Equations (1) and (2) have identical structure, differing only in the parameters involved. They represent the synthesis degradation and diluition of both the enzymes of the circuit, LuxI and AiiA, respectively in the first and second equation: in each of them both transcription and translation processes have been condensed.
These equations are composed of 2 parts:

1. 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 active even if there is no induction). 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 LuxR-HSL.
   In equation (2) only HSL seems to be the inducer, instead of the complex LuxR-HSL. This is motivated by 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 k and η stand respectively for the other parameter of the Hill relationship.
   
   
2. The second term in equations (1) and (2) is in turn composed of 2 parts. The first one (γ*LuxI or γ*AiiA respectively) describes, with a linear relation, the degradation rate per cell of the protein. The second one (μ*(Nmax-N)/Nmax)*LuxI or μ*(Nmax-N)/Nmax)*AiiA, respectively) takes into account the dilution term due to cell groth phase and is related to the cell replication process.

Here the kinetic of HSL is modeled, basicly through 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.
3 parts have been identified in thi equation:

1. The first term represents the production of HSL due to the LuxI expression. The parameters involved here are those of a classic Hill equation: V_max, the maximum transcription rate of the enzyme, and K_M, the dissociation constant of.... Moreover, this formula is multiplied by the number of cells N, due to the property of the lactone to diffuse free inside/outside bacteria.
   
   
2. The second term represents the degradation of HSL due to the AiiA expression. k_cat, KM1....For the same reason of phe first term, this is multiplied too by N.
   
   
3. The third term (γ_HSL*HSL) is similar to the corresponding ones present in the first two equations and describes the intrinsic protein degradation.

This is the common logistic cell growth, depending on the rate μ and the maximum number N_MAX of cells per well reachable.

In this section we examine the parameters of the model and justify the units of measure, relating them to the experiments performed for the characterization of the parts. We want to underline again our concept of modelling: beginning to caractherize simplier parts, we get their parameters and we try to predict the behaviour of the final engineerd closed-loop.

These are the first subparts tested. Firstly, in the figure above "RBSx" stands for, respectively, RBS30, RBS31, RBS32, RBS34. So we get 4 biobricks for each promoter, in order to investigate what happens in different conditions of RBS's efficiency. In this phase of the project we aim to increase our knowledge about promoter Ptet and Plux but it must be said that, here, it' s quite impossible to focus separately on the only activity of the promoter and RBSx; for this reason, when we "characterize promoters", we mean promoter and RBS together. We realize this by introducing the mRFP fuorescent protein (followed by a double terminator), and we make the assumption that the number of fluorescent protein produced, due to the concentration of induction (aTc, HSL for Ptet, Plux respectively) is exactly the same as the number given by any other protein that would be expressed instead of the mRFP. In other words, in our hypotesis, if we would substitute the mRFP coding region with a region coding for another protein, we would obtain the same synthesis rate: this is the reason why the strength of the complex promoter-RBSx is expressed in Arbitrary Units [AUr]. Clearly this is a strong hypotesis, however its level of approximation is considered to be adequate.
Keeping this idea in mind, let's talk about the steps to estimate parameters.
As shown in the box below, we consider a wide (more or less, depending on the type of test) range of induction and we monitor, during the time, absorbance (line1, line2) and fluorescence (line3); the two vertical segments for each figure highlight the exponential phase of bacteria' s groth. We are able to make these measurement due to the Tecan Infinite F200, spectrophotometer that allows to know the Scell (explained few lines below) as a function of inducer concentration, thereby providing the desired input-output relation (inducer concentration versus promoter+RBS activity), which was modelled as a Hill curve.
After that, we can calculate the Scell as: In the end, plotting Scell VS induction, we obtain the activation Hill curve of the promoter considered. As shown in the box above, α as already mentioned, represent the protein maximum synthesis rate, which is reached, in accordance with Hill's formalism, when the inducer concentration tends to infinite, and, more practically, for sufficently high concentrations of inducer, meanwhile the product α*δ stands for the leakage activity (induction=0ng/µL), liable for protein production (LuxI and AiiA respectively) even in the absence of autoinducer. The paramenter η is the Hill's cooperativity constant and it affects the rapidity and ripidity of the switch like curve relating Scell with the concentration of inducer. Lastly, k stands for the semi-saturation constant and, in case of a unity value for η, it indicates the concentration of substrate at which half the synthesis rate is achieved. The unities of the various parameters can be easily derived considering the hill equation and the unity of its left handed side (for more details see the Table of parameters above).

AiiA