Tryptophan (Trp) is converted to bromotryptophan (Br-Trp) by the action of Trp halogenase Fre-L-SttH, which follows Michaelis-Menten kinetics. Hence, the rate of Br-Trp synthesis from Trp is represented by equation (1):
$$
\frac{\text{d}\left[ Br-Trp \right]}{\text{dt}}=\frac{V_{Br}\left[ Trp \right]}{K_{Br}+\left[ Trp \right]}
$$
where [Br-Trp] and [Trp] represent the concentrations of bromotryptophan and tryptophan, respectively; \(K_Br\) is the Michaelis-Menten constant of Fre-L-SttH.
Since Br-Trp is oxidized to Br-Indo by Michaelis-Menten enzyme TnaA-FL-FMO (TLF), equation (1) is corrected by adding a consumption term：
$$
\frac{\text{d}\left[ Br-Trp \right]}{\text{dt}}=\frac{V_{Br}\left[ Trp \right]}{K_{Br}+\left[ Trp \right]}-\frac{V_{TLF}\left[ Br-Trp \right] \left[ TLF \right]}{K_{TLF}+\left[ Trp \right]}
$$
where [TLF] is the concentration of TnaA-FL-FMO; \(K_TLF\) is the Michaelis-Menten constant of TnaA-FL-FMO.
On the other hand, Trp is consumed by Fre-L-SttH catalyzed bromination reaction and TLF-catalyzed oxidation reaction. Trp consumption rate is represented in equation (3):
$$
\frac{\text{d}\left[ Trp \right]}{\text{dt}}=-\frac{V_{Br}\left[ Trp \right]}{K_{Br}+\left[ Trp \right]}-\frac{V_{TLF}\left[ Trp \right] \left[ TLF \right]}{K_{TLF}+\left[ Trp \right]}
$$
The synthesis process of tryptophan repressor (TrpR2) is divided into three steps: transcription, translation and dimerization.
Equation (4) represents the transcription process of TrpR2. [\(mRNA_{Trp-R}\)] is the concentration of mRNA of TrpR2. k_TC1 is the transcription rate of TrpR2 and k_DegM is the rate of mRNA degradation.
$$
\frac{\text{d}\left[ mRNA_{TrpR} \right]}{\text{dt}}=k_{TC1}DNA_{TrpR}-k_{DegM}\left[ mRNA_{TrpR} \right]
$$
Equation (5) represents the translation process of TrpR2. k_TL1 is the rate of TrpR2 translation and k_DegP is the rate of protein degradation.
$$
\frac{\text{d}\left[ TrpR \right]}{\text{dt}}=k_{TL1}\left[ mRNA_{TrpR} \right] -k_{DegP}\left[ TrpR \right]
$$
In addition to this, TrpR monomer concentration decreases due to dimerization while increases due to TrpR2 monomerization. Therefore, correcting the above equation, we obtain the equation describing TrpR translation as
$$
\frac{\text{d}\left[ TrpR \right]}{\text{dt}}=k_{TL1}\left[ mRNA_{TrpR} \right] -k_{DegP}\left[ TrpR \right] -k_{Di}\left[ TrpR \right] ^2+k_{Sepe1}\left[ TrpR2 \right]
$$
where k_Di is the parameter for TrpR dimerization and k_Sepe1 is the TrpR2 separation number.
The third equation is used to describe the dimerization process and represents the concentration of TrpR2. Firstly, as mentioned above, TrpR2 undergoes dimerization and separation.
$$
\frac{\text{d}\left[ TrpR2 \right]}{\text{dt}}=k_{Di}\left[ TrpR \right] ^2-k_{Sepe1}\left[ TrpR2 \right]
$$
In addition, by binding to Trp or Br-Trp, TrpR2 can form TrpR2-Trp (binding to two Trp) or TrpR2-BT (binding to two Br-Trp) , while Trp2-Trp , TrpR2-BT can both repress the transcription process of TLF. where k_AR_T is the association rate of TrpR2 and Trp. Therefore, we corrected the equation as follows:
$$
\frac{\text{d}\left[ TrpR2 \right]}{\text{dt}}=k_{Di}\left[ TrpR \right] ^2-k_{Sepe1}\left[ TrpR2 \right] -k_{AR\_T}\left[ TrpR2 \right] \left[ Trp \right] ^2-k_{AR\_BT}\left[ TrpR2 \right] \left[ Br-Trp \right] ^2
$$
Since TrpR2-Trp, and TrpR2-BT also dissociate and thus affect the concentration of TrpR2, we added the last term and obtained the final equation describing the concentration of TrpR2:
$$
\frac{\text{d}\left[ TrpR2 \right]}{\text{dt}}=k_{Di}\left[ TrpR \right] ^2-k_{Sepe1}\left[ TrpR2 \right]
$$
Dimerization leads to a conformational change of active site of each TrpR, enabling each monomer to bind to a molecule of either Trp or Br-Trp. Assuming that TrpR2 can bind with one type of molecule one time, hence TrpR2 can bind to two Trp or Br-Trp molecules, forming TrpR2-Trp or TrpR2-BT (Figure 3).
$$
\left[ TrpR2 \right] +2\left[ Trp \right] \left[ TrpR2-T \right]
$$
$$
\left[ TrpR2 \right] +2\left[ Br-Trp \right] \left[ TrpR2-TB \right]
$$
TrpR2 binding with two molecules of Trp or Br-Trp (W. Ellefson et al., 2018).
