Team:ETH Zurich/modeling/whole

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== Model ==
== Model ==
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The whole cell model is the combination of the Quorum sensing, Integrase and XOR modules. The model shows the behaviour of the a single cell in response to incoming signals. The model enables us to understand the effect of leakiness, cross-talk and their combinations on the whole system. All the simulations here are described for a red cell .i.e. a cell producing LasI and GFP when it is ON.
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The whole cell model is the combination of the Quorum sensing, Integrase and XOR modules. The model shows the behaviour of a single cell in response to incoming signals. The model enables us to understand the effect of leakiness, cross-talk and their combinations on the whole system. All the simulations here are described for a red cell .i.e. a cell producing LasI and GFP when it is ON.
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[[File:ETHZ_WholeCellRedCell.png|800px|center|thumb|The four cases with different combinations of input for a red cell.]]
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[[File:ETHZ_WholeCellRedCell.png|800px|center|thumb|'''Figure 1''' The four cases with different combinations of input for a red cell.]]
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$$\begin{align*}
$$\begin{align*}
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\frac{d[LuxAHL]}{dt} &= k_{-RLux}[R_{Lux}]-k_{RLux}[LuxAHL][LuxR]-d_{LuxAHL}[LuxAHL]\\
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\frac{d[LuxAHL]}{dt} &= -d_{LuxAHL}[LuxAHL]\\
\frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[LuxAHL][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\
\frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[LuxAHL][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\
\frac{d[RLux]}{dt} &=  k_{RLux}[LuxAHL][LuxR] - k_{-RLux}[RLux] - d_{RLux} [RLux] \\  
\frac{d[RLux]}{dt} &=  k_{RLux}[LuxAHL][LuxR] - k_{-RLux}[RLux] - d_{RLux} [RLux] \\  
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\frac{d[SF_{Bxb1}]}{dt}&=  2 k_{ToffBxb1} [SA_{Bxb1}]^2 [T_{on,i}] + 2 k_{-Toff\phi C31} [SA_{Bxb1}]^2 [T_{off\phi C31}]\\
\frac{d[SF_{Bxb1}]}{dt}&=  2 k_{ToffBxb1} [SA_{Bxb1}]^2 [T_{on,i}] + 2 k_{-Toff\phi C31} [SA_{Bxb1}]^2 [T_{off\phi C31}]\\
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\\
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\frac{d[LasAHL]}{dt} &= k_{-RLas}[R_{Las}]-k_{RLas}[LasAHL][LasR]-d_{LasAHL}[LasAHL] + k_{LasAHL} [LasI]\\
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\frac{d[LasAHL]}{dt} &= -d_{LasAHL}[LasAHL] + DF * k_{LasAHL} [LasI]\\
\frac{d[LasR]}{dt} &= \alpha_{LasR} -k_{RLas}[LasAHL][LasR] + k_{-RLas}[RLas] - d_{LasR}[LasR] \\
\frac{d[LasR]}{dt} &= \alpha_{LasR} -k_{RLas}[LasAHL][LasR] + k_{-RLas}[RLas] - d_{LasR}[LasR] \\
\frac{d[RLas]}{dt} &=  k_{RLas}[LasAHL][LasR] - k_{-RLas}[RLas]  - d_{RLas} [RLas] \\  
\frac{d[RLas]}{dt} &=  k_{RLas}[LasAHL][LasR] - k_{-RLas}[RLas]  - d_{RLas} [RLas] \\  
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=== Ideal case ===
=== Ideal case ===
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[[File:ETHZ_IdealCase.png|center|700px|thumb|The four cases for the whole cell under ideal conditions]]
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[[File:ETHZ_IdealCase.png|center|800px|thumb|'''Figure 2''' The four cases for the whole cell under ideal conditions. On [https://2014.igem.org/Team:ETH_Zurich/project/overview#Implementation_in_E._coli our project overview], you can see  see in action how the circuit responds to different inputs.]]
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Under ideal conditions, there is no leakiness or cross-talk. The figure shows the production of GFP as a function of four different combinations of inputs in an ideal whole cell. No GFP is produced when there is no LasAHL and LuxAHL or when both are present. GFP is produced only when one of the input AHLs are present, thus emulating an XOR system. In the case with 0.5 nM of LuxAHL and no LasAHL as input, the rate of production of GFP reduces after six hours. This could be attributed to the delayed production of LasAHL from LasI. In the case with only LasAHL as input, there is a positive feedback of LasAHL.
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Under ideal conditions, there is no leakiness or cross-talk. The Figure 2 shows the production of GFP as a function of four different combinations of inputs in an ideal whole cell. No GFP is produced when there is no LasAHL and LuxAHL or when both are present. GFP is produced only when one of the input AHLs are present, thus emulating an XOR system. In the case with 0.5 nM of LuxAHL and no LasAHL as input, the rate of production of GFP reduces after six hours. This could be attributed to the delayed production of LasAHL from LasI. In the case with only LasAHL as input, there is a positive feedback of LasAHL.
