Synthetic circuit for exact adaptation and fold-change detection

Biological organisms use their sensory systems to detect changes in their environment. The ability of sensory systems to adapt to static inputs allows wide dynamic range as well as sensitivity to input changes including fold-change detection, a response that depends only on fold changes in input, and not on absolute changes. This input scale invariance underlies an important strategy for search that depends solely on the spatial profile of the input. Synthetic efforts to reproduce the architecture and response of cellular circuits provide an important step to foster understanding at the molecular level. We report the bottom-up assembly of biochemical systems that show exact adaptation and fold-change detection. Using a malachite green aptamer as the output, a synthetic transcriptional circuit with the connectivity of an incoherent feed-forward loop motif exhibits pulse generation and exact adaptation. A simple mathematical model was used to assess the amplitude and duration of pulse response as well as the parameter regimes required for fold-change detection. Upon parameter tuning, this synthetic circuit exhibits fold-change detection for four successive rounds of two-fold input changes. The experimental realization of fold-change detection circuit highlights the programmability of transcriptional switches and the ability to obtain predictive dynamical systems in a cell-free environment for technological applications.


Methods
The sequence of DNA oligonucleotides and RNA outputs are listed as follows. See Figure S1 for sequence domains and predicted secondary structures.

Mathematical Model
To explore the phase space of circuit behavior and to build theoretical framework to understand experimental outcome, we constructed a simple mathematical model for the synthetic circuit. Table S1 lists the hybridization and branch migration reactions and the enzyme reactions as shown in Figure 1C. Here, we do not consider side reactions or incomplete transcription and degradation products. Also for the simple model presented here, the enzyme reactions are treated as approximately first-order reactions. The dynamics of this in vitro circuit can be described by the following five ordinary differential equations: (1) can be rewritten as follows:ẋ = α 1 u − β 1 x − kxy + γ 1 z, y = α 2 u − β 2 y − kxy + γ 2 z, where α 1 and α 2 are functions of k p 's (RNAP concentration), [TrMG tot ], and [TiMG tot ], β 1 , β 2 , γ 1 , and γ 2 are functions of RNase R concentration and the secondary structure of RNA molecules, and k is a function of the length of exposed toehold of rMG. Therefore, all of the rate constants are amenable to tuning.  Figure S1: DNA and RNA sequences of single-stranded species and complexes for the synthetic adapter circuit. The sequence domains are color-coded to indicate identical or complementary sequences: magenta indicates the input domain; dark blue indicates the T7 RNAP promoter; orange indicates the output domain. DNA strands are drawn as straight lines; RNA strands are illustrated as squiggly lines. The two DNA strands, TrMG-nt and TrMG-t, comprise the switch template TrMG and similarly TiMG-nt and TiMG-t form TiMG. For the RNA outputs, the predicted secondary structures are shown: iMG does not have a significant secondary structure, whereas rMG contains a central loop for MG dye with a single base of 3 end exposed. Note that the formation of rMG·iMG complex breaks open the central loop for MG dye and leaves a long domain of 3 end of rMG exposed while the previously exposed 3 end of iMG is hidden within the toehold binding domain of rMG.
Since TrMG and TiMG have identical input domains, the activator A will bind to TrMG and TiMG with the same affinity; the binding reaction will be fast and practically irreversible because of the large gain in thermodynamic energy upon binding. Therefore, we can calculate α 1 and α 2 as follows: The parameters α 1 and α 2 can be continuously tuned by adjusting template concentrations. Due to the binding requirements of RNase R [1], we expect that the degradation of rMG by RNase R would be negligible, i.e. β 2 0 (see Figure S6). Similarly, we expect that the degradation of iMG within rMG·iMG complex by RNase R would be neglibible, i.e. γ 2 0 (see Figure S7 and discussion thereof). Then, the dynamic equations can be rewritten as follows:ẋ The system has a unique equilibrium point if u > 0: The steady-state value of output y in this model is independent of input u, showing that the circuit exhibits property of exact adaptation. (When u = 0, the steady-states for x and z are determined to be zero, i.e.x =z = 0, butȳ can take any value.)

