Team:Paris/Network analysis and design/Core system/Model construction/Detailed justification
From 2008.igem.org
We shall present here a more detailed presentation of the choice we made as far as our model is concerned
Contents 
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 SUMgate 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 n_{EnvZ}=4 and θ_{EnvZ}=0.5.
Normalization
FliA, CFP, YFP, EnvZRFP
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 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
 This is highly interesting since normalization implies that β_{FlhDC}=1 , so that we do not need to find a value for β_{FlhDC}.
 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]:
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 halflife 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}  wetlab 
β_{FliA}  FlhDC activation coefficient  50  0.1429  min^{1}  [1] 
β'_{FliA}  FliA activation coefficient  300  0.8571  min^{1}  [1] 
β_{CFP}  FlhDC activation coefficient  1200  0.8276  min^{1}  [1] 
β'_{CFP}  FliA activation coefficient  250  0.1724  min^{1}  [1] 
β_{YFP}  FlhDC activation coefficient  150  0.3333  min^{1}  [1] 
β'_{YFP}  FliA activation coefficient  300  0.6667  min^{1}  [1] 
β_{EnvZRFP}  FlhDC activation coefficient  100  0.2222  min^{1}  [1] 
β'_{EnvZRFP}  FliA activation coefficient  350  0.7778  min^{1}  [1] 
β_{FlhDC}  Maximum production rate  1  min^{1}  ∅  
n_{envZ}  Hill coefficient  4  ¤  ∅  
θ_{envZ}  Hill characteristic concentration  0.5  c.u  ∅ 
c.u. being an arbitrary concentration unit.