We begin by showing how allelic manifolds can be used to measure the in vivo strength of TF binding to a specific DNA binding site. This measurement is accomplished by using the TF of interest as a transcriptional repressor. We place the TF binding site directly downstream of the RNAP binding site in a bacterial promoter so that the TF, when bound to DNA, sterically occludes the binding of RNAP. We then measure the rate of transcription from a few dozen variant RNAP binding sites. Transcription from each variant site is assayed in both the presence and in the absence of the TF.

Figure 1A illustrates a thermodynamic model (Shea and Ackers, 1985; Bintu et al., 2005; Sherman and Cohen, 2012) for this type of simple repression. In this model, promoter DNA can be in one of three states: unbound, bound by the TF, or bound by RNAP. Each of these three states is assumed to occur with a frequency that is consistent with thermal equilibrium, that is with a probability proportional to its Boltzmann weight.

Figure 1

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The energetics of protein-DNA binding determine the Boltzmann weight for each state. By convention we set the weight of the unbound state equal to 1. The weight of the TF-bound state is then given by $F=[\mathrm{TF}]K_F$ where $[\mathrm{TF}]$ is the concentration of the TF and $K_F$ is the affinity constant in inverse molar units. Similarly, the weight of the RNAP-bound state is $P=[\mathrm{RNAP}]K_P$. In what follows we refer to $F$ and $P$ as the ‘binding factors’ of the TF-DNA and RNAP-DNA interactions, respectively. We note that these binding factors can also be written as $F=e^{-\mathrm{\Delta }{G}_F/k_{B}T}$ and $P=e^{-\mathrm{\Delta }{G}_P/k_{B}T}$ where $k_B$ is Boltzmann’s constant, $T$ is temperature, and $\mathrm{\Delta }{G}_F$ and $\mathrm{\Delta }{G}_P$ respectively denote the Gibbs free energy of binding for the TF and RNAP. Note that each Gibbs free energy accounts for the entropic cost of pulling each protein out of solution. In what follows, we report $\mathrm{\Delta }G$ values in units of kcal/mol; note that 1 kcal/mol = $1.62k_{B}T$ at 37 °C.

The overall rate of transcription is computed by summing the amount of transcription produced by each state, weighting each state by the probability with which it occurs. In this case we assume the RNAP-bound state initiates at a rate of $t_{\mathrm{sat}}$, and that the other states produce no transcripts. We also add a term, $t_{\mathrm{bg}}$, to account for background transcription (e.g., from an unidentified promoter further upstream). The rate of transcription in the presence of the TF is thus given by 

In the absence of the TF ($F=0$), the rate of transcription becomes 

Our goal is to measure the TF-DNA binding factor $F$. To do this, we create a set of promoter sequences where the RNAP binding site is varied (thus generating an allelic series) but the TF binding site is kept fixed. We then measure transcription from these promoters in both the presence and absence of the TF, respectively denoting the resulting quantities by $t_{+}$ and $t_{-}$ (Figure 1B). Our rationale for doing this is that changing the RNAP binding site sequence should, according to our model, affect only the RNAP-DNA binding factor $P$. All of our measurements are therefore expected to lie along a one-dimensional allelic manifold residing within the two-dimensional space of ($t_{-}$, $t_{+}$) values. Moreover, this allelic manifold should follow the specific mathematical form implied by Equations 1 and 2 when $P$ is varied and the other parameters ($t_{\mathrm{sat}}$, $t_{\mathrm{bg}}$, $F$) are held fixed; see Figure 1C.

The geometry of this allelic manifold is nontrivial. Assuming $F\gg 1$ and $t_{\mathrm{bg}}\ll t_{\mathrm{sat}}$, there are five different regimes corresponding to different values of the RNAP binding factor $P$. These regimes are listed in Figure 1D and derived in Appendix 4. In regime 1, $P$ is so small that both $t_{+}$ and $t_{-}$ are dominated by background transcription, that is ${\displaystyle t_{+}\approx t_{-}\approx t_{\mathrm{b}\mathrm{g}}.}$ $P$ is somewhat larger in regime 2, causing $t_{-}$ to be proportional to $P$ while $t_{+}$ remains dominated by background. In regime 3, both $t_{+}$ and $t_{-}$ are proportional to $P$ with $t_{+}/t_{-}\approx 1/(1+F)$. In regime 4, $t_{-}$ saturates at $t_{\mathrm{sat}}$ while $t_{+}$ remains proportional to $P$. Regime five occurs when both $t_{+}$ and $t_{-}$ are saturated, that is $t_{+}\approx t_{-}\approx t_{\mathrm{sat}}$.

The placement of CRP immediately downstream of RNAP is known to repress transcription (Morita et al., 1988). We therefore reasoned that placing a DNA binding site for CRP downstream of RNAP would allow us to measure the binding factor of that site. Figure 2 illustrates measurements of the allelic manifold used to characterize the strength of CRP binding to the 22 bp site GAATGTGACCTAGATCACATTT. This site contains the well-known consensus site, which comprises two palindromic pentamers (underlined) separated by a 6 bp spacer (Gunasekera et al., 1992). We performed measurements using this CRP site centered at two different locations relative to the transcription start site (TSS): +0.5 bp and +4.5 bp. Note that the first transcribed base is, in this paper, assigned position 0 instead of the more conventional +1, and half-integer positions indicate centering between neighboring nucleotides. To avoid influencing CRP binding strength, the −10 region of the RNAP site was kept fixed in the promoters we assayed while the −35 region of the RNAP binding site was varied (Figure 2A). Promoter DNA sequences are shown in Appendix 1—figure 1.

