation , which for a zeroth-order kinetics enhances robustness of oscillations . In addition, the chemical IPTG (I) acts as inducer and binds to LacI inhibiting its repressive effect; the chemical AHL (A) is required by LuxR to activate transcription. The active form of LacI is a tetramer (X4), whereas for LuxR it consists of a dimmer of the complex LuxR-AHL ((Y:A)2). In addition, repression by LacI tetramer induces a DNA loop in the promoter region (1a) , which may introduce a delay (T) into the system. The processes of transcription, translation, folding and multimerization could also induce a delay, but this is neglected in this work. Indeed, transcription in eukaryotes dictates a delay because of splicing . However, this is not the case in prokaryotes. Herein, we assume that the reaction between two tetramers for making the DNA loop is not reversible, given that the looping structure remains even for low levels of LacI .
where κ is the synthesis constant of AHL by LuxI, δ A the thermodynamic degradation constant of AHL, and λ A the degradation rate by AiiA.
where kl/-lare the forward/reverse kinetic coefficients (l = 1, 2, 3, 4). By exploiting the different time scales in the dynamics and neglecting the amount of DNA-bound protein, we can define the dimensionless variables as x = X1/θ with , i = I/θ I , y = Y1/(K4θ Y )1/2, and a = A/K3. Notice that y and a depend on x. The reactions for multimerization can be assumed much faster than the ones for transcription and translation. Then the system (4) is reduced to the steady state. Being that, we obtain , Y:A = Y1A/K3, and , with K l = k-l/k l .
Furthermore, the total amounts of LacI and LuxR are X = X1 + 2X2 + 4X4 and Y = Y1 + Y:A + 2(Y:A)2 respectively. For simplicity in the notation, we denote X as a function of the dimensionless variable x, , and . In addition, we assume that the total amount of AHL is approximately equal to the free one, then it turns out that . Being that, Y:A = 0 when A = 0 and at very high levels of A. For the following, we denote x T = x(t -T). Time is also re-scaled by cell division, τ = μ0t/ln(2) = t/τ0.
Thereby, the equation of eigenvalues (λ) is λ = Φ -Γ + Ψe -λT . Let us define the following variables and U2 =(Γ - Φ)T to simplify the results of the spectral analysis. The oscillatory boundary condition is given by λ = jω, being j the imaginary unit. Then we obtain , the analytical equation for the Hopf bifurcation. On the contrary, the bistability boundary condition implies λ = 0, then we obtain U1 = -1. To further analyze the bistability of the system, we simplify the model at high levels of IPTG and externally introduced AHL. Without lost of generality, a dynamics governed by captures the principal features of the system, where α' ~Cα, θ' ~σ2K4θ Y /θ2, δ' ~δ, and Λ' ~Λ.
Numerical integration and stochasticity
To illustrate this point, let us consider a molecular system governed by the following general differential equation , where X is the protein amount. The transcription term is highly nonlinear and accounts for the system delay (T). In our case, the function f depends on both X(t) and X(t -T). For the enzymatic degradation, we have to notice that, as Λ is a low value, this becomes zeroth-order for high values of X, while first-order when X is close to 0.
To solve this equation we use the MATLAB routine dde23.
Dynamics at the population level
Since AHL is quickly degraded and Q ≃ 1 in a large population, we can take the quasi-steady state A = κZ/λ A when AHL is not externally introduced. In addition, we neglect the movement of cells when replicating because even for τ = 100 this displacement would be ~0.1 mm.
We thank J. Garcia-Ojalvo for reading the manuscript and useful suggestions. This work was supported by SYNTHBIOCLOCK (CNRS Interface Physique-Chimie-Biologie to AJ), FRM (INE20091218114 to AJ), the Generalitat Valenciana (BFPI-2007-160 to GR), the Spanish Ministry of Science and Innovation (TIN-2006-12860 to Vicente Hernandez and BFU2009-06993/BMC to SFE), the ATIGE Genopole to AJ, the Structural Funds ERDF, and the EU grants BioModularH2 (FP6-NEST-043340 to AJ), EMERGENCE (FP6-NEST-043338 to AJ) and TARPOL (FP7-KBBE-212894 to AJ).
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