Team:NTNU Trondheim/Modeling

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The pathway triggering the expression of mCherry upon reception of Glucose is shown in Figure 1

**Figure 1.** Biological molecules involved in the *P. Putida* signaling pathway for expressing mCherry.

Using the generalized mass action equations and the pathway shown in the figure above, we can write the kinetics of the systems in the following form:

\begin{matrix} \frac{d {KGN}_{i}}{dt} = k_{c} ({GLC}_{e} - {KGN}_{i}) - k_{1} {KGN}_{i}^{2} {PtxS}^{2} + k_{1} C_{1} \\ \frac{d C_{1}}{dt} = k_{1} {KGN}_{i} PtxS - k_{1} C_{1} k_{2} PGad \\ \frac{d C_{2}}{dt} = k_{2} C_{1} PGad - k_{2} C_{2} - k_{tr} C_{2} \\ \frac{d {KGN}_{2}}{dt} = k_{PtsX} - 2 C_{1} {-2C}_{1} \\ \frac{d {mCherry}_{i}}{dt} = k_{tr} C_{2} - k_{m} {mCherry}_{i} - k_{d} {mCherry}_{i} \\ \frac{d {mCherry}_{m}}{dt} = k_{m} G_{i} - k_{d} {mCherry}_{m} \end{matrix}

where ${KGN}_{i}$ is the internal ketogluconate concentration, ${GLC}_{e}$ is the external glucose concentration, $PtxS$ is the population of free PtxS, $C_{1}$ is the KGN-PtXs dimer, $C_{2}$ is the complex that involves the dimer bound to the promoter sequence, $PGad$ is the Gad promoter concentration, ${mCherry}_{i}$ is the immature mCherry, and ${mCherry}_{m}$ is mature mCherry. The rate constants are described below in Table 1.

Table 1 - System parameters

Description	Symbol	Value
Glucose diffusion
C1 formation/dissociation rate	$k_{1}$	$k_{2}$
PtxS production rate	$k_{PtxS}$
Protein production rate	$k_{tr}$
mCherry maturation rate	$k_{m}$
mCherry degradation rate	$k_{d}$