Figure 2

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We obtained $t_{-}$ and $t_{+}$ measurements for these constructs using a modified version of the colorimetric $\beta$-galactosidase assay of Lederberg (1950) and Miller (1972); see Appendix 2 for details. Our measurements are largely consistent with an allelic manifold having the expected mathematical form (Figure 2B). Moreover, the measurements for promoters with CRP sites at two different positions (+0.5 bp and +4.5 bp) appear consistent with each other, although the measurements for +4.5 bp promoters appear to have lower values for $P$ overall. A small number of data points do deviate substantially from this manifold, but the presence of such outliers is not surprising from a biological perspective (see Discussion). Fortunately, outliers appear at a rate small enough for us to identify them by inspection.

We quantitatively modeled the allelic manifold in Figure 2B by fitting $n+3$ parameters to our $2n$ measurements, where $n=39$ is the number of non-outlier promoters. The $n+3$ parameters were $t_{\mathrm{sat}}$, $t_{\mathrm{bg}}$, $F$, and $P_1$, $P_2$, …, $P_n$, where each $P_i$ is the RNAP binding factor of promoter $i$. Nonlinear least squares optimization was used to infer values for these parameters. Uncertainties in $t_{\mathrm{sat}}$, $t_{\mathrm{bg}}$, and $F$ were quantified by repeating this procedure on bootstrap-resampled data points. See Appendix 3 for details.

These results yielded highly uncertain values for $t_{\mathrm{sat}}$ because none of our measurements appear to fall within regime 4 or 5 of the allelic manifold. A reasonably precise value for $t_{\mathrm{bg}}$ was obtained, but substantial scatter about our model predictions in regime 1 and 2 remain. This scatter likely reflects some variation in $t_{\mathrm{bg}}$ from promoter to promoter, variation that is to be expected since the source of background transcription is not known and the appearance of even very weak promoters could lead to such fluctuations.

These data do, however, determine a highly precise value for the strength of CRP-DNA binding: $F={23.9}_{-2.5}^{+3.1}$ or, equivalently, $\mathrm{\Delta }{G}_F=-1.96\pm 0.07$ kcal/mol. This allelic manifold approach is thus able to measure the strength of TF-DNA binding with a precision of ~ 0.1 kcal/mol. For comparison, the typical strength of a hydrogen bond in liquid water is −1.9 kcal/mol (Markovitch and Agmon, 2007).

We note that CRP forms approximately 38 hydrogen bonds with DNA when it binds to a consensus DNA site (Parkinson et al., 1996). Our result indicates that, in living cells, the enthalpy resulting from these and other interactions is almost exactly canceled by entropic factors. We also note that our in vivo value for $F$ is far smaller than expected from experiments in aqueous solution. The consensus CRP binding site has been measured in vitro to have an affinity constant of $K_F\sim {10}^{11}{\mathrm{M}}^{-1}$ (Ebright et al., 1989). There are probably about 10³ CRP dimers per cell (Schmidt et al., 2016), giving a concentration ${\displaystyle [\mathrm{C}\mathrm{R}\mathrm{P}]\sim {10}^{-6}\mathrm{M}}$. Putting these numbers together gives a binding factor of $F\sim {10}^5$. The nonspecific binding of CRP to genomic DNA and other molecules in the cell, and perhaps limited DNA accessibility as well, might be responsible for this ~ 10⁵-fold disagreement with our in vivo measurements.

Varying cAMP concentrations in growth media changes the in vivo concentration of active CRP in the E. coli strain we assayed (JK10). Such variation is therefore expected to alter the CRP-DNA binding factor $F$. We tested whether this was indeed the case by measuring multiple allelic manifolds, each using a different concentration of ${\displaystyle [\mathrm{c}\mathrm{A}\mathrm{M}\mathrm{P}]}$${\displaystyle [\mathrm{c}\mathrm{A}\mathrm{M}\mathrm{P}]}$[cAMP] when measuring $t_{+}$. These measurements were performed on promoters with CRP binding sites at +0.5 bp (Figure 3A). The resulting data are shown in Figure 3B. To these data, we fit allelic manifolds having variable values for $F$, but fixed values for both $t_{\mathrm{bg}}$ and $t_{\mathrm{sat}}$ ($t_{\mathrm{bg}}=2.30\times {10}^{-3}$ a.u. was inferred in the prior analysis for Figure 2B; $t_{\mathrm{sat}}=15.1$ a.u. was inferred in the subsequent analysis for Figure 5C).

Figure 3

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This procedure allowed us to quantitatively measure changes in the RNAP binding factor $F$, and thus changes in the in vivo concentration of active CRP. Our results, shown in Figure 3C, suggest a nontrivial power law relationship between $F$ and [cAMP]. To quantify this relationship, we performed least squares regression ($\log F$ against $\log [\mathrm{cAMP}]$) using data for the four largest cAMP concentrations; measurements of $F$ for the three other cAMP concentrations have large asymmetric uncertainties and were therefore excluded. We found that $F\propto {[\mathrm{cAMP}]}^{1.41\pm 0.18}$, with error bars representing a 95% confidence interval. We emphasize, however that our data do not rule out a more complex relationship between [cAMP] and $F$.

There are multiple potential explanations for this deviation from proportionality. One possibility is cooperative binding of cAMP to the two binding sites within each CRP dimer. Such cooperativity could, for instance, result from allosteric effects like those described in Einav et al., 2018. Alternatively, this power law behavior might reflect unknown aspects of how cAMP is imported and exported from E. coli cells. It is worth comparing and contrasting this result to those reported in Kuhlman et al. (2007). JK10, the E. coli strain used in our experiments, is derived from strain TK310, which was developed in Kuhlman et al. (2007). In that work, the authors concluded that $F\propto [\mathrm{cAMP}]$, whereas our data leads us to reject this hypothesis. This illustrates one way in which using allelic manifolds to measure how in vivo TF concentrations vary with growth conditions can be useful.

