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. The corresponding mathematical formalism is analogous to that used by Pasotti et al 2011, Suppl. Inf., even if here we do not model LuxR-HSL complex formation, as explained below.
The 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 stands for the dissaciation constant of the promoter from the inducer (respectively aTc and HSL in eq. 1 and 2), while η is the cooperativity constant.
   
   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 growth and is related to the cell replication process.

Here the kinetics 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 appropriate expressions for HSL synthesis and degradation.
3 parts have been identified in this equation:

1. The first term represents the production of HSL due to LuxI expression. We model this process with saturation curve in which V_max is HSL maximum transcription rate, while K_M is the dissociation constant of LuxI from the substrate HSL and it represents the concentration of LuxI at which HSL synthesis rate is V_max/2.
   
   
2. The second term represents the degradation of HSL due to the AiiA expression. Similarly to LuxI, K_cat represents maximum degradation per unit of HSL concentration, while K_M1 is the concentration at which AiiA dependent HSL concentration rate is (K_cat*HSL)/2. The formalism is similar to that found in the Supplementary Information of Danino et al, 2010.
   
   
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.
   
   
   NOTE2: the whole equation, except for the term of intrinsic degradation of HSL, is multiplied by the number of cells N, due to the property of the lactone to diffuse free inside/outside bacteria. Notice that, in system equation, LuxI and AiiA amounts are expressed per cell.

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

The philosofy of the model is to predict the behavior of the final closed loop circuit starting from the characterization of single BioBrick parts through a set of well-designed ad hoc experiments. Relating to these, in this section the way parameters of the model have been identified is presented. As explained before in NOTE1, considering a set of 4 RBS for each subpart to caracterize increase their range of dynamics and helps us to understand deeplier the interactions between state variables and parameters.

These are the first subparts tested. In this phase of the project the target is to learn more about promoter pTet and pLux. Characterizing only promoter BioBrick is quite impossible: for this reason we consider promoter and the respective RBS from RBSx set together. Here a strong and reasonable hypothesis must be pointed out: 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].
As shown in the box below, we consider a 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. S_cell (explained few lines below) can be derived 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 S_cell as: In the end, plotting S_cell VS induction, we obtain the activation Hill curve of the considered promoter. As shown in the box above, α as already mentioned, represents 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 (at no induction), 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 S_cell with the concentration of inducer. Lastly, k stands for the semi-saturation constant and, in case of η=1, 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(for more details see the Table of parameters above).

AiiA & LuxI

From the above figures, it is also clear that HSL is not very sensitive to α_pLux, at least when this parameter presents values greater than unity, because this brings the two curves to intersect in their low slope regions.

Referring to the second RH, the only adjustable asymptote is the horizontal one, that we can move upward or downward by altering VLuxI, which in turn depends on LuxI.

A further deepening in the exponential phase analysis involves determining the characteristics of the input-output relation. Our closed loop system can be realized with two alternative purposes in mind:

1. realizing a circuit able to adapt HSL output depending on atc input concentration. This requires a good sensitivity between input and output.
2. designing a robust HSL concentration controller, which is immune to the input noise and offers a constant and defined amount of HSL. In this case HSL level should be appropriately tuned during the design stage, by choosing the correct strength of promoter-RBSx complexes.

Now,again considering the system of equations, it is easy to observe that HSL dependence on atc input passes through two Hill equations. The former describes atc driven LuxI synthesis, while the latter models LuxI dependent HSL synthesis rate. Therefore, in order to achieve a high atc sensitivity, it is advisable to tune atc and LuxI levels so that they place outside the saturation regions of their Hill curves. In this regard, it is possible to determine a closed form expression relating HSL to atc, if we hypothesize that both atc and LuxI are far lower than their respective half-saturation constant (k_ptet and Km), thus simplifying the Hills with a first order relation:

On the other side, in order to be more robust to the input variations, we should provide LuxI values corresponding to a saturation in LuxI dependent HSL synthesis rate.

References