Modified simple cell model (補完

We referenced and modified the model made by 2007 imperial iGEM team to our needs. This model simulate basic behavior of single cell, and assumptions are as follows:

1. All molecules, including proteins, protein complexes and small molecules are uniformly distributed in cell bodies.
2.  The diffusion rate constant $k16$ of AHL is determined only by the AHL gradient between cytoplasm (denoted as $[AHL_i]$) and extracellular matrix (denoted as $[AHL_e]$).
3. Initial $[AHL_i] = 0,$ and $[AHL_e] = \mbox{constant.}$
4. LuxR protein ($[LuxR]$) is produced either by housekeeping gene which is assumed to have a constant transcription rate $k1,$ or by positive feedback system discussed in 7. LuxR degrades with constant $k12.$
5. AHL binds to LuxR protein, thus forming complex $[C]$ at rate $k5.$ The complex degrades into AHL and LuxR at rate $k13.$ 
6. The aforementioned complex dimerize into dimer $[D]$ at rate $k6,$ and the dimer dissociates and forms two complex at rate $k14.$
7. The complex binds to inducible promoter of LuxR and GFP ($[GFP]$), which has three characteristics:
1. The promoter complies to Hill's equation with cooperativity 1.
2. Maximum transcription rate of this promoter is $k2.$
3. Hill's dissociation constant is $k3.$
8. The degradation rate of GFP is $k18.$ 
9. Translation and degradation rate of mRNA is $k4$ and $k11,$ respectively.
10. Concentration of every species of molecules is adjusted for cell growth (dilution):
\[\dfrac{dx}{dt} = - \dfrac{k8*x5}{k9+x5}*x\]
11. Parameters concerning bacterial growth:
1. $k7$  is growth yield of bacteria
2. $k8$ is maximum growth rate 
3. $k9$  is half-saturation constant
4. $k10$ is bacterial death fraction per time
5. [S] is nutrient in medium
6. [N] is cell density

\[\dfrac{d[S]}{dt} = -\dfrac{1}{k7}\cdot [N]\cdot\dfrac{k8*[S]}{k9+[S]}\]

\[\dfrac{d[N]}{dt} = [N]\cdot\dfrac{k8*[S]}{k9+[S]} - k10*[N]\]

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). 

Components of unknown variables:

x1 : [RHLR]

x2 : [Complex (AHL-RHLR)]

x3 : [dimer_of_complex]

x4 : [AHL_intracellular]

x5 : [S] (Nutrient in medium)

x6 : [N] (cell density)

x7 : [GFP]

System of ODEs:

\[\dfrac{d[LuxR]}{dt} =  (k4/k11)*(k1+(k2*x3/(k3+x3)))- k5*x1*x4 + k13*x2 - k12*x1 - (k8*x5/(k9+x5))*x1\]

\[\dfrac{d[Complex]}{dt} =   k5*x1*x4           - k13*x2           - 2*k6*(x2^2)          + 2*k14*x3          - (k8*x5/(k9+x5))*x2\]

\[\dfrac{d[dimer]}{dt} =   k6*(x2^2)- k14*x3- (k8*x5/(k9+x5))*x3\]

\[\dfrac{d[AHL_i]}{dt} =   k16*(AHLe - x4)  + k13*x2- k5*x1*x4 - k15*x4 - (k8*x5/(k9+x5))*x4\]

\[\dfrac{d[Nutrient]}{dt} =   (-1/k7)*x6*(k8*x5/(k9+x5))\]

\[\dfrac{d[Cell density]}{dt} =   x6*((k8*x5/(k9+x5)) - k10)\]

\[\dfrac{d[GFP]}{dt} =   (k17/k11)*((k2*x3/(k3+x3)))- k18*x7 - (k8*x5/(k9+x5))*x7\]