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 L 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 modelling reveals fundamental in the challenge of 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.

Synthetically, we can identify two major roles concerning mathematical models:

• 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 L 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 D et al. 2005; Canton B et al 2008).