Next we discuss how to measure an activating interaction between a DNA-bound TF and DNA-bound RNAP. A common mechanism of transcriptional activation is ‘stabilization’ (also called ‘recruitment’; see Ptashne, 2003). This occurs when a DNA-bound TF stabilizes the RNAP-DNA closed complex. Stabilization effectively increases the RNAP-DNA binding affinity $K_P$, and thus the binding factor $P$. It does not affect $t_{\mathrm{sat}}$, the rate of transcript initiation from RNAP-DNA closed complexes.

A thermodynamic model for activation by stabilization is illustrated in Figure 4A. Here promoter DNA can be in four states: unbound, TF-bound, RNAP-bound, or doubly bound. In the doubly bound state, a ‘cooperativity factor’ $\alpha$ contributes to the Boltzmann weight. This cooperativity factor is related to the TF-RNAP Gibbs free energy of interaction, $\mathrm{\Delta }{G}_{\alpha }$, via $\alpha =e^{-\mathrm{\Delta }{G}_{\alpha }/k_{B}T}$. Activation occurs when ${\displaystyle \alpha >1}$ (i.e., ${\displaystyle \mathrm{\Delta }{G}_{\alpha }<0}$). The resulting activated transcription rate is given by 

Figure 4

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This can be rewritten as

where

is a renormalized cooperativity that accounts for the strength of TF-DNA binding. As before, $t_{-}$ is given by Equation 2. Note that ${\alpha }^{\prime }\le \alpha$ and that ${\alpha }^{\prime }\approx \alpha$ when $F\gg 1$ and $\alpha \gg 1/F$.

As before, we measure both $t_{+}$ and $t_{-}$ for an allelic series of RNAP binding sites (Figure 4B). These measurements will, according to our model, lie along an allelic manifold resembling the one shown in Figure 4C. This allelic manifold exhibits five distinct regimes (when $t_{\mathrm{sat}}/t_{\mathrm{bg}}\gg {\alpha }^{\prime }\gg 1$), which are listed in Figure 4D.

CRP activates transcription at the lac promoter and at other promoters by binding to a 22 bp site centered at −61.5 bp relative to the TSS. This is an example of class I activation, which is mediated by an interaction between CRP and the C-terminal domain of one of the two RNAP $\alpha$ subunits (the $\alpha$CTDs) (Busby and Ebright, 1999). In vitro experiments have shown this class I CRP-RNAP interaction to activate transcription by stabilizing the RNAP-DNA closed complex.

We measured $t_{+}$ and $t_{-}$ for 47 variants of the lac* promoter (see Appendix 1—figure 1 for sequences). These promoters have the same CRP binding site assayed for Figure 2, but positioned at −61.5 bp relative to the TSS (Figure 5A). They differ from one another in the −10 or −35 regions of their RNAP binding sites. Figure 5B shows the resulting measurements. With the exception of 3 outlier points, these measurements appear consistent with stabilizing activation via a Gibbs free energy of $\mathrm{\Delta }{G}_{\alpha }=-4.05\pm 0.08$ kcal/mol, corresponding to a cooperativity of $\alpha ={712}_{-83}^{+102}$. We note that, with $F=23.9$ determined in Figure 2B, ${\alpha }^{\prime }=\alpha$ to 4% accuracy.

Figure 5

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This observed cooperativity is substantially stronger than suggested by previous work. Early in vivo experiments suggested a much lower cooperativity value, for example 50-fold (Beckwith et al., 1972), 20-fold (Ushida and Aiba, 1990), or even 10-fold (Gaston et al., 1990). These previous studies, however, only measured the ratio $t_{+}/t_{-}$ for a specific choice of RNAP binding site. This ratio is (by Equation 4) always less than $\alpha$ and the differences between these quantities can be substantial. However, even studies that have used explicit biophysical modeling have determined lower cooperativity values: Kuhlman et al. (2007) reported a cooperativity of $\alpha \approx 240$ ($\mathrm{\Delta }{G}_{\alpha }\approx -3.4$ kcal/mol), while Kinney et al. (2010) reported $\alpha \approx 220$ ($\mathrm{\Delta }{G}_{\alpha }\approx -3.3$ kcal/mol). Both of these studies, however, relied on the inference of complex biophysical models with many parameters. The allelic manifold in Figure 4, by contrast, is characterized by only three parameters ($t_{\mathrm{sat}}$, $t_{\mathrm{bg}}$, ${\alpha }^{\prime }$), all of which can be approximately determined by visual inspection.

To test the generality of this approach, we measured allelic manifolds for 11 other potential class I promoter architectures. At every one of these positions we clearly observed the collapse of data to a 1D allelic manifold of the expected shape (Figure 5C). We then modeled these data using values of $\alpha$ and $t_{\mathrm{bg}}$ that depend on CRP binding site location, as well as a single overall value for $t_{\mathrm{sat}}$. The resulting values for $\alpha$ (and equivalently $\mathrm{\Delta }{G}_{\alpha }$) are shown in Figure 5D and reported in Table 1. As first shown by Gaston et al. (1990) and Ushida and Aiba (1990), $\alpha$ depends strongly on the spacing between the CRP and RNAP binding sites. In particular, $\alpha$ exhibits a strong ~ 10.5 bp periodicity reflecting the helical twist of DNA. However, as with the measurement in Figure 5B, the $\alpha$ values we measure are far larger than the $t_{+}/t_{-}$ ratios previously reported by Gaston et al. (1990) and Ushida and Aiba (1990); see Table 1. We also find $t_{\mathrm{sat}}={15.1}_{-0.5}^{+0.6}$ a.u. The single-cell observations of So et al. (2011) suggest that this corresponds to $13.8\pm 6.6$ transcripts per minute. By pure coincidence, the ‘arbitrary unit’ (a.u.) units we use in this paper correspond very closely to ‘transcripts per minute’.

The measurement and modeling of allelic manifolds sidesteps the need to parametrically model how protein-DNA binding affinity depends on DNA sequence. In modeling the allelic manifolds in Figure 5C, we obtained values for the RNAP binding factor, $P=[\mathrm{RNAP}]K_P$, for each variant RNAP binding site from the position of the corresponding data point along the length of the manifold.

