# Team:Paris/Network analysis and design/Core system/Model construction/Detailed justification

We shall present here a more detailed presentation of the choice we made as far as our model is concerned

# Sum effect and linear modelling

• The flagella gene network has been thoroughly studied in [1]. We used two major results presented in this study. Firstly, Shiraz Kalir and Uri Alon came up with the fact that the promoters of class 2 genes, among which fliL, flgA and flhB, behaved like SUM-gate functions with flhDC and fliA inputs. Secondly, their experiments proved that these influences could be considered as linear. Thus the following model:

β and β’ represent the relative influence of flhDC and fliA respectively, the units of β and β’ being time-1.

• Furthermore, they came up with numerical values of β and β’ for each gene, which fitted quite well to their experiments. Since our experimental conditions are similar to those described in the article, we decided that we could use those values as well in our model.
• Thus the resulting equations

# Hill function

When we had no relevant information, we decided to model the promoter activity by a Hill function. This was the case for the effect of envZ over FlhDC :

Thus the dynamic equation for [FlhDC] :

As for the parameters, we decided to chose biologically feasible values, that is nEnvZ=4 and θEnvZ=0.5.

# Normalization

## FliA, CFP, YFP, EnvZ-RFP

We kept the β and β’ values found by S. Kalir and U. Alon, since they showed the relative influence of flhDC and fliA. To have the same order of magnitude between each specie, we normalized those parameters between 0 and 1 as following. We reasoned independently for each equation, wishing to set the equilibrium values of the concentration to 1 given input values of 1. With the values taken from S. Kalir and U. Alon, this gave:

• In fact, if we take CFP for example:

The maximum of [CFP] is reached when [fliA] = 1 and [flhDC] = 1 ; when we solve with these condidtions, we obtain :

Then setting the equilibrium value of [CFP] to 1 corresponds to setting

• The analysis of fliA is different, but not the result:

With an input of flhDC equal to 1, the solution of the differential equation is:

And the condition on the equilibrium imposes

• To conclude, we see that we always get the same condition:
• Finally, since we had imposed γ=1 we resulted with β+β'=1.

## FlhDC

• Likewise the previous analysis, we set γFlhDC to 1. Then, since FlhDC is fully expressed when envZ is not, we see that when solving under this conditions, we get

hence the need to set

• This is highly interesting since normalization implies that βFlhDC=1 , so that we do not need to find a value for βFlhDC.
• Furthermore, since [EnvZ] has been normalized, we have to do so for θEnvZ as well, since its role is to stand as a reference concentration for EnvZ. Therefore, we have to normalize it in the same way we did for [EnvZ]:
which means we have to impose :

# Time rescaling

We evaluated in the wet lab the half life time for our cells, and then calculated the degradation constants using :

The value for half-life time we found and used is 35min. Setting γ to one, gave us the time rescaling factor (0.0198).

# Parameters table

Parameter Table
Parameter Meaning Original Value Normalized Value Unit Source

γ Degradation rate 0.0198 1 min-1 wet-lab
βFliA FlhDC activation coefficient 50 0.1429 min-1 [1]
β'FliA FliA activation coefficient 300 0.8571 min-1 [1]
βCFP FlhDC activation coefficient 1200 0.8276 min-1 [1]
β'CFP FliA activation coefficient 250 0.1724 min-1 [1]
βYFP FlhDC activation coefficient 150 0.3333 min-1 [1]
β'YFP FliA activation coefficient 300 0.6667 min-1 [1]
βEnvZ-RFP FlhDC activation coefficient 100 0.2222 min-1 [1]
β'EnvZ-RFP FliA activation coefficient 350 0.7778 min-1 [1]
βFlhDC Maximum production rate 1 min-1
nenvZ Hill coefficient 4 ¤
θenvZ Hill characteristic concentration 0.5 c.u

c.u. being an arbitrary concentration unit.