Analog molecular circuits can exploit the nonlinear nature of biochemical reaction networks to compute low-precision outputs with fewer resources than digital circuits. This analog computation is similar to that employed by gene-regulation networks. Although digital systems have a tractable link between structure and function, the nonlinear and continuous nature of analog circuits yields an intricate functional landscape, which makes their design counter-intuitive, their characterization laborious and their analysis delicate. Here, using droplet-based microfluidics, we map with high resolution and dimensionality the bifurcation diagrams of two synthetic, out-of-equilibrium and nonlinear programs: a bistable DNA switch and a predator–prey DNA oscillator. The diagrams delineate where function is optimal, dynamics bifurcates and models fail. Inverse problem solving on these large-scale data sets indicates interference from enzymatic coupling. Additionally, data mining exposes the presence of rare, stochastically bursting oscillators near deterministic bifurcations.
In the past decade, DNA nanotechnology has produced a variety of circuits and functional devices, for example, smart drug delivery, molecular diagnostics or responsive plasmonic structures1,2,3. In general, such molecular programs have mimicked the Boolean paradigm of digital electronics4,5,6,7,8,9,10. However, the adequacy of computing digitally with biomolecules is not established11,12. First, each bit requires the synthesis of a dedicated reagent for its encoding and a digitization machinery for its restoration. Second, bimolecular reactions poorly approximate sharp digital transitions, as their rate varies linearly with each reagent. Third, the language of Boolean logic—which can express any discrete function given by its truth table—is not parsimonious for approximating continuous functions. For example, an essential computational primitive like the weighted mean of concentrations—which has a natural and concise chemical implementation10—would require an unwieldy digital circuit with analog-to-digital converters as well as numerous ripple-carry adders and multipliers.
By contrast, biological cells use analog designs and limited resources to achieve complex molecular operations, such as multistability for storing information or clocks to orchestrate metabolic processes13,14. Inspired by these natural examples, in vitro analog systems of various complexity have been demonstrated. Circuits that use nucleic acids as signalling species and implement oscillations15,16,17, bistability18,19, pattern recognition20, majority voting21 or molecular timers22 have been reported. Those out-of-equilibrium systems are either based solely on DNA (strand displacement drives the consumption of gates that release and sequester DNA signals20,21) or use a hybrid DNA–enzyme architecture in which enzymes that process nucleic acids continuously produce and degrade DNA/RNA signals16,17,18,19,22. Cell-free translation systems that rely on proteins for their signalling have also demonstrated dynamic behaviours23,24. DNA-only circuits tend to be more amenable to rational design, largely thanks to the predictable thermodynamics of DNA binding, whereas enzymatic circuits enjoy sharper nonlinearities, reduced leakage and larger turnover numbers. However, irrespective of their actuation, these analog circuits are often more delicate to conceive than their digital counterparts. Besides circuit topology, an additional layer of continuous parameters (concentrations, temperature and so on) governs their dynamics. Minute changes to these parameters may lead to a discontinuity in function, a counter-intuitive phenomenon known as bifurcation25.
Given the limited knowledge of the underlying molecular ‘hardware’, how can one then navigate within the functional landscape of an analog circuit? Combinatorial explosion is pervasive as the number of possible states grows exponentially with the number of parameters. Technologies such as liquid-handling robots, microfabrication, automated microscopy and integrated data processing have ushered the eras of high-content screening in drug discovery, digital pathology in medicine and next-generation sequencing in genomics. Yet similar high-throughput characterization has mostly eluded the communities of molecular programming and DNA nanotechnology26,27,28,29. One option to tackle the exponential size of parametric spaces in these fields is to take advantage of the inherent partitionability of solution-phase chemical systems—because their dynamics rely on intensive variables, they are unaffected by scaling and they can be split into many subsystems without affecting their functions. In particular, it was shown recently that synthetic DNA oscillators could be encapsulated into micrometric water-in-oil droplets at high rates and low dispersity and yet retain their oscillations30,31. Partitioning into an emulsion offers huge gains in throughput32,33: More than 105 reactions vessels with a size of 50 µm (∼100 pl) can be generated from a single bulk-reaction volume of 10 µl.
