Team:ETH Zurich/modeling/diffmodel

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=== Geometry and boundary conditions ===
=== Geometry and boundary conditions ===
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[[File:ETH Zurich Compartments Diffusion.png|float|500px|thumb|Millifluidic compartments used for the diffusion model]]
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[[File:ETH Zurich Compartments Diffusion.png|float|500px|thumb|'''Figure 1''' Millifluidic compartments used for the diffusion model]]
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[[File:ETHZ comsol geometry.png|center|610px|thumb|Geometry used for simulation on Comsol Multiphysics]]
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[[File:ETHZ comsol geometry.png|center|610px|thumb|'''Figure 2''' Geometry used for simulation on Comsol Multiphysics]]
<br>
<br>
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In order to fulfill these conditions in Comsol Multiphysics, the cells are not drawn but instead, beads are considered as compartments with a certain density of cells. Rates of diffusion through the membrane are added as reaction rates and intracellular species are contained in beads via a Newmann's boundary condition applied to the surface of the beads, that is :
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In order to fulfill these conditions in Comsol Multiphysics, the cells are not drawn but instead, beads are considered as compartments with a certain density of cells. Rates of diffusion through the membrane are added as reaction rates and intracellular species are contained in beads via a Neumann boundary condition applied to the surface of the beads, that is :
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$$ \nabla C . \textbf{n} =0$$
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$$ \nabla C \cdot \textbf{n} =0$$
$$\text{for all species except extracellular AHL}$$
$$\text{for all species except extracellular AHL}$$
$$\text{for every vector n normal to the surface of a bead}$$
$$\text{for every vector n normal to the surface of a bead}$$
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=== Reactions ===
=== Reactions ===
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These reactions happen in every bead of the pattern on the grid.
$$ \begin{align}
$$ \begin{align}
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RLux+RLux &\leftrightarrow DRLux\\
RLux+RLux &\leftrightarrow DRLux\\
DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\
DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\
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P_{luxON}&\rightarrow P_{luxON}+mRNA_{GFP}\\
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P_{luxON}&\rightarrow P_{luxON}+mRNA_{GFP}+mRNA_{LuxI}\\
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mRNA_{GFP}&\rightarrow GFP
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mRNA_{GFP}&\rightarrow GFP \\
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mRNA_{LuxI} &\rightarrow LuxI\\
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AHLint &\rightarrow \\
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AHLext &\rightarrow \\
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LuxR &\rightarrow  \\
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RLux &\rightarrow\\
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mRNA_{GFP} &\rightarrow\\
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mRNA_{LuxI} &\rightarrow\\
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GFP &\rightarrow \\
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LuxI &\rightarrow
\end{align}$$
\end{align}$$
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\begin{align*}
\begin{align*}
\frac{d[AHLext]}{dt} &= \frac{N_0 N_m e^{rt}}{N_m+N_0(e^{rt}-1)} \alpha\  D_m \ (AHL_{int}-AHL_{ext}) -d_{AHLext}[AHL_{ext}]\\
\frac{d[AHLext]}{dt} &= \frac{N_0 N_m e^{rt}}{N_m+N_0(e^{rt}-1)} \alpha\  D_m \ (AHL_{int}-AHL_{ext}) -d_{AHLext}[AHL_{ext}]\\
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\frac{d[AHLint]}{dt} &= Dm (AHL_{ext}-AHL_{int}) + k_{-RLux}[R_{Lux}]-k_{RLux}[LuxAHL_{int}][LuxR]-d_{AHLint}[AHL_{int}]\\
