Negative autoregulation matches production and demand in synthetic transcriptional networks

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1 Negative autoregulati matches producti and demand in synthetic transcriptial networks Experimental methods, data processing, and modeling Elisa Franco 1,*, Giulia Giordano, Per-Ola Forsberg 3, and Richard M. Murray 4 1 Mechanical Engineering, University of California at Riverside, Riverside, CA 951 Department of Mathematics and Computer Science, University of Udine, Italy 3 Kristianstad Central Hospital, 983 Tollarp, Sweden 4 Engineering and Applied Science, California Institute of Technology, Pasadena, CA 9115 * Correspding author 1

2 Ctents 1 Experimental implementati of a two-gene flux matching system based negative autoregulati: Materials and methods Reactis and domains design Oligucleotide sequences DNA oligucleotides and enzymes Transcripti protocol Data acquisiti and processing Characterizati assays Genelets in isolati Intercnected genelets Flux adaptati Data fitting Modeling and numerical analysis: two-gene flux matching system 1.1 Simple model system: derivati of nullclines and rate matching cditis Differential equatis modeling the experimental implementati Modeling and numerical analysis: Scalability of the negative feedback scheme for flux regulati 3.1 Simple model system Single product topology Handshake and neighbor topologies Parameters Performance overview of the different topologies as a functi of key parameters Positive feedback architecture for a two-gene system. Modeling and a viable experimental implementati Simple model system: derivati of nullclines and rate matching cditis A possible experimental implementati of a two-gene positive feedback scheme Modeling Numerical scalability analysis of our simplified positive feedback scheme model for flux regulati Single product topology Handshake and neighbor topologies Parameters Performance overview of the different topologies as a functi of key parameters 35

3 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 3 1 Experimental implementati of a two-gene flux matching system based negative autoregulati: Materials and methods 1.1 Reactis and domains design A graphical sketch of the domain-level design for the self-repressi intercnecti is shown in Figure S1 A. The RNA outputs of each genelet are designed so that: 1) Each RNA output has a domain complementary to its activator strand. ) The two RNA species are also complementary. These specificatis introduce a binding domain between T i and R j, which is csidered another off state, as shown in Figure S1 B. Such a complex is a substrate for RNase H and the RNA strand is degraded by the enzyme, releasing the genelet activati domain. We assume that the transcripti efficiency of an RNA-DNA promoter complex is very low. This hypothesis was not experimentally challenged for this specific system; however data shown in Franco et al. [11], Supplementary Informati, show that this assumpti is valid for other genelets with the same promoter domain. The design of a self-inhibiting genelet was first characterized in Kim [7]. The circuit design proposed here, with two-domain RNA transcripts, was originally presented in Franco et al. [8]. DNA strands were designed by thermodynamic analysis using the Winfree lab DNA design toolbox for MATLAB, Nupack Zadeh et al. [11] and Mfold Zuker and Stiegler [1981]. The strands were optimized to yield free energy gains favoring the desired reactis, and to avoid unwanted secdary structures and crosstalk. Further cstraints the length and structure of the strands, which can affect the transcripti efficiency and fidelity, were taken into account referring to Kim [7], Chapter Oligucleotide sequences Due to technical cstraints of the supplier IDT DNA, nt and nt were shortened with respect to the nominal design to have a length of 15 bases. The strands used in the experiments are those denoted below as Short. These modificatis did not alter the regulatory domains of the transcripts R 1 and R. Also the full length of the main transcripti products was not affected, as verified by gel electrophoresis in Figure S B. -nt Full (134-mer) 5 -CTA ATG AAC TAC TAC TAC ACA CTA ATA CGA CTC ACT ATA GGG AGA AAC AAG AAC GAC ACT AAT GAA CTA CTA CTA CAC ACC AAC CAC AAC TTT ACC TTA ACC TTA CTT ACC ACG GCA GCT GAC AAA GTC AGA AA-3 (not synthesized) -nt Short (15-mer) 5 -Tamra-CT AAT GAA CTA CTA CTA CAC ACT AAT ACG ACT CAC TAT AGG GAG AAA CAA GAA CGA CAC TAA TGA ACT ACT ACT ACA CAC CAA CCA CAA CTT TAC CTT AAC CTT ACT TAC CAC GGC AGC TGA CAA-3 -t (17-mer) 5 -TTT CTG ACT TTG TCA GCT GCC GTG GTA AGT AAG GTT AAG GTA AAG TTG TGG TTG GTG TGT AGT AGT AGT TCA TTA GTG TCG TTC TTG TTT CTC CCT ATA GTG AGT CG-3 A 1 (35-mer) 5 -TAT TAG TGT GTA GTA GTA GTT CAT TAG TGT CGT TC-3

4 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 4 A t1 a1 a t R1 P t t a a a1 t1 a1 t1 t a a1 t1 R a1 RNAP T1 a RNAP T t1 a1 T1 off t a T off t1 t1 a1 a1 R1 A1 t1 a1 A1 t a A t t a a R A RNaseH RNaseH B RNaseH t1 T1 off a1 a a1 a1 T1 off t a R a1 t1 RNaseH t a a T off T off R1 t1 a1 a t Figure S1: General reacti scheme representing a transcriptial circuit implementati of the two-gene negative feedback scheme for flux matching. Complementary domains have the same color. Promoters are in dark gray, terminator hairpin sequences in light gray. The RNA output of each genelet is designed to be complementary to its correspding activator strand. The two RNA species are also complementary. A. Desired self-inhibiti loops. B. Undesired cross-hybridizati and RNase H mediated degradati of the RNA-template complexes.

