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

Oscillations pervade biology—from metabolism and cell cycle to circadian rhythms, development, and ecological dynamics—yet fundamental nonlinear behaviors such as resonance, period-doubling, and chaos are difficult to probe experimentally in living systems due to complexity and noise. Prior evidence suggests resonance effects in circadian systems and chaotic dynamics in ecological and physiological settings, but controlled studies in cellular oscillators remain limited. Synthetic biology enables tractable, minimally confounded systems to study such dynamics. The repressilator, a three-node negative feedback loop (TetR-LacI-CI-TetR), inaugurated synthetic oscillators in vivo; improved versions exhibit robust oscillations and can form spatial ring patterns in growing E. coli colonies because only edge cells continue oscillating and growing. Many synthetic oscillators desynchronize without coupling or require microfluidics for controlled forcing. Natural oscillators commonly use periodic external cues for entrainment. This study redesigns the repressilator to create a light-activated ‘optoscillator’ to explore forced-oscillator phenomena and how temporal dynamics map onto spatial ring patterns. The goals are to test synchronisation, resonance, subharmonic resonance, period-doubling, and chaotic regimes using optogenetic forcing and to connect observations with mathematical modeling.

• Synthetic oscillators: Original repressilator exhibited irregular oscillations; later implementations improved robustness. Additional synthetic oscillators span cell-free, bacterial, and mammalian systems, with comparative analyses published.
• Spatial patterns: Improved repressilator produces concentric fluorescent rings in colonies, offering a simple readout of oscillatory dynamics.
• Forced/entrained oscillators: Natural clocks (e.g., circadian) use periodic cues for entrainment. In synthetic systems, entrainment and resonance were shown for a 2-node oscillator in E. coli (microfluidics), and period-doubling in a cell-free oscillator. Recent work used light to inhibit a repressilator to synchronise and detune single-cell oscillations. Prior approaches often required microfluidics to deliver periodic chemical inducers; optogenetic control circumvents this by enabling precise spatiotemporal induction without medium exchange.
• Nonlinear phenomena: Resonance, sub/superharmonics, period-doubling cascades, and chaos are classical in physics/chemistry (e.g., Duffing oscillator) and implicated in biology (e.g., ecological chaos, arrhythmicity in disease), but direct experimental demonstrations in synthetic gene circuits, particularly mapping to spatial patterns, are scarce.

Circuit design and optogenetic control:

• The optoscillator couples an improved repressilator (PLPT107 backbone) to a blue-light system adapted from the RGB optogenetic toolkit. The blue-light sensor YF1 is active in the dark and deactivated by blue light (470 nm); in the dark, YF1 phosphorylates FixJ, activating PhiF repressor expression, which blocks production of the T3 domain of split T7 RNAP. Under blue light, YF1 is off, PhiF repression ceases, the T3 domain is expressed and dimerizes with constitutively expressed T7 core to activate PT3 promoters.
• To render TetR light-inducible, the native PLlacO1 promoter upstream of tetR was replaced with a hybrid PT3lacO promoter that is repressed by LacI and indirectly activated by blue light via the YF1/FixJ/PhiF/T7 system. The CFP reporter retained PLlacO1 and is not light-induced; fluorescence readouts are from the CI/mVenus node downstream of TetR (displayed as green).

Promoter characterization:

• A reporter construct (mCherry under PT3lacO, with LacI) was used to test light and IPTG control. Colonies grown on agar under varying blue light intensities (LITOS device) with or without 1 mM IPTG were imaged after 96 h; fluorescence quantified showed tight LacI repression without IPTG and light-dependent induction with IPTG present.

Colony oscillation experiments:

• E. coli MK01 ΔclpXP strain carrying pLPT41 (sponge), pJP_Rep00 (light-inducible repressilator), and pJP_Azul01 (light system) was plated and preincubated 20 h in the dark to form isolated colonies. Constant light or light pulses were applied for 4 days at 21 °C using LITOS (blue LED only). Constant light intensity set to 30% (chosen as lowest robust intensity to avoid photobleaching/toxicity), corresponding to ~0.13 W/m^2. For pulsed forcing, the light intensity followed a sinusoid between 0–100% with period T_light; 8-bit resolution updated 20 times per period.
• T_light conditions spanned 12–56 h across experiments to probe resonance, entrainment, subharmonic resonance, period-doubling, and chaotic regimes.

Optogenetic setup and calibration:

• LITOS RGB LED matrix used (blue 465–470 nm). Relative intensities (15–100%) converted to absolute values with a calibrated light sensor; conversion curve provided in Supplementary Fig. 2.

Microscopy and image analysis:

• Fluorescence imaging with Zeiss AxioImager M1, 2.5× objective, Prime 95B sCMOS, 100 ms exposures in YFP and DsRed channels. Ring intensity measured radially from colony center to edge.
• Spatial-to-temporal transformation: Nonlinear colony growth was quantified (radius vs time) to convert radial fluorescence profiles (space vs intensity) into time vs intensity oscillation traces. Peaks and troughs were identified for each colony. Oscillation period T computed from intervals between consecutive maxima/minima; amplitude defined as max–min per ring. Data typically from n=3 colonies per condition; mean ± SE reported.

Mathematical modeling and simulations:

• Deterministic reaction-kinetics model: ODEs for three mRNAs and their proteins (TetR, LacI, CI), with regulatory interactions and light activation modeled via Langmuir-Hill functions. Parameterization used promoter characterization data and resonance amplitude data.
• Stochastic simulations: Chemical Langevin equation solved using fully composite Patankar methods to capture desynchronization/noise-induced effects.
• Spatial simulations: 2D reaction–diffusion-growth model with Fisher–KPP for colony expansion, coupled to local stochastic reaction kinetics, solved by finite-difference (FTCS) schemes. Simulations reproduced ring patterns under constant and pulsed light and explored noise effects.

