Every biological and physical processes including in the SBGN representation of the model was described mathematically using appropriate kinetic rate laws (passive diffusion, Michaelis-Menten enzyme kinetics, first order degradation of mRNAs, exponential growth law, etc). The final model contains 12 entities (genes, RNAs, proteins, metabolites, and cells) and 24 reactions from a broad range of processes (transcription, translation, RNAs and proteins degradation, growth and death of each micro-organism, metabolite poduction, transport between compartments, diffusion, etc). The dynamics of our microbial consortium was summed up in twelve differential equations as a system of ODEs.
For fearless people, our complete mathematical model and demonstration can be found there! The final set of ODEs is briefly described below.
SBGN representation of the model developed in the project
We need to consider the growth and death of the three microorganisms: Vibrio cholerae (Vc) in water (W), and 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 (AMPpeptide,Pp), produced by translation of antimicrobial peptides mRNA (AMPRNA). 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 between compartments: 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 (ALSenzyme). The als gene is first transcribed into ALSRNA, 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 and Parameters
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.
Estimation from yeast cell volume and ratio dry weight/volume for a bacterial cell
Ribosome density on a yeast cell
Nb/kb
6.5
BioNumbers ID 103026
Number of ribosome on AMP mRNA
Ribosome/RNA
Nb/RNA
2.34
Deduced from ribosome density and mRNA length
RNA polymerase density on a yeast gene
Nb/kb
6.5
BioNumbers ID 103026
Number of RNA polymerase on AMP DNA
RNA polymerase/DNA
Nb/DNA
4.95
BioNumbers ID 108308
Ribosome density on a bacterial mRNA
Nb/kb
6.6
BioNumbers ID 107727
Number of ribosome on als mRNA
Ribosomes/RNA
Nb/RNA
11
Deduced from ribosome density and mRNA length
RNA polymerase density on a bacterial gene
Nb/kb
7.6
Assuming the same density than yeast (BioNumbers ID 108308)
Number of RNA polymerase on als DNA
RNA Polymerase/DNA
Nb/DNA
13
Deduced from RNA polymerase density and DNA length
Pyruvate concentration in a bacterial cell
[S]
mol/L
3.9.10-4
BioNumbers ID 101192
Some preliminary experimental results were also expoited to refine important parameters of the model and verify some of our hypothesis. Indeed, we needed an experimental estimation of the growth rate and lag time of the two chassis microorganisms grown on a common medium, and these parameters were measured experimentally. The membrane diffusion tests we have performed also revealed a rapid molecular diffusion (equilibrium reached in less than one minute). This result confirmed the initial, arbitrary hypothesis of a high transfer coefficient (K) which would not slow down the system's dynamics.
Name
Notation
Unit
Value
Reference
Vibrio harveyi JMH626 maximum growth rate
μMAX,Vh
s-1
2.10-4
Experiment - 21/06/17
Pichia pastoris SMD1168 maximum growth rate
μMAX,Pp
s-1
4.10-5
Experiment - 21/06/17
Vibrio cholerae lag time
tl,Vc
s
1800
Experiment - 21/06/17, extrapolation from V. harveyi growth
Vibrio harveyi lag time
tl,Pp
s
1800
Experiment - 21/06/17
Pichia pastoris lag time
tl,Pp
s
1400
Experiment - 21/06/17
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 ode45 solver, very likely because the different time scales between some processes (e.g. slow growth - in the range of hours - vs fast signalling - in the range of milliseconds -) renders the problem partially stiff.17 The proposed implementation proved to be very efficient, with simulations performed in less than 1 second.
At the beginning of the project, we needed to know if our microbial synthetic consortium would work in practice and if the information transmission between the different modules was possible and sufficiently fast. We thus carried out simulations by solving the ODEs system to have a first estimation of the dynamics of our synthetic system.
The initial conditions, such as the concentrations of Vibrio harveyi and Pichia pastoris in the device and the device volume, were set to biologically plausible values. Vibrio cholerae initial concentration was set to 4.107 cell/L,
which is in the range of concentrations observed in contaminated water. 16
Temporal evolution of Vibrio cholerae concentration under realistic conditions (device size, initial concentrations of other microorganisms in the device)
The response time to reach a non-pathogenic concentration is estimated to 53.6 min. This result confirmed the feasibility of our project: our microbial consortium is predicted to efficiently and rapidly sense the presence of V. cholerae, transmit this information to P. Pastoris which, in response, may produce enough antimicrobial peptides to kill V. cholerae in less than one hour.
Evolution of the concentration of each entity considered in the model
This visual representation of the system's dynamics also allowed us to check that each variable evolves in a realistic range of concentrations, hence indicating the model predicts a consistent behavior. Further analyses were then perfomed to better understand the system functioning, and in particular to evaluate its sensibility and robustness. We first tested whether the synthetic 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.
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