Table 3.

Set of chemical master equations describing the metabolites interaction showed in Fig. 1

$\frac{\partial \left(0,0,t\right)}{\partial t}=\mathit{\mu p}\left(1,0,t\right)+\mathit{\mu p}\left(0,1,t\right)+\frac{k_3}{V}p\left(1,1,t\right)-V\left(k_1+k_2\right)p\left(0,0,t\right)$

$\begin{array}{c}\hfill \frac{\partial \left(0,b,t\right)}{\partial t}=V\frac{k_2}{1+\frac{b-1}{V{K}_2}}p\left(0,b-1,t\right)\hfill \\ \hfill +\mathit{\mu p}\left(1,b,t\right)+\mu \left(b+t\right)p\left(0,b+1,t\right)+\frac{k_3}{V}\left(b+1\right)p\left(1,b+1,t\right)-\hfill \\ \hfill -\left(V\left(k_1+\frac{k_2}{1+\frac{b}{V{K}_2}}\right)+\mathit{\mu b}\right)p\left(0,b,t\right)\hfill \end{array}$

$\begin{array}{c}\hfill \frac{\partial p\left(a,0,t\right)}{\partial t}=V\frac{k_1}{1+\frac{a-1}{V{K}_1}}p\left(a-1,0,t\right)+\hfill \\ \hfill +\mu \left(a+1\right)p\left(a+1,0,t\right)+\mathit{\mu p}\left(a,1,t\right)+\hfill \\ \hfill +\frac{k_3}{V}\left(a+1\right)p\left(a+1,1,t\right)-\hfill \\ \hfill -\left(V\left(\frac{k_1}{1+\frac{a}{V{K}_1}}\right)+\mathit{\mu a}\right)p\left(a,0,t\right)\hfill \end{array}$

$\begin{array}{c}\hfill \frac{\partial p\left(a,b,t\right)}{\partial t}=V\frac{k_1}{1+\frac{a-1}{V{K}_1}}p\left(a-1,b,t\right)+V\frac{k_2}{1+\frac{b-1}{V{K}_2}}p\left(a,b-1,t\right)+\hfill \\ \hfill +\mu \left(a+1\right)p\left(a+1,b,t\right)+\mu \left(b+1\right)p\left(a,b+1,t\right)+\hfill \\ \hfill +\frac{k_3}{V}\left(a+1\right)\left(b+1\right)p\left(a+1,b+1,t\right)-\hfill \\ \hfill -\left(V\left(\frac{k_1}{1+\frac{a}{V{K}_1}}+\frac{k_2}{1+\frac{b}{V{K}_2}}\right)+\mu \left(a+b\right)+\frac{k_3}{V}\mathit{ab}\right)p\left(a,b,t\right)\hfill \end{array}$