Design and analysis of a tunable synchronized oscillator
- Brendan M Ryback†1,
- Dorett I Odoni†1,
- Ruben GA van Heck1,
- Youri van Nuland1,
- Matthijn C Hesselman1,
- Vítor AP Martins dos Santos1Email author,
- Mark WJ van Passel1, 4 and
- Floor Hugenholtz2, 3
© Ryback et al.; licensee BioMed Central Ltd. 2013
Received: 19 April 2013
Accepted: 12 November 2013
Published: 18 November 2013
The use of in silico simulations as a basis for designing artificial biological systems (and experiments to characterize them) is one of the tangible differences between Synthetic Biology and “classical” Genetic Engineering. To this end, synthetic biologists have adopted approaches originating from the traditionally non-biological fields of Nonlinear Dynamics and Systems & Control Theory. However, due to the complex molecular interactions affecting the emergent properties of biological systems, mechanistic descriptions of even the simplest genetic circuits (transcriptional feedback oscillators, bi-stable switches) produced by these methods tend to be either oversimplified, or numerically intractable. More comprehensive and realistic models can be approximated by constructing “toy” genetic circuits that provide the experimenter with some degree of control over the transcriptional dynamics, and allow for experimental set-ups that generate reliable data reflecting the intracellular biochemical state in real time. To this end, we designed two genetic circuits (basic and tunable) capable of exhibiting synchronized oscillatory green fluorescent protein (GFP) expression in small populations of Escherichia coli cells. The functionality of the basic circuit was verified microscopically. High-level visualizations of computational simulations were analyzed to determine whether the reliability and utility of a synchronized transcriptional oscillator could be enhanced by the introduction of chemically inducible repressors.
Synchronized oscillations in GFP expression were repeatedly observed in chemically linked sub-populations of cells. Computational simulations predicted that the introduction of independently inducible repressors substantially broaden the range of conditions under which oscillations could occur, in addition to allowing the frequency of the oscillation to be tuned.
The genetic circuits described here may prove to be valuable research tools for the study of synchronized transcriptional feedback loops under a variety of conditions and experimental set-ups. We further demonstrate the benefit of using abstract visualizations to discover subtle non-linear trends in complex dynamic models with large parameter spaces.
KeywordsSynchronized tunable oscillator Genetic circuit Transcriptional feedback Delay differential equation
Synthetic genetic circuits as research tools
In order for synthetic genetic circuits to be technologically useful and modularly composable in higher order systems, their properties must be subject to formal mathematical descriptions that capture the salient features of a given circuit . One of the aims of synthetic biology is to develop models that are sufficiently accurate and comprehensive to provide a basis for predicting the emergent properties of newly built genetic circuits under varying conditions [2, 3]. Modeling approaches based on either a priori mechanistic descriptions (e.g. delay differential equations) , and to a lesser extent, data driven “black-box” model structure identification methods (e.g. NARMAX ) have become increasingly prominent. More recently, approaches attempting to consolidate models operating at varying levels of biological abstraction have also been proposed .
One way to improve the accuracy of models is to construct “toy” circuits which have externally controllable parameters. Such systems facilitate the rapid generation of a wide range of experimental conditions, which can be modeled in order to gain insights into potentially interesting dynamic behaviors. These systems should be complex enough to provide useful insights into the nonlinear dynamics of multi-component systems, without being so complex as to create indeterminable and/or intractable parameter spaces. Simple bi-stable memory switches have been studied extensively  as toy circuits for model building and parameter estimation methods, as they generally fulfill the aforementioned criteria and are amenable to study with conventional fluorescent plate-reader and FACS-based experimental set-ups. While bi-stable switches can provide insights into some non-linear regulatory interactions, they are - by their very design specifications - stable.