Michaelis-Menten enzyme kinetics
One of the limitations in the above mathematical model is the assumed first-order enzyme reactions. A more realistic model would use Michaelis-Menten enzyme reactions, where the available enzyme concentrations can be calculated as follows: where we assume high K M,T rM G , K M,T iM G , and K M,R,rM G (e.g. K M,R,rM G > 5µM) and similar K M for the two activated templates (since the sequences for the two templates around the promoter region are identical). The Michaelis constants, the affinity of substrates to the enzymes are calculated as K M = k OF F +kcat k ON . For instance, consider the production rate for rMG using TrMG·A as the template for RNAP: Note that the value for α 2 is identical to the previous first-order model if we take k p2 = Similarly, the production rate for iMG using TiMG·A as the template for RNAP is where f is as defined previous and α 1 is as follows: The degradation rates for iMG and rMG·iMG can be analogously transformed to the corresponding Michaelis-Menten enzyme reactions using an appropriate enzyme saturation term for RNase R: , and z = [rMG · iMG]. Then the dynamics for the system can be rewritten as follows similar to eqs. (3):ẋ Note that the derivations above used the standard Michaelis-Menten assumption that substrates are in excess of enzymes -enzymes used in excess of substrates help reduce the saturation of enzymes. For the rest of the paper, we will assume that the enzymes are in excess and/or the substrate concentrations are suitably lower than their respective Michaelis constants such that f (u) ≈ 1 and g(x, z) ≈ 1 -i.e. the dynamics of the system essentially follow eqs. (3).

Non-dimensional equations
We rescale variables with respect to their steady-state values, such that all normalized variables have values of 1 at steady-state. The timescale is normalized by the degradation rate of species x: τ = β 1 t. For example, equation for z is rewritten as follows: Upon change of variables to non-dimensional ones as above (we dropped theˆ's with a slight abuse of notations), eqs. (3) can be expressed as follows:ẋ where and p 3 = γ1 β1 .

Integral feedback
It is known that any system that perfectly adapts to a step input, including an incoherent feedforward loop can be transformed, possibly with a nonlinear state transformation, into an integral feedback form [2,3]. Accordingly, the set of eqs. (5) can be represented in the integral feedback form through the following choice of the coordinate transformation (x, y, z) → (I, y, z) where I = x − 1 p2 y + p1 p3 z. In particular, the dynamics of I satisfy the integral feedback form as follows:İ = x(y − 1).