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[[File:ETHZ_Terminator10.png|center|600px|thumb|The dynamics of flipping of the terminators when the cell receives 0.5 nM LuxAHL and 0 nM of LasAHL and produces GFP and LasI]]
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[[File:ETHZ_00TerminatorwithLeakiness.png|center|600px|thumb|'''Figure 3''' The dynamics of flipping of the terminators when the cell receives 0.5 nM LuxAHL and 0 nM of LasAHL and produces GFP and LasI]]
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In the case above, the cell receives only 0.5 nM of LuxAHL and no LasAHL but the cell produces LasI upon flipping of the terminator. The LasI produced in turn catalyses the production of LasAHL. The LasAHL activates P<sub>Las</sub> thus producing ΦC31. ΦC31 results in flipping of the terminator again thus, turning OFF the system. The dynamics of T<sub>oni</sub>, T<sub>offBxb1</sub>, T<sub>offΦC31</sub> and T<sub>onf</sub> is summarised in the figure above.  
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In the case above, the cell receives 0.5 nM of LuxAHL and 10 nM LasAHL. This results in production of Bxb1 and ΦC31 respectively. Each terminator is flanked by binding sites for each of these integrases. These integrases flip the terminators by binding to their respective sites. Since, both integrases are present, the terminator is flipped twice thus eventually turning the system OFF. This is summarized in the figure above.Then, they flip the terminators again thus turning the system OFF.  
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At first the systems are ON either due to flipping by ΦC31 or Bxb1. This results in the production of LasI which catalyses the production of LasAHL. The LasAHL further increases the production of ΦC31 which flips the already flipped terminators by Bxb1. The remaining Bxb1 flips the already flipped terminators by ΦC31. Thus, the system eventually turns OFF.Even though there is more Bxb1 flipped terminators than those flipped by ΦC31. The dynamics of T<sub>oni</sub>, T<sub>offBxb1</sub>, T<sub>offΦC31</sub> and T<sub>onf</sub> is summarised in the figure above.
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[[File:ETHZ_XORWholeCell.png|center|500px|thumb|Predicted XOR behaviour for the whole cell model without leakiness]]
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[[File:ETHZ_XORWholeCell.png|center|500px|thumb|'''Figure 4''' Predicted XOR behaviour for the whole cell model without leakiness]]
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[[File:ETHZ_Leakiness.png|center|800px|thumb|The four cases for the whole cell with basal leakiness]]
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[[File:ETHZ_Leakiness.png|center|800px|thumb|'''Figure 5''' The four cases for the whole cell with basal leakiness]]
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From our experimental data, we observed some basal leakiness for P<sub>Lux</sub> and P<sub>Las</sub> even after using riboregulators. From the model we see that, this small basal leakiness is amplified downstream. The basal leakiness results in production of integrases which further act on the XOR module and cause the switching of the terminator. Thus, there is GFP and LasI produced, and the LasI produced further catalyses the production of more LasAHL. Thus, we observe some GFP even without inputs.  
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From our experimental data, we observed some basal leakiness for P<sub>Lux</sub> and P<sub>Las</sub> even after using riboregulators. From the model we see that, this small basal leakiness is amplified downstream. The basal leakiness results in production of integrases which further act on the XOR module and cause the switching of the terminator. Thus, there is GFP and LasI produced, and the LasI produced further catalyses the production of more LasAHL. Thus, we observe some GFP even without inputs.
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<!--[[File:ETHZ_00TerminatorwithLeakiness.png|center|500px|thumb|No inputs and only basal leakiness]]-->
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<!--[[File:ETHZ_00TerminatorwithLeakiness.png|center|500px|thumb|'''Figure 6''' No inputs and only basal leakiness]]-->
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<!--The figure above summarises the predicted effect of basal leakiness on the flipping of the terminator. The basal leakiness results in production of Bxb1 and ΦC31 which result in flipping of the terminator. In this case, since the cell produces LasI there is increased production of LasAHL. The LasAHL produced induces the production of ΦC31 which further causes flipping of all terminators flanked by the ΦC31 sites. Thus, by 200 minutes almost all ΦC31 sites are inactive and the cell will stay ON. -->
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<!--The figure above summarizes the predicted effect of basal leakiness on the flipping of the terminator. The basal leakiness results in production of Bxb1 and ΦC31 which result in flipping of the terminator. In this case, since the cell produces LasI there is increased production of LasAHL. The LasAHL produced induces the production of ΦC31 which further causes flipping of all terminators flanked by the ΦC31 sites. Thus, by 200 minutes almost all ΦC31 sites are inactive and the cell will stay ON. -->
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However, if we measure the fluorescence at around 400 mins, the effect of leakiness becomes negligible: in the case with null inputs, the level of the output concentration, GFP, is negligible compared to the level of the GFP concentration produced when the cell receives only one type of AHL. At this point, our model behaves like an XOR gate. Therefore, one of the solutions we propose is to kill or freeze the cells in each row after 6.5 hours.
