Incoherent merger network for robust ratiometric gene expression response

Abstract A ratiometric response gives an output that is proportional to the ratio between the magnitudes of two inputs. Ratio computation has been observed in nature and is also needed in the development of smart probiotics and organoids. Here, we achieve ratiometric gene expression response in bacteria Escherichia coli with the incoherent merger network. In this network, one input molecule activates expression of the output protein while the other molecule activates an intermediate protein that enhances the output’s degradation. When degradation rate is first order and faster than dilution, the output responds linearly to the ratio between the input molecules’ levels over a wide range with R2 close to 1. Response sensitivity can be quantitatively tuned by varying the output’s translation rate. Furthermore, ratiometric responses are robust to global perturbations in cellular components that influence gene expression because such perturbations affect the output through an incoherent feedforward loop. This work demonstrates a new molecular signal processing mechanism for multiplexed sense-and-respond circuits that are robust to intra-cellular context.


INTRODUCTION
Ratiometric computation is common in biological systems since it allows cells to respond to relative, as opposed to absolute, changes in input stimuli ( Figure 1). For example, in the yeast sugar utilization network, the ratio between galactose and glucose controls the induction of galactose metabolic genes (1,2). In mammalian cells, the BMP signaling pathway comprises receptors that perform ratio sensing of multiple ligand concentrations, which ultimately leads to ratiometric responses of SMAD target genes (3,4). Finally, the ATP/ADP ratio drives the process of ATP hydrolysis and hence controls many reactions in the cell (5). Ratiometric biomolecule signatures have also been linked to cognitive performance in humans (6)(7)(8)(9). In particular, the ratio between norepinephrine and cortisol was shown to discriminate post-traumatic stress disorder (PTSD) patients from others (10,11) and the ratio between cortisol and DHEAS is predictive of stress level (12)(13)(14). For this reason, future smart probiotics could couple ratiometric biomarker sensors with genetic response systems that produce drugs to compensate for biomarker imbalances (15)(16)(17)(18)(19). Finally, in programmable organoids and directed cell differentiation, proper tissue composition requires the maintenance of specific ratios between different cell types' population sizes, and hence the ability of computing and responding to ratios is required (20,21). Similarly, microbial consortia also rely on the ability to compute ratios in order to maintain desired relative abundances of species in the consortium (22).
Although natural ratiometric computation has been identified upstream of gene expression, notably through competitive receptor-ligand or promoter-regulator binding, it ultimately affects cellular functions through gene expression responses (1)(2)(3)(4). These responses, in turn, have been shown to be robust to variations in the abundance of select cellular components in some natural networks (1). On the other hand, gene expression is sensitive to variations in the levels of cellular resources, whose availability changes when genes become dynamically activated and repressed in the cell (23)(24)(25)(26). Specifically, when a gene is activated, such as in response to one signaling molecule, any other gene's expression rate, possibly responding to a different molecule, will decrease due to reduction of translation resources in bacteria and of transcription resources in mammalian cells (23,27). As a result, although sensing through receptorligand or promoter-regulator binding may not be affected by changes in the availability of transcriptional and translational resources, the gene expression responses to these sensing events are influenced by them. Figure 1. Robust ratiometric response. (A) A ratiometric response takes two signaling molecules as inputs (X and Y) and provides a protein P Y as an output. The concentration of the output protein is proportional to the ratio between the concentrations of X and Y with proportionality constant c. R is a perturbation acting on the system that cannot be controlled and is unknown. If the output is independent of the perturbation R, the ratiometric response is robust to R. (B) Pictorial representation of the ratiometric input/output response. The output responds linearly to changes in the ratio Y/X.
In this paper, we introduce the incoherent merger network, a design that performs ratiometric sensing on the gene expression output response directly ( Figure 2A). In the incoherent merger network, one of the two input molecules (Y) activates the expression of the output protein (P Y ) while the other (X) activates an intermediate protein (P X ) that enhances the degradation of the output. Because global variations in cellular resources (R) implicated in gene expression equally affect both the output (P Y ) and intermediate protein (P X ), there is an incoherent feedforward loop (iFFL) formed by resource R, protein P X and protein P Y , wherein R enhances production of both proteins while P X degrades P Y (Figure 2A). This type of topology, the incoherent type I feedforward loop (28), is known to enable perfect adaptation of the output (P Y ) to changes in the input (R) (29)(30)(31)(32). As a consequence, the ratiometric output response can become robust to global changes in the cellular resource R.
We demonstrate this design with a bacterial genetic circuit where the intermediate molecule is a protease that targets the output protein for degradation. We develop a mathematical model of this system to determine the regime of biochemical parameters under which robust ratiometric response can be achieved. We experimentally demonstrate ratiometric computation and quantitative tuning of the response sensitivity based on model-driven part choices. We thus show that the sensitivity does not change when we vary the level of cellular resources through activation of a competitor gene. These results demonstrate a new mechanism to achieve ratio response, which is also robust to variability in global cellular components. More generally, our results can be used to engineer multiplexed genetic sense-and-respond systems that are robust to variable cellular context.

