Validation of the network

First of our modelling and experiments was to validate the work of our genetic network. Primary validate by modelling, the genetic network was validate with the construction of a toggle switch test.

Toggle switch and quorum sensing behavior

The toggle switch behavior

The first goal of the modelling team was to verify that the genetic circuit as conceived had the desired dynamical behavior. We divided the network into two modules, the Toggle switch and the Quorum Sensing modules, which we modeled separately to facilitate their modeling and dynamical analysis. Very early, the modelling of these modules gave promising results and we could rapidly conclude that our Toggle Switch system would be functional. Hence, with the models described in the deterministic modelling approach, we predicted the behaviour of our bacteria on the plate. Two regions could be distinguished on the plate: one region with bacteria in a state characterized by a high LacI concentration, while bacteria in the other region contain high TetR levels.

We show below a typical simulation result obtained with an IPTG gradient of $1.10^{−6} M$ to $1.10^{−2} M$ and an aTc concentration of $5.10{−6} M$. The X axis represents physical points on the plate, from the right side to the left side of the plate. Each point differs by the IPTG concentration. The second represent the concentration of both repressor on the plate.

Figure 1: Prediction of the switch on the plate for an aTc concentration of $5.10^{−6} M$. Upper panel: the logarithmic IPTG gradient is shown in green, the homogeneous aTc concentration, in red. Lower panel: LacI and TetR repressor concentrations are shown in blue and red, respectively.

As can be seen in Figure 1, the switch is not observed in the area where the aTc and IPTG concentrations are equal, but for an IPTG concentration of $1,5.10^{−4} M$. This result is due to the values chosen for the parameters in the ODE system: the dissociation constants of the repressors and their respective inhibitors are not the same. This is not affect how the system works. It just means that the aTc concentration can't be deduced directly from the IPTG concentration. We need to construct a law which gives us the aTc concentration depending on IPTG concentration.

Figure 2: Observation of the switch on the plate for a higher aTc concentration ($5.10^{-5} M$)

The results shown in Figure 1 and 2 show that, as expected, the boundary between the two different regions on the plate is shifted depending on the aTc concentration. For instance, a higher aTc concentration moves the interface to the right side of the plate as in Figure 2. We have therefore demonstrated that the Toggle switch behaviour is the one expected for our application.

In addition, the modelling results show that the presence of degradation tags is necessary to get the appropriate behaviour. If the degradation rates of LacI and TetR repressors were too long (even for the typical half life of about 10 hours), the concentration of each of these proteins would be too high and the switching between states would be way too long for our application. As a result we decided to only use LVA-tagged lacI and tetR genes, which reduce their half-life time to 10 minutes.

In summary:
Demonstration that the Toggle Switch has the expected behaviour
Use only unstable (LVA-tagged) lacI and tetR genes

Our models for the Quorum Sensing module were used to simulate the whole system functioning. This allowed us to verify the system has the expected behavior and to eventually obtain a visual representation of the entire device.

In a first , we observed the distribution of the protein involved in the quorum sensing system and the concentration of internal and external quorum sensing molecule (figure 3). The objective is to show that coupling toggle switch and quorum sensing modelling works well.

Figure 3: Predicted distribution of the Quorum sensing molecules on the plate. CinI and CinR concentrations are shown in green and red, respectively.

The upper panel in Fig. 3 shows the predicted CinI and CinR concentrations (the evolution of the latter follows approximately the same equation as the TetR concentration). The fact that CinR concentration is not as high as CinI concentration is due to the modelling: in the CinR equation, we needed to take into account the complexation of CinR with the quorum sensing molecule as a consumption term.

The other curves in the two lower panels show the intra- and extracellular concentrations of the quorum sensing molecules. We see that, due to the diffusion of quorum sensing molecules in the medium (third panel), the internal concentration of quorum sensing is not equal to zero when CinI is absent. This indicates that the quorum sensing molecules diffused correctly in the medium and were imported by receiving bacteria.

The following graphs show the complexation of CinR with the quorum sensing molecule.

Figure 4: Predicted complexation of the Quorum sensing molecule with the CinR receptor. The concentrations of the quorum sensing molecule and its receptor are shown in green and red, respectively.

The concentrations of internal quorum sensing molecules and their receptors are plotted in the upper panel. The complexation occurs in the area of the plate where the concentrations of the two differ from zero. The resulting complex concentration is given in the lower panel.

From the two previous figures, we confirm that quorum sensing molecules diffuse on the right side of the plate. They should be caught by receiving bacteria, resulting in the lycopene production and the appearance of a diffused coloration on the plate.

