Team:ETH Zurich/labblog/20140824mod

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$$\frac{d[GFP]}{dt}=\frac{k_{45}[P_{tot}]\Big(-1+\sqrt{1+8\frac{AHL_i}{K_{d3}}}\Big)^2}{16\frac{K_{d4}}{K_{d3}}+\Big(-1+\sqrt{1+8\frac{AHL_i}{K_{d3}}}\Big)^2}-d_{GFP}[GFP]$$.
$$\frac{d[GFP]}{dt}=\frac{k_{45}[P_{tot}]\Big(-1+\sqrt{1+8\frac{AHL_i}{K_{d3}}}\Big)^2}{16\frac{K_{d4}}{K_{d3}}+\Big(-1+\sqrt{1+8\frac{AHL_i}{K_{d3}}}\Big)^2}-d_{GFP}[GFP]$$.
For very high initial concentrations AHLi, we have  
For very high initial concentrations AHLi, we have  
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$$\frac{d[GFP]}{dt}={k_{45}[P_{tot}]-d_{GFP}[GFP]$$
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$$\frac{d[GFP]}{dt}=k_{45}[P_{tot}]-d_{GFP}[GFP]$$
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  so by taking $t_{1/2}=\frac{ln(2)}{d_{GFP}}$ from experimental curves, we are able to retrieve $$d_{GFP} = 4.9 . 10^{-3} min^{-1}$$.
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  so by taking  
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$$t_{1/2}=\frac{ln(2)}{d_{GFP}}$$
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from experimental curves, we are able to retrieve  
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$$d_{GFP} = 4.9 . 10^{-3} min^{-1}$$.
For very low initial concentrations of initial AHL and considering degradation, we have  
For very low initial concentrations of initial AHL and considering degradation, we have  
   
   
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$$\frac{d[GFP]}{dt}=\frac{k_{45}[P_{tot}][AHLi]^2e^{-2d_{AHL}t}}{K_{d4}K_{d3}}-d_{GFP}[GFP]$$
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$$ \frac{d[GFP]}{dt}=\frac{k_{45}[P_{tot}][AHLi]^2e^{-2d_{AHL}t}}{K_{d4}K_{d3}}-d_{GFP}[GFP].$$
   
   
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  We find $$[GFP]=\frac{k_{45}[P_{tot}][AHLi]^2}{K_{d4}K_{d3}(d_{GFP}-2d_{AHL})}(e^{-2d_{AHL}t}-e^{-d_{GFP}t})$$.
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  We find  
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$$ [GFP]=\frac{k_{45}[P_{tot}][AHLi]^2}{K_{d4}K_{d3}(d_{GFP}-2d_{AHL})}(e^{-2d_{AHL}t}-e^{-d_{GFP}t}).$$  
  This curve has a maximum at  
  This curve has a maximum at  
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$$t_{max}=\frac{1}{d_{GFP}-2d_{AHL}}ln\big(\frac{d_{GFP}}{2d_{AHL}}\big)$$
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$$ t_{max}=\frac{1}{d_{GFP}-2d_{AHL}}ln\big(\frac{d_{GFP}}{2d_{AHL}}\big).$$
   
   
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This way we can find from experimental curves $$d_{AHL}=4,0.10^{-3} min^{-1}$$
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This way we can find from experimental curves $$d_{AHL}=4,0.10^{-3} min^{-1}$$
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This AHL degradation rate is alumped parameter between internal and external degradation rates, equivalent to
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$$d_{AHLext}+\alpha d_{AHLint}$$  
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This AHL degradation rate is alumped parameter between internal and external degradation rates, equivalent to $$d_{AHLext}+\alpha d_{AHLint}$$ where $$\alpha$$ is the fraction of the volume occupied by cells in the whole culture. From that on we could find reasonable values for internal and external AHL degradation rates.  
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where &alpha; is the fraction of the volume occupied by cells in the whole culture. From that on we could find reasonable values for internal and external AHL degradation rates.  
Finally the curves simulated by our model with these parameters fit the experiments quite well:
Finally the curves simulated by our model with these parameters fit the experiments quite well:

Latest revision as of 20:04, 11 October 2014