We designed a microfluidic emulsification protocol to harness this throughput to chart the n-dimensional parametric space of a nonlinear system—that is, to generate and observe the evolutions of droplets with different chemical compositions (Fig. 1). First, we assembled a master mix with all common chemicals and used it to prepare n reaction tubes that each contained additionally one of the parameter species and its fluorescent barcode. The barcodes are spectrally distinct from each other, and are chemically orthogonal to the system (Supplementary Sections 2 and 6). Their fluorescence intensities are used later to report on the level of parameter species of interest in each droplet. An additional tube contained only the master mix and was used for volume compensation. Second, the n + 1 tubes were connected to the inlets of a microfluidic chip and pressurized individually by a controller. Varying the pressures according to a script generated monodisperse droplets with diverse parameters. An extra routine produced droplets that contained 100% of each parameter tube for calibration purposes. Third, a monolayer of droplets was enclosed into a chamber, incubated and imaged by confocal microscopy. Image processing and data analysis produced the desired bifurcation diagrams, which mapped the state of an observable against the system's parameters.
Using only ∼100 µl of reagents, this integrated approach yielded—within a day—high-resolution bifurcation diagrams with 104 data points. These diagrams not only pinpoint optimal regimes, but also visually enhance mechanistic understanding and suggest new usages of the circuits.
Results and discussions
Bistability is a prototypical dynamic behaviour that corresponds to the coexistence of two possible asymptotic states for one given set of parametric values. The system decides which state to take depending on its history, and the apparent violation of the second principle of thermodynamic is resolved by the constant dissipation of energy to ‘remember’ this decision25. At the molecular level, a variety of dissipative reaction networks can implement bistability18,19,34; we selected a symmetrical, non-cooperative architecture based on two autocatalytic and mutually inhibiting species (Figs 2b and 3a). This topology a priori leads to four fixed points (no species surviving, coexistence of both species, survival of one species and decay of the other, and vice versa), the stability of which depends, in particular, on the relative strengths of the feedback loops18.
We implemented this topology in vitro using a molecular programming language (the Polymerase Exonuclease Nickase (PEN) DNA toolbox35), that is, a set of basic DNA-processing biochemical reactions that can be connected arbitrarily using design rules based on Watson–Crick base pairing. Our bistable circuit comprises two self-replicating and mutually inhibiting single-strand DNA, α and β (the triggers). Their replication is mediated by polymerization/nicking reactions on two autocatalytic DNA templates, αtoα and βtoβ. The triggers bind their cognate templates and are elongated by a polymerase. The resulting duplexes are recognized and nicked by a nicking enzyme. The overall process generates new strands (either triggers or inhibiting strands) according to template-encoded rules. Inhibiting strands—produced through a similar mechanism—transiently repress replication by sequestering autocatalytic templates. An exonuclease enzyme continuously degrades single-stranded species (the templates are chemically protected).
The concentrations of the two autocatalytic templates αtoα and βtoβ define two natural design parameters, the strengths of the positive feedback loops (PFLs). We programmed our microfluidic platform to generate droplets that densely filled a rectangular area in the (αtoα, βtoβ) space (Supplementary Fig. 2 and Supplementary Movie 1). As the search was for multistability, the experiment was repeated twice, and each time it started in the vicinity of one of the expected fixed points αβ = 10 or 01. Bistable circuits should then converge to two different outcomes, whereas monostable circuits should end up in the same steady state irrespective of the initial conditions. Fluorescent reporters allow us to monitor steady states and visually assess the performance of each given parametric composition. Note that the relation between fluorescence and concentration of the trigger is monotonic but not linear (although it is consistent from droplet to droplet).
After incubation of the emulsion at 42 °C for 11 hours (typically ∼5–100 times longer than the time taken to reach a steady state18), the fluorescence of ∼104 droplets was recorded in four fluorescent channels (two barcodes, two reporters) using confocal imaging of a random, densely and orderly packed two-dimensional (2D) array of droplets (Supplementary Section 10). We then obtained the phase diagrams by simply replotting the spatial array of reporter intensities in barcode (parametric) space (Supplementary Section 10).
Bifurcations of the steady states are directly visible from these raw data (Fig. 3b). The relative positions and areas of the regions vary qualitatively as expected (for example, region 10 of high α and low β corresponds to circuits with high αtoα and low βtoβ, and expands when the initial α is increased). Satisfactorily, the superposition of the two plots delineates an extended bistable region in which the circuit does function as a memory (Fig. 3c). The frontiers' sharpness (as assessed by the magnitude of the gradient of α or β) also agrees with qualitative predictions from bifurcation theory36—the fold bifurcations that connect the bistable to monostable areas (sudden loss of stability of one state) appear as abrupt frontiers. By contrast, transcritical bifurcations (collision and exchange of stability between two fixed points, for example, the green-to-yellow transitions in Fig. 3b,c) materialize as a smooth gradient. To rule out artefacts caused by barcoding accuracy (∼5% relative error in each direction (Supplementary Section 4)) we performed independent 1D scans along two vertical lines (Fig. 3d), which confirmed these trends.
