We have condensed in a unique equation transcription and translation processes. Equations (1) and (2) have identical structure, differing only in the parameters involved. The first term describes, through Hill's equation formalism, the synthesis rate of the protein of interest (either LuxI or AiiA) depending on the concentration of the inducible protein (anhydrotetracicline -aTc- or HSL respectively). As can be seen in the parameters table (see below),α refers to the maximum activation of the promoter, δ stands for its leakage activity (this means that the promoter is quite induced even if there is no input). In particular, in equation (1), the quite total inhibition of pTet promoter is due to the constitutive production of TetR by our MGZ1 strain, while, in equation (2), Plux is almost repressed in the absence of the complex given by LuxR and HSL.
In the first term of equation (2) we have described the inducer as being represented only by HSL. This formalism stems from the fact that our final device offers a constitutive production of LuxR (due to the upstream constitutive promoter pLac), so that, assuming it abundant in the cytoplasm, we can derive the semplification of attributing pLux promoter induction only by HSL: this is the reason why we didn' t consider LuxR in the equations system as well as LuxI and AiiA.
Furthermore, in both equations (1) and (2) k and η stands respectively for the other parameter of the Hill relationship. The second term in equation (1) and (2) is composed of two parts. The first one (γ*LuxI/AiiA) describes with a linear relation the degradation rate per cell of the protein. The second one (μ*(Nmax-N)/Nmax)*LuxI/AiiA) takes into account the dilution term and is related to the cell replication process.

The processes described here are not those of transcription and translation, but in principle are enzymatic reactions either related to the production or the degradation of HSL. Based on the experiments performed, we derived Hill's equation in the case of η=1. They cannot be exactly defined Michaelis Menten's equations since that in our formalism, LuxI and AiiA aren't described as enzymes (since they appear also in the denominator). We simply derived empirical formulas relating either LuxI or AiiA to HSL, and treated them with the typical Michaelis Menten formalism since they presented the corresponding sigmoidal shape/switching like behaviour. Regarding to this, we believe that the saturation phenomenon observed either in HSL production rate due to LuxI, or HSL degradation rate due to AiiA, underlies limiting elements in cell metabolism. In the cell HSL binds to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the promoter Plux.
Intuitively, LuxI activity as an enzyme encounters an intrinsic limit in HSL synthesis depending on the finite and hypothetically fixed substrate concentration (namely SAM and hexanoyl-ACP, see ref.); this means that at a certain LuxI concentration, all the substrate forms activation complexes with LuxI, so that there is no more substrate available for the other LuxI produced. HSL degradation rate is limited by its availability; even if the concentration varies with time, there is always a corresponding limit in AiiA concentration, which determines a saturation in the degradation rate.
Moreover, both the formulas relating either LuxI or AiiA to HSL are multiplied by the number of cells N, due to the property of the lactone to diffuse free inside/outside bacteria. The third term in equation (3) is similar to the corresponding ones present in the first two equations and describes the intrinsic protein degradation.

Equation (4) is the common equation describing logistic cell growth, depending on the rate μ and the maximum number N_MAX of cells per well reachable.

According to the table above, the unit of the state variables are: