IGEM:IMPERIAL/2007/Projects/Biofilm Detector/Modelling

Biofilm Detection: Modelling

• Introduction
• Specifications
• Design
• Modelling
• Implementation
• Testing/Validation
• Notes
• References

Overview of Modelling

Welcome to our Portal Page for the modelling of Infector Detector.

Infector Dectector (ID) is based on the Quorum Sensing Pathway and our aim in modelling of ID is to determine the concentration of AHL in biofilm we can detect such that we report a visible signal .We are looking at two constructs to emulate the quorum sensing pathway:

Construct 1

Ptet promoting LuxR, Plux promoting GFP

The main feature of this construct is that it constitutively expresses LuxR

• Block Diagram Picture of how construct will work

• Chemical Equations

Construct 2

(image will appear here)

The main feature of this construct is that it does not constitutively expresses LuxR and therefore enables us to determine the initial concentration of LuxR

• Block Diagram Picture of how construct will work

• Chemical Equations

Previous work

Overall our modelling for this project will take the form of

[math]\displaystyle{ \frac{dFP}{dt}=k(t)-\delta_{FP}[FP] }[/math]

1. k : Function of Temperature. k is based on the promoter used as promoters take time to turn on.
2. dFP : Function of System. May be considered to be a function of temperature as proteins may degrade at high temperatures.
3. FP : The particular fluorescent protein employed e.g. GFP, DsRed, etc.

Two graphs of k vs. time (special pt is ko.) and [FP] vs. time key point is [FP]ss

• Our major problem at the moment is estimating the errors involved with our fluorometer and experimental procedures, including the use of pipettes; we hope to address these through calibration curves.

For each experiment we will do the following

1. Calibration curve to determine error in fluorometer
2. Decay Experiment @ varying temperatures
3. Plug together to find transient response and k
4. Find these parameters as a function of temperature(T)

Construct Specific Modelling

Ptet promoting LuxR, Plux promoting GFP

Modelled by 2006 Imperial Team, as part of their prey-sensing module of the molecular predation oscillator:
Test construct modelling & general derivation.
Parts page.
An operating/working version is quoted on the parts registry as T9002 - Link to MIT Parts Registry

The major problem with last year's derivation is that they assumed LuxR concentration to be constant and they didn't look into the co-operativity of the Plux Promoter e.g. the threshold of quorum.

• LuxR concentration:

LuxR will be constituitively expressed in our system, this means that the older our system is the more LuxR present and so the less sensitive our system will be to the amount of AHL around.

We can model this by having a initial condition or history of our system in addition to last year's path way.

[math]\displaystyle{ [LuxR]_{0}\; at\; t=t_{0} }[/math]

[math]\displaystyle{ LuxR+AHL\rightarrow A }[/math]

[math]\displaystyle{ A+P\leftarrow\rightarrow\;PA\;\rightarrow\;GFP }[/math]

[math]\displaystyle{ \frac{d[LuxR]}{dt}=k-d_{LuxR}[LuxR]-k_{1}[LuxR][AHL]+k_{2}[A] }[/math]

We will first consider the concentration of LuxR to be constant as per last year. Following this we will consider the history of the system and so the initial concentration of LuxR when AHL is detected.

• Cooperativity of Plux Promoter:

We do not know what the cooperativity of the Plux Promoter is and in effect we do not know what exactly the threshold of our system will be.

[math]\displaystyle{ \frac{d[GFP]}{dt}=\frac{X^a}{1+X^a} }[/math]

[math]\displaystyle{ X=[LuxR][AHL]\;complex }[/math]

[math]\displaystyle{ a\gt 1 }[/math] : initial gradient is zero and we see a step response
[math]\displaystyle{ a=1 }[/math] : response similar to first order eg. no threshold visible we have to define
[math]\displaystyle{ a\lt 1 }[/math] : initial gradient is infinite // to y-axis and mabye we get a step response ???? (is this true)

The threshold for biofilm detection, via AHL quorum sensing molecule is approximately 0.1nM.

