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.

Validation of our genetical network

Device specificities

With the statistical approaches described in the previous pages we could get through stochastic simulations the specificities ofour device.

We first consider the numbers of TetR and LacI proteins in the cells are random variables (X1 and X2). With the calculation described here we get the three following curves :

On this figure the µ_TetR*LacI curve is gaussian indeed. The curves should be much smoother, the statistical noise on the curves is due to the variance of our mean estimator. (The mean is calculated on a finite number of runs, and we had to compromise between precise estimation and time-consuming simulations of many millions of runs). From this curve we could get the minimum IPTG step of our gradient, i.e. the ΔIPTG between two wells. If the IPTG gradient minimal step (either for log gradient or linear gradient) is a lot smaller than the width of the gaussian curve obtained here, many wells will turn red. However if the IPTG gradient is bigger than the width of this curve, there is a chance that no well turns red. We then get a range of possible values for the IPTG step. In this curve the maximal IPTG step would be 6E-3 M. However, this result is not sufficient to precisely specify the requirements for the IPTG gradient on the plate. We first need to know if this curve is precise enough by estimating the variance of the TetR*LacI random variable at this point. Second, the width of the gaussian might not be the same if the interface was somewhere else, we need to process the same simulation with different parameters to get a proper estimation of the mean of the TetR*LacI variable with different values of IPTG and aTc concentration. Such a simulation would require a very important computational resource and/or a lot of time, but with the values obtained we can get an idea and an order of magnitude.

We also computed the variance for the TetR, LacI and TetR*LacI variables and got the variance for the TetR*LacI to get the needed number of molecules in the the wells : From this curve we get the maximum standard deviation at the interface which is here 1.9E4. According to central limit theorem we get the minimal number of cells per wells to get an error_68% of less than 10% (with µ_TetR*LacI = 1E6 proteins on the interface) that is here 9025 bacteria per well.

These values are just orders of magnitude now, we will need greater computational resource to have a precise idea of the device specifications but we know how to obtain them and extract them from our datasets, and have the scripts written already for this.