# Simple Deterministic Model

We modify the model which created in 2007 by the imperial iGEM team. The basic model simulate the behavior of single cell, and follow these assumptions:

1. All molecules, including proteins, protein complexes and small molecules are uniformly distributed in cell bodies.
2. The diffusion rate of AHL is determined only by the AHL gradient between cytoplasm and extracellular matrix.
3. The mechanism of GFP formation is shown below:
1. At the beginning, the AHL receptor, LuxR protein, is produced by housekeeping gene and has a constant producing rate:
2. $$luxR \overset{\text{k1}}{\Longrightarrow} LuxR$$
3. AHL molecules bind LuxR and then the complex "LuxR-AHL" are formed: Where k2 is the rate constant of binding of LuxR and AHL, and k3 is the rate constant of dissociation of complexes "LuxR-AHL"
4. $$LuxR+AHL \overset{\text{K2,K3}}\Longleftrightarrow (LuxR\_AHL)$$
5. The complexes "LuxR-AHL" bind the promoter and trigger the transcription and translation reaction to produce the GFP protein. The rate of GFP protein formation is determined by Hill function.
$$(LuxR\_AHL)\overset{\text{Hill function}}{\Longrightarrow} GFP+LuxR$$ $$Hill\ Function: k4 \times \frac{[LuxR\_AHL]^n}{Ke^n+[LuxR\_AHL]^n}$$
4. All molecules, including LuxR protein, AHL, LuxR-AHL complex and GFP are degraded on corresponded rate constant.
Implementation:

We implement the model in MATLAB. Users can simulate this deterministic model many times with different rate constant (by specifying the standard derivation of constants).

Figure. We simulate the model 100 times with different rate constant (by specifying the std derivation)