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Team:Cornell/project/drylab/modeling/time response
From 2012.igem.org
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<h5>Concentration of Analyte In Reactor</h5> | <h5>Concentration of Analyte In Reactor</h5> | ||
| - | Now that we have an analytical solution for the concentration of analyte in the reactor over time, we wish to model the current response given the input function $c(t)$. | + | Now that we have an analytical solution for the concentration of analyte in the reactor over time, we wish to model the current response given the input function $c(t)$. To accomplish this, we modeled the time rate of change of current of our arsenic sensor using a Hill function with a cooperativity coefficient of unity—fitting data presented in Fig. X of our <a href="https://2012.igem.org/Team:Cornell/project/wetlab/results/reactors">Current Response </a>characterization page—and lumping all transcriptional, translational, and post-translational processes: |
| - | $$\frac{dI}{dt} = \frac{\beta_1 \cdot c(t)}{c_{1/2}+c(t)}+\beta_0 - \alpha | + | $$\frac{dI}{dt} = \frac{\beta_1 \cdot c(t)}{c_{1/2}+c(t)}+\beta_0 - \alpha I, $$ |
| - | where $c(t)$ is as defined above. | + | where $c(t)$ is as defined above; $\beta_1$ is 1.4 $\mu$A/day, $\beta_0$ is 2.8 $\mu$A/day |
Revision as of 02:05, 26 October 2012
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Dry Lab
- Overview
- Specifications
- Design
- Modeling
Time Response
Motivation
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam sed nisl quis tellus convallis sagittis. Sed blandit metus at nulla mollis luctus. In eget turpis eros, eget lacinia sem. Curabitur ornare mauris nec lectus convallis vel ornare ligula viverra. Sed vel lectus mattis nisi auctor vestibulum vel nec ipsum. Integer pellentesque dolor lobortis elit viverra vitae vehicula dui cursus. Donec viverra, lectus eu faucibus rutrum, leo risus posuere justo, at rhoncus felis est in nibh. Pellentesque feugiat porta quam nec molestie. Morbi nunc dolor, consectetur in tempus in, hendrerit a augue. Phasellus ultrices volutpat diam vitae tincidunt. Mauris justo leo, blandit et tristique eu, lacinia in risus. Mauris elit eros, sollicitudin quis sodales quis, placerat a ante.Nam ac pulvinar felis. Mauris vitae erat at orci semper aliquet vitae quis urna. Donec sit amet tortor porttitor diam bibendum viverra. Nam dui nulla, viverra sed lacinia lobortis, ullamcorper et neque. Etiam rhoncus nibh a lacus varius vehicula convallis mi rutrum. Vestibulum vel nunc sit amet ipsum feugiat consectetur. Nulla nec ante vitae dui tristique accumsan. Morbi felis est, ornare a vestibulum vel, vulputate a eros. Nulla facilisi.
Problem Setup
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam sed nisl quis tellus convallis sagittis. Sed blandit metus at nulla mollis luctus. In eget turpis eros, eget lacinia sem. Curabitur ornare mauris nec lectus convallis vel ornare ligula viverra. Sed vel lectus mattis nisi auctor vestibulum vel nec ipsum. Integer pellentesque dolor lobortis elit viverra vitae vehicula dui cursus. Donec viverra, lectus eu faucibus rutrum, leo risus posuere justo, at rhoncus felis est in nibh. Pellentesque feugiat porta quam nec molestie. Morbi nunc dolor, consectetur in tempus in, hendrerit a augue. Phasellus ultrices volutpat diam vitae tincidunt. Mauris justo leo, blandit et tristique eu, lacinia in risus. Mauris elit eros, sollicitudin quis sodales quis, placerat a ante.Nam ac pulvinar felis. Mauris vitae erat at orci semper aliquet vitae quis urna. Donec sit amet tortor porttitor diam bibendum viverra. Nam dui nulla, viverra sed lacinia lobortis, ullamcorper et neque. Etiam rhoncus nibh a lacus varius vehicula convallis mi rutrum. Vestibulum vel nunc sit amet ipsum feugiat consectetur. Nulla nec ante vitae dui tristique accumsan. Morbi felis est, ornare a vestibulum vel, vulputate a eros. Nulla facilisi.
