As shown in Figure 1A and B, TetR can be generically divided into a LBD and a DBD, disregarding its homodimeric nature Leander et al. (2020); Takeuchi et al. (2019); Reichheld et al. (2009); Leander et al. (2022); Yuan et al. (2022); Scholz et al. (2004). In the same vein of the classic allostery models Bacon (1965); Koshland Jr et al. (1966), each domain features two (the relaxed/inactive and tense/active) conformations that differ in free energies and binding affinities. For the simplicity and mechanistic clarity of the model, we assume negligible binding affinity of the LBD to the lig- and and the DBD to DNA in their inactive conformations and consider competent binding only for the active ones. Each domain must overcome a free energy increase to transition from the inactive to the active conformation (ϵ_L for LBD and ϵ_D for DBD). Importantly, the two domains are allosterically coupled, in that there is a free energy penalty γ when both domains adopt the active conformations simultaneously. For example, when the ligand binds to the LBD, selecting its active conformation, it discourages the active conformation of the DBD and therefore DNA binding. This anti-cooperativity establishes the foundation for allosteric regulation in TetR.

Accordingly, there are four possible conformational states of TetR, namely L_I D_I, L_AD_I, L_I D_A and L_AD_A, with L∕D and I∕A denoting LBD/DBD and inactive/active conformation, respectively (see Figure 1B and the upper four states of Figure 1D). Binding of ligand/operator to the active LBD/DBD further lowers the free energy of the corresponding state in a concentration dependent manner following the standard formulation of binding equilibrium (see the lower three states of Figure 1D). The regulatory mechanism of TetR allostery can then be qualitatively explained by the schematics in Figure 1C and D. Without the ligand, wildtype (WT) repressor predominantly binds to the operator sequence, which obstructs the binding of RNA polymerase (RNAP) to the adjacent promoter of the regulated gene (Figure 2—figure Supplement 1A). In the presence of ligand (inducer) at a sufficiently high concentration, the ligand-bound L_AD_I state (L-L_AD_I) has the lowest free energy compared with other possible (DNA-bound) states, thus the repressor predominantly releases the operator upon ligand binding, enabling the expression of downstream genes (Figure 1C and Figure 2—figure Supplement 1A).

Mutations perturb the three biophysical parameters (ϵ_D, ϵ_L and γ) and hence the free energy landscape (Figure 1D), leading to changes in TetR function. The three limiting scenarios are depicted in Figure 1D: 1. when a mutation significantly increases ϵ_L, it raises the free energy of the L-L_AD_I state relative to the DNA-bound state L_I D_A-D, which results in a dead (noninducible) phenotype; 2. when a mutation substantially decreases ϵ_D, it raises the free energy of the L-L_AD_I state relative to both DNA-bound states (L_I D_A-D and L-L_AD_A-D), also leading to a dead mutant; 3. mutations that decrease γ greatly lower the free energy of the double-bound state (L-L_AD_A-D) rel-ative to L-L_AD_I, again leading to the suppression of induction. Therefore, these schemes highlight that hotspot mutations can disrupt allostery by perturbing either intra- (ϵ_D, ϵ_L) or inter-domain (γ) properties. Functional readouts like DMS can help distinguish these mechanistic differences at the biophysical level. Notably, mutations mainly perturbing ϵ_D or ϵ_L do not have to lie on the structural pathways linking the allosteric and active sites, which explains the broad hotspot distributions observed in the DMS measurement Leander et al. (2020); Reichheld et al. (2009); Leander et al. (2022); Scholz et al. (2004). In general, a dead mutation is likely of mixed nature, as long as its effect on the free energy landscape promotes the dominance of the DNA-bound states. Likewise, rescuing mutations may restore WT-like inducibility by modifying intra- and inter-domain energetics in various ways, as far as the dominance of the L-L_AD_I state is re-established, rationalizing the broad rescuing mutation distributions. Naturally, the rescuability of a dead mutation depends on which biophysical parameters it perturbs and by how much. At a qualitative level, the distributed nature 3of dead and rescuing mutations observed in the DMS measurement emerges naturally from the two-domain model Leander et al. (2020).

