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Model construction

Introduction

This preliminary approach is mainly based on a bibliographical work. In fact, it is important to understand that with this approach, we did not intend to design the most accurate model possible. We found essential to choose only parameters that could be found in literature. The main goal was to get quickly a first idea of the way our system could behave.
In this section, we will present the model we chose to describe the evolution of our system's concentrations.

Classical model and temporal rescaling

Classically we use the following equation to model gene interactions (see for example in [1]) :

where [Y] denotes the concentration of Y protein and γ its degradation rate (which unit is time^-1).

We did a lot of bibliography work and found many interesting expressions found in S.Kalir and U. Alon article. We also kept the parameters values. When we did not find relevant information, we chose a classical Hill function.

We normalized every concentration, so that their value would range between 0 and 1. Indeed, we found it necessary because we needed to be able to compare the respective influences of these concentrations.

Furthermore, it is important to note that this degradation rate represents both the influence of the degradation and dilution. We assume that the degradation can be neglected compared to the dilution caused by the cell growth. Thus, every degradation rates are equal. We kept the designation “degradation rate” for convenience, so as not to mix up with the dilution that might occur elsewhere.

We therefore wanted to have a proper time scale. We then set the degradation rates, γ ,to 1. Since we can know easily the value of the real half-time, we may know the real timescale out of our computations. Then we have:

Finally, the key problem with a dynamic system consists in the fact that adding a new equation gives a more detailed idea of the overall process, but one looses meaning by doing so since new undefined parameters appear. Hence, it becomes hard to link those parameters with a biological meaning and reality. In that respect, we chose not to introduce the mRNA state between transcription and translation in our model. We presented things as if a protein would act directly upon the following.

Conclusion

We finally obtained the following equations :

and the following parameters :

Parameter Table
Parameter	Meaning	Original Value	Normalized Value	Unit	Source
γ	Degradation rate	0.0198	1	min^-1	wet-lab
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_EnvZ-RFP	FlhDC activation coefficient	100	0.2222	min^-1	[1]
β'_EnvZ-RFP	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	∅

Now you have had a good overlook of our model, go see a more detailed justification] where our normalization choices are thouroughly explained :

↓ 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 n_EnvZ=4 and θ_EnvZ=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 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]:

we had

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
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_EnvZ-RFP	FlhDC activation coefficient	100	0.2222	min^-1	[1]
β'_EnvZ-RFP	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.

Navigator

Back to the overall presentation of our system and model Top of the page Have a look at our detailed justification! || Have a look at our Akaike criteria!

Bibliography

[1] Shiraz Kalir, Uri Alon. Using quantitative blueprint to reprogram the dynamics of the flagella network. Cell, June 11, 2004, Vol.117, 713-720.

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-Now you have had a good overlook of our model, go see a more [[Team:Paris/Network_analysis_and_design/Core_system/Model_construction/Detailed_justification|detailed justification]], where our normalization choices are thouroughly explained!
+Now you have had a good overlook of our model, go see a more detailed justification]  where our normalization choices are thouroughly explained :
+{{Paris/Toggle|Detailed justification|Team:Paris/Network_analysis_and_design/Core_system/Model_construction/Detailed_justification}}
 = Navigator =

Team:Paris/Analysis/Construction