RNAP has a very well established sequence motif (McClure et al., 1983). Indeed, its DNA binding requirements were among the first characterized for any DNA-binding protein (Pribnow, 1975). More recently, a high-resolution model for RNAP-DNA binding energy was determined using data from a massively parallel reporter assay called Sort-Seq (Kinney et al., 2010). This position-specific affinity matrix (PSAM) assumes that the nucleotide at each position contributes additively to the overall binding energy (Figure 6A). This model is consistent with previously described RNAP binding motifs but, unlike those motifs, it can predict binding energy in physically meaningful energy units (i.e., kcal/mol). In what follows we denote these binding energies as $\mathrm{\Delta }\mathrm{\Delta }{G}_P$, because they describe differences in the Gibbs free energy of binding between two DNA sites.

Figure 6

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There is good reason to believe this PSAM to be the most accurate current model of RNAP-DNA binding. However, subsequent work has suggested that the predictions of this model might still have substantial inaccuracies (Brewster et al., 2012). To investigate this possibility, we compared our measured values for the Gibbs free energy of RNAP-DNA binding ($\mathrm{\Delta }{G}_P=-k_{B}T\log P$) to binding energies (${\displaystyle \mathrm{\Delta }\mathrm{\Delta }{G}_P}$) predicted using the PSAM from Kinney et al. (2010). These values are plotted against one another in Figure 6B. Although there is a strong correlation between the predictions of the model and our measurements, deviations of 1 kcal/mol or larger (corresponding to variations in $P$ of 5-fold or greater) are not uncommon. Model predictions also systematically deviate from the diagonal, suggesting inaccuracy in the overall scale of the PSAM.

This finding is sobering: even for one of the best understood DNA-binding proteins in biology, our best sequence-based predictions of in vivo protein-DNA binding affinity are still quite crude. When used in conjunction with thermodynamic models, as in Kinney et al. (2010), the inaccuracies of these models can have major effects on predicted transcription rates. The measurement and modeling of allelic manifolds sidesteps the need to parametrically model such binding energies, enabling the direct inference of Gibbs free energy values for each assayed RNAP binding site.

E. coli TFs can regulate multiple different steps in the transcript initiation pathway (Lee et al., 2012; Browning and Busby, 2016). For example, instead of stabilizing RNAP binding to DNA, TFs can activate transcription by increasing the rate at which DNA-bound RNAP initiates transcription (Roy et al., 1998), a process we refer to as ‘acceleration’. CRP, in particular, has previously been reported to activate transcription in part by acceleration when positioned appropriately with respect to RNAP (Niu et al., 1996; Rhodius et al., 1997).

We investigated whether allelic manifolds might be used to distinguish activation by acceleration from activation by stabilization. First we generalized the thermodynamic model in Figure 4A to accommodate both $\alpha$-fold stabilization and $\beta$-fold acceleration (Figure 7A). This is accomplished by using the same set of states and Boltzmann weights as in the model for stabilization, but assigning a transcription rate $\beta t_{\mathrm{sat}}$ (rather than just $t_{\mathrm{sat}}$) to the TF-RNAP-DNA ternary complex. The resulting activated rate of transcription is given by

Figure 7

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This simplifies to

where ${\alpha }^{\prime }$ is the same as in Equation 5 and

is a renormalized version of the acceleration rate $\beta$. The resulting allelic manifold is illustrated in Figure 7C. Like the allelic manifold for stabilization, this manifold has up to five distinct regimes corresponding to different values of $P$ (Figure 7D). Unlike the stabilization manifold however, $t_{+}\ne t_{-}$ in the strong RNAP binding regime (regime 5); rather, $t_{+}\approx {\beta }^{\prime }{t}_{\mathrm{sat}}$ while $t_{-}\approx t_{\mathrm{sat}}$.

We asked whether class I activation by CRP has an acceleration component. Previous in vitro work had suggested that the answer is ‘no’ (Malan et al., 1984; Busby and Ebright, 1999), but our allelic manifold approach allows us to address this question in vivo. We proceeded by assaying promoters containing variant alleles of the consensus RNAP binding site (Figure 8A). Note that the consensus RNAP site is 1 bp shorter than the lac* RNAP site (Appendix 1—figure 1, panel C versus panel B). We therefore positioned the CRP binding site at −60.5 bp in order to realize the same spacing between CRP and the −35 element of the RNAP binding site that was realized in −61.5 bp non-consensus promoters.

Figure 8

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The resulting data (Figure 8B) are seen to largely fall along the previously measured all-stabilization allelic manifold in Figure 5B. In particular, many of these data points lie at the intersection of this manifold with the $t_{+}=t_{-}$ diagonal. We thus find that $\beta \approx 1$ for CRP at −61.5 bp. To further quantify possible $\beta$ values, we fit the acceleration model in Figure 7 to each dataset shown in Figure 5B, assuming a fixed value of $t_{\mathrm{sat}}=15.1$ a.u. The resulting inferred values for $\beta$, shown in Figure 8C, indicate little if any deviation from $\beta =1$. Our high-precision in vivo results therefore substantiate the previous in vitro results of Malan et al. (1984) regarding the mechanism of class I activation.

Many E. coli TFs participate in what is referred to as class II activation (Browning and Busby, 2016). This type of activation occurs when the TF binds to a site that overlaps the −35 element (often completely replacing it) and interacts directly with the main body of RNAP. CRP is known to participate in class II activation at many promoters (Keseler et al., 2011; Salgado et al., 2013), including the galP1 promoter, where it binds to a site centered at position −41.5 bp (Adhya, 1996). In vitro studies have shown CRP to activate transcription at −41.5 bp relative to the TSS through a combination of stabilization and acceleration (Niu et al., 1996; Rhodius et al., 1997).