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\frac{d[AHLint]}{dt} &= Dm (AHL_{ext}-AHL_{int}) + k_{-RLux}[R_{Lux}]-k_{RLux}[AHL_{int}][LuxR]-d_{AHLint}[AHL_{int}]\\
\frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[AHL_{int}][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\
\frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[AHL_{int}][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\
\frac{d[RLux]}{dt} &=  k_{RLux}[AHL_{int}][LuxR] - k_{-RLux}[RLux]  - d_{RLux} [RLux] \\  
\frac{d[RLux]}{dt} &=  k_{RLux}[AHL_{int}][LuxR] - k_{-RLux}[RLux]  - d_{RLux} [RLux] \\  
\frac{d[mRNA_{GFP}]}{dt} &=  \frac{k_{mRNA_{GFP}}[RLux]^2}{K_{mLux}^2 + [RLux]^2}- d_{mRNA_{GFP}} [mRNA_{GFP}]\\
\frac{d[mRNA_{GFP}]}{dt} &=  \frac{k_{mRNA_{GFP}}[RLux]^2}{K_{mLux}^2 + [RLux]^2}- d_{mRNA_{GFP}} [mRNA_{GFP}]\\
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\frac{d[GFP]}{dt} &=  k_{GFP} [mRNA_{GFP}] - d_{GFP}[GFP]\\
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\frac{d[GFP]}{dt} &=  k_{GFP} [mRNA_{GFP}] - d_{GFP}[GFP]
\end{align*}
\end{align*}
$$
$$
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$$ [LuxR]_i=\frac{a_{LuxR}}{d_{LuxR}} $$
$$ [LuxR]_i=\frac{a_{LuxR}}{d_{LuxR}} $$
The cells of the first row are induced by 10 &mu; M AHL just before they are encapsulated in alginate. Then, bead formation takes half-an-hour before these cells are added in the first well. During this half-an-hour AHL can diffuse out. Therefore initial conditions in the first cell completely depend on these previous steps. We have also simulated this bead preparation in order to get these initial conditions.
The cells of the first row are induced by 10 &mu; M AHL just before they are encapsulated in alginate. Then, bead formation takes half-an-hour before these cells are added in the first well. During this half-an-hour AHL can diffuse out. Therefore initial conditions in the first cell completely depend on these previous steps. We have also simulated this bead preparation in order to get these initial conditions.
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[[File:ETHZ Beads in storage solution.jpg|center|300px|thumb|'''Figure 3''' Beads stored in calcium chloride, 10mM]]
The geometry is a simple bead in a bigger compartment filled with calcium chloride. We look at the concentrations at the middle of the bead. Initial concentration of LuxR before bead diffusion starts is a classical steady state a<sub>LuxR</sub>/d<sub>LuxR</sub>
The geometry is a simple bead in a bigger compartment filled with calcium chloride. We look at the concentrations at the middle of the bead. Initial concentration of LuxR before bead diffusion starts is a classical steady state a<sub>LuxR</sub>/d<sub>LuxR</sub>
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[[File:ETHZ Signaling molecules.png|600px|center]]
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[[File:ETHZ Signaling molecules.png|600px|center|thumb|'''Figure 4''' Signaling molecules]]
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[[File:ETHZ Regulators and GFP.png|600px|center]]
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[[File:ETHZ Regulators and GFP.png|600px|center|thumb|'''Figure 5''' Regulators and GFP]]
Extracellular AHL diffuses very fast through the membrane, which makes intracellular AHL increase, until both reach the same value and diffusion through the membrane reaches an equilibrium. From this point on AHL starts to bind to LuxR to form RLux, so AHL decreases, until RLux triggers enough production of LuxI which catalyses again AHL production, and makes AHL concentration become stable.
Extracellular AHL diffuses very fast through the membrane, which makes intracellular AHL increase, until both reach the same value and diffusion through the membrane reaches an equilibrium. From this point on AHL starts to bind to LuxR to form RLux, so AHL decreases, until RLux triggers enough production of LuxI which catalyses again AHL production, and makes AHL concentration become stable.