A closer inspection of the diagrams' morphology reflects unexpected properties of the biomolecular circuit. First, except in a small area close to the origin, all the steady states of this system should yield at least one surviving trigger; provided the concentration of the corresponding trigger does not reach 0, the 00 zone is expected to be a rectangle with a shape that is independent of the initial conditions for α0 > 0 and β0 > 0 (Supplementary Section 7). The dramatic failure of this prediction (the black areas are highly asymmetrical and extend around the bistable zone (Fig. 3c)) signals that either convergence to the local steady state(s) is drastically slowed down (for example, by a catastrophic slowdown37) or even arrested (for example, by small number effects38), or that an alternative stable state emerges from an unsuspected mechanism39. In any case, these trapped states could provide alternative options for memories on practical time scales. Second, some frontiers show a pronounced non-monotonicity. For instance, in Fig. 3d, middle, it is counter-intuitive that βtoβ initially represses α for βtoβ between 150 and 200 nM, but eventually permits its survival for βtoβ over 200 nM. Similar non-monoticity is seen for the 10 and 01 frontier in Fig. 3c. This suggests a hidden mechanism that perhaps indirectly cross-activates α and β, whereas the designed circuit topology enforces cross-inhibition between them, and DNA sequences were specifically engineered to avoid such undesired interactions (Supplementary Section 5).
For a mechanistic elucidation, we turned to statistical inference based on these high-level data. A low-level chemical reaction network that comprises 26 species was compiled according to three nested models for the kinetics of enzymatic reactions (polymerization, nicking and degradation): model 1 (M1) without enzymatic saturation (first-order kinetics), model 2 (M2) with enzymatic saturation (Michaelis–Menten kinetics) and model 3 (M3) with enzymatic saturation and competition40. Uncertain parameters (stacking contributions, enzymatic activities and so on) were left as free parameters and fitted with a continuous black-box optimization algorithm41. After fitting, we defined a performance vector by combining fitness with a likelihood criterion (deviation of the optimized parameters from their starting values) to penalize a model that unrealistically stretches its parameters (Supplementary Section 9).
The resulting plot of performance (Fig. 3e) suggests that the model with saturation and competition (M3) is the most plausible on quantitative and qualitative grounds. Quantitatively, it is Pareto optimal: it best fits experiments with the least deviation from known parameters, as seen from the outer position of its Pareto front. Qualitatively, only M3 satisfyingly reproduces the positions, non-monoticity and smoothness of the experimental bifurcations (Fig. 3f–h). M1 and M2 fail to generate non-monotonic frontiers because cooperativity between α and β cannot emerge in their kinetics. Their diagrams are also unrealistically sharper (M1) or more blurred (M2) than experimental data, as both models attempt to interpolate linearly the non-monotonic frontiers with straight lines. Although M1 can produce linear frontiers thanks to its first-order kinetics, enzymatic saturation strongly constrains M2, which severely stretches its nicking Michaelis constant Km (∼5–10-fold increase) to recover the linear kinetic of M1. As a result, the Pareto front of the simplistic model M1 dominates almost entirely that of the more-complex model M2, which confirms that parsimony is an important aspect and that a model with more degrees of freedom does not necessarily better explain data when the parameters' deviation is accounted for.
This domination of M3 confirms predictions40,42,43 that the sharing of limited resources, such as enzymes, introduces global and nonlinear couplings between substrates, which qualitatively impacts the dynamics of biochemical regulation networks, the shape of their bifurcation diagrams and thus their properties, such as robustness and performance. Yet, none of the models captures the 00 buffer that lines the bistable area (Fig. 3c), which suggests that additional mechanisms or stochastic effects become magnified close to bifurcation frontiers.
We then extended the microfluidic mapping approach to higher dimensional spaces and unsteady dynamics. We reprogrammed the DNA chemistry to implement a predator–prey network15, an archetypical ecological system with limit cycles in addition to fixed points (Figs 2c and 4a). The mechanism of this molecular ecosystem entails three catalytic steps with rates that can be parameterized by the concentration of their exogenous catalysts: (1) the growth of DNA prey (catalysed by a polymerase and a DNA template), (2) predation (the conversion of prey into predators, catalysed by endogenous predators and a polymerase) and (3) degradation of predators and prey (catalysed by an exonuclease). We modified the pressure script to perform a microfluidic Monte Carlo sampling of the 3D parametric space (tem, pol, exo). We then recorded the time-resolved fluorescence induced by prey in ∼104 droplets upon incubation at 45 °C for a little over three days (Supplementary Section 11).
