Synthetic Genetic Oscillators Research Articles

Light-driven synchronization of optogenetic clocks

Synthetic genetic oscillators can serve as internal clocks within engineered cells to program periodic expression. However, cell-to-cell variability introduces a dispersion in the characteristics of these clocks that drives the population to complete desynchronization. Here, we introduce the optorepressilator, an optically controllable genetic clock that combines the repressilator, a three-node synthetic network in E. coli, with an optogenetic module enabling to reset, delay, or advance its phase using optical inputs. We demonstrate that a population of optorepressilators can be synchronized by transient green light exposure or entrained to oscillate indefinitely by a train of short pulses, through a mechanism reminiscent of natural circadian clocks. Furthermore, we investigate the system’s response to detuned external stimuli observing multiple regimes of global synchronization. Integrating experiments and mathematical modeling, we show that the entrainment mechanism is robust and can be understood quantitatively from single cell to population level.

Light-driven synchronization of optogenetic clocks.

Synthetic genetic oscillators can serve as internal clocks within engineered cells to program periodic expression. However, cell-to-cell variability introduces a dispersion in the characteristics of these clocks that drives the population to complete desynchronization. Here, we introduce the optorepressilator, an optically controllable genetic clock that combines the repressilator, a three-node synthetic network in E. coli, with an optogenetic module enabling to reset, delay, or advance its phase using optical inputs. We demonstrate that a population of optorepressilators can be synchronized by transient green light exposure or entrained to oscillate indefinitely by a train of short pulses, through a mechanism reminiscent of natural circadian clocks. Furthermore, we investigate the system's response to detuned external stimuli observing multiple regimes of global synchronization. Integrating experiments and mathematical modeling, we show that the entrainment mechanism is robust and can be understood quantitatively from single cell to population level.

Synthetic genetic oscillators demonstrate the functional importance of phenotypic variation in pneumococcal-host interactions

Phenotypic variation is the phenomenon in which clonal cells display different traits even under identical environmental conditions. This plasticity is thought to be important for processes including bacterial virulence, but direct evidence for its relevance is often lacking. For instance, variation in capsule production in the human pathogen Streptococcus pneumoniae has been linked to different clinical outcomes, but the exact relationship between variation and pathogenesis is not well understood due to complex natural regulation. In this study, we use synthetic oscillatory gene regulatory networks (GRNs) based on CRISPR interference (CRISPRi) together with live cell imaging and cell tracking within microfluidics devices to mimic and test the biological function of bacterial phenotypic variation. We provide a universally applicable approach for engineering intricate GRNs using only two components: dCas9 and extended sgRNAs (ext-sgRNAs). Our findings demonstrate that variation in capsule production is beneficial for pneumococcal fitness in traits associated with pathogenesis providing conclusive evidence for this longstanding question.

Synchronization of a genetic oscillator with the cell division cycle

Genetic circuits that control specific cellular functions are never fully insulated against influences of other parts of the cell. For example, they are subject to periodic modulation by the cell cycle through volume growth and gene doubling. To investigate possible effects of the cell cycle on oscillatory gene circuits dynamics, we modelled a simple synthetic genetic oscillator, the repressilator, and studied hallmarks of the resulting nonlinear dynamics. We found that the repressilator coupled to the cell cycle shows typical quasiperiodic motion with discrete Fourier spectra and windows in parameter space with synchronization of the two oscillators, with a devil’s stair case indicating the Arnold tongues of synchronization. In the case of identical parameters for the three genes of the repressilator and simultaneous gene duplication, we identify two classes of synchronization windows, symmetric and asymmetric, depending on whether the trajectories satisfy a discrete three-fold rotation symmetry, corresponding to cyclic permutation of the three genes. Unexpectedly changing the gene doubling time revealed that the width of the Arnold tongues is connected to that three-fold symmetry of the synchronization trajectories: non-simultaneous gene duplication increases the width of asymmetric synchronization regions, for some of them by an order of magnitude. By contrast, there is only a small or even a negative effect on the window size for symmetric synchronization. This observation points to a control mechanism of synchronization via the location of the genes on the chromosome.

Chaos-hyperchaos transition in three identical quorum-sensing mean-field coupled ring oscillators.