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However, if we measure the fluorescence at around 300 mins, we observe a good and acceptable XOR behaviour. Therefore, one of the solutions we propose is to kill or freeze the cells in each row after 3 hours.
 
<!-- Alternatively, we propose a modified construct, where production of LasI is regulated by a weaker promoter and is induced by a protein, whose production is coupled with production of GFP. -->
<!-- Alternatively, we propose a modified construct, where production of LasI is regulated by a weaker promoter and is induced by a protein, whose production is coupled with production of GFP. -->
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Initially from our model we observed that the feedback was rapid and hence, the amplification was much higher. However, from literature <sup>[[Team:ETH_Zurich/project/references|[9]]]</sup> we see that the XOR module is relatively slow. We were able to correct this by modelling transcription and translation steps. The delay introduced seems more reliable although we do not have our own experimental data to validate the same.
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Initially from our model we observed that the feedback was rapid and hence, the amplification was much higher. However, from literature <sup>[[Team:ETH_Zurich/project/references|[9]]]</sup> we see that the XOR module is relatively slow. We were able to correct this by modelling transcription and translation steps. The delay introduced seems more realistic although we do not have our own experimental data to validate the same. Further, we use a dilution factor (DF) which represents the density of the cells in the bead. By choosing and appropriate DF we can get a more delayed response.
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$$DF = \frac{No. of cells * V_{E.coli}}{V_{Bead}}$$
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[[File:ETHZ_XORWholeCellwithLeakiness.png|center|800px|thumb|Predicted XOR behaviour for the whole cell model with leakiness]]
 
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<!-- === With Leakiness and Crosstalk === -->
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[[File:ETHZ_XORWholeCellwithLeakiness.png|center|800px|thumb|'''Figure 6''' Predicted XOR behaviour for the whole cell model with leakiness]].
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=== With Leakiness and Crosstalk ===
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We modelled cross-talk by fitting Hill-functions to experimental data.  For each quorum sensing module, there are two levels of [https://2014.igem.org/Team:ETH_Zurich/expresults cross-talk]. At the first level, we could have any of the two AHLs binding to a given regulator thus activating the promoter while at the second level there is cross talk between regulators and their respective native promoters. Thus, for activation of a given promoter, we can have four Hill functions corresponding to the combinations of inducer and regulator interactions. Since we are using two quorum sensing modules activating two different promoters, we have eight Hill functions.
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As all the measurements were in terms of fluorescence, the V<sub>max</sub> for each Hill function was also in terms of fluorescence. In order to observe the effect of cross-talk we normalized the V<sub>max</sub> of non native interactions (in Lux system, P<sub>Lux</sub> being activated by LasAHL-LuxR, LasAHL-LasR or LuxAHL-LasR complexes) with the V<sub>max</sub> of the native interaction (LuxR binding to LuxAHL and activating P<sub>Lux</sub>). These ratios acted as weights for the effect of a non-native interaction on the promoter.
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[[File:ETHZ_DynamicsWithCrossTalk.png|center|800px|thumb|'''Figure 7''' The four cases for the whole cell with basal leakiness and cross-talk]]
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For an ideal whole cell model, it was necessary  to have no or minimum leakiness and cross-talk. However, in reality this was not the case. Although we were able to reduce leakiness significantly using riboregulators, we still had the issue of cross-talk. However, we observe that the system acts as an XOR gate till about 300 mins even with cross talk.
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In Figure 7 we observe the four cases with both cross-talk and leakiness.
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[[File:ETHZ_WikiXORCTDiffLeakiness.png|center|500px|thumb|'''Figure 8''' The four cases for the whole cell with basal leakiness and cross-talk]]
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Figure 8, shows the predicted XOR gate performance by a single cell model with basal leakiness and cross talk for different concentrations of LasAHL and LuxAHL at the end of 270 min. The system turns OFF when the input concentration of either AHLs is greater than 100 nM due to the effect of cross-talk.
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== Alternate Design ==
== Alternate Design ==
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Here, we present the simulation results. The four possibilities of the XOR gate are presented in the following figure. The system was the one with a feedback corresponding to the metabolite pathway. The metabolite parameter taken was the one of xylose, which is supposed to be generic for all models
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Here, we present the simulation results. The four possibilities of the XOR gate are presented in the following figure. The system was the one with a feedback corresponding to the metabolite pathway. Thus, when the only input is Lux-AHL, the cell will produce the metabolite B and create a feedback loop, which will switch off the gate. This behvior can indeed be observed on the following figure.
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[[File:ETH_Zurich_Collaboration_Four_Cases.jpg|800px|center|thumb|Dynamic simulations of the deterministic single-cell model. One input is a metabolite concentration. Cross-talk is then supposed to be assumed.]]
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[[File:ETH_Zurich_Collaboration_Four_Cases.jpg|800px|center|thumb|'''Figure 9''' Dynamic simulations of the deterministic single-cell model. One input is a metabolite concentration. Cross-talk is then supposed to be assumed.]]
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Latest revision as of 02:39, 18 October 2014

iGEM ETH Zurich 2014