In contrast, genetic oscillators exhibit unstable time-variant expression dynamics which can potentially provide insights into more complex (and subtle) emergent properties . Thus, genetic oscillators are excellent objects for the study of biological nonlinear dynamics, as is evidenced by the abundance of published theoretical work [9–12]. A drawback of genetic oscillators is that their experimental implementation poses non-trivial practical difficulties. Measuring the gene expression of individual bacterial cells in real time over time-spans relevant to transcriptional oscillators is technically challenging . Synchronization of populations, e.g. via quorum sensing, allows populations to be studied instead of individual cells, but imposes new constraints. The fact that systems governed by more unknown parameters than measurable (or controllable) variables are inherently underdetermined confounds the improvement of mechanistic models. Therefore, it is likely that the elucidation of the oscillatory dynamics emerging from (transcriptional) regulatory feedback loops could be facilitated by the introduction of simple control elements; assuming they remain orthogonal to the system’s basic circuitry and do not increase the complexity of the nonlinear interactions (or significantly affect the host cell’s metabolism). If implemented successfully, such control elements could substantially expand the range of experimental conditions used to characterize the circuit’s oscillatory dynamics.
Combining synchronization and tunability
Transcription of the luxR gene is regulated by a constitutive promoter, resulting in constant levels of the AHL-dependent transcription factor LuxR. Transcription of luxI and aiiA occurs at a basal rate when intracellular AHL concentrations are below the activation threshold of LuxR. As the cell density increases in the course of normal cell divisions in a constrained space, so too does the intracellular AHL concentration. AHL diffuses freely between the cells, which effectively synchronizes their internal states. When the AHL concentration reaches the activation threshold of LuxR, the rate of transcription of luxI and aiiA is greatly increased, initially giving rise to higher levels of the enzyme LuxI due to differences in transcription and maturation time as well as ribosomal saturation .
This positive feedback loop results in an exponential increase in AHL synthesis, and in turn, maximal expression of LuxI and AiiA. As catalytically active AiiA accumulates, AHL is rapidly degraded (negative feedback) to sub-LuxR activation levels, and transcription of aiiA and luxI recedes to the basal rate. Oscillations arise from the delayed interaction between these coupled positive and negative feedback loops and are contingent on the ability of the system to rapidly reset to its initial state. This property is dependent on the inclusion of LVA-degradation tags in all non-constitutively expressed proteins . Changes in the intracellular AHL concentration are visualized semi-quantitatively via the expression of a fluorescent reporter gene under control of a LuxR-AHL dependent promoter. Synchronization across a population of cells results from the rapid diffusion of AHL. This onset of synchronization via quorum sensing is not gradual, but sudden and a function of varying cell densities .
The designs used in this study differ from previously published work  due to three substantial modifications: (i) elimination of redundant regulatory and coding sequences, (ii) introduction of tunable hybrid promoters and (iii) consolidation of the circuit into a single DNA sequence conforming to the BioBrick assembly standard (i.e. BioBrick device) .
The introduction of these tuners is intended to provide an additional set of control variables that allow the kinetics of the circuit’s feedback loops to be influenced independently of one another by varying the inducer molecule concentrations within a dynamic range. This additional control may be exploited to compensate for external conditions that would prevent the basic, non-tunable circuit from producing oscillations, effectively increasing the oscillator’s robustness towards variations in cell density, and by extension, expand the range of experimental set-ups under which the circuit could be employed. The functionality of the basic, non-tunable circuit in E. coli was verified experimentally using a custom microbial growth chamber  in conjunction with a fluorescence microscope.
The dynamics of both circuits are described by a set of delay differential equations which served as the basis for deterministic simulations. A broad range of input values was chosen in order to elucidate the extent to which changes in the inducer molecule concentrations influence the cell-density dependent expression dynamics.
Results and discussion
The circuit represented in Figure 2 shows the oscillatory circuit with the aforementioned modifications: (i) elimination of two redundant copies of the sequences encoding the transcriptional regulator LuxR, (ii) replacement of natural bidirectional quorum sensing promoters with synthetic hybrid promoters containing repressor binding sites, the activity of which can be independently controlled via chemical inducers and (iii) consolidation of all the circuit’s components, including the tuner module, into a single device conforming to the BioBrick assembly standard.
