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.

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!

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