We investigate the dynamics of three identical three-dimensional ring synthetic genetic oscillators (repressilators) located in different cells and indirectly globally coupled by quorum sensing whereby it is meant that a mechanism in which special signal molecules are produced that, after the fast diffusion mixing and partial dilution in the environment, activate the expression of a target gene, which is different from the gene responsible for their production. Even at low coupling strengths, quorum sensing stimulates the formation of a stable limit cycle, known in the literature as a rotating wave (all variables have identical waveforms shifted by one third of the period), which, at higher coupling strengths, converts to complex tori. Further torus evolution is traced up to its destruction to chaos and the appearance of hyperchaos. We hypothesize that hyperchaos is the result of merging the saddle-focus periodic orbit (or limit cycle) corresponding to the rotating wave regime with chaos and present considerations in favor of this conclusion.

Emergence of multistability and strongly asymmetric collective modes in two quorum sensing coupled identical ring oscillators.

The simplest ring oscillator is made from three strongly nonlinear elements repressing each other unidirectionally, resulting in the emergence of a limit cycle. A popular implementation of this scheme uses repressor genes in bacteria, creating the synthetic genetic oscillator known as the Repressilator. We consider the main collective modes produced when two identical Repressilators are mean-field-coupled via the quorum-sensing mechanism. In-phase and anti-phase oscillations of the coupled oscillators emerge from two Andronov-Hopf bifurcations of the homogeneous steady state. Using the rate of the repressor's production and the value of coupling strength as the bifurcation parameters, we performed one-parameter continuations of limit cycles and two-parameter continuations of their bifurcations to show how bifurcations of the in-phase and anti-phase oscillations influence the dynamical behaviors for this system. Pitchfork bifurcation of the unstable in-phase cycle leads to the creation of novel inhomogeneous limit cycles with very different amplitudes, in contrast to the well-known asymmetrical limit cycles arising from oscillation death. The Neimark-Sacker bifurcation of the anti-phase cycle determines the border of an island in two-parameter space containing almost all the interesting regimes including the set of resonant limit cycles, the area with stable inhomogeneous cycle, and very large areas with chaotic regimes resulting from torus destruction and period doubling of resonant cycles and inhomogeneous cycles. We discuss the structure of the chaos skeleton to show the role of inhomogeneous cycles in its formation. Many regions of multistability and transitions between regimes are presented. These results provide new insights into the coupling-dependent mechanisms of multistability and collective regime symmetry breaking in populations of identical multidimensional oscillators.

Novel Tunable Spatio-Temporal Patterns From a Simple Genetic Oscillator Circuit.

Multicellularity, the coordinated collective behavior of cell populations, gives rise to the emergence of self-organized phenomena at many different spatio-temporal scales. At the genetic scale, oscillators are ubiquitous in regulation of multicellular systems, including during their development and regeneration. Synthetic biologists have successfully created simple synthetic genetic circuits that produce oscillations in single cells. Studying and engineering synthetic oscillators in a multicellular chassis can therefore give us valuable insights into how simple genetic circuits can encode complex multicellular behaviors at different scales. Here we develop a study of the coupling between the repressilator synthetic genetic ring oscillator and constraints on cell growth in colonies. We show in silico how mechanical constraints generate characteristic patterns of growth rate inhomogeneity in growing cell colonies. Next, we develop a simple one-dimensional model which predicts that coupling the repressilator to this pattern of growth rate via protein dilution generates traveling waves of gene expression. We show that the dynamics of these spatio-temporal patterns are determined by two parameters; the protein degradation and maximum expression rates of the repressors. We derive simple relations between these parameters and the key characteristics of the traveling wave patterns: firstly, wave speed is determined by protein degradation and secondly, wavelength is determined by maximum gene expression rate. Our analytical predictions and numerical results were in close quantitative agreement with detailed individual based simulations of growing cell colonies. Confirming published experimental results we also found that static ring patterns occur when protein stability is high. Our results show that this pattern can be induced simply by growth rate dilution and does not require transition to stationary phase as previously suggested. Our method generalizes easily to other genetic circuit architectures thus providing a framework for multi-scale rational design of spatio-temporal patterns from genetic circuits. We use this method to generate testable predictions for the synthetic biology design-build-test-learn cycle.

Comparison between Effects of Retroactivity and Resource Competition upon Change in Downstream Reporter Genes of Synthetic Genetic Circuits.

Reporter genes have contributed to advancements in molecular biology. Binding of an upstream regulatory protein to a downstream reporter promoter allows quantification of the activity of the upstream protein produced from the corresponding gene. In studies of synthetic biology, analyses of reporter gene activities ensure control of the cell with synthetic genetic circuits, as achieved using a combination of in silico and in vivo experiments. However, unexpected effects of downstream reporter genes on upstream regulatory genes may interfere with in vivo observations. This phenomenon is termed as retroactivity. Using in silico and in vivo experiments, we found that a different copy number of regulatory protein-binding sites in a downstream gene altered the upstream dynamics, suggesting retroactivity of reporters in this synthetic genetic oscillator. Furthermore, by separating the two sources of retroactivity (titration of the component and competition for degradation), we showed that, in the dual-feedback oscillator, the level of the fluorescent protein reporter competing for degradation with the circuits’ components is important for the stability of the oscillations. Altogether, our results indicate that the selection of reporter promoters using a combination of in silico and in vivo experiments is essential for the advanced design of genetic circuits.