Considering both association and dissociation of between TrpR2 and Trp or between TrpR2 and Br-Trp, we have the following equation:
$$
\frac{\text{d}\left[ TrpR2-Trp \right]}{\text{dt}}=k_{AR\_T}\left[ TrpR2 \right] \left[ Trp \right] ^2-k_{sepe2}\left[ TrpR2-Trp \right]
$$
$$
\frac{\text{d}\left[ TrpR2-TB \right]}{\text{dt}}=k_{AR\_BT}\left[ TrpR2 \right] \left[ Br-Trp \right] ^2-k_{sepe2}\left[ TrpR2-TB \right]
$$
Where the association and the dissociation rate of between TrpR2 and Trp (or between TrpR2 and Br-Trp) are k_AR_T (or k_AR_BT) and k_sepe2 respectively.
While during the formation of TrpR2-Trp or TrpR2-BT, the association between TrpR2 and Trp or Br-Trp only takes place when the Trp or Br-Trp concentration reaches a certain range. Therefore, a constant term [Trp_0] or [Br-Trp_0] is added. Hence we get the corrected equations:
$$
\frac{\text{d}\left[ TrpR2-T \right]}{\text{dt}}=k_{AR\_T}\left[ TrpR2 \right] \frac{\left[ Trp \right] ^2}{\left[ Trp \right] ^2+\left[ Trp_0 \right] ^2}-k_{sepe2}\left[ TrpR2-Trp \right]
$$
$$
\frac{\text{d}\left[ TrpR2-TB \right]}{\text{dt}}=k_{AR\_BT}\left[ TrpR2 \right] \frac{\left[ Br-Trp \right] ^2}{\left[ Br-Trp \right] ^2+\left[ Br-Trp_0 \right] ^2}-k_{sepe2}\left[ TrpR2-TB \right]
$$
As mentioned before, TLF refers to TnaA-FL-FMO, which catalyzes Reaction V presented at the following part. The transcription function of TLF is shown as follows:
$$
\frac{\text{d}\left[ mRNA_{TLF} \right]}{\text{dt}}=k_{TC2}DNA_{TLF}-k_{DegM}\left[ mRNA_{TLF} \right]
$$
When two molecules of Trp or Br-Trp are bound to TrpR2, TrpR2’s repression activity becomes activated. Trp-bound or Br-Trp-bound TrpR2 binds to the promoter of TLF and represses TLF transcription. Assuming that the repression efficiency of TrpR2-Trp and TrpR2-BT is the same, the transcription efficiency could be modified by a gene expression repression efficiency term (equation 17). Repression efficiency follows a Hill equation in relation to repressor concentration.
$$
\frac{\text{d}\left[ mRNA_{TLF} \right]}{\text{dt}}=\frac{k_{TC2}}{1+\left( \frac{\left[ TrpR2-T \right] +\left[ TrpR2-TB \right]}{K_{Hill}} \right) ^n}DNA_{TLF}-k_{DegM}\left[ mRNA_{TLF} \right]
$$
Finally, the amount of TLF is described in the following ODE equation, where k_TL2 and k_DegP are the parameter of TLF translation and TLF degradation respectively.
$$
\frac{\text{d}\left[ TLF \right]}{\text{dt}}=k_{TL2}\left[ mRNA_{TLF} \right] -k_{DegP}\left[ TLF \right]
$$
TLF indicates TnaA-Flexible Linker-FMO, which can catalyze Br-Trp transforming to Br-Indole in our design.
As one of the by-products, indigo can be transformed from Trp by TLF. In equation (19), [I] represents the concentration of indigo. We also used Michaelis-Menten equation to model the production of indigo, where V_TLF1 is the rate of indigo production at which TLF is saturated with Trp and K_TLF1 is the Michaelis-Menten constant of TLF.
$$
\frac{\text{d}\left[ I \right]}{\text{dt}}=\frac{V_{TLF1}\left[ Trp \right] \left[ TLF \right]}{K_{TLF1}+\left[ Trp \right]}
$$
The main function of TLF is to catalyze Br-Trp to transform to Br-Indole, which is the precursor of our target product TP-Indican.
$$
\frac{\text{d}\left[ Br-Indo \right]}{\text{dt}}=\frac{V_{TLF2}\left[ Br-Trp \right] \left[ TLF \right]}{K_{TLF2}+\left[ Trp \right]}
$$
Taking the consumption of Br-Indole into consideration, Br-Indole is transformed to TP-Indican by UGT, the enzyme that add a glucose moiety to tyrian purple as a biochemical protecting group. The equation (20) is modified as follows:
$$
\frac{\mathrm{d}\left[ Br-Indo \right]}{\mathrm{dt}}=\frac{V_{TLF2}\left[ Br-Trp \right] \left[ TLF \right]}{K_{TLF2}+\left[ Trp \right]}-\frac{V_{UGT}\left[ Br-Indo \right]}{K_{UGT}+\left[ Br-Indo \right]}-k_{O_2}\left[ Br-Indo \right] ^2
$$
The final reaction is about the production of our target product Br-Indi, which is catalyzed from Br-Indo by the enzyme UTG in equation (23). With the formation of Br-Indi, the byproduct TP which denotes the unmodified insoluble tyrian purple will also be producted as shown in equation (22).
$$
\frac{\text{d}\left[ TP \right]}{\text{dt}}=k_{O_2}\left[ TP-Indo \right] ^2
$$
$$
\frac{\text{d}\left[ TP-Indi \right]}{\text{dt}}=\frac{V_{UGT}\left[ Br-Indo \right]}{K_{UGT}+\left[ Br-Indo \right]}
$$