Strain and growth medium
Bacterial strain Escherichia coli Marionette DH10B (Addgene, #108251) was used to construct and characterize genetic circuits. The growth medium used for construction was LB broth Lennox. The growth medium used for characterization was M9 medium supplemented with 0.4% glucose, 0.2% casamino acids and 1 mM thiamine hydrochloride. Appropriate antibiotics were added according to the selection marker of the plasmid. The final concentration of ampicillin, kanamycin and chloramphenicol are 100, 25 and 12.5 g/ml, respectively.

Genetic circuit construction
The genetic circuit construction was based on Gibson assembly (33). DNA fragments to be assembled were amplified by PCR using Phusion High-Fidelity PCR Master Mix with GC Buffer (NEB, M0532S), purified with gel electrophoresis and Zymoclean Gel DNA Recovery Kit (Zymo Research, D4002), quantified with the nanophotometer (Implen, P330), and assembled with Gibson assembly protocol using NEBuilder HiFi DNA Assembly Master Mix (NEB, E2621S). Assembled DNA was transformed into competent cells prepared by the CCMB80 buffer (TekNova, C3132). Plasmid DNA was prepared by the plasmid miniprep kit (Zymo Research, D4015). The list of primers and constructs is in Supplementary Figures S1-S3 and Supplementary Table S1.

Microplate photometer
Overnight culture was prepared by inoculating a −80 • C glycerol stock in 1000 l M9 (+kanamycin) media in a 1.5 ml microcentrifuge tube and grown at 30 • C, 220 rpm in a horizontal orbiting shaker for 12 h. Overnight culture was first diluted to an initial optical density at 600 nm (OD 600 nm ) of 0.001 in 200 l growth medium per well in a 96-well plate (Falcon, 351172) and grown for 1.5 h to ensure exponential growth before induction. The 96-well plate was incubated at 30 • C in Tecan infinite M Nano+ microplate reader in static condition and was shaken at the 'fast' speed for 3 s right before taking OD and fluorescence measurements. The sampling interval was 5 min. Excitation and emission wavelengths to monitor GFP fluorescence were 488 nm and 548 nm, respectively. To ensure enough time to reach steady state GFP/OD signal while the cells were in exponential growth, the cell culture was diluted with fresh growth medium to OD 600nm of 0.005 when OD 600 nm approached 0.135 at the end of each batch. Three batches were conducted for a total experiment time of up to 10 h until GFP/OD reached steady state. The steady state GFP/OD value was computed from the second batch of each experiment by using the data point with an OD value most close Nucleic Acids Research, 2023, Vol. 51, No. 6 2965 Figure 2. The incoherent merger network and its genetic circuit implementation. (A) Merger network motif with inputs X and Y and disturbance R, which affects both P X and P Y with the same sign. Here, arrow '→' denotes upregulation and ' ' represents downregulation. The loop formed by R, P X and P Y is an incoherent feedforward loop. (B) Genetic implementation of the network motif in (A). Here, X is the first input, which is a negative inducer that binds to repressor protein R X and prevents it from binding to DNA. Molecule m X is the mRNA of P X . Signaling molecule Y is the second input, which is a positive inducer that binds to activator protein A Y and allows DNA to be transcribed. Molecule m Y is the mRNA of P Y . The half disks in front of the gene coding region represent RBS sequences. In panel (A), R is any cellular resource that is equally required for the expression of P X and P Y , including transcriptional and translational resources. In the specific genetic implementation in (b), R is a translational resource, such as the ribosome. (C) Green colored plot is obtained from the reduced model in Supplementary Equation (S32) with parameters in Supplementary Table S2 where assumptions (A0) -(A2) are satisfied. Blue and red colored plots are obtained from the full model in Supplementary Equation (S10) with parameters in Supplementary Table S2, in which perturbed R is 50% of nominal R. 50% change in R leads to only 5% change in the output, so the system attenuates the change. (D) Relative % error (=|P Y, Nomi nal R − P Y, Perturbed R |/P Y, Nomi nal R × 100 (%)) of incoherent merger network and broken merging, in which perturbed R is 50% of nominal R. In the broken merging, P X does not degrade P Y .