In summary, the interplay of intra- and inter-domain properties that govern the free energy landscape of TetR’s conformational and binding states, as elucidated by the two-domain model, aligns well with the unexpected DMS results on a qualitative level. To gain deeper insight into TetR allostery and the model itself, we aim to establish a quantitative framework with the model that accurately captures mutant induction curves in the next section.

The discussions in the previous section and Figure 1 are primarily meant to give an intuitive and qualitative understanding of the two-domain model and rationalization of recent DMS measurements Leander et al. (2020, 2022). In this section, we further establish a quantitative connection between the model and the induction curves of TetR variants, leveraging the recent success of linking sequence-level perturbations to system-level responses Daber et al. (2011); Chure et al. (2019); Garcia and Phillips (2011); Brewster et al. (2014); Weinert et al. (2014); Rydenfelt et al. (2014); Razo-Mejia et al. (2017). An induction curve describes the in vivo expression level of the TetR-regulated gene as a function of inducer concentration.

As described in the last section, the intra-domain parameters (ϵ_D, ϵ_L) and inter-domain parameter (γ) of the two-domain model collectively determine the free energy landscape of TetR (Figure 1). Consequently, a degenerate relationship arises between combinations of parameter values and the induction level at a specific ligand concentration as measured in the DMS study. To tease apart the allosteric effects of different parameters, we aim to formulate their connection to the full induction curve, which is characterized by (1) the expression level of the TF-regulated gene without ligand (leakiness), (2) gene expression level at saturating ligand concentration (saturation) and (3) the ligand concentration required for half-maximal expression (EC₅₀). As revealed in previous studies, these induction curve properties encode information for the key parameters of an allosteric regulation system, for example, the TF binding affinity to ligand and operator, equilibrium between different conformational states of the TF and the abundance of various essential molecules within the system. The establishment of a quantitative connection between the MWC model and the induction curve has yielded valuable insights into several allosteric systems Eaton (2022); Daber et al. (2011); Chure et al. (2019). However, the analogous aspects within a multi-domain thermodynamic model for allostery require further investigation.

Inspired by the pioneering works on transcription with MWC models Daber et al. (2011); Chure et al. (2019), we derived a quantitative relation between the gene expression level in cells and the three biophysical parameters of the two-domain model (Equation 1. Briefly, the ratio of the expression level of a TF-regulated gene to that of an unregulated gene, termed fold change (F C) and bound between 0 and 1, is used to quantify gene expression. The gene expression level on the other hand, is assumed to be proportional to the probability that a promoter in the system is bound by RNA polymerase, p(RNAP). Thus, F C is evaluated as the ratio of p(RNAP) in the presence of the TF to that in the absence of the TF. Intuitively, p(RNAP) can be calculated based on the equilibria among different binding and conformational states of the TF (Figure 2—figure Supplement 1 and Figure 2—figure Supplement 2), which are determined by the allosteric properties (ϵ_D, ϵ_L and γ) and several other parameters. Therefore, F C can be expressed as a function of these parameters, which under the assumption that intra- and inter-domain energetics adopt typical values of several k_BT, is simplified to Equation 1. Here, R^∗ is the rescaled TF copy number in the cell; c is ligand concentration, and K is the dissociation constant of ligand to the repressor with an active LBD. As we focus on understanding how sequence level perturbations are manifested in the allosteric properties of the protein (depicted by ϵ_D, ϵ_L and γ), the residues we choose for mutational study here are mostly not in direct contact with either the ligand or the operator (see Supplementary file for more discussions). Thus, K and R^∗, which is determined by the TF copy number in the cell and the affinity of the operator to the repressor with an active DBD, are taken to be constants across mutations in all subsequent discussions Chure et al. (2019). Detailed derivations of Equation 1 are provided in the Supplementary file.