Waveforms of molecular oscillations reveal circadian timekeeping mechanisms

Circadian clocks play a pivotal role in orchestrating numerous physiological and developmental events. Waveform shapes of the oscillations of protein abundances can be informative about the underlying biochemical processes of circadian clocks. We derive a mathematical framework where waveforms do reveal hidden biochemical mechanisms of circadian timekeeping. We find that the cost of synthesizing proteins with particular waveforms can be substantially reduced by rhythmic protein half-lives over time, as supported by previous plant and mammalian data, as well as our own seedling experiment. We also find that previously enigmatic, cyclic expression of positive arm components within the mammalian and insect clocks allows both a broad range of peak time differences between protein waveforms and the symmetries of the waveforms about the peak times. Such various peak-time differences may facilitate tissue-specific or developmental stage-specific multicellular processes. Our waveform-guided approach can be extended to various biological oscillators, including cell-cycle and synthetic genetic oscillators.

Sigma Factor-Mediated Tuning of Bacterial Cell-Free Synthetic Genetic Oscillators.

Cell-free transcription–translation provides a simplified prototyping environment to rapidly design and study synthetic networks. Despite the presence of a well characterized toolbox of genetic elements, examples of genetic networks that exhibit complex temporal behavior are scarce. Here, we present a genetic oscillator implemented in an E. coli-based cell-free system under steady-state conditions using microfluidic flow reactors. The oscillator has an activator–repressor motif that utilizes the native transcriptional machinery of E. coli: the RNAP and its associated sigma factors. We optimized a kinetic model with experimental data using an evolutionary algorithm to quantify the key regulatory model parameters. The functional modulation of the RNAP was investigated by coupling two oscillators driven by competing sigma factors, allowing the modification of network properties by means of passive transcriptional regulation.

Computational Re-design of Synthetic Genetic Oscillators for Independent Amplitude and Frequency Modulation.

To perform well in biotechnology applications, synthetic genetic oscillators must be engineered to allow independent modulation of amplitude and period. This need is currently unmet. Here, we demonstrate computationally how two classic genetic oscillators, the dual-feedback oscillator and the repressilator, can be re-designed to provide independent control of amplitude and period and improve tunability-that is, a broad dynamic range of periods and amplitudes accessible through the input "dials." Our approach decouples frequency and amplitude modulation by incorporating an orthogonal "sink module" where the key molecular species are channeled for enzymatic degradation. This sink module maintains fast oscillation cycles while alleviating the translational coupling between the oscillator's transcription factors and output. We characterize the behavior of our re-designed oscillators over a broad range of physiologically reasonable parameters, explain why this facilitates broader function and control, and provide general design principles for building synthetic genetic oscillators that are more precisely controllable.

Feedback loops interlocked at competitive binding sites amplify and facilitate genetic oscillations

Positive and negative feedback loops are often present in regulatory networks for genetic oscillations. Relative time scales and integration of these feedback loops are key to robust oscillations in expression levels. Using examples from the circadian clock and synthetic genetic oscillators, we study positive and negative feedback loops interlocked at competitive binding sites. In the mammalian circadian clock, a key clock gene Bmal1 is regulated by the activator ROR and the repressor REV-ERB. Conversely, Bmal1 activates both of them, forming interlocked feedback loops. Previous experiments indicate that the activator and repressor compete for the same binding sites in the Bmal1 promoter. Transcription patterns predict that ROR peaks later than REV-ERB and, moreover, the peak phase difference between them is small. Using mathematical modeling we reveal an optimal ratio of dissociation constants of an activator and a repressor for the competitive binding sites to enhance the amplitude of Bmal1 oscillations. This optimal ratio arises only when the amplitude of the repressor is larger than that of the activator. Secondly, we reveal that the preference of binding sites for an activator and a repressor depends on their relative time scales. A previous study demonstrated that noncompetitive binding sites are preferable for synthetic genetic oscillators that comprise a fast activator and a slow repressor with a large time scale separation. Here we show that when their time scales are similar, competitive binding sites are more likely to generate oscillation than noncompetitive sites. In contrast, for a slow activator and a fast repressor with a small phase difference as in Bmal1 regulation, noncompetitive binding sites are advantageous for amplifying oscillations. Our results, therefore, predict that additional mechanisms are necessary to compensate the disadvantage of the Bmal1 promoter and further facilitate amplification under the regulation by ROR and REV-ERB.