5 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 5 -nt Full (16-mer) 5 -GGT TAA GGT AAA GTT GTG GTT GTA ATA CGA CTC ACT ATA GGG AGA AAC AAG TAA GTA AGG TTA AGG TAA AGT TGT GGT TGG TGT GTA GTA GTA GTT CAT TAG TGT CGT TCC TGA CAA AGT CAG AAA-3 (not synthesized) -nt Short (16-mer) 5 -TexasRed-GG TTA AGG TAA AGT TGT GGT TGT AAT ACG ACT CAC TAT AGG GAG AAA CAA GTA AGT AAG GTT AAG GTA AAG TTG TGG TTG GTG TGT AGT AGT AGT TCA TTA GTG TCG TTC CTG ACA AAG TCA GAA-3 -t (99-mer) 5 -TTT CTG ACT TTG TCA GGA ACG ACA CTA ATG AAC TAC TAC TAC ACA CCA ACC ACA ACT TTA CCT TAA CCT TAC TTA CTT GTT TCT CCC TAT AGT GAG TCG-3 A (35-mer) 5 -TAT TAC AAC CAC AAC TTT ACC TTA ACC TTA CTT AC-3 R 1 (95-mer) 5 - GGG AGA AAC AAG AAC GAC ACU AAU GAA CUA CUA CUA CAC ACC AAC CAC AAC UUU ACC UUA ACC UUA CUU ACC ACG GCA GCU GAC AAA GUC AGA AA -3 R (87-mer) 5 -GGG AGA AAC AAG UAA GUA AGG UUA AGG UAA AGU UGU GGU UGG UGU GUA GUA GUA GUU CAU UAG UGU CGU UCC UGA CAA AGU CAG AAA DNA oligucleotides and enzymes All the strands were purchased from Integrated DNA Technologies, Coralville, IA IDT. nt is labeled with TAMRA at the 5 end, nt is labeled with Texas Red at the 5 end, both activators A 1 and A are labeled with the IOWA black RQ quencher at the 3 end. The transcripti buffer mix was prepared prior to each experiment run (two to four samples) using the T7 Megashortscript kit (#1354), Ambi, Austin, TX which includes the T7 RNA polymerase enzyme mix, the transcripti buffer, and rntps utilized in the experiments. E. coli RNase H was purchased from Ambi (#9). 1.4 Transcripti protocol The templates were annealed with 1% (v/v) 1 transcripti buffer from 9 C to 37 C for 1 h 3 min at a ccentrati 5 1 the target ccentrati. The DNA activators were added to the annealed templates from a higher ccentrati stock, in a soluti with 1% (v/v), 1 transcripti buffer, 7.5 mm each NTP, 4% (v/v) T7 RNA polymerase, and.44% (v/v) E. coli RNase H. Each transcripti experiment for fluorescence spectroscopy was prepared for a target volume of 7 µl. Samples for gel studies were quenched using a denaturing dye (8% formamide, 1 mm EDTA,.1g XCFF). 1.5 Data acquisiti and processing The fluorescence was measured at 37 C every two minutes with a Horiba/Jobin Yv Fluorolog 3 system. Excitati and emissi maxima for TAMRA were set to 559 nm and 583 nm, respectively, according to the IDT recommendati; for Texas Red the maxima for the spectrum were set to nm. Slit widths were set to nm for excitati and 4 nm for emissi. The raw fluorescence data Φ(t) were cverted to estimated switch activity by normalizing with respect to maximum fluorescence Φ max (measured before adding activators and enzymes) and to minimum

6 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 6 fluorescence Φ min (measured after adding activators and before adding enzymes): ( [T i A i ](t) = [Ti tot ] 1 Φ(t) Φ ) min. Φ max Φ min For the adaptati experiments, normalizati was de by measuring maximum and minimum fluorescence levels at the beginning of the experiment, and assuming that the maximum fluorescence level scales linearly with the change in fluorescently labeled strands, while the minimum is not significantly affected by that variati. We used the formula: [T i A i ](t) = α[t tot i ] ( 1 Φ(t) Φ min αφ max Φ min where α is a factor that scales the amount of template as it varies in the experiment. Denaturing polyacrylamide gels (8% 19:1 acrylamide:bis and 7 M urea in TBE buffer, 1 mm Tris, 9 mm boric acid, 1 mm EDTA) were run at 67 C for 45 min with 1 V/cm in TBE buffer. Samples were loaded using Xylene Cyanol FF dye. For quantitati, denaturing gels were stained with SYBR Gold (Molecular Probes, Eugene, OR; #S-11494). In the ctrol lane a 1-base DNA ladder (Invitrogen, Carlsbad, CA; #18-15) was utilized. The DNA ladder 1 bp band was used as a ctrol to roughly estimate the ccentratis of the RNA species in soluti in Figure S5 E and F. Gels were scanned using the Molecular Imager FX (Biorad, Hercules, CA) and analyzed using the Quantity One software (Biorad, Hercules, CA). 1.6 Characterizati assays This secti reports experimental results and numerical fits. All experiments were run in triplicates: mean and error bars (standard deviati) are shown in each figure, together with the simulated traces (dashed lines) from our fitted model. The full derivati for the model fitted to the data is in Secti Genelets in isolati Figure S A shows the behavior of the two genelets in isolati: we can verify that each genelet self-inhibits after the enzymes are added. (For details the data normalizati procedure, refer to Secti 1.5.) The ccentrati of RNA present in soluti can be measured through gel electrophoresis, as shown in Figure S B: lanes 1 and show that free RNA in soluti is effectively absent Intercnected genelets When the two genelets are present in soluti in stoichiometric amount, their RNA outputs bind quickly to form a double-stranded complex, and therefore the feedback loops become a secdary reacti (by design thermodynamically less favorable than the R 1 R complex formati). As shown in Figure S C, the two genelets ly moderately self-repress. The RNA ccentrati in soluti is high, as shown in the denaturing gel in Figure S B, lanes 3 and 4. When the templates and are in different ratios, the system behavior is shown in Figure S3. We can plot the resulting initial active template ratio (which correspds to the ),

7 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 7 A Ccentrati (nm) Ccentrati (nm) 1 8 Act RNAP =1 nm = nm = nm =1 nm B T 1nM T1 1nM T1 1nM T 1nM R1 R Ccentrati (nm) C =1 nm =1 nm Figure S: A. Experimental data showing the isolated active genelet ccentratis as a functi of time: the self-inhibiti reacti turns the switches off, and the RNA ccentrati in soluti is negligible, as verified in the gel electrophoresis data in panel B, lanes 1 and (samples taken at steady-state after h). Dashed lines represent numerical trajectories of equatis (5), using the fitted parameters in Table S. B. Denaturing gel image: lanes 1 and show that the switches in isolati self-inhibit and no significant transcripti is measured. Lanes 3 and 4 show the RNA amount in samples from the experiment shown at panel C, taken at steady-state after h. When the genelets are in stoichiometric amount, their flow rates are already balanced and there is ly moderate self-inhibiti.