Figure 5: Predicted observation of the red line on the plate

With this modelling part, we have shown that the system should work as expected. However we have also discovered a potential problem, that the diffusion of the quorum sensing molecule decreases the accuracy of the measure. Fixing this problem required to optimize our device

Experimental validation of the model

Construction of the toggle switch test

In order to test if our system could work experimentally, we constructed a toggle switch test based on Gardner's work [1].

• pTet : BBa_R0040
• RBS-LacI-oo-pLac : BBa_Q04121
• RBS-GFP : BBa_E0240
• RBS-TetR composed of RBS BBa_B0034 and TetR BBa_C0040

We first cloned RBS-GFP downstream of RBS-TetR, and Q04121 downstream of pTet. The following gel shows that both constructions have the expected size (1 500 bp). The constructions were confirmed by DNA sequencing.

Figure 1: Agarose gel electrophoresis of the fragments obtained with the first cloning step.

In the last step of cloning, we cloned RBS-tetR-RBS-GFP downstream of pTet-Q04121. The size is around 3 000 bp. The following gel shows that constructions was at the expected size. In addition to this test, transformation of bacteria have grown on plate with IPTG to block them in the fluorescence way. And some of the bacteria were fluorescent. Construction was also confirmed by sequencing.

Figure 2: Agarose gel electrophoresis of the fragments obtained with the last cloning step.

Figure 3: (A) Non fluorescent bacteria in medium with 500ng/ml aTc inducer. (B)Fluorescent bacteria in medium with 2mM IPTG inducer

Comparison of model predictions and experimental data

The expression of the fluorescent gene, put after the repressor TetR to measure its expression level, could be experimentally monitored. The presence of fluorescence will indicate that the system is in the state with a high TetR concentration while its absence of fluorescence will indicate that the system is in the LacI state.

We decided to compare the model with the experiments as follows:

• The bacteria are first blocked in the non-fluorescent pathway (LacI).
• Then placed in a 96-well plate with at different aTc and IPTG concentration.
• Measure of the fluorescence during 10 hours.

The experimental and modelling results are shown in Figure. 4 and 5,respectively.

Figure 4: The bacteria were first blocked in the non-fluorescent pathway (LacI) , before they were spread on a 96-well plate with different aTc concentrations and 1 mM of IPTG. The fluorescence signal was measured during 10 hours. The experimental and modelling results are shown in Figure. 4 and 5, respectively.

Figure 5:Modelling of tetR expression for two aTc concentrations: 50 ng/mL and 150 ng/mL

We conclude from the experimental graph that the switch occurs after 3 hours of experiment, at an aTc concentration between 50 and 150 ng/ml: fluorescence is obtained with 50ng/mL of aTc but not with 150ng/mL of aTc. From the modelling graph, we can see as well that TetR is produced at 50ng/mL of aTc, but not 150ng/mL. Therefore, we predict the same switch between these two concentrations, as in the experiment. However, we experimentally observe more fluorescence than expected: no fluorescence should be observed for the red curve at steady state. This is due to the high stability of the GFP protein (half life of 10 hours), which made it impossible to suppress the fluorescence signal in a ten-hour experiment.

We also did an experiment on bacteria grown in an IPTG preculture. But we did not see a switch because IPTG blocks bacteria in the fluorescent state. Because of the half life of GFP, it was only possible to detect a switch with bacteria grown with aTc.

To go further, it will be very interesting to put an LVA tag on GFP in order to control its degradation. In this case we will be abble to see the switch in both case and with higher magnitude. We also construct this toggle with the quorum sensing gene to get the proof of concept. In addition to the experimental results shown above, we also constructed this Toggle Switch together with the quorum sensing gene to get the proof of concept of our system functionality. The same construction has also been performed with the mercury repressor in place of TetR.

Stability Studies of the Toggle Switch

Nullclines studies

In order to predict the set point and the specifications of our system, we studied first the existence and the value of the steady state solutions of the set of ODE.

Isocline study is a classical study which implies a research of stationnary point in a system. These stationnary points are deduced from the equations of the differential system: when the variation of concentration of both repressors are equal to zero.

