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 inducer (anhydrotetracicline -aTc- or HSL respectively), responsible for the activation of the regulatory element composed of promoter and RBSx. In the parameter table (see below), α refers to the maximum activation of the promoter, while δ stands for its leakage activity (this means that the promoter is slightly active even if there is no induction). In particular, in equation (1), the almost entire inhibition of pTet promoter is given by the constitutive production of TetR by our MGZ1 strain. In equation (2), pLux is almost inactive in the absence of the complex LuxR-HSL.
   Furthermore, in both equations k stands for the dissociation 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 former one (γ*LuxI or γ*AiiA respectively) describes, with an exponential decay, the degradation rate per cell of the protein. The latter (μ*(Nmax-N)/Nmax)*LuxI or μ*(Nmax-N)/Nmax)*AiiA, respectively) takes into account the dilution factor against cell growth which is related to the cell replication process.

1. The first term represents the production of HSL due to LuxI expression. We modeled this process with a saturation curve in which V_max is the HSL maximum transcription rate, while k_M,LuxI is the dissociation constant of LuxI from the substrate HSL.
   
   
2. The second term represents the degradation of HSL due to the AiiA expression. Similarly to LuxI, k_cat represents the maximum degradation per unit of HSL concentration, while k_M,AiiA 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.

AiiA & LuxI

By now, parameter identification about promoters has already been performed. Furthermore, as explained before, the HP₄ is also valid in this case. Moreover, it's possible to quantify exactly the concentration of HSL, using the well-characterized BioBrick BBa_T9002.

This is a biosensor which receives HSL concentration as input and returns GFP intensity (more precisely S_cell) as output. (Canton et al, 2008). According to this, it is necessary to understand the input-output relationship: so, a T9002 "calibration" curve is plotted for each test performed.

So, our idea is to control the degradation of HSL in time. ATc activates pTet and, later, a certain concentration of HSL is introduced. Then, at fixed times, O.D.₆₀₀ and HSL concentration are monitored using Tecan and T9002 biosensor.

Referring to HP₅, in exponential growth enzymes equilibrium is conserved. Due to a known induction of aTc, the steady-state level per cell can be calculated:

Considering, for a determined promoter-RBSx couple, several induction of aTc and, for each of them, several samples of HSL concentration during time, parameters V_max, k_M,LuxI, k_cat and k_M,AiiA can be estimated, through numerous iterations of an algorithm implemented in MATLAB.

N

The parameters N_max and μ can be calculated from the analysis of the OD₆₀₀ produced by our MGZ1 culture. In particular, μ is derived as the slope of the log(O.D.₆₀₀) growth curve. Counting the number of cells of a saturated culture would be considerably complicated, so N_max is determined with a proper procedure. The aim here is to derive the linear proportional coefficient Θ between O.D'.₆₀₀ and N: this constant can be estimated as the ratio between absorbance (read from TECAN) and the respective number of CFU on a petri plate. Finally, N_max is calcultated as Θ*O.D'.₆₀₀. (Pasotti et al, 2010).

Degradation rates

The parameters γ_LuxI and γ_AiiA are taken from literature since they contain LVA tag for rapid degradation. Instead, approximating HSL kinetics as a decaying exponential, γ_HSL can be derived as the slope of the log(concentration), which can be monitored through BBa_T9002.

Simulations

On a biological level, the ability to control the concentration of a given molecule reveals itself as fundamental in limiting the metabolic burden of the cell; moreover, in the particular case of HSL signalling molecules, this would give the possibility to regulate quorum sensing-based population behaviours. In this section we first present the results of the simulations of the closed-loop circuit for feasible values of the parameters. The reported figures highlight some fundamental characteristics.

First of all, it is clear the validity of the steady state approximation in the exponential growth phase, since that LuxI, AiiA, and also HSL, undergo only minor changes in this phase (500>t<2500 min). Secondly, it can be noted that the circuit negative feedback rapidly activates above a proper amount of HSL, and after that it competes with LuxI synthesis term in defining HSL steady state value.

The concept of the closed-loop model we have discussed so far can be validated by the comparison with another circuit we implemented in Matlab, that is an open loop circuit, similar to CTRL+E, except for the constitutive production of AiiA. We can point out that, under the hypothesis of an equal amount of HSL production, the open-loop circuit requires a higher AiiA production level, thereby constituting a greater metabolic burden for the cell (see figure below), uncertain to be fulfilled in realistic conditions. Moreover, there is another major issue with the open loop circuit, that is the inability to monitor the output of the controlled system, so that it is not generally capable to adapt to changes in the system behavior.

References

D. Braun, S. Basu, R. Weiss, "Parameter Estimation for Two Synthetic Gene Networks: a Case Study", ICASSP 2005, vol. 5, v/769 - v/772.

B. Canton, A. Labno and D. Endy, "Refinement and Standardization of Synthetic Biological Parts and Devices", Volume 26, Number 7, July 2008.

T. Danino, O. Mondragon-Palomino, L. Tsimring & J. Hasty, "A Synchronized Quorum of Genetic Clocks", Nature vol. 463, pp. 326-330, January 2010.

L. Endler, N. Rodriguez, N. Juty, V. Chelliah, C. Laibe, C. Li and N. Le Novere, "Designing and Encoding Models for Synthetic Biology", Journal of the Royal Society, 2009 Aug 6;6 Suppl 4:S405-17. Epub 2009 Apr 1.

L. Pasotti, M. Quattrocelli, D. Galli, M.G. Cusella De Angelis, P. Magni, "Multiplexing and Demultiplexing Logic Functions for Computing Signal Processing Tasks in Synthetic Biology", Biotechnology Journal, Volume 6,pp. 784-795, 2011.

L. Pasotti, M. Quattrocelli, D. Galli, M.G. Cusella De Angelis, P. Magni, "Multiplexing and Demultiplexing Logic Functions for Computing Signal Processing Tasks in Synthetic Biology, Supporting Information", Biotechnology Journal, Volume 6,available online, 2011.