Preface.

This volume was inspired by the topics presented at the international conference "Micro and Macro Systems in Life Sciences" which was held on Jun 8-12, 2015 in Będlewo, Poland. System biology is an approach which tries to understand how micro systems, at the molecular and cellular levels, affect macro systems such as organs, tissue and populations. Thus it is not surprising that a major theme of this volume evolves around cancer and its treatment. Articles on this topic include models for tumor induced angiogenesis, without and with delays, metastatic niche of the bone marrow, drug resistance and metronomic chemotherapy, and virotherapy of glioma. Methods range from dynamical systems to optimal control. Another well represented topic of this volume is mathematical modeling in epidemiology. Mathematical approaches to modeling and control of more specific diseases like malaria, Ebola or human papillomavirus are discussed as well as a more general approaches to the SEIR, and even more general class of models in epidemiology, by using the tools of optimal control and optimization. The volume also brings up challenges in mathematical modeling of other diseases such as tuberculosis. Partial differential equations combined with numerical approaches are becoming important tools in modeling not only tumor growth and treatment, but also other diseases, such as fibrosis of the liver, and atherosclerosis and its associated blood flow dynamics, and our volume presents a state of the art approach on these topics. Understanding mathematics behind the cell motion, appearance of the special patterns in various cell populations, and age structured mutations are among topics addressed inour volume. A spatio-temporal models of synthetic genetic oscillators brings the analysis to the gene level which is the focus of much of current biological research. Mathematics can help biologists to explain the collective behavior of bacterial, a topic that is also presented here. Finally some more across the discipline topics are being addresses, which can appear as a challenge in studying problems in systems biology on all, macro, meso and micro levels. They include numerical approaches to stochastic wave equation arising in modeling Brownian motion, discrete velocity models, many particle approximations as well as very important aspect on the connection between discrete measurement and the construction of the models for various phenomena, particularly the one involving delays. With the variety of biological topics and their mathematical approaches we very much hope that the reader of the Mathematical Biosciences and Engineering will find this volume interesting and inspirational for their own research.

Impact of time delays on oscillatory dynamics of interlinked positive and negative feedback loops.

Interlinking a positive feedback loop (PFL) with a negative feedback loop (NFL) constitutes a typical motif in genetic networks, performing various functions in cell signaling. How time delay in feedback regulation affects the dynamics of such systems still remains unclear. Here, we investigate three systems of interlinked PFL and NFL with time delays: a synthetic genetic oscillator, a three-node circuit, and a simplified single-node model. The stability of steady states and the routes to oscillation in the single-node model are analyzed in detail. The amplitude and period of oscillations vary with a pointwise periodicity over a range of time delay. Larger-amplitude oscillations can be induced when the PFL has an appropriately long delay, in comparison with the PFL with no delay or short delay; this conclusion holds true for all the three systems. We unravel the underlying mechanism for the above effects via analytical derivation under a limiting condition. We also develop a stochastic algorithm for simulating a single reaction with two delays and show that robust oscillations can be maintained by the PFL with a properly long delay in the single-node system. This work presents an effective method for constructing robust large-amplitude oscillators and interprets why similar circuit architectures are engaged in timekeeping systems such as circadian clocks.

Synchronous long-term oscillations in a synthetic gene circuit.

Synthetically engineered genetic circuits can perform a wide range of tasks but generally with lower accuracy than natural systems. Here we revisited the first synthetic genetic oscillator, the repressilator1, and modified it based on principles from stochastic chemistry in single cells. Specifically, we sought to reduce error propagation and information losses, not by adding control loops, but by simply removing existing features. This created highly regular and robust oscillations. Some streamlined circuits kept 14 generation periods over a range of growth conditions and kept phase for hundreds of generations in single cells, allowing cells in flasks and colonies to oscillate synchronously without any coupling between them. Our results show that even the simplest synthetic genetic networks can achieve a precision that rivals natural systems, and emphasize the importance of noise analyses for circuit design in synthetic biology.

Spatio-temporal models of synthetic genetic oscillators.