Despite the considerable degeneracy in the activity (inducible and noninducible) within the parameter space of the two-domain model (Figure 1D), Equation 1 demonstrates that mutations affecting distinct biophysical parameters can be discerned based on their characteristic effects on the induction curve.

First, as ϵ_L and γ have no effect on the leakiness of the induction curve, only mutations that modify ϵ_D can lead to its changes (Figure 2 and Supplementary file for additional discussions). In addition, these mutations also change the level of saturation (F C value where the induction curve plateaus at large c). Thus, dead mutations that disrupt allostery by decreasing ϵ_D alone will uniquely feature a noticeably lower leakiness and a lower level of saturation compared with the WT (Figure 1D and Figure 2, see also Supplementary file and Figure 2—figure Supplement 3 for more discussions). Second, mutations that solely perturb the other intra-domain property ϵ_L, a crucial determinant of TetR’s ligand detection limit, primarily shift the EC₅₀ of the induction curve (see Equation 1). As ϵ_L increases/decreases from the WT value, the induction curve is right/left shifted with leakiness and the level of saturation remaining unchanged (see the blue and red curves of Figure 2A). However, when ϵ_L further increases, the induction curve loses the sigmoidal shape, with its sharply varying tail region being the characteristic of a dead mutation that disrupts allostery mainly by increasing ϵ_L (purple curve of Figure 2A, see Supplementary file and Figure 2—figure Supplement 3 for more discussions).

Finally, in the high concentration limit, Equation 1 converges to a constant value (see Equation 2). Hence, mutations affecting γ alone will tune the saturation of a sigmoidal induction curve, which increases/decreases as γ increases/decreases (see red and purple curves of Figure 2C). Therefore, in contrast to the two aforementioned scenarios, dead mutations that disrupt allostery through decreasing γ alone will feature a full sigmoidal induction curve with a low level of saturation and unchanged leakiness compared with the WT (with γ > 0, see the purple curve of Figure 2C). Furthermore, as γ dictates the inter-domain cooperativity, it thereby controls the sign of the monotonicity of F C as a function of c (see Supplementary file for detailed proof). Specifically, when γ > ∕ < 0, there is negative/positive cooperativity between ligand and operator binding, and F C increases/decreases monotonically with c (see Figure 2C). When γ = 0, however, the bindings of lig- and and operator become independent of each other, and c no longer affects F C (Figure 2—figure Supplement 3).

Having identified the distinctive impacts of the different allosteric parameters of the two-domain model on the induction curve, as outlined in Equation 1, we next apply the model to analyze actual experimental induction curves of TetR mutants. The analysis enables us to uncover the underlying biophysical factors contributing to diverse mutational effects. Additionally, it allows us to evaluate the model’s validity in capturing TetR allostery by assessing its accuracy in reproducing the experimental data.

With the diagnostic tools established, we mapped mutants to the parameter space of the two-domain model through fitting of their induction curves. We choose 15 mutants for analysis in this section, which contain mutations that span different regions in the sequence and structure of TetR (Figure 3—figure Supplement 1). Five of the 15 mutants consist of a dead mutation G102D and one of its rescuing mutations (see the third row of Figure 3A), while the other 10 contains the WT, 8 single mutants and a double mutant Q32A-E147G (see the first two rows of Figure 3A). In all cases, fitting of an induction curve is divided into two steps; first, at c = 0, Equation 1 can be rearranged to give

which enables the direct calculation of ϵ_D from the leakiness of the induction curve; second, ϵ_L and γ are inferred based on the remaining induction data using the method of Bayesian inference Chure et al. (2019). Briefly, given ϵ_D calculated in the first step, we select a set of ϵ_L and γ values from physical ranges of the parameters, based on the probability of observing the induction curve with the parameter values using Monte Carlo sampling. The medians of the selected sets of ϵ_L and γ values, which are known as posterior distributions, are reported as the inferred parameter values, and error bars show the 95% credible regions of the posterior distributions (Figure 3B and Figure 4—figure Supplement 2B). All details are provided in the “Materials and methods” section and Supplementary file.

