Team:Valencia Biocampus/modeling1

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== '''Modeling Bacteria''' ==
== '''Modeling Bacteria''' ==
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Here we modeled several of the Bacteria constructions.
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Let´s get a closer look to the modeling of our Bacteria constructions.
 +
 
 +
 
 +
==='''Are you hungry?'''===
 +
 
 +
The first model we did addressed our glucose-sensitive construction. We carried out three different experiments changing the carbon source, to see its influence.  First we used glucose (1) as the only carbon source. Then we added galactose (2) as an extra carbon source. Finally we put sodium acetate (3) as an extra carbon source instead of galactose.
 +
The use of alternatives to glucose is explained because  in previous tests we observed that low concentrations of glucose compromised cell growth.
 +
With the experimental data we fitted the following repressible Hill-like function.
 +
 
 +
[[Image:ecu_glu.png|300px|center]]
 +
 
 +
And we estimated the following parameters for the 3 different conditions.
 +
 
 +
[[Image:table_glu.png|600px|center]]
 +
 
 +
In figures 1, 2 and 3 we plot the models and experimental data for the three experiments with glucose (1), added galactose (2) and added sodium acetate (3) respectively. IPTG was added to all the experiments to induce expression.
 +
 
 +
<table border="0.01" style="background-color:transparent" align="center">
 +
    <tr>
 +
        <td width ="600" align="center">[[Image:figura_IPTG.png|center|600px]]</td>
 +
</tr>
 +
<tr>
 +
<td width ="600" align="justify"><b>Figure 1.</b><br>Model and experimental data for different glucose concentrations.</td></tr>
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</table>
 +
<table border="0.01" style="background-color:transparent" align="center">
 +
    <tr>
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        <td width ="600" align="center">[[Image:figura_GAL.png|center|600px]]</td>
 +
</tr>
 +
<tr>
 +
<td width ="600" align="justify"><b>      Figure 2.</b><br>        Model and experimental data for different glucose concentrations with 3 g/l of Galactose added to the medium.</td></tr>
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</table>
 +
 
 +
<table border="0.01" style="background-color:transparent" align="center">
 +
    <tr>
 +
        <td width ="600" align="center">[[Image:figura_ACE.png|center|600px]]</td>
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</tr>
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<tr>
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<td width ="600" align="justify"><b>Figure 3.</b><br>  Model and experimental data for different glucose concentrations with 3 g/l of Sodium Acetatee added to the medium.</td></tr>
 +
</table>
 +
 
 +
It is interesting to notice how the Hill coefficient increases  when adding Galatose and Acetate. From here we can hypothesize that having another source of carbon  makes the promoter more sensitive to changes in the glucose concentration, as the slope of the Hill function gets larger. Also the constant K decreases, which tell us that less glucose is necessary to de-activate the promoter when there is another carbon source.
 +
 
 +
Compare this results with [[Team:Valencia_Biocampus/Results1#Are you hungry?| Are you hungry? results]].
=== '''Are you hot?''' ===
=== '''Are you hot?''' ===
-
For the case of the Heat Shock promoter, we decided to model the static transfer function of the Fluorescent Intensity as a result of the Temperature with the following equation:
+
When analyzing the Heat Shock promoter, we decided to model the static transfer function of the Fluorescent Intensity as a result of the Temperature following the Hill-like equation:
[[Image:Ecu1.png|center]]
[[Image:Ecu1.png|center]]
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In order to estimate the parameters of this model we performed a heat shock experiment.
In order to estimate the parameters of this model we performed a heat shock experiment.
-
Analysis corresponding to a heat shock applied during 10 minutes at different temperatures ranging from 20ºC up to 51ºC. For each temperature three samples were tested. Fluorescence Intensity (FI) and optical density (OD) were measured at t1=60 minutes and t2=120 minutes after application of the heat pulse had finished.  
+
Our first analysis corresponds to a heat shock applied during 10 minutes at different temperatures, ranging from 20ºC to 51ºC. For each temperature three samples were tested. Fluorescence Intensity (FI) and optical density (OD) were measured at t<sub>1</sub>=60 min and t<sub>2</sub>=120 min after the application of the heat pulse had finished.  
   
