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. We then decided that we could use those values as well in our model.

Hill function

• When we had no relevant information, we decided to model the protein concentration evolution by a Hill function. We

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. 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 γ_FlhDC 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

• Furthermore, since [EnvZ] has been normalized, we have to do so for θ_EnvZ 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]:

Parameters table

Bibliography