As shown in Figure 3A, the fitted curves accurately capture the induction data of all the mutants studied here, enabling the quantification of their biophysical parameters with little uncertainty (see Figure 3B). These results support the general applicability of the two-domain model to describing TetR allostery, as these 15 mutants vary significantly in terms of location of mutations and phenotype. They are one to four mutations away from the WT, and the sites of mutations are distributed across LBD, DBD and the domain interface (Figure 3—figure Supplement 1). In terms of the phenotype, they fall into different classes as characterized by the F C value at c = 1000 nM (F C¹⁰⁰⁰), including dead (F C¹⁰⁰⁰ < 0.1), enhanced induction (F C¹⁰⁰⁰ > F C¹⁰⁰⁰) and neutral activity (the rest).

Besides inspecting the goodness of fit, a closer examination of the induction curves and the Bayesian inference results reveal additional features of the two-domain model of TetR allostery. First, the apparent binding affinity of TetR to the ligand (anhydrotetracycline; aTC) and operator (tetO2) used in our experiments are estimated to be about 17.4 k_BT and 16.4 k_BT, respectively, based on the parameter values inferred for the WT (see Supplementary file for detailed calculations); these agree well with the range of reported experimental values Scholz et al. (2000); Schubert et al. (2004); Kędracka-Krok and Wasylewski (1999); Kamionka et al. (2004); Bolintineanu et al. (2014); Normanno et al. (2015). Second, as shown in Figure 3B, the perturbations in ϵ_D are generally small in magnitude (< 1 k_BT) among the dead mutants, while substantially larger perturbations (up to ∼ 5 k_BT) are observed for ϵ_L and γ. The latter observation is in line with the predictions of the model (see the previous section) that two types of qualitatively different induction curves are expected for partially dead mutants (see the first and second rows of Figure 3A). Specifically, the induction curve of P176N-I174K-F177S (PIF) loses the sigmoidal shape, with its sharply varying tail region suggesting a significant increase of ϵ_L. This is confirmed by the parameter fitting result, which shows that the triple mutations of PIF, located in the core region of LBD (Supplementary file and Figure 3—figure Supplement 1), lead to the largest increase of ϵ_L among all mutants (4.5 k_BT higher than the WT, see Figure 3B). The induction curves of the other four dead mutants (R49G, D53H, P105M and G143M), however, maintain the sigmoidal shape yet with low levels of saturation. As they all exhibit a higher ϵ_D (leakiness) than the WT, the low levels of saturation have to result solely from weakened inter-domain couplings (Figure 3). In other words, these four dead mutations, located mostly at the domain interface (Figure 3—figure Supplement 1), disrupt allostery primarily through decreasing γ. Indeed, when the γ of these mutants is set to the WT value but keeping their respective ϵ_D and ϵ_L parameters unchanged, higher F C¹⁰⁰⁰ values than the WT are obtained instead (Figure 3—figure Supplement 7).

Despite the discussion of two limiting types of changes in the induction curves, it is interesting to observe in Figure 3B that the three biophysical parameters, especially ϵ_L and γ, are often perturbed together in the mutants. Indeed, although the three parameters are theoretically independent of each other, at the structural level, it is less likely to have a scenario where a mutation perturbs inter-domain coupling (γ) but leaves intra-domain properties unchanged. Lastly, while quantification of G102D from its flat induction curve incurs large uncertainties, (Figure 2—figure Supplement 3), the diversities of the induction curves and the biophysical parameters of the G102D-rescuing mutants (Figure 3B) provide clear support for the qualitative anticipation from the previous section; i.e., rescuing mutations of a dead mutant may restore inducibility through different combinations of tuning ϵ_D, ϵ_L and γ. For example, while Y42M-I57N and K98Q both rescue G102D to achieve similar F C¹⁰⁰⁰, they exert distinct influences on ϵ_D and ϵ_L, as reflected in the variations of leakiness and EC₅₀ between the corresponding induction curves. In another case, while the induction curves of G102D-T26A and G102D-L146A show comparable values of saturation and leakiness, their different EC₅₀ values suggest the different effects of the two rescuing mutations on ϵ_L, further quantified by the inferred parameter values. The diverse mechanistic origins of the rescuing mutations revealed here provide a rational basis for the broad distributions of such mutations.

