Model
As discussed in the Project, we can consider some basic variables of interest in our project: Synthetase (S), HSL (H), Receptor (R) and Protein of interest (R), which can be considered 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 equilibria, therefore we can consider it as a constant;
- Transcription, when constitutive, and translation events are very low effected by stochastic effects;
- Also, considering the transcription and translation can be coupled in a single term of an equation, due to a linear relation between mRNA and protein (I.e. 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, initially we can construct a graphic view of the model, applying mass action kinects law using now the terms of the equations:
This kind of figure makes 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
Also, the term above the line/in the right of the line, is the parameter that conducts the relation.
The index in the parameter represent in which variable is the parameter assigned. For a resume of the variables and parameters, see the table at the end of the page
With those data we can finally construct the Mathematical Model that will describe the 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 independent of DO 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 being able to estimate the alfa Yf 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 |