We sought to reproduce this finding in vivo by measuring allelic manifolds. We therefore placed a consensus CRP site at −41.5 bp, replacing much of the −35 element in the process, and partially mutated the −10 element of the RNAP binding site (Figure 9A). Surprisingly, we observed that the resulting allelic manifold saturates at the same $t_{\mathrm{sat}}$ value shared by all class I promoters. Thus, CRP appears to activate transcription in vivo solely through stabilization, and not at all through acceleration, when located at −41.5 bp relative to the TSS (Figure 9B).

Figure 9

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The genome-wide distribution of CRP binding sites suggests that CRP also participates in class II activation when centered at −40.5 bp (Keseler et al., 2011; Salgado et al., 2013). When assaying this promoter architecture, however, we obtained a 2D scatter of points that did not collapse to any discernible 1D allelic manifold (Figure 9D). Some of these promoters exhibit activation, some exhibit repression, and some exhibit no regulation by CRP.

These observations complicate the current understanding of class II regulation by CRP. Our in vivo measurements of CRP at −41.5 bp call into question the mechanism of activation previously discerned using in vitro techniques. The scatter observed when CRP is positioned at −40.5 bp suggests that, at this position, the −10 region of the RNAP binding site influences the values of at least two relevant biophysical parameters (not just $P$, as our model predicts). A potential explanation for both observations is that, because CRP and RNAP are so intimately positioned at class II promoters, even minor changes in their relative orientation caused by differences between in vivo and in vitro conditions or by changes in RNAP site sequence could have a major effect on CRP-RNAP interactions. Such sensitivity would not be expected to occur in class I activation, due to the flexibility with which the RNAP $\alpha$CTDs are tethered to the core complex of RNAP.

In each of our experiments, we quantitatively measured transcription from an allelic series of variant RNAP binding sites, each site embedded in a fixed promoter architecture. Two expression measurements were made for each variant promoter: $t_{+}$ was measured in the presence of the active form of CRP, while $t_{-}$ was measured in the absence of active CRP. This yielded a data point, $(t_{-},t_{+})$, in a two-dimensional measurement space. We had expected the data points thus obtained for each allelic series to collapse to a 1D curve (the allelic manifold), with different positions along this manifold corresponding to different values of RNAP-DNA binding affinity. Such collapse was indeed observed in all but one of the promoter architectures we studied. By fitting the parameters of quantitative biophysical models to these data, we obtained in vivo values for the Gibbs free energy ($\mathrm{\Delta }G$) of a variety of TF-DNA and TF-TF interactions.

In Part 1, we showed how measuring allelic manifolds for promoters in which a DNA-bound TF occludes RNAP can allow one to precisely measure the $\mathrm{\Delta }G$ of TF-DNA binding. We demonstrated this strategy on promoters where CRP occludes RNAP, thereby obtaining the $\mathrm{\Delta }G$ for a CRP binding site that was used in subsequent experiments. As an aside, we demonstrated how performing such measurements in different concentrations of the small molecule cAMP allowed us to quantitatively measure in vivo changes in active CRP concentration.

In Part 2, we showed how allelic manifolds can be used to measure the $\mathrm{\Delta }G$ of TF-RNAP interactions. We used this strategy to measure the stabilizing interactions by which CRP up-regulates transcription at a variety of class I promoter architectures. Our strategy consistently yielded $\mathrm{\Delta }G$ values with an estimated precision of $\sim 0.1$ kcal/mol. As an aside, we showed how $\mathrm{\Delta }G$ values for RNAP-DNA binding could also be obtained from these data. Notably, these $\mathrm{\Delta }G$ measurements for RNAP-DNA binding were seen to deviate substantially from sequence-based predictions using an established position-specific affinity matrix (PSAM) for RNAP. This highlights just how difficult it can be to accurately predict TF-DNA binding affinity from DNA sequence.

In Part 3, we showed how allelic manifolds can allow one to distinguish between two potential mechanisms of transcriptional activation: ‘stabilization’ (a.k.a. ‘recruitment’) and ‘acceleration’. Applying this approach to the data from Part 2, we confirmed (as expected) that class I activation by CRP does indeed occur through stabilization and not acceleration. As an aside, we pursued this approach at two class II promoters. In contrast to prior in vitro studies (Niu et al., 1996; Rhodius et al., 1997), no acceleration was observed when CRP was positioned at −41.5 bp relative to the TSS. Even more unexpectedly, no 1D allelic manifold was observed at all when CRP was positioned at −40.5 bp. This last finding indicates that the variant RNAP binding sites we assayed control at least one functionally important biophysical quantity in addition to RNAP-DNA binding affinity.

An important caveat is that our $\mathrm{\Delta }G$ measurements assume that the true transcription rates (of which we obtain only noisy measurements) exactly fall along a 1D allelic manifold of the hypothesized mathematical form. These assumptions are well-motivated by the data collapse that we observed for all except one promoter architecture. But for some promoter architectures, there were a small number of ‘outlier’ data points that we judged (by eye) to deviate substantially from the inferred allelic manifold. The presence of a few outliers makes sense biologically: the random mutations we introduced into variant RNAP binding sites will, with some nonzero probability, either shift the position of the RNAP site or create a new binding site for some other TF. However, even for promoters that exhibit clear clustering of 2D data around a 1D curve, the deviations of individual non-outlier data points from our inferred allelic manifold were often substantially larger than the experimental noise that we estimated from replicates. It may be that the biological cause of outliers is not qualitatively different from what causes these smaller but still detectable deviations from our assumed model.

