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 dynamic of the Kdp system working with PKdpF - 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. A plot with fluorescence intensity as dependent variable and potassium ions concentration as independent variable was obtained by solving the ordinary differential equations using 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, it contains the inserted plasmid of PKdpF plus green fluorescence protein generator (BBa_E0240) in a pSB3K3 backbone as well as the endogenous kdp operon, and both operons have the same promoter, PKdpF. As a matter of fact, titration by the endogenous kdp operon of the transcription inducer, phosphorylated KdpE which binds to PKdpF was expected initially. And the titration of phosphorylated KdpE is anticipated to lower the expression of GFP. However, when the DNA copy number of both endogenous and the inserted operons were determined, it was found that the number of endogenous kdp operon is 10 times smaller than that of the inserted one:
This implied that the number of endogenous promoter PKdpF is ten times smaller than that of the inserted operon and accounts only 8.33% of the total number of promoter PKdpF in the engineered E. coli. Therefore, the titration effect of phosphorylated KdpE becomes insignificant. As a result, the effect of endogenous Kdp system was negligible.
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 |
|
Kb: Dissociation constant (KdpE-DNA) |
0.0532 1/µM |
|
Kh: Dissociation constant (KdpF-KdpE dephosphorylation
system) |
0.0532 µM (adjustable) |
|
k0h: Reference kh value |
0.00023 1/µM hr
(adjustable) |
|
C0K+: Reference
concentration of potassium ions |
0.02mM |
|
ktr: transcription rate constant |
10600 hr-1 |
|
kz: mRNA degradation rate of kdpFABC
and kdpDE coding region |
21.74 hr-1 |
|
kd: degradation rate of KdpDE |
0.2 hr-1 |
|
kd,F: degradation of KdpF |
4.8 hr-1 |
|
ktlD: translation rate constant of KdpD |
160 hr-1 |
|
ktlE: translation rate constant of KdpE |
8100 hr-1 |
|
ktlF: translation rate constant of KdpF |
5.4 hr-1 |
|
u: growth rate of the cell (in K minimal medium,
2nd order) |
0.275466667 hr-1 (exp. obtained) |
|
Ψ: basal activity of the promoter |
0.0017 |
|
kdeg,GFP,mRNA: GFP mRNA degradation rate (1st
order) |
0.0048s-1 =17.28 /hr |
|
γI: Immature GFP degradation rate |
0.7 hr-1 |
|
γm: Mature GFP degradation rate |
0.021 hr-1 |
|
ρ: Immature GFP synthesis rate (trans rate, 1st order) |
1440 protein/hr/mRNA (0.4protein/s/mRNA) |
|
w: GFP maturation rate (1st
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 |
| [kdpDP] |
Concentration of phosphorylated KdpD |
µM |
| [kdpEP] |
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 |
- |
| kh |
Adjustable parameter |
- |
| cK+ |
Concentration of potassium ions |
mM |
| [kdpE - DNA] |
Concentration of KdpE bound promoter PkdpF |
µM |
| [DNAf] |
Concentration of Unbound promoter PkdpF |
µM |
| [DNA0] |
Total concentration of promoter PkdpF |
µM |
| [mRNAGFP] |
Concentration of mRNA molecules for green
fluorescence protein |
µM |
| [GFPI] |
Concentration of immature green fluorescent
protein |
µM |
| [GFPM] |
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.