# Kinetic model

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.

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). $$\frac{d[\textit{Vc}]_W}{dt} = V_{growth,Vc} - V_{death,Vc}$$ $$\frac{d[\textit{Vh}]_D}{dt} = V_{growth,Vh} - V_{death,Vh}$$ $$\frac{d[\textit{Pp}]_D}{dt} = V_{growth,Pp} - V_{death,Pp}$$

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

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

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

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. $$\frac{d[ALS_{RNA}]_{Vh}}{dt} = V_{transcription,ALS} - V_{degradation,ALS RNA}$$ $$\frac{d[ALS_{enzyme}]_{Vh}}{dt} = V_{translation,ALS} - V_{degradation,ALSenzyme}$$

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

# 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.

Name Notation Unit Value Reference
Vibrio cholerae maximum growth rate μMAX,Vc s-1 3.10-4 BioNumbers ID 112369 1
Leucrocine I MIC for V. cholerae MICLeucro,Vc mol/L 6.4.10-5 Pata et al., 2011 2
Leucrocine I MIC for V. harveyi MICLeucro,Vh mol/L 6.4.10-5 Pata et al., 2011 2 - Extrapolation from V. cholerae result
Leucrocine I MIC for P. pastoris MICLeucro,Pp mol/L 1015 Assuming no effects on Pichia pastoris
Leucrocine I IC50 for V. cholerae MICLeucro,Vc mol/L 1.92.10-4 Considering IC50 = 3.MIC - Extrapolation of data from standard antibiotics) 3
Leucrocine I IC50 for V. harveyi IC50Leucro,Vh mol/L 1.92.10-4 Considering IC50 = 3.MIC - Extrapolation of data from standard antibiotics 3
Leucrocine I IC50 for P. pastoris IC50Leucro,Pp mol/L 1015 Assuming no effects on Pichia pastoris
cOT2 MIC for V. cholerae MICcOT2,Vc mol/L 8.1.10-6 Prajanban et al., 2017 4
cOT2 MIC for V. harveyi MICcOT2,Vh mol/L 8.1.10-6 Prajanban et al., 2017 4 - Extrapolation from V. cholerae result
cOT2 MIC for P. pastoris MICcOT2,Pp mol/L 1015 Assuming no effects on Pichia pastoris
cOT2 IC50 for V. cholerae IC50cOT2,Vc mol/L 2.43.10-5 Considering IC50 = 3.MIC - Extrapolation of data from standard antibiotics 3
cOT2 IC50 for V. harveyi IC50cOT2,Vh mol/L 2.43.10-5 Considering IC50 = 3.MIC - Extrapolation of data from standard antibiotics 3
cOT2 IC50 for P. pastoris IC50cOT2,Pp mol/L 1015 Assuming no effects on Pichia pastoris
D-NY15 MIC for V. cholerae MICD-NY15,Vc mol/L 1.54.10-5 Yaraksa et al., 2014 5
D-NY15 MIC for V. harveyi MICD-NY15,Vh mol/L 1.54.10-5 Yaraksa et al., 2014 5 - Extrapolation from V. cholerae result
D-NY15 MIC for P. pastoris MICD-NY15,Pp mol/L 1015 Assuming no effects on Pichia pastoris
D-NY15 IC50 for V. cholerae IC50D-NY15,Vc mol/L 4.62.10-5 Considering IC50 = 3.MIC - Extrapolation of data from standard antibiotics 3
D-NY15 IC50 for V. harveyi IC50D-NY15,Vh mol/L 4.62.10-5 Considering IC50 = 3.MIC - Extrapolation of data from standard antibiotics 3
D-NY15 IC50 for P. pastoris IC50D-NY15,Pp mol/L 1015 Assuming no effects on Pichia pastoris
V. cholerae death rate with AMP kkill,Vc s-1 3.10-3 Yaraksa et al., 2014 5
V. harveyi death rate with AMP kkill,Vh s-1 3.10-3 Extrapolation from Yaraksa et al., 2014 5
P. pastoris death rate with AMP kkill,Pp s-1 0 Assuming no effects
Transfer coefficient through the membrane K s-1 1 Arbitrary value
Number of als gene per cell alsDNA,0 Nb/cell 15 Considering a low copy plasmid 6
Number of AMP gene per cell AMPDNA,0 Nb/cell 1 Protocol: genomic integration
V. harveyi transcription rate ktranscript,Vh nt/s 30 Molecular Biology course, Transcription - Faculté des Sciences - Rabat 7
Vibrio harveyi transcription rate ktranslation,Vh nt/s 15 Molecular Biology course, Translation - Faculté des Sciences - Rabat 8
als gene promoter influence kP,als / 1 Inductible promoter
AMP gene promoter influence kP,AMP / 1 Inductible promoter
mRNA degradation constant Kdeg,mRNA s-1 5.10-3 Esquerré et al., 2015 9
als gene length DNA length nucleotides 1662 UniProtKB - Q7DAV2 10
als mRNA length RNA length nucleotides 1730 Parts design
Number of CqsS* receptor per cell CqsS*/cell Nb/cell 1015 Arbitrary value
Number of Odr10 receptor per cell Odr10/cell Nb/cell 1015 Arbitrary value
Pichia pastoris transcription rate ktranscript,Pp nt/s 50 Molecular Biology course - INSA Toulouse(11)
Pichia pastoris translation rate ktranslation,Pp nt/s 48 Molecular Biology course, Translation - Faculté des Sciences - Rabat 8
AMP gene length DNA length nucleotides 360 Parts design
AMP mRNA length RNA length nucleotides 651 Parts design
Vibrio cholerae minimal pathogenic concentration [Vc]pathogenic cell/L 4.104 Medical Microbiology 4th edition (12)
CAI-1 initial concentration [CAI-1]0 mol/L 1.10-5 Ng et al., 2011 13
Michaelis-Menten constant of acetolactate synthase KM,ALS mol/L 1.36.10-2 Atsumi et al., 2009 14
Catalytic rate constant of acetolactate synthase kcat,ALS s-1 1.21.102 Atsumi et al., 2009 14
Degradation constant of acetolactate synthase Kdeg,ALS s-1 0 Assuming a negligeable value in our conditions
Degradation constant of antimicrobial peptides Kdeg,AMP s-1 0 Aleinein et al., 2013 15
Activation constant of CqsS*-CAI-1 complex Ka,CqsS*-CAI-1 mol/L 3.6.10-8 Ng et al., 2011 13
High Vibrio cholerae concentration in water [Vc]0 cell/L 107 Huq et al., 1984 16
Dry weight of a bacterial cell DWVh pg/cell 0.28 BioNumbers ID 100008
Volume of a bacterial cell Vintra,Vh μm3 1 BioNumbers ID 100004
Volume of a yeast cell Vintra,Pp μm3 66 BioNumbers ID 100452
Dry weight of a yeast cell DWPp pg/cell 18.48 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.

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 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

• [Vh]0,D = 1012 cell/L (OD600nm ≈ 1.5 18)
• [Pp]0,D = 1012 cell/L (OD600nm ≈ 1.5 18)
• [Vc]0,W = 4.107 cell/L
• VD = 20.10-3 L

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.

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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