During the first day, oscillations in a small percentage of the droplets were directly visible (Supplementary Fig. 18 and Supplementary Movie 2). We tracked the droplets, extracted their time traces and scored their oscillations (by summing the height of all the detected peaks) against their parameters. In the resulting 3D diagram, strong oscillations are confined to an asymmetric, low-dimensional manifold (Fig. 4c). A scaling argument on the programmed mechanism suggests that the manifold exo = constant × tem × pol should shape this oscillatory volume (Supplementary Section 8). Indeed, 2D slices of this region in the (pol, exo) plane (constant tem) are bordered by a line exo = constant × tem × pol (Fig. 4d); this reflects that the production needs to balance the stabilizing effect of degradation for the oscillations to persist. The oscillatory region shrinks with increasing tem; tem strengthens the growth of prey compared with predation, and eventually decouples prey and predators, and weakens the oscillations. The same scaling exo = constant × tem × pol is observed in the (tem, pol) slices (constant exo) for which the inner boundary approximates a hyperbolic curve. Increasing exo drags this shape to higher tem × pol levels, which by scaling shortens the period of oscillations and improves their score. Additionally, both sharp and smooth bifurcations are visible in Fig. 4d; with an initial low pol, oscillations start sharply but gradually weaken into damped oscillations (dashed line and inset). This suggests that the Hopf bifurcations that border the oscillatory regions lead to trajectories that spiral down to a stable coexistence focus on one side, but to an extinction state with no oscillatory potential on the other side.
All these observations agree with a simple analytical model of this molecular predator–prey system (Supplementary Section 8). However, stability analysis also predicts the existence of a very small region in which a stable steady state coexists with a stable limit cycle—a hybrid dynamics known as hard excitation, but rarely observed in experimental systems44. In this region, located precisely on the inner edge of the oscillatory zone, the noise of even a small amplitude (∼1% of the maximum concentration of preys) would be sufficient to induce rare, stochastic switching between steady and oscillatory behaviours, decorrelating the traces on long time scales (Supplementary Section 8).
The emergence of selfish DNA replicators generally forbids the long-term observation of the dynamics in amplification systems45. Compartmentalization in the emulsion mitigates this issue, because the stochastic parasite is confined to its original droplet46. Indeed, although some droplets showed signs of uncontrolled reaction after a day (Supplementary Section 12), in others we could follow the predator–prey dynamics for up to three days. We mined these long-term time traces for evidence of low autocorrelation (Fig. 4e). Although most either dampen (bottom right in Fig. 4e), oscillate sustainably (diagonal in Fig. 4e,f) or fail to oscillate at all (dark points in the bottom left of Fig. 4e), a small fraction of the surviving droplets exhibit ‘bursting’ oscillations that spontaneously switch ON or OFF (Fig. 4g and Supplementary Movie 3). Remarkably, these droplets originate, as predicted, from the small, well-defined region of parameter space that lines the (low pol, low temp) face of the oscillatory region (Fig. 4c). Spontaneous switching between the two attractors requires noise, which could originate from low copy numbers inherent to some stages of the predator–prey oscillations—in 100 pl droplets, the stochastic regime appears below the picomolar range, which matches the typical minimal concentrations of prey reached in simulations15 (Supplementary Section 13). This example, along with a report of extrinsic noise traced to enzymatic variability31, shows that small volumes could be instrumental to reveal and exploit alternative attractors, or generate and amplify chemical diversity among a population of droplets47.
As long as the parametric chemicals do not diffuse between droplets, our platform can provide scans of arbitrary circuits with 104 to 106 data points in 1D to 5D, and possibly more with advanced microscopy (Supplementary Section 3). We therefore expect this data-driven approach to be applicable to both fundamental and practical questions in molecular biosciences. First, it quickly provides a high-resolution functional snapshot that unveils the optimal parameters of a molecular network. As such, it will allow DNA nanotechnologists and synthetic biologists to expedite prototyping and engineering. It is also believed that biological networks exploit dynamically rich, but narrow, niches in their parameter space39,47. Such hypotheses could be finely tested using our in vitro modelling approach that recapitulates the flow of information in natural circuits. Second, exhaustive mapping has the potential to reveal unforeseen usages of the circuits—an alternative trapped state was discovered for the bistable switch, and stochastic bursts were exposed in a tiny parametric range of the oscillating circuit. Lastly, the diagrams, by their sheer precision, offer mechanistic insights that sharpen our understanding of the design principles in molecular systems. In particular, we found that enzymatic competition alters bifurcations in subtle ways, in line with recent works on the role of competition in cellular environments42.