Solution
Concentration of Analyte In Reactor
First, we need to solve for the concentration of analyte in the reactor as a function of time ($c(t)$) given an oscillating input of analyte ($[A](t)$) in the river. By performing a mass balance around the reactor (and ignoring any contribution from generation or consumption), we may write the following differential equation: $$\frac{dc}{dt} = [A](t)\cdot\frac{F}{V}-c\cdot\frac{F}{V}=D([A](t)-c),$$ $$ \mathrm{where} \ \ [A](t) = A_0\cdot\sin(2 \pi ft)+A_0$$Using $e^{Dt}$ as an integrating factor, we find $$c(t) = \frac{(A0 (D^2 \sin(2 \pi f t)+D^2-2 \pi D f \cos(2 \pi f t)+4 \pi^2 f^2))}{(D^2+4 \pi^2 f^2)}+k_1 e^{-D t}$$ $$ \mathrm{where} \ \ k_1 = - \frac{(A_0 (D^2-2 \pi D f+4 \pi^2))}{(D^2+4 \pi^2 f^2)} \ \ \mathrm{such\ that} \ c(0) =0 $$
Concentration of Analyte In Reactor
Now that we have an analytical solution for the concentration of analyte in the reactor over time, we wish to model the current response given the input function $c(t)$. To accomplish this, we modeled the time rate of change of current of our arsenic sensor using a Hill function with a cooperativity coefficient of unity—fitting data presented in Fig. X of our Current Response characterization page—and lumping all transcriptional, translational, and post-translational processes: $$\frac{dI}{dt} = \frac{\beta_1 \cdot c(t)}{c_{1/2}+c(t)}+\beta_0 - \alpha I, $$ where $c(t)$ is as defined above; $\beta_1$ is 1.4 $\mu$A/day, $\beta_0$ is 2.8 $\mu$A/dayLorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam sed nisl quis tellus convallis sagittis. Sed blandit metus at nulla mollis luctus. In eget turpis eros, eget lacinia sem. Curabitur ornare mauris nec lectus convallis vel ornare ligula viverra. Sed vel lectus mattis nisi auctor vestibulum vel nec ipsum. Integer pellentesque dolor lobortis elit viverra vitae vehicula dui cursus. Donec viverra, lectus eu faucibus rutrum, leo risus posuere justo, at rhoncus felis est in nibh. Pellentesque feugiat porta quam nec molestie. Morbi nunc dolor, consectetur in tempus in, hendrerit a augue. Phasellus ultrices volutpat diam vitae tincidunt. Mauris justo leo, blandit et tristique eu, lacinia in risus. Mauris elit eros, sollicitudin quis sodales quis, placerat a ante.
Nam ac pulvinar felis. Mauris vitae erat at orci semper aliquet vitae quis urna. Donec sit amet tortor porttitor diam bibendum viverra. Nam dui nulla, viverra sed lacinia lobortis, ullamcorper et neque. Etiam rhoncus nibh a lacus varius vehicula convallis mi rutrum. Vestibulum vel nunc sit amet ipsum feugiat consectetur. Nulla nec ante vitae dui tristique accumsan. Morbi felis est, ornare a vestibulum vel, vulputate a eros. Nulla facilisi.
Results: Time-Averaged Output for Rapidly Oscillating Analyte Concentrations
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam sed nisl quis tellus convallis sagittis. Sed blandit metus at nulla mollis luctus. In eget turpis eros, eget lacinia sem. Curabitur ornare mauris nec lectus convallis vel ornare ligula viverra. Sed vel lectus mattis nisi auctor vestibulum vel nec ipsum. Integer pellentesque dolor lobortis elit viverra vitae vehicula dui cursus. Donec viverra, lectus eu faucibus rutrum, leo risus posuere justo, at rhoncus felis est in nibh. Pellentesque feugiat porta quam nec molestie. Morbi nunc dolor, consectetur in tempus in, hendrerit a augue. Phasellus ultrices volutpat diam vitae tincidunt. Mauris justo leo, blandit et tristique eu, lacinia in risus. Mauris elit eros, sollicitudin quis sodales quis, placerat a ante.
Nam ac pulvinar felis. Mauris vitae erat at orci semper aliquet vitae quis urna. Donec sit amet tortor porttitor diam bibendum viverra. Nam dui nulla, viverra sed lacinia lobortis, ullamcorper et neque. Etiam rhoncus nibh a lacus varius vehicula convallis mi rutrum. Vestibulum vel nunc sit amet ipsum feugiat consectetur. Nulla nec ante vitae dui tristique accumsan. Morbi felis est, ornare a vestibulum vel, vulputate a eros. Nulla facilisi.
References
[1] Reference 1[2] Reference 2
[3] Reference 3
