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Why model the genetic circuit?
An accurate mathematical description of genetic circult behaviour is one of the foundations of synthetic biology. Such descriptions are an integral component of part submission to the registry, as exemplified by the canonical characterised part F2620. <citation needed>. The ability to capture part behaviour as a mathematical relationship between input and output is useful for future re-use of the part modification of integration into novel genetic circuits.
Modelling Constitutive Gene Expression
A simple synthesis-degradation model is assumed for the modelling of the expression of a protein under the control of a constitutive promoter, with the same model assumed for all four promoter-RBS constructs. The synthesis-degradation model assumes a steady state level of mRNA.
In this case, [protein] represents the concentration of GFP, k1 represents the rate of sythesis and d1 represents the degradation rate.
We can easily simulate this synthesis-degradation model using matlab:
We can also solve this ODE analytically.
Modelling Inducible Gene Expression
The repressor is constitutively expressed. Hence we can assume the constitutive expression model from the previous characterisation step.
When the inducer is added it binds reversibly to the repressor.
Repressor only binds to the promoter when it is in its unbound form, hence transcription will be a function of free repressor concentration.
And overall protein expression can be described as