The complete model uses 18 kinetic parameters and 11 biochemical reactions. We got 13 of these parameters researching the literature, and of the other 5 we estimated 2. The remaining 3 we adjusted to the observed results. Reaction kinetics were gotten from the literature, and if no evidence was found then we assumed it to be Law of Mass Action.
1. Degradation of AHL by AiiA
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AiiA + AHL → AiiA |
Kinetics: |
Michaelis-Menten1,2 |
Parameters: |
k1cat = 27.97 s-1
K1m = 3.723 mM = 224.20427E5 molecules |
Flux: |
![Equation 1](https://static.igem.org/mediawiki/2008/9/9c/Eq1a.PNG) |
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2. Complex formation and dissociation between AHL and LuxR
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AHL + LuxR ↔ AHL:LuxR |
Kinetics: |
Mass Action3 |
Parameters: |
k2 = 10 -5 molecules-1 s-1
k-2 = 3.33 x 10 -3 s-1
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Flux: |
![Equation 2](https://static.igem.org/mediawiki/2008/6/6b/Eq2a.PNG) |
Notes: |
The complex formation is slow and its dissociation is fast, so with few AHL and LuxR the complex concentration is negligible. |
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2.1. Dimer formation and dissociation between AHL:LuxR complexes
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2 AHL:LuxR ↔ (AHL:LuxR):(AHL:LuxR) |
Kinetics: |
Mass Action3 |
Parameters: |
k2.1 = 10 -5 molecules-1 s-1
k-2.1 = 10 -2 s-1
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Flux: |
![Equation 2.1](https://static.igem.org/mediawiki/2008/e/e7/Eq3a.PNG) |
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3.1. CI synthesis induced by AHL and LuxR complexes dimer
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ρcI + (AHL:LuxR):(AHL:LuxR) → ρcI + (AHL:LuxR):(AHL:LuxR) + CI |
Kinetics: |
Mass Action3 |
Parameters: |
k3on = 10 -2 molecules-1 s-1
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Flux: |
![Equation 3.1](https://static.igem.org/mediawiki/2008/3/3d/Eq4a.PNG) |
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3.2. Constitutive CI synthesis
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ρcI → ρcI + CI |
Kinetics: |
Mass Action3 |
Parameters: |
k3off = 4 x 10 -2 s-1
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Flux: |
![Equation 3.2](https://static.igem.org/mediawiki/2008/4/49/Eq5a.PNG) |
Notes: |
To give more stability to the off state in the model, the rate constant in the presence of the inducer is lower than the constitutive rate constant, regardless the implication of a greater threshold to achieve the on state3. |
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4. Natural degradation of CI
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CI → Ø |
Kinetics: |
Mass Action3 |
Parameters: |
k4 = 0.002888 s-1
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Flux: |
![Equation 4](https://static.igem.org/mediawiki/2008/e/e5/Eq6a.PNG) |
Notes: |
The half life of CI with LAA tail is 4 minutes8. Andersen JB et al.9 conclude that LAA tail and LVA tail modified the half life of GFP in a similar extent. Given this value, the rate constant was calculated.. |
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4.1. Dimer formation and dissociation between CI molecules
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2 CI ↔ CI:CI |
Kinetics: |
Mass Action3 |
Parameters: |
k4.1 = 0.00001 molecules-1 s-1
k-4.1 = 0.01 s-1
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Flux: |
![Equation 4.1](https://static.igem.org/mediawiki/2008/7/72/Eq7.PNG) |
Notes: |
Kenneth et al. estimated the change in free Gibbs energy in this reaction (with wildtype CI) as -11.1 kcal/mol,10 which leads to an equilibrium constant of 8.32186E16 molecules-1. This implies that the forward rate constant should be at least sixteen orders of magnitude greater than the reverse rate constant, which means a constant repression of RcnA even with the constitutive CI synthesis. A parameter scan was run to determine the range of values that gives the desired behavior and the rate constants were chosen arbitrarily within this range. These values are comparable to others typical biochemical parameters. It has been shown that kinetic parameters can be modified by changing amino acid sequences (for example, CI half life is reduced by adding a LVA tail in the C-terminal), it’s proposed that it’s possible to engineer the protein to reach an acceptable dissociation constant. |
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6. RcnA production
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ρ → ρ + RcnA |
Kinetics: |
Cooperative inhibition (Hill kinetics)4,5,6,7 |
Parameters: |
n5 =1.9
![](https://static.igem.org/mediawiki/2008/4/4b/Eq9.1.PNG)
![](https://static.igem.org/mediawiki/2008/5/5f/Eq9.2.PNG)
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Flux: |
![Equation 6](https://static.igem.org/mediawiki/2008/c/c6/Eq8.PNG) |
Notes: |
ΔGCI:CI-OR1=-11.6 kcal/mol
ΔGCI:CI-OR2=-10.1 kcal/mol
ΔGCI:CI-OR1-OR2=-23.8 kcal/mol
ν6(Pl)=20mM/h=3346.111 molecules/s with 20 promoter copies (ρ0)7.
