Validation of the network

Toggle switch and quorum sensing behavior

The toggle switch behavior

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. 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 chapter 3, 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.

The following simulation was realized for an IPTG gradient of $1x10^{-6} M$ to $1x10^{-2} M$ and an aTc concentration of $5x10^{-6} M$. The first graph present the logarithmic IPTG gradient in green and the homogeneous concentration of aTc in red. The second represent the concentration of both repressor on the plate.

Figure 1: Observation of the switch on the plate for an aTc concentration of $5x10^{-6}$

The first thing we could observe on this figure is that the switch doesn't appears at the equality of the concentration of aTc and IPTG but for an IPTG concentration of $1.5x10^{-4} M$. This is due to the value of the parameters in the ODE system presented previously. In fact, the dissociation constant of respective repressor and their inhibitor are not the same.

Figure 2: Observation of the switch on the plate for a higher aTc concentration of $5x10{-5}$

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]. Higher aTc concentration will move the interface to the right edge of the plate as in figure. We therefore demonstrated that the Toggle switch behaviour was the one we wanted for our application.

With this model, we also demonstrated that the presence of 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 which impose their half-life time of 10 minutes.

Demonstration that the Toggle switch behaviour was the one we wanted for our application.
Use only LVA tagged LacI and TetR genes which impose the half-life time of 10 minutes.

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.

In a first step, we observed the distribution of the protein acting in the quorum sensing system and the concentration of internal and external quorum sensing. The objective is to show that coupling toggle switch and quorum sensing modelling works.

Figure 3: Observation of the Quorum sensing molecule distribution on the plate

On the first graph of this figure we see(in green) the cinI concentration (which follows the same equation as lacI) and (in red) the cinR concentration (which follows approximately the same equation as tetR). If cinR concentration is not as high as cinI concentration, it's because in cinR equation we needed to take into account the complexation of cinR with the quorum sensing molecule as a disparition term.

Moreover, the two other curve in the first figure show the concentration of the quorum sensing molecule inside and outside the cells. And we see that, because of the diffusion of the quorum sensing molecule in the medium (third graph), the internal concentration of quorum sensing is not equal to zero where cinI is absent. Which indicate that quorum sensing well diffused in the medium and was caught by receiving bacterias.

In the following graphs we show the complexation of cinR with the quorum sensing molecule.

Figure 4: Observation of the Quorum sensing complexation with cinR receptor

On the first graph of this figure, intern quorum sensing concentration (in green) and cinR concentration (in red) are plotted. We can well see that there is an area on the plate where cinR concentration and intern quorum sensing concentration are both not equal to zero. This is predicting that a complexation between both of them could happen.
That's what it's shown is the second graph of this figure, the concentration of the cinR/Quorum Sensing complex in the bacterias.

With the two previous figures, we can confirm that the quorum sensing is diffusing on the right side of the plate. This quorum sensing should be caught by the receiving bacteria. This would produce lycopene and activate a diffused coloration on the plate.

Figure 5: Observation of the red stripe on the plate

With modelling we show that the system should work as expected. But we also hightlighted a problem: the diffusion of the quorum sensing which is decreasing the accuracy of the measure. To fixe this problem, we needed to optimize our device

Validation of the model

Construction of toggle switch test

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

For realized this toggle, we used 4 primary bricks :

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

First we put RBS-GFP behind RBS-TetR, and Q04121 behind pTet: both size are around 1 500 bp. The following gel shows that both constructions were at the expected size. Construction were confirmed by sequencing.

Figure 1: First step of cloning gel

In a last step of cloning, we put RBS-tetR-RBS-GFP behind 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: Last step of cloning gel

Figure 3: Fluorescence test picture

Validation of the model

The fluorescent gene, put after the repressor TetR to measure its expression level, could be experimentally measure. So, the presence of fluorescence will indicate that the system is in the TetR genetic pathway and the abscence of fluorescence will indicate that the system is in the lacI pathway.

We decided to compare the model with experience 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.

From this experiment we get the following curve(left curve) compared to the modelling curve(right):

Figure 4: Observation of an experimental switch at two aTc concentration: 50 ng/mL and 150 ng/mL

Results predicted by simulation are the following:

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

Both curve were obtained with an IPTG concentration of 1mM.
We can get from the experimental graph that between an aTc concentration of 50ng/mL and 150 ng/mL there is a switch after 3 hours of experiment. Fluorescence is produced with 50ng/mL of aTc and not produced with 150ng/mL of aTc. From the modelling graph we can see that with an aTc concentration of 50ng/mL, TetR is produced and with an aTc concentration of 150ng/mL, TetR is not produced. So between these two concentrations, we observe the same switch as in the experiment. However, we see more fluorescence than expected. In fact, in the steady state no fluorescence should be observed for the red curve. But the GFP protein has an half life of 10 hours and it was impossible to get no fluorescence in a 10 hours experiment.

We also did an experiment on bacterias which had grown in an IPTG preculture. But we did’nt see a switch because IPTG block bacterias in the fluorescence way. Because of the half life of GFP, it was possible to detect a switch only with bacteria which had 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 more magnitude. We also construct this toggle with the quorum sensing gene to get the proof of concept. And the construction with mercury repressor.

The model is validated by the experiment which shows a switch as predicted by the simulation. 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 more magnitude. We also construct this toggle with the quorum sensing gene to get the proof of concept.

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 this 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 $[aTc] = 5x10^{-6} M$ and $[IPTG] = 1.55x10^{-4} M$. These parameters reflect the situation of our system in the center of the plate in the presence of a logarithmic gradient of IPTG of $1x10^{-6} M$ to $1x10^{-2} M$.

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 $[aTc] = 5x10^{-6} M$ and for the left curve with $[IPTG] = 1x10^{-6} M$ and for the right curve with $[IPTG] = 1x10^{1} M$. 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.

Stochastic analysis of the stability

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 is normalized concentrations and the Y axis is number of runs that finished with the corresponding concentration (negative for LacI and positive for tetR)

This figure show the bacteria 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 show the bacteria 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