The low-throughput experimental approach we pursued here also has important limitations. Each of the 448 variant promoters for which we report data was individually catalogued, sequenced, and assayed for both $t_{+}$ and $t_{-}$ in at least three replicate experiments. We opted to use a low-throughput colorimetric assay of $\beta$-galactosidase activity (Lederberg, 1950; Miller, 1972) because this approach is well established in E. coli to produce a quantitative measure of transcription with high precision and high dynamic range. Such assays have also been used by other groups to develop sophisticated biophysical models of transcriptional regulation (Kuhlman et al., 2007; Cui et al., 2013). However, this low-throughput approach has limited utility because it cannot be readily scaled up.

Our reliance on cAMP as a small molecule effector of CRP presents a second limitation. In our experiments, we controlled the in vivo activity of CRP by growing a specially designed strain of E. coli in either the presence (for $t_{+}$) or absence ($t_{-}$) of cAMP. This mirrors the strategy used by Kuhlman et al. (2007), and the validity of this approach is attested to by the calibration data shown in Appendix 2—figure 1. However, controlling in vivo TF activity using small molecules has many limitations. Most TFs cannot be quantitatively controlled with small molecules, and those that can often require special host strains (e.g., see Kuhlman et al., 2007). Moreover, varying the in vivo concentration of a TF can affect cellular physiology in ways that can confound quantitative measurements.

MPRAs performed on array-synthesized promoter libraries should be able to overcome both of these experimental limitations. Current MPRA technology is able to quantitatively measure gene expression for ${\displaystyle \gtrsim }$10⁴ transcriptional regulatory sequences in parallel. We estimate that this would enable the simultaneous measurement of ~ 10² highly resolved allelic manifolds, each manifold representing a different promoter architecture. Moreover, by using array-synthesized promoters in conjunction with MPRAs, one can measure $t_{+}$ and $t_{-}$ by systematically altering the DNA sequence of TF binding sites, rather than relying on small molecule effectors of each TF. This capability would, among other things, enable biophysical studies of promoters that have multiple binding sites for the same TF; in such cases it might make sense to use measurement spaces having more than two dimensions.

Will allelic manifolds be useful for understanding transcriptional regulation in eukaryotes? Both Sort-Seq MPRAs (Sharon et al., 2012; Weingarten-Gabbay et al., 2017) and RNA-Seq MPRAs (Melnikov et al., 2012; Kwasnieski et al., 2012; Patwardhan et al., 2012) are well established in eukaryotes so, on a technical level, experiments analogous to those described here should be feasible. The bigger question, we believe, is whether the results of such experiments would be interpretable. Eukaryotic transcriptional regulation is far more complex than transcriptional regulation in bacteria. Still, we believe that pursuing the measurement and modeling of allelic manifolds in this context is worthwhile. Despite the underlying complexities, simple ‘effective’ biophysical models might work surprisingly well. Similar approaches might also be useful for studying other eukaryotic regulatory processes that are compatible with MPRAs, such as alternative splicing (Wong et al., 2018).

Based on these results, we advocate a very different approach to dissecting cis-regulatory grammar than has been pursued by other groups. Rather than attempting to identify a single quantitative model that can explain regulation by many different arrangements of TF binding sites (Gertz et al., 2009; Sharon et al., 2012; Mogno et al., 2013; Smith et al., 2013; Levo and Segal, 2014; White et al., 2016), we suggest focused studies of the biophysical interactions that result from specific TF binding site arrangements. The measurement and modeling of allelic manifolds provides a systematic and stereotyped way of doing this. By coupling this approach with MPRAs, it should be possible to perform such studies on hundreds of systematically varied regulatory sequence architectures in parallel. General rules governing cis-regulatory grammar might then be identified empirically. We suspect that this bottom-up strategy to studying cis-regulatory grammar is likely to reveal regulatory mechanisms that would be hard to anticipate in top-down studies.

Appendix 1—figure 1

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Expression measurements were performed on cells grown in rich defined media (RDM; purchased from Teknova) (Neidhardt et al., 1974) supplemented with 10 mM NaHCO${}_3$, 1 mM IPTG (Sigma), and 0.2% glucose. We refer to this media as RDM’. RDM’ was further supplemented with 50 µg/ml kanamycin (Sigma) when growing cells, as well as 250 µM cAMP (Sigma) when measuring $t_{+}$.

Expression measurements were performed in E. coli strain JK10, which has genotype ΔcyaA ΔcpdA ΔlacY ΔlacZ ΔdksA. JK10 is derived from strain TK310 (Kuhlman et al., 2007), which is ΔcyaA ΔcpdA ΔlacY. The ΔcyaA ΔcpdA mutations prevent TK310 from synthesizing or degrading cAMP, thus allowing in vivo cAMP concentrations to be quantitatively controlled by adding cAMP to the growth media. Into TK310 we introduced the ΔlacZ mutation, yielding strain DJ33; this mutation enables the use of $\beta$-galactosidase activity assays for measuring plasmid-based lacZ expression. In our initial experiments, we found that the growth rate of DJ33 in RDM’ varied strongly with the amount of cAMP added to the media. Fortunately, we isolated a spontaneous knock-out mutation in dksA (thus yielding JK10), which caused the growth rate (~ 30 min doubling time) in RDM’ to be independent of cAMP concentrations below ~500 µM. We note that JK10 will not grow in minimal media in the absence of cAMP. The TK310, DJ33, and JK10 genotypes were confirmed by whole genome sequencing using the PureLink Genomic DNA Mini Kit (ThermoFisher) for extracting genomic DNA from cultured cells and the Nextera XT DNA Library Preparation Kit (Illumina) for preparing whole-genome sequencing libraries.

