Table 1.

Mathematical model for the lac operon regulatory pathway in E. coli, which accounts for the operon simultaneous induction with lactose and TMG.

$$[\dot{M}]=k_M[D]p_{pc}([G_e]){\mathcal{P}}_R([A],[T])-({\gamma }_M+\mu )M$$ (1)

$$[\dot{E}]=k_E[M]-({\gamma }_E+\mu )[E]$$ (2)

$$[\dot{L}]=k_L{\beta }_L([L_e],[T_e]){\beta }_G([G_e])[Q]-2{ɸ}_Mℳ([L])[B]-\mu [L]$$ (3)

$$[\dot{T}]=k_T{\beta }_T([L_e],[T_e]){\beta }_G([G_e])[Q]-\mu [T]$$ (4)

$$\left[A\right]=\left[L\right]$$ (5)

$$\left[\text{Q}\right]=\left[E\right]$$ (6)

$$\left[\text{B}\right]=\left[E\right]/4$$ (7)

$$\text{μ}=\epsilon \left(J_G\left(\left[G_{\text{e}}\right]\right)+J_L\left(\left[L\right]\right)\right)$$ (8)

$$p_{pc}([G_e])=\frac{p_p(1+p_c([G_e])(k_{pc}-1))}{1+p_p{p}_c([G_e])(k_{pc}-1)}$$ (9)

$$p_{cp}([G_e])=\frac{p_c(1+p_p([G_e])(k_{pc}-1))}{1+p_p{p}_c([G_e])(k_{pc}-1)}$$ (10)

$$p_c([G_e])=\frac{K_G^{n_h}}{K_G^{n_h}+{[G_e]}^{n_h}}$$ (11)

$${\mathcal{P}}_R([A],[T])=\frac{(1+{\xi }_2{\rho }_1([A],[T]))(1+{\xi }_3{\rho }_1([A],[T]))}{\mathcal{Z}}$$ (12)

$$\begin{array}{c}\mathcal{Z}([A],[T])={\displaystyle \prod _{i=1,2,3}(1+{\text{ξ}}_i{\text{ρ}}_i([A],[T])}\\ +{\displaystyle \sum _{\sigma =\{\begin{array}{l}(1,2,3)\\ (2,1,3)\\ (3,1,2)\end{array}}\left(1+(p_{cp}-1){\delta }_{2{\sigma }_1}\right)\left(1+{\xi }_{{\sigma }_1}{\rho }_1([A],[T])\right){\xi }_{{\sigma }_2{\sigma }_3}{\rho }_1([A],[T])}\end{array}$$ (13)

$${\rho }_1([A],[T])={\left(\frac{1}{1+[A]/K_A+[T]/K_T}\right)}^2$$ (14)

$${\rho }_2([A],[T])={\left(\frac{1}{1+[A]/K_A+[T]/K_T}\right)}^4$$ (15)

$${\beta }_L([L_{\text{e}}],[T_{\text{e}}])=\frac{[L_{\text{e}}]/{\kappa }_L}{1+[L_{\text{e}}]/{\kappa }_L+[T_{\text{e}}]/{\kappa }_T}$$ (16)

$${\beta }_T([L_{\text{e}}],[T_{\text{e}}])=\frac{[T_{\text{e}}]/TL}{1+[L_{\text{e}}]/{\kappa }_L+[T_{\text{e}}]/{\kappa }_T}$$ (17)

$${\beta }_G([G_{\text{e}}])=1-{ɸ}_G\frac{[G_{\text{e}}]}{{\kappa }_G+[G_{\text{e}}]}$$ (18)

$$ℳ\left(\left[L\right]\right)=\frac{\left[L\right]}{{\kappa }_M+\left[L\right]}$$ (19)

$$J_G([G_{\text{e}}])=J_G^{\max }\frac{[G_{\text{e}}]}{[G_{\text{e}}]+{\Phi }_G}$$ (20)

$$J_L\left(\left[L\right]\right)=4{ɸ}_Mℳ\left(\left[L\right]\right)\left[B\right]$$ (21)