Validation of our genetical network

Validation of the principle

• First deterministic results

At early stage, the goal of the modelling team was to confirm the behaviour of the whole circuit. We divided the the network into two main models, Toggle switch and Quorum Sensing (see Our Equations). Very early the modelling results seemed promising and we could rapidly infer that our Toggle Switch design would be effective. Indeed, with the models described in Our Equations we can see the behaviour of our bacteria on the plate. On the plate, one whole region features bacteria in the LacI way and the rest of the plate features bacteria in the TetR way :



Figure 1: LacI and TetR concentrations on a 200 points plate; [IPTG] gradient linear 5E-7 5E-4 M; [aTc] = 1.5E-6 M

Figure 2: LacI and TetR concentrations on a 200 points plate; [IPTG] gradient linear 5E-7 5E-4 M; [aTc] = 1E-7 M

On the previous two figures X axis represents physical points on the plate, form left to right of the plate. In each of these points the only difference is the IPTG concentration, as we will apply on our plate an IPTG gradient. The interface between the two regions depends on [aTc]. Lower aTc concentration will move the interface to the left edge of the plate as in Figure 2. We therefore demonstrated that the Toggle switch behaviour was the one we wanted for our application.

With this model, we also demonstrated that degradation tags were necessary to get the appropriate behaviour. If the degradation rate of the LacI and TetR proteins were too long (typical half-time of 10 hours) the concentrations in each protein would be too high and the switching in one way or another would be way too long for our application. As a result we decided to use only lva tagged LacI and TetR genes.

GEOF

One last thing we could predict with the first deterministic models was that even though the toggle switch's purpose is to remain in one way or another, it was however possible to unswitch our system, i.e. basculate in one way even though the other way was already selected. Of course such reaction would require great concentrations of aTc or IPTG in the medium. This prediction was proved to be right later by experiments on our toggle switch.

• Quorum Sensing modelling

Our models for Quorum Sensing allowed us to simulate the behaviour of our whole system, confirm our expectations and finally have a visual representation of our entire device. With these deterministic models, we can validate the behaviour of our system.

Figure 3: CinI, CinR, QSi and QSe (AHL molecules inside and outside the cell) concentrations on the plate; [IPTG] gradient linear 5E-7 5E-4 M; [aTc] = 1.5E-6 M

Figure 4: Animation generated through MATLAB for visual representation of our models and the complete deterministic simulation
• Stochastic modelling

Even though deterministic modelling predicted a promising behaviour for our system, we modelled our system with Stochastic algorithms in order to check the robustness of our predictions with a highly stochastic medium and to get statistical information on our system. For biosensors the importance of stochastic modelling is clear, it gives a lot of information on the precision of the measure that is mainly caused by the inner randomness of the genetical network. We first worked on histograms, to get the probability density distrbution of our two ways. After each of the stochastic runs the concentration values were stored in histograms.


Figure 5: Histogram for several runs on the same point of the plate. We are far from interface and only the LacI way is transcripted. X axis is normalized concentrations and the Y axis is number of runs that finished with the corresponding concentration (negative for LacI and positive for tetR

Figure 6: Histogram for several runs on the same point of the plate. It is on one point of the interface between LacI area and TetR area of the plate. (LacI = green; TetR = blue)

As we were expecting, the probability distribution is bimodal at the interface. At this point the two ways are equally likely to be chosen in the cell, which is why we have an interface. To compare the results to what we obtained with deterministic modelling, we have to use the mean concentration of each species for each point of the plate. This kind of computation requires a great amount of runs (several tens of millions of runs for a proper analysis).


Figure 7: Mean of concentration in each point of the plate obtained through stochastic modelling. (Red = LacI; Blue = TetR; in number of proteins / cell)

On this curve we can see that our switch is still very efficient, but for a proper understanding of these results we needed a deeper statistical analysis of the dataset.