Team:ETH Zurich/Modeling/Analysis

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System Analysis
Parameter Sweeps Sensitivity Analysis
We wanted to analyse the effect of the parameters on the output of our system. We achieved this by looking at how the characteristics of the GFP band change when we explore the parameter space of a certain constant and at the sensitivity of GFP to the value of the toxic input substance (acetaldehyde or xylene) .

Parameter Sweeps

For the parameters that belong to the band detector module, we explored their parameter spaces to quantify how they affect the features of the GFP band . As we varied each parameter (while keeping the rest constant), we varied also the acetaldehyde input and monitored the GFP output. We analyzed the parameter space only for the model that uses acetaldehyde as an input parameter.

In both models the band is affected in the same way by the parameters of the band detector module. The sensor mechanism can only cause shifts in the band but the band detector module itself is unaffected by the input of the system. The parameter sweeps for the xylene model would therefore be similar to the acetaldehyde model, with differences only for the parameters involved in the sensor mechanism.

The following figures show how the band changes with the variation of protein production rates, repression coefficients and degradation rates.


Protein Synthesis Rates

Figure 1: Exploring the parameter space of TetR production rate
Figure 2: Exploring the parameter space of CI production rate

When the TetR production rate is zero, there is no TetR present in the cell, which means that LacI_{M1} is not repressed so it is free to block GFP production. This can be seen in the graph, since at this value of TetR production the band is absent. As soon as the production rate becomes non-zero, the band appears. The existence of the GFP band is not affected by the value of the TetR production rate, the band is present and the peak stays at a constant amplitude. TetR production rate only affects the width and position of the band which widens and shifts to higher acetaldehyde values as the production rate increases.

The GFP band is however very robust to the CI production rate, all its characteristics being preserved.

Figure 3: Exploring the parameter space of LacI production rate
Figure 4: Exploring the parameter space of LacIM1 production rate

The LacI and LacIM1 production rates have a more pronounced effect on the GFP band. When the values are very small the band seems to be wide and bounded only on one side. For larger values the desired band appears but its amplitude and width are decreasing with the further increase of the parameter values, up to the point where the band dissapears completely.

Figure 5: Exploring the parameter space of GFP production rate

For the GFP production rate, the band evolves as expected - it is absent for very small values and only its amplitude increases with increasing production rate.

Repression Coefficients

Figure 6: Exploring the parameter space of AlcR-AA repression coefficient
Figure 7: Exploring the parameter space of TetR repression coefficient

The AlcR-AA repression coefficient affects only the width and position of the GFP band, we can observe that the band widens and shifts to higher acetaldehyde concentrations as the value of the coefficient increases. The TetR repression coeffient seems to have an impact only on the amplitude of the peak which also exhibits positive correlation with the coefficient value.

Figure 8: Exploring the parameter space of CI repression coefficient
Figure 9: Exploring the parameter space of LacI repression coefficient

We observe that the GFP band is sensitive to the CI repression coefficent and appears only for low values of it. The LacI repression coefficient does not have a high impact on the band as it only widens it when the parameter value gets larger.

Protein Degradation Rates

Figure 10: Exploring the parameter space of TetR degradation rate
Figure 11: Exploring the parameter space of CI degradation rate

The degradation rate of TetR does not influence the existence of the band and the amplitude of its peak. The GFP band seems to be present at all parameter values, however its position and width change in the region of low degradation rates, but it is completely robust at higher concentration values. The CI degradation rate allows the band to appear only when its value is low, at higher rates the band formation is stopped.

Figure 12: Exploring the parameter space of LacI degradation rate
Figure 13: Exploring the parameter space of GFP degradation rate

We notice that the amplitude of the GFP peak and the bandwidth exhibit positive correlation with respect to the LacI degradation rate. As expected, there is a strong negative correlation between the height and width of the peak and the GFP degradation rate, since the GFP band can exist only of the GFP molecules are not degraded too quickly.

Sensitivity Analysis

Sensitivity analysis is a technique that studies the change of the output (or any observable) of a certain function with the variation of a certain parameter. It gives us an overview of how sensitive the model is with respect to the parameter, i.e. what the impact of the parameter is. The sensitivity is defined as the partial differential equation of the observable with respect to a certain parameter.

We performed sensitivity analysis with respect to acetaldehyde or xylene. For different input concentrations we monitored the change in the GFP output (dGFP/dAcetaldehyde or dGFP/dXylene).

For the acetaldehyde model, it can be seen from the figure below that the sensitivity is highest when GFP rises. For the peak itself (at [AA] = 1000uM), the sensitivity drops down and then rises again once GFP concentration starts decreasing. This tells us that the GFP concentration level is most sensitive to acetaldehyde at those concentrations where GFP rises and falls.


Figure 14: Sensitivity analysis with respect to acetaldehyde
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