# Numerical Analysis of Gene Regulatory Network Protein Synthesis and Decay

### + Parameters

• Gas Protein
group:
Gas
group:
Gas Protein Repressor
group:
AHLR
group:
AHL Synthesis
group:
- Graph group 1
- Graph group 2
- Graph group 3
- Graph group 4
- Graph group 5
• ## Governing equations

### Gas production/dissipation

The rate of production of gas is proportional to the amount of gas protein in the system, and some gas dissipates at a constant rate. Thus, $${dQ_{gas} \over dt} = P_{gas} \cdot Q_{gp} - D_{gas}$$

### Gas Protein Production/Decay

The production of the gas protein is affected negatively by its size, and positively by the promoter strength.

Given enough repressor, the production of the protein drops exponentially to zero.
Furthermore, it degrades with a given half-life.
Thus, $${dQ_{gp}\over dt} = \left\{ \begin{array}{l l} -{Q_{gp} \cdot ln(1/2) \over \tau_{\text{gp-halflife}} } + P_{prom\_str} \cdot P_{size\_fact} \cdot e ^ { {- ( Q_{rep} - P_{thresh} ) / P_{thresh\_sensitivity} }} & \quad \text{if Q_{rep} >= P_{thresh} }\\ -{Q_{gp} \cdot ln(1/2) \over \tau_{\text{gp-halflife}} } + P_{prom\_str} \cdot P_{size\_fact} & \quad \text{o.w. } \end{array} \right.$$ Alright. The first half of this stuff is the half-life. Tau itself is the half-life.
For this reason, the half-life is present regardless of the amount of repressor.
The second part is the production.

On the second equation, if the repressor is at small quantities, then the production is uninhibited.
On the first equation, if there is enough repressor, the production quickly drops to zero, at an exponential rate. How sensitive, or quick this "switch-off" occurs is controlled by the threshold sensitivity.
The larger this factor, the slower the switch-off/switch-on.

### Repressor Production/Decay

The repressor is promoted by the presence of gas. It will not be produced, if there is no gas.
There has to be enough gas for the repressor to be generated. This is the repressor's threshold.
The repressor itself has a half-life.
Thus, $${dQ_{rep}\over dt} = \left\{ \begin{array}{l l} -{Q_{rep} \cdot ln(1/2) \over \tau_{\text{rep-halflife}} } + R_{prom\_str} \cdot R_{size\_fact} - R_{prom\_str} \cdot R_{size\_fact} \cdot e ^ { {- ( Q_{gas} - R_{thresh} ) / R_{thresh\_sensitivity} }} & \quad \text{if Q_{gas} >= R_{thresh} }\\ -{Q_{rep} \cdot ln(1/2) \over \tau_{\text{rep-halflife}} } & \quad \text{o.w. } \end{array} \right.$$