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Mathematical modelling page
Mathematical modelling plays nowadays 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.
According to this, after a brief overview about the advantages that modelling engineered circuits can bring, we deeply analyze the system of equation formulas, underlining the role and the function of the parameters involved.
Then, experimental procedures for parameters estimation are presented and, finally, different types of circuit are discussed and their simulations performed, using ODE's with MATLAB and explainig the difference between a closed-loop model and an open one.<7div>
Then, experimental procedures for parameters estimation are presented and, finally, different types of circuit are discussed and their simulations performed, using ODE's with MATLAB and explainig the difference between a closed-loop model and an open one.<7div>
The importance of the mathematical model
Several motivations are strong enough to accept the idea that mathematical model is very useful in this study.
- Firstly in the initial steps of the project, beacuse of its capability to predict the kinetics of the enzymes (aiiA, Luxi) and HSL involved in our gene network, well realizing the a-priori identification in silico in order to understand if the complex circuit's structure and functioning could be achievable.
- Secondly, for the parametric identification. Using the lsqnonlin function of MATLAB it was possible to get all the parameters involved in the model and consequently to know, for example, the shape of the activation curve of the promoters (Plux, Ptet), according to the a-posteriori identification.
- Thirdly, the reproducibility. Studing and characterizing simple subparts can allow us not only to predict the behavior of the final circuit, but also it can be useful in other studies, facing with the same basic modules.
Equations for gene networks
Equations (1) and (2)
We have condensed in a unique equation transcription and translation processes. Equations (1) and (2) have identical structure, differing only in the parameters involved.
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 induced even if there is no input). 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 given by LuxR and HSL.
In the first term of equation (2) we have described the inducer as being represented only by HSL. This formalism stems from 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.
Furthermore, in both equations (1) and (2) k and η stands respectively for the other parameter of the Hill relationship. The second term in equation (1) and (2) is composed of two parts. The first one (gamma*LuxI/HSL) describes with a linear relation the degradation rate per cell of the protein. The second one (mu*(Nmax-N)/Nmax)*luxI/HSL) takes into account the dilution term and is related to the cell replication process.
In the first term of equation (2) we have described the inducer as being represented only by HSL. This formalism stems from 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.
Furthermore, in both equations (1) and (2) k and η stands respectively for the other parameter of the Hill relationship. The second term in equation (1) and (2) is composed of two parts. The first one (gamma*LuxI/HSL) describes with a linear relation the degradation rate per cell of the protein. The second one (mu*(Nmax-N)/Nmax)*luxI/HSL) takes into account the dilution term and is related to the cell replication process.
Equation (3)
Equation (4)
| Parameter | Description | Unit of Measurement | Value |
| αpTet | maximum transcription rate of Ptet | [(mRFP/min)/cell] | - |
| δpTet | leakage factor of promoter Ptet basic activity | [-] | - |
| ηpTet | Hill coe�fficient of Ptet | [-] | - |
| kpTet | dissociation costant of Ptet ? | [nM] | - |
| αpLux | maximum transcription rate of Plux | [(mRFP/min)/cell] | - |
| δpLux | leakage factor of promoter Plux basic activity | [-] | - |
| ηpLux | Hill coe�fficient of Plux | [-] | - |
| kpLux | dissociation costant of Plux ? | [nM] | - |
| γLuxI | LuxI costant degradation | [1/min] | - |
| γAiiA | AiiA costant degradation | [1/min] | - |
| γHSL | HSL costant degradation | [1/min] | - |
| Vmax_LuxI | maximum transcription rate of LuxI | [nM/(min*cell)] | - |
| km_LuxI | dissociation costant ? | [nM] | - |
| kCAT | ?? | [nM/(min*cell)] | - |
| km_AiiA | dissociation costant ? | [nM] | - |
| γHSL | HSL costant degradation | [1/min] | - |
| Nmax | maximum number of bacteria | [cell] | - |
| μ | rate of bacteria groth | [1/min] | - |