DNA strands were bought from Biomers in Germany. The polymerase and nickase were bought from New England Biolabs. The thermophilic exonuclease was expressed by our means in E. coli and purified by chromatography according to a published protocol48. We synthetized the surfactant from a perfluoropolyether carboxy-terminated polymer (Krytox, DuPont) and a polyetheramine (Jeffamine M1000, Huntsmann)49. Non-fluorescent species were barcoded by adding a chemically orthogonal and fluorescent dextran to the solution (Life Technologies). Details on the reagents, the sample preparation, the experimental conditions and the data analysis are given in the Supplementary Information.
Microfluidic generation of droplets
We generated droplets with radii of ∼50 µm in the flow-focusing junction of a polydimethylsiloxane microfluidic device by mixing aqueous channels that contained reaction mixes with an oil channel that contained HFE 7500 oil (3 M) with 2% (w/w) of surfactant. Channels were pressurized by a controller MFCS-EZ from Fluigent. Droplet generation was monitored with an epifluorescence microscope piloted with Micromanager50. Pressure profiles were scripted to explore uniformly a square in the space (αtoα, βtoβ) in the case of the bistable switch, or a cube in the space (tem, pol, exo) in the case of the predator–prey oscillator. The square was explored through a deterministic algorithm, whereas the cube was sampled with a scripted random walk with jumps for a quick and uniform exploration. In both case, we included a calibration routine to generate droplets with 100% of each barcoded solution.
After generation, a monolayer of droplets was sandwiched between two hydrophobic glass slides (coated with Cytop, Asahi Glass) and sealed into an airtight chamber with epoxy glue (Araldite) and Norland Optical Adhesive (Norland). The fluorescence of the chamber was imaged on a confocal laser microscope (Olympus Fluoview FV1000) equipped with an xy stage (Optosigma) and a transparent heat plate (Tokai Hit). For the bistable switch (endpoint scans), the droplets were imaged at room temperature after incubation at 42 °C for 11 hours. For the predator–prey oscillator (time-lapse imaging), the droplets were imaged periodically every six minutes during incubation at 45 °C for about three days.
Analysis and fit
Confocal images were then processed with Mathematica to extract the fluorescence of droplets in each channel. The centres were found by detecting local fluorescence maxima. We converted barcode fluorescence into parameter concentrations using calibration droplets and rescaled the fluorescence shifts of the observables. We wrote an algorithm to track droplets over time and assembled the fluorescence time traces of time-lapse images. Filtering criteria were applied to exclude droplets with outlier fluorescence or sudden displacements. Finally, continuous bifurcation diagrams were computed by locally smoothing the raw, discrete bifurcation diagrams.
For fitting the bifurcation diagram of the bistable switch, we modelled the biochemistry at the level of strand domains. We used the DACCAD framework to compile a list of reactions and compute the evolution of the system as a function of given kinetic and thermodynamic parameters. The optimization algorithm CMA-ES was used to optimize the parameters so as to minimize the fitting error.
This work was financially supported by the PHC Sakura program (project number 34171WG), implemented by the French Ministry of Foreign Affairs, the French Ministry Of Higher Education and Research and Japan Society for the Promotion of Science (JSPS), a Grant-in-Aid to Y.R. from the JSPS for Scientific Research on Innovative Areas ‘Synthetic Biology for Comprehension of Biomolecular Networks’ (project number 23119001), a l'Agence Nationale de la Recherche (ANR) Retour Postdoc grant (ANR-13-PDOC-0001) and a JSPS postdoctoral fellowship to A.J.G., and a PhD fellowship from Region Alsace to J.F.B. We thank H. Fujita and M. C. Tarhan for the loan of a microfluidic pump, A. Zadorin for discussions about the theory of bifurcations, K. Hasatani for preliminary work, E. Winfree for detailed comments on the manuscript and A. Estevez-Torres and Y. Tauran for expressing and purifying the exonuclease.