Difference between revisions of "Team:Toronto/Analysis"
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<h2 class="text-yellow">Introduction</h2> | <h2 class="text-yellow">Introduction</h2> | ||
| − | <p>Using the previously derived expressions from | + | <p>Using the previously derived expressions from our ODEs, we use the Mathworks Simulink package to derive solutions to our system for a range of parameters. </p> |
<figure> | <figure> | ||
<div class="figures"> | <div class="figures"> | ||
| − | <div class="image"><img src="https:// | + | <div class="image"><img src="https://2017.igem.org/File:T--Toronto--2017_CI.png" alt="data"></div> |
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<h2 class="text-yellow">ODE Solution</h2> | <h2 class="text-yellow">ODE Solution</h2> | ||
<p>Solving: </p> | <p>Solving: </p> | ||
| − | + | \begin{eqnarray} | |
| − | + | \frac{x_2}{dt} = \alpha - \gamma x_2 \\ | |
| − | + | \frac{x_2}{dt} + \gamma x_2 = = \alpha | |
| − | + | \end{eqnarray} | |
| − | + | ||
<p>Integrating Factor: </p> | <p>Integrating Factor: </p> | ||
| − | + | \begin{eqnarray} | |
| − | + | e^(\int \gamma dt) = e^(\gamma t) | |
| − | + | \end{eqnarray} | |
| − | + | ||
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<p>Multiplying both sides by our integrating factor: </p> | <p>Multiplying both sides by our integrating factor: </p> | ||
| − | + | \begin{eqnarray} | |
| − | + | (\frac{x_2}{dt} + \gamma x_2)e^(\gamma t) = \alpha e^(\gamma t) \\ | |
| − | + | \int (\frac{x_2}{dt} + \gamma x_2)e^(\gamma t) = \int \alpha e^(\gamma t) \\ | |
| − | + | x_2 = \frac{\alpha}{\gamma} + ce^(-\gamma t) | |
| − | + | \end{eqnarray} | |
</div> | </div> | ||
<div id="subsection-Plots" class="subsection"> | <div id="subsection-Plots" class="subsection"> | ||
<h2 class="text-yellow">R plots</h2> | <h2 class="text-yellow">R plots</h2> | ||
| − | + | Visit our <a href="https://github.com/igemuoftATG/drylab-matlab">GitHub repository</a> for our code. | |
| − | < | + | <figure> |
| − | + | <div class="figures"> | |
| − | + | <div class="image"><img src="https://static.igem.org/mediawiki/2017/6/66/T--Toronto--2017_mcherr_reg_log.png" alt="data"></div> | |
| − | + | </div> | |
| − | + | <figcaption>Figure 1.1: Log Linear transformation of RFU/OD600 vs Time, Regression Line (red) fitted to data</figcaption> | |
| − | + | </figure> | |
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<figure> | <figure> | ||
<div class="figures"> | <div class="figures"> | ||
<div class="image"><img src="https://static.igem.org/mediawiki/2017/4/42/T--Toronto--2017_mcherry-reg-norm.png" alt="data"></div> | <div class="image"><img src="https://static.igem.org/mediawiki/2017/4/42/T--Toronto--2017_mcherry-reg-norm.png" alt="data"></div> | ||
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</div> | </div> | ||
| + | <figcaption>Figure 1.1: RFU/OD600 vs Time with Transfromed Regression Line (red)</figcaption> | ||
</figure> | </figure> | ||
</div> | </div> | ||
Revision as of 01:40, 16 December 2017
Analysis
Introduction
Using the previously derived expressions from our ODEs, we use the Mathworks Simulink package to derive solutions to our system for a range of parameters.
ODE Solution
Solving:
\begin{eqnarray} \frac{x_2}{dt} = \alpha - \gamma x_2 \\ \frac{x_2}{dt} + \gamma x_2 = = \alpha \end{eqnarray}Integrating Factor:
\begin{eqnarray} e^(\int \gamma dt) = e^(\gamma t) \end{eqnarray}Multiplying both sides by our integrating factor:
\begin{eqnarray} (\frac{x_2}{dt} + \gamma x_2)e^(\gamma t) = \alpha e^(\gamma t) \\ \int (\frac{x_2}{dt} + \gamma x_2)e^(\gamma t) = \int \alpha e^(\gamma t) \\ x_2 = \frac{\alpha}{\gamma} + ce^(-\gamma t) \end{eqnarray}R plots
Visit our GitHub repository for our code.
R Analysis
R Analysis:
Call:
lm(formula = log(x) ~ c(time, time, time))
Residuals:
Min 1Q Median 3Q Max
-0.58853 -0.15536 0.01303 0.19867 0.44055
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.87199 0.21773 13.19 1.47e-15 ***
c(time, time, time) 0.15267 0.01142 13.37 9.74e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.2935 on 37 degrees of freedom
Multiple R-squared: 0.8285, Adjusted R-squared: 0.8238
F-statistic: 178.7 on 1 and 37 DF, p-value: 9.741e-16
Intercept represents the equilibrium value of LacILov, our intercept:
