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Early on, we decided to create a computational model of our pattern generation system. This would allow us to accomplish several important goals:
- Demonstrate that the circuit logic was sound
- Test the viability of our construct given realistic parameters
- Probe the effects that different parameters would have on the system
- Identify potential failure modes, and the events that might trigger them
- Evaluate alternative designs of the construct
With these goals in mind, we set about formalizing the system.
To start, we generate equations for all of the transcription reactions:
From there, we came up with the equations for each binding/unbinding reaction:
We can now see some interesting behaviors of our gene circuit. For instance, the transcription of lambda cI mRNA is dependent on the concentration of promoter I12006 that is bound with lambda cI. However, we can also see from the second set of equations that this promoter may be complexed with 434 cI, which acts as a repressor, removing some of the I12006 which would otherwise be available for transcription. It is important to note that the binding of the repressor "overrides" the activity of the activator; this has important effects on the system, and this behavior reflects the reality of the part.
After we had derived the necessary equations, we implemented a computer program that would execute the appropriate calculations in order to depict our system at different time steps beginning with the initial stimulus of the system.
This computer program generated the simulation by treating the blank screen of pixels as a representation of the lawn of cells we wanted to construct. Naturally, each pixel would then represent an individual cell in the system that has a certain degree of RFP expression.
In order to take into account the diffusion of the various substances in the system, we implemented a Gaussian blur at each pixel that would be recalculated at every time interval using the previous time interval's output as its input. In this way, we could properly represent the step-by-step concentration of substances at each time step in each pixel by calculating the change in concentration at each time step using the Gaussian blur algorithm at each pixel.
Pictured here is the initial output of the completed simulation. In this particular run, we initialized the system with a vertical stimulus down the center of the plate, resulting in linear oscillation outwards towards the edges.
From this, we can conclude that given our design, the system should function in accordance to our expectations.
To further test the simulation, we decided to try another type of initial stimulus and observe the results. The stimulus we chose was seeding the oscillation at the four corners of the "plate". The following is the result of this run, which again confirms the intended behavior of our system.
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