Reporting summary
Further information on experimental design is available in the Nature Research Reporting Summary linked to this article.

The incoherent merger network requirements for robust ratiometric response
We first demonstrate the mechanism for robust ratiometric response by the incoherent merger network shown in Figure 2A by using a simple ordinary differential equation (ODE) model that describes the rate of change of the species concentration. For any species S, we denote in italics S its concentration. Here, X is a signaling molecule that enables the production of protein P X and this production requires the cellular resource R. Resource R can model any transcriptional or translational resource, such as RNAP or ribosomes, and we leave it unspecified at this time. Similarly, Y is a signaling molecule that enables the production of the output protein P Y , which also requires the same resource R for its production. Assuming that the production of P X and of P Y can be written as one-step enzymatic reactions, we have: where k X and k Y are production rates constants. The interaction P X P Y in Figure 2(A) is implemented by having P X enhance the degradation of P Y . Thus, we choose P X as a protease, which binds to its target on protein P Y , forming a complex C, which in turn leads to degradation of P Y (31): We also consider dilution, P X , where ␥ is the dilution rate constant. With two assumptions (A1) P Y K and (A2) ␥ P XT (k + ␥ )/K, we obtain that the steady state value of the output protein P Y is given by where K is the Michaelis-Menten constant of the protease reaction (Supplementary Equation S3). Therefore, the steady state value of the output protein P Y is proportional to the ratio Y/X and this ratio is independent of the resource level R. In summary, the key design requirements to achieve robust ratiometric sensing are • (A0) The production rates of P X and P Y are linear with the levels of the signaling molecules X and Y, respectively, and with the resource level R (satisfied in this model by reactions (1)); • (A1) The protease enzymatic reaction is in the first order regime, that is, P Y K; • (A2) Dilution rate is negligible with respect to degradation rate, that is, ␥ P XT (k + ␥ )/K.
Regarding (A0), the linearity of the production rates of P X and P Y with the levels of X and Y, respectively, is needed for ratiometric computation, while the linearity of the production rates with the level of cellular resource R is needed to make the level of P Y independent of R. Assumptions (A1) and (A2) can be both satisfied by taking a sufficiently strong protease (large catalytic constant k). Finally, we can tune the sensitivity c by varying the production rate constant k Y of protein P Y , which can be easily accomplished by varying the ribosome binding site (RBS) strength for P Y .
We consider the specific genetic implementation of the incoherent merger network shown in Figure 2B. In this implementation, X and Y are sensed through transcriptional sensors, wherein X and Y each bind to a transcriptional regulator to modulate transcription of the regulator's target promoter. Transcriptional sensors have been developed for a variety of signaling molecules and therefore there is a large library to select from (34)(35)(36)(37)(38)(39). Although we focus our analysis on transcriptional sensors, translational sensors are also an option (40,41). For the sake of illustration, we choose X and Y as a negative and as a positive inducer, respectively. Specifically, a negative inducer is a molecule that binds to a transcriptional repressor and prevents it from binding to DNA and hence to obstruct RNAP binding. A positive inducer, instead, is a small molecule that binds to a transcriptional activator and allows this activator to bind to DNA/RNAP to start transcription (31). We also specialize resource R to a translational resource, such as ribosomes. In fact, it is well known that in bacterial synthetic genetic circuits translational resources are responsible for the observed interference among independently regulated genes (23,24). Our detailed mathematical model (we refer to it as the full model, Supplementary Equation S10) of the genetic circuit in Figure 2B reveals parameter conditions under which assumption (A0) can be satisfied (Supplementary Equations S12-S27). Specifically, the linearity of the production rates of P X and P Y in X and Y, respectively, is approximately satisfied within a range of X and Y concentrations. This range can be widened by suitable choices of biochemical parameters of the transcriptional sensors, such as the regulators' total concentration and target DNA copy number (Supplementary Equations S12-S14, S17-S19), and can be empirically determined by inspecting the experimental dose response curves of the sensors as we do in Section 2.2. The linearity in R of the production rates of P X and P Y is ensured when both the repressor's and activator's mRNAs are saturated by ribosomes while the mRNAs of P X and P Y are not. Suitable choices of the RBS strengths achieve these requirements, that is, sufficiently strong RBS for the repressor and activator proteins and weaker for the output protein P Y and protease P X (Supplementary Equations S23-S27). Assumptions (A1) and (A2) can be satisfied by taking a sufficiently strong protease, which is equivalent to large catalytic constant k. The slope c can also be quantitatively tuned by varying the strength of the RBS of the output protein P Y (Supplementary Equation S32). Satisfying the assumptions (A0) -(A2) (Supplementary Table S2) confirm that ratiometric sensing is robust to changes in the level of resource R, and that the steady state value of the output protein P Y from the full model can be approximated as Supplementary Equation (S32) ( Figure 2C). This equation has the same form as (3) and we refer to it as the reduced model. Also, degradation of P Y by P X is critical for the circuit's output is robust to changes in the level of resource R ( Figure 2D). For both Figure 2C and D, R perturbed is 50% of R nominal, which is reasonable assumption according to (23).

