Team:INSA-UPS France/Model/ODE

Croc'n Cholera
iGEM UPS-INSA Toulouse 2017

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System of ODEs

Equations

The dynamics of our microbial consortium was summed up in thirteen 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

Our model is mostly set by data from publications, because the majority of the required data necessitates complex experiments we could not perform during our project.

Name	Notation	Unit	Value	Reference
Vibrio cholerae maximum growth rate	μMAX,Vc	s-1	3.10-4	BioNumbers 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)
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)
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)
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)
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)
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 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
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 (11)
CAI-1 initial concentration	[CAI-1]0	mol/L	1.10-5	Ng et al., 2011 (12)
Michaelis-Menten constant of acetolactate synthase	KM,ALS	mol/L	1.36.10-2	Atsumi et al., 2009 (13)
Catalytic rate constant of acetolactate synthase	kcat,ALS	s-1	1.21.102	Atsumi et al., 2009 (13)
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 (14)
Activation constant of CqsS*-CAI-1 complex	Ka,CqsS*-CAI-1	mol/L	3.6.10-8	Ng et al., 2011 (12)
High Vibrio cholerae concentration in water	[Vc]0	cell/L	1.107	Huq et al., 1984 (15)
Dry weight of a bacterial cell	DWVh	pg/cell	0.28	BioNumbers 100008
Volume of a bacterial cell	Vintra,Vh	μm3	1	BioNumbers 100004
Volume of a yeast cell	Vintra,Pp	μm3	66	BioNumbers 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 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 103026
Number of RNA polymerase on AMP DNA	RNA polymerase/DNA	Nb/DNA	4.95	BioNumbers 108308
Ribosome density on a bacterial mRNA		Nb/kb	6.6	BioNumbers 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 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 101192

Even if we mostly describe our model as a predictive one, some preliminary experimental results could have been implemented into our model. Indeed, an experimental estimation of our two chassis behaviour on a common medium was needed.

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

The system of ODEs was solved using Matlab R2017a, thanks to the free offer from iGEM. Initially set with ode45 solver, the recommanded Matlab ODE solver, the final script uses ode15s because ode45 was not enough efficient. ode15s is the solver recommanded when having problems or inefficiency with ode45, and is adapted to stiff problems.(16)

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

References

• (6) : http://bitesizebio.com/22824/how-to-manipulate-plasmid-copy-number/
• (16) : https://fr.mathworks.com/help/matlab/math/choose-an-ode-solver.html