Gillespie mRNA Simulator

Dual Reporter Method

The following reaction scheme models a situation in which a modified gene produces two mRNAs which stochastically transcribe different reporter proteins, such as GFP and CFP, independently of one another forming the basis of a dual reporter method. The system features feedback from the mRNA given by its production rate being a function $f(x_{1})$.

$\text{Transcription: } \text{ } x_{1} \xrightarrow[]{f(x_{1})} x_{1} + 1$

$\text{Translation of } x_{2} \text{: } \text{ } x_{2} \xrightarrow[]{\lambda_{2} x_{1}} x_{2} + 1$

$\text{Translation of } x_{3} \text{: } x_{3} \xrightarrow[]{\lambda_{2} x_{1}} x_{3} + 1$

$\text{mRNA degradation: } \text{ } x_{1} \xrightarrow[]{\beta_{1} x_{1}} x_{1} - 1$

$\text{Degradation of } x_{2} \text{: } x_{2} \xrightarrow[]{\beta_{2} x_{2}} x_{2} - 1$

$\text{Degradation of } x_{3} \text{: } x_{3} \xrightarrow[]{\beta_{2} x_{3}} x_{3} - 1$

Gillespie Algorithm

The Gillespie algorithm was implemented in R and run to simulate two situations with the following functions:

1. $f(x_{1}) = \lambda_{1}$
• Whereby mRNA is transcribed at a constant rate
2. $f(x_{1}) = \lambda_{1} \frac{K}{K + x_{1}}$
• Whereby mRNA self-repressive as its transcription is inversly proportional to its own concentration $(x_{1})$ and a rate constant $K$

For both functions, rate constants were set with $\beta_{1} = \lambda_{2} = 1$ and $\beta_{2} = 0.1$. To ensure that $\left\langle x_{2} \right\rangle \simeq 20$, $\lambda_{1}$ and $K$ were determined below for functions (a) and (b). These were calculated using the principal that at stationarity, $\left\langle R^{+} \right\rangle = \left\langle R^{-} \right\rangle$.