The incoherent merger network achieves tunable ratio computation
We chose two transcriptional sensors from the Marionette strain (34) and a protein degradation tag from the library developed in (42). For demonstration, we chose transcriptional sensors that satisfy (A0) in a large range of X and Y concentrations (Supplementary Figure S7A). In general, for a given transcriptional sensor, there will be an acceptable linear range of the signaling molecule level. We can make this range larger by suitably changing key biochemical parameters, such as the total regulator (activator or repressor) level and the copy number of the regulator DNA target (Supplementary Note 1.1). We thus chose IPTG as X and Sal as Y and found a linear range from 0 to 50 M for the Sal sensor and from 50 to 400 M for the IPTG sensor (Supplementary Figure 7B). As P X , we chose the mf-Lon protease and picked the pdt#3 tag, which has the strongest protein degradation as experimentally determined in (42). Finally, we chose superfold GFP (sfGFP) as P Y  Table S1 and Supplementary Figure S1). Both LacI and NahR regulators are endogenously expressed from the host E. coli Marionette strain (34). ( Figure 3A). We transformed the constructs into the host cell E. coli Marionette strain in which the required regulators, LacI (R X ) and NahR (A Y ), are chromosomally integrated (34). We characterized the incoherent merger network response in the indicated linear range of the respective IPTG (X) and Sal (Y) concentrations ( Figures 3B and C). The linearity between P Y ([sfGFP]) and Y ([Sal]) is apparent from Figure 3B, and the required P Y ∝1/X trend between P Y and X ([IPTG]) can be appreciated by comparing the dotted lines, that is, equation (3) which has the trend 1/X, to the solid lines (experimental data) in Figure 3C. In particular, for both induction curves, experimental results match well the reduced model. A scatter plot of P Y versus the ratio Y/X shows a linear relationship with an R 2 value of 0.98347, demonstrating the expected performance of the ratiometric sensor ( Figure 3D). To further demonstrate that the output responds to changes in the ratio but not to changes in X or Y alone when the ratio is constant, we selected different pairs of X and Y that have the same ratio and plotted bar charts of the corresponding outputs ( Figure 3E). These data confirm that the output remains approximately constant when the ratio of X and Y stays the same.
It is well known that ratio computation is highly sensitive to noise in X especially when X is small (43). To determine how randomness due to intrinsic noise can affect ratiometric sensing, we conducted flow cytometry experiments (Supplementary Figures S4 and S5) and obtained scatter plots of P Y versus the ratio Y/X (Supplementary Figure S6), which also shows a linear relationship between P Y and Y/X with an R 2 value of 0.925. Violin plots of the output GFP fluorescence for decreasing values of IPTG show that in the tested inducer concentration ranges, the coefficient of variation (CV) is below 0.54 and remains approximately constant through the range of IPTG (X) concentrations ( Figure 3F).
To show the tunability of the response, we created a library of three constructs, where the strength of the GFP RBS measured by the translation initiation rate (TIR) using the RBS Calculator (44) is progressively decreased from 8875, to 4575, to 1770 (Supplementary Table S1, Supplementary Figures S1 and S2). According to (44), the TIR is proportional to parameter 1/K 6 = a 6 /(b 6 + k 6 + ␥ ) and the slope c is proportional to 1/K 6 (Supplementary Equation S16). Therefore, we expect that the slope should progressively decrease from 220,428, observed in Figure 3D, to 113 629 (=220 428 × 4575/8875), to 43 961 (=220 428 × 1770/8875), which is validated by the experimental data ( Figure 3G). Specifically, when the TIR was changed from 8875 to 4575, the slope of the best linear fit of the experimental data decreased from 220 428 to 104 594, which has only an 8% error gap compared to the reduced model-predicted slope, 113,629. Similarly, when the TIR was changed from 8875 to 1770, the slope of the best linear fit of the experimental data decreased from 220 428 to 44 846, which has only a 2% error gap compared to the reduced model-predicted slope, 43 961. These data indicate that the sensitivity of our ratiometric sensor is quantitatively tunable through modulation of the strength of the RBS of the output protein.
In order to demonstrate that ratio computation hinges on the incoherent merging of the X and Y signals onto P Y , we created a control construct in which the protease tag was removed from the GFP protein, thereby removing the enhanced degradation of GFP by the protease (Figure 4A and Supplementary Figure S3). In this case, we still expect that the level of GFP decreases as X is increased, due to sequestration of ribosomes by P X (23), but the model modifies to (see Supplementary Note 1.2): where c 1 and c 2 are defined in Supplementary Equation (S34) and R tot is the total concentration of ribosomes.
We also refer to it as the reduced model for the control construct, Figure 4A. This is not a ratio computation as confirmed by the experimental data of Figure 4. Specifically, for small Y, P Y linearly increases as Y increases, but it saturates for large Y ( Figure 4B). Dashed lines in Figure 4B and C show the reduced model given by (4) with parameter values in Supplementary Table S2. For both cases, experimental results match the theoretical predictions. Finally, different from Figure 3E, the output responds to changes in X or Y even if the ratio is constant since different pairs of X and Y that have the same ratio give different outputs ( Figure 4E). This confirms that, when the signal merging is broken, ratiometric computation is disrupted.