   
-
The first practical problem we encountered was the lack of enough heating coats. Because of these we took the culture, and we splitted the experiments in two batches. FI and OD were measured at t1=60, and t2=120 for batch number 1, while for batch number 2 the heat shock was delayed by 40 minutes with respect to the first batch. Therefore, measu-rements for batch no.2 were taken at t1=100, and t2=160 minutes respectively.
+
The first practical problem we encountered was the lack of enough heating coats. To try to solve it we took the culture and we split the experiments in two batches. FI and OD were measured at t<sub>1</sub>=60 min, and t<sub>2</sub>=120 min for batch number 1, while for batch number 2 the heat shock was delayed 40 min with respect to the first batch. Therefore, measurements for batch no.2 were taken at t<sub>1</sub>=100 min, and t<sub>2</sub>=160 min respectively.
-
During this 40 minutes delay, the microorganism was growing. Therefore, the heat shock in batch no. 2 was applied to a larger population than in the case of batch no.1. Consequently, the measurements of FI and DO gave much higher values. This affects the ratio FI/DO, were the two batches can be clearly distinguished (see figure 1).  
+
During the 40 min delay, the microorganism was growing. Therefore, the heat shock in batch no. 2 was applied to a larger population than in the case of batch no.1. Consequently, the measurements of FI and DO gave much higher values. This fact affects the ratio FI/DO, were the two batches can be clearly distinguished (see figure 4).  
<table border="0.01" style="background-color:transparent" align="center">
<table border="0.01" style="background-color:transparent" align="center">
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</tr>
</tr>
<tr>
<tr>
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<td width ="600" align="justify"><b>Figure 1.</b><br> Raw data from the heat shock experiment.</td></tr>
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<td width ="600" align="justify"><b>Figure 4.</b><br> Raw data from the heat shock experiment.</td></tr>
</table>
</table>
-
Our first modeling work was to perform data reconciliation. To this end,  we assume that during this 40 minutes delay the only thing that happended was that cells grew. Growth was approximately linear in the range analyzed, as seen in figure 1 (blue data). We fitted a simple model DO=p1+p2*t. The result can be seen in figure 2, were:
+
Our first modeling work was aimed to obtain data reconciliation. To this point,  we assumed that, during this 40 min delay, the only thing that happened was that cells grew. Growth was approximately linear in the range analyzed, as seen in figure 1 (blue data). We fitted a simple model DO=p1+p2*t. The result can be seen in figure 5, were:
p1 = 0.161573
p1 = 0.161573
p2 = 0.000866314
p2 = 0.000866314
-
Therefore, the ratio FI/DO above can be compensated by multiplying data by the frac-tions (p1+p2*t100)/(p1+p2*t), and (p1+p2*t160)/(p1+p2*t). Actually, what we did was to compensate all measurements to the same time base. The reconcialited result can be seen in figure 3.
+
Therefore, the above ratio FI/DO can be compensated by multiplying data by the fractions (p1+p2*t100)/(p1+p2*t), and (p1+p2*t160)/(p1+p2*t). Actually, what we did was to compensate all measurements to the same time base. The reconciled results can be seen in figure 6.
<table border="0.01" style="background-color:transparent" align="center">
<table border="0.01" style="background-color:transparent" align="center">
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</tr>
</tr>
<tr>
<tr>
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<td width ="600" align="justify"><b>Figure 2.</b><br> Model fitting for OD w.r.t. time.</td></tr>
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<td width ="600" align="justify"><b>Figure 5.</b><br>Model fitting for OD w.r.t. time.</td></tr>
</table>
</table>
<br>
<br>
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</tr>
</tr>
<tr>
<tr>
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<td width ="600" align="justify"><b>Figure 3.</b><br> Recoinciliated data for t=60min and t=120mins. </td></tr>
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<td width ="600" align="justify"><b>Figure 6.</b><br> Recoinciled data for t=60min and t=120min. </td></tr>
</table>
</table>
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Once we had all data reconciliated we fitted it to a static model relating the ratio be-ween fluorescence intensity (FI), and the culture optical density (OD) with the temperature applied during the heat shock.
+
Once we had all data reconciled, we fitted them to a static model relating the ratio between fluorescence intensity (FI) and the culture optical density (OD) with the temperature applied during the heat shock.
-
Since the model is nonlinear with respect to parameters K<sub>1</sub>, and n, we applied a global op-timization algorithm based on stochastic search. Actually we applied it to fit the data at t=60 minutes, and t=120 minutes independently. The estimated parameters were:
+
Since the model is nonlinear with respect to parameters K<sub>1</sub> and n, we applied a global optimization algorithm based on stochastic search. Actually we applied it to fit the data at t=60 min, and t=120 min independently. The estimated parameters were:
[[Image:Tabla_model_hs.png|500px|center]]
[[Image:Tabla_model_hs.png|500px|center]]
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The results can be seen in figures 4 (t60), and 5 (t120) respectively.
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The results can be seen in figures 7 (t<sub>60</sub>), and 8 (t<sub>120</sub>) respectively.
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</tr>
</tr>
<tr>
<tr>
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<td width ="600" align="justify"><b>Figure 4.</b><br> Model and experimental data for t=60min. </td></tr>
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<td width ="600" align="justify"><b>Figure 7.</b><br> Model and experimental data for t=60min. </td></tr>
</table>
</table>
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</tr>
</tr>
<tr>
<tr>
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<td width ="600" align="justify"><b>Figure 5.</b><br> Model and experimental data for t=120min. </td></tr>
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<td width ="600" align="justify"><b>Figure 8.</b><br>Model and experimental data for t=120min.</td></tr>
 +
</table>
 +
 
 +
Compare this results with [[Team:Valencia_Biocampus/Results1#Are you hot?| Are you hot? results]].
 +
 
 +
 
 +
One  question of interest is whether  the increment observed in the ratio IF/OD is caused by an incement of fluorescence during the time elapsed, or it is caused by a varying dilution effect due to a decrease in the specific groth rate during that period [1].
 +
 
 +
 
 +
To answer this question notice that the absolute growth rate can be obtained from the regression model OD(t)= p1+p2*t obtained above. Notice t is expressed in minutes in our model. Indeed, the specific growth rate at time instant t can be roughly estimated as mu(t)= [dOD(t)/dt]/OD(t). This gives mu(t) = p2*60/(p1+p2*t) if mu(t) expressed in h^(-1).
 +
 
 +
 
 +
Therefore, we took our models and weighted them by mu(t60) and mu(t120) respectively. The result can be seen in figure 6. Indeed, the increment observed in fluorescence from t=60 to t=120 minutes could have been caused by a dillution effect. Of course, the main point to keep is that the ratio IF/OD (or, alternatively IF/OD*mu(t)) increases with temperature. So...our construction works!
 +
 
 +
<table border="0.01" style="background-color:transparent" align="center">
 +
    <tr>
 +
        <td width ="600" align="center">[[Image:T60t120mu_jesust.png|center|660px]]</td>
 +
</tr>
 +
<tr>
 +
<td width ="600" align="justify"><b>Figure 9.</b><br> Model IF/OD*mu(t) w.r.t. temperature for t=60min, and t=120min.</td></tr>
</table>
</table>
-
Compare this results with [[Team:Valencia_Biocampus/Results1#Are you hot?| Are you hot? results]]
+
[1] S. Klumpp, Z. Zhang, T. Hwa, Growth Rate-Dependent Global Effects on Gene Expression in Bacteria, Cell 139, 1366-1375, 2009
== '''Modeling Yeast''' ==
== '''Modeling Yeast''' ==
-
For our yeast construct: ''How long have you been fermenting?'' we propose a phenomenological model to explain the relationship between glucose concentration and fluorescent protein production.
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For our yeast construct: ''How long have you been fermenting?'' We propose a phenomenological model to explain the relationship between glucose concentration and fluorescent protein production.
The following model was assumed:
The following model was assumed:
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<html>
<html>
<ul>
<ul>
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<il>x<sub>1</sub> = concentration of yeast<br>
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<il>x<sub>1</sub> = yeast concentration<br>
-
<il>x<sub>2</sub> = concentration of glucose<br>
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<il>x<sub>2</sub> = glucose concentration<br>
<il>x<sub>3</sub> = fluorescent protein<br>
<il>x<sub>3</sub> = fluorescent protein<br>
<il>γ<sub>3</sub> = degradation/dilution rate for the fluorescent protein<br>
<il>γ<sub>3</sub> = degradation/dilution rate for the fluorescent protein<br>
<il>k<sub>0</sub> = basal production of fluorescent protein<br>
<il>k<sub>0</sub> = basal production of fluorescent protein<br>
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<il>k<sub>p</sub><sup>n</sup> = Hill-like coefficient. Notice the exponent n that also appears associated to x<sub>2</sub>.<br>
+
<il>k<sub>p</sub><sup>n</sup> = Hill-like coefficient. Notice that the exponent n also appears associated to x<sub>2</sub>.<br>
</ul>
</ul>
</html>
</html>

Latest revision as of 23:37, 26 September 2012

Contents

Modeling Bacteria

Let´s get a closer look to the modeling of our Bacteria constructions.


Are you hungry?

The first model we did addressed our glucose-sensitive construction. We carried out three different experiments changing the carbon source, to see its influence. First we used glucose (1) as the only carbon source. Then we added galactose (2) as an extra carbon source. Finally we put sodium acetate (3) as an extra carbon source instead of galactose. The use of alternatives to glucose is explained because in previous tests we observed that low concentrations of glucose compromised cell growth. With the experimental data we fitted the following repressible Hill-like function.

Ecu glu.png

And we estimated the following parameters for the 3 different conditions.

Table glu.png

In figures 1, 2 and 3 we plot the models and experimental data for the three experiments with glucose (1), added galactose (2) and added sodium acetate (3) respectively. IPTG was added to all the experiments to induce expression.

Figura IPTG.png
Figure 1.
Model and experimental data for different glucose concentrations.
Figura GAL.png
Figure 2.
Model and experimental data for different glucose concentrations with 3 g/l of Galactose added to the medium.
Figura ACE.png
Figure 3.
Model and experimental data for different glucose concentrations with 3 g/l of Sodium Acetatee added to the medium.

It is interesting to notice how the Hill coefficient increases when adding Galatose and Acetate. From here we can hypothesize that having another source of carbon makes the promoter more sensitive to changes in the glucose concentration, as the slope of the Hill function gets larger. Also the constant K decreases, which tell us that less glucose is necessary to de-activate the promoter when there is another carbon source.

Compare this results with Are you hungry? results.

Are you hot?

When analyzing the Heat Shock promoter, we decided to model the static transfer function of the Fluorescent Intensity as a result of the Temperature following the Hill-like equation:

Ecu1.png

In order to estimate the parameters of this model we performed a heat shock experiment.

Our first analysis corresponds to a heat shock applied during 10 minutes at different temperatures, ranging from 20ºC to 51ºC. For each temperature three samples were tested. Fluorescence Intensity (FI) and optical density (OD) were measured at t1=60 min and t2=120 min after the application of the heat pulse had finished.


The first practical problem we encountered was the lack of enough heating coats. To try to solve it we took the culture and we split the experiments in two batches. FI and OD were measured at t1=60 min, and t2=120 min for batch number 1, while for batch number 2 the heat shock was delayed 40 min with respect to the first batch. Therefore, measurements for batch no.2 were taken at t1=100 min, and t2=160 min respectively.

During the 40 min delay, the microorganism was growing. Therefore, the heat shock in batch no. 2 was applied to a larger population than in the case of batch no.1. Consequently, the measurements of FI and DO gave much higher values. This fact affects the ratio FI/DO, were the two batches can be clearly distinguished (see figure 4).

HSrawData.png
Figure 4.
Raw data from the heat shock experiment.

Our first modeling work was aimed to obtain data reconciliation. To this point, we assumed that, during this 40 min delay, the only thing that happened was that cells grew. Growth was approximately linear in the range analyzed, as seen in figure 1 (blue data). We fitted a simple model DO=p1+p2*t. The result can be seen in figure 5, were: p1 = 0.161573 p2 = 0.000866314 Therefore, the above ratio FI/DO can be compensated by multiplying data by the fractions (p1+p2*t100)/(p1+p2*t), and (p1+p2*t160)/(p1+p2*t). Actually, what we did was to compensate all measurements to the same time base. The reconciled results can be seen in figure 6.

Hs do.png
Figure 5.
Model fitting for OD w.r.t. time.


HSreconcilData.png
Figure 6.
Recoinciled data for t=60min and t=120min.


Once we had all data reconciled, we fitted them to a static model relating the ratio between fluorescence intensity (FI) and the culture optical density (OD) with the temperature applied during the heat shock.

Since the model is nonlinear with respect to parameters K1 and n, we applied a global optimization algorithm based on stochastic search. Actually we applied it to fit the data at t=60 min, and t=120 min independently. The estimated parameters were:

Tabla model hs.png

The results can be seen in figures 7 (t60), and 8 (t120) respectively.


ModelHSt60.png
Figure 7.
Model and experimental data for t=60min.


ModelHSt120.png
Figure 8.
Model and experimental data for t=120min.

Compare this results with Are you hot? results.


One question of interest is whether the increment observed in the ratio IF/OD is caused by an incement of fluorescence during the time elapsed, or it is caused by a varying dilution effect due to a decrease in the specific groth rate during that period [1].


To answer this question notice that the absolute growth rate can be obtained from the regression model OD(t)= p1+p2*t obtained above. Notice t is expressed in minutes in our model. Indeed, the specific growth rate at time instant t can be roughly estimated as mu(t)= [dOD(t)/dt]/OD(t). This gives mu(t) = p2*60/(p1+p2*t) if mu(t) expressed in h^(-1).


Therefore, we took our models and weighted them by mu(t60) and mu(t120) respectively. The result can be seen in figure 6. Indeed, the increment observed in fluorescence from t=60 to t=120 minutes could have been caused by a dillution effect. Of course, the main point to keep is that the ratio IF/OD (or, alternatively IF/OD*mu(t)) increases with temperature. So...our construction works!

T60t120mu jesust.png
Figure 9.
Model IF/OD*mu(t) w.r.t. temperature for t=60min, and t=120min.

[1] S. Klumpp, Z. Zhang, T. Hwa, Growth Rate-Dependent Global Effects on Gene Expression in Bacteria, Cell 139, 1366-1375, 2009

Modeling Yeast

For our yeast construct: How long have you been fermenting? We propose a phenomenological model to explain the relationship between glucose concentration and fluorescent protein production. The following model was assumed:

Ecu yeast.png

where

    x1 = yeast concentration
    x2 = glucose concentration
    x3 = fluorescent protein
    γ3 = degradation/dilution rate for the fluorescent protein
    k0 = basal production of fluorescent protein
    kpn = Hill-like coefficient. Notice that the exponent n also appears associated to x2.

The graph below shows a simulated theoretical time evolution for the three variables.

Yeast evolution.png

As observed in the figure, once the glucose is depleted, the concentration of fluorescent protein increases from a basal value. Compare this results with Yeast fluorescence results