Modeling on the sensing device

In the mathematical modeling of quorum sensing, we formulated a system of ordinary equations representing the intracellular and extracellular interactions between the two Agr proteins and AIP (auto-inducing peptide) molecules. Along with numerical simulations, we performed an asymptotic analysis of the time-dependent model in order to characterize whether the AIP molecules produced by Staphylococcus aureus in the intestine would activate our sensing device.

To build the model, we first proposed the following assumptions:

• The agr mRNA contains all the information required for the translation of AgrC and AgrA. There are plentiful ribosomes for translation within the cells and the rates of translations of AgrC and AgrA are the same, and are proportional to the concentrations of their mRNA.
• Proteins and mRNA inside the cells are limited by natural degradation.
• Housekeeping phosphatases are able to dephosphorylate AgrA at rate αpidi.
• Receptor-bound AIP can dissociate spontaneously at rate α_unbind.
• When an AIP binds to AgrC, we assume that auto-phosphorylation of AgrC happens simultaneously because this process is sufficiently fast. When AgrC transfers its phosphate group to AgrA at rate α_pi, it is able to re-auto-phosphorylate.

The resulting equations, together with the definitions of the parameters and variables are shown below.

Table 1 Definitions of the parameters

Table 2 Definitions of the variables

The three Hill equations represent the rates of translation of AgrA, AgrC and sfGFP. Β₁ is the highest efficiency for the promoter pnisA to initiate the transcription of the agrC and agrA genes, and β₂ is the highest efficiency for the promoter P2 to initiate the transcription of the sfGFP gene. X is the concentration of nisin which is needed to activate the promoter pnisA, to this extent, k₁ equals to the concentration of A_pi when the rate of reaction is up to half of V_max. K₂, which is controlled by another regulatory factor, is the concentration of phosphorylated AgrA when the rate of reaction is up to half of V_max.

By assuming that 0.25 μM of AIP molecules is present in the intestine, we run the MATLAB script to check whether AIP molecules can successfully activate the promoter P2 by binding to AgrC and phosphorylating AgrA. We set the threshold concentration of sfGFP to be 0.5 μM, and at this point, we consider the promoter P2 is activated. The results are shown below.

Fig 1.    State values of AgrA, Cbind, AgrC, Cp, Api and sfGFP.

Fig 2.    Individual display of 6 variables

As it is shown in the second graph (values are hard to observe in the first one), concentration of sfGFP reaches 0.5 μM at the third hour. Therefore, we made a conclusion that the amount of AIP molecules can activate the promoter P2 to transcribe the genes downstream.

Modelling on peptide synthesis and cell lysis

Our design uses the tandem repeat strategy to express three copies of each peptide gene, LL-37, GF-17 and Grammistin-Pp1, aiming to producing peptides quickly and at a higher rate. To release the peptides to kill Staphylococcus aureus in the intestine, we choose lysis of the cells instead of secretion. A lysis gene is used to open up the cells, then all the peptides will surely be released into the guts. In addition, we plan to use a toggle switch to provide more time for peptide synthesis before lysis. When the cells are lysed, it will result in the release of intracellular proteins and stop all life activities. Therefore, we use modeling to identify:

1. How long does cell lysis take from the point of induction?
2. At this time point, how much peptides are produced by the gene circuit?

Results:
Inspired by the team TU-Delft (2013), we came up with the idea that the promoters P2, plac and ptet may serve as binary switches between the active and inactive promoter states instead of continuous activities from fully on to fully off. We used the parameter--s, a binary state descriptor, to refer to the situation when a promoter produces one of the two levels of activity: on or off.

Table 3 Definitions of parameters

By running the Matlab script, we obtained the results shown below.

Fig 3.    State values of LacIm, GFnm, Granm, LLnm, tetRm, AcmAm, LacI, tetR, AcmA, GFn, Gran, LLn.

Fig 4.    Individual display of transcribed LacIm, GFnm, Granm, LLnm, tetRm and AcmAm

Fig 5.    Individual display of translated LacI, tetR, AcmA, GFn, Gran and LLn.

From these graphs, we can make a general conclusion that the shift between the two states controlled by LacI and TetR takes at least 20 minutes. By the time the promoter P2 initiates the transcription and later efficiently translation of the mRNA of the tandem repeat genes (ll-37, gf-17, and grammistin-Pp1), the antimicrobial peptides are capable of being synthesized at high rates. When the repression of the promoter ptet (tetR) is relieved and the lysis gene acmA (AcmAm) starts to be transcribed, the antimicrobial peptides can be accumulated to high concentrations. Thereafter, enough amounts of antimicrobial peptides will be released to eradicate Staphylococcus aureus through the cell lysis.

References
[1] Z. Cai, et al. “A simulation of Synthetic agr System in E. coli,”in Bioinformatics Research and Applications. Charlotte, NC: Springer, 2013, pp76-86.
[2] NICE Expression System for Lactococcus lactis. MoBITec GmbH, Germany, 2010.
[3] Team: TU-Delft (2013). Timer Plus Sumo [Online]. Available: https://2013.igem.org/Team:TU-Delft/Timer_Plus_Sumo
[4] C. Wu, H. Lee, and B. Chen, "Robust synthetic gene network design via library-based search method," Bioinformatics, vol. 27, pp. 2700-2706, Oct. 2011.