From resonance to chaos by modulating spatiotemporal patterns through a synthetic optogenetic oscillator

Biological oscillations, rhythmic variations crucial for self-organization and complexity, are observed across scales in living systems. Examples range from cellular metabolic activities and cell cycle control to circadian rhythms, developmental pattern formation, and ecological population dynamics. While understanding these oscillations is essential, the complexity of biological systems makes experimental investigation challenging. Current knowledge of complex dynamic properties like resonance, period-doubling, and chaos mainly stems from physics and chemistry, although their biological relevance is increasingly recognized. For instance, resonance is suggested in circadian rhythms' correlation with longevity, and chaos is observed in ecological and microbial communities, even being linked to certain diseases. Synthetic biology offers a powerful alternative approach. By creating controllable synthetic circuits, it minimizes interference from the host regulatory network, enabling precise study of complex dynamics. The repressilator, a three-node negative feedback loop in *E. coli*, was a pioneering synthetic oscillator, though its early versions suffered from irregular oscillations. Subsequent improvements led to highly robust oscillations, and various other synthetic oscillatory circuits have since been developed in diverse systems, expanding our understanding of oscillatory systems and their applications in areas such as cancer treatment, bacterial growth monitoring, aging research, and spatial pattern generation. Most previous synthetic oscillators exhibited limited synchrony or relied on cell-cell communication for synchronization, unlike many natural oscillators that use periodic external signals (entrainment) for synchronization. This research aims to build upon previous work by constructing and investigating a forced synthetic optogenetic oscillator to explore the emergence of complex oscillatory dynamics and translate oscillations into spatial patterns.

The repressilator, a synthetic gene oscillator in *E. coli*, has been a key model for studying oscillatory systems. While initial versions showed irregular oscillations, improvements led to robust oscillations. Other synthetic oscillators with different topologies have been created, advancing our understanding of these systems and their potential applications. Studies have demonstrated entrainment and resonance in synthetic oscillators using microfluidics for precise control of chemical inducers. However, optogenetics offers advantages in controlling oscillatory behavior with precision in both space and time, eliminating the need for medium refreshing. Previous research on forced synthetic oscillators has primarily analyzed temporal patterns, leaving the investigation of spatial pattern translation unexplored. This study builds on these advancements, focusing on the spatiotemporal dynamics of a light-inducible repressilator.

The researchers constructed a light-inducible repressilator, the "optoscillator," by combining a blue light system from an RGB color vision circuitry in *E. coli* with an improved repressilator version. The blue light-inducible system utilizes a light-sensing histidine kinase (YFI) that is active in the dark and inhibited by blue light (470 nm). YFI phosphorylates FixJ, which activates PhiF repressor expression. Light exposure stops PhiF expression, allowing T7 RNA polymerase to activate downstream gene expression. To couple this system to the repressilator, the TetR node was modified using a hybrid promoter (PT3lacO) repressed by LacI and activated by blue light in the presence of the blue light activation system. The functionality of this promoter was characterized using a reporter construct (mCherry) with light and IPTG (LacI inhibitor). The improved repressilator had its TetR-controlling promoter replaced with PT3lacO. Experiments were conducted under constant light and with light pulses of varying periods (*T*_light). Colonies were grown on agar plates, and ring patterns were analyzed using fluorescence microscopy. The fluorescence intensity was measured radially, converting spatial data to temporal oscillations. A mathematical model of the genetic circuit, adapted from the original repressilator model, was developed to explore potential dynamic behaviors. The model included three mRNA species (one for each node) and their corresponding proteins, incorporating inhibition and light activation using Langmuir-Hill functions. Parameterization was done by fitting data from the promoter characterization and resonance experiment. Bifurcation diagrams were calculated to predict dynamic behaviors (resonance, subharmonic resonance, period doubling, chaos). Stochastic reaction kinetics simulations were also performed to capture desynchronization effects. Two-dimensional reaction-diffusion simulations, using the Fisher-KPP equation for colony growth, were employed to model ring pattern formation. Image analysis in MATLAB was used to quantify ring patterns from the experiments and simulations.

The optoscillator displayed various complex dynamics depending on the light regime. Under continuous light, rings were irregular due to desynchronization from stochastic effects. Periodic light pulses synchronized the cells, leading to sharper and stronger ring patterns. Resonance was observed when the light pulse period (*T*_light) matched the natural oscillation period (*T*_osc), resulting in higher fluorescence intensity. Entrainment was observed within a range of *T*_light values. The mathematical model accurately predicted the resonance peak and a second peak representing subharmonic resonance. The model also predicted period-doubling bifurcations (period-2 and period-N) and a chaotic regime. Experiments confirmed subharmonic resonance and period-2 oscillations, agreeing with the model. Evidence for a chaotic regime was observed in experiments at certain *T*_light values, characterized by rapid desynchronization and blurred, irregular ring patterns. Simulations showed good agreement with experimental observations across all regimes (constant light, resonance, period-2, period-N, and chaos). The model demonstrated the impact of stochasticity on synchrony and the emergence of chaotic patterns.

This study provides insights into the nonlinear dynamics of a simple synthetic oscillator, demonstrating the generation of complex spatiotemporal patterns. The observed phenomena (synchronization, resonance, subharmonic resonance, period-doubling, and chaos) highlight the richness of dynamic behaviors accessible in even simple biological systems. The ability to translate these dynamics into readily observable spatial patterns is a significant advancement. The findings resonate with the growing appreciation of chaos and other nonlinear phenomena's importance in biological systems, including their roles in increasing population heterogeneity and potentially providing advantages in variable environments. The optoscillator's capacity to switch between synchronized and desynchronized states depending on the forcing frequency is a remarkable feature. The limitations in observing a larger number of oscillations in the current agar plate setup necessitate future work using microfluidics or liquid culture with frequent dilutions to better characterize the period-N regime and provide stronger evidence for chaotic oscillations. This approach could also be beneficial in exploring higher-order bifurcations.

This research successfully demonstrates the creation of a synthetic optogenetic oscillator capable of exhibiting a range of complex dynamic behaviors, including chaos, which are translated into intricate spatial patterns. This work advances our understanding of biological oscillations and provides a valuable tool for investigating nonlinear dynamics in biological systems. Future research could focus on exploring a wider range of light intensities and spatial gradients, expanding pattern manipulation capabilities and furthering applications in biotechnology, medicine, and materials science.

The limited number of oscillations observable in the current agar plate setup restricts detailed analysis of the period-N and chaotic regimes. Future studies could utilize alternative methods, such as microfluidic devices or liquid culture with frequent dilutions, to observe a greater number of oscillations and provide more robust evidence of chaotic behavior. The current model could be further refined to better capture the intricacies of stochastic effects and improve the quantitative agreement with experimental observations.