Basic oscillator tested in a flow device
Work published by Prindle et al.  demonstrated synchronization of cells trapped in microfluidic chambers across a distance of 5 mm via diffusion of H2O2. Our findings are consistent with this to the extent that synchronization was observed across various spatially separated sub-populations. However, an important difference is that in our system populations are solely coupled by AHL diffusion, and therefore dependent on a fluid medium to travel. The diffusion kinetics of different signaling molecules therefore need to be taken into account when designing chemically coupled regulatory systems.
Computational simulations illustrating differential regulatory dynamics
It is clear from this representation that the relationships between the system inputs and the resulting waveforms are non-linear, giving rise to a number of localized trends within the 5-dimensional space. The most obvious feature is the relative sparseness of the space for IPTG values greater than 0.3 and cell density values smaller than 0.6. This space is populated exclusively by damped oscillations with few peaks. It is noteworthy that for small values of IPTG the frequency steadily increases as a function of cell density before decreasing (around 0.6 for IPTG = 0.2) and subsequently increasing again.
Furthermore, cell density is strongly correlated to the amplitude metric. This is related to the fact that sustained oscillations score far higher in this metric than damped oscillations, and sustained oscillations only occur outside the sparse parabolic region that spans the majority of the volume.
The aim of this study was to test the functionality of a refactored synchronized transcriptional oscillator and to investigate whether its reliability and utility could be enhanced by the introduction of chemically inducible repressors. The functionality of the basic circuit, assembled from BioBrick parts, was verified experimentally using a custom experimental platform. These experiments revealed synchronization at an unexpected scale between spatially separated but chemically linked populations of bacteria. Computational simulations of the tunable circuit design revealed a rich landscape of non-linear relationships between the oscillatory behavior of the circuit and the control variables. The simulations suggested that, while cell density is the primary determinant of gene expression dynamics in this system, the ability to tune transcriptional feedback kinetics via inducer molecules substantially broadens the range of waveforms that this circuit can generate. Assuming that the model upon which the simulations were based capture the actual dynamics, the tunable oscillator design described here should be highly versatile. While fluorescent plate reader experiments aimed at characterizing this circuit’s tunability repeatedly demonstrated dynamic gene expression, a lack of consistency between replicates was confounded by a low signal to noise ratio, ultimately yielding inconclusive results (data not shown).
These results offer a cursory glance at the type of methods that could be employed to study nonlinear transcriptional regulatory dynamics using this circuit. Future work on this system should aim to validate the model before exploring more rigorous analytical methods.
Due to its efficient single-plasmid design it also lends itself to the investigation of expression dynamics as a function of varying copy numbers using different plasmid backbones, or the effect of genomic integration. Such approaches could very well yield reliable, quantitative data if combined with advanced experimental platforms, such as fluorescence microscopy combined with microfluidics, fluorescence-based cell-sorting methods, or milliliter-scale continuous stirred-tank bioreactors. It is our hope that in the future, this circuit may be used by others as a tool for developing, and possibly benchmarking increasingly refined modeling approaches that shed light on the intricate and elusive properties of complex genetic circuits.
Induced by AHL-LuxR Repressed by LacI
Induced by AHL-LuxR, Repressed by TetR
Induced by AHL-LuxR
Repressor of aiiA
Repressor of luxI
Constitutive reporter molecule
Cloning & expression vector
Fluorescence measurements using microdish
Liquid cultures were made from single colonies which had grown on LB agar plates with ampicillin (50 μg/mL). The single colonies were grown over night at 37°C in 10-15 ml of LB ampicillin medium. The cultures were spun down and resuspended in 0.9% phosphate buffered saline (PBS), before inoculation in a custom made flow device  equipped with a microdish made from porous aluminium oxide containing 40 μm deep wells with a diameter of 180 μm . Since LB-amp medium was supplied from below the microdish, the growth of bacteria was restricted to the wells there nutrients could be obtained via diffusion through the porous material at the base. The visual output was measured using an Olympus fluorescence microscope BX41 with an exposure time of 200 ms and 100 × magnification. Measurements were taken in a time interval of 10 minutes by a Mindstorms Lego robot (http://mindstorms.lego.com). Data analysis and processing were done with ImageJ 1.45 (http://rsbweb.nih.gov/ij/index.html) and MATLAB (http://www.mathworks.com).