Expression of the lacZ gene was driven from variants of a plasmid we call pJK48. These reporter constructs were cloned as follows. We started with the vector pJK14 from Kinney et al. (2010). pJK14 contains a pSC101 origin of replication (~ 5 copies per cell; Thompson et al., 2018), a kanamycin resistance gene, and a ccdB cloning cassette positioned immediately upstream of a gfpmut2 reporter gene and flanked by outward-facing BsmBI restriction sites. First, the gfpmut2 gene in this vector was replaced with lacZ, yielding pJK47. Next, the ribosome binding site in the 5’ UTR of lacZ was weakened, yielding pJK47.419; this weakening prevents lacZ expression from substantially slowing cell growth in RDM’. pJK47.419 was propagated in DB3.1 E. coli (Invitrogen), which is resistant to the CcdB toxin. The promoters we assayed were variants of what we call the ‘lac*’ promoter. The lac* promoter is similar to the endogenous lac promoter of E. coli MG1655 except for (i) it contains a CRP binding site with a consensus right pentamer and (ii) it contains mutations that were introduced in an effort to remove previously reported cryptic promoters (Reznikoff, 1992). Promoter-containing insertion cassettes were created through overlap-extension PCR and flanked by outward-facing BsaI restriction sites. All primers were ordered from Integrated DNA Technologies. Note that some of the primers used to create these inserts were synthesized using pre-mixed phosphoramidites at specified positions; this is how a 24% mutation rate in the −10 or −35 regions of the RNAP binding site was achieved. The resulting promoter sequences are illustrated in Appendix 1—figure 1. To clone variants of pJK48, we separately digested the pJK47.419 vector with BsmBI (NEB) and the appropriate insert with BsaI (NEB). Digests were then cleaned up (Qiagen PCR purification kit) and ligated together in a 1:1 molar ratio for 1 hr using T4 DNA ligase (Invitrogen). After 90 min dialysis, plasmids were transformed into electrocompetent JK10 cells. Individual clones were plated on LB supplemented with kanamycin (50 µg/ml). After initial cloning and plating, each colony was re-streaked, grown in LB+kan, and stored as a catalogued glycerol stock. The promoter region of each clone was sequenced in both directions. Only plasmids with validated promoter sequences were used for the measurements presented in this paper. The promoter sequences of all 448 plasmids used in this study, as well as their measured $t_{+}$ and $t_{-}$ values, are provided at https://github.com/jbkinney/17_inducibility (copy archived at https://github.com/elifesciences-publications/17_inducibility).

Appendix 2—figure 1

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We obtained $t_{+}$ and $t_{-}$ measurements for each promoter as follows. First, the corresponding E. coli clone was streaked out on LB+kan agar and grown overnight. A colony was then picked and used to inoculate a 1.5 ml overnight LB+kan liquid culture. Either 8 µl, 6 µl, or 4 µl of the overnight culture were then diluted into 200 µl RDM’+kan. 25 µl of each dilution was then added to 175 µl RDM’+kan in a 96-well optical bottom plate and supplemented with either 0 µM cAMP (for $t_{-}^{\mathrm{raw}}$), 250 µM cAMP (for $t_{+}^{\mathrm{raw}}$), or another cAMP concentration (for some $t_{+}^{\mathrm{raw}}$ measurements in Figure 3). The plate was then covered with Breathe-Easier film (USA Scientific) and cells were cultured for $\sim 3$ hr at 37 °C, shaking at 900 RPM in a microplate shaker. During this time, 5.5 ml of lysis buffer was freshly prepared using 1.5 ml RDM’, 4.0 ml PopCulture reagent (Millipore), 114 µl of 35 mg/ml chloramphenicol (Sigma), and 44 µl of 40 U/µl rLysozyme (Sigma).

Microplate film was removed and cell density (quantified by $A_{600}$) was measured using an Epoch 2 Microplate Spectrophotometer (BioTek). Cells were then lysed by adding 25 µl lysis buffer to each microplate well, incubating the microplate at room temperature for 10 min without shaking, then cooling the microplate at 4 °C for a minimum of 15 min. In each well of a 96-well optical bottom plate, 50 µl of lysate was then added to 50 µl of pre-chilled Z-buffer (Miller, 1972) containing 1 mg/ml ONPG (Sigma). Samples were sealed with optical film and both $A_{420}$ and $A_{550}$ were periodically measured in the plate reader over an extended period of time (every 1.5 min for 1 hr or every 15 min for 10 hr, depending on the level of expression expected).

The raw expression levels were quantified from these absorbance data using the formula

where $V$ = 50 is the volume of lysate in µl added to the ONPG reaction, $\mathrm{\Delta }T$ is the change in time from the beginning of the measurement, and $\mathrm{\Delta }{A}_X$ indicates a change in absorbance at $X$ nm over this time interval. Only data from wells with ${\displaystyle A_{600}\lesssim 0.5}$ were analyzed. Note that the $A_{550}$ term in Equation 9 is not multiplied by 1.75 as it is in Miller (1972). This is because our $A_{550}$ measurements are used to compensate for condensation on the microplate film, not cellular debris as in Miller (1972); our lysis procedure produces no detectable cellular debris. In practice, Equation 9 was not evaluated using individual measurements, but was computed from the slope of a line fit to all of the non-saturated absorbance measurements. Raw $A_{420}$, $A_{550}$, and $A_{600}$ values, as well as our analysis scripts, are available at https://github.com/jbkinney/17_inducibility (copy archived at https://github.com/elifesciences-publications/17_inducibility). Median values from at least three independent Miller measurements (and often more) were used to define each measurement shown in the main figures.

Because we controlled the in vivo activity of CRP by supplementing media with or without cAMP, we tested whether CRP-independent promoters produce measurements that vary between these growth conditions. Specifically, we measured $t_{-}^{\mathrm{raw}}$ (in 0 µM cAMP) and $t_{+}^{\mathrm{raw}}$ (in 250 µM cAMP) for 39 promoters in which the CRP binding site was replaced with a ‘null’ site (see Appendix 1—figures 1B and C). These measurements are plotted in Appendix 2—figure 1, and show a slight bias. To correct for this bias, we use an unadjusted $t_{+}=t_{+}^{\mathrm{raw}}$ together with an adjusted $t_{-}=0.855\times t_{-}^{\mathrm{raw}}$ throughout the main text. Note that $t_{+}=t_{+}^{\mathrm{raw}}$ was used for all nonzero cAMP concentrations, including those in Figure 3B that differ from 250 µM. Some upward bias is therefore possible in these $t_{+}$ measurements, but we do not expect this to greatly affect our conclusions.

