Team:Imperial College/Genetic Circuit

An accurate mathematical description of the genetic circuit is essential for projects involving synthetic biology. Such descriptions are an integral component of part submission to the Registry, as exemplified by the canonical characterisation of part F2620 (1). The ability to capture part behaviour as a mathematical relationship between input and output is useful for future re-use of the part and modification of integration into novel genetic circuits.

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:
ODE
Simulation File

We can also solve this ODE analytically. Consider the steady-state behaviour of [protein].

This relationship can be seen in the parameter scan graphs on the right.

From the wetlab experiments it is likely that we will obtain steady-state data for each of the four promoter-RBS constructs. If we assume the same rate of degradation of GFP in each case, we can have some measure of the relative rate of transcription through each promoter which will help us with the selection of the most appropriate promoter to use for Phase 2. In order to obtain an absolute measure of transcription (as opposed to a relative measure of transcriptional strength) we require constitutive expression in terms of molecules per cell (as opposed to fluorescence in arbitrary units).
Note from the parameter scan graphs:

• In the case where k1 = 0, no GFP is sythesised.
• In the case where d1 = 0, the concentration of protein does not reach a steady state.

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

The ODEs and simulation may be found in the Appendices section of the Dry Lab hub.

A literature search revealed two models of IPTG-induced expression through the Plac promoter.

In the simpler model (2) IPTG competes with free promoter for LacI binding, but does not itself bind to the LacI-promoter complex.

Assuming this model, all else equal, the steady-state concentration of free promoter and hence the steady-state concentration of GFP are independent of the initial concentration of IPTG.

ODEs
Simulation File

Pre-steady-state time evolution of GFP using the simple model.

The pre-steady-state dynamic behaviour of the GFP concentration will differ with different initial concentrations of IPTG (but the steady-state behaviour will not). Hence, accuarate data collection during the pre-steady-state phase is crucial for paramater estimation.

A more sophisticated model allows for interaction between IPTG and the promoter-LacI complex (3). Under this model, the dynamic behaviour (whether or not [GFP] attains a maximum higher than its steady-state value) depends on the relative strengths of the kinetic constants describing the interactions underlying the model. Either way, all else equal the steady-state [GFP] will vary as a Hill-function dependent on the initial concentration of IPTG; this characteristic can be used to discriminate between the two models.

ODEs
Simulation File

Pre-steady-state time evolution of GFP using the more sophisticated model.

References

1. pmid=18612302
2. pmid=17376875
3. Uri Alon book
4. Author (Year) Title - JournalPublisher Page
5. Elowitz and Leibler (2000) A Synthetic Oscillatory Network of Transcriptional Regulators - Nature 403(6767) 335-8
6. Uri Alon (1999) A book - Cambridge Uni Press 45