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.
where ξ is a Gaussian-distributed random number with mean 0 and standard deviation .
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.
where, for small values of r0 (r0 << Dt), we have assumed that .
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).
- Benner SA, Sismour AM: Synthetic biology. Nat Rev Genet. 2005, 6: 533-543. 10.1038/nrg1637View ArticlePubMedGoogle Scholar
- Guido NJ, Wang X, Adalsteinsson D, McMillen D, Hasty J, Cantor CR, Elston TC, Collins JJ: A bottom-up approach to gene regulation. Nature. 2006, 439: 856-860. 10.1038/nature04473View ArticlePubMedGoogle Scholar
- Ellis T, Wang X, Collins JJ: Diversity-based, model-guided construction of synthetic gene networks with predicted functions. Nat Biotechnol. 2009, 27: 465-471. 10.1038/nbt.1536PubMed CentralView ArticlePubMedGoogle Scholar
- Cantone I, Marucci L, Iorio F, Ricci M, Belcastro V, Bansal M, Santini S, di Bernardo M, di Bernardo D, Cosma M: A yeast synthetic network for in vivo assessment of reverse-engineering and modeling approaches. Cell. 2009, 137: 172-181. 10.1016/j.cell.2009.01.055View ArticlePubMedGoogle Scholar
- Danino T, Mondragon-Palomino O, Tsimring L, Hasty J: A synchronized quorum of genetic clocks. Nature. 2010, 463: 326-330. 10.1038/nature08753PubMed CentralView ArticlePubMedGoogle Scholar
- Becskei A, Seraphin B, Serrano L: Positive feedback in eukaryotic gene networks: cell differentiation by graded to binary response conversion. EMBO J. 2001, 20: 2528-2535. 10.1093/emboj/20.10.2528PubMed CentralView ArticlePubMedGoogle Scholar
- Koch AJ, Meinhardt H: Biological pattern formation: from basic mechanisms to complex structures. Rev Mod Phys. 1994, 66: 1481-1507. 10.1103/RevModPhys.66.1481.View ArticleGoogle Scholar
- Hartmann D: Pattern formation in cultures of Bacillus subtilis. J Biol Syst. 2004, 12: 179-199. 10.1142/S0218339004001105.View ArticleGoogle Scholar
- Reeves GT, Muratov CB, Schupbach T, Shvartsman SY: Quantitative models of developmental pattern formation. Develop Cell. 2006, 11: 289-300. 10.1016/j.devcel.2006.08.006.View ArticleGoogle Scholar
- Basu S, Gerchman Y, Collins CH, Arnold FH, Weiss R: A synthetic multicellular system for programmed pattern formation. Nature. 2005, 434: 1130-1134. 10.1038/nature03461View ArticlePubMedGoogle Scholar
- Isalan M, Lemerle C, Serrano L: Engineering gene networks to emulate Drosophila embryonic pattern formation. PLoS Biol. 2005, 3: 488-496. 10.1371/journal.pbio.0030064.View ArticleGoogle Scholar
- Goldbeter A: Computational approaches to cellular rhythms. Nature. 2002, 420: 238-245. 10.1038/nature01259View ArticlePubMedGoogle Scholar
- Elowitz MB, Leibler S: A synthetic oscillatory network of transcriptional regulators. Nature. 2000, 403: 335-338. 10.1038/35002125View ArticlePubMedGoogle Scholar
- Atkinson MR, Savageau MA, Myers JT, Ninfa AJ: Development of genetic circuitry exhibiting toggle switch or oscillatory behavior in Escherichia coli. Cell. 2003, 113: 597-607. 10.1016/S0092-8674(03)00346-5View ArticlePubMedGoogle Scholar
- Becskei A, Serrano L: Engineering stability in gene networks by autoregulation. Nature. 2000, 405: 590-593. 10.1038/35014651View ArticlePubMedGoogle Scholar
- Lewis J: Autoinhibition with transcriptional delay: a simple mechanism for the Zebrafish somitogenesis oscillator. Curr Biol. 2003, 13: 1398-1408. 10.1016/S0960-9822(03)00534-7View ArticlePubMedGoogle Scholar
- Bratsun D, Volfson D, Tsimring LS, Hasty J: Delay-induced stochastic oscillations in gene regulation. Proc Natl Acad Sci USA. 2005, 102: 14593-14598. 10.1073/pnas.0503858102PubMed CentralView ArticlePubMedGoogle Scholar
