Team:TU-Delft/BandAid

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Band Aid

The Band Aid Modeling is related to the final application of our project. Specifically, we are going to use a band aid in which E. coli will be added. By locating the band aid on the wound, the S. aureus should be detected and killed (Figure 1).

                                                                        Figure 1: Band Aid Application

As, it is already known, S. aureus is a pathogenic bacterium that utilizes quorum sensing (QS), a cell-to-cell signalling mechanism, to enhance its ability to cause disease[1]. The communication is succeeded through small peptides known as AIPs. We engineered the receiver part of S. aureus to E. coli in order to be able for the last to detect AIPs produced by S. aureus.

The main purpose in the band aid modeling is to answer questions like these mentioned underneath:

1. If the MRSA is detected, is the amount of the produced peptide enough to kill it ?
2. How many pores are necessary in order to be possible for the peptide to be released?

Differential Equations

Our model consists of two Populations:

1. Sender: The S. aureus Population
2. Receiver: The E. coli Population

For our model we made the following assumptions:

1. The S. aureus population grows exponentially.
   
2. The E. coli population is constant.
3. The AIP concentration in both populations is modeled as homogeneous (perfectly mixed). This removes the need of a space model for the populations.
4. The membrane assumed in 47mm diameter and 0.1μm pore size.
5. No hydrophobic/hydrophilic interactions between membrane and the AIPs are taken into account.

The differential equations related to each population are represented below.

Sender: S. aureus Population

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Figure 2:Sender [1]

Receiver: E. coli Population

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Parameters

The references [1][2] used dimensionless variables and parameters. However, to be able to add equations to these models, units must be added. This is done by picking dimensionless values in or resulting from this model and use the known value with dimension to find out the used dimensions of the model.

In the model three units must be determined: the units of time and concentration. This is most easily done by plotting from the resulting model the AIPs over time. However, little is known on the typical concentration of AIPs and the time needed to reach these concentrations. Therefore, we used the quorum sensing system of gram negative bacteria (AHL molecules) as a reference and assume that the order of concentrations is the same for the AIP system.

From the model of [17] it is seen that a concentration of 200 nm is representative in a growth from 1 cell to 1 OD in 100 hours. This final concentration is reached in 130 hours.

Note that in the above equation of the permeability the distance is expressed in the ratio of the pore sizes. For example the distance between pores can be 5 times the pore diameter.

Parameter	Value	Description	Units	Reference
s	0.02	Ratio of basal to QS transcription	Molecules cells-1s-1	[1][2]
u	0.05	mRNA transcription rate	Molecules cells-1s-1	[1][2]
l	1	protein degradation	sec-1	[1][2]
f	1	agrA activation	Molecules-1cm3s-1	[1][2]
g1	1	unbinding AIP from AgrC	s-1	[1][2]
ks	0.1	AgrD loss through AIP production		[1][2]
ka	102	AIP production	Molecules-1cm3s-1	[1][2]
la	1	Natural AIP degradation	s-1	[1][2]
h	0.1	receptor loss through AIP binding		[1][2]
gij	1	AIP from Population i unbinding from Population j		[1][2]
bij	1	AIP binding from Population i to Population j		[1][2]
ba	1	Ratio of cognate AIP binding in Population 2 to Population 1		[1][2]
r	1/2/3600	growth rate		[1][2]
K	1	capacity		[1][2]
D	8e-4	diffusion coefficient of the membrane		[1][2]
A	20000	Area of the membrane		[1][2]
pore size	0.22	membrane pore size	0.001	[1][2]

Variables

The following variables were used in the equations:

Variable	Description	Reference
Mi	mRna concentration in Population i	[1][2]
Ai	agrA concentration in Population i	[1][2]
Si	agrD concentration in Population i	[1][2]
ai	AIP concentration in Population i	[1][2]
Ri	agrC concentration in Population i	[1][2]
Pi	Proportion of cells that are upregulated in S. aureus population	[1][2]

Results