Difference between revisions of "Team:ETH Zurich/Model/Environment Sensing/parameter fitting"

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<p>The rest of the model remain unchanged, as it concerns intracellular phenomena that are assumed to happen the same way in vitro than in vivo.
 
<p>The rest of the model remain unchanged, as it concerns intracellular phenomena that are assumed to happen the same way in vitro than in vivo.
 
</p>
 
</p>
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<h2>What experiment do we need to fit the expression of LuxR?</h2>
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<h1>Fitting of LuxR expression</h1>
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<h2>Strategie</h2>
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<p>LuxR is the regulatory protein binding AHL and subsequently activating a promoter where LuxR-AHL tetramers can bind:</p>
 
<p>LuxR is the regulatory protein binding AHL and subsequently activating a promoter where LuxR-AHL tetramers can bind:</p>
  
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<p>The amount of LuxR present in the cell is crucial to know how much AHL is necessary to fully activate the promoter. And inversely, knowing the AHL dose response curve of the very simple system constituted of the constitutively luxR gene and a fluorescent reporter under the Lux promoter is enough to deduce the amount of LuxR in the bacteria, and therefore their expression level. This is why we have designed and performed <a href="https://2017.igem.org/Team:ETH_Zurich/Experiments/MRI_Contrast_Agent#AHL_dose_resp">an experiment to precisely measure a dose response curve</a> of such a system to be able to fit our luxR expression parameter</p>
 
<p>The amount of LuxR present in the cell is crucial to know how much AHL is necessary to fully activate the promoter. And inversely, knowing the AHL dose response curve of the very simple system constituted of the constitutively luxR gene and a fluorescent reporter under the Lux promoter is enough to deduce the amount of LuxR in the bacteria, and therefore their expression level. This is why we have designed and performed <a href="https://2017.igem.org/Team:ETH_Zurich/Experiments/MRI_Contrast_Agent#AHL_dose_resp">an experiment to precisely measure a dose response curve</a> of such a system to be able to fit our luxR expression parameter</p>
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<details>
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    <summary>In vitro model extension</summary>
 
<h2>Fitting method</h2>
 
<h2>Fitting method</h2>
 
<p>We used the the following process to fit our model to experimental data:</p>
 
<p>We used the the following process to fit our model to experimental data:</p>
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<p>Thanks to the MEIGO toolbox already used for our <a href="https://2017.igem.org/Team:ETH_Zurich/Model/Environment_Sensing/parameter_space">parameter search</a>, we do not have to implement this whole process <it>de novo</it> but we can focus on the cost function that will be used for the fitting. We built it taking the chi2 test of goodness of fit as a basis, to be able to tell with precision whether our fit is significant or not, obtaining on top of that the uncertainty of the fit of our parameters which arise from experimental variability.</p>
 
<p>Thanks to the MEIGO toolbox already used for our <a href="https://2017.igem.org/Team:ETH_Zurich/Model/Environment_Sensing/parameter_space">parameter search</a>, we do not have to implement this whole process <it>de novo</it> but we can focus on the cost function that will be used for the fitting. We built it taking the chi2 test of goodness of fit as a basis, to be able to tell with precision whether our fit is significant or not, obtaining on top of that the uncertainty of the fit of our parameters which arise from experimental variability.</p>
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     <summary>Cost function based on the chi2 test</summary>
 
     <summary>Cost function based on the chi2 test</summary>
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<p>When this cost is below 1, the fit is considered to be significant, taking into account the experimental uncertainty of the measures. Therefore, the set of parameters corresponding to the lowest obtained cost value will be the center value of the parameter, while the maximum and minimum possible values of the parameter will be the maximal and minimal ones satisfying the condition that the cost is below 1.</p>
 
<p>When this cost is below 1, the fit is considered to be significant, taking into account the experimental uncertainty of the measures. Therefore, the set of parameters corresponding to the lowest obtained cost value will be the center value of the parameter, while the maximum and minimum possible values of the parameter will be the maximal and minimal ones satisfying the condition that the cost is below 1.</p>
 
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<h1>Result of the fit from our experimental data</h1>
 
  
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     <summary>Assumptions</summary>
 
     <summary>Assumptions</summary>

Revision as of 16:35, 31 October 2017

Fit the parameters of our system from experiments

Goal

Once the initial guidelines for the genetic design given in a first step, we need to measure on our obtained biological system whether it behaves as intended. Therefore, we will submit it to carefully designed experiments, in order to fit our model on the obtained data and retrieve crucial parameters. The tricky part here is that in a test tube, our system will behave differently than in a tumor, as the repartition of bacteria is not the same and diffusion of AHL does not take place in the lab experiments. We therefore have to extend our model to take this discrepancy into account.

In vitro model extension

Compared to the real-life situation happening in the tumor, during our experiments bacteria are in a very different state: they are in bulk in the liquid culture and not in an open solid medium like they would be around the tumor. Therefore, the diffusion of AHL in the test tube does not take place, and some of our equations used before should be adapted to describe the behavior of our circuit when bacteria are in the situation of our lab experiments: only AHL production and degradation happen in the test tubes.

\[\begin{aligned} \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &= P_{\text{AHL}} - k_{\text{deg}} [\text{AHL}]\end{aligned}\]

Therefore, once reached the steady state, and thanks to the still valid equation describing AHL production\begin{aligned}\[P_{\text{AHL}} &= d_{\text{cell}} a_{\text{AHL}} [\text{luxI}]\end{aligned} the level of AHL is now:

\[\begin{aligned} \text{[AHL]} &= \frac{d_{\text{cell}} a_{\text{AHL}} [\text{luxI}]} {k_{\text{deg}}}\end{aligned}\]

The rest of the model remain unchanged, as it concerns intracellular phenomena that are assumed to happen the same way in vitro than in vivo.

What experiment do we need to fit the expression of LuxR?

LuxR is the regulatory protein binding AHL and subsequently activating a promoter where LuxR-AHL tetramers can bind:

LuxR-AHL binding

The amount of LuxR present in the cell is crucial to know how much AHL is necessary to fully activate the promoter. And inversely, knowing the AHL dose response curve of the very simple system constituted of the constitutively luxR gene and a fluorescent reporter under the Lux promoter is enough to deduce the amount of LuxR in the bacteria, and therefore their expression level. This is why we have designed and performed an experiment to precisely measure a dose response curve of such a system to be able to fit our luxR expression parameter

In vitro model extension

Fitting method

We used the the following process to fit our model to experimental data:

  1. From an initial set of parameters for our model, we simulate the response of the system that we should obtain
  2. We quantify the difference (via a cost function) between the model-based response and the actual experimental data
  3. We change the set of parameters in a way that is expected to reduce the cost (with gradient-based method) the and go back to step 1. and iterate

Thanks to the MEIGO toolbox already used for our parameter search, we do not have to implement this whole process de novo but we can focus on the cost function that will be used for the fitting. We built it taking the chi2 test of goodness of fit as a basis, to be able to tell with precision whether our fit is significant or not, obtaining on top of that the uncertainty of the fit of our parameters which arise from experimental variability.

Cost function based on the chi2 test

The most simple way to measure the distance between simulated model data and experimental data is the sum of square of the difference between each couple of point. However, the statistical meaning of this value is not obvious, and it is hard to say from it whether the fit is statistically significant or not. This is why we have implemented a more sophisticated cost function based on the chi2 test of goodness of fit, using the the chi2inv function provided by Matlab. This cost function takes into account the experimental uncertainty to evaluate whether a fit is significant or not. The cost function is then, with theo being the vector of simulated data, exp the vector of experimental data, and \sigma exp the vector of the standard deviation for each data point (computed from triplicates):

\[\begin{aligned} cost(theo, exp, \sigma exp) = \frac{\sum_i^N\frac{(exp_i-theo_i)^2}{\sigma exp_i^2}}{chi2inv(0.95,N)} \end{aligned}\]

When this cost is below 1, the fit is considered to be significant, taking into account the experimental uncertainty of the measures. Therefore, the set of parameters corresponding to the lowest obtained cost value will be the center value of the parameter, while the maximum and minimum possible values of the parameter will be the maximal and minimal ones satisfying the condition that the cost is below 1.

Assumptions

We assumed fixed the following parameters of our model, which are considered known well enough from previous characterizations:

Constant Description Value Reference
K_LuxRAHL LuxR-AHL quadrimer binding constant 5.10-10 nM-3 [2]
d_luxR LuxR degradation rate 0.023 min-1 [3]
K_luxR Half-activation LuxR-AHL concentration of the hybrid promoter 5 nM iGEM ETH 2013

From the experimental data of the AHL dose response curve, we could fit two parameters of our model: the expression level of LuxR, and the leakiness of our promoter:

LuxR-AHL binding
Figure 1. Parameter space fitting the experimental data. Each point represent a parameter vector that significantly fit the experimental data. The blue points fit the data the best (least sum of square) while the yellow ones represent parameters combinations that barely fit the data (but still significant according to the chi2 test of goodness of fit). Fitted parameters are annotated in red.

NEEDS A BIG HIGHLIGHT THERE

From our statistically significant fit, we can gather with a 95% confidence that:

  1. The expression level of luxR in our system is equal to 41 ± 7 nM.min-1
  2. The leakiness of the PLux promoter in our system is equal to 3.9 ± 1.5 %

References

  1. ^ Vander Heiden, Matthew G., Lewis C. Cantley, and Craig B. Thompson. “Understanding the Warburg Effect: The Metabolic Requirements of Cell Proliferation.” Science (New York, N.Y.) 324.5930 (2009): 1029–1033. PMC. Web. 18 Oct. 2017.
  2. A synthetic multicellular system for programmed pattern formation Subhayu Basu, Yoram Gerchman, Cynthia H. Collins, Frances H. Arnold & Ron WeissNature 434, 1130-1134 (28 April 2005) | doi:10.1038/nature03461
  3. A.B. Goryachev, D.J. Toh T.Lee, Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants Biosystems, Volume 83, Issues 2–3, February–March 2006, Pages 178-187