In summary, Equation 1 accurately captures the induction curves of a comprehensive set of TetR mutants using a minimum set of parameters with realistic values (see the section “Discussion” and Figure 3—figure Supplement 1). The two-domain model thus provides a quantitative platform for investigating TetR allostery beyond an intuitive rationalization of the DMS results.

With the 15 mutants mapped to the parameter space of the two-domain model, we now explore the epistatic interactions between the relevant mutations. To do so, we start by assuming additivity in the perturbation of all three biophysical parameters (see Equation 4 and Equation 5, where p represents any one of ϵ_D, ϵ_L and γ).

We then evaluate how the induction curves generated by this additive model α_1,p = α_2,p = 1) deviate from the corresponding experimental results, and the magnitude of deviation quantifies the significance of epistasis. Eight mutation combinations were chosen for the analysis, where we pair up C203V and Y132A, the two single mutations that enhance the level of induction relative to the WT, with mutations from different structural regions of TetR.

Although predictions from the additive model are qualitatively correct for 5 of the 8 mutation combinations on phenotypic effects, none of them captures the corresponding induction curve accurately (Figure 4—figure Supplement 1). This again highlights the parameter space degeneracy of the phenotypes, and that mechanistic specificity is required of a model for making reliable predictions on combined mutants Li and Lehner (2020). For a deeper understanding of epistatic interactions between the mutations queried here, we next directly fit for the three biophysical parameters of the combined mutants using their induction curves, which are then compared with those obtained from the additive model.

As shown in Figure 4—figure Supplement 2A, the induction data of all eight combined mutants are well captured by the fitted curves, which reaffirms the applicability of the two-domain model. Interestingly, despite the discrepancy between the experimental induction curves and those predicted by the additive model, the fitted parameters of the combined mutants are comparable to those from the additive model in many cases (see Figure 4—figure Supplement 2B). In this regard, one noticeable example is the mutant C203V-G102D-L146A, for which the result of direct fitting is close to the additive model in terms of both ϵ_L and γ (with a difference of ⩽ 0.5 k_BT), while they differ more in ϵ_D (1.6 k_BT). Inspired by such observations, we then seek the minimal modifications to the additive model that can account for epistasis. As C203 is one of the most distant residues from DBD in TetR, we reason that its effect on ϵ_D may get dominated by mutations much closer to the DNA binding residues like G102D and L146A (Figure 3—figure Supplement 1). Remarkably, when we quench C203V’s contribution to ϵ_D in the additive model of C203V-G102D-L146A α_C203V,ϵ = 0, see Equation 4, the predicted induction curve well recapitulates the experimental data (Figure 4G). In another example, when C203V is combined with the triple mutations PIF, which are much closer to the DBD, the fitted parameters of the combined mutant also align well with the additive model except only for ϵ_D. Here, quenching C203V’s contribution to ϵ_D again leads to good agreement between the additive model and experiment in terms of the induction curve (Figure 4H).

The epistatic effects in other C203V-containing combined mutants can be largely accounted for by modifying the additive model following the same physical reasoning as above. For instance, mutation D53H is near the DNA-binding residues and located at the domain interface, which suggests its dominant role in defining the DBD energetics and inter-domain cooperativity when paired up with C203V Scholz et al. (2004). Accordingly, quenching C203V’s effect on both ϵ_D and γ in the additive model for mutant C203V-D53H leads to dramatic improvement in the induction curve prediction (Figure 4—figure Supplement 1F and Figure 4F). Along this line, success is also observed when accounting for the epistasis between C203V and R49G by the same approach (Figure 4E). These examples might also explain why C203V, although being able to enhance the induction of TetR, fails to rescue a range of dead mutations including R49A, D53V, G102D, N129D and G196D Leander et al. (2020). More broadly, these observations together indicate that epistasis in combined mutants (e.g., those containing C203V) can be captured by the additive two-domain model with modifications based on physical reasoning.

The epistasis in combined mutants containing Y132A can be understood in a similar manner. Noticeably, the fitted parameter values of Y132A-C203V agree very well with the additive model for

ϵ_D and γ (within a difference of 0.4 k_BT), while they differ in ϵ_L by 3.8 k_BT. Interestingly, such discrepancy in ϵ_L values can be essentially resolved by quenching Y132A’s contribution in the additive model (reduced to 0.4 k_BT). This likely suggests that the effect of Y132A on ϵ_L tends to be compromised when combined with other mutations that perturb ϵ_L. Indeed, for most of the Y132A containing mutants investigated here (Y132A-C203V/PIF/R49G), the accuracy of induction curve prediction by the additive model improves greatly when α_{Y 132A,ϵ} is tuned down (Figure 4B − D, see Supplementary file for more discussions).

The only exception to this trend is the mutant Y132A-G102D-T26A, for which we observe strong epistasis between Y132A and the dead-rescue mutation pair G102D-T26A. Although both mutations Y132A and G102D-T26A enhance the allosteric response of TetR (Figure 3), their combined effects radically change the sign of cooperativity between the two domains (γ), turning the ligand (aTC) from an inducer into a corepressor (Figure 4A and Figure 4—figure Supplement 2B). Here, large disparity exists between the additive model and the direct fitting result in all three biophysical parameters, which cannot be explained by simple modifications of the additive model.

Using a low-copy backbone (SC101 origin of replication) carrying spectinomycin resistance, we constructed a sensor plasmid with TetR(B) (Uniprot P04483). The tetRb gene was driven by a variant of promoter apFAB61 and Bba_J61132 RBS Kosuri et al. (2013). On a second reporter plasmid, superfolder GFP Pedelacq et al. (2006) was cloned into a high-copy backbone (ColE1 origin of replication) carrying kanamycin resistance. The expression of the superfolder GFP reporter was placed under the control of the ptetO promoter. To control for plasmid copy number, RFP was constitutively expressed with the BBa_J23106 promoter and Plotkin RBS Kosuri et al. (2013) in a divergent orientation to sfGFP.

A comprehensive single-mutant TetR library was generated by replacing wildtype residues at positions 2–207 of TetR to all other 19 canonical amino acids (3,914 total mutant sequences). Oligonucleotides encoding each single-point mutation were synthesized as single-stranded Oligo Pools from Twist Bioscience and organized into 6 subpools, spanning six segments of the tetRb gene, corresponding to residues 2–39, 40–77, 78–115, 116–153, 154–191, and 192–207 of TetR(B) respectively. Additional sequence diversity was observed in the library due to error rates in the synthesis of single-stranded Oligo Pools, leading to the downstream identification of some double and triple mutant TetR variants. Oligo pools were encoded as a concatemer of the forward priming sequence, a BasI restriction site (5^′-GGTCTC), six-base upstream constant region, TetR mutant sequence, sixbase downstream constant region, a BsaI site (5^′-GAGACC), and the reverse priming sequence. Subpools were resuspended in double-distilled water (ddH₂O) to a final molal concentration of 25 ng/μL and amplified using primers specific to each oligonucleotide subpool with KAPA SYBR FAST qPCR (KAPA Biosystems; 1-ng template). A second PCR amplification was performed with KAPA HiFi (KAPA Biosystems; 1-μL qPCR template, 15 cycles maximum). We amplified corresponding regions of pSC101_TetR_specR with primers that linearized the backbone, added a BsaI restriction site, and removed the replaced wildtype sequence. Vector backbones were further digested with DpnI, BsaI, and Antarctic phosphatase before library assembly.