8 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 8 template ratio) versus the steady-state e: we find that the system behaves symmetrically and the steady-state ratio is close to e across all the initial ratios. Therefore, given open loop transcripti rates that differ across a factor of 1 3, these results suggest that the system robustly matches the flux of R 1 and R Flux adaptati If the ccentrati of [Ti tot ] and [A tot i ] is changed over time, the steady-state ccentrati of active genelets adjusts as shown in Figure S5 A and B. Samples from this set of experiments were analyzed using a denaturing gel: the results are shown in Figure S5 C and D (correspding to the traces in Figure S5 A and B, respectively) and show the RNA amount in soluti and that [R1 tot ] [R tot ], as desired (Figure S5 E and F). The RNA ccentratis were estimated using the DNA ladder as a reference. We are aware that this method may result in inaccurate absolute ccentrati estimates for RNA: however, our objective here is to compare the evoluti over time of the relative RNA ccentratis. Thus, inaccuracy in the determinati of the absolute amount of RNA produced does not affect the measured outcome of our experiments. The adjustment of genelet activity becomes progressively slower over time: the third round of adaptati is csistently slower than the previous two. We attribute this slower adaptati to various phenomena: 1) Decrease of activity of enzymes over time; ) Accumulati of incomplete degradati products from RNase H hydrolizati of RNA in RNA-DNA hybrids: these products can be up to 7 8 bases lg, and may interfere with the desired inhibiti pathways; 3) Abortive transcripti of RNA, which could also potentially bind to regulatory domains of DNA activators. Our hypothesis of accumulati of short products over time is validated by the gels shown in Figure S5 C and D (below 6 bases, part of the gel that is not shown, a similar smear is visible) Data fitting We derived a system of ordinary differential equatis (ODEs) starting from mass acti kinetics, as described in Secti.. The ODE system was numerically fitted using MATLAB (The Math- Works) to fluorescence data in Figures S and S3. Only a subset of the parameters was fit using the MATLAB fminc routine. We fit the RNA polymerase and RNase H ccentratis and the rates k Ti A i, k Ti A i R i, k Ai R i, k R1 R, k Ri T j, and the parameters k catonii and k cathij. This specific subset of parameters was chosen because experimental outcomes are chiefly affected by branch migrati rates (which are tunable by design of the toehold lengths), enzyme ccentrati, and enzyme catalytic rates. The ccentrati and compositi of the transcripti enzyme mix for the T7 Megashortscript kit are not disclosed by Ambi, but available literature suggests that additial enzymes, such as pyrophosphatase, are present in the mix, Milburn et al. [U. S. Patent , 1993]. We neglected reactis associated with the possibly unknown amount of pyrophosphatase in the mix. The ccentrati of RNase H is also not disclosed by Ambi; we did not run separate experiments to fit exclusively the degradati rate parameters. A table reporting all the parameters is in Secti, Tables S1 and S.

9 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 9 Ccentrati (nm) =5 nm =1 nm 1 3 Ccentrati (nm) =1 nm =5 nm 1 3 Ccentrati (nm) =1 nm = nm 1 3 Ccentrati (nm) = nm =1 nm Ccentrati (nm) 1 =1 nm =3 nm Ccentrati (nm) 1 =3 nm =1 nm 1 3 Figure S3: Ccentrati of active genelets over time at different templates ccentrati. The ccentrati of activators is always stoichiometric to the amount of correspding template. Dashed lines in all the figures correspd to numerical simulatis for model (5), using the parameters in Table S.

10 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 1 A Final ratio of active genelets Numerical Model No feedback Data: T1 Varied, T Fixed Data: T Varied, T1 Fixed 1 3 Initial ratio of active genelets B Final ratio of active genelets Numerical Model No feedback 5 1 Initial ratio of active genelets Figure S4: A: Plot summarizing the data shown at Figure S3, overlaid with the predictis of the numerical model (5), using fitted parameters shown in Table S. B: Predicted initial versus final genelets activity for ratios up to 1:1, according to our model (5) and parameters at Table S. Modeling and numerical analysis: two-gene flux matching system.1 Simple model system: derivati of nullclines and rate matching cditis R We csider a system composed of two generating species and, whose products R 1 and R interact to form a complex P = R 1 R. We introduce negative autoregulati to minimize the ccentrati of product that is not used to form the output complex (Figure S6). Free molecules of R i, i = 1,, bind to active T i, thereby inactivating it: where T i R i + T i δ i T Ti α i T i, is an inactive complex. We assume that T tot i i, T1 R1 T Figure S6: Our two-gene negative feedback architecture = T i + T i, and that T i to its active state with a first-order rate α i. The amount of R i is [Ri tot The correspding differential equatis are: d[t i ] dt d[r i ] dt = α i ([T tot i ] [T i ]) δ i [R i ][T i ], naturally reverts ] = [R i ] + [Ti ] + [P ]. = β i [T i ] k [R i ][R j ] δ i [R i ][T i ]. (1) For illustrative purposes, these differential equatis are solved numerically. The parameters chosen are: α 1 = α = /s, β 1 = β =.1 /s, δ 1 = δ = 5 1 /M/s, and k = 1 3 /M/s. An imbalance in the producti rates of R 1 and R is created by setting

11 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 11 C A Ccentrati [nm] 1nM 5nM nM 1nM 15nM 1nM 1nM 5nM 1nM 5nM 1nM 1nM 1nM 1nM ON 15nM 1nM 15nM 1nM T ON Time [min] 1nM 5nM 1nM 1nM 15nM 1nM D B 5nM 1nM Ccentrati [nm] nM 1nM 5 1nM 15nM 5nM 1nM 5nM 1nM 1nM 1nM 1nM 1nM ON 1nM 15nM 1nM 5nM T ON Time [min] 5nM 1nM 1nM 1nM 1nM 15nM R1 R E [µg/µl] 3 1 R 1 Total R Total F [µg/µl] 3 1 R 1 Total R Total 4 6 Time [h] 4 6 Time [h] Figure S5: A and B. Fluorescent traces showing the adaptati of the active fracti of genelets, when the amount of templates is varied over time. C and D. Samples from the experiments shown in panels A and B, respectively, were analyzed with gel electrophoresis. E and F show the ccentratis of RNA species, estimated from the gel samples.

