Deconstructing a multiple antibiotic resistance regulation through the quantification of its input function

Many essential bacterial responses present complex transcriptional regulation of gene expression. To what extent can the study of these responses substantiate the logic of their regulation? Here, we show how the input function of the genes constituting the response, i.e., the information of how their transcription rates change as function of the signals acting on the regulators, can serve as a quantitative tool to deconstruct the corresponding regulatory logic. To demonstrate this approach, we consider the multiple antibiotic resistance (mar) response in Escherichia coli. By characterizing the input function of its representative genes in wild-type and mutant bacteria, we recognize a dual autoregulation motif as main determinant of the response, which is further adjusted by the interplay with other regulators. We show that basic attributes, like its reaction to a wide range of stress or its moderate expression change, are associated with a strong negative autoregulation, while others, like the buffering of metabolic signals or the lack of memory to previous stress, are related to a weak positive autoregulation. With a mathematical model of the input functions, we identify some constraints fixing the molecular attributes of the regulators, and also notice the relevance of the bicystronic architecture harboring the dual autoregulation that is unique in E. coli. The input function emerges then as a tool to disentangle the rationale behind most of the attributes defining the mar phenotype. Overall, the present study supports the value of characterizing input functions to deconstruct the complexity of regulatory architectures in prokaryotic and eukaryotic systems.

where  is the cell growth rate,  the degradation rate of MarA ( δ≫μ , as MarA is quickly degraded by Lon protease 10  mar and rob activity can be described in terms of Hill functions 11 , in which MarA and Rob act as monomers whereas MarR functions as dimer 6 .
The effect of MarA on Prob is not observable in physiological conditions 3 , so it is neglected to simplify the system of equations. Since MarB might not exert an observable repression on Pmar in presence of salicylate, it was also eliminated from the model. mar reads then as where  is the effective dissociation constant between salicylate (Sal) and MarR,  the Hill coefficient, and  a minimal fraction of free MarR.

Simplification of the mathematical model
The previous mathematical model can be simplified for a better analysis of the dynamics. We introduced the following normalized variables Here, MarA could be approximated to a quasi-steady state ( x ∝ π mar , a function of time). And Eq. (S4) now reads In case of maximal induction of the system with salicylate,  modulates the regulatory role of MarR ( 1 ≥ α S ≥ α , α = 1 represents absence of salicylate effect).

Analytical solutions of the model: steady state
In this section, we considered that a very strong the repressor acted on the system, MarR (i.e., y ≫ 1 ). We also assumed that the system was induced with high levels of salicylate, so that y 0 = α y , and that the activation term was simply reduced to the fold change (). This allowed us to simplify the model to just one equation, given by The steady state of this equation is given then by

Analytical solutions of the model: dynamic range
Our model to describe the dynamics of a self-repressed, self-activated operon (y), implemented with two regulatory genes (repressor and activator), can be rewritten as We used in this derivation Eq. (S6), and also assumed MarA to be a stable protein ( δ = μ ). This assumption leads to x = β y and does not change the dynamics of our protein of interest, MarR (y). The simplified model can be solved in steady state for a strong repressor ( y ≫ 1 ) and high activation fold ( ρ ≫ 1 ) to obtain For saturating levels of salicylate, we estimated with this model an output dynamic range of R out = α −2 /3 (i.e., the ratio between the highest and lowest values of y ∞ when varying salicylate), and an input dynamic range of R in = 9 2/ n in (i.e., the ratio between the salicylate values at which we have n in is an effective Hill coefficient; see also Goldbeter & Koshland, 1981 15 ).

Alternative models
For a constitutively expressed regulator ( y ), the dynamics of its regulated operon (ȳ ) can be written as where y is constant. The steady state solution for ȳ is straightforward to obtain. Moreover, a model to describe the dynamics of a self-repressed gene ( y ), with equal production rate as before, can be written as This model can be solved in steady state to obtain the same expression as before.
Finally, a model to describe the dynamics of a self-repressed gene ( y ) that becomes self-activator in presence of the inducer can be written as where we have assumed competitive binding between the repressor and activator [an extra term Ω ω ( (1−α S ) y ) n (α S y ) 2 in the denominator of (S13) includes the effect of independent binding, i.e., Ω = 0 for competitive binding between oxidized and non-oxidized MarR, or Ω = 1 for independent binding]. This model can be solved in steady state, also for a strong repressor ( y ≫ 1 ) and high activation fold ( ρ ≫ 1 ), as with θ H representing the Heaviside function. Note that in the expressions above ω measures the asymmetry in the binding affinities of the two forms of MarR: ω = 1 when binding is assumed similar (as in Fig. 6, main text), ω < 1 for stronger binding of the oxidized form, and ω > 1 when the non-oxidized form binding is stronger.). Moreover, the multimerization of the oxidized form could be considered to remain as a monomer ( n = 1 ), dimer ( n = 2 ), or tetramer ( n = 4 ).         to reveal that the mar core network presents a unique genetic architecture.