We assembled mutant libraries by combining the linearized sensor backbone with each oligo subpool at a molar ratio of 1:5 using Golden Gate Assembly Kit (New England Biolabs; 37 ^◦C for 5 min and 60 ^◦C for 5 min, repeated 30 times). Reactions were dialyzed with water on silica membranes (0.025-μm pores) for 1 h before transformed into DH10B cells (New England Biolabs). Library sizes of at least 100,000 colony-forming units (cfu) were considered successful. DH10B cells containing the reporter pColE1_sfGFP_RFP_kanR were transformed with extracted plasmids to obtain cultures of at least 100,000 cfu. Following co-transformation, the cultures for each subpool were stored as glycerol stocks and kept at -80 ^◦C.

The subpool library cultures were seeded from glycerol stocks into 3 mL lysogeny broth (LB) containing 50 μg/mL kanamycin (kan) and 50 μg/mL spectinomycin (spec) and grown for 16 h at 37 ^◦C. Each culture was then back-diluted into two wells of a 96-well plate containing LB kan/spec media using a dilution factor of 1:50 and grown for a period of 5 hours at 37 ^◦C. Following incubation, each well containing saturated library culture was diluted 1:75 into 1× phosphate saline buffer (PBS), and fluorescence intensity was measured on a SH800S Cell Sorter (Sony). We first gated cells to remove debris and doublets and selected for variants constitutively expressing RFP. Using this filter, we then proceeded to draw an additional gate to select for mutants that displayed low fluorescence intensities in the absence of anhydrotetracycline (aTC). This gate allowed us to select for mutants which retained the ability to repress GFP expression in the presence of an inducer, using prior measurements of WT TetR(B) as a reference for selecting repression competent mutants Leander et al. (2020). Utilizing these gates, we sorted 500,000 events for each gated population and recovered these cells in 5 mL of LB at 37 ^◦C before adding 50 μg/mL of kan and spec. Following the addition of antibiotics, the cultures were incubated at 37 ^◦C for 16 hours.

Following overnight growth, each culture was back diluted into two wells of a 96-well plate containing LB kan/spec media using a 1:50 dilution factor. Upon reaching an OD600 ∼ 0.2, aTC was added to one of the two wells at a final concentration of 1 μM, with the well not receiving aTC serving as the uninduced control population. Cells were then incubated for another 4-hour growth period at 37 ^◦C before being diluted into 1x PBS using a dilution factor of 1:75, and fluorescence intensity was measured on a SH800S Cell Sorter.

We first gated cells to remove debris and doublets and selected for variants constitutively expressing RFP before obtaining the distribution of fluorescence intensities (FITC-A) across the uninduced and induced populations of each of the 6 subpools (Figure 3—figure Supplement 8). Using the uninduced control population of each subpool as a reference, gates were drawn to capture mutants which displayed no response to the presence of aTC in the induced populations. Using these gates, 500,000 events were sorted from the induced populations of each subpool and cells were subsequently recovered at 37 ^◦C using 5 mL of LB media before being plated onto LB agar plates containing 50 μg/mL kan and spec.

To screen for functionally deficient variants of TetR, 100 individual colonies were picked from each subpool and grown to saturation in a 96-well plate for 6 hours. Saturated cultures were then diluted 1:50 in LB-kan/spec media and grown in the presence and absence of 1 μM aTC for 6 hours before OD600 and GFP fluorescence (Gain: 40; Excitation: 488/20; Emission: 525/20) were read on a multiplate reader (Synergy HTX, BioTek). Fluorescence was normalized to OD600, and the fold inductions of each clone were calculated by dividing its normalized fluorescence in the presence of inducer by the normalized fluorescence in the absence of inducer. The fold-induction of WT TetR was tested in parallel with these screens to serve as a benchmark for function, and clones which displayed < 50% activity to WT were replated and validated in triplicate before being sent for sanger sequencing (Functional Biosciences).