Signal transduction pathways play a major role in many important aspects of cellular function e.g. cell division, apoptosis. One important class of signal transduction pathways is gene regulatory networks (GRNs). In many GRNs, proteins bind to gene sites in the nucleus thereby altering the transcription rate. Such proteins are known as transcription factors. If the binding reduces the transcription rate there is a negative feedback leading to oscillatory behaviour in mRNA and protein levels, both spatially (e.g. by observing fluorescently labelled molecules in single cells) and temporally (e.g. by observing protein/mRNA levels over time). Recent computational modelling has demonstrated that spatial movement of the molecules is a vital component of GRNs and may cause the oscillations. These numerical findings have subsequently been proved rigorously i.e. the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. In this paper we first present a model of the canonical GRN (the Hes1 protein) and show the effect of varying the spatial location of gene and protein production sites on the oscillations. We then extend the approach to examine spatio-temporal models of synthetic gene regulatory networks e.g. n-gene repressilators and activator-repressor systems.

Host-aware modelling of a synthetic genetic oscillator.

Synthetic genetic circuits sometimes exhibit unexpected functionality or even fail entirely when implemented in vivo, due to the effects of interactions with the host cell that were not accounted for in the circuit's design. In this paper, we consider the effects that limitations in cellular resources have on the dynamics of a synthetic cellular oscillator. We show that incorporating these effects into a host-aware model of the synthetic oscillator results in significant changes in its dynamics, highlighting the need to take account of host-circuit interactions in mathematical models that are to be used as CAD tools for synthetic circuitry.

A simple model for Lutz and Bujard's controllable promoters and its application for analyzing a simple genetic oscillator.

We develop an exact and flexible mathematical model for Lutz and Bujard’s controllable promoters. It can be used as a building block for modeling genetic systems based on them. Special attention is paid to deduce all the model parameters from reported (in vitro) experimental data. We validate our model by comparing the regulatory ranges measured in vivo by Lutz and Bujard against the ranges predicted by the model, and which are calculated as the reporter activity obtained under inducing conditions divided by the activity measured under maximal repression. In particular, we verify Bond et al. assertion that the cooperativity between two lac operators can be assumed to be negligible when their central base pairs are separated by 22 or 32 bp [Gene repression by minimal lac loops in vivo, Nucleic Acids Res, 38 (2010) 8072–8082]. Moreover, we also find that the probability that two repressors LacI bind to these operators at the same time can be assumed to be negligible as well. We finally use the model for the promoter PLlacO-1 to analyze a synthetic genetic oscillator recently build by Stricker et al. [A fast, robust and tunable synthetic gene oscillator, Nature, 456 (2008) 516–519].

A Novel Synthesizing Genetic Logic Circuit: Frequency Multiplier.

This paper presents a novel synthesizing genetic logic circuit design based on an existing synthetic genetic oscillator, which provides a function of frequency multiplier to synthesize a clock signal whose frequency is a multiple of that of the genetic oscillator. In the renowned literature, the synthetic genetic oscillator, known as a repressilator, has been successfully built in Escherichia coli to generate a periodic oscillating phenomenon through three repressive genes repress each other in a chain. On the basis of this fact, our proposed genetic frequency multiplier circuit utilizes genetic Buffers in series with a waveform-shaping circuit to reshape the genetic oscillation signal into a crisp logic clock signal. By regulating different threshold levels in the Buffer, the time length of logic high/low levels in a fundamental sinusoidal wave can be engineered to pulse-width-modulated (PWM) signals with various duty cycles. Integrating some of genetic logic XOR gates and PWM signals from the output of the Buffers, a genetic frequency multiplier circuit can be created and the clock signal with the integer-fold of frequency of the genetic oscillator is generated. The synthesized signal can be used in triggering the downstream digital genetic logic circuits. Simulation results show the applicability of the proposed idea.

Dynamic Behavior of an Isolated Repressilator with Feedback

Dynamic behavior of the model describing the literature-known synthetic genetic oscillator (repressilator), which is constructed as a three-gene ring, is studied by the methods of numerical analysis of bifurcations. The protein products of each gene inhibit activity of the subsequent neighbor gene, which ensures the limit-cycle existence. One feedback caused by the production of the signal molecules activating expression of a ring gene is added to the repressilator ring. Such a system of four ordinary differential equations is shown to demonstrate diversified and unusual multistable behavior, i.e., coexistence of the stable limit cycle and stable stationary state. Homoclinic saddle–node bifurcations occur at the boundaries of parameter ranges where such a coexistence is observed. The number of homoclinic transitions increases with increasing cycle amplitude if the characteristic times of the protein and signal-molecule dynamics converge. In this case, the limit cycle undergoes forward and backward period-doubling bifurcations.