Feedback from Matty

Overview of modelling this construct:

• Experiment 1 : Get Data - k vs. [AHL]
• Analysis 1 : Determine whether minimum [AHL] neded to get visible level is below target concentration [AHL]t = 0.1nM
• Experiment 1 : Get Data - k vs. [LuxR]
• Analysis 2 : Determine how age of system affects its sensitivity

Breakdown of specific sections:

Experiment 1:

1. Determine constant LuxR concentrations to be used
   
2. Determine Po? (may not be possible)
   
3. Determine value of rate constants in below derivation
   
4. Determine Visible GFP level in terms of Fluoresence
   
5. Convert this level into a concentration
   
6. Determine error associated with this concentration
   
7. For constant luxR concentration determine range of AHL concentrations
   
8. For each AHL concentration record transient resonse esp. Steady State Value (this is the only real value we want) and time taken to reach steady state.
   

• Protocols said they can Get us either , k vs. [AHL] or [GFP]ss vs. [AHL] for both [LuxR] will be a constant.

Analysis 1:

1. We have determine the Menten Kinetics Equations describing our pathway excluding cooperativity to give us insight into what's going on:

[math]\displaystyle{ \frac{d[GFP]}{dt}=k_{5}\left\{\frac{[AHL][LuxR][P]_{0}}{K+[AHL][LuxR]}\right\}-\delta_{GFP}[GFP] }[/math]

[show][hide]

Media:Pathway.doc

1. plot steady state Value of GFP concentration [GFP] vs. concentration of AHL
2. Draw [GFP] level indicicative of Minimum visible concentration
3. Steady state value is equal to k/delta , using this we can find alpha by comparing to a family of curves
   1. [math]\displaystyle{ k=\frac{([AHL][LuxR])^\alpha}{1+([AHL][LuxR])^\alpha} }[/math]
      
4. With this plot we can also find the minimum [AHL] needed for a visible [GFP]
5. Compare this with desired value

Experiment 2:

• We have not considered this yet

Analysis 2:

• We have not considered this yet

Considerations

• Modelled deterministically; possibility for additional stochastic modelling.
• Degradation term in model lacks dilution term - look into amending the model to incorporate dilution of specific molecule due to cell growth (dilution)
• HRP system is employed as amplifier of low-level signal (where signal-to-noise ration is high)- amplification here probably amplifies the noise, leading to possible false positives for biofilm presence.
• A potential issue with the experiment is regarding using pTet as the promoter for LuxR expression. It may be necessary to add a gene for TetR expression to ensure control over this promoter. Issue of sensitivity control...

Diffusion coefficient of HSL in water - 4.9*10^-6 D_aq cm²/s
Diffusion coefficient is approximately half in biofilms [1]

Case1

Case 1 : Both initial concentrations of LuxR and AHl are controllled

[math]\displaystyle{ [A]=K_{\alpha}[AHL]_{0}[LuxR]_{0} }[/math]

[math]\displaystyle{ [AP]=\frac{K_{\alpha}K_{\beta}[AHL]_{0}[LuxR]_{0}[P]_{0}}{1+K_{\alpha}K_{\beta}[AHL]_{0}[LuxR]_{0}} }[/math]

[math]\displaystyle{ R_{y}(x)=\frac{[AP]}{[P]_{0}}=\frac{K_{\alpha}K_{\beta}xy}{1+K_{\alpha}K_{\beta}xy} }[/math]

[math]\displaystyle{ R_{y}(x)=1-\frac{1}{1+K_{\alpha}K_{\beta}xy} }[/math]

[math]\displaystyle{ R_{y}'(x)=\frac{K_{\alpha}K_{\beta}y}{\left(1+K_{\alpha}K_{\beta}xy\right)^2} }[/math]

Case2

Case 3 : Only y is controlled

[math]\displaystyle{ x=[AHL]_{0}=[AHL]+[A] }[/math]

[math]\displaystyle{ [A]=x-[AHL] }[/math]

but [math]\displaystyle{ [A]=K_{\alpha}[AHL]y }[/math]

therefore [math]\displaystyle{ x-[AHL]=K_{\alpha}y }[/math]

therefore [math]\displaystyle{ [AHL]=\frac{x}{1+K_{\alpha}y} }[/math]

[math]\displaystyle{ [A]=K_{\alpha}\left(\frac{xy}{1+K_{\alpha}y}\right) }[/math]

Now [math]\displaystyle{ Ry(x)=\frac{K_{\alpha}K_{\beta}xy}{1+yK_{\alpha}+K_{\alpha}K_{\beta}xy} }[/math]