### The native input function of the *araBAD* promoter has a broad input dynamic range

The input dynamic range is defined as the range of inputs over which the output changes significantly. Operationally, following Goldbeter and Koshland [36–38], we define the input dynamic range as the ratio *R* of input levels at which the system shows 90% and 10% of its maximal output (Figure 2). For a Hill curve with coefficient *n*, the input dynamic range is *R* = 81^{1/n}. Thus, Michaelis-Menten like curves with *n* = 1 show *R* = 81, steeper sigmoidal curves with *n* = 2 show *R* = 9, and very steep cooperative curves with *n* = 4 show *R* = 3.

In order to determine the input dynamic range of *E. coli* promoters we used a fluorescent reporter automated assay [35, 39], with strains from the comprehensive *E. coli* transcription reporter library [40]. Each strain bears a low-copy plasmid with a green fluorescent protein gene under the control of a full length copy of the promoter of interest. In this study we used reporters for the *araBAD* and *araC* promoters in *E. coli* strain MG1655 (see Methods).

Reporter strains were grown on glucose minimal medium containing saturating amount of cAMP (30 mM, to fully activate CRP) and increasing amounts of L-Arabinose [35, 41]. Promoter activity (PA) was defined as the rate of GFP production per OD (optical density) unit, PA = dGFP/dt/OD (see Methods). The input functions were derived from the promoter activity averaged over a window that spans 1-2 cell generations in exponential phase (5-7 hours after initial 1:600 inoculation). Over this time window, promoter activity was constant to a good approximation (see Additional File 1 for fluorescence and growth curves, p. 2-3, Figure S1 and S2 respectively).

The promoter activity of the *araBAD* in the parental strain (wild-type *araC* regulation, U424) as a function of arabinose concentration is shown in Figure 3a. At low arabinose levels (below about 10 μM arabinose) the fluorescence of the reporter is indistinguishable from the cells auto-fluorescence background. The input function reaches 10% of its maximal value at arabinose levels of about 0.1 mM, and 90% of its maximal value at about 10 mM. Fitting a Hill curve to the input function results in an apparent Hill coefficient of *n* = 1 ± 0.3 (s.e.), and halfway induction point of *K* = 1.1 ± 0.4 mM (s.e.). The input dynamic range is *R* = 100 ± 40 (s.e.). These results are similar to measurements of the input function of the *araBAD* reporter strain in wild-type MG1655 (U429) [35], and are consistent with the expected value for a curve with Hill coefficient equal to *n* = 1, in which *R* = 81.

### The *araC* gene is induced by arabinose

We also tested the dependence of the *araC* promoter activity on arabinose. Since AraC negatively regulates its own promoter, arabinose is expected to affect *araC* expression. Indeed, using an *araC* reporter strain (U428), we find that arabinose increases the activity of the *araC* promoter in a dose-dependent manner (Figure 4) [35].

### Disruption of negative auto-regulation of *araC* reduces the input dynamic range of *araBAD*

To study the role of the negative auto-regulation of *araC* on the input dynamic range of its downstream genes, we decoupled *araC* expression from its negative auto-regulation (Figure. 1c). For this purpose we deleted the *araC* open reading frame from the chromosome of the wild-type strain MG1655 and re-introduced *araC* on a plasmid (pZE11) which provides constitutive expression (strain U426, see Methods). The plasmid has a *tetR* controlled promoter, repressed by a chromosomal *tetR* gene. Without induction, this plasmid produces levels of AraC that are comparable to the induced wild-type AraC level, as assessed from the maximal promoter activity of the *araBAD* reporter. It should be noted that the parental strain in this study (U424) also contains chromosomal *tetR* as well as an emptly pZE11 vector, in order to preserve genotypic identity between the two strains.

We find that in the absence of NAR, the arabinose-dependent input function of *araBAD* is significantly steeper than the parental input function (Figure 3b,c), with an apparent Hill coefficient of *n* = 1.9 ± 0.4 (s.e.), and halfway induction point of *K* = 42 ± 0.6 mM (s.e.). The measured input dynamic range spans between 14 mM - 135 mM, and thus has *R* = 10 ± 3 (s.e.), in comparison to *R* = 100 ± 40 (s.e.) in the parental strain. Thus, decoupling *araC* from its negative auto-regulation reduces the input dynamic-range of its downstream genes by about an order of magnitude.

