Team:NTU-Taida/Modeling/Stochastic Modeling

Introduction to Stochastic Modeling

As we know, deterministic model for a biology system uses sets of ODEs (ordinary differential equation) to describe the system dynamic behaviour. These differential equation are treated under continuous manner, which is a approximation of large quantity of molecule number in the environment. Moreover, following the theory that each reaction is the collision with proper direction and energy of molecules, we can assume the molecules are spread uniformly about the environment and each reaction rate can be evaluated by classical chemical kinetic law. However, such assumption failed if molecule number is few.

When the molecule number is few, the effect of probability emerges. We should consider the probability of every effective collision. In this model, we define that a state is one or more species with a particular number of molecule for each species. Each reaction involves at most two molecules as reactant. The reactions lead to state transitions. Take our model into consideration. The species involving in this model are:

1. DNA
2. DNA with TF bound
3. mRNA of LuxR
4. mRNA of LuxR and GFP
5. LuxR
6. GFP
7. AHL (intra-cell or extra-cell)
8. AHL-LuxR complex

Assume that DNA is stable and AHL remains constant during the time we concern. All the reaction that will change the number of each species are:

Transcription of LuxR mRNA from DNA

$$ DNA_{bound}+DNA_{unbound} \longrightarrow DNA_{bound}+DNA_{unbound}+mRNA_{LuxR} $$

Transcription of LuxR and GFP mRNA from bound DNA

$$ DNA_{bound}\longrightarrow DNA_{bound}+mRNA_{LuxR-GFP} $$

Degradation of mRNA

$$ mRNA_{LuxR}\longrightarrow \emptyset $$

$$ mRNA_{LuxR-GFP}\longrightarrow \emptyset $$

Translation of LuxR

$$ mRNA_{LuxR}\longrightarrow mRNA_{LuxR}+LuxR $$

Translation of GFP

$$ mRNA_{LuxR-GFP}\longrightarrow LuxR+GFP $$

Degradation of LuxR

$$ LuxR\longrightarrow \emptyset $$

Degradation of GFP

$$ GFP\longrightarrow \emptyset $$

LuxR-AHL complex association

$$ LuxR+AHL\longrightarrow complex $$

LuxR-AHL complex dissociation

$$ complex\longrightarrow LuxR+AHL $$

Binding of complex and unbound DNA

$$ complex+DNA_{unbound} \longrightarrow DNA_{bound} $$

Unbinding of complex and unbound DNA

$$ DNA_{bound} \longrightarrow complex+DNA_{unbound} $$

Degradation of AHL

$$ AHL\longrightarrow \emptyset $$

Some of you might wonder "where is the Hill equation?" In fact, Hill equation is embedded in this model. Assume the binding rate of $$complex$$ and $$ DNA_{unbound} $$ is $$ k_1 $$ and unbinding rate is $$k_2$$. According to the reaction listed above, if $$complex$$ is constant, the equilibrium conditions are:

$$k_1\times complex \times DNA_{unbound} =k_2 DNA_{bound}$$ $$\frac{DNA_{bound}}{DNA_{unbound}}=\frac{k_1\times complex }{k_2}$$

Consider the total amount of DNA is a constant,

$$DNA_{bound}=DNA_{total}\frac{k_1\times complex}{k_2+k_1\times complex}$$

The generation of mRNA is proportion to DNA_bound. Hence, the equation becomes a Hill equation with order n=1.