12 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 1 [ ]() = = 1 nm and [ ]() = = nm, while [R 1 ]() = [R ]() =. The overall result of this feedback intercnecti is that the mismatch in the flow rate of R 1 and R is reduced, as shown in Figure S7. The flow rate is defined as the derivative of [Ri tot ]. The flow rate mismatch is defined as the absolute value of the difference between the two flows. The effect of changing the feedback strength, for simplicity chosen as δ 1 = δ, is shown in Figure S8: the figure shows the mean active fracti of [T i ] and the mean flow mismatch, averaged over the last two hours of a trajectory simulated for 1 hours. [µm] R 1 R [µm] Free R 1 Free R Flow mismatch [nm] [nm/min] Figure S7: Numerical simulati showing the soluti to the two-gene negative feedback architecture for flux matching modeled with equatis (1). The flow mismatch between R 1 and R is shown in the bottom-right panel. [nm] δ [/M/s] [nm]/min δ [/M/s] Figure S8: Numerical simulati showing genelet activity and mismatch over a range of values for the negative feedback parameter δ. It is possible to examine the nullclines relating and, and find the equilibria and

13 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 13 as intersecti of these nullclines: T i = = R i = α i(ti tot T i ), δ i T i Ṙ i = = R i = β i T i kr j + δ i T i. To simplify the derivati, we set δ 1 = δ = δ, β 1 = β = β, α 1 = α = α. Equating the two expressis for R i, we get the following equatis (for i = 1, and j = 1, ): ( α δ ) k ( T tot i T i T i ) ( T tot j ) T j T j + α(t tot i T i ) βt i =. ( ) T tot 1 ) k, φ = αt1 tot, ψ = αt tot, φ 3 = βt tot, and We can find an expressi of the nullclines by introducing a change of variables u = and v = ( ) T tot finally ψ 3 = βt tot 1 :, and defining φ 1 = ψ 1 = ( α δ u 1 (φ 1 v) + u(φ 1 v + φ φ v ) φ 1 3 =, 1 + v () v 1 (ψ 1 u) + v(φ 1 u + ψ ψ u ) ψ 1 3 =. 1 + u (3) The roots of the equatis above represent the nullclines of the system. Because all the parameters in these equatis are positive, there is always a single root. The nullclines are numerically solved, for varying δ, in Figure S9. A cditi for flow matching at steady-state can be derived as follows: Ṙ 1 Ṙ =, β 1 δ 1 R 1 = β δ R. Substituting the expressis for R 1 and R that can be derived by setting = =, we get: Taking α 1 = α = α, β 1 = β = β we get: β 1 T1 α 1 (T tot 1 ) = β T α (T tot ). = + α tot (T T1 tot ). (4) α + β The flow matching cditi above is shown in Figure S9, orange line (also shown in the main paper). If β α, i.e., the producti of R i is much faster than the generating species T i inactivati rate, then the cditi can be rewritten as:.

14 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 14 T [nm] 15 1 ( ) ( ) δ= 1 δ= 5 δ= 5 δ= 1 δ= T1 [nm] Figure S9: Nullclines computed for different values of negative feedback rate δ, and flux matching cditi (orange). Differential equatis modeling the experimental implementati Based our design specificatis and the resulting molecular interactis, we built a model for the system starting from the list of occurring chemical reactis. The switches T i and T j can have three possible states: the state where activator and template are bound and form the complex T i A i ; the off state given by free T i ; the off state represented by R j bound to T i forming T i R j. An off state still allows for RN AP weak binding and transcripti. Throughout this derivati, the dissociati cstants are omitted when assumed to be negligible. It is hypothesized that the ccentrati of enzymes is csiderably lower than that of the DNA molecules, allowing the classical steady-state assumpti for Michaelis-Menten kinetics. Branch migrati and hybridizati reactis amg nucleic acids are, for i {1, }, j {, 1}: Activati Inhibiti Annihilati Output formati Undesired hybridizati T i + A i k Ti A i T i A i R i + T i A i k Ti A i R i R i + A i k Ai R i R i + R j k Ri R j R i A i R i R j R j + T i k Rj T i R j T i. R i A i + T i

15 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 15 The enzymatic reactis are, for i {1, }, j {, 1}: Transcripti: state Transcripti: off state Transcripti: off state Degradati k + ONii RNAP + T i A i k ONii k + OF F ii RNAP + T i k OF F ii + kof F ji RNAP + R j T i k RNAP T i A catonii i RNAP + T i A i + R i k RNAP T catof F ii i RNAP + T i + R i k OF F ji k + Hii RNaseH + R i A i k Hii + khji RNaseH + R j T i k Hji k RNAP R j T catof F ji i RNAP + R j T i + R i k RNaseH R i A cathii i RNaseH + A i RNaseH R j T i k cathji RNaseH + T i. Using the law of mass acti, we derive the following ODEs: d dt [T i] = k Ti A i [T i ] [A i ] + k Ti A i R i [R i ] [T i A i ] k Rj T i [R j ] [T i ] + k cathji [RNaseH R j T i ], d dt [A i] = k Ti A i [T i ] [A i ] k Ai R i [R i ] [A i ] + k cathii [RNaseH R i A i ], d dt [R i] = k Ri R j [R i ] [R j ] k Ti A i R i [R i ] [T i A i ] k Ri T j [R i ] [T j ] k Ai R i [R i ] [A i ] + k catonii [RNAP T i A i ] + k catof F ii [RNAP T i ] + k catof F ji [RNAP R j T i ], d dt [R i R j ] = + k Ri R j [R i ] [R j ], d dt [R j T i ] = + k Rj T i [R j ] [T i ] k cathji [RNaseH R j T i ]. The molecular complexes appearing at the right-hand side of these equatis can be expressed using mass cservati: [T i A i ] = [T tot i ] [T i ] [R j T i ], [R i A i ] = [A tot i ] [A i ] [T i A i ]. We assume that binding of enzymes to their substrate is faster than the subsequent catalytic step, and that the substrate ccentrati is larger than the amount of enzyme. These assumptis allow us to use the standard Michaelis-Menten quasi-steady-state expressis. The Michaelis-Menten coefficients can be immediately defined; for instance, for the ON state of the template, define: k MONii = k ONii +k catonii. Then we find: k + ONii ( [RNAP tot ] =[RNAP ] 1 + [ A 1 ] [ ] [RNaseH tot ] =[RNaseH] + [] + [ A ] + + [R ] + [R 1 ] k MON11 k MOF F 11 k MON K MOF F k MOF F 1 ( 1 + [R 1 A 1 ] + [R A ] + [R ] + [R ) 1 ]. k MH11 k MH k MH1 k MH1 (5) k MOF F 1 ),

