IGEM:IMPERIAL/2008/New/Genetic Circuit
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In this case, [math]\displaystyle{ [protein] }[/math] represents the concentration of GFP, [math]\displaystyle{ k_{1} }[/math] represents the rate of sythesis and [math]\displaystyle{ d_{1} }[/math] represents the degradation rate.
We can easily simulate this synthesis-degradation model using matlab:
We can also solve this ODE analytically.
Modelling Inducible Gene ExpressionThe repressor is constitutively expressed. Hence we can assume the constitutive expression model from the previous characterisation step. [math]\displaystyle{ \frac{d[R]}{dt} = k_{1} - d_{1}[R] }[/math] When the inducer is added it binds reversibly to the repressor. [math]\displaystyle{ R + I \Leftrightarrow RI }[/math] Free repressor only binds to the promoter, we think this will show co-operative binding as there are two repressor binding sites on the promoter sequence. Then transcription will be a function of free repressor concentration. [math]\displaystyle{ Transcription = \frac{\beta.{[R]}^n}{{K_m}^n+[R]^n} }[/math] And overall protein expression can be described as [math]\displaystyle{ \frac{d[protein]}{dt} = Transcription - d_2[protein] }[/math]
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