Evolution and Dynamics of Regulatory Architectures Controlling Polymyxin B Resistance in Enteric Bacteria

Complex genetic networks consist of structural modules that determine the levels and timing of a cellular response. While the functional properties of the regulatory architectures that make up these modules have been extensively studied, the evolutionary history of regulatory architectures has remained largely unexplored. Here, we investigate the transition between direct and indirect regulatory pathways governing inducible resistance to the antibiotic polymyxin B in enteric bacteria. We identify a novel regulatory architecture—designated feedforward connector loop—that relies on a regulatory protein that connects signal transduction systems post-translationally, allowing one system to respond to a signal activating another system. The feedforward connector loop is characterized by rapid activation, slow deactivation, and elevated mRNA expression levels in comparison with the direct regulation circuit. Our results suggest that, both functionally and evolutionarily, the feedforward connector loop is the transitional stage between direct transcriptional control and indirect regulation.

sponse regulator protein were aligned using ClustalX [13] and subjected to maximum parsimony and nonparametric bootstrap resampling analysis as implemented in PAUP* (version 4.0b10).
Orthology of the PbgP proteins was assigned based on genomic context. All pbgP homologues were found to be the first gene in a seven-gene operon [14]. Promoter region of the pbgP orthologs were examined manually and with SOAR-TOOLS Binding Site Searching (http://soartools.wustl.edu) for the consensus binding sites for the PhoP [15][16][17] and PmrA [18][19][20] proteins.

Description of the mathematical models
The mathematical models for the feedforward connector loop (FCL), the feedforward loop    For the FFL model, the constituting ODEs are as follows: In Equations 6-8, the correspondence between the concentration variables and the chemical species is as follows: ~ pbgP mRNA; ~ the PmrA protein; ~ phosphorylated PmrA (PmrA-P) (here, pbgP corresponds to gene z, and PhoP and PmrA correspond to proteins X and Y, respectively -see main text, Figure 1D). The meaning of the rate and equilibrium constants is analogous to that of the corresponding parameters in Equations 1-5 (the corresponding parameters are the ones with identical subscripts).
For the direct regulation model, the constituting ODEs are as follows:

Derivation details for the mathematical models
We rely on the model-building methodology described earlier [5]; below we mention some notable details. We extend the previously published [5]  perimental data showed that the exponential depletion rate of PmrD/PmrA-P due to degradation and dilution is small compared to other parameter values [5]; thus, in Equation 5, we neglect the corresponding term. We also used this result to neglect the degradation+dilution term for PmrA-P in Equation 4.
In our models (Equations 1-11), the pbgP operon can be activated by PhoP-P and PmrA-P, which non-competitively bind to the promoter (we assume that PmrA-P/PmrD has the same binding properties as PmrA-P). For this type of control, the mRNA production rate can be expressed as kp , where is the maximum possible synthesis rate, and k p is the probability that the promoter is bound by at least one activator [5,21]. In the case of the FCL,  Equation 1). To derive this expression, we combined the general strategy for modeling transcriptional regulation by two proteins [21] and the previously developed approaches to modeling pbgP regulation [5]. The signal intensity for the second input (due to activation of the PmrA/PmrB twocomponent system) in the models is determined by the rates of phosphorylation and dephosphorylation of PmrA. We distinguish two situations: "strong activation" (higher phosphorylation, lower dephosphorylation rate) and "mild activation" (lower phosphorylation, higher dephosphorylation rate). We assume that both phosphorylation and dephosphorylation are performed by the PmrB protein [23]. At every moment,

Parameter sampling procedures for the models
The parameters for the FCL, FFL, and direct regulation circuit were sampled from uniform distributions over the appropriate intervals (Table S2). As described in the main text (Materials and Methods), a pair of parameter sets, one for the FCL and another one for direct regulation circuit, was accepted or rejected depending on whether the model outputs for these models satisfied the filtering criteria. For the FCL and the direct regulation circuit, we considered the case when there is activation of the PhoP input, and calculated the ratio of the output level at the reference time (30 min) and time 0 (activation ratio), both for strong and mild activation of the second input. For the pair of parameter sets to be accepted, all four ratios (2 regulation circuits, 2 states of the second input) were required to exceed the activation threshold (selected to equal 5). The pairs of parameter sets were generated randomly until the number of accepted pairs reached 1000. The pairs of parameter sets for the FFL and the direct regulation circuit were generated in a similar way; however, the filtering criteria were different. It is known that when the second input is inactive, two-component systems connected by a transcriptional cascade cannot be activated [24]. Therefore, we selected only the parameter sets for which, when the second input is under mild activation, at least one of the following conditions holds true for the transcriptional cascade (FFL lacking the direct activation branch): (a) the activation ratio is less than 1.5; (b) the output level at 30 min is over 10 times less than that for strong activation of the second input; (c) output level of the FFL is not greater than 0.005. Other filtering criteria were analogous to those described for the FCL.
In addition to independent parameter generation ( Figures 5A-D), we also implemented the "small-noise" strategy of parameter sampling (Figures S4-S6). According to this strategy, after generating a parameter set for the direct regulation circuit, we sampled the FCL parameters from intervals that depended on the values of the generated direct regulation parameters (the sampling distribution was uniform). For example, if the parameter value was , the value for the equivalent FCL parameter was sampled from the interval  Figures 5A-D).
The ODEs for the models with sampled and filtered parameters were solved numerically using MATLAB's ode15s function with relative tolerance and absolute tolerance .

Effect of the absence of the direct regulation branch in the FCL and FFL
In the versions of our models with no direct pbgP control by PhoP, we set the corresponding affinity ( ) equal to 0. Such a modification necessarily leads to a decrease in the output levels; we shall prove this statement for the FCL, and the FFL model can be treated in a similar way.

Equation 1 with
can be written in the form where It is easy to see that this inequality also holds if (and , are not necessarily steady-state concentrations).