Team:TUDelft/Modeling Cascade
From 2009.igem.org
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.
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 certain expression level 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:
Symbol  Definition 
k_{IPTGin}, k_{IPTGout}  rate constants 
k_{50IPTG}, k_{50LacI}, k_{50TetR}, k_{50CI}  dissociation constants 
d_{mRNA}  mRNA degradation rate 
d_{TetR}, d_{CI}, d_{RFP}, d_{GFP}  protein degradation rates 
a_{pLac}, a_{pTet}, a_{λp}  transcription leakage (%) 
c_{pLac}, c_{pTet}, c_{λp}  maximum transcription rates 
α_{1}, α_{2}, α_{3}, α_{4}  translation rates 
n_{IPTG}, n_{LacI}, n_{TetR}, n_{CI}  Hill coefficients 
[X]_{mRNA}  concentration of X mRNA 
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 1E8 M was used in our simulations, which corresponds to a handful of proteins within a cell, assuming a cell volume of around 1E15 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, ordered with the most sensitive parameter at the top.
Parameter  Description  Normalized Sensitivity 
d_{mRNA}  mRNA degradation rate  4.65 
d_{CI}  CI degradation rate  4.64 
a_{λp}  transcription leakage of λp  4.04 
α_{2}  CI translation rate (RBS strength)  1.06 
d_{TetR}  TetR degradation rate  0.53 
c_{λp}  λp promoter strength  0.42 
α_{4}  GFP translation rate (RBS strength)  0.38 
c_{pTet}  pTet promoter strength  0.36 
c_{pLac}  pLac promoter strength  0.22 
α_{1}  TetR translation rate (RBS strength)  0.14 
a_{pTet}  transcription leakage of pTet  0.12 
d_{GFP}  GFP degradation rate  0.04 
a_{pLac}  transcription leakage of pLac  0.00 
d_{LacI}  LacI degradation rate  0.00 
As we can see in the table above, the system is most affected by changes in d_{mRNA}, d_{CI}, and a_{λp}. This agrees well with the analysis done in the paper A general computational method for robustness analysis with applications to synthetic gene networks [9] on a similar cascade system. That analysis also found the sensitivity to be greatest for the degradation parameters.
The analysis also revealed that the leakage on the last stage of the cascade is also an important parameter. The plot below shows how the delay time quickly drops as the leakage passes 10^{4} %.
Parameter Sweeps
In the following plots the delay time of the cascade is shown as a function of two different parameters. A delay time of 800 is used to represent an infinite delay (maroon colour).

Stability
The stability of the system was investigated by determining the Jacobian of the system of ODEs analytically. The eigenvalues of the Jacobian can be seen below.
Since all of the values are real and negative we can conclude that our system will be stable.
As the parameter sweeps revealed, there are areas within the parameter space with a delay time of infinity. The plots below highlight the regions of the parameter space to avoid. Red areas correspond to a delay time of infinity, while blue areas correspond to a delay time less than 10 min (not enough time to conjugate the signal plasmid further). The cascade operates properly within the yellow areas.

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.
 Minimize the degradation rate of CI, do not use genes with degradation tags.
 Significant transcription leakages greatly shorten the delay time. Attempt to minimize leakages. Leakage of λp is a far bigger problem than pTet leakage.
 Use a weak promoter and a weak RBS on the last stage (λp) of the cascade.
 A weak pLac promoters is favorable.
 A strong pTet promoter is favorable.
 A strong RBS on CI gene is favorable.
 A weak RBS on TetR gene is favorable.
 A weak RBS on the endonuclease is favorable although a strong RBS can be used for the GFP gene.
 When choosing RBS and promoter strengths avoid the red and blue areas on the stability plots.
All of the source code used can be found on the downloads page.