$\frac{d[TetR]}{dt} = \frac{k_{pLac}.[pLac]_{tot}}{1 + (\frac{[lacI_{total}]}{K_{pLac} + \frac{K_{pLac}.[IPTG]}{K_{lacI-IPTG}}.})^\beta} - \delta_{TetR}.[TetR] = 0$

$\frac{d[lacI]}{dt} = \frac{k_{pTet}.[pTet]_{tot}}{1 + (\frac{[tetR_{total}]}{K_{pTet} + \frac{K_{pTet}.[aTc]}{K_{TetR-aTc}}.})^\gamma} - \delta_{lacI}.[lacI] = 0$

To facilitate the manipulation of the equation and reduced the number of parameters, we posed:

• $E_{TetR}$ = $k_{pLac}.[pLac]_{tot}$
• $R_{TetR}$ = $\frac{1}{1 + (\frac{[TetR_{total}]}{K_{pMerT} + \frac{K_{pTet}.[aTc]}{K_{TetR-aTc}}.})^\gamma}$
• $E_{LacI}$ = $k_{pTet}.[pTet]_{tot}$
• $R_{LacI}$ = $\frac{1}{1 + (\frac{[LacI_{total}]}{K_{pLac} + \frac{K_{pLac}.[IPTG]}{K_{LacI-IPTG}}.})^\beta}$
• $[TetR]_{r}$ = $R_{TetR}.[TetR]$ the relative concentration of TetR
• $[LacI]_{r}$ = $R_{LacI}.[LacI]$ the relative concentration of LacI
• $K$ = $\frac{R_{TetR}.E_{TetR}}{\delta_{TetR}}$
• $K_{prime}$ = $\frac{R_{LacI}.E_{LacI}}{\delta_{LacI}}$

After manipulation with these reduced parameters, we get the following equations:

$[TetR]_{r} = \frac{K}{1 + ([LacI]_{r})^\beta}$ (1)
$[LacI]_{r} = \frac{K_{prime}}{1 + ([TetR]_{r})^\gamma}$ (2)

From this equation we could see that, if $[LacI]_r$ >> 1, $[TetR]_r = 0$ and $[LacI]_r \approx K_{prime}$. In the other case if $[TetR]_r$ >> 1, $[LacI]_r = 0$ and $[TetR]_r \approx K$

From these equations, we get this figures:

Figure 6: Solution of the equation and emergence of three steady state

On this figure, the red lines represent the solution of the equation (1) and the green line the solution of (2). This figure was realized with $[aTc] = 5.10^{-6} M$ and $[IPTG] = 1,55.10^{-4} M$. These parameters reflect the situation of our system in the center of the plate in the presence of a logarithmic gradient of IPTG of $1.10^{-6} M$ to $1.10^{-2} M$.

Three stationary points emerge from this graph. These are the three points of intersection of two curves and represent the steady state of the system.
However, there is one of the three points which is an unstable steady state: the point 2. It represents the point when both relative concentration are equal. In a Toggle Switch, it's impossible to have concentration of both repressors equal because one repressed the other. So one of these should take the avantage on the other.

Figure 7: Nullclines for the left side of the plate

Figure 8: Nullclines for the right side of the plate

These figures were realized with $[aTc] = 5.10^{-6} M$ and for the left curve with $[IPTG] = 1.10^{-6} M$ and for the right curve with $[IPTG] = 1.10^{1} M$. The left graph represents the left side of the plate where aTc concentration is dominant and the right graph represents the right side of the plate where IPTG concentration is dominant.
These figures show that when the concentration of one of the repressor is too high, the system is no longer bistable but monostable.

By working on histograms, we get the distribution of bacteria's states(lacI or tetR pathway) along the plate.

Figure 9: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 represents in negative, the bacteria in the lacI pathway and in positive bacteria in the tetR pathway. The Y axis is number of runs that finished with the corresponding state for bacteria.

This figure shows the bacterial state distribution in the left of the plate, where aTc is predominent. The green peak indicates bacterias in the lacI pathway. Which is showing to us that in the left of the plate, bacteria could only be in the lacI genetic pathway. The distribution is monomodal.

Figure 10: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)

This figure shows the bacterial state distribution at the interface. The presence of two peaks indicates that bacterias are presents both in the lacI pathway and the tetR pathway as we were expecting. At this point the two ways are equally likely to be chosen in the cell, which is why we have an interface.

As we saw it with the nullcline study, stochastic modelling shows that on the edge of the plate, the toggle switch is monostable and at the interface it's bistable.

Conclusion about stability

According to the previous studies, we were able to predict(in fonction of aTc and IPTG concentration) where the system is monostable and where it's bistable.

Figure 11: Stability of the toggle switch on the plate

• On the extreme side of the plate, the system is monostable.
• On the switch area of the plate, the system is bistable.
• Bistability, in the switch area, allows us to obtain neighboring bacteria in different states. These bacterias could communicate together and give rise to the coloration

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