Equations

The dynamics of our microbial consortium was summed up in twelve differential equations. Every biological or physical process was described mathematically, and was gathered to constitute our system of ODEs. For fearless people, our complete mathematical model and demonstration can be found there!

We need to characterize the growth and death of the three microorganisms: Vibrio cholerae (Vc) in water (W), Vibrio harveyi (Vh) and Pichia pastoris (Pp) in the device (D). \begin{equation} \frac{d[\textit{Vc}]_W}{dt} = V_{growth,Vc} - V_{death,Vc} \end{equation} \begin{equation} \frac{d[\textit{Vh}]_D}{dt} = V_{growth,Vh} - V_{death,Vh} \end{equation} \begin{equation} \frac{d[\textit{Pp}]_D}{dt} = V_{growth,Pp} - V_{death,Pp} \end{equation}

Microorganisms death is impacted by antimicrobial peptides production(AMP_peptide,Pp), produced by translation of antimicrobial peptides mRNA (AMP_RNA). Peptides and mRNA are also degraded. \begin{equation} \frac{d[AMP_{RNA}]_{Pp}}{dt}=V_{transcription,AMP} - V_{degradation,AMP RNA} \end{equation} \begin{equation} \frac{d[AMP_{peptide}]_{D}}{dt}=V_{translation,AMP} - V_{degradation,AMP} + \frac{V_{diff,AMP,W \to D}}{\mathcal{V}_D} \end{equation}

These peptides are transfered from the device (D) to water (W). \begin{equation} \frac{d[AMP]_W}{dt} = -V_{diff,AMP,W\to D} \end{equation}

To produce antimicrobial peptides, an activation by diacetyl (dac) is needed. Diacetyl can freely diffuse from the device (D) to water (W). \begin{equation} \frac{d[dac]_D}{dt}=V_{prod,dac}+\frac{V_{diff,dac,W \to D}}{\mathcal{V}_D} \end{equation} \begin{equation} \frac{d[dac]_W}{dt}=- V_{diff,dac,W \to D} \end{equation}

Diacetyl is produced by the enzyme acetolactate synthase (ALS_enzyme). Als gene is first transcribed into ALS_RNA, which is then translated into the protein. Both the enzyme and its mRNA can be degraded. \begin{equation} \frac{d[ALS_{RNA}]_{Vh}}{dt} = V_{transcription,ALS} - V_{degradation,ALS RNA} \end{equation} \begin{equation} \frac{d[ALS_{enzyme}]_{Vh}}{dt} = V_{translation,ALS} - V_{degradation,ALSenzyme} \end{equation}

ALS production has to be activated by the quorum sensing molecule CAI-1, initially in water (W), after diffusing into the device (D). \begin{equation} \frac{d[CAI\text{-}1]_D}{dt} = \frac{V_{diff,CAI\text{-}1,W\to D}}{\mathcal{V}_D} \end{equation} \begin{equation} \frac{d[CAI\text{-}1]_W}{dt} = -V_{diff,CAI\text{-}1,W\to D} \end{equation}

Data

Model parameters were mostly collected from publications, because a large part of the required data necessitates a complex set of hundreds of experiments that could not have been performed in the course of the project.

Some preliminary experimental results were also expoited to refine important parameters of the model. Indeed, we needed an experimental estimation of the growth rate of the two chassis microorganisms grown on a common medium.

Solver

The system of ODEs was solved using Matlab R2017a, thanks to the free offer from iGEM. We used the ode15s solver, which was in this situation more efficient than the ode45s solver, very likely because the different time scales between some processes (e.g. growth vs signalling) rendered the problem partially stiff.(17)

You can freely re-use our code: General_resolution + System_of_ODEs.

Simulation results

At the beginning of the project, we needed to know if our microbial synthetic consortium would work and if the information transmission was possible. So we used our system of ODEs and our solver to have a first estimation of the functioning of our synthetic system.

Design parameters, such as Vibrio harveyi and Pichia pastoris initial concentration and the device volume, were set to biologically plausible values.

• [Vh]_0,D = 10¹² cell/L (OD_600nm ≈ 1.5 (18))
• [Pp]_0,D = 10¹² cell/L (OD_600nm ≈ 1.5 (18))
• V_D = 20.10^-3 L

This simulation confirmed the feasibility of our project: our microbial consortium would be enough effective to detect the presence of V. cholerae, transmit this information and, in response, produce enough antimicrobial peptides to kill V. cholerae. The response time to reach a non-pathogenic concentration is estimated to 53.6 min.

This visual representation of the system's dynamics allows us to check that every variables evolves in a realistic range of concentrations, hence indicating the model predicts a consistent behaviour. Further analyses were then perfomed to evaluate the sensibility and robustness of the synthetic system. We first tested whether the system would be robust to the variation of some parameters that could arise during the manufacturing or transport processes. Then, we identified parameters that control the response time (time to reach a non-pathogenic concentration) to guide the rational design of the system and of the device.

→ Sensitivity analysis methods and their results are described in our model analysis page

References

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