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After 30 minutes, almost all LuxR is bound to AHL and converted to RLux, so that we have  
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After 30 minutes, all LuxR is bound to AHL and converted to RLux, so that we have  
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$$ [LuxR]_i= 10nM$$
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$$ [LuxR]_i= 0nM$$
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$$ [RLux]_i= 160 nM$$
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$$ [RLux]_i= 200 nM$$
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$$ [AHLint]_i=[AHLint]_i=2 &mu; M$$
RLux has already started to trigger the production of LuxI and GFP, so that we have  
RLux has already started to trigger the production of LuxI and GFP, so that we have  
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$$ [mRNAGFP]_i =  24nM$$
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$$ [mRNAGFP]_i =  40nM$$
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$$ [GFP]_i =  180nM$$
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$$ [GFP]_i =  900nM$$
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$$ [mRNALuxI]_i = 24nM$$
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$$ [mRNALuxI]_i = 40nM$$
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$$ [LuxI]_i = 3300nM$$
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$$ [LuxI]_i = 16000nM$$
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<html><div id='Estimation'></html>
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<html><article id='Estimation'></html>
 
=== Estimation of parameters from literature ===
=== Estimation of parameters from literature ===
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The initial number of beads is 10 million. According to Lars Müller's master thesis<sup>[[Team:ETH_Zurich/project/references|[29]]]</sup>, in picoliter beads, cells doubling time is 30 minutes. Here we are using beads with a volume in the microliter range. Because of bead volume, oxygen and nutrients are much less accessible. Therefore, we multiplied this doubling time by 4. We have a rgowth rate of 0.006 min<sup>-1</sup> which is still above the growth rate in anaerobic conditions (0.004 min<sup>-1</sup> according to [http://bionumbers.hms.harvard.edu/search.aspx?log=y&task=searchbytrmorg&trm=growth+rate+e+coli&org= Bionumbers]) )
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The initial number of beads is 10 million. According to Lars Müller's master thesis<sup>[[Team:ETH_Zurich/project/references|[29]]]</sup>, in picoliter beads, cells doubling time is 30 minutes. Here we are using beads with a volume in the microliter range. Because of bead volume, oxygen and nutrients are much less accessible. Therefore, we multiplied this doubling time by 4. We have a growth rate of 0.006 min<sup>-1</sup> which is still above the growth rate in anaerobic conditions (0.004 min<sup>-1</sup> according to [http://bionumbers.hms.harvard.edu/search.aspx?log=y&task=searchbytrmorg&trm=growth+rate+e+coli&org= Bionumbers])  
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Lars Müller also mentions in his master thesis<sup>[[Team:ETH_Zurich/project/references|[29]]]</sup> that the maximum capacity of his 34 pL beads is 3000 cells, which would correspond to  maximum of N<sub>m</sub> = 8. 10<sup>8</sup> cells per bead in our case (10 &mu;L beads).  
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In the conditions of lack of oxygen and nutrients where our cells are, we consider that they might at maximum double or triple. So we take 8 . 10<sup>7</sup> cells per bead for N<sub>m</sub>.
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According to Kaplan's paper<sup>[[Team:ETH_Zurich/project/references|[28]]]</sup>, LuxAHL diffuses very fast through the membrane : ''"This report demonstrates that V. fischeri and E. coli are freely permeable to autoinducer. When autoinducer was added to cell suspensions, internal concentrations approximated external concentrations, and equilibration was rapid (within 20 s)."''
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According to Kaplan's paper<sup>[[Team:ETH_Zurich/project/references#Kaplan|[28]]]</sup>, LuxAHL diffuses very fast through the membrane : ''"This report demonstrates that V. fischeri and E. coli are freely permeable to autoinducer. When autoinducer was added to cell suspensions, internal concentrations approximated external concentrations, and equilibration was rapid (within 20 s)."''
Therefore we take 100 min<sup>-1</sup> for D<sub>m</sub>, and indeed external and internal concentrations become very quickly almost equal.
Therefore we take 100 min<sup>-1</sup> for D<sub>m</sub>, and indeed external and internal concentrations become very quickly almost equal.
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Cbeads
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According to Hannoun et al<sup>[[Team:ETH_Zurich/project/references#Hannoun|[30]]]</sup>,  diffusion coefficient of glucose in 2% calcium alginate (which is the material of our beads) is the same as in water. 3O-C6HSL and glucose have similar molecular weights (199.2 g/mol for 3O-C6HSL and 180.16 g/mol for glucose). Therefore we can consider that 3O-C6HSL will diffuse the same way and we took C<sub>bead</sub>=1.
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According to ... diffusion coefficient of glucose in alginate ... is almost the same as in water. 3O-C6HSL and glucose have similar molecular weights (199.2 g/mol for 3O-C6HSL and 180.16 g/mol for glucose). Therefore we can consider that 3O-C6HSL will diffuse the same way and we took C<sub>bead</sub>=1.
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<html></article></html>
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<html></div></html>
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The background in our experimental setup is very high and this is high we see actual fluorescence appear only after 11 hours. In order to account for this, we also set up a background in the simulated pattern by adjusting the scale.  