TetR mutants identified during clonal screening were reinoculated into 3 mL of LB Kan/Spec media and grown overnight. After an overnight growth period, each culture was added to 4 rows of 12 wells in a 96-well plate containing LB Kan/Spec media using a dilution factor of 1:50. Each mutant was tested across 12 different concentrations of aTC in quadruplicate fashion, with each row having an identical concentration gradient across the 12 wells. The aTC concentration gradient across the 12 wells were 0 nM, 10 nM, 20 nM, 40 nM, 60 nM, 80 nM, 100 nM, 150 nM, 250 nM, 500 nM, 750 nM, and 1μM aTC.

Following reinoculation into 96-well plates, selected mutants were placed on a plate shaker set to 900 RPM and incubated at 37 ^◦C for 5 hours. Following this 5-hour growth period, the plates were removed from the plate shaker and the OD600 and GFP fluorescence (Gain: 40; Excitation: 488/20; Emission: 525/20) of each well was read using a multiplate reader (Synergy HTX, BioTek). Fluorescence was normalized to OD600 for each well. The normalized fluorescence at each concentration was measured for 4 replicates of each mutant (unless otherwise stated). The fluorescence of GFP reporter-only control (no TetR), grown under identical conditions, was measured in the same way. This reporter-only control served as an upper bound to the fluorescence that could be measured in the experiment and was accompanied by a WT TetR-GFP reporter control, used to certify the precision of measurements made at different times.

To assess the predictive capability of the additive model, eight combined mutants were selected to undergo aTC dose response characterization. These combined mutants were synthesized using clonal DNA fragments from Twist Biosciences coding for the amino acid sequence of each of the selected combined mutants. The gene fragments were individually cloned into sensor plasmid backbones identical to those used in the previous identification of non-functional TetR(B) mutants following the manufacturers protocol for Gibson Assembly (New England Biolabs). The newly constructed plasmids carrying each of the combined mutants were then transformed into E. coli (DH10B) cells containing a GFP reporter plasmid identical to that used in the previous characterization of non-functional TetR(B) mutants. After a 1-hour recovery period, 100 μL of recovered cells were plated onto LB kan/spec agar plates for each of the combination mutants and incubated for 16 hours at 37 ^◦C.

Following overnight growth, individual colonies were picked and inoculated into 3 mL of LB kan/spec media to be grown overnight at 37 ^◦C. Following overnight growth, samples from each culture were submitted for sanger sequencing to confirm the identity of each of the synthesized combined mutants. Following sequence verification, the aTC dose-response behavior of each combined mutant was characterized using an experimental setup consistent with dose-response measurements made for the previously identified TetR(B) mutants.

As described above, the estimation of the three biophysical parameters (ϵ_D, ϵ_L and γ) for each mutant is divided into two steps. Firstly, as ϵ_L and γ do not affect F C at c = 0 (Equation 1), we directly calculate the value of ϵ_D from the leakiness of an induction curve, which is measured with high precision in our experiments. With the calculated ϵ_D (ϵ_D^{W T} is set to 0 as reference), we then fit the full induction curve to obtain the values of ϵ_L and γ using the method of Bayesian inference. Here, we first construct the statistical model that describes the data generating process based on Equation 1, and specify the prior distributions of the relevant parameters. This enables the derivation of the conditional probability of the parameter values of a mutant given its induction data, known as the posterior distribution. The posterior distribution is then sampled using Markov chain Monte Carlo (MCMC), from which we infer the values of ϵ_L and γ directly. The validity of the statistical model and computational algorithm is fully tested with several metrics, and all details regarding the parameter estimation process briefed here are provided in the Supplementary file.