Modeling

Potassium sensor

Phosphate sensor

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 dynamic of the Kdp system working with P_KdpF - GFP generator (BBa_E0240) in pSB3K3 backbone DH10B E.coli strain. Ordinary differential equations were derived to demonstrate how potassium ions concentration interact with the endogenous Kdp system in E.coli hence affecting the GFP expression of the cell. The whole modeling was done in MATLAB R2015a.

Coupled with that, with the use of the prediction model, users of our potassium biosensor can estimate the potassium concentration in cultures and mediums by obtaining per cell fluorescence intensity 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. And the 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 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 to assume the concentration of KdpD, KdpE and KdpF to be constant in the model.

It was assumed that the initial concentration of mRNA for GFP, immature GFP and mature GFP equal to 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:

Parameters and Variables list

Parameters:	Values:
k₁: Rate constant of forward auto-phosphorylation	0.23 1/µM hr
k_-1:Rate constant of backward auto-phosphorylation	0.0000051 1/µM hr
k₂: Rate constant of forward phosphyl-group transfer	0.00227 1/µM hr
k_-2: Rate constant of backward phosphyl-group transfer	0.00087 1/µM hr
α: Adjustable proportionality coefficient of ATP	0.001223
β: Adjustable proportionality coefficient of ADP	0.001223
K_b: Dissociation constant (KdpE-DNA)	0.0532 1/µM
K_h: Dissociation constant (KdpF-KdpE dephosphorylation system)	0.0532 µM (adjustable)
k⁰_h: Reference k_h value	0.00023 1/µM hr (adjustable)
C⁰_K+: Reference concentration of potassium ions	0.02mM
k_tr: transcription rate constant	10600 hr^-1
k_z: mRNA degradation rate of kdpFABC and kdpDE coding region	21.74 hr^-1
k_d: degradation rate of KdpDE	0.2 hr^-1
k_d,F: degradation of KdpF	4.8 hr^-1
k_tlD: translation rate constant of KdpD	160 hr^-1
k_tlE: translation rate constant of KdpE	8100 hr^-1
k_tlF: translation rate constant of KdpF	5.4 hr^-1
u: growth rate of the cell (in K minimal medium, 2^nd order)	0.275466667 hr^-1 (exp. obtained)
Ψ: basal activity of the promoter	0.0017
k_deg,GFP,mRNA: GFP mRNA degradation rate (1^st order)	0.0048s^-1 =17.28 /hr
γ_I: Immature GFP degradation rate	0.7 hr^-1
γ_m: Mature GFP degradation rate	0.021 hr^-1
ρ: Immature GFP synthesis rate (trans rate, 1^st order)	1440 protein/hr/mRNA (0.4protein/s/mRNA)
w: GFP maturation rate (1^st order)	6.48 hr^-1

Table 1. Parameters list 1.

Parameters	Values
ADP concentration	0.305 mM
ATP concentration	3.05 mM
Initial concentration of mRNA (GFP)	0/cell
Concentration of KdpF	10 µM
Concentration of KdpE	2.6 µM
Concentration of KdpD	2.6 µM
Concentration of inserted plasmid	0.0183 µM
Initial concentration of immature and mature GFP	0 /cell

Table 2. Parameters list 2.

Variables	Denotation	Unit
[kdpD]	Concentration of KdpD	µM
[kdpD^P]	Concentration of phosphorylated KdpD	µM
[kdpE^P]	Concentration of phosphorylated KdpE	µM
[ADP]	Concentration of ADP molecules	mM
[ATP]	Concentration of ATP molecules	mM
[kdpF]	Concentration of KdpF	µM
k₃	Rate constant of dephosphorylation of KdpE	-
k_h	Adjustable parameter	-
c_K⁺	Concentration of potassium ions	mM
[kdpE - DNA]	Concentration of KdpE bound promoter P_kdpF	µM
[DNA_f]	Concentration of Unbound promoter P_kdpF	µM
[DNA₀]	Total concentration of promoter P_kdpF	µM
[mRNA_GFP]	Concentration of mRNA molecules for green fluorescence protein	µM
[GFP_I]	Concentration of immature green fluorescent protein	µM
[GFP_M]	Concentration of mature green fluorescent protein	µM

Table 3. Variables list.

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.