Mathematical modelling plays a central role in Synthetic Biology, due to its ability to serve as a crucial link between the concept and realization of a biological circuit: what we propose in this page is a mathematical modelling approach to the entire project, which has proven extremely useful before and after the "wet lab" activities.

Thus, immediately at the beginning, when there was little knowledge, a mathematical model based on a system of differential equations was derived and implemented using a set of reasonable values of model parameters, to validate the feasibility of the project. Once this became clear, starting from the characterization of each simple subpart created in the wet lab, some of the parameters of the mathematical model were estimated thanks to several ad-hoc experiments we performed within the iGEM project (others were derived from literature) and they were used to predict the final behaviour of the whole engineered closed-loop circuit. This approach is consistent with the typical one adopted for the analysis and synthesis of a biological circuit, as exemplified by Pasotti et al 2011.

After a brief overview on the importance of the mathematical modelling approach, we deeply analyze the system of equations, underlining the role and function of the parameters involved.
Experimental procedures for parameter estimation are discussed and, finally, a different type of circuit is presented. Simulations were performed, using ODEs with MATLAB and used to explain the difference between a closed-loop control system model and an open one.

Mathematical modeling reveals fundamental in the challenge for understanding and engineering complex biological systems. Indeed, these are characterized by a high degree of interconnection among the single constituent parts, requiring a comprehensive analysis of their behavior through mathematical formalisms and computational tools

• Simulation: mathematical models allow to analyse complex system dynamics and to reveal the relationships between the involved variables, starting from the knowledge of the single subparts behavior and from simple hypotheses of their interconnection. (Endler et al, 2009)
• Knowledge elicitation: mathematical models summarize into a small set of parameters the results of several experiments (parameter identification), allowing a robust comparison among different experimental conditions and providing an efficient way to synthesize knowledge about biological processes. Then, through the simulation process, they make possible the re-usability of the knowledge coming from different experiments, engineering complex systems from the composition of its constituent subparts under appropriate experimental/environmental conditions (Braun et al, 2005; Canton et al, 2008).

Equations (1) and (2) have identical structure, differing only in the parameters involved. They represent the synthesis, degradation and diluition of both the enzymes in 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 mathematical formulation is analogous to the one used by Pasotti et al 2011, Suppl. Inf., even if we do not take LuxR-HSL complex formation into account, as explained below.

These equations are composed of 2 parts:

1. The first term describes, through Hill's equation, 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 RBS. 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.

   
   
2. 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.

Here the kinetics of HSL is modeled, through enzymatic reactions either related to the production or the degradation of HSL. This equation is composed of 3 parts:

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 LuxI dependent half-saturation constant.
   
   
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 degradation 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.

This is the typical cells growth equation, depending on the rate μ and the maximum number N_max of cells per well reachable (Pasotti et al, 2009).

The aim 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. This section presents the experiments performed. As explained before in NOTE, considering a set of 4 RBSs for each subpart expands the range of dynamics and helps us to better understand the interactions between state variables.

These are the first parts tested, with the target of learning more about pTet and pLux promoters. In particular, as previously explained in NOTE, for each promoter, we tested four different combinations of promoter-RBS, providing us a set of fundamental building blocks for the subsequent assebly of the closed-loop circuit.
As shown in the figure below, we considered a range of inductions and we monitored, in time, absorbance (O.D. stands for "optical density") and fluorescence; the two vertical segments for each graph highlight the exponential phase of bacterial growth. S_cell (namely, synthesis rate per cell) 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: However, also Relative Promoter Unit (RPU, Kelly 2009) has been calculated as a ratio of S_cell of promoter of interest and the S_cell of BBa_J23101 (reference to HP₃). As shown in the figure, α, as already mentioned, represents the protein maximum synthesis rate, which is reached, in accordance with Hill equation, when the inducer concentration tends to infinite, and, more practically, when the inducer concentration is sufficiently higher than the dissociation constant. 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 inducer. The paramenter η is the Hill's cooperativity constant and it affects the ripidity of transition from the lower and upper boundary of the curve relating S_cell to the inducer concentration. 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.

T9002 introduction

LuxI and AiiA tests have been always performed exploiting the well-characterized BioBrick BBa_T9002, by which it's possible to quantify exactly the concentration of HSL. This is a biosensor which receives HSL concentration as input and returns GFP intensity (more precisely Scell) 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.

AiiA & LuxI