# Team:HKUST-Rice/Modeling

# Modeling

# Introduction

To fully appreciate the mechanism of our biosensor and its behavior in an ideal situation, we explored the structure and dynamics
of our system by creating a mathematical model of the reaction kinetics. We studied the dynamics of the Kdp system using a *P _{KdpF}*
- GFP generator (BBa_E0240) in pSB3K3 backbone in a DH10B

*E. coli*strain. Ordinary differential equations were derived to demonstrate how potassium ions concentration interact with the endogenous Kdp system, thus affecting the GFP expression of the cell. Modeling was done using MATLAB R2015a.

In addition, by using the prediction model, users of our potassium biosensor can estimate the potassium concentration of cultures and mediums by obtaining fluorescence intensity per cell using flow cytometry.

# Prediction Model

# Model's Assumption

The effect of the endogenous Kdp system of *E. coli* was neglected.

In our engineered *E. coli*, titration by the endogenous *kdp* operon of the transcription regulator, phosphorylated KdpE which binds to *P _{KdpF}*, was expected initially. This titration of phosphorylated KdpE is anticipated to lower
the expression of GFP. However, since the native DNA copy number is only an 11

^{th}of the pSB3K3 plasmid copy number, the effect of endogenous Kdp system was neglected.

Level of KdpD, KdpE and KdpF were assumed to be constant.

In accordance to [Kremling A. 04], for the potassium ion concentration range which we were studying- 0 mM to 0.02 mM, the fluctuation of the concentration of KdpF as well as KdpD and KdpE was only within 10 μM and 3 μM respectively. Due to the small fluctuation range compared to the gene expression of GFP reporter, it was reasonable that KdpD, KdpE and KdpF concentrations were assumed to be constant in the model.

It was assumed that the initial concentration of mRNA for GFP, immature GFP and mature GFP was zero.

It was assumed that all reactions below were in steady state such that:

# Equations of the Model:

**Phosphorylation of KdpD:**

**Phosphyl-group Transfer:**

**Binding of KdpE to promoter PkdpF:**

**Transcription:**

**Translation:**

**Green Fluorescent Protein maturation:**

For parameters and variables, please click here

## References

Heermann R, Zigann K, Gayer S, Rodriguez-Fernandez M, Banga JR, et al. (2014) Dynamics of an Interactive Network Composed of a Bacterial Two- Component System, a Transporter and K+ as Mediator. PLoS ONE 9(2): e89671. doi:10.1371/journal.pone.0089671

Brewster RC, Jones DL, Phillips R (2012) Tuning Promoter Strength through RNA Polymerase Binding Site Design in Escherichia coli. PLoS Comput Biol 8(12): e1002811. doi:10.1371/journal.pcbi.1002811

Modeling, Simulation and Identification of the Dynamics of K Uptake in E. coli. (2014). Universitatsbibliothek der TU Munchen.

Kelly, Jason et al. “Measuring the activity of BioBrick promoters using an in vivo reference standard.” Journal of Biological Engineering 3.1 (2009): 4.

J. Gayer, Stefan. "Modeling, Simulation and Identification of the Dynamics of K Uptake in E. Coli." Technische Universitat Munchen Fachgebiet Fur Systembiotechnologie (2013). Print.

Kremling, A., Heermann, R., Centler, F., & Gilles, E. (2004). Analysis of two-component signal transduction by mathematical modeling using the KdpD/KdpE system of Escherichia coli.

Conboy, C., & Braff, J. (2013, May 29). Molecules of Equivalent GFP. Retrieved from http://openwetware.org/wiki/MEG

Epstein W (2003) The roles and regulation of potassium in bacteria. Prog Nucleic Acid Res Mol Biol 75: 293–320.