The incoherent merger network response is robust to perturbations in cellular context
Sophisticated genetic circuits typically include multiple sensors, logic computation/signal processing, and actuation (45,46). Because all of these components share cellular resources required for gene expression, independent circuit components can interfere with each other's functionality (23,24). Here, we investigate to what extent the incoherent merger network response is robust to depletion of gene expression resources due to activation of other sensors or circuit modules in the cell. We achieve this by adding to the genetic system of Figure 3A a resource competitor constituted of an inducible RFP gene ( Figure 5A). Referring to the incoherent merger network diagram of Figure 2A, reduced resource R decreases the production rate of both P X and P Y . Since the concentration of P X decreases, the degradation rate of P Y also decreases. If the network is in the proper parameter regime, this compensation can cancel the effect of a decrease in R on P Y . We conducted experiments to compare the steady state value of the output GFP protein of the circuit in show that the competitor and no competitor plots almost perfectly overlap (compare the cyan colored plot with the red colored plot). Both of these plots also overlap with the same plot obtained for the original circuit in Figure 3A (blue colored plot), in which the RFP gene expression cassette is absent. By contrast, when the degradation tag on the GFP gene is removed ('broken merging' in Figure 5A), not only ratiometric computation is lost, consistent with what observed in Figure 4D, but also the GFP output drops when the RFP gene is activated (compare the cyan and red colored plots in Figure 5C). These data demonstrate that the presence of an incoherent feedforward loop encompassing the resource results in an output that is robust to changes in cellular resources and, hence, also in robust ratiomteric response. These data are also consistent with our model predictions, according to Supplementary Equations (S32), (S34) and (S49).
To further show that this robustness to changes in cellular resources holds for every combination of the inputs X (IPTG) and Y (SAL), we also report a heat map in Figure 5D and E showing the fold-change of the steady state of the output GFP level when RFP is expressed for the incoherent merger network and for the case when GFP has no degradation tag (broken merging). While for the incoherent merger network the fold-change never goes above 1.2, for the system with broken merging, the fold-change reaches 2fold. These data show that ratiometric computation by the incoherent merger network is robust to changes in the level of cellular resources required for gene expression, which is not the case for the broken merging system.