- Mather W, Bennett MR, Hasty J, Tsimring LS: Delay-induced degrade-and-fire oscillations in small genetic circuits. Phys Rev Lett. 2009, 102: 068105- 10.1103/PhysRevLett.102.068105PubMed CentralView ArticlePubMedGoogle Scholar
- Isaacs FJ, Hasty J, Cantor CR, Collins JJ: Prediction and measurement of an autoregulatory genetic module. Proc Natl Acad Sci USA. 2003, 100: 7714-7719. 10.1073/pnas.1332628100PubMed CentralView ArticlePubMedGoogle Scholar
- McMillen D, Kopell N, Hasty J, Collins JJ: Synchronizing genetic relaxation oscillators by intercell signaling. Proc Natl Acad Sci USA. 2002, 99: 679-684. 10.1073/pnas.022642299PubMed CentralView ArticlePubMedGoogle Scholar
- Garcia-Ojalvo J, Elowitz MB, Strogatz SH: Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing. Proc Natl Acad Sci USA. 2003, 101: 10955-10960. 10.1073/pnas.0307095101.View ArticleGoogle Scholar
- Stricker J, Cookson S, Bennett MR, Mather WH, Tsimring LS, Hasty J: A fast, robust and tunable synthetic gene oscillator. Nature. 2008, 456: 516-519. 10.1038/nature07389View ArticlePubMedGoogle Scholar
- Kuhlman T, Zhang Z, H SJM, Hwa T: Combinatorial transcriptional control of the lactose operon of. Escherichia coli. 2007, 104: 6043-6048.Google Scholar
- Purnick PE, R W: The second wave of synthetic biology: from modules to systems. Nat Rev Mol Cell Biol. 2009, 10: 410-422. 10.1038/nrm2698View ArticlePubMedGoogle Scholar
- Wilkinson DJ: Stochastic modelling for quantitative description of heterogeneous biological systems. Nat Rev Genet. 2009, 10: 122-133. 10.1038/nrg2509View ArticlePubMedGoogle Scholar
- Hersch GL, Baker TA, Sauer RT: SspB delivery of substrates for ClpXP proteolysis probed by the design of improved degradation tags. Proc Natl Acad Sci USA. 2004, 101: 12136-12141. 10.1073/pnas.0404733101PubMed CentralView ArticlePubMedGoogle Scholar
- Wong WW, Tsai TY, Liao JC: Single-cell zeroth-order protein degradation enhances the robustness of synthetic oscillator. Mol Syst Biol. 2007, 3: 130- 10.1038/msb4100172PubMed CentralView ArticlePubMedGoogle Scholar
- Narang A: Effect of DNA looping on the induction kinetics of the lac operon. J Theor Biol. 2007, 247: 695-712. 10.1016/j.jtbi.2007.03.030View ArticlePubMedGoogle Scholar
- Bernstein J, Khodursky A, Lin PH, Lin-Chao S, Cohen S: Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two-color fluorescent DNA microarrays. Proc Natl Acad Sci USA. 2002, 99: 9697-9702. 10.1073/pnas.112318199PubMed CentralView ArticlePubMedGoogle Scholar
- Dong YH, Xu JL, Li XZ, Zhang LH: AiiA, an enzyme that inactivates the acylhomoserine lactone quorum-sensing signal and attenuates the virulence of. Erwinia carotovora. 2000, 97: 3526-3531.Google Scholar
- Verdugo A, Rand R: Hopf bifurcation in a DDE model of gene expression. Commun Nonlinear Sci Numer Simul. 2008, 13: 235-242. 10.1016/j.cnsns.2006.05.001.View ArticleGoogle Scholar
- Wang LH, Weng LX, Dong YH, Zhang LH: Specificity and enzyme kinetics of the quorum-quenching N-acyl homoserine lactone lactonase (AHL-lactonase). J Biol Chem. 2004, 279: 13645-13651. 10.1074/jbc.M311194200View ArticlePubMedGoogle Scholar
- Nikaido H, Rosenberg EY: Effect of solute size on diffusion rates through the transmembrane pores of the outer membrane of Escherichia coli. J Gen Physiol. 1981, 77: 121-135. 10.1085/jgp.77.2.121View ArticlePubMedGoogle Scholar
- Murray JD: Mathematical Biology: I. An Introduction. 2002, New York: Springer Verlag,Google Scholar
- Canton B, Labno A, Endy D: Refinement and standardization of synthetic biological parts and devices. Nat Biotechnol. 2008, 26: 787-793. 10.1038/nbt1413View ArticlePubMedGoogle Scholar
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