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Team:UC Davis/Signal Processing
From 2014.igem.org
Mathematical Approach
Mathematical Approach
Testing Our Model
Testing Our Model
Machine Learning
Machine Learning
Our signal processing data set can be downloaded here.
Mathematical Approach
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.
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.
Now our model has three unknown concentrations, but only one equation: this is why we need three enzymes. Now we consider the entire model:
Testing Our Model
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 E-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.
The catalytic matrix is inverted and multiplied by the observed velocity in each well and out comes our predicted concentrations!
In high concentrations, we found that aldehydes would crash 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 E-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?
Machine Learning
Instead of using the measured Kcat/Km, 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.
We found that with decreasing aldehyde concentrations, our model accuracy would increase. This is most likely due to the inherent non-linearity of michaelis Menten kinetics 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. 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!
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:
