Team:USP-Brazil/Model
With our project we aimed to create a system that could alternate between different states. And for that we created a circuit that utilizes a negative feedback loop combined with recombinases generating a switch (For further details go to our design page).
To show that this concept can work with the circuit that we have in mind, we develop a model to simulate the expected behaviour of the system. But also, we had another problem: Which inhibitors we should use? If we choose inhibitors too weak for our system, the entire inversion system will collapse If we use an inhibitor that is too strong, we can be stuck in only one state.
So, with that in mind, we develop a simple differential equation model, that allows us to predict the behaviour of our systems and, therefore, guiding our project.
The Model
At first, we had two initial choices of models: A boolean or a system of ODEs. The boolean could allow us to see the behaviour of our system, however since we are interested in finding not only the simulation of our circuit, but also with the behaviour in a continuous way, we choose to go with the ODEs. So, based in our design, we develop the following model:
We created a visual representation of our circuits in 3 different sub-topics:
Promoter inversion; Promoter repression; Light promoter repression
These diagrams represent our ODEs and makes easier to understand and visualize them, we these representations:
- 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
We already represented our system of equations, where the variables are pS_ for the promoter State 1 ou 2, L_ for the light promoters, pS_r and L_r are the repressed promoters, Rc_ are the recombinases, Rs_ are the repressors and Rp_ are the promoters. (For a quicker look at the variables and parameters, check the following table. So, from that we can write the ODEs.
And finally, our equations are the following:
| Parameter/Variable Name | Description |
|---|---|
| pS_ | Central promoter state (1 or 2) |
| L_ | Light promoter (1 or 2) |
| pS_r and L_r | Repressed promoters (1 or 2) |
| Rc_ | Recombinase (A or B) |
| Rp_ | Reporters of the system (1 or 2) |
| Rs_ | Repressors (1 or 2) |
| i | Recombinase reversion effect |
| delta | Dissociation rate of the repressor |
| fr_ | Repression strenght of the repressors (1 or 2) |
| l | Incidence of light |
| alpha | Expression rate of proteins |
| phi | Protein recovery after separation |
The Results
With our equations in hands, we were able to simulate the system. Utilizing data from the papers we learned found the recombinases we runned a script in python for simulate the behaviour of our system.
As we can see looking at the reporters, we have the expected behaviour for the system utilizing actual values. This shows that our system is very likely to work properly when the biological construct is created. Is important to denote that this time scale was used because is the same time scale used by Jesus&Voigt 2016.
For check, we decided to plot also the promoters. Although in real experiments we are not able to see them, correlating it with our model results for the reporters.
At the end, thinking about the effect of the repressors we simulated our model in two cases: With a very high repression and with a very low expression.
As we can see, if we have a repressor that is way to strong or a repressor that is not strong enough, our circuit breaks. In the first graph we have a very weak repression, so the whole dynamics is a mess. In the second we have repressors that are too strong, so the system just collapses and we have no more expression. So, based in these results, we analyzed different repressors. And dues to its values in the literature, we them choose to use the tet and the lac repressors. They are not only very well characterized, what can be very helpful in different analysis, but is also inside our range of effect!
Reference data for the simulations:
Jesus Fernandez-Rodriguez, Lei Yang, Thomas E. Gorochowski, D. Benjamin Gordon, and Christopher A. Voigt; Memory and Combinatorial Logic Based on DNA Inversions: Dynamics and Evolutionary Stability – 2015, American Chemical Society Synthetic Biology, 4, 1361-1372 DOI: 10.1021/acssynbio.5b00170