16 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 16 We can easily rewrite these equatis as [RNAP ] = [RNAP tot ] and [RNaseH] = [RNaseHtot ] P with a straightforward definiti of the coefficients P and H. Finally: [RNAP T i A i ] = [RNAP tot ] [T i A i ] P k MONii, [RNAP R j T i ] = [RNAP tot ] [R j T i ] P k MOF F ji, [RNAP T i ] = [RNAP tot ] [T i ] P k MOF F ii, [RNaseH R i A i ] = [RNaseHtot ] [R i A i ] H k MHii, [RNaseH R j T i ] = [RNaseHtot ] [R j T i ] H k MHji, H, which can be substituted in equatis (5). We note that our numerical fits result in an estimated RNAP ccentrati of about 1 nm: thus, in a subset of our experiments the substrate and enzyme ccentratis are actually comparable, breaking down e of the assumptis required for a quasi-steady-state approximati. Nevertheless, our model overall captures the system dynamics satisfactorily. The nlinear set of equatis (5) was solved numerically using MATLAB ode3 routine. Preliminary numerical analysis Prior to designing DNA strands and testing the system with wet lab experiments, we ran numerical simulatis using equatis (5) using parameters reported in Table S1. These parameters are csistent with those in Kim et al. [6], which were fitted from data obtained a transcriptial system with identical promoter/branch migrati design specificatis and sequence ctent; thus, we refer the reader to Kim et al. [6] for an accurate discussi and comparis to other branch migrati, transcripti, and degradati parameters found in the literature. Figure S1 shows the system trajectories that correspd to zero initial cditis for [A i ] and [R i ], while the complexes [ A 1 ] = = 1 nm, [ A ] = = 5 nm, [A tot 1 ] = 1 nm and [A tot ] = 5 nm. (The simulati first allows for equilibrati of all the DNA strands in the absence of enzymes. Only the porti of trajectories after additi of enzymes is shown.) The ccentrati of enzymes was assumed to be [RNAP tot ] = 8 nm and [RNaseH tot ] = 8.8 nm, csistent with typical volumes used in our experiments and with enzyme stock ccentratis of about µ M Kim and Winfree [11], Franco et al. [11]. An example of our numerical simulati results is shown in Figure S1. The behavior of the system proved to be csistent with the traces obtained for the simple model system shown at Figure S7. Data fitting results As already indicated in Secti 1.6.4, equatis (5) were fitted to all fluorescence data in Figures S and S3 simultaneously, using MATLAB routine fminc. Only a subset of the parameters was fit: the RNA polymerase and RNase H ccentratis, and the rates k Ti A i, k Ti A i R i, k Ai R i, k R1 R, k Ri T j, and the parameters k catonii and k cathij. Table S lists the results of the data fit; Table S3 reports the cstraints used in the fitting procedure. Our fits indicate that the hybridizati and branch migrati rates fitting these experiments are higher than what found in Kim et al. [6], Franco et al. [11]. In particular, the binding rate of the RNA species is higher than expected; hybridizati rates for complementary RNA strands of

17 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 17 1 R 1.1 Free R 1 [µm] 5 R [µm]..5 Free R [nm] [nm/min] Flow mismatch 1 3 Figure S1: Numerical simulati for equatis (5). Parameters are chosen as in Table S1. = [A tot 1 ] = 1 nm, = [A tot ] = 5 nm, [RNAP tot ] = 8 nm, and [RNaseH tot ] = 8.8 nm. These results are csistent with those of the simple model proposed in equatis (1), and analyzed numerically in Figure S7.

18 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 18 Table S1: Preliminary Simulati Parameters for Equatis (5) Units: [1/M/s] Units: [1/s] Units: [M] k Ti A i = k catonii =.6 k MONii = k Ti A i R i = k catof Fii = k MOF Fi = k Ai R i = k catof Fij = k MOF Fij = k Ri T j = k cathii =.1 k MHii = k Ri R j = k cathji =.1 k MHji = Units: [M] [RNAP tot ] = 8 nm Units: [M] [RNaseH tot ] = 8.8 nm similar length have (to our knowledge) not been assessed before. The expected ccentratis of RNA polymerase and RNase H and their k cat values are also higher than in previous studies Kim et al. [6], Franco et al. [11], where lower hybridizati rates were attributed to the presence of incomplete degradati products from RNase H hydrolizati of DNA/RNA hybrids. These short products, known to have length up to 7 8 bases, may interfere with desired regulatory pathways Kim and Winfree [11]. Because the activity and efficiency of off-the-shelf enzymes is known to csiderably vary from batch to batch Kim and Winfree [11], it is reasable to hypothesize that the RNA polymerase and RNase H batches used in this set of experiments had particularly high activity and low occurrence of incomplete transcripti/degradati which can slow down other reactis. Indeed, the accumulati of these incomplete products over time may be the reas for slower dynamics observed in our adaptati experiments in Figure S5. Table S: Fitted Parameters for (5); other parameters were left unvaried with respect to Table S1. Units: [1/M/s] Units: [1/s] k Ti A i = k caton11 =.1, k caton =.9 k T1 A 1 R 1 =.7 1 5, k T A R = k cathii =.9 k A1 R 1 = k A R = k cath1 =.3, k cath1 =. k Ri R j = Units: [M] [RNAP tot ] = 1 nm Units: [M] [RNaseH tot ] = nm

19 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 19 Table S3: Fitting cstraints for parameters in Table S. Parameter Lower Bound Upper Bound k Ti A i k Ti A i R i k Ai R i k Ri R j k catonii.1.1 k cathii.1.1 [RNAP tot ] [RNaseH tot ]

