# Team:NTNU Trondheim/Modeling

Modeling

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

### Glucose-mCherry response

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, the results are shown in Figure 2:

$\begin{array}{c}\frac{d{\mathit{KGN}}_{i}}{\mathit{dt}}={k}_{c}\left({\mathit{GLC}}_{e}-{\mathit{KGN}}_{i}\right)-{k}_{1}{\mathit{KGN}}_{i}^{2}{\mathit{PtxS}}^{2}+{k}_{1}{C}_{1}\\ \frac{d{C}_{1}}{\mathit{dt}}={k}_{1}{\mathit{KGN}}_{i}\mathit{PtxS}-{k}_{1}{C}_{1}{k}_{2}\mathit{PGad}\\ \frac{d{C}_{2}}{\mathit{dt}}={k}_{2}{C}_{1}\mathit{PGad}-{k}_{2}{C}_{2}-{k}_{\mathit{tr}}{C}_{2}\\ \frac{d{\mathit{KGN}}_{2}}{\mathit{dt}}={k}_{\mathit{PtxS}}-2{C}_{1}{\mathrm{-2C}}_{1}\\ \frac{d{\mathit{mCherry}}_{i}}{\mathit{dt}}={k}_{\mathit{tr}}{C}_{2}-{k}_{m}{\mathit{mCherry}}_{i}-{k}_{d}{\mathit{mCherry}}_{i}\\ \frac{d{\mathit{mCherry}}_{m}}{\mathit{dt}}={k}_{m}{G}_{i}-{k}_{d}{\mathit{mCherry}}_{m}\end{array}$

where ${\mathit{KGN}}_{i}$ is the internal ketogluconate concentration, ${\mathit{GLC}}_{e}$ is the external glucose concentration, $\mathit{PtxS}$ 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, $\mathit{PGad}$ is the Gad promoter concentration,${\mathit{mCherry}}_{i}$ is the immature mCherry, and ${\mathit{mCherry}}_{m}$ is mature mCherry. The rate constants are described below in Table 1.

Table 1 - System parameters
Description Symbol Value
Glucose diffusion rate ${k}_{e}$ 10 mmol/g/h [1]
C1 formation/dissociation rate ${k}_{1}$ 50/h (approximation)
C2 formation/dissociation rate ${k}_{2}$ 500/h (approximation)
PtxS production rate ${k}_{PtxS}$ 500 / h [2][3]
Protein production rate ${k}_{tr}$ 100 / h (approximation)
mCherry maturation rate ${k}_{m}$ 2/h [4]
mCherry degradation rate ${k}_{d}$ 40 / h [4]

### Brixells

The Brixells Modeling Tool, has been created to assist Team Warwick to predict the probability of bonding of 3D structures built with E. Coli expressing Zinc fingers and DNA strands. More details are available on the collaborations page.

### Conclusion

Modeling can be very useful to optimize the design of P. Putida as it is very sensitive to glucose toxicity. This framework can help in optimizing the lifetime of the glucose-sensing capsule and should be considered in the alginate formulation. This model is to be combined with a population level glucose toxicity model for agent-based modeling in BSim.

### References

[1] Rühl, Jana, Andreas Schmid, and Lars Mathias Blank. "Selected Pseudomonas putida strains able to grow in the presence of high butanol concentrations." Applied and environmental microbiology 75.13 (2009): 4653-4656.
[2] Colmer, J. A., and A. N. Hamood. "Characterization of ptxS, a Pseudomonas aeruginosa gene which interferes with the effect of the exotoxin A positive regulatory gene, ptxR." Molecular and General Genetics MGG 258.3 (1998): 250-259.
[3] Follonier, Stéphanie, Sven Panke, and Manfred Zinn. "A reduction in growth rate of Pseudomonas putida KT2442 counteracts productivity advances in medium-chain-length polyhydroxyalkanoate production from gluconate." Microb Cell Fact 10.1 (2011): 25.
[4] Piatkevich, Kiryl D., and Vladislav V. Verkhusha. "Guide to red fluorescent proteins and biosensors for flow cytometry." Methods in cell biology 102 (2011): 431.