Team:KULeuven/Model/FullModel

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Full Model

Describing the system

Dr. Coli produces a green fluorescent protein (drug) when it senses INPUT (certain disease signal in the human body). When Dr. Coli doesn't sense any INPUT anymore (the patient is cured), the invertimer will function as a molecular timer, counting towards the doctor's own demise: ccdB induced cell death. When INPUT (disease signal) reappers in time, the timer is reset. A filter ensures that the timer is not reset when only “noisy” INPUT signals are sensed. A memory device (stable switch) is included to ensure the clock only starts ticking when it is activated by the first input signal.

ODE's

Parameters

The parameters can be found in the sections about the simple parts of the system.

Models

CellDesigner (SBML file)

Matlab (file)

Simulations

Discussion of the simulation:

• In the beginning the memory has status "0" and the simulation clearly shows that the timer isn't activated. This means that the bacteria can grow without any problem (no ccdB-production and hence no cell death) when they haven't sensed any desease marker in the beginning.
• At t=50.000s the system is activated by a long lightpulse (simulating the insertion of the bacteria in the patient's body and sensing desease marker for the first time). This results in:
  • switching of the memory to status "1" --> activation of transcription of LuxR (green curve)
  • activation of the filter (desease marker is strong enough) which is reflected by the production of lactonase (blue curve), situated just after the filter mechanism.
• At t=75.000 the lightpulse is switched off (simulating that there's no more desease marker: the patient is cured). This results in:
  • stopping the production of lactonase, which starts to degrade naturally (blue curve)
  • stopping the production of LacI, which stops repressing the transcription of LuxI. In this way HSL can be produced, but, since there's still lactonase present in the cell, there won't be a visible increase in HSL-LuxR-complex because the HSL is being transformed into hydroxyacid.
• At t~=92.000 the lactonase is almost totally degraded which enables the HSL to form a complex with LuxR (yellow curve). This means that we have a timer that starts with a delay of +- 17.000s. At this point the HSL-LuxR-complex really starts to build up gradually (TIMER) for about 10.000s.
• At t~=102.000s the HSL-LuxR-complex peaks, with at the same time (free) LuxR decreasing to almost zero.
  • At this point all the newly produced LuxR will go into complex with HSL and no free LuxR will remain. From this point on, the HSL-LuxR-complex will decrease to a steady state value which is equal to the production rate of LuxR-proteins.
  • BUT you can see clearly that the ccdB (red curve) also peaks (to a value of 10 to 20 molecules) which is enough to result in cell death.

Full Model - Part 2

Extension to the previous model

During the summer we changed a lot of things to our grand scheme. This new system has quite some novelties:

• Cell Death has also gotten an overhaul. Transcription now begins at a new hybrid promoter we made: [http://partsregistry.org/wiki/index.php?title=Part:BBa_K145150 BBa_K145150]. This promoter is repressed by c2 P22, which is produced by the memory in the OFF state, making premature activation and cell death impossible. Besides this repression, the promoter is activated by the HSL-LuxR complex originating from a previously activated timer. The promoter behaves as shown schematically below.

• Second, LuxR is now no longer constitutively produced but is placed behind the beforementioned hybrid promoter. This construction mimicks more closely the natural system where LuxR is upregulated when a threshold amount of HSL is present. Plus it also increases the time it takes to activate ccdB, lengthening the timer. The system will now auto-activate if enough HSL is present and the memory is in the ON state.

• Third, the ccdB coding region is also downstream of the hybrid promoter and is thus also subject to the regulation explained above; c2 P22 repression and HSL-LuxR auto-activation. One difference is that the polymerase must first read through a bad terminator with about 60% efficiency before reaching this coding region. Another difference is in the ribosome binding sites preceding both coding regions. Where LuxR can be translated from a RBS with a relative efficiency of 1.00, the ccdB frame can only be read from a 0.01 efficiency RBS.

Because of this new organisation, we've also turned our attention towards multi cell interactions and diffusion of HSL in the medium again.

Describing the system

ODE's

todo!!!

Parameters

The parameters can be found in the sections about the simple parts of the system.

Extension: parameters of HSL-LuxR auto-activation can be found in Model:New Cell Death.

Models

CellDesigner (SBML file)

Attention: this CellDesigner model is no longer up to date. For the final version, use the MATLAB model below!

Matlab (file)

Simulations

This simulation can already be seen as a very basic multicell simulation, since it is performed for 1000 cells growing in a microchemostat with a volume of 1000 . For more deailed information, please see our MultiCell and Diffusion modeling pages. If time is of the essence, the Microchemostat subsection should be most useful.

Discussion of the simulation

1. In the absence of a disease market, the system starts of with a memory in the OFF state, this means the c2 P22 is produced which shuts down LuxR and ccdB production. LuxI production is also partially eliminated because the memory also outputs antisense LuxI. Put simple, the timer is not active and Dr. Coli can grow without worrying as there is no ccdB and hence no cell death.
2. At t=50.000s the system is activated by a long lightpulse (simulating the insertion of the doctor in the patient's body and sensing desease marker for the first time). This results in:
   1. the switching of the memory to the ON state. This means cI 434 instead of c2 P22 so no more antisense LuxI or repression of LuxR and ccdB. (more specific switching simulations can be found in the corresponding memory pages).
   2. With the memory in the ON state, LuxR as well as ccdB will rise to a background level. This level is extremely low for the latter but already significant for the former, because we want to make sure that auto-acivation is possible.
   3. the activation of the filter (long pulse, the disease marker is strong enough). This is reflected in the production of lactonase, situated just after the filter mechanism.
3. At t=130.000s the lightpulse is switched of (simulating that there's no more disease marker: the patient is cured). This results in:
   1. the shutting down of lactonase production, which continues to degrade naturally
   2. the shutting down of LacI production, which stops repressing the transcription of LuxI. Slowly, LuxI is produced and thus also 3OC6HSL.
4. At t~=150.000, all the lactonase and the LacI are gone and it is possible to see LuxI and HSL increase. This means that the actual 'counting' only starts about 17.000s after the input is switched off. Part of the HSL diffuses outside of the cell, another part remains free but the third part associates with the background of Lux receptors (free LuxR drops).
5. This third part will auto-induce more LuxR receptors and upregulate ccdB.
6. When the timer has counted to about 50.000s after the input has gone dead, ccdB has reached the lethal amount of around 10-14 molecules per cell and Dr. Coli calls it a day.

Sensitivity Analysis

Episode I: The MATLAB Menace

Sensitivity analysis for Dr. Coli has been a pain in the ***. Our first attempts at this were done with the Sensitivity Analysis Tool that is included in MATLAB Simbiology. The problem here is that our model calls MATLAB's power fuction. Sensitivity analysis currently uses complex-step differentiation, which is currently not supported for this function. The computed sensitivities may not even approximate true sensitivities. So the value of this analyisis is doubtful. Nevertheless, below are some results of the calculated sensitivities for species in the system that precede any reaction that calls the power function.

In these graphs, the 'parameters'-axis shows the parameters that are varied in the sensitivity analysis, while the ones along the states axis are the ones monitored as output. The z-axis is a measure for the sensitivity of the parameter under study.

Episode IV: A New Hope

We had come to the conclusion that the MATLAB sensitivity analysis just would not cut it for us, so we went on and started digging again. We tried a whole lot of different approaches, including something similar to ETH Zurich 2007, but nothing seemed to be appropriate for our system. (These attempts are episode II and III which we'll just skip here and forget about) Finally, we came up with the following. We define sensitivity in two different states, the ON state of the system and the OFF state The ON state receives maximal input during the entire simulation, while the OFF state only gets background signal. We come to the following formulae where X is the output species that we look at and P is the parameter that is changed to 110% or 90% of its own value. The second term is for normalisation purposes.

A third situation has also been analysed. In the above two cases we constantly give 0 or 1 respectively. Our system however has a third and very important state: 0 - 1 - 0. In this state, the memory is ON and the timer can function. To simulate this, we used the same formula but we also created an event during the simulation.

These attempts give us more helpful results. We can even interpret something stability-like for our memory now: how far can our parameterisation be off before the memory stops functioning as expected. The results obtained by using this method for sensitivity analysis can be seen depicted in the figures below.

Output

More as a control, the first figure depicts the sensitivity of our output GFP. It behaves completely as expected, clearly showing only dependencies for the degradation and synthesis rates of GFP while everything else is equal to zero. Also according to plan, the simulation gives exactly the same results for the two formulae above.

Memory

As mentioned before, we can use the sensitivity analysis of the memory to see how big our margins are on the arious parameters that make out our model. The first figure shows the sensitivity of c2 P22 for all system parameters in the OFF state. This means we use the second formula. The analysis shows all expected dependencies such as synthesis and degradation rates for c2 P22. But also for the affinity of cI 434 for the promoter in front of c2 P22, for the production/degradation rates of cI 434 with(out) LVA tag.

For cI 434 in the OFF state, we see similar but distinct results. We see that all sensitivities appear to be bit higher and that some extra factors have appeared describing repression by and degradation of c2 P22 repressor. Once again logical results.

The next figure shows the sensitivities for c2 P22 in the ON state of the memory (where it plays a similar role as cI 434 (LVA) in the OFF state). We can even see that the figure is very similar to the one just above, except that the affinity of c2 P22 for the promoter controlling cI 434 (LVA) doesn't seem to matter anymore. This can be easily understood if you remember that we're simulating a continuous 1 signal here, where a lot of cI 434 (without LVA) is coming from the TetR promoter, making small contributions to the cI 434 concentration by the c2 P22 repressible promoter quite irrelevant in this case.

cI 434 (without LVA tag) during a continuous 1 signal is only dependent on the obvious synthesis, degradation rates as can be expected with the total amount of cI 434 being high enough to remove all c2 P22 from the system.

A final figure shows the sensitivity of the same cI 434 without tag in a system that has been activated but is now without input. The switch appears to be so strong that we only detect the same sensitivities as in the above scenario of a continuous 1 signal.

Celldeath-related

The first figure shows the sensitivities for Lactonase in the ON state (the first formula). They can all be expected; we clearly see the dependence on the filter (T7 as well as key-lock characteristics) as well as more obvious synthesis and degradation rates of lactonase itself.

This next one shows the sensitivities of LuxI in a 0 - 1 - 0 event situation. We clearly see the filter effect again but this time it appears to be reinforced by the LacI repressor of the inverter. The parameters for LacI repression are also nicely visible.

By looking at HSL in the same 0 - 1 - 0 scenario, we see a very similar picture. All the same sensitivities are visible plus three significant new ones. The first is obvious, it is the rate of HSL synthesis by LuxI. The other two are less obvious, they are the cell volume and the volume of the medium. These are effects of the diffusion out of the cell and into the medium.

The final study is of ccdB, our final effector in the 0 - 1 - 0 scenario. We can see small sensitivities for the parameters characterising the HSL-LuxR complex such as affinity of the complex for the DNA, affinity of HSL for LuxR etc. The highest sensitivities we get for the properties of ccdB itself. The terminator efficiency and other synthesis and degradation rates.

Further conclusions

Our model seems to respond to these sensitivity analyses as suspected and at first glance appears to give us a bit of a margin on most of our parameters.

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