The background in our experimental setup is very high and this is high we see actual fluorescence appear only after 11 hours. In order to account for this, we also set up a background in the simulated pattern by adjusting the scale.  
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------------ Video  
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{|class="wikitable" style="background-color: white; text-align:center; width:auto; margin: auto; font-size:10pt;"
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|colspan="2" style='font-size:10pt';text-align:left|{{:Team:ETH_Zurich/Templates/Video|width=1080px|id=video3|ratio=1920/720|srcMP4=<html>https://static.igem.org/mediawiki/2014/b/b1/ETH_Zurich_2014_signal_propagation_with_simulation.mp4</html>|poster=<html>https://static.igem.org/mediawiki/2014/6/69/ETH_Zurich_2014_signal_propagation_with_simulation_preview.png</html>}}
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|colspan="2" style='font-size:10pt';text-align:left|'''Video 1''' '''Row wise, self-propagating [https://2014.igem.org/Team:ETH_Zurich/project/background/biotools#Quorum_Sensing cell-to-cell communication] of ''E. coli'' cells confined in [https://2014.igem.org/Team:ETH_Zurich/lab/bead alginate beads] (d=3 mm, initially 10<sup>7</sup> cells/bead) on a [https://2014.igem.org/Team:ETH_Zurich/lab/chip custom-made millifluidic PDMS chip].'''
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|style="width:50%"|All fluorescent cells contained [https://2014.igem.org/Team:ETH_Zurich/expresults/rr#Riboregulators riboregulated] sfGFP followed by [http://parts.igem.org/Part:BBa_C0161 LuxI (BBa_C0161)] together under the control of the [http://parts.igem.org/Part:BBa_R0062 pLux promoter (BBa_R0062)], and [http://parts.igem.org/Part:BBa_J23100 constitutively (BBa_J23100)] expressed [http://parts.igem.org/Part:BBa_C0062 LuxR (BBa_C0062)]. LuxI catalyzes the production of the autoinducer 3OC6-HSL, which is then diffusing from cell to cell. For initialization, the cells in one bead of the top row were induced with 3OC6-HSL before encapsulation. Imaging was implemented with a [https://2014.igem.org/Team:ETH_Zurich/lab/protocols#Biostep_Dark-Hood_DH-50.E2.84.A2__and_the_Argus-X1.E2.84.A2_software Biostep Dark-Hood DH-50 (Argus X1 software)] fitted with a Canon EOS 500D DSLR camera and a fluorescence filter (545 nm filter). Pictures were usually taken every 2 min at an excitation wavelength of 470 nm with the standard Canon EOS Utility software. Time-lapse movies were created with Adobe After Effects CC software. 1950x faster than real-time, the video shown starts 10 h after the initiation of the experiment (however the time scale shown corresponds to minutes after loading of the chip). For precise experimental setup, check the [https://2014.igem.org/Team:ETH_Zurich/expresults#Diffusion Results] page.
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||Simulation of the propagation of the pattern in the millifluidic chip. [http://www.comsol.com/comsol-multiphysics Comsol Multiphysics Simulation software] was used in order to simulate a detailed diffusion model including quorum sensing steps in colonies and cell growth. Overall GFP concentration in beads has been scaled in order to account for the high background of the experimental setup. Green Fluorescence Protein is produced earlier in the wells, but can be seen only above a certain threshold.<br>Accurate prediction of experimental data by the model has been achieved, with parameters from our own fittings or from the literature. Experimental observation combined with simulation enables to show that a pattern is able to develop in the millifluidic chip in a reasonable time scale.
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|}
=== Dynamics by row ===
=== Dynamics by row ===
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[[File:ETHZ AHL-regulators-pattern.png|center|900px|thumb|'''Figure 6''']]
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[[File:ETHZ GFP in beads.png|600px|center|thumb|'''Figure 7''' GFP in beads ]]
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[[File:ETHZ GFP in beads.png|600px|center]]
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[[File:ETHZ GFP in single cell.png|600px|center|thumb|'''Figure 8''' GFP in single cell]]
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[[File:ETHZ GFP in single cell.png|600px|center]]
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Latest revision as of 01:13, 18 October 2014

iGEM ETH Zurich 2014