Team:UC Davis/Signal Processing

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  <h2>Mathematical Approach</h2>
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  <h2>Testing Our Model</h2>
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  <h2>Olive Oil</h2>
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    <span><h2>Olive Oil</h2></span>
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<p>Mathematical Approach</p>
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To model our system, we first focused our attention on the linear range of our enzyme's Michaelis Menten plot. The linear range of this plot is governed by the above relationship. This was useful for describing single aldehydes, but olive oil contains many aldehydes and the enzymatic response is different for each one.</p><br>
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Our mathematical model consists of a simple 3x3 array which we call the catalytic matrix. Using a few tricks from linear algebra, we created a way of predicting the concentrations in a three-enzyme biosensor. The main assumption of the model is that the substrates involved do not competiviely inhibit each other. <br><br></p>
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<p align="center"><img src="https://static.igem.org/mediawiki/2014/b/b5/RelativeVelocity.png" style="margin-left:auto;margin-right:auto;float:center;"/></p>
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<p><b>To read more about our mathematical approach, click <a href="https://2014.igem.org/Team:UC_Davis/Signal_Math" class="brightlink">here</a></b>.</p>
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<p>To describe this mathematically we started with the assumption that none of the substrates would induce competitive inhibition. If this was the case, the modeling would be simple. We would consider the observed velocity to be a linear combination of the three singular responses to aldehyde.</p><br>
 
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Now our model has three unknown concentrations, but only one equation: <b>this is why we need three enzymes</b>. Now we consider the entire model:<br>
 
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<p>Testing Our Model</p>
<p>Testing Our Model</p>
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To test our model we built a combinatorial set of aldehydes. We chose three representative aldehydes from each bin, saturated medium, saturated long, and unsaturated. The three aldehydes were chosen such that the enzyme's response to each would best represent their respective groups. We created a total of 64 different combinations by mixing Pentanal, Decanal, and <i>E</i>-2-Decenal in four different values, 0 µM, 10 µM, 100 µM, & 1000 µM. Three combinatorial well plates were made and mixed with each enzyme separately. The observed velocity is recorded in each well.</p><br>
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The catalytic matrix is inverted and multiplied by the observed velocity in each well and out comes our predicted concentrations!<br>
 
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In high concentrations, we found that aldehydes would come out of solution. We chose to focus on only the data set below 1000 µM. A more serious issue was brought to light however. The observed velocities from the combinatorial plates suggested competitive inhibition was occurring when <i>E</i>-2-Decenal was present in solution. This obfuscated our model considerably. Our primary assumption was that competitive inhibition would not come into play. We needed to think more abstractly. We asked a simple question: If measured catalytic values would not work in our suggested model, was there still a set of numbers that would work?<br><br>
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To test our model, we built a combinatorial set of aldehydes, and compared our predicted concentrations with known values. The results suggested we needed to take a new approach. We taught our computer to solve the problem, and it worked. By randomizing the values in the catalytic matrix, we found that there was a vector space that could model our competitively inhibited system.</p><br><br>
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Instead of using the measured K<sub>cat</sub>/K<sub>m</sub>, we randomized the catalytic matrix and tested the variants for prediction accuracy. One million variants later and we were starting to produce consistently better predictions.</p><br>
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<p><b>To read more about our mathematical approach, click <a href="https://2014.igem.org/Team:UC_Davis/Signal_Test" class="brightlink">here</a></b>.</p>
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<p>With our success we redesigned the experiment with different concentrations. This time we would vary aldehyde in the following concentrations: 0 µM, 12.5 µM, 25 µM, & 50 µM. We would also train on an incomplete set and test the best matrices on the full set. This mimics a concept from protein crystallography, R-Free, where several atoms are removed from a model and later used to test the models accuracy. <br><br> 
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<p>Olive Oil</p>
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The results were astonishing! With our best mutant matrix, We were able to predict aldehyde concentration with an average error of only 6.25 µM!
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<p>With a working model, it was time for the ultimate test: Olive Oil<br><br>
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Nine samples of Extra Virgin Olive Oil were obtained and <a href="https://2014.igem.org/Team:UC_Davis/Protein_Engineering_Test" class="brightlink">prepared</a> for assay. The velocities were recorded with each enzyme for a total of 27 data points. We used the best catalytic matrix from our previous model and again inverted the matrix and multiplied by the observed velocity. The results are plotted below.</p><br>
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<p align="center"><img src="https://static.igem.org/mediawiki/2014/1/19/OliVEoil_Set.jpg" style="margin-left:auto;margin-right:auto;border:1.5px solid #212f20;"/></p><br>
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<p><b>To read more about how our model did when testing olive oil, click <a href="https://2014.igem.org/Team:UC_Davis/Signal_oil" class="brightlink">here</a></b>.</p>
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Latest revision as of 03:49, 18 October 2014

UC Davis iGEM 2014

Mathematical Approach

Mathematical Approach

Testing Our Model

Testing Our Model

Olive Oil

Olive Oil

Our signal processing data set can be downloaded here.

Mathematical Approach

Our mathematical model consists of a simple 3x3 array which we call the catalytic matrix. Using a few tricks from linear algebra, we created a way of predicting the concentrations in a three-enzyme biosensor. The main assumption of the model is that the substrates involved do not competiviely inhibit each other.

To read more about our mathematical approach, click here.

Testing Our Model


To test our model, we built a combinatorial set of aldehydes, and compared our predicted concentrations with known values. The results suggested we needed to take a new approach. We taught our computer to solve the problem, and it worked. By randomizing the values in the catalytic matrix, we found that there was a vector space that could model our competitively inhibited system.



To read more about our mathematical approach, click here.

Olive Oil

With a working model, it was time for the ultimate test: Olive Oil

Nine samples of Extra Virgin Olive Oil were obtained and prepared for assay. The velocities were recorded with each enzyme for a total of 27 data points. We used the best catalytic matrix from our previous model and again inverted the matrix and multiplied by the observed velocity. The results are plotted below.



To read more about how our model did when testing olive oil, click here.

Retrieved from "http://2014.igem.org/Team:UC_Davis/Signal_Processing"