The promoter in our construction is Pr, which is similar to Pl, the one used to estimated the parameter7. |
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7. Nickel efflux by RcnA
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RcnA + Niint → RcnA + Niext |
Kinetics: |
Mass Action3 |
Parameters: |
k7 = ?
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Flux: |
![](https://static.igem.org/mediawiki/2008/a/a6/Eq10.PNG) |
Notes: |
Experimentally measured |
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8. Natural degradation of RcnA
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RcnA → Ø |
Kinetics: |
Mass Action3 |
Parameters: |
k8 = 1.666E-4 s-1
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Flux: |
![](https://static.igem.org/mediawiki/2008/0/0f/Eq11.PNG) |
Notes: |
This kinetic parameter wasn’t found in our bibliographic search and personal communication with Peter T. Chivers (Washington University School of Medicine) confirmed that this parameter is unknown. The value used is the degradation rate of LacY, the lactose permease of E. coli, which is also a transmembran protein.11 |
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9. Nickel import by unknown channel
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Unk + Niext → Unk + Niint |
Kinetics: |
Mass Action3 |
Parameters: |
k9 = ?
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Flux: |
![](https://static.igem.org/mediawiki/2008/9/9b/Eq12.PNG) |
Notes: |
Experimentally measured |
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NOTE: The average volume of an E. coli cell is 10-15 liters.
Defining the initial state of the system
The initial concentrations of the constitutive proteins (AiiA, LuxR, CI -constitutive synthesis- and CI:CI -due to constitutive synthesis-) were estimated based on the efficiency rate of their promoters, number of promoters per cell, degradation rate of their mRNAs, translation efficiency and degradation rate of the proteins. Initial concentrations of AHL:LuxR complex, the dimer of complexes, CI and CI:CI due to complex activation were set to 0, given these are all due to the action of AHL. Number of copies of both cI and rcnA promoters are 10 based on plasmid copy number. RcnA and Unk were estimated experimentally and set consistent to the observed rate. Concentration of AHL and nickel is determined by us to obtain the desired results.
AHL: It’s an arbitrary and adjustable value. Different outcomes can be observed manipulating this initial value.
Nickel (total): It’s an arbitrary and adjustable value. Different outcomes can be observed manipulating this initial value.
Unk: Both the Unk concentration and its rate constant are unknown. They are arbitrarily defined in such a way that the flux of the reaction 9 is consistent with experimental measurements.
[Unk]= (SLOPE) molecules
Niint: The initial concentration of Nickel inside the cell is estimated based on experimental measurements in absence of AHL.
[Niint]= (SLOPE) molecules
ρ and ρCI: Their concentration is defined by the copy number of the plasmids that contain them.
[ρ]= 10 molecules
[ρcI]= 10 molecules
CI and CI:CI: Given the constitutive synthesis and degradation rate of CI, as well as its dimerization constant, CI and CI:CI concentrations are estimated in absence of AHL.
[CI]= 138 molecules
[CI:CI]= 19 molecules
RcnA: Given the synthesis and degradation rate of RcnA, as well as the constitutive concentration of CI:CI, RcnA concentration is estimated in absence of AHL.
[RcnA]= 33150 molecules
AiiA: The constant concentration of AiiA is calculated taking into account the following parameters retrieved from literature:
- pLac average transcription rate1,2: 0.003 s-1
- mRNA average degradation rate1,3: 0.00766 s-1
- Average translation rate1,3: 0.31333 s-1
- AiiA degration rate: 0.00012 s-1
The half life of RcnA with LVA tail is approximately 2 minutes14; Andersen JB et al. found that this tail reduces the half life of GFP forty-eight times.9 Therefore the half life of wildtype AiiA can be estimated to 96 minutes.
[AiiA]= 10000 molecules
LuxR: The constant concentration of LuxR is calculated taking into account the following parameters retrieved from literature:
- pTet average transcription rate12,15: 0.003 s-1
- mRNA average degradation rate13: 0.00766 s-1
- LuxR translation rate16: 0.556 s-1
- LuxR degration rate16: 9.627E-5 s-1
[LuxR]= 22000 molecules
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