Characterization of Elementary Reactions
We observed that the bleaching of free MG dye in the transcription reaction buffer is not negligible for long experiments. The bleaching of MG dye is reported to follow a pseudo-first order kinetics [4]; thus, an exponential curve was fit to the fluorescence data with a decay rate of 0.04/hr ( Figure S2A). However, the presence of MG aptamer is reported to suppress the bleaching of MG dye [5]. Since the MG dye in the system is in great excess as compared to the MG aptamer ([MG dye]=25 µM, [rMG]< 1 µM for all experiments), we applied the same correction for bleaching of MG dye to all fluorescence trajectories. The fluorescence increase upon addition of 1 µM of purified rMG was determined to be 500,000 counts in the presence of MG dye; this was used to convert the fluorescence reading to aptamer concentrations after background correction.
First, the transcription reactions for the two templates, TrMG and TiMG, are characterized through fluorescence measurements on spectrofluorometer. The template TrMG is formed by annealing the two DNA strands TrMG-nt and TrMG-t: the resulting template contains 5 bases of promoter region missing on the template side, and thus transcribes poorly [6] (cf. Figure S1). When the input A that complements the missing promoter region is added, the template TrMG·A can be transcribed well. The transcription of rMG is monitored in real time in the presence of 25 µM MG dye ( Figure S3). Except for the initial burst phase, which showed potential saturation for high template concentration To determine the bleaching rate of free MG dye, 25 µM MG dye was included for MG trajectories (green curve). An exponential decay curve (λ = 0.04/hr) was fit to the data (black curve). The background fluorescence did not significantly change when no MG dye was included in the reaction buffer (blue curve). The measurement conditions were identical to all the other spectrofluorometer measurements (excitation wavelength: 630 nm, emission wavelength: 655 nm, slit widths: 5 nm, integration time: 0.5 s, measurement interval: 1 min, temperature: 33 • C). (B) Using the initial measurement of baseline fluorescence as 0 nM rMG concentration without further baseline correction, the converted rMG concentration apparently became negative because the experimental trajectory went below the initial baseline fluorescence; the predicted steady-state of rMG was 0 nM in this case (cf. blue curve in Figure S9B). The bleaching curve for free MG dye was used to correct for the baseline: the corrected curve does not go below zero for converted rMG concentrations (black dashed line). However, the transcription rate was apparently not constant for the initial part of transcription reaction reminiscent of 'burst phase' in enzyme reaction [7]. (However, the timescale for typical burst phase is short for transcription studies with T7 RNAP [8].) It has been reported that strong secondary structure of RNA transcript reduces the transcription rate by T7 RNAP [9]. One hypothesis we can explore is that the change of transcription rate is dependent on the conformation change of DNA templates: the annealed fully duplex template serves as an efficient template for RNAP at the beginning of transcription; however, because the rMG transcript has a strong secondary structure, the duplex template may take a cruciform or hairpin structure at some rate upon transcription by RNAP which in turn reduces the efficiency of transcription. Thus, we use the following equation to fit the transcription rate for rMG: where B is the magnitude of increased transcription efficiency during the initial burst phase and τ b is the time constant for duration of burst phase. The fitting result showed k p2 = 0.006/s, B = 3.8 and τ b = 2500s for activator concentrations ranging from 20 to 80 nM ( Figure S3A). The highest activator concentration resulted in less than expected burst phase possibly due to saturation of enzyme dynamics. We note that the fit is phenomenological; a mechanistic understanding would require further characterization. Similarly, the template TiMG that codes for the inhibitor of rMG, iMG, contains the same input domain architecture as TrMG; therefore, the same input A can activate the transcription of iMG. The transcription of iMG can be monitored real time using fluorescence change; as iMG is transcribed, iMG binds to rMG and disrupts the binding pocket of rMG for MG dye (cf. Figure S1), which decreases the fluorescence ( Figure S4). For the case of iMG production, the 'burst phase' was not apparently observed possibly because the iMG transcript does not have a significant secondary structure.   Figure 3A of the main text. (B) The steady-state slopes of curves from (A) were determined as follows: the average slope were calculated between 2 hr and 3 hr points for all input concentrations to exclude the initial burst phase. The average slope for the initial 30 min prior to the addition of inputs was taken as the slope for no input. (C) The steady-state slopes in (B) were normalized by the amount of activator, i.e., the amount of active template TrMG·A. The normalized slopes (k p2 ) were approximately 0.5/min for activator concentrations ranging from 20 to 150 nM. Next, the binding reaction for the two RNA signals, rMG and iMG, is characterized through fluorescence measurements. Initially, 10 nM rMG and 25 µM MG dye are incubated in the cuvette to establish baseline fluorescence.