DISCUSSION
We have introduced the incoherent merger network to achieve robust ratiometric gene expression response. The output protein of the network responds to the ratio between the levels of two input molecules, but not to absolute changes that keep the ratio constant, and the response sensitivity can be quantitatively tuned (Figures 3 and 4). The response is further robust to changes in global cellular components that affect gene expression, such as due to dynamic activation or repression of other genes ( Figure 5).
A well known molecular mechanisms for ratiometric sensing identified in natural systems is competitive binding of two input molecules to an integrator molecule, such as ligands binding to a receptor or regulators binding to a promoter (1,3). This mechanism was leveraged to design a synthetic reporter that responds to the ratio ATP/ADP with a Michaelis-Menten function (5) and was later implemented with single guide RNAs (sgRNAs) for integrating the signals from two competing sgRNAs into one reporter output (47). In these designs, the molecule that integrates the signals (a promoter) is different from the molecule that responds to the signal ratio (the output protein), therefore the output response will vary as the level of global cellular resources implicated in gene expression change. In the incoherent merger network design, instead, the molecule that integrates the signals is the gene expression output protein itself and integration occurs through protein-protein interaction via modulation of protein activity. This ensures that the output's response is robust to any global perturbations in cellular components implicated in gene expression. Others have proposed analog circuits to compute logarithmic, as opposed to linear, ratios (48), but, just as noted above, these also do not integrate the signal on the response directly (gene expression), and hence the output response is not robust to changes in global cellular resources.
Although the incoherent merger network motif has not been linked before to ratiometric computation, it is found in all those incoherent feedforward loops (iFFL) of type I (28) where repression occurs through modulation of protein activity via protein-protein interaction. Many such iFFLs have been reported in natural networks and implicated in the control of cellular functions, such necroptotic cell death (49), initiation of mitotic exit (50), cell migration (51) and virulence in Salmonella,perez2021incoherent. In the necroptotic cell death iFFL, the output protein RIPK3 leads to plasma membrane rupture and cell death, and its activity is modulated by the A20 protein via protein-protein binding (49). In turn, RIPK1 is an upstream regulator of RIPK3 and the A20 protein has many upstream regulators (53). These A20 regulators and RIPK1 are inputs to the incoherent merger network and hence RIPK3 activity may respond to relative, but not to absolute, changes in the levels of these inputs. Similarly, in the iFFL implicated in cell cycle regulation, the output is the mitotic exit protein Dbf2, which initiates mitotic exit, and the Cdk1 kinase induces Dbf2 enhanced degradation by phosphorylation (50). Cdk1's ac-tivity is, in turn, modulated by the CDC25C phosphatase and Dbf2 has several upstream regulators, including Mob1 and CDC15 (54,55). Since the CDC25C phosphatase and Mob1/CDC15 enter the incoherent merger network as inputs, it is possible that Dbf2's activity responds to relative changes between CDC25C and Mob1/CDC15 levels but not as much to absolute variations. Finally, in the iFFL controlling virulence in Salmonella, the HilD protein controls the fine balance between growth cost and virulence benefit and its activity is regulated by the HilE protein through protein-protein interaction (52). Both HilD and HilE have many known regulators that enter the incoherent merger network as inputs. It is therefore possible that the balance between growth cost and virulence benefit is dictated by the relative concentrations among these regulators. Therefore, all the above systems could be re-analyzed in light of the ratiometric response that the incoherent merger network enables. This could shed light into previously unappreciated functional roles that the above iFFLs cover in their biological context.
IFFLs have been engineered in mammalian cells to achieve constant gene expression despite that transcriptional and translational resources vary in the cell (27,56). In particular, it was shown that such an iFFL design can stabilize the expression level of a gene of interest across cell types, thereby possibly enabling constant expression as cells undergo cell type changes, such as in directed differentiation (27). In the incoherent merger network, global changes in cellular components implicated in gene expression affect the response through an iFFL. This is the first realization of an iFFL that enables robustness of gene expression response to changes in intra-cellular components in bacterial cells. Although our genetic implementation of the incoherent merger network uses a protease as the intermediate species, in other implementations, the intermediate species could be a phosphatase or a scaffold protein that reduces the activity of the output protein.
The incoherent merger network can be widely used in all sensing applications where ratio response is required between any two small molecules for which transcriptional or translational sensors exist. Examples include future smart probiotics that sense stress level and release drugs to improve human performance (12)(13)(14)(15)(16) and programmable organoids where cells need to differentiate into different types while maintaining a desired population size ratio for proper tissue formation (20,21,(57)(58)(59)(60)(61)(62). Because in these applications, ratiometric computation operates in orchestration with other genetically encoded components, robustness to changes in cellular resources is critical for proper function. Our incoherent merger network design can therefore be used to engineer multiplexed genetic sense-and-respond systems that are robust to variable cellular context.

DATA AVAILABILITY
Fluorescence data generated or analyzed during this study are included in the paper and its Supplementary Information files. Essential DNA sequences are provided in Supplementary Figure S1. A reporting summary for this Article is available as a Supplementary Information file. Any other relevant data are available from the authors upon reasonable request.

SUPPLEMENTARY DATA
Supplementary Data are available at NAR Online.