Experimental regimes/examples:

• Constant light (30%): assesses free-running oscillations and desynchronization.
• Pulsed light: resonance near T_light ≈ 20 h; entrainment window 16–24 h; subharmonic resonance at T_light ≈ 44 h; period-2 and period-N regimes at selected T_light (e.g., 44 h and 28 h); chaotic regime near T_light ≈ 16 h. Additional control with higher-frequency square pulses (e.g., T_light = 12 h) tested ability to follow fast forcing.

• Successful construction of a light-inducible repressilator (optoscillator) that oscillates under blue light and is quiescent in the dark; ring patterns appear only with light.
• Synchronization by periodic forcing: Compared to constant light, pulsed light at T_light = 18 h produced sharper, higher-amplitude, more circular rings, indicating improved synchrony.
• Resonance and entrainment:
  • Resonance peak in average ring amplitude at T_light = 20 h, matching the natural period inferred from constant-light data.
  • Entrainment window observed for T_light between 16–24 h: oscillator period increased proportionally with T_light. For T_light > 24 h, the oscillator followed every other pulse (period halved relative to drive).
• Subharmonic resonance: Second amplitude peak at T_light = 44 h (approximately double the resonance period). Experimentally the peak was lower than at 20 h; simulations showed strong/weak alternating peaks whose average matched experimental amplitudes.
• Period-doubling and higher-order periodicity:
  • Period-2 dynamics at T_light = 44 h with alternating strong/faint rings.
  • Period-N regime exemplified at T_light = 28 h.
• Chaos evidence:
  • At T_light = 16 h, colonies exhibited very blurred and irregular rings, with rapid desynchronization and decreasing average amplitude, more severe than under constant light, consistent with chaotic behavior.
  • Controls with faster square-wave forcing (e.g., T_light = 12 h) still produced sharp rings, arguing against inability to follow high frequency as the cause of blurring at 16 h.
  • Modeling predicted chaotic regions in the bifurcation diagram and, via Poincaré maps, Lyapunov exponents, phase-space trajectories, and Fourier spectra, confirmed chaos; experimental observations aligned with predicted chaotic frequencies and dynamics.
• Modeling–experiment agreement: Calibrated model reproduced resonance peak location/intensity, subharmonic resonance, and period-2 oscillations; spatial simulations matched ring morphologies across regimes. Stochastic simulations captured amplitude decay under constant light due to desynchronization and maintained amplitude under resonance due to synchronization.
• Quantitative notes: Constant light at 30% (~0.13 W/m^2) chosen to minimize photobleaching/toxicity; amplitudes computed as ring max–min; n=3 colonies per condition; mean ± SE reported.

The optoscillator provides a minimal, controllable biological platform to study nonlinear dynamics characteristic of forced oscillators (analogous to Duffing-type behavior) and reveals how temporal dynamics map to spatial patterns in growing colonies. Periodic light forcing both synchronizes and entrains the oscillator within Arnold tongue regions, producing higher-amplitude, sharp rings at resonance (T_light ~20–24 h) and at subharmonic resonance (T_light ~44 h). Beyond entrainment zones, the system exhibits period-doubling (period-2 and period-N) and transitions consistent with chaos, where small differences in initial states or noise rapidly amplify, causing fast desynchronization and loss of ring clarity. The dual role of light—able to enhance synchrony near resonance but accelerate desynchronization near chaotic regimes—highlights the importance of forcing frequency in controlling collective cellular oscillations. Spatial period-doubling patterns bridge to morphogenetic wrinkling phenomena and suggest routes to engineer mechanical patterning via gene expression oscillations. The modeling framework, spanning deterministic, stochastic, and spatial dynamics, closely matches experimental findings, supporting the conclusion that the observed regimes include chaos. These insights advance understanding of biological oscillators and open opportunities to program spatiotemporal behaviors in living materials and biosystems.

This work introduces a light-inducible repressilator (optoscillator) that translates forced-oscillator dynamics into spatial ring patterns in E. coli colonies. The system demonstrates synchronization, resonance, subharmonic resonance, period-doubling (period-2 and period-N), and provides evidence for chaos, supported by both experiments and modeling. The approach circumvents microfluidic requirements for periodic chemical induction and enables precise, tunable spatiotemporal control. Future directions include: increasing the number of observable oscillations (e.g., larger colonies through optimized growth, or liquid cultures maintained in turbidostats/microfluidics); single-cell microfluidic measurements to quantify period-N cascades and chaos more rigorously; extending optogenetic forcing to other oscillator topologies; and exploring spatiotemporal light gradients and intensity modulation to engineer complex patterns for applications in engineered living materials, biotechnology, sensing, and biocomputing.

• Limited number of observable oscillations in colonies (typical diameter ~6 mm after 4 days; growth slows), constraining detailed analysis of successive period-doubling cascades and chaotic invariants.
• Distinguishing chaos from biological noise is challenging with few cycles and ensemble measurements; while modeling supports chaos, experimental confirmation is inherently difficult.
• Practical forcing-frequency range is constrained: very fast forcing effectively approximates constant light, very slow forcing yields too few rings within experimental time.
• Temperature sensitivity: At 37 °C the light system becomes leaky, leading to undesired induction; experiments performed at lower temperature to maintain control, potentially affecting growth rates.
• Spatial analysis aggregates over many cells; single-cell heterogeneity and coupling effects are inferred rather than directly measured in this plate-based setup.