### A model of NAR and increased input dynamic range

What is the main effect at play that allows negative auto-regulation to increase input dynamic range? To understand this, we analyzed a mathematical model of the NAR motif. We sought to make the model as simple as possible, in order to be able to understand it intuitively, and at the same time not too simple so as not to lose the essence of the problem. A more comprehensive model, based on mass-action kinetics, which includes a dual transcription factor that acts as both a repressor and an activator, is given in Additional File 1 (p. 5-7).

Consider a transcription factor whose concentration is *X*, that binds its inducer *s* with a dissociation constant *K*_{
s
}. The amount of *X* bound to *s*, which is the active form of the transcription factor *X**, is described by Michaelis-Menten binding:

The active transcription factor *X** binds the promoter of a downstream gene *Z* with Michaelis-Menten-like kinetics, so that the steady-state level of the *Z* gene product is:

Where *K*_{
z
}is the dissociation constant of *X*^{*}from the promoter of *Z*, *β*_{
Z
}is the maximal production rate of *Z*, and *α* is its degradation/dilution rate [1].

Without negative auto-regulation, the concentration of *X* is independent of the inducer levels. We denote this constant level *X*_{
0
}. Using Eq. (1) in Eq. (2) with *X = X*_{
0
}results in a sigmoidal regulation function with an input dynamic range of *R* = 9.

It is at this point that negative auto-regulation has an important effect: instead of a constitutive level of *X*, negative auto-regulation allows the signal *s* to modify the concentration of *X*, an effect termed direct coupling [42]. With negative auto-regulation of the type found in the *ara* system, the promoter that encodes *X* is repressed by free *X*, (denoted *X*_{
f
}) a repression which is relieved when *X* is bound to the signal.

To analyze this, consider the rate of production of *X* that is repressed by *X*_{
f
}[19], balanced by degradation/dilution of the protein at rate *α*, so that:

Where *K*_{
x
}is the dissociation constant of *X* from its own promoter, and the free *X* (*X*_{
f
}) is given by the unbound fraction, *X*_{
f
}*= X-X**:

Substituting Eq. (5) into Eq. (4) and assuming strong binding of the regulator to its own promoter *K*_{
x
}*<<X*_{
f
}, one finds that at steady-state the *level of X increases as the square root of the input signal s*:

Where *A*^{2}*= K*_{
x
}*β*_{
x
}*/α*. In other words, the transcription factor (*X*) levels increases with the signal (*s*) levels (see the relationship between AraC and L-Arabinose in Figure 4).

Using this expression for *X* instead of *X*_{
0
}in Eq. (2), results in an input-function that is less steep, because of the square-root dependence of *X* on *s*:

Where . This function has an input dynamic range of *R* = 81, which is 9 fold wider than that of Eq. (3). Thus, NAR increases the input dynamic range.

Note that the assumption *K*_{
x
}*<<X*_{
f
}is not crucial for the increased input dynamic range, and was used only for the sake of simplicity. In Additional File 1 we present a full mass-action model, without these assumptions, and show that the present considerations apply as well.

We further investigated the effect of NAR on input functions with different cooperativity in the binding of the TF to the promoter, as described by Hill equations. In the present system, the *araBAD* input function without NAR has an apparent Hill coefficient of about 2, suggesting that the AraC regulator is cooperative with *n* = 2. In Figure 5 we describe the results of the model with regulators with degrees of apparent cooperativity of the regulator ranging between *n* = 1 and *n* = 5. It is seen that NAR increases the input dynamics range in all cases. For example, at *n* = 2, the input dynamic range without NAR is *R* = 9, but can reach up to *R* ~ 1000 with NAR (the values observed above for the *ara* system are about *R* = 10 without NAR and *R* = 100 with NAR).

Furthermore, the model explains how the change in regulator levels caused by NAR can cause a shift in the halfway induction point *K* of downstream genes (relative to no NAR). The direction and size of the shift depends on the mode of regulation. For repressors, *K* generally increases with regulator levels, whereas activators show the converse dependence. Since AraC both activates and represses *araBAD*, the detailed model in Additional File 1 explains the observed increase in *K* shown in Figure 3 (Additional File 1, p.8-9, Figure S3).

To summarize the conclusions of this analysis, negative auto-regulation causes regulator levels to increase with inducer level. This enhances the input dynamic range by extending the range of inputs that can affect the downstream genes.