In our design, the DTE process is one of the most significant parts in manufacturing psicose. The main process of psicose manufacture is catalyzed by D-psicose 3-epimerase. The models of device A, B, C and D are as follows.

In device A, extracellular concentration of $psicose$ is higher than the intracellular concentration, so it can enter the cells by diffusion. As a small molecular, the $psicose$ inside the cell can be combined with $pPsiR$ to generate $CCI$. $pPsiR$ is a repressor, which can bind to promoters on DNA and block gene expression. After binding with $psicode$,$pPsiR$ falls off from the promoter and the gene starts expressing, And eventually produce the produce $EGFP$.

For device A, the dynamic equations can be listed as follows:

Concentrations of $psicose$ are different inside and outside the cell, so the diffusion rate of $psicose$ is proportional to the concentration difference between inside and outside of the cells. so the rate at which $psicose$ enters the cell by diffusion is $$-\gamma_F([PsiO]-[PsiI])$$
Where $\gamma_F$ is the diffusion coefficient.

Considering that the volume of the cells is small enough to be negligible relative to the external solution, so the amount of $psicose$ that diffuses into the cells is very small. We can think of the concentration of the external solution as constant, which means $$\frac{\text{d}[PsiO]}{ \text{d}t}=0$$ When reducer psicose combines with repressor, the process is $$\eta_1 PsiI+pPsiR\rightarrow CCI$$ Where $PsiI$ is the intracellular $psicose$, $pPsiR$ is $psicose$ dependent repressor, $CCI$ is $psicose$-repressor complex and $\eta_1$ is coordination number of $psicose$ for $pPsiR$.

The change in concentration of $PsiI$，$pPsiR$ and $CCI$ contains three influencing factors: the binding reaction of $pPsiR$ and $psicose$, and the degradation reaction of themselves and the reverse reaction.

Here we consider the binding reaction as a second order reaction and the degradation reaction and reverse reaction as first order reactions. According to law of mass action, reaction rate is proportional to the product of the reactant concentration, so we have $$\frac{\text{d}[PsiI]}{\text{d}t}=-\gamma_F([PsiI]-[PsiO])-m_{pPsiR,Psi}[PsiI]^{\eta_1}[pPsiR]+m_{CCI}[CCI]-\delta_{PsiI}[PsiI]$$ And considering $pPsiR$ is constantly expressing, we can get $$\frac{\text{d}[pPsiR]}{\text{d}t}=\alpha_{pPsiR}-m_{pPsiR,Psi}[PsiI]^{\eta_1}[pPsiR]+m_{CCI}[CCI]-\delta_{pPsiR}[pPsiR]$$ Where $m_{pPsiR,Psi}$ is the coefficient of reaction rate of the binding reaction, $\alpha_{pPsiR}$ is the rate of constant expression of $pPsiR$, $\delta_{pPsiR}$ and $\delta_{PsiI}$ are coefficients of reaction rate of the degradation reaction of $pPsiR$ and $PsicoseI$ respectively.
The concentration of inactivated repressor is $$\frac{\text{d}[CCI]}{\text{d}t}=m_{pPsiR,Psi}[PsiI]^{\eta_1}[pPsiR]-m_{CCI}[CCI]-\delta_{CCI}[CCI]$$ The change in $EGFP$ concentration depends on the concentration of its repressor, $pPsiR$ which can be described by Hill-equation. Considering $EGFP$ is also degrading, the equation is $$\frac{\text{d}[EGFP]}{\text{d}t}=H\frac{\beta_{EGFP}K^n}{K^n+[pPsiR]^n} -\delta_{EGFP}[EGFP]$$ Where $n$ is hill coefficient, $K$ is the ligand concentration producing half occupation, $\beta_{EGFP}$ is maximal transcription rate of gene $EGFP$, and $H$ is a constant used to indicate the deviation between the theoretical and actual values.

In device B, extracellular concentration of $IPTG$ is higher than which is intracellular, so it can enter the cells by diffusion. As a small molecular, the $IPTG$ inside the cell can be combined with $pLacR$ to generate $CC$. $pLacR$ is a repressor, which can bind to promoters on DNA and block gene expression. After binding with $IPTG$, $pLacR$ falls off from the promoter and the gene starts expressing, and eventually produce the produce $EGFP$. Since the gene of $ABR$ and the gene of $EGFP$ are connected in series, they are expressed together.

