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Effective translation of mRNA amber and tRNA amber diffusion

We already justified the hyptothesis of neglecting the time in takes for a molecule to reach any given point in a cell (see Modeling hypotheses). However in this design we want to have two tRNA amber reaching the amber codons on the same mRNA, exactly when the ribosome is on the first and second amber codon. This requires a little more analysis before validating the possibility of the translation of the mRNA amber into a fully functional T7 RNA polymerase.

Random walker model

This model is very similar to the one used in [1]. We consider that the tRNA amber diffusing in the cell is a random walker.

We want to see how long it takes for a tRNA to diffuse to any point of a cell. We use the following parameters:

• V volume of the cell (10^-18 m³)
• characteristic size of the particle (1.5x10^-8m [1])
• D diffusion coefficient of the particle (2.57x10^-11m².s^-1 [1])

We divide the cytoplasm volume V into for the walker. The characteristic time associated with the transition from one site to another is:[2]

If we have R walkers of this type, the probability that a molecule arrives at a given occupation site during the time interval is: .

Until this point, the model does not differ from what we used to do. But let's dive into the additions we had to make to explore the possibility of functional T7 RNA polymerase production.

The specificities of our design

What really interest us is the probability that two amber codons are successfully translated with the help of tRNA amber by the same ribosome. We have to monitor all the amber codons placed on our mRNA amber present in a cell. We therefore have two occupation sites of interest for each mRNA amber in the cell.

There are three important time scales in this model:

• The time scale for tRNA diffusion. An occupation site tRNA amber population is changed every 1.46x10^-5s. This means that you obtain a new tRNA population in an occupation site every 1.46x10^-5s with a probability of picking a tRNA amber.
• The time scale of ribosome movement on mRNA. The speed of ribosome is roughly 15 codon.s^-1 so the average time spent by a ribosome on a codon is:
• The time scale of ribosome movement between two amber codons. Since th

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