20 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 3 Modeling and numerical analysis: Scalability of the negative feedback scheme for flux regulati 3.1 Simple model system We csider now n generating species T i, outputting interacting products R i, and we explore different feedback intercnecti topologies. Initial studies scalability were outlined in Giordano et al. [13]. ODEs were derived using mass acti kinetics and used for numerical simulati of three and four compent networks. Negative autoregulati is implemented, as for smaller networks, with a self repressi scheme: when an output is in excess relative to the effectively used amount, it down regulates its own producti rate. R i + T i δ i T Ti α i T i, where Ti is an inactive complex. We assume that [Ti tot ] = [T i ] + [Ti ] and that Ti sptaneously reverts to its active state with a first-order rate α i. The correspding differential equati describing the template dynamics is the same regardless of the topology: d[t i ] = α i ([Ti tot ] [T i ]) δ i [R i ][T i ], i = 1,..., n. dt Depending the chosen interacti/binding topology for the products R i, we find that the system exhibits different behaviors, as shown in the following sectis Single product topology A single product topology occurs when a single complex P is formed by the simultaneous interacti of all the n outputs: n k R i P. The correspding differential equatis are d[r i ] dt d[p ] dt i=1 i, = β i [T i ] δ i [R i ][T i ] k = k n [R i ] i=1 n [R i ], and the amount of R i is [Ri tot ] = [R i ]+[Ti ]+[P ]. Figure S11 shows the numerical solutis to the ODEs for n = 3 and n = 4. Even though the initial amounts of T i are different, the ccentrati of active T i (bottom left panel) gradually decreases and the flow mismatches (namely the differences in absolute value between any two producti rates, shown in the bottom right panel) are csiderably reduced with a fast time respse. We can notice that the respse is slower in the case of 4 intercnected species. The quantity of produced R i (upper left panel) is of course increasing. With respect to the other topologies, as we will see, the single product topology leads to a much higher amount of free R i (upper right panel), which can be csidered waste because it is not used in the product formati. i=1

21 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 1 [μm] [nm] R 1 R R [μm] [nm/min] Free R 1 Free R Free R RNA Flux Mismatch M 1 M 3 M (a) single product, n = 3 [μm] 8 R 1 6 R 4 R 3 R [μm] 6 4 Free R 1 Free R Free R 3 Free R M 1 M 14 [nm] 1 T T 3 T 4 [nm/min] 1 5 M 3 M 13 M 4 M (b) single product, n = 4 Figure S11: Example traces from numerical simulatis: single product topology, negative feedback scheme.

22 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 3.1. Handshake and neighbor topologies A network of n generating species T i may be designed to produce different subcompents, that may later assemble into a larger product. In this scenario, we can take two extreme cases: the neighbor topology, when each output participates in at most two subcompents, and the handshake topology, when each output participates in n 1 subcompents. We thus have the generati of pairwise products P ij ; in the handshake case i, j = 1,..., n, j i, while in the neighbor case i = 1,..., n, j = i 1, i + 1 and when i = 1, i 1 = n, when i = n, i + 1 = 1, to close the loop. It is worth noticing that, in the case n = 3, the two topologies coincide. The reactis correspding to product generati are which lead to the following ODEs: d[r i ] dt d[p ij ] dt R i + R j k ij P ij, = β i [T i ] δ i [R i ][T i ] j = k ij [R i ][R j ]. k ij [R i ][R j ], The amount of R i is [Ri tot ] = [R i ] + [Ti ] + j [P ij]. Figure S1 shows the numerical solutis to the ODEs for n = 3, and for n = 4 in the handshake cnecti case. As for the single product topology, even though we initially have different amounts of active T i, the ccentrati of active T i decreases and the flux mismatches are csiderably reduced with a fast time respse. Although the quantity of produced R i is increasing, the feedback ctrol reduces and keeps bounded the amount of free R i, which can be csidered waste Parameters The parameters chosen in our simulatis are: k ij = 1 3 /M/s for the handshake/neighbor topology and k = /M/s for the single product topology, δ i = /M/s, α i = /s, β i = 1 1 /s, = 1 nm, = nm, [T3 tot ] = 3 nm, [T4 tot ] = 15 nm. An imbalance in the producti rates of R i is created by setting [T i ]() = [Ti tot ], while [R i ]() = Performance overview of the different topologies as a functi of key parameters We numerically explored the behavior of the different network topologies for n = 4 as a functi of the feedback parameter δ and of the rate of activati α. Figures S13, S14 and S15 show the network respse in terms of active percentage of T i ([T i ]/[Ti tot ] 1), flow mismatch (computed as in the previous cases) and respse time (defined as the time it takes for the active T i trajectory to go from [T i ()] 1% to [T i ()] 9%, where is the difference between its initial value [T i ()] and its steady state value). We solved the differential equatis for a time span of 1 hours and averaged the trajectories for active T i and for the computed mismatch over the last simulati hour. δ varies logarithmically from a tenth to a thousand times its nominal value; α varies from a hundredth to five times its nominal value. In each figure, pink squares mark the nominal behavior of the system (all parameters are identical to those listed in Secti 3.1.3).

23 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 3 [μm] 6 4 R 1 R R 3 [μm] Free R 1 Free R Free R [nm] 3 1 [nm/min] RNA Flux Mismatch M 1 M 3 M (a) handshake/neighbor, n = 3 [μm] 6 R 1 4 R R 3 R [μm] Free R 1 Free R Free R 3 Free R M 1 M 4 [nm] 1 T T 3 T 4 [nm/min] 1 5 M 3 M 13 M 14 M (b) handshake, n = 4 Figure S1: Example traces from numerical simulatis: handshake/neighbor topologies, negative feedback scheme.

24 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 4 % T i steady state activity δ [/M/s] T 4 Mean flow mismatch [nm/min] 1 M 1 1 M 3 8 M 13 6 M 14 4 M 4 M δ [/M/s] respse time [min] δ [/M/s] T 4 % T i steady state activity T α [/s] x 1 4 Mean flow mismatch [nm/min] α [/s] x 1 4 respse time [min] T T α [/s] x 1 4 Figure S13: Simulatis for the negative feedback, single product topology: parameter sensitivity analysis. In all network topologies, a large negative feedback parameter δ yields a lower mismatch and decreases the respse time; however, large δ clearly reduces the steady state activity of T i. In the handshake and neighbor topologies, a larger value of the sptaneous reactivati parameter α yields higher T i steady state activity, a larger mismatch, and a shorter respse time. On the ctrary, in the single product topology larger α, despite yielding higher T i steady state activity, dramatically increases the respse time, while the mismatch does not motically increase.

