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<h1>Equations</h1> | <h1>Equations</h1> | ||
− | <p>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 <a href="https://static.igem.org/mediawiki/2017/2/2c/T--INSA-UPS_France--Complete_model.pdf">there</a>!</p> | + | <p>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. The final model contains <b>12 entities (including genes, RNAs, proteins, and cells)</b> and <b>24 reactions from several processes (transcription, translation, RNAs and proteins degradation, growth and death of each micro-organism, transport between comportments)</b> described using kinetic rate laws. |
+ | </p><p>For fearless people, our complete mathematical model and demonstration can be found <a href="https://static.igem.org/mediawiki/2017/2/2c/T--INSA-UPS_France--Complete_model.pdf">there</a>! The final set of ODEs is briefly described below.</p> | ||
<p>We need to characterize the growth and death of the three microorganisms: <i><b>Vibrio cholerae</b></i> (<i>Vc</i>) in water (W), <i><b>Vibrio harveyi</b></i> (<i>Vh</i>) and <i><b>Pichia pastoris</b></i> (<i>Pp</i>) in the device (D). | <p>We need to characterize the growth and death of the three microorganisms: <i><b>Vibrio cholerae</b></i> (<i>Vc</i>) in water (W), <i><b>Vibrio harveyi</b></i> (<i>Vh</i>) and <i><b>Pichia pastoris</b></i> (<i>Pp</i>) in the device (D). | ||
Line 730: | Line 731: | ||
<section> | <section> | ||
<h1>Solver</h1> | <h1>Solver</h1> | ||
− | <p>The system of ODEs was solved using <b>Matlab R2017a</b>, thanks to the free offer from iGEM. We used the <i>ode15s</i> solver, which was in this situation more efficient than the <i>ode45s</i> solver, very likely because the different time scales between some processes (e.g. growth vs signalling) renders the problem partially stiff.(17)</p> | + | <p>The system of ODEs was solved using <b>Matlab R2017a</b>, thanks to the free offer from iGEM. We used the <i>ode15s</i> solver, which was in this situation more efficient than the <i>ode45s</i> solver, very likely because the different time scales between some processes (e.g. growth vs signalling) renders the problem partially stiff.(17) The proposed implementation proved to be very efficient, with simulations performed in less than 1 second.</p> |
<p>You can freely re-use our code: <a href="https://static.igem.org/mediawiki/2017/e/ec/T--INSA-UPS_France--iGEM_INSA-UPS_France_2017_Model.zip" alt="">General_resolution + System_of_ODEs</a>.</p> | <p>You can freely re-use our code: <a href="https://static.igem.org/mediawiki/2017/e/ec/T--INSA-UPS_France--iGEM_INSA-UPS_France_2017_Model.zip" alt="">General_resolution + System_of_ODEs</a>.</p> | ||
</section> | </section> |
Revision as of 09:55, 23 October 2017
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. The final model contains 12 entities (including genes, RNAs, proteins, and cells) and 24 reactions from several processes (transcription, translation, RNAs and proteins degradation, growth and death of each micro-organism, transport between comportments) described using kinetic rate laws.
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 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(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 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). 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}
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. 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) 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. At the beginning of the project, we needed to know if our microbial synthetic consortium would work in practice and if the information transmission was possible. We thus carried out simulations by solving the ODEs system to have a first estimation of the dynamics 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. Vibrio cholerae initial concentration was set to 4.107 cell/L,
which is slightly higher than the value from publication. (16) 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 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. 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 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.
Simulation
Equations
Data
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
∞
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 (Farrag et al., 2015 (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 (Farrag et al., 2015 (3))
Leucrocine I IC50 for P. pastoris
IC50Leucro,Pp
mol/L
∞
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
∞
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 (Farrag et al., 2015 (3))
cOT2 IC50 for V. harveyi
IC50cOT2,Vh
mol/L
2.43.10-5
Considering IC50 = 3.MIC - Extrapolation of data from standard antibiotics (Farrag et al., 2015 (3))
cOT2 IC50 for P. pastoris
IC50cOT2,Pp
mol/L
∞
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
∞
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 (Farrag et al., 2015 (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 (Farrag et al., 2015 (3))
D-NY15 IC50 for P. pastoris
IC50D-NY15,Pp
mol/L
∞
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
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
14 400
Experiment - 21/06/17
Solver
Simulation results
under realistic conditions (device size, initial concentrations in the device)
→ Sensitivity analysis methods and their results are described in our model analysis page
References