Modeling the Transcriptional Cascade

The modeling of the transcriptional cascade had several objectives:

• to provide the delay team with design guidelines which would maximize the delay time

• to asses the affect of parameter variation on the delay time

• to determine areas of instability in the parameter space

A schematic of the system to be modeled can be seen below. A full description of the Transcriptional Cascade can be found here.

Transcriptional cascade assembly and overview

The system shown above has one input: IPTG and one primary output: GFP. It is characterized by the delay time, which is measured as the time between the induction with IPTG and a significant expression in the final product (GFP).

ODEs

The kinetic equations were written out in a Matlab script. A total of ten equations were used: one for the diffusion of IPTG into the cell, one for the binding of IPTG to LacI, as well as four transcription equations, and four translation equations for the various levels of the cascade.

The notation in this system of equations can be seen in the table below:

The solution with the default parameters of the system of ODEs can be seen below:

A function was written to determine the point at which the concentration of the final product reached a certain threshold. A threshold of 1E-8 M was used in our simulations, which corresponds to less than a dozen molecules within a cell given a cell volume of around 1E-15 L. Using the default parameter values a delay time of 385 min was predicted.

Sensitivity

A sensitivity analysis was done on the system. This looked at how variations in each parameter would influence the delay time. Parameters were swept over a range of values while the change in delay time was observed. The normalized sensitivity for various parameters is shown below.

As we can see in the table above, the system is most affected by changes in a_λp, α₂, followed by the strengths of the promoters used.

Parameter Sweeps

In the following plots the delay time of the cascade is shown as a function of two different parameters. A delay time off 500 is used to represent an infinite delay (maroon colour). The delay is considered to be over once the concentration of the final product (in this case GFP) goes above 10nM.

Delay Time vs Transcription Leakage of λp. Translation Rate α1 (of TetR mRNA) vs Translation Rate α4 (of GFP mRNA). Maximum Transcription Rate of λp vs Translation Rate α4 (of GFP mRNA). Maximum Transcription Rate of pLac vs Maximum Transcription Rate of λp. Maximum Transcription Rate of pLac vs Maximum Transcription Rate of pTet. Maximum Transcription Rate of pTet vs Translation Rate α2 (of CI mRNA). Maximum Transcription Rate of pTet vs Maximum Transcription Rate of λp. Transcription Leakage of λp vs Transcription Leakage of pTet.

Stability

Maximum Transcription Rate of λp vs Translation Rate α4 (of GFP mRNA). Maximum Transcription Rate of pLac vs Maximum Transcription Rate of λp. Maximum Transcription Rate of pLac vs Maximum Transcription Rate of pTet. Maximum Transcription Rate of pTet vs Translation Rate α2 (of CI mRNA). Maximum Transcription Rate of pTet vs Maximum Transcription Rate of λp. Transcription Leakage of λp vs Transcription Leakage of pTet.

Design Recommendations

Based on the results of the simulations, a series of recommendations were given to the delay team to aid them in choosing parts which would maximize the delay time.

1. Significant transcription leakages greatly shorten the delay time. Attempt to minimize leakages. Leakage of λp is a far bigger problem than pTet leakage.
2. Use a weak promoter and a weak RBS on the last stage (λp) of the cascade.
3. A weak pLac promoters is favorable.
4. A strong pTet promoter is favorable.
5. A strong RBS on CI gene is favorable.
6. A weak RBS on TetR gene is favorable.
7. A weak RBS on the endonuclease is favorable although a strong RBS can be used for the GFP gene.
8. When choosing RBS and promoter strengths avoid the red areas on the stability plots.