The genetic circuit described above can be represented as a system of delay differential equations, which was adapted from Danino et al.  and expanded with Hill functions to represent the effect of the tuner repressors and their inducer molecules on the maximum expression level of the dynamically expressed components. The final model is presented as Equations 1, 2, 3, 4, 5.
The terms proportional to CA (1) and CL (2) represent the dependency of AiiA and LuxI expression on the cell density d. The hybrid promoters regulating luxI and aiiA were assumed to have the same response kinetics to LuxR-AHL as the natural lux promoter in the absence of the repressor proteins. To take possible differences between the hybrid promoters into account, the leakage constants for aiiA and luxI expression were replaced by the new leakage constants δ1 and δ2. The terms containing these leakage constants are the history functions present in the original model which consolidate the time delay resulting from gene transcription and translation into a single parameter τ. The term after that in equations (1) and (2) is the actual tuner term and represents the influence of the repressors on the system, which in turn is dependent on the presence of either IPTG for LacI or aTc for tetR, respectively. Equations (3) and (4) contain an additional term proportional to D, which shows the diffusion of AHL throughout the cells. Finally, the terms proportional to γ show the degradation of AiiA, LuxI and GFP in equations (1), (2) and (5), respectively.
In accordance with observations of most naturally occurring regulatory elements, all promoters are assumed to be “leaky”, and exhibit a basal expression level in the absence of activating TFs . The model is applicable to both the basic circuit and the tunable one, as the model for the latter can be reduced to represent the former simply by setting the concentration of the repressors to 0.
Deterministic simulations were performed using the MATLAB dde23 solver in order to elucidate the relationship between inducer molecule concentrations and their effect on GFP expression relative to cell density. The input values were chosen to cover the entirety of the controllable input space, ranging from full repression (IPTG and aTc set to 0) to full induction (both IPTG and aTc set to 1) in steps of 0.1. The cell density was also iterated from 0 to 0.85 in steps of 0.05, resulting in a total of 2178 simulated conditions.
Technically the expression of TetR and LacI is not constitutive due to regulation by the AraC/pBAD promoter and its corresponding inducer molecule arabinose. However, it is treated as such for the purposes of this study.
Equations1, 2, 3, 4, 5. Set of delay differential equations representing the interactions between the circuit’s components. The terms representing time and cell-density dependent changes in AiiA (1), LuxI (2), and GFP (5) all have the same basic features. The main difference is that the maximum expression levels of AiiA and LuxI are limited by Hill functions that take the concentrations of their respective repressors and corresponding inducer molecules into account. In contrast, the expression of GFP is only dependent on the cell density and intracellular AHL concentration. Changes in intracellular AHL (3) are a function of LuxI and AiiA levels as well as a diffusion term. A = AiiA, L = LuxI, Hi = internal AHL, He = external AHL, C = production constant, d = cell density, δ = promoter leakage, γ = degradation constant, D = diffusion rate, τ = time delay.
Green fluorescent protein
Fluorescence-activated cell sorting
We gratefully acknowledge the contributions of the Wageningen University 2011 iGEM team. Furthermore, we would like to thank Maria Suarez Diez for critically reading the manuscript. MWJvP is funded by the Netherlands Organization for Scientific Research (NWO) via a VENI grant. FH is funded by the Netherlands Consortium for Systems Biology (NCSB) which is part of the Netherlands Genomics Initiative and the NWO. MicroDish BV provided the cultivation chips used in the microscopy experiments. Additional resources were provided by Wageningen University.
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