We simulated the termination and anti-termination properties of our signal-terminator constructs with the Kinefold web server and used some standard estimations for diffusive terms. Our main goal was to prove that our constructs work and that termination is stopped efficiently, that is that the signal molecule binds and anti-terminations occurs before the RNA polymerase falls off.
The Kinefold webserver provides a web interface for stochastic folding simulations of nucleic acids and offers the choice of renaturation or co-transcriptional folding. The folding paths are simulated at the level of helix formation and dissociation as these stochastic formation and the removal of individual helices are known to be the limiting steps of RNA folding kinetics.
For our purposes co-transcriptional folding was the appropriate choice: Folding proceeds while the sequence is being synthesized from its 5' to 3' ends at a speed of 3ms per newly added base (for RNA polymerase T7 phage). Thus, the transcript starts to fold before the whole sequence is fully available.
Kinefold offers the possibility to include additional bases (X) which do not pair to model hybridization dynamics between two sequences. In order to simulate how the binding of the signal molecule prevents termination we linked the signal via a linker sequence consisting of 'X' bases to the sequence of the terminator.
For each signal-terminator pair we did batch simulations with various random seeds in order to guarantee accuracy.
We also varied signal length form two base pairs to full signal length which provides insight in how long the signal needs to be in order to bind to the terminator and how long this process takes at least.

The question whether anti-termination occurs is not only guided by the folding process of the signal-terminator pair, but also by how long the signal takes to diffuse to the terminator sequence. To account for the diffusion time, we estimated the hit rate τ (following 6.), which is the time until the signal meets the terminator sequence for the first time:

τ = 1/(3D*a/r³),

where D is the diffusion constant, a the radius of gyration of the signal molecule and r the radius of the cell.
For E.coli r is 1 μm. The radius of gyration a can be estimated using the worm-like-chain model by

a = (n*l)/3,

where n is the length of the signal which is 0,3 nm/monomer, l is the persistency length which is following (5.) 2nm for single-stranded RNA. Thus, for a signal of length 32 nt, a = 6,4 nm.

The diffusion constant D was obtained by

D = k_B T/ (6 π * 10*10^-3*a),

where k_B is the Boltzmann constant and T is the absolute temperature.
Using the a we calculated above, we get D = 3,4318 m²/s.

Thus, for a cell containing 100 signal molecules, the signal needs 0,1518 s until it first hits the terminator sequence.

As the folding time is significantly larger than the diffusion and thus much less relevant for modeling our signal-terminator constructs, we didn't employ more elaborate techniques to model diffusion.

We modeled the signal-terminator interaction for 2, 4-32 nt signal length and proved that the signal sequence binds to the terminator sequence and impedes the terminator folding, thus the polymerase doesn't fall off and the output signal is produced. As we observed a dependence of the folding time and minimal energy respectively from the linker length we used linkers of length 12, 16, 20, 24, 30, 34, 40, 50, 60, 120 'X' bases. For the minimal energy there is only a very small linker length dependence as one can see in figure 1. However, for the folding time linker length dependence is significant as one can see in figure 2. But, as, for us, the fact whether the terminator is folding or not is crucial and the videos below show that for appropriate signal length the terminator is not folding, the linker length dependence can be neglected.

RNA folding path videos:

For this video the signal is only 2 nt long. One can see that the terminator is folding and that the signal cannot bind to the terminator sequence due to its short length.

This video shows the folding of a terminator and a signal of length 20 nt. which shows that the terminator is already hindered from folding completely.

This video is done for full signal length of 32 nt. One can see that the terminator does not fold at all as the signal immediately binds to the terminator sequence as it is synthesized.