Model
As discussed in the Project, we can consider some basic points of interest in our project: Synthase (S), HSL (H), Receptor (R) and Protein of interest (R), which can be used as the variables of the systems. Also we can consider the complex formed between the HSL and the Receptor (C).
NOTE: For a simplified visualization, we adopt, during the whole discussion of the modelling, the letters indicated between parentheses as representation of the variables.
Before starting the model itself, we need to stablish some assumptions:
- The population, during the measurements, is at dynamic equilibrium, therefore we can consider it as a constant;
- Transcription, when constitutive, and translation events are very lowly affected by stochastic effects;
- Also, the transcription and translation can be coupled in a single term of an equation, due to a linear relation between mRNA and protein (therefore, the translation is not limited by ribosomes in the cell)
We can, then, propose the following reactions:
$$ \begin{align} &\rightarrow S \\ S &\rightarrow \\ S & \rightarrow H\\ H &\rightarrow \\ &\rightarrow R \\ R &\rightarrow\\ H + R &\leftrightharpoons C\\ C &\rightarrow\\ C & \rightarrow P\\ P &\rightarrow \\ \end{align}$$Based on this, we can initially construct a graphic view of the model, applying mass action kinects law using now the terms of the equations:
This kind of figure makes it easier to visualize the model and the relations between the classes of the model. Then, the way to read is the following:
- Sphere – Variable of the system
- Line – Relation between the systems
- Arrow - indicates the directionality of the system (back of the arrow decreases and front of the arrow grows
- Bar (In the line) – Indicates no decreasing of the variable with the line
The term above the line/on the right of the line represents the parameter that conducts the relation.
The index in the parameter represent in which variable is the parameter assigned. For a summary of the variables and parameters, see the table at the end of the page.
With this data we can finally construct the Mathematical Model that will describe our systems!
However, due to our design of the project, the protein of interest is going to be used as a relation between YFP and CFP.
This design allows us to do an easier comparison between the systems, having independence of OD measurement and reducing the deviations.
We then stablish the variable ρ as YFP/CFP.
Finally, looking at those representations, we can construct the following differential equations:
$$ \frac{dS}{dt} = \alpha_{S} - \mu_{S} S \\ \frac{dH}{dt} = \alpha_{H} S + \theta C - H( \mu_{H} H + \beta R)\\ \frac{dR}{dt} = \alpha_{R} + \theta C - R( \mu_{R} R + \beta H) \\ \frac{dC}{dt} = \beta R H - C (\mu_{C} + \theta)\\$$For ρ, there must be a more specifically mathematical work:
So as we define ρ as Yf/Cf, as Yf being YFP e Cf as CFP, we define the following derivation:
Using those general equations, we were able to estimate αYf from each system, establishing the crosstalk!
| Parameter/Variable Name | Description |
|---|---|
| S | System Synthase |
| H | System HSL (Produced bt the synthase) |
| R | System Receptor |
| C | Complex formed by the HSL and Receptor |
| Cf | CFP used as relative measure |
| $$\rho$$ | Relative Fluorescence (YFP/CFP) |
| $$\alpha_{N}$$ | Production/Expression of the variable, indicated in the index |
| $$\mu_{N}$$ | Degradation of the variable, indicated in the index |
| $$\beta$$ | Formation of the HSL/Receptor Complex |
| $$\theta$$ | Separation of the Complex HSL/Receptor |