25 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 5 % T i steady state activity δ [/M/s] T 4 Mean flow mismatch [nm/min] M 14 M 4 M δ [/M/s] M 1 M 3 M 13 respse time [min] δ [/M/s] T 4 % T i steady state activity T α [/s] x 1 4 Mean flow mismatch [nm/min] α [/s] x 1 4 respse time [min] T α [/s] x 1 4 Figure S14: Simulatis for the negative feedback, handshake topology: parameter sensitivity analysis. % T i steady state activity δ [/M/s] T 4 Mean flow mismatch [nm/min] M 14 M 4 M δ [/M/s] M 1 M 3 M 13 respse time [min] δ [/M/s] T 4 % T i steady state activity T 1 T T 3 T α [/s] x 1 4 Mean flow mismatch [nm/min] α [/s] x 1 4 respse time [min] T 1 T T α [/s] x 1 4 Figure S15: Simulatis for the negative feedback, neighbor topology: parameter sensitivity analysis.

26 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 6 4 Positive feedback architecture for a two-gene system. Modeling and a viable experimental implementati 4.1 Simple model system: derivati of nullclines and rate matching cditis As de for the negative feedback architecture, we csider a system composed of two generating species and, whose products R 1 and R interact to form a complex P = R 1 R. We devise a positive feedback intercnecti where product in excess upregulates the product in shortage (Figure S16). Free (and thus, in excess) molecules of R i bind to inactive T j and activate it: where again T i [R tot i R i + T j T i α i T i, δ ij T j T1 R1 R T Figure S16: Our two-gene positive feedback architecture is an inactive complex and [Ti tot ] = [T i ] + [Ti ]. The amount of R i is ] = [R i ] + [T j ] + [P ]. We now assume that T i naturally reverts to its inactive state with rate α i. The correspding differential equatis are d[t i ] dt d[r i ] dt = α i [T i ] + δ ji [R j ]([T tot i ] [T i ]), = β i [T i ] k [R i ][R j ] δ ij [R i ]([T tot j ] [T j ]). (6) This system was initially csidered in Franco [1]. The above differential equatis were solved numerically. The parameters were chosen for illustrative purposes as α 1 = α = /s, β 1 = β =.1 /s, δ 1 = δ = 5 1 /M/s, and k = 1 3 /M/s. The amount of templates was chosen as = 1 nm, = nm. The initial cditis of active [T i ] are set as [ ]() = 1 nm and [ ]() = 16 nm, while [R 1 ]() = [R ]() =. Example traces are shown in Figure S17 (a modified versi of this figure is also in the main paper). Each product s flux rate is defined again as the derivative of [Ri tot ]. The flux mismatch is defined as the absolute value of the difference between the two flux rates. The effect of changing the feedback strength, where for simplicity δ 1 = δ, is shown in Figure S17 B and C, which plots the active fracti of [T i ] and the flux mismatch averaged over the last e hour of a 1 hours simulati. The right panel in Figure S17 seems to indicate that the flux mismatch of the two circuits is minimized for a certain range of δ around the nominal value of δ = 5 1. The nullclines of the system in the - space can be calculated as de for the negative feedback design. Taking equatis (6), we find: T j = = R i = Ṙ i = = R i = α j T j δ ij (T tot j T j ), β i T i kr j + δ ij (T tot j T j ).

27 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 7 A [µm] R 1 R [µm] Free R 1 Free R [nm] [nm/min] Flow Flow mismatch B C Flow [nm] [nm]/min 4 mismatch δ [/M/s] δ [/M/s] Figure S17: A: Example numerical simulati showing the time evoluti of the source species in the positive feedback architecture (Figure S16 modeled with equatis (6). B: Active ccentrati of source species as a functi of the positive feedback rate δ. C: Flow mismatch between R 1 and R as a functi of δ. Dark circles indicate the value of δ used in panel A.

28 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 8 To simplify the derivati, we set δ 1 = δ 1 = δ, β 1 = β = β, α 1 = α = α. Equating the two expressis for R i, we get the following equatis (for i = 1, and j = 1, ): ( α ) ( ) ( ) T i T j k + αt δ Ti tot T i Tj tot i βt j =. (7) T j ( ) T We can find an expressi of the nullclines by introducing a change of variables z = 1 T1 tot T ( ) 1 T and w = T tot, and defining φ 1 = ψ 1 = ( ) α δ k, φ = αt1 tot, ψ = αt tot, φ 3 = βt tot, and finally ψ 3 = βt tot 1 : z w (φ 1 v) + z(φ 1 w + φ φ w ) φ w 3 =, (8) 1 + w w z (ψ 1 z) + w(φ 1 z + ψ ψ z ) ψ z 3 =. (9) 1 + z The roots of the equatis above represent the nullclines of the system. Because all the parameters in these equatis are positive, there is always a single root. The nullclines are numerically solved, for varying δ, in Figure S18. A cditi for flow matching at steady-state can be derived as follows: Ṙ 1 Ṙ =, β 1 δ 1 R 1 (T tot ) = β δ 1 R (T tot 1 ). Substituting the expressis for R 1 and R that can be derived by setting = =, we get: β 1 T1 δ 1 δ 1 α T = β T δ 1 δ 1 α 1 T1. Taking α 1 = α = α, β 1 = β = β, and δ 1 = δ 1 = δ we get: =. (1) This flow matching cditi is shown in Figure S18 in the red dashed line. Decreasing α (inactivati rate for the generating species) or increasing δ (speed of the positive feedback), with respect to the nominal values chosen here, causes the equilibrium of the system to be pushed toward the upper right corner of Figure S18. Moreover, when decreasing α or increasing δ the system reaches equilibrium a timescale in the order of several dozens of hours. Explicit tradeoffs the effects of α and δ may be found by further analysis the nullclines and the locus of equilibria in equati (7). 4. A possible experimental implementati of a two-gene positive feedback scheme The experimental implementati of our positive feedback scheme using transcriptial networks presents several challenges. Here we present its general idea. A viable strand design scheme is in