Similarly, we can get the function of device B by using the reaction rate equation and the diffusion function. First, $$\frac{\text{d}[IPTGO]}{\text{d}t}=0$$ According to the reaction between $IPTGI$ and $pLacR$ and the reaction rate equation, we can get $$\eta_2 IPTGI+pLacR\rightarrow CC$$ Where $\eta_2$ is coordination number of $IPTG$ for $pLacR$.

This reaction is similar with the reaction in device A, so we have $$\frac{\text{d}[IPTGI]}{\text{d}t}=-\gamma_{IPTG}([IPTGI]-[IPTGO])-m_{IPTG,pLacR}[IPTGI]^{\eta_2}[pLacR]+m_{CC}[CC]-\delta_{IPTG}[IPTGI]$$ Similarly, we can also get $$\frac{\text{d}[pLacR]}{\text{d}t}=\alpha_{pLacR}-m_{IPTG,pLacR}[IPTGI]^{\eta_2}[pLacR]+m_{CC}[CC]-\delta_{pLacR}[pLacR]$$ $$\frac{\text{d}[CC]}{\text{d}t}=m_{IPTG,pLacR}[IPTGI]^{\eta_2}[pLacR]-m_{CC}[CC]-\delta_{CC}[CC]$$ And $pLac$ is a transcription activator of gene $EGFP$, according to hill equation, the concentration of $EGFP$ is $$\frac{\text{d}[EGFP]}{\text{d}t}=H\frac{\beta_{EGFP}K^n}{K^n+[pLacR]^n} -\delta_{EGFP} [EGFP] $$ The concentration of $EGFP$ is the same as the concentration of antibiotics resistance due to the transcription and translation of antibiotic resistance gene combined with the gene of $EGFP$. $$\frac{\text{d}[EGFP]}{\text{d}t}=\frac{\text{d}[ABR]}{\text{d}t}$$ Where $[EGFP]$ is the concentration of $EGFP$ and $[ABR]$ is the concentration of antibiotic resistance protein expression.

Device C is the same with device B. The only difference is the hairpin between gene of $EGFP$ and gene of $RFP$. Similarly, we can get the function of hairpin and its coefficient.

The first few equations are the same as in device B: $$\frac{\text{d}[IPTGO]}{\text{d}t}=0$$ $$\frac{\text{d}[IPTGI]}{\text{d}t}=-\gamma_{IPTG}([IPTGI]-[IPTGO])-m_{IPTG,pLacR}[IPTGI][pLacR]+m_{CC}[CC]-\delta_{IPTG}[IPTGI]$$ $$\frac{\text{d}[pLacR]}{\text{d}t}=\alpha_{pLacR}-m_{IPTG,pLacR}[IPTGI][pLacR]+m_{CC}[CC]-\delta_{pLacR}[pLacR]$$ $$\frac{\text{d}[CC]}{\text{d}t}=m_{IPTG,pLacR}[IPTGI][pLacR]-m_{CC}[CC]-\delta_{CC}[CC]$$ $$\frac{\text{d}[EGFP]}{\text{d}t}=H\frac{\beta_{EGFP}K^n}{K^n+[pLacR]^n} -\delta_{EGFP} [EGFP]$$ The presence of the hairpin leads to a decrease in $RFP$ expression efficiency, so we have $$\frac{\text{d}[EGFP]}{\text{d}t}=k\frac{\text{d}[RFP]}{\text{d}t}$$ Where $[RFP]$ is the concentration of red fluorescence protein, $k$ is the coefficient of hairpin.

In device D, $IPTG$ gets in cells and bind with $pLacR$, which is a repressor for gene $DTE$. After $IPTG$ binding with $pLacR$, $DTE$ starts to express, and as an enzyme, to catalysis $fructose$ to turn into $psicose$. As more and more $psicose$ are produced, more and more the repressor of gene $EGFP$, $pLacR$ are inactivated, so expression of $EGPF$ increase. At the same time, expression of $ABR$ also increase since the gene of $ABR$ and the gene of $EGFP$ are connected in series by a hairpin.

For device D, $psicose$ and $fuctose$ get in cells by diffusion: $$\frac{\text{d}[PsiO]}{\text{d}t}=\frac{-\gamma_{F}([PsiO]-[PsiI])}{V_{outside}}=0 $$ $$\frac{\text{d}[FO]}{\text{d}t}=\frac{-\gamma_{F}([FO]-[FI])}{V_{outside}}=0$$ $$\frac{\text{d}[IPTGO]}{\text{d}t}=\frac{-\gamma_{F}([IPTGO]-[IPTGI])}{V_{outside}}=0$$ And device D consists of device A and device C connected by an extra step: