View source for Team:ETH Zurich/Model/Environment Sensing/parameter fitting

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<h1 class="headline">Fit the parameters of our system from experiments</h1>


<section class="first goal">
    <h1>Goal</h1>
    <p>Once the initial guidelines for the genetic design given <a href="https://2017.igem.org/Team:ETH_Zurich/Model/Environment_Sensing/parameter_space">in a first step</a>, 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.</p>
</section>


<div class="multi-summary">
<details>
    <summary>In vitro model extension</summary>
<section id="sec1">
<p>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.</p>
<span class="math">\[\begin{aligned}
        \frac{\mathrm{d} [\text{AHL}]}{\mathrm{d} t} &amp;= P_{\text{AHL}} - k_{\text{deg}} [\text{AHL}]\end{aligned}\]</span>
<p>Therefore, once reached the steady state, and thanks to the still valid equation describing AHL production<span class="math">\begin{aligned}\[P_{\text{AHL}} &amp;= d_{\text{cell}} a_{\text{AHL}} [\text{luxI}]\end{aligned}</span> the level of AHL is now:</p>
<span class="math">\[\begin{aligned}
        \text{[AHL]} &amp;= \frac{d_{\text{cell}} a_{\text{AHL}} [\text{luxI}]} {k_{\text{deg}}}\end{aligned}\]</span>
<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>
</section>
</details>

</div>
<section id="sec2">
<h1>Fitting of LuxR expression</h1>
<h2>Strategie</h2>
<p>LuxR is the regulatory protein binding AHL and subsequently activating a promoter where LuxR-AHL tetramers can bind:</p>

<figure class="fig-nonfloat" style="width: 400px;">
        <img alt="LuxR-AHL binding"
        src="https://static.igem.org/mediawiki/2017/1/10/T--ETH_Zurich--tumor_sensing_LuxRAHL_binding.png"/>
    </figure>

<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>
<h2>Fitting method</h2>
<p>We used the the following process to fit our model to experimental data:</p>
<ol>
     <li>From an initial set of parameters for our model, we simulate the response of the system that we should obtain</li>
     <li>We quantify the difference (via a cost function) between the model-based response and the actual experimental data</li>
     <li>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</li>
</ol>
<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>

<div class="multi-summary">
<details>
    <summary>Cost function based on the chi2 test</summary>
<p>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 <a href="https://en.wikipedia.org/wiki/Chi-squared_test">chi2 test</a> of goodness of fit, using the the <a href="https://ch.mathworks.com/help/stats/chi2inv.html">chi2inv function</a> 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 <span class="math">theo</span> being the vector of simulated data, <span class="math">exp</span> the vector of experimental data, and <span class="math">\sigma exp</span> the vector of the standard deviation for each data point (computed from triplicates):
</p>
<p><span class="math">\[\begin{aligned}
       cost(theo, exp, \sigma exp) = \frac{\sum_i^N\frac{(exp_i-theo_i)^2}{\sigma exp_i^2}}{chi2inv(0.95,N)}
       \end{aligned}\]</span>
    </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>
</details>
</section>
<section>
<h1>Result of the fit from our experimental data</h1>

<div class="multi-summary">
<details>
    <summary>Assumptions</summary>
    <p>We assumed fixed the following parameters of our model, which are considered known well enough from previous characterizations:</p>
    <table>
        <tr>
            <th>Constant</th>
            <th>Description</th>
            <th>Value</th>
            <th>Reference</th>
        </tr>
        <tr>
            <td>K_LuxRAHL</td>
            <td>LuxR-AHL quadrimer binding constant</td>
            <td>5.10<sup>-10</sup> nM<sup>-3</sup></td>
            <td><a href="#bib2" class="forward-ref">[2]</a></td>
        </tr>
        <tr>
            <td>d_luxR</td>
            <td>LuxR degradation rate</td>
            <td>0.023 min<sup>-1</sup></td>
            <td><a href="#bib3" class="forward-ref">[3]</a></td>
        </tr>
        <tr>
            <td>K_luxR</td>
            <td>Half-activation LuxR-AHL concentration of the hybrid promoter</td>
            <td>5 nM</td>
            <td><a href="https://2013.igem.org/Team:ETH_Zurich/Parameter">iGEM ETH 2013</a></td>
        </tr>
       
    </table>

</details>
<p>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:</p>

<figure class="fig-nonfloat" style="maxwidth: 800px;">
        <img alt="LuxR-AHL binding"
        src="https://static.igem.org/mediawiki/2017/9/9f/T--ETH_Zurich--luxR_fit_annotated.png"/>
    <figcaption>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.</figcaption>
    </figure>

<h1>NEEDS A BIG HIGHLIGHT THERE</h1>
<p>From our statistically significant fit, we can gather with a 95% confidence that:
<ol>
      <li>The expression level of luxR in our system is equal to 41 &#8723; 7 nM.min<sup>-1</sup></li>
      <li>The leakiness of the P<sub>Lux</sub> in our system is equal to 3.9 &#8723; 7 nM.min<sup>-1</sup></li>
</ol>
</section>

<section class="references">
    <h1>References</h1>
    <ol>
        <li id="bib1"><a href="#ref1">^ </a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2849637/">Vander Heiden, Matthew G., Lewis C. Cantley, and Craig B. Thompson. “Understanding the Warburg Effect: The Metabolic Requirements of Cell Proliferation.” <cite>Science</cite> (New York, N.Y.) 324.5930 (2009): 1029–1033. PMC. Web. 18 Oct. 2017.</a></li>
        <li id="bib2"><a href="http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html">A synthetic multicellular system for programmed pattern formation Subhayu Basu, Yoram Gerchman, Cynthia H. Collins, Frances H. Arnold & Ron Weiss<source>Nature</source> 434, 1130-1134 (28 April 2005) | doi:10.1038/nature03461</a></li>
        <li id="bib3"><a href="http://www.sciencedirect.com/science/article/pii/S0303264705001267">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 <source>Biosystems</source>, Volume 83, Issues 2–3, February–March 2006, Pages 178-187</a></li>
    </ol>
</section>
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