Then, 10 nM iMG is added to the cuvette and the decrease of fluorescence is monitored as rMG binds to iMG to form rMG·iMG complex. (Using only 10 nM rMG made the fluorescence signal a bit noisy due to overall low signal level; however, using high concentrations of rMG and iMG would make the binding reaction proceed too quickly to yield suitable number of data points.) Because the two single-stranded species have the same concentration, the following equations can be used to find k: and thus, d dt By integrating the above equation, we find where [rMG 0 ] is the initial concentration of rMG. Figure S5C illustrates the linear fit using the above relations for duplicate experiments. The fitted binding rate constant k is 4.2 × 10 4 /M/s, consistent with typical toehold-mediated branch migration reactions [10,11]. The resulting fit can be used to plot fit curves as shown in Figure S5B. Finally, we characterize the degradation of RNA signals catalyzed by RNase R. Due to the binding requirements of RNase R, RNA sequences with less than 4 nt of single-strand regions exposed at the 3 end would be degraded poorly by RNase R [1]. To minimize the degradation rate of rMG by RNase R (β 2 ), the initial design of rMG contained no available single-stranded region at its 3 end. However, we observed that having the 3 end perfectly paired to the 5 domain of rMG caused non-specific extension of RNA transcripts for rMG [12]. Therefore, we placed a single unpaired base at the 3 end of rMG (cf. Figure S1). (We typically place three unpaired bases at the 3 end of RNA signals used in transcriptional circuits [13,6].) The degradation rate of rMG by RNase R was monitored by fluorescence and gel. As shown in Figure S6, the degradation of rMG alone by RNase R was negligible (β 2 0).
On the other hand, the other RNA signal iMG is designed to have no significant secondary structure, presumably providing a long single-stranded region at its 3 end (cf. Figure S1). The degradation of iMG by RNase R was fast: 1 µM of iMG was completely degraded by 75 nM RNase R in less than 30 min as determined by gel (data not shown). The gel analysis did not lead to quantitative estimate of β 1 , but it is safe to conclude that β 1 β 2 . The remaining question is the degradation pathway of rMG·iMG complex. Figure S7A illustrates the potential degradation pathways for the rMG·iMG complex: either iMG or rMG within the complex can be processed first by RNase R and the resulting rMG or iMG will be further degraded by RNase R (or participate in a binding reaction). Gel analysis of preformed 1 µM rMG·iMG complex incubated with 75 nM RNase R showed that both rMG and iMG were almost completely degraded within 30 min. Because stoichiometric amount of rMG and iMG were used for gel analysis, if iMG were to be degraded first (γ 2 > γ 1 ), rMG would be left over which cannot be easily degraded (β 2 0). Thus, we conclude that the degradation of iMG within rMG·iMG complex is very slow (γ 2 0). To obtain estimates for parameter values, we performed a spectrofluorometer experiment where a substoichiometric amount of iMG was used with respect to rMG. Specifically, 100 nM of iMG was allowed to bind to 1 µM of rMG in the presence of MG dye. Upon addition of RNase R, the fluorescence signal from rMG decreased quickly to about 750 nM rMG. This suggests that the iMG molecule released upon degradation of rMG within the rMG·iMG complex can bind to free rMG molecules such that it catalyzes the degradation of multiple rMG molecules until the iMG within the system is exhausted. If the iMG within the rMG·iMG complex were to be degraded first (γ 2 > γ 1 ), the resulting rMG molecule will increase fluorescence. This was not observed in the spectrofluorometer experiment. Thus, we conclude that the design strategy that limits exposure of 3 end of iMG upon rMG·iMG complex formation while releasing the hidden 3 end of rMG results in fast degradation of rMG within rMG·iMG complex (γ 1 0 and γ 2 0). Unfortunately, it is difficult to separately characterize β 1 and γ 1 since both rMG·iMG complex and iMG molecule have similarly low fluorescence. Recognizing that the addition of RNase R may affect the hybridization reaction between the two RNA species possibly due to its interaction at the 3 ends of RNA molecules, we optimized the parameters for k, β 1 , and γ 1 simultaneously. The fitting result showed k = 3.5 × 10 4 /M/s, β 1 = 0.0176/s, and γ 1 = 0.0095/s ( Figure S7).  Figure 3C of the main text.     Table S2. For the simple model, the binding of input A to switch templates are instantaneous and hence are not modeled as separate states. The simulation results qualitatively captured the features of experimental results for RNAP variation (A), RNase R variation (B), and template ratio variation (C). (D-F) The simple model (eqs. 3) was used with optimized parameters listed in Table S2. These graphs are shown in Figure 4 of the main text. (G-I) The mathematical model for the transcriptional circuit (eqs. 1) was used for simulation where the input A and switch templates are modeled as separate species. The optimized k + was at the maximum allowed value (Table S2), and consequently, there were little differences between the simulation results of these two models. The experimental data are plotted as dots; the simulation results are plotted as lines.  Figure S10: Gel analysis of the transcription reaction. The transcription reaction for rMG and iMG was performed by adding 86 nM of RNAP to 50 nM of TrMG, 150 nM of TiMG, and 10 nM of A. The transcription products were analyzed using denaturing gel. Left most lane contains 10 bp ladder with 10 nt and 100 nt marked; other lanes contain samples incubated for 30 min increments from left to right. The DNA species and RNA transcripts were marked by black arrows with corresponding names: TrMG-t and TiMG-nt differ by a single nt in length and were not resolved separately; A was not clearly identified due to its low concentration. Major bands for unidentified products -possible truncation and non-specific extension products -were marked by grey arrows.