Allelic manifold parameters were fit to measured $t_{+}$ and $t_{-}$ values as follows. First, outlier data points were called by eye and excluded from the parameter fitting procedure. We denote the remaining measurements using $t_{+}^{i,\mathrm{data}}$ and $t_{-}^{i,\mathrm{data}}$, where $i=1,2,\mathrm{\dots }n$ indexes the $n$ non-outlier data points. Corresponding model predictions $t_{+}^i(\theta )$ and $t_{-}^i(\theta )$, where $\theta$ denotes model parameters, were then fit to these data using nonlinear least squares optimization. Specifically, we inferred parameters ${\displaystyle {\theta }^{\ast }={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{\theta }\mathcal{ℒ}(\theta )}$ where the loss function is given by

These optimal parameter values ${\displaystyle {\theta }^{\ast }}$ were used to generate the best-estimate allelic manifolds, which are plotted in black in the main figures. Uncertainties in $\theta$ were estimated by performing the same inference procedure on bootstrap-resampled data. For each variable ${\displaystyle X\in \{F,P,{\alpha }^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }},t_{\mathrm{s}\mathrm{a}\mathrm{t}},t_{\mathrm{b}\mathrm{g}}\}}$, we report

where $X_{50}$, $X_{84}$, and $X_{16}$ respectively denote the median, 84th percentile, and 16th percentile of $X$ values obtained from bootstrap resampling. In the case of $X\in \{F,P,\alpha \}$, we also report

where 1 kcal/mol = $1.62k_{B}T$ at 37 °C. We now describe each specific inference procedure in more detail.

Appendix 4—figure 1

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Each transcription rate modeled in this work is a sigmoidal function of the unitless RNAP-DNA binding factor $P$. As such, a log-log plot of transcription $t$ as a function of $P$ reveals a sigmoidal curve having three distinct regimes. The 'minimal' regime of this induction curve comprises values of $P$ that are sufficiently small for $t$ to be well-approximated by its smallest value ($t_{\mathrm{bg}}$ in all cases). The 'maximal' regime occurs when $P$ is so large that $t$ is well-approximated by its largest value (either $t_{\mathrm{sat}}$ or ${\beta }^{\prime }{t}_{\mathrm{sat}}$). Between these maximal and minimal regimes lies a 'linear' regime in which $t$ is approximately proportional to $P$.

For unregulated transcription, which in this paper is denoted $t_{-}$, these three regimes are given by

see Appendix 4—figure 1A. For transcription that is repressed by occlusion (with ${\displaystyle F\gg 1}$), which we denote here by ${\displaystyle t_{+}^{\mathrm{o}\mathrm{c}\mathrm{c}}}$, these three regimes are shifted (relative to ${\displaystyle t_{-}}$) to larger values of ${\displaystyle P}$ by a factor of approximately ${\displaystyle F}$. As a result,

see Appendix 4—figure 1B. By contrast, for transcription that is activated by stabilization, denoted here by ${\displaystyle t_{+}^{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}}$, these three regimes shift (relative to ${\displaystyle t_{-}}$) to lower values of ${\displaystyle P}$ by a factor of ${\displaystyle 1/{\alpha }^{\mathrm{\prime }}}$, giving

see Appendix 4—figure 1C. For transcription that is activated partially by acceleration and partially by stabilization, here denoted by ${\displaystyle t_{+}^{\mathrm{a}\mathrm{c}\mathrm{c}}}$, two parameters govern the shape of the induction curve. As a result, the boundary between the minimal and linear regimes are shifted (relative to ${\displaystyle t_{-}}$) to lower values of ${\displaystyle P}$ by a factor of ${\displaystyle 1/{\alpha }^{\mathrm{\prime }}{\beta }^{\mathrm{\prime }}}$, while the boundary between the linear regime and the maximal regime is shifted down by a factor of only ${\displaystyle 1/{\alpha }^{\mathrm{\prime }}}$. As a result,

see Appendix 4—figure 1D.

Each allelic manifold described in the main text has five distinct regimes. These arise from overlaps between the three regimes of ${\displaystyle t_{-}}$ and the three regimes of ${\displaystyle t_{+}}$. Specifically, the five regimes of the allelic manifold for repression by occlusion, which are listed in Figure 1D, arise from the overlaps between the three regimes for ${\displaystyle t_{-}}$ and the three regimes for ${\displaystyle t_{+}^{\mathrm{o}\mathrm{c}\mathrm{c}}}$. These overlaps are indicated in Appendix 4—figure 1E. Similarly, the five regimes of the allelic manifold for activation by stabilization (Figure 4D) arise from the overlaps between the regimes of ${\displaystyle t_{-}}$ and ${\displaystyle t_{+}^{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}}$, illustrated in Appendix 4—figure 1F, while the regimes of the manifold for activation by acceleration (Figure 7D) arise from overlaps between the regimes of ${\displaystyle t_{-}}$ and ${\displaystyle t_{+}^{\mathrm{a}\mathrm{c}\mathrm{c}}}$, illustrated in Appendix 4—figure 1G.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Stirling Churchman, Barak Cohen, David McCandlish, Bryce Nickels, and Saurabh Sinha for helpful discussions. We also thank Naama Barkai, Ulrich Gerland, Richard Neher, and one anonymous referee for reviewing this manuscript and providing helpful feedback. This work was supported by a CSHL/Northwell Health Alliance grant to JBK and by NIH Cancer Center Support Grant 5P30CA045508.

1. Received: July 31, 2018
2. Accepted: November 27, 2018
3. Version of Record published: December 20, 2018 (version 1)

© 2018, Forcier et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.