29 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 9 T [nm] Nullclines for varying δ ( ) ( ) δ= 15 δ= 5 δ= 5 δ= 1 δ= T1 [nm] Figure S18: Numerical simulati: nullclines of the positive feedback scheme (6) in the - plane, calculated for different values of δ finding the roots of equatis (8) and (9). The equilibrium correspding to the set of nominal parameters (trajectories in Figure S17 A) is circled in black. The flow matching cditi (1) is shown in the orange line. The flow matching cditi is satisfied by the equilibria and for δ =

30 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 3 Figure S19 A. Both genelets are cstitutively inhibited by a DNA inhibitor I i. Each RNA output R i is designed to bind to the inhibitor I j (domains indicated as q j -a j -t j ), thereby releasing the activator A j for binding to T j. Because R i should also cover the active domain of R j in the formati of P, then R i must also be complementary to A i (domains t i-a i-q i): therefore, this design is structurally affected by binding of RNA to templates (as for the self-repressing circuit), and by RNA-mediated self-inhibiti loops, as shown in the reacti scheme in Figure S19 C. The entity of these design pitfalls depends the length and sequences of the complementarity domains shared by R i and R j. For instance, we could avoid inserting in the RNA species the toehold sequences t 1, t 1, t, and t to minimize the self inhibiti; however, this would facilitate the formati of complexes A i I i R j that would slow down the release of A i. Preliminary experiments this design, reported in Franco [1], show that the issues described above are significant. In particular, the design could be improved if the self-inhibiti pathways were minimized: this was attempted, without cclusive success, by increasing the ccentrati of DNA inhibitors, the ccentrati of RNase H, and by lengthening the length of toeholds for A i and I i. Experiments also highlighted the possibility of leaky transcripti of inhibited switches. We refer the reader to Franco [1], Chapter 1, for further details. Here, we ly describe our numerical analysis, which suggests that the scheme has the ability to match transcripti rates of two cross-activating genelets when we choose plausible reacti parameters Modeling To cstruct a dynamic model for the cross-activating circuit represented in Figure S19 A, we start from a list of all the chemical reactis that can occur, Activati Inhibiti Annihilati Release Annihilati Output formati Undesired interactis k Ti A T i + A i i T i A i k Ti A T i A i + I i I i i k Ai I A i + I i i A i I i R i + A j I j k Ri A j I j R i + I j k Ri I j R i + R j k Ri R j k Ri A R i + A i i R i I j R i R j R i A i T i + I i A i R i I j + A i R i + T j k Ri T j R i T j

31 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 31 A P q a t t1 a1 q1 q a t t1 a1 q1 R1 q a t t1 a1 q1 q1 a1 t1 t a q R a1 RNAP T1 a RNAP T I1 t1 t1 a1 q1 a1 A1 T1 off I t t a q a a A T off RNaseH A1 RNaseH t1 a1 q1 t1 a1 A1 I1 t a q t a q1 a1 t1 R q a t A A I R1 q1 a1 q1 a1 t1 t1 R I1 q a t q a t R1 I B a1 T1 off a T off RNaseH a1 T1 off RNaseH a T off a1 a R R1 C RNaseH q1 a1 t A t1 a q1 R a1 t a q t t1 t a q R t a a RNaseH a q1 a1 t a q t t1 RNAP T off t a q a t a A A I T Figure S19: General reacti scheme of the transcriptial circuits implementati for the positive feedback scheme in Figure S16. Complementary domains are represented with the same color. Promoters are colored in dark gray, while hairpin terminator sequences are in light gray. A. Desired cross-activati loops. The activati reacti arrows are colored in red. B. Undesired crossactivati and RNase H-mediated degradati of the RNA-template complexes. C. Undesired self-inhibiti. The inhibiti pathway in cyan arrows nominally should not occur, since there is no exposed toehold to favor it. However, this reacti has been observed in preliminary experiments not shown in this manuscript and is therefore also included in the models. R

32 Supplementary Informati Appendix - Experiments, Data Processing and Modeling 3 Transcripti: state Transcripti: off state Degradati + kon ii k catonii RNAP + T i A i RNAP T i A i RNAP + T i A i + R i k ON ii + kof F i RNAP + T i RNAP T i kcatof Fi RNAP + T i + R i k OF F i + kof F ij k catof Fij RNAP + R i T j RNAP R i T j RNAP + R i T j + R j k OF F ij RNaseH + R i I j k + H Ij k cathij RNaseH R i I j RNaseH + I j k H Ij k + H Ai RNaseH + R i A i RNaseH R i A i k H Ai RNaseH + R i T j k cathai RNaseH + A i k + H Tj k cathtj RNaseH R i T j RNaseH + T j. k H Tj The resulting set of ordinary differential equatis is: d dt [T i] = k Ti A i [T i ] [A i ] k Rj T i [R j ] [T i ] + k Ti A i I i [T i A i ] [I i ] + k cathti [RNaseH R j T i ], d dt [A i] = k Ti A i [T i ] [A i ] k Ai I i [A i ] [I i ] k Ri A i [R i ] [A i ] + k cathai [RNaseH R i A i ], d dt [I i] = k Ai I i [A i ] [I i ] k Ti A i I i [T i A i ] [I i ] k Rj I i [R j ] [I i ] + k cathii [RNaseH R j I i ], d dt [R i] = k Ri A j I j [R i ] [A j I j ] k Ri R j [R i ] [R j ] k Ri T j [R i ] [T j ] k Ri I j [R i ] [I j ] k Ri A i [R i ] [A i ] + k catonii [RNAP T i A i ] + k catof Fi [RNAP T i ] + k catof Fji [RNAP R j T i ], d dt [R i T j ] = + k Ri T j [R i ] [T j ] k cathtj [RNaseH R i T j ], d dt [R i R j ] = + k Ri R j [R i ] [R j ]. As previously de for the self-inhibiting circuit model, we can express the enzyme-substrate complexes using the Michaelis-Menten approximati. For the RNAP substrate, for instance, we find: [RNAP T i A i ] = (11) [RNAP tot ] ( 1 + ). [T i A i ] i,j k MONii + [T i] k MOF Fi + [R i T j ] (1) k MOF Fij Analogous expressis can be derived for all other complexes. Equatis (11) are numerically solved using the MATLAB ode3s solver. Table S4 shows the parameters used for the simulatis. Such generic parameters are csistent with those in Kim